scholarly journals Representation of positive semidefinite elements as sum of squares in 2-dimensional local rings

2022 ◽  
Vol 116 (2) ◽  
Author(s):  
José F. Fernando
Keyword(s):  
Residue Field ◽  
Classical Problem ◽  
Positive Semidefinite ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Local Rings ◽  
Local Case ◽  
Real Geometry ◽  
Excellent Ring ◽  
The One

AbstractA classical problem in real geometry concerns the representation of positive semidefinite elements of a ring A as sums of squares of elements of A. If A is an excellent ring of dimension $$\ge 3$$ ≥ 3 , it is already known that it contains positive semidefinite elements that cannot be represented as sums of squares in A. The one dimensional local case has been afforded by Scheiderer (mainly when its residue field is real closed). In this work we focus on the 2-dimensional case and determine (under some mild conditions) which local excellent henselian rings A of embedding dimension 3 have the property that every positive semidefinite element of A is a sum of squares of elements of A.

scholarly journals On the Two-phase Fractional Stefan Problem

2020 ◽  
Vol 20 (2) ◽  
pp. 437-458 ◽  
Author(s):  
Félix del Teso ◽  
Jørgen Endal ◽  
Juan Luis Vázquez
Keyword(s):  
Free Boundary ◽  
Stefan Problem ◽  
Free Boundary Problems ◽  
Classical Problem ◽  
Nonlocal Diffusion ◽  
Two Phase ◽  
Boundary Problems ◽  
Numerical Studies ◽  
The One ◽  
Evolution Type

AbstractThe classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

scholarly journals Characterizations of regular local rings via syzygy modules of the residue field

2018 ◽  
Vol 10 (3) ◽  
pp. 327-337
Author(s):  
Dipankar Ghosh ◽  
Anjan Gupta ◽  
Tony J. Puthenpurakal
Keyword(s):  
Residue Field ◽  
Local Rings ◽  
Syzygy Modules ◽  
Regular Local Rings

scholarly journals VANISHING OF (CO)HOMOLOGY OVER DEFORMATIONS OF COHEN-MACAULAY LOCAL RINGS OF MINIMAL MULTIPLICITY

2018 ◽  
Vol 61 (03) ◽  
pp. 705-725
Author(s):  
DIPANKAR GHOSH ◽  
TONY J. PUTHENPURAKAL
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Regular Sequence ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Syzygy Modules ◽  
Tor Modules ◽  
Necessary And Sufficient ◽  
Minimal Multiplicity

AbstractLet R be a d-dimensional Cohen–Macaulay (CM) local ring of minimal multiplicity. Set S := R/(f), where f := f1,. . .,fc is an R-regular sequence. Suppose M and N are maximal CM S-modules. It is shown that if ExtSi(M, N) = 0 for some (d + c + 1) consecutive values of i ⩾ 2, then ExtSi(M, N) = 0 for all i ⩾ 1. Moreover, if this holds true, then either projdimR(M) or injdimR(N) is finite. In addition, a counterpart of this result for Tor-modules is provided. Furthermore, we give a number of necessary and sufficient conditions for a CM local ring of minimal multiplicity to be regular or Gorenstein. These conditions are based on vanishing of certain Exts or Tors involving homomorphic images of syzygy modules of the residue field.

Reduction numbers and Hilbert polynomials of ideals in higher dimensional CohenMacaulay local rings

1992 ◽  
Vol 111 (1) ◽  
pp. 47-56 ◽  
Author(s):  
Yinghwa Wu
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Primary Ideal ◽  
Local Rings ◽  
Hilbert Polynomials ◽  
Higher Dimensional ◽  
Reduction Number ◽  
Minimal Reduction ◽  
System Of Parameters

Throughout, (R, m) will denote a d-dimensional CohenMacaulay (CM for short) local ring having an infinite residue field and I an m-primary ideal in R. Recall that an ideal J I is said to be a reduction of I if Ir+1 = JIr for some r 0, and a reduction J of I is called a minimal reduction of I if J is generated by a system of parameters. The concepts of reduction and minimal reduction were first introduced by Northcott and Rees12. If J is a reduction of I, define the reduction number of I with respect to J, denoted by rj(I), to be min {r 0 Ir+1 = JIr}. The reduction number of I is defined as r(I) = min {rj(I)J is a minimal reduction of I}. The reduction number r(I) is said to be independent if r(I) = rj(I) for every minimal reduction J of I.

scholarly journals Linearity defect of the residue field of short local rings

2016 ◽  
Vol 16 (09) ◽  
pp. 1750163
Author(s):  
Rasoul Ahangari Maleki
Keyword(s):  
Local Ring ◽  
Maximal Ideal ◽  
Residue Field ◽  
Positive Answer ◽  
Local Rings ◽  
Finitely Generated ◽  
Ideal Formula ◽  
Noetherian Local Ring ◽  
Linear Resolution

Let [Formula: see text] be a Noetherian local ring with maximal ideal [Formula: see text] and residue field [Formula: see text]. The linearity defect of a finitely generated [Formula: see text]-module [Formula: see text], which is denoted [Formula: see text], is a numerical measure of how far [Formula: see text] is from having linear resolution. We study the linearity defect of the residue field. We give a positive answer to the question raised by Herzog and Iyengar of whether [Formula: see text] implies [Formula: see text], in the case when [Formula: see text].

