Mathematics
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Latest open access articles published in Mathematics at http://www.mdpi.com/journal/mathematics<![CDATA[Mathematics, Vol. 5, Pages 25: Discrete-Time Fractional Optimal Control]]>
http://www.mdpi.com/2227-7390/5/2/25
A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of α .Mathematics2017-04-1952Article10.3390/math5020025252227-73902017-04-19doi: 10.3390/math5020025Tirumalasetty ChiranjeeviRaj Biswas<![CDATA[Mathematics, Vol. 5, Pages 24: Fixed Points of Set Valued Mappings in Terms of Start Point on a Metric Space Endowed with a Directed Graph]]>
http://www.mdpi.com/2227-7390/5/2/24
In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster–Tarski theorem has also been established using our results.Mathematics2017-04-1952Article10.3390/math5020024242227-73902017-04-19doi: 10.3390/math5020024Murchana NeogPradip Debnath<![CDATA[Mathematics, Vol. 5, Pages 23: Best Proximity Point Results in Non-Archimedean Modular Metric Space]]>
http://www.mdpi.com/2227-7390/5/2/23
In this paper, we introduce the new notion of Suzuki-type ( α , β , θ , γ ) -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak P λ -property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature.Mathematics2017-04-0552Article10.3390/math5020023232227-73902017-04-05doi: 10.3390/math5020023Mohadeshe PaknazarManuel Sen<![CDATA[Mathematics, Vol. 5, Pages 22: On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces]]>
http://www.mdpi.com/2227-7390/5/2/22
The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings T integrated in a pair ( g , T ) of mappings. In particular, g is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and T is a p -cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings.Mathematics2017-04-0152Article10.3390/math5020022222227-73902017-04-01doi: 10.3390/math5020022Manuel SenMujahid AbbasNaeem Saleem<![CDATA[Mathematics, Vol. 5, Pages 21: On Some Extended Block Krylov Based Methods for Large Scale Nonsymmetric Stein Matrix Equations]]>
http://www.mdpi.com/2227-7390/5/2/21
In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term A X B − X + E F T = 0 . These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.Mathematics2017-03-2752Article10.3390/math5020021212227-73902017-03-27doi: 10.3390/math5020021Abdeslem BentbibKhalide JbilouEL Sadek<![CDATA[Mathematics, Vol. 5, Pages 19: A Generalization of b-Metric Space and Some Fixed Point Theorems]]>
http://www.mdpi.com/2227-7390/5/2/19
In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature.Mathematics2017-03-2352Article10.3390/math5020019192227-73902017-03-23doi: 10.3390/math5020019Tayyab KamranMaria SamreenQurat UL Ain<![CDATA[Mathematics, Vol. 5, Pages 20: F-Harmonic Maps between Doubly Warped Product Manifolds]]>
http://www.mdpi.com/2227-7390/5/2/20
In this paper, some properties of F -harmonic and conformal F -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic F -harmonic maps are constructed.Mathematics2017-03-2352Article10.3390/math5020020202227-73902017-03-23doi: 10.3390/math5020020Seyed TorbaghanMorteza Rezaii<![CDATA[Mathematics, Vol. 5, Pages 18: Characterization of the Minimizing Graph of the Connected Graphs Whose Complements Are Bicyclic]]>
http://www.mdpi.com/2227-7390/5/1/18
In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let G 1 , n c and G 2 , n c be the classes of the connected graphs of order n whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to G n c = G 1 , n c ∪ G 2 , n c , a class of the connected graphs of order n whose complements are bicyclic.Mathematics2017-03-1151Article10.3390/math5010018182227-73902017-03-11doi: 10.3390/math5010018Muhammad Javaid<![CDATA[Mathematics, Vol. 5, Pages 16: On the Additively Weighted Harary Index of Some Composite Graphs]]>
http://www.mdpi.com/2227-7390/5/1/16
The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index H A ( G ) is a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. This new invariant was introduced in (Alizadeh, Iranmanesh and Došlić. Additively weighted Harary index of some composite graphs, Discrete Math, 2013) and they posed the following question: What is the behavior of H A ( G ) when G is a composite graph resulting for example by: splice, link, corona and rooted product? We investigate the additively weighted Harary index for these standard graph products. Then we obtain lower and upper bounds for some of them.Mathematics2017-03-0751Article10.3390/math5010016162227-73902017-03-07doi: 10.3390/math5010016Behrooz KhosraviElnaz Ramezani<![CDATA[Mathematics, Vol. 5, Pages 17: Certain Concepts of Bipolar Fuzzy Directed Hypergraphs]]>
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A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with such types of problems, we use mathematical models that are based on bipolar fuzzy (BF) sets. In this research paper, we introduce the concept of BF directed hypergraphs. We describe certain operations on BF directed hypergraphs, including addition, multiplication, vertex-wise multiplication and structural subtraction. We introduce the concept of B = ( m + , m − ) -tempered BF directed hypergraphs and investigate some of their properties. We also present an algorithm to compute the minimum arc length of a BF directed hyperpath.Mathematics2017-03-0451Article10.3390/math5010017172227-73902017-03-04doi: 10.3390/math5010017Muhammad AkramAnam Luqman<![CDATA[Mathematics, Vol. 5, Pages 15: Dialectical Multivalued Logic and Probabilistic Theory]]>
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There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, one can derive a probability theory, adding the correspondence between truth value and fortuity.Mathematics2017-02-2351Article10.3390/math5010015152227-73902017-02-23doi: 10.3390/math5010015José Usó DoménechJosué Nescolarde-SelvaLorena Segura-Abad<![CDATA[Mathematics, Vol. 5, Pages 14: A Novel Iterative Algorithm Applied to Totally Asymptotically Nonexpansive Mappings in CAT(0) Spaces]]>
http://www.mdpi.com/2227-7390/5/1/14
In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature. We then provide some numerical examples to illustrate our main result and to display the efficiency of the proposed algorithm.Mathematics2017-02-2251Article10.3390/math5010014142227-73902017-02-22doi: 10.3390/math5010014Ali AbkarMohsen Shekarbaigi<![CDATA[Mathematics, Vol. 5, Pages 13: A Few Finite Trigonometric Sums]]>
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Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interesting properties, and can have amazingly simple values. However, only some of them are available in the literature. We obtain a number of such sums using the method of residues.Mathematics2017-02-1851Article10.3390/math5010013132227-73902017-02-18doi: 10.3390/math5010013Chandan DattaPankaj Agrawal<![CDATA[Mathematics, Vol. 5, Pages 12: Fractional Fokker-Planck Equation]]>
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We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain.Mathematics2017-02-1151Article10.3390/math5010012122227-73902017-02-11doi: 10.3390/math5010012Gerd BaumannFrank Stenger<![CDATA[Mathematics, Vol. 5, Pages 11: The Split Common Fixed Point Problem for a Family of Multivalued Quasinonexpansive Mappings and Totally Asymptotically Strictly Pseudocontractive Mappings in Banach Spaces]]>
http://www.mdpi.com/2227-7390/5/1/11
In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-ϕ-asymptotically nonexpansive mappings and k-quasi-strictly pseudocontractive mappings in the setting of Banach spaces. Our results improve and extend the results of Tang et al., Takahashi, Moudafi, Censor et al., and Byrne et al.Mathematics2017-02-1151Article10.3390/math5010011112227-73902017-02-11doi: 10.3390/math5010011Ali AbkarElahe ShahrosvandAzizollah Azizi<![CDATA[Mathematics, Vol. 5, Pages 10: Approximation in Müntz Spaces MΛ,p of Lp Functions for 1 < p < ∞ and Bases]]>
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Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces MΛ,p of Lp functions is studied, where 1 &lt; p &lt; ∞. It is proven that up to an isomorphism and a change of variables, these spaces are contained in Weil–Nagy’s class. Moreover, the existence of Schauder bases in the Müntz spaces MΛ,p is investigated.Mathematics2017-01-2551Article10.3390/math5010010102227-73902017-01-25doi: 10.3390/math5010010Sergey Ludkowski<![CDATA[Mathematics, Vol. 5, Pages 9: Existence of Mild Solutions for Impulsive Fractional Integro-Differential Inclusions with State-Dependent Delay]]>
http://www.mdpi.com/2227-7390/5/1/9
In this manuscript, we implement Bohnenblust–Karlin’s fixed point theorem to demonstrate the existence of mild solutions for a class of impulsive fractional integro-differential inclusions (IFIDI) with state-dependent delay (SDD) in Banach spaces. An example is provided to illustrate the obtained abstract results.Mathematics2017-01-2551Article10.3390/math501000992227-73902017-01-25doi: 10.3390/math5010009Selvaraj SuganyaMani Mallika Arjunan<![CDATA[Mathematics, Vol. 5, Pages 8: An Analysis of the Influence of Graph Theory When Preparing for Programming Contests]]>
http://www.mdpi.com/2227-7390/5/1/8
The subject known as Programming Contests in the Bachelor’s Degree in Computer Engineering course focuses on solving programming problems frequently met within contests such as the Southwest Europe Regional Contest (SWERC). In order to solve these problems one first needs to model the problem correctly, find the ideal solution, and then be able to program it without making any mistakes in a very short period of time. Leading multinationals such as Google, Apple, IBM, Facebook and Microsoft place a very high value on these abilities when selecting candidates for posts in their companies. In this communication we present some preliminary results of an analysis of the interaction between two optional subjects in the Computer Science Degree course: Programming Contests (PC) and Graphs, Models and Applications (GMA). The results of this analysis enabled us to make changes to some of the contents in GMA in order to better prepare the students to deal with the challenges they have to face in programming contests.Mathematics2017-01-2051Article10.3390/math501000882227-73902017-01-20doi: 10.3390/math5010008Cristina JordánJon GómezJ. Conejero<![CDATA[Mathematics, Vol. 5, Pages 7: Deterministic Seirs Epidemic Model for Modeling Vital Dynamics, Vaccinations, and Temporary Immunity]]>
http://www.mdpi.com/2227-7390/5/1/7
In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R0 for two cases of no epidemic (R0 ≤ 1) and epidemic (R0 &gt; 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X ¯ D F E ) and epidemic (EE X ¯ E E ).Mathematics2017-01-1751Article10.3390/math501000772227-73902017-01-17doi: 10.3390/math5010007Marek Trawicki<![CDATA[Mathematics, Vol. 5, Pages 6: Zoology of Atlas-Groups: Dessins D’enfants, Finite Geometries and Quantum Commutation]]>
http://www.mdpi.com/2227-7390/5/1/6
Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual, but significant to recognize that a P is a Grothendieck’s “dessin d’enfant” D and that a wealth of standard graphs and finite geometries G —such as near polygons and their generalizations—are stabilized by a D . In our paper, tripods P − D − G of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries G ′ s have a contextuality parameter close to its maximal value of one.Mathematics2017-01-1451Article10.3390/math501000662227-73902017-01-14doi: 10.3390/math5010006Michel PlanatHishamuddin Zainuddin<![CDATA[Mathematics, Vol. 5, Pages 4: Logical Entropy of Dynamical Systems—A General Model]]>
http://www.mdpi.com/2227-7390/5/1/4
In the paper by Riečan and Markechová (Fuzzy Sets Syst. 96, 1998), some fuzzy modifications of Shannon’s and Kolmogorov-Sinai’s entropy were studied and the general scheme involving the presented models was introduced. Our aim in this contribution is to provide analogies of these results for the case of the logical entropy. We define the logical entropy and logical mutual information of finite partitions on the appropriate algebraic structure and prove basic properties of these measures. It is shown that, as a special case, we obtain the logical entropy of fuzzy partitions studied by Markechová and Riečan (Entropy 18, 2016). Finally, using the suggested concept of entropy of partitions we define the logical entropy of a dynamical system and prove that it is the same for two dynamical systems that are isomorphic.Mathematics2017-01-0651Article10.3390/math501000442227-73902017-01-06doi: 10.3390/math5010004Abolfazl EbrahimzadehZahra GiskiDagmar Markechová<![CDATA[Mathematics, Vol. 5, Pages 5: Data Clustering with Quantum Mechanics]]>
http://www.mdpi.com/2227-7390/5/1/5
Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.Mathematics2017-01-0651Article10.3390/math501000552227-73902017-01-06doi: 10.3390/math5010005Tony ScottMadhusudan TheraniXing Wang<![CDATA[Mathematics, Vol. 5, Pages 3: From the Underdamped Generalized Elastic Model to the Single Particle Langevin Description]]>
http://www.mdpi.com/2227-7390/5/1/3
The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit.Mathematics2017-01-0651Article10.3390/math501000332227-73902017-01-06doi: 10.3390/math5010003Alessandro Taloni<![CDATA[Mathematics, Vol. 5, Pages 2: On Autonomy Imposition in Zero Interval Limit Perturbation Expansion for the Spectral Entities of Hilbert–Schmidt Integral Operators]]>
http://www.mdpi.com/2227-7390/5/1/2
In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but the dependence on this parameter arises from the coordinate change at the zero interval limit. Moreover, the related half interval length is utilized as the perturbation parameter in the perturbative analyses. Thus, the zero interval limit perturbation for solving the eigenproblem under consideration is developed. The aim of this work is to show that the autonomy imposition brings an important restriction on the kernel of the corresponding integral operator, and the constructed perturbation series is not capable of expressing the exact solution approximately unless a specific type of kernel is considered. The general structure for the encountered constraints is revealed, and the specific class of kernels is identified to this end. Error analysis of the developed method is given. These analyses are also supported by certain illustrative implementations involving the kernels, which are and are not in the specific class addressed above. Thus, the efficiency of the developed method is shown, and the relevant analyses are confirmed.Mathematics2017-01-0651Article10.3390/math501000222227-73902017-01-06doi: 10.3390/math5010002Süha TunaMetin Demiralp<![CDATA[Mathematics, Vol. 5, Pages 1: Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation]]>
http://www.mdpi.com/2227-7390/5/1/1
We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrödinger Equation.Mathematics2016-12-2251Article10.3390/math501000112227-73902016-12-22doi: 10.3390/math5010001R. SinuvasanAndronikos PaliathanasisRichard MorrisPeter Leach<![CDATA[Mathematics, Vol. 4, Pages 69: Proposal for the Formalization of Dialectical Logic]]>
http://www.mdpi.com/2227-7390/4/4/69
Classical logic is typically concerned with abstract analysis. The problem for a synthetic logic is to transcend and unify available data to reconstruct the object as a totality. Three rules are proposed to pass from classic logic to synthetic logic. We present the category logic of qualitative opposition using examples from various sciences. This logic has been defined to include the neuter as part of qualitative opposition. The application of these rules to qualitative opposition, and, in particular, its neuter, demonstrated that a synthetic logic allows the truth of some contradictions. This synthetic logic is dialectical with a multi-valued logic, which gives every proposition a truth value in the interval [0,1] that is the square of the modulus of a complex number. In this dialectical logic, contradictions of the neuter of an opposition may be true.Mathematics2016-12-1144Article10.3390/math4040069692227-73902016-12-11doi: 10.3390/math4040069José Usó-DoménechJosué Nescolarde-SelvaLorena Segura-Abad<![CDATA[Mathematics, Vol. 4, Pages 68: Results on Coincidence and Common Fixed Points for (ψ,φ)g-Generalized Weakly Contractive Mappings in Ordered Metric Spaces]]>
http://www.mdpi.com/2227-7390/4/4/68
Inspired by a metrical-fixed point theorem from Choudhury et al. (Nonlinear Anal. 2011, 74, 2116–2126), we prove some order-theoretic results which generalize several core results of the existing literature, especially the two main results of Harjani and Sadarangani (Nonlinear Anal. 2009, 71, 3403–3410 and 2010, 72, 1188–1197). We demonstrate the realized improvement obtained in our results by using a suitable example. As an application, we also prove a result for mappings satisfying integral type ( ψ , φ ) g -generalized weakly contractive conditions.Mathematics2016-12-1044Article10.3390/math4040068682227-73902016-12-10doi: 10.3390/math4040068Rqeeb GubranMohammad Imdad<![CDATA[Mathematics, Vol. 4, Pages 67: Continued-Fraction Expansion of Transport Coefficients with Fractional Calculus]]>
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The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have derived a space-time generalized telegrapher’s equation with a fractional nested hierarchy which can be used in separate developments for the mass transport, for the heat conduction and for the flux of ions. We have obtained a new formalism which includes the contribution of fast of higher-order fluxes in the mesoscopic and inhomogeneous media. The results take the form of continued fraction expansions. The balance equations are used in a scheme of continued fractions, and they appear as a closure condition. In this way the transport equation and its corresponding wave number-frequency relation are obtained, both of them in the mathematical structure of the continued fraction scheme. Numerical examples are included to show the dispersive nature of the solutions, and the generalized fractional transport equation in the same mathematical form, which can be applied to the mass transport, the heat conduction and the flux of ions.Mathematics2016-12-0944Article10.3390/math4040067672227-73902016-12-09doi: 10.3390/math4040067Abel Garcia-BernabéS. HernándezL. del CastilloDavid Jou<![CDATA[Mathematics, Vol. 4, Pages 66: Best Proximity Point Theorems in Partially Ordered b-Quasi Metric Spaces]]>
http://www.mdpi.com/2227-7390/4/4/66
In this paper, we introduce the notion of an ordered rational proximal contraction in partially ordered b-quasi metric spaces. We shall then prove some best proximity point theorems in partially ordered b-quasi metric spaces.Mathematics2016-11-2644Article10.3390/math4040066662227-73902016-11-26doi: 10.3390/math4040066Ali AbkarNarges MoezzifarAzizollah Azizi<![CDATA[Mathematics, Vol. 4, Pages 65: Some Determinantal Expressions and Recurrence Relations of the Bernoulli Polynomials]]>
http://www.mdpi.com/2227-7390/4/4/65
In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.Mathematics2016-11-2444Article10.3390/math4040065652227-73902016-11-24doi: 10.3390/math4040065Feng QiBai-Ni Guo<![CDATA[Mathematics, Vol. 4, Pages 64: Viability for Semilinear Differential Equations with Infinite Delay]]>
http://www.mdpi.com/2227-7390/4/4/64
Let X be a Banach space, A : D ( A ) ⊂ X → X the generator of a compact C 0 -semigroup S ( t ) : X → X , t ≥ 0 , D ( · ) : ( a , b ) → 2 X a tube in X, and f : ( a , b ) × B → X a function of Carathéodory type. The main result of this paper is that a necessary and sufficient condition in order that D ( · ) be viable of the semilinear differential equation with infinite delay u ′ ( t ) = A u ( t ) + f ( t , u t ) , t ∈ [ t 0 , t 0 + T ] , u t 0 = ϕ ∈ B is the tangency condition lim inf h ↓ 0 h − 1 d ( S ( h ) v ( 0 ) + h f ( t , v ) ; D ( t + h ) ) = 0 for almost every t ∈ ( a , b ) and every v ∈ B with v ( 0 ) ∈ D ( t ) .Mathematics2016-11-0844Article10.3390/math4040064642227-73902016-11-08doi: 10.3390/math4040064Qixiang DongGang Li<![CDATA[Mathematics, Vol. 4, Pages 63: Positive Solutions for Nonlinear Caputo Type Fractional q-Difference Equations with Integral Boundary Conditions]]>
http://www.mdpi.com/2227-7390/4/4/63
In this paper, by applying some well-known fixed point theorems, we investigate the existence of positive solutions for a class of nonlinear Caputo type fractional q-difference equations with integral boundary conditions. Finally, some interesting examples are presented to illustrate the main results.Mathematics2016-11-0244Article10.3390/math4040063632227-73902016-11-02doi: 10.3390/math4040063Wengui YangYaping Qin<![CDATA[Mathematics, Vol. 4, Pages 60: A Study of Controllability of Impulsive Neutral Evolution Integro-Differential Equations with State-Dependent Delay in Banach Spaces]]>
http://www.mdpi.com/2227-7390/4/4/60
In this paper, we study the problem of controllability of impulsive neutral evolution integro-differential equations with state-dependent delay in Banach spaces. The main results are completely new and are obtained by using Sadovskii’s fixed point theorem, theory of resolvent operators, and an abstract phase space. An example is given to illustrate the theory.Mathematics2016-10-1944Article10.3390/math4040060602227-73902016-10-19doi: 10.3390/math4040060Dimplekumar ChalishajarAnnamalai AngurajKandasamy MalarKulandhivel Karthikeyan<![CDATA[Mathematics, Vol. 4, Pages 62: Interval Type 2 Fuzzy Set in Fuzzy Shortest Path Problem]]>
http://www.mdpi.com/2227-7390/4/4/62
