On an Analog of the Plan’s Formula

In this paper we obtain an analog of the Plan’s formula, which plays an essential role in obtaining a functional relation for classical Riemann zeta-function.We provide examples of rational functions that satisfy a certain symmetry condition and admit a Maclaurin series expansion with coefficients equal to zero or one.     

Stanley’s Formula for Characters of the Symmetric Group

In his paper [9], Stanley finds a nice combinatorial formula for characters of irreducible representations of the symmetric group of rectangular shape. Then, in [10], he gives a conjectural generalisation for any shape. Here, we will prove this formula using shifted Schur functions and Jucys-Murphy elements.     

Euler’s Exponential Formula for Semigroups

The aim of this paper is to show that Eulers exponential formula $\lim_{n\rightarrow\infty}\linebreak[4] (I-tA/n)^{-n}x = e^{tA}x$, well known for $C_0$ semigroups in a Banach space $X\ni x$, can be used for semigroups not of class $C_0$, the sense of the convergence being related to the regularity of the semigroup for $t>0$. Although the strong convergence does not hold in general for not strongly continuous semigroups, an integrated version is stated for once integrated semigroups. Furthermore by replacing the initial topology on $X$ by some (coarser) locally convex topology $\tau$, the strong $\tau$-convergence takes place provided the semigroup is strongly $\tau$-continuous; in particular this applies to the class of bi-continuous semigroups. More generally if a $k$-times integrated semigroup $S(t)$ in a Banach space $X$ is strongly $k$-times $\tau$-differentiable, then Eulers formula holds in this topology with limit $S^{(k)}(t)$. On the other hand, for bounded holomorphic semigroups not necessarily of class $C_0$, Eulers formula is shown to hold in operator norm, with the error bound estimate ${\cal O}(\ln n/n)$, uniformly in $t>0$. All these results also concern degenerate semigroups.     

Pizzetti’s Formula for H-type Groups

Let ℍ be a H-type group. We provide a generalization of Pizzetti’s Formula for an orthogonal sub-Laplacian Δ_ℍ on ℍ. A formula expressing the k-th power of the operator Δ_ℍ is also proved. These results improve those contained in a former paper by the author, Bonfiglioli (Potential Anal 17:165–180, 2002).     

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.     

On Rayleigh’s Formula for the First Dirichlet Eigenvalue of a Radial Perturbation of a Ball

A formula going back to Rayleigh asserts that the first eigenvalue of the Dirichlet Laplacian on a perturbation of the disc in ?² is approximately the same as the one for the disc with the average radius of the perturbed disc. We prove this formula for a wide class of radial perturbations of the ball in ? ^m , and we obtain an estimate for the remainder.     

On Wilking’s criterion for the Ricci flow

Wilking has recently shown that one can associate a Ricci flow invariant cone of curvature operators $C(S)$ , which are nonnegative in a suitable sense, to every $Ad_{SO(n,\mathbb{C })}$ invariant subset $S \subset \mathbf{so}(n,\mathbb{C })$ . In this article we show that if $S$ is an $Ad_{SO(n,\mathbb{C })}$ invariant subset of $\mathbf{so}(n,\mathbb{C })$ such that $S\cup \{0\}$ is closed and $C_+(S)\subset C(S)$ denotes the cone of curvature operators which are positive in the appropriate sense then one of the two possibilities holds: (a) The connected sum of any two Riemannian manifolds with curvature operators in $C_+(S)$ also admits a metric with curvature operator in $C_+(S)$ (b) The normalized Ricci flow on any compact Riemannian manifold $M$ with curvature operator in $C_+(S)$ converges to a metric of constant positive sectional curvature. We also point out that if $S$ is an arbitrary $Ad_{SO(n,\mathbb{C })}$ subset, then $C(S)$ is contained in the cone of curvature operators with nonnegative isotropic curvature.     

On the Fedosov-Hörmander Formula for Differential Operators

The Fedosov-H?rmander formula gives the Fredholm index of some pseudodifferential operators of order 0 on L ². It is well known that it can be used to calculate the index of elliptic systems under the assumption that, among other things, the coefficients are smooth and their partial derivatives of all orders satisfy specific asymptotic conditions at infinity. We prove that the formula remains valid when the coefficients are only and bounded and have vanishing oscillation at infinity. In turn, this generalization is used to obtain a nonstandard invariance property of the index as well as various sufficient conditions for the index to be 0, when the coefficients are merely continuous and bounded with vanishing oscillation.      

