scholarly journals Uniform bounds in F-finite rings and lower semi-continuity of the F-signature

2017 ◽  
Vol 370 (5) ◽  
pp. 3147-3169 ◽  
Author(s):  
Thomas Polstra
Keyword(s):  
Finite Rings ◽  
Uniform Bounds ◽  
Lower Semi

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.

scholarly journals Stability of the Non-abelian X-ray Transform in Dimension $$\ge 3$$

2021 ◽  
Author(s):  
Jan Bohr
Keyword(s):  
Stability Estimate ◽  
Scattering Data ◽  
Regularity Conditions ◽  
Uniform Bounds ◽  
Ray Transform ◽  
Shallow Layer ◽  
Matrix Potential ◽  
Novel Method ◽  
Statistical Consistency ◽  
Appl Math

AbstractNon-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ M and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular, proving uniform bounds on layer depth and stability constants.

scholarly journals Quotient rings of Noetherian module finite rings

1994 ◽  
Vol 121 (2) ◽  
pp. 335-335 ◽  
Author(s):  
A. W. Chatters ◽  
C. R. Hajarnavis
Keyword(s):  
Finite Rings ◽  
Noetherian Module ◽  
Quotient Rings

scholarly journals Public-key cryptosystem based on invariants of diagonalizable groups

2017 ◽  
Vol 9 (1) ◽  
Author(s):  
František Marko ◽  
Alexandr N. Zubkov ◽  
Martin Juráš
Keyword(s):  
Linear Algebra ◽  
Finite Fields ◽  
Number Fields ◽  
Euclidean Algorithm ◽  
Public Key ◽  
Public Key Cryptosystem ◽  
Finite Rings

AbstractWe develop a public-key cryptosystem based on invariants of diagonalizable groups and investigate properties of such a cryptosystem first over finite fields, then over number fields and finally over finite rings. We consider the security of these cryptosystem and show that it is necessary to restrict the set of parameters of the system to prevent various attacks (including linear algebra attacks and attacks based on the Euclidean algorithm).

Finite rings with exactly two maximal subrings

2011 ◽  
Vol 55 (6) ◽  
pp. 46-52 ◽  
Author(s):  
S. S. Korobkov
Keyword(s):  
Finite Rings

Convex extensions and envelopes of lower semi-continuous functions

2002 ◽  
Vol 93 (2) ◽  
pp. 247-263 ◽  
Author(s):  
Mohit Tawarmalani ◽  
Nikolaos V Sahinidis
Keyword(s):  
Continuous Functions ◽  
Lower Semi

scholarly journals Finite rings in which commutativity is transitive

2009 ◽  
Vol 162 (2) ◽  
pp. 143-155 ◽  
Author(s):  
David Dolžan ◽  
Igor Klep ◽  
Primož Moravec
Keyword(s):  
Finite Rings
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