Related papers: The `Real' Schwarz Lemma
We develop a finite-dimensional, symmetric matrix framework associated with the Riemann zeta function for complex arguments s with Real(s) unequal 1/2.
We prove and test an efficient series representation for the European Black-Scholes call, which generalizes and refines previously known approximations, and works in every market configuration.
Zalcman's Lemma makes significant applications in normal families, complex dynamics and related problems in complex analysis. In the present paper, we are devoted to generalizing the classical Zalcman's lemma to complex Lie groups by means…
The converse of Fortin's Lemma in Banach spaces is established in this Note.
We generalize Dirichlet's diophantine approximation theorem to approximating any real number $\alpha$ by a sum of two rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2}$ with denominators $1 \leq q_1, q_2 \leq N$. This turns out to be…
We survey the classical results of the Dirichlet Approximation Theorem.
It is known that almost all approaches to quantum gravity produce a logarithmic correction term to the entropy of a black hole, but the exact coefficient of such a term varies between the different approach to quantum gravity. Such…
Schur's inequality states that the sum of three special terms is always nonnegative. This note is a short review of inequalities for the sum of the reciprocals of these terms and of extensions of the latter inequalities to an arbitrary…
The direct and inverse spectral problems are solved for a wide subclass of the class of Schwarz matrices. A connection between the Schwarz matrices and the so-called generalized Hurwitz polynomials is found. The known results due to H. Wall…
We examine versions of the classical inequalities of Paley and Zygmund for functions of several variables. A sharp multiplier inclusion theorem and variants on the real line are obtained.
The aim of this paper is twofold. The first is to give a quantitative version of Schmidt's subspace theorem for arbitrary families of higher degree polynomials. The second is to give a generalization of the subspace theorem for arbitrary…
This set of notes re-proves known results on weighted automata (over a field, also known as multiplicity automata). The text offers a unified view on theorems and proofs that have appeared in the literature over decades and were written in…
The aim of this short note is to present an elementary, self-contained, and direct proof for the classical Lebesgue decomposition theorem.
In this note, we derive a Leibniz rule for difference quotient.
Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…
We prove a polynomial bound in the "true complexity" problem of Gowers and Wolf. The proof uses only repeated applications of the Cauchy--Schwarz inequality, answering negatively a question posed by Gowers and Wolf. To choose and reason…
The purpose of this paper is to generalize the classical Mazur's lemma from the classical convex analysis to the framework of locally $L^0$-convex modules. In this version an extra condition of countable concatenation is included. We…
This document is both a synthesis of current notions about complex systems, and a practical approach description. A disambiguation is proposed and exposes possible reasons for controversies related to causation and emergence. Theoretical…
Present notes can be viewed as an attempt to extend the notion of Schubert/Grothendieck polynomial to the context of an arbitrary algebraic oriented cohomology theory and, hence, of a commutative one-dimensional formal group law.
In this paper, we will generalize the definition of partially random or complex reals, and then show the duality of random and complex, i.e., a generalized version of Levin-Schnorr's theorem. We also study randomness from the view point of…