Mathematics
http://www.mdpi.com/journal/mathematics
Latest open access articles published in Mathematics at http://www.mdpi.com/journal/mathematics<![CDATA[Mathematics, Vol. 3, Pages 1083-1094: The San Francisco MSM Epidemic: A Retrospective Analysis]]>
http://www.mdpi.com/2227-7390/3/4/1083
We investigate various scenarios for ending the San Francisco MSM (men having sex with men) HIV/AIDS epidemic (1978–1984). We use our previously developed model and explore changes due to prevention strategies such as testing, treatment and reduction of the number of contacts. Here we consider a “what-if” scenario, by comparing different treatment strategies, to determine which factor has the greatest impact on reducing the HIV/AIDS epidemic. The factor determining the future of the epidemic is the reproduction number R0; if R0 &lt; 1, the epidemic is stopped. We show that treatment significantly reduces the total number of infected people. We also investigate the effect a reduction in the number of contacts after seven years, when the HIV/AIDS threat became known, would have had in the population. Both reduction of contacts and treatment alone, however, would not have been enough to bring R0 below one; but when combined, we show that the effective R0 becomes less than one, and therefore the epidemic would have been eradicated.Mathematics2015-11-2434Article10.3390/math3041083108310942227-73902015-11-24doi: 10.3390/math3041083Brandy RapatskiJuan Tolosa<![CDATA[Mathematics, Vol. 3, Pages 1069-1082: A Class of Extended Mittag–Leffler Functions and Their Properties Related to Integral Transforms and Fractional Calculus]]>
http://www.mdpi.com/2227-7390/3/4/1069
In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function, including, for example, certain basic properties, Laplace transform, Mellin transform and Euler-Beta transform. Further, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag–Leffler function are investigated. Some interesting special cases of our main results are also pointed out.Mathematics2015-11-0634Article10.3390/math3041069106910822227-73902015-11-06doi: 10.3390/math3041069Rakesh Parmar<![CDATA[Mathematics, Vol. 3, Pages 1045-1068: Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure]]>
http://www.mdpi.com/2227-7390/3/4/1045
Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrödinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states (generated through the power moments) can define an L2 convergent, non-orthonormal, basis expansion with sufficient pointwise convergent behaviors, enabling the direct coupling of the global (power moments) and local (Taylor series) expansions in configuration space. Our formulation is elaborated within the orthogonal polynomial projection quantization (OPPQ) configuration space representation previously developed The quantization approach pursued here defines an alternative strategy emphasizing the relevance of OPPQ to the reconstruction of the local structure of the physical states.Mathematics2015-11-0534Article10.3390/math3041045104510682227-73902015-11-05doi: 10.3390/math3041045Carlos HandyDaniel VrinceanuCarl MarthHarold Brooks<![CDATA[Mathematics, Vol. 3, Pages 1032-1044: A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation]]>
http://www.mdpi.com/2227-7390/3/4/1032
This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.Mathematics2015-10-2734Article10.3390/math3041032103210442227-73902015-10-27doi: 10.3390/math3041032Treena Basu<![CDATA[Mathematics, Vol. 3, Pages 1001-1031: A Cohomology Theory for Commutative Monoids]]>
http://www.mdpi.com/2227-7390/3/4/1001
Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.Mathematics2015-10-2734Article10.3390/math3041001100110312227-73902015-10-27doi: 10.3390/math3041001María Calvo-CerveraAntonio Cegarra<![CDATA[Mathematics, Vol. 3, Pages 984-1000: Gauge Invariance and Symmetry Breaking by Topology and Energy Gap]]>
http://www.mdpi.com/2227-7390/3/4/984
For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.Mathematics2015-10-2234Article10.3390/math304098498410002227-73902015-10-22doi: 10.3390/math3040984Franco StrocchiCarlo Heissenberg<![CDATA[Mathematics, Vol. 3, Pages 961-983: Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics]]>
http://www.mdpi.com/2227-7390/3/4/961
A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle’s theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided.Mathematics2015-10-2134Article10.3390/math30409619619832227-73902015-10-21doi: 10.3390/math3040961Ellina GrigorievaEvgenii Khailov<![CDATA[Mathematics, Vol. 3, Pages 945-960: Reformulated First Zagreb Index of Some Graph Operations]]>
http://www.mdpi.com/2227-7390/3/4/945
The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper, we study the behavior of the reformulated first Zagreb index and apply our results to different chemically interesting molecular graphs and nano-structures.Mathematics2015-10-1634Article10.3390/math30409459459602227-73902015-10-16doi: 10.3390/math3040945Nilanjan DeSk. NayeemAnita Pal<![CDATA[Mathematics, Vol. 3, Pages 913-944: Understanding Visceral Leishmaniasis Disease Transmission and its Control—A Study Based on Mathematical Modeling]]>
http://www.mdpi.com/2227-7390/3/3/913
Understanding the transmission and control of visceral leishmaniasis, a neglected tropical disease that manifests in human and animals, still remains a challenging problem globally. To study the nature of disease spread, we have developed a compartment-based mathematical model of zoonotic visceral leishmaniasis transmission among three different populations—human, animal and sandfly; dividing the human class into asymptomatic, symptomatic, post-kala-azar dermal leishmaniasis and transiently infected. We analyzed this large model for positivity, boundedness and stability around steady states in different diseased and disease-free scenarios and derived the analytical expression for basic reproduction number (R0). Sensitive parameters for each infected population were identified and varied to observe their effects on the steady state. Epidemic threshold R0 was calculated for every parameter variation. Animal population was identified to play a protective role in absorbing infection, thereby controlling the disease spread in human. To test the predictive ability of the model, seasonal fluctuation was incorporated in the birth rate of the sandflies to compare the model predictions with real data. Control scenarios on this real population data were created to predict the degree of control that can be exerted on the sensitive parameters so as to effectively reduce the infected populations.Mathematics2015-09-2333Article10.3390/math30309139139442227-73902015-09-23doi: 10.3390/math3030913Abhishek SubramanianVidhi SinghRam Sarkar<![CDATA[Mathematics, Vol. 3, Pages 897-912: Photon Localization Revisited]]>
http://www.mdpi.com/2227-7390/3/3/897
In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as quantum field ontology associated with the eventualization principle, which enables us to explain the mutual relations, back and forth, between quantum fields and various forms of particles in the localized form of the former.Mathematics2015-09-2333Article10.3390/math30308978979122227-73902015-09-23doi: 10.3390/math3030897Izumi OjimaHayato Saigo<![CDATA[Mathematics, Vol. 3, Pages 891-896: Smooth K-groups for Monoid Algebras and K-regularity]]>
http://www.mdpi.com/2227-7390/3/3/891
