Symmetry
http://www.mdpi.com/journal/symmetry
Latest open access articles published in Symmetry at http://www.mdpi.com/journal/symmetry<![CDATA[Symmetry, Vol. 7, Pages 294-304: Live Mobile Distance Learning System for Smart Devices]]>
http://www.mdpi.com/2073-8994/7/2/294
In recent years, mobile and ubiquitous computing has emerged in our daily lives, and extensive studies have been conducted in various areas using smart devices, such as tablets, smartphones, smart TVs, smart refrigerators, and smart media devices, in order to realize this computing technology. Especially, the integration of mobile networking technology and intelligent mobile devices has made it possible to develop the advanced mobile distance learning system that supports portable smart devices such as smartphones and tablets for the future IT environment. We present a synchronous mobile learning system that enables both instructor and student to participate in distance learning with their tablets. When an instructor gives a lecture using a tablet with front-face camera by bringing up slides and making annotations on them, students in the distance can watch the instructor and those slides with annotation on their own tablets in real time. A student can also ask a question or have a discussion together using the text chat feature of the system during a learning session. We also show the user evaluation of the system. A user survey shows that about 67% are in favor of the prototype of the system.Symmetry2015-03-2572Article10.3390/sym70202942943042073-89942015-03-25doi: 10.3390/sym7020294Jang LeeDoo-Soon ParkYoung-Sik JeongJong Park<![CDATA[Symmetry, Vol. 7, Pages 284-293: Variation in Measurement Error in Asymmetry Studies: A New Model, Simulations and Application]]>
http://www.mdpi.com/2073-8994/7/2/284
The importance of measurement error in studies of asymmetry has been acknowledged for a long time. It is now common practice to acquire independent repeated measurements of trait values and to estimate the degree of measurement error relative to the amount of asymmetry. Methods also allow obtaining unbiased estimates of asymmetry, both at the population and individual level. One aspect that has been ignored is potential between-individual variation in measurement error. In this paper, I develop a new method to investigate this variation in measurement error and to generate unbiased estimates of individual asymmetries. Simulations show that variation in measurement error can indeed result in biased estimates of individual asymmetry and that the proposed method adequately provides unbiased estimates. The method is applied to two empirical datasets and shows that, at least in some traits, substantial variations in measurement occur. The limitations of the model are discussed.Symmetry2015-03-2472Article10.3390/sym70202842842932073-89942015-03-24doi: 10.3390/sym7020284Stefan Van Dongen<![CDATA[Symmetry, Vol. 7, Pages 269-283: Study on User Authority Management for Safe Data Protection in Cloud Computing Environments]]>
http://www.mdpi.com/2073-8994/7/1/269
In cloud computing environments, user data are encrypted using numerous distributed servers before storing such data. Global Internet service companies, such as Google and Yahoo, recognized the importance of Internet service platforms and conducted self-research and development to create and utilize large cluster-based cloud computing platform technology based on low-priced commercial nodes. As diverse data services become possible in distributed computing environments, high-capacity distributed management is emerging as a major issue. Meanwhile, because of the diverse forms of using high-capacity data, security vulnerability and privacy invasion by malicious attackers or internal users can occur. As such, when various sensitive data are stored in cloud servers and used from there, the problem of data spill might occur because of external attackers or the poor management of internal users. Data can be managed through encryption to prevent such problems. However, existing simple encryption methods involve problems associated with the management of access to data stored in cloud environments. Therefore, in the present paper, a technique for data access management by user authority, based on Attribute-Based Encryption (ABE) and secret distribution techniques, is proposed.Symmetry2015-03-1971Technical Note10.3390/sym70102692692832073-89942015-03-19doi: 10.3390/sym7010269Su-Hyun KimIm-Yeong Lee<![CDATA[Symmetry, Vol. 7, Pages 255-268: Fluctuating Asymmetry of Plant Leaves: Batch Processing with LAMINA and Continuous Symmetry Measures]]>
http://www.mdpi.com/2073-8994/7/1/255
Unlike landmark methods for estimating object asymmetry, continuous symmetry measures (CSM) can be used to measure the symmetry distance (ds) of inconsistent objects, such as plant leaves. Inconsistent objects have no homologous landmarks, no consistent topology, no quantitative consistency, and sometimes no matching points. When CSM is used in conjugation with LAMINA Leaf Shape Determination software, one can quickly and efficiently process a large number of scanned leaves. LAMINA automatically generates equally-spaced points around the perimeter of each leaf and the resulting x-y coordinates are normalized to average centroid size prior to estimating ds using a fold, average, unfold algorithm. We estimated shape asymmetry of leaves of three species of flowering plants: Ligustrum sinense (Chinese Privet), Rubus cuneifolius (blackberry), and Perilla frutescens (Perilla), as well as individual leaves from a few species of oaks (Quercus) and maples (Acer). We found that 100 to 200 equally-spaced points worked well for all three of the main species. Measurement error accounted for a small proportion of the asymmetry variation. Nevertheless, measurement error was great enough to generate some negative size scaling after normalization to average centroid size.Symmetry2015-03-1371Article10.3390/sym70102552552682073-89942015-03-13doi: 10.3390/sym7010255John GrahamMattie WhitesellMark IIHagit Hel-OrEviatar NevoShmuel Raz<![CDATA[Symmetry, Vol. 7, Pages 241-254: Evidence for Obliqueness of Angles as a Cue to Planar Surface Slant Found in Extremely Simple Symmetrical Shapes]]>
http://www.mdpi.com/2073-8994/7/1/241
The Necker cube is a striking example for perceptual dominance of 3D over 2D. Object symmetry and obliqueness of angles are co-varying cues that may underlie the perceived slant of Necker cubes. To investigate the power of the oblique-angle cue, slants were judged of extremely simple symmetrical shapes. Slant computations based on an assumption of orthogonality were made for two abutting lines as a function of vertex angle and the slant of the screen. Computed slants were compared with slants judged by six subjects under binocular viewing conditions. Judged slant was highly correlated with slant specified by the oblique angles under an assumption of orthogonality. The contributions of screen cues, including binocular disparity, were negligible. The consistency of the judgments across subjects indicates the assumption of orthogonality as one of the principles underlying slant perception. Necker cubes illustrate that the visual system can disengage unambiguous cues in favor of ambiguous object-symmetry and oblique-angle cues, if the latter indicate very different slants. Selective disengagement of cues may be the mechanism that underlies the success of 2D images in ancient, as well as modern civilizations.Symmetry2015-03-0971Article10.3390/sym70102412412542073-89942015-03-09doi: 10.3390/sym7010241Casper Erkelens<![CDATA[Symmetry, Vol. 7, Pages 220-240: Inflationary Cosmology in Modified Gravity Theories]]>
http://www.mdpi.com/2073-8994/7/1/220
We review inflationary cosmology in modified gravity such as R2 gravity with its extensions in order to generalize the Starobinsky inflation model. In particular, we explore inflation realized by three kinds of effects: modification of gravity, the quantum anomaly, and the R2 term in loop quantum cosmology. It is explicitly demonstrated that in these inflationary models, the spectral index of scalar modes of the density perturbations and the tensor-to-scalar ratio can be consistent with the Planck results. Bounce cosmology in F(R) gravity is also explained.Symmetry2015-03-0971Review10.3390/sym70102202202402073-89942015-03-09doi: 10.3390/sym7010220Kazuharu BambaSergei Odintsov<![CDATA[Symmetry, Vol. 7, Pages 206-219: Teaching-Learning Activity Modeling Based on Data Analysis]]>
http://www.mdpi.com/2073-8994/7/1/206
Numerous studies are currently being carried out on personalized services based on data analysis to find and provide valuable information about information overload. Furthermore, the number of studies on data analysis of teaching-learning activities for personalized services in the field of teaching-learning is increasing, too. This paper proposes a learning style recency-frequency-durability (LS-RFD) model for quantified analysis on the level of activities of learners, to provide the elements of teaching-learning activities according to the learning style of the learner among various parameters for personalized service. This is to measure preferences as to teaching-learning activity according to recency, frequency and durability of such activities. Based on the results, user characteristics can be classified into groups for teaching-learning activity by categorizing the level of preference and activity of the learner.Symmetry2015-03-0571Article10.3390/sym70102062062192073-89942015-03-05doi: 10.3390/sym7010206Kyungrog KimYoo-Joo ChoiMihui KimJung-Won LeeDoo-Soon ParkNammee Moon<![CDATA[Symmetry, Vol. 7, Pages 193-205: 126 GeV Higgs Boson Associated with D-term Triggered Dynamical Supersymmetry Breaking]]>
http://www.mdpi.com/2073-8994/7/1/193
Continuing with our previous work on \(D\)-term triggered dynamical supersymmetry breaking, , we consider a system in which our generic \({\cal N}=1\) action is minimally extended to include the pair of Higgs doublet superfields charged under the overall \(U(1)\) together with μ and Bμ terms. The gauge group is taken to be \(SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)\). We point out, among other things, that the Higgs mass less than the \(\textit{Z}\)-boson mass at tree level can be pushed up to be around 126 GeV by \(D\)-term contributions of the overall \(U(1)\). This is readily realized by taking a \(U(1)\) gauge coupling to be \({\cal O}(1)\).Symmetry2015-03-0371Article10.3390/sym70101931932052073-89942015-03-03doi: 10.3390/sym7010193Hiroshi ItoyamaNobuhito Maru<![CDATA[Symmetry, Vol. 7, Pages 182-192: Real-Time Projection-Based Augmented Reality System for Dynamic Objects in the Performing Arts]]>
http://www.mdpi.com/2073-8994/7/1/182
This paper describes the case study of applying projection-based augmented reality, especially for dynamic objects in live performing shows, such as plays, dancing, or musicals. Our study aims to project imagery correctly inside the silhouettes of flexible objects, in other words, live actors or the surface of actor’s costumes; the silhouette transforms its own shape frequently. To realize this work, we implemented a special projection system based on the real-time masking technique, that is to say real-time projection-based augmented reality system for dynamic objects in performing arts. We installed the sets on a stage for live performance, and rehearsed particular scenes of a musical. In live performance, using projection-based augmented reality technology enhances technical and theatrical aspects which were not possible with existing video projection techniques. The projected images on the surfaces of actor’s costume could not only express the particular scene of a performance more effectively, but also lead the audience to an extraordinary visual experience.Symmetry2015-02-2771Article10.3390/sym70101821821922073-89942015-02-27doi: 10.3390/sym7010182Jaewoon LeeYeonjin KimMyeong-Hyeon HeoDongho KimByeong-Seok Shin<![CDATA[Symmetry, Vol. 7, Pages 164-181: Symmetry of “Twins”]]>
http://www.mdpi.com/2073-8994/7/1/164
