Mathematics
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Latest open access articles published in Mathematics at http://www.mdpi.com/journal/mathematics<![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]]>
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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]]>
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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]]>
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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]]>
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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]]>
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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]]>
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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]]>
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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]]>
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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]]>
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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]]>
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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]]>
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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]]>
http://www.mdpi.com/2227-7390/4/2/24
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<![CDATA[Mathematics, Vol. 3, Pages 1095-1138: Free W*-Dynamical Systems From p-Adic Number Fields and the Euler Totient Function]]>
http://www.mdpi.com/2227-7390/3/4/1095
In this paper, we study relations between free probability on crossed product W * -algebras with a von Neumann algebra over p-adic number fields ℚp (for primes p), and free probability on the subalgebra Φ, generated by the Euler totient function ϕ, of the arithmetic algebra A , consisting of all arithmetic functions. In particular, we apply such free probability to consider operator-theoretic and operator-algebraic properties of W * -dynamical systems induced by ℚp under free-probabilistic (and hence, spectral-theoretic) techniques.Mathematics2015-12-0234Article10.3390/math3041095109511382227-73902015-12-02doi: 10.3390/math3041095Ilwoo ChoPalle Jorgensen<![CDATA[Mathematics, Vol. 3, Pages 1083-1094: The San Francisco MSM Epidemic: A Retrospective Analysis]]>
http://www.mdpi.com/2227-7390/3/4/1083
We investigate various scenarios for ending the San Francisco MSM (men having sex with men) HIV/AIDS epidemic (1978–1984). We use our previously developed model and explore changes due to prevention strategies such as testing, treatment and reduction of the number of contacts. Here we consider a “what-if” scenario, by comparing different treatment strategies, to determine which factor has the greatest impact on reducing the HIV/AIDS epidemic. The factor determining the future of the epidemic is the reproduction number R0; if R0 &lt; 1, the epidemic is stopped. We show that treatment significantly reduces the total number of infected people. We also investigate the effect a reduction in the number of contacts after seven years, when the HIV/AIDS threat became known, would have had in the population. Both reduction of contacts and treatment alone, however, would not have been enough to bring R0 below one; but when combined, we show that the effective R0 becomes less than one, and therefore the epidemic would have been eradicated.Mathematics2015-11-2434Article10.3390/math3041083108310942227-73902015-11-24doi: 10.3390/math3041083Brandy RapatskiJuan Tolosa<![CDATA[Mathematics, Vol. 3, Pages 1069-1082: A Class of Extended Mittag–Leffler Functions and Their Properties Related to Integral Transforms and Fractional Calculus]]>
http://www.mdpi.com/2227-7390/3/4/1069
In a joint paper with Srivastava and Chopra, we introduced far-reaching generalizations of the extended Gammafunction, extended Beta function and the extended Gauss hypergeometric function. In this present paper, we extend the generalized Mittag–Leffler function by means of the extended Beta function. We then systematically investigate several properties of the extended Mittag–Leffler function, including, for example, certain basic properties, Laplace transform, Mellin transform and Euler-Beta transform. Further, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag–Leffler function are investigated. Some interesting special cases of our main results are also pointed out.Mathematics2015-11-0634Article10.3390/math3041069106910822227-73902015-11-06doi: 10.3390/math3041069Rakesh Parmar<![CDATA[Mathematics, Vol. 3, Pages 1045-1068: Pointwise Reconstruction of Wave Functions from Their Moments through Weighted Polynomial Expansions: An Alternative Global-Local Quantization Procedure]]>
