Arbitrary Norm Research Articles

On Some Properties of Superreflexive Besov Spaces

This paper contains results concerning superreflective Besov spaces $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ . Namely, expressions for convexity moduli and smoothness moduli with respect to the “canonical” norms are derived, and properties related to the finite representability of Banach spaces and linear compact operators in $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ are examined. Additionally, inequalities of the Prus–Smarzewski type for arbitrary equivalent norms and inequalities of the James–Gurariy type are presented. Based on the latter, two-sided estimates for the norms of elements in $$B_{{p,q}}^{s}({{\mathbb{R}}^{n}})$$ can be obtained in terms of the expansion coefficients of these elements in unconditional normalized Schauder bases.

FUZZY LOCATION PROBLEMS WITH TRIANGULAR NORM : EXISTENCE AND STABILITY

A fuzzy max-T location problem is considered. The fuzzy max-T location problem is a generalization of a fuzzy maximin location problem by allowing in the objective function arbitrary triangular norms instead of the triangular norm defined by the minimum operation. Then we give conditions for the existence of its optimal solutions, and derive a relationship between its optimal solutions and efficient solutions of a fuzzy multicriteria location problem. Furthermore, we give some properties of triangular norms, and for triangular norms, we investigate the stability of optimal solutions of the fuzzy max-T location problem. 1 Introduction and preliminaries In a general continuous location model, finitely many points called demand points in R n , modeling existing facilities or customers, are given. Let di ∈ R n , i =1 , 2, ··· , � (≥ 2) be demand points. We put I ≡{ 1, 2, ··· , � }. Then a problem to locate a new facility in R n is called a single facility location problem. If one prefers the location of the facility near demand points, then the problem is formulated as follows: (1) min x∈Rn f (γ1(x − d1) ,γ 2(x − d2), ··· ,γ � (x − d� ))

On curvature of surfaces immersed in normed spaces

The normal map given by Birkhoff orthogonality yields extensions of principal, Gaussian and mean curvatures to surfaces immersed in three-dimensional spaces whose geometry is given by an arbitrary norm and which are also called Minkowski spaces. The relations of this setting to the field of relative differential geometry are clarified. We obtain characterizations of the Minkowski Gaussian curvature in terms of surface areas, and respective generalizations of the classical theorems of Huber, Willmore, Alexandrov, and Bertrand–Diguet–Puiseux are derived. A generalization of Weyl’s formula for the volume of tubes and some estimates for volumes and areas in terms of curvature are obtained, and in addition we discuss also two-dimensional subcases of the results in more detail.

Social Conventions and Boundary Work in an Online Q&A: The Example of Vegetarianism and Veganism

The aim of this article is to explore the regulation of social conventions (more or less arbitrary norms of how one ought to act that are based on joint commitments) in online interaction. As the empirical case, the article scrutinises discussions about what it means to exercise vegetarianism or veganism. Drawing on ‘naturally occurring’ data from the question-and-answer platform Yahoo! Answers, conventions of vegetarianism and veganism are argued to be protected and upheld through symbolic boundaries, primarily in three forms: incompatibility, inauthenticity and noncommitment. Moreover, by utilising the platform’s system of thumbs-up and thumbs-down, it is argued that we are provided evidence of social approvals and sanctions in real time. The findings are suggested to have implications for continued theorising on conventions, studies of boundaries in the social sciences and analyses of social media as an arena for re-orientations of established sociological problems.

A note on norms of signed sums of vectors

Abstract Improving a result of Hajela, we show for every function f with lim n→∞ f(n) = ∞ and f(n) = o(n) that there exists n 0 = n 0(f) such that for every n ⩾ n 0 and any S ⊆ {–1, 1} n with cardinality |S| ⩽ 2 n/f(n) one can find orthonormal vectors x 1, …, xn ∈ ℝ n satisfying ∥ ε 1 x 1 + ⋯ + ε n x n ∥ ∞ ⩾ c log ⁡ f ( n ) $\begin{array}{} \displaystyle \|\varepsilon_1x_1+\dots+\varepsilon_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)} \end{array}$ for all (ε 1, …, εn ) ∈ S. We obtain analogous results in the case where x 1, …, xn are independent random points uniformly distributed in the Euclidean unit ball B 2 n $\begin{array}{} \displaystyle B_2^n \end{array}$ or in any symmetric convex body, and the ℓ ∞ n $\begin{array}{} \displaystyle \ell_{\infty }^n \end{array}$ -norm is replaced by an arbitrary norm on ℝ n .

