Showing papers in "Open Mathematics in 2022"

Strong convergence of a self-adaptive inertial Tseng's extragradient method for pseudomonotone variational inequalities and fixed point problems

Abstract: Abstract In this paper, we study the problem of finding a common solution of the pseudomonotone variational inequality problem and fixed point problem for demicontractive mappings. We introduce a new inertial iterative scheme that combines Tseng’s extragradient method with the viscosity method together with the adaptive step size technique for finding a common solution of the investigated problem. We prove a strong convergence result for our proposed algorithm under mild conditions and without prior knowledge of the Lipschitz constant of the pseudomonotone operator in Hilbert spaces. Finally, we present some numerical experiments to show the efficiency of our method in comparison with some of the existing methods in the literature.

Geometry of conformal η-Ricci solitons and conformal η-Ricci almost solitons on paracontact geometry

Abstract: Abstract We prove that if an η \eta -Einstein para-Kenmotsu manifold admits a conformal η \eta -Ricci soliton then it is Einstein. Next, we proved that a para-Kenmotsu metric as a conformal η \eta -Ricci soliton is Einstein if its potential vector field V V is infinitesimal paracontact transformation or collinear with the Reeb vector field. Furthermore, we prove that if a para-Kenmotsu manifold admits a gradient conformal η \eta -Ricci almost soliton and the Reeb vector field leaves the scalar curvature invariant then it is Einstein. We also construct an example of para-Kenmotsu manifold that admits conformal η \eta -Ricci soliton and satisfy our results. We also have studied conformal η \eta -Ricci soliton in three-dimensional para-cosymplectic manifolds.

On the general position number of two classes of graphs

Abstract: Abstract The general position problem is to find the cardinality of the largest vertex subset S S such that no triple of vertices of S S lies on a common geodesic. For a connected graph G G , the cardinality of S S is denoted by gp ( G ) {\rm{gp}}\left(G) and called the gp {\rm{gp}} -number (or general position number) of G G . In the paper, we obtain an upper bound and a lower bound regarding the gp {\rm{gp}} -number in all cacti with k k cycles and t t pendant edges. Furthermore, the exact value of the gp {\rm{gp}} -number on wheel graphs is determined.

On Hahn-Banach theorem and some of its applications

Abstract: Abstract First, this work provides an overview of some of the Hahn-Banach type theorems. Of note, some of these extension results for linear operators found recent applications to isotonicity of convex operators on a convex cone. Next, the work investigates applications of the Krein-Milman theorem to representation theory and elements of Choquet theory. A sandwich theorem of intercalating an affine function h h between f f and g , g, where f f\hspace{.25em} and – g \mbox{--}g are convex, f ≤ g f\le g on a finite-simplicial set, is recalled. Its recent topological version is also noted. Here, the novelty is that a finite-simplicial set may be unbounded in any locally convex topology on the domain space. Third, the paper summarizes and comments on recently published applications of a Hahn-Banach extension result for positive linear operators, combined with polynomial approximation on unbounded subsets, to the Markov moment problem. Some applications to concrete spaces are detailed as well. Finally, this work provides a characterization of a finite-dimensional convex bounded subset in terms of the property that any convex function defined on that subset is bounded below. This last property remains valid for a large class of convex operators.

Dynamical analysis of a Lotka Volterra commensalism model with additive Allee effect

Abstract: Abstract We propose and analyze a Lotka-Volterra commensal model with an additive Allee effect in this article. First, we study the existence and local stability of possible equilibria. Second, the conditions for the existence of saddle-node bifurcations and transcritical bifurcations are derived by using Sotomayor’s theorem. Third, we give sufficient conditions for the global stability of the boundary equilibrium and positive equilibrium. Finally, we use numerical simulations to verify the above theoretical results. This study shows that for the weak Allee effect case, the additive Allee effect has a negative effect on the final density of both species, with increasing Allee effect, the densities of both species are decreasing. For the strong Allee effect case, the additive Allee effect is one of the most important factors that leads to the extinction of the second species. The additive Allee effect leads to the complex dynamic behaviors of the system.

The influence of the noise on the exact solutions of a Kuramoto-Sivashinsky equation

Abstract: Abstract In this article, we take into account the stochastic Kuramoto-Sivashinsky equation forced by multiplicative noise in the Itô sense. To obtain the exact stochastic solutions of the stochastic Kuramoto-Sivashinsky equation, we apply the G ′ G \frac{{G}^{^{\prime} }}{G} -expansion method. Furthermore, we extend some previous results where this equation has not been previously studied in the presence of multiplicative noise. Also, we show the influence of multiplicative noise on the analytical solutions of the stochastic Kuramoto-Sivashinsky equation.