http://www.mdpi.com/2227-7390/3/4/1045
Many quantum systems admit an explicit analytic Fourier space expansion, besides the usual analytic Schrödinger configuration space representation. We argue that the use of weighted orthonormal polynomial expansions for the physical states (generated through the power moments) can define an L2 convergent, non-orthonormal, basis expansion with sufficient pointwise convergent behaviors, enabling the direct coupling of the global (power moments) and local (Taylor series) expansions in configuration space. Our formulation is elaborated within the orthogonal polynomial projection quantization (OPPQ) configuration space representation previously developed The quantization approach pursued here defines an alternative strategy emphasizing the relevance of OPPQ to the reconstruction of the local structure of the physical states.Mathematics2015-11-0534Article10.3390/math3041045104510682227-73902015-11-05doi: 10.3390/math3041045Carlos HandyDaniel VrinceanuCarl MarthHarold Brooks<![CDATA[Mathematics, Vol. 3, Pages 1032-1044: A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation]]>
http://www.mdpi.com/2227-7390/3/4/1032
This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O(N2) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O(N3) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O(N) and a computational cost of O(N logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.Mathematics2015-10-2734Article10.3390/math3041032103210442227-73902015-10-27doi: 10.3390/math3041032Treena Basu<![CDATA[Mathematics, Vol. 3, Pages 1001-1031: A Cohomology Theory for Commutative Monoids]]>
http://www.mdpi.com/2227-7390/3/4/1001
Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.Mathematics2015-10-2734Article10.3390/math3041001100110312227-73902015-10-27doi: 10.3390/math3041001María Calvo-CerveraAntonio Cegarra<![CDATA[Mathematics, Vol. 3, Pages 984-1000: Gauge Invariance and Symmetry Breaking by Topology and Energy Gap]]>
http://www.mdpi.com/2227-7390/3/4/984
For the description of observables and states of a quantum system, it may be convenient to use a canonical Weyl algebra of which only a subalgebra A, with a non-trivial center Z, describes observables, the other Weyl operators playing the role of intertwiners between inequivalent representations of A. In particular, this gives rise to a gauge symmetry described by the action of Z. A distinguished case is when the center of the observables arises from the fundamental group of the manifold of the positions of the quantum system. Symmetries that do not commute with the topological invariants represented by elements of Z are then spontaneously broken in each irreducible representation of the observable algebra, compatibly with an energy gap; such a breaking exhibits a mechanism radically different from Goldstone and Higgs mechanisms. This is clearly displayed by the quantum particle on a circle, the Bloch electron and the two body problem.Mathematics2015-10-2234Article10.3390/math304098498410002227-73902015-10-22doi: 10.3390/math3040984Franco StrocchiCarlo Heissenberg<![CDATA[Mathematics, Vol. 3, Pages 961-983: Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics]]>
http://www.mdpi.com/2227-7390/3/4/961
A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle’s theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided.Mathematics2015-10-2134Article10.3390/math30409619619832227-73902015-10-21doi: 10.3390/math3040961Ellina GrigorievaEvgenii Khailov<![CDATA[Mathematics, Vol. 3, Pages 945-960: Reformulated First Zagreb Index of Some Graph Operations]]>
http://www.mdpi.com/2227-7390/3/4/945
The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper, we study the behavior of the reformulated first Zagreb index and apply our results to different chemically interesting molecular graphs and nano-structures.Mathematics2015-10-1634Article10.3390/math30409459459602227-73902015-10-16doi: 10.3390/math3040945Nilanjan DeSk. NayeemAnita Pal<![CDATA[Mathematics, Vol. 3, Pages 913-944: Understanding Visceral Leishmaniasis Disease Transmission and its Control—A Study Based on Mathematical Modeling]]>
http://www.mdpi.com/2227-7390/3/3/913
