Abstract computability and its relation to the general purpose analog computer (some connections between logic, differential equations and analog computers)
Marian Boykan
Pour-el
1-28
Abstract: Our aim is to study computability from the viewpoint of the analog computer. We present a mathematical definition of an analog generable function of a real variable. This definition is formulated in terms of a simultaneous set of nonlinear differential equations possessing a ``domain of generation.'' (The latter concept is explained in the text.) Our definition includes functions generated by existing general-purpose analog computers. Using it we prove two theorems which provide a characterization of analog generable functions in terms of solutions of algebraic differential polynomials. The characterization has two consequences. First we show that there are entire functions which are computable (in the sense of recursive analysis) but which cannot be generated by any analog computer in any interval--e.g. $1/\Gamma (x)$ and $\Sigma _{n = 1}^\infty ({x^n}/{n^{({n^3})}})$. Second we note that the class of analog generable functions is very large: it includes special functions which arise as solutions to algebraic differential polynomials. Although not all computable functions are analog generable, a kind of converse holds. For entire functions, $f(x) = \Sigma _{i = 0}^\infty {b_i}{x^i}$, the theorem takes the following form. If $f(x)$ is analog generable on some closed, bounded interval then there is a finite number of $ {b_k}$ such that, on every closed bounded interval, $f(x)$ is computable relative to these $ {b_k}$. A somewhat similar theorem holds if $f$ is not entire. Although the results are stated and proved for functions of a real variable, they hold with minor modifications for functions of a complex variable.
Perturbed semigroup limit theorems with applications to discontinuous random evolutions
Robert P.
Kertz
29-53
Abstract: For $\varepsilon > 0$ small, let ${U^\varepsilon }(t)$ and $S(t)$ be strongly continuous semigroups of linear contractions on a Banach space $L$ with infinitesimal operators $A(\varepsilon )$ and $B$ respectively, where $ A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o()$ as $\varepsilon \to 0$. Let $\{ B(u);u \geqslant 0\} $ be a family of linear operators on $L$ satisfying $ B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2})$ as $\varepsilon \to 0$. Assume that $A(\varepsilon ) + {\varepsilon ^{ - 1}}B()$ is the infinitesimal operator of a strongly continuous contraction semigroup ${T_\varepsilon }(t)$ on $L$ and that for each $f \in L,{\lim _{\lambda \to 0}}\lambda \int_0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf$ exists. We give conditions under which ${T_\varepsilon }(t)$ converges as $\to 0$ to the semigroup generated by the closure of $P({A^{(1)}} + {\Pi ^{(1)}})$ on $\mathcal{R}(P) \cap \mathcal{D}({A^{(1)}}) \cap \mathcal{D}({\Pi ^{(1)}})$. If $P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f$, and we let $ \hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h$, then we show that ${T_\varepsilon }(t/\varepsilon )f$ converges as $ \varepsilon \to 0$ to the strongly continuous contraction semigroup generated by the closure of ${V^{(2)}} + \hat V$. From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.
Asymptotic properties of $U$-statistics
Raymond N.
Sproule
55-64
Abstract: Let $r$ be a fixed positive integer. A $ U$-statistic $ {U_n}$ is an average of a symmetric measurable function of $r$ arguments over a random sample of size $ n$. Such a statistic may be expressed as an average of independent and identically distributed random variables plus a remainder term. We develop a Kolmogorov-like inequality for this remainder term as well as examine some of its (a.s.) convergence properties. We then relate these properties to the $U$-statistic. In addition, the asymptotic normality of ${U_N}$, where $N$ is a positive integer-valued random variable, is established under certain conditions.
Chapman-Enskog-Hilbert expansion for the Ornstein-Uhlenbeck process and the approximation of Brownian motion
Richard S.
Ellis
65-74
Abstract: Let $(x(t),\upsilon (t))$ denote the joint Ornstein-Uhlenbeck position-velocity process. Special solutions of the backward equation of this process are studied by a technique used in statistical mechanics. This leads to a new proof of the fact that, as $\varepsilon \downarrow 0,\varepsilon x(t/{\varepsilon ^2})$ tends weakly to Brownian motion. The same problem is then considered for $ \upsilon (t)$ belonging to a large class of diffusion processes.
