Some inequalities about the covering radius of Reed-Muller codes

Let R(r, m) be the rth order Reed-Muller code of length 2 ^m , and let (r, m) be its covering radius. We prove that if 2 k m - r - 1, then (r + k, m + k) (r, m + 2(k - 1). We also prove that if m - r 4, 2 k m - r - 1, and R(r, m) has a coset with minimal weight (r, m) which does not contain any vector of weight (r, m) + 2, then (r + k, m + k) (r, m) + 2k(. These inequalities improve repeated use of the known result (r + 1, m + 1) (r, m).This work was supported by a grant from the Research Council of Wright State University.     

Properties of solutions of linear functional-differential equations depending on a parameter

On the segment 0 t1 we study the equation A(d/dt, )x(t) + [F()x](t)=f(t), whereA (d/dt, ) x=x( ⁿ )+A ₁ x(^n–1 +...+ ⁿ A _n x, the matrices A₁,...,A_n are of size m × m, x is an unknown and f a given function with values in the m-dimensional space ^m , F() is a linear operator acting from a Hölder space to a Lebesgue space of vectorfunctions with values in ^m and depending on a complex parameter . We find the set of those at which a one-to-one correspondence is established between the solutions of the given equation and the solutions of the equation A(d/dt, )x(t)=0.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1213–1231, September, 1991.     

Approximation of the viability kernel

We study recursive inclusionsx ⁿ⁺¹ G(x ⁿ ). For instance, such systems appear for discrete finite-difference inclusionsx ⁿ⁺¹ G (x ⁿ) whereG :=1+F. The discrete viability kernel ofG , i.e., the largest discrete viability domain, can be an internal approximation of the viability kernel ofK underF. We study discrete and finite dynamical systems. In the Lipschitz case we get a generalization to differential inclusions of the Euler and Runge-Kutta methods. We prove first that the viability kernel ofK underF can be approached by a sequence of discrete viability kernels associated withx ⁿ⁺¹ (xⁿ) where (x) =x + F(x) + (ML/2) ². Secondly, we show that it can be approached by finite viability kernels associated withx _h ⁿ⁺¹ ( (x _h ⁿ⁺¹ ) +(h) X _h .     

Further results on the covering radii of the Reed-Muller codes

LetR(r, m) by therth order Reed-Muller code of length2 ^m , and let (r, m) be its covering radius. We obtain the following new results on the covering radius ofR(r, m): 1. (r+1,m+2) 2(r, m)+2 if 0rm–2. This improves the successive use of the known inequalities (r+1,m+2)2(r+1,m+1) and (r+1,m+1) (r, m).2.(2, 7)44. Previously best known upper bound for (2, 7) was 46. 3. The covering radius ofR(1,m) inR(m–1,m) is the same as the covering radius ofR(1,m) inR(m–2,m) form4.     

An Extension of the Pizzetti Formula for Polyharmonic Functions

We represent the integral over the unit ball B in R ⁿ of any poly-harmonic function u(x) of degree m as a linear combination with constant coefficients of the integrals of its Laplacians ^j u (j = 0,...,m - 1) over any fixed(n - 1)-dimensional hypersphere S() of radius (0 1). In case = 0 theformula reduces to the classical Pizzetti formula. In particular, the cubature formula derived here integrates exactly all algebraic polynomials of degree 2m - 1.     

A solution to one of Kano's conjectures concerningk-th maximal spanning trees

Let (G) be the collection of all spanning trees of a connected and weighted graphG, andF ₁,F ₂, ...,F _m the partition of (G) such thatF _n the set ofi-th maximal spanning trees ofG. Kano^[1] conjectured that for anyA F ₁ and every integerk, 1km, there existsT F _k such that |T/A|k–1. This paper gives the conjecture a very simple proof, and related results.     

Some observations on the distribution of values of continued fractions

Summary There have been many studies of the values taken on by continued fractionsK(a _n /1) when its elements are all in a prescribed setE. The set of all values taken on is the limit regionV(E). It has been conjectured that the values inV(E), are taken on with varying probabilities even when the elementsa _n are uniformly distributed overE. In this article, we present the first concrete evidence that this is indeed so. We consider two types of element regions: (A)E is an interval on the real axis. Our best results are for intervals [–(1–), (1–)], 0 <1/2. (B)E is a disk in the complex plane defined byE={z:|z|(1–)}., 0<1/2.     

Some Asymptotic Results for the M/M/∞ Queue with Ranked Servers

We consider an M/M/ model with m primary servers and infinitely many secondary ones. An arriving customer takes a primary server, if one is available. We derive integral representations for the joint steady state distribution of the number of occupied primary and secondary servers. Letting =/ be the ratio of arrival and service rates (all servers work at rate ), we study the joint distribution asymptotically for . We consider both m=O(1) and m scaled to be of the same order as . We also give results for the marginal distribution of the number of secondary servers that are occupied.     

Les classes de Chern des représentations galoisiennes complexes

Let F be a field, G _F its absolute Galois group, : G _FGL(C) a continuous complex representation of G _F and c _i() H_2i(F, Z) its Chern classes. We show, under a mild assumption on F. that c _i ()=0 for all i2. For general F, one has that 2c_i ()=0 for all i 2.

Cette dernière condition résulte en fait de la continuité de .     

