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Measuring similarities/dissimilarities between atomic structures is important for the exploration of potential energy landscapes. However, the cell vectors together with the coordinates of the atoms, which are generally used to describe periodic systems, are quantities not directly suitable as fingerprints to distinguish structures. Based on a characterization of the local environment of all atoms in a cell, we introduce crystal fingerprints that can be calculated easily and define configurational distances between crystalline structures that satisfy the mathematical properties of a metric. This distance between two configurations is a measure of their similarity/dissimilarity and it allows in particular to distinguish structures. The new method can be a useful tool within various energy landscape exploration schemes, such as minima hopping, random search, swarm intelligence algorithms, and high-throughput screenings. [ABSTRACT FROM AUTHOR]

Abstract The coexistence of subharmonic periodic solutions of various orders is investigated to the first-order vector system of impulsive (upper-) Carathéodory differential equations and inclusions on tori. As the main tool, our recent Sharkovsky-type results for multivalued maps on tori are applied via the associated Poincaré translation operators along the trajectories of given systems. The solvability criteria are formulated, under natural bi-periodicity assumptions imposed on the right-hand sides, in terms of the Lefschetz numbers of admissible impulsive maps. Since the criteria become effective on the circle, the main general theorem can be improved and reformulated there in a more transparent way. The obtained results can be regarded in a certain sense as a nontrivial extension of those due to Poincaré [28] , Denjoy [17] and van Kampen [24]. [ABSTRACT FROM AUTHOR]

In this paper, we study mappings, which approximately preserve angles between inner product spaces. We also introduce a notion of angle in normed spaces. The notion of angle, considered in this part, relates to the well-known Birkhoff-James orthogonality. Based on it, we express a characterization for approximate Birkhoff-James orthogonality, introduced in the literature, through this notion of angle. Then we return to the issue of mappings which approximately preserve angle stating some results in normed spaces. [ABSTRACT FROM AUTHOR]

A general conjecture is stated on the cone of automorphic vector bundles admitting nonzero global sections on schemes endowed with a smooth, surjective morphism to a stack of G-zips of connected Hodge type; such schemes should include all Hodge-type Shimura varieties with hyperspecial level. We prove our conjecture for groups of type A1n, C2, and Fp-split groups of type A2 (this includes all Hilbert–Blumenthal varieties and should also apply to Siegel modular 3-folds and Picard modular surfaces). An example is given to show that our conjecture can fail for zip data not of connected Hodge type. [ABSTRACT FROM AUTHOR]

In this paper, we study a vector optimization problem (VOP) with both inequality and equality constraints. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point of study. By virtue of a second-order constraint qualification of Abadie type, we provide second-order Karush-Kuhn-Tucker type necessary optimality conditions for the VOP. Moreover, we also obtain second-order sufficient optimality conditions for a kind of strict local efficiency. Both the necessary conditions and the sufficient conditions are shown in equivalent pairs of primal and dual formulations by using theorems of the alternative for the VOP. [ABSTRACT FROM AUTHOR]

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function f:Z+n→R+ is given via an evaluation oracle. Assume further that f satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time (1-1/e-ϵ)-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a (1-1/e-ϵ)-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property. [ABSTRACT FROM AUTHOR]

We study the Specht property for L-varieties of vector spaces embedded in associative algebras over an arbitrary field. An L-variety with no finite basis of identities over a field, which is the join of two Spechtian L-varieties, is exemplified. A condition under which L-varieties will have the Specht property is found. [ABSTRACT FROM AUTHOR]

Abstract Primordial non-Gaussianities enhanced at small wavevectors can induce a power spectrum of the galaxy overdensity that differs greatly from that of the matter overdensity at large length scales. In previous work, it was shown that "squeezed" three-point and "collapsed" four-point functions of the curvature perturbation ζ can generate these non-Gaussianities and give rise to so-called scale-dependent and stochastic bias in the galaxy overdensity power spectrum. We explore a third way to generate non-Gaussianities enhanced at small wavevectors: the infrared behavior of quantum loop contributions to the four-point correlations of ζ. We show that these loop effects can give the largest contributions to the four-point function of ζ in the collapsed limit and be observable in the context of quasi-single field inflation. [ABSTRACT FROM AUTHOR]

