A SURJECTIVE HOMOMORPHISM OF LOCAL COHOMOLOGY MODULES

Let R be a local ring and M a finitely generated generalized Cohen-Macaulay R-module such that dim _R M = dim _R M/αM + height_Mα a for all ideals α of R. Suppose that H_I ^j(M) ≠ 0 for an ideal I of R and an integer j > height_M I. We show that there exists an ideal J ? such that a. height_M J = j;

b. the natural homomorphism H_I ^j(M) → H_I ^j(M) is an isomorphism, for all i > j; and,

c. the natural homomorphism H_I ^j(M) → H_I ^j(M) is surjective.

By using this theorem, we obtain some results about Betti numbers, coassociated primes, and support of local cohomology modules.     

A difference scheme with anti-diffusion terms to improve the computation of contact discontinuities and the study of weak discontinuities in compressible flow

The investigation of Mach reflection formed after the impingement of a weak plane shock wave on a wedge with shock Mach number M_s near 1, is still an open problem[12]. It's difficult for shock tube experiments with interferometer to detect contact discontinuities if it is too weak; also difficult to catch with due accuracy the transition condition between Mach reflection and regular reflection. The interest to this phenomenon is continuing, especially for weak shocks, because there was systematic discrepancy between simplified three shock theory of von Neumann [8] and shock tube results [15] which was named by G. Birkhoff as “von Neumann Paradox on three shock theory” [18].In 1972, K.O.Friedrichs called for more computational efforts on this problem. Recently it is known that for weak impinging shocks it's still difficult to get contact discontinuities and curved Mach stem with satisfactory accuracy. Recent numerical computation sometimes even fails to show reflected shock wave[6]. These explain why von Neumann paradox of the three shock theory in case of weak discontinuities is still a problem of interesting [9,12,14]. In this paper, on one hand, we investigate the numerical methods for Euler's equation for compressible inviscid flow, aiming at improving the computation of contact discontinuities, on the other hand, a methodology is suggested to correctly plot flow data from the massive information in storage. On this basis, all the reflected shock wave , contact discontinuities and the curved Mach stem are determined. We get Mach reflection under the condition when over-simplified shock theory predicts no such configuration[5].     

Expanding the additive reduct of a model of Peano arithmetic

Let M be a model of first order Peano arithmetic ( PA ) and I an initial segment of M that is closed under multiplication. LetM₀ be the {0, 1,+}‐reduct ofM. We show that there is another model N of PA that is also an expansion of M₀ such that a · ^M a = a · ^N a if and only if a ∈ I for all a ∈ M.     

Aerodynamic Sound Emission as a Singular Perturbation Problem

Sound emission from an eddy region involves three length scales: the eddy size I, wavelength λ of the sound, and a dimension L ofthe region. They are related by the Mach number M = l/λ, small for nearly incompressible eddies, and a parameter Λ = L/λ which plays no apparent role in current theories of aerodynamic sound. The theories of Lighthill and Ribner are examined in the case M ? 1, Λ ? 1. Ribner's result is found to contain an unacceptable improper integral. The utility of Lighthill's solution is found to depend on properties of the quadrupole moment T_ij that can be established only by studying the flow in more detail than Lighthill's theory allows. The general problem is posed in the form: given the body force f and vorticity ω find the density ρ and potential φ of the velocity u = ? × ψ{ ω } + ?φ The problem is solved for M ? 1, Λ ? 1 by matching a compressible eddy core scaled on I to a surrounding acoustic field scaled on λ. Lighthill's solution for ρ is shown to be adequate in both regions if T_ij is approximated by ρ₀υ_iυ_j, with v = ? × ψ. The situation M ? 1, Λ ? 1 is studied, and the conclusion is reached that sound emission from large bodies of turbulence is an open problem, Lighthill's theory notwithstanding.     

Brownian motion in a wedge with oblique reflection

This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties:

• (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion.
• (ii) The process reflects instantaneously at the boundary of the wedge, the angle of reflection being constant along each side.
• (iii) The amount of time that the process spends at the comer of the wedge is zero (i.e., the set of times for which the process is at the comer has Lebesgue measure zero).

