Upper bound for the product of nonhomogeneous linear forms

It is proved that for any unimodular lattice Λ with homogeneous minimum L>0 and any set of real numbers α₁, α₂,..., α_n there exists a point (y₁, y₂,..., y_n) of Λ such that $$\Pi _{1 \leqslant i \leqslant n} |y_i + \alpha _i | \leqslant 2^{ - n/2_\gamma n} (1 + 3L^{8/(3n)/(\gamma ^{2/3} - 2L^{8/(3n)} )} )^{ - n/2} ,$$ where γⁿ= n^n/(n?1).     

Estimates of the nonhomogeneous arithmetic minimum of the product of linear forms

Further refinements of Chebotarev type estimates are obtained for the inhomogeneous arithmetic minimum M_n of a lattice Λ of determinant d(Λ) in the inhomogeneous Minkowski conjecture. In particular, it is proved that for every n₀≥2 there exists an effectively computed constant c=c(n₀) for which $$M_n \leqslant 2^{ - n/2} (cn^{ - 1/2} log^{1/2} n) d (\Lambda )$$ .     

Refinement of estimates for some nonhomogeneous linear forms

In this paper, estimates for nonhomogeneous linear forms in values of hypergeometric functions with irrational parameters are refined. This refinement is carried out by means of an optimal choice of the degree of the zeroth polynomial appearing in the construction of approximate functional forms.Translated from Matematicheskie Zametki, vol. 77, no. 4, 2005, pp. 515–521.Original Russian Text Copyright © 2005 by P. L. Ivankov.This revised version was published online in April 2005 with a corrected issue number.     

A refinement of an estimate of the arithmetic minimum of the product of nonhomogeneous linear forms (regarding Minkowski's nonhomogeneous conjecture)

One refines an estimate of B. F. Skubenko (Tr. Mat. Inst. Akad. Nauk 148, 218–224 (1978)). Let be a point lattice of determinant d() in the n-dimensional Euclidean space ⁿ. and let L^ñ. We consider the nonhomogeneous One proves that there exists an effectively computable constant n_o such that if nn_o, then.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 82, pp. 88–94, 1979.     

An upper estimate for the product linear nonhomogeneous forms for lattices with a small homogeneous minimum

One obtains the following transfer theorem, related to Minkowski's nonhomogeneous conjecture. Let Λ ? ?ⁿ be a point lattice, det Λ=1. We consider the nonhomogeneous Π(Λ) and the homogeneous L(Λ)=бⁿ. б ? 0, arithmetical minima of the lattice Λ. Then for sufficiently largen, if , then .     

A refinement of the estimates of the arithmetic minimum of the product of nonhomogeneous linear forms for large dimensions

Let M _n be the nonhomogeneous arithmetic minimum of the lattice Λ ? ?ⁿ of determinant Δ>0. It is proved that forn?1,1·n⁵, we have . One obtains other estimates of a similar type in the nonhomogeneous Minkowski conjecture, obtained for the first time by N. G. Chebotarev. These estimates strengthen the results of B. P. Skubenko.     

The existence of linear factors in the direct sum of forms and in the tensor product of forms

The tensor product f⊗g of two forms of degree r has a linear factor if and only if both fand g have a linear factor and the sum of the multiplicities of these factors is larger than r. In fact, if (V,ϕ) and (Wθ) are symmetric spaces corresponding to fand g with linear factors U and T respectively the sum of whose multiplicities is larger than r, then the linear factor of (V,ϕ)⊗(Wθ)is U⊗W+V⊗T. Conversely, each linear factor of (V,ϕ)⊗(Wθ)is of this form. Additionally, the direct sum of two nondegenerate symmetric spaces has a linear factor if and only if each is one dimensional and isomorphic to the negative of the other.     

Theory of nonhomogeneous linear ordinary differential equations

One constructs a general formula for the solution of the Cauchy problem in the case of a general linear ordinary differential equation of an arbitrary order.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 6–12, 1983.     

非齐次线性差分方程的Cm解[英文]

设c0,c1,…,cn均为实的常数，F（x)是个从R到R的C^m映射。本文讨论了非齐次线性差分方程∑i=1ncif(x i)=F(x)的C^m(m≥0）的存在性和唯一性。     

Oscillatory solutions of nonhomogeneous linear differential equations

The complex oscillation of nonhomogeneous linear differential equations with transcendental coefficients is discussed. Results concerning the equation f ^(k)+a _k−1 f ^(k−1)+...+a ₀ f=F where a ₀,...,a _k−i and Fare entire functions, possessing an oscillatory solution subspace in which all solutions (with at most one exception) have infinite exponent of convergence of zeros are obtained. All solutions of the equation are also characterized when the coefficients a ₀,a ₁,...,a _k−1 are polynomials and F=h exp (p ₀), where p ₀ is a polynomial and h is an entire function. Author supported by Max-Planck-Gesellschaft and by NSFC.     

A linear algorithm for nonhomogeneous spectra of numbers

Given a sequence of integers [a_i]_i=1ⁿ, an O(n) iterative algorithm is presented which decides whether there exist real numbers α and β such that a_i = [iα + β] (1 ? i ? n). In fact, the linear algorithm computes the partial quotients of the continued fraction expansions of $\text{d}$ and $\text{d}$ such that $\text{d < α <}\text{d}$ if and only if a_i = [iα + β] (1 ? i ? n) for suitable β = β(α).     

高阶亚纯系数非齐次线性微分方程的复振荡

对一类高阶亚纯系数非齐次线性微分方程的复振荡进行了研究,应用值分布的理论和方法,得到了它的亚纯解的超级,二级不同零点收剑指数的精确估计.     

An isolation theorem for the arithmetic minimum of the product of linear forms with complex coefficients

A strong isolation theorem is stated for the case of m3 pairs of complex conjugate linear forms L₁(x), ¯L₁(x), ..., L_m(x), ¯L_m(x), which define a form of degree n=2m indecomposable in 2e. This theorem is a direct analog of Skubenko's result for real linear forms [2].Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 151, pp. 5–6, 1986.     

一类高阶变系数线性非齐次微分方程的解

利用高阶变系数之间的关系,通过适当的线性变换,得到了五阶变系数线性非齐次方程常系数化的条件,给出了一类高阶变系数线性非齐次微分方程的新解法.     

On the solutions of certain linear nonhomogeneous second-order differential equations

The system Nω=(N-α)ω+y, N= bN+aωω^T, N(t)?R^m×m, ω(t)?R^m which originally arose from a model for the pathological behavior of neural networks, is studied. Similar equations can arise in a variety of applications. It is shown that if N(0) is positive definite, then solutions exist for all time. Equilibrium points are determined. N is found to be singular at the equilibrium points, making the analysis of the asymptotic properties of the system non-trivial. The asymptotic behavior when y = 0 is completely described. Some results are proven on the asymptotic behavior of N and ω when y≠0