Some recurrence relations in finite topologies

In a number of papers (see, e.g., RZhMat, 1977, 11B586) there is given for the number T_o(n) of labeled topologies on n points satisfying the T_o separation axiom the formula $$T_0 (n) = \sum {\frac{{n!}}{{p_1 ! \ldots p_k !}}V(p_1 , \ldots p_k ),} $$ where the summation extends over all ordered sets (p₁,...,P_k) of natural numbers such that p₁+...+P_k=n. In the present paper there is found a relation for calculating, whenn?2, the sum of all terms in this formula for which p₂=1 in terms of the values V(q₁,...,q_t) withq₁+...+q_t?n-2. This permits the determination (with the aid of a computer) of the new value T_o(12)=414 864 951 055 853 499     

On the Vinogradov additive problem

Let us state the main result of the paper. Suppose that the collection N ₁, ..., N _n is admissible. Then, in the representation $$ \left\{ \begin{gathered} p_1 + p_2 + \cdots + p_k = N_1 , \hfill \\ \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \hfill \\ p_1^n + p_2^n + \cdots + p_k^n = N_n , \hfill \\ \end{gathered} \right. $$ where the unknowns p ₁, p ₂, ..., p _k take prime values under the condition p _s > n+ 1, s = 1, ..., k, the number k is of the form $$ k = k_0 + b\left( n \right)s, $$ where s is a nonnegative integer. Further, if k ₀ ≥ a, then, in the representation for k, we can set s = 0, but if k ₀ ≤ a ? 1, then, for a given k ₀ there exist admissible collections (N ₁, ..., N _n ) that cannot be expressed as k ₀ summands of the required form, but can be expressed as k ₀ + b(n) summands.     

A Supplement to Hölder’s Inequality. The Resonance Case. I

Suppose that m ≥ 2, numbers p₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ₁ ∈ \({L^{{p_1}}}\)(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγ_k ∈ \({L^{{p_k}}}\)(?¹) under which the resonance set of each function γ_k + Δγ_k coincides with that of γ_k for 1 ≤ k ≤ m, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces L ^p (?¹), p ∈ (1, +∞], were introduced by the author in his previous papers.     

Многомерные вариаци и множеств и их контин генции

LetE be a compact set inR ⁿ (n≧2), and denote byV ₀(E) the number of the components ofE. Letp=1,2, ...,n?1;k=0,1, ...,p, and $$V_k (E;n,p) = \int\limits_{\Omega _k^n } {V_0 (E \cap \tau )^{{{(n - p)} \mathord{\left/ {\vphantom {{(n - p)} {(n - k)}}} \right. \kern-\nulldelimiterspace} {(n - k)}}} d\mu _\tau ,}$$ whereΩ _k ⁿ is the set of all (n-k)-dimensional hyperplanesτ?R ⁿ anddμ _τ is the Haar measure on the spaceΩ _k ⁿ ; furthermore, let $$V_n (E;n,n - 1) = mes_n E.$$ . Theorem. Let E?Rⁿ, p=1, 2 ..., n?1, V_p+1(E;n,p)=0, and V_k(E; n, p)<∞ for k= =0,1, ..., p. Then the contingency of the set E at a point x∈E coincides with a certain p-dimensional hyperplane for almost all points x∈E in the sense of Hausdorff p-measure.     

On estimating the hidden periodicities in linear time series models

In this paper the tollowing modelX(n)=sum from j=1 to p α_je~(inλj)+ξ_nis considered,where p,λ_1,λ_2,…,λ_p,are constants α=(α_1,α_2,…,α_p) is a random vector and {ξ_n;n=0,±1,±2,…} is a wide-sense stationary sequence with zero means.In [4],theorem about thestrong consistent estimates of λ_1,λ_2.…,λ_p and α are proved under the assumption that α is a constantvector and p and δ are known constants such that0<δ<(?){λ_i-λ_j}.The main purpose of the present paper is to prove theorems on the strong consistent estimates ofparameters p,λ_1,λ_2,…,λ_p and random vector α without knowing p and δ.Numerical examples arealso given to illustrate our method of estimation.     