CIAO⁎: MPC-based Safe Motion Planning in Predictable Dynamic Environments

Robots have been operating in dynamic environments and shared workspaces for decades. Most optimization based motion planning methods, however, do not consider the movement of other agents, e.g. humans or other robots, and therefore do not guarantee collision avoidance in such scenarios. This paper builds upon the Convex Inner ApprOximation (CIAO) method and proposes a motion planning algorithm that guarantees collision avoidance in predictable dynamic environments. Furthermore, it generalizes CIAO’s free region concept to arbitrary norms and proposes a cost function to approximate time optimal motion planning. The proposed method, CIAO*, finds kinodynamically feasible and collision free trajectories for constrained single body robots using model predictive control (MPC). It optimizes the motion of one agent and accounts for the predicted movement of surrounding agents and obstacles. The experimental evaluation shows that CIAO* reaches close to time optimal behavior.

ℓ q -Norm Sample-Adaptive Multiple Kernel Learning

Existing multiple kernel learning (MKL) algorithms indiscriminately apply the same set of kernel combination weights to all samples by pre-specifying a group of base kernels. Sample-adaptive MKL learning (SAMKL) overcomes this limitation by adaptively switching on/off the base kernels with respect to each sample. However, it restricts to solving MKL problems with pre-specified kernels. And, the formulation of existing SAMKL falls to an $\ell _{1}$ -norm MKL which is not flexible. To allow for robust kernel mixtures that generalize well in practical applications, we extend SAMKL to the arbitrary norm and apply it to image classification. In this paper, we formulate a closed-form solution for optimizing the kernel weights based on the equivalence between group-lasso and MKL, and derive an efficient $\ell _{q}$ -norm ( $q\geq 1$ and denoting the $\ell _{q}$ -norm of kernel weights) SAMKL algorithm. The cutting plane method is used to solve this margin maximization problem. Besides, we propose a framework for solving MKL problems in image classification. Experimental results on multiple data sets show the promising performance of the proposed solution compared with other competitive methods.

Approximation and Loewner theory of holomorphic covering mappings

We give conditions in order to approximate locally uniformly holomorphic covering mappings of the unit ball of $${{\mathbb {C}}^{n}}$$ with respect to an arbitrary norm, with entire holomorphic covering mappings. The results rely on a generalization of the Loewner theory for covering mappings which we develop in the paper.

Arbitrary norm semi-supervised extreme learning machine

Arbitrary norm semi-supervised extreme learning machine

The least distance between extremums and minimal period of solutions to autonomous vector differential equations

Solutions x(t) of the Lipschitz equation x = f(x) with an arbitrary vector norm are considered. It is proved that the sharp lower bound for the distances between successive extremums of xk(t) equals π/L where L is the Lipschitz constant. For non-constant periodic solutions, the lower bound for the periods is 2π/L. These estimates are achieved for norms that are invariant with respect to permutation of the indices.

Data-Dependent Generalization Bounds for Multi-Class Classification

In this paper, we study data-dependent generalization error bounds exhibiting a mild dependency on the number of classes, making them suitable for multi-class learning with a large number of label classes. The bounds generally hold for empirical multi-class risk minimization algorithms using an arbitrary norm as regularizer. Key to our analysis are new structural results for multi-class Gaussian complexities and empirical $\ell_\infty$-norm covering numbers, which exploit the Lipschitz continuity of the loss function with respect to the $\ell_2$- and $\ell_\infty$-norm, respectively. We establish data-dependent error bounds in terms of complexities of a linear function class defined on a finite set induced by training examples, for which we show tight lower and upper bounds. We apply the results to several prominent multi-class learning machines, exhibiting a tighter dependency on the number of classes than the state of the art. For instance, for the multi-class SVM by Crammer and Singer (2002), we obtain a data-dependent bound with a logarithmic dependency which significantly improves the previous square-root dependency. Experimental results are reported to verify the effectiveness of our theoretical findings.

