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Highly automated theorem provers like Dafny allow users to prove simple properties with little effort, making it easy to quickly sketch proofs. The drawback is that such provers leave users with little control about the proof search,…
A classical theorem of Wendroff shows that one may reconstructs a sequence of orthogonal polynomials on the real line from two non-constant polynomials of consecutive degrees whose zeros strictly interlace on the real line. In this note we…
A definition of a probabilistic automaton is formulated in which its prime decomposition follows as a direct consequence of Krohn-Rhodes theorem. We first characterize the local structure of probabilistic automata. The prime decomposition…
An integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built. The solutions from a wide class can be used as building blocks for the representation. Considerations are based on…
We give a modern account of Agafonov's original proof of his eponymous theorem. The original proof was only reported in Russian in a journal not widely available, and the work most commonly cited in western literature is instead the English…
Combining a standard proof search method, such as resolution or tableaux, and rewriting is a powerful way to cut off search space in automated theorem proving, but proving the completeness of such combined methods may be challenging. It may…
The article starts with generalizations of some classical results and new truncation error upper bounds in the sampling theorem for bandlimited stochastic processes. Then, it investigates $L_p([0,T])$ and uniform approximations of…
In a series of papers, we have shown that from the representatio theory of a compact groupoid one can reconstruct the groupoid using the procedure similar to the Tannaka-Krein duality for compact groups. In this part we study continuous…
We describe an inductive machinery to prove various properties of representations of a category equipped with a generic shift functor. Specifically, we show that if a property (P) of representations of the category behaves well under the…
This paper presents a comprehensive formalization of the von Neumann-Morgenstern (vNM) expected utility theorem using the Lean 4 interactive theorem prover. We implement the classical axioms of preference-completeness, transitivity,…
We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work [3].
Dynamical sampling deals with representations of a frame $\{ f_k \}_{k=1}^\infty$ as an orbit $\{ T^n \varphi \}_{n=0}^\infty$ of a linear and possibly bounded operator $T$ acting on the underlying Hilbert space. It is known that the desire…
We prove the constructive version of Birkhoff's ergodic theorem following Vyugin but trying to separate and state explicitly the combinatorial statement on which this proof is based. We pose some questions related to this statement (and the…
We consider the Deduction Theorem used in the literature of game theory to run a purported proof by contradiction. In the context of game theory, it is stated that if we have a proof of $\phi \vdash \varphi$, then we also have a proof of…
We present a comprehensive analysis of the convergence properties of the frame operators of Weyl-Heisenberg systems and shift-invariant systems, and relate these to the convergence of the Walnut representation. We give a deep analysis of…
We prove a fixed-point theorem that generalises and simplifies a number of results in the theory of $F$-contractions. We show that all of the previously imposed conditions on the operator can be either omitted or relaxed. Furthermore, our…
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…
Pecan is an automated theorem prover for reasoning about properties of Sturmian words, an important object in the field of combinatorics on words. It is capable of efficiently proving non-trivial mathematical theorems about all Sturmian…
Let \phi be a first order formula and M be a countable model. \phi^M denotes the set of all assignments that satisfy \phi in M. Let M, N be countable models. A formula \phi distinguishes these models if |\phi^M|\neq |\phi^N|. We show that…
Two new representations for Ramanujan's function $\sigma(q)$ are obtained. The proof of the first one uses the three-variable reciprocity theorem due to Soon-Yi Kang and a transformation due to R.P. Agarwal while that of the second uses the…