The idea of construction of twin buildings is as old as architecture itself, and yet there is hardly any study emphasizing their specificity. Most frequently there are two objects or elements in an architectural composition of “twins” in which there may be various symmetry relations, mostly bilateral symmetries. The classification of “twins” symmetry in this paper is based on the existence of bilateral symmetry, in terms of the perception of an observer. The classification includes both, 2D and 3D perception analyses. We start analyzing a pair of twin buildings with projection of the architectural composition elements in 2D picture plane (plane of the composition) and we distinguish four 2D keyframe cases based on the relation between the bilateral symmetry of the twin composition and the bilateral symmetry of each element. In 3D perception for each 2D keyframe case there are two sub-variants, with and without a symmetry plane parallel to the picture plane. The bilateral symmetry is dominant if the corresponding symmetry plane is orthogonal to the picture plane. The essence of the complete classification is relation between the bilateral (dominant) symmetry of the architectural composition and the bilateral symmetry of each element of that composition.Symmetry2015-02-1371Article10.3390/sym70101641641812073-89942015-02-13doi: 10.3390/sym7010164Vladan NikolićLjiljana RadovićBiserka Marković<![CDATA[Symmetry, Vol. 7, Pages 146-163: The Genetic Architecture of Fluctuating Asymmetry of Mandible Size and Shape in a Population of Mice: Another Look]]>
http://www.mdpi.com/2073-8994/7/1/146
Fluctuating asymmetry (FA), typically measured by variation in the differences between right and left sides of bilateral traits, is commonly used to assess developmental instability (DI) in populations. A previous quantitative trait locus (QTL) investigation using an F2 intercross mouse population found little evidence of individual loci affecting FA in mandible size, but an abundance of epistatic interactions between loci. Here we extend this work by testing whether these patterns replicate in an F3 population derived from the same intercross. Using a large number of molecular markers genotyped in over 1200 mice, we uncovered significant interactions between loci (QTLs) affecting FA in mandible size (and shape). Epistasis contributed roughly 20% of the variation in FASIZE and 19% of the variation in FASHAPE at the 0.0001 probability level alone, and was comparable to that previously estimated for the F2 mice, and much greater than that generated from the few single-locus QTLs affecting the mandible FA traits. The positions of the single-locus and epistatic QTLs for FA that we discovered suggested that logical candidate genes for DI are those controlling size or shape in the traits themselves, and that they may be interacting with genes for heat shock proteins.Symmetry2015-02-1171Article10.3390/sym70101461461632073-89942015-02-11doi: 10.3390/sym7010146Larry LeamyChristian KlingenbergEmma SherrattJason WolfJames Cheverud<![CDATA[Symmetry, Vol. 7, Pages 125-145: Crystallography and Magnetic Phenomena]]>
http://www.mdpi.com/2073-8994/7/1/125
This essay describes the development of groups used for the specification of symmetries from ordinary and magnetic point groups to Fedorov and magnetic space groups, as well as other varieties of groups useful in the study of symmetric objects. In particular, we consider the problem of some incorrectness in Vol. A of the International Tables for Crystallography. Some results of tensor calculus are presented in connection with magnetoelectric phenomena, where we demonstrate the use of Ascher’s trinities and Opechowski’s magic relations and their connection. Specific tensor decomposition calculations on the grounds of Clebsch Gordan products are illustrated.Symmetry2015-02-0271Article10.3390/sym70101251251452073-89942015-02-02doi: 10.3390/sym7010125Vojtěch Kopský<![CDATA[Symmetry, Vol. 7, Pages 105-124: Two-Round Password-Only Authenticated Key Exchange in the Three-Party Setting]]>
http://www.mdpi.com/2073-8994/7/1/105
We present the first provably-secure three-party password-only authenticated key exchange (PAKE) protocol that can run in only two communication rounds. Our protocol is generic in the sense that it can be constructed from any two-party PAKE protocol. The protocol is proven secure in a variant of the widely-accepted model of Bellare, Pointcheval and Rogaway (2000) without any idealized assumptions on the cryptographic primitives used. We also investigate the security of the two-round, three-party PAKE protocol of Wang, Hu and Li (2010) and demonstrate that this protocol cannot achieve implicit key authentication in the presence of an active adversary.Symmetry2015-01-2771Article10.3390/sym70101051051242073-89942015-01-27doi: 10.3390/sym7010105Junghyun NamKim-Kwang ChooSangchul HanJuryon PaikDongho Won<![CDATA[Symmetry, Vol. 7, Pages 89-104: Can the Comparisons of Feature Locations Explain the Difficulty in Discriminating Mirror-Reflected Pairs of Geometrical Figures from Disoriented Identical Pairs?]]>
http://www.mdpi.com/2073-8994/7/1/89
The present experiment investigates whether patterns of shifts of feature locations could affect the same/different decisions of simultaneously presented pairs of geometrical figures. A shift of locations was defined as the angular distance from the location of a feature in one figure to the location of the same feature in another figure. It was hypothesized that the difficulty in discriminating mirror-reflected (or axisymmetric) pairs from disoriented identical pairs was caused by complex shifting patterns inherent in axisymmetric pairs. According to the shifts of the locations of the four structural features, five pair types were prepared. They could be ordered from completely identical to completely different in their shifts: identical 0/4 pairs, non-identical 1/4 pairs, non-identical 2/4 pairs = axisymmetric 2/4 pairs and non-identical 4/4 pairs. The latencies for non-identical pairs decreased with the increase of difference in the shifts of feature locations, indicating that serial, self-terminating comparisons of the shifts were applied to the discrimination of non-identical pairs from identical pairs. However, the longer latencies in axisymmetric 2/4 pairs than in non-identical 2/4 pairs suggested that the difficulty for axisymmetric pairs was not caused by the complex shifting patterns, and the difficulty was not satisfactorily explained by the comparisons of feature locations.Symmetry2015-01-2371Article10.3390/sym7010089891042073-89942015-01-23doi: 10.3390/sym7010089Fumio Kanbe<![CDATA[Symmetry, Vol. 7, Pages 67-88: Dihedral Reductions of Cyclic DNA Sequences]]>
http://www.mdpi.com/2073-8994/7/1/67
The data-analytic methodology of dihedral reductions for cyclic orbits of distinct-base codons is described both in terms of Fourier analysis over the dihedral groups and in (algebraically equivalent) terms of canonical projections. Numerical evaluations are presented for discrete and continuous scalar data indexed by cyclic orbits.Symmetry2015-01-1671Article10.3390/sym701006767882073-89942015-01-16doi: 10.3390/sym7010067Marlos Viana<![CDATA[Symmetry, Vol. 7, Pages 53-66: Output Effect Evaluation Based on Input Features in Neural Incremental Attribute Learning for Better Classification Performance]]>
http://www.mdpi.com/2073-8994/7/1/53
Machine learning is a very important approach to pattern classification. This paper provides a better insight into Incremental Attribute Learning (IAL) with further analysis as to why it can exhibit better performance than conventional batch training. IAL is a novel supervised machine learning strategy, which gradually trains features in one or more chunks. Previous research showed that IAL can obtain lower classification error rates than a conventional batch training approach. Yet the reason for that is still not very clear. In this study, the feasibility of IAL is verified by mathematical approaches. Moreover, experimental results derived by IAL neural networks on benchmarks also confirm the mathematical validation.Symmetry2015-01-1471Article10.3390/sym701005353662073-89942015-01-14doi: 10.3390/sym7010053Ting WangSheng-Uei GuanKa ManJong ParkHui-Huang Hsu<![CDATA[Symmetry, Vol. 7, Pages 43-52: Development of Guanidine-Bisurea Bifunctional Organocatalyst Bearing Chirality at the Inner and Outer Sides of the Urea Groups, and Application to Enantioselective α-Hydroxylation of Pyranoindolizine Intermediate for Camptothecin Synthesis]]>
http://www.mdpi.com/2073-8994/7/1/43
Pyranoindolizine is a tricyclic structure found in various biologically active compounds, such as camptothecin (CPT) and its derivatives. In the case of CPTs, the chirality at the α-position in the α-hydroxyl lactone moiety of pyranoindolizine is important for the antitumor activity. This paper deals with enantioselective oxidation of the α-position in pyranoindolizine lactone, which corresponds at C20 in CPT, with cumene hydroperoxide (CHP) in the presence of newly synthesized guanidine-bisurea bifunctional organocatalysts bearing chirality on both the inner and outer sides of the urea groups.Symmetry2015-01-1471Article10.3390/sym701004343522073-89942015-01-14doi: 10.3390/sym7010043Minami OdagiTatsuya WatanabeKazuo Nagasawa<![CDATA[Symmetry, Vol. 7, Pages 40-42: Acknowledgement to Reviewers of Symmetry in 2014]]>
http://www.mdpi.com/2073-8994/7/1/40
The editors of Symmetry would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2014:[...]Symmetry2015-01-0771Editorial10.3390/sym701004040422073-89942015-01-07doi: 10.3390/sym7010040 Symmetry Editorial Office<![CDATA[Symmetry, Vol. 7, Pages 32-39: Topology and the Visualization of Space]]>
http://www.mdpi.com/2073-8994/7/1/32
Overlapping patterns provide the diagrammatics for four-dimensional space. If these patterns are three-dimensional lattices, and if one imagines them extended in three-dimensional space, then the diagram makes a model of physical space.Symmetry2014-12-3071Article10.3390/sym701003232392073-89942014-12-30doi: 10.3390/sym7010032Tony Robbin<![CDATA[Symmetry, Vol. 7, Pages 15-31: Effect of Tensor Correlations on the Density Dependence of the Nuclear Symmetry Energy]]>
http://www.mdpi.com/2073-8994/7/1/15
We analyze the effect of the tensor force and other components of the nucleon-nucleon interaction on the nuclear symmetry energy and its density dependence by using the Hellmann–Feynman theorem. The analysis is performed within the microscopic Brueckner–Hartree–Fock approach using the Argonne V18 potential plus a Urbana IX three-nucleon force. Our results show that the potential part of the nuclear Hamiltonian, and in particular its tensor component, gives the largest contribution to the symmetry energy. The decomposition of the symmetry energy into a kinetic part and a potential energy part provides physical insight on the correlated nature of the system, indicating that pure neutron matter is less correlated than symmetric nuclear matter.Symmetry2014-12-2971Article10.3390/sym701001515312073-89942014-12-29doi: 10.3390/sym7010015Isaac VidañaConstança ProvidênciaArtur Polls<![CDATA[Symmetry, Vol. 7, Pages 1-14: A Study on Electronic-Money Technology Using Near Field Communication]]>
http://www.mdpi.com/2073-8994/7/1/1
Recently, due to the introduction of NFC (Near Field Communication), it has become possible to make easy electronic payments. Therefore, a secure communication method is necessary in these environments. NFC can be said to be relatively safe compared to other communication methods, because it carries out communications within 10 cm. However, it has made possible the risk of impersonation attacks by a disguised reader, leaving user information on the reader. In order to solve these problems, in this paper, we propose an authentication scheme that can reduce the weight of computation by using only a hash function and XOR (eXclusive OR) operation algorithms. This paper also shows that our method is safe, since it leaves no information with the other party.Symmetry2014-12-2671Article10.3390/sym70100011142073-89942014-12-26doi: 10.3390/sym7010001Min Jung<![CDATA[Symmetry, Vol. 6, Pages 1037-1057: Coulomb Solutions from Improper Pseudo-Unitary Free Gauge Field Operator Translations]]>
