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Let G be a semiabelian variety defined over an algebraically closed field K of characteristic 0. Let Φ : G ⇢ G be a dominant rational self-map. Assume that an iterate Φ m : G → G is regular for some m ⩾ 1 and that there exists no non-constant homomorphism τ : G ⟶ G 0 of semiabelian varieties such that τ ∘ Φ m k = τ for some k ⩾ 1. We show that under these assumptions Φ itself must be a regular. We also prove a variant of this assertion in prime characteristic and present examples showing that our results are sharp. [ABSTRACT FROM AUTHOR]

Let X be an algebraic variety equipped with a dominant rational map ϕ : X ⤏ X . A new quantity measuring the interaction of (X , ϕ) with trivial dynamical systems is introduced; the stabilised algebraic dimension of (X , ϕ) captures the maximum number of new algebraically independent invariant rational functions on (X × Y , ϕ × ψ) , as ψ : Y ⤏ Y ranges over all dominant rational maps on algebraic varieties. It is shown that this birational invariant agrees with the maximum dim X ′ where (X , ϕ) ⤏ (X ′ , ϕ ′) is a dominant rational equivariant map and ϕ ′ is part of an algebraic group action on X ′ . As a consequence, it is deduced that if some cartesian power of (X , ϕ) admits a nonconstant invariant rational function, then already the second cartesian power does. [ABSTRACT FROM AUTHOR]

Let G$G$ be a semiabelian variety defined over an algebraically closed field K$K$, endowed with a rational self‐map Φ$\Phi$. Let α∈G(K)$\alpha \in G(K)$ and let Γ⊆G(K)$\Gamma \subseteq G(K)$ be a finitely generated subgroup. We show that the set {n∈N:Φn(α)∈Γ}$\lbrace n\in \mathbb {N} \colon \Phi ^n(\alpha)\in \Gamma \rbrace$ is a union of finitely many arithmetic progressions along with a set S$S$ of Banach density equal to 0$\hskip.001pt 0$. In addition, assuming that Φ$\Phi$ is regular, we prove that the set S$S$ must be finite. [ABSTRACT FROM AUTHOR]

We give examples of birational selfmaps of Pd,d⩾3$\mathbb {P}^d, d \geqslant 3$, whose dynamical degree is a transcendental number. This contradicts a conjecture by Bellon and Viallet. The proof uses a combination of techniques from algebraic dynamics and diophantine approximation. [ABSTRACT FROM AUTHOR]

Let L be a field of characteristic zero, let h:\mathbb {P}^1\to \mathbb {P}^1 be a rational map defined over L, and let c\in \mathbb {P}^1(L). We show that there exists a finitely generated subfield K of L over which both c and h are defined along with an infinite set of inequivalent non-archimedean completions K_{\mathfrak {p}} for which there exists a positive integer a=a(\mathfrak {p}) with the property that for i\in \{0,\ldots,a-1\} there exists a power series g_i(t)\in K_{\mathfrak {p}}[[t]] that converges on the closed unit disc of K_{\mathfrak {p}} such that h^{an+i}(c)=g_i(n) for all sufficiently large n. As a consequence we show that the dynamical Mordell-Lang conjecture holds for split self-maps (h,g) of \mathbb {P}^1 \times X with g an étale self-map of a quasiprojective variety X. [ABSTRACT FROM AUTHOR]

We study the asymptotic growth of coefficients of Mahler power series with algebraic coefficients, as measured by their logarithmic Weil height. We show that there are five different growth behaviors, all of which being reached. Thus, there are gaps in the possible growths. In proving this height gap theorem, we find that a k-Mahler function is k-regular if and only if its coefficients have height in O.log n/. Moreover, we deduce that, over an arbitrary ground field of characteristic 0, a k-Mahler function is k-automatic if and only if its coefficients belong to a finite set. As a by-product of our results, we also recover a conjecture of Becker which was recently settled by Bell, Chyzak, Coons, and Dumas. [ABSTRACT FROM AUTHOR]

