Unique Factorization Theorem Research Articles

Hurwitz enumeration problem through the perspective of Cayley graph

The Hurwitz enumeration problem studies how to determine the Hurwitz number for a branch profile, which counts the number of ways a per mutation can be factored into transpositions. In this paper, we consider the length of strictly monotone factorization from the perspective of Cayley graph theory. We represent the factorization problem using a Cayley graph, where vertices are permutations and edges are transpositions. Our focus is proving the unique monotone factorization theorem, which states that for a given permutation, there is only one monotone factorization of minimal length. To prove this, we employ inductive arguments on the structure of the Cayley graph. The key insight is using the connectivity of the graph and properties of shortest paths to characterize the uniqueness of the minimal factorization. This inductive approach allows us to rigorously connect the combinatorial Hurwitz problem to foundational graph concepts. Overall, this paper makes important theoretical advances in enumerating Hurwitz numbers by using Cayley graphs and induction to prove the novel unique monotone factorization theorem. The connections drawn between combinatorics, graphs, and inductive proofs are technically innovative. This theoretical foundation will hopefully stimulate further research into the deep links between the Hurwitz problem and other branches of mathematics.

ON HYPERBOLIC INTEGERS

The purpose of this paper is to investigate an ℓ-ring structure of algebraic integers from an arithmetic point of view. We endow the algebra D of hyperbolic numbers with its standard f-algebra structure [7]. We introduce the ring of hyperbolic integers Zh as a sub f-ring of the ring ZD of integers of D. Next, we prove that Zh is the unique, up to ring isomorphism, Archimedean f-ring of quadratic integers. Our study focuses on arithmetic properties of Zh related to its latticeordered structure. We show that many basic properties of the ring of integers Z such as primes, unique factorization theorem and the notions of floor and ceiling functions can be extended to Zh. A surprising fact is that prime numbers seen as hyperbolic integers are semiprimes. We also obtain some properties of hyperbolic Gaussian integers. As an application, we discuss the Dirichlet divisor problem using hyperbolic intervals.

Higher order Kirillov--Reshetikhin modules for 𝐔 q (A n (1)), imaginary modules and monoidal categorification

Abstract We study the family of irreducible modules for quantum affine 𝔰 ⁢ 𝔩 n + 1 {\mathfrak{sl}_{n+1}} whose Drinfeld polynomials are supported on just one node of the Dynkin diagram. We identify all the prime modules in this family and prove a unique factorization theorem. The Drinfeld polynomials of the prime modules encode information coming from the points of reducibility of tensor products of the fundamental modules associated to A m {A_{m}} with m ≤ n {m\leq n} . These prime modules are a special class of the snake modules studied by Mukhin and Young. We relate our modules to the work of Hernandez and Leclerc and define generalizations of the category 𝒞 - {\mathscr{C}^{-}} . This leads naturally to the notion of an inflation of the corresponding Grothendieck ring. In the last section we show that the tensor product of a (higher order) Kirillov–Reshetikhin module with its dual always contains an imaginary module in its Jordan–Hölder series and give an explicit formula for its Drinfeld polynomial. Together with the results of [D. Hernandez and B. Leclerc, A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules, J. Eur. Math. Soc. (JEMS) 18 2016, 5, 1113–1159] this gives examples of a product of cluster variables which are not in the span of cluster monomials. We also discuss the connection of our work with the examples arising from the work of [E. Lapid and A. Mínguez, Geometric conditions for □ \square -irreducibility of certain representations of the general linear group over a non-archimedean local field, Adv. Math. 339 2018, 113–190]. Finally, we use our methods to give a family of imaginary modules in type D 4 {D_{4}} which do not arise from an embedding of A r {A_{r}} with r ≤ 3 {r\leq 3} in D 4 {D_{4}} .

