Mathematics doi: 10.3390/math5040053

Authors: Yang-Hi Lee

In this paper, we prove the stability of the following functional equation ∑ i = 0 n n C i ( − 1 ) n − i f ( i x + y ) − n ! f ( x ) = 0 on a restricted domain by employing the direct method in the sense of Hyers.

]]>Mathematics doi: 10.3390/math5040052

Authors: Alexander Varchenko

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz.

]]>Mathematics doi: 10.3390/math5040051

Authors: Bang-Yen Chen

The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (t) is usually the most sought-after quantity because the position vector field defines the motion of a particle (i.e., a point mass): its location relative to a given coordinate system at some time variable t. This article is a survey article. The purpose of this article is to survey recent results of Euclidean submanifolds associated with the tangential components of their position vector fields. In the last section, we present some interactions between torqued vector fields and Ricci solitons.

]]>Mathematics doi: 10.3390/math5040050

Authors: André Erhardt

In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation ∂ t u − div a ( x , t , ∇ u ) + λ ( | u | p ( x , t ) − 2 u ) = 0 in Ω T , where λ ≥ 0 and ∂ t u denote the partial derivative of u with respect to the time variable t, while ∇ u denotes the one with respect to the space variable x. Moreover, the vector-field a ( x , t , · ) satisfies certain nonstandard p ( x , t ) -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values.

]]>Mathematics doi: 10.3390/math5040049

Authors: Adam Glick Antonio Mastroberardino

Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is a concern. Drug therapies have been developed that attack proliferating endothelial cells instead of the tumor in an attempt to create a therapy that is resistant to resistance in contrast to other forms of treatment such as chemotherapy and radiation therapy. In this study, a two-compartment model in terms of differential equations is presented in order to determine the optimal protocol for the delivery of anti-angiogenesis therapy. Optimal control theory is applied to the model with a range of anti-angiogenesis doses to determine optimal doses to minimize tumor volume at the end of a two week treatment and minimize drug toxicity to the patient. Applying a continuous optimal control protocol to our model of angiogenesis and tumor cell growth shows promising results for tumor control while minimizing the toxicity to the patients. By investigating a variety of doses, we determine that the optimal angiogenesis inhibitor dose is in the range of 10–20 mg/kg. In this clinically useful range of doses, good tumor control is achieved for a two week treatment period. This work shows that varying the toxicity of the treatment to the patient will change the optimal dosing scheme but tumor control can still be achieved.

]]>Mathematics doi: 10.3390/math5040048

Authors: Daniele Mortari

This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [ − 1 , + 1 ] . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.

]]>Mathematics doi: 10.3390/math5040047

Authors: Hayman Thabet Subhash Kendre Dimplekumar Chalishajar

This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods.

]]>Mathematics doi: 10.3390/math5030046

Authors: Timothy Sands

Electric vehicles have undergone a recent faddy trend in the United States and Europe, and several recent publications trumpet the continued rise of electric vehicles citing steadily-climbing monthly vehicle sales. The broad purpose of this study is to examine this optimism with some degree of mathematical rigor. Specifically, the methodology will use catastrophe theory to explore the possibility of a sudden, seemingly-unexplainable crash in vehicle sales. The study begins by defining optimal system equations that well-model the available sales data. Next, these optimal models are used to investigate the potential response to a slow dynamic acting on the relatively faster dynamic of the optimal system equations. Catastrophe theory indicates a potential sudden crash in sales when a slow dynamic is at-work. It is noteworthy that the prediction can be made even while sales are increasing.

]]>Mathematics doi: 10.3390/math5030045

Authors: Raquel Caballero-Águila Aurora Hermoso-Carazo Josefa Linares-Pérez

In this paper, the information fusion estimation problem is investigated for a class of multisensor linear systems affected by different kinds of stochastic uncertainties, using both the distributed and the centralized fusion methodologies. It is assumed that the measured outputs are perturbed by one-step autocorrelated and cross-correlated additive noises, and also stochastic uncertainties caused by multiplicative noises and randomly missing measurements in the sensor outputs are considered. At each sampling time, every sensor output is sent to a local processor and, due to some kind of transmission failures, one-step correlated random delays may occur. Using only covariance information, without requiring the evolution model of the signal process, a local least-squares (LS) filter based on the measurements received from each sensor is designed by an innovation approach. All these local filters are then fused to generate an optimal distributed fusion filter by a matrix-weighted linear combination, using the LS optimality criterion. Moreover, a recursive algorithm for the centralized fusion filter is also proposed and the accuracy of the proposed estimators, which is measured by the estimation error covariances, is analyzed by a simulation example.

]]>Mathematics doi: 10.3390/math5030044

Authors: José Velhinho

This short review is devoted to measures on infinite dimensional spaces. We start by discussing product measures and projective techniques. Special attention is paid to measures on linear spaces, and in particular to Gaussian measures. Transformation properties of measures are considered, as well as fundamental results concerning the support of the measure.