We prove a general criterion for an irrational power series f(z)=\sum _{n=0}^{\infty }a_nz^n with coefficients in a number field K to admit the unit circle as a natural boundary. As an application, let F be a finite field, let d be a positive integer, let A\in M_d(F[t]) be a d\times d-matrix with entries in F[t], and let \zeta _A(z) be the Artin-Mazur zeta function associated to the multiplication-by-A map on the compact abelian group F((1/t))^d/F[t]^d. We provide a complete characterization of when \zeta _A(z) is algebraic and prove that it admits the circle of convergence as a natural boundary in the transcendence case. This is in stark contrast to the case of linear endomorphisms on \mathbb {R}^d/\mathbb {Z}^d in which Baake, Lau, and Paskunas [Monatsh. Math. 161 (2010), pp. 33–42] prove that the zeta function is always rational. Some connections to earlier work of Bell, Byszewski, Cornelissen, Miles, Royals, and Ward are discussed. Our method uses a similar technique in recent work of Bell, Nguyen, and Zannier [Amer. Math. Soc. 373 (2020), pp. 4889–4906] together with certain patching arguments involving linear recurrence sequences. [ABSTRACT FROM AUTHOR]

A ring has bounded factorizations if every cancellative nonunit a ∈ R can be written as a product of atoms and there is a bound λ (a) on the lengths of such factorizations. The bounded factorization property is one of the most basic finiteness properties in the study of non-unique factorizations. Every commutative noetherian domain has bounded factorizations, but it is open whether such a result holds in the noncommutative setting. We provide sufficient conditions for a noncommutative noetherian prime ring to have bounded factorizations. Moreover, we construct a (noncommutative) finitely presented semigroup algebra that is an atomic domain but does not satisfy the ascending chain condition on principal right or left ideals (ACCP), whence it does not have bounded factorizations. [ABSTRACT FROM AUTHOR]

Mutations in the breast cancer susceptibility gene, BRCA2, greatly increase an individual’s lifetime risk of developing breast and ovarian cancers. BRCA2 suppresses tumor formation by potentiating DNA repair via homologous recombination. Central to recombination is the assembly of a RAD51 nucleoprotein filament, which forms on single-stranded DNA (ssDNA) generated at or near the site of chromosomal damage. However, replication protein-A (RPA) rapidly binds to and continuously sequesters this ssDNA, imposing a kinetic barrier to RAD51 filament assembly that suppresses unregulated recombination. Recombination mediator proteins—of which BRCA2 is the defining member in humans—alleviate this kinetic barrier to catalyze RAD51 filament formation. We combined microfluidics, microscopy, and micromanipulation to directly measure both the binding of full-length BRCA2 to— and the assembly of RAD51 filaments on—a region of RPA-coated ssDNA within individual DNA molecules designed to mimic a resected DNA lesion common in replication-coupled recombinational repair. We demonstrate that a dimer of RAD51 is minimally required for spontaneous nucleation; however, growth self-terminates below the diffraction limit. BRCA2 accelerates nucleation of RAD51 to a rate that approaches the rapid association of RAD51 to naked ssDNA, thereby overcoming the kinetic block imposed by RPA. Furthermore, BRCA2 eliminates the need for the rate-limiting nucleation of RAD51 by chaperoning a short preassembled RAD51 filament onto the ssDNA complexed with RPA. Therefore, BRCA2 regulates recombination by initiating RAD51 filament formation. [ABSTRACT FROM AUTHOR]

Let |$X$| be a projective variety and let |$f$| be a dominant endomorphism of |$X$|⁠ , both of which are defined over a number field |$K$|⁠. We consider a question of the 2nd author, Meng, Shibata, and Zhang, who asks whether the tower of |$K$| -points |$Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots $| eventually stabilizes, where |$Y\subset X$| is a subvariety invariant under |$f$|⁠. We show this question has an affirmative answer when the map |$f$| is étale. We also look at a related problem of showing that there is some integer |$s_0$|⁠ , depending only on |$X$| and |$K$|⁠ , such that whenever |$x, y \in X(K)$| have the property that |$f^{s}(x) = f^{s}(y)$| for some |$s \geqslant 0$|⁠ , we necessarily have |$f^{s_{0}}(x) = f^{s_{0}}(y)$|⁠. We prove this holds for étale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on |${\mathbb {P}}^1$| where we allow for composition by multiple different maps |$f_1,\dots ,f_r$|⁠. [ABSTRACT FROM AUTHOR]

