Non-homogeneous non-linear damped wave equations in unbounded domains

We present a global existence theorem for solutions of u^tt ? ?_ia_ik (x)?_ku + u_t = ?(t, x, u, u_t, ?u, ?u_t, ?²u), u(t = 0) = u⁰, u(=0)=u¹, u(t, x), t ? 0, x?Ω.Ω equals ?³ or Ω is an exterior domain in ?³ with smoothly bounded star-shaped complement. In the latter case the boundary condition u|_?Ω = 0 will be studied. The main theorem is obtained for small data (u⁰, u¹) under certain conditions on the coefficients a_ik. The L^p - L^q decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.     

On the product of sign vectors and unit vectors

For a fixed unit vectora=(a ₁,...,a _n )S ^n-1, consider the 2 ⁿ sign vectors=(₁,..., _n ){±1{ ⁿ and the corresponding scalar products^·a = _n ⁱ⁼¹ = _i a _i . The question that we address is: for how many of the sign vectors must^.a lie between–1 and 1. Besides the straightforward interpretation in terms of the sums ±a ₂ , this question has appealing reformulations using the language of probability theory or of geometry.The natural conjectures are that at least 1/2 the sign vectors yield |.a|1 and at least 3/8 of the sign vectors yield |.a|<1 (the latter excluding the case when |a _i |=1 for somei). These conjectured lower bounds are easily seen to be the best possible. Here we prove a lower bound of 3/8 for both versions of the problem, thus completely solving the version with strict inequality. The main part of the proof is cast in a more general probabilistic framework: it establishes a sharp lower bound of 3/8 for the probability that |X+Y|<1, whereX andY are independent random variables, each having a symmetric distribution with variance 1/2.We also consider an asymptotic version of the question, wheren along a sequence of instances of the problem satisfying ||a||0. Our result, best expressed in probabilistic terms, is that the distribution of .a converges to the standard normal distribution, and in particular the fraction of sign vectors yielding .a between –1 and 1 tends to 68%.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Concerning algebraic independence of some transcendental numbers

Given the three numbers of,a ^{1 2} , and Ina ₂/Ina ₁, wherea ₁ anda ₂ are algebraic numbers whose logarithms are linearly independent in a rational field and is a quadratic irrationality, it is shown that they are not all expressible algebraically in terms of one of them.Translated from Matematicheskie Zametki, Vol.3, No. 1, pp. 51–58, January, 1968.     

Another Note on the Greatest Prime Factors of Fermat Numbers

For every positive integer k > 1, let P(k) be the largest prime divisor of k. In this note, we show that if F_m = 2^2m + 1 is the mth Fermat number, then P(F_m) 2^m+2(4m + 9) + 1 for all m 4. We also give a lower bound of a similar type for P(F_a,m), where F_a,m = a^2m + 1 whenever a is even and m a¹⁸.AMS Subject Classification (1991) 11A51 11J86     

Eine Ω-Abschätzung zu einem zweidimensionalen Gitterpunktproblem

LetG ₂ (R) denote the number of lattice points (x, y) (i. e. points of the plane with integer coordinates) in the domainx ^a +y ^a R,x0,y0 (with weight 1/2 ifxy=0) and letV ₂ (a)R ^2/a be the area of this region. Then for 0<a<1/3 it is known [2] thatG ₂ (R)–V ₂ (a)R ^2/a K(a)R ^(1/a)–1 . Combining a method due toBleicher andKnopp [1] with a result proved elsewhere [4], [5] by the author it is shown here thatG ₂ (R)=V ₂ (a)R ^2/a +(R ^(1/a)–1 ) for any fixed rational numbera with 1/3<a<1/2.     

On extending backwards positive definite sequences

The two-sided Hamburger moment problem¹, also called the strong one [4], has been extensively studied in recent years in connection with rational approximation. We propose to consider the question of when a sequence, say {a _n } _n=0 can be extended backwards so that the resulting sequence {a _n } _n=–N has an integral representation of the Hamburger type. This was settled (without any proof) under different circumstances in [6]. Here we wish to discuss this completely, as well as the possibility of extending {a _n } _n=0 to {a _n } _n– .     

Distributive multiplication rings

A ringR is said to be a left (right)n-distributive multiplication ring, n>1 a positive integer, if aa₁a₂...a_n=aa₁aa₂...aa_n (a₁a₂...a_na=a₁aa₂a...a_na) for all a, a₁,...,a_n R. It will be shown that the semi-primitive left (right)n-distributive rings are precisely the generalized boolean ringsA satisfying aⁿ=a for all a A. An arbitrary left (right)n-distributive multiplication ring will be seen to be an extension of a nilpotent ringN satisfyingN ⁿ⁺¹=0 by a generalized boolean ring described above. Under certain circumstances it will be shown that this extension splits.     

