Computing a family of reproducing kernels for statistical applications

For an open subset of , an integer,m, and a positive real parameter , the Sobolev spacesH ^m () equipped with the norms: u²=u(t)²dt+(1/^2m u ^(m)(t)² constitute a family of reproducing kernel Hilbert spaces. When is an open interval of the real line, we describe the computation of their reproducing kernels. We derive explicit formulas for these kernels for all values ofm in the case of the whole real line, and form=1 andm=2 in the case of a bounded open interval.This research was partly supported by NSF Grant DMS-9002566.     

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.     

Long paths and cycles in tough graphs

For a graphG, letp(G) andc(G) denote the length of a longest path and cycle, respectively. Let (t,n) be the minimum ofp(G), whereG ranges over allt-tough connected graphs onn vertices. Similarly, let (t,n) be the minimum ofc(G), whereG ranges over allt-tough 2-connected graphs onn vertices. It is shown that for fixedt>0 there exist constantsA, B such that (t,n)A·log(n) and (t,n)·log((t,n))B·log(n). Examples are presented showing that fort1 there exist constantsA, B such that (t,n)A·log(n) and (t,n)B· log(n). It is conjectured that (t,n) B·log(n) for some constantB. This conjecture is shown to be valid within the class of 3-connected graphs and, as conjectured in Bondy [1] forl=3, within the class of 2-connectedK _1.l-free graphs, wherel is fixed.     

Solving systems of polynomial inequalities over a real closed field in subexponential time

Let _m= (₁,..., _m) denote an ordered field, where _i+1>0 is infinitesimal relative to the elements of _i, 0 < –i < m (by definition, ₀= ). Given a system of inequalities f₁ > 0, ..., f_s > 0, f_s+1 0, ..., f_k 0, where f_{j m} [X₁,..., X_n] are polynomials such that, and the absolute value of any integer occurring in the coefficients of the f_js is at most 2^M. An algorithm is constructed which tests the above system of inequalities for solvability over the real closure of _m in polynomial time with respect to M, ((d)ⁿd₀)^n+m. In the case _m=, the algorithm explicitly constructs a family of real solutions of the system (provided the latter is consistent). Previously known algorithms for this problem had complexity of the order ofM(d d ₀ ^{m 2U(n)} .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Maternaticheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 174, pp. 3–36, 1988.     

The Ramsey property for collections of sequences not containing all arithmetic progressions

A family of sequences has the Ramsey property if for every positive integerk, there exists a least positive integerf (k) such that for every 2-coloring of {1,2, ...,f (k)} there is a monochromatick-term member of . For fixed integersm > 1 and 0 q < m, let _q(m) be the collection of those increasing sequences of positive integers {x ₁,..., x_k} such thatx _i+1 – x_i q(modm) for 1 i k – 1. Fort a fixed positive integer, denote byA _t the collection of those arithmetic progressions having constant differencet. Landman and Long showed that for allm 2 and 1 q < m, _q(m) does not have the Ramsey property, while _q(m) A _m does. We extend these results to various finite unions of _q(m) 's andA _t 's. We show that for allm 2, _q=1 ^m–1 _q(m) does not have the Ramsey property. We give necessary and sufficient conditions for collections of the form _q(m) ( _{t T} A _t) to have the Ramsey property. We determine when collections of the form _a(m₁) _b(m₂) have the Ramsey property. We extend this to the study of arbitrary finite unions of _q(m)'s. In all cases considered for which has the Ramsey property, upper bounds are given forf .     

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.     

On Localization of Connective Covers

A result of Neisendorfer says that, for every connected p-complete finite complex Y with ₂Y torsion, the p-completion of P_{K(/p, 1)} (Ym) and Y are of the same homotopy type for any positive integer m. Here, P_{K(/p, 1)}(Ym) is the periodization functor of Bousfield and Ym) is the m-connective cover of the space Y. The proof of this result depends on Millers Theorem of Sullivans conjecture. The aim in this paper is to study the phenomenon without the use of Millers Theorem.AMS Subject Classification (2000): 55P60     

Characterization of regular singular linear systems of difference equations

This paper deals with linear systems of difference equations whose coefficients admit generalized factorial series representations atz=. We are concerned with the behavior of solutions near the pointz= (the only fixed singularity for difference equations). It is important to know whether a system of linear difference equations has a regular singularity or an irregular singularity. To a given system () we can assign a number , called the Moser's invariant of (), so that the system is regular singular if and only if 1. We shall develop an algorithm, implementable in a computer algebra system, which reduces in a finite number of steps the system of difference equations to an irreducible form. The computation ot the number can be done explicitly from this irreducible form.     

Characterization,structure and analysis on abelian ℒ∞ groups

An abelian topological group is an group if and only if it is a locally -compactk-space and every compact subset in it is contained in a compactly generated locally compact subgroup. Every abelian groupG is topologically isomorphic to G ₀ where ₀ andG ₀ is an abelian group where every compact subset is contained in a compact subgroup. Intrinsic definitions of measures, convolution of measures, measure algebra,L ₁-algebra, Fourier transforms of abelian groups are given and their properties are studied.     

