Integrability of infinite sums of independent vector-valued random variables
Naresh C.
Jain;
Michael B.
Marcus
1-36
Abstract: Let B be a normed vector space (possibly a Banach space, but it could be more general) and $ \{ {X_n}\}$ a sequence of B-valued independent random variables on some probability space. Let $ {S_n} = \Sigma _{j = 1}^n{X_j},M = {\sup _n}\vert{S_n}\vert$ and $S = {\lim _n}{S_n}$ is norm, whenever it exists. Assuming that S exists or $M < \infty$ a.s. and given certain nondecreasing functions $\varphi$, we find conditions in terms of the distributions of $\left\Vert {{X_n}} \right\Vert$ such that $E(\varphi (M))$ or $E(\varphi (\left\Vert S \right\Vert))$ is finite. Let $\{ {u_n}\}$ be a sequence of elements in B and $ \{ {\varepsilon _n}\}$ a sequence of independent, identically distributed random variables such that $P\{ {\varepsilon _1} = 1\} = P\{ {\varepsilon _1} = - 1\} = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}$. We prove some comparison theorems which generalize the following contraction principle of Kahane: If $\{ {\lambda _n}\}$ is a bounded sequence of scalars, then $ \Sigma {\varepsilon _n}{u_n}$ converges in norm a.s. (or is bounded a.s.) implies the corresponding conclusion for the series $\Sigma {\lambda _n}{\varepsilon _n}{u_n}$. Some generalizations of this contraction principle have already been carried out by Hoffmann-Jørgensen. All these earlier results are subsumed by ours. Applications of our results are made to Gaussian processes, random Fourier series and other random series of functions.
Well-ordering of certain numerical polynomials
William Yu
Sit
37-45
Abstract: An ordering is introduced in the set of numerical polynomials (in one variable) which is then shown to induce a well-ordering on a certain subset of numerical polynomials, namely those which occur as the differential dimension polynomials of differential algebraic varieties, or equivalently, those which come from initial subsets of ${{\text{N}}^m}$.
Finite projections in tensor product von Neumann algebras
George A.
Elliott
47-60
Abstract: The work of Bures, Moore, Takenouchi, Hill and Størmer on the type classification of infinite tensor products of factors is extended to the nonfactor case.
Integration of functions with values in locally convex Suslin spaces
G. Erik F.
Thomas
61-81
Abstract: The main purpose of the paper is to give some easily applicable criteria for summability of vector valued functions with respect to scalar measures. One of these is the following: If E is a quasi-complete locally convex Suslin space (e.g. a separable Banach or Fréchet space), $\sigma (E,H)$. f is actually Pettis summable for the given topology. (Thus any E-valued function for which the integrals over measurable subsets can be reasonably defined as elements of E is Pettis summable.) A class of ``totally summable'' functions, generalising the Bochner integrable functions, is introduced. For these Fubini's theorem, in the case of a product measure, and the differentiation theorem, in the case of Lebesgue measure, are valid. It is shown that weakly summable functions with values in the spaces
The structure of semiprimary and Noetherian hereditary rings
John
Fuelberth;
James
Kuzmanovich
83-111
Abstract: In the first portion of this paper a structure theorem for semiprimary hereditary rings is given in terms of $M \times M$ ``triangular'' row-finite matrices over a division ring D. This structure theorem differs from previous theorems of this type in that the representation is explicit in terms of matrices over a division ring. In the second portion of this paper we are able to apply the results of Gordon and Small to obtain a structure theorem for semihereditary and left hereditary rings which are left orders in a semiprimary ring. In the case of the left hereditary rings, the representation is explicit in terms of matrices over left hereditary Goldie prime rings and their respective classical left quotient rings. As an application we obtain, by a different method, a non-Noetherian generalization of a result of Chatters which states that a two-sided hereditary Noetherian ring is a ring direct sum of an Artinian ring and a semiprime ring.
Symplectic homogeneous spaces
Shlomo
Sternberg
113-130
Abstract: In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [4] and Souriau [5] and was recently developed from a more general point of view by Chu [2]. The first part of this paper is devoted to reviewing the Kostant, Souriau, Chu results and deriving from them a generalization of the Cartan conjugacy theorem. In the second part of this paper we apply these results to Lie algebras admitting a generalized (k, p) decomposition.
Minimal complementary sets
Gerald
Weinstein
131-137
Abstract: Let G be a group on which a measure m is defined. If $A,B \subset G$ we define $A \oplus B = C = \{ c\vert c = a + b,a \in A,b \in B\}$. By $ {A_k} \subset G$ we denote a subset of G consisting of k elements. Given ${A_k}$ we define $s({A_k}) = \inf m\{ B\vert B \subset G,{A_k} \oplus B = G\}$ and ${c_k} = {\sup _{{A_k} \subset G}}s({A_k})$. Theorems 1, 2, and 3 deal with the problem of determining $ {c_k}$. In the dual problem we are given B, $m(B) > 0$, and required to find minimal A such that $A \oplus B = G$ or, sometimes, $m(A \oplus B) = m(G)$. Theorems 5 and 6 deal with this problem.