]]>Mathematics doi: 10.3390/math5030043

Authors: Ruth Kellerhals

We study lattices with a non-compact fundamental domain of small volume in hyperbolic space H n . First, we identify the arithmetic lattices in Isom + H n of minimal covolume for even n up to 18. Then, we discuss the related problem in higher odd dimensions and provide solutions for n = 11 and n = 13 in terms of the rotation subgroup of certain Coxeter pyramid groups found by Tumarkin. The results depend on the work of Belolipetsky and Emery, as well as on the Euler characteristic computation for hyperbolic Coxeter polyhedra with few facets by means of the program CoxIter developed by Guglielmetti. This work complements the survey about hyperbolic orbifolds of minimal volume.

]]>Mathematics doi: 10.3390/math5030042

Authors: Rahman Ullah Xiao-Min Li Faiz Faizullah Hong-Xun Yi Riaz Khan

This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference polynomials of these functions have the same fixed points or share a nonzero value. This extends the research work of Qi, Yang and Liu, where they used the finite ordered meromorphic mappings.

]]>Mathematics doi: 10.3390/math5030041

Authors: Jens Fischer

In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are also inverse of each other. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.

]]>Mathematics doi: 10.3390/math5030040

Authors: Feng Qi Bai-Ni Guo

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.

]]>Mathematics doi: 10.3390/math5030039

Authors: Warisa Thangjai Suparat Niwitpong Sa-Aat Niwitpong

This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single normal mean with unknown CV. These confidence intervals were compared with existing confidence interval for the single normal mean based on the Student’s t-distribution (small sample size case) and the z-distribution (large sample size case). Furthermore, the confidence intervals for the difference between two normal means with unknown CVs were constructed based on the GCI approach, the method of variance estimates recovery (MOVER) approach and the LS approach and then compared with the Welch–Satterthwaite (WS) approach. The coverage probability and average length of the proposed confidence intervals were evaluated via Monte Carlo simulation. The results indicated that the GCIs for the single normal mean and the difference of two normal means with unknown CVs are better than the other confidence intervals. Finally, three datasets are given to illustrate the proposed confidence intervals.

]]>Mathematics doi: 10.3390/math5030038

Authors: Hananeh Nojavan Saeid Abbasbandy Tofigh Allahviranloo

This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For time-dependent partial differential equations (PDEs), it would be changed to a system of ordinary differential equations (ODEs). Here, we intended to decrease the error through utilizing variable shape parameter (VSP) strategies. This method was an appropriate way to solve the two-dimensional nonlinear coupled Burgers’ equations comprised of Dirichlet and mixed boundary conditions. Numerical examples indicated that the variable shape parameter strategies were more efficient than constant ones for various values of the Reynolds number.

]]>Mathematics doi: 10.3390/math5030036

Authors: Evelina Viada

In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank N - 1 on transverse curves in E N , where E is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when E has Q -rank ≤ N - 1 . We also give some explicit examples. This result generalises from rank 1 to rank N - 1 previous results of S. Checcoli, F. Veneziano and the author.

]]>Mathematics doi: 10.3390/math5030037

Authors: Rémy Tuyéras

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/ Ω -spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allow one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an elimination of quotients and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak ( ω , n ) -categories, weak ω -groupoids and higher moduli stacks.

]]>Mathematics doi: 10.3390/math5030035

Authors: Sergey Ludkowski

In this article, Müntz spaces M Λ , C of continuous functions supplied with the absolute maximum norm are considered. An existence of Schauder bases in Müntz spaces M Λ , C is investigated. Moreover, Fourier series approximation of functions in Müntz spaces M Λ , C is studied.

]]>Mathematics doi: 10.3390/math5020034

Authors: Shadi Al-Omari Fiazuddin Zaman Hassan Azad

Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived. All possible invariant variables of the optimal system are obtained. The corresponding reduced systems of ordinary differential equations (ODEs) are also provided. All possible non-similar invariant conditions prescribed on invariant surfaces under symmetry transformations are given. As an application, explicit solutions of the system are given where the beam is hinged at one end and free at the other end.

]]>Mathematics doi: 10.3390/math5020033

Authors: Din Prathumwan Wannika Sawangtong Panumart Sawangtong

The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We prove the existence and uniqueness of a solution for the FAFDEs. Furthermore, the stability analysis of the model is investigated and the numerical simulation is accordingly performed to support the proposed model.

]]>Mathematics doi: 10.3390/math5020032

Authors: Dagmar Markechová Peter Vrábel Beáta Stehlíková

In this paper, we have proved a metrization theorem that gives the sufficient conditions for a uniform IP-loop X to be metrizable by a left-invariant metric. It is shown that by consideration of topological IP-loop dual to X we obtain an analogical theorem for the case of the right-invariant metric.

]]>Mathematics doi: 10.3390/math5020031

Authors: Weimin Wang Yuqiang Feng Yuanyuan Wang

In this paper, some nonlinear Gronwall–Bellman type inequalities are established. Then, the obtained results are applied to study the Hyers–Ulam stability of a fractional differential equation and the boundedness of solutions to an integral equation, respectively.

