scholarly journals Irredundant Decomposition of Algebras into One-Dimensional Factors

2016 ◽  
Vol 45 (3/4) ◽  
Author(s):  
Bogdan Staruch
Keyword(s):  
Congruence Lattice ◽  
Subdirect Product ◽  
Star Product ◽  
Special Kind ◽  
Boolean Algebras ◽  
Algebraic Lattice ◽  
Linear Spaces ◽  
One Dimensional ◽  
Irredundant Decomposition ◽  
Decomposition Theorems

We introduce a notion of dimension of an algebraic lattice and, treating such a lattice as the congruence lattice of an algebra, we introduce the dimension of an algebra, too. We define a star-product as a special kind of subdirect product. We obtain the star-decomposition of algebras into one-dimensional factors, which generalizes the known decomposition theorems e.g. for Abelian groups, linear spaces, Boolean algebras.

scholarly journals Topological correlators and surface defects from equivariant cohomology

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Rodolfo Panerai ◽  
Antonio Pittelli ◽  
Konstantina Polydorou
Keyword(s):  
Quantum Mechanics ◽  
Equivariant Cohomology ◽  
Surface Defects ◽  
Star Product ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional System ◽  
The Novel ◽  
One Dimensional ◽  
Dual Quantum

Abstract We find a one-dimensional protected subsector of $$ \mathcal{N} $$ N = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S3. Then, we apply it to the novel case of S2× S1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncom- mutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general $$ \mathcal{N} $$ N = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

One Dimensional Experimental Setup to Study the Heating of Nanoparticle Laden Systems

2010 ◽  
Author(s):  
Zhenpeng Qin ◽  
John C. Bischof
Keyword(s):  
Small Volume ◽  
Laser Energy ◽  
Special Kind ◽  
Gold Nanoshells ◽  
Experimental Setup ◽  
Heat Absorption ◽  
One Dimensional ◽  
Lumped Models

Intensive efforts have been put into the use of gold nanoparticles (GNPs) for the enhancement of hyperthermia using laser in recent years since the groundbreaking work of Hirsh et al.(1) using gold nanoshells (GNS). Both in vitro (2), and in vivo (3) studies show promising results. For example, GNS, a special kind of GNP, are being manufactured and are in clinical trials (Nanospectra Bioscience, Inc). While the data is compelling, unfortunately the fundamentals of GNP heating are not entirely understood. For example, there are large discrepancies in the experimentally measured photothermal efficiency of GNPs (4, 5). Furthermore, lumped models of GNP heating in solution, by using small volume of GNP solution (4, 5), or stirring the solution (6), neglecting the variation of heat absorption throughout a system require improvement. In reality, the GNPs will attenuate the laser beam as it passes through the GNP host medium. GNPs at different locations will absorb different amount of laser energy and hence have different heat generation.

Discrete space theory of radiative transfer I. Fundamentals

1969 ◽  
Vol 313 (1513) ◽  
pp. 183-197 ◽  
Keyword(s):  
Radiative Transfer ◽  
Star Product ◽  
Formal Theory ◽  
Space Theory ◽  
Scattering Media ◽  
One Dimensional ◽  
Infinitesimal Generators ◽  
Arbitrary Thickness ◽  
Internal Sources ◽  
Physical Constitution

This paper sets out a formal theory of radiative transfer in one-dimensional scattering media of arbitrary physical constitution. The theory is based on an extension of the treatment of Redheffer, in which the response of a layer of arbitrary thickness to fluxes incident on its boundaries is described by a certain linear operator. Juxtaposition of two such layers gives a third layer, whose operator can be related to those of its constituents by an operation designated as the star product. It is shown that this set of operators constitutes a semigroup under the star product, and that the infinitesimal generators of the semigroup can be computed in term s of the physical properties of the medium , point by point. This makes it possible to write equivalent discrete and differential equations from both of which transmission and reflexion operators, the emission due to internal sources, and the internal fluxes at prescribed levels in the medium can be obtained.

CONSTRUCTING CHAOTIC POLYNOMIAL MAPS

2009 ◽  
Vol 19 (02) ◽  
pp. 531-543 ◽  
Author(s):  
XU ZHANG ◽  
YUMING SHI ◽  
GUANRONG CHEN
Keyword(s):  
Logistic Map ◽  
Special Kind ◽  
Quadratic Polynomials ◽  
One Dimensional ◽  
Polynomial Maps ◽  
Polynomial Map ◽  
Expansion Theory ◽  
Chaotic Parameter ◽  
The Given ◽  
Nonzero Polynomial

This paper studies the construction of one-dimensional real chaotic polynomial maps. Given an arbitrary nonzero polynomial of degree m (≥ 0), two methods are derived for constructing chaotic polynomial maps of degree m + 2 by simply multiplying the given polynomial with suitably designed quadratic polynomials. Moreover, for m + 2 arbitrarily given different positive constants, a method is given to construct a chaotic polynomial map of degree 2m based on the coupled-expansion theory. Furthermore, by multiplying a real parameter to a special kind of polynomial, which has at least two different non-negative or nonpositive zeros, the chaotic parameter region of the polynomial is analyzed based on the snap-back repeller theory. As a consequence, for any given integer n ≥ 2, at least one polynomial of degree n can be constructed so that it is chaotic in the sense of both Li–Yorke and Devaney. In addition, two natural ways of generalizing the logistic map to higher-degree chaotic logistic-like maps are given. Finally, an illustrative example is provided with computer simulations for illustration.

