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In the last decade, the Poincaré conjecture has probably been the most famous statement among all the contributions of Poincaré to the mathematics community. There have been many papers and books that describe various attempts and the final works of Perelman leading to a positive solution to the conjecture, but the evolution of Poincaré's works leading to this conjecture has not been carefully discussed or described, and some other historical aspects about it have not been addressed either. For example, one question is how it fits into the overall work of Poincaré in topology, and what are some other related questions that he had raised. Since Poincaré did not state the Poincaré conjecture as a conjecture but rather raised it as a question, one natural question is why he did this. In order to address these issues, in this paper, we examine Poincaré's works in topology in the framework of classifying manifolds through numerical and algebraic invariants. Consequently, we also provide a full history of the formulation of the Poincaré conjecture which is richer than what is usually described and accepted and hence gain a better understanding of overall works of Poincaré in topology. In addition, this analysis clarifies a puzzling question on the relation between Poincaré's stated motivations for topology and the Poincaré conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we give an affirmative answer to two conjectures on generalized (m, n)-Jordan derivations and generalized (m, n)-Jordan centralizers raised in Ali and Fošner (Algebra Colloq 21:411–420, 2014) and Fošner (Demonstr Math 46:254–262, 2013). Precisely, when R is a semiprime ring, we prove, under some suitable torsion restrictions, that every nonzero generalized (m, n)-Jordan derivation (resp., a generalized (m, n)-Jordan centralizer) is a derivation (resp., a two-sided centralizer). [ABSTRACT FROM AUTHOR]

The present paper continues a cycle of papers, in which the Yoneda algebras were calculated for several families of algebras of dihedral and semidihedral type in the classification by K. Erdmann. Using the technique of a previous paper, a description of the Yoneda algebras for both families of local algebras occurring in this classification is given. Namely, a conjecture about the structure of the minimal free resolution of a (unique) simple module is stated, which is based on some empirical observations, and after establishing this conjecture, “cohomology information" is derived from the resolution discovered, and, as a result, this allows us to describe the Yoneda algebras of the algebras under consideration, It is noted that a similar technique was applied in computation of the Hochschild cohomology algebra for some finite-dimensional algebras. Bibliography: 23 titles. [ABSTRACT FROM AUTHOR]

In an earlier paper we proved some results concerning Gross's conjecture on tori. This conjecture, which we call the Minus Conjecture, is closely related to a conjecture of Burns, which is now known to hold generally in the absolutely abelian setting; however Burns' conjecture does not directly imply the Minus Conjecture. The result proved in the earlier paper was concerned with imaginary absolutely abelian extensions K/ℚ of the form K = FK+, with F imaginary quadratic and K+/ℚ being tame, l-elementary and ramified at most at two primes. In the present paper we complement these results by proving the Minus Conjecture for extensions K/ℚ as above but without any restriction on the number s of ramified primes. The price we have to pay for this generality is that our proof only works if the odd prime l is large enough, more precisely if l ≧ 3( s + 1). There is one more restriction, namely . (A similar restriction was already needed in our previous paper.) [ABSTRACT FROM AUTHOR]

Let ${\mathfrak{g}}$ be a finite dimensional simple Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic 0. Let ${\mathfrak{g}}_{{\mathbb{Z}}}$ be a Chevalley ℤ-form of ${\mathfrak{g}}$ and ${\mathfrak{g}}_{\Bbbk}={\mathfrak{g}}_{{\mathbb{Z}}}\otimes _{{\mathbb{Z}}}\Bbbk$, where $\Bbbk$ is the algebraic closure of ${\mathbb{F}}_{p}$. Let $G_{\Bbbk}$ be a simple, simply connected algebraic $\Bbbk$-group with $\operatorname{Lie}(G_{\Bbbk})={\mathfrak{g}}_{\Bbbk}$. In this paper, we apply recent results of Rudolf Tange on the fraction field of the centre of the universal enveloping algebra $U({\mathfrak{g}}_{\Bbbk})$ to show that if the Gelfand–Kirillov conjecture (from 1966) holds for ${\mathfrak{g}}$, then for all p≫0 the field of rational functions $\Bbbk ({\mathfrak{g}}_{\Bbbk})$ is purely transcendental over its subfield $\Bbbk({\mathfrak{g}}_{\Bbbk})^{G_{\Bbbk}}$. Very recently, it was proved by Colliot-Thélène, Kunyavskiĭ, Popov, and Reichstein that the field of rational functions ${\mathbb{K}}({\mathfrak{g}})$ is not purely transcendental over its subfield ${\mathbb{K}}({\mathfrak{g}})^{\mathfrak{g}}$ if ${\mathfrak{g}}$ is of type B n, n≥3, D n, n≥4, E6, E7, E8 or F4. We prove a modular version of this result (valid for p≫0) and use it to show that, in characteristic 0, the Gelfand–Kirillov conjecture fails for the simple Lie algebras of the above types. In other words, if ${\mathfrak{g}}$ is of type B n, n≥3, D n, n≥4, E6, E7, E8 or F4, then the Lie field of ${\mathfrak{g}}$ is more complicated than expected. [ABSTRACT FROM AUTHOR]