Jensen-type inequalities for m-convex functions

Abstract: Abstract Inequalities play an important role in pure and applied mathematics. In particular, Jensen’s inequality, one of the most famous inequalities, plays the main role in the study of the existence and uniqueness of initial and boundary value problems for differential equations. In this work, we prove some new Jensen-type inequalities for m-convex functions and apply them to generalized Riemann-Liouville-type integral operators. Furthermore, as a remarkable consequence, some new inequalities for convex functions are obtained.

Dynamical behaviors of a k-order fuzzy difference equation

Abstract: Abstract Difference equations are often used to create discrete mathematical models. In this paper, we mainly study the dynamical behaviors of positive solutions of a nonlinear fuzzy difference equation: x n + 1 = x n A + B x n − k ( n = 0 , 1 , 2 , … ) , {x}_{n+1}=\frac{{x}_{n}}{A+B{x}_{n-k}}\hspace{0.33em}\left(n=0,1,2,\ldots ), where parameters A , B A,B and initial value x − k , x − k + 1 , … , x − 1 , x 0 {x}_{-k},{x}_{-k+1},\ldots ,{x}_{-1},{x}_{0} , k ∈ { 0 , 1 , … } k\in \{0,1,\ldots \} are positive fuzzy numbers. We investigate the existence, boundedness, convergence, and asymptotic stability of the positive solutions of the fuzzy difference equation. At last, we give numerical examples to intuitively reflect the global behavior. The conclusion of the global stability of this paper can be applied directly to production practice.

Global weak solution of 3D-NSE with exponential damping

Abstract: Abstract In this paper, we prove the global existence of incompressible Navier-Stokes equations with damping α ( e β ∣ u ∣ 2 − 1 ) u \alpha \left({e}^{\beta | u{| }^{2}}-1)u , where we use the Friedrich method and some new tools. The delicate problem in the construction of a global solution is the passage to the limit in exponential nonlinear term. To solve this problem, we use a polynomial approximation of the damping part and a new type of interpolation between L ∞ ( R + , L 2 ( R 3 ) ) {L}^{\infty }\left({{\mathbb{R}}}^{+},{L}^{2}\left({{\mathbb{R}}}^{3})) and the space of functions f f such that ( e β ∣ f ∣ 2 − 1 ) ∣ f ∣ 2 ∈ L 1 ( R + × R 3 ) \left({e}^{\beta | f{| }^{2}}-1)| f{| }^{2}\in {L}^{1}\left({{\mathbb{R}}}^{+}\times {{\mathbb{R}}}^{3}) . Fourier analysis and standard techniques are used.

Convergence rate of the modified Levenberg-Marquardt method under Hölderian local error bound

Abstract: Abstract In this article, we analyze the convergence rate of the modified Levenberg-Marquardt (MLM) method under the Hölderian local error bound condition and the Hölderian continuity of the Jacobian, which are more general than the local error bound condition and the Lipschitz continuity of the Jacobian. Under special circumstances, the convergence rate of the MLM method coincides with the results presented by Fan. A globally convergent MLM algorithm by the trust region technique will also be given.

Representations by degenerate Daehee polynomials

Abstract: Abstract In this paper, we consider the problem of representing any polynomial in terms of the degenerate Daehee polynomials and more generally of the higher-order degenerate Daehee polynomials. We derive explicit formulas with the help of umbral calculus and illustrate our results with some examples.

On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems

Abstract: Abstract In a real Banach space, let the VI indicate a variational inclusion for two accretive operators and let the CFPP denote a common fixed point problem of countably many nonexpansive mappings. In this article, we introduce a generalized extragradient implicit method for solving a general system of variational inequalities (GSVI) with the VI and CFPP constraints. Strong convergence of the suggested method to a solution of the GSVI with the VI and CFPP constraints under some suitable assumptions is established.

Multilevel MC method for weak approximation of stochastic differential equation with the exact coupling scheme

Abstract: Abstract Davie’s exact coupling technique for stochastic differential equations may be used to enhance the convergence of the multilevel Monte Carlo (MC) methodology. Giles developed the multilevel MC technique, which is based on executing the MC method several times with various time increments. It cuts computing costs significantly by executing most simulations at a low cost. The essential concept behind the multilevel MC approach with the exact coupling is discussed in this article. Numerical implementation reveals significant computational savings, which supports the analysis.