The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quasi stable locally convex algebras is established. We prove that the Quillen K-groups are isomorphic to smooth K-groups for monoid algebras over quasi-stable Frechet algebras having a properly uniformly bounded approximate unit and not necessarily m-convex. Based on these results the K-regularity property for quasi-stable Frechet algebras having a properly uniformly bounded approximate unit is established.Mathematics2015-09-1033Article10.3390/math30308918918962227-73902015-09-10doi: 10.3390/math3030891Hvedri Inassaridze<![CDATA[Mathematics, Vol. 3, Pages 880-890: A Note on Necessary Optimality Conditions for a Model with Differential Infectivity in a Closed Population]]>
http://www.mdpi.com/2227-7390/3/3/880
The aim of this note is to present the necessary optimality conditions for a model (in closed population) of an immunizing disease similar to hepatitis B following. We study the impact of medical tests and controls involved in curing this kind of immunizing disease and deduced a well posed adjoint system if there exists an optimal control.Mathematics2015-08-2133Article10.3390/math30308808808902227-73902015-08-21doi: 10.3390/math3030880Yannick Kouakep<![CDATA[Mathematics, Vol. 3, Pages 843-879: Chern-Simons Path Integrals in S2 × S1]]>
http://www.mdpi.com/2227-7390/3/3/843
Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\).Mathematics2015-08-2133Article10.3390/math30308438438792227-73902015-08-21doi: 10.3390/math3030843Adrian Lim<![CDATA[Mathematics, Vol. 3, Pages 781-842: Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics]]>
http://www.mdpi.com/2227-7390/3/3/781
A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.Mathematics2015-08-2033Article10.3390/math30307817818422227-73902015-08-20doi: 10.3390/math3030781Kundeti Muralidhar<![CDATA[Mathematics, Vol. 3, Pages 758-780: The Segal–Bargmann Transform for Odd-Dimensional Hyperbolic Spaces]]>
http://www.mdpi.com/2227-7390/3/3/758
We develop isometry and inversion formulas for the Segal–Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.Mathematics2015-08-1833Article10.3390/math30307587587802227-73902015-08-18doi: 10.3390/math3030758Brian HallJeffrey Mitchell<![CDATA[Mathematics, Vol. 3, Pages 746-757: A Moonshine Dialogue in Mathematical Physics]]>
http://www.mdpi.com/2227-7390/3/3/746
Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive at dessins d’enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell’s theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincaré would have been happy to see it.Mathematics2015-08-1433Essay10.3390/math30307467467572227-73902015-08-14doi: 10.3390/math3030746Michel Planat<![CDATA[Mathematics, Vol. 3, Pages 727-745: From Classical to Discrete Gravity through Exponential Non-Standard Lagrangians in General Relativity]]>
http://www.mdpi.com/2227-7390/3/3/727
Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this paper is to fill the gap. Though non-standard Lagrangians may be defined by a multitude form, in this paper, we considered the exponential type. One basic feature of exponential non-standard Lagrangians concerns the modified Euler-Lagrange equation obtained from the standard variational analysis. Accordingly, when applied to spacetime geometries, one unsurprisingly expects modified geodesic equations. However, when taking into account the time-like paths parameterization constraint, remarkably, it was observed that mutually discrete gravity and discrete spacetime emerge in the theory. Two different independent cases were obtained: A geometrical manifold with new spacetime coordinates augmented by a metric signature change and a geometrical manifold characterized by a discretized spacetime metric. Both cases give raise to Einstein’s field equations yet the gravity is discretized and originated from “spacetime discreteness”. A number of mathematical and physical implications of these results were discussed though this paper and perspectives are given accordingly.Mathematics2015-08-1433Article10.3390/math30307277277452227-73902015-08-14doi: 10.3390/math3030727Rami El-Nabulsi<![CDATA[Mathematics, Vol. 3, Pages 690-726: Root Operators and “Evolution” Equations]]>
http://www.mdpi.com/2227-7390/3/3/690
Root-operator factorization à la Dirac provides an effective tool to deal with equations, which are not of evolution type, or are ruled by fractional differential operators, thus eventually yielding evolution-like equations although for a multicomponent vector. We will review the method along with its extension to root operators of degree higher than two. Also, we will show the results obtained by the Dirac-method as well as results from other methods, specifically in connection with evolution-like equations ruled by square-root operators, that we will address to as relativistic evolution equations.Mathematics2015-08-1333Article10.3390/math30306906907262227-73902015-08-13doi: 10.3390/math3030690Giuseppe DattoliAmalia Torre<![CDATA[Mathematics, Vol. 3, Pages 666-689: Evaluation of Interpolants in Their Ability to Fit Seismometric Time Series]]>
http://www.mdpi.com/2227-7390/3/3/666
This article is devoted to the study of the ASARCO demolition seismic data. Two different classes of modeling techniques are explored: First, mathematical interpolation methods and second statistical smoothing approaches for curve fitting. We estimate the characteristic parameters of the propagation medium for seismic waves with multiple mathematical and statistical techniques, and provide the relative advantages of each approach to address fitting of such data. We conclude that mathematical interpolation techniques and statistical curve fitting techniques complement each other and can add value to the study of one dimensional time series seismographic data: they can be use to add more data to the system in case the data set is not large enough to perform standard statistical tests.Mathematics2015-08-0733Article10.3390/math30306666666892227-73902015-08-07doi: 10.3390/math3030666Kanadpriya BasuMaria MarianiLaura SerpaRitwik Sinha<![CDATA[Mathematics, Vol. 3, Pages 653-665: Zeta Function Expression of Spin Partition Functions on Thermal AdS3]]>
http://www.mdpi.com/2227-7390/3/3/653
We find a Selberg zeta function expression of certain one-loop spin partition functions on three-dimensional thermal anti-de Sitter space. Of particular interest is the partition function of higher spin fermionic particles. We also set up, in the presence of spin, a Patterson-type formula involving the logarithmic derivative of zeta.Mathematics2015-07-2833Article10.3390/math30306536536652227-73902015-07-28doi: 10.3390/math3030653Floyd L.Williams<![CDATA[Mathematics, Vol. 3, Pages 644-652: On the Nature of the Tsallis–Fourier Transform]]>
http://www.mdpi.com/2227-7390/3/3/644
By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis’ q-statistics may revolve around equivalence classes of distributions and not individual ones, as orthodox statistics does. We solve here the qFT’s non-invertibility issue, but discover a problem that remains open.Mathematics2015-07-2133Article10.3390/math30306446446522227-73902015-07-21doi: 10.3390/math3030644A. PlastinoMario Rocca<![CDATA[Mathematics, Vol. 3, Pages 626-643: Time Automorphisms on C*-Algebras]]>
http://www.mdpi.com/2227-7390/3/3/626
Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C*-algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.Mathematics2015-07-1633Article10.3390/math30306266266432227-73902015-07-16doi: 10.3390/math3030626R. Hilfer<![CDATA[Mathematics, Vol. 3, Pages 615-625: Reproducing Kernel Hilbert Space vs. Frame Estimates]]>
http://www.mdpi.com/2227-7390/3/3/615
