On the elgenvectors of Schur's matrix

A basis of eigenvectors is given for the matrix $\text{U}$ = (e^2πimn/q), (1 ≤ m, n ≤ q). The eigenvectors arise from the characters on the reduced residue class group (mod q).     

Generalized bent functions and their properties

Jet J_q^m denote the set of m-tuples over the integers modulo q and set $\text{i=}\text{?1}$, $\text{w = e}{}^{\mathrm{<mtext>i(</mtext><mtext>2\pi </mtext><mtext>q</mtext><mtext>)</mtext>}}$. As an extension of Rothaus' notion of a bent function, a function f, f: J_q^m → J_q¹ is called bent if all the Fourier coefficients of w^f have unit magnitude. An important feature of these functions is that their out-of-phase autocorrelation value is identically zero. The nature of the Fourier coefficients of a bent function is examined and a proof for the non-existence of bent functions over J_q^m, m odd, is given for many values of q of the form q = 2 (mod 4). For every possible value of q and m (other than m odd and q = 2 (mod 4)), constructions of bent functions are provided.     

Operators as spectral integrals of operator-valued functions from the study of multivariate stationary stochastic processes

P. Masani and the author have previously answered the question, “When is an operator on a Hilbert space $\text{H}$ the integral of a complex-valued function with respect to a given spectral (projection-valued) measure?” In this paper answers are given to the question, “When is a linear operator from $\text{H}$^q to $\text{H}$^p the integral of a spectral measure?”; here the values of the integrand are linear operators from the square-summable q-tuples of complex numbers to the square-summable p-tuples of complex numbers, and our spectral measure for $\text{H}$^q is the “inflation” of a spectral measure for $\text{H}$. In the course of this paper, we make available tools for handling the spectral analysis of q-variate weakly stationary processes, 1 ≤ q ≤ ∞, which should enable researchers to deal in the future with the case q = ∞. We show as one application of our theory that if U = ∫(in0, 2π]e^?iθE(dθ) is a unitary operator on $\text{H}$ and if T is a bounded linear operator from $\text{H}$^q to $\text{H}$^q (1 ≤ q ≤ ∞) which is a prediction operator for each stationary process (Uⁿx)_?∞^∞ ?$\text{H}$^q (for each x = (xⁱ)_i^j ∈ $\text{H}$^q, Uⁿx = (Uⁿxⁱ)_i=1^q), then T is a spectral integral, ∫(_0,2π)]Φ(θ) E(dθ), and the Banach norm of T, |T|_B = ess sup |Φ(θ)|_B.     

On Diophantine equations over the ring of all algebraic integers

For each odd prime q an integer NH_q (NH₃ = ?1, NH₅ = ?1, NH₇ = 97, NH₁₁ = ?243, …) is defined as the norm from L to $\text{Q}$ of the Heilbronn sum H_q = Tr_I^{$\text{Q}$(ζ)}(ζ), where ζ is a primitive q²th root of unity and L ?- $\text{Q}$(ζ) the subfield of degree q. Various properties are proved relating the congruence properties of H_q and NH_q modulo p (p ≠ q prime) to the Fermat quotient $\text{(p}{}^{\mathrm{q\; ?\; 1}}\text{? 1)}\text{q}\text{(}\text{mod}\text{q)}$; in particular, it is shown that NH_q is even iff 2^{q ? 1} ≡ 1 (mod q²).     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

A family of difference sets in non-cyclic groups

A construction is given for difference sets in certain non-cyclic groups with the parameters $\text{v = q}{}^{\mathrm{s+1}}\text{{[}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}\text{] + 1}}$, $\text{k = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s+1}}\text{? 1)}\text{(q ? 1)}$, $\text{λ = q}{}^{\mathrm{s}}\text{(q}{}^{\mathrm{s}}\text{? 1)}\text{(q ? 1)}$, n = q^2s for every prime power q and every positive integer s. If q^s is odd, the construction yields at least $\text{1}\text{2}\text{(q}{}^{\mathrm{s}}\text{+ 1)}$ inequivalent difference sets in the same group. For q = 5, s = 2 a difference set is obtained with the parameters (v, k, λ, n) = (4000, 775, 150, 625), which has minus one as a multiplier.     