]]>Mathematics doi: 10.3390/math5020030

Authors: Muhammad Khan Akbar Azam Nayyar Mehmood

In this article the coincidence points of a self map and a sequence of multivalued maps are found in the settings of complete metric space endowed with a graph. A novel result of Asrifa and Vetrivel is generalized and as an application we obtain an existence theorem for a special type of fractional integral equation. Moreover, we establish a result on the convergence of successive approximation of a system of Bernstein operators on a Banach space.

]]>Mathematics doi: 10.3390/math5020029

Authors: Amrik Sen Raymond Aschheim Klee Irwin

We present the emergence of a root system in six dimensions from the tetrahedra of an icosahedral core known as the 20-group (20G) within the framework of Clifford’s geometric algebra. Consequently, we establish a connection between a three-dimensional icosahedral seed, a six-dimensional (6D) Dirichlet quantized host and a higher dimensional lattice structure. The 20G, owing to its icosahedral symmetry, bears the signature of a 6D lattice that manifests in the Dirichlet integer representation. We present an interpretation whereby the three-dimensional 20G can be regarded as the core substratum from which the higher dimensional lattices emerge. This emergent geometry is based on an induction principle supported by the Clifford multi-vector formalism of three-dimensional (3D) Euclidean space. This lays a geometric framework for understanding several physics theories related to S U ( 5 ) , E 6 , E 8 Lie algebras and their composition with the algebra associated with the even unimodular lattice in R 3 , 1 . The construction presented here is inspired by Penrose’s three world model.

]]>Mathematics doi: 10.3390/math5020028

Authors: Fu-Rong Lin Shi-Wei Yang

A numerical method is proposed for estimating piecewise-constant solutions for Fredholm integral equations of the first kind. Two functionals, namely the weighted total variation (WTV) functional and the simplified Modica-Mortola (MM) functional, are introduced. The solution procedure consists of two stages. In the first stage, the WTV functional is minimized to obtain an approximate solution f TV * . In the second stage, the simplified MM functional is minimized to obtain the final result by using the damped Newton (DN) method with f TV * as the initial guess. The numerical implementation is given in detail, and numerical results of two examples are presented to illustrate the efficiency of the proposed approach.

]]>Mathematics doi: 10.3390/math5020027

Authors: Zeeshan Khan Muhammad Khan Saeed Islam Bilal Jan Fawad Hussain Haroon Ur Rasheed Waris Khan

Wire coating process is a continuous extrusion process for primary insulation of conducting wires with molten polymers for mechanical strength and protection in aggressive environments. Nylon, polysulfide, low/high density polyethylene (LDPE/HDPE) and plastic polyvinyl chloride (PVC) are the common and important plastic resin used for wire coating. In the current study, wire coating is performed using viscoelastic third grade fluid in the presence of applied magnetic field and porous medium. The governing equations are first modeled and then solved analytically by utilizing the homotopy analysis method (HAM). The convergence of the series solution is established. A numerical technique called ND-solve method is used for comparison and found good agreement. The effect of pertinent parameters on the velocity field and temperature profile is shown with the help of graphs. It is observed that the velocity profiles increase as the value of viscoelastic third grade parameter β increase and decrease as the magnetic parameter M and permeability parameter K increase. It is also observed that the temperature profiles increases as the Brinkman number B r , permeability parameter K , magnetic parameter M and viscoelastic third grade parameter (non-Newtonian parameter) β increase.

]]>Mathematics doi: 10.3390/math5020026

Authors: Mohammad Abolhasani Saeid Abbasbandy Tofigh Allahviranloo

In this paper, we introduced a new generalization method to solve fractional convection–diffusion equations based on the well-known variational iteration method (VIM) improved by an auxiliary parameter. The suggested method was highly effective in controlling the convergence region of the approximate solution. By solving some fractional convection–diffusion equations with a propounded method and comparing it with standard VIM, it was concluded that complete reliability, efficiency, and accuracy of this method are guaranteed. Additionally, we studied and investigated the convergence of the proposed method, namely the VIM with an auxiliary parameter. We also offered the optimal choice of the auxiliary parameter in the proposed method. It was noticed that the approach could be applied to other models of physics.

]]>Mathematics doi: 10.3390/math5020025

Authors: Tirumalasetty Chiranjeevi Raj Biswas

A formulation and solution of the discrete-time fractional optimal control problem in terms of the Caputo fractional derivative is presented in this paper. The performance index (PI) is considered in a quadratic form. The necessary and transversality conditions are obtained using a Hamiltonian approach. Both the free and fixed final state cases have been considered. Numerical examples are taken up and their solution technique is presented. Results are produced for different values of α .

]]>Mathematics doi: 10.3390/math5020024

Authors: Murchana Neog Pradip Debnath

In the present article, we introduce the new concept of start point in a directed graph and provide the characterizations required for a directed graph to have a start point. We also define the notion of a self path set valued map and establish its relation with start point in the setting of a metric space endowed with a directed graph. Further, some fixed point theorems for set valued maps have been proven in this context. A version of the Knaster–Tarski theorem has also been established using our results.

