Striking patterns in natural magic squares’ associated electrostatic potentials: Matrices of the 4th and 5th order

A magic square is a square matrix whereby the sum of any row, column, or any one of the two principal diagonals is equal. A surrogate of this abstract mathematical construct, introduced in 2012 by Fahimi and Jaleh, is the “electrostatic potential (ESP)” that results from treating the matrix elements of the magic square as electric charges. The overarching idea is to characterize patterns associated with these matrices that can possibly be used, in the future, in reverse to generate these squares. This study focuses on squares of order 4 and 5 with 880 and 275,305,224 distinct (irreducible/unique) realizations, respectively. It is shown that characteristic patterns emerge from plots of the ESPs of the matrices representing the studied squares. The electrostatic potentials for natural magic squares exhibit a striking pattern of maxima and minima in all distinct 880 of the 4th order and all distinct 275,305,224 of the 5th order matrices. The minimum values of ESP of Dudeney groups are discussed. Equipotential points and certain constants are found among the ESP sums along horizontal and vertical lines on the square lattice. These findings may help to open a new perspective regarding magic squares unsolved problems. While mathematics often leads discovery in physics, the latter (physics) is used here to detect otherwise invisible patterns in a mathematical object such as magic squares.     

Temporal asymptotics for the p’th power Newtonian fluid in one space dimension

The balance laws of mass, momentum and energy are considered for a pth power Newtonian fluid undergoing one dimensional longitudinal motions. For initial-boundary value problems involving fixed endpoints held at a prescribed temperature or insulated, we prove exponential convergence of solutions to equilibria for generic initial data. The estimates for different boundary conditions are presented in a unified manner by utilising the thermodynamic concept of availability.     

Temporal asymptotics for the p’th power Newtonian fluid in one space dimension

The balance laws of mass, momentum and energy are considered for a pth power Newtonian fluid undergoing one dimensional longitudinal motions. For initial-boundary value problems involving fixed endpoints held at a prescribed temperature or insulated, we prove exponential convergence of solutions to equilibria for generic initial data. The estimates for different boundary conditions are presented in a unified manner by utilising the thermodynamic concept of availability.     

Proceedings of the 9th Seminar “Programs and Algorithms of Numerical Mathematics”

Preface

Proceedings of the 9th Seminar Programs and Algorithms of Numerical Mathematics     

A specialissue dedicated to Professor LU QiKeng on the occasion of his 80th birthday

The Guest EditorS of the ISSUe afC Siu Yum一Tongy乞u StePhen 5.T. Liu Kefeng Zhou XiangyuA specialissue dedicated to Professor LU QiKeng on the occasion of his 80th birthday~~     

How do the positive embeddings of Banach lattices depend on the αth derivatives of K?

Let K and S be locally compact Hausdorff spaces and X be an abstract space. Suppose that T is a positive Banach lattice isomorphism from into . Then for each ordinal α the cardinalities of the αth derivatives and satisfy the following inequality Moreover, if then is a continuous image of a subset of which can be taken closed when K is compact. The first statement of this result for is a vector‐valued extension of a Cengiz's theorem and the second one is vector‐valued version of a Holsztyński's theorem. A simple example shows that the number is sharp in these vector‐valued theorems.     

Version Carleman du théorème de l'« edge of the wedge å

J.-P. Rosay has given a C^∞ version of “the edge of the wedge” theorem for non-smooth edges. We give now a “Carleman” version of this theorem, taking on one hand, an edge Rⁿ, and on the other hand, a C³ totally real submanifold of Cⁿ.     

Remarks on the rank properties of formal CR maps Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We prove several new transversality results for formal CR maps between formal real hypersurfaces in complex space. Both cases of finite and infinite type hypersurfaces are tackled in this note.     

Coefficient multipliers of mixed norm space in the ball Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0相似文献   

The Teichmuller distance on the space of spherical CR structures Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

We define the Teichmuller pseudodistance on the space of spherical CR structures on a fixed compact manifold by using quasiconformal mappings between spherical CR manifolds. The pseudodistance is shown to be a complete distance.     