The shortest path problem (SPP) is one of the most important combinatorial optimization problems in graph theory due to its various applications. The uncertainty existing in the real world problems makes it difficult to determine the arc lengths exactly. The fuzzy set is one of the popular tools to represent and handle uncertainty in information due to incompleteness or inexactness. In most cases, the SPP in fuzzy graph, called the fuzzy shortest path problem (FSPP) uses type-1 fuzzy set (T1FS) as arc length. Uncertainty in the evaluation of membership degrees due to inexactness of human perception is not considered in T1FS. An interval type-2 fuzzy set (IT2FS) is able to tackle this uncertainty. In this paper, we use IT2FSs to represent the arc lengths of a fuzzy graph for FSPP. We call this problem an interval type-2 fuzzy shortest path problem (IT2FSPP). We describe the utility of IT2FSs as arc lengths and its application in different real world shortest path problems. Here, we propose an algorithm for IT2FSPP. In the proposed algorithm, we incorporate the uncertainty in Dijkstra’s algorithm for SPP using IT2FS as arc length. The path algebra corresponding to the proposed algorithm and the generalized algorithm based on the path algebra are also presented here. Numerical examples are used to illustrate the effectiveness of the proposed approach.Mathematics2016-10-0944Article10.3390/math4040062622227-73902016-10-09doi: 10.3390/math4040062Arindam DeyAnita PalTandra Pal<![CDATA[Mathematics, Vol. 4, Pages 61: Nuclear Space Facts, Strange and Plain]]>
http://www.mdpi.com/2227-7390/4/4/61
We present a scenic but practical guide through nuclear spaces and their dual spaces, examining useful, unexpected, and often unfamiliar results both for nuclear spaces and their strong and weak duals.Mathematics2016-10-0944Article10.3390/math4040061612227-73902016-10-09doi: 10.3390/math4040061Jeremy BecnelAmbar Sengupta<![CDATA[Mathematics, Vol. 4, Pages 59: Effective Potential from the Generalized Time-Dependent Schrödinger Equation]]>
http://www.mdpi.com/2227-7390/4/4/59
We analyze the generalized time-dependent Schrödinger equation for the force free case, as a generalization, for example, of the standard time-dependent Schrödinger equation, time fractional Schrödinger equation, distributed order time fractional Schrödinger equation, and tempered in time Schrödinger equation. We relate it to the corresponding standard Schrödinger equation with effective potential. The general form of the effective potential that leads to a standard time-dependent Schrodinger equation with the same solution as the generalized one is derived explicitly. Further, effective potentials for several special cases, such as Dirac delta, power-law, Mittag-Leffler and truncated power-law memory kernels, are expressed in terms of the Mittag-Leffler functions. Such complex potentials have been used in the transport simulations in quantum dots, and in simulation of resonant tunneling diode.Mathematics2016-09-2844Article10.3390/math4040059592227-73902016-09-28doi: 10.3390/math4040059Trifce SandevIrina PetreskaErvin Lenzi<![CDATA[Mathematics, Vol. 4, Pages 58: Finite-Time Stabilization of Homogeneous Non-Lipschitz Systems]]>
http://www.mdpi.com/2227-7390/4/4/58
This paper focuses on the problem of finite-time stabilization of homogeneous, non-Lipschitz systems with dilations. A key contribution of this paper is the design of a virtual recursive Hölder, non-Lipschitz state feedback, which renders the non-Lipschitz systems in the special case dominated by a lower-triangular nonlinear system finite-time stable. The proof is based on a recursive design algorithm developed recently to construct the virtual Hölder continuous, finite-time stabilizer as well as a C1 positive definite and proper Lyapunov function that guarantees finite-time stability of the non-Lipschitz nonlinear systems.Mathematics2016-09-2444Article10.3390/math4040058582227-73902016-09-24doi: 10.3390/math4040058Nawel KhelilMartin Otis<![CDATA[Mathematics, Vol. 4, Pages 57: Analysis of Dynamics in Multiphysics Modelling of Active Faults]]>
http://www.mdpi.com/2227-7390/4/4/57
Instabilities in Geomechanics appear on multiple scales involving multiple physical processes. They appear often as planar features of localised deformation (faults), which can be relatively stable creep or display rich dynamics, sometimes culminating in earthquakes. To study those features, we propose a fundamental physics-based approach that overcomes the current limitations of statistical rule-based methods and allows a physical understanding of the nucleation and temporal evolution of such faults. In particular, we formulate the coupling between temperature and pressure evolution in the faults through their multiphysics energetic process(es). We analyse their multiple steady states using numerical continuation methods and characterise their transient dynamics by studying the time-dependent problem near the critical Hopf points. We find that the global system can be characterised by a homoclinic bifurcation that depends on the two main dimensionless groups of the underlying physical system. The Gruntfest number determines the onset of the localisation phenomenon, while the dynamics are mainly controlled by the Lewis number, which is the ratio of energy diffusion over mass diffusion. Here, we show that the Lewis number is the critical parameter for dynamics of the system as it controls the time evolution of the system for a given energy supply (Gruntfest number).Mathematics2016-09-2244Article10.3390/math4040057572227-73902016-09-22doi: 10.3390/math4040057Sotiris AlevizosThomas PouletManolis VeveakisKlaus Regenauer-Lieb<![CDATA[Mathematics, Vol. 4, Pages 56: Quantum Measurements, Stochastic Networks, the Uncertainty Principle, and the Not So Strange “Weak Values”]]>
http://www.mdpi.com/2227-7390/4/3/56
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.Mathematics2016-09-1543Article10.3390/math4030056562227-73902016-09-15doi: 10.3390/math4030056Dmitri Sokolovski<![CDATA[Mathematics, Vol. 4, Pages 55: Amenability Modulo an Ideal of Second Duals of Semigroup Algebras]]>
http://www.mdpi.com/2227-7390/4/3/55
The aim of this paper is to investigate the amenability modulo, an ideal of Banach algebras with emphasis on applications to homological algebras. In doing so, we show that amenability modulo, an ideal of A * * implies amenability modulo, an ideal of A. Finally, for a large class of semigroups, we prove that l 1 ( S ) * * is amenable modulo I σ * * if and only if an appropriate group homomorphic image of S is finite, where I σ is the closed ideal induced by the least group congruence σ .Mathematics2016-09-1343Article10.3390/math4030055552227-73902016-09-13doi: 10.3390/math4030055Hamidreza RahimiKhalil Nabizadeh<![CDATA[Mathematics, Vol. 4, Pages 54: Quantum Incompatibility in Collective Measurements]]>
http://www.mdpi.com/2227-7390/4/3/54
We study the compatibility (or joint measurability) of quantum observables in a setting where the experimenter has access to multiple copies of a given quantum system, rather than performing the experiments on each individual copy separately. We introduce the index of incompatibility as a quantifier of incompatibility in this multi-copy setting, as well as the notion of the compatibility stack representing various compatibility relations present in a given set of observables. We then prove a general structure theorem for multi-copy joint observables and use it to prove that all abstract compatibility stacks with three vertices have realizations in terms of quantum observables.Mathematics2016-09-1043Article10.3390/math4030054542227-73902016-09-10doi: 10.3390/math4030054Claudio CarmeliTeiko HeinosaariDaniel ReitznerJussi SchultzAlessandro Toigo<![CDATA[Mathematics, Vol. 4, Pages 53: Solution for Rational Systems of Difference Equations of Order Three]]>
http://www.mdpi.com/2227-7390/4/3/53
In this paper, we consider the solution and periodicity of the following systems of difference equations: x n + 1 = y n − 2 − 1 + y n − 2 x n − 1 y n , y n + 1 = x n − 2 ± 1 ± x n − 2 y n − 1 x n , with initial conditions x − 2 , x − 1 , x 0 , y − 2 , y − 1 , and y 0 are nonzero real numbers.Mathematics2016-09-0343Article10.3390/math4030053532227-73902016-09-03doi: 10.3390/math4030053Mohamed El-Dessoky<![CDATA[Mathematics, Vol. 4, Pages 52: Role of Measurement Incompatibility and Uncertainty in Determining Nonlocality]]>
http://www.mdpi.com/2227-7390/4/3/52
It has been recently shown that measurement incompatibility and fine grained uncertainty—a particular form of preparation uncertainty relation—are deeply related to the nonlocal feature of quantum mechanics. In particular, the degree of measurement incompatibility in a no-signaling theory determines the bound on the violation of Bell-CHSH inequality, and a similar role is also played by (fine-grained) uncertainty along with steering, a subtle non-local phenomenon. We review these connections, along with comments on the difference in the roles played by measurement incompatibility and uncertainty. We also discuss why the toy model of Spekkens (Phys. Rev. A 75, 032110 (2007)) shows no nonlocal feature even though steering is present in this theory.Mathematics2016-08-1543Article10.3390/math4030052522227-73902016-08-15doi: 10.3390/math4030052Guruprasad KarSibasish GhoshSujit ChoudharyManik Banik<![CDATA[Mathematics, Vol. 4, Pages 50: Complete Classification of Cylindrically Symmetric Static Spacetimes and the Corresponding Conservation Laws]]>
http://www.mdpi.com/2227-7390/4/3/50
In this paper we find the Noether symmetries of the Lagrangian of cylindrically symmetric static spacetimes. Using this approach we recover all cylindrically symmetric static spacetimes appeared in the classification by isometries and homotheties. We give different classes of cylindrically symmetric static spacetimes along with the Noether symmetries of the corresponding Lagrangians and conservation laws.Mathematics2016-08-0843Article10.3390/math4030050502227-73902016-08-08doi: 10.3390/math4030050Farhad AliTooba Feroze<![CDATA[Mathematics, Vol. 4, Pages 51: A New Approach to Study Fixed Point of Multivalued Mappings in Modular Metric Spaces and Applications]]>
http://www.mdpi.com/2227-7390/4/3/51
The purpose of this paper is to present a new approach to study the existence of fixed points for multivalued F-contraction in the setting of modular metric spaces. In establishing this connection, we introduce the notion of multivalued F-contraction and prove corresponding fixed point theorems in complete modular metric space with some specific assumption on the modular. Then we apply our results to establish the existence of solutions for a certain type of non-linear integral equations.Mathematics2016-08-0843Article10.3390/math4030051512227-73902016-08-08doi: 10.3390/math4030051Dilip JainAnantachai PadcharoenPoom KumamDhananjay Gopal<![CDATA[Mathematics, Vol. 4, Pages 49: Preparational Uncertainty Relations for N Continuous Variables]]>