Understanding the transmission and control of visceral leishmaniasis, a neglected tropical disease that manifests in human and animals, still remains a challenging problem globally. To study the nature of disease spread, we have developed a compartment-based mathematical model of zoonotic visceral leishmaniasis transmission among three different populations—human, animal and sandfly; dividing the human class into asymptomatic, symptomatic, post-kala-azar dermal leishmaniasis and transiently infected. We analyzed this large model for positivity, boundedness and stability around steady states in different diseased and disease-free scenarios and derived the analytical expression for basic reproduction number (R0). Sensitive parameters for each infected population were identified and varied to observe their effects on the steady state. Epidemic threshold R0 was calculated for every parameter variation. Animal population was identified to play a protective role in absorbing infection, thereby controlling the disease spread in human. To test the predictive ability of the model, seasonal fluctuation was incorporated in the birth rate of the sandflies to compare the model predictions with real data. Control scenarios on this real population data were created to predict the degree of control that can be exerted on the sensitive parameters so as to effectively reduce the infected populations.Mathematics2015-09-2333Article10.3390/math30309139139442227-73902015-09-23doi: 10.3390/math3030913Abhishek SubramanianVidhi SinghRam Sarkar<![CDATA[Mathematics, Vol. 3, Pages 897-912: Photon Localization Revisited]]>
http://www.mdpi.com/2227-7390/3/3/897
In the light of the Newton–Wigner–Wightman theorem of localizability question, we have proposed before a typical generation mechanism of effective mass for photons to be localized in the form of polaritons owing to photon-media interactions. In this paper, the general essence of this example model is extracted in such a form as quantum field ontology associated with the eventualization principle, which enables us to explain the mutual relations, back and forth, between quantum fields and various forms of particles in the localized form of the former.Mathematics2015-09-2333Article10.3390/math30308978979122227-73902015-09-23doi: 10.3390/math3030897Izumi OjimaHayato Saigo<![CDATA[Mathematics, Vol. 3, Pages 891-896: Smooth K-groups for Monoid Algebras and K-regularity]]>
http://www.mdpi.com/2227-7390/3/3/891
The isomorphism of Karoubi-Villamayor K-groups with smooth K-groups for monoid algebras over quasi stable locally convex algebras is established. We prove that the Quillen K-groups are isomorphic to smooth K-groups for monoid algebras over quasi-stable Frechet algebras having a properly uniformly bounded approximate unit and not necessarily m-convex. Based on these results the K-regularity property for quasi-stable Frechet algebras having a properly uniformly bounded approximate unit is established.Mathematics2015-09-1033Article10.3390/math30308918918962227-73902015-09-10doi: 10.3390/math3030891Hvedri Inassaridze<![CDATA[Mathematics, Vol. 3, Pages 880-890: A Note on Necessary Optimality Conditions for a Model with Differential Infectivity in a Closed Population]]>
http://www.mdpi.com/2227-7390/3/3/880
The aim of this note is to present the necessary optimality conditions for a model (in closed population) of an immunizing disease similar to hepatitis B following. We study the impact of medical tests and controls involved in curing this kind of immunizing disease and deduced a well posed adjoint system if there exists an optimal control.Mathematics2015-08-2133Article10.3390/math30308808808902227-73902015-08-21doi: 10.3390/math3030880Yannick Kouakep<![CDATA[Mathematics, Vol. 3, Pages 843-879: Chern-Simons Path Integrals in S2 × S1]]>
http://www.mdpi.com/2227-7390/3/3/843
Using torus gauge fixing, Hahn in 2008 wrote down an expression for a Chern-Simons path integral to compute the Wilson Loop observable, using the Chern-Simons action \(S_{CS}^\kappa\), \(\kappa\) is some parameter. Instead of making sense of the path integral over the space of \(\mathfrak{g}\)-valued smooth 1-forms on \(S^2 \times S^1\), we use the Segal Bargmann transform to define the path integral over \(B_i\), the space of \(\mathfrak{g}\)-valued holomorphic functions over \(\mathbb{C}^2 \times \mathbb{C}^{i-1}\). This approach was first used by us in 2011. The main tool used is Abstract Wiener measure and applying analytic continuation to the Wiener integral. Using the above approach, we will show that the Chern-Simons path integral can be written as a linear functional defined on \(C(B_1^{\times^4} \times