The Selberg trace formula for ${\rm SL}(3,{\bf Z})\backslash{\rm SL}(3,{\bf R})/{\rm SO}(3,{\bf R})$
D. I.
Wallace
1-36
Abstract: In this paper we compute the trace formula for $SL(3,\mathbb{Z})$ in detail and refine it to a greater extent than has previously been done. We show that massive cancellation occurs in the parabolic terms, leading to a far simpler formula than had been thought possible.
Finite Group Actions on Siegel Modular Spaces
K. F.
Lai;
Ronnie
Lee
37-45
Abstract: The theory of nonabelian cohomology is used to show that the set of fixed points of a finite group acting on a Siegel modular space is a union of Shimura varieties
Differential equations for symmetric generalized ultraspherical polynomials
Roelof
Koekoek
47-72
Abstract: We look for differential equations satisfied by the generalized Jacobi polynomials $\{ P_n^{\alpha ,\beta ,M,N}(x)\} _{n = 0}^\infty$ which are orthogonal on the interval $[- 1,1]$ with respect to the weight function $\displaystyle \frac{{\Gamma (\alpha + \beta + 2)}}{{{2^{\alpha + \beta + 1}}\Ga... ...ta + 1)}}{(1 - x)^\alpha }{(1 + x)^\beta } + M\delta (x + 1) + N\delta (x - 1),$ where $ \alpha > - 1$, $\beta > - 1$, $M \geq 0$, and $N \geq 0$. In the special case that $\beta = \alpha$ and $N = M$ we find all differential equations of the form $\displaystyle \sum\limits_{i = 0}^\infty {{c_i}(x){y^{(i)}}(x) = 0,\quad y(x) = P_n^{\alpha ,\alpha ,M,M}(x),}$ where the coefficients $\{ {c_i}(x)\} _{i = 1}^\infty$ are independent of the degree n. We show that if $ M > 0$ only for nonnegative integer values of $\alpha$ there exists exactly one differential equation which is of finite order $2\alpha + 4$. By using quadratic transformations we also obtain differential equations for the polynomials $\{ P_n^{\alpha, \pm 1/2,0,N}(x)\} _{n = 0}^\infty$ for all $ \alpha > - 1$ and $ N \geq 0$.
Nonlinear quantum fields in $\geq 4$ dimensions and cohomology of the infinite Heisenberg group
J.
Pedersen;
I. E.
Segal;
Z.
Zhou
73-95
Abstract: Aspects of the cohomology of the infinite-dimensional Heisenberg group as represented on the free boson field over a given Hilbert space are treated. The 1-cohomology is shown to be trivial in certain spaces of generalized vectors. From this derives a canonical quantization mapping from classical (unquantized) forms to generalized operators on the boson field. An example, applied here to scalar relativistic fields, is the quantization of a given classical interaction Lagrangian or Hamiltonian, i.e., the establishment and characterization of corresponding boson field operators. For example, if $\phi$ denotes the free massless scalar field in d-dimensional Minkowski space ($d \geq 4$, even) and if q is an even integer greater than or equal to 4, then ${\smallint _{{{\mathbf{M}}_0}}}:\phi {(X)^q}:dX$ exists as a nonvanishing, Poincaré invariant, hermitian, selfadjointly extendable operator, where : $ \phi {(X)^q}$ : denotes the Wick power. Applications are also made to the rigorous establishment of basic symbolic operators in heuristic quantum field theory, including certain massive field theories; to a class of pseudo-interacting fields obtained by substituting the free field into desingularized expressions for the total Hamiltonian in the conformally invariant case $d = q = 4$ and to corresponding scattering theory.
Eigenvalues and eigenspaces for the twisted Dirac operator over ${\rm SU}(N,1)$ and ${\rm Spin}(2N,1)$
Esther
Galina;
Jorge
Vargas
97-113
Abstract: Let X be a symmetric space of noncompact type whose isometry group is either $SU(n,1)$ or $ Spin(2n,1)$. Then the Dirac operator D is defined on $ {L^2}$-sections of certain homogeneous vector bundles over X. Using representation theory we obtain explicitly the eigenvalues of D and describe the eigenspaces in terms of the discrete series.
