Rational chebyshev approximations of elementary functions

Explicit formulæ which should be useful in practical computations, are given for the functions e^–x, log x, (1 + x), arctg x/x, sin x/x, cos x and (2/x) arcsin (x/2) in the interval 0x1 except for log x, where the interval is 1/2x1 instead.     

Asymptotic Density and the Asymptotics of Partition Functions

Let A be a set of positive integers with gcd (A) = 1, and let p _A (n) be the partition function of A. Let c ₀ = 2/3. If A has lower asymptotic density and upper asymptotic density , then lim inf log p _A (n)/c ₀ n and lim sup log p _A (n)/c ₀ n . In particular, if A has asymptotic density > 0, then log p _A (n) c₀n. Conversely, if > 0 and log p _A (n) c ₀ n, then the set A has asymptotic density .     

Cutting disjoint disks by straight lines

Fork>0 letf(k) denote the minimum integerf such that, for any family ofk pairwise disjoint congruent disks in the plane, there is a direction such that any line having direction intersects at mostf of the disks. We determine the exact asymptotic behavior off(k) by proving that there are two positive constantsd ₁,d ₂ such thatd ₁k logkf(k)d ₂k logk. This result has been motivated by problems dealing with the separation of convex sets by straight lines.The work of the first author was supported in part by the Allon Fellowship, by the Bat Sheva de Rothschild Foundation, by the Fund for Basic Research administered by the Israel Academy of Sciences, and by the Center for Absorbtion in Science. Work by the second author was supported by the Technion V. P.R. Fund, Grant No. 100-0679. The third author's work was supported by the Natural Sciences and Engineering Research Council, Canada, and the joint project Combinatorial Optimization of the Natural Science and Engineering Research Council (NSERC), Canada, and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 303).     

Order of the best spline approximations of some classes of functions

The rate of decrease of the upper bounds of the best spline approximations E_m,n(f)p with undetermined n nodes in the metric of the space L_p(0, 1) (1p) is studied in a class of functionsf(x) for which f ^m+1 (x)L_q(0, 1)1(1qt8) or var {f(m) (x); 0, 1}1 (m=1, 2, ..., the preceding derivative is assumed absolutely continuous). An exact order of decrease of the mentioned bounds is found as n , and asymptotic formulas are obtained for p= and 1q in the case of an approximation by broken lines (m=1). The simultaneous approximation of the function and its derivatives by spline functions and their appropriate derivatives is also studied.Translated from Matematicheskie Zametki, Vol. 7, No. 1, pp. 31–42, January, 1970.     

The Logarithmic Frequency of Values of Additive Functions

We consider the weak convergence of distribution functions (_mx 1/ m)^-1 _{m x},f_x(m)x is a set (x 2) of strongly additive functions such that f_x(p){0,1} for each prime number p.     

The tree number of a graph with a given girth

A family of subtrees of a graphG whose edge sets form a partition of the edge set ofG is called atree decomposition ofG. The minimum number of trees in a tree decomposition ofG is called thetree number ofG and is denoted by(G). It is known that ifG is connected then(G) |G|/2. In this paper we show that ifG is connected and has girthg 5 then(G) |G|/g + 1. Surprisingly, the case wheng = 4 seems to be more difficult. We conjecture that in this case(G) |G|/4 + 1 and show a wide class of graphs that satisfy it. Also, some special graphs like complete bipartite graphs andn-dimensional cubes, for which we determine their tree numbers, satisfy it. In the general case we prove the weaker inequality(G) (|G| – 1)/3 + 1.     

Continuous convolution semigroups integrating a submultiplicative function

Let be a submultiplicative function on a locally compact group G and let S be a convolution semigroup on G with Lévy measure . It is shown that the measures of S integrate if and only if integrates outside some neighbourhood of the identity of G (Theorem 1). Moreover if (X(t);t0) is the G-valued process with independent increments associated with the semigroup S it is shown that the measures of S integrate if and only if the random variable sup { (X(t)):0t1} is integrable (Theorem 2).     

Application of a computer to the proof of a conjecture of Minkowski in the geometry of numbers

A computer-assisted proof is given of Minkowski's conjecture on the critical determinant of the region x^p+y^p<1 in the cases 1.03p 1.9745, p2.40, p2.577.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 71, pp. 163–180, 1977.     

Approximation of the Distribution of the Supremum of a Centered Random Walk. Application to the Local Score

Let (X _n ) _{n 0} be a real random walk starting at 0, with centered increments bounded by a constant K. The main result of this study is: |P(S _n n x)–P( sup_{0 u 1} B _u x)| C(n,K) n/n, where x 0, ² is the variance of the increments, S _n is the supremum at time n of the random walk, (B _u ,u 0) is a standard linear Brownian motion and C(n,K) is an explicit constant. We also prove that in the previous inequality S _n can be replaced by the local score and sup₀ u 1 B _u by sup₀ u 1|B _u |.     