http://www.mdpi.com/2073-8994/6/4/1037
Fundamental problems of quantum field theory related to the representation problem of canonical commutation relations are discussed within a gauge field version of a van Hove-type model. The Coulomb field generated by a static charge distribution is described as a formal superposition of time-like pseudo-photons in Fock space with a Krein structure. In this context, a generalization of operator gauge transformations is introduced to generate coherent states of Abelian gauge fields interacting with a charged background.Symmetry2014-12-1564Article10.3390/sym6041037103710572073-89942014-12-15doi: 10.3390/sym6041037Andreas Aste<![CDATA[Symmetry, Vol. 6, Pages 1026-1036: Benjamin–Bona–Mahony Equation with Variable Coefficients: Conservation Laws]]>
http://www.mdpi.com/2073-8994/6/4/1026
This paper aims to construct conservation laws for a Benjamin–Bona–Mahony equation with variable coefficients, which is a third-order partial differential equation. This equation does not have a Lagrangian and so we transform it to a fourth-order partial differential equation, which has a Lagrangian. The Noether approach is then employed to construct the conservation laws. It so happens that the derived conserved quantities fail to satisfy the divergence criterion and so one needs to make adjustments to the derived conserved quantities in order to satisfy the divergence condition. The conservation laws are then expressed in the original variable. Finally, a conservation law is used to obtain exact solution of a special case of the Benjamin–Bona–Mahony equation.Symmetry2014-12-1564Article10.3390/sym6041026102610362073-89942014-12-15doi: 10.3390/sym6041026Ben MuatjetjejaChaudry Khalique<![CDATA[Symmetry, Vol. 6, Pages 1011-1025: Privacy-Enhancing Security Protocol in LTE Initial Attack]]>
http://www.mdpi.com/2073-8994/6/4/1011
Long-Term Evolution (LTE) is a fourth-generation mobile communication technology implemented throughout the world. It is the communication means of smartphones that send and receive all of the private date of individuals. M2M, IOT, etc., are the base technologies of mobile communication that will be used in the future cyber world. However, identification parameters, such as International Mobile Subscriber Identity (IMSI), Radio Network Temporary Identities (RNTI), etc., in the initial attach section for accessing the LTE network are presented with the vulnerability of being exposed as clear text. Such vulnerability does not end in a mere identification parameter, but can lead to a secondary attack using the identification parameter, such as replication of the smartphone, illegal use of the mobile communication network, etc. This paper proposes a security protocol to safely transmit identification parameters in different cases of the initial attach. The proposed security protocol solves the exposed vulnerability by encrypting the parameters in transmission. Using an OPNET simulator, it is shown that the average rate of delay and processing ratio are efficient in comparison to the existing process.Symmetry2014-12-1264Article10.3390/sym6041011101110252073-89942014-12-12doi: 10.3390/sym6041011Uijin JangHyungmin LimHyungjoo Kim<![CDATA[Symmetry, Vol. 6, Pages 997-1010: MLDS: Multi-Layer Defense System for Preventing Advanced Persistent Threats]]>
http://www.mdpi.com/2073-8994/6/4/997
Here we report on the issue of Advanced Persistent Threats (APT), which use malware for the purpose of leaking the data of large corporations and government agencies. APT attacks target systems continuously by utilizing intelligent and complex technologies. To overthrow the elaborate security network of target systems, it conducts an attack after undergoing a pre-reconnaissance phase. An APT attack causes financial loss, information leakage, etc. They can easily bypass the antivirus system of a target system. In this paper, we propose a Multi-Layer Defense System (MLDS) that can defend against APT. This system applies a reinforced defense system by collecting and analyzing log information and various information from devices, by installing the agent on the network appliance, server and end-user. It also discusses how to detect an APT attack when one cannot block the initial intrusion while continuing to conduct other activities. Thus, this system is able to minimize the possibility of initial intrusion and damages of the system by promptly responding through rapid detection of an attack when the target system is attacked.Symmetry2014-12-0364Article10.3390/sym604099799710102073-89942014-12-03doi: 10.3390/sym6040997Daesung MoonHyungjin ImJae LeeJong Park<![CDATA[Symmetry, Vol. 6, Pages 975-996: Brain Activity in Response to Visual Symmetry]]>
http://www.mdpi.com/2073-8994/6/4/975
A number of studies have explored visual symmetry processing by measuring event related potentials and neural oscillatory activity. There is a sustained posterior negativity (SPN) related to the presence of symmetry. There is also functional magnetic resonance imaging (MRI) activity in extrastriate visual areas and in the lateral occipital complex. We summarise the evidence by answering six questions. (1) Is there an automatic and sustained response to symmetry in visual areas? Answer: Yes, and this suggests automatic processing of symmetry. (2) Which brain areas are involved in symmetry perception? Answer: There is an extended network from extrastriate areas to higher areas. (3) Is reflection special? Answer: Reflection is the optimal stimulus for a more general regularity-sensitive network. (4) Is the response to symmetry independent of view angle? Answer: When people classify patterns as symmetrical or random, the response to symmetry is view-invariant. When people attend to other dimensions, the network responds to residual regularity in the image. (5) How are brain rhythms in the two hemispheres altered during symmetry perception? Answer: Symmetry processing (rather than presence) produces more alpha desynchronization in the right posterior regions. Finally, (6) does symmetry processing produce positive affect? Answer: Not in the strongest sense, but behavioural measures reveal implicit positive evaluation of abstract symmetry.Symmetry2014-12-0264Review10.3390/sym60409759759962073-89942014-12-02doi: 10.3390/sym6040975Marco BertaminiAlexis Makin<![CDATA[Symmetry, Vol. 6, Pages 954-974: On the Self-Mobility of Point-Symmetric Hexapods]]>
http://www.mdpi.com/2073-8994/6/4/954
In this article, we study necessary and sufficient conditions for the self-mobility of point symmetric hexapods (PSHs). Specifically, we investigate orthogonal PSHs and equiform PSHs. For the latter ones, we can show that they can have non-translational self-motions only if they are architecturally singular or congruent. In the case of congruency, we are even able to classify all types of existing self-motions. Finally, we determine a new set of PSHs, which have so-called generalized Dietmaier self-motions. We close the paper with some comments on the self-mobility of hexapods with global/local symmetries.Symmetry2014-11-1864Article10.3390/sym60409549549742073-89942014-11-18doi: 10.3390/sym6040954Georg Nawratil<![CDATA[Symmetry, Vol. 6, Pages 938-953: Design of a Secure System Considering Quality of Service]]>
http://www.mdpi.com/2073-8994/6/4/938
Improvements in networking technologies have provided users with useful information services. Such information services may bring convenience and efficiency, but might be accompanied by vulnerabilities to a variety of attacks. Therefore, a variety of research to enhance the security of the systems and get the services at the same time has been carried out. Especially, research on intrusion-tolerant systems (ITSs) has been conducted in order to survive against every intrusion, rather than to detect and prevent them. In this paper, an ITS based on effective resource conversion (ERC) is presented to achieve the goal of intrusion-tolerance. Instead of using the fixed number of virtual machines (VMs) to process requests and recover as in conventional approaches, the ITS based on ERC can transform the assigned resources depending on the system status. This scheme is proved to maintain a certain level of quality of service (QoS) and quality of security service (QoSS) in threatening environments. The performance of ERC is compared with previous studies on ITS by CSIM 20, and it is verified that the proposed scheme is more effective in retaining a specific level of QoS and QoSS.Symmetry2014-11-1364Article10.3390/sym60409389389532073-89942014-11-13doi: 10.3390/sym6040938Seondong HeoSoojin LeeSeokjoo DooHyunsoo Yoon<![CDATA[Symmetry, Vol. 6, Pages 926-937: Keyframe Extraction from Human Motion Capture Data Based on a Multiple Population Genetic Algorithm]]>
http://www.mdpi.com/2073-8994/6/4/926
To reduce reconstruction errors during keyframe extraction and to control the optimal compression ratio, this study proposes a method for keyframe extraction from human motion capture data based on a multiple population genetic algorithm. The fitness function is defined to meet the goals of minimal reconstruction errors and the optimal compression rate, where multiple initial populations are subjected to co-evolution. The multiple population genetic algorithm considers global and local search. Experimental results showed that the algorithm can effectively extract the keyframe from motion capture data and it satisfied the desired reconstruction error.Symmetry2014-11-1264Article10.3390/sym60409269269372073-89942014-11-12doi: 10.3390/sym6040926Qiang ZhangShulu ZhangDongsheng Zhou<![CDATA[Symmetry, Vol. 6, Pages 909-925: Determining Amino Acid Chirality in the Supernova Neutrino Processing Model]]>
http://www.mdpi.com/2073-8994/6/4/909
A model is described that can be used to estimate the bulk polarization of largerotating meteoroids in the magnetic field of a neutron star. The results of this model areapplicable to the Supernova Neutrino Amino Acid Processing model, which describes onepossible way in which the amino acids, known in nearly all cases to exhibit supramolecularchirality, could have become enantiomeric.Symmetry2014-11-0364Article10.3390/sym60409099099252073-89942014-11-03doi: 10.3390/sym6040909Michael FamianoRichard BoydToshitaka KajinoTakashi OnakaKatrina KoehlerSarah Hulbert<![CDATA[Symmetry, Vol. 6, Pages 896-908: Dynamical Response of Particles in Asymmetric Ratchet Potential]]>
http://www.mdpi.com/2073-8994/6/4/896
The dynamics and response of particles in an asymmetric potential of the ratchet type is considered. An approximate asymmetric ratchet potential, which allows us to apply an analytical approach to investigate the systems’ response, is proposed based on the power series interpolation scheme. The method of multiple scales (MMS) is then employed to obtain an approximate solution at the principal resonance. Comparison between results obtained from the approximated and exact potentials show excellent agreement. The frequency response curves (FRCs) for different forcing are presented. The numerically computed FRCs were compared with the theoretical result to show the extent of agreement. Furthermore, by using perturbation methods, the stable and unstable domains of the solution were obtained.Symmetry2014-10-2764Article10.3390/sym60408968969082073-89942014-10-27doi: 10.3390/sym6040896Usman MarteUchechukwu VincentAbdulahi NjahBiodun Badmus<![CDATA[Symmetry, Vol. 6, Pages 880-895: Supramolecular Chirality in Dynamic Coordination Chemistry]]>
http://www.mdpi.com/2073-8994/6/4/880
Labile metal complexes have a useful coordination bond; which is weaker than a covalent C–C bond and is reversibly and dynamically formed and dissociated. Such labile metal complexes also can be used to construct chiral shapes and offer dynamic conversion of chiral molecular shapes in response to external stimuli. This review provides recent examples of chirality induction and describes the dynamic conversion systems produced by chiral metal complexes including labile metal centers, most of which respond to external stimuli by exhibiting sophisticated conversion phenomena.Symmetry2014-10-2464Review10.3390/sym60408808808952073-89942014-10-24doi: 10.3390/sym6040880Hiroyuki Miyake<![CDATA[Symmetry, Vol. 6, Pages 851-879: Recent Advances in Microscopic Approaches to Nuclear Matter and Symmetry Energy]]>