On some new Hermite-Hadamard and Ostrowski type inequalities for s-convex functions in (p, q)-calculus with applications

Abstract: Abstract In this study, we establish some new Hermite-Hadamard type inequalities for s-convex functions in the second sense using the post-quantum calculus. Moreover, we prove a new ( p , q ) \left(p,q) -integral identity to prove some new Ostrowski type inequalities for ( p , q ) \left(p,q) -differentiable functions. We also show that the newly discovered results are generalizations of comparable results in the literature. Finally, we give application to special means of real numbers using the newly proved inequalities.

Basic inequalities for statistical submanifolds in Golden-like statistical manifolds

Abstract: Abstract In this paper, we introduce and study Golden-like statistical manifolds. We obtain some basic inequalities for curvature invariants of statistical submanifolds in Golden-like statistical manifolds. Also, in support of our definition, we provide a couple of examples.

The dual index and dual core generalized inverse

Abstract: Abstract In this article, we introduce the dual index and dual core generalized inverse (DCGI). By applying rank equation, generalized inverse, and matrix decomposition, we give several characterizations of the dual index when it is equal to 1. We realize that if DCGI exists, then it is unique. We derive a compact formula for DCGI and a series of equivalent characterizations of the existence of the inverse. It is worth noting that the dual index of A ^ \widehat{A} is equal to 1 if and only if its DCGI exists. When the dual index of A ^ \widehat{A} is equal to 1, we study dual Moore-Penrose generalized inverse (DMPGI) and dual group generalized inverse (DGGI) and consider the relationships among DCGI, DMPGI, DGGI, Moore-Penrose dual generalized inverse, and other dual generalized inverses. In addition, we consider symmetric dual matrix and its dual generalized inverses. Finally, two examples are given to illustrate the application of DCGI in linear dual equations.

Bifurcation and chaos in a discrete predator-prey system of Leslie type with Michaelis-Menten prey harvesting

Abstract: Abstract In this paper, a discrete Leslie-Gower predator-prey system with Michaelis-Menten type harvesting is studied. Conditions on the existence and stability of fixed points are obtained. It is shown that the system can undergo fold bifurcation, flip bifurcation, and Neimark-Sacker bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations are presented to illustrate the main theoretical results. Compared to the continuous analog, the discrete system here possesses much richer dynamical behaviors including orbits of period-16, 21, 35, 49, 54, invariant cycles, cascades of period-doubling bifurcation in orbits of period-2, 4, 8, and chaotic sets.

F4 and PSp (8, ℂ)-Higgs pairs understood as fixed points of the moduli space of E6-Higgs bundles over a compact Riemann surface

Abstract: Abstract Let X X be a compact Riemann surface of genus g ≥ 2 g\ge 2 and ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E 6 {E}_{6} -Higgs bundles over X X . We consider the automorphisms σ + {\sigma }_{+} of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ + ( E , φ ) = ( E ∗ , − φ t ) {\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}) , induced by the action of the outer involution of E 6 {E}_{6} in ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , and σ − {\sigma }_{-} defined by σ − ( E , φ ) = ( E ∗ , φ t ) {\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}) , which results from the combination of σ + {\sigma }_{+} with the involution of ℳ ( E 6 ) {\mathcal{ {\mathcal M} }}\left({E}_{6}) , which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ + {\sigma }_{+} and σ − {\sigma }_{-} , as F 4 {F}_{4} -Higgs bundles, F 4 {F}_{4} -Higgs pairs associated with the fundamental irreducible representation of F 4 {F}_{4} , and PSp ( 8 , C ) {\rm{PSp}}\left(8,{\mathbb{C}}) -Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp ( 8 , C ) {\rm{Sp}}\left(8,{\mathbb{C}}) . Finally, we describe the reduced notions of semistability and polystability for these objects.

The multinomial convolution sum of a generalized divisor function

Abstract: Abstract The main theorem of this article is to evaluate and express the multinomial convolution sum of the divisor function σ r ♯ ( n ; N / 4 , N ) {\sigma }_{r}^{\sharp }\left(n;\hspace{0.33em}N\hspace{-0.08em}\text{/}\hspace{-0.08em}4,N) in as a simple form as possible, where N / 4 N\hspace{-0.08em}\text{/}\hspace{-0.08em}4 is an arbitrary odd positive integer. This generalizes previous result in combination with Cho and Kim, which is about the case N = 4 N=4 . While obtaining our main theorem, we derive some generalizations of other identities to the case that we are dealing with.