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A smooth function of the second moments of N continuous variables gives rise to an uncertainty relation if it is bounded from below. We present a method to systematically derive such bounds by generalizing an approach applied previously to a single continuous variable. New uncertainty relations are obtained for multi-partite systems that allow one to distinguish entangled from separable states. We also investigate the geometry of the “uncertainty region” in the N ( 2 N + 1 ) -dimensional space of moments. It is shown to be a convex set, and the points on its boundary are found to be in one-to-one correspondence with pure Gaussian states of minimal uncertainty. For a single degree of freedom, the boundary can be visualized as one sheet of a “Lorentz-invariant” hyperboloid in the three-dimensional space of second moments.Mathematics2016-07-1943Article10.3390/math4030049492227-73902016-07-19doi: 10.3390/math4030049Spiros KechrimparisStefan Weigert<![CDATA[Mathematics, Vol. 4, Pages 47: Uncertainty Relations for Quantum Coherence]]>
http://www.mdpi.com/2227-7390/4/3/47
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.Mathematics2016-07-1643Article10.3390/math4030047472227-73902016-07-16doi: 10.3390/math4030047Uttam SinghArun PatiManabendra Bera<![CDATA[Mathematics, Vol. 4, Pages 48: Sharing of Nonlocality of a Single Member of an Entangled Pair of Qubits Is Not Possible by More than Two Unbiased Observers on the Other Wing]]>
http://www.mdpi.com/2227-7390/4/3/48
We address the recently posed question as to whether the nonlocality of a single member of an entangled pair of spin 1 / 2 particles can be shared among multiple observers on the other wing who act sequentially and independently of each other. We first show that the optimality condition for the trade-off between information gain and disturbance in the context of weak or non-ideal measurements emerges naturally when one employs a one-parameter class of positive operator valued measures (POVMs). Using this formalism we then prove analytically that it is impossible to obtain violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality by more than two Bobs in one of the two wings using unbiased input settings with an Alice in the other wing.Mathematics2016-07-1643Article10.3390/math4030048482227-73902016-07-16doi: 10.3390/math4030048Shiladitya MalArchan MajumdarDipankar Home<![CDATA[Mathematics, Vol. 4, Pages 46: Geometrical Inverse Preconditioning for Symmetric Positive Definite Matrices]]>
http://www.mdpi.com/2227-7390/4/3/46
We focus on inverse preconditioners based on minimizing F ( X ) = 1 − cos ( X A , I ) , where X A is the preconditioned matrix and A is symmetric and positive definite. We present and analyze gradient-type methods to minimize F ( X ) on a suitable compact set. For this, we use the geometrical properties of the non-polyhedral cone of symmetric and positive definite matrices, and also the special properties of F ( X ) on the feasible set. Preliminary and encouraging numerical results are also presented in which dense and sparse approximations are included.Mathematics2016-07-0943Article10.3390/math4030046462227-73902016-07-09doi: 10.3390/math4030046Jean-Paul ChehabMarcos Raydan<![CDATA[Mathematics, Vol. 4, Pages 45: Fourier Spectral Methods for Some Linear Stochastic Space-Fractional Partial Differential Equations]]>
http://www.mdpi.com/2227-7390/4/3/45
Fourier spectral methods for solving some linear stochastic space-fractional partial differential equations perturbed by space-time white noises in the one-dimensional case are introduced and analysed. The space-fractional derivative is defined by using the eigenvalues and eigenfunctions of the Laplacian subject to some boundary conditions. We approximate the space-time white noise by using piecewise constant functions and obtain the approximated stochastic space-fractional partial differential equations. The approximated stochastic space-fractional partial differential equations are then solved by using Fourier spectral methods. Error estimates in the L 2 -norm are obtained, and numerical examples are given.Mathematics2016-07-0143Article10.3390/math4030045452227-73902016-07-01doi: 10.3390/math4030045Yanmei LiuMonzorul KhanYubin Yan<![CDATA[Mathematics, Vol. 4, Pages 43: Cohen Macaulayness and Arithmetical Rank of Generalized Theta Graphs]]>
http://www.mdpi.com/2227-7390/4/3/43
In this paper, we study some algebraic invariants of the edge ideal of generalized theta graphs, such as arithmetical rank, big height and height. We give an upper bound for the difference between the arithmetical rank and big height. Moreover, all Cohen-Macaulay (and unmixed) graphs of this type will be characterized.Mathematics2016-06-2943Article10.3390/math4030043432227-73902016-06-29doi: 10.3390/math4030043Seyyede SeyyediFarhad Rahmati<![CDATA[Mathematics, Vol. 4, Pages 44: Exact Discrete Analogs of Canonical Commutation and Uncertainty Relations]]>
http://www.mdpi.com/2227-7390/4/3/44
An exact discretization of the canonical commutation and corresponding uncertainty relations are suggested. We prove that the canonical commutation relations of discrete quantum mechanics, which is based on standard finite difference, holds for constant wave functions only. In this paper, we use the recently proposed exact discretization of derivatives, which is based on differences that are represented by infinite series. This new mathematical tool allows us to build sensible discrete quantum mechanics based on the suggested differences and includes the correct canonical commutation and uncertainty relations.Mathematics2016-06-2843Article10.3390/math4030044442227-73902016-06-28doi: 10.3390/math4030044Vasily Tarasov<![CDATA[Mathematics, Vol. 4, Pages 42: Exponential Energy Decay of Solutions for a Transmission Problem With Viscoelastic Term and Delay]]>
http://www.mdpi.com/2227-7390/4/2/42
In our previous work (Journal of Nonlinear Science and Applications 9: 1202–1215, 2016), we studied the well-posedness and general decay rate for a transmission problem in a bounded domain with a viscoelastic term and a delay term. In this paper, we continue to study the similar problem but without the frictional damping term. The main difficulty arises since we have no frictional damping term to control the delay term in the estimate of the energy decay. By introducing suitable energy and Lyapunov functionals, we establish an exponential decay result for the energy.Mathematics2016-06-0942Article10.3390/math4020042422227-73902016-06-09doi: 10.3390/math4020042Danhua WangGang LiBiqing Zhu<![CDATA[Mathematics, Vol. 4, Pages 41: Entropic Uncertainty Relations for Successive Generalized Measurements]]>
http://www.mdpi.com/2227-7390/4/2/41
We derive entropic uncertainty relations for successive generalized measurements by using general descriptions of quantum measurement within two distinctive operational scenarios. In the first scenario, by merging two successive measurements into one we consider successive measurement scheme as a method to perform an overall composite measurement. In the second scenario, on the other hand, we consider it as a method to measure a pair of jointly measurable observables by marginalizing over the distribution obtained in this scheme. In the course of this work, we identify that limits on one’s ability to measure with low uncertainty via this scheme come from intrinsic unsharpness of observables obtained in each scenario. In particular, for the Lüders instrument, disturbance caused by the first measurement to the second one gives rise to the unsharpness at least as much as incompatibility of the observables composing successive measurement.Mathematics2016-06-0742Article10.3390/math4020041412227-73902016-06-07doi: 10.3390/math4020041Kyunghyun BaekWonmin Son<![CDATA[Mathematics, Vol. 4, Pages 39: Morphisms and Order Ideals of Toric Posets]]>
http://www.mdpi.com/2227-7390/4/2/39
Toric posets are in some sense a natural “cyclic” version of finite posets in that they capture the fundamental features of a partial order but without the notion of minimal or maximal elements. They can be thought of combinatorially as equivalence classes of acyclic orientations under the equivalence relation generated by converting sources into sinks, or geometrically as chambers of toric graphic hyperplane arrangements. In this paper, we define toric intervals and toric order-preserving maps, which lead to toric analogues of poset morphisms and order ideals. We develop this theory, discuss some fundamental differences between the toric and ordinary cases, and outline some areas for future research. Additionally, we provide a connection to cyclic reducibility and conjugacy in Coxeter groups.Mathematics2016-06-0442Article10.3390/math4020039392227-73902016-06-04doi: 10.3390/math4020039Matthew Macauley<![CDATA[Mathematics, Vol. 4, Pages 40: Uncertainty Relations and Possible Experience]]>
http://www.mdpi.com/2227-7390/4/2/40
The uncertainty principle can be understood as a condition of joint indeterminacy of classes of properties in quantum theory. The mathematical expressions most closely associated with this principle have been the uncertainty relations, various inequalities exemplified by the well known expression regarding position and momentum introduced by Heisenberg. Here, recent work involving a new sort of “logical” indeterminacy principle and associated relations introduced by Pitowsky, expressable directly in terms of probabilities of outcomes of measurements of sharp quantum observables, is reviewed and its quantum nature is discussed. These novel relations are derivable from Boolean “conditions of possible experience” of the quantum realm and have been considered both as fundamentally logical and as fundamentally geometrical. This work focuses on the relationship of indeterminacy to the propositions regarding the values of discrete, sharp observables of quantum systems. Here, reasons for favoring each of these two positions are considered. Finally, with an eye toward future research related to indeterminacy relations, further novel approaches grounded in category theory and intended to capture and reconceptualize the complementarity characteristics of quantum propositions are discussed in relation to the former.Mathematics2016-06-0342Review10.3390/math4020040402227-73902016-06-03doi: 10.3390/math4020040Gregg Jaeger<![CDATA[Mathematics, Vol. 4, Pages 38: Measurement Uncertainty for Finite Quantum Observables]]>
http://www.mdpi.com/2227-7390/4/2/38
Measurement uncertainty relations are lower bounds on the errors of any approximate joint measurement of two or more quantum observables. The aim of this paper is to provide methods to compute optimal bounds of this type. The basic method is semidefinite programming, which we apply to arbitrary finite collections of projective observables on a finite dimensional Hilbert space. The quantification of errors is based on an arbitrary cost function, which assigns a penalty to getting result x rather than y, for any pair ( x , y ) . This induces a notion of optimal transport cost for a pair of probability distributions, and we include an Appendix with a short summary of optimal transport theory as needed in our context. There are then different ways to form an overall figure of merit from the comparison of distributions. We consider three, which are related to different physical testing scenarios. The most thorough test compares the transport distances between the marginals of a joint measurement and the reference observables for every input state. Less demanding is a test just on the states for which a “true value” is known in the sense that the reference observable yields a definite outcome. Finally, we can measure a deviation as a single expectation value by comparing the two observables on the two parts of a maximally-entangled state. All three error quantities have the property that they vanish if and only if the tested observable is equal to the reference. The theory is illustrated with some characteristic examples.Mathematics2016-06-0242Article10.3390/math4020038382227-73902016-06-02doi: 10.3390/math4020038René SchwonnekDavid ReebReinhard Werner<![CDATA[Mathematics, Vol. 4, Pages 37: Smoothness in Binomial Edge Ideals]]>
