scholarly journals The lower semicontinuity of the Frobenius splitting numbers

2010 ◽  
Vol 150 (1) ◽  
pp. 35-46 ◽  
Author(s):  
FLORIAN ENESCU ◽  
YONGWEI YAO
Keyword(s):  
Local Ring ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Prime Characteristic ◽  
Frobenius Splitting ◽  
Mild Conditions ◽  
Splitting Numbers

AbstractWe show that, under mild conditions, the (normalized) Frobenius splitting numbers of a local ring of prime characteristic are lower semicontinuous.

scholarly journals Uniform bounds in F-finite rings and their applications

2017 ◽  
Author(s):  
◽  
Thomas Marion Polstra
Keyword(s):  
Finite Type ◽  
Local Ring ◽  
Lower Semicontinuous ◽  
Multiplicity Function ◽  
Characteristic P ◽  
Finite Rings ◽  
Uniform Bounds ◽  
Frobenius Splitting ◽  
Lower Semi ◽  
Splitting Numbers

This dissertation establishes uniform bounds in characteristic p rings which are either F-finite or essentially of finite type over an excellent local ring. These uniform bounds are then used to show that the Hilbert-Kunz length functions and the normalized Frobenius splitting numbers defined on the spectrum of a ring converge uniformly to their limits, namely the Hilbert-Kunz multiplicity function and the Fsignature function. From this we establish that the F-signature function is lower semicontinuous. Lower semi-continuity of the F-signature of a pair is also established. We also give a new proof of the upper semi-continuity of Hilbert-Kunz multiplicity, a result originally proven by Ilya Smirnov.

Frobenius closure and prime characteristic singularities

2020 ◽  
Author(s):  
◽  
Kyle Logan Maddox
Keyword(s):  
Local Ring ◽  
Local Cohomology ◽  
Sufficient Condition ◽  
Zariski Topology ◽  
Prime Characteristic ◽  
Joint Work ◽  
University Of Missouri ◽  
Mild Conditions ◽  
Flat Maps ◽  
The University

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT REQUEST OF AUTHOR.] This dissertation outlines several results about prime characteristic singularities for which the nilpotent part under the induced Frobenius action on local cohomology is either finite colength or the entire module, collectively referred to here as nilpotent singularities. First, we establish a sufficient condition for the finiteness of the Frobenius test exponent for a local ring and apply it to conclude that nilpotent singularities have finite Frobenius test exponent. In joint work with Jennifer Kenkel, Thomas Polstra, and Austyn Simpson, we show that under mild conditions nilpotent singularities descend and ascend along faithfully flat maps. Consequently, we then prove that the loci of primes which are weakly F-nilpotent and F-nilpotent are open in the Zariski topology for rings which are either F-finite or essentially of fiiite type over an excellent local ring.

scholarly journals Equivalence of viscosity and weak solutions for a p-parabolic equation

2021 ◽  
Author(s):  
Jarkko Siltakoski
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Relationship Of ◽  
The Relationship

AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .

scholarly journals A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 890
Author(s):  
Suthep Suantai ◽  
Kunrada Kankam ◽  
Prasit Cholamjiak
Keyword(s):  
Image Processing ◽  
Minimization Problem ◽  
Convex Functions ◽  
Lower Semicontinuous ◽  
Image Inpainting ◽  
Constrained Minimization ◽  
Convex Minimization Problem ◽  
Mild Conditions ◽  
Constrained Minimization Problem ◽  
Weak Convergence Theorem

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

scholarly journals Hilbert-Kunz Multiplicity of Three-Dimensional Local Rings

2005 ◽  
Vol 177 ◽  
pp. 47-75 ◽  
Author(s):  
Kei-ichi Watanabe ◽  
Ken-ichi Yoshida
Keyword(s):  
Lower Bound ◽  
Local Ring ◽  
Three Dimensional ◽  
Krull Dimension ◽  
Local Rings ◽  
Mild Conditions ◽  
Quadric Hypersurface ◽  
Complete Local Ring

