Rational Subsets Research Articles

Algebraic and context-free subsets of subgroups

We study the relation between the structure of algebraic and context-free subsets of a group G and that of a finite index subgroup H. Using these results, we prove that a kind of Fatou property, previously studied by Berstel and Sakarovitch in the context of rational subsets and by Herbst in the context of algebraic subsets, holds for context-free subsets if and only if the group is virtually free. We also exhibit a counterexample to a question of Herbst concerning this property for algebraic subsets.

On the word problem for weakly compressible monoids

We study the language-theoretic properties of the word problem, in the sense of Duncan & Gilman, of weakly compressible monoids, as defined by Adian & Oganesian. We show that if C is a reversal-closed super-, as defined by Greibach, then M has word problem in C if and only if its compressed left monoid L(M) has word problem in C. As a special case, we may take C to be the class of context-free or indexed languages. As a corollary, we find many new classes of monoids with decidable rational subset membership problem. Finally, we show that it is decidable whether a one-relation monoid containing a non-trivial idempotent has context-free word problem. This answers a generalization of a question first asked by Zhang in 1992.

On the rational subsets of the monogenic free inverse monoid

We prove that the equality problem is decidable for rational subsets of the monogenic free inverse monoid F. It is also decidable whether or not a rational subset of F is recognizable. We prove that a submonoid of F is rational if and only if it is finitely generated. We also prove that the membership problem for rational subsets of a finite J-above monoid is decidable, covering the case of free inverse monoids.

Currents on cusped hyperbolic surfaces and denseness property

The space \operatorname{GC}(\Sigma) of geodesic currents on a hyperbolic surface \Sigma can be considered as a completion of the set of weighted closed geodesics on \Sigma when \Sigma is compact, since the set of rational geodesic currents on \Sigma , which correspond to weighted closed geodesics, is a dense subset of \operatorname{GC}(\Sigma) . We prove that even when \Sigma is a cusped hyperbolic surface with finite area, \operatorname{GC}(\Sigma) has the denseness property of rational geodesic currents, which correspond not only to weighted closed geodesics on \Sigma but also to weighted geodesics connecting two cusps. In addition, we present an example in which a sequence of weighted closed geodesics converges to a geodesic connecting two cusps, which is an obstruction for the intersection number to extend continuously to \operatorname{GC}(\Sigma) . To construct the example, we use the notion of subset currents. Finally, we prove that the space of subset currents on a cusped hyperbolic surface has the denseness property of rational subset currents.

Engineering off-the-shelf universal CAR T cells: A silver lining in the cloud.

CAR therapy holds promise in treating aggressive hematological malignancies. Nonetheless, the present autologous CAR therapy regimen makes multiple patients ineligible for the therapy due to inadequate quantity, quality and purity of the product. Furthermore, timely manufacturing of benchmarked cell products is logistically challenging and unaffordable. Extensive genetic modifications may be required to overcome the biological, clinical and manufacturing limitations of the autologous CAR therapy. n the light of the numerous configurations of CAR therapy, engineering "off-the-shelf" universal CAR T cells (UCART) is emerging as a safer, effective and affordable alternative to conventional CAR T cells With UCART therapy, batch production of a quality-controlled product with multiplex genetic modification can be feasible in a shorter period of time. Currently vast majority of the UCART programs target CD19 followed by BCMA and CD70. In order to make universal CAR T cell therapy possible, it is imperative to have engineering strategies to curb graft versus host disease (GvHD) and graft rejection (GR). Moreover, approaches to offer alternate strategies for intense preparative chemotherapy, infection control and CAR T cell persistence need to be optimized. An ideal universal immune receptor (UIR) design should counter the antigen escape and further the therapeutic value and affordability. UIRs would allow flexibility to personalize the therapy based on the specific malignancy characteristics as well. With the innovations in the inducible molecular switch, split CAR design, CRISPR/Cas9 mediated gene targeting, rational subset composition and cryopreservation, the strategies to engineer universal CAR T therapy is fast advancing from bench to bedside.