Given a finitely generated group with generating set S , we study the cogrowth sequence, which is the number of words of length n over the alphabet S that are equal to the identity in the group. This is related to the probability of return for walks on the corresponding Cayley graph. Muller and Schupp proved the generating function of the sequence is algebraic when G has a finite-index-free subgroup (using a result of Dunwoody). In this work, we make this result effective for free products of finite groups: we determine bounds for the degree and height of the minimal polynomial of the generating function, and determine the minimal polynomial explicitly for some families of free products. Using these results we are able to prove that a gap theorem holds: if S is a finite symmetric generating set for a group G and if a n denotes the number of words of length n over the alphabet S that are equal to 1 then lim sup n a n 1 / n is either 1 , 2 or at least 2 2 . [ABSTRACT FROM AUTHOR]

Cognitive-communication disorders resulting from traumatic brain injury may cause speakers to produce excessive verbal output, i.e., to be verbose. There is little evidence on the specific behaviours through which speakers achieve verbosity in conversation. Specifying verbosity-related behaviours can provide an improved basis for diagnosis, treatment, and measuring change in communication over the course of recovery. This study explores whether people with verbosity caused by traumatic brain injury adopt behaviours that violate the normative organisation of turn-taking in conversation. Using conversation analysis, this study examines 1 hour and 40 minutes of conversations involving two participants with cognitive communication disorders characterised by verbosity following severe traumatic brain injury. Overlapping talk, self-initiated self-repair, parenthetical inserts, and practices for turn continuation are identified as candidate signs of verbosity. There is also evidence that verbose speakers employ practices to manage topical discontinuity caused by verbosity. Verbose behaviour involves a complex interplay between cognitive impairment, communicative activity, and the communicative environment. Clinical measures for verbosity should be developed with reference to empirical findings about typical conversation. [ABSTRACT FROM AUTHOR]

Given a finite alphabet Σ and a right-infinite word w over Σ, we define the Lie complexity function L w : N → N , whose value at n is the number of conjugacy classes (under cyclic shift) of length- n factors x of w with the property that every element of the conjugacy class appears in w. We show that the Lie complexity function is uniformly bounded for words with linear factor complexity. As a result, we show that words of linear factor complexity have at most finitely many primitive factors y with the property that y n is again a factor for every n. We then look at automatic sequences and show that the Lie complexity function of a k -automatic sequence is also k -automatic. [ABSTRACT FROM AUTHOR]

We study an open question at the interplay between the classical and the dynamical Mordell-Lang conjectures in positive characteristic. Let K be an algebraically closed field of positive characteristic, let G be a finitely generated subgroup of the multiplicative group of K, and let X be an (irreducible) quasiprojective variety defined over K. We consider K-valued sequences of the form an : = f(φn(x0)), where φ : X → X and f : X → P¹ are rational maps defined over K and x0 ∈ X is a point whose forward orbit avoids the indeterminacy loci of φ and f. We show that the set of n for which an ∈ G is a finite union of arithmetic progressions along with a set of Banach density zero. [ABSTRACT FROM AUTHOR]

According to Winthrop Bell's analysis of the idea of Canada, a pioneering account of Canadian philosophy written during World War I, Canada's essence involves intercultural interpretation. In terms familiar to the phenomenology and pragmatism in which he was trained, Bell shows how Canada, by creating a new culture founded upon intercultural respect, could replace monocultural imperialism and materialism with something better: a culturally diverse nation, serving wisdom from a unique conversation among Canadian peoples, but that is for all people. In this article, we describe Bell's philosophy of Canada as interpretation and further apply it to the idea, practice, and end of reparations to Canada's Indigenous peoples as an expression of loyalty to Canadian philosophy today. [ABSTRACT FROM AUTHOR]