In this paper, we investigate the lower bound sHK(p, d) of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension d containing a field of characteristic p > 0. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that sHK (p, 3) = 4/3 and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degenerate quadric hypersurface k[[X, Y, Z,W]]/(X2 + Y2 + Z2 + W2) under mild conditions.Furthermore, we pose a generalization of the main theorem to the case of dim A ≥ 4 as a conjecture, and show that it is also true in case dim A = 4 using the similar method as in the proof of the main theorem.

scholarly journals Tight closure of powers of ideals and tight hilbert polynomials

2019 ◽  
Vol 169 (2) ◽  
pp. 335-355
Author(s):  
KRITI GOEL ◽  
J. K. VERMA ◽  
VIVEK MUKUNDAN
Keyword(s):  
Local Ring ◽  
Hilbert Function ◽  
Simplicial Complexes ◽  
Primary Ideal ◽  
Hilbert Polynomial ◽  
Prime Characteristic ◽  
Tight Closure ◽  
Powers Of Ideals ◽  
Hilbert Polynomials

AbstractLet (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.

Lower semicontinuity of mass under C0{C^{0}} convergence and Huisken’s isoperimetric mass

2019 ◽  
Vol 2019 (756) ◽  
pp. 227-257 ◽  
Author(s):  
Jeffrey L. Jauregui ◽  
Dan A. Lee
Keyword(s):  
Scalar Curvature ◽  
Lower Semicontinuity ◽  
Mean Curvature ◽  
Total Mass ◽  
Mean Curvature Flow ◽  
Lower Semicontinuous ◽  
Curvature Flow ◽  
Three Dimensions ◽  
Asymptotically Flat ◽  
Limit Space

AbstractGiven a sequence of asymptotically flat 3-manifolds of nonnegative scalar curvature with outermost minimal boundary, converging in the pointed {C^{0}} Cheeger–Gromov sense to an asymptotically flat limit space, we show that the total mass of the limit is bounded above by the liminf of the total masses of the sequence. In other words, total mass is lower semicontinuous under such convergence. In order to prove this, we use Huisken’s isoperimetric mass concept, together with a modified weak mean curvature flow argument. We include a brief discussion of Huisken’s work before explaining our extension of that work. The results are all specific to three dimensions.

scholarly journals Rings of Frobenius operators

2014 ◽  
Vol 157 (1) ◽  
pp. 151-167 ◽  
Author(s):  
MORDECHAI KATZMAN ◽  
KARL SCHWEDE ◽  
ANURAG K. SINGH ◽  
WENLIANG ZHANG
Keyword(s):  
Local Ring ◽  
Residue Field ◽  
Injective Hull ◽  
Degree Zero ◽  
Prime Characteristic ◽  
Finitely Generated ◽  
Finite Generation

AbstractLet R be a local ring of prime characteristic. We study the ring of Frobenius operators ${\mathcal F}(E)$, where E is the injective hull of the residue field of R. In particular, we examine the finite generation of ${\mathcal F}(E)$ over its degree zero component ${\mathcal F}^0(E)$, and show that ${\mathcal F}(E)$ need not be finitely generated when R is a determinantal ring; nonetheless, we obtain concrete descriptions of ${\mathcal F}(E)$ in good generality that we use, for example, to prove the discreteness of F-jumping numbers for arbitrary ideals in determinantal rings.

scholarly journals Coepi-analysis of lower semicontinuity property of vector-valued mappings

2021 ◽  
Vol 18 ◽  
pp. 110
Author(s):  
A.V. Dovzhenko ◽  
P.I. Kogut
Keyword(s):  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Topological Properties ◽  
Variational Representation ◽  
Semicontinuity Property ◽  
Vector Valued

Topological properties of coepigraphs of vector-valued mappings are investigated. Lower semicontinuous regularization of such mappings and its variational representation are received with the help of coepigraphs.

scholarly journals CONTINUITY OF HILBERT–KUNZ MULTIPLICITY AND F-SIGNATURE

2018 ◽  
Vol 239 ◽  
pp. 322-345 ◽  
Author(s):  
THOMAS POLSTRA ◽  
ILYA SMIRNOV
Keyword(s):  
Local Ring ◽  
Prime Characteristic ◽  
Real Numbers ◽  
The Real

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring $(R,\mathfrak{m},k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\mathfrak{m}$-adic topology.

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