http://www.mdpi.com/2073-8994/6/4/851
Nuclear matter is a convenient theoretical laboratory to test many-body theories. When neutron and proton densities are different, the isospin dependence of the nuclear force gives rise to the symmetry energy term in the equation of state. This quantity is a crucial mechanism in the formation of the neutron skin in nuclei, as well as in other systems and phenomena involved in the dynamics of neutrons and protons in neutron-rich systems, such as isospin-asymmetric heavy-ion collisions. In this article, we will review phenomenological facts about the symmetry energy and recent experimental efforts to constrain its density dependence and related quantities. We will then review our microscopic approach to the equation of state of symmetric and asymmetric nuclear matter and present a corresponding set of predictions. Our calculations utilize the Dirac–Brueckner–Hartree–Fock method and realistic meson-theoretic nucleon-nucleon potentials. Chiral perturbation theory is an alternative approach, based on a well-defined scheme, which allows one to develop nuclear forces at each order of the chiral expansion. We will present and discuss predictions based on chiral perturbation theory, where we employ consistent two- and three-body chiral interactions. Throughout the article, one of the focal points is the importance of pursuing ab initio methods towards a deeper understanding of the many-body system.Symmetry2014-10-2064Review10.3390/sym60408518518792073-89942014-10-20doi: 10.3390/sym6040851Francesca Sammarruca<![CDATA[Symmetry, Vol. 6, Pages 844-850: Polar Vector Property of the Stationary State of Condensed Molecular Matter]]>
http://www.mdpi.com/2073-8994/6/4/844
Crystalline phases undergoing 180\(^{\circ}\) orientational disorder of dipolar entities in the seed or at growing (hkl) faces will show a polar vector property described by \(\infty\) /mm symmetry. Seeds and crystals develop a bi-polar state (\(\infty\)/mm), where domains related by a mirror plane m allow for a \(\infty\) m symmetry in each domain. The polarity of domains is due to energetic favorable interactions at the object-to-nutrient interface. Such interactions are well reproduced by an Ising Hamiltonian. Two-dimensional Monte Carlo simulations performed for real molecules with full long-range interactions allow us to calculate the spatial distribution of the electrical polarization Pel. The investigation has been extended to liquid droplets made of dipolar entities by molecular dynamics simulations. We demonstrate the development of an m\(\bar{\infty}\) quasi bi-polar state leading to a charged surface.Symmetry2014-10-1364Article10.3390/sym60408448448502073-89942014-10-13doi: 10.3390/sym6040844Jürg HulligerLuigi CannavacciuoloMathias Rech<![CDATA[Symmetry, Vol. 6, Pages 781-843: Chlorophylls, Symmetry, Chirality, and Photosynthesis]]>
http://www.mdpi.com/2073-8994/6/3/781
Chlorophylls are a fundamental class of tetrapyrroles and function as the central reaction center, accessory and photoprotective pigments in photosynthesis. Their unique individual photochemical properties are a consequence of the tetrapyrrole macrocycle, the structural chemistry and coordination behavior of the phytochlorin system, and specific substituent pattern. They achieve their full potential in solar energy conversion by working in concert in highly complex, supramolecular structures such as the reaction centers and light-harvesting complexes of photobiology. The biochemical function of these structures depends on the controlled interplay of structural and functional principles of the apoprotein and pigment cofactors. Chlorophylls and bacteriochlorophylls are optically active molecules with several chiral centers, which are necessary for their natural biological function and the assembly of their supramolecular complexes. However, in many cases the exact role of chromophore stereochemistry in the biological context is unknown. This review gives an overview of chlorophyll research in terms of basic function, biosynthesis and their functional and structural role in photosynthesis. It highlights aspects of chirality and symmetry of chlorophylls to elicit further interest in their role in nature.Symmetry2014-09-1063Review10.3390/sym60307817818432073-89942014-09-10doi: 10.3390/sym6030781Mathias SengeAoife RyanKristie LetchfordStuart MacGowanTamara Mielke<![CDATA[Symmetry, Vol. 6, Pages 758-780: Twinning of Polymer Crystals Suppressed by Entropy]]>
http://www.mdpi.com/2073-8994/6/3/758
We propose an entropic argument as partial explanation of the observed scarcity of twinned structures in crystalline samples of synthetic organic polymeric materials. Polymeric molecules possess a much larger number of conformational degrees of freedom than low molecular weight substances. The preferred conformations of polymer chains in the bulk of a single crystal are often incompatible with the conformations imposed by the symmetry of a growth twin, both at the composition surfaces and in the twin axis. We calculate the differences in conformational entropy between chains in single crystals and chains in twinned crystals, and find that the reduction in chain conformational entropy in the twin is sufficient to make the single crystal the stable thermodynamic phase. The formation of cyclic twins in molecular dynamics simulations of chains of hard spheres must thus be attributed to kinetic factors. In more realistic polymers this entropic contribution to the free energy can be canceled or dominated by nonbonded and torsional energetics.Symmetry2014-09-0463Article10.3390/sym60307587587802073-89942014-09-04doi: 10.3390/sym6030758Nikos KarayiannisKaterina FoteinopoulouManuel Laso<![CDATA[Symmetry, Vol. 6, Pages 722-757: Recognition of Symmetric 3D Bodies]]>
http://www.mdpi.com/2073-8994/6/3/722
The paper deals with the recognition of symmetric three-dimensional (3D) bodies that can be rotated and translated. We provide a complete list of all existing combinations of rotation and reflection symmetries in 3D. We define 3D complex moments by means of spherical harmonics, and the influence of individual symmetry groups on complex moment values is studied. Each particular symmetry pre-defines certain moment values. These moments can no longer differentiate between two objects of the same symmetry, which decreases the recognition power of the feature set. They should not be included when constructing the invariants. Translation and rotation invariants up to the fourth order are presented and their performance is studied on both artificial and real data.Symmetry2014-09-0163Article10.3390/sym60307227227572073-89942014-09-01doi: 10.3390/sym6030722Tomáš SukJan Flusser<![CDATA[Symmetry, Vol. 6, Pages 704-721: Adhesive/Repulsive Codes in Vertebrate Forebrain Morphogenesis]]>
http://www.mdpi.com/2073-8994/6/3/704
The last fifteen years have seen the identification of some of the mechanisms involved in anterior neural plate specification, patterning, and morphogenesis, which constitute the first stages in the formation of the forebrain. These studies have provided us with a glimpse into the molecular mechanisms that drive the development of an embryonic structure, and have resulted in the realization that cell segregation in the anterior neural plate is essential for the accurate progression of forebrain morphogenesis. This review summarizes the latest advances in our understanding of mechanisms of cell segregation during forebrain development, with and emphasis on the impact of this process on the morphogenesis of one of the anterior neural plate derivatives, the eyes.Symmetry2014-08-1463Review10.3390/sym60307047047212073-89942014-08-14doi: 10.3390/sym6030704Florencia Cavodeassi<![CDATA[Symmetry, Vol. 6, Pages 677-703: Supramolecular Chirality: Solvent Chirality Transfer in Molecular Chemistry and Polymer Chemistry]]>
http://www.mdpi.com/2073-8994/6/3/677
Controlled mirror symmetry breaking arising from chemical and physical origin is currently one of the hottest issues in the field of supramolecular chirality. The dynamic twisting abilities of solvent molecules are often ignored and unknown, although the targeted molecules and polymers in a fluid solution are surrounded by solvent molecules. We should pay more attention to the facts that mostly all of the chemical and physical properties of these molecules and polymers in the ground and photoexcited states are significantly influenced by the surrounding solvent molecules with much conformational freedom through non-covalent supramolecular interactions between these substances and solvent molecules. This review highlights a series of studies that include: (i) historical background, covering chiral NaClO3 crystallization in the presence of d-sugars in the late 19th century; (ii) early solvent chirality effects for optically inactive chromophores/fluorophores in the 1960s–1980s; and (iii) the recent development of mirror symmetry breaking from the corresponding achiral or optically inactive molecules and polymers with the help of molecular chirality as the solvent use quantity.Symmetry2014-08-1363Review10.3390/sym60306776777032073-89942014-08-13doi: 10.3390/sym6030677Michiya Fujiki<![CDATA[Symmetry, Vol. 6, Pages 655-676: Lubricating and Waxy Esters. VI. Effect of Symmetry about Ester on Crystallization of Linear Monoester Isomers]]>
http://www.mdpi.com/2073-8994/6/3/655
The crystal structure development of jojoba-like esters incorporating either 1-decenoic acid and/or 1-decenol, namely octadec-9-enyl dec-9-enoate (JLE-281), and its isomer dec-9-enyl oleate (JLE-282) was investigated to reveal the effect of symmetry about the ester group on crystallization of aliphatic fatty monoesters. The phase transformation path was investigated with temperature-time resolved X-ray diffraction during stepped isothermal crystallization, and while cooling from the melt at a fixed rate. Startling differences in phase behavior were uncovered between the isomers. When stepped isothermals were used, selective extinctions occurred at a transition temperature for JLE-281 but not for JLE-282. The extinctions, which are due to dramatic changes in the electronic density of certain families of planes, indicate a phase transition attributed to a brusque rearrangement of the oxygen atoms in the crystal subcell. The phase transition did not occur when the JLEs were cooled continuously. The crucial role played by the position of the alkyl chain and its orientation relative to the easy rotation site of the C–O bond in the phase trajectories of the JLEs was particularly highlighted.Symmetry2014-08-0763Article10.3390/sym60306556556762073-89942014-08-07doi: 10.3390/sym6030655Laziz BouzidiShaojun LiSuresh Narine<![CDATA[Symmetry, Vol. 6, Pages 622-654: Symmetry Breaking in NMR Spectroscopy: The Elucidation of Hidden Molecular Rearrangement Processes]]>
http://www.mdpi.com/2073-8994/6/3/622
Variable-temperature NMR spectroscopy is probably the most convenient and sensitive technique to monitor changes in molecular structure in solution. Rearrangements that are rapid on the NMR time-scale exhibit simplified spectra, whereby non-equivalent nuclear environments yield time-averaged resonances. At lower temperatures, when the rate of exchange is sufficiently reduced, these degeneracies are split and the underlying “static” molecular symmetry, as seen by X-ray crystallography, becomes apparent. Frequently, however, such rearrangement processes are hidden, even when they become slow on the NMR time-scale, because the molecular point group remains unchanged. Judicious symmetry breaking, such as by substitution of a molecular fragment by a similar, but not identical moiety, or by the incorporation of potentially diastereotopic (chemically non-equivalent) nuclei, allows the elucidation of the kinetics and energetics of such processes. Examples are chosen that include a wide range of rotations, migrations and other rearrangements in organic, inorganic and organometallic chemistry.Symmetry2014-08-0463Review10.3390/sym60306226226542073-89942014-08-04doi: 10.3390/sym6030622Michael McGlinchey<![CDATA[Symmetry, Vol. 6, Pages 589-621: Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures]]>