On Large Values of the Divisor Function

Let d(n) denote the divisor function, and let D(X) denote the maximal value of d(n) for n X. For 0 < z 1, both lower and upper bounds are given for the number of integersn with n X, zD(X) d(n).     

Some functional inequalities and their Baire category properties

Summary In the paper we consider, from a topological point of view, the set of all continuous functionsf:I I for which the unique continuous solution:I – [0, ) of(f(x)) (x, (x)) and(x, (x)) (f(x)) (x, (x)), respectively, is the zero function. We obtain also some corollaries on the qualitative theory of the functional equation(f(x)) = g(x, (x)). No assumption on the iterative behaviour off is imposed.     

Best approximation and best unilateral approximation of the kernel of a biharmonic equation and optimal renewal of the values of operators

For the classB _p , 0 < 1, 1p , of 2-periodic functions of the form f(t)=u(,t), whereu (,t) is a biharmonic function in the unit disk, we obtain the exact values of the best approximation and best unilateral approximation of the kernel K(t) of the convolution f= K *g, gl, with respect to the metric of L₁. We also consider the problem of renewal of the values of the convolution operator by using the information about the values of the boundary functions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol.47, No. 11, pp. 1549–1557, November, 1995.     

A Characterization of Some Minihypers and its Application to Linear Codes

Any {f,r- 2+s; r,q}-minihyper includes a hyperplane in PG(r, q) if fr-1 + s 1 + q – 1 for 1 s q – 1, q 3, r 4, where i = (qi + 1 – 1)/ (q – 1 ). A lower bound on f for which an {f, r – 2 + 1; r, q}-minihyper with q 3, r 4 exists is also given. As an application to coding theory, we show the nonexistence of [ n, k, n + 1 – qk – 2 ]q codes for k 5, q 3 for qk – 1 – 2q – 1 < n qk – 1 – q – 1 when k > q – q - \sqrt q + 2$$ " align="middle" border="0"> and for when , which is a generalization of [18, Them. 2.4].     

Eine Bemerkung zu den höheren Pythagoraszahlen reeller Körper

Becker has shown in [1] that for the 4-th Pythagoras number of the field (X) the inequality P₄ ((X)) 36 holds. In this paper we will show P₄ ((X)) 24 and P₄ (K) 3 for all real pythagorean fields K.     

On the Approximation by Modified Interpolation Polynomials in Spaces L p

We consider certain modified interpolation polynomials for functions from the space L _p[0, 2], 1 p . An estimate for the rate of approximation of an original function f by these polynomials in terms of its modulus of continuity is obtained. We establish that these polynomials converge almost everywhere to f.     

Fredholm integro-differential equation

We consider numerical solution of an integro-differential equation with nonsmooth initspaial values. Unique solvability in Sobolev spaceW ₂ (0, 1), =1,2, is proved. We establish the rate of convergence of the approximate solution to the exact solution in fractional spacesW ₂ ⁺¹ , 01, with approximation order O(h ^++1/2 ) for 01/2 andO(h ⁺¹ |ln h|1/2, for 1/2 #x2264;1.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 64, pp. 8–16, 1988.     

Measure ratio of equidecomposable sets

Let n 2. There are Lebesgue measurable sets A and B in ³ such that (B)/(A)=r and A _n B if and only if 2/n r n/2.     

Boolean Term Orders and the Root System Bn

A Boolean term order is a total order on subsets of [n] ={1,..., n} such that for all [n], , and for all with ( ) = . Boolean term orders arise in several different areas of mathematics, including Gröbner basis theory for the exterior algebra, and comparative probability.The main result of this paper is that Boolean term orders correspond to one-element extensions of the oriented matroid M(B_n), where B_n is the root system {e_i : 1 i n} {e_i ± e_j : 1 i < j n}. This establishes Boolean term orders in the framework of the Baues problem, in the sense of (Reiner, 1998). We also define a notion of coherence for a Boolean term order, and a flip relation between different term orders. Other results include examples of noncoherent term orders, including an example exhibiting flip deficiency, and enumeration of Boolean term orders for small values of n.     

Distribution of the number of nonappearing lengths of cycles in a random mapping

One-to-one random mappings of the set 1, 2,..., n onto itself are considered. Limit theorems are proved for the quantities _i, 0in, max _i, min _i, where _i is the number of 0in components of the vector ( ₁, ₂,..., _n) which are equal to i, 0< i< n, and ar is the number of components of dimension r of the random mapping.Translated from Matematicheskle Zametki, Vol. 23, No. 6, pp. 895–898, June, 1978.The author is grateful to V. P. Chistyakov and V. E. Stepanov for many useful remarks.