Arithmetic in generalized Artin-Schreier extensions of k(x)

If k is a perfect field of characteristic p ≠ 0 and k(x) is the rational function field over k, it is possible to construct cyclic extensions K_n over k(x) such that [K : k(x)] = pⁿ using the concept of Witt vectors. This is accomplished in the following way; if [β₁, β₂,…, β_n] is a Witt vector over k(x) = K₀, then the Witt equation $\text{y}{}^{\mathrm{p}}\text{?}\text{y}\text{= β}$ generates a tower of extensions through $\text{K}{}_{\mathrm{i}}\text{= K}{}_{\mathrm{i?1}}\text{(}\text{y}{}_{\mathrm{i}}\text{)}$ where $\text{y}\text{= [}\text{y}{}_1\text{,}\text{y}{}_2\text{,…,}\text{y}{}_{\mathrm{n}}\text{]}$. In this paper, it is shown that there exists an alternate method of generating this tower which lends itself better for further constructions in K_n. This alternate generation has the form K_i = K_i?1(y_i); y_i^p ? y_i = B_i, where, as a divisor in K_i?1, B_i has the form . In this form q is prime to Πp_j^λ_j and each λ_j is positive and prime to p. As an application of this, the alternate generation is used to construct a lower-triangular form of the Hasse-Witt matrix of such a field K_n over an algebraically closed field of constants.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,II

Let L = ∑_{j = 1}^mX_j² be sum of squares of vector fields in $\text{R}$ⁿ satisfying a Hörmander condition of order 2: span{X_j, [X_i, X_j]} is the full tangent space at each point. A point x??D of a smooth domain D is characteristic if X₁,…, X_m are all tangent to ?D at x. We prove sharp estimates in non-isotropic Lipschitz classes for the Dirichlet problem near (generic) isolated characteristic points in two special cases: (a) The Grushin operator $\text{?}{}^2\text{?x}{}^2\text{+ x}{}^2\text{?}{}^2\text{?t}{}^2$ in $\text{R}$². (b) The real part of the Kohn Laplacian on the Heisenberg group $\text{∑}{}_{\mathrm{j\; ?\; 1}}{}^{\mathrm{n}}\text{(}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2\text{+ (}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}\text{)}{}^2$ in $\text{R}$^{2n + 1}. In contrast to non-characteristic points, C^∞ regularity may fail at a characteristic point. The precise order of regularity depends on the shape of ?D at x.     

The Dirichlet problem for the Kohn Laplacian on the Heisenberg group,I

For (x,y,t)∈$\text{R}$ⁿ × $\text{R}$ⁿ × $\text{R}$, denote $\text{X}{}_{\mathrm{j}}\text{=}\text{?}\text{?x}{}_{\mathrm{j}}\text{+ 2y}{}_{\mathrm{j}}\text{?}\text{?t}\text{, y}{}_{\mathrm{j}}\text{=}\text{?}\text{?y}{}_{\mathrm{j}}\text{? 2x}{}_{\mathrm{j}}\text{?}\text{?t}$ and $\text{L}{}_{\mathrm{\alpha }}\text{=?}\text{1}\text{4}\text{∑}\text{j=1}\text{n}\text{X}{}_{\mathrm{j}}{}^2\text{+ Y}{}_{\mathrm{j}}{}^2\text{+}\text{?}\text{?t}$. When α = n ? 2q, $\text{L}$_a represents the action of the Kohn Laplacian □_b on q-forms on the Heisenberg group. For ?n < α < n, we construct a parametrix for the Dirichlet problem in smooth domains D near non-characteristic points of ?D. A point w of ?D is non-characteristic if one of X₁,…, X_n, Y₁,…, Y_n is transverse to ?D at w. This yields sharp local estimates in the Dirichlet problem in the appropriate non-isotropic Lipschitz classes. The main new tool is a “convolution calculus” of pseudo-differential operators that can be applied to the relevant layer potentials, for which the usual asymptotic composition formula is false. Characteristic points are treated in Part II.     

