An analogical study of wave equations,physical quantities,conservation and reciprocity equations between electromagnetic and elastic waves

This paper presents an analogical study between electromagnetic and elastic wave fields,with a one-to-one correspondence principle established regarding the basic wave equations,the physical quantities and the differential operations.Using the electromagnetic-to-elastic substitution,the analogous relations of the conservation laws of energy and momentum are investigated between these two physical fields.Moreover,the energy-based and momentum-based reciprocity theorems for an elastic wave are also derived in the time-harmonic state,which describe the interaction between two elastic wave systems from the perspectives of energy and momentum,respectively.The theoretical results obtained in this analysis can not only improve our understanding of the similarities of these two linear systems,but also find potential applications in relevant fields such as medical imaging,non-destructive evaluation,acoustic microscopy,seismology and exploratory geophysics.

Magnetic Nonreciprocity in a Hybrid Device of Asymmetric Artificial Spin-Ice-Superconductors

Controlling the size and distribution of potential barriers within a medium of interacting particles can unveil unique collective behaviors and innovative functionalities.We introduce a unique superconducting hybrid device using a novel artificial spin ice structure composed of asymmetric nanomagnets.This structure forms a distinctive superconducting pinning potential that steers unconventional motion of superconducting vortices,thereby inducing a magnetic nonreciprocal effect,in contrast to the electric nonreciprocal effect commonly observed in superconducting diodes.Furthermore,the polarity of the magnetic nonreciprocity is in situ reversible through the tunable magnetic patterns of artificial spin ice.Our findings demonstrate that artificial spin ice not only precisely modulates superconducting characteristics but also opens the door to novel functionalities,offering a groundbreaking paradigm for superconducting electronics.

Dilation,discrimination and Uhlmann's theorem of link products of quantum channels

We establish the Stinespring dilation theorem of the link product of quantum channels in two different ways,discuss the discrimination of quantum channels,and show that the distinguishability can be improved by self-linking each quantum channel n times as n grows.We also find that the maximum value of Uhlmann's theorem can be achieved for diagonal channels.

New Asymptotic Results on Fermat-Wiles Theorem

We analyse the Diophantine equation of Fermat xp yp = zp with p > 2 a prime, x, y, z positive nonzero integers. We consider the hypothetical solution (a, b, c) of previous equation. We use Fermat main divisors, Diophantine remainders of (a, b, c), an asymptotic approach based on Balzano Weierstrass Analysis Theorem as tools. We construct convergent infinite sequences and establish asymptotic results including the following surprising one. If z y = 1 then there exists a tight bound N such that, for all prime exponents p > N , we have xp yp zp.

Diophantine equations and Fermat's last theorem for multivariate(skew-)polynomials

Fermat’s Last Theorem is a famous theorem in number theory which is difficult to prove.However,it is known that the version of polynomials with one variable of Fermat’s Last Theorem over C can be proved very concisely.The aim of this paper is to study the similar problems about Fermat’s Last Theorem for multivariate(skew)-polynomials with any characteristic.

AAK Theorem and a Design of Multidimensional BIBO Stable Filters

Rational approximation theory occupies a significant place in signal processing and systems theory. This research paper proposes an optimal design of BIBO stable multidimensional Infinite Impulse Response filters with a realizable (rational) transfer function thanks to the Adamjan, Arov and Krein (AAK) theorem. It is well known that the one dimensional AAK results give the best approximation of a polynomial as a rational function in the Hankel semi norm. We suppose that the Hankel matrix associated to the transfer function has a finite rank.

Some convergence theorems of fuzzy concave integral on fuzzyσ-algebra

In this paper,we consider the extension of the concave integral from classical crispσ-algebra to fuzzyσ-algebra of fuzzy sets.Firstly,the concept of fuzzy concave integral on a fuzzy set is introduced.Secondly,some important properties of such integral are discussed.Finally,various kinds of convergence theorems of a sequence of fuzzy concave integrals are proved.