]]>Mathematics doi: 10.3390/math5020023

Authors: Mohadeshe Paknazar Manuel Sen

In this paper, we introduce the new notion of Suzuki-type ( α , β , θ , γ ) -contractive mapping and investigate the existence and uniqueness of the best proximity point for such mappings in non-Archimedean modular metric space using the weak P λ -property. Meanwhile, we present an illustrative example to emphasize the realized improvements. These obtained results extend and improve certain well-known results in the literature.

]]>Mathematics doi: 10.3390/math5020022

Authors: Manuel Sen Mujahid Abbas Naeem Saleem

The main objective of this paper is to deal with some properties of interest in two types of fuzzy ordered proximal contractions of cyclic self-mappings T integrated in a pair ( g , T ) of mappings. In particular, g is a non-contractive fuzzy self-mapping, in the framework of non-Archimedean ordered fuzzy complete metric spaces and T is a p -cyclic proximal contraction. Two types of such contractions (so called of type I and of type II) are dealt with. In particular, the existence, uniqueness and limit properties for sequences to optimal fuzzy best proximity coincidence points are investigated for such pairs of mappings.

]]>Mathematics doi: 10.3390/math5020021

Authors: Abdeslem Bentbib Khalide Jbilou EL Sadek

In the present paper, we consider the large scale Stein matrix equation with a low-rank constant term A X B − X + E F T = 0 . These matrix equations appear in many applications in discrete-time control problems, filtering and image restoration and others. The proposed methods are based on projection onto the extended block Krylov subspace with a Galerkin approach (GA) or with the minimization of the norm of the residual. We give some results on the residual and error norms and report some numerical experiments.

]]>Mathematics doi: 10.3390/math5020019

Authors: Tayyab Kamran Maria Samreen Qurat UL Ain

In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature.

]]>Mathematics doi: 10.3390/math5020020

Authors: Seyed Torbaghan Morteza Rezaii

In this paper, some properties of F -harmonic and conformal F -harmonic maps between doubly warped product manifolds are studied and new examples of non-harmonic F -harmonic maps are constructed.

]]>Mathematics doi: 10.3390/math5010018

Authors: Muhammad Javaid

In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let G 1 , n c and G 2 , n c be the classes of the connected graphs of order n whose complements are bicyclic with exactly two and three cycles, respectively. In this paper, we characterize the unique minimizing graph among all the graphs which belong to G n c = G 1 , n c ∪ G 2 , n c , a class of the connected graphs of order n whose complements are bicyclic.

]]>Mathematics doi: 10.3390/math5010016

Authors: Behrooz Khosravi Elnaz Ramezani

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. The additively weighted Harary index H A ( G ) is a modification of the Harary index in which the contributions of vertex pairs are weighted by the sum of their degrees. This new invariant was introduced in (Alizadeh, Iranmanesh and Došlić. Additively weighted Harary index of some composite graphs, Discrete Math, 2013) and they posed the following question: What is the behavior of H A ( G ) when G is a composite graph resulting for example by: splice, link, corona and rooted product? We investigate the additively weighted Harary index for these standard graph products. Then we obtain lower and upper bounds for some of them.

]]>Mathematics doi: 10.3390/math5010017

Authors: Muhammad Akram Anam Luqman

A hypergraph is the most developed tool for modeling various practical problems in different fields, including computer sciences, biological sciences, social networks and psychology. Sometimes, given data in a network model are based on bipolar information rather than one sided. To deal with such types of problems, we use mathematical models that are based on bipolar fuzzy (BF) sets. In this research paper, we introduce the concept of BF directed hypergraphs. We describe certain operations on BF directed hypergraphs, including addition, multiplication, vertex-wise multiplication and structural subtraction. We introduce the concept of B = ( m + , m − ) -tempered BF directed hypergraphs and investigate some of their properties. We also present an algorithm to compute the minimum arc length of a BF directed hyperpath.

]]>Mathematics doi: 10.3390/math5010015

Authors: José Usó Doménech Josué Nescolarde-Selva Lorena Segura-Abad

There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, one can derive a probability theory, adding the correspondence between truth value and fortuity.

]]>Mathematics doi: 10.3390/math5010014

Authors: Ali Abkar Mohsen Shekarbaigi

In this paper we introduce a new iterative algorithm for approximating fixed points of totally asymptotically quasi-nonexpansive mappings on CAT(0) spaces. We prove a strong convergence theorem under suitable conditions. The result we obtain improves and extends several recent results stated by many others; they also complement many known recent results in the literature. We then provide some numerical examples to illustrate our main result and to display the efficiency of the proposed algorithm.