http://www.mdpi.com/2227-7390/4/2/37
In this paper we study some geometric properties of the algebraic set associated to the binomial edge ideal of a graph. We study the singularity and smoothness of the algebraic set associated to the binomial edge ideal of a graph. Some of these algebraic sets are irreducible and some of them are reducible. If every irreducible component of the algebraic set is smooth we call the graph an edge smooth graph, otherwise it is called an edge singular graph. We show that complete graphs are edge smooth and introduce two conditions such that the graph G is edge singular if and only if it satisfies these conditions. Then, it is shown that cycles and most of trees are edge singular. In addition, it is proved that complete bipartite graphs are edge smooth.Mathematics2016-06-0142Article10.3390/math4020037372227-73902016-06-01doi: 10.3390/math4020037Hamid DamadiFarhad Rahmati<![CDATA[Mathematics, Vol. 4, Pages 36: SIC-POVMs and Compatibility among Quantum States]]>
http://www.mdpi.com/2227-7390/4/2/36
An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.Mathematics2016-06-0142Article10.3390/math4020036362227-73902016-06-01doi: 10.3390/math4020036Blake Stacey<![CDATA[Mathematics, Vol. 4, Pages 35: Three Identities of the Catalan-Qi Numbers]]>
http://www.mdpi.com/2227-7390/4/2/35
In the paper, the authors find three new identities of the Catalan-Qi numbers and provide alternative proofs of two identities of the Catalan numbers. The three identities of the Catalan-Qi numbers generalize three identities of the Catalan numbers.Mathematics2016-05-2642Article10.3390/math4020035352227-73902016-05-26doi: 10.3390/math4020035Mansour MahmoudFeng Qi<![CDATA[Mathematics, Vol. 4, Pages 34: Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics]]>
http://www.mdpi.com/2227-7390/4/2/34
We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation.Mathematics2016-05-1342Article10.3390/math4020034342227-73902016-05-13doi: 10.3390/math4020034Andronikos PaliathanasisRichard MorrisPeter Leach<![CDATA[Mathematics, Vol. 4, Pages 33: Chaos Control in Three Dimensional Cancer Model by State Space Exact Linearization Based on Lie Algebra]]>
http://www.mdpi.com/2227-7390/4/2/33
This study deals with the control of chaotic dynamics of tumor cells, healthy host cells, and effector immune cells in a chaotic Three Dimensional Cancer Model (TDCM) by State Space Exact Linearization (SSEL) technique based on Lie algebra. A non-linear feedback control law is designed which induces a coordinate transformation thereby changing the original chaotic TDCM system into a controlled one linear system. Numerical simulation has been carried using Mathematica that witness the robustness of the technique implemented on the chosen chaotic system.Mathematics2016-05-1042Article10.3390/math4020033332227-73902016-05-10doi: 10.3390/math4020033Mohammad Shahzad<![CDATA[Mathematics, Vol. 4, Pages 32: On the Dimension of Algebraic-Geometric Trace Codes]]>
http://www.mdpi.com/2227-7390/4/2/32
We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X.Mathematics2016-05-0742Article10.3390/math4020032322227-73902016-05-07doi: 10.3390/math4020032Phong LeSunil Chetty<![CDATA[Mathematics, Vol. 4, Pages 30: New Approach for Fractional Order Derivatives: Fundamentals and Analytic Properties]]>
http://www.mdpi.com/2227-7390/4/2/30
The rate of change of any function versus its independent variables was defined as a derivative. The fundamentals of the derivative concept were constructed by Newton and l’Hôpital. The followers of Newton and l’Hôpital defined fractional order derivative concepts. We express the derivative defined by Newton and l’Hôpital as an ordinary derivative, and there are also fractional order derivatives. So, the derivative concept was handled in this paper, and a new definition for derivative based on indefinite limit and l’Hôpital’s rule was expressed. This new approach illustrated that a derivative operator may be non-linear. Based on this idea, the asymptotic behaviors of functions were analyzed and it was observed that the rates of changes of any function attain maximum value at inflection points in the positive direction and minimum value (negative) at inflection points in the negative direction. This case brought out the fact that the derivative operator does not have to be linear; it may be non-linear. Another important result of this paper is the relationships between complex numbers and derivative concepts, since both concepts have directions and magnitudes.Mathematics2016-05-0442Article10.3390/math4020030302227-73902016-05-04doi: 10.3390/math4020030Ali Karcı<![CDATA[Mathematics, Vol. 4, Pages 31: Fractional Schrödinger Equation in the Presence of the Linear Potential]]>
http://www.mdpi.com/2227-7390/4/2/31
In this paper, we consider the time-dependent Schrödinger equation: i ∂ ψ ( x , t ) ∂ t = 1 2 ( − Δ ) α 2 ψ ( x , t ) + V ( x ) ψ ( x , t ) , x ∈ R , t &gt; 0 with the Riesz space-fractional derivative of order 0 &lt; α ≤ 2 in the presence of the linear potential V ( x ) = β x . The wave function to the one-dimensional Schrödinger equation in momentum space is given in closed form allowing the determination of other measurable quantities such as the mean square displacement. Analytical solutions are derived for the relevant case of α = 1 , which are useable for studying the propagation of wave packets that undergo spreading and splitting. We furthermore address the two-dimensional space-fractional Schrödinger equation under consideration of the potential V ( ρ ) = F · ρ including the free particle case. The derived equations are illustrated in different ways and verified by comparisons with a recently proposed numerical approach.Mathematics2016-05-0442Article10.3390/math4020031312227-73902016-05-04doi: 10.3390/math4020031André LiemertAlwin Kienle<![CDATA[Mathematics, Vol. 4, Pages 28: Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility]]>
http://www.mdpi.com/2227-7390/4/2/28
We perform a classification of the Lie point symmetries for the Black-Scholes-Merton Model for European options with stochastic volatility, σ, in which the last is defined by a stochastic differential equation with an Orstein-Uhlenbeck term. In this model, the value of the option is given by a linear (1 + 2) evolution partial differential equation in which the price of the option depends upon two independent variables, the value of the underlying asset, S, and a new variable, y. We find that for arbitrary functional form of the volatility, σ ( y ) , the (1 + 2) evolution equation always admits two Lie point symmetries in addition to the automatic linear symmetry and the infinite number of solution symmetries. However, when σ ( y ) = σ 0 and as the price of the option depends upon the second Brownian motion in which the volatility is defined, the (1 + 2) evolution is not reduced to the Black-Scholes-Merton Equation, the model admits five Lie point symmetries in addition to the linear symmetry and the infinite number of solution symmetries. We apply the zeroth-order invariants of the Lie symmetries and we reduce the (1 + 2) evolution equation to a linear second-order ordinary differential equation. Finally, we study two models of special interest, the Heston model and the Stein-Stein model.Mathematics2016-05-0342Article10.3390/math4020028282227-73902016-05-03doi: 10.3390/math4020028Andronikos PaliathanasisK. KrishnakumarK.M. TamizhmaniPeter Leach<![CDATA[Mathematics, Vol. 4, Pages 29: An Adaptive WENO Collocation Method for Differential Equations with Random Coefficients]]>
http://www.mdpi.com/2227-7390/4/2/29
The stochastic collocation method for solving differential equations with random inputs has gained lots of popularity in many applications, since such a scheme exhibits exponential convergence with smooth solutions in the random space. However, in some circumstance the solutions do not fulfill the smoothness requirement; thus a direct application of the method will cause poor performance and slow convergence rate due to the well known Gibbs phenomenon. To address the issue, we propose an adaptive high-order multi-element stochastic collocation scheme by incorporating a WENO (Weighted Essentially non-oscillatory) interpolation procedure and an adaptive mesh refinement (AMR) strategy. The proposed multi-element stochastic collocation scheme requires only repetitive runs of an existing deterministic solver at each interpolation point, similar to the Monte Carlo method. Furthermore, the scheme takes advantage of robustness and the high-order nature of the WENO interpolation procedure, and efficacy and efficiency of the AMR strategy. When the proposed scheme is applied to stochastic problems with non-smooth solutions, the Gibbs phenomenon is mitigated by the WENO methodology in the random space, and the errors around discontinuities in the stochastic space are significantly reduced by the AMR strategy. The numerical experiments for some benchmark stochastic problems, such as the Kraichnan-Orszag problem and Burgers’ equation with random initial conditions, demonstrate the reliability, efficiency and efficacy of the proposed scheme.Mathematics2016-05-0342Article10.3390/math4020029292227-73902016-05-03doi: 10.3390/math4020029Wei GuoGuang LinAndrew ChristliebJingmei Qiu<![CDATA[Mathematics, Vol. 4, Pages 27: Stagnation-Point Flow towards a Stretching Vertical Sheet with Slip Effects]]>
http://www.mdpi.com/2227-7390/4/2/27
The effects of partial slip on stagnation-point flow and heat transfer due to a stretching vertical sheet is investigated. Using a similarity transformation, the governing partial differential equations are reduced into a system of nonlinear ordinary differential equations. The resulting equations are solved numerically using a shooting method. The effect of slip and buoyancy parameters on the velocity, temperature, skin friction coefficient and the local Nusselt number are graphically presented and discussed. It is found that dual solutions exist in a certain range of slip and buoyancy parameters. The skin friction coefficient decreases while the Nusselt number increases as the slip parameter increases.Mathematics2016-04-2142Article10.3390/math4020027272227-73902016-04-21doi: 10.3390/math4020027Khairy ZaimiAnuar Ishak<![CDATA[Mathematics, Vol. 4, Pages 26: POD-Based Constrained Sensor Placement and Field Reconstruction from Noisy Wind Measurements: A Perturbation Study]]>
