Diffusion with interactions between two types of particles and Pressureless gas equations

We construct two d-dimensional independent diffusions $\text{X}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{=a+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{a}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{,}\text{X}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{=b+∫}{}_0{}^{\mathrm{t}}\text{u(X}{}_{\mathrm{s}}{}^{\mathrm{b}}\text{,s)}\text{d}\text{s+νB}{}_{\mathrm{t}}{}^{\mathrm{b}}$, with the same viscosity ν≠0 and the same drift u(x,t)=(pρ_t^a(x)v₁+(1?p)ρ_t^b(x)v₂)/(pρ_t^a(x)+(1?p)ρ_t^b(x)), where ρ_t^a,ρ_t^b are respectively the density of X_t^a and X_t^b. Here $\text{a,b,v}{}_1\text{,v}{}_2\text{∈}\text{R}{}^{\mathrm{d}}$ and p∈(0,1) are given. We show that $\text{(ρ}{}_{\mathrm{t}}\text{(x)=pρ}{}_{\mathrm{t}}{}^{\mathrm{a}}\text{(x)+(1?p)ρ}{}_{\mathrm{t}}{}^{\mathrm{b}}\text{(x),u(x,t):}\text{t?0,}\text{x∈}\text{R}{}^{\mathrm{d}}\text{)}$ is the unique weak solution of the following pressureless gas system

such that $\text{ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pδ}{}_{\mathrm{a}}\text{+(1?p)δ}{}_{\mathrm{b}}\text{,}\text{u(x,t)ρ}{}_{\mathrm{t}}\text{(x)}\text{d}\text{x→pv}{}_1\text{δ}{}_{\mathrm{a}}\text{+(1?p)v}{}_2\text{δ}{}_{\mathrm{b}}$ as t→0+. To cite this article: A. Dermoune, S. Filali, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

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 Feynman-Kac gauge for solvability of the Schrödinger equation

Let {X_t, t ≥ 0} be Brownian motion in $\text{R}$^d (d ≥ 1). Let D be a bounded domain in $\text{R}$^d with C² boundary, ?D, and let q be a continuous (if d = 1), Hölder continuous (if d ≥ 2) function in D?. If the Feynman-Kac “gauge” E_x{exp(∝₀^τ_Dq(X_t)dt)1_A(X_τD)}, where τ_D is the first exit time from D, is finite for some non-empty open set A on ?D and some x?D, then for any $\text{? ? C}{}^0\text{(?D}$), is the unique solution in $\text{C}{}^2\text{(D) ∩ C}{}^0\text{(}\text{D}\text{?}\text{)}$ of the Schrödinger boundary value problem $\text{(}\text{1}\text{2}\text{Δ}\text{+ q)φ = 0}\text{in}\text{D, φ = ?}\text{on}\text{?D}$.     

Algebras containing a divisible unimodular sequence with dense self adjoint span

We characterize the uniform algebras A on a compact Hausdorff space X which contain a sequence {u_j}_{j = 0}^∞ of unimodular elements with $\text{u}{}_{\mathrm{j}}\text{u}{}_{\mathrm{j\; ?\; 1}}\text{? A}$ and closed span $\text{{u}{}_{\mathrm{j}}\text{u}\text{}}{}_{\mathrm{j\; =\; 0}}{}^{\mathrm{\infty }}\text{= C(X)}$ in terms of the maximal ideal space of A. Roughly, the essential set of A looks like (at most) countably many copies of the boundary of the unit disk, and A looks like the disk algebra on each.     

