scholarly journals Some Algebraic Properties of Polynomial Rings

2016 ◽  
Vol 24 (3) ◽  
pp. 227-237 ◽  
Author(s):  
Christoph Schwarzweller ◽  
Artur Korniłowicz ◽  
Agnieszka Rowinska-Schwarzweller
Keyword(s):  
Algebraic Theory ◽  
Polynomial Rings ◽  
Canonical Embedding ◽  
Irreducible Polynomials ◽  
Irreducible Elements ◽  
Algebraic Properties ◽  
The One

Abstract In this article we extend the algebraic theory of polynomial rings, formalized in Mizar [1], based on [2], [3]. After introducing constant and monic polynomials we present the canonical embedding of R into R[X] and deal with both unit and irreducible elements. We also define polynomial GCDs and show that for fields F and irreducible polynomials p the field F[X]/<p> is isomorphic to the field of polynomials with degree smaller than the one of p.

scholarly journals Irreducible polynomials in Int(ℤ)

2018 ◽  
Vol 20 ◽  
pp. 01004 ◽  
Author(s):  
Austin Antoniou ◽  
Sarah Nakato ◽  
Roswitha Rissner
Keyword(s):  
Computational Method ◽  
Irreducible Polynomials ◽  
Open Questions ◽  
Irreducible Elements

In order to fully understand the factorization behavior of the ring Int(ℤ) = {f ∈ ℚ[x] | f (ℤ) ⊆ ℤ} of integer-valued polynomials on ℤ, it is crucial to identify the irreducible elements. Peruginelli [8] gives an algorithmic criterion to recognize whether an integer-valued polynomial g/d] is irreducible in the case where d is a square-free integer and g ∈ ℤ[x] has fixed divisor d. For integer-valued polynomials with arbitrary composite denominators, so far there is no algorithmic criterion known to recognize whether they are irreducible. We describe a computational method which allows us to recognize all irreduciblexc polynomials in Int(ℤ). We present some known facts, preliminary new results and open questions.

scholarly journals On Roots of Polynomials and Algebraically Closed Fields

2017 ◽  
Vol 25 (3) ◽  
pp. 185-195 ◽  
Author(s):  
Christoph Schwarzweller
Keyword(s):  
Algebraic Theory ◽  
Polynomial Rings ◽  
Multiple Roots ◽  
Real Numbers ◽  
The Real ◽  
Closed Fields ◽  
Roots Of Polynomials ◽  
Algebraically Closed Fields ◽  
Finite Domains ◽  
Identity Theorem

Summary In this article we further extend the algebraic theory of polynomial rings in Mizar [1, 2, 3]. We deal with roots and multiple roots of polynomials and show that both the real numbers and finite domains are not algebraically closed [5, 7]. We also prove the identity theorem for polynomials and that the number of multiple roots is bounded by the polynomial’s degree [4, 6].

scholarly journals A REVIEW OF THE ONE-PARAMETER DIVISION UNDISTORTION MODEL

2021 ◽  
Vol V-1-2021 ◽  
pp. 89-96
Author(s):  
B. Erdnüß
Keyword(s):  
Image Plane ◽  
Distortion Parameter ◽  
Distortion Model ◽  
Radial Distortion ◽  
Projective Invariant ◽  
Algebraic Properties ◽  
The One ◽  
Metric Distance ◽  
Straight Lines ◽  
Line Space

Abstract. The one-parameter division undistortion model by (Lenz, 1987) and (Fitzgibbon, 2001) is a simple radial distortion model with beneficial algebraic properties that allows to reason about some problems analytically that can only be handled numerically in other distortion models. One property of this distortion model is that straight lines in the undistorted image correspond to circles in the distorted image. These circles are fully described by their center point, as the radius can be calculated from the position of the center and the distortion parameter only. This publication collects the properties of this distortion model from several sources and reviews them. Moreover, we show in this publication that the space of this center is projectively isomorphic to the dual space of the undistorted image plane, i.e. its line space. Therefore, projective invariant measurements on the undistorted lines are possible by the according measurements on the centers of the distorted circles. As an example of application, we use this to find the metric distance of two parallel straight rails with known track gauge in a single uncalibrated camera image with significant radial distortion.

scholarly journals The costructure–cosemantics adjunction for comodels for computational effects