B_2^{\times^2}, \mathbb{C})\) and this linear functional is similar to the Chern-Simons linear functional defined by us in 2011, for the Chern-Simons path integral in the case of \(\mathbb{R}^3\). We will define the Wilson Loop observable using this linear functional and explicitly compute it, and the expression is dependent on the parameter \(\kappa\). The second half of the article concentrates on taking \(\kappa\) goes to infinity for the Wilson Loop observable, to obtain link invariants. As an application, we will compute the Wilson Loop observable in the case of \(SU(N)\) and \(SO(N)\). In these cases, the Wilson Loop observable reduces to a state model. We will show that the state models satisfy a Jones type skein relation in the case of \(SU(N)\) and a Conway type skein relation in the case of \(SO(N)\). By imposing quantization condition on the charge of the link \(L\), we will show that the state models are invariant under the Reidemeister Moves and hence the Wilson Loop observables indeed define a framed link invariant. This approach follows that used in an article written by us in 2012, for the case of \(\mathbb{R}^3\).Mathematics2015-08-2133Article10.3390/math30308438438792227-73902015-08-21doi: 10.3390/math3030843Adrian Lim<![CDATA[Mathematics, Vol. 3, Pages 781-842: Algebra of Complex Vectors and Applications in Electromagnetic Theory and Quantum Mechanics]]>
http://www.mdpi.com/2227-7390/3/3/781
A complex vector is a sum of a vector and a bivector and forms a natural extension of a vector. The complex vectors have certain special geometric properties and considered as algebraic entities. These represent rotations along with specified orientation and direction in space. It has been shown that the association of complex vector with its conjugate generates complex vector space and the corresponding basis elements defined from the complex vector and its conjugate form a closed complex four dimensional linear space. The complexification process in complex vector space allows the generation of higher n-dimensional geometric algebra from (n — 1)-dimensional algebra by considering the unit pseudoscalar identification with square root of minus one. The spacetime algebra can be generated from the geometric algebra by considering a vector equal to square root of plus one. The applications of complex vector algebra are discussed mainly in the electromagnetic theory and in the dynamics of an elementary particle with extended structure. Complex vector formalism simplifies the expressions and elucidates geometrical understanding of the basic concepts. The analysis shows that the existence of spin transforms a classical oscillator into a quantum oscillator. In conclusion the classical mechanics combined with zeropoint field leads to quantum mechanics.Mathematics2015-08-2033Article10.3390/math30307817818422227-73902015-08-20doi: 10.3390/math3030781Kundeti Muralidhar<![CDATA[Mathematics, Vol. 3, Pages 758-780: The Segal–Bargmann Transform for Odd-Dimensional Hyperbolic Spaces]]>
http://www.mdpi.com/2227-7390/3/3/758
We develop isometry and inversion formulas for the Segal–Bargmann transform on odd-dimensional hyperbolic spaces that are as parallel as possible to the dual case of odd-dimensional spheres.Mathematics2015-08-1833Article10.3390/math30307587587802227-73902015-08-18doi: 10.3390/math3030758Brian HallJeffrey Mitchell<![CDATA[Mathematics, Vol. 3, Pages 746-757: A Moonshine Dialogue in Mathematical Physics]]>
http://www.mdpi.com/2227-7390/3/3/746
Phys and Math are two colleagues at the University of Saçenbon (Crefan Kingdom), dialoguing about the remarkable efficiency of mathematics for physics. They talk about the notches on the Ishango bone and the various uses of psi in maths and physics; they arrive at dessins d’enfants, moonshine concepts, Rademacher sums and their significance in the quantum world. You should not miss their eccentric proposal of relating Bell’s theorem to the Baby Monster group. Their hyperbolic polygons show a considerable singularity/cusp structure that our modern age of computers is able to capture. Henri Poincaré would have been happy to see it.Mathematics2015-08-1433Essay10.3390/math30307467467572227-73902015-08-14doi: 10.3390/math3030746Michel Planat<![CDATA[Mathematics, Vol. 3, Pages 727-745: From Classical to Discrete Gravity through Exponential Non-Standard Lagrangians in General Relativity]]>
http://www.mdpi.com/2227-7390/3/3/727