Perturbations of Supinf Problems with Constraints

We investigate constrained supinf problems for functions of two variables. Conditions are given which assure that the objective function can be perturbed by continuous functions with arbitrary small norms in such a way that the supinf problem for the perturbed function has a solution. We also give a characterization of the notion of well-posedness for such problems.

Stability Aspects of Multicriteria Integer Linear Programming Problems

Under consideration are the multicriteria integer linear programming problems with finitely many feasible solutions. The problem itself consists in finding a set of extremal solutions. We derive some lower and upper bounds for the T1-stability radius under assumption that arbitrary Holder norms are given in the solution and criteria spaces. A class of the problems with an infinitely large stability radius is specified. We also consider the case of the multicriteria linear Boolean problem.

DICHOTOMY PROPERTY FOR MAXIMAL OPERATORS IN A NONDOUBLING SETTING

We investigate a dichotomy property for Hardy–Littlewood maximal operators, noncentred $M$ and centred $M^{c}$ , that was noticed by Bennett et al. [‘Weak- $L^{\infty }$ and BMO’, Ann. of Math. (2) 113 (1981), 601–611]. We illustrate the full spectrum of possible cases related to the occurrence or not of this property for $M$ and $M^{c}$ in the context of nondoubling metric measure spaces $(X,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D707})$ . In addition, if $X=\mathbb{R}^{d}$ , $d\geq 1$ , and $\unicode[STIX]{x1D70C}$ is the metric induced by an arbitrary norm on $\mathbb{R}^{d}$ , then we give the exact characterisation (in terms of $\unicode[STIX]{x1D707}$ ) of situations in which $M^{c}$ possesses the dichotomy property provided that $\unicode[STIX]{x1D707}$ satisfies some very mild assumptions.

Even arbitrary norms influence moral decision-making.

It is well known that individuals tend to copy behaviours that are common among other people-a phenomenon known as the descriptive norm effect1-3. This effect has been successfully used to encourage a range of real-world prosocial decisions4-7, such as increasing organ donor registrations8. However, it is still unclear why it occurs. Here, we show that people conform to social norms, even when they understand that the norms in question are arbitrary and do not reflect the actual preferences of other people. These results hold across multiple contexts and when controlling for confounds such as anchoring or mere-exposure effects. Moreover, we demonstrate that the degree to which participants conform to an arbitrary norm is determined by the degree to which they self-identify with the group that exhibits the norm. Two prominent explanations of norm adherence-the informational and social sanction accounts2,9-11-cannot explain these results, suggesting that these theories need to be supplemented by an additional mechanism that takes into account self-identity.

Sufficient Conditions for Triangular Norms Preserving ⊗-Convexity

The convexity in triangular norm (for short, ⊗−convexity) is a generalization of Zadeh’s quasiconvexity. The aggregation of two ⊗−convex sets is under the aggregation operator ⊗ is also ⊗−convex, but the aggregation operator ⊗ is not unique. To solve it in complexity, in the present paper, we give some sufficient conditions for aggregation operators preserve ⊗−convexity. In particular, when aggregation operators are triangular norms, we have that several results such as arbitrary triangular norm preserve ⊗ D − convexity and ⊗ a − convexity on bounded lattices, ⊗ M preserves ⊗ H − convexity in the real unite interval [ 0 , 1 ] .

