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## ‣ Uma justificava da validade do teorema fundamental da álgebra para o ensino médio

Nicacio, Nilson Herminio
Fonte: Universidade Federal do Rio Grande do Norte; BR; UFRN; Programa de Pós-Graduação em Matemática em Rede Nacional; Álgebra; Análise matemática; Ensino de matemática; Geometria e topologia; Matemática aplicada Publicador: Universidade Federal do Rio Grande do Norte; BR; UFRN; Programa de Pós-Graduação em Matemática em Rede Nacional; Álgebra; Análise matemática; Ensino de matemática; Geometria e topologia; Matemática aplicada
Tipo: Dissertação Formato: application/pdf
Português
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Among several theorems which are taught in basic education some of them can be proved in the classroom and others do not, because the degree of difficulty of its formal proof. A classic example is the Fundamental Theorem of Algebra which is not proved, it is necessary higher-level knowledge in mathematics. In this paper, we justify the validity of this theorem intuitively using the software Geogebra. And, based on [2] we will present a clear formal proof of this theorem that is addressed to school teachers and undergraduate students in mathematics; Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Dentre os vários teoremas que são ensinados na educação básica, alguns podem ser demonstrados em sala de aula e outros não, devido o grau de dificuldade de sua prova formal. Um exemplo clássico e o Teorema Fundamental da Algébra, que não é demonstrado, pois é necessário conhecimentos em Matemática de nível superior. Neste trabalho, justicamos intuitivamente a validade do Teorema Fundamental da Algebra usando o software Geogebra. E, baseados em [2], apresentamos uma clara demonstração formal desse teorema que está endereçada aos professores do ensino básico e alunos de licenciatura em Matemática

## ‣ Causal attributions for success or failure by passing and failing students in College Algebra

Cortes-Suarez, Georgina
Fonte: FIU Digital Commons Publicador: FIU Digital Commons
Tipo: Artigo de Revista Científica
Português
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Success in mathematics has been identified as a predictor of baccalaureate degree completion. Within the coursework of college mathematics, College Algebra has been identified as a high-risk course due to its low success rates. ^ Research in the field of attribution theory and academic achievement suggests a relationship between a student's attributional style and achievement. Theorists and researchers contend that attributions influence individual reactions to success and failure. They also report that individuals use attributions to explain and justify their performance. Studies in mathematics education identify attribution theory as the theoretical orientation most suited to explain academic performance in mathematics. This study focused on the relationship among a high risk course, low success rates, and attribution by examining the difference in the attributions passing and failing students gave for their performance in College Algebra. ^ The methods for the study included a pilot administration of the Causal Dimension Scale (CDSII) which was used to conduct reliability and principal component analyses. Then, students (n = 410) self-reported their performance on an in-class test and attributed their performance along the dimensions of locus of causality...

## ‣ An investigation into the effects of introducing algebra using a function-based approach

Sterritt, Nicola
Fonte: University of Limerick Publicador: University of Limerick
Tipo: info:eu-repo/semantics/masterThesis; all_ul_research; ul_published_reviewed; ul_theses_dissertations
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peer-reviewed; Ireland is currently witnessing a major overhaul of its mathematics syllabus for second level education. This syllabus is known as ‘Project Maths’ and came about as a results of concerns relating to the mathematics performance of students in Ireland in international comparative studies such as the PISA (Program for International Student Assessment) tests (Close & Oldham 2005; Cosgrove, Shiel, Sofroniou, Zastrutzki & Shortt 2005; Perkins, Moran, Cosgrove and Shiel 2010; Oldham 2002, 2006). The author found inspiration for this research when she identified concerns in her own classroom. These concerns were two-fold; firstly the author found that first year students began secondary school with a poor attitude towards mathematics and secondly, the author found that first year students had a lot of difficulty grasping and retaining basic algebraic concepts. The author followed an action research approach to implementing an intervention in her classroom aimed at overcoming these problems. In the first phase of this research, the author carried out a comprehensive review of literature on affect pertaining to mathematics education and on the teaching and learning of algebra. As a result of this review, the author decided to use a function-based approach to teaching algebra as a means of improving students understanding of basic algebra. A collaborative peer learning environment was chosen as the main pedagogical tool for improving attitude towards mathematics. The second phase of this research saw the development and implementation of an intervention in the author’s classroom during which fourth year students tutored first year students. Quantitative and qualitative data was gathered during this phase. The third phase comprised of an analysis of data...

