Singular perturbations in foliations

Summary We define a constraint system , [0,₀), which is a kind of family of vector fields on a manifold. This is a generalized version of the family of the equations , [0,₀),x ^m ,y ⁿ . Finally, we prove a singular perturbation theorem for the system , [0,₀).Dedicated to Professor Kenichi Shiraiwa on his 60th birthday     

Interior Regularity for Free and Constrained Local Minimizers of Variational Integrals Under General Growth and Ellipticity Conditions

We consider strictly convex energy densities f: (x) under nonstandard growth conditions. More precisely, we assume that for some constants , and for all Z, Y ⁿ the inequality

Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)

We solve Tikhomirov's problem on the explicit computation of sharp constants in the Kolmogorov type inequalities

The Law of the Logarithm for Weighted Sums of Independent Random Variables

Let X,X _n ;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b _n =B(n),n1. If

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

Fehlerabschätzungen für Gauß-Quadraturformeln

Summary We consider Gauss quadrature formulaeQ _n ,n, approximating the integral ,w an even weight function. Let be analytic inK _r :={z:|z|<r},r>1, and . The error functionalR _n :=I-Q _n is continuous with respect to |·|_r and the relation , q_2k (x):=x ^2k holds.In this paper estimates for R _n are given. To this end we first derive two new representations of R _n which are essential for our further investigations. The R _n =r ² R _n (), with (x):=1/(r ²-x ²), is estimated in various ways by using the best uniform approximation of in P_2n-1, and also the expansion of with respect to Chebyshe polynomials of the first and second kind. Forw(x)=(1-x ²), =±1/2, R _n is calculated. The asymptotic behaviour, forr1+, of R _n and of the derived error bounds is also discussed. Finally, we compare different error bounds and give numerical examples.

     

Blow-up of solutions of nonlinear wave equations in three space dimensions

Let u(x,t) be a solution, uA|u|^p for xIR₃, t0 where is the d'Alembertian, and A, p are constants with A>0, 10–|x–x⁰|, if the initial data u(x,0), u_t(x,0) have their support in the ball |x–x⁰|t₀. In particular global solutions of u=A|u|^p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|^p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u^–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.     

Problem of the extremal decomposition of a closed plane

One considers the problem of the maximum of the product of powers of conformal radii of nonoverlapping domains in the following formulation. Let A=a₁, ..., a_n and B=(b₁, ..., b_m be systems of distinguished points in ¯C and let ={₁,..., _m} be a system of positive numbers. ByU(D,b ) we denote the reduced modulus of the simply connected domain D relative to the pointb D. Find the maximum of the sum in the familyD of all systems of nonoverlapping simply connected domains D_j, j=1, ..., m, satisfying the following condition: the domain D_j does not contain points b_i B, different from b_j, and some collection A_j, for each domain, of points from A, _j=1 ^m A _j =A. The solution of this problem is obtained by the simultaneous use of the method of variation and of the method of the moduli of families of curves and is given by Theorem 1 of the present paper. As consequences of Theorem 1 one obtains Theorems 2 and 3, strengthening the corresponding results of a previous paper of the author.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 144, pp. 149–154, 1985.     

Numerical computation of least constants for the Sobolev inequality

Summary Least constantsc for the well-known Sobolev inequality fcf _{m, G} ,fH ^m (G) are obtained in closed form by a reproducing kernel technique, where the Sobolev spaceH ^m (G) for a domainG in ⁿ is defined as the completion ofC ^m (G) with respect to the Sobolev norm given by , where is the norm ofL ₂ (G) and is the supremum norm onG. Numerical values for the case whereG is the ⁿ are given.     

Limits of certain iterates of Post-Widder operators

The paper is a study of the limiting behaviour of the [n t]-th iterates of the well-known Post-Widder operatorsL _{n, x} used in the real inversion of the Laplace transform. It is shown that the limiting operators constitute a semigroup T _t;t0 of class (C ₀) on a family C _,; , >0 of Banach spaces. Applications of the semigroup structure lead to a pointwise saturation theorem forL _{n, x} and a characterization of convex functions inC _, through an inequality involving the action ofL _{n, x}.     

A remark on the strong law of large numbers

The strong law of large numbers for independent and identically distributed random variablesX _i ,i=1, 2, 3,... with finite expectationE|X ₁| can be stated as, for any >0, the number of integersn such that \varepsilon $$ " align="middle" border="0"> ,N is finite a. s. It is known thatEN < iffEX ₁ ² < and that ² EN var X₁ as 0, ifE X ₁ ² <. Here we consider the asymptotic behaviour ofEN (n) asn, whereN (n) is the number of integerskn such that \varepsilon $$ " align="middle" border="0"> andE N ₁ ² =.     

On the Boundedness of a Recurrence Sequence in a Banach Space

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: where |y _n} and | _n } are sequences bounded in B, and A _k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition is satisfied, then the sequence |x _n} is bounded for arbitrary bounded sequences |y _n} and | _n } if and only if the operator has the continuous inverse for every z C, |z| 1.     

