In this paper, k-harmonic submanifolds with parallel mean curvature vector in pseudo-Riemannian space forms are studied. We first investigate the fundamental distinction between biharmonic and k-harmonic submanifolds, proving that any nonminimal biharmonic submanifold with non light-like mean curvature vector in a nonflat pseudo-Riemannian space form can be not k-harmonic ( k≥ 3 ). Then, we give a partial classification of these pseudo-umbilical submanifolds.
In this paper, we prove that s-perspective, a-perspective and ad-perspective conditions are equivalent conditions, for finite dimensional left R-modules, where R is an K-algebra. We construct more examples of s-perspective modules and hence a-perspective modules which do not have essentially unique direct complements. We introduce a stronger variation of s-perspective modules named as ss-perspective. Let R be any ring and let M be any left R-module. We will say that M is ss-perspective if for any isomorphic two direct summands A and B of M, the equalities M=A⊕ X = B⊕ Y imply that A=B or X=Y . We prove that the class of ss-perspective modules is contained properly in the class of s-perspective modules. We also prove that for commutative rings, s-perspectivity and ss-perspectivity coincide. Finally, we prove that for a left R-module M with S=End_R(M) , if S_S is ss-perspective then M is ss-perspective, while we prove that if _SS is ss-perspective and M is weak duo, then M is ss-perspective.
We study a general class of nonlinear complementarity problems including the subclass of polynomial ones. As a solution method, we propose an algorithm based on branch-reduce-and-bound paradigm. In contrast to most existing solution methods, our algorithm works under very general conditions. Computational experiments on a number of complementarity problems show the practicability of our solution approach.
The aim of this paper is to investigate type-II E_∞ -Benson properly efficient solutions with respect to the recession cone for the vector optimization problem (in short, VOP) in real locally convex Hausdorff topological linear spaces. We establish both linear and nonlinear scalarization theorems for VOP in the sense of type-II E_∞ -Benson properly efficient solutions, employing the Gerstewitz functional, the Hiriart-Urruty functional, and augmented dual cones. The results presented in this paper generalize and improve several known results in the literature.
Let (R, 𝔪) be a Noetherian local ring, I an ideal of R and M a finitely generated R-module. In this paper, we define and study E-depth and Ef-depth of M in I. We prove that E-depth (resp. Ef-depth under mild conditions) of M in I is the common length of all maximal sequential sequences (resp. maximal sequential f-sequences) of M in I. We show that E-depth and Ef-depth do not decrease under small perturbations. We describe the non sequentially Cohen–Macaulay locus nSCM(M) of M in terms of support and non Cohen–Macaulay locus of deficiency modules of M. Using this description, we prove that the dimension of non-sequentially Cohen–Macaulay locus with respect to a sequential f-sequence does not increase under small pertubations.
In this paper, we consider a roughly convex multiobjective optimization problem and use the concept of the outer γ -convexity (resp., the γ -convexlikeness) of the objective mapping, where γ >0 is the roughness degree of the objective one, to show that every γ -local weak efficient solution (resp., γ -local efficient solution) of the considered problem is also a global weak efficient one (resp., global efficient one). Necessary and sufficient conditions for efficiency solutions are established by using γ -subdifferentials. Examples are also given to illustrate the obtained results.
In this paper, we introduce and analyze several new classes of Stepanov almost periodic type functions F: Λ× X → Y , where ∅Λ⊆ℝ^n , X is an arbitrary non-empty set and Y is a sequentially complete locally convex space. We present various applications of the introduced notion in the analysis of the existence and uniqueness of (Stepanov) almost periodic type solutions to the abstract Volterra integro-differential inclusions in locally convex spaces.
The k-Pell sequence is a generalization of the classical Pell sequence obtained by extending the order of its defining linear recurrence from the second order to an arbitrary order k ≥ 2 . Motivated by the works of Gómez and Luca (Glas. Mat. III 50, 17–24, 2015; Math. Slovaca 68, 939–949, 2018) on Diophantine quadruples with values in generalized Fibonacci sequences, we investigate whether there exist quadruples of positive integers a_1
We study a class of bilevel variational inequality problems over the solution sets of split variational inequality problems with multiple output sets in real Hilbert spaces, where the cost operators are quasimonotone. To solve this class of problems, we propose a strongly convergent algorithm that combines alternated inertial extrapolation with a self-adaptive step-size strategy. The proposed method guarantees strong convergence without requiring line search procedures or prior knowledge of problem-dependent parameters such as Lipschitz constants or strong monotonicity coefficients of the upper-level cost operator. Numerical experiments confirm the robustness and computational efficiency of the algorithm.
In the whole space of low spatial dimensions, namely d ≤ 2 , we study the minimizers of an energy functional with an attractive cubic nonlinearity and a repulsive quintic nonlinearity, which describes a quantum Bose gas with a two-body attraction and a three-body repulsion. We prove the existence of minimizers at fixed effective statistics parameter. In the limit of a large effective statistics parameter of the mass-critical nonlinearity (with respect to the kinetic energy), we derive an effective Thomas–Fermi-like model for the homogeneous Bose gas. We also consider other limit regimes depending on the mass-critical nonlinearity.
Step systems of connected graphs were introduced by Ladislav Nebeský as a means of describing shortest paths solely in terms of local information. A ternary relation T encodes, for each point u, the first step x towards a target v. Eight first-order axioms characterize the ternary relation T that are step systems. Here we show that one of Nebeský’s eight axioms is in fact redundant. Moreover, we characterize the step systems of bipartite graphs, partial cubes, and weakly modular graphs in terms of first-order axioms. In each case we show that the corresponding sets of axioms are non-redundant.
