Asymptotics of Jack polynomials as the number of variables goes to infinity

A 高分子化学和高分子物理UNIT 1 What are Polymer?第一单元什么是高聚物？What are polymers? For one thing, they are complex and giant molecules and are different from low molecular weight compounds like, say, common salt. To contrast the difference, the molecular weight of common salt is only 58.5, while that of a polymer can be as high as several hundred thousand, even more than thousand thousands. These big molecules or ‘macro-molecules’are made up of much smaller molecules, can be of one or more chemical compounds. To illustrate, imagine that a set of rings has the same size and is made of the same material. When these things are interlinked, the chain formed can be considered as representing a polymer from molecules of the same compound. Alternatively, individual rings could be of different sizes and materials, and interlinked to represent a polymer from molecules of different compounds.什么是高聚物？首先，他们是合成物和大分子，而且不同于低分子化合物，譬如说普通的盐。

克隆巴赫系数的英文English:The Bernstein–Sato polynomial of a germ of a complex analytic function at a point is an object of central importance in singularity theory and complex analysis. It encodes crucial information about the singular behavior of the function near the given point. The Bernstein–Sato polynomial is intimately related to the Bernstein–Sato ideal, which is a fundamental concept in the study of D-modules and p-adic differential equations. The study of Bernstein–Sato polynomials and ideals has deep connections to various areas of mathematics, including algebraic geometry, representation theory, and harmonic analysis. These polynomials have been extensively studied in the context of local zeta functions, where they play a crucial role in understanding the structure of singularities and in the computation of zeta functions associated with singularities. In recent years, there has been significant progress in understanding the properties and applications of Bernstein–Sato polynomials, leading to new insights into the nature of singularities and their interactions with other mathematical objects.中文翻译:克隆巴赫多项式是复解析函数在某一点处的一个基本概念，它在奇点理论和复分析中具有重要的地位。

a r X i v :h e p -t h /9212152v 1 26 D e c 1992Unimodality of generalized Gaussian coefficients.Anatol N.KirillovSteklov Mathematical Institute,Fontanka 27,St.Petersburg,191011,RussiaJanuary 1991AbstractA combinatorial proof of the unimodality of the generalizedq -Gaussian coefficients Nλ qbased on the explicit formula forKostka-Foulkes polynomials is given.10.Let us mention that the proof of the unimodality of the general-ized Gaussian coefficients based on theoretic-representation considerations was given by E.B.Dynkin [1](see also [2],[10],[11]).Recently K.O’Hara [6]gave a constructive proof of the unimodality of the Gaussian coefficient n +knq=s (k )(1,···,q k ),and D.Zeilberger [12]derived some identitywhich may be consider as an “algebraization”of O’Hara’s construction.By induction this identity immediately implies the unimodality of n +knq.Using the observation (see Lemma 1)that the generalized Gaussian coef-ficient nλ′qmay be identified (up to degree q )with the Kostka-Foulkespolynomial K λ,µ(q )(see Lemma 1),the proof of the unimodality of nλ′qis a simple consequence of the exact formula for Kostka-Foulkes polynomials contained in [4].Furthermore the expression for K λ,µ(q )in the case λ=(k )coincides with identity (KOH)from [8].So we obtain a generalization and a combinatorial proof of (KOH)for arbitrary partition λ.120.Let us recallsome well known facts which will be used later.Webase ourselves[9]and[5].Letλ=(λ1≥λ2≥···)be a partition,|λ|be the sum of its partsλi,n(λ)= i(i−1)λi and nλ q be the generalized Gaussian coefficient.Recall thatsλ(1,···,q n)=q n(λ) nλ′ q= x∈λ1−q n+c(x)2·|λ|+n(λ) nλ′ q=K λ,µ(q).(2)Proof.We use the description of the polynomial q|λ|+n(λ)· nλ′ q as a generating function for the standard Young tableaux of the shapeλfilled with numbers from the interval[1,···,n].Let us denote this set of Young tableaux by ST Y(λ,≤n).Thenq|λ|+n(λ) nλ′ q= T∈ST Y(λ,≤n)q|T|.