A density property and applications
Richard J.
O’Malley
75-87
Abstract: An unexpected metric density property of a certain type of ${F_\sigma }$ set is proven. An immediate application is a characterization of monotone functions similar to a well-known result by Zygmund. Several corollaries of this characterization are given as well as a simple proof of a theorem due to Tolstoff.
The Riemann problem for general $2\times 2$ conservation laws
Tai Ping
Liu
89-112
Abstract: The Riemann Problem for a system of hyperbolic conservation laws of form $\displaystyle (1)\quad \begin{array}{*{20}{c}} {{u_t} + f{{(u,\upsilon )}_x} = 0,} {{\upsilon _t} + g{{(u,\upsilon )}_x} = 0} \end{array}$ with arbitrary initial constant states $\displaystyle (2)\quad ({u_0}(x),{v_0}(x)) = \left\{ {\begin{array}{*{20}{c}} {... ...},{v_l}),\quad x < 0,} {({u_r},{v_r}),\quad x > 0,} \end{array} } \right.$ is considered. We assume that ${f_\upsilon } < 0,{g_u} < 0$. Let ${l_i}({r_i})$ be the left (right) eigenvectors of $dF \equiv d(f,g)$ for eigenvalues ${\lambda _1} < {\lambda _2}$. Instead of assuming the usual convexity condition $d{\lambda _i}({r_i}) \ne 0,i = 1,2$ we assume that $d{\lambda _i}({r_i}) = 0$ on disjoint union of $ 1$-dim manifolds in the $(u,\upsilon )$ plane. Oleinik's condition (E) for single equation is extended to system (1); again call this new condition (E). Our condition (E) implies Lax's shock inequalities and, in case $d{\lambda _i}({r_i}) \ne 0$, the two are equivalent. We then prove that there exists a unique solution to the Riemann Problem (1) and (2) in the class of shocks, rarefaction waves and contact discontinuities which satisfies condition (E).
The oriented bordism of $Z\sb{2\sp{k}}$ actions
E. R.
Wheeler
113-121
Abstract: Let ${R_2}$ be the subring of the rationals given by ${R_2} = Z[1/2]$. It is shown that for $G = {Z_{{2^k}}}$ the bordism group of orientation preserving $G$ actions on oriented manifolds tensored with ${R_2}$ is a free ${\Omega _ \ast } \otimes {R_2}$ module on even dimensional generators (where ${\Omega _ \ast }$ is the oriented bordism ring).
Examples of nonsolvable partial differential equations
Robert
Rubinstein
123-129
Abstract: Two examples of nonsolvable partial differential operators with multiple characteristics are presented. They illustrate the possibility that certain terms in the principal part may play no role in determining the solvability properties of the operator. This situation cannot occur for simple characteristics, where solvability is determined by the principal part.
An isomorphism and isometry theorem for a class of linear functionals
William D. L.
Appling
131-140
Abstract: Suppose $ U$ is a set, ${\mathbf{F}}$ is a field of subsets of $U$ and ${\mathfrak{p}_{AB}}$ is the set of all real-valued, finitely additive functions defined on ${\mathbf{F}}$. Two principal notions are considered in this paper. The first of these is that of a subset of $ {\mathfrak{p}_{AB}}$, defined by certain closure properties and called a $ C$-set. The second is that of a collection $ \mathcal{C}$ of linear transformations from $ {\mathfrak{p}_{AB}}$ into $ {\mathfrak{p}_{AB}}$ with special boundedness properties. Given a $ C$-set $M$ which is a linear space, an isometric isomorphism is established from the dual of $ M$ onto the set of all elements of $ \mathcal{C}$ with range a subset of $M$. As a corollary it is demonstrated that the above-mentioned isomorphism and isometry theorem, together with a previous representation theorem of the author (J. London Math. Soc. 44 (1969), pp. 385-396), imply an analogue of a dual representation theorem of Edwards and Wayment (Trans. Amer. Math. Soc. 154 (1971), pp. 251-265). Finally, a ``pseudo-representation theorem'' for the dual of ${\mathfrak{p}_{AB}}$ is demonstrated.