http://www.mdpi.com/2227-7390/4/2/26
It is shown in literature that sensor placement at the extrema of Proper Orthogonal Decomposition (POD) modes is efficient and leads to accurate reconstruction of the field of quantity of interest (velocity, pressure, salinity, etc.) from a limited number of measurements in the oceanography study. In this paper, we extend this approach of sensor placement and take into account measurement errors and detect possible malfunctioning sensors. We use the 24 hourly spatial wind field simulation data sets simulated using the Weather Research and Forecasting (WRF) model applied to the Maine Bay to evaluate the performances of our methods. Specifically, we use an exclusion disk strategy to distribute sensors when the extrema of POD modes are close. We demonstrate that this strategy can improve the accuracy of the reconstruction of the velocity field. It is also capable of reducing the standard deviation of the reconstruction from noisy measurements. Moreover, by a cross-validation technique, we successfully locate the malfunctioning sensors.Mathematics2016-04-1442Article10.3390/math4020026262227-73902016-04-14doi: 10.3390/math4020026Zhongqiang ZhangXiu YangGuang Lin<![CDATA[Mathematics, Vol. 4, Pages 25: Recurrence Relations for Orthogonal Polynomials on Triangular Domains]]>
http://www.mdpi.com/2227-7390/4/2/25
In Farouki et al, 2003, Legendre-weighted orthogonal polynomials P n , r ( u , v , w ) , r = 0 , 1 , … , n , n ≥ 0 on the triangular domain T = { ( u , v , w ) : u , v , w ≥ 0 , u + v + w = 1 } are constructed, where u , v , w are the barycentric coordinates. Unfortunately, evaluating the explicit formulas requires many operations and is not very practical from an algorithmic point of view. Hence, there is a need for a more efficient alternative. A very convenient method for computing orthogonal polynomials is based on recurrence relations. Such recurrence relations are described in this paper for the triangular orthogonal polynomials, providing a simple and fast algorithm for their evaluation.Mathematics2016-04-1242Article10.3390/math4020025252227-73902016-04-12doi: 10.3390/math4020025Abedallah Rababah<![CDATA[Mathematics, Vol. 4, Pages 24: Qualitative Properties of Difference Equation of Order Six]]>
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In this paper we study the qualitative properties and the periodic nature of the solutions of the difference equation x n + 1 = α x n - 2 + β x n - 2 2 γ x n - 2 + δ x n - 5 , n = 0 , 1 , . . . , where the initial conditions x - 5 , x - 4 , x - 3 , x - 2 , x - 1 , x 0 are arbitrary positive real numbers and α , β , γ , δ are positive constants. In addition, we derive the form of the solutions of some special cases of this equation.Mathematics2016-04-1242Article10.3390/math4020024242227-73902016-04-12doi: 10.3390/math4020024Abdul KhaliqE.M. Elsayed<![CDATA[Mathematics, Vol. 4, Pages 23: Existence of Semi Linear Impulsive Neutral Evolution Inclusions with Infinite Delay in Frechet Spaces]]>
http://www.mdpi.com/2227-7390/4/2/23
In this paper, sufficient conditions are given to investigate the existence of mild solutions on a semi-infinite interval for first order semi linear impulsive neutral functional differential evolution inclusions with infinite delay using a recently developed nonlinear alternative for contractive multivalued maps in Frechet spaces due to Frigon combined with semigroup theory. The existence result has been proved without assumption of compactness of the semigroup. We introduced a new phase space for impulsive system with infinite delay and claim that the phase space considered by different authors are not correct.Mathematics2016-04-0642Article10.3390/math4020023232227-73902016-04-06doi: 10.3390/math4020023Dimplekumar ChalishajarKulandhivel KarthikeyanAnnamalai Anguraj<![CDATA[Mathematics, Vol. 4, Pages 21: Optimal Control and Treatment of Infectious Diseases. The Case of Huge Treatment Costs]]>
http://www.mdpi.com/2227-7390/4/2/21
The representation of the cost of a therapy is a key element in the formulation of the optimal control problem for the treatment of infectious diseases. The cost of the treatment is usually modeled by a function of the price and quantity of drugs administered; this function should be the cost as subjectively perceived by the decision-maker. Nevertheless, in literature, the choice of the cost function is often simply done to make the problem more tractable. A specific problem is also given by very expensive therapies in the presence of a very high number of patients to be treated. Firstly, we investigate the optimal treatment of infectious diseases in the simplest case of a two-class population (susceptible and infectious people) and compare the results coming from five different shapes of cost functions. Finally, a model for the treatment of the HCV virus using the blowing-up cost function is investigated. Some numerical simulations are also given.Mathematics2016-04-0142Article10.3390/math4020021212227-73902016-04-01doi: 10.3390/math4020021Andrea Di Liddo<![CDATA[Mathematics, Vol. 4, Pages 22: Higher Order Methods for Nonlinear Equations and Their Basins of Attraction]]>
http://www.mdpi.com/2227-7390/4/2/22
In this paper, we have presented a family of fourth order iterative methods, which uses weight functions. This new family requires three function evaluations to get fourth order accuracy. By the Kung–Traub hypothesis this family of methods is optimal and has an efficiency index of 1.587. Furthermore, we have extended one of the methods to sixth and twelfth order methods whose efficiency indices are 1.565 and 1.644, respectively. Some numerical examples are tested to demonstrate the performance of the proposed methods, which verifies the theoretical results. Further, we discuss the extraneous fixed points and basins of attraction for a few existing methods, such as Newton’s method and the proposed family of fourth order methods. An application problem arising from Planck’s radiation law has been verified using our methods.Mathematics2016-04-0142Article10.3390/math4020022222227-73902016-04-01doi: 10.3390/math4020022Kalyanasundaram MadhuJayakumar Jayaraman<![CDATA[Mathematics, Vol. 4, Pages 20: Birkhoff Normal Forms, KAM Theory and Time Reversal Symmetry for Certain Rational Map]]>
http://www.mdpi.com/2227-7390/4/1/20
By using the KAM(Kolmogorov-Arnold-Moser) theory and time reversal symmetries, we investigate the stability of the equilibrium solutions of the system:
x
n
+
1
=
1
y
n
,
y
n
+
1
=
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x
n
1
+
y
n
,
n
=
0
,
1
,
2
,
&#x2026;
,
where the parameter
&#x3B2;
&gt;
0
,
and initial conditions
x
0
and
y
0
are positive numbers. We obtain the Birkhoff normal form for this system and prove the existence of periodic points with arbitrarily large periods in every neighborhood of the unique positive equilibrium. We use invariants to find a Lyapunov function and Morse&#x2019;s lemma to prove closedness of invariants. We also use the time reversal symmetry method to effectively find some feasible periods and the corresponding periodic orbits.Mathematics2016-03-1841Article10.3390/math4010020202227-73902016-03-18doi: 10.3390/math4010020Erin DenetteMustafa KulenovićEsmir Pilav<![CDATA[Mathematics, Vol. 4, Pages 19: Solution of Differential Equations with Polynomial Coefficients with the Aid of an Analytic Continuation of Laplace Transform]]>
http://www.mdpi.com/2227-7390/4/1/19
In a series of papers, we discussed the solution of Laplace’s differential equation (DE) by using fractional calculus, operational calculus in the framework of distribution theory, and Laplace transform. The solutions of Kummer’s DE, which are expressed by the confluent hypergeometric functions, are obtained with the aid of the analytic continuation (AC) of Riemann–Liouville fractional derivative (fD) and the distribution theory in the space D′R or the AC of Laplace transform. We now obtain the solutions of the hypergeometric DE, which are expressed by the hypergeometric functions, with the aid of the AC of Riemann–Liouville fD, and the distribution theory in the space D′r,R, which is introduced in this paper, or by the term-by-term inverse Laplace transform of AC of Laplace transform of the solution expressed by a series.Mathematics2016-03-1741Article10.3390/math4010019192227-73902016-03-17doi: 10.3390/math4010019Tohru MoritaKen-ichi Sato<![CDATA[Mathematics, Vol. 4, Pages 17: Skew Continuous Morphisms of Ordered Lattice Ringoids]]>
http://www.mdpi.com/2227-7390/4/1/17
Skew continuous morphisms of ordered lattice semirings and ringoids are studied. Different associative semirings and non-associative ringoids are considered. Theorems about properties of skew morphisms are proved. Examples are given. One of the main similarities between them is related to cones in algebras of non locally compact groups.Mathematics2016-03-1641Article10.3390/math4010017172227-73902016-03-16doi: 10.3390/math4010017Sergey Ludkowski<![CDATA[Mathematics, Vol. 4, Pages 18: Dynamics and the Cohomology of Measured Laminations]]>
http://www.mdpi.com/2227-7390/4/1/18
In this paper, the interconnection between the cohomology of measured group actions and the cohomology of measured laminations is explored, the latter being a generalization of the former for the case of discrete group actions and cocycles evaluated on abelian groups. This relation gives a rich interplay between these concepts. Several results can be adapted to this setting—for instance, Zimmer’s reduction of the coefficient group of bounded cocycles or Fustenberg’s cohomological obstruction for extending the ergodicity \(\mathbb{Z}\)-action to a skew product relative to an \(S^{1}\) evaluated cocycle. Another way to think about foliated cocycles is also shown, and a particular application is the characterization of the existence of certain classes of invariant measures for smooth foliations in terms of the \(L^{\infty}\)-cohomology class of the infinitesimal holonomy.Mathematics2016-03-1541Article10.3390/math4010018182227-73902016-03-15doi: 10.3390/math4010018Carlos Meniño Cotón<![CDATA[Mathematics, Vol. 4, Pages 16: New Method of Randomized Forecasting Using Entropy-Robust Estimation: Application to the World Population Prediction]]>
http://www.mdpi.com/2227-7390/4/1/16
We propose a new method of randomized forecasting (RF-method), which operates with models described by systems of linear ordinary differential equations with random parameters. The RF-method is based on entropy-robust estimation of the probability density functions (PDFs) of model parameters and measurement noises. The entropy-optimal estimator uses a limited amount of data. The method of randomized forecasting is applied to World population prediction. Ensembles of entropy-optimal prognostic trajectories of World population and their probability characteristics are generated. We show potential preferences of the proposed method in comparison with existing methods.Mathematics2016-03-1141Article10.3390/math4010016162227-73902016-03-11doi: 10.3390/math4010016Yuri PopkovYuri DubnovAlexey Popkov<![CDATA[Mathematics, Vol. 4, Pages 14: Cost Effectiveness Analysis of Optimal Malaria Control Strategies in Kenya]]>
