In this paper, we establish the regularity of strong solutions to nondivergence parabolic equations with BMO coefficients in nondoubling weighted spaces.

In a recent paper [2], Bushard established and applied a comparison theorem for positive solutions to the equation:

in an arbitrary bounded domain D of Euclidean w-space Rn. The proof of these results depended on the absence of mixed derivatives of u in the equation considered.

We say that two elements e and f of a lattice are moderately separated provided e ∧ f = 0 and both (e′, f′) and (f′, e′) are modular pairs for all e′ ≦ e and f′ ≦ f. Here (e′, f′) a modular pair means that, for all g ≧ e′,

In the lattice of projections of a factor we show that e and f, with e ∧ f = 0, are modularly separated if and only if ‖(e – k)f‖ < 1 for some finite projection k ≦ e. From there we can show that a kind of “independence property” holds for modular separation in this case: if e and f are modularly separated and if e ∨ f and g are modularly separated, then e and f ∧ g are modularly separated.

Let M be a von Neumann algebra acting on a Hilbert space and assume that M has a separating and cyclic vector ω in . Then it can happen that M contains a proper von Neumann subalgebra N for which ω is still cyclic. Such an example was given by Kadison in [4]. He considered and acting on where is a separable Hilbert space. In fact by a result of Dixmier and Maréchal, M, M′ and N have a joint cyclic vector [3]. Also Bratteli and Haagerup constructed such an example ([2], example 4.2) to illustrate the necessity of one of the conditions in the main result of their paper. In fact this situation seems to occur rather often in quantum field theory (see [1] Section 24.2, [3] and [4]).

In a recent interesting paper (3) H. Schwerdtfeger answered a question of W. R. Utz (4) on the structure of the real solutions A of A* = B, where A is skew-symmetric. (Utz and Schwerdtfeger call A* the "adjugate" of A ; A* is the n-square matrix whose (i, j) entry is (—1)i+j times the determinant of the (n — 1)-square matrix obtained by deleting row i and column j of A. The word "adjugate," however, is more usually applied to the matrix (AT)*, where AT denotes the transposed matrix of A ; cf. (1, 2).)

The object of the present paper is to find all real n-square skew-symmetric solutions A to the equation

A quantization of a fixed classical mechanical system is firstly an association between quantum mechanical observables (preferably self-adjoint operators on Hilbert space) and classical mechanical observables (i.e. real-valued functions on phase space). Secondly, a quantization should permit an interpretation of the correspondence principle that ‘classical mechanics is the limit of quantum mechanics as Planck's constant approaches zero'. With these two underlying precepts, Section 2 states the four basic requirements, I to IV, of a quantization along with an additional requirement V that characterizes the subclass of special quantizations.

Let $\pi $ be a square integrable representation of ${G}'=\text{S}{{\text{L}}_{n}}(D)$, with $D$ a central division algebra of finite dimension over a local field $F$of non-zero characteristic. We prove that, on the elliptic set, the character of $\pi $ equals the complex conjugate of the orbital integral of one of the pseudocoefficients of $\pi $. We prove also the orthogonality relations for characters of square integrable representations of ${G}'$. We prove the stable transfer of orbital integrals between $\text{S}{{\text{L}}_{n}}(F)$ and its inner forms.

In this paper, we are concerned with the basic problem defined in [9]. Formulas for δV(0)and δ∞V(0),respectively the generalized and asymptotic gradient of the value function at zero, corresponding to an L2 -additive perturbation of dynamics are given. Under the normality condition, δV(0)turns out to be a compact subset of L2, formed entirely of arcs, and V is locally finite and Lipschitz at 0. Moreover, estimations of the generalized directional derivative and Dini's derivative of V at 0 are derived. Supplementary conditions imply that Dini's derivative of V at 0 exists, and V is actually strictly differentiate at this point.

Let A be a uniform algebra on a compact Hausdorff space X. The spectrum, or the maximal ideal space, MA, of A is given by

We define the measure spectrum, SA, of A by

SA is the set of all representing measures on X for all Φ ∈ MA. (A representing measure for Φ ∈ MA is a probability measure μ on X satisfying

The concept of representing measure continues to be an effective tool in the study of uniform algebras. See for example [12, Chapters 2 and 3], [5, pp. 15-22] and [3]. Most of the known results on the subject of representing measures, however, concern measures associated with a single homomorphism.

In this note we are concerned with normed algebras over a non-discrete field with absolute value. The norm, N, of a normed algebra A is said to satisfy a polynomial identity on a subset B if there is a polynomial such that P(N(x1), … , N(xr)) = N(P(x1 … , xr)) whenever x1, … , xr are in B, where the polynomial has rational integer coefficients, degree greater than 1, constant term zero, and non-negative coefficients for each term of highest degree. It is shown in Theorem 1, following a method of proof used by Kadison in (6, § 7), that if the norm of a normed algebra satisfies a polynomial identity on the entire algebra, then the norm is power multiplicative. (That is, then N(x)2 = N(x2) for all x.)

1. Introduction. As is well known, the functions

(1.1)

have various applications in theoretical physics and engineering. It is thus worthwhile to study their behaviour for real and complex values of the argument.

