Unique Factorization Research Articles

The bilateral birth–death chain generated by the associated Jacobi polynomials

AbstractWe give a probabilistic interpretation of the associated Jacobi polynomials, which can be constructed from the three‐term recurrence relation for the classical Jacobi polynomials by shifting the integer index n by a real number t. Under certain restrictions, this will give rise to a doubly infinite tridiagonal stochastic matrix, which can be interpreted as the one‐step transition probability matrix of a discrete‐time bilateral birth–death chain with state space on . We also study the unique UL and LU stochastic factorizations of the transition probability matrix, as well as the discrete Darboux transformations and corresponding spectral matrices. Finally, we use all these results to provide an urn model on the integers for the associated Jacobi polynomials.

Unique Symbolic Factorization for Fast Contingency Analysis Using Full Newton–Raphson Method

Contingency analysis plays an important role in assessing the static security of a network. Its purpose is to check whether a system can operate safely when some elements are out of service. In a real-time application, the computational time required to perform the calculation is paramount for operators to take immediate actions to prevent cascading outages. Therefore, the numerical performance of the contingency analysis is the main focus of this current research. In power flow calculation, when solving the network equations with a sparse matrix solver, most of the time is spent factorizing the Jacobian matrix. In terms of computation time, the symbolic factorization is the costliest operation in the LU (Lower-upper) factorization process. This paper proposes a novel method to perform the calculation with only one symbolic factorization using a full Newton–Raphson-based generic formulation and modular approach (GFMA). The symbolic factorization retained can be used during the iterations of any power flow contingency scenario. A computer study demonstrates that reusing the same symbolic factorization greatly reduces computation time and improves numerical performance. Power system security assessment under N-1 and N-2 contingency conditions is performed for the IEEE standard 54-bus and 108-bus to evaluate the numerical performance of the proposed method. A comparison with the conventional power flow method shows that the time required for the analysis is shortened considerably, with a minimum gain of 228%. The comparative analysis demonstrates that the proposed solution has better numerical performance for large-scale networks.

Factorization in additive monoids of evaluation polynomial semirings

For a positive real α, we can consider the additive submonoid M of the real line that is generated by the nonnegative powers of α. When α is transcendental, M is a unique factorization monoid. However, when α is algebraic, M may not be atomic, and even when M is atomic, it may contain elements having more than one factorization (i.e., decomposition as a sum of irreducibles). The main purpose of this paper is to study the phenomenon of multiple factorizations inside M. When α is algebraic but not rational, the arithmetic of factorizations in M is highly interesting and complex. In order to arrive to that conclusion, we investigate various factorization invariants of M, including the sets of lengths, sets of Betti elements, and catenary degrees. Our investigation gives continuity to recent studies carried out by Chapman et al. in 2020 and by Correa-Morris and Gotti in 2022.

Length-factoriality and pure irreducibility

An atomic monoid M is called length-factorial if for every non-invertible element , no two distinct factorizations of x into irreducibles have the same length (i.e., number of irreducible factors, counting repetitions). The notion of length-factoriality was introduced by J. Coykendall and W. Smith in 2011 under the term “other-half-factoriality”: they used length-factoriality to provide a characterization of unique factorization domains. In this paper, we study length-factoriality in the more general context of commutative, cancellative monoids. In addition, we study factorization properties related to length-factoriality, namely, the PLS property (recently introduced by Chapman et al.) and bi-length-factoriality in the context of semirings.

Unique Factorization for Ideals in the Ring 𝑍[√−5

Unique Factorization for Ideals in the Ring 𝑍[√−5

Topological Explorations on the Fundamental Theorem of Arithmetic

In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and investigate non-standard questions. The ideas are arranged as a series of questions for students to explore, with expositions to the instructor on how to guide them in looking for answers. The investigations culminate in a satisfying outcome: that partition topologies are the same as unique factorization topologies, those that satisfy our topological version of the Fundamental Theorem of Arithmetic.

Efficient Identification of Butterfly Sparse Matrix Factorizations

Fast transforms correspond to factorizations of the form , where each factor is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any matrix having the so-called butterfly structure admits an essentially unique factorization into butterfly factors (where ), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size . Computing such factorizations costs , which is on the order of dense matrix-vector multiplication, while the obtained factorizations enable fast matrix-vector multiplications and have the potential to be applied to compress deep neural networks.