http://www.mdpi.com/2227-7390/4/1/14
Malaria remains a leading cause of mortality and morbidity among the children under five and pregnant women in sub-Saharan Africa, but it is preventable and controllable provided current recommended interventions are properly implemented. Better utilization of malaria intervention strategies will ensure the gain for the value for money and producing health improvements in the most cost effective way. The purpose of the value for money drive is to develop a better understanding (and better articulation) of costs and results so that more informed, evidence-based choices could be made. Cost effectiveness analysis is carried out to inform decision makers on how to determine where to allocate resources for malaria interventions. This study carries out cost effective analysis of one or all possible combinations of the optimal malaria control strategies (Insecticide Treated Bednets—ITNs, Treatment, Indoor Residual Spray—IRS and Intermittent Preventive Treatment for Pregnant Women—IPTp) for the four different transmission settings in order to assess the extent to which the intervention strategies are beneficial and cost effective. For the four different transmission settings in Kenya the optimal solution for the 15 strategies and their associated effectiveness are computed. Cost-effective analysis using Incremental Cost Effectiveness Ratio (ICER) was done after ranking the strategies in order of the increasing effectiveness (total infections averted). The findings shows that for the endemic regions the combination of ITNs, IRS, and IPTp was the most cost-effective of all the combined strategies developed in this study for malaria disease control and prevention; for the epidemic prone areas is the combination of the treatment and IRS; for seasonal areas is the use of ITNs plus treatment; and for the low risk areas is the use of treatment only. Malaria transmission in Kenya can be minimized through tailor-made intervention strategies for malaria control which produces health improvements in the most cost effective way for different epidemiological zones. This offers the good value for money for the public health programs and can guide in the allocation of malaria control resources for the post-2015 malaria eradication strategies and the achievement of the Sustainable Development Goals.Mathematics2016-03-0941Article10.3390/math4010014142227-73902016-03-09doi: 10.3390/math4010014Gabriel OtienoJoseph KoskeJohn Mutiso<![CDATA[Mathematics, Vol. 4, Pages 15: Conformal Maps, Biharmonic Maps, and the Warped Product]]>
http://www.mdpi.com/2227-7390/4/1/15
In this paper we study some properties of conformal maps between equidimensional manifolds, we construct new example of non-harmonic biharmonic maps and we characterize the biharmonicity of some maps on the warped product manifolds.Mathematics2016-03-0841Article10.3390/math4010015152227-73902016-03-08doi: 10.3390/math4010015Seddik OuakkasDjelloul Djebbouri<![CDATA[Mathematics, Vol. 4, Pages 13: Existence Results for a New Class of Boundary Value Problems of Nonlinear Fractional Differential Equations]]>
http://www.mdpi.com/2227-7390/4/1/13
In this article, we study the following fractional boundary value problem
D
0
+
α
c
u
(
t
)
+
2
r
D
0
+
α
−
1
c
u
(
t
)
+
r
2
D
0
+
α
−
2
c
u
(
t
)
=
f
(
t
,
u
(
t
)
)
,
r
>
0
,
0
Mathematics2016-03-0441Article10.3390/math4010013132227-73902016-03-04doi: 10.3390/math4010013Meysam AlvanRahmat DarziAmin Mahmoodi<![CDATA[Mathematics, Vol. 4, Pages 12: Inverse Eigenvalue Problems for Two Special Acyclic Matrices]]>
http://www.mdpi.com/2227-7390/4/1/12
In this paper, we study two inverse eigenvalue problems (IEPs) of constructing two special acyclic matrices. The first problem involves the reconstruction of matrices whose graph is a path, from given information on one eigenvector of the required matrix and one eigenvalue of each of its leading principal submatrices. The second problem involves reconstruction of matrices whose graph is a broom, the eigen data being the maximum and minimum eigenvalues of each of the leading principal submatrices of the required matrix. In order to solve the problems, we use the recurrence relations among leading principal minors and the property of simplicity of the extremal eigenvalues of acyclic matrices.Mathematics2016-03-0341Communication10.3390/math4010012122227-73902016-03-03doi: 10.3390/math4010012Debashish SharmaMausumi Sen<![CDATA[Mathematics, Vol. 4, Pages 11: Solution of Excited Non-Linear Oscillators under Damping Effects Using the Modified Differential Transform Method]]>
http://www.mdpi.com/2227-7390/4/1/11
The modified differential transform method (MDTM), Laplace transform and Padé approximants are used to investigate a semi-analytic form of solutions of nonlinear oscillators in a large time domain. Forced Duffing and forced van der Pol oscillators under damping effect are studied to investigate semi-analytic forms of solutions. Moreover, solutions of the suggested nonlinear oscillators are obtained using the fourth-order Runge-Kutta numerical solution method. A comparison of the result by the numerical Runge-Kutta fourth-order accuracy method is compared with the result by the MDTM and plotted in a long time domain.Mathematics2016-03-0241Article10.3390/math4010011112227-73902016-03-02doi: 10.3390/math4010011H. Abdelhafez<![CDATA[Mathematics, Vol. 4, Pages 10: A Note on Burg’s Modified Entropy in Statistical Mechanics]]>
http://www.mdpi.com/2227-7390/4/1/10
Burg’s entropy plays an important role in this age of information euphoria, particularly in understanding the emergent behavior of a complex system such as statistical mechanics. For discrete or continuous variable, maximization of Burg’s Entropy subject to its only natural and mean constraint always provide us a positive density function though the Entropy is always negative. On the other hand, Burg’s modified entropy is a better measure than the standard Burg’s entropy measure since this is always positive and there is no computational problem for small probabilistic values. Moreover, the maximum value of Burg’s modified entropy increases with the number of possible outcomes. In this paper, a premium has been put on the fact that if Burg’s modified entropy is used instead of conventional Burg’s entropy in a maximum entropy probability density (MEPD) function, the result yields a better approximation of the probability distribution. An important lemma in basic algebra and a suitable example with tables and graphs in statistical mechanics have been given to illustrate the whole idea appropriately.Mathematics2016-02-2741Article10.3390/math4010010102227-73902016-02-27doi: 10.3390/math4010010Amritansu RayS. Majumder<![CDATA[Mathematics, Vol. 4, Pages 9: Coefficient Inequalities of Second Hankel Determinants for Some Classes of Bi-Univalent Functions]]>
http://www.mdpi.com/2227-7390/4/1/9
In this paper, we investigate two sub-classes S∗ (θ, β) and K∗ (θ, β) of bi-univalent functions in the open unit disc Δ that are subordinate to certain analytic functions. For functions belonging to these classes, we obtain an upper bound for the second Hankel determinant H2 (2).Mathematics2016-02-2541Article10.3390/math401000992227-73902016-02-25doi: 10.3390/math4010009Rayaprolu Bharavi SharmaKalikota Rajya Laxmi<![CDATA[Mathematics, Vol. 4, Pages 8: Tight State-Independent Uncertainty Relations for Qubits]]>
http://www.mdpi.com/2227-7390/4/1/8
The well-known Robertson–Schrödinger uncertainty relations have state-dependent lower bounds, which are trivial for certain states. We present a general approach to deriving tight state-independent uncertainty relations for qubit measurements that completely characterise the obtainable uncertainty values. This approach can give such relations for any number of observables, and we do so explicitly for arbitrary pairs and triples of qubit measurements. We show how these relations can be transformed into equivalent tight entropic uncertainty relations. More generally, they can be expressed in terms of any measure of uncertainty that can be written as a function of the expectation value of the observable for a given state.Mathematics2016-02-2441Article10.3390/math401000882227-73902016-02-24doi: 10.3390/math4010008Alastair AbbottPierre-Louis AlzieuMichael HallCyril Branciard<![CDATA[Mathematics, Vol. 4, Pages 7: Nevanlinna’s Five Values Theorem on Annuli]]>
http://www.mdpi.com/2227-7390/4/1/7
By using the second main theorem of the meromorphic function on annuli, we investigate the problem on two meromorphic functions partially sharing five or more values and obtain some theorems that improve and generalize the previous results given by Cao and Yi.Mathematics2016-02-1841Article10.3390/math401000772227-73902016-02-18doi: 10.3390/math4010007Hong-Yan XuHua Wang<![CDATA[Mathematics, Vol. 4, Pages 6: Microtubules Nonlinear Models Dynamics Investigations through the exp(−Φ(ξ))-Expansion Method Implementation]]>
http://www.mdpi.com/2227-7390/4/1/6
In this research article, we present exact solutions with parameters for two nonlinear model partial differential equations(PDEs) describing microtubules, by implementing the exp(−Φ(ξ))-Expansion Method. The considered models, describing highly nonlinear dynamics of microtubules, can be reduced to nonlinear ordinary differential equations. While the first PDE describes the longitudinal model of nonlinear dynamics of microtubules, the second one describes the nonlinear model of dynamics of radial dislocations in microtubules. The acquired solutions are then graphically presented, and their distinct properties are enumerated in respect to the corresponding dynamic behavior of the microtubules they model. Various patterns, including but not limited to regular, singular kink-like, as well as periodicity exhibiting ones, are detected. Being the method of choice herein, the exp(−Φ(ξ))-Expansion Method not disappointing in the least, is found and declared highly efficient.Mathematics2016-02-0441Article10.3390/math401000662227-73902016-02-04doi: 10.3390/math4010006Nur AlamFethi Belgacem<![CDATA[Mathematics, Vol. 4, Pages 5: Modular Forms and Weierstrass Mock Modular Forms]]>
http://www.mdpi.com/2227-7390/4/1/5
Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass ζ-functions associated to modular elliptic curves “encode” the vanishing and nonvanishing for central values and derivatives of twisted Hasse-Weil L-functions for elliptic curves. Previously, Martin and Ono proved that there are exactly five weight 2 newforms with complex multiplication that are eta-quotients. In this paper, we construct a canonical harmonic Maass form for these five curves with complex multiplication. The holomorphic part of this harmonic Maass form arises from the Weierstrass ζ-function and is referred to as the Weierstrass mock modular form. We prove that the Weierstrass mock modular form for these five curves is itself an eta-quotient or a twist of one. Using this construction, we also obtain p-adic formulas for the corresponding weight 2 newform using Atkin’s U-operator.Mathematics2016-02-0241Article10.3390/math401000552227-73902016-02-02doi: 10.3390/math4010005Amanda Clemm<![CDATA[Mathematics, Vol. 4, Pages 4: Acknowledgement to Reviewers of Mathematics in 2015]]>