Singular perturbation problems for systems of partial differential equations of elliptic type

Elliptic boundary value problems for systems of nonlinear partial differential equations of the form $\text{F}{}_{\mathrm{i}}\text{(x, u}{}_1\text{, u}{}_2\text{,…, u}{}_{\mathrm{N}}\text{,}\text{?u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{, ?}{}^{\mathrm{p}}{}_{\mathrm{i}}\text{?}{}^2\text{u}{}_{\mathrm{i}}\text{?x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{) = ?}{}_{\mathrm{i}}\text{(x), x ? R}{}^{\mathrm{n}}$, i = 1(1)N, j, k = 1(1)n, p_i ? 0, ? being a small parameter, with Dirichlet boundary conditions are considered. It is supposed that a formal approximation Z is given which satisfies the boundary conditions and the differential equations upto the order χ(?) = o(1) in some norm. Then, using the theory of differential inequalities, it is shown that under certain conditions the difference between the exact solution u of the boundary value problem and the formal approximation Z, taken in the sense of a suitable norm, can be made small.     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Numerical approximation of product integrals

A technique for the numerical approximation of matrix-valued Riemann product integrals is developed. For a ? x < y ? b, I_m(x, y) denotes

$\text{∫}\text{χ}\text{y}\text{∫}\text{χ}\text{v}{}_2\text{?}\text{∫}\text{χ}\text{v}{}_2\text{∏}\text{i=1}\text{m}\text{F(ν}{}_{\mathrm{i}}\text{)dν}{}_1\text{dν}{}_2\text{?dν}{}_{\mathrm{m}}$

, and A_m(x, y) denotes an approximation of I_m(x, y) of the form

$\text{(y?x)}{}^{\mathrm{m}}\text{∑}\text{k=1}\text{n}\text{a}{}_{\mathrm{k}}\text{∏}\text{i=1}\text{m}\text{F(χ}{}_{\mathrm{ik}}\text{)}$

, where a_k and y_ik are fixed numbers for i = 1, 2,…, m and k = 1, 2,…, N and x_ik = x + (y ? x)y_ik. The following result is established. If p is a positive integer, F is a function from the real numbers to the set of w × w matrices with real elements and F⁽¹⁾ exists and is continuous on [a, b], then there exists a bounded interval function H such that, if n, r, and s are positive integers, $\text{(b ? a)}\text{n}\text{= h < 1, x}{}_{\mathrm{i}}\text{= a + hi}\text{for}\text{i = 0, 1,…, n}\text{and}\text{0 < r ? s ? n}$, then

$\text{∏}\text{χ}{}_{\mathrm{r?}}\text{χ}{}_{\mathrm{s}}\text{(I+F dχ)?}\text{∏}\text{i=r}\text{s}\text{I+}\text{∑}\text{j=1}\text{p}\text{I}{}_{\mathrm{j}}\text{(χ}{}_{\mathrm{i?1}}\text{,χ}{}_{\mathrm{i}}\text{)}$

$\text{=h}{}^{\mathrm{p}}\text{H(χ}{}_{\mathrm{r?1}}\text{,χ}{}_{\mathrm{s}}\text{)+O(h}{}^{\mathrm{p+1}}\text{)}$

Further, if F^(j) exists and is continuous on [a, b] for j = 1, 2,…, p + 1 and A is exact for polynomials of degree less than p + 1 ? j for j = 1, 2,…, p, then the preceding result remains valid when A_j is substituted for I_j.     

A central limit theorem for the sojourn times of strongly ergodic Markov chains

Let X_n be an irreducible aperiodic recurrent Markov chain with countable state space I and with the mean recurrence times having second moments. There is proved a global central limit theorem for the properly normalized sojourn times. More precisely, if $\text{t}{}^{\mathrm{(n}}\text{)}{}_{\mathrm{i}}\text{=Σ}{}^{\mathrm{n}}{}_{\mathrm{k=1}}\text{i}\text{?}{}_{\mathrm{i}}\text{(X}{}_{\mathrm{k}}\text{)}$, then the probability measures induced by {t⁽ⁿ⁾_i/√n?√nπ_i}_i?I(π_i being the ergotic distribution) on the Hilbert-space of square summable I-sequences converge weakly in this space to a Gaussian measure determined by a certain weak potential operator.     