A (noncommutative) Pólya series over a field K is a formal power series whose nonzero coefficients are contained in a finitely generated subgroup of K × . We show that rational Pólya series are unambiguous rational series, proving a 40 year old conjecture of Reutenauer. The proof combines methods from noncommutative algebra, automata theory, and number theory (specifically, unit equations). As a corollary, a rational series is a Pólya series if and only if it is Hadamard sub-invertible. Phrased differently, we show that every weighted finite automaton taking values in a finitely generated subgroup of a field (and zero) is equivalent to an unambiguous weighted finite automaton. [ABSTRACT FROM AUTHOR]

We prove a uniform version of the Dynamical Mordell–Lang Conjecture for étale maps; also, we obtain a gap result for the growth rate of heights of points in an orbit along an arbitrary endomorphism of a quasiprojective variety defined over a number field. More precisely, for our 1st result, we assume |$X$| is a quasi-projective variety defined over a field |$K$| of characteristic |$0$|⁠ , endowed with the action of an étale endomorphism |$\Phi $|⁠ , and |$f\colon X\longrightarrow Y$| is a morphism with |$Y$| a quasi-projective variety defined over |$K$|⁠. Then for any |$x\in X(K)$|⁠ , if for each |$y\in Y(K)$|⁠ , the set |$S_{x,y}:=\{n\in{\mathbb{N}}\colon f(\Phi ^n(x))=y\}$| is finite, then there exists a positive integer |$N_x$| such that |$\sharp S_{x,y}\le N_x$| for each |$y\in Y(K)$|⁠. For our 2nd result, we let |$K$| be a number field, |$f:X\dashrightarrow{\mathbb{P}}^1$| is a rational map, and |$\Phi $| is an arbitrary endomorphism of |$X$|⁠. If |${\mathcal{O}}_{\Phi }(x)$| denotes the forward orbit of |$x$| under the action of |$\Phi $|⁠ , then either |$f({\mathcal{O}}_{\Phi }(x))$| is finite, or |$\limsup _{n\to \infty } h(f(\Phi ^n(x)))/\log (n)>0$|⁠ , where |$h(\cdot)$| represents the usual logarithmic Weil height for algebraic points. [ABSTRACT FROM AUTHOR]

We determine the possible functions that can occur, up to asymptotic equivalence, as growth functions of semigroups, hereditary languages, and algebras. [ABSTRACT FROM AUTHOR]

A theorem of Cobham says that if k and ℓ are two multiplicatively independent natural numbers then a subset of the natural numbers that is both k - and ℓ -automatic is eventually periodic. A multidimensional extension was later given by Semenov. In this paper, we give a quantitative version of the Cobham-Semenov theorem for sparse automatic sets, showing that the intersection of a sparse k -automatic subset of N d and a sparse ℓ -automatic subset of N d is finite with size that can be explicitly bounded in terms of data from the automata that accept these sets. [ABSTRACT FROM AUTHOR]

Motivated by a result of van der Poorten and Shparlinski for univariate power series, Bell and Chen prove that if a multivariate power series over a field of characteristic 0 is D-finite and its coefficients belong to a finite set, then it is a rational function. We extend and strengthen their results to certain power series whose coefficients may form an infinite set. We also prove that if the coefficients of a univariate D-finite power series ''look like'' the coefficients of a rational function, then the power series is rational. Our work relies on the theory of Weil heights, the Manin-Mumford theorem for tori, an application of the Subspace Theorem, and various combinatorial arguments involving heights, power series, and linear recurrence sequences. [ABSTRACT FROM AUTHOR]

Let |$M$| be a representable matroid on |$n$| elements. We give bounds, in terms of |$n$|⁠ , on the least positive characteristic and smallest field over which |$M$| is representable. [ABSTRACT FROM AUTHOR]

Let K be a field, let σ be an automorphism, and let δ be a σ-derivation of K. We show that the multiplicative group of nonzero elements of the division ring D = K(x;σ,δ) contains a free noncyclic subgroup unless D is commutative, answering a special case of a conjecture of Lichtman. As an application, we show that division algebras formed by taking the Goldie ring of quotients of group algebras of torsion-free nonabelian solvable-by-finite groups always contain free noncyclic subgroups. [ABSTRACT FROM AUTHOR]