Суперпозиции свободных производных Фокса

Fox derivations are an effective tool for studying free groups and their group rings. Let Fr be a free group of finite rank r with basis {f1,..., fr}. For every i, the partial Fox derivations ∂/∂fi и ∂/∂fi-1 are defined on the group ring ℤ[Fr]. For k / 2, their superpositions Dfϵi = = ∂/∂fϵki о ... о ∂/∂fϵk1, ϵ = (ϵ1,..., ϵk) Є{±1}k, are not Fox derivations. In this paper, we study the properties of superpositions Dfϵi. It is shown that the restrictions of such superpositions to the commutant F′r are Fox derivations. As an application of the obtained results, it is established that for any rational subset R of F′r and any i there are parameters k and ϵ such that R is annihilated by Dfϵi.

DFT based QSAR study on quinolone-triazole derivatives as antibacterial agents

QSAR modeling was performed on 39 quinolone-triazole derivatives against gram-positive Staphylococcus aureus and gram-negative Pseudomonas aeruginosa bacteria. The molecular structures were optimized using the DFT/B3LYP method and 6–31 G basis set. Molecular descriptors were extracted using quantum mechanical calculations. The hierarchical cluster analysis was performed for a rational subset division. The initial dataset was divided into calibration and validation sets, and modeling was done by stepwise MLR method for each of the two bacteria. Internal and external validation methods confirmed the robustness and predictability of the obtained models. According to the obtained model for S. aureus (R 2 = 0.889, R 2 ext = 0.938, Q 2 LOO = 0.853), the four descriptors- partial atomic charges for the N1 atom in triazole and C7 of the quinolone nucleus, 4-carbonyl bond length, and 13C-NMR chemical shift of 3-carboxylic acid- were found to be the descriptors controlling the activity. According to the obtained model for P. aeruginosa (R 2 = 0.957, R 2 ext = 0.923, Q 2 LOO = 0.909), the O atom’s partial charge in carbonyl, LUMO-HOMO energy gap, and logP were found to be the descriptors having the highest correlation with the antibacterial activity. Finally, some new compounds with higher activities were designed and proposed.

Two Problems for Solvable and Nilpotent Groups

Section 1 gives a brief review of known results on embeddings of solvable, nilpotent, and polycyclic groups in 2-generated groups from these classes, including the description of the author’s recently obtained solution to the Mikaelian–Ol’schanskii problem on embeddings of finitely generated solvable groups of derived length l in solvable groups of derived length l + 1 with a fixed small number of generators. Section 2 contains a somewhat more extensive review of known results on the rational subset membership problem for groups, including the presentation of the author’s recently obtained solution to the Laurie–Steinberg–Kambites–Silva–Zetsche problem of whether the membership problem is decidable for finitely generated submonoids of free nilpotent groups.

Алгоритмическая теория разрешимых групп

The purpose of this survey is to give some picture of what is known about algorithmic and decision problems in the theory of solvable groups. We will provide a number of references to various results, which are presented without proof. Naturally, the choice of the material reported on reflects the author’s interests and many worthy contributions to the field will unfortunately go without mentioning. In addition to achievements in solving classical algorithmic problems, the survey presents results on other issues. Attention is paid to various aspects of modern theory related to the complexity of algorithms, their practical implementation, random choice, asymptotic properties. Results are given on various issues related to mathematical logic and model theory. In particular, a special section of the survey is devoted to elementary and universal theories of solvable groups. Special attention is paid to algorithmic questions regarding rational subsets of groups. Results on algorithmic problems related to homomorphisms, automorphisms, and endomorphisms of groups are presented in sufficient detail.

A group-theoretical interpretation of the word problem for free idempotent generated semigroups

The set of idempotents of any semigroup carries the structure of a biordered set, which contains a great deal of information concerning the idempotent generated subsemigroup of the semigroup in question. This leads to the construction of a free idempotent generated semigroup IG(E) – the ‘free-est’ semigroup with a given biordered set E of idempotents. We show that when E is finite, the word problem for IG(E) is equivalent to a family of constraint satisfaction problems involving rational subsets of direct products of pairs of maximal subgroups of IG(E). As an application, we obtain decidability of the word problem for an important class of examples. Also, we prove that for finite E, IG(E) is always a weakly abundant semigroup satisfying the congruence condition.

