Symmetric hyperbolic difference schemes and matrix problems

Sufficient conditions for the stability of multidimensional finite difference schemes are difficult to obtain. It is shown that for special families of amplification matrices G (A, B) a sufficient condition for power boundedness can be obtained by replacing the matrices by appropriate scalars, and so the problem is reduced to a scalar one. As one application it is shown that the Lax-Wendroff scheme in two dimensions is stable if |Au|^{$\text{2}\text{3}$} + |Bu|^{$\text{2}\text{3}$} ? 1 for all real unit vectors u. The Lax- Wendroff scheme with stabilizer does not always permit such large time steps. It is conjectured that the analysis for all symmetric hyperbolic schemes can be reduced to the scalar case.     

Fourier analysis on the Heisenberg group. I. Schwartz space

We derive a usable characterization of the group FT (Fourier Transform) of Schwartz space on the Heisenberg group $\text{H}$ⁿ, in terms of certain asymptotic series. To accomplish this we study in detail the FT of multiplication and differentiation operators on $\text{H}$ⁿ, the relation of multiple Fourier series to the FT, and the process of group contraction on $\text{H}$ⁿ. We use our characterization to solve a form of the division problem for convolution of $\text{H}$ⁿ, which has application to Hardy space theory.     

Asymptotic expansions for conditional distributions

It is shown that—under appropriate regularity conditions—the conditional distribution of the first p components of a normalized sum of i.i.d. m-dimensional random vectors, given the complementary subvector, admits a Chebyshev-Cramér asymptotic expansion of order $\text{o(n}{}^{\mathrm{<mtext>?(s?2)</mtext><mtext>2</mtext>}}\text{)}$, uniformly over all Borelsets in $\text{R}$^p and uniformly in a region of the conditioning subvector that includes moderate deviations.     

Quantization of the action of U(k,l) on R2(K + l)

The natural action of U(k, l) on $\text{C}$^{k + l} leaves invariant a real skew non-degenerate bilinear form B, which turns $\text{C}$^{k + l} into a symplectic manifold (M, ω). The polarization F of M defined by the complex structure of $\text{C}$^{k + l} is non-positive. If L is the prequantization complex line bundle carried by (M, ω), then U(k, l) acts on the space $\text{U}$ of square-integrable $\text{L ? ΛF}{}^1$ forms on M, leaving invariant the natural non-degenerate, but non-definite, inner product ((·, ·)) on $\text{U}$. The polarization F also defines a closed, densely defined covariant differential ?? on $\text{U}$ which is U(k, l)-invariant. Let $\text{⊥}$ denote orthocomplementation with respect to ((·, ·)). It is shown that the restriction of ((·, ·)) to the U(k, l)-stable subspace ? (Ker ??) ∩ (Im ??)^$\text{⊥}$ is semi-definite and that the unitary representation of Uk, l on the Hilbert space $\text{H}$ arising from ? by dividing out null vectors is unitarily equivalent to the representation of U(k, l) obtained from the tensor product of the metap ectic and $\text{Det}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ representations of MU(k, l), the double cover of U(k, l).     

Über die Wurzelschranke für das Minimalgewicht von codes

Let d be the minimum distance of an (n, k) code $\text{C}$, invariant under an abelian group acting transitively on the basis of the ambient space over a field F with char F × n. Assume that $\text{C}$ contains the repetition code, that dim($\text{C}$ ∏ $\text{C}$^⊥) = k ? 1 and that the supports of the minimal weight vectors of $\text{C}$ form a 2-design. Then d² ? d + 1 ? n with equality if and only if the design is a projective plane of order d ? 1. The case d² ? d + 1 = n can often be excluded with Hall's multiplier theorem on projective planes, a theorem which follows easily from the tools developed in this paper Moreover, if d² ? d + 1 > n and F = GF(2) then (d ? 1)² ? n. Examples are the generalized quadratic residue codes.     

Algebras of subnormal operators

The paper deals with the following: (I) If S is a subnormal operator on $\text{H}$, then $\text{Ol}$(S) = $\text{W}$(S) = Alg Lat S. (II) If L ∈ ($\text{Ol}$(S), σ-wot)¹, then there exist vectors a and b in $\text{H}$ such that L(T) = 〈Ta, b〉 for every T in $\text{Ol}$. (III) In addition to I the map i(T) = T is a homeomorphism from ($\text{Ol}$, σ-wot) onto ($\text{W}$(S), wot). (IV) If S is not a reductive normal operator, then there exists a cyclic invariant subspace for S that has an open set of bounded point evaluations. (This open set can be constructed to be as large as possible.)     