(3)Here|T|is the sum of all numbersfilling the boxes of T.For any tableau T (or diagramλ)let us denote by T[k](orλ[k])the part of T(orλ)consisting of rows starting from the(k+1)-st one.Given tableau T∈ST Y(λ,≤n), then consider tableau T∈ST Y( λ,µ)such that T[1]=T+suppλ[1],and wefill thefirst row of˜T with all remaining numbers in increasing order from left to right.Here for any diagramλwe denote by suppλthe plane partition of the shapeλand content(1|λ|).It is easy to see thatc( T)=|T|+(n+1)(n−2)2Let us consider an explanatory example.Assumeλ=(2,1),n=3. Then λ=(9,2,1),µ=(34).It is easy to see that|ST Y(λ,≤3)|=8. T|T| T c( T)30.First let us recall some definitions from[4].Given a partitionλand compositionµ,a configuration{ν}of the type(λ,µ)is,by definition,a collection of partitionsν(1),ν(2),···such that1)|ν(k)|= j≥k+1λj;2)P(k)n(λ,µ):=Q n(ν(k−1))−2Q n(ν(k))+Q n(ν(k+1))≥0for all k,n≥1, where Q n(λ):= j≤nλ′j,andν(0)=µ.Proposition1[4]Letλbe a partition andµbe a composition,then Kλ,µ(q)= {ν}q c(ν) k,n P(k)n(λ,µ)+m n(ν(k))m n(ν(k)) q,(4) where the summation in(4)is taken over all configurations of{ν}of the type(λ,µ),m n(ν(k))=(ν(k))′n−(ν(k))′n+1.From Lemma1and Proposition1we deduceTheorem1Letλbe a partition.ThenNλ′q = {ν}q c0(ν) k,n P(k)n(λ,N)+m n(ν(k))m n(ν(k)) q,(5)where the summation in(5)is taken over all collections{ν}of partitions {ν}={ν(1),ν(2),···}such that1)|ν(k)|= j≥kλj,k≥1,|ν(0)|=0,2)P(k)n(ν,N):=n(N+1)·δk,1+Q n(ν(k−1))−2Q n(ν(k))+Q n(ν(k+1))≥0, for all k,n≥1.Herec0(ν)=n(ν(1))−n(λ)+ k,n≥1 α(k)n−α(k+1)n2 ,α(k)n:=(ν(k))′n(6) and by definition α2 :=α(α−1)Proof.First,it is well known(e.g.[10],[11])that the product of symmet-ric and unimodal polynomials is again symmetric and unimodal.Secondly, we use a well known fact(e.g.[10]),that the ordinary q-Gaussian coefficientm+n nq is a symmetric and unimodal polynomial of degree mn.So inorder to prove Corollary1,it is sufficient to show that the sum2c0(ν)+ k,n m n(ν(k))P(k)n(ν,N)(7)is the same for all collections of partitions{ν}which satisfy the conditions 1)and2)of the Theorem1.In oder to compute the sum(7),we use the following result(see[4]):Lemma2Assume{ν}to be a configuration of the type(λ,ν).Then k,n m n(ν(k))P(k)n(ν,µ)=2n(µ)−2c(ν)− n≥1µ′n·α(1)n.(8)Using Lemma2,it is easy to see that the sum(7)is equal to(N−1)|λ|−n(λ).This concludes the proof.References[1]Dynkin E.B.,Some properties of the weight system of a linear repre-sentation of a semisimple Lie group(in Russian).Dokl.Akad.Nauk USSR,1950,71,221-224.[2]Hughes J.W.,Lie algebraic proofs of some theorems on partitions.InNumber Theory and Algebra,Ed.H.Zassenhaus,Academic Press,NY, 1977,135-155.[3]Kirillov A.N.,Completeness of the Bethe vectors for generalized Heisen-berg magnet.Zap.Nauch.Sem.LOMI(in Russian),1984,134,169-189.[4]Kirillov A.N.,On the Kostka-Green-Foulkes polynomials and Clebsch-Gordan numbers.Journ.Geom.and Phys.,1988,5,365-389.[5]Macdonald I.G.,Symmetric Functions and Hall Polynomials.OxfordUniversity Press,1979.[6]O’Hara K.,Unimodality of Gaussian coefficients:a constructive proof.b.Theory A,1990,53,29-52.[7]Goodman F.,O’Hara K.,Stanton D.,A unimodality identity for aSchur function.Preprint1990/1991.[8]Stanton D.,Zeilberger D.,The Odlyzko conjecture and O’Hara’s uni-modality proof.Bull.Amer.Math.Soc.,1989,107,39-42.[9]Stanley R.,Theory and application of plane partitions I,II.StudiesAppl.Math.,1971,50,167-188,259-279.[10]Stanley R.,Unimodal sequences arising from Lie algebras.InYoung Day Proceedings,Eds.J.V.Narayama,R.M.Mathsen,and J.G.Williams,Dekker,New York/Basel,1980,127-136.[11]Stanley R.,Log-concave and unimodal sequences in algebra,combina-torics,and geometry.Annals of the New York Academy of Sciences, 1989,576,500-535.[12]Zeilberger D.,Kathy O’Hara’s constructive proof of the unimodality ofthe Gaussian polynomials.Amer.Math.Monthly,1989,96,590-602.6。

