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Inspired by a problem of invariance of power series families of probability distributions under binomial thinning, Baron and Wesołowski (Res Math 76:168, 2021) investigated a functional equation being a unification of the Cauchy and the Goła̧b-Schinzel type equations. They determined the solutions of the equation with no or mild regularity assumptions imposed on unknown functions. In this paper we extend their results onto a multi-dimensional case. [ABSTRACT FROM AUTHOR]

Let K be an n-dimensional equiangular cone in a Euclidean vector space. Let 0 < ϕ < π be the common angle between any pair of vectors taken among the generators of the cone. We obtain an explicit formula for the maximum angle of K in terms of n and ϕ . We explain also how to identify a pair { u , v } of unit vectors in K achieving the maximum angle. Finally, we analyze a counterintuitive phenomenon that occurs when n is large: the equiangular cone K can be constructed so that u and v are almost opposite, whereas ϕ is near π / 2 . [ABSTRACT FROM AUTHOR]

In this paper, some concepts of convex analysis on hyperbolic spaces are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first- and second-order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented. [ABSTRACT FROM AUTHOR]

In this study, we describe higher-order (p; r)-ρ-cone arcwise connectedness as an entirely new type of generalized higher-order arcwise connectedness for set-valued optimization problems. Under the higher-order contingent epiderivative and higher-order (p; r)-ρ-cone arcwise connectedness suppositions, we demonstrate the higher-order sufficient Karush-Kuhn-Tucker (KKT) optimality requirements. We also investigate higherorder Mond-Weir (??) form of duality and establish the corresponding higher-order weak, strong, and converse theorems of duality between the primary (??) and the corresponding dual problems by employing the higher-order (p; r)-ρ-cone arcwise connectedness supposition. In order to demonstrate that higher-order (p; r)-ρ-cone arcwise connectedness is more widespread than higher-order cone arcwise connectedness, an example is also constructed. [ABSTRACT FROM AUTHOR]

We present existing results and properties for the solutions of some vector equilibrium problems with set-valued functions in the case of a vector space ordered by a cone with some "interiority" properties. Some applications concerning the existence of equilibrium for abstract economies and vector optimization problems are given. [ABSTRACT FROM AUTHOR]

This paper addresses the numerical computation of critical angles between two convex cones in Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary points of a fractional programming problem. To efficiently compute these stationary points, we introduce a partial linearization-like algorithm that offers significant computational advantages. Solving a sequence of strictly convex subproblems with straightforward solutions in several settings gives the proposed algorithm high computational efficiency while delivering reliable results: our theoretical analysis demonstrates that the proposed algorithm asymptotically computes critical angles. Numerical experiments validate the efficiency of our approach, even when dealing with problems of relatively large dimensions: only a few seconds are necessary to compute critical angles between different types of cones in spaces of dimension 1000. [ABSTRACT FROM AUTHOR]

In this paper, an optimization problem (DP) is studied where the objective maps and the constraints are the difference of set-valued maps (abbreviated as SVMs). The higher-order σ-cone arcwise connectedness is described as an entirely new type of generalized higher-order arcwise connectedness for set-valued optimization problems. Under the higher-order contingent epiderivative and higher-order σ-cone arcwise connectedness suppositions, the higher-order sufficient Karush–Kuhn–Tucker (KKT) optimality requirements are demonstrated for the problem (DP). The higher-order Wolfe (WD) form of duality is investigated and the corresponding higher-order weak, strong, and converse theorems of duality are established between the primary (DP) and the corresponding dual problem by employing the higher-order σ-cone arcwise connectedness supposition. In order to demonstrate that higher-order σ-cone arcwise connectedness is more generalized than higher-order cone arcwise connectedness, an example is also constructed. As a special case, the results coincide with the existing ones available in the literature.