2021 ◽  
pp. 1-46
Author(s):  
Richard Garner
Keyword(s):  
Dynamic Properties ◽  
Algebraic Theory ◽  
The Other ◽  
Algebraic Theories ◽  
And Mathematics ◽  
The One ◽  
Suitable Environment ◽  
The Way

Abstract It is well established that equational algebraic theories and the monads they generate can be used to encode computational effects. An important insight of Power and Shkaravska is that comodels of an algebraic theory $\mathbb{T}$ – i.e., models in the opposite category $\mathcal{S}\mathrm{et}^{\mathrm{op}}$ – provide a suitable environment for evaluating the computational effects encoded by $\mathbb{T}$ . As already noted by Power and Shkaravska, taking comodels yields a functor from accessible monads to accessible comonads on $\mathcal{S}\mathrm{et}$ . In this paper, we show that this functor is part of an adjunction – the “costructure–cosemantics adjunction” of the title – and undertake a thorough investigation of its properties. We show that, on the one hand, the cosemantics functor takes its image in what we term the presheaf comonads induced by small categories; and that, on the other, costructure takes its image in the presheaf monads induced by small categories. In particular, the cosemantics comonad of an accessible monad will be induced by an explicitly-described category called its behaviour category that encodes the static and dynamic properties of the comodels. Similarly, the costructure monad of an accessible comonad will be induced by a behaviour category encoding static and dynamic properties of the comonad coalgebras. We tie these results together by showing that the costructure–cosemantics adjunction is idempotent, with fixpoints to either side given precisely by the presheaf monads and comonads. Along the way, we illustrate the value of our results with numerous examples drawn from computation and mathematics.

scholarly journals THE LOGIC OF RESOURCES AND CAPABILITIES

2018 ◽  
Vol 11 (2) ◽  
pp. 371-410 ◽  
Author(s):  
MARTA BÍLKOVÁ ◽  
GIUSEPPE GRECO ◽  
ALESSANDRA PALMIGIANO ◽  
APOSTOLOS TZIMOULIS ◽  
NACHOEM WIJNBERG
Keyword(s):  
Algebraic Theory ◽  
Cut Elimination ◽  
Disjunction Property ◽  
Resources And Capabilities ◽  
The One ◽  
Key Aspects ◽  
Planning Problems ◽  
Mutually Independent ◽  
Structural Proof Theory ◽  
Subformula Property

AbstractWe introduce the logic LRC, designed to describe and reason about agents’ abilities and capabilities in using resources. The proposed framework bridges two—up to now—mutually independent strands of literature: the one on logics of abilities and capabilities, developed within the theory of agency, and the one on logics of resources, motivated by program semantics. The logic LRC is suitable to describe and reason about key aspects of social behaviour in organizations. We prove a number of properties enjoyed by LRC (soundness, completeness, canonicity, and disjunction property) and its associated analytic calculus (conservativity, cut elimination, and subformula property). These results lay at the intersection of the algebraic theory of unified correspondence and the theory of multitype calculi in structural proof theory. Case studies are discussed which showcase several ways in which this framework can be extended and enriched while retaining its basic properties, so as to model an array of issues, both practically and theoretically relevant, spanning from planning problems to the logical foundations of the theory of organizations.

scholarly journals The First Isomorphism Theorem and Other Properties of Rings

2014 ◽  
Vol 22 (4) ◽  
pp. 291-301 ◽  
Author(s):  
Artur Korniłowicz ◽  
Christoph Schwarzweller
Keyword(s):  
Principal Ideal ◽  
Polynomial Rings ◽  
Isomorphism Theorem ◽  
Principal Ideal Domain ◽  
Ideal Domain ◽  
Ring Homomorphisms ◽  
Irreducible Elements

Summary Different properties of rings and fields are discussed [12], [41] and [17]. We introduce ring homomorphisms, their kernels and images, and prove the First Isomorphism Theorem, namely that for a homomorphism f : R → S we have R/ker(f) ≅ Im(f). Then we define prime and irreducible elements and show that every principal ideal domain is factorial. Finally we show that polynomial rings over fields are Euclidean and hence also factorial

