Related papers: An Introduction to Multisets
In the present paper, several properties concerning generalized derivatives of multifunctions implicitly defined by set-valued inclusions are studied by techniques of variational analysis. Set-valued inclusions are problems formalizing the…
Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and…
Hypergraphs are used to model higher-order interactions amongst agents and there exist many practically relevant instances of hypergraph datasets. To enable efficient processing of hypergraph-structured data, several hypergraph neural…
Every one knows that an equation is equivalent to a multivariate function. Generally speaking, there are more than one unknown x in this multivariate function and it is not easy to reduce the number of unknown x to one. In this paper we…
This is a tutorial introduction to the functional analysis mathematics needed in many physical problems, such as in waves in continuous media. Functional analysis takes us beyond finite matrices, allowing us to work with infinite sets of…
Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the…
Diversities are an extension of the concept of a metric space which assign a non-negative value to every finite set of points, rather than just pairs. A general theory of diversities has been developed which exhibits many deep analogies to…
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum…
Using multisets, we develop novel techniques for mechanizing the proofs of the synthesis conjectures for list-sorting algorithms, and we demonstrate them in the Theorema system. We use the classical principle of extracting the algorithm as…
A functional Menger $\cap$-algebra is a set of n-place functions containing n projections and closed under the so-called Menger's compositions of n-place functions and the set-theoretic intersection of functions. We give the abstract…
A finite subset $M \subset \mathbb{R}^d$ is basic, if for any function $f \colon M \to \mathbb{R}$ there exists a collection of functions $f_1, \ldots, f_d \colon \mathbb{R} \to \mathbb{R}$ such that for each element $(x_1, \ldots, x_d)\in…
The compactness phenomenon is one of the featured aspects of structuralism in mathematics. In simple and broad words, a compactness property holds in a structure if a related property is satisfied by sufficiently many substructures of that…
Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with…
We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In…
The decomposition into interaction subspaces is a hierarchical decomposition of the spaces of cylindrical functions of a finite product space, also called factor spaces. It is an important construction in graphical models and a standard way…
The set of answers to a query may be very large, potentially overwhelming users when presented with the entire set. In such cases, presenting only a small subset of the answers to the user may be preferable. A natural requirement for this…
Integral properties of multifunctions determined by vector valued functions are presented. Such multifunctions quite often serve as examples and counterexamples. In particular it can be observed that the properties of being integrable in…
The usual nonnegative modulus function is based on addition. A natural different modulus function on the set of positive reals is introduced. Arguments for results for series through the usual modulus function are transformed to arguments…
Choice functions constitute a simple, direct and very general mathematical framework for modelling choice under uncertainty. In particular, they are able to represent the set-valued choices that appear in imprecise-probabilistic decision…
We propose learning flexible but interpretable functions that aggregate a variable-length set of permutation-invariant feature vectors to predict a label. We use a deep lattice network model so we can architect the model structure to…