A remark on the tail probability of a distribution

Let {X_n}_n≥1 be a sequence of independent and identically distributed random variables. For each integer n ≥ 1 and positive constants r, t, and ?, let S_n = Σ_j=1ⁿX_j and $\text{E{N}{}_{\mathrm{\infty }}\text{(r, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{r?2}}\text{P{|S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>r</mtext><mtext>t</mtext>}}\text{}}$. In this paper, we prove that (1) lim_?→0+?^α(r?1)E{N_∞(r, t, ?)} = K(r, t) if E(X₁) = 0, Var(X₁) = 1, and E(| X₁ |^t) < ∞, where 2 ≤ t < 2r ≤ 2t, $\text{K(r, t) =}\text{{2}{}^{\mathrm{<mtext>\alpha (r?1)</mtext><mtext>2</mtext>}}\text{Γ(}\text{(1 + α(r ? 1))}\text{2}\text{)}}\text{{(r ? 1) Γ(}\text{1}\text{2}\text{)}}$, and $\text{α =}\text{2t}\text{(2r ? t)}$; (2) $\text{lim}{}_{\mathrm{?\to 0+}}\text{G(t, ?)}\text{H(t, ?)}\text{= 0}$ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(|X₁|^t) < ∞, where G(t, ?) = E{N_∞(t, t, ?)} = Σ_n=1^∞n^t?2P{| S_n | > ?n} → ∞ as ? → 0+ and $\text{H(t, ?) = E{N}{}_{\mathrm{\infty }}\text{(t, t, ?)} = Σ}{}_{\mathrm{n=1}}{}^{\mathrm{\infty }}\text{n}{}^{\mathrm{t?2}}\text{P{| S}{}_{\mathrm{n}}\text{| > ?n}{}^{\mathrm{<mtext>2</mtext><mtext>t</mtext>}}\text{} → ∞}\text{as}\text{? → 0+}$, i.e., H(t, ?) goes to infinity much faster than G(t, ?) as ? → 0+ if 2 < t < 4, E(X₁) = 0, Var(X₁) > 0, and E(| X₁ |^t) < ∞. Our results provide us with a much better and deeper understanding of the tail probability of a distribution.     

Abstract equipartition of energy theorems

Let H be a self-adjoint operator on a complex Hilbert space $\text{H}$. The solution of the abstract Schrödinger equation $\text{idu}\text{dt}\text{= Hu}$ is given by u(t) = exp(?itH)u(0). The energy E = ∥u(t)∥² is independent of t. When does the energy break up into different kinds of energy E = ∑_{j = 1}^NE_j(t) which become asymptotically equipartitioned ? (That is, $\text{E}{}_{\mathrm{j}}\text{(t) →}\text{E}\text{N}\text{as}\text{t → ± ∞}$ for all j and all data u(0).) The “classical” case is the abstract wave equation $\text{d}{}^2\text{v}\text{dt}{}^2\text{+ A}{}^2\text{v = 0}\text{with}\text{A}$ self-adjoint on $\text{H}$₁. This becomes a Schrödinger equation in a Hilbert space $\text{H}$ (essentially $\text{H}$ is two copies of $\text{H}$₁), and there are two kinds of associated energy, viz., kinetic and potential. Two kinds of results are obtained. (1) Equipartition of energy is related to the C¹-algebra approach to quantum field theory and statistical mechanics. (2) Let A₁,…, A_N be commuting self-adjoint operators with N = 2 or 4. Then the equation $\text{Π}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{N}}\text{(}\text{d}\text{dt}\text{? iA}{}_{\mathrm{j}}\text{) u(t) = 0}$ admits equipartition of energy if and only if exp(it(A_j ? A_k)) → 0 in the weak operator topology as t → ± ∞ for j ≠ k.     

Measuring flatness in linear spaces

Let L be a finite-dimensional normed linear space and let M be a compact subset of L lying on one side of a hyperplane through 0. A measure of flatness for M is the number $\text{D(M) =}\text{inf}\text{{}\text{sup}\text{f(x)}\text{f(y)}\text{: x, y ? M}}$, where the infimum is over all f in $\text{L}{}^1$ which are positive on M. Thus D(M) = 1 if M is flat, but otherwise D(M) > 1. On the other hand, let E(M) be a second measure on M defined as follows: If M is linearly independent, E(M) = 1. If M is linearly dependent, then (1) let Z be a minimal, linearly dependent subset of M; (2) partition Z into mutually exclusive subsets U = {u₁, …, u_p} and V = {v₁, …, v_q} such that there exist positive coefficients a_i and b_i for which Σ_{i = 1}^pa_iu_i = Σ_{i = 1}^qb_iv_i; (3) let $\text{r =}\text{max}\text{{}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{a}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{b}{}_{\mathrm{i}}\text{,}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{p}}\text{b}{}_{\mathrm{i}}\text{Σ}{}_{\mathrm{i\; =\; 1}}{}^{\mathrm{q}}\text{a}{}_{\mathrm{i}}\text{}}$; (4) let E(M) be the supremum of all ratios r which can be formed by steps (1), (2) and (3). The main result of this paper is that these two measures are the same: D(M) = E(M). This result is then used to obtain results concerning the Banach distance-coefficient between an arbitrary finite-dimensional normed linear space and Hilbert space.     

