Search results for "predicate logic"

Extension maps in ultradifferentiable and ultraholomorphic function spaces

2000

A note on multiple summing operators and applications

2018

We prove a new result on multiple summing operators and, among other results and applications, we provide a new extension of Littlewood’s 4 / 3 inequality to m-linear forms.

Product of extension domains is still an extension domain

2018

We prove the product of the Sobolev-extension domains is still a Sobolev-extension domain.

Basic Measure Theory

2020

In this chapter, we lay the measure theoretic foundations of probability theory. We introduce the classes of sets (semirings, rings, algebras, σ-algebras) that allow for a systematic treatment of events and random observations. Using the measure extension theorem, we construct measures, in particular probability measures on σ-algebras. Finally, we define random variables as measurable maps and study the σ-algebras generated by certain maps.

Nonlocalization Properties of Time Operators Transformations

2014

It is presented a general approach to the problem of extension of time operators and the associated Lambda transformations on singular measures. It is also shown that Lambda transformations defined on function spaces having the Urysohn property are non localized. Particular attention has been devoted to time and Lambda operators associated with the Walsh-Paley system and to a characterization of their domain and non locality.

Dimension estimates for the boundary of planar Sobolev extension domains

2020

We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev $W^{1,p}$-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.

Itô calculus extended to systems driven by -stable Lévy white noises (a novel clip on the tails of Lévy motion)

2007

Abstract The paper deals with probabilistic characterization of the response of non-linear systems under α -stable Levy white noise input. It is shown that, by properly selecting a clip in the probability density function of the input, the moments of the increments of Levy motion process remain all of the same order ( d t ) , like the increments of the Compound Poisson process. It follows that the Ito calculus extended to Poissonian input, may also be used for α -stable Levy white noise input processes. It is also shown that, when the clip on the tails of the probability of the increments of the Levy motion approaches to infinity, the Einstein–Smoluchowsky equation is restored. Once these c…

Neural Networks with Multidimensional Cross-Entropy Loss Functions

2019

Deep neural networks have emerged as an effective machine learning tool successfully applied for many tasks, such as misinformation detection, natural language processing, image recognition, machine translation, etc. Neural networks are often applied to binary or multi-class classification problems. In these settings, cross-entropy is used as a loss function for neural network training. In this short note, we propose an extension of the concept of cross-entropy, referred to as multidimensional cross-entropy, and its application as a loss function for classification using neural networks. The presented computational experiments on a benchmark dataset suggest that the proposed approaches may …

Rotation topological factors of minimal $\mathbb {Z}^{d}$-actions on the Cantor set

2006

In this paper we study conditions under which a free minimal Z d -action on the Cantor set is a topological extension of the action of d rotations, either on the product T d of d 1-tori or on a single 1-torus T 1 . We extend the notion of linearly recurrent systems defined for Z-actions on the Cantor set to Z d -actions, and we derive in this more general setting a necessary and sufficient condition, which involves a natural combinatorial data associated with the action, allowing the existence of a rotation topological factor of one of these two types.

THE CAUCHY DUAL AND 2-ISOMETRIC LIFTINGS OF CONCAVE OPERATORS

2018

We present some 2-isometric lifting and extension results for Hilbert space concave operators. For a special class of concave operators we study their Cauchy dual operators and discuss conditions under which these operators are subnormal. In particular, the quasinormality of compressions of such operators is studied.