]]>Mathematics doi: 10.3390/math5010013

Authors: Chandan Datta Pankaj Agrawal

Finite trigonometric sums occur in various branches of physics, mathematics, and their applications. These sums may contain various powers of one or more trigonometric functions. Sums with one trigonometric function are known; however, sums with products of trigonometric functions can become complicated, and may not have a simple expression in a number of cases. Some of these sums have interesting properties, and can have amazingly simple values. However, only some of them are available in the literature. We obtain a number of such sums using the method of residues.

]]>Mathematics doi: 10.3390/math5010012

Authors: Gerd Baumann Frank Stenger

We shall discuss the numerical solution of the Cauchy problem for the fully fractional Fokker-Planck (fFP) equation in connection with Sinc convolution methods. The numerical approximation is based on Caputo and Riesz-Feller fractional derivatives. The use of the transfer function in Laplace and Fourier spaces in connection with Sinc convolutions allow to find exponentially converging computing schemes. Examples using different initial conditions demonstrate the effective computations with a small number of grid points on an infinite spatial domain.

]]>Mathematics doi: 10.3390/math5010011

Authors: Ali Abkar Elahe Shahrosvand Azizollah Azizi

In this paper, we introduce an iterative algorithm for solving the split common fixed point problem for a family of multi-valued quasinonexpansive mappings and totally asymptotically strictly pseudocontractive mappings, as well as for a family of totally quasi-ϕ-asymptotically nonexpansive mappings and k-quasi-strictly pseudocontractive mappings in the setting of Banach spaces. Our results improve and extend the results of Tang et al., Takahashi, Moudafi, Censor et al., and Byrne et al.

]]>Mathematics doi: 10.3390/math5010010

Authors: Sergey Ludkowski

Müntz spaces satisfying the Müntz and gap conditions are considered. A Fourier approximation of functions in the Müntz spaces MΛ,p of Lp functions is studied, where 1 &lt; p &lt; ∞. It is proven that up to an isomorphism and a change of variables, these spaces are contained in Weil–Nagy’s class. Moreover, the existence of Schauder bases in the Müntz spaces MΛ,p is investigated.

]]>Mathematics doi: 10.3390/math5010009

Authors: Selvaraj Suganya Mani Mallika Arjunan

In this manuscript, we implement Bohnenblust–Karlin’s fixed point theorem to demonstrate the existence of mild solutions for a class of impulsive fractional integro-differential inclusions (IFIDI) with state-dependent delay (SDD) in Banach spaces. An example is provided to illustrate the obtained abstract results.

]]>Mathematics doi: 10.3390/math5010008

Authors: Cristina Jordán Jon Gómez J. Conejero

The subject known as Programming Contests in the Bachelor’s Degree in Computer Engineering course focuses on solving programming problems frequently met within contests such as the Southwest Europe Regional Contest (SWERC). In order to solve these problems one first needs to model the problem correctly, find the ideal solution, and then be able to program it without making any mistakes in a very short period of time. Leading multinationals such as Google, Apple, IBM, Facebook and Microsoft place a very high value on these abilities when selecting candidates for posts in their companies. In this communication we present some preliminary results of an analysis of the interaction between two optional subjects in the Computer Science Degree course: Programming Contests (PC) and Graphs, Models and Applications (GMA). The results of this analysis enabled us to make changes to some of the contents in GMA in order to better prepare the students to deal with the challenges they have to face in programming contests.

]]>Mathematics doi: 10.3390/math5010007

Authors: Marek Trawicki

In this paper, the author proposes a new SEIRS model that generalizes several classical deterministic epidemic models (e.g., SIR and SIS and SEIR and SEIRS) involving the relationships between the susceptible S, exposed E, infected I, and recovered R individuals for understanding the proliferation of infectious diseases. As a way to incorporate the most important features of the previous models under the assumption of homogeneous mixing (mass-action principle) of the individuals in the population N, the SEIRS model utilizes vital dynamics with unequal birth and death rates, vaccinations for newborns and non-newborns, and temporary immunity. In order to determine the equilibrium points, namely the disease-free and endemic equilibrium points, and study their local stability behaviors, the SEIRS model is rescaled with the total time-varying population and analyzed according to its epidemic condition R0 for two cases of no epidemic (R0 ≤ 1) and epidemic (R0 &gt; 1) using the time-series and phase portraits of the susceptible s, exposed e, infected i, and recovered r individuals. Based on the experimental results using a set of arbitrarily-defined parameters for horizontal transmission of the infectious diseases, the proportional population of the SEIRS model consisted primarily of the recovered r (0.7–0.9) individuals and susceptible s (0.0–0.1) individuals (epidemic) and recovered r (0.9) individuals with only a small proportional population for the susceptible s (0.1) individuals (no epidemic). Overall, the initial conditions for the susceptible s, exposed e, infected i, and recovered r individuals reached the corresponding equilibrium point for local stability: no epidemic (DFE X ¯ D F E ) and epidemic (EE X ¯ E E ).