The Mechanics of Crystal GrowthCrystal growth is the process of growing a crystal from a small seed. The process is slow and controlled, allowing the crystal to form in a manner that is both consistent and uniform. There are many factors that can influence crystal growth, including the temperature, pressure, and chemical concentration of the solution in which the crystal is growing. Understanding the mechanics of crystal growth is critical in a wide range of fields, from materials science to pharmaceuticals.One important factor in crystal growth is the rate of nucleation. This is the process by which a seed crystal is formed in a solution, and is often the first step in the crystal growth process. The rate of nucleation is critical, as it can determine both the size and quality of the resulting crystal. In general, slower rates of nucleation lead to larger, higher-quality crystals, while faster rates of nucleation lead to smaller, lower-quality crystals.Another important factor in crystal growth is the rate of crystal growth itself. This is influenced by a number of factors, including the concentration of the solution, the temperature at which the crystal is growing, and the presence of any impurities in the solution. In general, slower rates of crystal growth lead to larger, more uniform crystals, while faster rates of growth can result in smaller, more irregular crystals.Crystal growth can be influenced by a wide range of factors, including the physical properties of the solution in which the crystal is growing. For example, the viscosity of the solution can affect crystal growth, as more viscous solutions may be more likely to trap impurities and inhibit crystal growth. Similarly, the surface tension of the solution can affect crystal growth, as higher surface tension may lead to more uniform crystal growth.The shape of the crystal itself can also be influenced by the mechanics of crystal growth. For example, crystals that grow in a cylindrical shape are often shaped by the flow of the solution around the crystal. This can lead to a number of interesting shapes, including hexagonal and octagonal crystals. Other crystal shapes are influenced by theunderlying crystal structure, and can result in a wide range of unique and interesting shapes.Overall, the mechanics of crystal growth are incredibly complex and multifaceted, influenced by a wide range of variables and factors. Understanding these mechanics is essential in a wide range of fields, from materials science to pharmaceuticals, and can help to unlock new and exciting developments in these fields. By studying the mechanics of crystal growth, researchers can develop new methods for controlling and directing crystal growth, leading to advances in a wide range of fields.。

a rXiv:mat h-ph/03652v227Nov25On a problem with nonperiodic frequent alternation of boundary condition imposed on fast oscillating sets Denis I.Borisov ∗Bashkir State Pedagogical University,October Revolution,St.3a,450000Ufa,Russia,E-mail:BorisovDI@ic.bashedu.ru ,BorisovDI@bspu.ru ,URL:http://borisovdi.narod.ru Abstract We consider singular perturbed eigenvalue problem for Laplace operator in a cylinder with frequent and nonperiodic alternation of boundary con-ditions imposed on narrow strips lying in the lateral surface.The width of strips depends on a small parameter in a arbitrary way and may oscil-late fast,moreover,the nature of oscillation is arbitrary,too.We obtain two-sided estimates for degree of convergences of the perturbed eigenvalues.1Introduction This paper is devoted to the study of eigenvalue problem with frequent alternation of boundary conditions.The main feature of such boundary conditions is as follows.One should define the subset of a boundary consisting of a great number of disjoint closely-spaced parts of small measure.The boundary condition of one type is imposed on this subset,while one of another kind is settled on the rest part ofthe