Weakly Associated Prime Filtration

Let R be a (not necessarily Noetherian) commutative ring and let M be a (not necessarily finitely generated) R-module. We characterize the modules with only finitely many weakly associated primes as those modules M admitting a chain 0 = M ₀ M ₁ ... M _n = M of submodules together with prime ideals p₁, p₂,...,p _n such that the set of weakly associated primes of M _i /M _i-1 is equal to {p _i } for all 1 i n. Let M = gr_a(M) = _n0a ⁿ M/a ⁿ⁺¹ M be the corresponding graded module over the graded ring R = gr_a(R) = _n0a ⁿ /a ⁿ⁺¹. It is shown that the union of the set of weakly associated primes of.....     

Groups with a class of Frobenius-Abelian elements

Let G be a group, a G ^#, and let F_a be the set of all Frobenius subgroups with noninvariant factor a in G. In Theorems 1–3, we show that if a ² 1and G has sufficiently many subgroups a, a ^g F _a. then a^G F _a.An element a is called (almost) Frobenius if, for (almost) all elements a ^g,the subgroup a, a ^g either belongs to F _a or is Abelian. In Theorems 4–5, we investigate the structure of a ^G in G for the case where a is an (almost) Frobenius-Abelian element of order 2.In Theorem 6, we prove that a binary factorable group is locally completely factorable. Translated fromAlgebra i Logika, Vol. 34, No. 5, pp. 531–549, September-October, 1995.     

Proof of Oppenheim's area inequalities for triangles and quadrilaterals

Leta ₁,b ₁,c ₁,A ₁ anda ₂,b ₂,c ₂,A ₂ be the sides and areas of two triangles. Ifa=(a ₁ ^p +a ₂ ^p )^1/p ,b=(b ₁ ^p +b ₂ ^p )^1/p ,c=(c ₁ ^p +c ₂ ^p )^1/p , and 1p4, thena, b, c are the sides of a triangle and its area satisfiesA ^p/2A ₁ ^p/2 +A ₂ ^p/2 . If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaA _f, which satisfies (4A _f/3)^1/2f((4A/3)^1/2).     

Density of Gabor Frames

A Gabor system is a set of time-frequency shifts S(g, Λ) ={e^{2 π ibx}g(x − a)}_{(a, b) Λ} of a function g L²(R^d). We prove that if a finite union of Gabor systems _{k = 1}^rS(g_k, Λ_k) forms a frame for L²(R^d) then the lower and upper Beurling densities of Λ = _{k = 1}^r Λ_k satisfy D⁻(Λ) ≥ 1 and D ⁺ (Λ) < ∞. This extends recent work of Ramanathan and Steger. Additionally, we prove the conjecture that no collection _{k = 1}^r{g_k(x − a)}_{a Γk} of pure translates can form a frame for L²(R^d).     

The algebraic and geometric classification of associative algebras of dimension five

The scheme Alg₅ of associative, unitary algebra structures on k⁵, k an algebraically closed field with char (k)2 is investigated. We establish the list of GL₅-orbits on Alg₅ under the action of structural transport. The numberalg₅ of irreducible components of Alg₅ is 10; a list of generic structures is included. We exhibit upper and lower bounds for the asymptotic behaviour of the numberalg_n.     

Two Conditions for Infinite Groups to Satisfy Certain Laws

Let G be an infinite group and m {2^k | k N^*}. In this paper, we prove that G satisfies the law [x_m, y_m] = 1 if and only if in any two infinite subsets X and Y of G, there exist a X and b Y such that [a^m,b^m] = 1. We also prove that G satisfies the law (x₁^mx₂^m x_n^m)² = 1 if and only if in any n infinite subsets X₁, X₂,..., X_n, there exist a_i X_i (i = 1,..., n) such that (a₁^ma₂^m a_n^m)² = 1.2000 Mathematics Subject Classification: 20F99     

On the Problem of Describing Sequences of Best Trigonometric Rational Approximations

For a strictly decreasing sequence a_{n n=0} of nonnegative real numbers converging to zero, we construct a continuous 2-periodic function f such that R^T _n(f) = a_n, n=0,1,2,..., where R^T _n(f) are best approximations of the function f in uniform norm by trigonometric rational functions of degree at most n.     

