On the independence of Heegner points associated to distinct quadratic imaginary fields

Let E/Q be an elliptic curve with no CM and a fixed modular parametrization and let be Heegner points attached to the rings of integers of distinct quadratic imaginary fields k₁,…,kr. We prove that if the odd parts of the class numbers of k₁,…,kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in .     

Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup

We consider the Laplacian ΔR subject to Robin boundary conditions on the space , where Ω is a smooth, bounded, open subset of RN. It is known that ΔR generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C₀(RN) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from to C₀(RN) which maps an operator core for ΔR into the domain of the generator of the Gaussian semigroup.     

On the independence of Heegner points in the function field case

Let ∞ be a fixed place of a global function field k. Let E be an elliptic curve defined over k which has split multiplicative reduction at ∞ and fix a modular parametrization ΦE:X₀(N)→E. Let be Heegner points associated to the rings of integers of distinct quadratic “imaginary” fields K₁,…,Kr over (k,∞). We prove that if the “prime-to-2p” part of the ideal class numbers of ring of integers of K₁,…,Kr are larger than a constant C=C(E,ΦE) depending only on E and ΦE, then the points P₁,…,Pr are independent in . Moreover, when k is rational, we show that there are infinitely many imaginary quadratic fields for which the prime-to-2p part of the class numbers are larger than C.     

A construction of covers of arithmetic schemes

Let X be a regular arithmetic scheme, i.e. a regular integral separated scheme flat and of finite type over SpecZ. Assume that for all closed irreducible subschemes C⊆X of dimension 1 with normalisation there are given open normal subgroups NC of , which fulfil the following compatibility condition: For all the pre-images of NC₁ and NC₂ in coincide. If the indices of the NC are bounded, then these data uniquely determine an open normal subgroup of π₁(X), whose pre-image in is NC for all C.     

Rank determines semi-stable conductor

Suppose that E₁ and E₂ are elliptic curves over the rational field, , such that for all quadratic fields . We prove that their conductors N(E₁), and N(E₂) are equal up to squares. If for all quadratic fields , then the same conclusion holds, provided the 2-parts of their Tate-Shafarevich groups are finite.     

On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices

Let X_n be n×N containing i.i.d. complex entries and unit variance (sum of variances of real and imaginary parts equals 1), σ>0 constant, and R_n an n×N random matrix independent of X_n. Assume, almost surely, as n→∞, the empirical distribution function (e.d.f.) of the eigenvalues of converges in distribution to a nonrandom probability distribution function (p.d.f.), and the ratio tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation.     

On the index of circular units in the full group of units of a compositum of quadratic fields

For a compositum of quadratic fields , where d₁,…,ds are square-free odd integers and , we study the group C of circular units of k. We construct a basis of C, compute the index of C in the full group of units of k and derive a lower bound for the divisibility of this index by a power of 2. These results give a lower bound for the divisibility of the class number of the maximal real subfield of k by a power of 2.     

No eigenvalues outside the support of the limiting empirical spectral distribution of a separable covariance matrix

We consider a class of matrices of the form , where X_n is an n×N matrix consisting of i.i.d. standardized complex entries, is a nonnegative definite square root of the nonnegative definite Hermitian matrix A_n, and B_n is diagonal with nonnegative diagonal entries. Under the assumption that the distributions of the eigenvalues of A_n and B_n converge to proper probability distributions as , the empirical spectral distribution of C_n converges a.s. to a non-random limit. We show that, under appropriate conditions on the eigenvalues of A_n and B_n, with probability 1, there will be no eigenvalues in any closed interval outside the support of the limiting distribution, for sufficiently large n. The problem is motivated by applications in spatio-temporal statistics and wireless communications.     

Area and the inradius of lemniscates

We study geometric properties of filled lemniscates of a complex polynomial p(z) of degree n. In particular, we answer a question raised by H.H. Cuenya and F.E. Levis (2007) by showing that there is a constant C(n) such that for every lemniscate E(p,c). Here μ(E(p,c)) and r(E(p,c)) denote the area and the inradius of E(p,c).     

Strong continuity of generalized Feynman-Kac semigroups: Necessary and sufficient conditions

Let (E,D(E)) be a strongly local, quasi-regular symmetric Dirichlet form on L²(E;m) and ((Xt)_t?0,(Px)_x∈E) the diffusion process associated with (E,D(E)). For u∈De(E), u has a quasi-continuous version and has Fukushima's decomposition: , where is the martingale part and is the zero energy part. In this paper, we study the strong continuity of the generalized Feynman-Kac semigroup defined by , t?0. Two necessary and sufficient conditions for to be strongly continuous are obtained by considering the quadratic form (Qu,Db(E)), where Qu(f,f):=E(f,f)+E(u,f²) for f∈Db(E), and the energy measure μ〈u〉 of u, respectively. An example is also given to show that is strongly continuous when μ〈u〉 is not a measure of the Kato class but of the Hardy class with the constant (cf. Definition 4.5).     