http://www.mdpi.com/2073-8994/6/3/589
Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a2, b2 and a · b scalar products of the lattice basis vectors with rational coefficients. To characterize OD families and to check their validity, these lattice restrictions are expressed in the bases of different layers and combined. For a more familiar notation, they can be expressed in terms of the lattice parameters a, b and . Alternatively, the description of the lattice restrictions may be simplified by using centered lattices. The representation of the lattice restrictions in terms of scalar products is dependent on the chosen basis. A basis-independent classification of the lattice restrictions is outlined.Symmetry2014-07-2163Article10.3390/sym60305895896212073-89942014-07-21doi: 10.3390/sym6030589Berthold Stöger<![CDATA[Symmetry, Vol. 6, Pages 578-588: Coset Group Construction of Multidimensional Number Systems]]>
http://www.mdpi.com/2073-8994/6/3/578
Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical construction of complex and multicomplex numbers multiplication rules are postulated instead of being derived from a general principle. A more transparent and systematic approach is proposed here based on the concept of coset product from group theory. It is shown that extensions of real numbers in two or more dimensions follow naturally from the closure property of finite coset groups adding insight into the utility of multidimensional number systems in describing symmetries in nature.Symmetry2014-07-1163Article10.3390/sym60305785785882073-89942014-07-11doi: 10.3390/sym6030578Horia Petrache<![CDATA[Symmetry, Vol. 6, Pages 566-577: Gestalt Algebra—A Proposal for the Formalization of Gestalt Perception and Rendering]]>
http://www.mdpi.com/2073-8994/6/3/566
Gestalt Algebra gives a formal structure suitable for describing complex patterns in the image plain. This can be useful for recognizing hidden structure in images. The work at hand refers to the laws of perceptual psychology. A manifold called the Gestalt Domain is defined. Next to the position in 2D it also contains an orientation and a scale component. Algebraic operations on it are given for mirror symmetry as well as organization into rows. Additionally the Gestalt Domain contains an assessment component, and all the meaning of the operations implementing the Gestalt-laws is realized in the functions giving this component. The operation for mirror symmetry is binary, combining two parts into one aggregate as usual in standard algebra. The operation for organization into rows, however, combines n parts into an aggregate, where n may well be more than two. This is algebra in its more general sense. For recognition, primitives are extracted from digital raster images by Lowe’s Scale Invariant Feature Transform (SIFT). Lowe’s key-point descriptors can also be utilized. Experiments are reported with a set of images put forth for the Computer Vision and Pattern Recognition Workshops (CVPR) 2013 symmetry contest.Symmetry2014-07-0763Article10.3390/sym60305665665772073-89942014-07-07doi: 10.3390/sym6030566Eckart Michaelsen<![CDATA[Symmetry, Vol. 6, Pages 551-565: Autosolvation: Architecture and Selection of Chiral Conformers in Alkylcobalt Carbonyl Molecular Clocks]]>
http://www.mdpi.com/2073-8994/6/3/551
Autosolvation is an important factor in stabilizing the architecture of medium complicated molecules. It is a kind of “supramolecular force” acting in intramolecular manner, consisting of orbital-orbital interactions between polar groups, separated by more than one covalent bonds within the same molecule. This effect facilitates also the development of chiral conformations. Two typical alkylcobalt carbonyl type molecules are discussed here as examples of autosolvating intramolecular interactions, leading to dramatic selection of chiral conformers and indicating also to the limits of the effect. The conformers stabilized by autosolvation and their interconversion are excellent examples of a “molecular clockwork”. Operation mode of these molecular clockworks gives some insight into the intramolecular transfer of chiral information.Symmetry2014-07-0463Review10.3390/sym60305515515652073-89942014-07-04doi: 10.3390/sym6030551Róbert KurdiAttila TáborosiClaudia ZucchiGyula Pályi<![CDATA[Symmetry, Vol. 6, Pages 516-550: Symmetry Adapted Assur Decompositions]]>
http://www.mdpi.com/2073-8994/6/3/516
Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools.Symmetry2014-06-2763Article10.3390/sym60305165165502073-89942014-06-27doi: 10.3390/sym6030516Anthony NixonBernd SchulzeAdnan SljokaWalter Whiteley<![CDATA[Symmetry, Vol. 6, Pages 473-515: Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere]]>
http://www.mdpi.com/2073-8994/6/3/473
The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown that this oscillator system contains the essential features of Wigner’s little groups dictating the internal space-time symmetries of particles in the Lorentz-covariant world. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. It is shown that the damping modes of the oscillator correspond to the little groups for massive and imaginary-mass particles respectively. When the system makes the transition from the oscillation to damping mode, it corresponds to the little group for massless particles. Rotations around the momentum leave the four-momentum invariant. This degree of freedom extends the Sp(2) symmetry to that of SL(2, c) corresponding to the Lorentz group applicable to the four-dimensional Minkowski space. The Poincaré sphere contains the SL(2, c) symmetry. In addition, it has a non-Lorentzian parameter allowing us to reduce the mass continuously to zero. It is thus possible to construct the little group for massless particles from that of the massive particle by reducing its mass to zero. Spin-1/2 particles and spin-1 particles are discussed in detail.Symmetry2014-06-2563Article10.3390/sym60304734735152073-89942014-06-25doi: 10.3390/sym6030473Sibel BaşkalYoung KimMarilyn Noz<![CDATA[Symmetry, Vol. 6, Pages 444-472: Chiral Liquid Crystals: Structures, Phases, Effects]]>
http://www.mdpi.com/2073-8994/6/2/444
The introduction of chirality, i.e., the lack of mirror symmetry, has a profound effect on liquid crystals, not only on the molecular scale but also on the supermolecular scale and phase. I review these effects, which are related to the formation of supermolecular helicity, the occurrence of novel thermodynamic phases, as well as electro-optic effects which can only be observed in chiral liquid crystalline materials. In particular, I will discuss the formation of helical superstructures in cholesteric, Twist Grain Boundary and ferroelectric phases. As examples for the occurrence of novel phases the Blue Phases and Twist Grain Boundary phases are introduced. Chirality related effects are demonstrated through the occurrence of ferroelectricity in both thermotropic as well as lyotropic liquid crystals. Lack of mirror symmetry is also discussed briefly for some biopolymers such as cellulose and DNA, together with its influence on liquid crystalline behavior.Symmetry2014-06-1662Review10.3390/sym60204444444722073-89942014-06-16doi: 10.3390/sym6020444Ingo Dierking<![CDATA[Symmetry, Vol. 6, Pages 427-443: Symmetry Detection in Visual Impairment: Behavioral Evidence and Neural Correlates]]>
http://www.mdpi.com/2073-8994/6/2/427
Bilateral symmetry is an extremely salient feature for the human visual system. An interesting issue is whether the perceptual salience of symmetry is rooted in normal visual development. In this review, we discuss empirical work on visual and tactile symmetry detection in normally sighted and visually impaired individuals. On the one hand, available evidence suggests that efficient visual symmetry detection may need normal binocular vision development. On the other hand, converging evidence suggests that symmetry can develop as a principle of haptic perceptual organization in individuals lacking visual experience. Certain features of visual symmetry detection, however, such as the higher salience of the patterns containing a vertical axis of symmetry, do not systematically apply to the haptic modality. The neural correlates (revealed with neuroimaging) associated with visual and haptic symmetry detection are also discussed.Symmetry2014-05-2662Review10.3390/sym60204274274432073-89942014-05-26doi: 10.3390/sym6020427Zaira CattaneoSilvia BonaCorinna BauerJuha SilvantoAndrew HerbertTomaso VecchiLotfi Merabet<![CDATA[Symmetry, Vol. 6, Pages 409-426: The Symmetric-Partitioning and Incremental-Relearning Classification and Back-Propagation-Network Tree Approach for Cycle Time Estimation in Wafer Fabrication]]>
http://www.mdpi.com/2073-8994/6/2/409
An innovative classification and back-propagation-network tree (CABPN tree) approach is proposed in this study to estimate the cycle time of a job in a wafer fabrication factory, which is one of the most important tasks in controlling the wafer fabrication factory. The CABPN tree approach is an extension from the traditional classification and regression tree (CART) approach. In CART, the cycle times of jobs of the same branch are estimated with the same value, which is far from accurate. To tackle this problem, the CABPN tree approach replaces the constant estimate with variant estimates. To this end, the cycle times of jobs of the same branch are estimated with a BPN, and may be different. In this way, the estimation accuracy can be improved. In addition, to determine the optimal location of the splitting point on a node, the symmetric partition with incremental re-learning (SP-IR) algorithm is proposed and illustrated with an example. The applicability of the CABPN tree approach is shown with a real case. The experimental results supported its effectiveness over several existing methods.Symmetry2014-05-2362Case Report10.3390/sym60204094094262073-89942014-05-23doi: 10.3390/sym6020409Toly Chen<![CDATA[Symmetry, Vol. 6, Pages 396-408: Invisibility and PT Symmetry: A Simple Geometrical Viewpoint]]>
http://www.mdpi.com/2073-8994/6/2/396
We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via Möbius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.Symmetry2014-05-2262Article10.3390/sym60203963964082073-89942014-05-22doi: 10.3390/sym6020396Luis Sánchez-SotoJuan Monzón<![CDATA[Symmetry, Vol. 6, Pages 383-395: Hydrodynamic Helical Orientations of Nanofibers in a Vortex]]>
http://www.mdpi.com/2073-8994/6/2/383
In this review article, I report our recent studies on spectroscopic visualizations of macroscopic helical alignments of nanofibers in vortex flows. Our designed supramolecular nanofibers, formed through self-assemblies of dye molecules, helically align in torsional flows of a vortex generated by mechanical rotary stirring of the sample solutions. The nanofiber, formed through bundling of linear supramolecular polymers, aligns equally in right- and left-handed vortex flows. However, in contrast, a one-handedly twisted nanofiber, formed through helical bundling of the supramolecular polymers, shows unequal helical alignments in these torsional flows. When the helical handedness of the nanofiber matches that of the vortex flow, the nanofiber aligns more efficiently in the flowing fluid. Such phenomena are observed not only with the artificial helical supramolecular nanofibers but also with biological nanofibers such as double-stranded DNA.Symmetry2014-05-2062Review10.3390/sym60203833833952073-89942014-05-20doi: 10.3390/sym6020383Akihiko Tsuda<![CDATA[Symmetry, Vol. 6, Pages 368-382: Symmetry Perspectives on Some Auxetic Body-Bar Frameworks]]>
http://www.mdpi.com/2073-8994/6/2/368
