Tensor products of Drinfeld modules andv-adic representations

We define the tensor product ϕ ⊗ ψ and relatedt-modules Sym²(ϕ), and ∧²(ϕ) for Drinfeld modules ϕ, ψ defined over the rational function fieldK=F _q (T), and describe thev-adic Tate modules of theset-modules by using those of ϕ, ψ.     

A simplification of the Kovarik formula

Let P,Q be two idempotents on a Hilbert space. Z.V. Kovarik (Z.V. Kovarik, Similarity and interpolation between projectors, Acta Sci. Math. (Szeged) 39 (1977) 341-351) showed that when P+Q−I is invertible, the formula K(P,Q)=P⁻²(P+Q−I)Q gives the only idempotent such that R(K)=R(P), N(K)=N(Q), where N(T) and R(T) denote the nullspace and the range of a bounded linear operator T on a Hilbert space, respectively. This formula was later extended to the context of Banach algebras and used in 1983 by J. Esterle to show that two homotopic idempotents may always be connected by a polynomial idempotent valued path. In the present paper, we give a simplification of Kovarik's original formula and one natural generalization of it.     

Two-dimensional Fuchsian systems and the Chebyshev property

Let (x(t),y(t))^? be a solution of a Fuchsian system of order two with three singular points. The vector space of functions of the form P(t)x(t)+Q(t)y(t), where P,Q are real polynomials, has a natural filtration of vector spaces, according to the asymptotic behavior of the functions at infinity. We describe a two-parameter class of Fuchsian systems, for which the corresponding vector spaces obey the Chebyshev property (the maximal number of isolated zeros of each function is less than the dimension of the vector space). Up to now, only a few particular systems were known to possess such a non-oscillation property. It is remarkable that most of these systems are of the type studied in the present paper. We apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic Hamiltonian systems in the plane.     

Existence and comparison theorems for nonlinear diffusion systems

In this paper we prove existence, uniqueness, and regularity results for systems of nonlinear second order parabolic equations with boundary conditions of the Dirichlet, Neumann, and regular oblique derivative types. Let K(t) consist of all functions (v¹(x), v²(x),…, v^m(x)) from $\text{Ω ? R}{}^{\mathrm{n}}$ into R^m which satisfy ψⁱ(x, t) ? vⁱ(x) ? θⁱ(x, t) for all $\text{x ? Ω}\text{and}\text{1 ? i ? m}$, where ψⁱand θⁱ are extended real-valued functions on $\text{}\text{?}\text{gW × [0, T)}$. We find conditions which will ensure that a solution U(x, t) ≡ (u¹(x, t), u²(x, t),…, u^m(x, t)) which satisfies U(x, 0) ?K(0) will also satisfy U(x, t) ?K(t) for all 0 ? t < T. This result, which has some similarity to the Gronwall Inequality, is then used to prove a global existence theorem.     

An algebraic equation for linear operators

Let T be a linear operator on a vector space V, possibly of infinite dimension, over a general field K. We solve the functional equation p(T) = F where p ∈ K[x] and F, an algebraic operator on V, are given. For nilpotent F we give an explicit linear system which determines the solutions by their similarity classes. The method is based on a canonical decomposition theorem.     

Root numbers and ranks in positive characteristic

For a global field K and an elliptic curve E_η over K(T), Silverman's specialization theorem implies rank(E_η(K(T)))?rank(E_t(K)) for all but finitely many t∈P¹(K). If this inequality is strict for all but finitely many t, the elliptic curve E_η is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial.Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K=κ(u) over any finite field κ with characteristic ≠2, we construct an explicit 2-parameter family E_c,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c,d∈κ^×) such that, under the parity conjecture, each E_c,d has elevated rank.     

Homogeneous tri-additive forms and derivations

Gleason [A.M. Gleason, The definition of a quadratic form, Amer. Math. Monthly 73 (1966) 1049-1066] determined all functionals Q on K-vector spaces satisfying the parallelogram law Q(x+y)+Q(x-y)=2Q(x)+2Q(y) and the homogeneity Q(λx)=λ²Q(x). Associated with Q is a unique symmetric bi-additive form S such that Q(x)=S(x,x) and 4S(x,y)=Q(x+y)-Q(x-y). Homogeneity of Q corresponds to that of S: S(λx,λy)=λ²S(x,y). The associated S is not necessarily bi-linear.Let V be a vector space over a field K, char(K)≠2,3. A tri-additive form T on V is a map of V³ into K that is additive in each of its three variables. T is homogeneous of degree 3 if T(λx,λy,λz)=λ³T(x,y,z) for all .We determine the structure of tri-additive forms that are homogeneous of degree 3. One of the keys to this investigation is to find the general solution of the functional equation

F(t)+t³G(1/t)=0,     

Estimation of the Gauss–Markov process from observation of its sign

Let X(t) be the ergodic Gauss–Markov process with mean zero and covariance function e^?|τ|. Let D(t) be +1, 0 or ?1 according as X(t) is positive, zero or negative. We determine the non-linear estimator of X(t₁) based solely on D(t), ?T ? t ? 0, that has minimal mean–squared error ε²(t₁, T). We present formulae for ε²(t₁, T) and compare it numerically for a range of values of t₁ and T with the best linear estimator of X(t₁) based on the same data.     

Non-uniform Turán-type problems

Given positive integers n,k,t, with 2?k?n, and t<2^k, let m(n,k,t) be the minimum size of a family F of (nonempty distinct) subsets of [n] such that every k-subset of [n] contains at least t members of F, and every (k-1)-subset of [n] contains at most t-1 members of F. For fixed k and t, we determine the order of magnitude of m(n,k,t). We also consider related Turán numbers T_?r(n,k,t) and T_r(n,k,t), where T_?r(n,k,t) (T_r(n,k,t)) denotes the minimum size of a family such that every k-subset of [n] contains at least t members of F. We prove that T_?r(n,k,t)=(1+o(1))T_r(n,k,t) for fixed r,k,t with and n→∞.     

