Generalizations of girstmair’s formulas

In 1994 and 1995 GIRSTMAIR gave (relative) class number formulas for the imaginary quadratic field $\mathbb{Q}(\sqrt { - p} )$ , P an odd prime with p ≡ 3 (mod 4) and p ≥ 7, using the coefficients of the digit expression of 1/p and z/p, respectively, where z is an integer with 1 ≤ z ≤p - 1. We extend the formulas to an imaginary abelian number field.     

内∑群

<正> 根据子群的特性来研究原群的性质无疑是群论的一个重要方面.我们考虑这样的问题:设已给群论性质∑,如果 G 的每一真子群都具有性质∑,那么群 G 若何呢?当∑为正常且 G 非交换时,我们已知 G 是 Hamilton 群.正常性是群的一种相对性质,即与包含它的群有关.现设∑是群的一个绝对性质,即一群之是否具有性质∑仅由该群本身所决定.具有性质∑的群简称∑群.     

Golusin–Krylov formulas in complex analysis

Abstract

This is a brief survey of a constructive technique of analytic continuation related to an explicit integral formula of Golusin and Krylov (1933). It goes far beyond complex analysis and applies to the Cauchy problem for elliptic partial differential equations as well. As started in the classical papers, the technique is elaborated in generalised Hardy spaces also called Hardy–Smirnov spaces.     

Integral formulas and the Ohsawa-Takegoshi extension theorem

We construct a semiexplicit integral representation of the canonical solution to the (?)-equation with respect to a plurisubharmonic weight function in a pseudoconvex domain. The construction is based on a construction related to the Ohsawa-Takegoshi extension theorem combined with a method to construct weighted integral representations due to M. Andersson.     

Effective formulas for the Carathéodory distance

For a class of Reinhardt domains we give formulas for the Carathéodory distance. As an application we discuss the product property of the Carathéodory distance when one factor domain is a Reinhardt domain of special type.     

Graphing formulas: Unraveling experts’ recognition processes

An instantly graphable formula (IGF) is a formula that a person can instantly visualize using a graph. These IGFs are personal and serve as building blocks for graphing formulas by hand. The questions addressed in this paper are what experts’ repertoires of IGFs are and what experts attend to while recognizing these formulas. Three tasks were designed and administered to five experts. The data analysis, which was based on Barsalou and Schwarz and Hershkowitz, showed that experts’ repertoires of IGFs could be described using function families that reflect the basic functions in secondary school curricula and revealed that experts’ recognition could be described in terms of prototype, attribute, and part-whole reasoning. We give suggestions for teaching graphing formulas to students.     

Approximation formulas for Landauʼs constants

In this paper, we establish approximation formulas for evaluating Landau?s constants.     

Euler–MacLaurin formulas via differential operators

Recently there has been a renewed interest in asymptotic Euler–MacLaurin formulas, because of their applications to spectral theory of differential operators. Using elementary means, we recover such formulas for compactly supported smooth functions on intervals, polygons, and three-dimensional polytopes, where the coefficients in the asymptotic expansion are sums of differential operators involving only derivatives of the function in directions normal to the faces of the polytope. Our formulas apply to wedges of any dimension.     

Effective formulas for the local ?ojasiewicz exponent

We give an effective formula for the local ?ojasiewicz exponent of a polynomial mapping. Moreover, we give an algorithm for computing the local dimension of an algebraic variety.     

关于∑sin~3A—∑cos~3A的下界

关于∑ｓｉｎ~３Ａ—∑ｃｏｓ~３Ａ的下界宁波大学数学系陈计陈宽宏在［１］中讨论了关于ＡＢＣ中界，他给出了如下结论：但未能给出（３）不平凡的下界．粗略探索之后，估计（３）在（Ａ，Ｂ，Ｃ）→（π，０，０）时取下确界－１；然而这并不成立：令看来，（３）的下确...     

Whitney’s formulas for curves on surfaces

The classical Whitney formula relates the number of times an oriented plane curve cuts itself to its rotation number and the index of a base point. In this paper we generalize Whitney’s formula to curves on an oriented punctured surface Σ _{m, n} , obtaining a family of identities indexed by elements of π ₁(Σ _{m, n} ). To define analogs of the rotation number and the index of a base point of a curve γ, we fix an arbitrary vector field on Σ _{m, n} . Similar formulas are obtained for non-based curves.     

关于“∑”和

“∑”和(是sum(和)的头一个字母S,是希腊字母,读作Sigma)是在通用教材初中数学课本第六册第七章开始引入的。它的概念、性质及其应用,在教材中都未作讨论。本文对此谈点看法,供参考。     

Riesz–Kantorovich formulas for operators on multi-wedged spaces

We introduce the notions of multi-suprema and multi-infima for vector spaces equipped with a collection of wedges, generalizing the notions of suprema and infima in ordered vector spaces. Multi-lattices are vector spaces that are closed under multi-suprema and multi-infima and are thus an abstraction of vector lattices. The Riesz decomposition property in the multi-wedged setting is also introduced, leading to Riesz–Kantorovich formulas for multi-suprema and multi-infima in certain spaces of operators.     

Effective formulas for the ?ojasiewicz exponent at infinity

We give effective formulas for the ?ojasiewicz exponent at infinity of an arbitrary complex polynomial mapping.     

Rademacher-type formulas for restricted partition and overpartition functions

A collection of Hardy-Ramanujan-Rademacher type formulas for restricted partition and overpartition functions is presented, framed by several biographical anecdotes.