The distribution of sublattices of ℤ m

Lattices , are similar if one can be transformed into the other by an angle-preserving linear map. Similarity classes of lattices of rankn may be parametrized by a fundamental domain of the action ofGL _n () on the generalized upper half-plane _n . Given 1<nm and, letN(D,T) be the number of sublattices of _n which have rankn, similarity class inD, and determinant T. Our most basic result will be thatN(D,T)c ₁(m, n)(D)T ^m asT for suitable setsD, where is the invariant measure on _n . The casen=2 had been dealt with by Roelcke and by Maass using the theory of modular forms.Herrn Professor Hlawka zum achtzigsten Geburtstag gewidmetSupported in part by NSF-DMS-9401426     

Upper and lower bounds for the frequencies of rectangular free plates

Zusammenfassung Eine neuentwickelte Methode für untere Schranken und das Rayleigh-Ritzverfahren für obere Schranken werden von den Verfassern dazu angewandt, die Eigenfrequenzen der Schwingungen von dünnen gleichförmigen, rechteckigen Platten mit freien Rändern abzuschätzen. Die Anwendbarkeit des Verfahrens für untere Schranken wird hervorgehoben, und es werden Berechnungen für eine Symmetrieklasse von rechteckigen sowie für eine Unterklasse von quadratischen Platten angegeben. Die dadurch entstehenden Schranken führen zu einer Verbesserung bisher veröffentlichter Resultate und weisen auf die Brauchbarkeit der Methode für untere Schranken hin.

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Nomenclature The following symbols are not defined in the text ² Laplacian differential operator (= ²/x ² + ²/y ²) - ⁴ Biharmonic differential operator (= ⁴/x ⁴ + 2 ⁴/x ² y ² + ⁴/y ⁴) - u Generic name for displacement functions - Generic name for eigenvalues - a,b Plate side lengths - Circular frequency of free vibration - Mass density of plate material - h Plate thickness - D Plate flexural rigidity - First variation - d Element of surface area - _ij Kronecker's delta - {} Matrix or vector of the included elements The research reported in this article has been sponsored by the Department of the Navy under Contract NOw-62-0604-c with the Bureau of Naval Weapons.     

Flag-Transitive 2-Designs Arising from Line-Spreads in PG(2n-1,2)

A Singer cycle in GL(n,q) is an element of order q permuting cyclically all the nonzero vectors. Let be a Singer cycle in GL(2n,2). In this note we shall count the number of lines in PG (2n-1,2) whose orbit under the subgroup of index 3 in the Singer group is a spread. The lines constituting such a spread are permuted cyclically by the group ³, hence gives rise to a flag-transitive 2-(2²ⁿ ,4,1) design.     

The Skorohod oblique reflection problem in domains with corners and application to stochastic differential equations

Summary The Skorohod oblique reflection problem for (D, , w) (D a general domain in ^d , (x),xD, a convex cone of directions of reflection,w a function inD(⁺, ^d )) is considered. It is first proved, under a condition on (D, ), corresponding to (x) not being simultaneously too large and too much skewed with respect to D, that given a sequence {w ⁿ} of functions converging in the Skorohod topology tow, any sequence {(x ⁿ, ⁿ)} of solutions to the Skorohod problem for (D, , w ⁿ) is relatively compact and any of its limit points is a solution to the Skorohod problem for (D, , w). Next it is shown that if (D, ) satisfies the uniform exterior sphere condition and another requirement, then solutions to the Skorohod problem for (D, , w) exist for everywD(⁺, ^d ) with small enough jump size. The requirement is met in the case when D is piecewiseC _b ¹ , is generated by continuous vector fields on the faces ofD and (x) makes and angle (in a suitable sense) of less than /2 with the cone of inward normals atD, for everyxD. Existence of obliquely reflecting Brownian motion and of weak solutions to stochastic differential equations with oblique reflection boundary conditions is derived.     

On Birational Maps and Jacobian Matrices

One is concerned with Cremona-like transformations, i.e., rational maps from ⁿ to ^m that are birational onto the image Y ^m and, moreover, the inverse map from Y to ⁿ lifts to ^m . We establish a handy criterion of birationality in terms of certain syzygies and ranks of appropriate matrices and, moreover, give an effective method to explicitly obtaining the inverse map. A handful of classes of Cremona and Cremona-like transformations follow as applications.     