Let k be a field and let R be a left noetherian k -algebra. The algebra R satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the Spec (R) and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if R and S are two left noetherian k -algebras with dim k (R) , dim k (S) < | k | then if R and S have homeomorphic spectra then R satisfies the Dixmier-Moeglin equivalence if and only if S does. In particular, the topology of Spec (R) can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if k is uncountable and R is affine noetherian and its prime spectrum is a finite disjoint union of locally closed subspaces that are each homeomorphic to the spectrum of an affine commutative ring then R satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if k is countable and R is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that R and S are left noetherian k -algebras that satisfy the Dixmier-Moeglin equivalence then R ⊗ k S does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that eRe also satisfies the Dixmier-Moeglin equivalence, where e is a nonzero idempotent of R. [ABSTRACT FROM AUTHOR]

In 1994, Becker conjectured that if F(z) is a k-regular power series, then there exists a k-regular rational function R(z) such that F(z)/R(z) satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies a0(z) = 1. In this paper, we prove Becker’s conjecture in the best-possible form; we show that the rational function R(z) can be taken to be a polynomial zγQ(z) for some explicit nonnegative integer γ and such that 1/Q(z) is k-regular. [ABSTRACT FROM AUTHOR]

Let G be a semiabelian variety defined over a field of characteristic 0, endowed with an endomorphism Φ. We prove there is no proper subvariety Y ⊂ G which intersects the orbit of each periodic point of G under the action of Φ. As an application, we are able to give a topological characterization of the annihilator ideals of irreducible representations in certain skew polynomial algebras. [ABSTRACT FROM AUTHOR]

We give a brief survey of primitivity in ring theory and in particular look at characterizations of primitive ideals in the prime spectrum for various classes of rings. [ABSTRACT FROM AUTHOR]

We characterize those k-automatic sets S of natural numbers that form an additive basis for the natural numbers, and we show that this characterization is effective. In addition, we give an algorithm to determine the smallest j such that S forms an additive basis of order j, if it exists. [ABSTRACT FROM AUTHOR]

This work is motivated by the articles [9] and [19] in which the following two problems are solved. Let O be a finitely generated ℤ-algebra that is an integrally closed domain of characteristic zero, consider the following problems: (A) Fix s that is integral over O, describe all t such that O[s]= O[t]. (B) Fix s and t that are integral over O, describe all pairs (m, n) ϵ ℕ² such that O[sm]= O[tn]. In this article, we solve these problems and provide a uniform bound for a certain "discriminant form equation" that is closely related to Problem (A) when O has characteristic p > 0. While our general strategy roughly follows [9] and [19], many new delicate issues arise due to the presence of the Frobenius automorphism x → xp. Recent advances in unit equations over fields of positive characteristic together with classical results in characteristic zero play an important role in this article. [ABSTRACT FROM AUTHOR]

Previous psychophysical studies at durations greater than 1000 ms have confirmed the anecdotal reports of an increase in the perceived duration of both positively and negatively valenced emotive stimuli; however, the results of studies at durations less than 1000 ms have been inconsistent. This study further investigated the effect of valence on the perception of durations less than 1000 ms. We used both positively and negatively valenced stimuli in order to compare their effects on the distortion of duration, and we tested multiple data points within the sub-one-second range. We found an increase in the perceived duration of both positively and negatively valenced emotional stimuli at all data points. This is consistent with studies at durations longer than 1000 ms and also with models of temporal processing. We also confirmed that Weber fractions, within the range tested, followed the generalized form of Weber’s law. [ABSTRACT FROM AUTHOR]