A relation in finite topologies

For the number T₀(n) of labeled topologies on n points with the separation axiom T₀, a series of authors have obtained the formula $$T_0 (n) = \sum {\frac{{n!}}{{P_1 ! \cdot \cdot \cdot P_m !}}} V(P_1 , \cdot \cdot \cdot P_m ),$$ in which the summation is taken over all ordered partitions n=p₁ + ...+ p_m ^into natural summands (cf., e.g., RZhMat, 1977, 11B586). In this note it is proved that in the cited formula the sum of all those summands for which p₁=2 is equal to n(n?1)/2·T₀(n?1).     

Supplement to Hölder’s inequality. II

Suppose that m ≥ 2, numbers p ₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + \cdots + \frac{1}{{{p_m}}} < 1\), and functions \({\gamma _1} \in {L^{{p_1}}}\left( {{?^1}} \right), \cdots ,{\gamma _m} \in {L^{{p_m}}}\left( {{?^1}} \right)\) are given. It is proved that if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both notions were defined by the author for functions in L ^p (?¹), p ∈ (1, +∞]), then \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\mathop \smallint \limits_a^b \prod\limits_{k = 1}^m {[{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)]} d\tau } \right| \leqslant C\prod\limits_{k = 1}^m {{{\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|}_{L_{ak}^{pk}\left( {{R^1}} \right)}}} \) where the constant C > 0 is independent of the functions \(\Delta {\gamma _k} \in L_{ak}^{pk}\left( {{?^1}} \right)\) and \(L_{ak}^{pk}\left( {{?^1}} \right) \subset {L^{pk}}\left( {{?^1}} \right)\), 1 ≤ k ≤ m, are special normed spaces. A condition for the integral over ?¹ of a product of functions to be bounded is also given.     

Algebraic integers with discriminants containing fixed prime divisors

It is proved that any algebraic integer a of degree n 2whose discriminant is a product of powers of prescribed primes p₁, ..., pr has the form , where , V₁, ..., v_r are rational integers and is an integer whose height does not exceed an effectively defined bound depending on max (p₁, ..., pr), r, and n.Translated from Matematicheskie Zametki, Vol. 21, No. 3, pp. 289–296, March, 1977.The author would like to thank V. G. Sprindzhuk for his assistance and unfailing interest.     

Minimizing and maximizing the Euclidean norm of the product of two polynomials

We consider the problem of minimizing or maximizing the quotient $$f_{m,n}(p,q):=\frac{\|{pq}\|}{\|{p}\|\|{q}\|} \ ,$$ where $p=p_0+p_1x+\dots+p_mx^m$ , $q=q_0+q_1x+\dots+q_nx^n\in{\mathbb K}[x]$ , ${\mathbb K}\in\{{\mathbb R},{\mathbb C}\}$ , are non-zero real or complex polynomials of maximum degree $m,n\in{\mathbb N}$ respectively and $\|{p}\|:=(|p_0|^2+\dots+|p_m|^2)^{\frac{1}{2}}$ is simply the Euclidean norm of the polynomial coefficients. Clearly f _m,n is bounded and assumes its maximum and minimum values min f _m,n ?=?f _m,n (p _min, q _min) and max f _m,n ?=?f(p _max, q _max). We prove that minimizers p _min, q _min for ${\mathbb K}={\mathbb C}$ and maximizers p _max, q _max for arbitrary ${\mathbb K}$ fulfill $\deg(p_{\min})=m=\deg(p_{\max})$ , $\deg(q_{\min})=n=\deg(q_{\max})$ and all roots of p _min, q _min, p _max, q _max have modulus one and are simple. For ${\mathbb K}={\mathbb R}$ we can only prove the existence of minimizers p _min, q _min of full degree m and n respectively having roots of modulus one. These results are obtained by transferring the optimization problem to that of determining extremal eigenvalues and corresponding eigenvectors of autocorrelation Toeplitz matrices. By the way we give lower bounds for min f _m,n for real polynomials which are slightly better than the known ones and inclusions for max f _m,n .     