In this paper, some concepts related to the intrinsic convexity of non-homogeneous quadratic functions on the hyperbolic space are studied. Unlike in the Euclidean space, the study of intrinsic convexity of non-homogeneous quadratic functions in the hyperbolic space is more elaborate than that of homogeneous quadratic functions. Several characterizations that allow the construction of many examples will be presented. [ABSTRACT FROM AUTHOR]

We have recently introduced and studied the concept of singular value of a rectangular matrix relative to a pair of closed convex cones. Such cones act as constraint sets for the left-singular and right-singular vectors. This work extends the theory of cone-constrained singular value problems from rectangular matrices to linear maps between Euclidean spaces. [ABSTRACT FROM AUTHOR]

Starting from the principle of locality in quantum field theory, which states that an object is influenced directly only by its immediate surroundings, we review some features of the notion of locality arising in physics and mathematics. We encode these in locality relations, given by symmetric binary relations, and locality morphisms, namely maps that factorise on products of pairs in the graph of such locality relations. This factorisation is a key property in the context of renormalisation, as illustrated on the factorisation of an exponential sum on convex cones, discussed at the end of the paper. The subject of locality is so vast and the issues it raises are so subtle, that this brief and modest presentation can only offer a small glimpse into this fascinating topic. [ABSTRACT FROM AUTHOR]

In this paper, we determine the sufficient Karush-Kuhn-Tucker (KKT) conditions of optimality of a set-valued fractional programming problem (in short, SVFP) $\rm (FP)$ under the suppositions of contingent epidifferentiation and $ \sigma $-arcwisely connectivity. We additionally explore the results of duality of parametric $\rm (PD)$, Mond-Weir $\rm (MWD)$, Wolfe $\rm (WD)$, and mixed $\rm (MD)$ kinds for the problem $\rm (FP)$.

A set-valued minimax programming problem (in short, SVMP) is taken into consideration in this study. We present the idea of σ-arcwisely connectivity of set-valued maps (in short, SVM) in the broader sense of arcwisely connected SVMs. The sufficient criteria for Karush-Kuhn-Tucker (KKT) optimality are constituted for the problem (MP) under contingent epidifferentiation and σ-arcwisely connectivity suppositions. In addition, we develop the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) kinds models of duality and verify the associated strong, weak, and converse theorems of duality among the primal (MP) and the associated figures of duals under σ-arcwisely connectivity supposition.

A new area of research called intuitionistic fuzzy soft expert set is expected to overcome the drawbacks of an intuitionistic fuzzy soft set in terms of eligibility for soft expert-argument approximate function. This type of function views the power set of the universe as its co-domain and the cartesian product of attributes, experts, and their opinions as its domain. The domain of this function is larger as compared to the domain of a soft approximation function. It can manage a situation in which several expert opinions are taken into account by a single model. For the soft expert-argument approximate function with intuitionistic fuzzy setting, concepts such as set inclusion, [Formula: see text]-convexity(concave) sets, strongly [Formula: see text]-convexity (concave) sets, strictly [Formula: see text]-convexity (concave) sets, convex hull, and convex cone are conceived in this paper. Some set-theoretic inequalities are established with generalized properties and results on the basis of these specified notions. Additionally, by using a theoretic cum analytical approach, various elements of computational geometry, such as convex hull and convex cone, are theorized and some pertinent results are generalized.

Let C be a closed convex subset of a real Hilbert space containing the origin, and assume that K is the homogenization cone of C, i.e., the smallest closed convex cone containing C × { 1 } . Homogenization cones play an important role in optimization for the construction of examples and counterexamples. A famous examples is the second-order/Lorentz/“ice cream” cone which is the homogenization cone of the unit ball. In this paper, we discuss the polar cone of K as well as an algorithm for finding the projection onto K provided that the projection onto C is available. Various examples illustrate our results. [ABSTRACT FROM AUTHOR]

In the first part of the paper, we present a theorem of type "partition of unity", and three of its consequences, i.e., for algebras, linear subspaces and convex cones. In the second part, some theorems of localizability (or density) in a weighted space are presented. We mention that the weighted spaces are classes of continuous scalar functions on a locally compact space (for example, the space of function with compact support, the space of bounded functions, the space of functions vanishing at infinity, the space of functions rapidly decreasing at infinity). [ABSTRACT FROM AUTHOR]

Let P be a finitely generated commutative semiring with a unity. It was shown recently that if the multiplicative reduct of P is a group then P is additively idempotent. We extend this result by showing that P is additively idempotent, provided that P is additively divisible. We further generalize this result using a weaker form of divisibility (almost-divisibility) as follows. Let S be a semiring that is a factor of a monoid semiring N [ C ] where C is a submonoid of a free commutative monoid of finite rank. Then the semiring S is additively almost-divisible if and only if S is torsion. In particular, we show that if S is a ring then S contains no non-finitely generated subring of Q . [ABSTRACT FROM AUTHOR]

