Re-Formulation of Mean King’s Problem Using Shannon’s Entropy

Mean King’s problem is formulated as a retrodiction problem among noncommutative observables. In this paper, we reformulate Mean King’s problem using Shannon’s entropy as a first step of introducing quantum uncertainty relation with delayed classical information. As a result, we give informational and statistical meanings to the estimation on Mean King problem. As its application, we give an alternative proof of nonexistence of solutions of Mean King’s problem for qubit system without using entanglement.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part III. An Original Solution of Hilbert’s Fourth Problem

This article refers to the “Mathematics of Harmony” by Alexey Stakhov [1], a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries—New Geometric Theory of Phyl-lotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci -Goniometry ( is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scien-tific ideas—The “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—The “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s Fourth Problem—Part II. A New Geometric Theory of Phyllotaxis (Bodnar’s Geometry)

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discoveries–New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry (λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas-the “golden mean,” which had been introduced by Euclid in his Elements, and its generalization—the “metallic means,” which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

用修正的King-Werner方法求解奇异问题

本文利用Hilbert空间几何特征,修正了King-Werner方法来求解奇异问题.在几乎不增加计算量的前提下,使其渐近收敛速率由原来的0.430 2提高到0.300 8.

Hyperbolic Fibonacci and Lucas Functions, “Golden” Fibonacci Goniometry, Bodnar’s Geometry, and Hilbert’s——Part I. Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci Goniometry

This article refers to the “Mathematics of Harmony” by Alexey Stakhov in 2009, a new interdisciplinary direction of modern science. The main goal of the article is to describe two modern scientific discove-ries—New Geometric Theory of Phyllotaxis (Bodnar’s Geometry) and Hilbert’s Fourth Problem based on the Hyperbolic Fibonacci and Lucas Functions and “Golden” Fibonacci λ-Goniometry ( λ > 0 is a given positive real number). Although these discoveries refer to different areas of science (mathematics and theoretical botany), however they are based on one and the same scientific ideas—the “golden mean”, which had been introduced by Euclid in his Elements, and its generalization—the “metallic means”, which have been studied recently by Argentinian mathematician Vera Spinadel. The article is a confirmation of interdisciplinary character of the “Mathematics of Harmony”, which originates from Euclid’s Elements.

A Procedure for the Squaring of a Circle (of Any Radius)

This paper presents a graphical procedure for the squaring of a circle of any radius. This procedure, which is based on a novel application of the involute profile, when applied to a circle of arbitrary radius (using only an unmarked ruler and a compass), produced a square equal in area to the given circle, which is 50 cm2. This result was a clear demonstration that not only is the construction valid for the squaring of a circle of any radius, but it is also capable of achieving absolute results (independent of the number pi (π), in a finite number of steps), when carried out with precision.

A Method for the Squaring of a Circle

This paper presents a Method for the squaring of a circle (i.e., constructing a square having an area equal to that of a given circle). The construction, when applied to a given circle having an area of 12.7 cm2, it produced a square having an area of 12.7 cm2, using only an unmarked ruler and a compass. This result was a clear demonstration that not only is the construction valid for the squaring of a circle but also for achieving absolute results (independent of the number pi (π) and in a finite number of steps) when carried out with precision.

Recent progress on the Dirichlet divisor problem and the mean square of the Riemann zeta-function

Let Δ(x) and E(t) denote respectively the remainder terms in the Dirichlet divisor problem and the mean square formula for the Riemann zeta-function on the critical line.This article is a survey of recent developments on the research of these famous error terms in number theory.These include upper bounds,Ω-results,sign changes,moments and distribution,etc.A few open problems are also discussed.

宋玉《对楚王问》唱和描写研究

文章讨论宋玉《对楚王问》中的关于楚歌唱和的问题。通过对其音乐的研究,文章认为:其演唱形式是一人领唱,众人相和;其歌曲性质属于先秦楚国民歌;从《下里》《巴人》到《阳春》《白雪》,曲调由低而高,唱者与和者先在嗓音音高方面进行了比较,而“引商刻羽”的转调行腔描写,则是两者演唱技巧的比较。通过对其内容的分析,文章认为:宋玉的楚歌唱和描写的喻义在于,说明自己既有极高的天赋,又有深厚的后天修为,而所谓的“遗行”,就是自己的“瑰意琦行”的人生追求,不能被士民众庶所理解,所接受。

多元平均值不等式的建立和Mitrinovi'c问题2的解决

本文构造了几个重要平均ｎ元情况的统一形式，利用函数的单调性建立系列ｎ元平均不等式，最后顺便解决了Ｍｉｔｒｉｎｏｖｉ＇ｃ问题２．

曲面上的曲率在理论物理中的一些应用

In this survey article,we present two applications of surface curvatures in theoretical physics.The first application arises from biophysics in the study of the shape of cell vesicles involving the minimization of a mean curvature type energy called the Helfrich bending energy.In this formalism,the equilibrium shape of a cell vesicle may present itself in a rich variety of geometric and topological characteristics.We first show that there is an obstruction,arising from the spontaneous curvature,to the existence of a minimizer of the Helfrich energy over the set of embedded ring tori.We then propose a scale-invariant anisotropic bending energy,which extends the Canham energy,and show that it possesses a unique toroidal energy minimizer,up to rescaling,in all parameter regime.Furthermore,we establish some genus-dependent topological lower and upper bounds,which are known to be lacking with the Helfrich energy,for the proposed energy.We also present the shape equation in our context,which extends the Helfrich shape equation.The second application arises from astrophysics in the search for a mechanism for matter accretion in the early universe in the context of cosmic strings.In this formalism,gravitation may simply be stored over a two-surface so that the Einstein tensor is given in terms of the Gauss curvature of the surface which relates itself directly to the Hamiltonian energy density of the matter sector.This setting provides a lucid exhibition of the interplay of the underlying geometry,matter energy,and topological characterization of the system.In both areas of applications,we encounter highly challenging nonlinear partial differential equation problems.We demonstrate that studies on these equations help us to gain understanding of the theoretical physics problems considered.

浅析反证法在物理学中的应用

浅析反证法在物理学中的应用杜娟“反证法”是一种科学的论证方法，在数学中常见其应用，若把它巧妙、灵活地运用于物理学中，将会使问题大为简化，且便于理解。特别是对于一些结论、性质的证明、判断等运用起来更加方便。下面笔者就结合几个具体实例来加以讨论：一、结论...

On a conjecture of Chowla and Walum

We show the asymptotic behaviour of the mean square of the sum n c√x naPk(x/n),where Pk(x) = Bk({x}) and Bk(x) denotes the Bernoulli polynomial of degree k and c 】 0 is a real number such that c2 is rational.Our result implies that a conjecture of Chowla and Walum is true on average.