Stochastic control problems and spherical functions on symmetric spaces
T. E.
Duncan;
H.
Upmeier
1083-1130
Abstract: A family of explicitly solvable stochastic control problems is formulated and solved in noncompact symmetric spaces. The symmetric spaces include all of the classical spaces and four of the exceptional spaces. The stochastic control problems are the control of Brownian motion in these symmetric spaces by a drift vector field. For each symmetric space a family of stochastic control problems is formulated by using spherical functions in the cost functionals. These spherical functions are explicitly described and are polynomials in suitable coordinates. A generalization to abstract root systems is given.
On a quadratic-trigonometric functional equation and some applications
J. K.
Chung;
B. R.
Ebanks;
C. T.
Ng;
P. K.
Sahoo
1131-1161
Abstract: Our main goal is to determine the general solution of the functional equation \begin{displaymath}\begin{array}{*{20}{c}} {{f_1}(xy) + {f_2}(x{y^{ - 1}}) = {f_... ...,} {{f_i}(txy) = {f_i}(tyx)\qquad (i = 1,2)} \end{array} \end{displaymath} where ${f_i}$ are complex-valued functions defined on a group. This equation contains, among others, an equation of H. Swiatak whose general solution was not known until now and an equation studied by K.S. Lau in connection with a characterization of Rao's quadratic entropies. Special cases of this equation also include the Pexider, quadratic, d'Alembert and Wilson equations.
Norm estimates for radially symmetric solutions of semilinear elliptic equations
Ryuji
Kajikiya
1163-1199
Abstract: The semilinear elliptic equation $ \Delta u + f(u) = 0$ in $ {R^n}$ with the condition $ {\lim _{\vert x\vert \to \infty }}u(x) = 0$ is studied, where $n \geqslant 2$ and $f(u)$ has a superlinear and subcritical growth at $u = \pm \infty$. For example, the functions $f(u) = \vert u{\vert^{p - 1}}u - u\;(1 < p < \infty \;{\text{if}}\;n = 2,\;1 < p < (n + 2)/(n - 2)\;{\text{if}}\;n \geqslant 3)$ and $f(u) = u\log \vert u\vert$ are treated. The $ {L^2}$ and ${H^1}$ norm estimates $ {C_1}{(k + 1)^{n/2}} \leqslant \vert\vert u\vert{\vert _{{L^2}}} \leqslant \vert\vert u\vert{\vert _{{H^1}}} \leqslant {C_2}{(k + 1)^{n/2}}$ are established for any radially symmetric solution $u$ which has exactly $k \geqslant 0$ zeros in the interval $0 \leqslant \vert x\vert < \infty$. Here ${C_1},\;{C_2} > 0$ are independent of $ u$ and $k$.
Hopf constructions and higher projective planes for iterated loop spaces
Nicholas J.
Kuhn;
Michael
Slack;
Frank
Williams
1201-1238
Abstract: We define a category, $ \mathcal{H}_p^n$ (for each $ n$ and $p$), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the $\bmod p$ Dyer-Lashof operations for $ n$-fold loop spaces. The category $ \mathcal{H}_p^n$ is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, $P_p^n$, from $ \mathcal{H}_p^n$ to the homotopy category of spaces and show $P_p^n$ to be left adjoint to the $ n$-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.
Attractors in inhomogeneous conservation laws and parabolic regularizations
Hai Tao
Fan;
Jack K.
Hale
1239-1254
Abstract: The asymptotic behavior of inhomogeneous conservation laws is considered. The attractor of the equation is characterized. The relationship between attractors of the equation and that of its parabolic regularization is studied.
Verifiable conditions for openness and regularity of multivalued mappings in Banach spaces
A.
Jourani;
L.
Thibault
1255-1268
Abstract: This paper establishes verifiable conditions in terms of approximate subdifferentials implying openness and metric regularity of multivalued mappings in Banach spaces. The results are then applied to derive Lagrange multipliers for general nonsmooth vector optimization problems.
Partial extensions of Attouch's theorem with applications to proto-derivatives of subgradient mappings
A. B.
Levy;
R.
Poliquin;
L.
Thibault
1269-1294
Abstract: Attouch's Theorem, which gives on a reflexive Banach space the equivalence between the Mosco epi-convergence of a sequence of convex functions and the graph convergence of the associated sequence of subgradients, has many important applications in convex optimization. In particular, generalized derivatives have been defined in terms of the epi-convergence or graph convergence of certain difference quotient mappings, and Attouch's Theorem has been used to relate these various generalized derivatives. These relations can then be used to study the stability of the solution mapping associated with a parameterized family of optimization problems. We prove in a Hilbert space several "partial extensions" of Attouch's Theorem to functions more general than convex; these functions are called primal-lower-nice. Furthermore, we use our extensions to derive a relationship between the second-order epi-derivatives of primal-lower-nice functions and the proto-derivative of their associated subgradient mappings.
