Basic Commutators in Weights Six and Seven as Relators

The goal of this chapter is to determine normal forms for elements in finitely generated free groups and free nilpotent groups. This is done by using a collection process which we discuss in Sect. 3.1. A study of weighted commutators and basic commutators in a group relative to a given generating set also appears in this section. The highlight of Sect. 3.1 is a fundamental result stating that if G is any group generated by a set X and γ i G denotes the ith lower central subgroup, then each quotient γ n G/γ n+1G is generated, modulo γ n+1G, by a sequence of basic commutators on X of weight n. Section 3.2 is devoted to the so-called collection formula. This formula expresses a positive power of a product of elements x1, …, x r of a group as a product of positive powers of basic commutators in x1, …, x r . The collection process developed in Sect. 3.1 plays a key role here. In Sect. 3.3, we investigate basic commutators in finitely generated free groups and free nilpotent groups. We prove a major result which states that a finitely generated free nilpotent group, freely generated by a set X, has a “basis” consisting of basic commutators in X. The techniques used in this section involve groupoids, Lie rings, and the Magnus embedding. We end the chapter with Sect. 3.4, which is devoted to the rather technical proof of the collection formula obtained in Sect. 3.2.

We prove here that a finitely presented group with a free quotient of rank n is an HNN-extension with n stable letters of a finitely generated group where the associated subgroups are finitely generated. This theorem has a number of consequences. In particular, in the event that the free quotient is cyclic it reduces to an elementary and quick proof of a theorem of Bieri and Strebel. 1. Finitely presented groups with free quotients. Our main objective in this note is to prove the following Theorem 1. Let G be a finitely presented group with a free quotient of rank n. Then G is an HNN extension with n stable letters, of a finitely generated group B and finitely generated associated subgroups. It is worth noting that if N is the normal closure in G of B, then N is the fundamental group π(Γ) of a graph Γ of groups, where Γ is the Cayley graph of a free group of rank n. The vertex groups are isomorphic to B and the edge groups are isomorphic to the associated subgroups involved in the decomposition of G as an HNN–extension and are therefore all finitely generated. There are a number of consequences of Theorem 1. The first of these is the following theorem of Bieri and Strebel [3] which is a special case of Theorem 1. Corollary 1. A finitely presented group with an infinite cyclic quotient is an HNN extension of a finitely generated group with a single stable letter and finitely generated associated subgroups. In order to formulate the second consequence of Theorem 1, we recall that a group is termed coherent if all of its finitely generated subgroups Date: November 26, 2006. 1991 Mathematics Subject Classification. Primary 20.

For an arbitrary class of groups M we denote by L(M) the class of all groups G in which the normal closure of every element of G belongs to M. The class L(M) is called the Levi class generated by M. A study of Levi classes should be considered as a step towards studying the structure of groups covered by a system of normal subgroups. Levi classes were introduced under the influence of Levi’s article in which Levi gave the classification of groups with abelian normal closures. R.F. Morse proved that if the class M is a variety of groups then L(M)is also a variety of groups. It follows from the works of A.I. Budkin that if M is a quasivariety, then L(M) is a quasivariety of groups, too. We consider that the quasivariety qH2 is generated by the relatively free group in the class of nilpotent groups of length at most 2 with the exponent commutant of 2. Earlier we found descriptions of the Levi classes generated by the almost abelian quasivarieties of nilpotent groups (i.e. the nonabelian quasivarieties of nilpotent groups in which all proper subquasivarieties are abelian) except L(qH2). In this paper, we continue to explore the Levi classes generated by the almost abelian quasivarieties of nilpotent groups. It is proved that the class L(qH2) contains the nilpotent group of lenght 3.DOI 10.14258/izvasu(2016)1-26

Groups are very often described as quotient groups of free groups: G = F/N. If F is free with basis X and N is the normal closure in F of a set R, we say that the pair (X; R) is a presentation for G, and, by a mild abuse of language, we write G = (X; R).KeywordsWord ProblemFree ProductFree ResolutionNormal ClosureConjugacy ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper aims at solving several open questions in current neutrino physics: the neutrino mass hierarchy, the Dirac CP violating phase, the absolute mass of neutrinos, the nature of neutrinos (Dirac or Majorana), the Majorana matrix and the absolute value of the effective Majorana neutrino mass. In the research presented in this paper, we have shown that the precise definition of the mass splittings between neutrino mass eigenstates, done in the latest analysis of experimental data, can be of crucial importance for defining the nature of neutrino mass hierarchy. The Standard Model has three generations of fundamental matter particles. Three generations of the charged lepton mass show a hierarchical structure: mτ > mμ > me. Owing to that, there is a belief and it is considered that neutrinos may follow such hierarchical structure. In our calculations, we have also included the latest data obtained, based on the processing of measurement results, which showed that even with such data, obtained results favor the normal neutrino mass hierarchy. As for the individual neutrino mass calculated in this paper, in today’s neutrino physics it is only known that neutrino mass scale is bounded only from above, and both the Dirac and the Majorana character of neutrinos are compatible with all observations. Among some of the questions resolved in this paper, which are related to the properties of neutrinos, a positive answer was also given to the question of whether light neutrinos are self-conjugate particles or not.