Generation of analytic semigroups by strongly elliptic operators
H. Bruce
Stewart
141-162
Abstract: Strongly elliptic operators realized under Dirichlet boundary conditions in unbounded domains are shown to generate analytic semigroups in the topology of uniform convergence. This fact is applied to initial-boundary value problems for temporally homogeneous and temporally inhomogeneous parabolic equations.
Semigroups of operators on locally convex spaces
V. A.
Babalola
163-179
Abstract: Let $X$ be a complex Hausdorff locally convex topological linear space and $L(X)$ the family of all continuous linear operators on $X$. This paper discusses the generation and perturbation theory for ${C_0}$ semigroups $\{ S(\xi ):\xi \geqslant 0\} \subset L(X)$ such that for each continuous seminorm $p$ on $X$ there exist a positive number ${\sigma _p}$ and a continuous seminorm $ q$ on $X$ with $p(S(\xi )x) \leqslant {e^{^\sigma {p^\xi }}}q(x)$ for all $ \xi \geqslant 0$ and $ x \in X$. These semigroups are studied by means of a realization of $ X$ as a projective limit of Banach spaces, using certain naturally-defined operators and ${C_0}$ semigroups on these Banach spaces to connect the present results to the classical Hille-Yosida-Phillips theory.
Conjugate points, triangular matrices, and Riccati equations
Zeev
Nehari
181-198
Abstract: Let $A$ be a real continuous $n \times n$ matrix on an interval $\Gamma$, and let the $n$-vector $x$ be a solution of the differential equation $ x' = Ax$ on $\Gamma$. If $[\alpha ,\beta ] \in \Gamma ,\beta$ is called a conjugate point of $\alpha$ if the equation has a nontrivial solution vector $x = ({x_1},{\kern 1pt} \ldots ,{x_n})$ such that $ {x_1}(\alpha ) = \ldots = {x_k}(\alpha ) = {x_{k + 1}}(\beta ) = \ldots = {x_n}(\beta ) = 0$ for some $k \in [1,n - 1]$. It is shown that the absence on $ ({t_1},{t_2})$ of a point conjugate to ${t_1}$ with respect to the equation $x' = Ax$ is equivalent to the existence on $ ({t_1},{t_2})$ of a continuous matrix solution $L$ of the nonlinear differential equation $L({t_1}) = I$, where ${[B]_{{\tau _0}}}$ denotes the matrix obtained from the $n \times n$ matrix $B$ by replacing the elements on and above the main diagonal by zeros. This nonlinear equation--which may be regarded as a generalization of the Riccati equation, to which it reduces for $n = 2$--can be used to derive criteria for the presence or absence of conjugate points on a given interval.
Killing vector fields and harmonic forms
Edward T.
Wright
199-202
Abstract: The paper is concerned with harmonic $(p,q)$-forms on compact Kähler manifolds which admit Killing vector fields with discrete zero sets. Let $ {h^{p,q}}$ denote the dimension of the space of harmonic $(p,q)$-forms. The main theorem states that ${h^{p,q}} = 0,p \ne q$.
A Pr\"ufer transformation for the equation of the vibrating beam
D. O.
Banks;
G. J.
Kurowski
203-222
Abstract: In this paper, the oscillatory properties of the eigenfunctions of an elastically constrained beam are studied. The method is as follows. The eigenfunction and its first three derivatives are considered as a four-dimensional vector, $ (u,u',pu'',(pu'')')$. This vector is projected onto two independent planes and polar coordinates are introduced in each of these two planes. The resulting transformation is then used to study the oscillatory properties of the eigenfunctions and their derivatives in a manner analogous to the use of the Prüfer transformation in the study of second order Sturm-Liouville systems. This analysis yields, for a given set of boundary conditions, the number of zeros of each of the derivatives, $u',pu'',(pu'')'$ and the relation of these zeros to the $n - 1$ zeros of the $n$th eigenfunction. The method also can be used to establish comparison theorems of a given type.
