Class groups of imaginary function fields: The inert case

Let be a finite field and a transcendental element over . An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers and , there are infinitely many function fields of fixed degree such that the class group of contains a subgroup isomorphic to and the prime at infinity is inert.

     

Wecken property for roots

We show that maps from a compact space into a topological manifold which have geometric Nielsen root number zero satisfy the Wecken property, i.e., such that .

     

Symmetrically approximately continuous functions, consistent density theorems, and Fubini type inequalities

Using the continuum hypothesis, Sierpinski constructed a nonmeasurable function such that is countable for every Clearly, such a function is symmetrically approximately continuous everywhere.

Here we to show that Sierpinski's example cannot be constructed in ZFC. Moreover, we show it is consistent with ZFC that if a function is symmetrically approximately continuous almost everywhere, then it is measurable.

     

Prime Producing Quadratic Polynomials and Class Number One or Two

Let d≡ 5 mod 8 be a positive square-free integer and let h(d) be the class number of the real quadratic field ℚ(√d). Let p be a divisor of d = pq and let . Assume that is prime or equal to 1 for all integers x with 0≤x<W. Under the assumption that the Riemann hypothesis is true, we prove that if , then h(d) < 2. Furthermore we show that h(d)< 2 implies d < 4245. In the case when there exists at least one split prime less than W, we prove the following results without any assumptions on the Riemann hypothesis. If then h< 2 or h = 4. If , then h≤ 2, h = 4 or h = 2^t−2, where t is the number of primes dividing d. In the case when h = 2^t−2 we have , where φ = 2 or 4. 2000 Mathematics Subject Classification: Primary–11R29     

Generalized Zeta Regularizations,Quantum Class Number Formulas,and Appell's <Emphasis Type="Italic">O</Emphasis>-Functions

We introduce a generalized notion of the zeta regularization. As applications we show the quantum analogue of the Lerch formula and of the Dirichlet class number formulas. We also give some further examples related to Appell's O-functions. 2000 Mathematics Subject Classification: Primary—11M36 Work in part supported by Grant-in Aid for Scientific Research (B) No. 11440010, and by Grant-in Aid for Exploratory Research No. 13874004, Japan Society for the Promotion of Science.     

Isomorphism criterion for monomial graphs

Let q be a prime power, ??_q be the field of q elements, and k, m be positive integers. A bipartite graph G = G_q(k, m) is defined as follows. The vertex set of G is a union of two copies P and L of two‐dimensional vector spaces over ??_q, with two vertices (p₁, p₂) ∈ P and [ l₁, l₂] ∈ L being adjacent if and only if p₂ + l₂ = pl. We prove that graphs G_q(k, m) and G_q′(k′, m′) are isomorphic if and only if q = q′ and {gcd (k, q ? 1), gcd (m, q ? 1)} = {gcd (k′, q ? 1),gcd (m′, q ? 1)} as multisets. The proof is based on counting the number of complete bipartite INFgraphs in the graphs. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 322–328, 2005     

Stability of Preconditioned Finite Volume Schemes at Low Mach Numbers

A finite volume method for inviscid unsteady flows at low Mach numbers is studied. The method uses a preconditioning of the dissipation term within the numerical flux function only. It can be observed by numerical experiments that the preconditioned scheme combined with an explicit time integrator is unstable if the time step Δt does not satisfy the requirement to be as the Mach number M tends to zero, whereas the corresponding standard method remains stable up to , M → 0, though producing unphysical results. A comprehensive mathematical substantiation of this numerical phenomenon by means of a von Neumann stability analysis is presented, which reveals that in contrast to the standard approach, the dissipation matrix of the preconditioned numerical flux function possesses an eigenvalue growing like M^–2 as M tends to zero, thus causing the diminishment of the stability region of the explicit scheme. The theoretical results are afterwards confirmed by numerical experiments. AMS subject classification (2000) 35L65, 35C20, 76G25     

Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.     

Method for construction of operators in Fock space

It is shown, by providing a general method for the construction that any Fock space linear operator defined on the dense linear manifold spanned by the particle number representation basis can be represented in terms of the annihilation and creation operators. The normal form of the representation is unique.     

On the Intersection Problem for Steiner Triple Systems of Different Orders

A Steiner triple system of order v, or STS(v), is a pair (V, ) with V a set of v points and a set of 3-subsets of V called blocks or triples, such that every pair of distinct elements of V occurs in exactly one triple. The intersection problem for STS is to determine the possible numbers of blocks common to two Steiner triple systems STS(u), (U, ), and STS(v), (V, ), with U⊆V. The case where U=V was solved by Lindner and Rosa in 1975. Here, we let U⊂V and completely solve this question for v−u=2,4 and for v≥2u−3. supported by NSERC research grant #OGP0170220. supported by NSERC postdoctoral fellowship. supported by NSERC research grant #OGP007621.