We discuss the enumeration of function fields and number fields by discriminant. We show that Malle’s conjectures agree with heuristics arising naturally from geometric computations on Hurwitz schemes. These heuristics also suggest further questions in the number field setting.

The necessary and sufficient condition for stability of K-symmetrizable interval matrix is given. An algorithm for checking stability of K-symmetrizable interval matrix is proposed.

Let Σ denote the class of analytic functions in the punctured unit disc U∗ = {z ∈ C : 0 < |z| < 1} . In this paper, we introduce several new subclasses of meromorphic functions defined by means of the linear operator d(a, c; z). Inclusion properties of these classes and some applications involving integral operator are also considered. 2000 Mathematics Subject Classification: 30C45.

We extend the Eneström-Kakeya theorem and its refinement by Hurwitz to polynomial matrices H(z) with positive semidefinite coefficients. We determine an annular region containing the zeros of detH(z). A stability result for systems of linear difference equations is given as an application.

We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as the Frobenius character formula for counting triples in a finite group, pointing out some common traps and misconceptions associated with them. Alt...

For distinct primes / and p , the Iwasawa invariant XT stabilizes in the cyclotomic Zp -tower over a complex abelian base field. We calculate this stable invariant for p = 3 and various / and K. Our motivation was to search for a formula of Riemann-Hurwitz type for XT that might hold in a p-extension. From our numerical results, it is clear that no formula of such a simple kind can hold. In the...

Between 1968 and 1978 four negative income tax experiments were conducted; they were designed to measure labor supply and earnings. The experiments were not designed to measure the effects of government programs on such demographic behavior as marital dissolution, fertility, family composition, or the decision to marry or remarry. Nevertheless, the data from the experiments have been used to an...

We show that the Hurwitz zeta function, ζ(ν, a), and the Legendre chi function, χν(z), defined by ζ(ν, a) = ∞ ∑ k=0 1 (k + a)ν , 0 < a ≤ 1, Re ν > 1, and χν(z) = ∞ ∑ k=0 z2k+1 (2k + 1)ν , |z| ≤ 1, Re ν > 1 with ν = 2, 3, 4, . . . , respectively, form a discrete Fourier transform pair. Many formulae involving the values of these functions at rational arguments, most of them unknown, are obtained...

Consider the polynomial tr (A + tB)m in t for positive hermitian matrices A and B with m ∈ N. The Bessis-Moussa-Villani conjecture (in the equivalent form of Lieb and Seiringer) states that this polynomial has nonnegative coefficients only. We prove that they are at least asymptotically positive, for the nontrivial case of AB 6= 0. More precisely, we show —once complex-analytically, once combin...

In the design of decentralized networked systems, it is useful to know whether a given network topology can sustain stable dynamics. We consider a basic version of this problem here: given a vector space of sparse real matrices, does it contain a stable (Hurwitz) matrix? Said differently, is a feedback channel (corresponding to a non-zero entry) necessary for stabilization or can it be done wit...