Unbounded derivations commuting with compact group actions. II

Let ($\text{A}$, G, α) be a $\text{C}{}^1$ dynamical system and let δ be a closed ¹ derivation in $\text{A}$ which commutes with α and satisfies $\text{A}$^G ? ker(δ). If $\text{A}$ is a separable Type I $\text{C}{}^1$ algebra and G is a second countable compact group, then δ generates a strongly continuous one parameter group of ¹ automorphisms of $\text{A}$.     

The bit-complexity of arithmetic algorithms

Time-complexity and space-complexity of arithmetic algorithms without divisions measured by the number of binary bits processed in computations are estimated for algorithms for discrete Fourier transform (DFT) and polynomial multiplication (PM, or convolution of vectors). It is proved that $\text{Ω(N}{}^2\text{)}$ bit-operations are required in an evaluation-interpolation algorithm for PM over the field of real constants while O(N log²N) arithmetic operations suffice.     

Bessel functions and representation theory. I

In this paper a general theory of operator-valued Bessel functions is presented. These functions arise naturally in representation theory in the context of metaplectic representations, discrete series, and limits of discrete series for certain semi-simple Lie groups. In general, Bessel functions J_λ are associated to the action by automorphisms of a compact group U on a locally compact abelian group X, and are indexed by the irreducible representations λ of U that appear in the primary decomposition of the regular representation of U on L²(X). Then on the λ-primary constituent of L²(X), the Fourier transform is described by the Hankel transform corresponding to J_λ. More detailed information is available in the case in which (U, X) is an orthogonal transformation group which possesses a system of polar coordinates. In particular, when X=$\text{F}$^k×n,$\text{F}$ a real finite-dimensional division algebra, with k ? 2n and $\text{O}$(k, $\text{F}$), the representations λ of U are induced in a certain sense from representations π of GL(n, $\text{F}$). This leads to a characterization of J_λ as a reduced Bessel function defined on the component of 1 in GL(n, $\text{F}$) and to the connection between metaplectic representations and holomorphic discrete series for the group of biholomorphic automorphisms of the Siegel upper half-plane in the complexification of $\text{F}$^{n × n}.     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

Orthogonality and the numerical range. II

For a continuous linear operator A on a Hilbert space X and unit vectors x and y, an investigation of the set $\text{W[x,y]={z}{}^1\text{Az:z}{}^1\text{z=1}$ and z?span{x,y}} reveals several new results about W(A), the numerical range of A. W[x,y] is an elliptical disk (possibly degenerate), and several conditions are given which imply that W[x,y] is a line segment. In particular if x is a reducing eigenvector of A, then W[x,y] is a line segment. A unit vector is called interior (boundary) if x¹Ax is in the interior (boundary) of W(A). It is shown that interior reducing eigenvectorsare orthogonal to all boundary vectors and that boundary eigenvectors are orthogonal to all other boundary vectors y [except possibly when y¹Ay is interior to a line segment in the boundary of W(A) through the given eigenvalue].     

Equivalence of Room designs. II

The relation of equivalence of Room designs is investigated with respect to the class of Room designs formed from starters with adders in the conventional way. It is shown that, if p^k = 3 + 4t, $\text{s = t ? 2[}\text{t}\text{2}\text{]}$, and $\text{I = (}\text{1}\text{k}\text{) ∑}{}_{\mathrm{<mtext>d</mtext><mtext>k</mtext>}}\text{P}{}^{\mathrm{d}}\text{φ}$ ($\text{k}\text{d}$), then it is possible to construct, without duplication, at least $\text{(I + (?3)}{}^{\mathrm{1–8}}\text{)}\text{8}$ inequivalent patterned Room designs of side p^k. This result implies, for instance, the existence of at least 11 equivalence classes of Room designs of side 87 which contain a patterned Room design; it also implies the existence of at least 5 · 10⁵ equivalence classes of Room designs of side 7⁹ which contain a patterned Room design. Some additional results are obtained.     

Representations of differential operators on a Lie group

In this paper we apply the theory of second-order partial differential operators with nonnegative characteristic form to representations of Lie groups. We are concerned with a continuous representation U of a Lie group G in a Banach space $\text{B}$. Let $\text{E}$ be the enveloping algebra of G, and let dU be the infinitesimal homomorphism of $\text{E}$ into operators with the Gårding vectors as a common invariant domain. We study elements in $\text{E}$ of the form