In this study, we investigated the effect of ecological flower planting measures on the control of insects in plants using two-time scales. Because of the different characteristics of each stage of the insect species, we employed a model with a population structure consisting of two stages: eggs and adults. We assumed the model was built in a heterogeneous environment, specifically on two adjoining fields. To reduce the dimension of the model, we utilized the aggregation of variables method and obtained an aggregated model with fewer dimensions. The interaction between species has been carefully analyzed. Outstanding in our results are the local stability analysis of equilibrium points and the global stability of the ideal equilibrium point. From a biological viewpoint, we provide helpful insights for pest control and crop yield enhancement as follows: two best-case scenarios can occur to achieve optimal crop yield. Firstly, we can divide the land into smaller fields to obtain maximum yield in both fields and eliminate the insects. Secondly, growing crops over a larger land area and rationally planting flowers like increasing the number of flowers and situating them near the crops, will create conditions for natural enemies to move around and support crop pollination. This, in turn, will boost crop production and restrict the growth of harmful insects.
Let t_1,… ,t_n be non zero complex numbers. We consider Wada’s representation φ _n(t_1,… ,t_n) :P_n→Gl_n-1(ℂ) when it is reducible. We then determine the irreducible representation φ̂_n(t_1,… ,t_n):P_n→Gl_n-2(ℂ) and show that it is the extension of the irreducible representation φ _n-1(t_1,… ,t_n-1) to P_n .
Consider the problem of minimizing a lower semicontinuous, semialgebraic function f :ℝ^n →ℝ∪{∞} over ℝ^n . In this paper, we first show that the set of tangency values (at infinity) of f is a finite set. Then we present necessary and sufficient conditions for the existence of optimal solutions of the problem as well as the boundedness from below and coercivity of the objective function. Finally, we obtain a full characterization of the infimum value at infinity of f and show some relationships between Łojasiewicz exponents at infinity of f for this value.
This paper studies the properness of polynomial selfmaps F:ℝ^m →ℝ^m or F:ℂ^m→ℂ^m . By results of Druzkowski, this question is reduced to that of properness of maps of the following special form, which we call identity plus linear powers. For two vectors x,y∈ℝ^m , we use the notation x *y =(x_1y_1,… ,x_my_m) , and if x=y we also use the notation x^2=x*x and by induction x^k=x*(x^k-1) . We use ⟨ , ⟩ for the usual inner product on ℝ^m . For A an m× m matrix with coefficients in ℝ , we can assign a map F_A(x)=x+(Ax)^3: ℝ^m→ℝ^m . A matrix A is Druzkowski if and only if (JF_A(x))=1 for all x∈ℝ^m . In this paper we research on the question of to what extent the above maps F_A(x) can be proper, and obtain various necessary conditions and sufficient conditions which suit very well the special form of the maps F_A . A complete characterisation of the properness, in terms of the existence of non-zero solutions to a system of polynomial equations of degree at most 3, in the case where A has corank 1, is obtained. Extending this, we propose a new conjecture, and discuss some applications to the (real) Jacobian conjecture. We also consider the properness of more general maps x± (Ax)^k or x± A(x^k) . The advantage of our method, compared to existing methods such as Newton’s polyhedron, is that our criteria, which are tailored for the maps F_A , are easy to check and applicable to parametrised families of matrices. Moreover, we can work on ℝ , while other known results requiring working over algebraically closed fields such as ℂ .
We establish a second main theorem with truncated counting functions for algebraically nondegenerate meromorphic mappings into a projective variety and a family of hypersurfaces in subgeneral position. The above bound of the total defect obtained from our result is better than that of the previous results. Moreover, in our result, the truncation level of the counting functions is estimated explicitly and independently of the number of hypersurfaces. Especially, we do not need the assumption that the family of hypersurfaces must satisfy the Bezout properties as imposed in some earlier studies, but only a weak Bezout property.
We employ the reduced Burau representation of the braid group to compute the Alexander polynomial and verify Fox’s Trapezoidal Conjecture for certain families of generalized weaving links with three and four strands. Furthermore, we propose an approach to addressing the conjecture for closed alternating braids by analyzing the reduced Burau representation.
We investigate the existence of almost p-standard systems of parameters under flat base change. For a faithfully flat homomorphism of Noetherian local rings R→ S with a Cohen–Macaulay special fiber, we show that the existence of such a system of parameters on R implies its existence on S. As applications, we give precise relationships between various invariants such as the Cohen–Macaulay defect, length function, and Hilbert coefficients relative to almost p-standard systems of parameters of a module over R and its base change over S.
We provide a complete description of the dynamics of isometric uniformly differentiable functions on ℤ_2 using some new representations of the isometric transformations including permutations on the set {0, 1} . We obtain some ergodicity criteria on compact subsets of ℤ_2 by proving the transitivity of finite orders.
In this paper we prove that the set of robustly quasiconvex functions is dense in the set of quasiconvex functions that obtain global minimum values on a closed bounded interval of X=ℝ and a real-valued function defined on a convex set in a normed linear space X is quasiconvex iff the robustness radius of its epigraph is non-negative. In addition, a function is robustly quasiconvex if and only if its epigraph in X×ℝ is robustly horizontally convex. A preservation property of convexity for robustly horizontally convex sets is established.
