Hermite-Fejer Interpolation With Moved End Points

Let Z_n={z_(kn)=cosθ_(kn):θ_(kn)=(2k-1)/(2n)π,k=1,2…,n}be the zeros of T_n(x)=cosnθ(x=cosθ,θ∈[0,π]).For 0≤ε≤1,let α_n=:α_n(ε)=:cos(1-ε)/(2n)π,β_n=:β_n(ε)=:cos(2n-1+ε)/(2n)π=-α_n,X_n~(1)=(Z_n-{z_(1z)})∪{α_n},X_n~(2)=(Zn-{z_(nn)})∪{β_n},X_n~(3)=(Z_n-{z_(1n),z_(nn)})∪{α_n,β_n},Y_n~(1)=Z_n∪{α_n},Y_n~(2)=Z_n∪{β_n},Y_n~(3)=Z_n∪{α_nβ_n}.     

The mKdV Hierarchy With Integral Source

1 IntroductionThe nonlinear Schrodinger equation with integral source (NLSEIS) and the KdV equationwith integral source (KdVIS) were integrated by inverse scattering method in [1,2] and -methodin [3]. However, since the evolution equations for eigenfunction in Lax representation for NLSEISand KdVIS were not obtained, the determining scatterging data in [1-3] was quite complicatedand required some skill.In the present paper we discuss the new mKdV hierarchy with integral source (mKdV…     

Universal Unfolding of Equivariant Bifurcation Problems With Multiparameter

SupposethatthecompactLiegroupTmayactontwodifferentspacesRnandRm,wheren≥m.LetAbifurcationproblemwithsymmetrygrouprisamap-germgEE.,*,m(r)whichsatisfiesthefollowingconditionsg(0,0)=0and(D.g)(0,0)=0.HereD.gdenotesthederiVativeofgwithrespecttox.Letg(x,A)Et.,*,.(r)bear-equivariantbifurcationproblem.Ar-parameterr-unfoldingofgisar-equlvarialltmopgermG(x,A,a)EE.,k,.:m(r),wherea=(al,'',a.)6R'isunfoldingparameterandG(x,A,0)~g(x,A).LetH(x,A,g)Et.,*,.:.(r)beans-parameterrullfoldingofg.Wesaythat…     

Probabilistic Treatment of Schrodinger Equation With Infinite Gauge

Consider Dirichlet problem of Schrodinger equation     

Kramer’s Sampling Theorem With Discontinuous Kernels

Kramer’s sampling theorem provides an algorithm for reconstructing a function ?, in the form $$ f(t)=\int_{a}^{b}\ F(x)K(x,t)dx,\qquad {\rm for\ some}\ F\ \in\ L^{2}(a,b), $$ from its values at a discrete set of points. In all the known examples, the kernel of the transform, K(x,t) is continuous in x and entire in t, even though the proof of the theorem shows that the continuity in x is not essential. This raises the question of whether it is possible to find an example of Kramer’s theorem with a discontinuous kernel. The aim of the paper is to answer this question in the affirmative. We show how one can construct a family of discontinuous kernels for which Kramer’s theorem holds and, in addition, each member of this family arises from a Sturm-Liouville problem, but with discontinuous initial conditions.     

Continuum Schrödinger Operators Associated With Aperiodic Subshifts

We study Schrödinger operators on the real line whose potentials are generated by an underlying ergodic subshift over a finite alphabet and a rule that replaces symbols by compactly supported potential pieces. We first develop the standard theory that shows that the spectrum and the spectral type are almost surely constant and that identifies the almost sure absolutely continuous spectrum with the essential closure of the set of energies with vanishing Lyapunov exponent. Using results of Damanik–Lenz and Klassert–Lenz–Stollmann, we also show that the spectrum is a Cantor set of zero Lebesgue measure if the subshift satisfies the Boshernitzan condition and the potentials are aperiodic and irreducible. We then study the case of the Fibonacci subshift in detail and prove results for the local Hausdorff dimension of the spectrum at a given energy in terms of the value of the associated Fricke–Vogt invariant. These results are elucidated for some simple choices of the local potential pieces, such as piecewise constant ones and local point interactions. In the latter special case, our results explain the occurrence of so-called pseudo bands, which have been pointed out in the physics literature.     

Finite Groups With Small Automizers for Their Nonabelian Subgroups

Ｔｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒ，Ｇｉｓａｆｉｎｉｔｅｇｒｏｕｐ．ＳｅｔＡｕｔＧ（Ｈ）＝ＮＧ（Ｈ）／ＣＧ（Ｈ）ｆｏｒＨ≤Ｇ．Ｉｎ［ＤＭ］，ａｕｔｈｏｒｓｇａｖｅｔｈｅｆｏｌｏｗｉｎｇｄｅｆｉｎｉｔｉｏｎｓ：（１）ＬｅｔＨ≤Ｇ，ＡｕｔＧ（Ｈ）＝Ｎ...     