M\"obius transformations of the disc and one-parameter groups of isometries of $H\sp{p}$
Earl
Berkson;
Robert
Kaufman;
Horacio
Porta
223-239
Abstract: Let $\{ {T_t}\}$ be a strongly continuous one-parameter group of isometries in ${H^p}(1 \leqslant p < \infty ,p \ne 2)$ with unbounded generator. There is a uniquely determined one-parameter group $\{ {\phi _t}\} ,t \in {\mathbf{R}}$, of Möbius transformations of the (open) disc $D$ corresponding to $\{ {T_t}\}$ by way of Forelli's theorem. The interplay between $\{ {T_t}\}$ and $\{ {\phi _t}\}$ is studied, and the spectral properties of the generator $A$ of $\{ {T_t}\}$ are analyzed in this context. The nature of the set $S$ of common fixed points of the functions $ {\phi _t}$ plays a crucial role in determining the behavior of $A$. The spectrum of $A$, which is a subset of $i{\mathbf{R}}$, can be a discrete set, a translate of $i{{\mathbf{R}}_ + }$ or of $i{{\mathbf{R}}_ - }$, or all of $i{\mathbf{R}}$. If $S$ is not a doubleton subset of the unit circle, $\{ {T_t}\}$ can be extended to a holomorphic semigroup of ${H^p}$-operators, the semigroup being defined on a half-plane. The treatment of $\{ {T_t}\}$ is facilitated by developing appropriate properties of one-parameter groups of Möbius transformations of $D$. In particular, such groups are in one-to-one correspondence (via an initial-value problem) with the nonzero polynomials $q$, of degree at most 2, such that Re$[\bar zq(z)] = 0$ for all unimodular $ z$. A has an explicit description (in terms of the polynomial corresponding to $\{ {\phi _t}\} $) as a differential operator.
Series inequalities involving convex functions
Christopher O.
Imoru
241-252
Abstract: The object of this paper is to extend some recent generalizations of Petrović's inequality by Vasić and others. We shall also use our technique to obtain some results which have interesting applications in the theory of Fourier series as well as the theory of approximations.
``Image of a Hausdorff arc'' is cyclically extensible and reducible
J. L.
Cornette
253-267
Abstract: It is shown that a Hausdorff continuum $S$ is the continuous image of an arc (respectively arcwise connected) if and only if each cyclic element of $S$ is the continuous image of an arc (respectively, arcwise connected). Also, there is given an analogue to the metric space cyclic chain approximation theorem of G. T. Whyburn which applies to locally connected Hausdorff continua.
On the constructibility of prime characteristic periodic associative and Jordan rings
J. A.
Loustau
269-279
Abstract: The object of this paper is to show that any periodic associative ring of prime characteristic can be embedded in a periodic associative ring of prime characteristic which is constructible from a relatively complemented, distributive lattic and a family of periodic fields. Further, it will be proved that any periodic Jordan ring of prime characteristic is also embeddable in a periodic Jordan ring which is constructible from a lattice of the above type and a family of periodic Jordan rings of a symmetric bilinear form.
The category of generalized Lie groups
Su Shing
Chen;
Richard W.
Yoh
281-294
Abstract: We consider the category $\Gamma$ of generalized Lie groups. A generalized Lie group is a topological group $G$ such that the set $LG = Hom({\mathbf{R}},G)$ of continuous homomorphisms from the reals $ {\mathbf{R}}$ into $ G$ has certain Lie algebra and locally convex topological vector space structures. The full subcategory ${\Gamma ^r}$ of $r$-bounded ($r$ positive real number) generalized Lie groups is shown to be left complete. The class of locally compact groups is contained in $\Gamma$. Various properties of generalized Lie groups $G$ and their locally convex topological Lie algebras $ LG = Hom({\mathbf{R}},G)$ are investigated.
On the asymptotic distribution of eigenvalues for semi-elliptic operators
Akira
Tsutsumi;
Chung Lie
Wang
295-315
Abstract: This paper is focused on the asymptotic distribution of eigenvalues for semielliptic operators under weaker smoothness assumptions on coefficients of operators than those of F. E. Browder [3] and Y. Kannai [8] by applying the method of Maruo-Tanabe [9].