How big are the increments of a multi-parameter Wiener process?

Summary Let _T = (2a _T(log Ta _T ^–1 +log log T))^–1/2, 0< a _T T< and let R^* be the set of sub-rectangles of the square [0, T ^1/2]x[0, T^1/2], having an area a _T . This paper studies the almost sure limiting behaviour of as T, where W is a two-time parameter Wiener process. With a _T =T, our results give the well-known law of iterated logarithm and a generalization of the latter is also attained. The multi-time parameter analogues of our twotime parameter Wiener process results are also stated in the text.Research partially supported by a Canadian NRC grant and a Canada Council Leave Fellowship     

An application of an optimal behaviour of the greedy solution in number theory

Leta ₁,a ₂, ...,a _n be relative prime positive integers. The Frobenius problem is to determine the greatest integer not belonging to the set { _j=1 ⁿ a _j x _j :xZ ₊ ⁿ }. The Frobenius problem belongs to the combinatorial number theory, which is very rich in methods. In this paper the Frobenius problem is handled by integer programming which is a new tool in this field. Some new upper bounds and exact solutions of subproblems are provided. A lot of earlier results obtained with very different methods can be discussed in a unified way.     

Compact operators whose singular numbers have powerlike asymptotics

For a compact operator in a Hilbert space, let s_n(A), n =1, 2,... be the singular numbers and let N(s; A) =card{n N:s_n(A)>, s>0. For 0

a _p and not on the individual elementAa, (H. Weyl's lemma); this allows us to write _p (a), _pp (a), a_p. One obtains certain results regarding the functionals _p, _p (and about the analogous functionals for the positive and negative eigenvalues in the casea=a ^*=A ^*:A a. In particular: I. Ifa ₁ a _2p, then. II.Let a ₁,a _{2 pP} ,.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matetmaticheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 21–30, 1983.     

Approximation by Fourier sums of classes of functions with several bounded derivatives

An ordered estimate is obtained for the approximation by Fourier sums, in the metric ofd=(d ₁ , ...,d _n ), 1<dj<,j=1, ...,_n of classes of periodic functions of several variables with zero means with respect to all their arguments, having m mixed derivatives of order a¹..., a^m., aⁱ rⁿ. which are bounded in the metrics ofp ⁱ =p ₁ ⁱ , ..., p _n ⁱ , i

j ⁱ <,i=i, ...,n, j=1, ...,n by the constants ₁, ., _m, respectively.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 197–212, February, 1978.     

Roots under convolution of sequences

The convolution a * b of the sequences a = a₀, a₁, a₂, and b is the sequence with elements ∑₀ⁿ a_kb_{n − k}. One sets 1, 1, 1, equal to σ. Given that a * a with a ≥ 0 is close to σ * σ, how close is a to σ? More generally, one asks how close a is to σ if the p-th convolution power, a*P with a ≥ 0, is close to σ*P. Power series and complex analysis form a natural tool to estimate the ‘summed deviation’ ρ = σ * (a — σ) in terms of b = a * a — σ * σ or b = a*P − σ*P. Optimal estimates are found under the condition ∑_k=0ⁿ b_k² = %plane1D;512;(n^{2β + 1}) whenever −½ < β < p − 1. It is not known what the optimal estimates are for the special case b_n = %plane1D;512;(n^β).     

Cross-monotone subsequences

Two finite real sequences (a ₁,...,a _k ) and (b ₁,...,b _k ) are cross-monotone if each is nondecreasing anda _i+1–a _i b _i+1–b _i for alli. A sequence (₁,..., _n ) of nondecreasing reals is in class CM(k) if it has disjointk-term subsequences that are cross-monotone. The paper shows thatf(k), the smallestn such that every nondecreasing (₁,..., _n ) is in CM(k), is bounded between aboutk ²/4 andk ₂/2. It also shows thatg(k), the smallestn for which all (₁,..., _n ) are in CM(k)and eithera _k b ₁ orb _k a ₁, equalsk(k–1)+2, and thath(k), the smallestn for which all (₁,..., _n ) are in CM(_k)and eithera ₁b ₁...a _k b _k orb ₁a ₁...b _k a _k , equals 2(k–1)²+2.The results forf andg rely on new theorems for regular patterns in (0, 1)-matrices that are of interest in their own right. An example is: Every upper-triangulark ²×k ² (0, 1)-matrix has eitherk 1's in consecutive columns, each below its predecessor, ork 0's in consecutive rows, each to the right of its predecessor, and the same conclusion is false whenk ² is replaced byk ²–1.