On solutions of functional-differential equations f ′( x ) = a ( x ) f ( g ( x )) + b ( x ) f ( x ) + c ( x ) in the large

We consider solutions of functional-differential equations

Cauchy problem F(t, x(t), x′(t)) = 0, x(0) = 0: solvability and asymptotics of solutions

For the singular Cauchy problem, the authors find some sufficient conditions for the existence of continuously differentiable solutions x: (0, ρ] → ℝ (ρ > 0 is sufficiently small) of the form

唯一g(x)-clean环

本文研究了唯一g(x)-clean环的性质与结构.利用g(x)-clean环的方法,得到了唯一g(x)-clean环与g(x)-clean环的关系,唯一g(x)-clean环与一类特殊的生成环的等价条件,以及斜Hurwitz级数环的g(x)-clean性,推广了g(x)-clean环的研究结果.     

On the Diophantine Equation G n ( x )?= G m ( P ( x ))

 Let K be a field of characteristic 0 and let p, q, G ₀ , G ₁ , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G _n (x)) _n=0 ^∞ be defined by the second order linear recurring sequence

On the Diophantine Equation G n ( x )?= G m ( P ( x ))

 Let K be a field of characteristic 0 and let p, q, G ₀ , G ₁ , P ∈K[x], deg P ⩾ 1. Further, let the sequence of polynomials (G _n (x)) _n=0 ^∞ be defined by the second order linear recurring sequence

Topological hochschild homology of extensions of ?/p? by polynomials, and of ?[x]/(xn) and ?[x]/(xn?1)

Topological Hochschild homology is calculated for the rings /p[x]/(f(x)) (where p is prime and f(x) /p[x] any polynomial), [x]/(x ⁿ) and [x]/(x ⁿ–1). A spectral sequence argument is used for calculating the homology of the topological Hochschild homology spectrum, from which its stable homotopy structure can be read off since the spectrum is known for a priori reasons to be a restricted product of Eilenberg-MacLane spectra.     

Lie Symmetries of the Equation ut(x, t)?+?g(u)ux(x, t) = 0

In this paper the invariance criterion is applied for the nonlinear equation

方程x'(t)=ax(t)+bx(3[(t+1)/3])的数值稳定性分析

本文研究了分段连续型微分方程x''(t)=ax(t)+bx(3[(t+1)/3]) Euler-Maclaurin方法的数值稳定性问题.利用特征分析的方法,获得了数值解稳定的充分条件,进而证明了Euler-Maclaurin方法保持了精确解的稳定性.最后给出了一些数值例子.     

Asymptotics of the Best Polynomial Approximation of?|x|p and?of?the Best Laurent Polynomial Approximation of sgn(x) on Two Symmetric Intervals

We present a new method that allows us to get a direct proof of the classical Bernstein asymptotics for the error of the best uniform polynomial approximation of |x| ^p on two symmetric intervals. Note that, in addition, we get asymptotics for the polynomials themselves under a certain renormalization. Also, we solve a problem on asymptotics of the best approximation of sgn(x) on [−1,−a]∪[a,1] by Laurent polynomials.      

On the subspace L((x ∧ y)m) of Sm(∧2?4)

We take the exterior power ℝ⁴ ∧ ℝ⁴ of the space ℝ⁴, its mth symmetric power V = S ^m (∧²ℝ⁴) = (ℝ⁴ ∧ ℝ⁴) ∨ (ℝ⁴ ∧ ℝ⁴) ∨ ... ∨(ℝ⁴ ∧ ℝ⁴), and put V ₀ = L((x ∧ y)∨ ... ∨(x ∧ y): x, y ∈ ℝ⁴). We find the dimension of V ₀ and an algorithm for distinguishing a basis for V ₀ efficiently. This problem arose in vector tomography for the purpose of reconstructing the solenoidal part of a symmetric tensor field. Original Russian Text Copyright ? 2009 Gubarev V. Yu. The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-344.2008.1). __________ Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 3, pp. 503–514, May–June, 2009.     

Sharp estimates for the convergence rate of double Fourier series in terms of orthogonal polynomials in the space L2((a, b) × (c, d); p(x)q(y))

Sharp estimates are obtained for the convergence rate of double Fourier series in terms of general orthogonal polynomials in some classes of functions and for the Kolmogorov N-widths of these classes. These results find applications in numerical analysis. Original Russian Text ? V.A. Abilov, M.K. Kerimov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 8, pp. 1364–1368.     