The derivatives of homotopy theory
Brenda
Johnson
1295-1321
Abstract: We construct a functor of spaces, ${M_n}$, and show that its multilinearization is equivalent to the $n$th layer of the Taylor tower of the identity functor of spaces. This allows us to identify the derivatives of the identity functor and determine their homotopy type.
When does unique local support ensure convexity?
Donald Francis
Young
1323-1329
Abstract: A basic theorem of convex analysis states that a real-valued function on an open interval of the real line is convex and differentiable if at each point of its domain there exists a unique supporting line. In this paper we show that the same conclusion can be drawn under the weaker hypothesis that there exists a unique locally supporting line at each point. We also show by counterexample that convexity cannot be concluded under analogous circumstances for $ f:S \to \mathbb{R}$, where $S \subset {\mathbb{R}^n}$ is open and convex, if $n > 1$.
Cauchy-Green type formulae in Clifford analysis
John
Ryan
1331-1341
Abstract: A Cauchy integral formula is constructed for solutions to the polynomial Dirac equation $({D^k} + \sum\nolimits_{m = 0}^{k - 1} {{b_m}{D^m})f = 0}$, where each ${b_m}$ is a complex number, $ D$ is the Dirac operator in ${R^n}$, and $f$ is defined on a domain in $^{{R^n}}$ and takes values in a complex Clifford algebra. Some basic properties for the solutions to this equation, arising from the integral formula, are described, including an approximation theorem. We also introduce a Bergman kernel for square integrable solutions to $ (D + \lambda )f = 0$ over bounded domains with piecewise ${C^1}$, or Lipschitz, boundary.
A convergence theorem for Riemannian submanifolds
Zhong Min
Shen
1343-1350
Abstract: In this paper we study the convergence of Riemannian submanifolds. In particular, we prove that any sequence of closed submanifolds with bounded normal curvature and volume in a closed Riemannian manifold subconverge to a closed submanifold in the $ {C^{1,\alpha }}$ topology. We also obtain some applications to irreducible homogeneous manifolds and pseudo-holomorphic curves in symplectic manifolds.
A discrete transform and Triebel-Lizorkin spaces on the bidisc
Wei
Wang
1351-1364
Abstract: We use a discrete transform to study the Triebel-Lizorkin spaces on bidisc $\dot F_p^{\alpha q},\,\dot f_p^{\alpha q}$ and establishes the boundedness of transform $ {S_\phi }:\dot F_p^{\alpha q} \to \dot f_p^{\alpha q}$ and $ {T_\psi }:\dot f_p^{\alpha q} \to \dot F_p^{\alpha q}$. We also define the almost diagonal operator and prove its boundedness. With the use of discrete transform and Journé lemma, we get the atomic decomposition of $\dot f_p^{\alpha q}$ for $0 < p \leqslant 1,\,p \leqslant q < \infty$. The atom supports on an open set, not a rectangle. Duality $ {(\dot f_1^{\alpha q})^{\ast}} = \dot f_\infty ^{ - \alpha q'},\,\tfrac{1} {q} + \tfrac{1} {{q'}} = 1,\,q > 1,\,\alpha \in R$, is established, too. The case for $\dot F_p^{\alpha q}$ is similar.
A norm convergence result on random products of relaxed projections in Hilbert space
H. H.
Bauschke
1365-1373
Abstract: Suppose $ X$ is a Hilbert space and ${C_1}, \ldots ,{C_N}$ are closed convex intersecting subsets with projections ${P_1}, \ldots ,{P_N}$. Suppose further $ r$ is a mapping from $\mathbb{N}$ onto $ \{ 1, \ldots ,N\}$ that assumes every value infinitely often. We prove (a more general version of) the following result: If the $ N$-tuple $({C_1}, \ldots ,{C_N})$ is "innately boundedly regular", then the sequence $({x_n})$, defined by $\displaystyle {x_0} \in X\;{\text{arbitrary,}}\quad {x_{n + 1}}: = {P_{r(n)}}{x_n},\quad {\text{for all}}\;n \geqslant 0,$ converges in norm to some point in $\cap _{i = 1}^N{C_i}$. Examples without the usual assumptions on compactness are given. Methods of this type have been used in areas like computerized tomography and signal processing.