Hypersurface With Constant Mean Curvature in Hyperbolic Space Form

In [1], S. T. Yau proved the following theorems: (1) If M is compact hyper-surface with constant mean curvature and non-negative Ricci curvature in the Eucli-dean space, then M is umbilical. (2) If M is compact hypersurface with constantscalar curvature in hyperbolic space form and M has positive sectional curvature, thenM is totally umbilical. In this paper, we shall generalize the theorems as follows     

On a Singular Dirichlet Problem With a Convection Term

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Some Bivariate Inverse Relations With Applications to Interpolation Formulae

This paper investigates the bivariate analogs of Gould-Hsu's inversions and obtains an inverse relation including five sets of free parameters which can be seen as a partial solution for Hsu's open problem. Some instances and applications to interpolation theory are remarked.     

Entire Functions That Share A Value With Their Derivatives

This paper obtained a unicity theorem on entire functions that share a value with their derivatives, which extered and improved some results of Gundersen and L.Z.Yang.     

Solving ℓ1 Regularization Problems With Piecewise Linear Losses

This article is concerned with the computational aspect of ?₁ regularization problems with a certain class of piecewise linear loss functions. The problem of computing the ?₁ regularization path for a piecewise linear loss can be formalized as a parametric linear programming problem. We propose an efficient implementation method of the parametric simplex algorithm for such a problem. We also conduct a simulation study to investigate the behavior of the number of “breakpoints” of the regularization path when both the number of observations and the number of explanatory variables vary. Our method is also applicable to the computation of the regularization path for a piecewise linear loss and the blockwise ?_∞ penalty. This article has supplementary material online.     

On Critical Fujita Exponents for a Degenerate Parabolic Equation With

Our interest is to determine the critical Fujita exponent concerned withthe following initial-boundary value problem ut= Δum, x ∈RN+, t>0, (1) u(x,0)=u0(x), x∈RN+, (2) -(um)/(x1)=up, x1=0, t>0, (3) where RN+=(x1, x′)| x′∈R{N-1, x1>0,m>1, p>0, and u0(x) is a nonnegative bounded function with compact supportsatisfying thecompatibility condition -(um0(x))/(x1)=up0(x), x1=0. We call p0 the critical global existence exponent if ithas the following property: if p>p0, there always exist nonglobalsolutions of the problem (1)--(3) while if 0pc small data solutionsexist globally in time while large data solutions are nonglobal.     

Stochastic Reaction-Diffusion Systems With Hölder Continuous Multiplicative Noise

We prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with a locally Lipschitz continuous reaction term of polynomial growth and Hölder continuous multiplicative noise. Under additional assumptions on the coefficients, we also prove positivity of the solutions.     

Boundary Unique Continuation Properties for Elliptic Operators With Singular Potentials

0 Introduction Because of applications to control theory, see Schmit and Week's work in [1], the following unique continuation question (UCP) was raised in [2-3]: if u is a harmonic function in a connected Lipschitz domainΩ, vanishing continuously on an open subsetΓof the boundary (?)Ωand whose normal derivative vanishes on a subset of Γof positive surface measure, does it follows that u is identically zero inΩ? On the other hand, the unique continuation problem has been focusing on second order equations in which the coefficients of the lower-order terms are allowed to be     

Periodic Solution of Nonlinear Parabolic Differential Equation With Time Lag

ＰｅｒｉｏｄｉｃＳｏｌｕｔｉｏｎｏｆＮｏｎｌｉｎｅａｒＰａｒａｂｏｌｉｃＤｉｆｆｅｒｅｎｔｉａｌＥｑｕａｔｉｏｎＷｉｔｈＴｉｍｅＬａｇ￥（王克）ＷａｎｇＫｅ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＮｏｒｔｈｅａｓｔＮｏｒｍａｌＵｎｉｖｅｒｓ...     

All Lambda-Designs With λ = 2 p are Type-1

A-design is a family B ₁,B ₂,...,B _v of subsets of X={1, 2,..., v} such that B _i B _j = for all i jand not all B _i are of the same size. Ryser's andWoodall's -design conjecture states thateach -design can be obtained from a symmetricblock design by a certain complementation procedure. Our mainresult is that the conjecture is true when is twice a prime number.     

Kähler–Ricci Flow With Small Initial Energy

In this paper, we prove that the K?hler–Ricci flow converges to a K?hler–Einstein metric when E ₁ energy is small. We also prove that E ₁ is bounded from below if and only if the K-energy is bounded from below in the canonical class. The first named author is partially supported by a NSF grant, while the third author was partially supported by a NSF supplement grant.     

Stokes Flow in Cylindrical Containers With Rotating Ends北大核心CSCD

该文以端部旋转的圆柱形容器内的Stokes流为研究对象,根据流动的特点,将轴向坐标模拟为时间,则问题归结为Hamilton对偶方程的本征值和本征解问题.利用本征解空间的完备性和本征解之间的共轭辛正交关系,给出了问题解的展开形式,并建立了展开系数的数值求解方法.采用该方法研究了单端旋转、两端以相同或相反角速度旋转时不同外形比(容器的高度与半径之比)时圆柱形容器内流动速度和应力的分布情况,展示了不同边界条件下流场的一些特点.