Given an abelian group G and a mapping θ that maps a subgroup A of G homomorphically onto another subgroup B of G, then it is known (3) that there always exists an embedding group G* ⊇ G which is abelian and possesses an endomorphism θ* which coincides with θ on A, i.e. aθ = aθ* whenever aθ is defined. θ is called a partial endomorphism of G and θη a total endomorphism or simply an endomorphism that extends or continues θ.

Let $G$ be a symmetrizable indefinite Kac-Moody group over $\mathbb{C}$. Let $\text{T}{{\text{r}}_{{{\Lambda }_{1}}\,,\ldots ,\,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}}$ be the characters of the fundamental irreducible representations of $G$, defined as convergent series on a certain part ${{G}^{\text{tr}-\text{alg}}}\,\subseteq \,G$. Following Steinberg in the classical case and Brüchert in the affine case, we define the Steinberg map $\chi \,:=\,\left( \text{T}{{\text{r}}_{{{\Lambda }_{1}},\ldots ,}}\text{T}{{\text{r}}_{{{\Lambda }_{2n-l}}}} \right)$ as well as the Steinberg cross section $C$, together with a natural parametrisation $\omega :{{\mathbb{C}}^{n}}\times {{\left( {{\mathbb{C}}^{\times }} \right)}^{n-l}}\to C$. We investigate the local behaviour of $\text{ }\!\!\chi\!\!\text{ }$ on $C$ near $\omega \left( \,\left( 0,\ldots 0 \right)\,\times \,\left( 1,\ldots ,1 \right)\, \right)$, and we show that there exists a neighborhood of $\left( 0,...,0 \right)\,\,\times \,\,\left( 1,...,1 \right)$, on which $\text{ }\!\!\chi\!\!\text{ }\,\circ \,\omega $ is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of $G$ on $C$.

Let be a W*-algebra, A ( ) the group of all automorphisms of . In this paper we have determined the commutant G’ of a subgroup G of A () for certain classes of G and . The main results are as follows.

Theorem 1. If G is a locally compact abelian group acting by translation on the W*-algebra L∞(G), then the commutant of a dense subgroup of G is G itself.

1. Introduction. M. Cohen [1] denned a polyhedron K to be n-collapsible if K × In PL collapses. He proved that any spine of the cube B3 is 3-collapsible. This was a step directed toward the Zeeman Conjecture [4], which asserts that every compact contractible 2-polyhedron is 1-collapsible. In this paper we improve the result of Cohen by one dimension (Theorem 3): Any spine of the cube is 2-collapsible. The central question of 1-collapsibility remains unanswered.

Gillman and Rolfsen [3] have shown that any standard spine of the cube is 1-collapsible. Conjecture: If K is any spine of the cube, then K × I collapses to a standard spine of the cube. This would imply our main theorem. Lacking a proof of this conjecture, we must resort to an argument independent of [3].

THEOREM 1. Let A1, A2, …, Anbe a finite collection of pairwise disjoint contractible PL subsets of the cube. Then the decomposition obtained by shrinking each Ai to a point is 1-collapsible.

The Kronecker modules, $\mathbb{V}\left( m,h,\alpha \right)$, where $m$ is a positive integer, $h$ is a height function, and $\alpha $ is a $K$-linear functional on the space $K(X)$ of rational functions in one variable $X$ over an algebraically closed field $K$, are models for the family of all torsion-free rank-2 modules that are extensions of finite-dimensional rank-1 modules. Every such module comes with a regulating polynomial $f$ in $K(X)[Y]$. When the endomorphism algebra of $\mathbb{V}\left( m,h,\alpha \right)$ is commutative and non-trivial, the regulator $f$ must be quadratic in $Y$. If $f$ has one repeated root in $K(X)$, the endomorphism algebra is the trivial extension $K\ltimes S$ for some vector space $S$. If $f$ has distinct roots in $K(X)$, then the endomorphisms form a structure that we call a bridge. These include the coordinate rings of some curves. Regardless of the number of roots in the regulator, those End $\mathbb{V}\left( m,h,\alpha \right)$ that are domains have zero radical. In addition, each semi-local End $\mathbb{V}\left( m,h,\alpha \right)$ must be either a trivial extension $K\ltimes S$ or the product $K\times K$.

Consider the nth order linear equation

and particularly the third order equation

A nontrivial solution of (1)n is said to be oscillatory or nonoscillatory depending on whether it has infinitely many or finitely many zeros on [a, ∞). Let denote respectively the set of all solutions, oscillatory solutions, nonoscillatory solutions of (1)n. is an n-dimensional linear space. A subspace is said to be nonoscillatory or strongly oscillatory respectively if every nontrivial solution of is nonoscillatory or oscillatory. If contains both oscillatory and nonoscillatory solutions then is said to be weakly oscillatory.

Let ωn(x) denote the Jacobi polynomials with the weight Function

If we denote the corresponding normalized Jacobi polynomials by we Have

(1.1)

Now let

be the nth partial sum of the Fourier series of Jacobi polynomials of a function f(x).

We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.

Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).