Segmental solutions of Boolean equations

Davio and Deschamps have shown that the solution set, K, of a consistent Boolean equation $\text{?(x}{}_1\text{, …, x}{}_{\mathrm{n}}\text{)=0}$ over a finite Boolean algebra B may be expressed as the union of a collection of subsets of Bⁿ, each of the form {(x₁, …, x_n) | a_i≤x_i≤b_i, a_i?B, b_i?B, i = 1, …, n}. We refer to such subsets of Bⁿ as segments and to the collection as a segmental cover of K. We show that $\text{?(x}{}_1\text{, …, x}{}_{\mathrm{n}}\text{) = 1}$ is consistent if and only if ? can be expressed by one of a class of sum-of-products expressions which we call segmental formulas. The object of this paper is to relate segmental covers of K to segmental formulas for ?.     

Interval linear systems with prescribed column sums

Nonnegative solutions of an interval linear system A^Ix=b^I (A^I being an interval matrix and b^I an interval vector) with additional column sum restrictions of the type $\text{σ}{}_{\mathrm{i}}\text{α}{}_{\mathrm{ij}}\text{a}{}_{\mathrm{ij}}\text{∈[}\text{c}{}_{\mathrm{-j}}\text{,}\text{c}{}_{\mathrm{j}}\text{] ?j}$ are described by a system of linear inequalities with auxiliary variables.     

A complete set of invariants for finite groups and other results

For a finite group G and a set I ? {1, 2,…, n} let

${}_{\mathrm{G}}\text{(n,I) = ∑}{}_{\mathrm{g\; \in \; G}}\text{ε}{}_1\text{(g)?ε}{}_2\text{(g)???ε}{}_{\mathrm{n}}\text{(g)}$

,where

$\text{ε}{}_{\mathrm{i}}\text{(g)=g}\text{if}\text{i=∈ I,}$

$\text{ε}{}_{\mathrm{l}}\text{(g)=l}\text{if}\text{i=∈ I.}$

We prove, among other results, that the positive integers

$\text{tr}\text{(e}{}_{\mathrm{G}}\text{(n,I}{}_1\text{)+?+e}{}_{\mathrm{G}}\text{(n,I}{}_{\mathrm{r}}\text{))}{}^{\mathrm{k}}\text{:n,r,k,?1, I}{}_{\mathrm{j}}\text{?{1,…,n}, 1?|i}{}_{\mathrm{j}}\text{|?3}$

for 1 ? j ? r, I_j1 ∩ I_j2 ∩ I_j3 ∩ I_j4 = Ø for any 1 ? j₁ <j₂ <j₃ <j₄ ? r, determine G up to isomorphism. We also show that under certain assumptions finite groups are determined up to isomorphism by the number of their subgroups.     

Existence of stable equilibrium point for dynamical systems of Volterra type

This paper presents sufficient conditions for the existence of a nonnegative and stable equilibrium point of a dynamical system of Volterra type, (1) $\text{(}\text{d}\text{dt}\text{) x}{}_{\mathrm{i}}\text{(t) = ?x}{}_{\mathrm{i}}\text{(t)[f}{}_{\mathrm{i}}\text{(x}{}_1\text{(t),…, x}{}_{\mathrm{n}}\text{(t)) ? q}{}_{\mathrm{i}}\text{], i = 1,…, n}$, for every q = (q₁,…, q_n)^T?Rⁿ. Results of a nonlinear complementarity problem are applied to obtain the conditions. System (1) has a nonnegative and stable equilibrium point if (i) f(x) = (f₁(x),…,f_n(x))^T is a continuous and differentiable M-function and it satisfies a certain surjectivity property, or (ii), f(x) is continuous and strongly monotone on R₊₀ⁿ.     