An algebraic input–output equation for planar RRRP and PRRP linkages

2020 ◽  
Vol 44 (4) ◽  
pp. 520-529 ◽  
Author(s):  
Mirja Rotzoll ◽  
M. John D. Hayes ◽  
Manfred L. Husty
Keyword(s):  
Input Output ◽  
Yield Information ◽  
Algebraic Properties ◽  
Analysis And Synthesis ◽  
Half Angle ◽  
Approximate Synthesis ◽  
Prismatic Joints ◽  
The One ◽  
Displacement Constraints ◽  
Insight Into

In this paper, the algebraic input–output (IO) equations for planar RRRP and PRRP linkages are derived by mapping the linkage displacement constraints into Study’s soma coordinates and then using tangent half-angle substitutions to transform the trigonometric into algebraic expressions. Both equations are found to be equivalent to the one that has already been derived for RRRR linkages, giving exciting new insight into kinematic analysis and synthesis of planar four-bar linkages. The algebraic properties of the IO curve equations yield information regarding the topology of the linkage, such as the sliding position limits of the prismatic joints and (or) the angle limits of the rotational joints. Additionally, the utility of the equations is successfully demonstrated with two approximate synthesis examples.

Point transfer matrices for the Schrödinger equation: the algebraic theory

1999 ◽  
Vol 129 (4) ◽  
pp. 717-732 ◽  
Author(s):  
N. A. Gordon ◽  
D. B. Pearson
Keyword(s):  
Schrödinger Equation ◽  
Transfer Matrix ◽  
Schrodinger Equation ◽  
Algebraic Theory ◽  
Complete Characterization ◽  
Transfer Matrices ◽  
Point Interactions ◽  
Singularity Point ◽  
The One ◽  
Image Position

This paper deals with the theory of point interactions for the one-dimensional Schrödinger equation. The familiar example of the δ-potential V(x) = gδ(x−x0), for which the transfer matrix across the singularity (point transfer matrix) is given byis extended to cover cases in which the transfer matrix M(z) is dependent on the (complex) spectral parameter z, and which can be obtained as limits of transfer matrices across finite intervals for sequences of approximating potentials Vn.The case of point transfer matrices polynomially dependent on z is treated in detail, with a complete characterization of such matrices and a proof of their factorization as products of point transfer matrices linearly dependent on z.The theory presented here has applications to the study of point interactions in quantum mechanics, and provides new classes of point interactions which can be obtained as limiting cases of regular potentials.

scholarly journals Factorization properties of subrings in trigonometric polynomial rings

2009 ◽  
Vol 86 (100) ◽  
pp. 123-131
Author(s):  
Tariq Shah ◽  
Ehsan Ullah
Keyword(s):  
Trigonometric Polynomial ◽  
Commutative Ring ◽  
Polynomial Rings ◽  
Trigonometric Polynomials ◽  
Ring Extension ◽  
Irreducible Elements ◽  
Euclidean Domain

We explore the subrings in trigonometric polynomial rings and their factorization properties. Consider the ring S' of complex trigonometric polynomials over the field Q(i) (see [11]). We construct the subrings S'1 , S'0 of S' such that S'1 ?S'0 ?S'. Then S'1 is a Euclidean domain, whereas S'0 is a Noetherian HFD. We also characterize the irreducible elements of S'1, S'0 and discuss among these structures the condition: Let A ?B be a unitary (commutative) ring extension. For each x ? B there exist x' ?U(B) and x'' ? A such that x = x'x''. .

ON A GROUPOID CONSTRUCTION FOR ACTIONS OF CERTAIN INVERSE SEMIGROUPS

1994 ◽  
Vol 05 (03) ◽  
pp. 349-372 ◽  
Author(s):  
ALEXANDRU NICA
Keyword(s):  
Inverse Semigroup ◽  
Compact Space ◽  
Transformation Group ◽  
Algebraic Theory ◽  
Inverse Semigroups ◽  
Discrete Groups ◽  
Crossed Products ◽  
Locally Compact ◽  
Locally Compact Space ◽  
The One

We consider a version of the notion of F-inverse semigroup (studied in the algebraic theory of inverse semigroups). We point out that an action of such an inverse semigroup on a locally compact space has associated a natural groupoid construction, very similar to the one of a transformation group. We discuss examples related to Toeplitz algebras on subsemigroups of discrete groups, to Cuntz-Krieger algebras, and to crossed-products by partial automorphisms in the sense of Exel.

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