We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn H into a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.Mathematics2015-07-0833Article10.3390/math30306156156252227-73902015-07-08doi: 10.3390/math3030615Palle JorgensenMyung-Sin Song<![CDATA[Mathematics, Vol. 3, Pages 604-614: Topological Integer Additive Set-Sequential Graphs]]>
http://www.mdpi.com/2227-7390/3/3/604
Let \(\mathbb{N}_0\) denote the set of all non-negative integers and \(X\) be any non-empty subset of \(\mathbb{N}_0\). Denote the power set of \(X\) by \(\mathcal{P}(X)\). An integer additive set-labeling (IASL) of a graph \(G\) is an injective function \(f : V (G) \to P(X)\) such that the image of the induced function \(f^+: E(G) \to \mathcal{P}(\mathbb{N}_0)\), defined by \(f^+(uv)=f(u)+f(v)\), is contained in \(\mathcal{P}(X)\), where \(f(u) + f(v)\) is the sumset of \(f(u)\) and \(f(v)\). If the associated set-valued edge function \(f^+\) is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL \(f\) is said to be a topological IASL (TIASL) if \(f(V(G))\cup \{\emptyset\}\) is a topology of the ground set \(X\). An IASL is said to be an integer additive set-sequential labeling (IASSL) if \(f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}\). An IASL of a given graph \(G\) is said to be a topological integer additive set-sequential labeling of \(G\), if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of \(G\). In this paper, we study the conditions required for a graph \(G\) to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.Mathematics2015-07-0333Article10.3390/math30306046046142227-73902015-07-03doi: 10.3390/math3030604Sudev NaduvathGermina AugustineChithra Sudev<![CDATA[Mathematics, Vol. 3, Pages 563-603: Singular Bilinear Integrals in Quantum Physics]]>
http://www.mdpi.com/2227-7390/3/3/563
Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.Mathematics2015-06-2933Article10.3390/math30305635636032227-73902015-06-29doi: 10.3390/math3030563Brian Jefferies<![CDATA[Mathematics, Vol. 3, Pages 527-562: The Schwartz Space: Tools for Quantum Mechanics and Infinite Dimensional Analysis]]>
http://www.mdpi.com/2227-7390/3/2/527
An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory.Mathematics2015-06-1632Article10.3390/math30205275275622227-73902015-06-16doi: 10.3390/math3020527Jeremy BecnelAmbar Sengupta<![CDATA[Mathematics, Vol. 3, Pages 510-526: Effective Summation and Interpolation of Series by Self-Similar Root Approximants]]>
http://www.mdpi.com/2227-7390/3/2/510
We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that of Padé approximants, when the latter can be defined.Mathematics2015-06-1532Article10.3390/math30205105105262227-73902015-06-15doi: 10.3390/math3020510Simon GluzmanVyacheslav Yukalov<![CDATA[Mathematics, Vol. 3, Pages 487-509: The Fractional Orthogonal Difference with Applications]]>
http://www.mdpi.com/2227-7390/3/2/487
This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolutel value of the modulus of the frequency response. These make clear that for a good insight into the behavior of a fractional differentiating filter, one has to look for the modulus of its frequency response in a log-log plot, rather than for plots in the time domain.Mathematics2015-06-1232Article10.3390/math30204874875092227-73902015-06-12doi: 10.3390/math3020487Enno Diekema<![CDATA[Mathematics, Vol. 3, Pages 481-486: The Complement of Binary Klein Quadric as a Combinatorial Grassmannian]]>
http://www.mdpi.com/2227-7390/3/2/481
Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286; 563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).Mathematics2015-06-0832Letter10.3390/math30204814814862227-73902015-06-08doi: 10.3390/math3020481Metod Saniga<![CDATA[Mathematics, Vol. 3, Pages 444-480: Sinc-Approximations of Fractional Operators: A Computing Approach]]>
http://www.mdpi.com/2227-7390/3/2/444
We discuss a new approach to represent fractional operators by Sinc approximation using convolution integrals. A spin off of the convolution representation is an effective inverse Laplace transform. Several examples demonstrate the application of the method to different practical problems.Mathematics2015-06-0532Article10.3390/math30204444444802227-73902015-06-05doi: 10.3390/math3020444Gerd BaumannFrank Stenger<![CDATA[Mathematics, Vol. 3, Pages 428-443: The 1st Law of Thermodynamics for the Mean Energy of a Closed Quantum System in the Aharonov-Vaidman Gauge]]>
http://www.mdpi.com/2227-7390/3/2/428
The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law of Thermodynamics that is applicable to the mean energy of a closed quantum system when the mean energy is expressed in the Aharonov-Vaidman gauge, i.e., when the system’s energy is weak valued. This is achieved by identifying the generalized heat and work exchange terms that appear in the equation of motion for weak valued energy. The complex valued contributions of the additive gauge term to these generalized exchange terms are discussed and this extended 1st Law is shown to subsume the usual 1st Law that is applicable for the mean energy of a closed quantum system. It is found that the gauge transformation introduces an additional energy uncertainty exchange term that—while it is neither a heat nor a work exchange term—is necessary for the conservation of weak valued energy. A spin-1/2 particle in a uniform magnetic field is used to illustrate aspects of the theory. It is demonstrated for this case that the extended 1st Law implies the existence of a gauge potential ω and that it generates a non-vanishing gauge field F. It is also shown for this case that the energy uncertainty exchange accumulated during the evolution of the system along a closed evolutionary cycle C in an associated parameter space is a geometric phase. This phase is equal to both the path integral of ω along C and the integral of the flux of F through the area enclosed by C.Mathematics2015-06-0132Article10.3390/math30204284284432227-73902015-06-01doi: 10.3390/math3020428Allen Parks<![CDATA[Mathematics, Vol. 3, Pages 412-427: Subordination Principle for a Class of Fractional Order Differential Equations]]>
http://www.mdpi.com/2227-7390/3/2/412
The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t&gt;0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alpha}\) is the Riemann–Liouville fractional derivative of order \(\alpha \in (0,1)\), \(\gamma&gt;0\) and \(f\) is an \(X\)-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the \(C_{0}\)-semigroup, generated by the operator \(A\). As an example, the Rayleigh–Stokes problem for a generalized second-grade fluid is considered.Mathematics2015-05-2632Article10.3390/math30204124124272227-73902015-05-26doi: 10.3390/math3020412Emilia Bazhlekova<![CDATA[Mathematics, Vol. 3, Pages 398-411: Implicit Fractional Differential Equations via the Liouville–Caputo Derivative]]>
http://www.mdpi.com/2227-7390/3/2/398
We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results.Mathematics2015-05-2532Article10.3390/math30203983984112227-73902015-05-25doi: 10.3390/math3020398Juan NietoAbelghani OuahabVenktesh Venktesh<![CDATA[Mathematics, Vol. 3, Pages 382-397: The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials]]>
http://www.mdpi.com/2227-7390/3/2/382
The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).Mathematics2015-05-1432Article10.3390/math30203823823972227-73902015-05-14doi: 10.3390/math3020382Tom BellaJenna Reis<![CDATA[Mathematics, Vol. 3, Pages 368-381: The Role of the Mittag-Leffler Function in Fractional Modeling]]>
http://www.mdpi.com/2227-7390/3/2/368
