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Generalizing a formula of Stanley, we prove combinatorially that the probability that $1, 2, \dots, k$ are contained in the same cycle of a product of two random $n$-cycles is \[\frac{1}{k} + \frac{4 (-1)^n}{ \binom{2k}{k}}…
In this study, we obtain some new integral inequalities for different classes of convex functions by using some elementary inequalities and classical inequalities like general Cauchy inequality and Minkowski inequality.
Similarity metric which is not positive definite, and present a general theorem which provides a large family of similarity metrics which are positive definite.
We estimate the expected number of limit cycles situated in a neighbourhood of the origin of a planar polynomial vector field. Our main tool is a distributional inequality for the number of zeros of some families of univariate holomorphic…
In this note we consider inequalities involving the error function $\phi$. Our methodes give new proofs of some known inequalities of Komatsu, and of Szarek and Werner, and also produce two families of inequalities that give upper and lower…
Problems in additive number theory related to sum and difference sets, more general binary linear forms, and representation functions of additive bases for the integers and nonnegative integers.
Let $$ T(q)=\sum_{k=1}^\infty d(k) q^k, \quad |q|<1, $$ where $d(k)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we proved…
In this paper we propose a method for proving some exponential inequalities based on power series expansion and analysis of derivations of the corresponding functions. Our approach provides a simple proof and generates a new class of…
Presentation of set matrices and demonstration of their efficiency as a tool using the path/cycle problem.
We prove the cyclic sum formulas for certain two-parameter multiple series. These are new and non-trivial generalizations of the cyclic sum formulas for multiple zeta values and multiple zeta-star values.
The aim of this work is to expose some asymptotic series associated to some expressions involving the volume of the n-dimensional unit ball. All proofs and the methods used for improving the classical inequalities announced in the final…
Equivalencies of many basic elementary inequalities are given
We give a historical perspective on the role of the cyclic category in the development of cyclic theory. This involves a continuous interplay between the extension in characteristic one and in S-algebras, of the traditional development of…
We establish a set of relations between several quite diverse types of weighted inequalities involving various integral operators and fairly general quasinorm-like functionals which we call sub-monotone. The main result enables one to solve…
For an analytic function f(z)=z+\sum_{n=2}^\infty a_n z^n satisfying the inequality \sum_{n=2}^\infty n(n-1)|a_n|\leq \beta, sharp bound on $\beta$ is determined so that $f$ is either starlike or convex of order $\alpha$. Several other…
We prove the main rules of causal calculus (also called do-calculus) for i/o structural causal models (ioSCMs), a generalization of a recently proposed general class of non-/linear structural causal models that allow for cycles, latent…
A {\em cyclic graph} is a graph with at each vertex a cyclic order of the edges incident with it specified. We characterize which real-valued functions on the collection of cubic cyclic graphs are partition functions of a real vertex model…
In this paper, we prove Newton-Maclaurin type inequalities for functions obtained by linear combination of two neighboring primary symmetry functions, which is a generalization of the classical Newton-Maclaurin inequality.
Matrix inequalities that extend certain scalar ones have been at the center of numerous researchers' attention. In this article, we explore the celebrated subadditive inequality for matrices via concave functions and present a reversed…
We study the triangle inequalities for angles (with different definitions) and present inequalities concerning the entries of correlation matrices through the positivity of $3\times 3$ matrices. We extend our discussions to the inequalities…