The Wedderburn principal theorem for generalized alternative algebras. I
Harry F.
Smith
139-148
Abstract: A generalized alternative ring I is a nonassociative ring R in which the identities $(wx,y,z) + (w,x,(y,z)) - w(x,y,z) - (w,y,z)x;((w,x),y,z) + (w,x,yz) - y(w,x,z) - (w,x,y)z$; and $(x,x,x)$ are identically zero. Let A be a finite-dimensional algebra of this type over a field F of characteristic $\ne 2,3$. Then it is established that (1) A cannot be a nodal algebra, and (2) the standard Wedderburn principal theorem is valid for A.
Inequalities for a complex matrix whose real part is positive definite
Charles R.
Johnson
149-154
Abstract: Denote the real part of $A \in {M_n}(C)$ by $ H(A) = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}(A + {A^\ast})$. We provide dual inequalities relating $H({A^{ - 1}})$ and $ H{(A)^{ - 1}}$ and an identity between two functions of A when A satisfies $H(A) > 0$. As an application we give an inequality (for matrices A satisfying $H(A) > 0$) which generalizes Hadamard's determinantal inequality for positive definite matrices.
Equivariant homology theories on $G$-complexes
Stephen J.
Willson
155-171
Abstract: A definition is given for a ``cellular'' equivariant homology theory on G-complexes. The definition is shown to generalize to G-complexes with prescribed isotropy subgroups. A ring I is introduced to deal with the general definition. One obtains a universal coefficient theorem and studies the universal coefficients.
On the existence of a universal germ of deformations for elliptic pseudogroup structures on compact manifolds
Suresh H.
Moolgavkar
173-197
Abstract: The purpose of this paper is to prove the existence of a versal germ of deformations for elliptic pseudogroup structures on compact manifolds. Under suitable restrictions, the versal germ is shown to be universal.
$C\sp*$-algebras with Hausdorff spectrum
John W.
Bunce;
James A.
Deddens
199-217
Abstract: By the spectrum of a ${C^\ast}$-algebra we mean the set of unitary equivalence classes of irreducible representations equipped with the hull-kernel topology. We are concerned with characterizing the $ {C^\ast}$-algebras with identity which have Hausdorff spectrum. We characterize the ${C^\ast}$-algebras with identity and bounded representation dimension which have Hausdorff spectrum. Our results are more natural when the $ {C^\ast}$-algebra is singly generated. For singly generated $ {C^\ast}$-algebras with unbounded representation dimension, we reduce the problem to the case when the generator is an infinite direct sum of irreducible finite scalar matrices, and we have partial results in this case.
Semigroups with a dense subgroup
W. S.
Owen
219-228
Abstract: The purpose of this paper is twofold. First, it is shown that the ideal structure of a semigroup with dense subgroup is closely related to its transformation group structure. That is, if a left orbit through a given point is locally compact, then the members of this orbit are precisely those elements which generate the same left ideal as the given point. Secondly, the author gives a number of theorems which have as their goal the establishment of a natural product structure near a nonzero idempotent. Specifically the work of F. Knowles [11] is improved upon to include (1) the possibility of a nonconnected group; (2) the possibility of a nonsimply connected orbit; and (3) the case in which the boundary of the group is more than a single orbit.
Singularity subschemes and generic projections
Joel
Roberts
229-268
Abstract: Corresponding to a morphism $f:V \to W$ of algebraic varieties (such that $\dim (V) \leqslant \dim (W)$), we construct a family of subschemes $S_1^{(q)}(f) \subset V$. When V and W are nonsingular, the $S_1^{(q)},q \geqslant 1$, induce a filtration of the set of closed points $x \in V$ such that the tangent space map $ d{f_x}:T{(V)_x} \to T{(W)_{f(x)}}$ has rank $ = \dim (V) - 1$. We prove that if V is a suitably embedded nonsingular projective variety and $\pi :V \to {{\mathbf{P}}^m}$ is a generic projection, then the $ S_1^{(q)}(f)$ and certain fibre products of several of the $S_1^{(q)}(f)$ are either empty or smooth and of the smallest possible dimension, except in cases where $ q + 1$ is divisible by the characteristic of the ground field. We apply this result to describe explicitly the ring homomorphisms ${\pi ^\ast}:{\hat{\mathcal{O}}_{{{\mathbf{P}}^m}\pi (x)}} \to {\hat{\mathcal{O}}_{V,x}}$ and (when $ m \geqslant r + 1$) to study the local structure of the image
The spectral sequence of a finite group extension stops
Leonard
Evens
269-277
Abstract: It is proved that if G is a finite group, H a normal subgroup, and A a finitely generated G-module, then both the cohomology and homology spectral sequences for the group extension stop in a finite number of stops. A lemma about $ {\operatorname{Tor}}(M,N)$ as a module over $R \otimes S$ is proved. Two spectral sequences of Hochschild and Serre are shown to be the same.