The Parameters of Bipartite Q-polynomial Distance-Regular Graphs

Let denote a bipartite distance-regular graph with diameter D 3 and valency k 3. Suppose ₀, ₁, ..., _D is a Q-polynomial ordering of the eigenvalues of . This sequence is known to satisfy the recurrence _{i – 1} – _i + _{i + 1} = 0 (0 > i > D), for some real scalar . Let q denote a complex scalar such that q + q ^–1 = . Bannai and Ito have conjectured that q is real if the diameter D is sufficiently large.We settle this conjecture in the bipartite case by showing that q is real if the diameter D 4. Moreover, if D = 3, then q is not real if and only if ₁ is the second largest eigenvalue and the pair (, k) is one of the following: (1, 3), (1, 4), (1, 5), (1, 6), (2, 4), or (2, 5). We observe that each of these pairs has a unique realization by a known bipartite distance-regular graph of diameter 3.     

Random k-Sat: A Tight Threshold For Moderately Growing k

We consider a random instance I of k-SAT with n variables and m clauses, where k=k(n) satisfies k—log₂ n. Let m ₀=2 ^k nln2 and let =(n)>0 be such that n. We prove that

The functional equation of multiplicative derivation is superstable on standard operator algebras

LetX be a real or complex infinite dimensional Banach space andA a standard operator algebra onX. Denote byB(X) the algebra of all bounded linear operators onX. Let : _{+ +} be a function with the property lim _t (t)t ^–1=0. Assume that a mappingD:A B(X) satisfies D(AB)–AD(B)–D(A)B<(A B) for all operatorsA, B D (no linearity or continuity ofD is assumed). ThenD is of the formD(A)=AT–TA for someTB(X).This work was supported by the Research Council of Slovenia     

The solubility of certain diophantine inequalities

Let ₁, ₂, ..., _s bes non-zero real numbers not all of the same sign and not all in rational ratio, andk be a natural number; letD(k) be the leasts for which the inequality < (maxn _j )^– has infinitely many solutions. In this paper, we give some new estimations ofD(k) fork 6.Supported by the National Natural Science Foundation of China     

Two congruence lattices of completely simple semigroups

For a Rees matrix semigroupS with normalized sandwich matrix and C(S), the congruence lattice ofS, we consider the lattice generated by {itpT_l, pK, pT_r, pt_l, pk, pt_r}. HerepT ₁ andpt _l are the upper and lower ends of the interval which makes up the _i -class of , _i being the left trace relation onC(S). The remaining symbols have the analogous meaning relative to the kernel and the right trace relations. We also consider the lattice generated by {T _l, K, T_r, t_l, _k, t_r} where and are the equality and the universal relations onS, respectively. In both cases, we find lattices freest relative to these lattices and represent them as distributive lattices with generators and relations.With 3 Figures     

Existenzaussagen für gewisse Zweipunkt-Randwertaufgaben

Zusammenfassung Für Randwertaufgaben der Form–u–l ₀ ...u–l ₀ ...u=f(x, u) mitl ₀R,lR,f definiert und stetig auf {a<-x<-b, |u|<} wird eine Existenzaussage gewonnen, fallsf inu linear durch die aufeinanderfolgenden Eigenwerte der zugehörigen linearen Aufgabe beschränkt ist. Zum Beweis betrachtet man die äquivalente Hammersteinsche Integralgleichung mit nichtsymmetrischem Kern. Mit Hilfe des Schauderschen Fixpunktsatzes erhält man für diese Integralgleichung Existenzaussagen, welche Ergebnisse von Dolph verallgemeinern.

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Summary This note contains an existence theorem for a two-point boundary value problem of the form–u–l ₀ u–l ₀ u=f(x, u) wherel ₀R,l ₀R,f defined and continuous on {a<-xb, |u|<} iff is linear bounded inu by the successive eigenvalues of the corresponding linear problem. To prove this result we consider the equivalent Hammerstein integral equation with non-symmetric kernel. Schauders fixpoint principle supplies existence theorems for integral equations of this type which generalize results of Dolph in some sense.

     

Point systems with extremal properties and conformal mapping

Summary LetD be the unit disk. It is a well-known fact that by use of simply connected domain methods the general conformal mapping problem of doubly connected domains can be reduced to the special case of a regionD bounded by the unit circle and a Jordan curve inD, where . Here we treat this special case and assume to be piecewise analytic without cusps. Let be the conformal mapping of {<|w|<1} onto the doubly connected domainD with (1)=1. We approximate by interpolation with finite Laurent series using point systems with extremal properties. Numerical results for four examples are given.     

A Bernstein result for energy minimizing hypersurfaces

We show that there are no entire, positive, stable solutions in ⁿ of the Euler equation corresponding to the singular variational integral ,>0, if+n<5.236.... Furthermore we prove a related result for smooth boundaries of least-energy |x _n+1||D _U | in ⁿ⁺¹.