Rational Dyck Paths in the Non Relatively Prime Case

We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb{Z},$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$–Dyck paths and $d$-tuples of $(n,m)$-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.

Inverse subsemigroups of finite index in finitely generated inverse semigroups

We study some aspects of Schein’s theory of cosets for closed inverse subsemigroups of inverse semigroups. We establish an index formula for chains of subsemigroups, and an analogue of M. Hall’s Theorem on the number of cosets of a fixed finite index. We then investigate the relationships between the following properties of a closed inverse submonoid of an inverse monoid: having finite index; being a recognizable subset; being a rational subset; being finitely generated (as a closed inverse submonoid). A remarkable result of Margolis and Meakin shows that these properties are equivalent for a closed inverse submonoid of a free inverse monoid.

The pro-nilpotent group topology on a free group

In this paper, we study the pro-nilpotent group topology on a free group. First we describe the closure of the product of finitely many finitely generated subgroups of a free group in the pro-nilpotent group topology and then present an algorithm to compute it. We deduce that the nil-closure of a rational subset of a free group is an effectively constructible rational subset and hence has decidable membership. We also prove that the Gnil-kernel of a finite monoid is computable and hence pseudovarieties of the form VⓜGnil have decidable membership problem, for every decidable pseudovariety of monoids V. Finally, we prove that the semidirect product J⁎Gnil has a decidable membership problem.

POLYCYCLIC, METABELIAN OR SOLUBLE OF TYPE (FP)∞ GROUPS WITH BOOLEAN ALGEBRA OF RATIONAL SETS AND BIAUTOMATIC SOLUBLE GROUPS ARE VIRTUALLY ABELIAN

AbstractLet G be a polycyclic, metabelian or soluble of type (FP)∞ group such that the class Rat(G) of all rational subsets of G is a Boolean algebra. Then, G is virtually abelian. Every soluble biautomatic group is virtually abelian.

The monoid of queue actions

We model the behavior of a fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. We describe this monoid by means of a confluent and terminating semi-Thue system and study some of its basic algebraic properties such as conjugacy. Moreover, we show that while several properties concerning its rational subsets are undecidable, their uniform membership problem is $${{\mathsf {N}}}{{\mathsf {L}}}$$ -complete. Furthermore, we present an algebraic characterization of this monoid’s recognizable subsets. Finally, we prove that it is not Thurston-automatic.

Genetic Classification of Gliomas: Refining Histopathology

Newly published studies validate prior reports that specific combinations of genetic alternations in IDH1/2, ATRX, TERT, TP53, and co-deletion of 1p/19q have the ability to reclassify gliomas into rational subsets, defining a glioma's biological and clinical behavior more accurately than stratifications based solely on histopathology.

Knapsack problems in products of groups

The classic knapsack and related problems have natural generalizations to arbitrary (non-commutative) groups, collectively called knapsack-type problems in groups. We study the effect of free and direct products on their time complexity. We show that free products in certain sense preserve time complexity of knapsack-type problems, while direct products may amplify it. Our methods allow to obtain complexity results for rational subset membership problem in amalgamated free products over finite subgroups.

Rational subsets of unitriangular groups

It is shown that there exist d, ℓ ≥ 3 and a sequence C1, …, Cℓ of cyclic subgroups of the d-dimensional unitriangluar matrix group over ℤ (which is finitely generated nilpotent) such that membership in the product C1 C2 ⋯ Cℓ is undecidable.