Double domination in maximal outerplanar graphs

Abstract: Abstract In graph G G , a vertex dominates itself and its neighbors. A subset S ⊆ V ( G ) S\subseteq V\left(G) is said to be a double-dominating set of G G if S S dominates every vertex of G G at least twice. The double domination number γ × 2 ( G ) {\gamma }_{\times 2}\left(G) is the minimum cardinality of a double dominating set of G G . We show that if G G is a maximal outerplanar graph on n ≥ 3 n\ge 3 vertices, then γ × 2 ( G ) ≤ 2 n 3 {\gamma }_{\times 2}\left(G)\le ⌊\frac{2n}{3}⌋ . Further, if n ≥ 4 n\ge 4 , then γ × 2 ( G ) ≤ min n + t 2 , n − t {\gamma }_{\times 2}\left(G)\le \min \left\{⌊\frac{n+t}{2}⌋,n-t\right\} , where t t is the number of vertices of degree 2 in G G . These bounds are shown to be tight. In addition, we also study the case that G G is a striped maximal outerplanar graph.

Non-binary quantum codes from constacyclic codes over 𝔽q[u1, u2,…,uk]/⟨ui3 = ui, uiuj = ujui⟩

Abstract: Abstract Let q = p m q={p}^{m} , p p be an odd prime, and R k = F q [ u 1 , u 2 , … , u k ] / ⟨ u i 3 = u i , u i u j = u j u i ⟩ {R}_{k}={{\mathbb{F}}}_{q}\left[{u}_{1},{u}_{2},\ldots ,{u}_{k}]\hspace{-0.08em}\text{/}\hspace{-0.08em}\langle {u}_{i}^{3}={u}_{i},{u}_{i}{u}_{j}={u}_{j}{u}_{i}\rangle , where k ≥ 1 k\ge 1 and 1 ≤ i , j ≤ k 1\le i,j\le k . In this article, we define a Gray map from R k n {R}_{k}^{n} to F q 3 k n {{\mathbb{F}}}_{q}^{{3}^{k}n} . We study constacyclic codes over R k {R}_{k} and construct non-binary quantum codes over F q {{\mathbb{F}}}_{q} .

On a more accurate half-discrete Hilbert-type inequality involving hyperbolic functions

Abstract: Abstract In this work, by the introduction of a new kernel function composed of exponent functions with several parameters, and using the method of weight coefficient, Hermite-Hadamard’s inequality, and some other techniques of real analysis, a more accurate half-discrete Hilbert-type inequality including both the homogeneous and non-homogeneous cases is established. Furthermore, by introducing the Bernoulli number and the rational fraction expansion of tangent function, some special and interesting Hilbert-type inequalities and their equivalent hardy-type inequalities are presented at the end of the paper.

Global existence and blow up of the solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term

Abstract: Abstract This article is devoted to study the existence of global solutions and finite time blow-up of local solution for nonlinear Klein-Gordon equation with variable coefficient nonlinear source term. By applying the potential well and energy estimation method, in low initial energy and critical initial energy, we derive some sufficient conditions which are global existence and explosion of the solutions for this type of Klein-Gordon equation.

A class of p1(x, ⋅) & p2(x, ⋅)-fractional Kirchhoff-type problem with variable s(x, ⋅)-order and without the Ambrosetti-Rabinowitz condition in ℝN

Abstract: Abstract In this article, we study a class of Kirchhoff-type equation driven by the variable s(x, ⋅)-order fractional p1(x, ⋅) & p2(x, ⋅)-Laplacian. With the help of three different critical point theories, we obtain the existence and multiplicity of solutions in an appropriate space of functions. The main difficulties and innovations are the Kirchhoff functions with double Laplace operators in the whole space ℝN. Moreover, the approach is variational, but we do not impose any Ambrosetti-Rabinowitz condition for the nonlinear term.