]]>Mathematics doi: 10.3390/math5010006

Authors: Michel Planat Hishamuddin Zainuddin

Every finite simple group P can be generated by two of its elements. Pairs of generators for P are available in the Atlas of finite group representations as (not necessarily minimal) permutation representations P . It is unusual, but significant to recognize that a P is a Grothendieck’s “dessin d’enfant” D and that a wealth of standard graphs and finite geometries G —such as near polygons and their generalizations—are stabilized by a D . In our paper, tripods P − D − G of rank larger than two, corresponding to simple groups, are organized into classes, e.g., symplectic, unitary, sporadic, etc. (as in the Atlas). An exhaustive search and characterization of non-trivial point-line configurations defined from small index representations of simple groups is performed, with the goal to recognize their quantum physical significance. All of the defined geometries G ′ s have a contextuality parameter close to its maximal value of one.

]]>Mathematics doi: 10.3390/math5010004

Authors: Abolfazl Ebrahimzadeh Zahra Giski Dagmar Markechová

In the paper by Riečan and Markechová (Fuzzy Sets Syst. 96, 1998), some fuzzy modifications of Shannon’s and Kolmogorov-Sinai’s entropy were studied and the general scheme involving the presented models was introduced. Our aim in this contribution is to provide analogies of these results for the case of the logical entropy. We define the logical entropy and logical mutual information of finite partitions on the appropriate algebraic structure and prove basic properties of these measures. It is shown that, as a special case, we obtain the logical entropy of fuzzy partitions studied by Markechová and Riečan (Entropy 18, 2016). Finally, using the suggested concept of entropy of partitions we define the logical entropy of a dynamical system and prove that it is the same for two dynamical systems that are isomorphic.

]]>Mathematics doi: 10.3390/math5010005

Authors: Tony Scott Madhusudan Therani Xing Wang

Data clustering is a vital tool for data analysis. This work shows that some existing useful methods in data clustering are actually based on quantum mechanics and can be assembled into a powerful and accurate data clustering method where the efficiency of computational quantum chemistry eigenvalue methods is therefore applicable. These methods can be applied to scientific data, engineering data and even text.

]]>Mathematics doi: 10.3390/math5010003

Authors: Alessandro Taloni

The generalized elastic model encompasses several linear stochastic models describing the dynamics of polymers, membranes, rough surfaces, and fluctuating interfaces. While usually defined in the overdamped case, in this paper we formally include the inertial term to account for the initial diffusive stages of the stochastic dynamics. We derive the generalized Langevin equation for a probe particle and we show that this equation reduces to the usual Langevin equation for Brownian motion, and to the fractional Langevin equation on the long-time limit.

]]>Mathematics doi: 10.3390/math5010002

Authors: Süha Tuna Metin Demiralp

In this work, we deal with the autonomy issue in the perturbation expansion for the eigenfunctions of a compact Hilbert–Schmidt integral operator. Here, the autonomy points to the perturbation expansion coefficients of the relevant eigenfunction not depending on the perturbation parameter explicitly, but the dependence on this parameter arises from the coordinate change at the zero interval limit. Moreover, the related half interval length is utilized as the perturbation parameter in the perturbative analyses. Thus, the zero interval limit perturbation for solving the eigenproblem under consideration is developed. The aim of this work is to show that the autonomy imposition brings an important restriction on the kernel of the corresponding integral operator, and the constructed perturbation series is not capable of expressing the exact solution approximately unless a specific type of kernel is considered. The general structure for the encountered constraints is revealed, and the specific class of kernels is identified to this end. Error analysis of the developed method is given. These analyses are also supported by certain illustrative implementations involving the kernels, which are and are not in the specific class addressed above. Thus, the efficiency of the developed method is shown, and the relevant analyses are confirmed.

]]>Mathematics doi: 10.3390/math5010001

Authors: R. Sinuvasan Andronikos Paliathanasis Richard Morris Peter Leach

We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrödinger Equation.

]]>Mathematics doi: 10.3390/math4040069

Authors: José Usó-Doménech Josué Nescolarde-Selva Lorena Segura-Abad

Classical logic is typically concerned with abstract analysis. The problem for a synthetic logic is to transcend and unify available data to reconstruct the object as a totality. Three rules are proposed to pass from classic logic to synthetic logic. We present the category logic of qualitative opposition using examples from various sciences. This logic has been defined to include the neuter as part of qualitative opposition. The application of these rules to qualitative opposition, and, in particular, its neuter, demonstrated that a synthetic logic allows the truth of some contradictions. This synthetic logic is dialectical with a multi-valued logic, which gives every proposition a truth value in the interval [0,1] that is the square of the modulus of a complex number. In this dialectical logic, contradictions of the neuter of an opposition may be true.

]]>Mathematics doi: 10.3390/math4040068

Authors: Rqeeb Gubran Mohammad Imdad

Inspired by a metrical-fixed point theorem from Choudhury et al. (Nonlinear Anal. 2011, 74, 2116–2126), we prove some order-theoretic results which generalize several core results of the existing literature, especially the two main results of Harjani and Sadarangani (Nonlinear Anal. 2009, 71, 3403–3410 and 2010, 72, 1188–1197). We demonstrate the realized improvement obtained in our results by using a suitable example. As an application, we also prove a result for mappings satisfying integral type ( ψ , φ ) g -generalized weakly contractive conditions.