Dixmier and Moeglin gave an algebraic condition and a topological condition for recognising the primitive ideals among the prime ideals of the universal enveloping algebra of a finite-dimensional complex Lie algebra; they showed that the primitive, rational, and locally closed ideals coincide. In modern terminology, they showed that the universal enveloping algebra of a finite-dimensional complex Lie algebra satisfies the Dixmier–Moeglin equivalence . We define quantities which measure how “close” an arbitrary prime ideal of a noetherian algebra is to being primitive, rational, and locally closed; if every prime ideal is equally “close” to satisfying each of these three properties, then we say that the algebra satisfies the strong Dixmier–Moeglin equivalence . Using the example of the universal enveloping algebra of sl 2 ( C ) , we show that the strong Dixmier–Moeglin equivalence is strictly stronger than the Dixmier–Moeglin equivalence. For a simple complex Lie algebra g , a non-root of unity q ≠ 0 in an infinite field K , and an element w of the Weyl group of g , De Concini, Kac, and Procesi have constructed a subalgebra U q [ w ] of the quantised enveloping K -algebra U q ( g ) . These quantum Schubert cells are known to satisfy the Dixmier–Moeglin equivalence and we show that they in fact satisfy the strong Dixmier–Moeglin equivalence. Along the way, we show that commutative affine domains, uniparameter quantum tori, and uniparameter quantum affine spaces satisfy the strong Dixmier–Moeglin equivalence. [ABSTRACT FROM AUTHOR]

We study how well Fekete polynomials F p ( X ) = ∑ n = 0 p − 1 n p X n ∈ Z [ X ] with the coefficients given by Legendre symbols modulo a prime p can be approximated by power series representing algebraic functions of a given degree. We also obtain some explicit results describing polynomial recurrence relations which are satisfied by the coefficients of such algebraic functions. [ABSTRACT FROM AUTHOR]

We prove in this paper that a multivariate D-finite power series with coefficients from a finite set is rational. This generalizes a rationality theorem of van der Poorten and Shparlinski in 1996. [ABSTRACT FROM AUTHOR]

Let k be a field and let A = ⨁ n ≥ 1 A n be a positively graded k -algebra. We recall that A is graded nilpotent if for every d ≥ 1 , the subalgebra of A generated by elements of degree d is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field k there exists a finitely generated graded nilpotent algebra that contains a free k -subalgebra on two generators. [ABSTRACT FROM AUTHOR]

We introduce a dynamical Mordell-Lang-type conjecture for coherent sheaves. When the sheaves are structure sheaves of closed subschemes, our conjecture becomes a statement about unlikely intersections. We prove an analogue of this conjecture for affinoid spaces, which we then use to prove our conjecture in the case of surfaces. These results rely on a module-theoretic variant of Strassman's theorem that we prove in the appendix. [ABSTRACT FROM AUTHOR]

Brown and Gordon asked whether the Poisson Dixmier-Moeglin equivalence holds for any complex affine Poisson algebra, that is, whether the sets of Poisson rational ideals, Poisson primitive ideals, and Poisson locally closed ideals coincide. In this article a complete answer is given to this question using techniques from differential-algebraic geometry and model theory. In particular, it is shown that while the sets of Poisson rational and Poisson primitive ideals do coincide, in every Krull dimension at least four there are complex affine Poisson algebras with Poisson rational ideals that are not Poisson locally closed. These counterexamples also give rise to counterexamples to the classical (noncommutative) Dixmier-Moeglin equivalence in finite GK dimension. A weaker version of the Poisson Dixmier-Moeglin equivalence is proven for all complex affine Poisson algebras, from which it follows that the full equivalence holds in Krull dimension three or less. Finally, it is shown that everything, except possibly that rationality implies primitivity, can be done over an arbitrary base field of characteristic zero. [ABSTRACT FROM AUTHOR]

A noncommutative analogue of the Zariski cancellation problem asks whether $$A[x]\cong B[x]$$ implies $$A\cong B$$ when A and B are noncommutative algebras. We resolve this affirmatively in the case when A is a noncommutative finitely generated domain over the complex field of Gelfand-Kirillov dimension two. In addition, we resolve the Zariski cancellation problem for several classes of Artin-Schelter regular algebras of higher Gelfand-Kirillov dimension. [ABSTRACT FROM AUTHOR]