Generalized 4-connectivity of hierarchical star networks

Abstract: Abstract The connectivity is an important measurement for the fault-tolerance of a network. The generalized connectivity is a natural generalization of the classical connectivity. An S S -tree of a connected graph G G is a tree T = ( V ′ , E ′ ) T=\left(V^{\prime} ,E^{\prime} ) that contains all the vertices in S S subject to S ⊆ V ( G ) S\subseteq V\left(G) . Two S S -trees T T and T ′ T^{\prime} are internally disjoint if and only if E ( T ) ∩ E ( T ′ ) = ∅ E\left(T)\cap E\left(T^{\prime} )=\varnothing and V ( T ) ∩ V ( T ′ ) = S V\left(T)\cap V\left(T^{\prime} )=S . Denote by κ ( S ) \kappa \left(S) the maximum number of internally disjoint S S -trees in graph G G . The generalized k k -connectivity is defined as κ k ( G ) = min { κ ( S ) ∣ S ⊆ V ( G ) and ∣ S ∣ = k } {\kappa }_{k}\left(G)=\min \left\{\kappa \left(S)| S\subseteq V\left(G)\hspace{0.33em}\hspace{0.1em}\text{and}\hspace{0.1em}\hspace{0.33em}| S| \hspace{0.33em}=\hspace{0.33em}k\right\} . Clearly, κ 2 ( G ) = κ ( G ) {\kappa }_{2}\left(G)=\kappa \left(G) . In this article, we show that κ 4 ( H S n ) = n − 1 {\kappa }_{4}\left(H{S}_{n})=n-1 , where H S n H{S}_{n} is the hierarchical star network.

Construction of special soliton solutions to the stochastic Riccati equation

Abstract: Abstract A scheme for the analytical stochastization of ordinary differential equations (ODEs) is presented in this article. Using Itô calculus, an ODE is transformed into a stochastic differential equation (SDE) in such a way that the analytical solutions of the obtained equation can be constructed. Furthermore, the constructed stochastic trajectories remain bounded in the same interval as the deterministic solutions. The proposed approach is in a stark contrast to methods based on the randomization of solution trajectories and is not focused on the analysis of martingales. This article extends the theory of Itô calculus by directly implementing it into analytical schemes for the solution of differential equations based on the generalized operator of differentiation. The efficacy of the presented analytical stochastization techniques is demonstrated by deriving stochastic soliton solutions to the Riccati differential equation. The presented semi-analytical stochastization scheme is relevant for the investigation of the global dynamics of different biological and biomedical processes where the variation interval of the stochastic solution is predetermined by the rationale of the model.

Extension of isometries in real Hilbert spaces

Abstract: Abstract The main purpose of this article is to develop a theory that extends the domain of any local isometry to the whole space containing the domain, where a local isometry is an isometry between two proper subsets. In fact, the main purpose of this article consists of the following three detailed objectives: The first objective is to extend the bounded domain of any local isometry to the first-order generalized linear span. The second one is to extend the bounded domain of any local isometry to the second-order generalized linear span. The third objective of this article is to extend the bounded domain of any local isometry to the whole Hilbert space.

On the p-integrable trajectories of the nonlinear control system described by the Urysohn-type integral equation

Abstract: Abstract The control systems described by the Urysohn-type integral equations and integral constraints on the control functions are considered. The functions from the closed ball of the space L p {L}_{p} , p > 1 , p\gt 1, with radius r r , are chosen as admissible control functions. The trajectory of the system is defined as a p-integrable function, satisfying the system’s equation almost everywhere. The boundedness and path-connectedness of the set of p-integrable trajectories are discussed. It is illustrated that the set of trajectories, in general, is not a closed subset of the space L p {L}_{p} . The robustness of a trajectory with respect to the fast consumption of the remaining control resource is established, and it is proved that every trajectory of the system can be approximated by the trajectory obtained by the full consumption of the control resource.

Approximations related to the complete p-elliptic integrals

Abstract: Abstract In this paper, the authors present some monotonicity properties for certain functions involving the complete p-elliptic integrals of the first and second kinds, by showing the monotonicity and concavity-convexity properties of certain combinations defined in terms of K p {{\mathscr{K}}}_{p} , E p {{\mathscr{E}}}_{p} and the inverse hyperbolic tangent arth p {{\rm{arth}}}_{p} , which is of importance in the computation of the generalized pi and in the elementary proof of Ramanujan’s cubic transformation. By these results, several well-known results for the classical complete elliptic integrals including its bounds and logarithmic inequalities are remarkably improved.

Explicit construction of mock modular forms from weakly holomorphic Hecke eigenforms

Abstract: Abstract Extending our previous work we construct weakly holomorphic Hecke eigenforms whose period polynomials correspond to elements in a basis consisting of odd and even Hecke eigenpolynomials induced by only cusp forms. As an application of our results, we give an explicit construction of the holomorphic parts of harmonic weak Maass forms that are good for Hecke eigenforms. Moreover, we give an explicit construction of the Hecke-equivariant map between the space of weakly holomorphic cusp forms and two copies of the spaces of cusp forms, and show that the map is compatible with the corresponding map on the spaces of period polynomials.