We prove the Mercat Conjecture for rank 2 vector bundles on generic curves of every genus. For odd genus, we identify the effective divisor in Mg where the Mercat Conjecture fails and compute its slope. [ABSTRACT FROM AUTHOR]

In this paper, we establish compactness and existence results to a Branson–Paneitz type problem on a bounded domain of R n with Navier boundary condition. [ABSTRACT FROM AUTHOR]

We extend the univariate α-quantile residual life function to multivariate setting preserving its dynamic feature. Principal attributes of this function are derived and their relationship to the dynamic multivariate hazard rate function is discussed. A corresponding ordering, namely, α-quantile residual life order, for random vectors of lifetimes is introduced and studied. Based on the proposed ordering, a notion of positive dependency is presented. Finally, a discussion about conditions characterizing the class of decreasing multivariate α-quantile residual life functions is pointed out. [ABSTRACT FROM AUTHOR]

We study the cyclicity of vectors u in ℓ p ( Z ) . It is known that a vector u is cyclic in ℓ 2 ( Z ) if and only if the zero set, Z ( u ˆ ) , of its Fourier transform, u ˆ , has Lebesgue measure zero and log ⁡ | u ˆ | ∉ L 1 ( T ) , where T is the unit circle. Here we show that, unlike ℓ 2 ( Z ) , there is no characterization of the cyclicity of u in ℓ p ( Z ) , 1 < p < 2 , in terms of Z ( u ˆ ) and the divergence of the integral ∫ T log ⁡ | u ˆ | . Moreover we give both necessary conditions and sufficient conditions for u to be cyclic in ℓ p ( Z ) , 1 < p < 2 . [ABSTRACT FROM AUTHOR]

In this paper, we derive the Bishop frames of the pseudo null curve and show that its normal Bishop vectors N 1 and N 2 can be obtained by applying the hyperbolic rotation and the composition of three rotations about two lightlike and one spacelike axis to the Frenet vectors N and B respectively. We also derive Bishop frame of the null Cartan curve and show that among all null Cartan curves in E 1 3 , only the null Cartan cubic has two Bishop frames, one of which coincides with its Cartan frame. As an application, we obtain some solutions of the Da Rios vortex filament equation in terms of the Bishop frames of the pseudo null curves and null Cartan cubic. [ABSTRACT FROM AUTHOR]

An alternating sign matrix, or ASM, is a ( 0 , ± 1 ) -matrix where the nonzero entries in each row and column alternate in sign. We generalize this notion to hypermatrices: an n × n × n hypermatrix A = [ a i j k ] is an alternating sign hypermatrix , or ASHM, if each of its planes, obtained by fixing one of the three indices, is an ASM. Several results concerning ASHMs are shown, such as finding the maximum number of nonzeros of an n × n × n ASHM, and properties related to Latin squares. Moreover, we investigate completion problems, in which one asks if a subhypermatrix can be completed (extended) into an ASHM. We show several theorems of this type. [ABSTRACT FROM AUTHOR]

In our paper we will continue the comparison which was started by Vsevolod I. Ivanov [Nonlinear Analysis 125 (2015), 270--289], where he compared scalar optimality conditions stated in terms of Hadamard derivatives for arbitrary functions and those which was stated for ℓ-stable functions in terms of Dini derivatives. We will study the vector optimization problem and we show that also in this case the optimality condition stated in terms of Hadamard derivatives is more advantageous. [ABSTRACT FROM AUTHOR]

Given a graph G , the cut polytope is the convex hull of its cut vectors. The latter objects are the incidence vectors associated to all cuts of G . Especially motivated by related conjectures of Sturmfels and Sullivant, we study various properties and invariants of the toric algebra of the cut polytope, called its cut algebra. In particular, we characterize those cut algebras which are complete intersections, have linear resolutions or have Castelnuovo–Mumford regularity equal to 2. The key idea of our approach is to consider suitable algebra retracts of cut algebras. Additionally, combinatorial retracts of the graph are defined and investigated, which are special minors whose algebraic properties can be compared in a very pleasant way with the corresponding ones of the original graph. Moreover, we discuss several examples and pose new problems as well. [ABSTRACT FROM AUTHOR]