On properties of the approximate Peano derivatives
Bruce S.
Babcock
279-294
Abstract: The notion of kth approximate Peano differentiation not only generalizes kth ordinary differentiation but also kth Peano differentiation and kth ${L_p}$ differentiation. Recently, M. Evans has shown that a kth approximate Peano derivative at least shares with these other derivatives the property of belonging to Baire class one. In this paper the author extends the properties possessed by a kth approximate Peano derivative by showing that it is like the above derivatives in that it also possesses the following properties: Darboux, Denjoy, Zahorski, and a new property stronger than the Zahorski property, Property Z.
Units and periodic Jacobi-Perron algorithms in real algebraic number fields of degree $3$
Leon
Bernstein
295-306
Abstract: It is not known whether or not the Jacobi-Perron Algorithm of a vector in $ {R_{n - 1}},n \geqslant 3$, whose components are algebraic irrationals, always becomes periodic. The author enumerates, from his previous papers, a few infinite classes of real algebraic number fields of any degree for which this is the case. Periodic Jacobi-Perron Algorithms are important, because they can be applied, inter alia, to calculate units in the corresponding algebraic number fields. The main result of this paper is expressed in the following theorem: There are infinitely many real cubic fields $ Q(w),{w^3}$ cubefree, a and T natural numbers, such that the Jacobi-Perron Algorithm of the vector $(w,{w^2})$ becomes periodic; the length of the primitive preperiod is four, the length of the primitive period is three; a fundamental unit of $ Q(w)$ is given by $e = {a^3}T + 1 - aw$.
On sum-free subsequences
S. L. G.
Choi;
J.
Komlós;
E.
Szemerédi
307-313
Abstract: A subsequence of a sequence of n distinct integers is said to be sum-free if no integer in it is the sum of distinct integers in it. Let $f(n)$ denote the largest quantity so that every sequence of n distinct integers has a sum-free subsequence consisting of $f(n)$ integers. In this paper we strengthen previous results by Erdös, Choi and Cantor by proving $\displaystyle {(n\;\log \;n/\log \;\log \;n)^{{\raise0.5ex\hbox{$\scriptstyle 1... ....15em \lower0.25ex\hbox{$\scriptstyle 2$}}}} \ll f(n) \ll n{(\log \;n)^{ - 1}}.$
On commutators of singular integrals and bilinear singular integrals
R. R.
Coifman;
Yves
Meyer
315-331
Abstract: ${L^p}$ estimates for multilinear singular integrals generalizing Calderón's commutator integral are obtained. The methods introduced involve Fourier and Mellin analysis.
Stable equivalence for some categories with radical square zero
Idun
Reiten
333-345
Abstract: For certain abelian categories with radical square zero, containing artin rings with radical square zero as a special case, we give a way of constructing hereditary abelian categories stably equivalent to them, i.e. such that their categories modulo projectives are equivalent categories.
On finite Hilbert transforms
Kevin F.
Clancey
347-354
Abstract: Let E be a bounded measurable subset of the real line. The finite Hilbert transform is the operator ${H_E}$ defined on one of the spaces ${L^p}(E)(1 < p < \infty )$ by $\displaystyle {H_E}f(x) = {(\pi i)^{ - 1}}\int_E {f(t){{(t - x)}^{ - 1}}\;dt;}$ here, the singular integral is interpreted as a Cauchy principal value. The main result establishes that for ${H_E}$ to be Fredholm on ${L^p}(E)$, when $p \ne 2$, it is necessary and sufficient that E be equal almost everywhere to a finite union of intervals. The sufficiency of this condition was established in 1960 by H. Widom. In the case where E is not a finite union of intervals and $p < 2$ it is shown that the operator ${H_E}$ has an infinite dimensional null space. The method of proof is constructive.
On some real hypersurfaces of a complex projective space
Masafumi
Okumura
355-364
Abstract: A principal circle bundle over a real hypersurface of a complex projective space $C{P^n}$ can be regarded as a hypersurface of an odd-dimensional sphere. From this standpoint we can establish a method to translate conditions imposed on a hypersurface of $C{P^n}$ into those imposed on a hypersurface of ${S^{2n + 1}}$. Some fundamental relations between the second fundamental tensor of a hypersurface of $C{P^n}$ and that of a hypersurface of ${S^{2n + 1}}$ are given.