The upper and lower second order directional derivatives of a sup-type function

The purpose of this paper is to give a formula for expressing the second order directional derivatives of the sup-type functionS(x) = sup{f(x, t); t T} in terms of the first and second derivatives off(x, t), whereT is a compact set in a metric space and we assume thatf, f/x and ² f/x ² are continuous on ⁿ × T. We will give a geometrical meaning of the formula. We will moreover give a sufficient condition forS(x) to be directionally twice differentiable.     

On a Generalization of Nikolskij's Extension Theorem in the Case of Two Variables

A modification of the Nikolskij extension theorem for functions from Sobolev spaces H ^k() is presented. This modification requires the boundary to be only Lipschitz continuous for an arbitrary k however, it is restricted to the case of two-dimensional bounded domains.     

On pseudo-differential operators and smoothness of special Lie-group representations

Two algebras of global pseudo-differential operators over ⁿ are investigated, with corresponding classes of symbols A⁰=CB (all (x, )-derivatives bounded over ²ⁿ), and A₁ (all finite applications of _xj, _j, and _pq=_pq–_px_p on the symbol are in A₀). The class A₁ consists of classical symbols, i.e., ^x a= 0((1+||)–^||) for x Kc ;ⁿ, K, compact. It is shown that a bounded operator A of 210C=L²(Rⁿ) is a pseudo-differential operator with symbol aA_j if and only if the map AG^–1AG, G g_j is infinitely differentiable, from a certain Lie-group g_j c GL(210C) to (210C) with operator norm. g₀ is the Weyl (or Heisenberg) group. Extensions to operators of arbitrary order are discussed. Applications to follow in a subsequent paper.Dedicated to Hans Lewy and Charles B. Morrey, Jr.     

Analogues of hadwiger's theorem in space ℝ n —Sufficient conditions for a convex domain to enclose another

We estimate the kinematic measure of one convex domain moving to another under the groupG of rigid motions in ⁿ . We first estimate the kinematic formula for the total scalar curvature D ₀gD ₁ Rdv of then–2 dimensional intersection submanifold D ₀gD ₁. Then we use Chern and Yen's kinematic fundamental formula and our integral inequality to obtain a sufficient condition for one convex domain to contain another in ⁿ (4). Forn=4, we directly obtain another sufficient condition in ⁴.     

On the Uniqueness ofL 2-solutions in half-space of certain differential equations

Square integrable solutions to the equation{– ²/y² + P(D_x)+b(y)–}u(x, y) = f(x, y) are considered in the half-spacey>0, x ⁿ , whereP(D _x) is a constant coefficient operator. Under suitable conditions on lim_y0u(x, y), b(y), f(x, y) and , it is shown that suppu = suppf. This generalizes a result due to Walter Littman.Research partially supported by USNSF Grant 79-02538-A02.     

A Remark on Bernstein,Prokhorov, Bennett,Hoeffding, and Talagrand Inequalities

Let X ₁,..., X_n be independent random variables such that {X_j 1}=1 and E X _j=0 for all j. We prove an upper bound for the tail probabilities of the sum M _n=X₁+...+ X_n. Namely, we prove the inequality {M _nx} 3.7 {S_n x}, where S _n=₁+...+ _n is a sum of centered independent identically distributed Bernoulli random variables such that E S _n ² =ME M _n ² and {_k=1}=E S _n ² /(n+E S _n ² ) for all k (we call a random variable Bernoulli if it assumes at most two values). The inequality holds for x at which the survival function x{S _nx} has a jump down. For remaining x, the inequality still holds provided that we interpolate the function between the adjacent jump points linearly or log-linearly. If necessary, in order to estimate {S _nx} one can use special bounds for binomial probabilities. Up to the factor at most 2.375, the inequality is final. The inequality improves the classical Bernstein, Prokhorov, Bennett, Hoeffding, Talagrand, and other bounds.     

Perturbation of the poles of the scattering matrix under variation of the boundary condition

The behavior of the poles z_n(), n=1,2,... of the scattering matrix of the operatorl u=–u(x), x , (u/n)+(x)u|=0 as 0 is considered. It is proved that |z_n()–z_n|=0(^(1/2)qn), where q_n is the order of the pole of the scattering matrix for the operator ₀u=–u, u/=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 117, pp. 183–191, 1981.     