Recently, non-standard Lagrangians have gained a growing importance in theoretical physics and in the theory of non-linear differential equations. However, their formulations and implications in general relativity are still in their infancies despite some advances in contemporary cosmology. The main aim of this paper is to fill the gap. Though non-standard Lagrangians may be defined by a multitude form, in this paper, we considered the exponential type. One basic feature of exponential non-standard Lagrangians concerns the modified Euler-Lagrange equation obtained from the standard variational analysis. Accordingly, when applied to spacetime geometries, one unsurprisingly expects modified geodesic equations. However, when taking into account the time-like paths parameterization constraint, remarkably, it was observed that mutually discrete gravity and discrete spacetime emerge in the theory. Two different independent cases were obtained: A geometrical manifold with new spacetime coordinates augmented by a metric signature change and a geometrical manifold characterized by a discretized spacetime metric. Both cases give raise to Einstein’s field equations yet the gravity is discretized and originated from “spacetime discreteness”. A number of mathematical and physical implications of these results were discussed though this paper and perspectives are given accordingly.Mathematics2015-08-1433Article10.3390/math30307277277452227-73902015-08-14doi: 10.3390/math3030727Rami El-Nabulsi<![CDATA[Mathematics, Vol. 3, Pages 690-726: Root Operators and “Evolution” Equations]]>
http://www.mdpi.com/2227-7390/3/3/690
Root-operator factorization à la Dirac provides an effective tool to deal with equations, which are not of evolution type, or are ruled by fractional differential operators, thus eventually yielding evolution-like equations although for a multicomponent vector. We will review the method along with its extension to root operators of degree higher than two. Also, we will show the results obtained by the Dirac-method as well as results from other methods, specifically in connection with evolution-like equations ruled by square-root operators, that we will address to as relativistic evolution equations.Mathematics2015-08-1333Article10.3390/math30306906907262227-73902015-08-13doi: 10.3390/math3030690Giuseppe DattoliAmalia Torre<![CDATA[Mathematics, Vol. 3, Pages 666-689: Evaluation of Interpolants in Their Ability to Fit Seismometric Time Series]]>
http://www.mdpi.com/2227-7390/3/3/666
This article is devoted to the study of the ASARCO demolition seismic data. Two different classes of modeling techniques are explored: First, mathematical interpolation methods and second statistical smoothing approaches for curve fitting. We estimate the characteristic parameters of the propagation medium for seismic waves with multiple mathematical and statistical techniques, and provide the relative advantages of each approach to address fitting of such data. We conclude that mathematical interpolation techniques and statistical curve fitting techniques complement each other and can add value to the study of one dimensional time series seismographic data: they can be use to add more data to the system in case the data set is not large enough to perform standard statistical tests.Mathematics2015-08-0733Article10.3390/math30306666666892227-73902015-08-07doi: 10.3390/math3030666Kanadpriya BasuMaria MarianiLaura SerpaRitwik Sinha<![CDATA[Mathematics, Vol. 3, Pages 653-665: Zeta Function Expression of Spin Partition Functions on Thermal AdS3]]>
http://www.mdpi.com/2227-7390/3/3/653
We find a Selberg zeta function expression of certain one-loop spin partition functions on three-dimensional thermal anti-de Sitter space. Of particular interest is the partition function of higher spin fermionic particles. We also set up, in the presence of spin, a Patterson-type formula involving the logarithmic derivative of zeta.Mathematics2015-07-2833Article10.3390/math30306536536652227-73902015-07-28doi: 10.3390/math3030653Floyd L.Williams<![CDATA[Mathematics, Vol. 3, Pages 644-652: On the Nature of the Tsallis–Fourier Transform]]>
http://www.mdpi.com/2227-7390/3/3/644
By recourse to tempered ultradistributions, we show here that the effect of a q-Fourier transform (qFT) is to map equivalence classes of functions into other classes in a one-to-one fashion. This suggests that Tsallis’ q-statistics may revolve around equivalence classes of distributions and not individual ones, as orthodox statistics does. We solve here the qFT’s non-invertibility issue, but discover a problem that remains open.Mathematics2015-07-2133Article10.3390/math30306446446522227-73902015-07-21doi: 10.3390/math3030644A. PlastinoMario Rocca<![CDATA[Mathematics, Vol. 3, Pages 626-643: Time Automorphisms on C*-Algebras]]>