Direct Product of Finite Abelian Group

In this project, finite abelian groups with some theoretical and algebraic structures are considered. The order of each group is factorized completely with factor of higher multiplicity where necessary. This unique factorization will allow for a way of building new groups and understanding a given group better. Essentially, it provides a way of relating the given group to the direct products of some of its subgroups. Finally, it also reveals how a group of a finite order is isomorphically related to one of the direct products satisfying certain relatively prime condition.

On a certain determinant for a U.F.D.

Let $R$ be a unique factorization domain (U.F.D.). Let $P=\{p_i\}$ be a prime system of $R$, that is, $P$ is a well-ordered set of non-associate prime elements of $R$ and $P$ is complete. An element $x\in R$ is called a $P$-number if $x$ is a product of e

Least-Squares Methods for Nonnegative Matrix Factorization Over Rational Functions

Nonnegative matrix factorization (NMF) models are widely used to analyze linearly mixed nonnegative data. When the data is made of samplings of continuous signals, the factors in NMF can be constrained to be samples of nonnegative rational functions. This leads to a fairly general model referred to as NMF using rational functions (R-NMF). We first show that, unlike NMF, R-NMF possesses an essentially unique factorization under mild assumptions, which is crucial in applications where the ground-truth factors need to be recovered, as in blind source separation problems. Then we present different approaches to solve R-NMF : the R-HANLS, R-ANLS and R-NLS methods. In our tests, no method significantly outperforms the others in all cases, and all three methods offer a different trade-off between solution accuracy and computational requirements. Indeed, while R-HANLS is fast and accurate for large problems, R-ANLS is more accurate, but also more resources demanding, both in time and memory and R-NLS is even more accurate but only for small problems. Then, crucially we show that R-NMF models outperforms NMF in various tasks including the recovery of semi-synthetic continuous signals, and a classification problem of real hyperspectral signals.

Transverse Λ polarization in e+e− processes within a TMD factorization approach and the polarizing fragmentation function

We perform a re-analysis of Belle data for the transverse Λ and overline{Lambda} polarization in e+e− annihilation processes within a transverse momentum dependent (TMD) factorization approach. We consider two data sets, one referring to the associated production of Λ’s with a light unpolarized hadron in an almost back-to-back configuration, and one for the inclusive Λ production, with the reconstruction of the thrust axis. We adopt the Collins-Soper-Sterman framework and employ the recent formulation on the factorization of single-inclusive hadron production. This extends a previous phenomenological study carried out in a more simplified TMD approach, leading to a new extraction of the polarizing fragmentation function (FF). While confirming several features of the previous analysis, here we include the proper QCD scale dependence of this TMD FF and carefully exploit the role of its nonperturbative component. The compatibility, within a unique TMD factorization scheme, of double and single-inclusive hadron production is discussed in detail. Moreover, we consider another data set for inclusive Λ production, at much larger energies, from the OPAL Collaboration, in order to test the consistency of the entire approach. Finally, we elaborate on some fundamental issues like the role of SU(2) isospin symmetry and the heavy quark contributions.

Generalized power series with a limited number of factorizations

Several past authors have studied questions related to unique factorization of generalized power series. Here we examine the broader topic of generalized power series that (in a sense we will make precise) have a limited number of factorizations. Special cases of our general results include new results about “limited factorization” in (Laurent) power series rings, (Laurent) polynomial rings, and the “large polynomial rings” of Halter-Koch. Along the way to our main results, we study Krull domains and Cohen–Kaplansky rings of generalized power series and give several slight extensions to the fundamental ring theory of generalized power series.

The Green-Tao Theorem and the Infinitude of Primes in Domains

We first prove an elementary analogue of the Green-Tao Theorem. The celebrated Green-Tao Theorem states that there are arbitrarily long arithmetic progressions in the set of prime numbers. In fact, we show the Green-Tao Theorem for polynomial rings over integral domains with several variables. Using the Generalized Polynomial van der Waerden Theorem, we also prove that in an infinite unique factorization domain, if the cardinality of the set of units is strictly less than that of the domain, then there are infinitely many prime elements. Moreover, we deduce the infinitude of prime numbers in the positive integers using polynomial progressions of length three. In addition, using unit equations, we provide two more proofs of the infinitude of prime numbers. Finally, we give a new proof of the divergence of the sum of reciprocals of all prime numbers.