boundary.Papers devoted to homogenization of the elliptic problem with frequent alternation of boundary condition appeared in the second half of the last century (see,for instance,[8,9,10,12,16,19]).Nonlinear elliptic boundary value problems involving frequent alternation of boundary conditions were treated in[1,13].In [1]they solved the problem numerically,while in [13]the homogenization theorems were proved.In the papers [14,15]initial-boundary parabolic problemwith frequent alternation of boundary condition were considered and,in particular, homogenization theorems were proved.Asymptotics expansion for the solutions to the elliptic problems with frequently alternating boundary condition are constructed last decade.Two-dimensional case for periodic structure of alternation was studied well enough(see[6,7,17,18]and References of these works),nonperiodic case was considered in[5].Asymptotics expansions for three-dimensional problems with periodic structure of alternation were established in[2,3,4].The leading terms of asymptotics expansions for the solution to a parabolic problem with frequent alternation of boundary conditions were obtained in[14,15].In the present paper we study eigenvalue problem for Laplace operator in a cylinder.The lateral surface is partitioned into great number of narrow strips on those the Dirichlet and Neumann condition are imposed in turns.We deal with essentially nonperiodic alternation,moreover,the width of the strips may varies arbitrarily,including the cases of slow and fast oscillations of complicated nature. Under minimal restrictions to the structure of alternations we give best possible two-sided estimates for degree of convergences of the perturbed eigenvalues.We also note that estimates for degrees of convergence for another structure of alternation were obtained in[5,11,14,15,19],where parts of the boundary with different boundary condition were assumed to shrink to points.2Formulation of the problem and the main re-sultsLet x=(x′,x3),x′=(x1,x2)be Cartesian coordinates,ωbe an arbitrary bounded simply-connected domain in R2having infinitely differentiable boundary,Ω=ω×[0,H]be a cylinder of height H>0with upper and lower basisω1andω2 respectively.By s we denote the natural parameter of the curve∂ω,whileεis a small positive parameter:ε=H/(πN),where N≫1is a great integer.We define a subset of the lateral surfaceΣof the cylinderΩ,consisting of N narrow strips (cf.figure):γε={x:x′∈∂ω,−εa j(s,ε)<x3−επ(j+1/2)<εb j(s,ε),j=0,...,N−1}. Here a j(s,ε)and b j(s,ε)are arbitrary functions belonging to C∞(∂ω)and satisfy-ing uniform onεand j estimates:0<a j(s,ε)<π2.(1)From geometrical point of view these estimates means that strips of the subsetγεassociated with different values of j do not intersect.2Figure1:We study singularly perturbed eigenvalue problem:−∆ψε=λεψε,x∈Ω,(2)ψε=0,x∈ω1∪γε,∂γε),(3)whereνis the outward normal.The object of the paper is tofind out the limits to the eigenvaluesλεand to estimate the degree of convergences asε→0under minimal restrictions for the setγε.Earlier elliptic problem with boundary conditions(3)were studied in [2,3,4,9].In the paper[9]they studied the homogenization of the elliptic problem in a circular cylinder with boundary conditions(3)under additional assumption that a j(s,ε)=b j(s,ε)=η(ε),j=0,...,N−1,whereη(ε)is a some function. Geometrically this assumption means that the setγεis periodic and all strips have same constant width.In[3,4]under the same assumptions complete asymptotics expansions for the problem(2),(3)were constructed.The case of arbitrary section and periodic distribution of the strips of slowly varying width was treated in[2,3]. In[2,3]it was assumed that a j(s,ε)=b j(s,ε)=η(ε)g(s),where g(s)∈C∞(∂ω), 0<c0≤g(s)≤1,and the functionη(ε)satisfies the estimate0<η(ε)<π/2.The leading terms of the asymptotics expansions for the eigenelements were constructed,moreover,the definition of a j and b j mentioned above played essential