Relationship between the Hadamard Theorem on Three Disks and Certain Problems of Polynomial Approximation of Analytic Functions

By using the classical Hadamard theorem, we obtain an exact (in a certain sense) inequality for the best polynomial approximations of an analytic function f(z) from the Hardy space H _p, p 1, in disks of radii , ₁, and ₂, 0 < ₁ < < ₂ < 1.     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

The broken sample problem

Suppose that (X_i,Y_i),i=1,2, ... ,n, are iid. random vectors with uniform marginals and a certain joint distribution F, where is a parameter with =_o corresponds to the independence case. However, the Xs and Ys are observed separately so that the pairing information is missing. Can be consistently estimated? This is an extension of a problem considered in (1980) which focused on the bivariate normal distribution with being the correlation. In this paper we show that consistent discrimination between two distinct parameter values ₁ and ₂ is impossible if the density f of F is square integrable and the second largest singular value of the linear operator is strictly less than 1 for =₁ and ₂. We also consider this result from the perspective of a bivariate empirical process which contains information equivalent to that of the broken sample.Dedicated to Professor Xiru Chen on His 70th BirthdayMathematics Subject Classification (2000): primary: 60F99, 62F12Research supported by NSFC Grant 201471000 and the NUS Grant R-155-000-040-112.Research supported by the Texas Advanced Research Program.     

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.     

Problem of the maximum of the n-th diameter in the hyperbolic plane

Let E be a compactum in the circle and let d_n(E) be the n-th Euclidean diameter of E: Let K^(h)() be the family of all continua in U of hyperbolic capacity , 0<<1. Let E ₁ ^* ()=[0, R₁()] and let for n=2,3,..., where R_n() is the solution of the equation K(k) being the elliptic integral of the first kind with module k and. In Sec. 1 of the paper one shows that for all even n=2m 4- and all 0<<1, the symmetric continuum E _n ^* () does not yield a maximum for d _n ^(h) (E) in the family K^(h)(). This complements the known result of a negative character in the problem of the maximum of the n-th Euclidean diameter in the family of all continua of a prescribed capacity. In Sec. 2 one shows that for any 0<<1, the maximum of d₃(E) in the family K^(h)() is attained only by the continua of the form, being a real number.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 100, pp. 113–130, 1980.     

Improved asymptotic analysis of the average number of steps performed by the self-dual simplex algorithm

In this paper we analyze the average number of steps performed by the self-dual simplex algorithm for linear programming, under the probabilistic model of spherical symmetry. The model was proposed by Smale. Consider a problem ofn variables withm constraints. Smale established that for every number of constraintsm, there is a constantc(m) such that the number of pivot steps of the self-dual algorithm,(m, n), is less thanc(m)(lnn) ^m(m+1) . We improve upon this estimate by showing that(m, n) is bounded by a function ofm only. The symmetry of the function inm andn implies that(m, n) is in fact bounded by a function of the smaller ofm andn. Parts of this research were done while the author was visiting Stanford University, XEROX- PARC, Carnegie-Mellon University and Northwestern University and was supported in part by the National Science Foundation under Grants MCS-8300984, ECS-8218181 and ECS-8121741.     

The eigenvalues of the Sinyukov mapping for geodesically equivalent metrics are globally ordered

Suppose all geodesics of two Riemannian metrics g and defined on a (connected, geodesically complete) manifold M ⁿ coincide. At each point x M ⁿ , consider the common eigenvalues ₁, ₂, ... , _n of the two metrics (we assume that _{1 2 n}) and the numbers . We show that the numbers _i are ordered over the entire manifold: for any two points x and y in M the number _k(x) is not greater than _k+1(y). If _k(x)= _k+1(y), then there is a point z M _n such that _k(z)= _k+1(z). If the manifold is closed and all the common eigenvalues of the metrics are pairwise distinct at each point, then the manifold can be covered by the torus.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 412–423.Original Russian Text Copyright © 2005 by V. S. Matveev.This revised version was published online in April 2005 with a corrected issue number.     

Asymptotic Behavior of Loss Probability in GI/M/1/K Queue as K Tends to Infinity

We obtain an asymptotic behavior of the loss probability for the GI/M/1/K queue as K for cases of <1, >1 and =1.     

On steiner ratio conjectures

LetM be a metric space andP a finite set of points inM. The Steiner ratio inM is defined to be(M)=inf{L _s(P)/L _m(P) |P M}, whereL _s(P) andL _m(P) are the lengths of the Steiner minimal tree and the minimal spanning tree onP, respectively. In this paper, we study various conjectures on(M). In particular, we show that forn-dimensional Euclidean space _n ,( _n )>0.615.Supported in part by the National Science Foundation of China.     

Indices of Maximal Subgroups of Finite Soluble Groups

We look at the structure of a soluble group G depending on the value of a function m(G)= max m _p G), where m _p(G)=max{log_p|G:M| | M< G, |G:M|=p ^a}, p (G). Theorem 1 states that for a soluble group G, (1) r(G/ (G))= m(G); (2) d(G/ (G)) 1+ (m(G)) 3+m(G); (3) l _p(G) 1+t, where 2^t-1<m _p(G) 2^t. Here, (G) is the Frattini subgroup of G, and r(G), d(G), and l _p(G) are, respectively, the principal rank, the derived length, and the p-length of G. The maximum of derived lengths of completely reducible soluble subgroups of a general linear group GL(n,F) of degree n, where F is a field, is denoted by (n). The function m(G) allows us to establish the existence of a new class of conjugate subgroups in soluble groups. Namely, Theorem 2 maintains that for any natural k, every soluble group G contains a subgroup K possessing the following properties: (1) m(K); k; (2) if T and H are subgroups of G such that K T <_max <_max H G then |H:T|=p ^t for some prime p and for t>k. Moreover, every two subgroups of G enjoying (1) and (2) are mutually conjugate.     

The connectivities of adjacent tree graphs

The connectivity and the circuit rank of a graphG are denoted byx(G) and, respectively. It is shown that ifH is the adjacent tree graph of a simple connected graphG, thenx(H)=2.