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Many classical planning frameworks are built on first-order languages. The first-order expressive power is desirable for compactly representing actions via schemas, and for specifying quantified conditions such as $\neg\exists…
A "numerical set-expression" is a term specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. If these operations are confined to the usual Boolean operations together with the result of…
From the integration of non-symmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. We show that functions characterizing…
In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…
Using appropriate notation systems for proofs, cut-reduction can often be rendered feasible on these notations, and explicit bounds can be given. Developing a suitable notation system for Bounded Arithmetic, and applying these bounds, all…
This paper continues the author's previous work on a limit-free algebraic-geometric construction of the derivative in the class of polynomial functions and extends the proposed framework to elementary functions. Derivatives of rational…
Entire functions in one complex variable are extremely relevant in several areas ranging from the study of convolution equations to special functions. An analog of entire functions in the quaternionic setting can be defined in the slice…
We employ tools from complex analysis to construct the $*$-logarithm of a quaternionic slice regular function. Our approach enables us to achieve three main objectives: we compute the monodromy associated with the $*$-exponential; we…
We contribute to the refined understanding of the language-logic-algebra interplay in the context of first-order properties of countable words. We establish decidable algebraic characterizations of one variable fragment of FO as well as…
An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving $-\infty$ and/or $+\infty$, so-called residuations. Based on this,…
Holomorphic functions of exponential type on a complex Lie group $G$ (introduced by Akbarov) form a locally convex algebra, which is denoted by $\cO_{exp}(G)$. Our aim is to describe the structure of $\cO_{exp}(G)$ in the case when $G$ is…
We investigate continuous functions definable in a definably complete uniformly locally o-minimal expansion of the second kind of a densely linearly ordered abelian group (DCULOAS structure). We prove a variant of the Arzela-Ascoli theorem…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
We examine the class of languages that can be defined entirely in terms of provability in an extension of the sorted type theory (Ty_n) by embedding the logic of phonologies, without introduction of special types for syntactic entities.…
Let the randomized query complexity of a relation for error probability $\epsilon$ be denoted by $R_\epsilon(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow…
The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations.…
Through a cascade of generalizations, we develop a theory of motivic integration which works uniformly in all non-archimedean local fields of characteristic zero, overcoming some of the difficulties related to ramification and small residue…
In our previous work, we have constructed explicit smooth real algebraic functions which may have both compact and non-compact preimages on smooth real algebraic manifolds. This paper presents its variant. Our result is new in obtaining…
Let $R$ be a ring, let $G$ be an amenable group and let $R\ast G$ be a crossed product. The goal of this paper is to construct, starting with a suitable additive function $L$ on the category of left modules over $R$, an additive function on…
We study a class of Landau-de Gennes energy functionals in the asymptotic regime of small elastic constant $L>0$. We revisit and sharpen the results in [18] on the convergence to the limit Oseen-Frank functional. We examine how the…