Temporally inhomogeneous scattering theory

A theory of scattering for the time dependent evolution equations $\text{du}\text{dt}\text{= iH}{}_{\mathrm{j}}\text{(t)u, j = 0, 1}$ (¹) is developed. The wave operators are defined in terms of the evolution operators U_j(t, s), which govern (¹). The scattering operator remains unitary. Sufficient conditions for existence and completeness of the wave operators are obtained; these are the main results. General properties, such as the chain rule and various intertwining relations, are also established. Applications include potential scattering (H₀(t) = ?Δ, Δ denoting the Laplacian, and H₁(t) = ?Δ + q(t, ·)) and scattering for second-order differential operators with coefficients constant in the spatial variable ($\text{H}{}_{\mathrm{j}}\text{(t) = ∑}{}_{\mathrm{m,\; k\; =\; 1}}{}^{\mathrm{n}}\text{a}{}_{\mathrm{mk}}{}^{\mathrm{(j)}}\text{(t)(}\text{?}{}^2\text{?x}{}_{\mathrm{m}}\text{?x}{}_{\mathrm{k}}\text{) + b}{}_{\mathrm{j}}\text{(t)}\text{for}\text{j = 0, 1}$).     

Asymptotic properties of general symmetric hyperbolic systems

Results on partition of energy and on energy decay are derived for solutions of the Cauchy problem $\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0, u(0, x) = ?(x)}$. Here the A_j's are constant, k × k Hermitian matrices, x = (x₁,…, x_n), t represents time, and u = u(t, x) is a k-vector. It is shown that the energy of Mu approaches a limit $\text{E}{}_{\mathrm{M}}\text{(?)}\text{as}\text{¦ t ¦ → ∞}$, where M is an arbitrary matrix; that there exists a sufficiently large subspace of data ?, which is invariant under the solution group U₀(t) and such that $\text{U}{}_0\text{(t)? = 0}\text{for}\text{¦ x ¦ ? a ¦ t ¦ ? R, a}\text{and}\text{R}$ depending on ? and that the local energy of nonstatic solutions decays as $\text{¦ t ¦ → ∞}$. More refined results on energy decay are also given and the existence of wave operators is established, considering a perturbed equation $\text{E(x)}\text{?u}\text{?t}\text{+ ∑}{}_{\mathrm{j\; =\; 1}}{}^{\mathrm{n}}\text{A}{}_{\mathrm{j}}\text{?u}\text{?x}{}_{\mathrm{j}}\text{= 0,}\text{where}\text{¦ E(x) ? I ¦ = O(¦ x ¦}{}^{\mathrm{?1\; ?\; ?}}\text{)}$ at infinity.     

A completeness theorem for a nonlinear multiparameter eigenvalue problem

In this paper we study the linked nonlinear multiparameter system

$\text{y}{}_{\mathrm{r}}{}^{\mathrm{n}}\text{(X}{}_{\mathrm{r}}\text{) + M}{}_{\mathrm{r}}\text{Y}{}_{\mathrm{r}}\text{+}\text{∑}\text{s=1}\text{k}\text{λ}{}_{\mathrm{s}}\text{(a}{}_{\mathrm{rs}}\text{(X}{}_{\mathrm{r}}\text{) + P}{}_{\mathrm{rs}}\text{) Y}{}_{\mathrm{r}}\text{(X}{}_{\mathrm{r}}\text{) = 0, r = l,…, k}$

, where x_r? [a_r, b_r], y_r is subject to Sturm-Liouville boundary conditions, and the continuous functions a_rs satisfy ¦ $\text{A ¦ (x) =}\text{det}\text{a}{}_{\mathrm{rs}}\text{(x}{}_{\mathrm{r}}\text{) > 0}$. Conditions on the polynomial operators M_r, P_rs are produced which guarantee a sequence of eigenfunctions for this problem yⁿ(x) = Π_r=1^ky_rⁿ(x_r), n ? 1, which form a basis in $\text{L}{}^2\text{([a, b], ¦ A ¦)}$. Here [a, b] = [a₁, b₁ × … × [a_k, b_k].     

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.     