Scalar mobility counting rules and their symmetry extensions are reviewed for finite frameworks and also for infinite periodic frameworks of the bar-and-joint, body-joint and body-bar types. A recently published symmetry criterion for the existence of equiauxetic character of an infinite framework is applied to two long known but apparently little studied hinged-hexagon frameworks, and is shown to detect auxetic behaviour in both. In contrast, for double-link frameworks based on triangular and square tessellations, other affine deformations can mix with the isotropic expansion mode.Symmetry2014-05-1562Article10.3390/sym60203683683822073-89942014-05-15doi: 10.3390/sym6020368Patrick FowlerSimon GuestTibor Tarnai<![CDATA[Symmetry, Vol. 6, Pages 345-367: Dynamic Processes in Prochiral Solvating Agents (pro-CSAs) Studied by NMR Spectroscopy]]>
http://www.mdpi.com/2073-8994/6/2/345
Several dynamic processes, including tautomerism and macrocyclic inversion, in 1H-NMR prochiral solvating agents (pro-CSAs) are investigated. Various features of pro-CSA, including modes of interaction for complex formation, stoichiometry, binding strength and temperature effects were compared for three representative pro-CSA molecules. Structural effects of conjugated tetrapyrrole pro-CSA on the mechanism of enantiomeric excess determination are also discussed. Detailed analysis of species (complexes) and dynamic processes occurring in solution and their 1H-NMR spectral manifestations at various temperatures is presented.Symmetry2014-05-1462Article10.3390/sym60203453453672073-89942014-05-14doi: 10.3390/sym6020345Jan LabutaShinsuke IshiharaKatsuhiko ArigaJonathan Hill<![CDATA[Symmetry, Vol. 6, Pages 329-344: Closed-Form Expressions for the Matrix Exponential]]>
http://www.mdpi.com/2073-8994/6/2/329
We discuss a method to obtain closed-form expressions of f(A), where f is an analytic function and A a square, diagonalizable matrix. The method exploits the Cayley–Hamilton theorem and has been previously reported using tools that are perhaps not sufficiently appealing to physicists. Here, we derive the results on which the method is based by using tools most commonly employed by physicists. We show the advantages of the method in comparison with standard approaches, especially when dealing with the exponential of low-dimensional matrices. In contrast to other approaches that require, e.g., solving differential equations, the present method only requires the construction of the inverse of the Vandermonde matrix. We show the advantages of the method by applying it to different cases, mostly restricting the calculational effort to the handling of two-by-two matrices.Symmetry2014-04-2962Article10.3390/sym60203293293442073-89942014-04-29doi: 10.3390/sym6020329F. De Zela<![CDATA[Symmetry, Vol. 6, Pages 308-328: The Almost Periodic Rigidity of Crystallographic Bar-Joint Frameworks]]>
http://www.mdpi.com/2073-8994/6/2/308
A crystallographic bar-joint framework, C in Rd, is shown to be almost periodically infinitesimally rigid if and only if it is strictly periodically infinitesimally rigid and the rigid unit mode (RUM) spectrum, Ω (C), is a singleton. Moreover, the almost periodic infinitesimal flexes of C are characterised in terms of a matrix-valued function, ΦC(z), on the d-torus, Td, determined by a full rank translation symmetry group and an associated motif of joints and bars.Symmetry2014-04-2462Article10.3390/sym60203083083282073-89942014-04-24doi: 10.3390/sym6020308Ghada BadriDerek KitsonStephen Power<![CDATA[Symmetry, Vol. 6, Pages 295-307: Dynamical Relation between Quantum Squeezing and Entanglement in Coupled Harmonic Oscillator System]]>
http://www.mdpi.com/2073-8994/6/2/295
In this paper, we investigate into the numerical and analytical relationship between the dynamically generated quadrature squeezing and entanglement within a coupled harmonic oscillator system. The dynamical relation between these two quantum features is observed to vary monotically, such that an enhancement in entanglement is attained at a fixed squeezing for a larger coupling constant. Surprisingly, the maximum attainable values of these two quantum entities are found to consistently equal to the squeezing and entanglement of the system ground state. In addition, we demonstrate that the inclusion of a small anharmonic perturbation has the effect of modifying the squeezing versus entanglement relation into a nonunique form and also extending the maximum squeezing to a value beyond the system ground state.Symmetry2014-04-2362Article10.3390/sym60202952953072073-89942014-04-23doi: 10.3390/sym6020295Lock ChewNing Chung<![CDATA[Symmetry, Vol. 6, Pages 256-294: Supramolecular Chirality in Porphyrin Chemistry]]>
http://www.mdpi.com/2073-8994/6/2/256
Supramolecular chirality, being an intelligent combination of supramolecular chemistry and chiral science, plays a decisive role in the functioning of various natural assemblies and has attracted much attention from the scientific community, due to different applications in modern technologies, medicine, pharmacology, catalysis and biomimetic research. Porphyrin molecules are of particular interest to study this phenomenon owing to their unique spectral, physico-chemical and synthetic properties. This review highlights the most important types of chiral porphyrin structures by using the best-suited representative examples, which are frequently used in the area of supramolecular chirality.Symmetry2014-04-2362Review10.3390/sym60202562562942073-89942014-04-23doi: 10.3390/sym6020256Victor Borovkov<![CDATA[Symmetry, Vol. 6, Pages 234-255: Quantum Local Symmetry of the D-Dimensional Non-Linear Sigma Model: A Functional Approach]]>
http://www.mdpi.com/2073-8994/6/2/234
We summarize recent progress on the symmetric subtraction of the Non-Linear Sigma Model in D dimensions, based on the validity of a certain Local Functional Equation (LFE) encoding the invariance of the SU(2) Haar measure under local left transformations. The deformation of the classical non-linearly realized symmetry at the quantum level is analyzed by cohomological tools. It is shown that all the divergences of the one-particle irreducible (1-PI) amplitudes (both on-shell and off-shell) can be classified according to the solutions of the LFE. Applications to the non-linearly realized Yang-Mills theory and to the electroweak theory, which is directly relevant to the model-independent analysis of LHC data, are briefly addressed.Symmetry2014-04-1762Review10.3390/sym60202342342552073-89942014-04-17doi: 10.3390/sym6020234Andrea Quadri<![CDATA[Symmetry, Vol. 6, Pages 222-233: Domain Specificity in Human Symmetry Preferences: Symmetry is Most Pleasant When Looking at Human Faces]]>
http://www.mdpi.com/2073-8994/6/2/222
Visual symmetry has been found to be preferred to asymmetry in a variety of domains and across species. A number of theories propose to explain why symmetry is preferred. In this article, I compare a perceptual bias view, in which symmetry is preferred due to factors inherit to the visual system, and an evolutionary advantage view, in which symmetry is preferred due to selection pressures on partner preference. Preferences for symmetry in three stimulus types were determined by having symmetric and asymmetric versions of the same images rated for pleasantness: human female faces, macaque monkey faces, and abstract art. It was found that preferences for symmetry were strongest for human female faces and weakest for art. This finding builds on previous research suggesting that symmetry preferences for human faces are different from symmetry preferences in other domains and that simple perceptual bias explanations do not wholly explain human visual face symmetry preferences. While consistent with an evolutionary advantage view, these data are also potentially explainable via a perceptual bias view which accounts for experience of stimuli. The interplay between these two views is discussed in the context of the current study.Symmetry2014-04-1762Article10.3390/sym60202222222332073-89942014-04-17doi: 10.3390/sym6020222Anthony Little<![CDATA[Symmetry, Vol. 6, Pages 210-221: Metalloporphyrin Symmetry in Chiral Recognition and Enantioselective Catalysis]]>
http://www.mdpi.com/2073-8994/6/2/210
Symmetry plays a fundamental role in chiral recognition and enantioselective catalysis. Porphyrins possess a number of structural features that make them attractive for the stereocontrol of chiral recognition and metal-catalyzed asymmetric reactions. This article is a brief account of our studies on chiral recognition and enantioselective catalysis by optically active metalloporphyrins. Some of the studies on chiral recognition and asymmetric catalysis by metalloporphyrins performed by others have also been included when useful.Symmetry2014-04-1062Review10.3390/sym60202102102212073-89942014-04-10doi: 10.3390/sym6020210Gérard SimonneauxHassan SrourPaul MauxSoizic ChevanceDaniel Carrie<![CDATA[Symmetry, Vol. 6, Pages 189-209: Topological Symmetry Groups of Small Complete Graphs]]>
http://www.mdpi.com/2073-8994/6/2/189
Topological symmetry groups were originally introduced to study the symmetries of non-rigid molecules, but have since been used to study the symmetries of any graph embedded in R3. In this paper, we determine for each complete graph Kn with n ≤ 6, what groups can occur as topological symmetry groups or orientation preserving topological symmetry groups of some embedding of the graph in R3.Symmetry2014-04-0862Article10.3390/sym60201891892092073-89942014-04-08doi: 10.3390/sym6020189Dwayne ChambersErica Flapan<![CDATA[Symmetry, Vol. 6, Pages 171-188: Development of Symmetry Concepts for Aperiodic Crystals]]>
http://www.mdpi.com/2073-8994/6/2/171
An overview is given of the use of symmetry considerations for aperiodic crystals. Superspace groups were introduced in the seventies for the description of incommensurate modulated phases with one modulation vector. Later, these groups were also used for quasi-periodic crystals of arbitrary rank. Further extensions use time reversal and time translation operations on magnetic and electrodynamic systems. An alternative description of magnetic structures to that with symmetry groups, the Shubnikov groups, is using representations of space groups. The same can be done for aperiodic crystals. A discussion of the relation between the two approaches is given. Representations of space groups and superspace groups play a role in the study of physical properties. These, and generalizations of them, are discussed for aperiodic crystals. They are used, in particular, for the characterization of phase transitions between aperiodic crystal phases.Symmetry2014-03-3162Article10.3390/sym60201711711882073-89942014-03-31doi: 10.3390/sym6020171Ted Janssen<![CDATA[Symmetry, Vol. 6, Pages 164-170: Spacetime Metrics from Gauge Potentials]]>
http://www.mdpi.com/2073-8994/6/2/164
I present an approach to gravity in which the spacetime metric is constructed from a non-Abelian gauge potential with values in the Lie algebra of the group U(2) (or the Lie algebra of quaternions). If the curvature of this potential vanishes, the metric reduces to a canonical curved background form reminiscent of the Friedmann S3 cosmological metric.Symmetry2014-03-2762Article10.3390/sym60201641641702073-89942014-03-27doi: 10.3390/sym6020164Ettore Minguzzi<![CDATA[Symmetry, Vol. 6, Pages 148-163: Symmetry Aspects of Dislocation-Effected Crystal Properties: Material Strength Levels and X-ray Topographic Imaging]]>
http://www.mdpi.com/2073-8994/6/1/148
Several materials science type research topics are described in which advantageous use of crystal symmetry considerations has been helpful in ferreting the essential elements of dislocation behavior in determining material properties or for characterizing crystal/polycrystalline structural relationships; for example: (1) the mechanical strengthening produced by a symmetrical bicrystal grain boundary; (2) cleavage crack formation at the intersection within a crystal of symmetrical dislocation pile-ups; (3) symmetry aspects of anisotropic crystal indentation hardness measurements; (4) X-ray diffraction topography imaging of dislocation strains and subgrain boundary misorientations; and (5) point and space group aspects of twinning. Several applications are described in relation to the strengthening of grain boundaries in nanopolycrystals and of multiply-oriented crystal grains in polysilicon photovoltaic solar cell materials. A number of crystallographic aspects of the different topics are illustrated with a stereographic method of presentation.Symmetry2014-03-2061Review10.3390/sym60101481481632073-89942014-03-20doi: 10.3390/sym6010148Ronald Armstrong<![CDATA[Symmetry, Vol. 6, Pages 111-147: The Hunt for Supersymmetry at the Tevatron]]>