We consider D-finite power series f (z) = ∑ n ≥ 0 a n z n with coefficients in a number field K. We show that there is a dichotomy governing the behaviour of h (a n) as a function of n , where h is the absolute logarithmic Weil height. As an immediate consequence of our results, we have that either f (z) is rational or h (a n) > [ K : Q ] − 1 ⋅ log ⁡ (n) + O (1) for n in a set of positive upper density and this is best possible when K = Q. [ABSTRACT FROM AUTHOR]

A domain R is said to have the finite factorization property if every nonzero nonunit element of R has at least one and at most finitely many distinct factorizations up to multiplication of irreducible factors by central units. Let k be an algebraically closed field and let A be a k-algebra. We show that if A has an associated graded ring that is a domain with the property that the dimension of each homogeneous component is finite, then A is a finite factorization domain. As a corollary, we show that many classes of algebras have the finite factorization property, including Weyl algebras, enveloping algebras of finite-dimensional Lie algebras, quantum affine spaces and shift algebras. This provides a termination criterion for factorization procedures over these algebras. In addition, we give explicit upper bounds on the number of distinct factorizations of an element in terms of data from the filtration. [ABSTRACT FROM AUTHOR]

We give two tests for transcendence of Mahler functions. For our first, we introduce the notion of the eigenvalue λF of a Mahler function F(z) and develop a quick test for the transcendence of F(z) over C(z), which is determined by the value of the eigenvalue λF. While our first test is quick and applicable for a large class of functions, our second test, while a bit slower than our first, is universal; it depends on the rank of a certain Hankel matrix determined by the initial coefficients of F(z). We note that these are the first transcendence tests for Mahler functions of arbitrary degree. Several examples and applications are given. [ABSTRACT FROM AUTHOR]

Let A and B be two connected graded algebras finitely generated in degree one. If A is isomorphic to B as ungraded algebras, then they are also isomorphic to each other as graded algebras. [ABSTRACT FROM AUTHOR]

Given a finite alphabet Σ and a right-infinite word w over the alphabet Σ, we construct a topological space Rec (w) consisting of all right-infinite recurrent words whose factors are all factors of w , where we work up to an equivalence in which two words are equivalent if they have the exact same set of factors (finite contiguous subwords). We show that Rec (w) can be endowed with a natural topology and we show that if w is word of linear factor complexity then Rec (w) is a finite topological space. In addition, we note that there are examples which show that if f : N → N is a function that tends to infinity as n → ∞ then there is a word whose factor complexity function is O (n f (n)) such that Rec (w) is an infinite set. Finally, we pose a realization problem: which finite topological spaces can arise as Rec (w) for a word of linear factor complexity? [ABSTRACT FROM AUTHOR]

Homologous recombination maintains genomic integrity by repairing broken chromosomes. The broken chromosome is partially resected to produce single-stranded DNA (ssDNA) that is used to search for homologous double-stranded DNA (dsDNA). This homology driven ‘search and rescue’ is catalyzed by a class of DNA strand exchange proteins that are defined in relation to Escherichia coli RecA, which forms a filament on ssDNA. Here, we review the regulation of RecA filament assembly and the mechanism by which RecA quickly and efficiently searches for and identifies a unique homologous sequence among a vast excess of heterologous DNA. Given that RecA is the prototypic DNA strand exchange protein, its behavior affords insight into the actions of eukaryotic RAD51 orthologs and their regulators, BRCA2 and other tumor suppressors. [ABSTRACT FROM AUTHOR]

Leavitt path algebras of an arbitrary graph over a field satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When is a finite graph, satisfying a polynomial identity is shown to be equivalent to the Gelfand-Kirillov dimension of being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph , the Leavitt path algebra has Gelfand-Kirillov dimension zero if and only if has no cycles. Likewise, has Gelfand-Kirillov dimension one if and only if contains at least one cycle, but no cycle in has an exit. [ABSTRACT FROM AUTHOR]