Adiabatic invariants for linear Hamiltonian systems

n independent adiabatic invariants in involution are found for a slowly varying Hamiltonian system of order 2n × 2n. The Hamiltonian system considered is $\text{?}\text{u}\text{?}\text{= A(t)u}\text{as}\text{? → 0}{}^{\mathrm{+}}$, where A(t) is a 2n × 2n real matrix with distinct, pure imaginary eigen values for each t? [?∞, ∞], and $\text{d}{}^{\mathrm{(j)}}\text{A}\text{dt}{}^{\mathrm{(j)}}\text{? L}{}_{\mathrm{j}}\text{(?∞, ∞)}$, for all j > 0. The adiabatic invariants I_s(u, t), s = 1,…, n are expressed in terms of the eigen vectors of A(t). Approximate solutions for the system to arbitrary order of ? are obtained uniformly for t? [?∞, ∞].     

Radial bounds for perturbations of elliptic operators

Elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$, α a multi-index, with leading term positive and constant coefficient, and with lower order coefficients defined on $\text{R}{}^{\mathrm{<mi\; mathvariant="italic"\; is="true">n</mi>}}$ or a quotient space $\text{R}{}^{\mathrm{n}}\text{R}{}^{\mathrm{n}}\text{U}{}_{\mathrm{\alpha }}\text{, U}{}_{\mathrm{\alpha }}\text{? R}{}^{\mathrm{n}}$ are considered. It is shown that the L^p-spectrum of A is contained in a “parabolic region” Ω of the complex plane enclosing the positive real axis, uniformly in p. Outside Ω, the kernel of the resolvent of A is shown to be uniformly bounded by an L¹ radial convolution kernel. Some consequences are: A can be closed in all L^p (1 ? p ? ∞), and is essentially self-adjoint in L² if it is symmetric; A generates an analytic semigroup e^?tA in the right half plane, strongly L^p and pointwise continuous at t = 0. A priori estimates relating the leading term and remainder are obtained, and summability $\text{φ(εA)?→}{}_{\mathrm{\epsilon \; \to \; 0}}\text{φ(0) ?}$, with φ analytic, is proved for $\text{? ? L}{}^{\mathrm{p}}$, with convergence in L^p and on the Lebesgue set of ?. More comprehensive summability results are obtained when A has constant coefficients.     

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.     

On Lp-spectrum of elliptic operators

For elliptic operators $\text{A = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{a}{}_{\mathrm{\alpha }}\text{(x) D}{}^{\mathrm{\alpha }}$ on Rⁿ and certain of their singular perturbations $\text{B = ∑}{}_{\mathrm{¦\alpha ¦\; ?\; m}}\text{b}{}_{\mathrm{\alpha }}\text{(x)D}{}^{\mathrm{\alpha }}$ relative compactness of B with respect to A is established. This result applies to the study of L^p-spectra of elliptic operators for different p.     

On the rate for uniform strong consistency of empirical distributions of independent nonidentically distributed multivariate random variables

Let X_j = (X_1j ,…, X_pj), j = 1,…, n be n independent random vectors. For x = (x₁ ,…, x_p) in R^p and for α in [0, 1], let F_j${}^1$(x) = αI(X_1j < x₁ ,…, X_pj < x_p) + (1 ? α) I(X_1j ≤ x₁ ,…, X_pj ≤ x_p), where I(A) is the indicator random variable of the event A. Let F_j(x) = E(F_j${}^1$(x)) and D_n = sup_{x, α} max_{1 ≤ N ≤ n} |Σ₀ⁿ(F_j${}^1$(x) ? F_j(x))|. It is shown that P[D_n ≥ L] < 4pL exp{?2(L²n^?1 ? 1)} for each positive integer n and for all L² ≥ n; and, as n → ∞, $\text{D}{}_{\mathrm{n}}\text{= 0((n}\text{log}\text{n)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ with probability one.     

Matrix identities of the fast fourier transform

Let A_n(ω) be the nxn matrix A_n(ω)=(a_ij with a_ij=ω^ij, 0?i,j?n?1, ωⁿ=1. For n=rs we show

$\text{A}{}_{\mathrm{n}}\text{(w)P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{P}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{∵}\text{0}\text{s?1}\text{A}{}_{\mathrm{r}}\text{(w}{}^{\mathrm{s}}\text{)}\text{P}{}_{\mathrm{s}}{}^{\mathrm{r}}\text{{T}{}_{\mathrm{r}}{}^{\mathrm{s}}\text{(w)}}\text{∵}\text{0}\text{r?1}\text{A}{}_{\mathrm{s}}\text{(w}{}^{\mathrm{r}}\text{)}$

=(A_r?I_s)T^s_r(I_r?A_s). When r and s are relatively prime this identity implies a wide class of identities of the form PA_n(ω)Q^T=A_r(ω^αs)?A_s(ω^βr). The matrices P^s_r, P^r_s, P, and Q are permutation matrices corresponding to the “data shuffling” required in a computer implementation of the FFT, and T^s_r is a diagonal matrix whose nonzeros are called “twiddle factors.” We establish these identities and discuss their algorithmic significance.     