The knaster problem: More counterexamples

Given a continuous function\(f:\mathbb{S}^{n - 1} \to \mathbb{R}^m \) andn ?m + 1 pointsp ₁, …,p _{n?m + 1} ε\(p_1 ,...,p_{n - m + 1} \in \mathbb{S}^{n - 1} \), does there exist a rotation ? εSO(n) such thatf(?(p ₁)) = … =f(?(p _n?m+1))? We give a negative answer to this question form = 1 ifn ε {61, 63, 65} orn≥67 and form=2 ifn≥5.     

On Marcus-Wang conjecture

LetA _m ,B _m ,m=1, ...,p, be linear operators on ann-dimensional unitary space \(V.L = \sum\limits_{m = 1}^p {A_m \otimes B_m } \) is a linear operator on ?² V, the tensor product space with the customarily induced inner product. The numerical range ofL is defined as $$W\tfrac{1}{2}(L) = \left\{ {(L)x \otimes y,x \otimes y):x,y o.n.} \right\}$$ where “o.n.” means “orthonormal”. In [1], M.Marcus and B.Y. Wang conjecture: There exists no non-zero operatorL of minimum length less thann for whichW ₂ ¹ (L)=0. In this paper, we prove that this conjecture is true.     

On the positive nonoscillatory solutions of the difference equation Xn＋1=α＋（xn-k/xn-m）^p

The aim of this paper is to show that the following difference equation：Xn＋1=α＋（xn-k/xn-m）^p, n=0,1,2,…,
where α 〉 -1, p 〉 O, k,m ∈ N are fixed, 0 ≤ m 〈 k, x-k, x-k＋1,…,x-m,…,X-1, x0 are positive, has positive nonoscillatory solutions which converge to the positive equilibrium x=α＋1. It is interesting that the method described in the paper, in some cases can also be applied when the parameter α is variable.     

On the inverse problem relative to dynamics of the <Emphasis Type="Italic">w</Emphasis> function

Let P be the set of prime numbers and P(n) denote the largest prime factor of integer n > 1. Write

K 0 of hypersurfaces defined by x 1 2 + ... + x n 2 = ±1

Let k be a field of characteristic ≠ 2 and let Q _n,m (x ₁, ..., x _n , y ₁, ..., y _m ) = x ₁ ² +...+x _n ² ? (y ₁ ² +...+y _m ² ) be a quadratic form over k. Let R(Q _n,m ) = R _n,m = k[x ₁, ..., x _n , y ₁, ..., y _m ]/(Q _n,m ? 1). In this note we will calculate $\tilde K_0 \left( {R_{n,m} } \right)$ for every n,m ≥ 0. We will also calculate CH ₀(R _n,m ) and the Euler class group of R _n,m when k = ?.     

Stability of solutions of functional equations connected with problems of characterizing probability distributions

The work contains some results pertaining to the solution ψ_j(x) of the functional equation $$\left| {\Sigma \Psi _j \left( {a_j^T t} \right)} \right| \leqslant \varepsilon ,$$ where a _j ^T =(a_1j, a_2j, ..., a_pj)∈ ?^p, all the coefficients a_ij are constant, t=(t₁, t₂, ..., t_p) ∈ ?^p, \(a_j^T t = \sum\limits_{i - 1}^p {a_{ij} t_i } ,p \geqslant 2\) and the relation (*) is satisfied for all Inequality (*) is connected with certain characterization theorems of probability theory and statistics. For simplicity, it is assumed that the ψ_j(x) are continuous functions, x∈?¹. The following basic ressult is obtained.     

On one complement to the Hölder inequality: I

Let m ≥ 2, the numbers p ₁,…, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ...\frac{1}{{{p_m}}} < 1\), and γ₁ ∈ L ^p1(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹). We prove that, if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both concepts have been introduced by the author for functions of spaces L ^p (?¹), p ∈ (1, +∞]), we have the inequality \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\int\limits_a^b {\prod\limits_{k = 1}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]} d\tau } } \right| \leqslant C{\prod\limits_{k = 1}^m {\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|} _{L_{{a_k}}^{{p_k}}}}\left( {{\mathbb{R}^1}} \right)\), where the constant C > 0 is independent of functions \(\Delta {\gamma _k} \in L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right)\) and \(L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right) \subset {L^{{p_k}}}\left( {{\mathbb{R}^1}} \right)\), 1 ≤ k ≤ m are some specially constructed normed spaces. In addition, we give a boundedness condition for the integral of the product of functions over a subset of ?¹.     