A set-valued minimax programming problem (in short, SVMP) is taken into consideration in this study. We present the idea of s-arcwisely connectivity of set-valued maps (in short, SVM) in the broader sense of arcwisely connected SVMs. The sufficient criteria for Karush-Kuhn-Tucker (KKT) optimality are constituted for the problem (MP) under contingent epidifferentiation and s-arcwisely connectivity suppositions. In addition, we develop the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) kinds models of duality and verify the associated strong, weak, and converse theorems of duality among the primal (MP) and the associated figures of duals under s-arcwisely connectivity supposition. [ABSTRACT FROM AUTHOR]

Using the geometric approach, we formulate a quantum theory in terms of Jordan algebras. We analyze the notion of a (quasi)particle (=elementary excitation of translation-invariant stationary state) and the scattering of (quasi)particles in this framework. [ABSTRACT FROM AUTHOR]

This paper presents some variants of minimal point theorem together with corresponding variants of Ekeland variational principle. In the second part of this article, there is a discussion on Ekeland variational principle and minimal point theorem relative to it in uniform spaces. The aim of these series of important results is to highlight relations between them, some improvements and specific applications.

Abstract Within the inquiry about work, we explore a parametric set-valued optimization problem, where the objective as well as constraint maps are set-valued. A generalization of cone arcwise associated set-valued maps is presented named ρ-cone arcwise connectedness of set-valued maps. We set up adequate Karush–Kuhn–Tucker optimality conditions for the problem beneath contingent epiderivative and ρ-cone arcwise connectedness presumptions. Assist, Mond–Weir, Wolfe, and blended sorts duality models are examined. We demonstrate the related theorems between the primal and the comparing dual problems beneath the presumption.

We obtain the radial symmetry of the solution to a partially overdetermined boundary value problem in a convex cone in space forms by using the maximum principle for a suitable subharmonic function P and integral identities. In dimension 2, we prove Serrin‐type results for partially overdetermined problems outside a convex cone. Furthermore, we obtain a Rellich identity for an eigenvalue problem with mixed boundary conditions in a cone. [ABSTRACT FROM AUTHOR]

Let K be a closed convex cone in the state space \mathbb {R}^n. This note characterizes the K-monotonicity of a non-autonomous dynamical system \dot x (t) = f(t,x(t)) governed by a locally Lipschitz velocity field. We deviate from the classical literature in two important ways. Firstly, the velocity field f is not required to be differentiable with respect to the state variables. And, secondly, the closed convex cone K is allowed to be absolutely general. In particular, we impose neither pointedness, nor solidity. [ABSTRACT FROM AUTHOR]

A new area of research called intuitionistic fuzzy soft expert set is expected to overcome the drawbacks of an intuitionistic fuzzy soft set in terms of eligibility for soft expert-argument approximate function. This type of function views the power set of the universe as its co-domain and the cartesian product of attributes, experts, and their opinions as its domain. The domain of this function is larger as compared to the domain of a soft approximation function. It can manage a situation in which several expert opinions are taken into account by a single model. For the soft expert-argument approximate function with intuitionistic fuzzy setting, concepts such as set inclusion, (α , v) -convexity(concave) sets, strongly (α , v) -convexity (concave) sets, strictly (α , v) -convexity (concave) sets, convex hull, and convex cone are conceived in this paper. Some set-theoretic inequalities are established with generalized properties and results on the basis of these specified notions. Additionally, by using a theoretic cum analytical approach, various elements of computational geometry, such as convex hull and convex cone, are theorized and some pertinent results are generalized. [ABSTRACT FROM AUTHOR]

In this paper, random vectors following the multivariate generalized hyperbolic (GH) distribution are compared using the hessian stochastic order. This family includes the classes of symmetric and asymmetric distributions by which different behaviors of kurtosis in skewed and heavy tail data can be captured. By considering some closed convex cones and their duals, we derive some necessary and sufficient conditions for some important applied stochastic orderings. The linear convex orderings are shown to be equivalent with a certain kind of hessian orderings. Based on copulas generated by the GH distributions, it is revealed that the ordering of the GH distributions in terms of their dependence structures corresponds to some hessian stochastic orderings being satisfied. The results are shown to be relevant to some insurance and economic applications.

In this paper, we establish second-order sufficient KKT optimality conditions of a set-valued fractional programming problem under second-order generalized cone convexity assumptions. We also prove duality results between the primal problem and second-order dual problems of parametric, Mond-Weir, Wolfe, and mixed types via the notion of second-order contingent epiderivative.