Unique factorization theorem for pure quantum states

In this paper we establish a unique factorization theorem for pure quantum states expressed in computational basis. We show that there always exists unique factorization for any given N-qubit pure quantum state in terms of the tensor product of non-factorable or ``prime'' pure quantum states. This result is based on a simple criterion: Given N-qubit pure quantum state in computational basis can be factorized as the tensor product of an m-qubit pure quantum state and an n-qubit pure quantum state, where (m + n) = N, if and only if the rank of the certain associated matrix is equal to one. This simple criterion leads to a factorization algorithm which when applied to an N-qubit pure quantum state factorizes that state into the tensor product of non-factorable or ``prime'' pure quantum states. This paper shows that for any given N-qubit pure quantum state the said factorization always ``exists'' and is ``unique''. We demonstrated our work here on a computational basis. PACS Number: 03.67.Mn, 03.65.Ca, 03.65.Ud

Prime II1 factors arising from actions of product groups

We prove that any II1 factor arising from a free ergodic probability measure preserving action Γ↷X of a product Γ=Γ1×…×Γn of icc hyperbolic, free product or wreath product groups is prime, provided Γi↷X is ergodic, for any 1≤i≤n. We also completely classify all the tensor product decompositions of a II1 factor associated to a free ergodic probability measure preserving action of a product of icc, hyperbolic, property (T) groups. As a consequence, we derive a unique prime factorization result for such II1 factors. Finally, we obtain a unique prime factorization theorem for a large class of II1 factors which have property Gamma.

Ozawa's class𝒮for locally compact groups and unique prime factorization of group von Neumann algebras

AbstractWe study class𝒮for locally compact groups. We characterize locally compact groups in this class as groups having an amenable action on a boundary that is small at infinity, generalizing a theorem of Ozawa. Using this characterization, we provide new examples of groups in class𝒮and prove a unique prime factorization theorem for group von Neumann algebras of products of locally compact groups in this class. We also prove that class𝒮is a measure equivalence invariant.

Products of abstract polytopes

Given two convex polytopes, the join, the cartesian product and the direct sum of them are well understood. In this paper we extend these three kinds of products to abstract polytopes and introduce a new product, called the topological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of the automorphism groups of the factors and show that (non trivial) products are almost never regular or two-orbit polytopes. We finish the paper by studying the monodromy group of a product, show that such a group is always an extension of a symmetric group, and give some examples in which this extension splits.

The Cartesian product of graphs with loops

We extend the definition of the Cartesian product to graphs with loops and show that the Sabidussi–Vizing unique factorization theorem for connected finite simple graphs still holds in this context for all connected finite graphs with at least one unlooped vertex. We also prove that this factorization can be computed in O(m) time, where m is the number of edges of the given graph.

Strong products of hypergraphs: Unique prime factorization theorems and algorithms

It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation.In this contribution, we show that every connected thin hypergraph H has a unique prime factorization with respect to the normal and strong (hypergraph) product. Both products coincide with the usual strong graph product whenever H is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as a natural generalization of the Cartesian skeleton of graphs and prove that it is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian skeleton of hypergraphs can be determined in O(|E|2) time and that the PFD can be computed in O(|V|2|E|) time, for hypergraphs H=(V,E) with bounded degree and bounded rank.

Integrally closed ideals in regular local rings of dimension two

Let I be an integrally closed m -primary ideal of a two-dimensional regular local ring ( R , m , k ) . By Zariski’s unique factorization theorem, the ideal I factors as I 1 s 1 ⋯ I h s h , where the s i are positive integers and the I i are distinct simple integrally closed ideals. For each integer i with 1 ≤ i ≤ h , let V i denote the unique Rees valuation ring associated with I i , and let v i denote the Rees valuation associated with V i . Assume that the field k is relatively algebraically closed in the residue field k ( v i ) of V i . Let J = ( a , b ) R be a reduction of I and let ( a b ) v i ¯ denote the image of a b in the residue field k ( v i ) of V i . We prove that [ k ( v i ) : k ( ( a b ) v i ¯ ) ] = s i . Assume that ( e , f ) R is a reduction of I i . We prove that there exists a minimal generating set { g e s i , g e s i − 1 f , … , g f s i , ξ s i + 1 , … , ξ r } for I , where g ∈ ∏ j ≠ i I j s j with g V i = ( ∏ j ≠ i I j s j ) V i and v i ( ξ p ) > v i ( I ) for p with s i + 1 ≤ p ≤ r . There exists Q i ∈ Min ( m R [ I t ] ) such that V i = R [ I t ] Q i ∩ Q ( R ) . We prove that the quotient ring R [ I t ] / Q i is normal Cohen–Macaulay with minimal multiplicity at its maximal homogeneous ideal with this multiplicity being s i . In particular, R [ I t ] / Q i is regular if and only if s i = 1 .