Completing an extension of Tunnell's theorem

We look at a special case of a familiar problem: Given a locally compact group G, a subgroup H and a complex representation π₊ of G how does π₊ decompose on restriction to H. Here G is GL⁺(2,F), where F is a nonarchimedian local field of characteristic not two, K a separable quadratic extension of F, GL⁺(2,F) the subgroup of index 2 in GL(2,F) consisting of those matrices whose determinant is in NK/F(K^∗), π₊ is an irreducible, admissible supercuspidal representation of GL⁺(2,F) and H=K^∗ under an embedding of K^∗ into GL(2,F).     

Second order nonlinear multivalued boundary problems in Hilbert spaces

In this paper we consider second order differential inclusions in real Hilbert space, namely p(t)⋅x^″(t)+r(t)⋅x^′(t)∈Ax(t)+F(t,x(t)), a.e. on [0,T], under the nonlinear boundary conditions. Using techniques from multivalued analysis and the theory of operators of monotone type, we prove the existence of solutions for both the ‘convex’ and ‘nonconvex’ problems. Finally, we present a special case of interest, which fit into our framework, illustrating the generality of our results.     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

Boundary representations and rigidity of submodules

This paper considers two problems on the Fock type spaces Fs (0<s?1). Firstly, it is shown that on the space Fs (0<s<1), the identity representation of C^∗(I,T₁,…,Tn) is a boundary representation for the Banach subalgebra B(I,T₁,…,Tn), while on the space F₁, it is not. Secondly, it is shown that all the submodules of F₁ are rigid.     

A fragmentation process connected to Brownian motion

Let (B _s , s≥ 0) be a standard Brownian motion and T ₁ its first passage time at level 1. For every t≥ 0, we consider ladder time set ℒ ^(t) of the Brownian motion with drift t, B ^(t) _s = B _s + ts, and the decreasing sequence F(t) = (F ₁(t), F ₂(t), …) of lengths of the intervals of the random partition of [0, T ₁] induced by ℒ ^(t) . The main result of this work is that (F(t), t≥ 0) is a fragmentation process, in the sense that for 0 ≤t < t′, F(t′) is obtained from F(t) by breaking randomly into pieces each component of F(t) according to a law that only depends on the length of this component, and independently of the others. We identify the fragmentation law with the one that appears in the construction of the standard additive coalescent by Aldous and Pitman [3]. Received: 19 February 1999 / Revised version: 17 September 1999 /?Published online: 31 May 2000     

Determination of the unknown source term in a linear parabolic problem from the measured data at the final time

The problem of determining the source term F(x, t) in the linear parabolic equation u _t = (k(x)u _x (x, t)) _x + F(x, t) from the measured data at the final time u(x, T) = µ(x) is formulated. It is proved that the Fréchet derivative of the cost functional J(F) = ‖µ _T (x) ? u(x, T)‖ ₀ ² can be formulated via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is proved. An existence result for a quasi solution of the considered inverse problem is proved. A monotone iteration scheme is obtained based on the gradient method. Convergence rate is proved.     

On the solution set of an equation of the type f(t, Φ(u(t)) = 0

We investigate the solution set Γ of an equation of the type f(t, Φ(u(t)) = 0, where Φ is a linear homeomorphism from a topological vector space X onto L ¹(T) and f: T×R → R is a Carathéodory function. More precisely, we characterize the property of Γ of intersecting each closed hyperplane of X.     

Densities for 4-class ranks of quadratic function fields

Let F=Fq(T) be a rational function field of odd characteristic, and fix a positive integer t. In this article we study the family of quadratic function fields , where D is a polynomial over Fq of odd degree having t distinct irreducible factors. The 4-class rank r₄(K) is the rank of the 4-torsion of the group of divisor classes of K, and it is known that 0?r₄(K)?t−1. For fixed r we compute the proportion of such fields K satisfying r₄(K)=r, and in particular we determine the behaviour of this value as t→∞. We will need some asymptotic results for these computations, in particular the number of polynomials D as above whose irreducible factors fulfill certain parity and quadratic residue conditions.     

Identification of the unknown diffusion coefficient in a linear parabolic equation by the semigroup approach

In this article, we study the semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k(x) in the linear parabolic equation ut(x,t)=(k(x)uxx(x,t)), with Dirichlet boundary conditions u(0,t)=ψ₀, u(1,t)=ψ₁. Main goal of this study is to investigate the distinguishability of the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] via semigroup theory. In this paper, we show that if the null space of the semigroup T(t) consists of only zero function, then the input-output mappings Φ[⋅] and Ψ[⋅] have the distinguishability property. Moreover, the values k(0) and k(1) of the unknown diffusion coefficient k(x) at x=0 and x=1, respectively, can be determined explicitly by making use of measured output data (boundary observations) f(t):=k(0)ux(0,t) or/and h(t):=k(1)ux(1,t). In addition to these, the values k^′(0) and k^′(1) of the unknown coefficient k(x) at x=0 and x=1, respectively, are also determined via the input data. Furthermore, it is shown that measured output dataf(t) and h(t) can be determined analytically, by an integral representation. Hence the input-output mappings Φ[⋅]:K→C¹[0,T], Ψ[⋅]:K→C¹[0,T] are given explicitly in terms of the semigroup. Finally by using all these results, we construct the local representations of the unknown coefficient k(x) at the end points x=0 and x=1.