Religion, Evolution, and the Basis of Institutions: The Institutional Cognition Model of Religion

Abstract Few outstanding questions in the human behavioral sciences are timelier or more urgently debated than the evolutionary source of religious behaviors and beliefs. Byproduct theorists locate the origins of religion in evolved cognitive defaults and transmission biases. Others have argued that cultural evolutionary processes integrated non-adaptive cognitive byproducts into coherent networks of supernatural beliefs and ritual that encouraged in-group cooperativeness, while adaptationist models assert that the cognitive and behavioral foundations of religion have been selected for at more basic levels. Here, we survey these differing approaches, noting their respective strengths and weaknesses. We then advance a novel model that centers on the ability of language to generate alternative worlds independent of immediate empirical facts, and thus highlight the similarities between religious belief and the modes of cognition that underlie institutions in general. The institutional cognition model of religion accounts for some of the shortcomings of extant approaches and draws attention to the human ability to create non-empirical worlds; that is, worlds that are imaginary. Both religious beliefs and institutional facts—such as jurisdictional borders—are non-empirical assertions, yet they are socially accepted as truths and reified through ritual and behavior. One type of non-empirical, linguistically generated belief—supernatural agent belief—is particularly effective for stabilizing systems of arbitrary norms by rooting them in deontic rather than utilitarian reasoning. The evolutionary roots and continued persistence of religion are thus functions of the capacity for humans to generate cognitive alternatives to empirical reality, and the need to stably coordinate those alternative conceptions.

THE ROSENTHAL–SZASZ INEQUALITY FOR NORMED PLANES

We study the classical Rosenthal–Szasz inequality for a plane whose geometry is determined by a norm. This inequality states that the bodies of constant width have the largest perimeter among all planar convex bodies of given diameter. In the case where the unit circle of the norm is given by a Radon curve, we obtain an inequality which is completely analogous to the Euclidean case. For arbitrary norms we obtain an upper bound for the perimeter calculated in the anti-norm, yielding an analogous characterisation of all curves of constant width. To derive these results, we use methods from the differential geometry of curves in normed planes.

A Characterization of Unitary-Antiunitary Similarity Transformations via Kubo–Ando Means

In this note, we characterize all those bijective transformations on the set of positive semidefinite matrices which preserve an arbitrary unitarily invariant norm of any given Kubo–Ando mean corresponding to an n-monotone strictly concave generating function. This result strengthens a former theorem of Molnar and Szokol.

Evolutionary Origins of Morality: Insights From Non-human Primates

The aim of this contribution is to explore the origins of moral behavior and its underlying moral preferences and intuitions from an evolutionary perspective. Such a perspective encompasses both the ultimate, adaptive function of morality in our own species, as well as the phylogenetic distribution of morality and its key elements across primates. First, with regard to the ultimate function, we argue that human moral preferences are best construed as adaptations to the affordances of the fundamentally interdependent hunter-gatherer lifestyle of our hominin ancestors. Second, with regard to the phylogenetic origin, we show that even though full-blown human morality is unique to humans, several of its key elements are not. Furthermore, a review of evidence from nonhuman primates regarding prosocial concern, conformity, and the potential presence of universal, biologically anchored and arbitrary cultural norms shows that these elements of morality are not distributed evenly across primate species. This suggests that they have evolved along separate evolutionary trajectories. In particular, the element of prosocial concern most likely evolved in the context of shared infant care, which can be found in humans and some New World monkeys. Strikingly, many if not all of the elements of morality found in nonhuman primates are only evident in individualistic or dyadic contexts, but not as third-party reactions by truly uninvolved bystanders. We discuss several potential explanations for the unique presence of a systematic third-party perspective in humans, but focus particularly on mentalizing ability and language. Whereas both play an important role in present day, full-blown human morality, it appears unlikely that they played a causal role for the original emergence of morality. Rather, we suggest that the most plausible scenario to date is that human morality emerged because our hominin ancestors, equipped on the one hand with large and powerful brains inherited from their ape-like ancestor, and on the other hand with strong prosocial concern as a result of cooperative breeding, could evolve into an ever more interdependent social niche.