Unique factorization theorem for object-systems

Unique factorization theorem for object-systems

Factorisation of Snarks

We develop a theory of factorisation of snarks — cubic graphs with edge-chromatic number $4$ — based on the classical concept of the dot product. Our main concern are irreducible snarks, those where the removal of every nontrivial edge-cut yields a $3$-edge-colourable graph. We show that if an irreducible snark can be expressed as a dot product of two smaller snarks, then both of them are irreducible. This result constitutes the first step towards the proof of the following "unique-factorisation" theorem: Every irreducible snark $G$ can be factorised into a collection $\{H_1,\dots,H_n\}$ of cyclically $5$-connected irreducible snarks such that $G$ can be reconstructed from them by iterated dot products. Moreover, such a collection is unique up to isomorphism and ordering of the factors regardless of the way in which the decomposition was performed. The result is best possible in the sense that it fails for snarks that are close to being irreducible but themselves are not irreducible. Besides this theorem, a number of other results are proved. For example, the unique-factorisation theorem is extended to the case of factorisation with respect to a preassigned subgraph $K$ which is required to stay intact during the whole factorisation process. We show that if $K$ has order at least $3$, then the theorem holds, but is false when $K$ has order $2$.

On uniqueness of a general factorization of graph properties

A graph property is any class of simple graphs, which is closed under isomorphisms. Let H be a given graph on vertices v1, …, vn. For graph properties 𝒫1, …, 𝒫n, we denote by H[𝒫1, …, 𝒫n] the class of those (𝒫1, …, 𝒫n) -partitionable graphs G, with a corresponding vertex partition (V1, …, Vn), for which an edge {xi, xj} with xi∈Vi and xj∈Vj implies the existence of the edge {vi, vj} in the graph H. The problem of the unique description of a graph property 𝒫 in the form H[𝒫1, …, 𝒫n] is investigated for 𝒫, 𝒫1, …, 𝒫n being from the class La of all graph properties closed under taking disjoint unions and subgraphs. The unique factorization theorems obtained in the paper generalize known results of this type bringing together ∘ -reducibility over La and ∨ -reducibility in the lattice (La, ⊆). There is also offered a new insight into the modular decomposition tree for a graph. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 48–64, 2009

A UNIQUE FACTORIZATION IN COMMUTATIVE MÖBIUS MONOIDS

We show that any commutative Möbius monoid satisfies a unique factorization theorem and thus possesses arithmetical properties similar to those of the multiplicative semigroup of positive integers. Particular attention is paid to standard examples, which arise from the bicyclic semigroup and the multiplicative analogue of the bicyclic semigroup. The second example shows that the Fundamental Theorem of Arithmetic is a special case of the unique factorization theorem in commutative Möbius monoids. As an application, we study generalized arithmetical functions defined on an arbitrary commutative Möbius monoid.

Adjacent integrally closed ideals in 2-dimensional regular local rings

Let ( A , m ) be a 2-dimensional regular local ring with algebraically closed residue field. Zariski's Unique Factorization Theorem asserts that every integrally closed (complete) m -primary ideal I is uniquely factored into a product of powers of simple complete ideals I = P 1 a 1 P 2 a 2 ⋯ P n a n , where P i is a simple complete ideal for a i ⩾ 1 and n ⩾ 1 . In this paper, we give a new characterization for a simple complete ideal in terms of adjacent complete ideals. We also give a characterization for a complete ideal I to have finitely many adjacent complete m -primary over-ideals. Namely, we show that I is simple if and only if it has a unique adjacent over-ideal and that I = P 1 a 1 P 2 a 2 ⋯ P n a n has only finitely many complete adjacent over-ideals if and only if a i = 1 for every i and there are no proximity relations among P i .