role in construction of these asymptotics.At the same time,the question on behaviour of the eigenvalues of the problem(2),(3)for arbitrary functions a j and b j is open.Clear,the arbitrary choice of the functions a j and b j leads,generally saying,to the setγεof nonperiodic structure.Moreover,the functions a j(s,ε)and b j(s,ε)may oscillate fast on s.Such oscillation takes place under,for instance, following choice of the functions a j and b j:a j(s,ε)=b j(s,ε)=η(ε)g(s/α(ε)), whereη(ε)is a rough width of the strip,g∈C∞(∂ω),α(ε)−−→ε→00.We stress,3that in the example shown the leading terms of the asymptotics expansions for the eigenvalues of the problem(2),(3)will differ from the similar results of the papers [2,3,4]and will depend essentially on the functionα(ε).If the setγεis nonperiodic and the width of all or some strips oscillates fast,and also,these oscillations have arbitrary nonperiodic on s structure(cf.figure),the question of constructing the asymptotics expansions for the eigenvalues of the problem(2),(3)becomes very complicated and can be solved only under some additional constraints for the functions a j and b j.That’s why to estimate the degree of convergence for the eigenvalues of the(2),(3)is a topical question.Exactly this question is solved in the present paper.Now we proceed to the formulation of the main results.We take the perturbed eigenvalues in ascending order counting multiplicity:λ1ε≤λ2ε≤...λkε≤... Limiting eigenvalues defined below are assumed to be taken in the same order:λ10≤λ20≤...λk0≤...Theorem2.1.Suppose the inequalities(1)andη(ε)≤a j(s,ε),η(ε)≤b j(s,ε),hold,whereη(ε)is an arbitrary function obeying the estimate0<η(ε)<π/2and the equality:εlnη(ε)=0.limε→0Then uniform onεandηestimates−c kε(|lnη|+1)≤λkε−λk0≤0(4) hold true.Here c k>0are constants,λk0are eigenvalues of the problem−∆ψ0=λ0ψ0,x∈Ω,∂ψ0=0,x∈ω1∪Σ,=−A,A=const>0.εlnη(ε)Then uniform onεandηestimates−c k µ(ε)+ε2+εlnη0(ε) ≤λkε−λk0≤c k µ(ε)+ε2)(6) hold true.Here c k>0are constants,1µ=µ(ε)=−A−λk0are eigenvalues of the problem−∆ψ0=λ0ψ0,x∈Ω,ψ0=0,x∈ω1,∂+A ψ0=0,x∈Σ.(7)∂νTheorem2.3.Suppose the inequalities0<a j(s,ε)≤η(ε),0<b j(s,ε)≤η(ε)hold,whereη(ε)is an arbitrary function obeying an equality:1limε→0,εlnη(ε)λk0are eigenvalue of the problem(7)as A=0.Remark2.1.We note that the hypotheses of the Theorems2.1-2.3admit the non-periodic setγε.Moreover,this set may contain strips of width varying arbitrary, in particular,strips of fast oscillating width.For instance,the case shown in the figure is included under consideration.Remark2.2.We stress that the estimates(4),(6),(8)are best possible.More con-cretely,under hypothesis of each theorem there exist function a j and b j,for those the degree of convergence has the smallness order exactly as given in the estimates (4),(6),(8).This statement will be established in the proofs of Theorems2.1-2.3.3Proof of Theorems2.1-2.3The proof of Theorems2.1-2.3is based on the following auxiliary statement. Lemma3.1.Suppose the subsetsγε,∗andγ∗εof the lateral surfaceΣsatisfyγε,∗⊆γε⊆γ∗ε,(9) and letλkε,∗andλ∗,kεbe eigenvalues of the problem(2),(3)withγεreplaced byγε,∗andγ∗ε,respectively.Then the estimatesλkε,∗≤λkε≤λ∗,kε(10) are valid.5This lemma can be proved in a standard way on the base of minimax property of the eigenvalues of the problem(2),(3).Proof of Theorem2.1.Letγε,∗={x:x′∈∂ω,|x3−επ(j+1/2)|<εη(ε),j=0,...,N−1},γ∗ε=Σ.In accordance with hypothesis,these sets meet the inclusions(9).The setγε,∗being periodic,the asymptoticsλkε,∗=λk0+ελk1ln sinη+o ε(|lnη|+1) ,(11) hold.Hereλk1are some constants,λk0are eigenvalues of the problem(5).In the case of simple eigenvalueλk0this asymptotics was constructed formally in[2]. The case of eigenvalueλk0of arbitrary multiplicity was studied rigorously in[4] under assumptionη=const,ωis a unit circle.These additional assumptions are not essential and the asymptotics(11)can be proved completely by analogy with[2,4];the rigorous two-parametrical estimates for