On the tautological ring of ℳ g

We prove that any product of tautological classes of _g of degreed (in the Chow ring of _g with rational coefficients) vanishes ford>g–2 and is proportional to the class of the hyperelliptic locus in degreeg–2.Oblatum 28-I-1195 & 15-IV-1995In memory of Nicolaas H. Kuiper (1920–1994)     

Group of automorphisms of the ring ℤA4

The structure of the group of automorphisms of the integer group ring of the group A₄ is studied in terms of a semidirect product. We show that the Zassenhaus conjecture on the structure of automorphisms of integer group rings of finite groups for the group Aut ?A₄ holds.     

The κ ring of the moduli of curves of compact type

The subalgebra of the tautological ring of the moduli of curves of compact type generated by the κ classes is studied in all genera. Relations, constructed via the virtual geometry of the moduli of stable quotients, are used to obtain minimal sets of generators. Bases and Betti numbers of the κ rings are computed. A universality property relating the higher genus κ rings to the genus 0 rings is proven using the virtual geometry of the moduli space of stable maps. The λ_g-formula for Hodge integrals arises as the simplest consequence.     

Structure of H0R characters the ring R

For a commutative ringR, its related characterizations are given by investigating the structure and properties ofH ₀ R. Furthermore, by virtue ofH ₀ structure, some important characterizations of CPF properties and connected properties onK ₀ R are obtained from related rings.     

The Gröbner ring conjecture in the lexicographic order case

We prove that a valuation domain $\mathbf{V}$ has Krull dimension $\le $ 1 if and only if, for any $n$ , fixing the lexicographic order as monomial order in $\mathbf{V}[X_1,\ldots ,X_n]$ , for every finitely generated ideal $I$ of $\mathbf{V}[X_1,\ldots ,X_n]$ , the ideal generated by the leading terms of the elements of $I$ is also finitely generated. This proves the Gröbner ring conjecture in the lexicographic order case. The proof we give is both simple and constructive. The same result is valid for Prüfer domains. As a “scoop”, contrary to the common idea that Gröbner bases can be computed exclusively on Noetherian ground, we prove that computing Gröbner bases over $\mathbf{R}[X_1,\ldots , X_n]$ , where $\mathbf{R}$ is a Prüfer domain, has nothing to do with Noetherianity, it is only related to the fact that the Krull dimension of $\mathbf{R}$ is $\le $ 1.     

An algebraic construction of the ring of¶piecewise polynomial functions

In this note we will prove an algebraic characterization of the piecewiese polynomial functions of a real ℚ-algebra A. This characterization is related to the realm of investigations concerning the Pierce–Birkhoff conjecture. Received: 23 June 1997 / Revised version: 26 May 1998     

A Scaling Limit of the Becker-Döring Equations in the Regime of Small Excess Density

The Becker-Doring equations serve as a model for the nucleation of a new thermodynamic phase in a first-order phase transformation. This corresponds to the case when the total density of monomers exceeds a critical value and the excess density is contained in larger and larger clusters as time proceeds. It has been derived in Penrose [J. Stat. Phys. 89:1/2 (1997), 305-320] and Niethammer [J. Nonlin. Sci. 13:1 (2003), 115-155] that the evolution of these large clusters can on a certain large time scale be described by a nonlocal transport equation coupled with the constraint that the total volume of new phase is conserved. For specific coefficients this equation is well known as a classical mean-field model for coarsening. In the present paper we consider the regime of small excess density on a large time scale, but not as large as in Penrose (1997) or Niethammer (2003). We show rigorously that the leading order dynamics are governed by another variant of the classical mean-field model in which total mass is preserved.     

Computation of the number representations of the elements of the ringℤ/dℤ as a sum of squares

The number of representations of the elements of the ringℤ/dℤ as a sum of invertible squares is computed, provided that each square occurs in the sum no more tha a fized number of times. For prime d an exhaustive answer is given in term of the class number and the fundamental unit of the real quadratic field . Biblography: 5 titles. Translted fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 5–8.     

Codismantlability and projective dimension of the Stanley–Reisner ring of special hypergraphs

In this paper, we generalize the concept of codismantlable graphs to hypergraphs and show that some special vertex decomposable hypergraphs are codismantlable. Then we generalize the concept of bouquet in graphs to hypergraphs to extend some combinatorial invariants of graphs about disjointness of a set of bouquets. We use these invariants to characterize the projective dimension of Stanley–Reisner ring of special hypergraphs in some sense.     

Branching rules in the ring of superclass functions of unipotent upper-triangular matrices

It is becoming increasingly clear that the supercharacter theory of the finite group of unipotent upper-triangular matrices has a rich combinatorial structure built on set-partitions that is analogous to the partition combinatorics of the classical representation theory of the symmetric group. This paper begins by exploring a connection to the ring of symmetric functions in non-commuting variables that mirrors the symmetric group’s relationship with the ring of symmetric functions. It then also investigates some of the representation theoretic structure constants arising from the restriction, tensor products and superinduction of supercharacters in this context.     

Right Bézout ring with waist is a right Hermite ring

We study noncommutative rings in which the Jacobson radical contains a completely prime ideal. It is proved that a right Bézout ring in which the Jacobson radical contains a completely prime ideal is a right Hermite ring. We describe a new class of Bézout rings that are not elementary divisor rings.     

Invariant ultrametrics and Markov processes on the finite adèle ring of ℚ

This article introduces several rotation and additive invariant ultrametrics on the finite adèle ring A _f of the rational numbers ?. Symmetry, regularity and uniqueness properties of these ultrametrics are provided. With these non-Archimedean metrics at hand it is possible to define a wide class of rotation and additive invariant Markov processes on A _f .     

A “quite superfluous” characterization of the Jacobson radical of an associative ring

It is well-known that the Jacobson radical of a unital ring R is the largest superfluous right ideal of R. While a simple example shows this to be false for non-unital rings, it is shown that the result remains valid in the absence of a unit provided the notion of “superfluous” is taken relative to all regular right ideals. The purpose of this note is to provide a proof for this little-known fact.     

Exponential convergence to equilibrium for subcritical solutions of the Becker–Döring equations

We prove that any subcritical solution to the Becker–Döring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin and Niethammer (2003) [17]. Our approach is based on a careful spectral analysis of the linearized Becker–Döring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted ?¹${?}^1$ spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation.     

Homological determinacy of representations of a group of type P∞ over the ring of p-adic integers

It is proved that homologically equivalent p-adic representations of a group of type p are isomorphic to within imbedding, under which the factor module is homologically trivial.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 164–171, 1983.