We define inclusive scattering matrix in the framework of a geometric approach to quantum field theory. We review the definitions of scattering theory in the algebraic approach and relate them to the definitions in the geometric approach. [ABSTRACT FROM AUTHOR]

In this paper, we establish two new versions of generalized set-valued Ekeland variational principles related to the so-called null set in a hyperspace ordered by a convex cone. Moreover, we propose some interval versions of Ekeland variational principles. As applications, the obtained theoretical results are applied to prove a set-valued fixed point theorem, to solve a set-valued optimization problem, and to explore a set-valued noncooperative game. [ABSTRACT FROM AUTHOR]

In this paper, some stochastic comparison results are developed for the class of multivariate normal variance-mean mixture (NVM) distributions. These comparisons are done based on Hessian and increasing Hessian orderings as well as several of their special cases. Necessary and/or sufficient conditions of the orderings are provided simply based on a comparison of the underlying model parameters. Furthermore, some linear stochastic orderings are expressed as equivalent to some multivariate orderings. The adequacy region of correlation-based ordering for the dependence structure ordering is extended from multivariate normal to NVM family by expressing supermodular order as equivalent to pairwise correlation order in NVM distributions. Some interpretations and applications of the results to actuarial science, finance and reliability are also provided. The results are finally illustrated with two real data sets. [ABSTRACT FROM AUTHOR]

We consider the simplest optimal control problem with one nonregular mixed constraint G(x, u) ≤ 0, i.e., such a constraint that the gradient Gu(x, u) can vanish on the surface G = 0. Using the Dubovitskii-Milyutin theorem on the approximate separation of convex cones, we prove a first order necessary condition for a weak minimum in the form of the so-called local minimum principle, which is formulated in terms of functions of bounded variation, integrable functions, and Lebesgue-Stieltjes measures and does not use functionals from (L∞)*. Two illustrative examples are provided. The work is based on the book by Milyutin [Maximum Principle in the General Problem of Optimal Control, Fizmatlit, Moscow, 2001 (in Russian)]. [ABSTRACT FROM AUTHOR]

In this paper, we establish second-order KKT conditions of a set-valued optimization problem and study second-order Mond-Weir, Wolfe, and mixed types duals with the help of second-order contingent epiderivative and second-order generalized cone convexity assumptions.

Within the inquiry about work, we explore a parametric set-valued optimization problem, where the objective as well as constraint maps are set-valued. A generalization of cone arcwise associated set-valued maps is presented named ρ-cone arcwise connectedness of set-valued maps. We set up adequate Karush–Kuhn–Tucker optimality conditions for the problem beneath contingent epiderivative and ρ-cone arcwise connectedness presumptions. Assist, Mond–Weir, Wolfe, and blended sorts duality models are examined. We demonstrate the related theorems between the primal and the comparing dual problems beneath the presumption. [ABSTRACT FROM AUTHOR]

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic geometry can be expressed through these conic intrinsic volumes. A list of such quantities includes internal and external solid angles of regular simplices and crosspolytopes, the probability that a (symmetric) Gaussian random polytope or the Gaussian zonotope contains a given point, the expected number of faces of the intersection of a regular polytope with a random linear subspace passing through its centre, and the expected number of faces of the projection of a regular polytope onto a random linear subspace. [ABSTRACT FROM AUTHOR]

This paper is devoted to provide sufficient Karush Kuhn Tucker (in short, KKT) conditions of optimality of second-order for a set-valued fractional minimax problem. In addition, we define duals of the types Mond-Weir and Wolfe of second-order for the problem. Further we obtain the theorems of duality under contingent epi-derivative together with generalized cone convexity suppositions of second-order. [ABSTRACT FROM AUTHOR]

In this paper, we consider a set-valued minimax fractional programming problem (MFP), where the objective as well as constraint maps are set-valued. We introduce the notion of ρ-cone arcwise connectedness of set-valued maps as a generalization of cone arcwise connected set-valued maps. We establish the sufficient Karush-Kuhn-Tucker (KKT) conditions for the existence of minimizers of the problem (MFP) under ρ-cone arcwise connectedness assumption. Further, we study the Mond-Weir (MWD), Wolfe (WD), and mixed (MD) types of duality models and prove the corresponding weak, strong, and converse duality theorems between the primal (MFP) and the corresponding dual problems under ρ-cone arcwise connectedness assumption. [ABSTRACT FROM AUTHOR]