the error term can be easily established on the base of the results of[6].Sinceγ∗ε=Σ,it follows thatλ∗,kε=λk0. Substituting these equalities and asymptotics(11)into(10),we getλk0+c kε(|lnη|+1)≤λkε≤λk0,where c k>0are some constants.This implies the needed two-sided estimates. These estimates are best possible.Indeed,taking a j(s,ε)=b j(s,ε)=η(ε),we see thatλkεmeet asymptotics(11),i.e.,λkε−λk0=O(ε(|lnη|+1).Proof of Theorem2.2.The scheme of the proof is same with one of Theorem2.1. The setsγε,∗andγ∗εshould be defined as followsγε,∗={x:x′∈∂ω,|x3−επ(j+1/2)|<εη0(ε)η(ε),j=0,...,N−1},γ∗ε={x:x′∈∂ω,|x3−επ(j+1/2)|<εη(ε),j=0,...,N−1}.By the hypothesis,these sets obey inclusions(9).The corresponding eigenvalues λkε,∗andλ∗,kεhave the asymptoticsλkε,∗=λk0+ µλk0,1+ε2λk1,0+o | µ|+ε2 ,λ∗,kε=λk0+µλk0,1+ε2λk1,0+o |µ|+ε2 .Hereλk i,j are some constants,λk0are eigenvalues of the problem(7),µ= µ(ε)=−A−1For the case of simple eigenvalueλk0and arbitrary sectionω,as well as for the case of multiply eigenvalueλk0and circular sectionωthese asymptotics were found in[2,3].The technique of these two papers can be easily applied to the case of arbitrary sectionωand multiply eigenvalueλk0.The asymptotics forλkε,∗and λ∗,kε,definition of µand inequalities(10)yield needed estimates for degree of convergence.Unimprovability for these estimates is established by analogy with Theorem2.1.The proof of Theorem2.3is similar to proof of Theorems2.1,2.2.Here sets γε,∗andγ∗εand asymptotics for corresponding eigenvaluesλkε,∗,λ∗,kε([2,3])are as follows:γε,∗=∅,γ∗ε={x:x′∈∂ω,|x3−επ(j+1/2)|<εη(ε),j=0,...,N−1},λkε,∗=λk0,λ∗,kε=λk0+λk0,1µ+o(µ),whereλk0,1are some constants,λk0are eigenvalues of the problem(7)with A=0. References[1]Barenbaltt G.I.,Bell J.B.,and Crutchfiled W.Y.,The thermal explosion revis-ited,A95,(1998)13384-13386.[2]Borisov,D.I.The asymptotics for the eigenelements of the Laplacian in a cylin-der with frequently alternating boundary conditions,C.R.Acad.Sci.Paris, S´e rie IIb.329,(2001)717-721.[3]Borisov,D.I.Boundary value problem in a cylinder with frequently changing type of boundary condition,Sb.Math.193,(2002)977-1008.[4]Borisov,D.I.On a singularly perturbed boundary value problem for Laplacian in a cylinder,Differ.Equations38,(2002)1140-1148.[5]Borisov, D.I.On a Laplacian with frequently nonperiodically alternating boundary conditions,Dokl.Math.65,(2002)224-226.[6]Borisov,D.I.Two-parametrical asymptotics for the eigenevalues of the Lapla-cian with frequent alternation of the boundary conditions,Vestnik Molodyh Uchenyh.Seriya Prikladnaya matematika i mehanika1,(2002)36-52.(in Rus-sian)[7]Borisov,D.I.On a model boundary value problem for Laplacian with frequently alternating type of boundary condition,Asymptotic Anal.35,(2003)1-26. [8]Brillard A.,Lobo M.,P`e rez E.Homog´e n´e isation de fronti`e res par´e pi-convergence en´e lasticit´e lin´e aire Mod´e lisation math´e matique et Analyse num´e rique.24,(1990)5-26.7[9]Lobo M.,P´e rez E.Boundary homogenization of certain elliptic problems for cylindrical bodies,Bull.Sc.math.116,(1992)399-426.[10]Chechkin,G.A.Averaging of boundary value problem with singular pertur-bation of boundary conditions,Sb.Math.79,(1994)191-222.[11]Chechkin,G.A.,Doronina,E.I.On asymptotics of spetrum of boundary value problem with nonperiodic rapidly alternating boundary conditions,Functional Differential Equations,8,(2001)111-122.[12]Damlamian A.and Li Ta-Tsien(Li Daqian).Boundary homogenization for ellpitic problems,J.Math.Pure et Appl.66,(1987)351-361.[13]D´a vila,J.A nonlinear elliptic equation with rapidly oscillating boundary con-ditions,Asymptotic Anal.28,(2001)279-307.[14]Filo J.,Luckhaus S.Asymptotic expansion for a periodic boundary condition, J.Diff.Equations,120,(1995)133-173.[15]Filo J.,Luckhaus S.Homogenization of a boundary condition for the heat equation,J.Eur.Math.Soc.2,(2000)217-258.[16]Friedman A.,Huang Ch.and Yong J.Effective permeability of the boundary of a domain,Comm.Partial Differential Equations,20,(1995)59-102.[17]Gadyl’shin,R.R.On the eigenvalue asymptotics for periodically clamped membranes,St.Peterburg Math.J.,10,(1999)1-14.[18]Gadyl’shin,R.R.Homogenization and asymptotics for a membrane with closely spaced clamping points,Comp.Math.and Math.Phys.41,(2001)1765-1776.[19]Oleinik O.A.,Chechkin G.A.Solutions and eigenvalues of the boundary value problems with rapidly alternating boundary conditions for the system of elas-ticity,Rendiconti Lincei:Mathematica e Applicazioni.Serie IX.7,(1996)5-15.8。