This paper is motivated by the open question whether the union of two disjoint NP-complete sets always is NP-complete. We discover that such unions retain much of the complexity of their single components. More precisely, they are complete with respect to more general reducibilities.Moreover, we approach the main question in a more general way: We analyze the scope of the complexity of unions of m-equivalent disjoint sets. Under the hypothesis that NE ≠ coNE, we construct degrees in NP where our main question has a positive answer, i.e., these degrees are closed under unions of disjoint sets.KeywordsChromatic NumberSIAM JournalPolynomial HierarchyMathematical System TheoryPolynomial Time ReducibilityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Comments on Mark Seidenfeld’s “Revisiting Congressional Delegation of Interpretive Primacy as Foundation for <i>Chevron</i> Deference”

Randomizing polynomials represent a function f(x) by a low-degree randomized mapping p(x, r) over a finite field F such that, for any input x, the output distribution of p(x, r) depends only on the value of f(x). We study the class of functions f which admit an efficient representation by constant-degree randomizing polynomials. It is known that this class contains NC1 as well as log-space classes contained in NC2. Whether it contains all polynomial-time computable functions is a wide open question. A positive answer would have major and unexpected consequences, including the existence of efficient constant-round multiparty protocols with unconditional security, and the equivalence of (polynomial-time) cryptography and cryptography in NC0.We obtain evidence for the limited power of randomizing polynomials by showing that a useful subclass of constant-degree randomizing polynomials cannot efficiently capture functions beyond NC. Concretely, we consider randomizing polynomials over fields F of a small characteristic in which each monomial has degree (at most) 2 in the random inputs r and constant degree in x. This subclass captures most constructions of randomizing polynomials from the literature. Our main result is that all functions f which can be efficiently represented by such randomizing polynomials over fields of a small characteristic are in non-uniform NC. (The same holds over arbitrary fields given a quadratic residuosity oracle.) This result is obtained in two steps: (1) we observe that computing f as above reduces to counting roots of degree-2 multivariate polynomials; (2) we design parallel algorithms for the latter problem. These parallel root counting algorithms may be of independent interest.On the flip side, our main result provides an avenue for obtaining new parallel algorithms via the construction of randomizing polynomials. This gives an unexpected application of cryptography to algorithm design. We provide several examples for the potential usefulness of this approach.

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2 k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we obtain a lower bound of $$\Omega \left( {\sqrt {\log k/\log \log k} } \right)$$ for flow sparsifiers and a lower bound of $$\Omega \left( {\sqrt {\log k} /\log \log k} \right)$$ for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist $$\tilde O\left( {\sqrt {\log k} } \right)$$ cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than $$\tilde \Omega \left( {\sqrt {\log k} } \right)$$ would imply a negative answer to this question.

Consider a class of periodic discrete Schrödinger equations urn:x-wiley:mma:media:mma5183:mma5183-math-0001 Under the condition of , by applying the linking theorem, Pankov obtained the existence of gap solitons. Pankov's result was then generalized to various type of nonlinear function fn, but most of studying works were concentrated on the case of as t→0 . In this paper, by ingeniously using Ekeland variational principle, for the nonlinearity which meets as t→0 (s1 ∈ (0,1)), we obtain the existence of gap solitons, which supplements the existing ones and gives a positive answer in part to the open problem proposed by Pankov.

The question whether every subnormal tuple \({S}\,=\,({S}_1, \ldots , {S}_n)\) on a complex Hilbert space is reflexive is one of the major open problems in multivariable invariant subspace theory. Positive answers have been given for subnormal tuples with rich spectrum in the unit polydisc or the unit ball. The ball case has been extended by Didas [6] to strictly pseudoconvex domains. In the present note we extend the polydisc case by showing that every subnormal tuple with pure components and rich Taylor spectrum in a bounded polydomain \(U = U_1 \times \ldots \times U_n \subset {\mathbb{C}}^{n}\) is reflexive.

One of the methods for studying the solvability of infinite systems of integral or differential equations is the application of various fixed-point theorems to operators acting in appropriate functional Banach spaces. This method is fairly well developed, frequently used, and effective in many situations. However, there are cases in which certain infinite systems of differential equations arise—linked to the modeling of significant real-world phenomena—where this method, based on situating considerations within Banach spaces, fails and cannot be applied. In this paper, we propose a slightly different approach, which involves conducting the analysis within appropriate functional Fréchet spaces. We discuss the fundamental properties of these spaces and formulate compactness criteria. The main result of this paper is a positive answer, using the proposed method, to an open problem concerning the modeling of a stochastic birth-and-death process, as formulated in one of the cited publications. The most important conclusion is that the presented computational technique, based on functional Fréchet spaces, can be regarded as a more effective alternative to methods based on Banach spaces.

Polyhedra for which every homotopy domination over itself is a homotopy equivalence

This paper is concerned with the algorithmic learning where the learner is allowed to make finite but bounded number of mind changes. Briefly, in our learning paradigm, a learner is given examples from a recursive function, which the learner attempts to learn by producing programs to compute that function. We say that a team is successful if at least one member of the team learns the target function. The problem, given two teams with bounded number of learners and mind changes whether one team can provably learn more than the other team has been open for the last fifteen years. This paper makes significant progress toward a complete solution of this problem. In the case of error-free learning, this paper solves the open problem. Finally, in the case of EX learning our result shows that there is no team with a≥0 mind changes whose learning power is exactly equal to a single learner with bounded b(≠ a) number of mind changes. In the case of PEX learning we have a positive answer.

In this work, we provide a new approach to address the solvability of a class of partial integrodifferential equations with nonlocal conditions and two linear parts, the first one being the generator of a strongly continuous semigroup, while the second one is of integral type. We introduce new ideas to handle the measure of noncompactness of a class of integral operators. The methods developed in this paper are of independent interest and enable us to evolve a general existence theory for a large class integrodifferential equations and to give positive answers to some questions raised in earlier works. In our considerations the semigroup generated by the first linear part is not necessarily compact or equicontinuous, which is an extra interesting feature. To demonstrate the usefulness of our results we illustrate three examples.