http://www.mdpi.com/2227-7390/4/1/4
The editors of Mathematics would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2015. [...]Mathematics2016-01-2541Editorial10.3390/math401000442227-73902016-01-25doi: 10.3390/math4010004 Mathematics Editorial Office<![CDATA[Mathematics, Vol. 4, Pages 3: Multiplicative Expression for the Coefficient in Fermionic 3–3 Relation]]>
http://www.mdpi.com/2227-7390/4/1/3
Recently, a family of fermionic relations were discovered corresponding to Pachner move 3–3 and parameterized by complex-valued 2-cocycles, where the weight of a pentachoron (4-simplex) is a Grassmann–Gaussian exponent. Here, the proportionality coefficient between Berezin integrals in the l.h.s. and r.h.s. of such relations is written in a form multiplicative over simplices.Mathematics2016-01-2041Article10.3390/math401000332227-73902016-01-20doi: 10.3390/math4010003Igor Korepanov<![CDATA[Mathematics, Vol. 4, Pages 2: Barrier Option Under Lévy Model : A PIDE and Mellin Transform Approach]]>
http://www.mdpi.com/2227-7390/4/1/2
We propose a stochastic model to develop a partial integro-differential equation (PIDE) for pricing and pricing expression for fixed type single Barrier options based on the Itô-Lévy calculus with the help of Mellin transform. The stock price is driven by a class of infinite activity Lévy processes leading to the market inherently incomplete, and dynamic hedging is no longer risk free. We first develop a PIDE for fixed type Barrier options, and apply the Mellin transform to derive a pricing expression. Our main contribution is to develop a PIDE with its closed form pricing expression for the contract. The procedure is easy to implement for all class of Lévy processes numerically. Finally, the algorithm for computing numerically is presented with results for a set of Lévy processes.Mathematics2016-01-0441Article10.3390/math401000222227-73902016-01-04doi: 10.3390/math4010002Sudip ChandraDiganta Mukherjee<![CDATA[Mathematics, Vol. 4, Pages 1: On Diff(M)-Pseudo-Differential Operators and the Geometry of Non Linear Grassmannians]]>
http://www.mdpi.com/2227-7390/4/1/1
We consider two principal bundles of embeddings with total space E m b ( M , N ) , with structure groups D i f f ( M ) and D i f f + ( M ) , where D i f f + ( M ) is the groups of orientation preserving diffeomorphisms. The aim of this paper is to describe the structure group of the tangent bundle of the two base manifolds: B ( M , N ) = E m b ( M , N ) / D i f f ( M ) and B + ( M , N ) = E m b ( M , N ) / D i f f + ( M ) from the various properties described, an adequate group seems to be a group of Fourier integral operators, which is carefully studied. It is the main goal of this paper to analyze this group, which is a central extension of a group of diffeomorphisms by a group of pseudo-differential operators which is slightly different from the one developped in the mathematical litterature e.g. by H. Omori and by T. Ratiu. We show that these groups are regular, and develop the necessary properties for applications to the geometry of B ( M , N ) . A case of particular interest is M = S 1 , where connected components of B + ( S 1 , N ) are deeply linked with homotopy classes of oriented knots. In this example, the structure group of the tangent space T B + ( S 1 , N ) is a subgroup of some group G L r e s , following the classical notations of (Pressley, A., 1988). These constructions suggest some approaches in the spirit of one of our previous works on Chern-Weil theory that could lead to knot invariants through a theory of Chern-Weil forms.Mathematics2015-12-2541Article10.3390/math401000112227-73902015-12-25doi: 10.3390/math4010001Jean-Pierre Magnot<![CDATA[Mathematics, Vol. 3, Pages 1255-1273: Two Dimensional Temperature Distributions in Plate Heat Exchangers: An Analytical Approach]]>
http://www.mdpi.com/2227-7390/3/4/1255
Analytical solutions are developed to work out the two-dimensional (2D) temperature changes of flow in the passages of a plate heat exchanger in parallel flow and counter flow arrangements. Two different flow regimes, namely, the plug flow and the turbulent flow are considered. The mathematical formulation of problems coupled at boundary conditions are presented, the solution procedure is then obtained as a special case of the two region Sturm-Liouville problem. The results obtained for two different flow regimes are then compared with experimental results and with each other. The agreement between the analytical and experimental results is an indication of the accuracy of solution method.Mathematics2015-12-1634Article10.3390/math3041255125512732227-73902015-12-16doi: 10.3390/math3041255Amir Ansari DezfoliMozaffar MehrabianMohamad Saffaripour<![CDATA[Mathematics, Vol. 3, Pages 1241-1254: Modeling ITNs Usage: Optimal Promotion Programs Versus Pure Voluntary Adoptions]]>
http://www.mdpi.com/2227-7390/3/4/1241
We consider a mosquito-borne epidemic model, where the adoption by individuals of insecticide–treated bed–nets (ITNs) is taken into account. Motivated by the well documented strong influence of behavioral factors in ITNs usage, we propose a mathematical approach based on the idea of information–dependent epidemic models. We consider the feedback produced by the actions taken by individuals as a consequence of: (i) the information available on the status of the disease in the community where they live; (ii) an optimal health-promotion campaign aimed at encouraging people to use ITNs. The effects on the epidemic dynamics of each of these feedback are assessed and compared with the output of classical models. We show that behavioral changes of individuals may sensibly affect the epidemic dynamics.Mathematics2015-12-1134Article10.3390/math3041241124112542227-73902015-12-11doi: 10.3390/math3041241Bruno Buonomo<![CDATA[Mathematics, Vol. 3, Pages 1222-1240: Robust Finite-Time Anti-Synchronization of Chaotic Systems with Different Dimensions]]>
http://www.mdpi.com/2227-7390/3/4/1222
In this paper, we demonstrate that anti-synchronization (AS) phenomena of chaotic systems with different dimensions can coexist in the finite-time with under the effect of both unknown model uncertainty and external disturbance. Based on the finite-time stability theory and using the master-slave system AS scheme, a generalized approach for the finite-time AS is proposed that guarantee the global stability of the closed-loop for reduced order and increased order AS in the finite time. Numerical simulation results further verify the robustness and effectiveness of the proposed finite-time reduced order and increased order AS schemes.Mathematics2015-12-0834Article10.3390/math3041222122212402227-73902015-12-08doi: 10.3390/math3041222Israr AhmadAzizan SaabanAdyda IbrahimMohammad Shahzad<![CDATA[Mathematics, Vol. 3, Pages 1192-1221: From Cayley-Dickson Algebras to Combinatorial Grassmannians]]>
http://www.mdpi.com/2227-7390/3/4/1192
Given a 2N -dimensional Cayley-Dickson algebra, where 3 ≤ N ≤ 6 , we first observe that the multiplication table of its imaginary units ea , 1 ≤ a ≤ 2N - 1 , is encoded in the properties of the projective space PG(N - 1,2) if these imaginary units are regarded as points and distinguished triads of them {ea, eb , ec} , 1 ≤ a &lt; b &lt; c ≤ 2N - 1 and eaeb = ±ec , as lines. This projective space is seen to feature two distinct kinds of lines according as a + b = c or a + b ≠ c . Consequently, it also exhibits (at least two) different types of points in dependence on how many lines of either kind pass through each of them. In order to account for such partition of the PG(N - 1,2) , the concept of Veldkamp space of a finite point-line incidence structure is employed. The corresponding point-line incidence structure is found to be a specific binomial configuration CN; in particular, C3 (octonions) is isomorphic to the Pasch (62, 43) -configuration, C4 (sedenions) is the famous Desargues (103) -configuration, C5 (32-nions) coincides with the Cayley-Salmon (154, 203) -configuration found in the well-known Pascal mystic hexagram and C6 (64-nions) is identical with a particular (215, 353) -configuration that can be viewed as four triangles in perspective from a line where the points of perspectivity of six pairs of them form a Pasch configuration. Finally, a brief examination of the structure of generic CN leads to a conjecture that CN is isomorphic to a combinatorial Grassmannian of type G2(N + 1).Mathematics2015-12-0434Article10.3390/math3041192119212212227-73902015-12-04doi: 10.3390/math3041192Metod SanigaFrédéric HolweckPetr Pracna<![CDATA[Mathematics, Vol. 3, Pages 1139-1170: HIV vs. the Immune System: A Differential Game]]>
http://www.mdpi.com/2227-7390/3/4/1139
A differential game is formulated in order to model the interaction between the immune system and the HIV virus. One player is represented by the immune system of a patient subject to a therapeutic treatment and the other player is the HIV virus. The aim of our study is to determine the optimal therapy that allows to prevent viral replication inside the body, so as to reduce the damage caused to the immune system, and allow greater survival and quality of life. We propose a model that considers all the most common classes of antiretroviral drugs taking into account different immune cells dynamics. We validate the model with numerical simulations, and determine optimal structured treatment interruption (STI) schedules for medications.Mathematics2015-12-0334Article10.3390/math3041139113911702227-73902015-12-03doi: 10.3390/math3041139Alessandra BurattoRudy CesarettoRita Zamarchi<![CDATA[Mathematics, Vol. 3, Pages 1171-1191: Construction of Periodic Wavelet Frames Generated by the Walsh Polynomials]]>
http://www.mdpi.com/2227-7390/3/4/1171
An explicit method for the construction of a tight wavelet frame generated by the Walsh polynomials with the help of extension principles was presented by Shah (Shah, 2013). In this article, we extend the notion of wavelet frames to periodic wavelet frames generated by the Walsh polynomials on R+ by using extension principles. We first show that under some mild conditions, the periodization of any wavelet frame constructed by the unitary extension principle is still a periodic wavelet frame on R + . Then, we construct a pair of dual periodic wavelet frames generated by the Walsh polynomials on R + using the machinery of the mixed extension principle and Walsh–Fourier transforms.Mathematics2015-12-0334Article10.3390/math3041171117111912227-73902015-12-03doi: 10.3390/math3041171Sunita GoyalFirdous Shah