Small Modular Solutions to Fermat’s Last Theorem

The proof by Andrew Wiles of Fermat’s Last Theorem in 1995 resolved the existence question for non-trivial solutions in integers x,y,zto the equation xn+yn=znfor n>2. There are none. Surprisingly, there are infinitely many solutions if the problem is recast in terms of modular arithmetic. Over a hundred years ago Issai Schur was able to show that for any n there is always a sufficiently large prime p0such that for all primes p≥p0the congruence xn+yn≡zn(modp)has a non-trivial solution. Schur’s argument wasnon-constructive, and there is no systematic method available at present to construct specific examples for small primes. We offer a simple method for constructing all possible solutions to a large class of congruences of this type.

Reciprocity Relation between Alternative Gravity Formulas

We compare Newton’s force law of universal gravitation with a corrected simple approach based on Bhandari’s recently presented work, where the gravitation constant G is maintained. A reciprocity relation exists between both alternative gravity formulas with respect to the distances between mass centers. We conclude a one-to-one mapping of the two gravitational formulas. We don’t need Einstein’s construct of spacetime bending by matter.

Whole Perfect Vectors and Fermat’s Last Theorem

A naïve discussion of Fermat’s last theorem conundrum is described. The present theorem’s proof is grounded on the well-known properties of sums of powers of the sine and cosine functions, the Minkowski norm definition, and some vector-specific structures.

Vanishing Theorems for p-Harmonic Forms on Submanifolds in Spheres

In this paper,we present some vanishing theorems for p-harmonic forms on-super stable complete submanifold M immersed in sphere Sn+m.When 2≤1≤n-2,M has a flat normal bundle.Assuming that M is a minimal submanifold andδ>1(n-1)p2/4n[p-1+(p-1)2kp],we prove a vanishing theorem for p-harmonicℓ-forms.

A Study on High School Mathematics Teaching Design Based on Teaching for Robust Understanding: Taking the Cosine Theorem as an Example

Teachers’teaching behavior plays a crucial role in students’development,and there are problems in the current teaching behavior of mathematics teachers such as ignoring students’cognitive needs,lack of equal opportunities for students’classroom performance as well as lack of formative evaluation of students.In order to solve the phenomenon,this paper analyzes and explains how to promote teaching based on the Teaching for Robust Understanding(TRU)evaluation framework with the goal of focusing on the development of all students,taking the teaching design of The Cosine Theorem as an example,and provides ideas and methods for first-line high school mathematics teachers.

The Njiki’s Fundamental Theorem-Definition on Fractions in the Mathematical Set ℚ and by Extension in ℝ and ℂ, for the Purpose of Leading to the Construction of Some Algebraic Structures as Its Theoretical Applications and for the Practical Ones

The purpose of the research in the NJIKI’s fundamental THEOREM-DEFINITION on fractions in the mathematical set ℚand by extension in ℝand ℂand in order to construct some algebraic structures is about the proved EXISTENCE and the DEFINITION by NJIKI of two INNOVATIVE, IMPORTANT and TEACHABLE operations of addition or additive operations, in ℚ, marked ⊕and +α,β, and taken as VECTORIAL, TRIANGULAR, of THREE or PROPORTIONAL operations and in order to make THEM not be different from the RATIONAL ONE, +, but to bring much more and new information on fractions, and, by extension in ℝand ℂ. And the very NJIKI’s fundamental THEOREM-DEFINITION having many APPLICATIONS in the everyday life of the HUMAN BEINGS and without talking about computer sciences, henceforth being supplied with very interesting new ALGORITHMS. And as for the work done in the research, it will be waiting for its extension to be done after publication and along with the research results concerned.