The mysteries of the atom QuantummechanicsQuantum mechanics is a branch of physics that deals with the behavior of very small particles, such as atoms and subatomic particles. It is a fascinating and complex field that has puzzled scientists for decades, and continues to challenge our understanding of the universe. The mysteries of the atom, as revealed through quantum mechanics, have sparked both wonder and confusion among scientists and laypeople alike. One of the most perplexing aspects of quantum mechanics is the concept of superposition. This principle states that particles can exist in multiple states at once, until they are observed or measured. This means that an electron, for example, can be in multiple places at the same time, until it is observed and its position is determined. This idea goes against our everyday experience of the world, where objects exist in specific states at any given time. The idea of superposition challenges our understanding of reality and raises deep philosophical questions about the nature of existence. Another mind-boggling concept in quantum mechanics is entanglement. This phenomenon occurs whenparticles become linked in such a way that the state of one particle is instantly correlated with the state of another, no matter how far apart they are. This seemingly instantaneous connection defies our classical understanding of cause and effect, and has been described by Albert Einstein as "spooky action at a distance." The implications of entanglement are profound, and have led to the development of technologies such as quantum computing and quantum cryptography. The uncertainty principle, formulated by Werner Heisenberg, is another fundamental concept in quantum mechanics. This principle states that certain pairs of physical properties, such as position and momentum, cannot be simultaneously known to arbitrary precision. In other words, the more precisely we know one of these properties, the less precisely we can know the other. This inherent uncertainty at the quantum level challenges our classical intuition and has profound implications for the nature of reality and our ability to predict the behavior of particles. The mysteries of the atom, as uncovered by quantum mechanics, have not only confounded scientists, but also inspired awe and wonder. The strange andcounterintuitive behavior of particles at the quantum level has led to a reevaluation of our understanding of the universe and our place within it. The profound implications of quantum mechanics have sparked philosophical andspiritual discussions about the nature of reality, consciousness, and the interconnectedness of all things. In conclusion, the mysteries of the atom, as revealed through quantum mechanics, continue to challenge our understanding of the universe and our place within it. The concepts of superposition, entanglement, and the uncertainty principle have led to profound philosophical and scientific discussions about the nature of reality and the fundamental building blocks of the universe. While quantum mechanics may be perplexing and counterintuitive, it has also inspired awe and wonder, and has led to the development of groundbreaking technologies that have the potential to revolutionize the way we live. The mysteries of the atom continue to captivate and perplex scientists and laypeople alike, and are likely to remain a source of fascination for years to come.。