On the additive structure of algebraic valuations of polynomial semirings

In this paper, we study factorizations in the additive monoids of positive algebraic valuations N0[α] of the semiring of polynomials N0[X] using a methodology introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in 1990. A cancellative commutative monoid is atomic if every non-invertible element factors into irreducibles. We begin by determining when N0[α] is atomic, and we give an explicit description of its set of irreducibles. An atomic monoid is a finite factorization monoid (FFM) if every element has only finitely many factorizations (up to order and associates), and it is a bounded factorization monoid (BFM) if for every element there is a bound for the number of irreducibles (counting repetitions) in each of its factorizations. We show that, for the monoid N0[α], the property of being a BFM and the property of being an FFM are equivalent to the ascending chain condition on principal ideals (ACCP). Finally, we give various characterizations for N0[α] to be a unique factorization monoid (UFM), two of them in terms of the minimal polynomial of α. The properties of being finitely generated, half-factorial, and length-factorial are also investigated along the way.

Arithmetical functions associated with conjugate pairs of sets under regular convolutions

Two subsets P and Q of the set of positive integers is said to form a conjugate pair if each positive integer n possesses a unique factorization of the form n = ab, a ∈ P, b ∈ Q. In this paper we generalize conjugate pairs of sets to the setting of regular convolutions and study associated arithmetical functions. Particular attention is paid to arithmetical functions associated with k-free integers and k-th powers under regular convolution.

Irreducibility of integer-valued polynomials in several variables

Let \(\underline{S}\) be an arbitrary subset of \(R^n\) where R is a domain with the field of fractions \(\mathbb {K}\). Denote the ring of polynomials in n variables over \(\mathbb {K}\) by \(\mathbb {K}[\underline{x}] \). The ring of integer-valued polynomials over \(\underline{S}\), denoted by Int\((\underline{S},R)\), is defined as the set of the polynomials of \(\mathbb {K}[\underline{x}] \), which maps \(\underline{S}\) to R. In this article, we study the irreducibility of the polynomials of Int\((\underline{S},R)\) for the first time in the case when R is a Unique Factorization Domain. We also show that our results remain valid when R is a Dedekind domain or, sometimes, any domain.

Constellations with range and IS-categories

Constellations are asymmetric generalisations of categories. Although they are not required to possess a notion of range, many natural examples do. These include commonly occurring constellations related to concrete categories (since they model surjective morphisms), and also others arising from quite different sources, including from well-studied classes of semigroups . We show how constellations with a well-behaved range operation are nothing but ordered categories with restrictions. We characterise abstractly those categories that are canonical extensions of constellations with range, as so-called IS-categories. Such categories contain distinguished subcategories of insertions (which are monomorphisms) and surjections (in general different to the epimorphisms) such that each morphism admits a unique factorisation into a surjection followed by an insertion. Most familiar concrete categories are IS-categories, and we show how some of the well-known properties of these categories arise from the fact that they are IS-categories. For appropriate choices of morphisms in each, the category of IS-categories is shown to be equivalent to the category of constellations with range.

On Unique Factorization Modules: A Submodule Approach

Let M be a torsion-free module over an integral domain D. We define a concept of a unique factorization module in terms of v-submodules of M. If M is a unique factorization module (UFM), then D is a unique factorization domain. However, the converse situation is not necessarily to be held, and we give four different characterizations of unique factorization modules. Further, it is shown that the concept of the UFM is equivalent to Nicolas’s UFM, which is defined in terms of irreducible elements of D and M.

Ternary arithmetic, factorization, and the class number one problem

Ordinary multiplication of natural numbers can be generalized to a ternary operation by considering discrete volumes of lattice hexagons. With this operation, a natural notion of ‘3-primality’ -primality with respect to ternary multiplication- is defined, and it turns out that there are very few 3-primes. They correspond to imaginary quadratic fields Q(√-n), n > 0, with odd discriminant and whose ring of integers admits unique factorization. We also describe how to determine representations of numbers as ternary products and related algorithms for usual primality testing and integer factorization.

Modular Categories with Transitive Galois Actions

In this paper, we study modular categories whose Galois group actions on their simple objects are transitive. We show that such modular categories admit unique factorization into prime transitive factors. The representations of $SL_2(\mathbb{Z})$ associated with transitive modular categories are proven to be minimal and irreducible. Together with the Verlinde formula, we characterize prime transitive modular categories as the Galois conjugates of the adjoint subcategory of the quantum group modular category $\mathcal{C}(\mathfrak{sl}_2,p-2)$ for some prime $p > 3$. As a consequence, we completely classify transitive modular categories. Transitivity of super-modular categories can be similarly defined. A unique factorization of any transitive super-modular category into s-simple transitive factors is obtained, and the split transitive super-modular categories are completely classified.