$\text{P=}\text{∑}\text{1}\text{r}\text{X}{}^2{}_{\mathrm{j}}\text{|X}{}_0$

with the X_j,'s in the Lie algebra $\text{G}$.If the elements X₀, X₁,…, X_r generate $\text{G}$ as a Lie algebra then we show that the space of C^∞-vectors for U is precisely equal to the C^∞-vectors for the closure $\text{dU(P)}\text{,}\text{of}\text{dU(P)}$. This result is applied to obtain estimates for differential operators.The operator $\text{dU(P)}$ is the infinitesimal generator of a strongly continuous semigroup of operators in $\text{B}$. If X₀ = 0 we show that this semigroup can be analytically continued to complex time ζ with Re ζ > 0. The generalized heat kernels of these semigroups are computed. A space of rapidly decreasing functions on G is introduced in order to treat the heat kernels.For unitary representations we show essential self-adjointness of all operators $\text{dU(Σ}{}_1{}^{\mathrm{r}}\text{X}{}_{\mathrm{j}}{}^2\text{+ (?1)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{X}{}_0$ with X₀ in the real linear span of the X_j's. An application to quantum field theory is given.Finally, the new characterization of the C^∞-vectors is applied to a construction of a counterexample to a conjecture on exponentiation of operator Lie algebras.Our results on semigroups of exponential growth, and on the space of C^∞ vectors for a group representation can be viewed as generalizations of various results due to Nelson-Stinespring [18], and Poulsen [19], who prove essential self-adjointness and a priori estimates, respectively, for the sum of the squares of elements in a basis for $\text{G}$ (the Laplace operator). The work of Hörmander [11] and Bony [3] on degenerate-elliptic (hypoelliptic) operators supplies the technical basis for this generalization. The important feature is that elliptic regularity is too crude a tool for controlling commutators. With the aid of the above-mentioned hypoellipticity results we are able to “control” the (finite dimensional) Lie algebra generated by a given set of differential operators.     

Convex polyhedra of doubly stochastic matrices. I. Applications of the permanent function

The permanent function is used to determine geometrical properties of the set $\text{Ω}{}_{\mathrm{n}}$ of all n × n nonnegative doubly stochastic matrices. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$, then $\text{F}$ corresponds to an n × n (0, 1)-matrix A, where the permanent of A is the number of vertices of $\text{F}$. If A is fully indecomposable, then the dimension of $\text{F}$ equals σ(A) ? 2n + 1, where σ(A) is the number of 1's in A. The only two-dimensional faces of $\text{Ω}{}_{\mathrm{n}}$ are triangles and rectangles. For n ? 6, $\text{Ω}{}_{\mathrm{n}}$ has four types of three-dimensional faces. The facets of the faces of $\text{Ω}{}_{\mathrm{n}}$ are characterized. Faces of $\text{Ω}{}_{\mathrm{n}}$ which are simplices are determined. If $\text{F}$ is a face of $\text{Ω}{}_{\mathrm{n}}$ which is two-neighborly but not a simplex, then $\text{F}$ has dimension 4 and six vertices. All k-dimensional faces with k + 2 vertices are determined. The maximum number of vertices of a k-dimensional face is 2^k. All k-dimensional faces with at least 2^k?1 + 1 vertices are determined.     

Bounds for the uniform deviation of empirical measures

If X₁,…,X_n are independent identically distributed R^d-valued random vectors with probability measure μ and empirical probability measure μ_n, and if $\text{a}$ is a subset of the Borel sets on R^d, then we show that P{sup_{A∈$\text{a}$}|μ_n(A)?μ(A)|≥ε} ≤ cs($\text{a}$, n²)e^?2n∈2, where c is an explicitly given constant, and s($\text{a}$, n) is the maximum over all (x₁,…,x_n) ∈ R^dn of the number of different sets in {{x₁…,x_n}∩A|A ∈$\text{a}$}. The bound strengthens a result due to Vapnik and Chervonenkis.     

The spherical Bochner theorem on semisimple Lie groups

Let G be a connected semisimple Lie group with finite center and K a maximal compact subgroup. Denote (i) Harish-Chandra's Schwartz spaces by $\text{C}$^p(G)(0<p?2), (ii) the K-biinvariant elements in $\text{C}$^p(G) by $\text{I}$^p(G), (iii) the positive definite (zonal) spherical functions by $\text{P}$, and (iv) the spherical transform on $\text{C}$^p(G) by ? → $\text{}\text{?}$gj. For T a positive definite distribution on G it is established that (i) T extends uniquely onto $\text{C}$^l(G), (ii) there exists a unique measure μ of polynomial growth on $\text{P}$ such that T[ψ]=∫pψdμ for all ψ?I¹(G) (iii) all measures μ of polynomial growth on $\text{P}$ are obtained in this way, and (iv) T may be extended to a particular $\text{I}$^p(G) space (1 ? p ? 2) if and only if the support of μ lies in a certain easily defined subset of $\text{P}$. These results generalize a well-known theorem of Godement, and the proofs rely heavily on the recent harmonic analysis results of Trombi and Varadarajan.     