Problem of instability in the first approximation

Let E be a Banach space, A be a continuous linear operator such that (A) ; Re>0 Ø, and F(t, x) be a continuous function on [0, )×E satisfying the condition F(t, x) q x (q= const). An example of a system dx/dt=Ax + F(t, x) is given which has an exponentially stable zero solution for certain F(t, x) with arbitrarily small q.Translated from Matematicheskie Zametki, Vol. 23, No. 5, pp. 721–723, May, 1978.     

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

Minimal Fine Limits for a Class of Potentials

We consider potentials G _k associated with the Weinstein equation with parameter k in , _j=1 ⁿ (² u/ x ² _j ) + (k/x _n ) ( u/ x _n ) = 0, on the upper half space in ⁿ . We show that if the representing measure satisfies the growth condition y _n /(1+|y|) ^n-k < , where max(k, 2 – n) < 1, then G _k has a minimal fine limit of 0 at every boundary point except for a subset of vanishing (n – 2 + ) dimensional Hausdorff measure. We also prove this exceptional set is best possible.     

An intersection property of the simple random walks inZ d

Let {S _d (n)} _n0 be the simple random walk inZ ^d , and ^(d)(a,b)={S _d (n)Z ^d :anb}. Supposef(n) is an integer-valued function and increases to infinity asn tends to infinity, andE _n ^(d) ={^(d)(0,n)^(d)(n+f(n),)}. In this paper, a necessary and sufficient condition to ensureP(E _n ^d) ,i.o.)=0, or 1 is derived ford=3, 4. This problem was first studied by P. Erdös and S.J. Taylor.This work is partly supported by the National Natural Sciences Foundation of China.     

On the existence of wave operators for the Klein-Gordon equation

The problem of existence of wave operators for the Klein-Gordon equation ( _t ² –+²+iV_1t+V₂)u(x,t)=0 (x R ⁿ,t R, n3, >0) is studied where V₁ and V₂ are symmetric operators in L₂(R ⁿ) and it is shown that conditions similar to those of Veseli-Weidmann (Journal Functional Analysis 17, 61–77 (1974)) for a different class of operators are also sufficient for the Klein-Gordon equation.     

Zur Lösung parameterabhängiger nichtlinearer Gleichungen mit singulären Jacobi-Matrizen

Summary For solving the nonlinear systemG(x, t)=0,G| ⁿ × ^{1 n} , which is assumed to have a smooth curve of solutions a continuation method with self-choosing stepsize is proposed. It is based on a PC-principle using an Euler-Cauchy-predictor and Newton's iteration as corrector. Under the assumption thatG is sufficiently smooth and the total derivative (₁ G(x, t)₂ G(x, t)) has full rankn along the method is proven to terminate with a solution (x ^N , 1) of the system fort=1. It works succesfully, too, if the Jacobians ₁ G(x, t) become singular at some points of , e.g., if has turning points. The method is especially able to give a point-wise approximation of the curve implicitly defined as solution of the system mentioned above.

     

The Darboux-Type Theorems for ∂-Symplectic and ∂-Contact Structures

A -symplectic structure on a complex manifold M of complex dimension2n is given by a smooth -closed (2, 0)-form such that ⁿ is nonvanishing. We prove that a version of the Darboux theorem isvalid for such a structure: locally can be represented as _i=1 ⁿ f _i f _{n + i} for appropriate smooth complex valuedfunctions f ₁, ..., f _2n . We also present a contact version of this theorem.     

Unique solvability of certain matrix partial differential equations

A class of matrix-valued functions is picked out, invariant relative to the operator, where t=(t₁, ..., t_n) are complex variables, x is a real parameter, A(x) is a matrix, {_i(x)} ₁ ⁿ =(A(x)). It is shown that the operator is normally solvable in the class picked out and a uniqueness theorem is proved for the solution of a nonstandard problem: the desired matrix-valued function Z(x, t) is known only at a point andz/x Ker ^*. Such problems arise naturally when developing the general theory of singular perturbations.Translated from Matematicheskie Zametki, Vol. 21, No. 4, pp. 525–530, April, 1977.