The trigonometric Hermite-Birkhoff interpolation problem
Darell J.
Johnson
365-374
Abstract: The classical Hermite-Birkhoff interpolation problem, which has recently been generalized to a special class of Haar subspaces, is here considered for trigonometric polynomials. It is shown that a slight weakening of the result (conservativity and Pólya conditions) established for those special Haar subspaces also holds for trigonometric polynomials after one rephrases the statement of the problem, the underlying assumptions, and the result itself appropriately to reflect the inherent differences between algebraic polynomials (which the special class of Haar subspaces essentially are) and the periodic trigonometric polynomials. Furthermore, simple necessary and sufficient conditions for poisedness of one-rowed incidence matrices analogous to the Pólya conditions for two-rowed incidence matrices in the algebraic version are proved, and an elementary necessary condition for the poisedness of an arbitrary (trigonometric) incidence matrix stated.
Existence and uniqueness theorems for Riemann problems
Tai Ping
Liu
375-382
Abstract: In [2] the author proposed the entropy condition (E) and solved the Riemann problem for general $2 \times 2$ conservation laws ${u_t} + f{(u,v)_x} = 0,{v_t} + g{(u,v)_x} = 0$, under the assumptions that the system is hyperbolic, and ${f_u} \geqslant 0$ and ${g_v} \leqslant 0$. The purpose of this paper is to extend the above results to a much wider class of $2 \times 2$ conservation laws. Instead of assuming that $ {f_u} \geqslant 0$ and ${g_v} \leqslant 0$, we assume that the characteristic speed is not equal to the shock speed of different family. This assumption is motivated by the works of Lax [1] and Smoller [4].
Smooth locally convex spaces
John
Lloyd
383-392
Abstract: The main theorem is Let E be a separable (real) Fréchet space with a nonseparable strong dual. Then E is not strongly $D_F^1$-smooth. It follows that if X is uncountable, locally compact, $\sigma $-compact, metric space, then $C(X)$ (with the topology of compact convergence) does not have a class of seminorms which generate its topology and are Fréchet differentiable (away from their null-spaces).
On all kinds of homogeneous spaces
Gerald S.
Ungar
393-400
Abstract: Several open questions on homogeneous spaces are answered. A few of the results are: (1) An n-homogeneous metric continuum, which is not the circle, is strongly n-homogeneous. (2) A 2-homogeneous metric continuum is locally connected. (3) If X is a homogeneous compact metric space or a homogeneous locally compact, locally connected separable metric space, then X is a coset space. (4) If G is a complete separable metric topological group with is n-connected, then G is locally n-connected.
Iterated nontangential limits
K.
Gowrisankaran
401-402
Abstract: For functions f in the Nevanlinna class of ${U^n}$ it is proved that the iterated nontangential limits are equal up to a set of measure zero.
On the fixed point indices and Nielsen numbers of fiber maps on Jiang spaces
Jingyal
Pak
403-415
Abstract: Let $T = \{ E,P,B\}$ be a locally trivial fiber space, where E, B and $ {P^{ - 1}}(b)$ for each $b \in B$ are compact, connected ANR's (absolute neighborhood retracts). If $ f:E \to E$ is a fiber (preserving) map then f induces ${f_b}:{P^{ - 1}}(b) \to {P^{ - 1}}(b)$ for each $b \in B$ such that $ Pf = f'P$. If E, B and ${P^{ - 1}}(b)$ for each $ b \in B$ satisfy the Jiang condition then $i:{P^{ - 1}}(b) \to E$ induces a monomorphism ${i_\char93 }:{\pi _1}({P^{ - 1}}(b)) \to {\pi _1}(E)$ and $f'$ induces a fixed point free homomorphism $T = \{ E,P,CP(n)\}$ be a principal torus bundle over an n-dimensional complex projective space $ CP(n)$. If $f:E \to E$ is a fiber map such that for some $b \in CP(n),{f_b}:{P^{ - 1}}(b) \to {P^{ - 1}}(b)$ is homotopic to a fixed point free map, then there exists a map $g:E \to E$ homotopic to f and fixed point free.
Addendum to: ``Behnke-Stein theorem for analytic spaces'' (Trans. Amer. Math. Soc. {\bf 199} (1974), 317--326)
Alessandro
Silva
417-418
Abstract: A very simple argument shows that Theorem 3.1 in my paper Behnke-Stein theorem for analytic spaces, (these Transactions, 199 (1974), pp. 317 326) is enough, via a Narasimhan result, to obtain information about the torsion of the homology groups of a Runge pair of Stein spaces.