∣x∣在调整的第二类Chebyshev结点组的有理插值

本文研究了Newman型有理算子逼近|x|的收敛速度,插值结点组X取调整的第二类Chebyshev结点组.利用上界估计得到确切的逼近阶为O 1n2.这个结果优于结点组取作第一、二类Chebyshev结点组、等距结点组和正切结点组.     

On the description of overgroups of E(m,R)?E(n,R)

The subgroups E(m,R) ⊗ E(n,R) ≤ H ≤ G = GL(mn,R) are studied under the assumption that the ring R is commutative and m, n ≥ 3. The group GL _m ⊗GL _n is defined by equations, the normalizer of the group E(m,R) ⊗ E(n,R) is calculated, and with each intermediate subgroup H it is associated a uniquely determined lower level (A,B,C), where A,B,C are ideals in R such that mA,A ² ≤ B ≤ A and nA,A ² ≤ C ≤ A. The lower level specifies the largest elementary subgroup satisfying the condition E(m, n,R, A,B,C) ≤ H. The standard answer to this problem asserts that H is contained in the normalizer N _G (E(m,n,R, A,B,C)). Bibliography: 46 titles.     

Balanced colombeau products of the distributions x±-p and x-p

Results on singular products of the distributions x _± ^-p and x ^-p for natural p are derived, when the products are balanced so that their sum exists in the distribution space. These results follow the pattern of a known distributional product published by Jan Mikusiski in 1966. The results are obtained in the Colombeau algebra of generalized functions, which is the most relevant algebraic construction for tackling nonlinear problems of Schwartz distributions.     

Complete description of invariant Einstein metrics on the generalized flag manifold SO(2n)/U(p)?× U(n ? p)

We find the precise number of non-K?hler SO(2n)-invariant Einstein metrics on the generalized flag manifold M = SO(2n)/U(p)×U(n−p) with n ≥ 4 and 2 ≤ p ≤ n−2. We use an analysis on parametric systems of polynomial equations and we give some insight towards the study of such systems. We also examine the isometric problem for these Einstein metrics.     

Subexponential ultracontractivity and L p ? L q functional calculus

 We prove that if a symmetric submarkovian semigroup (T _t ) _t>0 satisfies an estimate of the form

The generalized bisymmetric solutions of the matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 + ? + A l X l B l = C and its optimal approximation

For fixed generalized reflection matrix P, i.e. P ^T  = P, P ² = I, then matrix X is said to be generalized bisymmetric, if X = X ^T  = PXP. In this paper, an iterative method is constructed to find the generalized bisymmetric solutions of the matrix equation A ₁ X ₁ B ₁ + A ₂ X ₂ B ₂ + ⋯ + A _l X _l B _l  = C where [X ₁,X ₂, ⋯ ,X _l ] is real matrices group. By this iterative method, the solvability of the matrix equation can be judged automatically. When the matrix equation is consistent, for any initial generalized bisymmetric matrix group , a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors, and the least norm generalized bisymmetric solution group can be obtained by choosing a special kind of initial generalized bisymmetric matrix group. In addition, the optimal approximation generalized bisymmetric solution group to a given generalized bisymmetric matrix group in Frobenius norm can be obtained by finding the least norm generalized bisymmetric solution group of the new matrix equation , where . Given numerical examples show that the algorithm is efficient. Research supported by: (1) the National Natural Science Foundation of China (10571047) and (10771058), (2) Natural Science Foundation of Hunan Province (06JJ2053), (3) Scientific Research Fund of Hunan Provincial Education Department(06A017).     

矩阵方程AX=B,XC=D的定秩解

本文研究了一类矩阵方程组解的秩的范围.利用矩阵的奇异值分解以及Frobenius范数的特征,得到了解的极值秩以及解的通式,并就这些问题的特殊情况进行了讨论,得到了一些结果.     

Sum of sequence spaces and matrix transformations mapping in s α 0 ((Δ ? λ I ) h ) + s β ( c ) ((Δ ? μ I ) l )

We deal with the sum of sequence spaces. Then we apply these results to characterize matrix transformations mapping between s ^h,l (λ, μ) = s _α ⁰((Δ − λI) ^h ) + s _β ^(c)((Δ − μI) ^l ) and s _γ . Among other things the aim of this paper is to reduce the set (s ^h,l (λ, μ), s _γ to a set of the form S _τ,γ .      

Gaussian Estimates of Order α and L p -Spectral Independence of Generators of C 0-Semigroups

We prove L^p-spectral independence for generators of C₀-semigroups estimated by the positive C₀-semigroup . In the preliminary process of the proof, we obtain the asymptotic expansion formula for the integral kernel of the C₀-semigroup .