Asymptotic stability in functional-differential equations by Liapunov functionals
Bo
Zhang
1375-1382
Abstract: We consider the asymptotic stability in a system of functional differential equations $ {I_n} = [{S_n},{t_n}]$. We also show that it is not necessary to require a uniform upper bound on $V$ for nonuniform asymptotic stability.
The de Branges-Rovnyak model with finite-dimensional coefficients
James
Guyker
1383-1389
Abstract: A characterization in terms of the canonical model spaces of L. de Branges and J. Rovnyak is obtained for Hilbert spaces of formal power series with vector coefficients which satisfy a difference-quotient inequality, thereby extending the closed ideal theorems of A. Beurling and P. D. Lax.
Remarks on some integrals and series involving the Stirling numbers and $\zeta(n)$
Li-Chien
Shen
1391-1399
Abstract: From the perspective of the well-known identity $\displaystyle {}_2{F_1}(a,b;c;1) = \frac{{\Gamma (c)\Gamma (c - a - b)}} {{\Gamma (c - a)\Gamma (c - b)}},$ we clarify the connections between the Stirling numbers $s_k^n$ and the Riemann zeta function $\zeta (n)$. As a consequence, certain series and integrals can be evaluated in terms of $ \zeta (n)$ and $ s_k^n$.
The exposed points of the set of invariant means
Tianxuan
Miao
1401-1408
Abstract: Let $G$ be a $\sigma$-compact infinite locally compact group, and let $LIM$ be the set of left invariant means on ${L^\infty }(G)$. We prove in this paper that if $ G$ is amenable as a discrete group, then $LIM$ has no exposed points. We also give another proof of the Granirer theorem that the set $ LIM(X,G)$ of $ G$-invariant means on ${L^\infty }(X,\beta ,p)$ has no exposed points, where $G$ is an amenable countable group acting ergodically as measure-preserving transformations on a nonatomic probability space $ (X,\beta ,p)$.
Inverse theorems for subset sums
Melvyn B.
Nathanson
1409-1418
Abstract: Let $A$ be a finite set of integers. For $h \geqslant 1$, let ${S_h}(A)$ denote the set of all sums of $ h$ distinct elements of $ A$. Let $S(A)$ denote the set of all nonempty sums of distinct elements of $A$. The direct problem for subset sums is to find lower bounds for $ \vert{S_h}(A)\vert$ and $ \vert S(A)\vert$ in terms of $\vert A\vert$. The inverse problem for subset sums is to determine the structure of the extremal sets $A$ of integers for which $ \vert{S_h}(A)\vert$ and $ \vert S(A)\vert$ are minimal. In this paper both the direct and the inverse problem for subset sums are solved.
Groups with no free subsemigroups
P.
Longobardi;
M.
Maj;
A. H.
Rhemtulla
1419-1427
Abstract: We look at groups which have no (nonabelian) free subsemigroups. It is known that a finitely generated solvable group with no free subsemigroup is nilpotent-by-finite. Conversely nilpotent-by-finite groups have no free subsemigroups. Torsion-free residually finite- $ p$ groups with no free subsemigroups can have very complicated structure, but with some extra condition on the subsemigroups of such a group one obtains satisfactory results. These results are applied to ordered groups, right-ordered groups, and locally indicable groups.
Matrix variate $\theta$-generalized normal distribution
A. K.
Gupta;
T.
Varga
1429-1437
Abstract: In this paper, the matrix variate $\theta$-generalized normal distribution is introduced. Then its properties are studied. In particular, it is proved that this distribution has maximal entropy in a certain class of distributions.
Differential operators, $n$-branch curve singularities and the $n$-subspace problem
R. C.
Cannings;
M. P.
Holland
1439-1451
Abstract: Let $R$ be the coordinate ring of a smooth affine curve over an algebraically closed field of characteristic zero $k$. For $S$ a subalgebra of $R$ with integral closure $R$ denote by $ \mathcal{D}(S)$ the ring of differential operators on $S$ and by $H(S)$ the finite-dimensional factor of $ \mathcal{D}(S)$ by its unique minimal ideal. The theory of diagonal $ n$-subspace systems is introduced. This is used to show that if $A$ is a finite-dimensional $ k$-algebra and $t \geqslant 1$ is any integer there exists such an $ S$ with $\displaystyle H(S) \cong \left( {\begin{array}{*{20}{c}} A & {\ast} 0 & {{M_t}(k)} \end{array} } \right).$ Further, the Morita classes of $H(S)$ are classified for curves with few branches, and it is shown how to lift Morita equivalences from $ H(S)$ to $\mathcal{D}(S)$.