It is well known that a systematical way to construct finite dimensional filter is to classify all finite dimensional estimation algebras. Mitter conjecture, which states that all functions in any finite dimensional estimation algebra are necessarily degree one polynomial, plays a crucial role in classifying all finite dimensional estimation algebras. The purpose of this paper is to prove the Mitter Conjecture for estimation algebra of dimension at most 5 with arbitrary state space dimension. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we compute the congruences of Catalan and Motzkin numbers modulo 4 and 8. In particular, we prove the conjecture proposed by Deutsch and Sagan that no Motzkin number is a multiple of 8. [Copyright &y& Elsevier]

We consider the affine action of a nilpotent Lie group on ℝn. Lipsman (1995) conjectured that such an action is proper in the sense of Palais if and only if the action is (CI) in the sense of Kobayashi. The present paper gives a counterexample to Lipsman's conjecture for n ≥ 5. [ABSTRACT FROM AUTHOR]

Let G be a finite group. Order components of G were introduced in Chen [5]. Let OC ( G ) be the set of order components of G . Some finite groups are characterizable by their order components. This assertion was proved for the simple groups PSU ( p,q ), where p =3, 5, 7 and 11. In this paper, we prove that the simple groups PSU ( p , q ) can be uniquely determined by their order components, where p≥ 13 is a prime number. Main consequences of our results are the validity of a conjecture of J. G. Thompson and another conjecture of W. Shi and J. Bi for the groups under consideration. [ABSTRACT FROM AUTHOR]

We use the class of representation-finite algebras to investigate the finitistic dimension conjecture. In this way we obtain a large class of algebras for which the finitistic dimension conjecture holds. The main results in this paper are: (1) Let A be an artin algebra and let Ij,1⩽j⩽n be a family of ideals in A with I1I2⋯In=0, such that proj.dim(AIj)<∞ and proj.dim(Ij)A=0 for all j⩾3. If A/I1 and A/I2 are representation-finite and if A/Ij has finite finitistic dimension for j⩾3, then the finitistic dimension of A is finite. In particular, the finitistic dimension conjecture is true for algebras obtained from representation-finite algebras by forming dual extensions, trivially twisted extensions, Hochschild extensions, matrix algebras and tensor products with algebras of radical-square-zero. (2) Let A,B and C be three artin algebras with the same identity such that (i) C⊆B⊆A, and (ii) the Jacobson radical of C is a left ideal of B and the Jacobson radical of B is a left ideal of A. If A is representation-finite, then C has finite finitistic dimension. We also provide a way to construct algebras satisfying all conditions in (2), and this leads to a new reformulation of the finitistic dimension conjecture. [Copyright &y& Elsevier]

In this paper, we show first that any prime (or semiprime) element of a commutative Banach algebra must have closed range. As a corollary, we find that in a commutative radical Banach algebra, all primes are zero divisors; indeed, all semiprimes are zero divisors (see below for the definition of semiprimeness). Our result is also true of a semiprime that is in the centre of a noncommutative Banach algebra.The proof is fairly simple and entertaining, and we obtain a result that is helpful for the ambitious classification of elements in commutative radical Banach algebras being attempted by Marc Thomas. It is also related to the unbounded Kleinecke–Shirov conjecture. [ABSTRACT FROM PUBLISHER]

The Andrews--Curtis conjecture claims that every balanced presentation of the trivial group can be transformed into the trivial presentation by a finite sequence of "elementary transformations" which are Nielsen transformations together with an arbitrary conjugation of a relator. It is believed that the Andrews--Curtis conjecture is false; however, not so many possible counterexamples are known. It is not a trivial matter to verify whether the conjecture holds for a given balanced presentation or not. The purpose of this paper is to describe some nondeterministic methods, called Genetic Algorithms, designed to test the validity of the Andrews-Curtis conjecture. Using such algorithm we have been able to prove that all known (to us) balanced presentations of the trivial group where the total length of the relators is at most 12 satisfy the conjecture. In particular, the Andrews-Curtis conjecture holds for the presentation 〈x, y&verbar;xyx = yxy, x[SUP2] = y[SUP3]〉 which was one of the well known potential counterexamples. [ABSTRACT FROM AUTHOR]