A class of extension properties that are not simply generated

Let $\text{P}$ be a closed-hereditary topological property preserved by products. Call a space $\text{P}$-regular if it is homeomorphic to a subspace of a product of spaces with $\text{P}$. Suppose that each $\text{P}$-regular space possesses a $\text{P}$-regular compactification. It is well-known that each $\text{P}$-regular space X is densely embedded in a unique space γ_scPX with $\text{P}$ such that if f: X → Y is continuous and Y has $\text{P}$, then f extends continuously to γ_scPX. Call $\text{P}$-pseudocompact if γ_scPX is compact.Associated with $\text{P}$ is another topological property $\text{P}$^#, possessing all the properties hypothesized for $\text{P}$ above, defined as follows: a $\text{P}$-regular space X has $\text{P}$^# if each $\text{P}$-pseudocompact closed subspace of X is compact. It is known that the $\text{P}$-pseudocompact spaces coincide with the $\text{P}$^#-pseudocompact spaces, and that $\text{P}$^# is the largest closed-hereditary, productive property for which this is the case. In this paper we prove that if $\text{P}$ is not the property of being compact and $\text{P}$-regular, then $\text{P}$^# is not simply generated; in other words, there does not exist a space E such that the spaces with $\text{P}$^# are precisely those spaces homeomorphic to closed subspaces of powers of E.     

Localization in fine potential theory and uniform approximation by subharmonic functions

It is shown how the cone $\text{l}$(U) of superharmonic functions ?0 on an open set U in $\text{R}$ⁿ, n ? 3, can be recovered from the cone $\text{l}$ of superharmonic functions ?0 on the whole of $\text{R}$ⁿ by a process involving the operator of localization associated with U. Actually we treat the more general case where U is open in the Cartan-Brelot fine topology on $\text{R}$ⁿ. As an application we obtain a new proof of a theorem of J. Bliedtner and W. Hansen on uniform approximation by continuous subharmonic functions in open sets containing a given compact set K in $\text{R}$ⁿ.     

Spectral integrals of operator-valued functions. II. From the study of stationary processes

This paper is a continuation of the study made in [38]. Using Douglas' operator range theorem and Crimmins' corollary we obtain several new results on the “square-integrability of operator-valued functions with respect to a nonnegative hermitian measure”. Using these facts we are able to extend in an important way theorems on the “spectral integral of an operator-valued function” which were obtained in [38], to wit, we are able to drop assumptions that functions are closed operator-valued. We apply these results to Wiener-Masani type infinite-dimensional stationary processes, representing a purely non-deterministic process as a “moving average” and obtaining a “factorization” of its spectral density. Next, anticipating global applications of our tools, we investigate the adjoint and generalized inverse of spectral integrals. Our definition of measurability for closed-operator-valued functions plays a key role here. Finally, we partially prove a conjecture (J. Multivariate Anal. (1974), 166–209) on simpler necessary and sufficient conditions on “when is a closed densely defined operator T from $\text{H}$^q to $\text{H}$^p a spectral integral T = fΦdE?”: Let q be finite and E be of countable multiplicity for $\text{H}$. Then (i) Tx ∈ $\text{S}$_x^p each x ∈ $\text{D}$_T (T is E-subordinate), and (ii) E(B)T ? TE(B) each B ∈ $\text{B}$ (T is E-commutative) implies L_x^pT ? TL_x^q each x ∈ $\text{H}$^q (T commutes with all the cyclic projections), and thus T = fΦdE.     

Partitions of large multipartites with congruence conditions. II

For 1 ≦ l ≦ j, let $\text{a}$_l = ?_h=1^q(l){a_lh + Mv: v = 0, 1, 2,…}, where j, M, q(l) and the a_lh are positive integers such that j > 1, a_l1 < … < a_lq(2) ≦ M, and let $\text{a}$_l^′ = $\text{a}$_l ∪ {0}. Let p(n : $\text{B}$) be the number of partitions of n = (n₁,…,n_j) where, for 1 ≦ l ≦ j, the lth component of each part belongs to $\text{B}$_l and let p¹(n : $\text{B}$) be the number of partitions of n into different parts where again the lth component of each part belongs to $\text{B}$_l. Asymptotic formulas are obtained for p(n : $\text{a}$), p¹(n : $\text{a}$) where all but one n_l tend to infinity much more rapidly than that n_l, and asymptotic formulas are also obtained for p(n : $\text{a}$′), p¹(n ; $\text{a}$′), where one n_l tends to infinity much more rapidly than every other n_l. These formulas contrast with those of a recent paper (Robertson and Spencer, Trans. Amer. Math. Soc., to appear) in which all the n_l tend to infinity at approximately the same rate.