Let K be a field of characteristic zero, let σ be an automorphism of K and let δ be a σ -derivation of K . We show that the division ring D = K ( x ; σ , δ ) either has the property that every finitely generated subring satisfies a polynomial identity or D contains a free algebra on two generators over its center. In the case when K is finitely generated over a subfield k we then see that for σ a k -algebra automorphism of K and δ a k -linear derivation of K , K ( x ; σ ) having a free subalgebra on two generators is equivalent to σ having infinite order, and K ( x ; δ ) having a free subalgebra is equivalent to δ being nonzero. As an application, we show that if D is a division ring with center k of characteristic zero and D ⁎ contains a solvable subgroup that is not locally abelian-by-finite, then D contains a free k -algebra on two generators. Moreover, if we assume that k is uncountable, without any restrictions on the characteristic of k , then D contains the k -group algebra of the free group of rank two. [ABSTRACT FROM AUTHOR]

Let $${\mathcal {A}}$$ be a finite set of $$d\times d$$ matrices with integer entries and let $$m_n({\mathcal {A}})$$ be the maximum norm of a product of $$n$$ elements of $${\mathcal {A}}$$ . In this paper, we classify gaps in the growth of $$m_n({\mathcal {A}})$$ ; specifically, we prove that $$\lim _{n\rightarrow \infty } \log m_n({\mathcal {A}})/\log n\in \mathbb {Z}_{\geqslant 0}\cup \{\infty \}.$$ This has applications to the growth of regular sequences as defined by Allouche and Shallit. [ABSTRACT FROM AUTHOR]

The repair of DNA by homologous recombination is an essential, efficient, and high-fidelity process that mends DNA lesions formed during cellular metabolism; these lesions include double-stranded DNA breaks, daughter-strand gaps, and DNA cross-links. Genetic defects in the homologous recombination pathway undermine genomic integrity and cause the accumulation of gross chromosomal abnormalities-including rearrangements, deletions, and aneuploidy-that contribute to cancer formation. Recombination proceeds through the formation of joint DNA molecules-homologously paired but metastable DNA intermediates that are processed by several alternative subpathways-making recombination a versatile and robust mechanism to repair damaged chromosomes. Modern biophysical methods make it possible to visualize, probe, and manipulate the individual molecules participating in the intermediate steps of recombination, revealing new details about the mechanics of genetic recombination. We review and discuss the individual stages of homologous recombination, focusing on common pathways in bacteria, yeast, and humans, and place particular emphasis on the molecular mechanisms illuminated by single-molecule methods. [ABSTRACT FROM AUTHOR]

Let k be a field and let R be a countable-dimensional prime von Neumann regular k-algebra. We show that R is primitive, answering a special case of a question of Kaplansky. [ABSTRACT FROM AUTHOR]

Using methods of p-adic analysis, we obtain effective bounds for the length of the orbit of a preperiodic subvariety Y ⊂X under the action of an étale endomorphism of X. As a corollary of our result, we obtain effective bounds for the size of torsion of any semiabelian variety over a finitely generated field of characteristic 0. Our method allows us to show that any finitely generated torsion subgroup of Aut(X) is finite. This yields a different proof of Burnside's problem for automorphisms of quasiprojective varieties X defined over a field of characteristic 0. [ABSTRACT FROM AUTHOR]

Suppose that F(x)∈Z[[x]] is a Mahler function and that 1/b is in the radius of convergence of F(x). In this paper, we consider the approximation of F(1/b) by algebraic numbers. In particular, we prove that F(1/b) cannot be a Liouville number. If F(x) is also regular, we show that F(1/b) is either rational or transcendental, and in the latter case that F(1/b) is an S-number or a T-number. [ABSTRACT FROM AUTHOR]

Let R be a ring satisfying a polynomial identity and let δ be a derivation of R . We show that if R is locally nilpotent then R [ x ; δ ] is locally nilpotent. This affirmatively answers a question of Smoktunowicz and Ziembowski. As a consequence we have that if R is a unital PI algebra over a field of characteristic zero then the Jacobson radical of R [ x ; δ ] is equal to N [ x ; δ ] , where N is the nil radical of R . [ABSTRACT FROM AUTHOR]