Hereafter, let ξ be the angle of the wedge (0 < ξ < 2π), let θ₁ and θ₂ be the angles of reflection on the two sides of the wedge, measured from the inward normals, the positive angles being toward the corner (-½π < θ₁, θ₂ ½π), and set α = (θ₁ + θ₂)/ξ. The question of existence and uniqueness is recast as a submartingale problem in the style used by Stroock and Varadhan (Diffusion processes with boundary conditions, Comm. Pure Appl. Math. 24, 1971, pp. 147-225), for diffusions on smooth domains with smooth boundary conditions. It is shown that no solution exists if α ≧ 2. In this case, there is a unique continuous strong Markov process satisfying (i)-(ii) above; it reaches the corner of the wedge almost surely and it remains there. If α < 2, however, then there is a unique continuous strong Markov process statisfying (i)-(iii). It is shown that starting away from the corner this process does not reach the corner of the wedge if α ≦ 0, and does reach the corner if 0 < α < 2. The general theory of multi-dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner. For some values of α, the process arises from diffusion approximations to storage systems and queueing networks. (i) The state space is an infinite two-dimensional wedge, and the process behaves in the interior of the wedge like an ordinary Brownian motion. (ii) The process reflects instantaneously at the boundary of the wedge, and the angle of reflection being constant along each side. (iii) The amount of time that the process spends at the corner of the wedge is zero (i.e., the set of times for which the process is at the corner has Lebesgue measure zero).     

The associated primes of local cohomology modules over rings of small dimension

Let R be a commutative Noetherian local ring of dimension d, I an ideal of R, and M a finitely generated R-module. We prove that the set of associated primes of the local cohomology module H ⁱ _I (M) is finite for all i≥ 0 in the following cases: (1) d≤ 3; (2) d= 4 and $R$ is regular on the punctured spectrum; (3) d= 5, R is an unramified regular local ring, and M is torsion-free. In addition, if $d>0$ then H ^{d − 1} _I (M) has finite support for arbitrary R, I, and M. Received: 31 October 2000 / Revised version: 8 January 2001     

Potential theory for regular and mach reflection of a shock at a wedge

If a plane shock hits a wedge, a self-similar pattern of reflected shocks travels outward as the shock moves forward in time. The nature of the pattern is explored for weak incident shocks (strength b) and small wedge angles 2θ_w through potential theory, a number of different scalings, some study of mixed equations and matching asymptotics for the different scalings. The self-similar equations are of mixed type. A linearization gives a linear mixed flow valid away from a sonic curve. Near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used. The parameter β = c₁θ²_w(γ + 1)b ranges from 0 to ∞. Here γ is the polytropic constant and C₁ is the sound speed behind the incident shock. For β > 2 regular reflection (weak or strong) can occur and the whole pattern is reconstructed to lowest order in shock strength. For β < 1/2 Mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle (with an open elliptic problem) and matched. The case β = 0 can be solved. For 1/2 < β < 2 or even larger β the flow behind a Mach reflection may be transonic and further investigation must be made to determine what happens. The basic pattern of reflection is an almost semi-circular shock issuing, for regular reflection, from the reflection point on the wedge and for Mach reflection, matched with a local interaction flow. Assuming their nature, choosing the least entropy generation, the weak regular reflection will occur for β sufficiently large (von Neumann paradox). An accumulation point of vorticity occurs on the wedge above the leading point. © 1994 John Wiley & Sons, Inc.     

Quotient finite dimensional modules with acc on subdirectly irreducible submodules are noetherian

A right R-module M is (Goldie) finite dimensional (= f.d.) if M contains no infinite direct sums of submodules.M is quotient f.d. (= q.f.d.) if M/K is f.d. for all submodules K.A submodule I of M is subdirectly irreducible (= SDI) if V is the intersection of all submodules S _α of M that properly contain I, then V ≠ I, equivalentlyM/I has simple essential socle V/I. A theorem of Shock [74] states that a q.f.d. right module M is Noether-ian iff every proper submodule of M is contained in a maximal submodule. Camillo [77], proved a companion theorem: M is q.f.d. iff every submodule A ≠ 0 contains a finitely generated (= f.g) submodule S such that A/S has no maximal submodules. Using these two results, and an idea of Camillo [75], we prove the theorem stated in the title.     

Noetherian rings with almost injective simple modules

We characterize right Noetherian rings over which all simple modules are almost injective. It is proved that R is such a ring, if and only if, the complements of semisimple submodules of every R-module M are direct summands of M, if and only if, R is a finite direct sum of right ideals I_r, where I_r is either a Noetherian V-module with zero socle, or a simple module, or an injective module of length 2. A commutative Noetherian ring for which all simple modules are almost injective is precisely a finite direct product of rings R_i, where R_i is either a field or a quasi-Frobenius ring of length 2. We show that for commutative rings whose all simple modules are almost injective, the properties of Kasch, (semi)perfect, semilocal, quasi-Frobenius, Artinian, and Noetherian coincide.     

Self-dual hopf algebras

We introduce the notions of self-dual (graded) Hopf algebras and of structurally simple (graded) Hopf algebras. We prove that the self-dual Hopf algebras are structurally simple and provide a construction of self-dual Hopf algebras. Finally, we classify the self-dual quotients of the form T_B (M)/I, where T_B (M) is a path algebra with a graded Hopf algebra structure, and I is a graded admissible Hopf ideal.     

