For a Banach space X, the Lidskii trace property is equivalent to the nest approximation property; that is, for every nuclear operator on X that has summable eigenvalues, the trace of the operator is equal to the sum of the eigenvalues if and only if for every nest N of closed subspaces of X, there is a net of finite rank operators on X, each of which leaves invariant all subspaces in N , that ...

Of course, if we take S to be the set of columns of a matrix, and the independent sets to be the ones that are linearly independent, we get a first example, called a linear matroid. Another important class consists of the graphic ones – here S is the set of edges of a graph G and a subset is independent if it forms a forest. Whitney established some equivalent versions of the axioms, highlighte...

This paper addresses the general problem of modelling and learning rank data with ties. We propose a probabilistic generative model, that models the process as permutations over partitions. This results in super-exponential combinatorial state space with unknown numbers of partitions and unknown ordering among them. We approach the problem from the discrete choice theory, where subsets are chos...

Selecting the best items in a dataset is common task data exploration. However, concept of “best” lies eyes beholder: Different users may consider different attributes more important and, hence, arrive at rankings. Nevertheless, one can remove “dominated” and create “representative” subset data, comprising “best items” it. A Pareto-optimal representative guaranteed to contain item each possible...

This paper formalizes de Finetti’s book-making principle as a static individual preference condition. It thus avoids the confounding strategic and dynamic effects of modern formulations that consider games with sequential moves between a bookmaker and a bettor. This paper next shows that the book-making principle, commonly used to justify additive subjective probabilities, can be modified to ag...

We investigate the existence of perfect homogeneous sets for analytic colorings. An analytic coloring of X is an analytic subset of [X] , where N > 1 is a natural number. We define an absolute rank function on trees representing analytic colorings, which gives an upper bound for possible cardinalities of homogeneous sets and which decides whether there exists a perfect homogeneous set. We const...

This paper introduces combinatorial representations, which generalise the notion of linear representations of matroids. We show that any family of subsets of the same cardinality has a combinatorial representation via matrices. We then prove that any graph is representable over all alphabets of size larger than some number depending on the graph. We also provide a characterisation of families r...

Multi-subtype tumor diagnosis based on gene expression profiles is promising in clinical medicine application. Therefore, a great deal of research on tumor classification based on gene expression profiles has been developed, where various machine learning approaches were applied to constructing the best tumor classification model to improve the classification performance as much as possible. To...

Introduction: Quantitative evaluation is recommended to improve diagnostic ability and serial assessment of dopamine transporter (DAT) density scans. We decided to compare the ordered subsets expectation-maximization (OSEM) with filtered back-projection (FBP), and to investigate the impact of different iteration and cut-off frequencies on SBR values. Methods</stro...

In this paper we study the structure of standard Einstein solvmanifolds of arbitrary rank. Also the validity of a variational method for finding standard Einstein solvmanifolds is proved.