Detecting algebraic (in)dependence of explicitly presented functions (some applications of Nevanlinna theory to mathematical logic)
R. H.
Gurevič
1-67
Abstract: We consider algebraic relations between explicitly presented analytic functions with particular emphasis on Tarski's high school algebra problem. The part not related directly to Tarski's high school algebra problem. Let $ U$ be a connected complex-analytic manifold. Denote by $\mathcal{F}(U)$ the minimal field containing all functions meromorphic on $U$ and closed under exponentiation $f \mapsto {e^f}$. Let ${f_j} \in \mathcal{F}(U)$, ${p_j} \in \mathcal{M}(U) - \{ 0\}$ for $1 \leq j \leq m$, and ${g_k} \in \mathcal{F}(U)$, ${q_k} \in \mathcal{M}(U) - \{ 0\}$ for $1 \leq k \leq n$ (where $\mathcal{M}(U)$ is the field of functions meromorphic on $U$). Let ${f_i} - {f_j} \notin \mathcal{H}(U)$ for $ i \ne j$ and $ {g_k} - {g_l} \notin \mathcal{H}(U)$ for $k \ne 1$ (where $ \mathcal{H}(U)$ is the ring of functions holomorphic on $U$). If all zeros and singularities of $\displaystyle h = \frac{{\sum\nolimits_{j = 1}^m {{p_j}{e^{{f_j}}}} }} {{\sum\nolimits_{k = 1}^n {{q_k}{e^{{g_k}}}} }}$ are contained in an analytic subset of $ U$ then $m = n$ and there exists a permutation $ \sigma$ of $\{ 1, \ldots ,m\}$ such that $h = ({p_j}/{q_{\sigma (j)}}) \cdot {e^{{f_j} - {g_{\sigma (j)}}}}$ for $ 1 \leq j \leq m$. When $ h \in \mathcal{M}(U)$, additionally ${f_j} - {g_{\sigma (j)}} \in \mathcal{H}(U)$ for all $j$ . On Tarski's high school algebra problem. Consider $L = \{$-terms in variables and $1$, $+$, $\cdot$, $ \uparrow \}$ , where $\uparrow :a$, $ b \mapsto {a^b}$ for positive $a$, $b$. Each term $t \in L$ naturally determines a function $ \bar t$ : $ {({{\mathbf{R}}_ + })^n} \to {{\mathbf{R}}_ - }$ , where $n$ is the number of variables involved. For $S \subset L$ put $\bar S = \{ \bar t\vert t \in S\}$ . (i) We describe the algebraic structure of $\bar \Lambda$ and $ \bar{\mathcal{L}}$ , where $\Lambda = \{ t \in L\vert$ if $u \uparrow v$ occurs as a subterm of $t$ then either $u$ is a variable or $u$ contains no variables at all, and $\mathcal{L} = \{ t \in L\vert$ if $u \uparrow v$ occurs as a subterm of $t$ then $ u \in \Lambda \}$. Of these, $\bar \Lambda $ is a free semiring with respect to addition and multiplication but $\bar{\mathcal{L}}$ is free only as a semigroup with respect to addition. A function $\bar t \in \bar S$ is called $+$-prime in $\bar S$ if $\bar t \ne \bar u + \bar v$ for all $u$, $v \in S$ and is called multiplicatively prime in $ \bar S$ if $ \bar t = \bar u \cdot \bar v \Rightarrow \bar u = 1$ or $\bar v = 1$ for $u$, $v \in S$. A function is called $( + , \cdot )$-prime in $\bar S$ if it is both $+$-prime and multiplicatively prime in $ \bar S$. A function in $\bar \Lambda$ is said to have content $ 1$ if it is not divisible by constants in $ {\mathbf{N}} - \{ 1\}$ or by $\ne 1 ( + , \cdot )$-primes of $\bar \Lambda$ . The product of functions of content $1$ has content $1$ . Let $P$ be the multiplicative subsemigroup of $\bar \Lambda$ of functions of content $ 1$ . Then $\bar{\mathcal{L}}$ as a semiring is isomorphic to the semigroup semiring $\bar \Lambda ({ \oplus _f}{P_f})$, where each $ {P_f}$ is a copy of $ P$ and $f$ ranges over the $\ne 1\; + $-primes of $\bar{\mathcal{L}}$. (ii) We prove that if $ t$, $u \in \mathcal{L}$ and ${{\mathbf{R}}_ + } \vdash t = u$ (i.e., if $\bar t = \bar u$) then Tarski's "high school algebra" identities $ \vdash t = u$. This result covers a conjecture of C. W. Henson and L. A. Rubel. (Note: this result does not generalize to arbitrary $ t$, $u \in L$ . Moreover, the equational theory of $({{\mathbf{R}}_ + };\;1, + , \cdot , \uparrow )$ is not finitely axiomatizable.