This is a survey paper illuminating the distinguished role of the Mittag-Leffler function and its generalizations in fractional analysis and fractional modeling. The content of the paper is connected to the recently published monograph by Rudolf Gorenflo, Anatoly Kilbas, Francesco Mainardi and Sergei Rogosin.Mathematics2015-05-1332Article10.3390/math30203683683812227-73902015-05-13doi: 10.3390/math3020368Sergei Rogosin<![CDATA[Mathematics, Vol. 3, Pages 337-367: High-Precision Arithmetic in Mathematical Physics]]>
http://www.mdpi.com/2227-7390/3/2/337
For many scientific calculations, particularly those involving empirical data, IEEE 32-bit floating-point arithmetic produces results of sufficient accuracy, while for other applications IEEE 64-bit floating-point is more appropriate. But for some very demanding applications, even higher levels of precision are often required. This article discusses the challenge of high-precision computation, in the context of mathematical physics, and highlights what facilities are required to support future computation, in light of emerging developments in computer architecture.Mathematics2015-05-1232Article10.3390/math30203373373672227-73902015-05-12doi: 10.3390/math3020337David BaileyJonathan Borwein<![CDATA[Mathematics, Vol. 3, Pages 329-336: Action at a Distance in Quantum Theory]]>
http://www.mdpi.com/2227-7390/3/2/329
The purpose of this paper is to present a consistent mathematical framework that shows how the EPR (Einstein. Podolsky, Rosen) phenomenon fits into our view of space time. To resolve the differences between the Hilbert space structure of quantum theory and the manifold structure of classical physics, the manifold is taken as a partial representation of the Hilbert space. It is the partial nature of the representation that allows for action at a distance and the failure of the manifold picture.Mathematics2015-05-0632Article10.3390/math30203293293362227-73902015-05-06doi: 10.3390/math3020329Jerome Blackman<![CDATA[Mathematics, Vol. 3, Pages 319-328: There Are Quantum Jumps]]>
http://www.mdpi.com/2227-7390/3/2/319
In this communication we take up the age-old problem of the possibility to incorporate quantum jumps. Unusually, we investigate quantum jumps in an extended quantum setting, but one of rigorous mathematical significance. The general background for this formulation originates in the Balslev-Combes theorem for dilatation analytic Hamiltonians and associated complex symmetric representations. The actual jump is mapped into a Jordan block of order two and a detailed derivation is discussed for the case of the emission of a photon by an atom. The result can be easily reassigned to analogous cases as well as generalized to Segrè characteristics of arbitrary order.Mathematics2015-05-0532Article10.3390/math30203193193282227-73902015-05-05doi: 10.3390/math3020319Erkki Brändas<![CDATA[Mathematics, Vol. 3, Pages 299-318: On the Duality of Discrete and Periodic Functions]]>
http://www.mdpi.com/2227-7390/3/2/299
Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Periodization Theorem on generalized functions. We define discretization and periodization on generalized functions and show that the Fourier transform of periodic functions are discrete functions and, vice versa, the Fourier transform of discrete functions are periodic functions.Mathematics2015-04-3032Article10.3390/math30202992993182227-73902015-04-30doi: 10.3390/math3020299Jens Fischer<![CDATA[Mathematics, Vol. 3, Pages 273-298: The Fractional Orthogonal Derivative]]>
http://www.mdpi.com/2227-7390/3/2/273
This paper builds on the notion of the so-called orthogonal derivative, where an n-th order derivative is approximated by an integral involving an orthogonal polynomial of degree n. This notion was reviewed in great detail in a paper by the author and Koornwinder in 2012. Here, an approximation of the Weyl or Riemann–Liouville fractional derivative is considered by replacing the n-th derivative by its approximation in the formula for the fractional derivative. In the case of, for instance, Jacobi polynomials, an explicit formula for the kernel of this approximate fractional derivative can be given. Next, we consider the fractional derivative as a filter and compute the frequency response in the continuous case for the Jacobi polynomials and in the discrete case for the Hahn polynomials. The frequency response in this case is a confluent hypergeometric function. A different approach is discussed, which starts with this explicit frequency response and then obtains the approximate fractional derivative by taking the inverse Fourier transform.Mathematics2015-04-2232Article10.3390/math30202732732982227-73902015-04-22doi: 10.3390/math3020273Enno Diekema<![CDATA[Mathematics, Vol. 3, Pages 258-272: Fractional Euler-Lagrange Equations Applied to Oscillatory Systems]]>
http://www.mdpi.com/2227-7390/3/2/258
In this paper, we applied the Riemann-Liouville approach and the fractional Euler-Lagrange equations in order to obtain the fractional nonlinear dynamic equations involving two classical physical applications: “Simple Pendulum” and the “Spring-Mass-Damper System” to both integer order calculus (IOC) and fractional order calculus (FOC) approaches. The numerical simulations were conducted and the time histories and pseudo-phase portraits presented. Both systems, the one that already had a damping behavior (Spring-Mass-Damper) and the system that did not present any sort of damping behavior (Simple Pendulum), showed signs indicating a possible better capacity of attenuation of their respective oscillation amplitudes. This implication could mean that if the selection of the order of the derivative is conveniently made, systems that need greater intensities of damping or vibrating absorbers may benefit from using fractional order in dynamics and possibly in control of the aforementioned systems. Thereafter, we believe that the results described in this paper may offer greater insights into the complex behavior of these systems, and thus instigate more research efforts in this direction.Mathematics2015-04-2032Article10.3390/math30202582582722227-73902015-04-20doi: 10.3390/math3020258Sergio DavidCarlos Valentim<![CDATA[Mathematics, Vol. 3, Pages 190-257: Maxwell–Lorentz Electrodynamics Revisited via the Lagrangian Formalism and Feynman Proper Time Paradigm]]>
http://www.mdpi.com/2227-7390/3/2/190
We review new electrodynamics models of interacting charged point particles and related fundamental physical aspects, motivated by the classical A.M. Ampère magnetic and H. Lorentz force laws electromagnetic field expressions. Based on the Feynman proper time paradigm and a recently devised vacuum field theory approach to the Lagrangian and Hamiltonian, the formulations of alternative classical electrodynamics models are analyzed in detail and their Dirac type quantization is suggested. Problems closely related to the radiation reaction force and electron mass inertia are analyzed. The validity of the Abraham-Lorentz electromagnetic electron mass origin hypothesis is argued. The related electromagnetic Dirac–Fock–Podolsky problem and symplectic properties of the Maxwell and Yang–Mills type dynamical systems are analyzed. The crucial importance of the remaining reference systems, with respect to which the dynamics of charged point particles is framed, is explained and emphasized.Mathematics2015-04-1732Article10.3390/math30201901902572227-73902015-04-17doi: 10.3390/math3020190Nikolai BogolubovAnatolij PrykarpatskiDenis Blackmore<![CDATA[Mathematics, Vol. 3, Pages 171-189: Asymptotic Expansions of Fractional Derivatives andTheir Applications]]>
http://www.mdpi.com/2227-7390/3/2/171