An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. Though they arise in many applications, only a few methods for constructing them are known. Motivated by the connection between real ETFs and graph theory, we introduce the notion of ETFs that are symmetric about their centroid. We then discuss how well-known constructions, such as harmonic ETFs and Steiner ETFs, can have centroidal symmetry. Finally, we establish a new equivalence between centroid-symmetric real ETFs and certain types of strongly regular graphs (SRGs). Together, these results give the first proof of the existence of certain SRGs, as well as the disproofs of the existence of others. [ABSTRACT FROM AUTHOR]

Provost and Cheong (2000) show the importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts. Thec-characteristic function has been shown to be very useful and more practical in some distributions that are hard to manage with the traditional characteristic functions. The importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts, is well known. We first provide its inversion formula which is practical in determining the distribution function of a random variable when itsc-characteristic function is known. We then use this inversion formula to find an expression of probability density function of linear combinations of components of any Dirichlet vector. This would generalize the currently well known results. [ABSTRACT FROM PUBLISHER]

Numerical algorithms for elliptic partial differential equations frequently employ error estimators and adaptive mesh refinement strategies in order to reduce the computational cost. We can extend these techniques to general vectors by splitting the vectors into a hierarchically organized partition of subsets and using appropriate bases to represent the corresponding parts of the vectors. This leads to the concept of hierarchical vectors. A hierarchical vector with m subsets and bases of rank k requires mk units of storage, and typical operations like the evaluation of norms and inner products or linear updates can be carried out in O(mk2) operations. Using an auxiliary basis, the product of a hierarchical vector and an H2-matrix can also be computed in O(mk2) operations, and if the result admits an approximation with m˜ subsets in the original basis, this approximation can be obtained in O((m+ m˜)k2) operations. Since it is possible to compute the corresponding approximation error exactly, sophisticated error control strategies can be used to ensure the optimal compression. Possible applications of hierarchical vectors include the approximation of eigenvectors, optimal control problems, and time-dependent partial differential equations with moving local irregularities. [ABSTRACT FROM AUTHOR]

Updating a given a matrix Amxn by a rank-one matrix B = cdT, where c and d are appropriately sized column vectors, is a common practice throughout all applied areas of mathematics, science, and engineering. Because rank is often tied to the number of degrees of freedom or the level of independence in underlying models or data, it can be imperative to know exactly how the update term affects rank. While it is well known that a rank-one update can only increase or decrease rank by at most one, there is not a widely known formula for exactly how this occurs. This note presents an expression in simply stated terms for the exact rank of a rank-one updated matrix. [ABSTRACT FROM AUTHOR]

We consider the testing of mutual independence among all entries in a d-dimensional random vector based on n independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and we propose tests that control the Type I error in the high-dimensional setting where d > n. We further show that the two tests are rate-optimal in terms of power against sparse alternatives and that they outperform competitors in simulations, especially when d is large. [ABSTRACT FROM AUTHOR]

Calgero-Bogoyavlenskii-Schiff (CBS) equation is analytically solved through two successive reductions into an ordinary differential equation (ODE) through a set of optimal Lie vectors. During the second reduction step, CBS equation is reduced using hidden vectors. The resulting ODE is then analytically solved through the singular manifold method in three steps; First, a Bäcklund truncated series is obtained. Second, this series is inserted into the ODE, and finally, a seminal analysis leads to a Schwarzian differential equation in the eigenfunction φ( η). Solving this differential equation leads to new analytical solutions. Then, through two backward substitution steps, the original dependent variable is recovered. The obtained results are plotted for several Lie hidden vectors and compared with previous work on CBS equation using Lie transformations. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper proposes a new measure for recommendation through integrating Triangle and Jaccard similarities. The Triangle similarity considers both the length and the angle of rating vectors between them, while the Jaccard similarity considers non co-rating users. We compare the new similarity measure with eight state-of-the-art ones on four popular datasets under the leave-one-out scenario. Results show that the new measure outperforms all the counterparts in terms of the mean absolute error and the root mean square error. [ABSTRACT FROM AUTHOR]

