3,471 research outputs found

    Symmetric chain decomposition for cyclic quotients of Boolean algebras and relation to cyclic crystals

    Full text link
    The quotient of a Boolean algebra by a cyclic group is proven to have a symmetric chain decomposition. This generalizes earlier work of Griggs, Killian and Savage on the case of prime order, giving an explicit construction for any order, prime or composite. The combinatorial map specifying how to proceed downward in a symmetric chain is shown to be a natural cyclic analogue of the sl2\mathfrak{sl}_2 lowering operator in the theory of crystal bases.Comment: minor revisions; to appear in IMR

    Speeding up Cylindrical Algebraic Decomposition by Gr\"obner Bases

    Get PDF
    Gr\"obner Bases and Cylindrical Algebraic Decomposition are generally thought of as two, rather different, methods of looking at systems of equations and, in the case of Cylindrical Algebraic Decomposition, inequalities. However, even for a mixed system of equalities and inequalities, it is possible to apply Gr\"obner bases to the (conjoined) equalities before invoking CAD. We see that this is, quite often but not always, a beneficial preconditioning of the CAD problem. It is also possible to precondition the (conjoined) inequalities with respect to the equalities, and this can also be useful in many cases.Comment: To appear in Proc. CICM 2012, LNCS 736

    Random walks on rings and modules

    Get PDF
    We consider two natural models of random walks on a module VV over a finite commutative ring RR driven simultaneously by addition of random elements in VV, and multiplication by random elements in RR. In the coin-toss walk, either one of the two operations is performed depending on the flip of a coin. In the affine walk, random elements a∈R,b∈Va \in R,b \in V are sampled independently, and the current state xx is taken to ax+bax+b. For both models, we obtain the complete spectrum of the transition matrix from the representation theory of the monoid of all affine maps on VV under a suitable hypothesis on the measure on VV (the measure on RR can be arbitrary).Comment: 26 pages, 1 figure, minor improvements, final versio

    Finite Boolean Algebras for Solid Geometry using Julia's Sparse Arrays

    Full text link
    The goal of this paper is to introduce a new method in computer-aided geometry of solid modeling. We put forth a novel algebraic technique to evaluate any variadic expression between polyhedral d-solids (d = 2, 3) with regularized operators of union, intersection, and difference, i.e., any CSG tree. The result is obtained in three steps: first, by computing an independent set of generators for the d-space partition induced by the input; then, by reducing the solid expression to an equivalent logical formula between Boolean terms made by zeros and ones; and, finally, by evaluating this expression using bitwise operators. This method is implemented in Julia using sparse arrays. The computational evaluation of every possible solid expression, usually denoted as CSG (Constructive Solid Geometry), is reduced to an equivalent logical expression of a finite set algebra over the cells of a space partition, and solved by native bitwise operators.Comment: revised version submitted to Computer-Aided Geometric Desig

    Information-Based Physics: An Observer-Centric Foundation

    Full text link
    It is generally believed that physical laws, reflecting an inherent order in the universe, are ordained by nature. However, in modern physics the observer plays a central role raising questions about how an observer-centric physics can result in laws apparently worthy of a universal nature-centric physics. Over the last decade, we have found that the consistent apt quantification of algebraic and order-theoretic structures results in calculi that possess constraint equations taking the form of what are often considered to be physical laws. I review recent derivations of the formal relations among relevant variables central to special relativity, probability theory and quantum mechanics in this context by considering a problem where two observers form consistent descriptions of and make optimal inferences about a free particle that simply influences them. I show that this approach to describing such a particle based only on available information leads to the mathematics of relativistic quantum mechanics as well as a description of a free particle that reproduces many of the basic properties of a fermion. The result is an approach to foundational physics where laws derive from both consistent descriptions and optimal information-based inferences made by embedded observers.Comment: To be published in Contemporary Physics. The manuscript consists of 43 pages and 9 Figure
    • …
    corecore