Volumes of vector fields on spheres
Sharon L.
Pedersen
69-78
Abstract: In this paper we study the problem: What is the unit vector field of smallest volume on an odd-dimensional sphere? We exhibit on each sphere a unit vector field with singularity which has exceptionally small volume on spheres of dimension greater than four. We conjecture that this volume is the infimum for volumes of bona fide unit vector fields, and is only achieved by the singular vector field. We generalize the construction of the singular vector field to give a family of cycles in Stiefel manifolds, each of which is a smooth manifold except for one singular point. Except for some low-dimensional cases, the tangent cones at these singular points are volume-minimizing; and half of the cones are nonorientable. Thus, we obtain a new family of nonorientable volume-minimizing cones.
Reye constructions for nodal Enriques surfaces
A.
Conte;
A.
Verra
79-100
Abstract: A classical Reye congruence $X$ is an Enriques surface of rational equivalence class $(3,7)$ in the grassmannian $G(1,3)$ of lines of ${{\mathbf{P}}^3}$. $X$ is the locus of lines of ${{\mathbf{P}}^3}$ which are included in two quadrics of $W=$ web of quadrics. A generalization to $ G(1,t)$ is given (1) for each $t > 2$ there exist Enriques surfaces $ X$ of class $(t,3t - 2)$ in $G(1,t)$, (2) the determinant of the dual of the universal bundle on $X$ is ${\mathcal{O}_X}(2E + R + {K_X})$, with $ E=$ isolated elliptic curve, ${R^2} = - 2$, $E \cdot R = t$, (3) $X$ parameterizes lines of ${{\mathbf{P}}^t}$ which are included in a codimension $2$ subsystem of $W$, $W=$ linear system of quadrics of dimension $\left( \begin{array}{*{20}{c}} t 2 \end{array} \right)$. The paper includes a description of the variety of trisecant lines to a smooth Enriques surface of degree $10$ in $ {{\mathbf{P}}^5}$ .
Nonstandard topology on function spaces with applications to hyperspaces
Hermann
Render
101-119
Abstract: In this paper the techniques of Nonstandard Analysis are used to study topologies on the set of all continuous functions. We obtain nonstandard characterizations for conjoining and splitting topologies and we give a complete description of the monads of the compact-open topology which leads to very elegant and simple proofs of some important results. For example we prove a generalized Ascoli Theorem where the image space is only Hausdorff or regular. Then we apply our results to the hyperspace and solve questions of Arens and Dugundji, Wattenberg and Topsøe. Finally we discuss real compact spaces and the continuity of the diagonal function.
Algebraic shift equivalence and primitive matrices
Mike
Boyle;
David
Handelman
121-149
Abstract: Motivated by symbolic dynamics, we study the problem, given a unital subring $S$ of the reals, when is a matrix $A$ algebraically shift equivalent over $S$ to a primitive matrix? We conjecture that simple necessary conditions on the nonzero spectrum of $ A$ are sufficient, and establish the conjecture in many cases. If $ S$ is the integers, we give some lower bounds on sizes of realizing primitive matrices. For Dedekind domains, we prove that algebraic shift equivalence implies algebraic strong shift equivalence.
Zero-equivalence in function fields defined by algebraic differential equations
John
Shackell
151-171
Abstract: We consider function fields obtained as towers over the field of rational functions, each extension being by a solution of an algebraic differential equation. On the assumption that an oracle exists for the constants, we present two algorithms for determining whether a given expression is functionally equivalent to zero in such a field. The first, which uses Gröbner bases, has the advantage of theoretical simplicity, but is liable to involve unnecessary computations. The second method is designed with a view to eliminating these.
Geometry of weight diagrams for ${\rm U}(n)$
Eng-Chye
Tan
173-192
Abstract: We study the geometry of the weight diagrams for irreducible representations of $U(n)$. Multiplicity-one weights are shown to have nice geometric characterizations. We then apply our results to study multiplicity-one $ K$-types of principal representations of $U(n,n)$.