Solution of scattering from rough surface with a 2D target above it by a hybrid method based on the reciprocity theorem and the forward-backward method

This paper proposes a hybrid method based on the forward-backward method （FBM） and the reciprocity theorem （RT） for evaluating the scattering field from dielectric rough surface with a 2D target above it. Here, the equivalent electric/magnetic current densities on the rough surface as well as the scattering field from it are numerically calculated by FBM, and the scattered field from the isolated target is obtained utilizing the method of moments （MOM）. Meanwhile, the rescattered coupling interactions between the target and the surface are evaluated employing the combination of FBM and RT. Our hybrid method is first validated by available MOM results. Then, the functional dependences of bistatic and monostatic scattering from the target above rough surface upon the target altitude, incident and scattering angles are numerically simulated and discussed. This study presents a numerical description for the scattering mechanism associated with rescattered coupling interactions between a target and an underlying randomly rough surface.

Uniqueness,reciprocity theorem,and plane waves in thermoelastic diffusion with a fractional order derivative

In this work, a theory of thermoelasticity with diffusion is taken into consideration by using the methodology of fractional calculus. The governing equations for particle motion in a homogeneous anisotropic fractional order generalized thermoelastic diffusive medium are presented. Uniqueness and reciprocity theorems are proved. The plane wave propagation in the homogeneous transversely isotropic thermoelastic diffusive medium with fractional order derivative is studied. For the two-dimensional problem, there exist a quasi-longitudinal wave, a quasi-transverse wave, a quasi-mass diffusion wave, and a quasi-thermal wave. From the obtained results, the different characteristics of waves, like phase velocity, attenuation coefficient, specific loss, and penetration depth, are computed numerically and presented graphically. Some special cases are also discussed.

A Sound Field Separation and Reconstruction Technique Based on Reciprocity Theorem and Fourier Transform

We show a method to separate the sound field radiated by a signal source from the sound field radiated by noise sources and to reconstruct the sound field radiated by the signal source. The proposed method is based on reciprocity theorem and the Fourier transform. Both the sound field and its gradient on a measurement surface are needed in the method. Evanescent waves are considered in the method, which ensures a high resolution reconstruction in the near field region of the signal source when evanescent waves can be measured. A simulation is given to verify the method and the influence of measurement noise on the method is discussed.

Uniqueness theorem,theorem of reciprocity,and eigenvalue problems in linear theory of porous piezoelectricity

The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is stud- ied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special ease is also given.

A nonlinear creep model for surrounding rocks of tunnels based on kinetic energy theorem

The initiating condition for the accelerated creep of rocks has caused difficulty in analyzing the whole creep process.Moreover,the existing Nishihara model has evident shortcomings in describing the accelerated creep characteristics of the viscoplastic stage from the perspective of internal energy to analyze the mechanism of rock creep failure and determine the threshold of accelerated creep initiation.Based on the kinetic energy theorem,Perzyna viscoplastic theory,and the Nishihara model,a unified creep constitutive model that can describe the whole process of decaying creep,stable creep,and accelerated creep is established.Results reveal that the energy consumption and creep damage in the process of creep loading mainly come from the internal energy changes of geotechnical materials.The established creep model can not only describe the viscoelasticeplastic creep characteristics of rock,but also reflect the relationship between rock energy and creep deformation change.In addition,the research results provide a new method for determining the critical point of creep deformation and a new idea for studying the creep model and creep mechanical properties.

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics.The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given.The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given.The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail,and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.

DISTORTION THEOREMS FOR CLASSES OF g-PARAMETRIC STARLIKE MAPPINGS OF REAL ORDER IN C^(n)

In this paper,we define the class S_(g)^(BX)of g-parametric starlike mappings of real order γ on the unit ball BX in a complex Banach space X,where g is analytic and satisfies certain conditions.By establishing the distortion theorem of the Fr´echet-derivative type of S_(g)^(BX)with a weak restrictive condition,we further obtain the distortion results of the Jacobi-determinant type and the Fr´echet-derivative type for the corresponding classes(compared with S_(g)^(BX))defined on the unit polydisc(resp.unit ball with the arbitrary norm)in the space of n-dimensional complex variables,n≥2.Our results extend the classic distortion theorem of holomorphic functions from the case in one-dimensional complex space to the case in the higher dimensional complex space.The main theorems also generalize and improve some recent works.