Related papers: Rockafellar's Sum Theorem
We prove the boundedness of $n$-complements for surface pairs in a generalized case without restrictions on multiplicities or the Fano type assumption.
We exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of…
We consider the general Wigner function for a particle confined to a finite interval and subject to Dirichlet boundary conditions. We derive the boundary corrections to the "star-genvalue" equation and to the time evolution equation. These…
We study a new approach to the problem of transparent boundary conditions for the Helmholtz equation in unbounded domains. Our approach is based on the minimization of an integral functional arising from a volume integral formulation of the…
The most important open problem in Monotone Operator Theory concerns the maximal monotonicity of the sum of two maximally monotone operators provided that the classical Rockafellar's constraint qualification holds. In this paper, we…
We give a sharpened form of Siegel Lemma's w. r. t. the maximum norm. This implies a new lower bound on the greatest element of a sum-distinct set of positive integers (Erd\"os-Moser problem). The main tools are Minkowski's theorem on…
We solve an elementary number theory problem on sums of fractional parts, using methods from group theory. We apply our result to deduce the finiteness of certain monodromy representations.
We give a new, elementary proof of a key inequality used by Rudelson in the derivation of his well-known bound for random sums of rank-one operators. Our approach is based on Ahlswede and Winter's technique for proving operator Chernoff…
We provide a direct proof of Weyl's law for the buckling eigenvalues of the biharmonic operator on a wide class of domains of $\mathbb R^d$ including bounded Lipschitz domains. The proof relies on asymptotically sharp lower and upper bounds…
We propose a systematic method for analyzing Rohrlich-type divisor sums for arbitrary congruence subgroups $\Gamma_0(N)$. Our main theorem unifies various results from the literature, and its significance is illustrated through the…
We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/\varphi (a_{n}))^{s}$, where $\varphi$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some…
We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…
The Kuznecov sum formula, proved by Zelditch in the Riemannian setting, is an asymptotic sum formula $$N(\lambda) := \sum_{\lambda_j \leq \lambda} \left| \int_H e_j \, dV_H \right|^2 = C_{H,M} \lambda^{\operatorname{codim} H} +…
We give a linking theorem that strengthens and unifies some many minimax theorems including Ambrosetti-Rabinowitz ``mountain pass theorem'', Rabinowitz ``multidimensional mountain pass theorem'', Rabinowitz ``saddle point theorem'' and…
We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem.
We formalize the proof of Reingold's Theorem that SL=L [Rei05] in the theory of bounded arithmetic VL, which corresponds to ``logspace reasoning''. As a consequence, we get that VL=VSL, where VSL is the theory of bounded arithmetic for…
Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…
We establish two results regarding the query complexity of bounded-error randomized algorithms. * Bounded-error separation theorem. There exists a total function $f : \{0,1\}^n \to \{0,1\}$ whose $\epsilon$-error randomized query complexity…
Associated Legendre functions arise in many problems of mathematical physics. By using the generalized Abel-Plana formula, in this paper we derive a summation formula for the series over the zeros of the associated Legendre function of the…
We prove boundedness and inverse logarithmic decay in time of solutions to the Teukolsky equations on Schwarzschild-Anti-de Sitter backgrounds with standard boundary conditions originating from fixing the conformal class of the non-linear…