]]>Mathematics doi: 10.3390/math4040067

Authors: Abel Garcia-Bernabé S. Hernández L. del Castillo David Jou

The main objective of this paper is to generalize the Extended Irreversible Thermodynamics in order to include the anomalous transport in systems in non-equilibrium conditions. Considering the generalized entropy, the corresponding flux and entropy production, and using the time fractional derivative, we have derived a space-time generalized telegrapher’s equation with a fractional nested hierarchy which can be used in separate developments for the mass transport, for the heat conduction and for the flux of ions. We have obtained a new formalism which includes the contribution of fast of higher-order fluxes in the mesoscopic and inhomogeneous media. The results take the form of continued fraction expansions. The balance equations are used in a scheme of continued fractions, and they appear as a closure condition. In this way the transport equation and its corresponding wave number-frequency relation are obtained, both of them in the mathematical structure of the continued fraction scheme. Numerical examples are included to show the dispersive nature of the solutions, and the generalized fractional transport equation in the same mathematical form, which can be applied to the mass transport, the heat conduction and the flux of ions.

]]>Mathematics doi: 10.3390/math4040066

Authors: Ali Abkar Narges Moezzifar Azizollah Azizi

In this paper, we introduce the notion of an ordered rational proximal contraction in partially ordered b-quasi metric spaces. We shall then prove some best proximity point theorems in partially ordered b-quasi metric spaces.

]]>Mathematics doi: 10.3390/math4040065

Authors: Feng Qi Bai-Ni Guo

In the paper, the authors recall some known determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, find alternative determinantal expressions in terms of the Hessenberg determinants for the Bernoulli numbers and polynomials, and present several new recurrence relations for the Bernoulli numbers and polynomials.

]]>Mathematics doi: 10.3390/math4040064

Authors: Qixiang Dong Gang Li

Let X be a Banach space, A : D ( A ) ⊂ X → X the generator of a compact C 0 -semigroup S ( t ) : X → X , t ≥ 0 , D ( · ) : ( a , b ) → 2 X a tube in X, and f : ( a , b ) × B → X a function of Carathéodory type. The main result of this paper is that a necessary and sufficient condition in order that D ( · ) be viable of the semilinear differential equation with infinite delay u ′ ( t ) = A u ( t ) + f ( t , u t ) , t ∈ [ t 0 , t 0 + T ] , u t 0 = ϕ ∈ B is the tangency condition lim inf h ↓ 0 h − 1 d ( S ( h ) v ( 0 ) + h f ( t , v ) ; D ( t + h ) ) = 0 for almost every t ∈ ( a , b ) and every v ∈ B with v ( 0 ) ∈ D ( t ) .

]]>Mathematics doi: 10.3390/math4040063

Authors: Wengui Yang Yaping Qin

In this paper, by applying some well-known fixed point theorems, we investigate the existence of positive solutions for a class of nonlinear Caputo type fractional q-difference equations with integral boundary conditions. Finally, some interesting examples are presented to illustrate the main results.

]]>Mathematics doi: 10.3390/math4040060

Authors: Dimplekumar Chalishajar Annamalai Anguraj Kandasamy Malar Kulandhivel Karthikeyan

In this paper, we study the problem of controllability of impulsive neutral evolution integro-differential equations with state-dependent delay in Banach spaces. The main results are completely new and are obtained by using Sadovskii’s fixed point theorem, theory of resolvent operators, and an abstract phase space. An example is given to illustrate the theory.

]]>Mathematics doi: 10.3390/math4040062

Authors: Arindam Dey Anita Pal Tandra Pal

The shortest path problem (SPP) is one of the most important combinatorial optimization problems in graph theory due to its various applications. The uncertainty existing in the real world problems makes it difficult to determine the arc lengths exactly. The fuzzy set is one of the popular tools to represent and handle uncertainty in information due to incompleteness or inexactness. In most cases, the SPP in fuzzy graph, called the fuzzy shortest path problem (FSPP) uses type-1 fuzzy set (T1FS) as arc length. Uncertainty in the evaluation of membership degrees due to inexactness of human perception is not considered in T1FS. An interval type-2 fuzzy set (IT2FS) is able to tackle this uncertainty. In this paper, we use IT2FSs to represent the arc lengths of a fuzzy graph for FSPP. We call this problem an interval type-2 fuzzy shortest path problem (IT2FSPP). We describe the utility of IT2FSs as arc lengths and its application in different real world shortest path problems. Here, we propose an algorithm for IT2FSPP. In the proposed algorithm, we incorporate the uncertainty in Dijkstra’s algorithm for SPP using IT2FS as arc length. The path algebra corresponding to the proposed algorithm and the generalized algorithm based on the path algebra are also presented here. Numerical examples are used to illustrate the effectiveness of the proposed approach.