On the Scope of Averaging for Frankl’s Conjecture

Let be a union-closed family of subsets of an m-element set A. Let . For b ∈ A let w(b) denote the number of sets in containing b minus the number of sets in not containing b. Frankl’s conjecture from 1979, also known as the union-closed sets conjecture, states that there exists an element b ∈ A with w(b) ≥ 0. The present paper deals with the average of the w(b), computed over all b ∈ A. is said to satisfy the averaged Frankl’s property if this average is non-negative. Although this much stronger property does not hold for all union-closed families, the first author (Czédli, J Comb Theory, Ser A, 2008) verified the averaged Frankl’s property whenever n ≥ 2 ^m  − 2 ^m/2 and m ≥ 3. The main result of this paper shows that (1) we cannot replace 2 ^m/2 with the upper integer part of 2 ^m /3, and (2) if Frankl’s conjecture is true (at least for m-element base sets) and then the averaged Frankl’s property holds (i.e., 2 ^m/2 can be replaced with the lower integer part of 2 ^m /3). The proof combines elementary facts from combinatorics and lattice theory. The paper is self-contained, and the reader is assumed to be familiar neither with lattices nor with combinatorics. This research was partially supported by the NFSR of Hungary (OTKA), grant no. T 049433, T 48809 and K 60148.     

On the semilocal convergence behavior for Halley’s method

The present paper is concerned with the semilocal convergence problems of Halley’s method for solving nonlinear operator equation in Banach space. Under some so-called majorant conditions, a new semilocal convergence analysis for Halley’s method is presented. This analysis enables us to drop out the assumption of existence of a second root for the majorizing function, but still guarantee Q-cubic convergence rate. Moreover, a new error estimate based on a directional derivative of the twice derivative of the majorizing function is also obtained. This analysis also allows us to obtain two important special cases about the convergence results based on the premises of Kantorovich and Smale types.     

Analogs of Schmidt’s Formula for Polyorthogonal Polynomials of the First Type

Mathematical Notes - Given any system of Laurent-type power series, a criterion for the uniqueness of polyorthogonal polynomials of first type associated with this system is stated and proved, and...     

On waring’s problem for cubes

Proceedings - Mathematical Sciences -     

On the Exceptional Set for Goldbach’s Problemin Short Intervals

 Let θ be a constant satisfying . We prove that there exists , such that the number of even integers in the interval which cannot be written as a sum of two primes is . References (Received 15 May 2000; in revised form 11 October 2000)     

On the Ostrowski’s inequality for Riemann-Stieltjes integral and applications

An Ostrowski type integral inequality for the Riemann-Stieltjes integral ∫_a^b ƒ (t) du (t), where ƒ is assumed to be of bounded variation on [a,b] andu is ofr-H- H?lder type on the same interval, is given. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.     

On Ramanujan’s large argument formula for the Gamma function

The aim of this paper is to improve some approximation formulas of Ramanujan type discussed by E.A. Karatsuba [J. Comput. Appl. Math. 135 (2001), 225–240].     

On Huppert’s Conjecture for the Monster and Baby Monster

Let G be a finite group. Let cd(G) be the set of all complex irreducible character degrees of G. In this paper, we will show that if cd(G)?=?cd(H), where H is the Monster or the Baby Monster simple sporadic groups, then ${G\cong H\times A,}$ where A is an abelian group.     

On the Exceptional Set for Goldbach’s Problemin Short Intervals

 Let θ be a constant satisfying . We prove that there exists , such that the number of even integers in the interval which cannot be written as a sum of two primes is . References     

Chambers’s Formula for the Graphene and the Hou Model with Kagome Periodicity and Applications

The aim of this article is to prove that for the graphene model like for a model considered by the physicist Hou on a kagome lattice, there exists a formula which is similar to the one obtained by Chambers for the Harper model in the case of the rational flux. As an application, we propose a semi-classical analysis of the spectrum of the Hou butterfly near a flat band.