Behnke-Stein theorem for analytic spaces
Alessandro
Silva
317-326
Abstract: The notion of $ q$-Runge pair is extended to reduced complex analytic spaces. A necessary and sufficient condition for a pair of $n$-dimensional analytic spaces to be an $ n$-Runge pair is proved and it is shown that this result extends a Behnke-Stein theorem when $n = 1$. A topological property of $q$-Runge pairs of spaces is also proved.
The unitary representations of the generalized Lorentz groups
Ernest A.
Thieleker
327-367
Abstract: For $n \geqslant 2$, let $G(n)$ denote the two-fold covering group of ${\text{SO} _e}(1,n)$. In case $n \geqslant 3,G(n)$ is isomorphic to $\operatorname{Spin} (1,n)$ and is simply connected. In a previous paper we determined all the irreducible quasi-simple representations of these groups, up to infinitesimal equivalence. The main purpose of the present paper is to determine which of these representations are unitarizable. Thus, with the aid of some results of Harish-Chandra and Nelson we determine all the irreducible unitary representations of $G(n)$, up to unitary equivalence. One by-product of our analysis is the explicit construction of the infinitesimal equivalences, which are known to exist from our previous work, between the various subquotient representations and certain subrepresentations in the nonirreducible cases of the nonunitary principal series representations of $G(n)$.
Interpolating sequences in polydisks
Eric P.
Kronstadt
369-398
Abstract: Let ${D^n}$ be the unit polydisk in ${{\mathbf{C}}^n}$, $A$ be a uniform algebra, ${H^\infty }({D^n},A)$, the space of bounded analytic $A$ valued functions on ${D^n}$, $ {l^\infty }A$, the space of bounded sequences of elements in $A$. A sequence, $ S = \{ {a_i}\} _{i = 1}^\infty \subset {D^n}$ will be called an interpolating sequence with respect to $A$ if the map $T:{H^\infty }({D^n},A) \to {l^\infty }A$ given by $ T(f) = \{ f({a_i})\} _{i = 1}^\infty$ is surjective. In 1958, L. Carleson showed that for $n = 1,S$ is interpolating w.r.t. ${\mathbf{C}}$ iff $S$ satisfies a certain zero-one interpolation property called uniform separation. We generalize this result to cases where $n > 1$ and $ A \ne {\mathbf{C}}$. In particular, we show that if $S \subset {D^n}$ is uniformly separated and $S \subset {W_1} \times {W_2} \times \cdots \times {W_n}$ (where each ${W_j}$ is a region in $D$ lying between two circular arcs which intersect twice on the boundary of $D$) then $S$ is an interpolating sequence w.r.t. any uniform algebra. If $ S \subset {D^n}$ is uniformly separated and $S \subset D \times {W_2} \times \cdots \times {W_n}$ then $S$ is interpolating w.r.t. ${\mathbf{C}}$. Other examples and generalizations of interpolating sequences are discussed.
Smooth complex projective space bundles and $B{\rm U}(n)$
R. Paul
Beem
399-411
Abstract: Smooth fiberings with complex projective and Dold manifold fibers are studied and a bordism classification for even complex projective space bundles is given. The $ {Z_2}$-cohomology of $B\tilde U(n)$ is computed with its Steenrod algebra action.
Semi-isotopies and the lattice of inner ideals of certain quadratic Jordan algebras
Jerome M.
Katz
413-427
Abstract: The concept of isotopy plays an extremely important role in the structure theory of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals. We take Koecher's characterization of isotopy and use it as the basis of a definition of semi-isotopy. It is clear that semi-isotopies induce, in a natural way, automorphisms of the lattice of inner ideals. We concern ourselves with the converse problem; namely, if $\eta$ is a semilinear bijection of a quadratic Jordan algebra such that $\eta$ induces an automorphism of the lattice of inner ideals, is $\eta$ necessarily a semi-isotopy? We answer the above question in the affirmative for a large class of simple quadratic Jordan algebras satisfying the minimum condition on principal inner ideals (said class includes all such algebras of capacity at least three over fields of characteristic unequal to two). Moreover, we prove that the only such maps which induce the identity automorphism on the lattice are the scalar multiplications.
Erratum to: ``Essential spectrum for a Hilbert space operator'' (Trans. Amer. Math. Soc. {\bf 163} (1972), 437--445)
Richard
Bouldin
429