We compare the Riemann–Liouville fractional integral (fI) of a function f(z)with the Liouville fI of the same function and show that there are cases in which theasymptotic expansion of the former is obtained from those of the latter and the differenceof the two fIs. When this happens, this fact occurs also for the fractional derivative (fD).This method is applied to the derivation of the asymptotic expansion of the confluenthypergeometric function, which is a solution of Kummer’s differential equation. In thepresent paper, the solutions of the equation in the forms of the Riemann–Liouville fI orfD and the Liouville fI or fD are obtained by using the method, which Nishimoto used insolving the hypergeometric differential equation in terms of the Liouville fD.Mathematics2015-04-1532Article10.3390/math30201711711892227-73902015-04-15doi: 10.3390/math3020171Tohru MoritaKen-ichi Sato<![CDATA[Mathematics, Vol. 3, Pages 153-170: Analytical Solution of Generalized Space-Time Fractional Cable Equation]]>
http://www.mdpi.com/2227-7390/3/2/153
In this paper, we consider generalized space-time fractional cable equation in presence of external source. By using the Fourier-Laplace transform we obtain the Green function in terms of infinite series in H-functions. The fractional moments of the fundamental solution are derived and their asymptotic behavior in the short and long time limit is analyzed. Some previously obtained results are compared with those presented in this paper. By using the Bernstein characterization theorem we find the conditions under which the even moments are non-negative.Mathematics2015-04-0932Article10.3390/math30201531531702227-73902015-04-09doi: 10.3390/math3020153Ram SaxenaZivorad TomovskiTrifce Sandev<![CDATA[Mathematics, Vol. 3, Pages 131-152: Fractional Diffusion in Gaussian Noisy Environment]]>
http://www.mdpi.com/2227-7390/3/2/131
We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the time variable \(t\), \(\textit{B}\) is a second order elliptic operator with respect to the space variable \(x\in\mathbb{R}^d\) and \(\dot W^H\) a time homogeneous fractional Gaussian noise of Hurst parameter \(H=(H_1, \cdots, H_d)\). We obtain conditions satisfied by \(\alpha\) and \(H\), so that the square integrable solution \(u\) exists uniquely.Mathematics2015-03-3132Article10.3390/math30201311311522227-73902015-03-31doi: 10.3390/math3020131Guannan HuYaozhong Hu<![CDATA[Mathematics, Vol. 3, Pages 119-130: Multiple q-Zeta Brackets]]>
http://www.mdpi.com/2227-7390/3/1/119
The multiple zeta values (MZVs) possess a rich algebraic structure of algebraic relations, which is conjecturally determined by two different (shuffle and stuffle) products of a certain algebra of noncommutative words. In a recent work, Bachmann constructed a q-analogue of the MZVs—the so-called bi-brackets—for which the two products are dual to each other, in a very natural way. We overview Bachmann’s construction and discuss the radial asymptotics of the bi-brackets, its links to the MZVs, and related linear (in)dependence questions of the q-analogue.Mathematics2015-03-2031Article10.3390/math30101191191302227-73902015-03-20doi: 10.3390/math3010119Wadim Zudilin<![CDATA[Mathematics, Vol. 3, Pages 92-118: Quantum Measurements of Scattered Particles]]>
http://www.mdpi.com/2227-7390/3/1/92
We investigate the process of quantum measurements on scattered probes. Before scattering, the probes are independent, but they become entangled afterwards, due to the interaction with the scatterer. The collection of measurement results (the history) is a stochastic process of dependent random variables. We link the asymptotic properties of this process to spectral characteristics of the dynamics. We show that the process has decaying time correlations and that a zero-one law holds. We deduce that if the incoming probes are not sharply localized with respect to the spectrum of the measurement operator, then the process does not converge. Nevertheless, the scattering modifies the measurement outcome frequencies, which are shown to be the average of the measurement projection operator, evolved for one interaction period, in an asymptotic state. We illustrate the results on a truncated Jaynes–Cummings model.Mathematics2015-03-1931Article10.3390/math3010092921182227-73902015-03-19doi: 10.3390/math3010092Marco MerkliMark Penney<![CDATA[Mathematics, Vol. 3, Pages 76-91: Basic Results for Sequential Caputo Fractional Differential Equations]]>
http://www.mdpi.com/2227-7390/3/1/76
We have developed a representation form for the linear fractional differential equation of order q when 0 &lt; q &lt; 1, with variable coefficients. We have also obtained a closed form of the solution for sequential Caputo fractional differential equation of order 2q, with initial and boundary conditions, for 0 &lt; 2q &lt; 1. The solutions are in terms of Mittag–Leffler functions of order q only. Our results yield the known results of integer order when q = 1. We have also presented some numerical results to bring the salient features of sequential fractional differential equations.Mathematics2015-03-1931Article10.3390/math301007676912227-73902015-03-19doi: 10.3390/math3010076Bhuvaneswari SambandhamAghalaya Vatsala<![CDATA[Mathematics, Vol. 3, Pages 47-75: Twistor Interpretation of Harmonic Spheres and Yang–Mills Fields]]>
http://www.mdpi.com/2227-7390/3/1/47
We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space \(\Omega G\) of a compact Lie group \(G\) and the moduli space of Yang–Mills \(G\)-fields on \(\mathbb R^4\).Mathematics2015-03-1631Review10.3390/math301004747752227-73902015-03-16doi: 10.3390/math3010047Armen Sergeev<![CDATA[Mathematics, Vol. 3, Pages 40-46: Analyticity and the Global Information Field]]>
http://www.mdpi.com/2227-7390/3/1/40
The relation between analyticity in mathematics and the concept of a global information field in physics is reviewed. Mathematics is complete in the complex plane only. In the complex plane, a very powerful tool appears—analyticity. According to this property, if an analytic function is known on the countable set of points having an accumulation point, then it is known everywhere. This mysterious property has profound consequences in quantum physics. Analyticity allows one to obtain asymptotic (approximate) results in terms of some singular points in the complex plane which accumulate all necessary data on a given process. As an example, slow atomic collisions are presented, where the cross-sections of inelastic transitions are determined by branch-points of the adiabatic energy surface at a complex internuclear distance. Common aspects of the non-local nature of analyticity and a recently introduced interpretation of classical electrodynamics and quantum physics as theories of a global information field are discussed.Mathematics2015-03-1331Review10.3390/math301004040462227-73902015-03-13doi: 10.3390/math3010040Evgeni Solov'ev<![CDATA[Mathematics, Vol. 3, Pages 29-39: A Study on the Nourishing Number of Graphs and Graph Powers]]>
http://www.mdpi.com/2227-7390/3/1/29
Let \(\mathbb{N}_{0}\) be the set of all non-negative integers and \(\mathcal{P}(\mathbb{N}_{0})\) be its power set. Then, an integer additive set-indexer (IASI) of a given graph \(G\) is defined as an injective function \(f:V(G)\to \mathcal{P}(\mathbb{N}_{0})\) such that the induced edge-function \(f^+:E(G) \to\mathcal{P}(\mathbb{N}_{0})\) defined by \(f^+ (uv) = f(u)+ f(v)\) is also injective, where \(f(u)+f(v)\) is the sumset of \(f(u)\) and \(f(v)\). An IASI \(f\) of \(G\) is said to be a strong IASI of \(G\) if \(|f^+(uv)|=|f(u)|\,|f(v)|\) for all \(uv\in E(G)\). The nourishing number of a graph \(G\) is the minimum order of the maximal complete subgraph of \(G\) so that \(G\) admits a strong IASI. In this paper, we study the characteristics of certain graph classes and graph powers that admit strong integer additive set-indexers and determine their corresponding nourishing numbers.Mathematics2015-03-0631Article10.3390/math301002929392227-73902015-03-06doi: 10.3390/math3010029Sudev NaduvathGermina Augustine<![CDATA[Mathematics, Vol. 3, Pages 16-28: Existence Results for Fractional Neutral Functional Differential Equations with Random Impulses]]>
