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This report defines (plain) Dag-like derivations in the purely implicational fragment of minimal logic $M_{\supset}$. Introduce the horizontal collapsing set of rules and the algorithm {\bf HC}. Explain why {\bf HC} can transform any…

The technique of symmetric extensions is derived from forcing and it is one of the most important tools for studying models without the Axiom of Choice. Despite being incredibly successful since the 1960s, our understanding of the technique…

We introduce a hierarchy of conditions necessarily satisfied by any distribution P(ab) representing the probabilities for two separate observers to obtain outcomes a and b when making local measurements on a shared quantum state. Each…

The apparent difficulty of efficient distribution-free PAC learning has led to a large body of work on distribution-specific learning. Distributional assumptions facilitate the design of efficient algorithms but also limit their reach and…

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in $\mathbb{R}^d$-the existence of $k$ mutually orthogonal…

For an integer $m\ge 2$, a partition $\lambda=(\lambda_1,\lambda_2,\ldots)$ is called $m$-falling, a notion introduced by Keith, if the least nonnegative residues mod $m$ of $\lambda_i$'s form a nonincreasing sequence. We extend a bijection…

In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function's non…

A simplicial complex of dimension $d-1$ is said to be balanced if its graph is $d$-colorable. Juhnke-Kubitzke and Murai proved an analogue of the generalized lower bound theorem for balanced simplicial polytopes. We establish a…

We propose Bell inequalities for discrete or continuous quantum systems which test the compatibility of quantum physics with an interpretation in terms of deterministic hidden-variable theories. The wave function collapse that occurs in a…

Lifting theorems are theorems that relate the query complexity of a function $f:\{0,1\}^{n}\to\{0,1\}$ to the communication complexity of the composed function $f \circ g^{n}$, for some "gadget" $g:\{0,1\}^{b}\times\{0,1\}^{b}\to\{0,1\}$.…

For a $k$-uniform hypergraph $H$, let $\delta_1(H)$ denote the minimum vertex degree of $H$, and $\nu(H)$ denote the size of the largest matching in $H$. In this paper, we show that for any $k\geq 3$ and $\beta>0$, there exists an integer…

New bounds on the number of similar or directly similar copies of a pattern within a finite subset of the line or the plane are proved. The number of equilateral triangles whose vertices all lie within an $n$-point subset of the plane is…

The Church-Rosser theorem in the type-free lambda-calculus is well investigated both for beta-equality and beta-reduction. We provide a new proof of the theorem for beta-equality with no use of parallel reductions, but simply with…

In the framework of computable queries in Database Theory, there are many examples of queries to (properties of) relational database instances that can be expressed by simple and elegant third order logic ($\mathrm{TO}$) formulae. In many…

We proved in a previous work that Cattani-Sassone's higher dimensional transition systems can be interpreted as a small-orthogonality class of a topological locally finitely presentable category of weak higher dimensional transition…

The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver and Harrington-Kechris-Louveau show that with respect to Borel reducibility, any…

The polynomial method by Beals, Buhrman, Cleve, Mosca, and de Wolf (FOCS 1998, J. ACM 2001), the adversary method by Ambainis (STOC 2000, J. Comput. Syst. Sci. 2002), and the compressed oracle method by Zhandry (CRYPTO 2019) have been shown…

In this paper, we study quantum query complexity of the following rather natural tripartite generalisations (in the spirit of the 3-sum problem) of the hidden shift and the set equality problems, which we call the 3-shift-sum and the…

Let $d \geq 3$ be a natural number. We show that for all finite, non-empty sets $A \subseteq \mathbb{R}^d$ that are not contained in a translate of a hyperplane, we have \[ |A-A| \geq (2d-2)|A| - O_d(|A|^{1- \delta}),\] where $\delta >0$ is…

Proving that there are problems in $\mathsf{P}^\mathsf{NP}$ that require boolean circuits of super-linear size is a major frontier in complexity theory. While such lower bounds are known for larger complexity classes, existing results only…