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In this note, we show that for an ``$F$-crystal" (the equal characteristic analogue of $F$-crystals), its {\it isomorphism number} and its {\it level torsion} coincide. This confirms a conjure of Vasiu \cite{Va} in the equal characteristic…
Roman logarithmic binomial formula analogue has been found . It is presented here also for the case of fibonomial coefficients which recently have been given a combinatorial interpretation by the present author.
Thanks to recent results on ring homomorphisms of Azumaya algebras and to the following ones about endomorphisms of canonical Poisson algebras and Dirac quantum algebras, and about the reformulation in positive characteristic of these…
This paper presents a model for linguistic description based on group theory. A grammar in this model, or "G-grammar", is a collection of lexical expressions which are products of logical forms, phonological forms, and their inverses.…
Explaining decisions of black-box classifiers is both important and computationally challenging. In this paper, we scrutinize explainers that generate feature-based explanations from samples or datasets. We start by presenting a set of…
In what follows we improve an inequality related to matrix theory. T. Laffey proved (2013) a weaker form of this inequality [2].
We develop a Glivenko--Cantelli theory for monotone, almost additive functions of i.\,i.\,d.\ sequences of random variables indexed by~$\Z^d$. Under certain conditions on the random sequence, short range correlations are allowed as well. We…
We consider one problem that was largely left open by Rudolf Carnap in his work on inductive logic, the problem of analogical inference. After discussing some previous attempts to solve this problem, we propose a new solution that is based…
A $4^-$-power is a non-empty word of the form $XXXX^-$, where $X^-$ is obtained from $X$ by erasing the last letter. A binary word is called {\em faux-bonacci} if it contains no $4^-$-powers, and no factor 11. We show that faux-bonacci…
The paper presents a linguistic and computational model aiming at making the morphological structure of the lexicon emerge from the formal and semantic regularities of the words it contains. The model is word-based. The proposed…
We derive some identities and relations and extremal problems and minimization and Fourier development involving of integral Legendre polynomials.
Post-hoc explanations for black box models have been studied extensively in classification and regression settings. However, explanations for models that output similarity between two inputs have received comparatively lesser attention. In…
We confirm that the diagonal elements of the Gell-Mann and Hartle's decoherence decoherence functional are equal to the relative frequencies of the results of many identical experiments, when a set of alternative histories decoheres. We…
Based on ideas of quantum theory of open systems we propose the consistent approach to the formulation of logic of plausible propositions. To this end we associate with every plausible proposition diagonal matrix of its likelihood and…
An approach to induction is presented, based on the idea of analysing the context of a given problem into `circumstances'. This approach, fully Bayesian in form and meaning, provides a complement or in some cases an alternative to that…
Entropy is being used in physics, mathematics, informatics and in related areas to describe equilibration, dissipation, maximal probability states and optimal compression of information. The Gini index on the other hand is an established…
Li showed that the Riemann Hypothesis is equivalent to the nonnegativity of a certain sequence of numbers. Bombieri and Lagarias gave an arithmetic formula for the number sequence based on the Guinand-Weil explicit formula and showed that…
We study the complementary behaviors of external and internal examples in image restoration, and are motivated to formulate a composite dictionary design framework. The composite dictionary consists of the global part learned from external…
Alexia Auff\`eves and Philippe Grangier proposed to modify the quantum ontology by requiring that physical properties are attributed jointly to the system and to the context in which it is embedded. Comparing the formal structures of…
We initiate a systematic development of $F_N(a, b; t)$, a finite analogue of Fine's function $F(a, b; t)$. Our results are transformations between $F_N(a, b; t)$ and $F_N(aq^{\ell}, bq^{m}; tq^{n})$, where $\ell,m$ and $n$ take the values…