A new form of trigonometric orthogonality and Gaussian-type quadrature

Gauss's (2n+1)-point trigonometric interpolation formula, based upon f(x_i), i = 1(1)2n+1, gives a trigonometric sum of the n^th order, S_2n+1(x = a₀ + ∑_jⁿ = 1^(a_jcos jx + b_jsin jx), which may be integrated to provide formulas for either direct quadrature or stepwise integration of differential equations having periodic (or near-periodic) solutions. An “orthogonal” trigonometric sum S_2r+1(x) is one that satisfies

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{2r+1}}\text{(x)S}{}_{\mathrm{2r\prime +1}}\text{(x)dx=0, r′<r}$

and two other arbitrarily imposable conditions needed to make S_2r1(x) unique. Two proofs are given of a fundamental factor theorem for any S_2n+1(x) (somewhat different from that for polynomials) from which we derive 2r-point Gaussian-type quadrature formulas, r = [n/2] + 1, which are exact for any S_4r?1(x). We have

$\text{∫}{}_{\mathrm{a}}{}^{\mathrm{b}}\text{S}{}_{\mathrm{4r?1(x)}}\text{dx=∑}{}_{\mathrm{j=1}}{}^{\mathrm{2r}}\text{A}{}_{\mathrm{j}}\text{S}{}_{\mathrm{4r?1}}\text{(x}{}_{\mathrm{j}}\text{)}$

where the nodes x_j, j = 1(1)2r, are the zeros of the orthogonal S_2r+1(x). It is proven that A_j > 0 and that 2r-1 of the nodes must lie within the interval [a,b], and the remaining node (which may or may not be in [a,b]) must be real. Unlike Legendre polynomials, any [a′,b′] other than a translation of [a,b], requires different and unrelated sets of nodes and weights. Gaussian-type quadrature formulas are applicable to the numerical integration of the Gauss (2n+1)-point interpolation formulas, with extra efficiency when the latter are expressed in barycentric form. S_2r+1(x), x_jandA_j, j = 1(1)2r, were calculated for [a,b] = [0, π/4], 2r = 2 and 4, to single-precision accuracy.     

A method of feasible directions using function approximations,with applications to min max problems

This paper presents a demonstrably convergent method of feasible directions for solving the problem min{φ(ξ)| gⁱ(ξ)?0i=1,2,…,m}, which approximates, adaptively, both φ(x) and ▽φ(x). These approximations are necessitated by the fact that in certain problems, such as when $\text{φ(x) =}\text{max}\text{{f(x, y) ¦ y ? Ω}{}_{\mathrm{y}}\text{}}$, a precise evaluation of φ(x) and ▽φ(x) is extremely costly. The adaptive procedure progressively refines the precision of the approximations as an optimum is approached and as a result should be much more efficient than fixed precision algorithms.It is outlined how this new algorithm can be used for solving problems of the form $\text{min}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{x}}\text{max}{}_{\mathrm{y}}\text{? Ω}{}_{\mathrm{y}}\text{f(x, y)}$ under the assumption that Ωm_ξ={x|gⁱ(x)?0, j=1,…,s} ∩$\text{R}$ⁿ, Ω_y={y|ζ_i(y)?0, i-1,…,t} ∩ $\text{R}$^m, with f, g^j, ζⁱ continuously differentiable, f(x, ·) concave, ζⁱ convex for $\text{i = 1,…, t,}\text{and}\text{Ω}{}_{\mathrm{x}}\text{, Ω}{}_{\mathrm{y}}$ compact.     

Doubly stochastic matrices with equal subpermanents

It has been conjectured that if A is a doubly stochastic n>× n matrix such that per A(i, j)≥perA for all i, j, then either A = J_n, then n × n matrix with each entry equal to $\text{1}\text{n}$, or, up to permutations of rows and columns, $\text{A =}\text{1}\text{2}\text{(I}{}_{\mathrm{n}}\text{+ P}{}_{\mathrm{n}}\text{)}$, where P_n is a full cycle permutation matrix. This conjecture is proved.     