Let F be an ordered topological vector space (over R ) whose positive cone F + is weakly closed, and let E ⊆ F be a subspace. We prove that the set of positive continuous linear functionals on E that can be extended (positively and continuously) to F is weak- ∗ dense in the topological dual wedge E + ′ . Furthermore, we show that this result cannot be generalized to arbitrary positive operators, even in finite-dimensional spaces. [ABSTRACT FROM AUTHOR]

In this note we show the anti-isotonicity of a certain operator function induced by the eigenvalue map λ(·) on the space of Hermitian matrices. In consequence, we obtain some majorization refinements of the Ky Fan's eigenvalue inequality. Thus we extend Maligranda's inequalities from a norm to the eigenvalue map, and from the usual order on ℝ to the majorization preorder on ℝn. [ABSTRACT FROM AUTHOR]

It is naturally useful to compare two normally or elliptically symmetric distributed random vectors by some stochastic orders. Stochastic orderings of multivariate normal random vectors have been carried out by Mosler (1984), Scarsini (1998), Müller (2001) and Arlotto and Scarsini (2009) among others. Landsman and Tsanakas (2006) studied conditions under which bivariate elliptical distributions are classified. Recently, Pan et al. (2016) provided more general results for the multivariate elliptical distributions. There are numerous situations where the assumption of normality or the property of elliptically symmetric is violated in terms of the fluctuations exhibited in the data histogram. Several examples of such cases have been presented by Genton (2004). Due to the deficiency of the elliptical distributions in capturing the skewness arisen out of any data set in practice, consideration of more general flexible distributions is thought to be impressive. The multivariate normal distribution holds at the disposal of the huge class of the mean-variance mixtures of multivariate normal distribution in the case when the mixing distribution is allowed to be a special one. The Generalized Hyperbolic (GH) family of distributions, introduced by Barndorff-Nielsen and Blaesild (1981), comes up with the Normal mean-variance mixture distribution as a particular case. Recently, the GH distributions have been applied by many researchers to fit financial data; see, for example, Rydberg (1999), Bibby and Sorensen (2003) and Cont and Tankov (2004). Material and methods In this paper, using some hessian stochastic orderings, we compare random vectors having generalized hyperbolic distributions. By considering the random vectors from the generalized hyperbolic distributions, we derive necessary and/or sufficient conditions for their comparisons on the basis of the hessian stochastic orderings. By virtue of some concepts established on the basis of closed convex cones and their duals, a number of necessary and sufficient conditions for the convex, the supermodular, the directionally convex, the componentwise convex, the copositive and the completely positive stochastic orderings are acquired. Results and discussion It is verified that the linear convex order and the positive linear convex order correspond with some kinds of hessian orderings. The linear orderings are considered to be appropriate because of the substantial reduction in the dimension they make as to their multivariate counterparts. The results can be potentially applied to the Normal Mean-variance mixture family of distributions. In a special case, stochastic orders for the symmetric GH and the scale mixture of the multivariate normal (SMN) distributions are characterized. It is noteworthy that some of the achieved results related to the SMN family are new as they have not been attained by Pan et al. (2016) for the elliptical families. In terms of copulas induced by the GH family, dependence orderings of GH distributions with fixed marginal distributions are fulfilled by the hessian stochastic orderings. Interpretations of the results in a number of insurance and economic applications have been rendered. We characterize the riskiness of portfolios by ordering the average of their correlations, in the sense of the stop-loss order. The elliptical distributions do not inspire such property as the frequency curves of components exhibit skewed trends. Dhaene and Goovaerts (1996) showed that riskiness of a portfolio consisting of two positive random variables increases, in the sense of the stop-loss order when two risks become more correlated. Landsman and Tsanakas (2006) derived an analogues result for the bivariate elliptical distributions. The results undertaken can be useful for slightly broader audiences as the considered the family of distributions entertain arbitrary values in their domain, the dimension, the skewness and the kurtosis. Conclusion The following conclusions were drawn from this research. • The positive linear convex order and the supermodular order are equivalent. • The riskiness of a generalized hyperbolic distributed portfolio, in the sense of the stop-loss order, increases (decreases) as the average of correlations of portfolio increases (decreases) and vice versa. • In terms of copulas generated by the GH family, dependence orderings among GH distributions with fixed marginal are characterized by the hessian stochastic orderings. • By considering two random bundles of commodities sharing the GH-generated copula signalized by a common unknown price vector, the bundle with less correlated components is preferred, as a result of the less riskiness it brings into account. [ABSTRACT FROM AUTHOR]