http://www.mdpi.com/2227-7390/3/1/16
In this paper, we investigate the existence of solutions for the fractional neutral differential equations with random impulses. The results are obtained by using Krasnoselskii’s fixed point theorem. Examples are added to show applications of the main results.Mathematics2015-01-2131Communication10.3390/math301001616282227-73902015-01-21doi: 10.3390/math3010016Annamalai AngurajMullarithodi RanjiniMargarita RiveroJuan Trujillo<![CDATA[Mathematics, Vol. 3, Pages 2-15: On θ-Congruent Numbers, Rational Squares in Arithmetic Progressions, Concordant Forms and Elliptic Curves]]>
http://www.mdpi.com/2227-7390/3/1/2
The correspondence between right triangles with rational sides, triplets of rational squares in arithmetic succession and integral solutions of certain quadratic forms is well-known. We show how this correspondence can be extended to the generalized notions of rational θ-triangles, rational squares occurring in arithmetic progressions and concordant forms. In our approach we establish one-to-one mappings to rational points on certain elliptic curves and examine in detail the role of solutions of the θ-congruent number problem and the concordant form problem associated with nontrivial torsion points on the corresponding elliptic curves. This approach allows us to combine and extend some disjoint results obtained by a number of authors, to clarify some statements in the literature and to answer some hitherto open questions.Mathematics2015-01-1931Article10.3390/math30100022152227-73902015-01-19doi: 10.3390/math3010002Erich SelderKarlheinz Spindler<![CDATA[Mathematics, Vol. 3, Pages 1: Acknowledgement to Reviewers of Mathematics in 2014]]>
http://www.mdpi.com/2227-7390/3/1/1
The editors of Mathematics would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2014:[...]Mathematics2015-01-0931Editorial10.3390/math3010001112227-73902015-01-09doi: 10.3390/math3010001 Mathematics Editorial Office<![CDATA[Mathematics, Vol. 2, Pages 232-239: A Conjecture of Han on 3-Cores and Modular Forms]]>
http://www.mdpi.com/2227-7390/2/4/232
In his study of Nekrasov–Okounkov type formulas on “partition theoretic” expressions for families of infinite products, Han discovered seemingly unrelated q-series that are supported on precisely the same terms as these infinite products. In collaboration with Ono, Han proved one instance of this occurrence that exhibited a relation between the numbers a(n) that are given in terms of hook lengths of partitions, with the numbers b(n) that equal the number of 3-core partitions of n. Recently Han revisited the q-series with coefficients a(n) and b(n), and numerically found a third q-series whose coefficients appear to be supported on the same terms. Here we prove Han’s conjecture about this third series by proving a general theorem about this phenomenon.Mathematics2014-12-1924Article10.3390/math20402322322392227-73902014-12-19doi: 10.3390/math2040232Amanda Clemm<![CDATA[Mathematics, Vol. 2, Pages 218-231: Characteristic Variety of the Gauss–Manin Differential Equations of a Generic Parallelly Translated Arrangement]]>
http://www.mdpi.com/2227-7390/2/4/218
We consider a weighted family of \(n\) generic parallelly translated hyperplanes in \(\mathbb{C}^k\) and describe the characteristic variety of the Gauss–Manin differential equations for associated hypergeometric integrals. The characteristic variety is given as the zero set of Laurent polynomials, whose coefficients are determined by weights and the Plücker coordinates of the associated point in the Grassmannian Gr\((k,n)\). The Laurent polynomials are in involution.Mathematics2014-10-1624Article10.3390/math20402182182312227-73902014-10-16doi: 10.3390/math2040218Alexander Varchenko<![CDATA[Mathematics, Vol. 2, Pages 196-217: The Second-Order Shape Derivative of Kohn–Vogelius-Type Cost Functional Using the Boundary Differentiation Approach]]>
http://www.mdpi.com/2227-7390/2/4/196
A shape optimization method is used to study the exterior Bernoulli free boundaryproblem. We minimize the Kohn–Vogelius-type cost functional over a class of admissibledomains subject to two boundary value problems. The first-order shape derivative of the costfunctional is recalled and its second-order shape derivative for general domains is computedvia the boundary differentiation scheme. Additionally, the second-order shape derivative ofJ at the solution of the Bernoulli problem is computed using Tiihonen’s approach.Mathematics2014-09-2624Article10.3390/math20401961962172227-73902014-09-26doi: 10.3390/math2040196Jerico BacaniGunther Peichl<![CDATA[Mathematics, Vol. 2, Pages 172-195: Dynamics of a Parametrically Excited System with Two Forcing Terms]]>
http://www.mdpi.com/2227-7390/2/3/172
Motivated by the dynamics of a trimaran, an investigation of the dynamic behaviour of a double forcing parametrically excited system is carried out. Initially, we provide an outline of the stability regions, both numerically and analytically, for the undamped linear, extended version of the Mathieu equation. This paper then examines the anticipated form of response of our proposed nonlinear damped double forcing system, where periodic and quasiperiodic routes to chaos are graphically demonstrated and compared with the case of the single vertically-driven pendulum.Mathematics2014-09-2223Article10.3390/math20301721721952227-73902014-09-22doi: 10.3390/math2030172Anastasia SofroniouSteven Bishop<![CDATA[Mathematics, Vol. 2, Pages 136-171: Modeling the Influence of Environment and Intervention onCholera in Haiti]]>
http://www.mdpi.com/2227-7390/2/3/136
We propose a simple model with two infective classes in order to model the cholera epidemic in Haiti. We include the impact of environmental events (rainfall, temperature and tidal range) on the epidemic in the Artibonite and Ouest regions by introducing terms in the transmission rate that vary with environmental conditions. We fit the model on weekly data from the beginning of the epidemic until December 2013, including the vaccination programs that were recently undertaken in the Ouest and Artibonite regions. We then modified these projections excluding vaccination to assess the programs’ effectiveness. Using real-time daily rainfall, we found lag times between precipitation events and new cases that range from 3:4 to 8:4 weeks in Artibonite and 5:1 to 7:4 in Ouest. In addition, it appears that, in the Ouest region, tidal influences play a significant role in the dynamics of the disease. Intervention efforts of all types have reduced case numbers in both regions; however, persistent outbreaks continue. In Ouest, where the population at risk seems particularly besieged and the overall population is larger, vaccination efforts seem to be taking hold more slowly than in Artibonite, where a smaller core population was vaccinated. The models including the vaccination programs predicted that a year and six months later, the mean number of cases in Artibonite would be reduced by about two thousand cases, and in Ouest by twenty four hundred cases below that predicted by the models without vaccination. We also found that vaccination is best when done in the early spring, and as early as possible in the epidemic. Comparing vaccination between the first spring and the second, there is a drop of about 40% in the case reduction due to the vaccine and about 10% per year after that.Mathematics2014-09-0523Article10.3390/math20301361361712227-73902014-09-05doi: 10.3390/math2030136Stephen TennenbaumCaroline FreitagSvetlana Roudenko<![CDATA[Mathematics, Vol. 2, Pages 119-135: A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter]]>