Fast Growing Solutions of Linear Differential Equations in the Unit Disc

For $n \in \mathbb{N}$ , the n-order of an analytic function f in the unit disc D is defined by $$\sigma _{{{M,n}}} (f) = {\mathop {\lim \sup }\limits_{r \to 1^{ - } } }\frac{{\log ^{ + }_{{n + 1}} M(r,f)}} {{ - \log (1 - r)}},$$ where log⁺ x  =  max{log x, 0}, log ⁺ ₁ x  =  log ⁺ x, log ⁺ _n+1 x  =  log ⁺ log ⁺ _n x, and M(r, f) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equation $$f^{{(k)}} + a_{{k - 1}} (z)f^{{(k - 1)}} + \cdots + a_{1} (z)f^{\prime} + a_{0} (z)f = 0,\quad \quad \quad (\dag)$$ where the coefficients are analytic in D, satisfy σ _M,n+1(f)  ≤  α if and only if σ _M,n (a _j )  ≤  α for all j  =  0, ..., k ? 1. Moreover, if q ∈{0, ..., k ? 1} is the largest index for which $\sigma _{M,n} ( a_{q}) = {\mathop {\max }\limits_{0 \leq j \leq k - 1} }{\left\{ {\sigma _{{M,n}} {\left( {a_{j} } \right)}} \right\}}$ , then there are at least k ? q linearly independent solutions f of ( $\dag$ ) such that σ _M,n+1(f) = σ _M,n (a _q ). Some refinements of these results in terms of the n-type of an analytic function in D are also given.     

Extremal property of some surfaces in n-dimensional Euclidean space

A surface Γ=(f ₁(X₁,..., x_m),...,f _n(x₁,..., x_m)) is said to be extremal if for almost all points of Γ the inequality $$\parallel a_1 f_1 (x_1 , \ldots ,x_m ) + \ldots + a_n f_n (x_1 , \ldots ,x_m )\parallel< H^{ - n - \varepsilon } ,$$ , where H=max(¦a _i¦) (i=1, 2, ..., n), has only a finite number of solutions in the integersa ₁, ...,a _n. In this note we prove, for a specific relationship between m and n and a functional condition on the functionsf ₁, ...,f _n, the extremality of a class of surfaces in n-dimensional Euclidean space.     

Note on the local growth of iterated polynomials

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_i, for i=1, 2, ..., n+2, be defined recursively by x_i+1=f(x_i), where x₁ is an arbitrary real number and f is a polynomial of degree n. Let x_i+1?x_i≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1?i, $$ - \frac{{2^{k - 1} }}{{k!}}< f\left[ {x_1 ,... + x_{i + k} } \right]< \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ where f[x_i, ..., x_i+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {x_i}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_i+1=f(x_i) where x₁ is an arbitrarily assigned real number and f is the polynomial \(f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2\) . Let δ be positive with δ^n?1=|2^n?1/n!a_n|. Then, if n is even and a_n<0, there do not exist n + 1 consecutive increments Δx_i=x_i+1?x_i in the solution {x_i} with Δx_i≧δ. The special case in which the iterated polynomial has integer coefficients leads to a “nice” upper bound on a generalization of the van der Waerden numbers. Ap _k -sequence of length n is defined to be a strictly increasing sequence of positive integers {x ₁, ...,x _n } for which there exists a polynomial of degree at mostk with integer coefficients and satisfyingf(x _j )=x _j+1 forj=1, 2, ...,n?1. Definep _k (n) to be the least positive integer such that if {1, 2, ...,p _k (n)} is partitioned into two sets, then one of the two sets must contain ap _k -sequence of lengthn. THEOREM. p_n?2(n)≦(n!)^(n?2)!/2.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.