Braid groups and left distributive operations
Patrick
Dehornoy
115-150
Abstract: The decidability of the word problem for the free left distributive law is proved by introducing a structure group which describes the underlying identities. This group is closely connected with Artin's braid group ${B_\infty }$. Braid colourings associated with free left distributive structures are used to show the existence of a unique ordering on the braids which is compatible with left translation and such that every generator $ {\sigma _i}$ is preponderant over all $ {\sigma _k}$ with $k > i$. This ordering is a linear ordering.
Analysis and applications of holomorphic functions in higher dimensions
R. Z.
Yeh
151-177
Abstract: Holomorphic functions in ${R^n}$ are defined to generalize those in $ {R^2}$. A Taylor formula and a Cauchy integral formula are presented. An application of the Taylor formula to the kernel of the Cauchy integral formula results in Taylor series expansions of holomorphic functions. Real harmonic functions are expanded in series of homogeneous harmonic polynomials.
Th\'eor\`eme de Ney-Spitzer sur le dual de ${\rm SU}(2)$
Philippe
Biane
179-194
Abstract: Let $\phi$ be a central, noneven, positive type function on $ {\text{SU}}(2)$ with $\phi (e) < 1$. For any polynomial function p on $ {\text{SU}}(2)$, let $ V(p)$ be the left convolution operator by $ p/(1 - \phi )$ on $ {L^2}({\text{SU}}(2))$, we prove that $V(p)/V(1)$ is a pseudodifferential operator of order 0 and give an explicit formula for its principal symbol. This is interpreted in terms of Martin compactification of a quantum random walk.
Lattice-ordered algebras that are subdirect products of valuation domains
Melvin
Henriksen;
Suzanne
Larson;
Jorge
Martinez;
R. G.
Woods
195-221
Abstract: An f-ring (i.e., a lattice-ordered ring that is a subdirect product of totally ordered rings) A is called an SV-ring if $A/P$ is a valuation domain for every prime ideal P of A . If M is a maximal $ \ell$-ideal of A, then the rank of A at M is the number of minimal prime ideals of A contained in M , rank of A is the sup of the ranks of A at each of its maximal $\ell$-ideals. If the latter is a positive integer, then A is said to have finite rank, and if $A = C(X)$ is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank of the f-ring $ C(X)$, and X is called an SV-space if $C(X)$ is an ST-ring. X has finite rank k iff k is the maximal number of pairwise disjoint cozero sets with a point common to all of their closures. In general f-rings these two concepts are unrelated, but if A is uniformly complete (in particular, if $ A = C(X)$) then if A is an SV-ring then it has finite rank. Showing that this latter holds makes use of the theory of finite-valued lattice-ordered (abelian) groups. These two kinds of rings are investigated with an emphasis on the uniformly complete case. Fairly powerful machinery seems to have to be used, and even then, we do not know if there is a compact space X of finite rank that fails to be an SV-space.
Semigroups and stability of nonautonomous differential equations in Banach spaces
Nguyen Van Minh
223-241
Abstract: This paper is concerned with nonautonomous differential equations in Banach spaces. Using the theory of semigroups of linear and nonlinear operators one investigates the semigroups of weighted translation operators associated with the underlying equations. Necessary and sufficient conditions for different types of stability are given in terms of spectral properties of the translation operators and the differential operators associated with the equations.
Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger's theorem in ${\bf R}\sp {2n}$
Jia Zu
Zhou
243-262
Abstract: We first discuss the theory of hypersurfaces and submanifolds in the m-dimensional Euclidean space leading up to high dimensional analogues of the classical Euler's and Meusnier's theorems. Then we deduce the kinematic formulas for powers of mean curvature of the $ (m - 2)$-dimensional intersection submanifold $ {S_0} \cap g{S_1}$ of two $ {C^2}$-smooth hypersurfaces ${S_0}$, ${S_1}$, i.e., ${\smallint _G}({\smallint _{{S_0} \cap g{S_1}}}{H^{2k}}d\sigma )dg$. Many well-known results, for example, the C-S. Chen kinematic formula and Crofton type formulas are easy consequences of our kinematic formulas. As direct applications of our formulas, we obtain analogues of Hadwiger's theorem in ${\mathbb{R}^{2n}}$, i.e., sufficient conditions for one domain $ {K_\beta }$ to contain, or to be contained in, another domain ${K_\alpha }$.