http://www.mdpi.com/2227-7390/3/3/626
Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C*-algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.Mathematics2015-07-1633Article10.3390/math30306266266432227-73902015-07-16doi: 10.3390/math3030626R. Hilfer<![CDATA[Mathematics, Vol. 3, Pages 615-625: Reproducing Kernel Hilbert Space vs. Frame Estimates]]>
http://www.mdpi.com/2227-7390/3/3/615
We consider conditions on a given system F of vectors in Hilbert space H, forming a frame, which turn H into a reproducing kernel Hilbert space. It is assumed that the vectors in F are functions on some set Ω . We then identify conditions on these functions which automatically give H the structure of a reproducing kernel Hilbert space of functions on Ω. We further give an explicit formula for the kernel, and for the corresponding isometric isomorphism. Applications are given to Hilbert spaces associated to families of Gaussian processes.Mathematics2015-07-0833Article10.3390/math30306156156252227-73902015-07-08doi: 10.3390/math3030615Palle JorgensenMyung-Sin Song<![CDATA[Mathematics, Vol. 3, Pages 604-614: Topological Integer Additive Set-Sequential Graphs]]>
http://www.mdpi.com/2227-7390/3/3/604
Let \(\mathbb{N}_0\) denote the set of all non-negative integers and \(X\) be any non-empty subset of \(\mathbb{N}_0\). Denote the power set of \(X\) by \(\mathcal{P}(X)\). An integer additive set-labeling (IASL) of a graph \(G\) is an injective function \(f : V (G) \to P(X)\) such that the image of the induced function \(f^+: E(G) \to \mathcal{P}(\mathbb{N}_0)\), defined by \(f^+(uv)=f(u)+f(v)\), is contained in \(\mathcal{P}(X)\), where \(f(u) + f(v)\) is the sumset of \(f(u)\) and \(f(v)\). If the associated set-valued edge function \(f^+\) is also injective, then such an IASL is called an integer additive set-indexer (IASI). An IASL \(f\) is said to be a topological IASL (TIASL) if \(f(V(G))\cup \{\emptyset\}\) is a topology of the ground set \(X\). An IASL is said to be an integer additive set-sequential labeling (IASSL) if \(f(V(G))\cup f^+(E(G))= \mathcal{P}(X)-\{\emptyset\}\). An IASL of a given graph \(G\) is said to be a topological integer additive set-sequential labeling of \(G\), if it is a topological integer additive set-labeling as well as an integer additive set-sequential labeling of \(G\). In this paper, we study the conditions required for a graph \(G\) to admit this type of IASL and propose some important characteristics of the graphs which admit this type of IASLs.Mathematics2015-07-0333Article10.3390/math30306046046142227-73902015-07-03doi: 10.3390/math3030604Sudev NaduvathGermina AugustineChithra Sudev<![CDATA[Mathematics, Vol. 3, Pages 563-603: Singular Bilinear Integrals in Quantum Physics]]>
http://www.mdpi.com/2227-7390/3/3/563
Bilinear integrals of operator-valued functions with respect to spectral measures and integrals of scalar functions with respect to the product of two spectral measures arise in many problems in scattering theory and spectral analysis. Unfortunately, the theory of bilinear integration with respect to a vector measure originating from the work of Bartle cannot be applied due to the singular variational properties of spectral measures. In this work, it is shown how ``decoupled'' bilinear integration may be used to find solutions \(X\) of operator equations \(AX-XB=Y\) with respect to the spectral measure of \(A\) and to apply such representations to the spectral decomposition of block operator matrices. A new proof is given of Peller's characterisation of the space \(L^1((P\otimes Q)_{\mathcal L(\mathcal H)})\) of double operator integrable functions for spectral measures \(P\), \(Q\) acting in a Hilbert space \(\mathcal H\) and applied to the representation of the trace of \(\int_{\Lambda\times\Lambda}\varphi\,d(PTP)\) for a trace class operator \(T\). The method of double operator integrals due to Birman and Solomyak is used to obtain an elementary proof of the existence of Krein's spectral shift function.Mathematics2015-06-2933Article10.3390/math30305635636032227-73902015-06-29doi: 10.3390/math3030563Brian Jefferies<![CDATA[Mathematics, Vol. 3, Pages 527-562: The Schwartz Space: Tools for Quantum Mechanics and Infinite Dimensional Analysis]]>
http://www.mdpi.com/2227-7390/3/2/527