Rationale approximation mit Hilfe pythagoräischer Zahlen

The wealth of Pythagorean number triples is demonstrated afresh by showing that for every rational number $\text{p}\text{q}$, p, q ∈ $\text{N}$ there exist infinitely many Pythagorean number triples (a, b, c) which satisfy

$\text{|}\text{a}\text{b}\text{?}\text{p}\text{q}\text{| ?}\text{1}\text{b}$

where a is odd, b is even and a² + b² = c². The special cases where p = 1 or where q = 1 are considered first as they illustrate the method and yield additional results. Rational approximations are also possible by means of the quotients $\text{c}\text{b}$, provided p > q. The results generalize Pythagorean number triples (a, b, c) where a and b differ by a constant investigated by S. Pignataro.     

Quaternionic compositum genus

A quaternionic field over the rationals contains three quadratic subfields with a compositum genus relation of the type described in the author's paper in Volume 9 of this journal, involving the representation of a prime as norm in these subfields. These representations had previously been only partially exlored by the transfer of class structure from the rational to the quadratic fields. Here a full exposition is given by constructing the Artin characters when the subfields are $\text{Q}$(2^1/2), $\text{Q}$(q^1/2), and $\text{Q}$(2q)^1/2 (q prime). A special role belongs to q = A² + 32b².     

Infinite-variate wide-sense Markov processes and functional analysis for bounded operator-forming vectors

Let p, q be arbitrary parameter sets, and let $\text{H}$ be a Hilbert space. We say that x = (xⁱ)_i?q, xⁱ ? $\text{H}$, is a bounded operator-forming vector (?$\text{H}$_F^q) if the Gram matrix 〈x, x〉 = [(xⁱ, x^j)]_i?q,j?q is the matrix of a bounded (necessarily ≥ 0) operator on $\text{l}{}_{\mathrm{q}}{}^2$, the Hilbert space of square-summable complex-valued functions on q. Let A be p × q, i.e., let A be a linear operator from $\text{l}{}_{\mathrm{q}}{}^2$ to $\text{l}{}_{\mathrm{p}}{}^2$. Then exists a linear operator ǎ from (the Banach space) $\text{H}$_F^q to $\text{H}$_F^p on $\text{D}$(A) = {x:x ? $\text{H}$_F^q, $\text{A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is p × q bounded on $\text{l}{}_{\mathrm{q}}{}^2$} such that y = ǎx satisfies y^j?σ(x) = {space spanned by the xⁱ}, 〈y, x〉 = A〈x, x〉 and $\text{〈y, y〉 = A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{(A〈x, x〉}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}{}^1$. This is a generalization of our earlier [J. Multivariate Anal.4 (1974), 166–209; 6 (1976), 538–571] results for the case of a spectral measure concentrated on one point. We apply these tools to investigate q-variate wide-sense Markov processes.     

Path length in the covering graph of a lattice

Here it is proved that a cyclic (n, k) code over GF(q) is equidistant if and only if its parity check polynomial is irreducible and has exponent $\text{e =}\text{(q}{}^{\mathrm{k}}\text{? 1)}\text{a}$ where a divides q ? 1 and (a, k) = 1. The length n may be any multiple of e. The proof of this theorem also shows that if a cyclic (n,k) code over GF(q) is not a repetition of a shorter code and the average weight of its nonzero code words is integral, then its parity check polynomial is irreducible over GF(q) with exponent $\text{n =}\text{(q}{}^{\mathrm{k}}\text{? 1)}\text{a}$ where a divides q ? 1.     

Product formulas for semigroups with elliptic generators

Real constant coefficient nth order elliptic operators, Q, which generate strongly continuous semigroups on L²($\text{R}$^k) are analyzed in terms of the elementary generator,

$\text{A = (?n)}{}^{\mathrm{<mtext>(</mtext><mtext>n</mtext><mtext>2</mtext><mtext>\; ?\; 1</mtext>}}\text{)(n!)}{}^{\mathrm{?1}}\text{∑}\text{k}\text{j = 1}\text{?}{}^{\mathrm{n}}\text{?x}{}_{\mathrm{j}}{}^{\mathrm{n}}$

, for n even. Integral operators are defined using the fundamental solutions p_n(x, t) to u_t = Au and using real polynomials q_l,…, q_k on $\text{R}$^m by the formula, for q = (q_l,…, q_k),

$\text{(F(t)?)(x) = ∫}$

_$\text{R}$m

$\text{?(x + q(z)) P}{}_{\mathrm{n}}\text{(z, t)dz}$

. It is determined when, strongly on L²($\text{R}$^k),

$\text{e}{}^{\mathrm{tQ}}\text{=}\text{lim}\text{j → ∞}\text{F}\text{t}\text{j}{}^{\mathrm{j}}$

. If n = 2 or k = 1, this can always be done. Otherwise the symbol of Q must have a special form.     