Asymptotic measures for skew products of Bernoulli shifts with generalized north pole--south pole diffeomorphisms
D. K.
Molinek
263-291
Abstract: We study asymptotic measures for a certain class of dynamical systems. In particular, for $T:{\Sigma _2} \times M \to {\Sigma _2} \times M$, a skew product of the Bernoulli shift with a generalized north pole-south pole diffeomorphism, we describe the limits of the following two sequences of measures: (1) iterates under T of the product of Bernoulli measure with Lebesgue measure, $T_\ast ^n(\mu \times m)$, and (2) the averages of iterates of point mass measures, $\frac{1}{n}\Sigma _{k = 0}^{n - 1}{\delta _{{T^k}(w,x)}}$. We give conditions for the limit of each sequence to exist. We also determine the subsequential limits in case the sequence does not converge. We exploit several properties of null recurrent Markov Chains and apply them to the symmetric random walk on the integers. We also make use of Strassen's Theorem as an aid in determining subsequential limits.
Connections with exotic holonomy
Lorenz J.
Schwachhöfer
293-321
Abstract: Berger [Ber] partially classified the possible irreducible holonomy representations of torsion free connections on the tangent bundle of a manifold. However, it was shown by Bryant [Bry] that Berger's list is incomplete. Connections whose holonomy is not contained on Berger's list are called exotic. We investigate a certain 4-dimensional exotic holonomy representation of $Sl(2,\mathbb{R})$. We show that connections with this holonomy are never complete and do not exist on compact manifolds. We give explicit descriptions of these connections on an open dense set and compute their groups of symmetry.
On the additive formulae of the theta functions and a collection of Lambert series pertaining to the modular equations of degree $5$
Li-Chien
Shen
323-345
Abstract: We examine the connection between the additive formulae of the theta functions, the Fourier series expansion of the 12 elliptic functions, and the logarithmic derivatives of the theta functions. As an application, we study the Lambert series related to the modular equations of degree 5 and many interesting identities of Ramanujan are found in this process.
A homotopy invariance theorem in coarse cohomology and $K$-theory
Nigel
Higson;
John
Roe
347-365
Abstract: We introduce a notion of homotopy which is appropriate to the coarse geometry and topology studied by the second author in [7]. We prove the homotopy invariance of coarse cohomology, and of the K-theory of the $ {C^\ast}$-algebra associated to a coarse structure on a space. We apply our homotopy invariance results to show that if M is a Hadamard manifold then the inverse of the exponential map at any point 0 induces an isomorphism between the K-theory groups of the ${C^\ast}$-algebras associated to M and its tangent space at 0 (see Theorem 7.9). This result is consistent with a coarse version of the Baum-Connes conjecture.
Writing integers as sums of products
Charles E.
Chace
367-379
Abstract: In this paper we obtain an asymptotic expression for the number of ways of writing an integer N as a sum of k products of l factors, valid for $k \geq 3$ and $l \geq 2$. The proof is an application of the Hardy-Littlewood method, and uses recent results from the divisor problem for arithmetic progressions.
The spectra of random pseudo-differential operators
Jingbo
Xia
381-411
Abstract: We study the spectra of random pseudo-differential operators generated by the same symbol function on different $ {L^2}$-spaces. Our results generalize the spectral coincidence theorem of S. Kozlov and M. Shubin (Math. USSRSb. 51 (1985), 455-471) for elliptic operators of positive order associated with ergodic systems. Because of our new approach, we are able to treat operators of arbitrary order and associated with arbitrary dynamical systems. Furthermore, we characterize the spectra of these operators in terms of certain naturally obtained Borel measures on R.
A family of real $2\sp n$-tic fields
Yuan Yuan
Shen;
Lawrence C.
Washington
413-434
Abstract: We study the family of polynomials $\displaystyle {P_n}(X;a) = \Re ({(X + i)^{{2^n}}}) - \frac{a}{{{2^n}}}\Im ({(X + i)^{{2^n}}})$ and determine when ${P_n}(X;a)$, $a \in \mathbb{Z}$, is irreducible. The roots are all real and are permuted cyclically by a linear fractional transformation defined over the real subfield of the ${2^n}$th cyclotomic field. The families of fields we obtain are natural extensions of those studied by M.-N. Gras and Y.-Y. Shen, but in general the present fields are non-Galois for $n \geq 4$. From the roots we obtain a set of independent units for the Galois closure that generate an "almost fundamental piece" of the full group of units. Finally, we discuss the two examples where our fields are Galois, namely $a = \pm {2^n}$ and $a = \pm {2^4} \bullet 239$.