A Note of a G.S. Petrov's Result About the Weadened 16th Hilbert Probelm

ＡＮｏｔｅｏｆａＧ．Ｓ．Ｐｅｔｒｏｖ’ｓＲｅｓｕｌｔＡｂｏｕｔｔｈｅＷｅａｋｅｎｅｄ１６ｔｈＨｉｌｂｅｒｔＰｒｏｂｌｅｍＬｉＢａｏｙｉ（李宝毅）（Ｄｅｐｔ．ｏｆＭａｔｈ．ＴｉａｎｊｉｎＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｔｉａｎｊｉｎ，３００１９３，Ｐ...     

Very weak solutions to the two-dimensional Monge-Ampére equation Dedicated to Professor Jean-Yves Chemin on the Occasion of His 60th Birthday

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with H?lder-continuous ?rst derivatives of exponent β 1/5. Our approach is based on combining the approach of Lewicka and Pakzad(2017) with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by De Lellis et al.(2018) for the isometric immersion problem.     

Is reliability a new science? A paper from the panel session held at the 10th International Conference on Mathematical Methods in Reliability

The 10th International Conference on Mathematical Methods in Reliability, MMR 2017, held in Grenoble, France during July 3‐7, entailed a panel discussion entitled “Is reliability a new science?” with Mark Brown, Regina Liu, William Meeker, Sheldon Ross, and Nozer Singpurwalla as panelists. Bill Meeker also doubled as a chair and as a moderator of the panel. The panel discussion was spawned by the recent appearance of a book by Professor Paolo Rocchi, Docent Emeritus of IBM, titled “Reliability is a new science: Gnedenko was right” published by Springer in 2017. The panel discussion was well attended and enthusiastically received and could serve as a forerunner to other such panel discussions at future MMR conferences. This paper presents some elements from the lively debate generated by this discussion.     

Laguerre expansion on the Heisenberg group and Fourier-Bessel transform on C~n Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

Given a principal value convolution on the Heisenberg group Hn = Cn×R, we study the relation between its Laguerre expansion and the Fourier-Bessel expansion of its limit on Cn. We also calculate the Dirichlet kernel for the Laguerre expansion on the group Hn.     

Sur la théorie des demi-groupes

Sans résumé Dédié à Monsieur S. Bays pour son soixante-dixième anniversaire     

Stochastic Hamiltonian flows with singular coefficients Dedicated to the 60th Birthday of Professor Michael R¨ockner

In this paper, we study the following stochastic Hamiltonian system in R~(2d)(a second order stochastic differential equation):dX_t = b(X_t,X_t)dt + σ(X_t,X_t)dW_t,(X_0,X_0) =(x, v) ∈ R~(2d),where b(x, v) : R~(2d)→ R~d and σ(x, v) : R~(2d)→ R~d ? R~d are two Borel measurable functions. We show that if σ is bounded and uniformly non-degenerate, and b ∈ H_p~(2/3,0) and ?σ∈ L~p for some p 2(2 d + 1), where H_p~(α,β)is the Bessel potential space with differentiability indices α in x and β in v, then the above stochastic equation admits a unique strong solution so that(x, v) → Z_t(x, v) :=(Xt,Xt)(x, v) forms a stochastic homeomorphism flow,and(x, v) → Z_t(x, v) is weakly differentiable with ess.sup_(x,v)E(sup_(t∈[0,T])|?Z_t(x, v)|~q) ∞ for all q ≥ 1 and T≥ 0. Moreover, we also show the uniqueness of probability measure-valued solutions for kinetic Fokker-Planck equations with rough coefficients by showing the well-posedness of the associated martingale problem and using the superposition principle established by Figalli(2008) and Trevisan(2016).     

Carpenter,tractors and microbes for the development of logical–mathematical thinking – the way 10th graders and pre-service teachers solve thinking challenges

The objective of this case study was to investigate the ability of 10th graders and pre-service teachers to solve logical–mathematical thinking challenges. The challenges do not require mathematical knowledge beyond that of primary school but rather an informed use of the problem representation. The percentage of correct answers given by the 10th graders was higher than that of the pre-service teachers. Unlike the 10th graders, some of whom used various strategies for representing the problem, most of the pre-service teachers’ answers were based on a technical algorithm, without using control processes. The obvious conclusion drawn from the findings supports and recommends expanding and enhancing the development of logical–mathematical thinking, both in specific lessons and as an integral part of other lessons in pre-service frameworks.