On the solutions to x′(t) = A(t) x (t) over all A(t), where P ⩽ A(t) ⩽ Q

Let P_ij and q_ij be positive numbers for i ≠ j, i, j = 1, …, n, and consider the set of matrix differential equations x′(t) = A(t) x(t) over all A(t), where a_ij(t) is piecewise continuous, a_ij(t) = ?∑_{i ≠ j}a_ij(t), and p_ij ? a_ij(t) ? q_ij all t. A solution x is also to satisfy ∑_{i = 1}ⁿx_i(0) = 1. Let C_t denote the set of all solutions, evaluated at t to equations described above. It is shown that $\text{C}{}_{\mathrm{t}}$, the topological closure of C_t, is a compact convex set for each t. Further, the set valued function $\text{C}{}_{\mathrm{t}}$, of t is continuous and $\text{limit}{}_{\mathrm{t\; \to \; \infty }}\text{C}\text{?}{}_{\mathrm{t}}\text{= ∩}\text{C}\text{?}{}_{\mathrm{t}}$.     

Isospectral matrices and classical polynomials

The matrices of order n defined, in terms of the n arbitrary numbers x_j, by the formulae $\text{X=}\text{diag}\text{(x}{}_{\mathrm{j}}\text{)}\text{and}\text{Z}{}_{\mathrm{jk}}\text{=δ}{}_{\mathrm{jk}}\text{∑′}\text{l=1}\text{n}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{l}}\text{)}{}^{\mathrm{?1}}\text{+(1?δ}{}_{\mathrm{jk}}\text{(x}{}_{\mathrm{j}}\text{?x}{}_{\mathrm{k}}\text{)}{}^{\mathrm{?1}}$, are representations of the multiplicative operator ξ and of the differential operator d/dξ in a space spanned by the polynomials in ξ of degree less than n. This elementary fact implies a number of remarkable formulae involving these matrices, including novel representations of the classical polynomials.     

“Integer-making” theorems

Let X = {x₁, x₂,…} be a finite set and associate to every x_i a real number α_i. Let f(n) [g (n)] be the least value such that given any family $\text{F}$ of subsets of X having maximum degree n [cardinality n], one can find integers α_i, i=1,2,… so that α_i ? α_i|<1 and

$\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤?(n)}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{a}{}_{\mathrm{i}}\text{?}\text{∑}\text{x}{}_{\mathrm{i\; ?\; E}}\text{α}{}_{\mathrm{i}}\text{≤}\text{g(n)}$

for all $\text{E ?}\text{F}$. We prove

$\text{f(n)≤n ? 1 and g(n)≤c(n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$

.     

Best pseudo-isolated Gerschgorin disks for Eigenvalues

Let A be an arbitrary n×n matrix, partitioned so that if A=[A_ij], then all submatrices A_ii are square. If x is a positive vector, it is well-known that $\text{G(}\text{x}\text{) =∪}{}^{\mathrm{N}}{}_{\mathrm{i=1}}\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{)}$, where

$\text{G}{}_{\mathrm{i}}\text{(}\text{x}\text{) =}\text{z}\text{6(zI ? A}{}_{\mathrm{ii}}\text{)}{}^{\mathrm{?1}}\text{6}{}^{\mathrm{?1}}\text{?}\text{1}\text{x}{}_{\mathrm{i}}\text{∑}\text{j = 1}\text{j ≠ i}\text{N}\text{`6A}{}_{\mathrm{ij}}\text{6x}{}_{\mathrm{j}}$

, contains all the eigenvalues of A. The purpose of this paper is to give a new definition of the concept of an isolated subregion of G(x). An algorithm is given for obtaining the best such isolated subregion in a certain sense, and examples are given to show that tighter bounds for some eigenvalues of A may be obtained than with previous algorithms. For ease of computation, each subregion G_i(x) is replaced by the union of circular disks centered at the eigenvalues of A_ii.