]]>Mathematics doi: 10.3390/math4040061

Authors: Jeremy Becnel Ambar Sengupta

We present a scenic but practical guide through nuclear spaces and their dual spaces, examining useful, unexpected, and often unfamiliar results both for nuclear spaces and their strong and weak duals.

]]>Mathematics doi: 10.3390/math4040059

Authors: Trifce Sandev Irina Petreska Ervin Lenzi

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.

]]>Mathematics doi: 10.3390/math4040058

Authors: Nawel Khelil Martin Otis

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.

]]>Mathematics doi: 10.3390/math4040057

Authors: Sotiris Alevizos Thomas Poulet Manolis Veveakis Klaus Regenauer-Lieb

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).

]]>Mathematics doi: 10.3390/math4030056

Authors: Dmitri Sokolovski

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.

]]>Mathematics doi: 10.3390/math4030055

Authors: Hamidreza Rahimi Khalil Nabizadeh

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 .

]]>Mathematics doi: 10.3390/math4030054

Authors: Claudio Carmeli Teiko Heinosaari Daniel Reitzner Jussi Schultz Alessandro Toigo

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.

]]>Mathematics doi: 10.3390/math4030053

Authors: Mohamed El-Dessoky

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.

]]>Mathematics doi: 10.3390/math4030052

Authors: Guruprasad Kar Sibasish Ghosh Sujit Choudhary Manik Banik

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.

]]>Mathematics doi: 10.3390/math4030050

Authors: Farhad Ali Tooba Feroze

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.

]]>Mathematics doi: 10.3390/math4030051

Authors: Dilip Jain Anantachai Padcharoen Poom Kumam Dhananjay Gopal

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.

]]>Mathematics doi: 10.3390/math4030049

Authors: Spiros Kechrimparis Stefan Weigert

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.

]]>Mathematics doi: 10.3390/math4030047

Authors: Uttam Singh Arun Pati Manabendra Bera

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.

]]>Mathematics doi: 10.3390/math4030048

Authors: Shiladitya Mal Archan Majumdar Dipankar Home

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.

]]>Mathematics doi: 10.3390/math4030046

Authors: Jean-Paul Chehab Marcos Raydan

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.

]]>Mathematics doi: 10.3390/math4030045

Authors: Yanmei Liu Monzorul Khan Yubin Yan

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.

]]>Mathematics doi: 10.3390/math4030043

Authors: Seyyede Seyyedi Farhad Rahmati

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.

]]>Mathematics doi: 10.3390/math4030044

Authors: Vasily Tarasov

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.

]]>Mathematics doi: 10.3390/math4020042

Authors: Danhua Wang Gang Li Biqing Zhu

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.

]]>Mathematics doi: 10.3390/math4020041

Authors: Kyunghyun Baek Wonmin Son

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.

]]>Mathematics doi: 10.3390/math4020039

Authors: Matthew Macauley

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.

]]>Mathematics doi: 10.3390/math4020040

Authors: Gregg Jaeger

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.

]]>Mathematics doi: 10.3390/math4020038

Authors: René Schwonnek David Reeb Reinhard Werner

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.

]]>Mathematics doi: 10.3390/math4020037

Authors: Hamid Damadi Farhad Rahmati

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.

]]>Mathematics doi: 10.3390/math4020036

Authors: Blake Stacey

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.

]]>Mathematics doi: 10.3390/math4020035

Authors: Mansour Mahmoud Feng Qi

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.

]]>Mathematics doi: 10.3390/math4020034

Authors: Andronikos Paliathanasis Richard Morris Peter Leach

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.

]]>Mathematics doi: 10.3390/math4020033

Authors: Mohammad Shahzad

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.

]]>Mathematics doi: 10.3390/math4020032

Authors: Phong Le Sunil Chetty

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.

]]>Mathematics doi: 10.3390/math4020030

Authors: Ali Karcı

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.

]]>Mathematics doi: 10.3390/math4020031

Authors: André Liemert Alwin Kienle

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.

]]>Mathematics doi: 10.3390/math4020028

Authors: Andronikos Paliathanasis K. Krishnakumar K.M. Tamizhmani Peter Leach

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.

]]>Mathematics doi: 10.3390/math4020029

Authors: Wei Guo Guang Lin Andrew Christlieb Jingmei Qiu

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.

]]>Mathematics doi: 10.3390/math4020027

Authors: Khairy Zaimi Anuar Ishak

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.

]]>Mathematics doi: 10.3390/math4020026

Authors: Zhongqiang Zhang Xiu Yang Guang Lin

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.

]]>Mathematics doi: 10.3390/math4020025

Authors: Abedallah Rababah

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.

]]>Mathematics doi: 10.3390/math4020024

Authors: Abdul Khaliq E.M. Elsayed

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.

]]>Mathematics doi: 10.3390/math4020023

Authors: Dimplekumar Chalishajar Kulandhivel Karthikeyan Annamalai Anguraj

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.

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