Algebra

Misyurov D.A. Dialectical formulas based on the binary notation as the development formulas // Credo New. 2012. №2

by Dmitry Misyurov

The article suggests dialectical formulas based on the binary notation as the development formulas: formula with... more The article suggests dialectical formulas based on the binary notation as the development formulas: formula with dominant and the non-dominant elements; universal formula; formula with symbolic weight of elements; tautological formula. For example, it suggests an opportunity to use the dialectical formulas for modeling and artificial intelligence creation, etc.

« Alfred North Whitehead (1861–1947) » (2002)

by Michel Weber

« Alfred North Whitehead (1861–1947) », in Mander, W. J. and Sell, A. P. F. (Senior Editors), Dictionary of Nineteenth-Century British Philosophers, Bristol, Thoemmes Press, 2002, Vol. II, pp. 1236-1241.

Mathematical philosopher, born on 15 February 1861, at Ramsgate (Kent) and deceased 30 December 1947, at Cambridge... more Mathematical philosopher, born on 15 February 1861, at Ramsgate (Kent) and deceased 30 December 1947, at Cambridge (Massachusetts, United States). Whitehead entered Trinity College in 1880 with a scholarship in mathematics; in 1884, he was elected Fellow in Mathematics with a dissertation (now lost) on Maxwell’s Treatise on Electricity and Magnetism and started teaching mathematics and mathematical physics.

Apartness on strongly regular rings

by Bassel Mannaa

A note on square-free polynomials

by Bassel Mannaa

Dynamic Newton-Puiseux Theorem

by Bassel Mannaa

Dynamic Newton Theorem

by Bassel Mannaa

Z-relation and homometry in musical distributions

by John Mandereau

Co-authored with Daniele Ghisi, Emmanuel Amiot, Moreno Andreatta, Carlos Agon. Published in Journal of Mathematics and Music.

This paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases... more This paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientific areas. The link between these two approaches was only made recently, suggesting new interesting musical applications and opening new theoretical problems. We present some old and new results on homometry, and give perspective on future research assisted by computational methods. We assume from the reader basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

Discrete Phase Retrieval in Musical Structures

by John Mandereau

Co-authored with Daniele Ghisi, Emmanuel Amiot, Moreno Andreatta, Carlos Agon. Preprint. Published in Journal of Mathematics and Music.

This paper describes phase retrieval approaches in music by focusing on the particular case of the cyclic groups... more This paper describes phase retrieval approaches in music by focusing on the particular case of the cyclic groups (beltway problem). After presenting some old and new results on phase retrieval, we introduce the extended phase retrieval for generalized musical Z-relation. This concept is accompanied by mathematical definitions and motivations from computer-aided composition. We assume from the reader basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

Safety and Hazard Analysis in Concurrent Systems

by Shrisha Rao

Ph.D. thesis, University of Iowa, 2005.

Safety is a well-known and important class of property of software programs, and of systems in general. The basic... more Safety is a well-known and important class of property of software programs, and of systems in general. The basic notion that informs this work is that the time to think about safety is when it still exists but could be lost. The notion is not just to analyse safety as existing or not with a given system state, but also in the sense that a system is presently safe but becoming less so. Safety as considered here is not restricted to one type of property, and indeed for much of the discussion it does not matter what types of measures are used to assess safety.

The work done here is for the purpose of laying a theoretical and
mathematical foundation for allowing static analyses of systems to further safety. This is done using tools from lattice theory applied to the poset of system states partially ordered by reachability. Such analyses are common (e.g., with abstract interpretations of software functioning) with respect to other kinds of systems, but there does not seem to exist a formalism that permits them specifically for safety.

Using the basic analytical tools developed, a study is made of the problem of composing systems from components. Three types of composition: direct sum, direct product, and exponentiation---are noted, and the first two are treated in some depth. It is shown that the set of all systems formed with the direct sum and direct product operators can be specified by a BNF grammar. A three-valued ``safety logic'' is specified, using which the safety and fault-tolerance of composed systems can be computed given the system composition. It is
also shown that the set of all systems also forms separate monoids (in the sense familiar to mathematicians), and that other monoids can be derived based on equivalence classes of systems.

The example of a train approaching a railroad crossing, where a gate
must be closed prior to the train's arrival and opened after its exit,
is considered and analysed as an example.