Graphs without quadrilaterals

Let f(n) denote the maximum number of edges of a graph on n vertices not containing a circuit of length 4. It is well known that $\text{f(n) ～}\text{1}\text{2}\text{n}\text{n}$. The old conjecture $\text{f(q}{}^2\text{+ q + 1) =}\text{1}\text{2}\text{q(q + 1)}{}^2$ is proved for infinitely many q (whenever q = 2^k).     

The quartic characters of certain quadratic units

Criteria are obtained for the quartic residue character of the fundamental unit of the real quadratic field $\text{Q((2q)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where q is prime and either q ≡ 7(mod 8), or q ≡ 1(mod 8) and X² ? 2qY² = ?2 is solvable in integers X and Y.     

On maximum principles for a class of nonlinear second-order elliptic equations

In a recent paper [3] the authors derived maximum principles which involved $\text{u(x)}\text{and}\text{q = ¦}\text{grad}\text{u¦}$, where u(x) is a classical solution of an alliptic differential equation of the form ($\text{g(q}{}^2\text{)u}{}_{\mathrm{,i}}\text{)}{}_{\mathrm{,i}}\text{+ ?(u) ?(q}{}^2\text{) = 0}$. In this paper these results are extended to the more general case in which $\text{g = g(u, q}{}^2\text{)}\text{and}\text{?(u) ?(q}{}^2\text{)}$ is replaced by h(u, q²).     

nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems

This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in $\text{L}$_n^p[0,1] × $\text{L}$_n^p[0,1], 1 ? p < ∞. The adjoint relation in $\text{L}$_n^q[0,1] × $\text{L}$_n^q[0,1], $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1}$, is derived. Green's formula is also found. Self-adjoint relations are found in $\text{L}$_n²[0,1] × $\text{L}$_n²[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of $\text{L}$_n^p[0,1] × $\text{L}$_n²[0,1] is established.     

On the cubic character of quadratic units

An elementary proof is given of the theorem: If D = ?3q or ?27q is the discriminant of a cubic field, where q ≡ 1 (mod 4) is a prime, and if p or 4p is represented by c² + ∥ D ∥ d², then the fundamental unit in the field $\text{Q(q}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is a cubic residue of the prime p. In special cases necessary and sufficient conditions are derived.     

Uncertainty principle and Lp–Lq-sufficient pairs on noncompact real symmetric spaces

We consider a real semi-simple Lie group G with finite center and a maximal compact sub-group K of G. Let $\text{G=K}\text{exp}\text{(}\text{a}{}_{\mathrm{+}}\text{)K}$ be a Cartan decomposition of G. For x∈G denote ∥x∥ the norm of the $\text{a}{}_{\mathrm{+}}$-component of x in the Cartan decomposition of G. Let $\text{a>0,}\text{b>0}$ and 1?p,q?∞. In this Note we give necessary and sufficient conditions on $\text{a,}\text{b}$ such that for all K-bi-invariant measurable function f on G, if e^a∥x∥2f∈L^p(G) and then f=0 almost everywhere. To cite this article: S. Ben Farah, K. Mokni, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Graphes connexes représentation des entiers et équirépartition

Let q be an integer ≥2 and Ω a suitable subset of {0,…,q ? 1}²; $\text{C}$(q; Ω) denotes the set of natural integers, the pairs of successive q-adic digits of which are in Ω. If P is an irrational polynomial, the sequence (P(n): n ∈ $\text{C}$(q; Ω)) is uniformly distributed modulo one.     

Specialized Möbius inversion

Let Φ denote the set of arithmetic functions $\text{f}\text{(n)}$ such that $\text{f}\text{(n) = f(e}{}_1\text{, e}{}_2\text{,…, e}{}_{\mathrm{r}}\text{)}$, where n = p₁^e₁p₂^e₂…p_r^e_r and f(e₁, e₂,…,e_r) depends on the e_j but is independent of the primes p_j. The general Möbius inversion theorem then becomes the equivalence of (1.11) and (1.12) below. This in turn leads to relations between exponential generating functions as well as several applications.