We develop interactive algorithms to find a strict total order for a set of discrete alternatives for two different value functions: linear and quasiconcave. The algorithms first construct a preference matrix and then find a strict total order. Based on the ordering, they select a meaningful pair of alternatives to present the decision maker (DM) for comparison. We employ methods to find all implied preferences of the DM, after he or she makes a preference. Considering all the preferences of the DM, the preference matrix is updated and a new strict total order is obtained until the termination conditions are met. We test the algorithms on several instances. The algorithms show fast convergence to the exact total order for both value functions, and eliciting preference information progressively proves to be efficient. [ABSTRACT FROM AUTHOR]

We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA 1 and NA 1 may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs. [ABSTRACT FROM AUTHOR]

Let R be a commutative noetherian ring. Let H (R) be the quotient of the Grothendieck group of finitely generated R-modules by the subgroup generated by pseudo-zero modules. Suppose that the R -vector space H (R) R = H (R) ⊗ Z R has finite dimension. Let C (R) (resp. C r (R) ) be the convex cone in H (R) R spanned by maximal Cohen–Macaulay R-modules (resp. maximal Cohen–Macaulay R-modules of rank r). We explore the interior, closure and boundary, and convex polyhedral subcones of C (R) . We provide various equivalent conditions for R to have only finitely many rank r maximal Cohen–Macaulay points in C r (R) in terms of topological properties of C r (R) . Finally, we consider maximal Cohen–Macaulay modules of rank one as elements of the divisor class group Cl (R) . [ABSTRACT FROM AUTHOR]

Applying basic facts of linear algebra, we present new simpler and much shorter proofs of results presented by Cherednik in the paper [The non-negative basis of a lattice, Diskret. Mat. 26(3) 2014, 127-135]. Recall, Cherednik proved that each lattice of dimension n in the linear space Rn has a basis consisting non-negative vectors, (i.e., vectors which contain only non-negative coordinates). Applying this theorem, he also showed that an arbitrary (not necessarily of the maximal dimension) lattice has such a basis if and only if it is generated by all its non-negative vectors. Next, these results are generalized for arbitrary convex cones (note that the set of all non-negative vectors is a convex cone). Finally, he showed that each lattice of dimension n ≥ 2 in Rn has a basis in any translation of every convex cone of dimension n. [ABSTRACT FROM AUTHOR]

This paper develops a uniformly valid and asymptotically nonconservative test based on projection for a class of shape restrictions. The key insight we exploit is that these restrictions form convex cones, a simple and yet elegant structure that has been barely harnessed in the literature. Based on a monotonicity property afforded by such a geometric structure, we construct a bootstrap procedure that, unlike many studies in nonstandard settings, dispenses with estimation of local parameter spaces, and the critical values are obtained in a way as simple as computing the test statistic. Moreover, by appealing to strong approximations, our framework accommodates nonparametric regression models as well as distributional/density‐related and structural settings. Since the test entails a tuning parameter (due to the nonstandard nature of the problem), we propose a data‐driven choice and prove its validity. Monte Carlo simulations confirm that our test works well. [ABSTRACT FROM AUTHOR]

Minkowski's classical existence theorem provides necessary and sufficient conditions for a Borel measure on the unit sphere of Euclidean space to be the surface area measure of a convex body. The solution is unique up to a translation. We deal with corresponding questions for unbounded convex sets, whose behavior at infinity is determined by a given closed convex cone. We provide an existence theorem and a stability result. [ABSTRACT FROM AUTHOR]

In this paper, we present the duality theory for general weighted space of vector functions. We mention that a characterization of the dual of a weighted space of vector functions in the particular case V ϲ C+(X) is mentioned by J. B. Prolla in [6]. Also, we extend de Branges lemma in this new setting for convex cones of a weighted spaces of vector functions (Theorem 4.2). Using this theorem, we find various approximations results for weighted spaces of vector functions: Theorems 4.2-4.6 as well as Corollary 4.3. We mention also that a brief version of this paper, in the particular case V ϲ C+(X), is presented in [3], Chapter 2, subparagraph 2.5. [ABSTRACT FROM AUTHOR]