Countable closed ${\rm LFC}$-groups with $p$-torsion
Felix
Leinen
193-217
Abstract: Let $LFC$ be the class of all locally $ FC$-groups. We study the existentially closed groups in the class $LF{C_p}$ of all $LFC$-groups $H$ whose torsion subgroup $T(H)$ is a $p$-group. Differently from the situation in $ LFC$, every existentially closed $LF{C_p}$-group is already closed in $ LF{C_p}$, and there exist ${2^{{\aleph _0}}}$ countable closed $ LF{C_P}$-groups $ G$. However, in the countable case, $T(G)$ is up to isomorphism always a unique locally finite $p$-group with similar properties as the unique countable existentially closed locally finite $ p$-group ${E_p}$.
Conformal metrics with prescribed Gaussian curvature on $S\sp 2$
Kuo-Shung
Cheng;
Joel A.
Smoller
219-251
Abstract: We consider on $ {S^2}$ the problem of which functions $K$ can be the scalar curvature of a metric conformal to the standard metric on ${S^2}$. We assume that $K$ is a function of one variable, and we obtain a necessary and sufficient condition for the problem to be solvable. We also obtain several new sufficient conditions on $k$ (which are easy to check), in order that the problem be solvable.
${\rm GL}(4,{\bf R})$-Whittaker functions and ${}\sb 4F\sb 3(1)$ hypergeometric series
Eric
Stade
253-264
Abstract: In this paper we consider spaces of $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions, which are special functions that arise in the study of $ {\text{GL}}(4,\mathbb{R})$ automorphic forms. Our main result is to determine explicitly the series expansion for a $ {\text{GL}}(4,\mathbb{R})$-Whittaker function that is "fundamental," in that it may be used to generate a basis for the space of all $ {\text{GL}}(4,\mathbb{R})$-Whittaker functions of fixed eigenvalues. The series that we find in the case of $ {\text{GL}}(4,\mathbb{R})$ is particularly interesting in that its coefficients are not merely ratios of Gamma functions, as they are in the lower-rank cases. Rather, these coefficients are themselves certain series-- namely, they are finite hypergeometric series of unit argument. We suspect that this is a fair indication of what will happen in the general case of $ {\text{GL}}(n,\mathbb{R})$.
On Klein's combination theorem. IV
Bernard
Maskit
265-294
Abstract: This paper contains an expansion of the combination theorems to cover the following problems. New rank $1$ parabolic subgroups are produced, while, as in previous versions, all elliptic and parabolic elements are tracked. A proof is given that the combined group is analytically finite if and only if the original groups are; in the analytically finite case, we also give a formula for the hyperbolic area of the combined group (i.e., the hyperbolic area of the set of discontinuity on the $2$-sphere modulo $G$) in terms of the hyperbolic areas of the original groups. There is also a new variation on the first combination theorem in which the common subgroup has finite index in one of the two groups.
Artinian subrings of a commutative ring
Robert
Gilmer;
William
Heinzer
295-310
Abstract: Given a commutative ring $R$, we investigate the structure of the set of Artinian subrings of $R$. We also consider the family of zero-dimensional subrings of $R$. Necessary and sufficient conditions are given in order that every zero-dimensional subring of a ring be Artinian. We also consider closure properties of the set of Artinian subrings of a ring with respect to intersection or finite intersection, and the condition that the set of Artinian subrings of a ring forms a directed family.
Decidable discriminator varieties from unary classes
Ross
Willard
311-333
Abstract: Let $\mathcal{K}$ be a class of (universal) algebras of fixed type. $ {\mathcal{K}^t}$ denotes the class obtained by augmenting each member of $\mathcal{K}$ by the ternary discriminator function $(t(x,y,z) = x$ if $x \ne y,t(x,x,z) = z)$, while $\vee ({\mathcal{K}^t})$ is the closure of ${\mathcal{K}^t}$ under the formation of subalgebras, homomorphic images, and arbitrary Cartesian products. For example, the class of Boolean algebras is definitionally equivalent to $\vee ({\mathcal{K}^t})$ where $\mathcal{K}$ consists of a two-element algebra whose only operations are the two constants. Any equationally defined class (that is, variety) of algebras which is equivalent to some $\vee ({\mathcal{K}^t})$ is known as a discriminator variety. Building on recent work of S. Burris, R. McKenzie, and M. Valeriote, we characterize those locally finite universal classes $ \mathcal{K}$ of unary algebras of finite type for which the first-order theory of $ \vee ({\mathcal{K}^t})$ is decidable.