http://www.mdpi.com/2227-7390/2/3/119
Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.Mathematics2014-07-0923Article10.3390/math20301191191352227-73902014-07-09doi: 10.3390/math2030119Marco Herrera-ValdezSergei SuslovJosé Vega-Guzmán<![CDATA[Mathematics, Vol. 2, Pages 96-118: The Riccati System and a Diffusion-Type Equation]]>
http://www.mdpi.com/2227-7390/2/2/96
We discuss a method of constructing solutions of the initial value problem for diffusion-type equations in terms of solutions of certain Riccati and Ermakov-type systems. A nonautonomous Burgers-type equation is also considered. Examples include, but are not limited to the Fokker-Planck equation in physics, the Black-Scholes equation and the Hull-White model in finance.Mathematics2014-05-1522Article10.3390/math2020096961182227-73902014-05-15doi: 10.3390/math2020096Erwin SuazoSergei SuslovJosé Vega-Guzmán<![CDATA[Mathematics, Vol. 2, Pages 83-95: Traveling Wave Solutions of Reaction-Diffusion Equations Arising in Atherosclerosis Models]]>
http://www.mdpi.com/2227-7390/2/2/83
In this short review article, two atherosclerosis models are presented, one as a scalar equation and the other one as a system of two equations. They are given in terms of reaction-diffusion equations in an infinite strip with nonlinear boundary conditions. The existence of traveling wave solutions is studied for these models. The monostable and bistable cases are introduced and analyzed.Mathematics2014-05-0822Article10.3390/math202008383952227-73902014-05-08doi: 10.3390/math2020083Narcisa Apreutesei<![CDATA[Mathematics, Vol. 2, Pages 68-82: Numerical Construction of Viable Sets for Autonomous Conflict Control Systems]]>
http://www.mdpi.com/2227-7390/2/2/68
A conflict control system with state constraints is under consideration. A method for finding viability kernels (the largest subsets of state constraints where the system can be confined) is proposed. The method is related to differential games theory essentially developed by N. N. Krasovskii and A. I. Subbotin. The viability kernel is constructed as the limit of sets generated by a Pontryagin-like backward procedure. This method is implemented in the framework of a level set technique based on the computation of limiting viscosity solutions of an appropriate Hamilton–Jacobi equation. To fulfill this, the authors adapt their numerical methods formerly developed for solving time-dependent Hamilton–Jacobi equations arising from problems with state constraints. Examples of computing viability sets are given.Mathematics2014-04-1122Article10.3390/math202006868822227-73902014-04-11doi: 10.3390/math2020068Nikolai BotkinVarvara Turova<![CDATA[Mathematics, Vol. 2, Pages 53-67: Convergence of the Quadrature-Differences Method for Singular Integro-Differential Equations on the Interval]]>
http://www.mdpi.com/2227-7390/2/1/53
In this paper, we propose and justify the quadrature-differences method for the full linear singular integro-differential equations with the Cauchy kernel on the interval (–1,1). We consider equations of zero, positive and negative indices. It is shown that the method converges to an exact solution, and the error estimation depends on the sharpness of derivative approximations and on the smoothness of the coefficients and the right-hand side of the equation.Mathematics2014-03-0421Article10.3390/math201005353672227-73902014-03-04doi: 10.3390/math2010053Alexander Fedotov<![CDATA[Mathematics, Vol. 2, Pages 37-52: Bounded Gaps between Products of Special Primes]]>
http://www.mdpi.com/2227-7390/2/1/37
In their breakthrough paper in 2006, Goldston, Graham, Pintz and Yıldırım proved several results about bounded gaps between products of two distinct primes. Frank Thorne expanded on this result, proving bounded gaps in the set of square-free numbers with r prime factors for any r ≥ 2, all of which are in a given set of primes. His results yield applications to the divisibility of class numbers and the triviality of ranks of elliptic curves. In this paper, we relax the condition on the number of prime factors and prove an analogous result using a modified approach. We then revisit Thorne’s applications and give a better bound in each case.Mathematics2014-03-0321Article10.3390/math201003737522227-73902014-03-03doi: 10.3390/math2010037Ping ChungShiyu Li<![CDATA[Mathematics, Vol. 2, Pages 29-36: Some New Integral Identities for Solenoidal Fields and Applications]]>
http://www.mdpi.com/2227-7390/2/1/29
In spaces Rn, n ≥ 2, it has been proved that a solenoidal vector field and its rotor satisfy the series of new integral identities which have covariant form. The interest in them is explained by hydrodynamics problems for an ideal fluid.Mathematics2014-03-0321Article10.3390/math201002929362227-73902014-03-03doi: 10.3390/math2010029Vladimir Semenov<![CDATA[Mathematics, Vol. 2, Pages 12-28: On the Folded Normal Distribution]]>
http://www.mdpi.com/2227-7390/2/1/12
The characteristic function of the folded normal distribution and its moment function are derived. The entropy of the folded normal distribution and the Kullback–Leibler from the normal and half normal distributions are approximated using Taylor series. The accuracy of the results are also assessed using different criteria. The maximum likelihood estimates and confidence intervals for the parameters are obtained using the asymptotic theory and bootstrap method. The coverage of the confidence intervals is also examined.Mathematics2014-02-1421Article10.3390/math201001212282227-73902014-02-14doi: 10.3390/math2010012Michail TsagrisChristina BenekiHossein Hassani<![CDATA[Mathematics, Vol. 2, Pages 1-11: One-Dimensional Nonlinear Stefan Problems in Storm’s Materials]]>
http://www.mdpi.com/2227-7390/2/1/1
We consider two one-phase nonlinear one-dimensional Stefan problems for a semi-infinite material x &gt; 0; with phase change temperature Tf : We assume that the heat capacity and the thermal conductivity satisfy a Storm’s condition. In the first case, we assume a heat flux boundary condition of the type q(t) = q 0 t , and in the second case, we assume a temperature boundary condition T = Ts &lt; Tf at the fixed face. Solutions of similarity type are obtained in both cases, and the equivalence of the two problems is demonstrated. We also give procedures in order to compute the explicit solution.Mathematics2013-12-2721Article10.3390/math20100011112227-73902013-12-27doi: 10.3390/math2010001Adriana BriozzoMaría Natale<![CDATA[Mathematics, Vol. 1, Pages 111-118: Sign-Periodicity of Traces of Singular Moduli]]>
http://www.mdpi.com/2227-7390/1/4/111
Zagier proved that the generating functions of traces of singular values of Jm(z) are weight 3 2 weakly holomorphic modular forms. In this paper we prove that there is the sign-periodicity of traces of singular values of Jm(z).Mathematics2013-10-1514Article10.3390/math10401111111182227-73902013-10-15doi: 10.3390/math1040111Dohoon ChoiByungchan KimSubong Lim<![CDATA[Mathematics, Vol. 1, Pages 100-110: Effective Congruences for Mock Theta Functions]]>