On the eigenvalues of the matrix pencilA+μB

Summary The number of independent invariants ofn×n matricesA, B and their products on which the eigenvalues () of the matrix pencilA+B depend is determined by means of the theory of algebraic invariants and combinatorial analysis. Formulas are displayed for coefficients for the calculation of () forn5.

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Zusammenfassung Wir bestimmen die Anzahl der unabhängigen Invarianten dern×n MatrizenA, B und ihrer Produkte, von denen die Eigenwerte () der MatrixbüschelA+B abhängen, mittels der Theorie der algebraischen Invarianten und mittels kombinatorischer Analyse. Formeln für Koeffizienten zur Berechnung von () werden angegeben fürn5.

     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Hybrid programs: Linear and least-distance

We consider a quadratic program equivalent to the general problem of minimizing a convex quadratic function of many variables subject to linear inequality constraints.In previous work [12], one of us presented an algorithm related to such problems, and classified them under combinatorial equivalence. The classification contained linear programs at one end (where the function has zero quadratic part) and least-distance programs at the other. A least-distance program is a problem of finding a point of a convex polyhedron which is at least distance from a given point; such programs have been studied by one of us [15, 17] and were shown to correspond to that case of the general problem where the function has positive definite quadratic part.We now extend the work on least-distance programs to those programs intermediate between linear and least-distance (called essentially bisymmetric in [12]), and show that such programs are really hybrids, with traits inherited from both parent programs: linear and leastdistance.This paper was presented at the 7th Mathematical Programming Symposium 1970, The Hague, The Netherlands.This work was supported in part by Office of Naval Research Contract No. N00014-67-A-0151-0010 (Princeton University).     

On Remez-type inequalities for polynomials in R m and C m

Remez-type inequalities provide estimates for the size of polynomials on given sets KR ^m (or C ^m ) when the magnitude of polynomials on largeldquo subsets of K is known. We shall study this question on smooth sets K in R ^m and C ^m and show how the smoothness of K effects the estimates.     

Integrability of multiple Walsh series and Parseval's formula

f(x,y) _jk . , {c _jk} , f(x, )(, ) [0,1)&#x0445;[0,1) , - (0,0). , , f, - f. , , , [1] . . - [5] [6].

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This research is supported by National Science Council, Taipei, R.O.C. under Grant #NSC 84-2121-M-007-026.     

m-Systems and Partial m-Systemsof Polar Spaces

LetP be a finite classical polar space of rankr, withr 2. A partialm-systemM ofP, with 0 m r - 1, is any set (₁), ₂,..., _k ofk ( 0) totally singularm-spaces ofP such that no maximal totally singular space containing _i has a point in common with (_{1 2} ... _k) — _i,i = 1, 2,...,k. In a previous paper an upper bound for ¦M¦ was obtained (Theorem 1). If ¦M¦ = , thenM is called anm-system ofP. Form = 0 them-systems are the ovoids ofP; form =r - 1 them-systems are the spreads ofP. In this paper we improve in many cases the upper bound for the number of elements of a partialm-system, thus proving the nonexistence of several classes ofm-systems.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

Waterman classes and spherical partial sums of double Fourier series

f(x,y) ₀BV(T²), ={1/n} _n=1 .

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Dedicated to Professor Károly Tandori, the outstanding mathematician and academician on his seventieth birthday

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This work was done under the financial support of the Russian Foundation for Fundamental Scientific Research, Grant 93-01-00240.     

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 .     

On the asymptotic equivalence ofL pmetrics for convergence to normality

Summary This paper is concerned with the rate of convergence to zero of theL _pmetrics _np1p, constructed out of differences between distribution functions, for departure from normality for normed sums of independent and identically distributed random variables with zero mean and unit variance. It is shown that the _np are, under broad conditions, asymptotically equivalent in the strong sense that, for 1p, p, _np/_np is universally bounded away from zero and infinity asn.     

Some extremal results on circles containing points

We define (n) to be the largest number such that for every setP ofn points in the plane, there exist two pointsx, y P, where every circle containingx andy contains (n) points ofP. We establish lower and upper bounds for (n) and show that [n/27]+2(n)[n/4]+1. We define for the special case where then points are restricted to be the vertices of a convex polygon. We show that .     

Conic decompositions of a class of lattices and universal algebras

We call an elementa of a finite-height latticeL conic if all coversb _i , ofa form coverings a,b _i which are related suitably (by strict projective equivalence) inL. This leads to existence and uniqueness theorems forconic decompositions. All of this is defined in general lattices, the class of lattices with a.c.c. and a generalized form of semimodularity calledConesemimodularity being the natural framework for most of the results. In a varietyV of universal algebras, we call an algebraA conic if it is a homomorphic image of some algebraA L with kernel, say, such that is conic in ConA. Conic decomposition of the 0 congruence leads to a subdirect product decomposition with conic algebras as factors. Special properties of conic algebras are given. We also consider a dual notion, which in suitable lattices leads to join-decompositions of lattice elements.Presented by B. Jonsson.