Almost split sequences for commutative Artinian rings

by Dag Oskar Madsen

Published in: "Algebras and modules, II" (Geiranger, 1996), 403-408, CMS Conference Proceedings, 24, American Mathematical Society, Providence, RI, 1998
Math Reviews: MR1648641

Quasi-hereditary algebras and generalized Koszul duality

by Dag Oskar Madsen

We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to... more We present an easily applicable sufficient condition for standard Koszul algebras to be Koszul with respect to $\Delta$. If a quasi-hereditary algebra $\L$ is Koszul with respect to $\Delta$, then $\L$ and the extension algebra of $\Delta$ are Koszul dual in a sense explained below, implying in particular that their bounded derived categories of finitely generated graded modules are equivalent. We also prove that the extension algebra of $\Delta$ is Koszul in the classical sense.

Filtrations in abelian categories with a tilting object of homological dimension two

by Dag Oskar Madsen

Co-authored with Bernt Tore Jensen and Xiuping Su
Accepted for publication in Journal of Algebra and Its Applications

We consider filtrations of objects in an abelian category $\catA$ induced by
a tilting object $T$ of homological... more We consider filtrations of objects in an abelian category $\catA$ induced by
a tilting object $T$ of homological dimension at most two. We define three pairwise
disjoint extension closed subcategories $\mathcal{E}^0, \mathcal{E}^1$ and $\mathcal{E}^2$
with $Hom(\mathcal{E}^i,\mathcal{E}^j)=0$
for $j>i$, such that each object in $\catA$ has a unique filtation with factors in these
categories. In dimension one, this filtration coincides with the classical two-step
filtration induced by the torsion pair. We also give a refined filtration, using the derived
equivalence between the derived categories of $\catA$ and the module category of
$End_\catA (T)^{op}$.

(m, n)-Semirings and a Generalized Fault Tolerance Algebra of Systems

by Shrisha Rao

Co-authored with Syed Eqbal Alam and Bijan Davvaz.

We propose a new class of mathematical structures called (m,n)-semirings (which generalize the usual semirings), and... more We propose a new class of mathematical structures called (m,n)-semirings (which generalize the usual semirings), and describe their basic properties. We also define partial ordering, and generalize the concepts of congruence, homomorphism, ideals, etc., for (m,n)-semirings. Following earlier work by Rao, we consider a system as made up of several components whose failures may cause it to fail, and represent the set of systems algebraically as an (m,n)-semiring. Based on the characteristics of these components we present a formalism to compare the fault tolerance behaviour of two systems using our framework of a partially ordered (m,n)-semiring.

A systems algebra and its applications

by Shrisha Rao

IEEE Systems Conference 2008, Montreal, Canada

Since every system of any significant size is created by composition from
smaller sub-systems or components, an... more Since every system of any significant size is created by composition from
smaller sub-systems or components, an attempt is made to analyze the
properties of a system as a function of its composition. Using a
partial ordering of system states based on reachability, system states
are classified in the abstract into bad, hazardous, unsafe and safe
states, and a safety function that separates these is derived.
Two basic types of system composition are described, and an algebra to
describe the safety and fault tolerance of composed systems is obtained.
The set of systems forms monoids under the two composition
operators, and a semiring when both are concerned. A partial
ordering relation between systems is used to compare their
fault-tolerance behaviors.

An Algebra of Fault Tolerance

by Shrisha Rao

The Semiring Properties of Boolean Propositional Algebras

by Shrisha Rao

Hochschild homology and truncated cycles

by Dag Oskar Madsen

Co-authored with Petter Andreas Berg and Yang Han
Published in: Proceedings of the American Mathematical Society 140 (2012), 1133-1139
DOI: 10.1090/S0002-9939-2011-10942-0

We study algebras having 2-truncated cycles, and show that these algebras have infinitely many nonzero Hochschild... more We study algebras having 2-truncated cycles, and show that these algebras have infinitely many nonzero Hochschild homology groups. Consequently, algebras of finite global dimension have no 2-truncated cycles, and therefore satisfy a higher version of the ``no loops conjecture".

Projective dimensions and almost split sequences

by Dag Oskar Madsen

Published in: Journal of Algebra 271 (2004), no. 2, 652--672
DOI: 10.1016/j.jalgebra.2003.09.015
Math Reviews: MR2025545

Let Λ be an Artin algebra and let 0→A→B→C→0 be an almost split sequence. In this paper we discuss under which... more Let Λ be an Artin algebra and let 0→A→B→C→0 be an almost split sequence. In this paper we discuss under which conditions the inequality pdB⩽max{pdA,pdC} is strict.

Secant Square Theorem

by Cody Johnson

Projective dimensions and Nakayama algebras

by Dag Oskar Madsen

Published in: "Representations of algebras and related topics", 247–265, Fields Institute Communications, 45, American Mathematical Society, Providence, RI, 2005.
Math Reviews: MR2146655