Edenic Freedoms

Edenic Freedoms Joshua Scodel Celebrating the “freedom” of prelapsarian Adam and Eve,1 Paradise Lost exploits the polysemous ambiguities in the term and concept of “freedom.” In his attempt to provide a psychologically plausible, emotionally compelling, and theodicy-satisfying version of his biblical narrative, Milton adapts various ethical, theological, legal, political, and poetic ways of characterizing freedom and its value from both classical and Christian traditions. Disparate visions of freedom contribute to the joys, tensions, and tragedy in Milton’s representation of Edenic life. I distinguish four major kinds of Edenic freedom: (1) voluntary motion; (2) divinely permitted choices; (3) best or correct rational choices; and (4) the choice to obey or disobey God. The first three kinds of freedom embody distinct conceptions of Edenic blessedness only partially harmonized. The free choice of obedience or disobedience, the crucial kind of freedom for the epic’s narrative arc, is not only the most difficult to reconcile with other conceptions of freedom in Paradise Lost, it is also deeply paradoxical because of obedience’s connection to fear of punishment. I conclude by showing how the tension between freedom and fear in Miltonic obedience troubles his central notion of the Fall as a free choice. [End Page 153] I. Adam’s Voluntary Motion Let me contextualize Paradise Lost’s representations of voluntariness and choice by juxtaposing Milton’s discussions of choice in Ars logicae (1672) and of unfallen Adam and Eve in De doctrina Christiana.2 While Milton’s Ars logicae borrows heavily from Petrus Ramus’s 1572 Dialecticae and George Downame’s 1601 Ramist commentary, Milton deviates from these works to express his philosophical and theological views.3 Adding to his sources’ contrasts between natura and consilium as efficient causes, chapter 5 argues that unlike animals, who act solely by “nature,” uncoerced humans can act either by internal “natural necessity” (necessitas naturae) or “freely” (libere) with consilium.4 Consilium renders Aristotle’s central ethical term in his Nicomachean Ethics, proairesis, which can be translated as “choice,” “decision,” “deliberate choice,” or “rational choice.”5 Downame identifies consilium with proairesis, and consilium is the most common rendering of proairesis in Renaissance Aristotle translations and the Aristotle commentaries studied at Milton’s Cambridge.6 The Nicomachean Ethics argues that proairesis, which results from deliberation (3.2.17, 3.3.19, 5.8.5), is an efficient cause of human action (6.2.4), concerns actions up to us to do or not (3.5.2–3), and is constitutive of virtue, which is a habit involving choice (2.4.3, 2.6.15).7 Early modern commentaries interpret (and distort) Aristotle with patristic and Scholastic concepts. Though Aristotle never directly connects proairesis to his (political) conception of freedom, eleutheria, commentators treat proairesis as a manifestation of “free choice” (liberum arbitrium) conceived as what Scholastics term libertas indifferens, the liberty to perform an act or not.8 Milton likewise claims that God created humans and angels as rational beings able “freely” (libere) to do “this or that by choice” (hoc vel illud pro arbitrio) with the “power … of doing or not doing” (potestas … agendi vel non agendi) (CM 11:40–42). Aristotle treats proairesis as a distinctively rational subset of to hekousion or voluntariness, arguing that animals and children behave voluntarily but only adult humans act through rational choice (NE 3.2.2–4). Milton’s contrast between acting by “natural necessity” and freely with consilium identifies Aristotle’s merely [End Page 154] voluntary with natural necessity. His association of “appetite” (appetitus) with “nature” (CM 11:38) underscores the difference between humans, who can reason about their appetites to make free choices, and appetite-impelled animals. Milton’s Ars logicae is broadly consistent with his anti-Calvinist, Arminian commitment in both De doctrina Christiana and Paradise Lost to human freedom as the ability to choose between alternative possibilities.9 Yet there are tensions. Following Augustine’s City of God, De doctrina argues that God’s prohibition turned a tree otherwise neither good nor bad into a test of human obedience.10 Without the prohibition, unfallen humankind, virtuous “by nature” (naturâ), “natural inclination” (naturali … ductu), and “voluntarily” (sponte), could not be tested (DDC 358–59). Milton’s...

Rational subsets and submonoids of wreath products

It is shown that membership in rational subsets of wreath products H≀V with H a finite group and V a virtually free group is decidable. On the other hand, it is shown that there exists a fixed finitely generated submonoid in the wreath product Z≀Z with an undecidable membership problem.