In this paper, we revisit the classical semidiscrete heat equation using probabilistic techniques. Our main objective is to elucidate the probabilistic structures underlying its analytic properties and to establish explicit connections between the two frameworks. We analyze the moments, the moment generating function, the characteristic function, and cumulants of the stochastic process associated with the semidiscrete heat equation. A key component of our approach is the use of Bell polynomials, which provide the combinatorial structure necessary to express higher-order moments and cumulants. Moreover, we show that the distribution of the process is uniquely determined by its sequence of moments, a fact ensured by the Carleman condition.
In this work, we focus on the study of the components of semilevel sets of pseudoperiodic functions. Specifically, we provide partial proofs of a conjecture proposed by V.I. Arnol’d in 1991, concerning the boundedness of such components. In pursuit of a complete answer to this conjecture, we also present an estimate for the length of a path connecting points within components that satisfy a key property. This study can be considered a sequel to a work published by the author in 2024. Nevertheless, we have endeavored to make the text as self-contained as possible by maintaining consistent notation throughout.
This paper contains two theorems concerning the vanishing of natural transformations of (co)homology functors. Precisely, assume that R is an associative ring with identity and f: M→ N is a morphism of finitely presented right R-modules. The first theorem proves that the natural transformation ^1(f, -) vanishes over the category of all right R-modules if and only if _1(f, -) vanishes over the category of all left R-modules. As a corollary of this result, we establish that ^1(f, -) is epic if and only if _1(f, -) is monic. The second theorem shows that if R is left and right Noetherian and M, N are Gorenstein projective, then the natural transformations _1(f, -) , ^1(-, f) , and ^1(f, -) vanish over the category of all Gorenstein projective modules, simultaneously. This, in particular, yields that over Gorenstein projective modules, the notions of phantom morphisms and -phantom morphisms coincide. Also, it is proved that if R is n-Gorenstein, then for any integer i>n , the natural transformations ^i(f, -) , ^i(-, f) , and _i(f, -) vanish over all modules, simultaneously. As an interesting consequence, we show that under the same assumptions, ^i(-, f) is epic (resp. monic) if and only if ^i(f, -) is monic (resp. epic) if and only if _i(f, -) is epic (resp. monic).
In this article, we prove that for a strictly increasing sequence (a_n) of natural numbers, if the additive energy of {a_n:n≤ N} is less than N^3/(log N)^C for some C≥ 14.71, then ({a_nα}) has Poissonian pair correlation for almost all α∈ℝ. This provides a lower bound for the exponent C in the additive energy bound established by Bloom and Walker [4].