An account of the Schwartz space of rapidly decreasing functions as a topological vector space with additional special structures is presented in a manner that provides all the essential background ideas for some areas of quantum mechanics along with infinite-dimensional distribution theory.Mathematics2015-06-1632Article10.3390/math30205275275622227-73902015-06-16doi: 10.3390/math3020527Jeremy BecnelAmbar Sengupta<![CDATA[Mathematics, Vol. 3, Pages 510-526: Effective Summation and Interpolation of Series by Self-Similar Root Approximants]]>
http://www.mdpi.com/2227-7390/3/2/510
We describe a simple analytical method for effective summation of series, including divergent series. The method is based on self-similar approximation theory resulting in self-similar root approximants. The method is shown to be general and applicable to different problems, as is illustrated by a number of examples. The accuracy of the method is not worse, and in many cases better, than that of Padé approximants, when the latter can be defined.Mathematics2015-06-1532Article10.3390/math30205105105262227-73902015-06-15doi: 10.3390/math3020510Simon GluzmanVyacheslav Yukalov<![CDATA[Mathematics, Vol. 3, Pages 487-509: The Fractional Orthogonal Difference with Applications]]>
http://www.mdpi.com/2227-7390/3/2/487
This paper is a follow-up of a previous paper of the author published in Mathematics journal in 2015, which treats the so-called continuous fractional orthogonal derivative. In this paper, we treat the discrete case using the fractional orthogonal difference. The theory is illustrated with an application of a fractional differentiating filter. In particular, graphs are presented of the absolutel value of the modulus of the frequency response. These make clear that for a good insight into the behavior of a fractional differentiating filter, one has to look for the modulus of its frequency response in a log-log plot, rather than for plots in the time domain.Mathematics2015-06-1232Article10.3390/math30204874875092227-73902015-06-12doi: 10.3390/math3020487Enno Diekema<![CDATA[Mathematics, Vol. 3, Pages 481-486: The Complement of Binary Klein Quadric as a Combinatorial Grassmannian]]>
http://www.mdpi.com/2227-7390/3/2/481
Given a hyperbolic quadric of PG(5, 2), there are 28 points off this quadric and 56 lines skew to it. It is shown that the (286; 563)-configuration formed by these points and lines is isomorphic to the combinatorial Grassmannian of type G2(8). It is also pointed out that a set of seven points of G2(8) whose labels share a mark corresponds to a Conwell heptad of PG(5, 2). Gradual removal of Conwell heptads from the (286; 563)-configuration yields a nested sequence of binomial configurations identical with part of that found to be associated with Cayley-Dickson algebras (arXiv:1405.6888).Mathematics2015-06-0832Letter10.3390/math30204814814862227-73902015-06-08doi: 10.3390/math3020481Metod Saniga<![CDATA[Mathematics, Vol. 3, Pages 444-480: Sinc-Approximations of Fractional Operators: A Computing Approach]]>
http://www.mdpi.com/2227-7390/3/2/444
We discuss a new approach to represent fractional operators by Sinc approximation using convolution integrals. A spin off of the convolution representation is an effective inverse Laplace transform. Several examples demonstrate the application of the method to different practical problems.Mathematics2015-06-0532Article10.3390/math30204444444802227-73902015-06-05doi: 10.3390/math3020444Gerd BaumannFrank Stenger<![CDATA[Mathematics, Vol. 3, Pages 428-443: The 1st Law of Thermodynamics for the Mean Energy of a Closed Quantum System in the Aharonov-Vaidman Gauge]]>
http://www.mdpi.com/2227-7390/3/2/428
The Aharonov-Vaidman gauge additively transforms the mean energy of a quantum mechanical system into a weak valued system energy. In this paper, the equation of motion of this weak valued energy is used to provide a mathematical statement of an extended 1st Law of Thermodynamics that is applicable to the mean energy of a closed quantum system when the mean energy is expressed in the Aharonov-Vaidman gauge, i.e., when the system’s energy is weak valued. This is achieved by identifying the generalized heat and work exchange terms that appear in the equation of motion for weak valued energy. The complex valued contributions of the additive gauge term to these generalized exchange terms are discussed and this extended 1st Law is shown to subsume the usual 1st Law that is applicable for the mean energy of a closed quantum system. It is found that the gauge transformation introduces an additional energy uncertainty exchange term that—while it is neither a heat nor a work exchange term—is necessary for the conservation of weak