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It is a classical observation that lacunary function systems exhibit many properties which are typical for systems of independent random variables. However, it had already been observed by Erd\H{o}s and Fortet in the 1950s that probability…
Non-classical probability (along with its underlying logic) is a defining feature of quantum mechanics. A formulation that incorporates them, inherently and directly, would promise a unified description of seemingly different prescriptions…
Equilibrium statistics of Hamiltonian systems is correctly described by the microcanonical ensemble, whereas canonical ones fail in the most interesting, mostly inhomogeneous, situations like phase separations or away from the thermodynamic…
We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative…
It is known that the nonnegativity of Li coefficients is a necessary and sufficient condition for the Riemann hypothesis. We show that it is a necessary and sufficient condition for the Riemann hypothesis that all Li coefficients are norms…
The Green-Tao Theorem, one of the most celebrated theorems in modern number theory, states that there exist arbitrarily long arithmetic progressions of prime numbers. In a related but different direction, a recent theorem of Shiu proves…
Abstract numeration systems encode natural numbers using radix ordered words of an infinite regular language and linear recurrence sequences play a key role in their valuation. Sequence automata, which are deterministic finite automata with…
Despite their popularity, many questions about the algebraic constraints imposed by linear structural equation models remain open problems. For causal discovery, two of these problems are especially important: the enumeration of the…
We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for canonical $U$-statistics of arbitrary order $d$, extending the previously known results for $d=2$. The nasc's are expressed as growth conditions…
We study existence of random elements with partially specified distributions. The technique relies on the existence of a positive extension for linear functionals accompanied by additional conditions that ensure the regularity of the…
A rigorous geometric proof of the Lie's Theorem on nonlinear superposition rules for solutions of non-autonomous ordinary differential equations is given filling in all the gaps present in the existing literature. The proof is based on an…
A classical question in combinatorial number theory asks whether an equation has a solution inside a particular subset of its domain. The Rado's Theorem gives a set of necessary and sufficient conditions for a systems of linear equations to…
We provide an axiomatic system modeling conditional preference orders which is based on conditional set theory. Conditional numerical representations are introduced, and a conditional version of the theorems of Debreu on the existence of…
Axiomatizing mathematical structures and theories is an objective of Mathematical Logic. Some axiomatic systems are nowadays mere definitions, such as the axioms of Group Theory; but some systems are much deeper, such as the axioms of…
We propose to extend ``invertibility'' to ``regularity'' for categories in general abstract algebraic manner. Higher regularity conditions and ``semicommutative'' diagrams are introduced. Distinction between commutative and…
The transference principle of Green and Tao enabled various authors to transfer Szemer\'edi's theorem on long arithmetic progressions in dense sets to various sparse sets of integers, mostly sparse sets of primes. In this paper, we provide…
We establish a central limit theorem for the unnormalized linear statistic of the Gaussian Unitary Ensemble under optimal conditions: the linear statistics converges if and only if the expression for the limiting variance is finite.
Nonadditive composition rules for several physical quantities are treated in thermodynamics. It is argued that the zeroth law defines the existence of their additive forms, the formal logarithms. A further principle, the universal…
When regularity lemmas were first developed in the 1970s, they were described as results that promise a partition of any graph into a ``small'' number of parts, such that the graph looks ``similar'' to a random graph on its edge subsets…
Finsler's lemma is a classic mathematical result with applications in control and optimization. When the lemma is applied to parameter-dependent LMIs, as such those that arise from problems of robust stability, the extra variables…