Synthesis of DoAll loops is a key aspect of High Level Synthesis since they allow to easily exploit the potential parallelism provided by programmable devices. This type of parallelism can be implemented in several ways: by duplicating the implementation of body loop, by exploiting loop pipelining or by applying vectorization. In this paper a methodology for the synthesis of nested irregular DoAll loops based on outer vectorization is proposed. The methodology transforms the intermediate representation of the DoAll loop to introduce vectorization and it can be easily integrated in existing state of the art High Level Synthesis flows since does not require any modification in the rest of the flow. Vectorization is not limited to perfectly nested countable loops: conditional constructs and loops with variable number of iterations are supported. Experimental results on parallel benchmarks show that the generated parallel accelerators have significant speed-up with limited penalties in terms of resource usage and frequency decrement. [ABSTRACT FROM AUTHOR]

We study the problem of using mobile guards to defend the vertices of a graph against a single attack on its vertices. A vertex cover of a graph is a set such that for each edge , at least one of or is in . The minimum cardinality of a vertex cover is termed the vertex covering number and it is denoted by . In this context, we introduce a new protection strategy called the secure vertex cover SVC problem, where the guards are placed at the vertices of a graph, in order to protect the graph against a single attack on its vertices. We are concerned with the protection of against a single attack, using at most one guard per vertex and require the set of guarded vertices to be a vertex cover. In addition, if any guard moves along an edge to deal with an attack to an unguarded vertex, then the resulting placement of guards must also form a vertex cover. Formally, this protection strategy defends the vertices of a graph against a single attack and simultaneously protects the edges. We define a SVC to be a set such that is a vertex cover and for each , there exists a such that is a vertex cover. The minimum cardinality of an SVC is called the secure vertex covering number and it is denoted by . In this paper, a few properties of SVC of a graph are studied and specific values of for few classes of well-known graphs are evaluated. [ABSTRACT FROM AUTHOR]

This paper investigates regularity properties of two non-negative sparsity sets: non-negative sparse vectors, and low-rank positive semi-definite matrices. Novel formulae for their Mordukhovich normal cones are given and used to formulate sufficient conditions for non-convex notions of regularity to hold. Our results provide a useful tool for justifying the application of projection methods to certain rank constrained feasibility problems. [ABSTRACT FROM AUTHOR]

Let A be a normal operator in a Hilbert space H , and let G ⊂ H be a countable set of vectors. We investigate the relations between A , G and L that make the system of iterations { A n g : g ∈ G , 0 ≤ n < L ( g ) } complete, Bessel, a basis, or a frame for H . The problem is motivated by the dynamical sampling problem and is connected to several topics in functional analysis, including, frame theory and spectral theory. It also has relations to topics in applied harmonic analysis including, wavelet theory and time-frequency analysis. [ABSTRACT FROM AUTHOR]

We use supernatural bundles to build $\mathbf{GL}$-equivariant resolutions supported on the diagonal of $\mathbb{P}^{n}\times \mathbb{P}^{n}$, in a way that extends Beilinson’s resolution of the diagonal. We thus obtain results about supernatural bundles that largely parallel known results about exceptional collections. We apply this construction to Boij–Söderberg decompositions of cohomology tables of vector bundles, yielding a proof of concept for the idea that those positive rational decompositions should admit meaningful categorifications. [ABSTRACT FROM AUTHOR]

Consider a two-dimensional delayed renewal risk model with a constant interest rate, where the claim sizes of the two classes form a sequence of independent and identically distributed random vectors following a common bivariate Sarmanov distribution. In the presence of heavy-tailed claim sizes, some asymptotic formulas are derived for the finite-time and infinite-time ruin probabilities. [ABSTRACT FROM AUTHOR]

A random vector X = ( X 1 , … , X n ) with the X i taking values in an arbitrary measurable space ( S , S ) is exchangeable if its law is the same as that of ( X σ ( 1 ) , … , X σ ( n ) ) for any permutation σ . We give an alternative and shorter proof of the representation result (Jaynes [6] and Kerns and Székely [9] ) stating that the law of X is a mixture of product probability measures with respect to a signed mixing measure. The result is “finitistic” in nature meaning that it is a matter of linear algebra for finite S . The passing from finite S to an arbitrary one may pose some measure-theoretic difficulties which are avoided by our proof. The mixing signed measure is not unique (examples are given), but we pay more attention to the one constructed in the proof (“canonical mixing measure”) by pointing out some of its characteristics. The mixing measure is, in general, defined on the space of probability measures on S ; but for S = R , one can choose a mixing measure on R n . [ABSTRACT FROM AUTHOR]