Complete solution to a problem on the maximal energy of unicyclic bipartite graphs

The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by C_n the cycle, and the unicyclic graph obtained by connecting a vertex of C₆ with a leaf of P_n-6. Caporossi et al. conjectured that the unicyclic graph with maximal energy is for n=8,12,14 and n≥16. In Hou et al. (2002) [Y. Hou, I. Gutman, C. Woo, Unicyclic graphs with maximal energy, Linear Algebra Appl. 356 (2002) 27-36], the authors proved that is maximal within the class of the unicyclic bipartite n-vertex graphs differing from C_n. And they also claimed that the energies of C_n and is quasi-order incomparable and left this as an open problem. In this paper, by utilizing the Coulson integral formula and some knowledge of real analysis, especially by employing certain combinatorial techniques, we show that the energy of is greater than that of C_n for n=8,12,14 and n≥16, which completely solves this open problem and partially solves the above conjecture.     

Hecke curves and Hitchin discriminant

Let C be a smooth projective curve of genus g?4 over the complex numbers and be the moduli space of stable vector bundles of rank r with a fixed determinant of degree d. In the projectivized cotangent space at a general point E of , there exists a distinguished hypersurface SE consisting of cotangent vectors with singular spectral curves. In the projectivized tangent space at E, there exists a distinguished subvariety CE consisting of vectors tangent to Hecke curves in through E. Our main result establishes that the hypersurface SE and the variety CE are dual to each other. As an application of this duality relation, we prove that any surjective morphism , where C^′ is another curve of genus g, is biregular. This confirms, for , the general expectation that a Fano variety of Picard number 1, excepting the projective space, has no non-trivial self-morphism and that morphisms between Fano varieties of Picard number 1 are rare. The duality relation also gives simple proofs of the non-abelian Torelli theorem and the result of Kouvidakis-Pantev on the automorphisms of .     

Strong convergence of the composite Halpern iteration

Let C be a closed convex subset of a uniformly smooth Banach space E and let T:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈C, the initial guess x₀∈C is chosen arbitrarily and given sequences , and in (0,1), the following conditions are satisfied:

(i)

;

(ii)

αn→0, βn→0 and 0<a?γn, for some a∈(0,1);

(iii)

, and . Let be a composite iteration process defined by

     

Universality in complex Wishart ensembles for general covariance matrices with 2 distinct eigenvalues

We considered N×N Wishart ensembles in the class W_C(Σ_N,M) (complex Wishart matrices with M degrees of freedom and covariance matrix Σ_N) such that N₀ eigenvalues of Σ_N are 1 and N₁=N−N₀ of them are a. We studied the limit as M, N, N₀ and N₁ all go to infinity such that , and 0<c,β<1. In this case, the limiting eigenvalue density can either be supported on 1 or 2 disjoint intervals in R₊, and a phase transition occurs when the support changes from 1 interval to 2 intervals. By using the Riemann-Hilbert analysis, we have shown that when the phase transition occurs, the eigenvalue distribution is described by the Pearcey kernel near the critical point where the support splits.     

An explicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of an infinite family of nonexpansive mappings

Let H be a Hilbert space and C be a nonempty closed convex subset of H, {T_i}_i∈N be a family of nonexpansive mappings from C into H, G_i:C×C→R be a finite family of equilibrium functions (i∈{1,2,…,K}), A be a strongly positive bounded linear operator with a coefficient and -Lipschitzian, relaxed (μ,ν)-cocoercive map of C into H. Moreover, let , {α_n} satisfy appropriate conditions and ; we introduce an explicit scheme which defines a suitable sequence as follows:

     

The case of equality in the Dobrushin-Deutsch-Zenger bound

Suppose that A=(a_i,j) is an n×n real matrix with constant row sums μ. Then the Dobrushin-Deutsch-Zenger (DDZ) bound on the eigenvalues of A other than μ is given by . When A a transition matrix of a finite homogeneous Markov chain so that μ=1,Z(A) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, , of the iteration , to the stationary distribution vector of the chain.In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which |λ|=Z(A) for some stochastic matrix A.     

On real-analytic recurrence relations for cardinal exponential B-splines

Let L_N+1 be a linear differential operator of order N+1 with constant coefficients and real eigenvalues λ₁,…,λ_N+1, let E(Λ_N+1) be the space of all C^∞-solutions of L_N+1 on the real line. We show that for N2 and n=2,…,N, there is a recurrence relation from suitable subspaces to involving real-analytic functions, and with if and only if contiguous eigenvalues are equally spaced.