Self-similar measures and their Fourier transforms. II
Robert S.
Strichartz
335-361
Abstract: A self-similar measure on $ {{\mathbf{R}}^n}$ was defined by Hutchinson to be a probability measure satisfying $({\ast})$ $\displaystyle \mu = \sum\limits_{j = 1}^m {{a_j}\mu \circ S_j^{ - 1}}$ , where $ {S_j}x = {\rho _j}{R_j}x + {b_j}$ is a contractive similarity $(0 < {\rho _j} < 1,{R_j}$ orthogonal) and the weights ${a_j}$ satisfy $0 < {a_j} < 1,\sum\nolimits_{j = 1}^m {{a_j} = 1}$. By analogy, we define a self-similar distribution by the same identity $( {\ast} )$ but allowing the weights ${a_j}$ to be arbitrary complex numbers. We give necessary and sufficient conditions for the existence of a solution to $ ( {\ast} )$ among distributions of compact support, and show that the space of such solutions is always finite dimensional. If $ F$ denotes the Fourier transformation of a self-similar distribution of compact support, let $\displaystyle H(R) = \frac{1}{{{R^{n - \beta }}}}\int_{\vert x\vert \leq R} {\vert F(x){\vert^2}dx,}$ where $\beta$ is defined by the equation $\sum\nolimits_{j = 1}^m {\rho _j^{ - \beta }\vert{a_j}{\vert^2} = 1}$. If $\rho _j^{{\nu _j}} = \rho $ for some fixed $ \rho$ and ${\nu _j}$ positive integers we say the $\{ {\rho _j}\} $ are exponentially commensurable. In this case we prove (under some additional hypotheses) that $H(R)$ is asymptotic (in a suitable sense) to a bounded function $ \tilde H(R)$ that is bounded away from zero and periodic in the sense that $ \tilde H(\rho R) = \tilde H(R)$ for all $R > 0$. If the $ \{ {\rho _j}\}$ are exponentially incommensurable then ${\lim _{R \to \infty }}H(R)$ exists and is nonzero.
Rational approximations to the dilogarithm
Masayoshi
Hata
363-387
Abstract: The irrationality proof of the values of the dilogarithmic function $ {L_2}(z)$ at rational points $z = 1/k$ for every integer $k \in ( - \infty , - 5] \cup [7,\infty )$ is given. To show this we develop the method of Padé-type approximations using Legendre-type polynomials, which also derives good irrationality measures of $ {L_2}(1/k)$. Moreover, the linear independence over $ {\mathbf{Q}}$ of the numbers $1$, $\log (1 - 1/k)$, and ${L_2}(1/k)$ is also obtained for each integer $ k \in ( - \infty , - 5] \cup [7,\infty )$ .
An extension theorem for closing maps of shifts of finite type
Jonathan
Ashley
389-420
Abstract: If there exists some right-closing factor map $\pi :{\Sigma _A} \to {\Sigma _B}$ between aperiodic shifts of finite type, then any right-closing map $ \varphi :X \to {\Sigma _B}$ from any shift of finite type $X$ contained in ${\Sigma _A}$ can be extended to a right-closing factor map from all of $ {\Sigma _A}$ onto ${\Sigma _B}$. We prove this and give some consequences.
Hyperbolic structures for surfaces of infinite type
Ara
Basmajian
421-444
Abstract: Our main objective is to understand the geometry of hyperbolic structures on surfaces of infinite type. In particular, we investigate the properties of surfaces called flute spaces which are constructed from infinite sequences of "pairs of pants," each glued to the next along a common boundary geodesic. Necessary and sufficient conditions are supplied for a flute space to be constructed using only "tight pants," along with sufficient conditions on when the hyperbolic structure is complete. An infinite version of the Klein-Maskit combination theorem is derived. Finally, using the above constructions a number of applications to the deformation theory of infinite type hyperbolic surfaces are examined.
Witt's extension theorem for mod four valued quadratic forms
Jay A.
Wood
445-461
Abstract: The $\bmod \,4$ valued quadratic forms defined by E. H. Brown, Jr. are studied. A classification theorem is proven which states that these forms are determined by two things: whether or not their associated bilinear form is alternating, and the $\sigma $-invariant of Brown (which generalizes the Arf invariant of an ordinary quadratic form). Particular attention is paid to a generalization of Witt's extension theorem for quadratic forms. Some applications to selforthogonal codes are sketched, and an exposition of some unpublished work of E. Prange on Witt's theorem is provided in an appendix.