The product of linear nonhomogeneous forms

We show that for an arbitrary unimodular lattice Λ of dimension n and an arbitrary point C =(c₁, c₂...c_n) ? Rⁿ a point Y = (y₁, y₂,..., y_n) ε Λ can be found and also a number h, satisfying the condition 1 ?h ? 2^?n/2 θ^?1 + 1 (0 < θ ? 2^?n/2), such that the inequality $$\prod\nolimits_{i = 1}^n {\left| {Y_i + hc_i } \right|}< \theta $$ will be satisfied.     

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 $$M_n \leqslant 2^{ - n/2} (cn^{ - 1/2} log^{1/2} n) d (\Lambda )$$ .     

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 .     

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.     

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.     

A bound for the representability of large numbers by ternary quadratic forms and nonhomogeneous waring equations

The solvability of the equation n = x ² + y ² + 6pz ² (p is a fixed large prime) is proved under some natural congruential conditions and the assumption nm ¹² > p ²¹. As an implication, the solvability of the equation n = x ² + y ² + u ³ + v ³ + z ⁴ + w ¹⁶ + t ^4k+1 for all sufficiently large n is established. Bibliography: 13 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 357, 2008, pp. 5–21.     

A lower bound for linear forms in values of a continued fraction

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β ₁ y ₁ + β ₂ y ₂ + α| is given, where y ₁ = y(x ₁), y ₂ = y(x ₂), x ₁ and x ₂ are positive integers, α, β ₁ and β ₂ are algebraic irrational numbers.     

A lower bound for linear forms in values of certain hypergeometric functions

Lower bounds are found for linear forms in values of certain hypergeometric functions and their derivatives, analogous to the bounds obtained by K. Mahler for the linear form of the numbers 1, e, e²,..., e^m.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 19–28, July, 1970.We wish to thank A. B. Shidlovskii for his interest in this work and for his many useful suggestions.     

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).     

Upper bound multigraphs for posets

The study of upper bound graphs of posets can be extended naturally to multigraphs. This paper characterizes such upper bound multigraphs, shows they determine the associated posets up to isomorphism, and extends results of D. Scott to characterize posets having chordal or interval upper bound multigraphs.Research partially supported by Office of Naval Research contract N00014-88-K-0163.     

A lower bound for the first eigenvalue of a vibrating nonhomogeneous plate

Summary In this note we obtain a lower bound for the first eigenvalue of a nonhomogeneous plate problem. As a consequence we obtain an inequality of Barta-type.

---

Résumé En cette note nous obtenons une borne inférieure pour la première valeur propre d'une plaque inhomogène. En conséquence nous obtenons une inégalité de type Barta.

     

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.     

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.     

Upper bound of weak-limitation for the Gross-Pitaevskii equation

This paper is concerned with the blow-up solutions of Gross-Pitaevskii equation. We obtain the upper bound of weak-limitation for the blow-up solutions by using the method of Cazenave (2003) [3] as well as the concentration compact principle.     

Upper estimates for the energy of solutions of nonhomogeneous boundary value problems

We establish upper bounds for the energy of critical levels of the functional associated to a perturbed superlinear elliptic boundary value problem. We show that the perturbed problem satisfies the estimates obtained by Bahri and Lions (1988) for the symmetric problem. We use these estimates to prove the existence of nonradial solutions to a radial elliptic boundary value problem. Our results fill a gap in an earlier paper by Aduén and Castro.