CONSTRUCTION OF LATTICES OF BALANCED EQUIVALENCE RELATIONS FOR REGULAR HOMOGENEOUS NETWORKS USING LATTICE GENERATORS AND LATTICE INDICES

2009 ◽  
Vol 19 (11) ◽  
pp. 3691-3705 ◽  
Author(s):  
HIROKO KAMEI
Keyword(s):  
Special Kind ◽  
Building Blocks ◽  
Equivalence Relations ◽  
Lattice Structures ◽  
Internal Dynamics ◽  
One Dimensional ◽  
Cell Network ◽  
Coupled Cell Networks ◽  
Cell Networks ◽  
Number Of Cells

Regular homogeneous networks are a class of coupled cell network, which comprises one type of cell (node) with one type of coupling (arrow), and each cell has the same number of input arrows (called the valency of the network). In coupled cell networks, robust synchrony (a flow-invariant polydiagonal) corresponds to a special kind of partition of cells, called a balanced equivalence relation. Balanced equivalence relations are determined solely by the network structure. It is well known that the set of balanced equivalence relations on a given finite network forms a complete lattice. In this paper, we consider regular homogeneous networks in which the internal dynamics of each cell is one-dimensional, and whose associated adjacency matrices have simple eigenvalues (real or complex). We construct explicit forms of lattices of balanced equivalence relations for such networks by introducing key building blocks, called lattice generators, along with integer numbers called lattice indices. The properties of lattice indices allow construction of all possible lattice structures for balanced equivalence relations of regular homogeneous networks of any number of cells with any valency. As an illustration, we show all 14 possible lattice structures of balanced equivalence relations for four-cell regular homogeneous networks.

scholarly journals On Decompositions of Matrices over Distributive Lattices

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Yizhi Chen ◽  
Xianzhong Zhao
Keyword(s):  
Distributive Lattice ◽  
Direct Product ◽  
Boolean Algebras ◽  
Distributive Lattices ◽  
The Matrix ◽  
Nilpotent Matrices ◽  
Decomposition Theorems ◽  
Finite Distributive Lattices ◽  
Matrix Semigroup

LetLbe a distributive lattice andMn,q(L)(Mn(L), resp.) the semigroup (semiring, resp.) ofn×q(n×n, resp.) matrices overL. In this paper, we show that if there is a subdirect embedding from distributive latticeLto the direct product∏i=1m‍Liof distributive latticesL1,L2, …,Lm, then there will be a corresponding subdirect embedding from the matrix semigroupMn,q(L)(semiringMn(L), resp.) to semigroup∏i=1m‍Mn,q(Li)(semiring∏i=1m‍Mn(Li), resp.). Further, it is proved that a matrix over a distributive lattice can be decomposed into the sum of matrices over some of its special subchains. This generalizes and extends the decomposition theorems of matrices over finite distributive lattices, chain semirings, fuzzy semirings, and so forth. Finally, as some applications, we present a method to calculate the indices and periods of the matrices over a distributive lattice and characterize the structures of idempotent and nilpotent matrices over it. We translate the characterizations of idempotent and nilpotent matrices over a distributive lattice into the corresponding ones of the binary Boolean cases, which also generalize the corresponding structures of idempotent and nilpotent matrices over general Boolean algebras, chain semirings, fuzzy semirings, and so forth.

CALCULATING ANCESTORS IN ONE-DIMENSIONAL CELLULAR AUTOMATA

2004 ◽  
Vol 15 (08) ◽  
pp. 1151-1169 ◽  
Author(s):  
JUAN CARLOS SECK TUOH MORA ◽  
GENARO JUÁREZ MARTÍNEZ ◽  
HAROLD V. MCINTOSH
Keyword(s):  
Dynamical Systems ◽  
Cellular Automata ◽  
Cellular Automaton ◽  
Special Kind ◽  
Local Interaction ◽  
Garden Of Eden ◽  
One Dimensional ◽  
Space And Time

One-dimensional cellular automata are dynamical systems characterized by discreteness (in space and time), determinism and local interaction. We present a procedure to calculate the ancestors for a given sequence of states, which is based on a special kind of graph called subset diagram. We use this diagram to specify subset tables for calculating ancestors which are not Garden-of-Eden sequences, hence the process is able to yield ancestors in several generations. Some examples are illustrated using the cellular automaton Rule 110 which is the most interesting automaton of two states and three neighbors.

scholarly journals Universally Incomparable Ring-Homomorphisms

1984 ◽  
Vol 29 (3) ◽  
pp. 289-302 ◽  
Author(s):  
David E. Dobbs ◽  
Marco Fontana
Keyword(s):  
Prime Ideal ◽  
Special Kind ◽  
Commutative Rings ◽  
Base Change ◽  
Primitive Elements ◽  
Algebraic Field ◽  
One Dimensional ◽  
Ring Homomorphisms ◽  
Field Extensions ◽  
Noetherian Domain

A homomorphism f: R → T of (commutative) rings is said to be universally incomparable in case each base change R → S induces an incomparable map S → S⊗RT. The most natural examples of universally incomparable homomorphisms are the integral maps and radiciel maps. It is proved that a homomorphism f: R → T is universally incomparable if and only if f is an incomparable map which induces algebraic field extensions of fibres, k(f-1(Q))→k(Q), for each prime ideal Q of T. In several cases (f algebra-finite, T generated as R-algebra by primitive elements, T an overring of a one-dimensional Noetherian domain R), each universally incomparable map is shown to factor as a composite of an integral map and a special kind of radiciel.

scholarly journals The congruence lattice of a combinatorial strict inverse semigroup

1994 ◽  
Vol 37 (1) ◽  
pp. 25-37 ◽  
Author(s):  
Karl Auinger
Keyword(s):  
Inverse Semigroup ◽  
Congruence Lattice ◽  
Subdirect Product ◽  
Congruence Lattices

The congruence lattice of a combinatorial strict inverse semigroup is shown to be isomorphic to a complete subdirect product of congruence lattices of semilattices preserving pseudocomplements.

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