The p-center problem is one of the most important models in location theory. Its objective is to place a fixed number of facilities so that the maximum service distance for all customers is as small as possible. This article develops a reliable p-center problem that can account for system vulnerability and facility failure. A basic assumption is that located centers can fail with a given probability and a customer will fall back to the closest nonfailing center for service. The proposed model seeks to minimize the expected value of the maximum service distance for a service system. In addition, the proposed model is general and can be used to solve other fault-tolerant center location problems such as the (p, q)-center problem using appropriate assignment vectors. I present an integer programming formulation of the model and computational experiments, and then conclude with a summary of findings and point out possible future work. [ABSTRACT FROM AUTHOR]

For the weighted k-out-of-n system with statistically dependent component lifetimes, we have a discussion on the system performance through investigating the ordering properties of the total system weight with respect to component weight vector. Applications of the present ordering results to signature of coherent systems are presented as well. [ABSTRACT FROM PUBLISHER]

The main result of this paper states that a universally complete real vector lattice admits a band preserving complex structure if and only if it contains no locally one-dimensional bands. A description of order bounded disjointness preserving complex structures on Archimedean semiprime f -algebras is also given. [ABSTRACT FROM AUTHOR]

We investigate the optimum correction of an absolute value equation by minimally changing the coefficient matrix and right-hand side vector using Tikhonov regularization. Solving this problem is equivalent to minimizing the sum of fractional quadratic and quadratic functions. The primary difficulty with this problem is its nonconvexity. Nonetheless, we show that a global optimal solution to this problem can be found by solving an equation on a closed interval using the subgradient method. Some examples are provided to illustrate the efficiency and validity of the proposed method. [ABSTRACT FROM PUBLISHER]

A complex-valued convolutional network (convnet) implements the repeated application of the following composition of three operations, recursively applying the composition to an input vector of nonnegative real numbers: (1) convolution with complex-valued vectors, followed by (2) taking the absolute value of every entry of the resulting vectors, followed by (3) local averaging. For processing real-valued random vectors, complex-valued convnets can be viewed as data-driven multiscale windowed power spectra, data-driven multiscale windowed absolute spectra, data-driven multiwavelet absolute values, or (in their most general configuration) data-driven nonlinear multiwavelet packets. Indeed, complex-valued convnets can calculate multiscale windowed spectra when the convnet filters are windowed complex-valued exponentials. Standard real-valued convnets, using rectified linear units (ReLUs), sigmoidal (e.g., logistic or tanh) nonlinearities, or max pooling, for example, do not obviously exhibit the same exact correspondence with data-driven wavelets (whereas for complex-valued convnets, the correspondence ismuchmore than just a vague analogy). Courtesy of the exact correspondence, the remarkably rich and rigorous body of mathematical analysis for wavelets applies directly to (complex-valued) convnets. [ABSTRACT FROM AUTHOR]

In this study, firstly, the darboux vector ... of the natural lift ... of a curve a are calculated in terms of those of a in ℝ³1. Secondly, we obtained striction lines and distribution parameters of ruled surface pair generated by by Darboux vectors of the curve a and its natural lift .... Finally, for α and ... those notions are compared with each other. [ABSTRACT FROM AUTHOR]

The goal of this paper is to explicitly detect all the arithmetic genera of arithmetically Cohen–Macaulay projective curves with a given degree d . It is well-known that the arithmetic genus g of a curve C can be easily deduced from the h -vector of the curve; in the case where C is arithmetically Cohen–Macaulay of degree d , g must belong to the range of integers { 0 , … , ( d − 1 2 ) } . We develop an algorithmic procedure that allows one to avoid constructing most of the possible h -vectors of C . The essential tools are a combinatorial description of the finite O-sequences of multiplicity d , and a sort of continuity result regarding the generation of the genera. The efficiency of our method is supported by computational evidence. As a consequence, we single out the minimal possible Castelnuovo–Mumford regularity of a curve with Cohen–Macaulay postulation and given degree and genus. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some conditions to compare multivariate mixture models with respect to some well-known multivariate stochastic orders. We also utilize the established results in reliability theory to compare the vectors of residual life-lengths of live components of ( n − k + 1 ) -out-of- n systems in both one sample and two samples situations. [ABSTRACT FROM AUTHOR]

