Nonvanishing of Hecke <Emphasis Type=Italic>L</Emphasis>-functions and the Bloch–Kato conjecture

In this paper we study the central values of L-functions associated to a large class of algebraic Hecke characters of imaginary quadratic fields. When these central values are nonzero, the Bloch–Kato conjecture predicts an exact formula for the algebraic parts of the central values in terms of periods and arithmetic data, most notably the Selmer groups corresponding to the Hecke characters. We investigate the nonvanishing of these central values, and prove the p-part of the Bloch–Kato conjecture in these cases for primes p which split in K.     

On Li’s coefficients for the Rankin–Selberg <Emphasis Type="Italic">L</Emphasis>-functions

We define a generalized Li coefficient for the L-functions attached to the Rankin–Selberg convolution of two cuspidal unitary automorphic representations π and π ^′ of GL_m(\mathbbA_F)GL_{m}(\mathbb{A}_{F}) and GL_m^￠(\mathbbA_F)GL_{m^{\prime }}(\mathbb{A}_{F}) . Using the explicit formula, we obtain an arithmetic representation of the n th Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) attached to L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) . Then, we deduce a full asymptotic expansion of the archimedean contribution to l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of L(s,p_f×[(p)\tilde]_f^￠)L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime }) , the nth Li coefficient l_p,p^￠(n)\lambda _{\pi ,\pi ^{\prime }}(n) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μ _π (v,j) of π. Namely, we prove that under GRH for L(s,p_f×[(p)\tilde]_f)L(s,\pi _{f}\times \widetilde{\pi}_{f}) one has |Rem_p(v,j)| ￡ \frac14|\mathop {\mathrm {Re}}\mu_{\pi}(v,j)|\leq \frac{1}{4} for all archimedean places v at which π is unramified and all j=1,…,m.     

Modular-type relations associated to the Rankin–Selberg <Emphasis Type="Italic">L</Emphasis>-function

Hafner and Stopple proved a conjecture of Zagier relating to the asymptotic behaviour of the inverse Mellin transform of the symmetric square L-function associated with the Ramanujan tau function. In this paper, we prove a similar result for any cusp form over the full modular group.     

Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

Strengthening of the Kneser theorem on zeros of solutions of the equation <Emphasis Type=Italic>u</Emphasis>″ + <Emphasis Type=Italic>q</Emphasis>(<Emphasis Type=Italic>t</Emphasis>)<Emphasis Type=Italic>u</Emphasis> = 0 using one functional equation

We present conditions under which a linear homogeneous second-order equation is nonoscillatory on a semiaxis and conditions under which its solutions have infinitely many zeros.     

On the basis property of eigenelements of ordinary linear differential operators in a space of the type <Emphasis Type=Italic>L</Emphasis><Subscript><Emphasis Type=Italic>p</Emphasis></Subscript>, <Emphasis Type=Italic>p</Emphasis> ≥ 2

For a linear differential expression with matrix coefficients in the class L _p , p ≥ 2, and with a parameter λ, we consider a boundary value problem with boundary conditions at the endpoints of the interval [a, b]. Under the condition that the problem is regular, we obtain a formula for the Fourier series expansion of an arbitrary vector function of the class L _p in the root functions of the problem.     

On the asymptotic optimality of the <Emphasis Type=Italic>cμ</Emphasis>/<Emphasis Type=Italic>θ</Emphasis> rule under ergodic cost

We consider an overloaded multi-server multi-class queueing model where customers may abandon while waiting to be served. For class i, service is provided at rate μ _i , and abandonment occurs at rate θ _i . In a many-server fluid regime, we show that prioritizing the classes in decreasing order of c _i μ _i /θ _i asymptotically minimizes an ergodic holding cost, where c _i denotes the equivalent holding cost per unit time for class i.     

Sur la rigidité des groupes de triangles (<Emphasis Type=Italic>r</Emphasis>, <Emphasis Type=Italic>p</Emphasis>, <Emphasis Type=Italic>q</Emphasis>)

We show the length spectral rigidity for the family of fuchsian triangular groups.     

Calibration relations for <Emphasis Type=Italic>B</Emphasis><Subscript><Emphasis Type=Italic>φ</Emphasis></Subscript>-splines of fourth order

We study necessary and sufficient conditions for the smoothness of (not necessarily polynomial) splines of fourth order. We establish the uniqueness of a space of splines of maximal smoothness and prove embeddings on refined grids. We also obtain the corresponding calibration relations. Bibliography: 12 titles.     

<Emphasis Type=Italic>Eω</Emphasis>-Clifford congruences and <Emphasis Type=Italic>Eω</Emphasis>-<Emphasis Type=Italic>E</Emphasis>-reflexive congruences on an inverse semigroup

This paper enlarges the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower T and lower K. Two classes of inverse semigroups, namely Eω-Clifford semigroups and Eω-E-reflexive semigroups, are studied. ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are found to be the least Eω-Clifford and Eω-E-reflexive congruences on S, respectively. Characterizations of the congruences ((ω _T ) _K ) _T and (((ω _T ) _K ) _T ) _K are developed. The lattices of all Eω-Clifford and Eω-E-reflexive congruences are also studied.     

Generating <Emphasis Type=Italic>Sym</Emphasis>(<Emphasis Type=Italic>ω</Emphasis>) by Permutations Preserving Dense Orders

We show how generic permutations of ω can be presented by products of permutations preserving dense orders.     

On the identification <Emphasis Type=Italic>H</Emphasis> = <Emphasis Type=Italic>H</Emphasis>′ in the Lions theorem and a related inaccuracy

We show that the common identification of H with its dual space in the famous functional frame V ⊂ H = H′ ⊂ V′ is incompatible with the distributional frame for some standard pdes and we recall an error related to this unnecessary identification which is repeated from years.