http://www.mdpi.com/2073-8994/6/1/111
During the Tevatron data-taking period from April 2001 to September 2011 (Run-II), several searches for supersymmetric particles were performed. The results from searches by the CDF and DØ collaborations are concisely reviewed. This includes results up to the summer conferences of 2013. Model-independent and model-dependent limits on new particle production are set, and interpretations in supersymmetric models are given. Several limits from the Large Electron Positron (LEP) era have been extended. Specific results are placed into the context of the Tevatron performance expectations and a few of the current results from searches at the Large Hadron Collider (LHC).Symmetry2014-03-1961Review10.3390/sym60101111111472073-89942014-03-19doi: 10.3390/sym6010111André Sopczak<![CDATA[Symmetry, Vol. 6, Pages 103-110: Pseudo Hermitian Interactions in the Dirac Equation]]>
http://www.mdpi.com/2073-8994/6/1/103
We consider a (2 + 1)-dimensional massless Dirac equation in the presence of complex vector potentials. It is shown that such vector potentials (leading to complex magnetic fields) can produce bound states, and the Dirac Hamiltonians are η-pseudo Hermitian. Some examples have been explicitly worked out.Symmetry2014-03-1761Article10.3390/sym60101031031102073-89942014-03-17doi: 10.3390/sym6010103Orlando PanellaPinaki Roy<![CDATA[Symmetry, Vol. 6, Pages 90-102: Using Symmetries (Beyond Geometric Symmetries) in Chemical Computations: Computing Parameters of Multiple Binding Sites]]>
http://www.mdpi.com/2073-8994/6/1/90
We show how transformation group ideas can be naturally used to generate efficient algorithms for scientific computations. The general approach is illustrated on the example of determining, from the experimental data, the dissociation constants related to multiple binding sites. We also explain how the general transformation group approach is related to the standard (backpropagation) neural networks; this relation justifies the potential universal applicability of the group-related approach.Symmetry2014-02-2561Article10.3390/sym6010090901022073-89942014-02-25doi: 10.3390/sym6010090Andres OrtizVladik Kreinovich<![CDATA[Symmetry, Vol. 6, Pages 89: Acknowledgement to Reviewers of Symmetry in 2013]]>
http://www.mdpi.com/2073-8994/6/1/89
The editors of Symmetry would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2013. [...]Symmetry2014-02-2461Editorial10.3390/sym601008989892073-89942014-02-24doi: 10.3390/sym6010089 Symmetry Editorial Office<![CDATA[Symmetry, Vol. 6, Pages 67-88: On General Off-Shell Representations of World Line (1D) Supersymmetry]]>
http://www.mdpi.com/2073-8994/6/1/67
Every finite-dimensional unitary representation of the N-extended world line supersymmetry without central charges may be obtained by a sequence of differential transformations from a direct sum of minimal Adinkras, simple supermultiplets that are identifiable with representations of the Clifford algebra. The data specifying this procedure is a sequence of subspaces of the direct sum of Adinkras, which then opens an avenue for the classification of the continuum of the so-constructed off-shell supermultiplets.Symmetry2014-02-0361Article10.3390/sym601006767882073-89942014-02-03doi: 10.3390/sym6010067Charles DoranTristan HübschKevin IgaGregory Landweber<![CDATA[Symmetry, Vol. 6, Pages 23-66: Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator]]>
http://www.mdpi.com/2073-8994/6/1/23
The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Ζ2 x Ζ2 symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks.Symmetry2014-01-0361Article10.3390/sym601002323662073-89942014-01-03doi: 10.3390/sym6010023Ian Stewart<![CDATA[Symmetry, Vol. 6, Pages 1-22: Peripheral Contour Grouping and Saccade Targeting: The Role of Mirror Symmetry]]>
http://www.mdpi.com/2073-8994/6/1/1
Integrating shape contours in the visual periphery is vital to our ability to locate objects and thus make targeted saccadic eye movements to efficiently explore our surroundings. We tested whether global shape symmetry facilitates peripheral contour integration and saccade targeting in three experiments, in which observers responded to a successful peripheral contour detection by making a saccade towards the target shape. The target contours were horizontally (Experiment 1) or vertically (Experiments 2 and 3) mirror symmetric. Observers responded by making a horizontal (Experiments 1 and 2) or vertical (Experiment 3) eye movement. Based on an analysis of the saccadic latency and accuracy, we conclude that the figure-ground cue of global mirror symmetry in the periphery has little effect on contour integration or on the speed and precision with which saccades are targeted towards objects. The role of mirror symmetry may be more apparent under natural viewing conditions with multiple objects competing for attention, where symmetric regions in the visual field can pre-attentively signal the presence of objects, and thus attract eye movements.Symmetry2014-01-0261Article10.3390/sym60100011222073-89942014-01-02doi: 10.3390/sym6010001Michaël SassiMaarten DemeyerJohan Wagemans<![CDATA[Symmetry, Vol. 5, Pages 344-354: Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points]]>
http://www.mdpi.com/2073-8994/5/4/344
Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group.Symmetry2013-12-0954Article10.3390/sym50403443443542073-89942013-12-09doi: 10.3390/sym5040344Guillaume VasseurYannick Fagot-RevuratBertrand KierrenMuriel SicotDaniel Malterre<![CDATA[Symmetry, Vol. 5, Pages 313-343: Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models]]>
http://www.mdpi.com/2073-8994/5/4/313
We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group.Symmetry2013-11-1154Article10.3390/sym50403133133432073-89942013-11-11doi: 10.3390/sym5040313Stefan MaierCarsten HonerkampQiang-Hua Wang<![CDATA[Symmetry, Vol. 5, Pages 287-312: Multiple Solutions to Implicit Symmetric Boundary Value Problems for Second Order Ordinary Differential Equations (ODEs): Equivariant Degree Approach]]>
http://www.mdpi.com/2073-8994/5/4/287
In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D4-symmetries.Symmetry2013-11-0754Article10.3390/sym50402872873122073-89942013-11-07doi: 10.3390/sym5040287Zalman BalanovWieslaw KrawcewiczZhichao LiMylinh Nguyen<![CDATA[Symmetry, Vol. 5, Pages 271-286: A New Route to the Majorana Equation]]>
http://www.mdpi.com/2073-8994/5/4/271
In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group.Symmetry2013-09-2654Article10.3390/sym50402712712862073-89942013-09-26doi: 10.3390/sym5040271Eckart Marsch<![CDATA[Symmetry, Vol. 5, Pages 253-270: Supersymmetric Version of the Euler System and Its Invariant Solutions]]>
http://www.mdpi.com/2073-8994/5/3/253
In this paper, we formulate a supersymmetric extension of the Euler system of equations. We compute a superalgebra of Lie symmetries of the supersymmetric system. Next, we classify the one-dimensional subalgebras of this superalgebra into 49 equivalence conjugation classes. For some of the subalgebras, the invariants have a non-standard structure. For nine selected subalgebras, we use the symmetry reduction method to find invariants, orbits and reduced systems. Through the solutions of these reduced systems, we obtain solutions of the supersymmetric Euler system. The obtained solutions include bumps, kinks, multiple wave solutions and solutions expressed in terms of arbitrary functions.Symmetry2013-07-1253Article10.3390/sym50302532532702073-89942013-07-12doi: 10.3390/sym5030253A. GrundlandAlexander Hariton<![CDATA[Symmetry, Vol. 5, Pages 233-252: Symmetries Shared by the Poincaré Group and the Poincaré Sphere]]>
http://www.mdpi.com/2073-8994/5/3/233
Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner’s little groups for internal space-time symmetries are studied in detail. While the particle mass is a Lorentz-invariant quantity, it is shown to be possible to address its variations in terms of the decoherence mechanism in polarization optics.Symmetry2013-06-2753Article10.3390/sym50302332332522073-89942013-06-27doi: 10.3390/sym5030233Young KimMarilyn Noz<![CDATA[Symmetry, Vol. 5, Pages 215-232: Fermi Surface Reconstruction due to Hidden Rotating Antiferromagnetism in N and P-Type High-TC Cuprates]]>
http://www.mdpi.com/2073-8994/5/2/215
The Fermi surface calculated within the rotating antiferromagnetism theory undergoes a topological change when doping changes from p-type to n-type, in qualitative agreement with experimental data for n-type cuprate Nd2−xCexCuO4 and p-type La2−xSrxCuO4. Also, the reconstruction of the Fermi surface, observed experimentally close to optimal doping in p-type cuprates, and slightly higher than optimal doping in the overdoped regime for this n-type high-TC cuprate, is well accounted for in this theory. This reconstruction is a consequence of the quantum criticality caused by the disappearance of rotating antiferromagnetism. The present results are in qualitative agreement with recently observed quantum oscillations in some high-TC cuprates. This paper presents new results about the application of the rotating antiferromagnetism theory to the study of the electronic structure for n-type materials.Symmetry2013-05-0752Article10.3390/sym50202152152322073-89942013-05-07doi: 10.3390/sym5020215Mohamed Azzouz<![CDATA[Symmetry, Vol. 5, Pages 119-214: Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry]]>
http://www.mdpi.com/2073-8994/5/2/119
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking, but most typically manifest a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate, regardless of the parity of bulk (super)spins. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.Symmetry2013-04-2652Review10.3390/sym50201191192142073-89942013-04-26doi: 10.3390/sym5020119Kazuki HasebeKeisuke Totsuka<![CDATA[Symmetry, Vol. 5, Pages 86-118: Frame Transformation Relations and Symmetry Analysis of Fluxional Symmetric Rotor Dimers]]>
http://www.mdpi.com/2073-8994/5/1/86
The theory of Frame transformation relations between the states of Born Oppenheimer and the weak coupling approximations is developed for polyatomic molecules. The symmetry relations are a generalization of the frame transformation relations derived by Harter and Crogman for coupled rotor molecules. A key internal symmetry label (named “soul”) is defined so that it remains a constant label for frame transformation relations, and is conserved during vibronic transitions, ionization, and even dissociation provided the nuclear spin-rotation interaction is relatively small. Simplified procedures are given for obtaining selection rules, statistical weights, and matrix elements of multipole operators for common molecules having various point symmetries.Symmetry2013-02-0551Article10.3390/sym5010086861182073-89942013-02-05doi: 10.3390/sym5010086Horace CrogmanBumgyunmiga ChoiHarrison ChenWilliam Harter<![CDATA[Symmetry, Vol. 5, Pages 81-85: Perlman and Wellner’s Circular and Transformed Circular Copulas are Particular Beta and t Copulas]]>
http://www.mdpi.com/2073-8994/5/1/81
All but one of the copulas in a recent paper in Symmetry by Perlman and Wellner can be identified as particular members of either the beta or t families of elliptical copulas.Symmetry2013-01-3151Short Note10.3390/sym501008181852073-89942013-01-31doi: 10.3390/sym5010081M. C. Jones<![CDATA[Symmetry, Vol. 5, Pages 54-80: Non-Crystallographic Symmetry in Packing Spaces]]>
http://www.mdpi.com/2073-8994/5/1/54