Two different nondiagonal tetrad spaces reproducing spherically symmetric spacetime are applied to the field equations of higher-order torsion scalar theories. Assuming the existence of conformal Killing vector, two isotropic solutions are derived. We show that the first solution is not stable while the second one confirms a stable behavior. We also discuss the construction of the stellar model and show that one of our solutions is capable of such construction while the other is not. Finally, we discuss the generalized Tolman-Oppenheimer-Volkoff and show that one of our models has a tendency to equilibrium. [ABSTRACT FROM AUTHOR]

Vector-valued fractional maximal inequalities on variable Morrey spaces are proved. Applying atomic decomposition of variable Hardy-Morrey spaces, we obtain the boundedness of fractional integrals on variable Hardy-Morrey spaces, which extends the Taibleson-Weiss's results for the boundedness of fractional integrals on Hardy spaces. The corresponding boundedness for the fractional type integrals is also considered. [ABSTRACT FROM AUTHOR]

In this paper we continue the study (initiated in a previous paper) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting and prove its compatibility with morphisms of vector bundles and base change. [ABSTRACT FROM AUTHOR]

We completely classify redundant blow-ups appearing in the theory of rational surfaces with big anticanonical divisor due to Sakai. In particular, we construct a rational surface with big anticanonical divisor which is not a minimal resolution of a del Pezzo surface with only rational singularities, which gives a negative answer to a question raised in a paper by Testa, Várilly-Alvarado, and Velasco. [ABSTRACT FROM AUTHOR]

In this note we show that the sheaf R 1 f ⁎ O X is an anti-nef vector bundle (i.e., its dual is nef), where f : X → Y is a family of Du Bois schemes of pure dimension. [ABSTRACT FROM AUTHOR]

Let $$X_1,\ldots ,X_n$$ be a random sample from a distribution function $$F$$ that denote lifetimes of $$n$$ components of a coherent system. Suppose the system fails at $$X_{k:n}$$ , the $$k$$ th order statistic of $$X$$ 's, since we are not aware of the exact time at which the system has been failed, the residual lifetimes of the remaining $$n-k$$ components, denoted by $$X^{(k)}_1,\ldots ,X^{(k)}_{n-k}$$ , are no longer independent but exchangeable. In this paper, multivariate stochastic comparisons of two vectors of lifetimes of the remaining components in the two sample problems are studied. Some sufficient conditions under which multivariate mixture models are compared stochastically with respect to the multivariate likelihood ratio ordering, the multivariate hazard rate ordering and the multivariate reversed hazard rate ordering are provided. These comparisons are done for different choices of the mixed distributions as well as mixing distributions. The new results obtained are applied to compare multivariate mixtures of location models. [ABSTRACT FROM AUTHOR]

In this paper, we propose a class of generalized proximal-type method by the virtue of Bregman functions to solve weak vector variational inequality problems in Banach spaces. We carry out a convergence analysis on the method and prove the weak convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions. Our results extend some known results to more general cases. [ABSTRACT FROM AUTHOR]

We define covariant Lie derivatives acting on vector-valued forms on Lie algebroids and study their properties. This allows us to obtain a concise formula for the Frölicher-Nijenhuis bracket on Lie algebroids. [ABSTRACT FROM AUTHOR]

In the present paper, we consider the inequality constrained vector problem with continuously Fréchet differentiable objective functions and constraints. We obtain second-order necessary optimality conditions of Karush-Kuhn-Tucker type for weak efficiency. A new second-order constraint qualification of Zangwill type is introduced. It is applied in the optimality conditions. Some connections with other constraint qualifications are established. [ABSTRACT FROM AUTHOR]

We consider sequences of vectors from Nd. Each coordinate of a vector can be reset or incremented by 1 with respect to the same coordinate of the preceding vector. We give an example of non-dominating sequence, like in Dickson's Lemma, of length 22θ(n), what matches the previously known upper bound. [ABSTRACT FROM AUTHOR]