Mordell exceptional locus for subvarieties of the additive group

AbstractIn this article, we prove the generalized Hyers-Ulam stability for the following additive-quartic functional equation:f(x+3y)+f(x-3y)+f(x+2y)+f(x-2y)+22f(x)+24f(y)=13{[}f(x+y)+f(x-y)]+12f(2y),where f maps from an additive group to a complete non-Archimedean normed space.

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On Multi-Scalar Multiplication Algorithms for Register-Constrained Environments

Electronics ◽

10.3390/electronics10050605 ◽

2021 ◽

Vol 10 (5) ◽

pp. 605

Author(s):

Da-Zhi Sun ◽

Ji-Dong Zhong ◽

Hong-De Zhang ◽

Xiang-Yu Guo

Keyword(s):

Electronic Devices ◽

Additive Group ◽

Fixed Number ◽

Scalar Multiplication ◽

Computation Cost ◽

Public Key Cryptosystems ◽

Adaptive Window ◽

Computation Efficiency ◽

Window Method ◽

Constrained Environments

A basic but expensive operation in the implementations of several famous public-key cryptosystems is the computation of the multi-scalar multiplication in a certain finite additive group defined by an elliptic curve. We propose an adaptive window method for the multi-scalar multiplication, which aims to balance the computation cost and the memory cost under register-constrained environments. That is, our method can maximize the computation efficiency of multi-scalar multiplication according to any small, fixed number of registers provided by electronic devices. We further demonstrate that our method is efficient when five registers are available. Our method is further studied in detail in the case where it is combined with the non-adjacent form (NAF) representation and the joint sparse form (JSF) representation. One efficiency result is that our method with the proposed improved NAF n-bit representation on average requires 209n/432 point additions. To the best of our knowledge, this efficiency result is optimal compared with those of similar methods using five registers. Unlike the previous window methods, which store all possible values in the window, our method stores those with comparatively high probabilities to reduce the number of required registers.

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A STRENGTHENING OF RESOLUTION OF SINGULARITIES IN CHARACTERISTIC ZERO

Proceedings of the London Mathematical Society ◽

10.1112/s0024611502013801 ◽

2003 ◽

Vol 86 (2) ◽

pp. 327-357 ◽

Cited By ~ 6

Author(s):

A. BRAVO ◽

O. VILLAMAYOR U.

Keyword(s):

Characteristic Zero ◽

Resolution Of Singularities ◽

Closed Subscheme ◽

Subject Classification ◽

Invertible Sheaf ◽

Simple Manner ◽

Birational Morphism ◽

Exceptional Locus

Let $X$ be a closed subscheme embedded in a scheme $W$, smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I} (X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $\pi : W_r \longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r \subset W_r$ denotes the strict transform of $X \subset W$ then:(1) the morphism $\pi : W_r \longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem);(2) the total transform of ${\mathcal I} (X)$ in ${\mathcal O}_{W_r}$ factors as a product of an invertible sheaf of ideals ${\mathcal L}$ supported on the exceptional locus, and the sheaf of ideals defining the strict transform of $X$ (that is, ${\mathcal I}(X){\mathcal O}_{W_r} = {\mathcal L} \cdot {\mathcal I}(X_r)$).Thus (2) asserts that we can obtain, in a simple manner, the equations defining the desingularization of $X$.2000 Mathematical Subject Classification: 14E15.

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SMALL VALUE ESTIMATES FOR THE ADDITIVE GROUP

International Journal of Number Theory ◽

10.1142/s179304211000323x ◽

2010 ◽

Vol 06 (04) ◽

pp. 919-956 ◽

Cited By ~ 1

Author(s):

DAMIEN ROY

Keyword(s):

Algebraic Independence ◽

Additive Group

We generalize Gel'fond's criterion for algebraic independence to the context of a sequence of polynomials whose first derivatives take small values on large subsets of a fixed subgroup of ℂ, instead of just one point (one extension deals with a subgroup of ℂ×).

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CANCELLATION LAW AND UNIQUE FACTORIZATION THEOREM FOR STRING OPERATIONS

Journal of Algebra and Its Applications ◽

10.1142/s0219498806001739 ◽

2006 ◽

Vol 05 (02) ◽

pp. 231-243

Author(s):

DONGVU TONIEN

Keyword(s):

Abelian Group ◽

Algebraic Structure ◽

Additive Group ◽

New Product ◽

Binary String ◽

Factorization Theorem ◽

General Criterion ◽

Unique Factorization ◽

Product Operation ◽

Associative Law

Recently, Hoit introduced arithmetic on blocks, which extends the binary string operation by Jacobs and Keane. A string of elements from the Abelian additive group of residues modulo m, (Zm, ⊕), is called an m-block. The set of m-blocks together with Hoit's new product operation form an interesting algebraic structure where associative law and cancellation law hold. A weaker form of unique factorization and criteria for two indecomposable blocks to commute are also proved. In this paper, we extend Hoit's results by replacing the Abelian group (Zm, ⊕) by an arbitrary monoid (A, ◦). The set of strings built up from the alphabet A is denoted by String(A). We extend the operation ◦ on the alphabet set A to the string set String(A). We show that (String(A), ◦) is a monoid if and only if (A, ◦) is a monoid. When (A, ◦) is a group, we prove that stronger versions of a cancellation law and unique factorization hold for (String(A), ◦). A general criterion for two irreducible strings to commute is also presented.

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Subgroups of finite index in an additive group of a ring

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171201010274 ◽

2001 ◽

Vol 27 (2) ◽

pp. 83-89

Author(s):

Doostali Mojdeh ◽

S. Hassan Hashemi

Keyword(s):

Finite Index ◽

Additive Group ◽

Infinite Field ◽

Division Rings ◽

Invertible Elements

IfKis an infinite field andG⫅Kis a subgroup of finite index in an additive group, thenK∗=G∗G∗−1whereG∗denotes the set of all invertible elements inGandG∗−1denotes all inverses of elements ofG∗. Similar results hold for various fields, division rings and rings.

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Eigenvalues and strong orbit equivalence

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2015.26 ◽

2015 ◽

Vol 36 (8) ◽

pp. 2419-2440 ◽

Cited By ~ 1

Author(s):

MARÍA ISABEL CORTEZ ◽

FABIEN DURAND ◽

SAMUEL PETITE

Keyword(s):

Equivalence Class ◽

Quotient Group ◽

Additive Group ◽

Orbit Equivalence ◽

Dimension Group ◽

Torsion Free

We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues $E(X,T)$ of the minimal Cantor system $(X,T)$ is a subgroup of the intersection $I(X,T)$ of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated with $(X,T)$ is trivial, the quotient group $I(X,T)/E(X,T)$ is torsion free. We give examples with non-trivial infinitesimal subgroups where this property fails. We also provide some realization results.

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