Sum of Squares Rings

1977 ◽  
Vol 29 (1) ◽  
pp. 155-160 ◽  
Author(s):  
Benjamin Fine
Keyword(s):  
Number Theory ◽  
Sum Of Squares ◽  
Sums Of Squares ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Quadratic Residues

One of the nicest results in elementary number theory is the following, giving the relation between quadratic residues and sums of squares.

scholarly journals On value groups and residue fields of some valued function fields

1994 ◽  
Vol 37 (3) ◽  
pp. 445-454
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Cyclic Group ◽  
Function Field ◽  
Function Fields ◽  
Residue Field ◽  
Torsion Group ◽  
Transcendence Degree ◽  
Manuscripta Math ◽  
Direct Sum ◽  
Residue Fields ◽  
The One

Let K = K0(x, y) be a function field of transcendence degree one over a field K0 with x, y satisfying y2 = F(x), F(x) being any polynomial over K0. Let υ0 be a valuation of K0 having a residue field k0 and υ be a prolongation of υ to K with residue field k. In the present paper, it is proved that if G0⊆G are the value groups of υ0 and υ, then either G/G0 is a torsion group or there exists an (explicitly constructible) subgroup G1 of G containing G0 with [G1:G0]<∞ together with an element γ of G such that G is the direct sum of G1 and the cyclic group ℤγ. As regards the residue fields, a method of explicitly determining k has been described in case k/k0 is a non-algebraic extension and char k0≠2. The description leads to an inequality relating the genus of K/K0 with that of k/k0: this inequality is slightly stronger than the one implied by the well-known genus inequality (cf. [Manuscripta Math.65 (1989), 357–376’, [Manuscripta Math.58 (1987), 179–214]).

The Momentum Imbalance Paradox Revisited

2005 ◽  
Vol 35 (10) ◽  
pp. 1928-1939 ◽  
Author(s):  
Doron Nof
Keyword(s):  
Momentum Flux ◽  
Classical Problem ◽  
Relative Vorticity ◽  
Rossby Number ◽  
Loop Current ◽  
Coriolis Parameter ◽  
Southern Boundary ◽  
Steady Current ◽  
The One ◽  
The Right

Abstract The classical problem of a point source situated along a southern boundary emptying buoyant water into a (β plane) ocean is revisited. Pichevin and Nof (PN) have shown that, in contrast to the view prevailing at the time, such an inviscid outflow does not simply turn to the right. Rather, it bifurcates into two branches: a steady branch that does turn to the right (eastward) and an unsteady branch that periodically sheds eddies to the left (westward). This partition is because a simple turn to the right of the entire outflow leaves the outflow’s long-shore momentum flux unbalanced, creating a paradox. In contrast, the branching allows the westward-drifting eddies (westward branch) to balance the momentum flux of the steady current (eastward branch). Although the analytical PN solution is useful and informative, it is cumbersome and difficult to apply to actual outflows. Here, a considerably simpler nonlinear analytical solution is presented. Using the idea that the eddies grow slowly relative to their rotation rate, it is shown that an intense (i.e., large Rossby number) and large relative vorticity outflow dumps most of its mass flux (Q) into the eddies (66%). (The remaining 33% goes into the eastward long-shore current.) By contrast, a weak outflow (i.e., an outflow with weak anticyclonic vorticity −αf, where α is analogous to the Rossby number and is much smaller than unity and f is the Coriolis parameter) dumps most of its water into the downstream current [(1–2α)Q]. Unexpectedly, this partition of mass turns out to be the same as the one taking place on an f plane. (Note that this is not at all the case for southward outflow nor is it the case for either eastward or westward outflow, where β alters the balance drastically.) Although the above partition of mass is independent of β, the size of the eddies generated by the above process is a function of β. It is given by [768g′Q/βπf 2α(2 − α)(1 + 2α)]1/5, where g′ is the reduced gravity. This gives a reasonable estimate for the Loop Current eddies’ size and generation frequency. Numerical simulations are in agreement with the above nonlinear solution, though the agreement is not necessarily any better than that of PN.

scholarly journals Sally’s question and a conjecture of shimoda

2013 ◽  
Vol 211 ◽  
pp. 137-161 ◽  
Author(s):  
Shiro Goto ◽  
Liam O’carroll ◽  
Francesc Planas-Vilanova
Keyword(s):  
Residue Field ◽  
Complete Intersections ◽  
Prime Ideals ◽  
Local Rings ◽  
Geometrical Approach ◽  
Graded Algebras ◽  
Definitive Answer ◽  
Number Of Generators ◽  
Excessive Number ◽  
Standing Question

AbstractIn 2007, Shimoda, in connection with a long-standing question of Sally, asked whether a Noetherian local ring, such that all its prime ideals different from the maximal ideal are complete intersections, has Krull dimension at most 2. In this paper, having reduced the conjecture to the case of dimension 3, if the ring is regular and local of dimension 3, we explicitly describe a family of prime ideals of height 2 minimally generated by three elements. Weakening the hypothesis of regularity, we find that, to achieve the same end, we need to add extra hypotheses, such as completeness, infiniteness of the residue field, and the multiplicity of the ring being at most 3. In the second part of the paper, we turn our attention to the category of standard graded algebras. A geometrical approach via a double use of a Bertini theorem, together with a result of Simis, Ulrich, and Vasconcelos, allows us to obtain a definitive answer in this setting. Finally, by adapting work of Miller on prime Bourbaki ideals in local rings, we detail some more technical results concerning the existence in standard graded algebras of homogeneous prime ideals with an (as it were) excessive number of generators.

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