http://www.mdpi.com/2227-7390/1/3/100
Let M(q) =∑ c(n)q n be one of Ramanujan’s mock theta functions. We establish the existence of infinitely many linear congruences of the form: c(An + B) ≡ 0 (mod l j ) where A is a multiple of l and an auxiliary prime, p. Moreover, we give an effectively computable upper bound on the smallest such p for which these congruences hold. The effective nature of our results is based on the prior works of Lichtenstein [1] and Treneer [2].Mathematics2013-09-0413Article10.3390/math10301001001102227-73902013-09-04doi: 10.3390/math1030100Nickolas AndersenHolley FriedlanderJeremy FullerHeidi Goodson<![CDATA[Mathematics, Vol. 1, Pages 89-99: Scattering of Electromagnetic Waves by Many Nano-Wires]]>
http://www.mdpi.com/2227-7390/1/3/89
Electromagnetic wave scattering by many parallel to the z−axis, thin, impedance, parallel, infinite cylinders is studied asymptotically as a → 0. Let Dm be the cross-section of the m−th cylinder, a be its radius and x ^ m = (x m1 , x m2 ) be its center, 1 ≤ m ≤ M , M = M (a). It is assumed that the points, x ^ m , are distributed, so that N(Δ)= 1 2πa ∫ Δ N ( x ^ )d x ^ [1+o(1)] where N (∆) is the number of points, x ^ m , in an arbitrary open subset, ∆, of the plane, xoy. The function, N( x ^ ) ≥0 , is a continuous function, which an experimentalist can choose. An equation for the self-consistent (effective) field is derived as a → 0. A formula is derived for the refraction coefficient in the medium in which many thin impedance cylinders are distributed. These cylinders may model nano-wires embedded in the medium. One can produce a desired refraction coefficient of the new medium by choosing a suitable boundary impedance of the thin cylinders and their distribution law.Mathematics2013-07-1813Article10.3390/math103008989992227-73902013-07-18doi: 10.3390/math1030089Alexander Ramm<![CDATA[Mathematics, Vol. 1, Pages 76-88: On the Distribution of the spt-Crank]]>
http://www.mdpi.com/2227-7390/1/3/76
Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence { N S (m , n) } m is unimodal, where N S (m , n) is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conjecture is true and allows us to asymptotically establish the conjecture. Additionally, we give an asymptotic study for the distribution of the spt-crank statistic. Finally, we give some speculations about a definition for the spt-crank in terms of “marked” partitions. A “marked” partition is an unrestricted integer partition where each part is marked with a multiplicity number. It remains an interesting and apparently challenging problem to interpret the spt-crank in terms of ordinary integer partitions.Mathematics2013-06-2813Article10.3390/math103007676882227-73902013-06-28doi: 10.3390/math1030076George AndrewsFreeman DysonRobert Rhoades<![CDATA[Mathematics, Vol. 1, Pages 65-75: On the Class of Dominant and Subordinate Products]]>
http://www.mdpi.com/2227-7390/1/2/65
In this paper we provide proofs of two new theorems that provide a broad class of partition inequalities and that illustrate a na¨ıve version of Andrews’ anti-telescoping technique quite well. These new theorems also put to rest any notion that including parts of size 1 is somehow necessary in order to have a valid irreducible partition inequality. In addition, we prove (as a lemma to one of the theorems) a rather nontrivial class of rational functions of three variables has entirely nonnegative power series coefficients.Mathematics2013-05-1512Article10.3390/math102006565752227-73902013-05-15doi: 10.3390/math1020065Alexander BerkovichKeith Grizzell<![CDATA[Mathematics, Vol. 1, Pages 46-64: Stability of Solutions to Evolution Problems]]>
http://www.mdpi.com/2227-7390/1/2/46
Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as t → ∞ , in particular, sufficient conditions for this limit to be zero. The evolution problem is: u ˙ = A(t)u + F(t, u) + b(t), t ≥ 0; u(0) = u 0 . (*) Here u ˙ := du dt , u = u(t) ∈ H, H is a Hilbert space, t ∈ R + := [0,∞), A(t) is a linear dissipative operator: Re(A(t)u,u) ≤−γ(t)(u, u) where F(t, u) is a nonlinear operator, ‖ F(t, u) ‖ ≤ c 0 ‖ u ‖ p , p &gt; 1, c 0 and p are positive constants, ‖ b(t) ‖ ≤ β(t) , and β(t)≥0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case γ(t) ≤ 0 is also treated.Mathematics2013-05-1312Article10.3390/math102004646642227-73902013-05-13doi: 10.3390/math1020046Alexander Ramm<![CDATA[Mathematics, Vol. 1, Pages 31-45: A Converse to a Theorem of Oka and Sakamoto for Complex Line Arrangements]]>
http://www.mdpi.com/2227-7390/1/1/31
Let C1 and C2 be algebraic plane curves in ℂ 2 such that the curves intersect in d1 · d2 points where d1, d2 are the degrees of the curves respectively. Oka and Sakamoto proved that π1( ℂ 2 \ C1 U C2)) ≅ π1 ( ℂ 2 \ C1) × π1 ( ℂ 2 \ C2) [1]. In this paper we prove the converse of Oka and Sakamoto’s result for line arrangements. Let A1 and A2 be non-empty arrangements of lines in ℂ 2 such that π1 (M(A1 U A2)) ≅ π1 (M(A1)) × π1 (M(A2)) Then, the intersection of A1 and A2 consists of /A1/ · /A2/ points of multiplicity two.Mathematics2013-03-1411Article10.3390/math101003131452227-73902013-03-14doi: 10.3390/math1010031Kristopher Williams<![CDATA[Mathematics, Vol. 1, Pages 9-30: ρ — Adic Analogues of Ramanujan Type Formulas for 1/π]]>
http://www.mdpi.com/2227-7390/1/1/9
Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form Following Ramanujan's work on modular equations and approximations of π, there are formulas for 1/π of the form ∑ k = 0 ∞ ( 1 2 ) k ( 1 d ) k ( d - 1 d ) k k ! 3 ( a k + 1 ) ( λ d ) k = δ π for d=2,3,4,6, where łd are singular values that correspond to elliptic curves with complex multiplication, and a,δ are explicit algebraic numbers. In this paper we prove a p-adic version of this formula in terms of the so-called Ramanujan type congruence. In addition, we obtain a new supercongruence result for elliptic curves with complex multiplication.Mathematics2013-03-1311Article10.3390/math10100099302227-73902013-03-13doi: 10.3390/math1010009Sarah ChisholmAlyson DeinesLing LongGabriele NebeHolly Swisher<![CDATA[Mathematics, Vol. 1, Pages 3-8: On Matrices Arising in the Finite Field Analogue of Euler’s Integral Transform]]>
http://www.mdpi.com/2227-7390/1/1/3
In his 1984 Ph.D. thesis, J. Greene defined an analogue of the Euler integral transform for finite field hypergeometric series. Here we consider a special family of matrices which arise naturally in the study of this transform and prove a conjecture of Ono about the decomposition of certain finite field hypergeometric functions into functions of lower dimension.Mathematics2013-02-0511Article10.3390/math1010003382227-73902013-02-05doi: 10.3390/math1010003Michael GriffinLarry Rolen<![CDATA[Mathematics, Vol. 1, Pages 1-2: Mathematics—An Open Access Journal]]>
http://www.mdpi.com/2227-7390/1/1/1
As is widely known, mathematics plays a unique role in all natural sciences as a refined scientific language and powerful research tool. Indeed, most of the fundamental laws of nature are written in mathematical terms and we study their consequences by numerous mathematical methods (and vice versa, any essential progress in a natural science has been accompanied by fruitful developments in mathematics). In addition, the mathematical modeling in various interdisciplinary problems and logical development of mathematics on its own should be taken into account. [...]Mathematics2012-12-2811Editorial10.3390/math1010001122227-73902012-12-28doi: 10.3390/math1010001Sergei Suslov