Zeta matrices of elliptic curves

Let $\text{O =}\text{lim}{}_{\mathrm{n}}\text{Z/p}{}^{\mathrm{n}}\text{Z}$, let $\text{A}\text{= O[g}{}_2\text{, g}{}_3\text{]}{}_{\mathrm{\Delta }}$, where g₂ and g₃ are coefficients of the elliptic curve: Y² = 4X³ ? g₂X ? g₃ over a finite field and Δ = g₂³ ? 27g₃² and let $\text{B}\text{=}\text{A}\text{[X, Y]}\text{(Y}{}^2\text{? 4X}{}^3\text{+ g}{}_2\text{X + g}{}_3\text{)}$. Then the p-adic cohomology theory will be applied to compute explicitly the zeta matrices of the elliptic curves, induced by the pth power map on the free $\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q}$-module $\text{H}{}^1\text{(X,}\text{A}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$. Main results are; Theorem 1.1: X²dY and YdX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{, Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$; Theorem 1.2: YdX, X²dY, Y^?1dX, Y^?2dX and XY^?2dX are basis elements for $\text{H}{}^1\text{(}\text{X}\text{? (Y = 0), Γ}{}_{\mathrm{<mtext>A</mtext>}}{}^1\text{(}\text{X}\text{)}{}^2\text{?}{}_{\mathrm{Z}}\text{Q)}$, where $\text{X}$ is a lifting of X, and all the necessary recursive formulas for this explicit computation are given.     

Deterministic version of Wigner's semicircle law for the distribution of matrix eigenvalues

It is proved that Wigner's semicircle law for the distribution of eigenvalues of random matrices, which is important in the statistical theory of energy levels of heavy nuclei, possesses the following completely deterministic version. Let A_n=(a_ij), 1?i, ?n, be the nth section of an infinite Hermitian matrix, {λ⁽ⁿ⁾}_1?k?n its eigenvalues, and {u_k⁽ⁿ⁾}_1?k?n the corresponding (orthonormalized column) eigenvectors. Let $\text{v}{}^1{}_{\mathrm{n}}\text{=(a}{}_{\mathrm{n1}}\text{,a}{}_{\mathrm{n2}}\text{,?,a}{}_{\mathrm{n,n?1}}\text{)}$, put

$\text{X}{}_{\mathrm{n}}\text{(t)=[n(n-1)]}{}^{\mathrm{<mtext>-1</mtext><mtext>2</mtext>}}\text{∑}\text{k=1}\text{[(n-1)t]}\text{|v}{}_{\mathrm{n}}{}^1\text{u}{}_{\mathrm{f}}{}^{\mathrm{(n-1)}}\text{|}{}^2\text{,}\text{0?t?1}$

(bookeeping function for the length of the projections of the new row v¹_n of A_n onto the eigenvectors of the preceding matrix A_n?1), and let finally

$\text{F}{}_{\mathrm{n}}\text{(x)=n}{}^{-1}\text{(}\text{number of}\text{λ}{}_{\mathrm{k}}{}^{\mathrm{(n)}}\text{?x}\text{n}\text{,1?k?n)}$

(empirical distribution function of the eigenvalues of $\text{A}{}_{\mathrm{n}}\text{n}$. Suppose (i) $\text{lim}{}_{\mathrm{n}}\text{a}{}_{\mathrm{nn}}\text{n}\text{=0}$, (ii) lim_nX_n(t)=Ct(0<C<∞,0?t?1). Then

$\text{F}{}_{\mathrm{n}}\text{?W(·,C)}\text{(n→∞)}$

,where W is absolutely continuous with (semicircle) density

$\text{w(x,C)=}\text{(2Cπ)}{}^{-1}\text{(4C-x}{}^2{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{for}\text{|x|?2}\text{C}\text{0}\text{for}\text{|x|?2}\text{C}$