On the Best Approximation in Non-Archimedean Normed Linear Spaces

ＯｎｔｈｅＢｅｓｔＡｐｐｒｏｘｉｍａｔｉｏｎｉｎＮｏｎ－ＡｒｃｈｉｍｅｄｅａｎＮｏｒｍｅｄＬｉｎｅａｒＳｐａｃｅｓｏｎＷｅｉｄｕｎ（邱维敦）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＬｏｎｇｙｏｎＮｏｒｍａｌＣｏｌｌｅｇｅ，Ｆｕｊｉａｎ，３６...     

On the Order of Best Approximation by Polynomials in H_q~p Spaces

ＯｎｔｈｅＯｒｄｅｒｏｆＢｅｓｔＡｐｐｒｏｘｉｍａｔｉｏｎｂｙＰｏｌｙｎｏｍｉａｌｓｉｎＨ■ＳｐａｃｅｓＸｉｎｇＦｕｃｈｏｎｇ（邢富冲）（ＣｅｎｔｒａｌＵｎｉｖｅｒｓｉｔｙｆｏｒＮａｔｉｏｎａｌｉｔｉｅｓ，Ｂｅｉｊｉｎｇ，１０００８１）Ｃｏｍｍｕｎｉ...     

On Numerical Landau Damping for Splitting Methods Applied to the Vlasov–HMF Model

We consider time discretizations of the Vlasov–HMF (Hamiltonian mean-field) equation based on splitting methods between the linear and nonlinear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time step size.     

On the Wave Equation Associated to the Hermite and the Twisted Laplacian

The dispersive properties of the wave equation u _tt +Au=0 are considered, where A is either the Hermite operator −Δ+|x|² or the twisted Laplacian −(∇ _x −iy)²/2−(∇ _y +ix)²/2. In both cases we prove optimal L ¹−L ^∞ dispersive estimates. More generally, we give some partial results concerning the flows exp (itL ^ν ) associated to fractional powers of the twisted Laplacian for 0<ν<1.     

On a serious mathematical error in the “Mathematical Encyclopedia” related to the solution of the Gibbs paradox

The solution of the Gibbs paradox in the “Mathematical Encyclopedia” [1] contained an error. It is also contained in many physics textbooks. A correct solution is given in this paper. New thermodynamic quantities taking into account the occurrence of dimers and clusters are introduced.     

Inequalities for Two-Point Padé Approximants to the Expansions of Stieltjes Functions in a Real Domain

General inequalities satisfied in real domain by two-point Padé approximants (2PPA) to two formal power expansions of Stieltjes functions have been derived. The obtained formulae: (i) generalize the existing inequalities for one-point Padé approximants, (ii) extend the validity of the inequalities reported in J. Comp. Appl. Math. 67 (1996), 59–72, from the particular case 0 k 2 to the general case 0 k , where (u,k) denotes a given number of coefficients of a power expansion of Stieltjes functions at infinity, (iii) clarify the properties of 2PPA estimations of the effective conductivity for regular composites.     

On the “log rank”-conjecture in communication complexity

We show the existence of a non-constant gap between the communication complexity of a function and the logarithm of the rank of its input matrix. We consider the following problem: each of two players gets a perfect matching between twon-element sets of vertices. Their goal is to decide whether or not the union of the two matcliings forms a Hamiltonian cycle. We prove:

1. The rank of the input matrix over the reals for this problem is 2 ^O(n) .
2. The non-deterministic communication complexity of the problem is Ω(nloglogn).

Our result also supplies a superpolynomial gap between the chromatic number of a graph and the rank of its adjacency matrix. Another conclusion from the second result is an Ω(nloglogn) lower bound for the graph connectivity problem in the non-deterministic case. We make use of the theory of group representations for the first result. The second result is proved by an information theoretic argument.     

First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ? iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.     

On the Suitable Weak Solutions to the Boussinesq Equations in a Bounded Domain

ＯｎｔｈｅＳｕｉｔａｂｌｅＷｅａｋＳｏｌｕｔｉｏｎｓｔｏｔｈｅＢｏｕｓｓｉｎｅｓｑＥｑｕａｔｉｏｎｓｉｎａＢｏｕｎｄｅｄＤｏｍａｉｎＧｕｏＢｏｌｉｎｇ（郭柏灵）ＹｕａｎＧｕａｎｇｗｅｉ（袁光伟）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＰｈｙｓｉｃｓａｎ...     

Counterexample to the paper “On the Gorenstein injective dimension and Bass formula”

In this note, we give a counterexample for Theorem 2.3 of the above mentioned paper that is a generalization of the Grothendieck non-vanishing theorem to a class of modules larger than finitely generated modules.     

On Basic Features of the “Mathematics in Nine Sections”

The "Mathematics in Nine Sections" is a famous mathematical treatise trans mitted down from ancient China. The first author of the treatise is not know to us, nor can the date of its composition be exactly determined. Liu Hui, who lived in the third century of the Christian era, points out in the preface of his book, "All books were burned in the reign of the despotic emperor Shi Huang     

On “Ultraproducts in topology”

We correct an error in the above-mentioned paper and provide a solution to an open problem contained therein.     

Best “Continuous” n-Term Approximations in the Spaces $$S_\phi ^p $$

We find exact values of upper bounds for the best approximations of q-ellipsoids by polynomials of degree n in the spaces in the case where the approximating polynomials are constructed on the basis of n-dimensional subsystems chosen successively from a given orthonormal system .     

On the Blow-up Criterion of Smooth Solutions to the MHD System in BMO Space

In this paper we study the blow-up criterion of smooth solutions to the incompressible magnetohydrodynamics system in BMO space. Let （u（x,t）,b（x,t）） be smooth solutions in （0, T）. It is shown that the solution （u（x, t）, b（x, t）） can be extended beyond t = T if （u（x,t）, b（x, t）） ∈ L^1 （0, T; BMO） or the vorticity （rot u（x, t）, rot b（x, t）） ∈ L^1 （0, T; BMO） or the deformation （Def u（x, t）, Def b（x, t）） ∈ L^1 （0, T; BMO）.     

On the normal bundle to abelian surfaces embedded in ℙ4(ℂ)

In this note one computes the cohomology of the normal bundle N_X (and its twists) to any abelian surface X in ⁴() and one shows that N_X is simple. As a by-product we reobtain the results of Decker about the smoothness of the irreducible component of the moduli scheme M(–1,4) of rank 2 stable vector bundles on ⁴ with c₁=–1,c₂=4, along the orbit of the Horrocks-Mumford bundle by the action of SL₅ () (cf. [2]).     

On Homogenization of Problems in Domains of the “Infusorium” Type

In the paper, we consider problems in perforated domains with rapidly oscillating boundary. We assume that the oscillating boundary has a special structure. The width of the hills has the same size as the cell of periodicity inside the domain and the order of the height of the hills is bigger. Under these assumptions we construct the homogenized problem, prove the homogenization theorem, construct the correctors, and obtain estimates of the deviation of the solution to the initial problem from the leading terms of the asymptotic series.