Non‐well‐founded extensions of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbf {V}$\end{document}

This paper describes a new and user‐friendly method for constructing models of non‐well‐founded set theory. Given a sufficiently well‐behaved system θ of non‐well‐founded set‐theoretic equations, we describe how to construct a model M_θ for $\mathsf {ZFC}^-$ in which θ has a non‐degenerate solution. We shall prove that this M_θ is the smallest model for $\mathsf {ZFC}^-$ which contains $\mathbf {V}$ and has a non‐degenerate solution of θ.     

Some <Emphasis Type="Italic">I</Emphasis>-convergent sequence spaces defined by Orlicz functions

In this article we introduce the sequence spaces cI(M),c0I(M),mI(M) and m0I(M) using the Orlicz function M.We study some of the properties like solid,symmetric,sequence algebra,etc and prove some inclusion relations.     

Ideals of analytic deviation one with respect to a Cohen-Macaulay module

Let (A, m) be a Cohen-Macaulay local ring, M a Cohen-Macaulay A-module of dimension d ≥ 1 and I a proper ideal of analytic deviation one with respect to M. In this paper we study the Cohen-Macaulayness of associated graded module of a Cohen-Macaulay module. We show that if I is generically a complete intersection of analytic deviation one and reduction number at most one with respect to M then G _I (M) is Cohen-Macaulay. When analytic spread of I with respect to M equals d we prove a similar result when reduction number of an ideal is atmost two.     

A unified quantum <Emphasis Type="Italic">SO</Emphasis>(3) invariant for rational homology 3-spheres

Given a rational homology 3-sphere M with |H ₁(M,ℤ)|=b and a link L inside M, colored by odd numbers, we construct a unified invariant I _M,L belonging to a modification of the Habiro ring where b is inverted. Our unified invariant dominates the whole set of the SO(3) Witten–Reshetikhin–Turaev invariants of the pair (M,L). If b=1 and L=∅, I _M coincides with Habiro’s invariant of integral homology 3-spheres. For b>1, the unified invariant defined by the third author is determined by I _M . Important applications are the new Ohtsuki series (perturbative expansions of I _M ) dominating quantum SO(3) invariants at roots of unity whose order is not a power of a prime. These series are not known to be determined by the LMO invariant.     

The Dual Hilbert–Samuel Function of a Maximal Cohen-Macaulay Module

Let R be a Cohen–Macaulay local ring with a canonical module ω _R . Let I be an 𝔪-primary ideal of R and M, a maximal Cohen–Macaulay R-module. We call the function n??(Hom _R (M, ω _R /I ⁿ⁺¹ω _R )) the dual Hilbert–Samuel function of M with respect to I . By a result of Theodorescu, this function is of polynomial type. We study its first two normalized coefficients. In particular, we analyze the case when R is Gorenstein.     

Maximal Holonomy of Almost Bieberbach Groups for Heis5

Let Heis _2n+1 be the Heisenberg group of dimension 2n + 1 and M an infra-nilmanifold with Heis _2n+1-geometry. The fundamental group of M contains a cocompact lattice of Heis _2n+1 with index bounded above by a universal constant I _n+1, i.e., I _n+1 is the maximal order of the holonomy groups. We prove that I ₃ = 24. As an application we give an estimate for the volumes of finite volume non-compact complex hyperbolic 3-manifolds.     

Tournament matrices and their generalizations,I.

If M is any complex matrix with rank (M + M ^* + I) = 1, we show that any eigenvalue of M that is not geometrically simple has 1/2 for its real part. This generalizes a recent finding of de Caen and Hoffman: the rank of any n × n tournament matrix is at least n ? 1. We extend several spectral properties of tournament matrices to this and related types of matrices. For example, we characterize the singular real matrices M with 0 diagonal for which rank (M + M^T + I) = 1 and we characterize the vectors that can be in the kernels of such matrices. We show that singular, irreducible n × n tournament matrices exist if and only n? {2,3,4,5} and exhibit many infinite families of such matrices. Connections with signed digraphs are explored and several open problems are presented.     

The cardinality of certain -orthogonal exponentials

The self-affine measures μ_M,D corresponding to the case (i) M=pI₃, D={0,e₁,e₂,e₃} in the space and the case (ii) M=pI₂, D={0,e₁,e₂,e₁+e₂} in the plane are non-spectral, where p>1 is odd, I_n is the n×n identity matrix, and e₁,…,e_n are the standard basis of unit column vectors in . One of the non-spectral problem on μ_M,D is to estimate the number of orthogonal exponentials in L²(μ_M,D) and to find them. In the present paper we show that, in both cases (i) and (ii), there are at most 4 mutually orthogonal exponentials in L²(μ_M,D) each, and the number 4 is the best.