We compute the critical L-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral K. In doing so, we prove closed form formulas for some moments of K ′3. Many of our L-values can be expressed in terms of Gamma functions, and this also gives new lattice sum evaluations.

In this paper we prove a version of Deligne’s conjecture for potentially automorphic motives, twisted by certain algebraic Hecke characters. The Hecke characters are chosen in such a way that we can use automorphic methods in the context of totally definite unitary groups.

This article considers the problem of multiple hypothesis testing using t-tests. The observed data are assumed to be independently generated conditional on an underlying and unknown two-state hidden model. We propose an asymptotically valid data-driven procedure to find critical values for rejection regions controlling k-family wise error rate (k-FWER), false discovery rate (FDR) and the tail p...

We describe how to compute topological objects associated to a complex polynomial map of n > 2 variables with isolated singularities. These objects are: the affine critical values, the affine Milnor numbers for all irregular fibers, the critical values at infinity, and the Milnor numbers at infinity for all irregular fibers. Then for a family of polynomials we detect parameters where the topolo...

Studying a critical value function φ in parametric nonlinear programming, we recall conditions guaranteeing that φ is a C function and derive second order Taylor expansion formulas including second-order terms in the form of certain generalized derivatives of Dφ. Several specializations and applications are discussed. These results are understood as supplements to the well–developed theory of f...

Let K be an imaginary quadratic eld and let O be the ring of integers of K. Let E be an elliptic curve deened over Q with complex multiplication by O. Let be the Grr ossencharacter attached to the curve E over K by the theory of complex multiplication, and let L(k ; s) be the complex Hecke L-function attached to the powers of , k = 1; 2; ; here we have xed an embedding of K in C. In a previous ...

We investigate numerically the critical behaviour of a onedimensional non-attractive lattice gas model that is the continuoustime version of the Domany-Kinzel cellular automaton in one of its parameter subspaces. The model shows a phase transition in the directed percolation universality class of critical behaviour, as expected.

Many multivariate statistical methods call upon the assumption of multivariate normality. However, many researchers fail to test this assumption. This omission could be due either to ignorance of the existence of tests of multivariate normality or confusion about which test to use. Although at least 50 tests of multivariate normality exist, relatively little is known about the power of these pr...

We introduce the “sample” technique to generate robust critical values for the Jarque and Bera (JB) Lagrangian Multiplier (LM) test for normality, JBCV( 1 2 , k k ), by using improved critical values the true size of the test approaches its nominal value. Monte Carlo methods are used to study the size, and the power of the JB normality test with the “sample” critical values and compare with thr...

Enders and Granger provide critical values to test the null hypothesis of a unit-root against the alternative of threshold adjustment. However, in obtaining their critical values, Enders and Granger did not use a consistent estimate of the threshold nor did they use a lag-augmented data generating process. This note remedies both of these problems. The power of the test statistics using the con...