valued energy. A spin-1/2 particle in a uniform magnetic field is used to illustrate aspects of the theory. It is demonstrated for this case that the extended 1st Law implies the existence of a gauge potential ω and that it generates a non-vanishing gauge field F. It is also shown for this case that the energy uncertainty exchange accumulated during the evolution of the system along a closed evolutionary cycle C in an associated parameter space is a geometric phase. This phase is equal to both the path integral of ω along C and the integral of the flux of F through the area enclosed by C.Mathematics2015-06-0132Article10.3390/math30204284284432227-73902015-06-01doi: 10.3390/math3020428Allen Parks<![CDATA[Mathematics, Vol. 3, Pages 412-427: Subordination Principle for a Class of Fractional Order Differential Equations]]>
http://www.mdpi.com/2227-7390/3/2/412
The fractional order differential equation \(u'(t)=Au(t)+\gamma D_t^{\alpha} Au(t)+f(t), \ t&gt;0\), \(u(0)=a\in X\) is studied, where \(A\) is an operator generating a strongly continuous one-parameter semigroup on a Banach space \(X\), \(D_t^{\alpha}\) is the Riemann–Liouville fractional derivative of order \(\alpha \in (0,1)\), \(\gamma&gt;0\) and \(f\) is an \(X\)-valued function. Equations of this type appear in the modeling of unidirectional viscoelastic flows. Well-posedness is proven, and a subordination identity is obtained relating the solution operator of the considered problem and the \(C_{0}\)-semigroup, generated by the operator \(A\). As an example, the Rayleigh–Stokes problem for a generalized second-grade fluid is considered.Mathematics2015-05-2632Article10.3390/math30204124124272227-73902015-05-26doi: 10.3390/math3020412Emilia Bazhlekova<![CDATA[Mathematics, Vol. 3, Pages 398-411: Implicit Fractional Differential Equations via the Liouville–Caputo Derivative]]>
http://www.mdpi.com/2227-7390/3/2/398
We study an initial value problem for an implicit fractional differential equation with the Liouville–Caputo fractional derivative. By using fixed point theory and an approximation method, we obtain some existence and uniqueness results.Mathematics2015-05-2532Article10.3390/math30203983984112227-73902015-05-25doi: 10.3390/math3020398Juan NietoAbelghani OuahabVenktesh Venktesh<![CDATA[Mathematics, Vol. 3, Pages 382-397: The Spectral Connection Matrix for Any Change of Basis within the Classical Real Orthogonal Polynomials]]>
http://www.mdpi.com/2227-7390/3/2/382
The connection problem for orthogonal polynomials is, given a polynomial expressed in the basis of one set of orthogonal polynomials, computing the coefficients with respect to a different set of orthogonal polynomials. Expansions in terms of orthogonal polynomials are very common in many applications. While the connection problem may be solved by directly computing the change–of–basis matrix, this approach is computationally expensive. A recent approach to solving the connection problem involves the use of the spectral connection matrix, which is a matrix whose eigenvector matrix is the desired change–of–basis matrix. In Bella and Reis (2014), it is shown that for the connection problem between any two different classical real orthogonal polynomials of the Hermite, Laguerre, and Gegenbauer families, the related spectral connection matrix has quasiseparable structure. This result is limited to the case where both the source and target families are one of the Hermite, Laguerre, or Gegenbauer families, which are each defined by at most a single parameter. In particular, this excludes the large and common class of Jacobi polynomials, defined by two parameters, both as a source and as a target family. In this paper, we continue the study of the spectral connection matrix for connections between real orthogonal polynomial families. In particular, for the connection problem between any two families of the Hermite, Laguerre, or Jacobi type (including Chebyshev, Legendre, and Gegenbauer), we prove that the spectral connection matrix has quasiseparable structure. In addition, our results also show the quasiseparable structure of the spectral connection matrix from the Bessel polynomials, which are orthogonal on the unit circle, to any of the Hermite, Laguerre, and Jacobi types. Additionally, the generators of the spectral connection matrix are provided explicitly for each of these cases, allowing a fast algorithm to be implemented following that in Bella and Reis (2014).Mathematics2015-05-1432Article10.3390/math30203823823972227-73902015-05-14doi: 10.3390/math3020382Tom BellaJenna Reis