A unique factorization theorem for matroids

We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids M and N is maximal with respect to the weak order among matroids having M as a submatroid, with complementary contraction equal to N . Any minor of the free product of M and N is a free product of a repeated truncation of the corresponding minor of M with a repeated Higgs lift of the corresponding minor of N . We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra K { M } of a family of matroids M that is closed under formation of minors and free products: namely, K { M } is cofree, cogenerated by the set of irreducible matroids belonging to M .

Graded rings associated with contracted ideals

The study of the ideals in a regular local ring (R,m) of dimension 2 has a long and important tradition dating back to the fundamental work of Zariski [ZS]. More recent contributions are due to several authors including Cutkosky, Huneke, Lipman, Rees, Sally, and Teissier among others, see [C1,C2,H,HS,L,LT,R]. One of the main result in this setting is the unique factorization theorem for complete (i.e., integrally closed) ideals proved originally by Zariski [ZS, Theorem 3, Appendix 5]. It asserts that any complete ideal can be factorized as a product of simple complete ideals in a unique way (up to the order of the factors). By definition, an ideal is simple if it cannot be written as a product of two proper ideals. Another important property of a complete ideal I is that its reduction number is 1 which in turns implies that the associated graded ring grI (R) is Cohen–Macaulay and its Hilbert series is well-understood; this is due to Lipman and Teissier [LT], see also [HS]. The class of contracted ideals plays an important role in the original work of Zariski as well as in the work of Huneke [H]. An ideal I of R is contracted if I = R ∩ IR[m/x] for some x ∈ m \m2. Any complete ideal is contracted but not the other way round. The associated graded ring grI (R) of a contracted ideal I need not be Cohen–Macaulay and its Hilbert series can be very complicated.

Deformation and rigidity of simplicial group actions on trees

We study a notion of deformation for simplicial trees with group actions (G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a deformation if there is a finite sequence of collapse and expansion moves joining them. We show that this relation on the set of G-trees has several characterizations, in terms of dynamics, coarse geometry, and length functions. Next we study the deformation space of a fixed G-tree X. We show that if X is `strongly slide-free' then it is the unique reduced tree in its deformation space. These methods allow us to extend the rigidity theorem of Bass and Lubotzky to trees that are not locally finite. This yields a unique factorization theorem for certain graphs of groups. We apply the theory to generalized Baumslag-Solitar groups and show that many have canonical decompositions. We also prove a quasi-isometric rigidity theorem for strongly slide-free G-trees.

CONES OF CURVES AND OF LINE BUNDLES ON SURFACES ASSOCIATED WITH CURVES HAVING ONE PLACE AT INFINITY

Let V be a pencil of curves in ${\bf P}^2$ with one place at infinity, and $X \longrightarrow {\bf P}^2$ the minimal composition of point blow-ups eliminating its base locus. We study the cone of curves and the cones of numerically effective and globally generated line bundles on X. It is proved that all of these cones are regular. In particular, this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves. The surfaces in this class have non-numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic. An application is a global version on X of Zariski's unique factorization theorem for complete ideals. We also define invariants of the semigroup of globally generated line bundles on X depending only on the topology of V at infinity.2000 Mathematical Subject Classification: primary 14C20; secondary 14E05.

Fastfinger: A study into the use of compressed residue pair separation matrices for protein sequence comparison

Protein sequences are diverse in size and in content meaningful to researchers. They are rich in what seems to be “noise,” or aspects of lesser interest that obscure clearer core features required to establish true relatedness and function. This paper represents part of a larger study that explores the possible efficient use and storage of “fingers” for protein sequence analysis, i.e., matrices of uniform size and shape that can “stand for” protein sequences by making more explicit the essential aspects of protein sequence pattern information. The essence of the study relates to data compression. Compression invokes an interesting alternative idea of pattern—the concept of “primeness” as in number theory is used to create the notion of an irreducible and potentially recurrent pattern element, and then this philosophy is mapped onto number theory by the unique factorization theorem, in order to define a novel measure of pattern difference. Other possible approaches are also discussed. Because compression and other approximations involve information loss, this is also a study of performance in the face of such loss. Because of the effects of this loss, no claims are made that encourage replacement of established sequence comparison methods, but the concept may have value in a number of applications within, and outside, molecular biology.