In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure.Symmetry2013-01-0951Article10.3390/sym501005454802073-89942013-01-09doi: 10.3390/sym5010054Valery RauLeonty LomtevTamara Rau<![CDATA[Symmetry, Vol. 5, Pages 47-53: A Note on Lower Bounds for Colourful Simplicial Depth]]>
http://www.mdpi.com/2073-8994/5/1/47
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.Symmetry2013-01-0751Short Note10.3390/sym501004747532073-89942013-01-07doi: 10.3390/sym5010047Antoine DezaTamon StephenFeng Xie<![CDATA[Symmetry, Vol. 5, Pages 1-46: Taylor–Socolar Hexagonal Tilings as Model Sets]]>
http://www.mdpi.com/2073-8994/5/1/1
The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice Q in the plane. If XQ denotes the set of all Taylor–Socolar tilings with centers on Q, then XQ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of Q.The Q-adic completion Q of Q is a natural factor of XQ and the natural mapping XQ → Q is bijective except at a dense set of points of measure 0 in /Q. We show that XQ consists of three LI classes under translation. Two of these LI classes are very small, namely countable Q-orbits in XQ. The other is a minimal dynamical system, which maps surjectively to /Q and which is variously 2 : 1, 6 : 1, and 12 : 1 at the singular points. We further develop the formula of what determines the parity of the tiles of a tiling in terms of the coordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull XQ; more precisely the two hulls are mutually locally derivable.Symmetry2012-12-2851Article10.3390/sym50100011462073-89942012-12-28doi: 10.3390/sym5010001Jeong-Yup LeeRobert Moody<![CDATA[Symmetry, Vol. 4, Pages 667-685: Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries]]>
http://www.mdpi.com/2073-8994/4/4/667
The eigenfunction approach used for discrete symmetries is deduced from the concept of quantum numbers. We show that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table). The need of a canonical chain of groups to establish a complete set of commuting operators is emphasized. This analysis allows us to establish in natural form the connection between the quantum numbers and the eigenfunction method proposed by J.Q. Chen to obtain symmetry adapted functions. We then proceed to present a friendly version of the eigenfunction method to project functions.Symmetry2012-11-3044Article10.3390/sym40406676676852073-89942012-11-30doi: 10.3390/sym4040667Renato Lemus<![CDATA[Symmetry, Vol. 4, Pages 644-666: A Peculiarly Cerebroid Convex Zygo-Dodecahedron is an Axiomatically Balanced “House of Blues”: The Circle of Fifths to the Circle of Willis to Cadherin Cadenzas]]>
http://www.mdpi.com/2073-8994/4/4/644
A bilaterally symmetrical convex dodecahedron consisting of twelve quadrilateral faces is derived from the icosahedron via a process akin to Fuller’s Jitterbug Transformation. The unusual zygomorphic dodecahedron so obtained is shown to harbor a bilaterally symmetrical jazz/blues harmonic code on its twelve faces that is related to such fundamental music theoretical constructs as the Circle of Fifths and Euler’s tonnetz. Curiously, the patterning within the aforementioned zygo-dodecahedron is discernibly similar to that observed in a ventral view of the human brain. Moreover, this same pattern is arguably evident during development of the embryonic pharynx. A possible role for the featured zygo-dodecahedron in cephalogenesis is considered. Recent studies concerning type II cadherins, an important class of proteins that promote cell adhesion, have generated data that is demonstrated to conform to this zygo-dodecahedral brain model in a substantially congruous manner.Symmetry2012-11-1544Article10.3390/sym40406446446662073-89942012-11-15doi: 10.3390/sym4040644David Becker<![CDATA[Symmetry, Vol. 4, Pages 626-643: Dirac Matrices and Feynman’s Rest of the Universe]]>
http://www.mdpi.com/2073-8994/4/4/626
There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four γ matrices. These fifteen matrices can also serve as the generators of the group SL(4, r). The second set consists of ten generators of the Sp(4) group which Dirac derived from two coupled harmonic oscillators. It is shown possible to extend the symmetry of Sp(4) to that of SL(4, r) if the area of the phase space of one of the oscillators is allowed to become smaller without a lower limit. While there are no restrictions on the size of phase space in classical mechanics, Feynman’s rest of the universe makes this Sp(4)-to-SL(4, r) transition possible. The ten generators are for the world where quantum mechanics is valid. The remaining five generators belong to the rest of the universe. It is noted that the groups SL(4, r) and Sp(4) are locally isomorphic to the Lorentz groups O(3, 3) and O(3, 2) respectively. This allows us to interpret Feynman’s rest of the universe in terms of space-time symmetry.Symmetry2012-10-3044Article10.3390/sym40406266266432073-89942012-10-30doi: 10.3390/sym4040626Young S. KimMarilyn E. Noz<![CDATA[Symmetry, Vol. 4, Pages 603-625: N = (4,4) Supersymmetry and T-Duality]]>
http://www.mdpi.com/2073-8994/4/4/603
A sigma model with four-dimensional target space parametrized by chiral and twisted chiral N =(2,2) superfields can be extended to N =(4,4) supersymmetry off-shell, but this is not true for a model of semichiral fields, where the N = (4,4) supersymmetry can only be realized on-shell. The two models can be related to each other by T-duality. In this paper we perform a duality transformation from a chiral and twisted chiral model with off-shell N = (4,4) supersymmetry to a semichiral model. We find that additional non-linear terms must be added to the original transformations to obtain a semichiral model with N =(4,4) supersymmetry, and that the algebra closes on-shell as a direct consequence of the T-duality.Symmetry2012-10-2444Article10.3390/sym40406036036252073-89942012-10-24doi: 10.3390/sym4040603Malin Göteman<![CDATA[Symmetry, Vol. 4, Pages 581-602: Hexagonal Inflation Tilings and Planar Monotiles]]>
http://www.mdpi.com/2073-8994/4/4/581
Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces.Symmetry2012-10-2244Article10.3390/sym40405815816022073-89942012-10-22doi: 10.3390/sym4040581Michael BaakeFranz GählerUwe Grimm<![CDATA[Symmetry, Vol. 4, Pages 566-580: On the Notions of Symmetry and Aperiodicity for Delone Sets]]>
http://www.mdpi.com/2073-8994/4/4/566
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions.Symmetry2012-10-1044Article10.3390/sym40405665665802073-89942012-10-10doi: 10.3390/sym4040566Michael BaakeUwe Grimm<![CDATA[Symmetry, Vol. 4, Pages 545-565: Barrel Pseudotilings]]>
http://www.mdpi.com/2073-8994/4/3/545
This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel tiles or congruent apeirogonal pseudotiles. No physical space is occupied by the apeirogonal pseudotiles. Many interesting pseudotilings arise from plane tilings with high symmetry. As combinatorial structures, these are abstract polytopes of rank 4 with both finite and infinite 2-faces and facets.Symmetry2012-08-3043Article10.3390/sym40305455455652073-89942012-08-30doi: 10.3390/sym4030545Undine Leopold<![CDATA[Symmetry, Vol. 4, Pages 537-544: A Higher Dimensional Description of the Structure of β-Mn]]>
http://www.mdpi.com/2073-8994/4/3/537
The structure of β-Mn crystallizes in space group P4132. The pseudo 8-fold nature of the 41 axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular 8-fold axes cannot generate P4132 on projection to 3-d space and how this may instead be accomplished from a six-dimensional model. A procedure for generating the actual structure of β-Mn lifted to six-dimensional space is given.Symmetry2012-08-2743Article10.3390/sym40305375375442073-89942012-08-27doi: 10.3390/sym4030537Sven LidinDaniel Fredrickson<![CDATA[Symmetry, Vol. 4, Pages 517-536: Supersymmetric Extensions of Non-Relativistic Scaling Algebras]]>
http://www.mdpi.com/2073-8994/4/3/517
An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent z plays an important role in constructing the gravity duals for non-relativistic field theories. Supersymmetric extensions of symmetry algebras including the anisotropic scaling are very helpful to consider holographic relations accurately. We give a short summary on the classification of superalgebras with the anisotropic scaling as subalgebras of the following Lie superalgebras, psu(2,2|4), osp(8|4) and osp (8*|4), which appear in the study of AdS/CFT in type IIB string and M theories. It contains supersymmetric extensions of Schrödinger algebra and Lifshitz algebra.Symmetry2012-08-2443Review10.3390/sym40305175175362073-89942012-08-24doi: 10.3390/sym4030517Makoto SakaguchiKentaroh Yoshida<![CDATA[Symmetry, Vol. 4, Pages 507-516: Flexibility of Hydrogen Bond and Lowering of Symmetry in Proton Conductor]]>
http://www.mdpi.com/2073-8994/4/3/507
In order to investigate why crystal symmetry lowers with increasing temperature by phase transition of TII–III (=369 K) in Cs3H(SeO4)2, in spite of the fact that crystal symmetry in the high-temperature phase of many ionic conductors becomes higher by the phase transition, we have studied the relation between the change in crystal symmetry and the appearance of proton motion. It was found from the analysis of domains based on crystal structure that the number of possible geometrical arrangement of hydrogen bond in phase II becomes two times larger than that in phase III, derived from the lowering of crystal symmetry with increasing temperature. These results indicate that the lowering of crystal symmetry in phase II appears by the increase of the number of geometrical arrangements and by the enhancement of the flexibility of hydrogen bond. Considering that the enhancement of the flexibility of hydrogen bond yields mobile proton in phase II, it is deduced that mobile proton in phase II appears in exchange for the lowering of crystal symmetry at II–III phase transition.Symmetry2012-08-2343Article10.3390/sym40305075075162073-89942012-08-23doi: 10.3390/sym4030507Yukihiko YoshidaJunko HatoriHinako KawakamiYasumitsu MatsuoSeiichiro Ikehata<![CDATA[Symmetry, Vol. 4, Pages 474-506: Supersymmetric Sigma Model Geometry]]>
http://www.mdpi.com/2073-8994/4/3/474
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces.Symmetry2012-08-2343Article10.3390/sym40304744745062073-89942012-08-23doi: 10.3390/sym4030474Ulf Lindström<![CDATA[Symmetry, Vol. 4, Pages 452-473: Supersymmetric Quantum Mechanics and Solvable Models]]>
http://www.mdpi.com/2073-8994/4/3/452
We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ħ-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ħ.Symmetry2012-08-1643Review10.3390/sym40304524524732073-89942012-08-16doi: 10.3390/sym4030452Jonathan BougieAsim GangopadhyayaJeffry MallowConstantin Rasinariu<![CDATA[Symmetry, Vol. 4, Pages 441-451: Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations]]>
http://www.mdpi.com/2073-8994/4/3/441
We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts.Symmetry2012-08-0843Article10.3390/sym40304414414512073-89942012-08-08doi: 10.3390/sym4030441Laurent DelisleVéronique Hussin<![CDATA[Symmetry, Vol. 4, Pages 427-440: Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection]]>
http://www.mdpi.com/2073-8994/4/3/427
In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 x 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 x 3 subequations which are thus a supersymmetric link between fermionic and bosonicdegrees of freedom.Symmetry2012-08-0743Article10.3390/sym40304274274402073-89942012-08-07doi: 10.3390/sym4030427Andrzej Okniński<![CDATA[Symmetry, Vol. 4, Pages 379-426: Symmetry-Adapted Fourier Series for the Wallpaper Groups]]>
http://www.mdpi.com/2073-8994/4/3/379
Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the Fourier coefficients. Visualising the equivalencies of Fourier coefficients by means of discrete 2D maps reveals how direct-space symmetry is transformed into coefficient-space symmetry. Explicit expressions are given for the Fourier series and Fourier coefficient maps of both real and complex functions, readily applicable to the description of the properties of 2D materials like graphene or boron-nitride.Symmetry2012-07-1743Article10.3390/sym40303793794262073-89942012-07-17doi: 10.3390/sym4030379Bart Verberck