## ‣ The Lie algebra perturbation lemma

Huebschmann, Johannes
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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Let g be a differential graded Lie algebra and suppose given a contraction of chain complexes of g onto a general chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S" to the given Lie algebra g, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on g onto S" which is natural in the data. This extends a result established in a joint paper of the author with J. Stashef [Forum math. 14 (2002), 847-868, math.AG/9906036] where only the particular where M is the homology of g has been explored.; Comment: 20 pages; in view of a number of comments of J. Stasheff, the exposition has been improved

## ‣ Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

De Loera, Jesús A.; Petrović, Sonja; Stasi, Despina
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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This paper transfers a randomized algorithm originally used in geometric optimization to computational commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems, for which we utilize a Helly-type result for algebraic varieties, and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of G\"artner et al.\ applies to these polynomial ideal problems. The resulting algorithms have expected runtime linear in the number of input polynomials, making the method particularly interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables is constant).; Comment: 14 pages; corrected Example 3.2; added some references; results unchanged

## ‣ Factorization in $SL_n(R)$ with elementary matrices when $R$ is the disk algebra and the Wiener algebra

Sasane, Amol
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 20/05/2014 Português
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Let $R$ be the polydisc algebra or the Wiener algebra. It is shown that the group $SL_n(R)$ is generated by the subgroup of elementary matrices with all diagonal entries $1$ and at most one nonzero off-diagonal entry. The result an easy consequence of the deep result due to Ivarsson and Kutzschebauch (Ann. of Math. 2012).; Comment: 5 pages

## ‣ The sh-Lie algebra perturbation Lemma

Huebschmann, Johannes
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
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Let R be a commutative ring which contains the rationals as a subring and let g be a chain complex. Suppose given an sh-Lie algebra structure on g, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra T' on the suspension of g and write the perturbed coalgebra as T". Suppose, furthermore, given a contraction of g onto a chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S" to the loop Lie algebra L on the perturbed coalgebra T", and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on L onto S" which is natural in the data. For the special case where M and g are connected we also construct an explicit extension of the perturbed retraction to an sh-Lie map. This approach includes a very general solution of the master equation.; Comment: 20 pages

## ‣ Algebra retracts and Stanley-Reisner rings

Epstein, Neil; Nguyen, Hop D.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field $k$ are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner rings over a field $k$ are again Stanley-Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over $k$ descend along graded algebra retracts.; Comment: Incorporating several useful suggestions due to a referee. To appear in Journal of Pure and Applied Algebra

## ‣ Torus actions, combinatorial topology and homological algebra

Buchstaber, Victor M.; Panov, Taras E.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 07/10/2000 Português
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The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and, especially, spheres. We describe important constructions which allow to study all these combinatorial objects by means of methods of commutative and homological algebra. The proposed approach to combinatorial problems relies on the theory of moment-angle complexes, currently being developed by the authors. The theory centres around the construction that assigns to each simplicial complex $K$ with $m$ vertices a $T^m$-space $\zk$ with a special bigraded cellular decomposition. In the framework of this theory, the well-known non-singular toric varieties arise as orbit spaces of maximally free actions of subtori on moment-angle complexes corresponding to simplicial spheres. We express different invariants of simplicial complexes and related combinatorial-geometrical objects in terms of the bigraded cohomology rings of the corresponding moment-angle complexes. Finally, we show that the new relationships between combinatorics, geometry and topology result in solutions to some well-known topological problems.; Comment: 87 pages...

## ‣ Evolution algebra of a "chicken" population

Ladra, M.; Rozikov, U. A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 18/07/2013 Português
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We consider an evolution algebra which corresponds to a bisexual population with a set of females partitioned into finitely many different types and the males having only one type. We study basic properties of the algebra. This algebra is commutative (and hence flexible), not associative and not necessarily power associative, in general. Moreover it is not unital. A condition is found on the structural constants of the algebra under which the algebra is associative, alternative, power associative, nilpotent, satisfies Jacobi and Jordan identities. In a general case, we describe the full set of idempotent elements and the full set of absolute nilpotent elements. The set of all operators of left (right) multiplications is described. Under some conditions on the structural constants it is proved that the corresponding algebra is centroidal. Moreover the classification of 2-dimensional and some 3-dimensional algebras are obtained.; Comment: 15 pages

## ‣ Modular decomposition of the Orlik-Terao algebra of a hyperplane arrangement

Denham, Graham; Garrousian, Mehdi; Tohaneanu, Stefan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/11/2012 Português
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Let A be a collection of n linear hyperplanes in k^l, where k is an algebraically closed field. The Orlik-Terao algebra of A is the subalgebra R(A) of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of A. It determines an irreducible subvariety of projective space. We show that a flat X of A is modular if and only if R(A) is a split extension of the Orlik-Terao algebra of the subarrangement A_X. This provides another refinement of Stanley's modular factorization theorem and a new characterization of modularity, similar in spirit to the modular fibration theorem of Paris. We deduce that if A is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.; Comment: 23 pages

## ‣ Some curiosities of the algebra of bounded Dirichlet series

Mortini, Raymond; Sasane, Amol
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/11/2015 Português
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It is shown that the algebra of bounded Dirichlet series is not a coherent ring, and has infinite Bass stable rank. As corollaries of the latter result, it is derived that the algebra of bounded Dirichlet series has infinite topological stable rank and infinite Krull dimension.; Comment: 8 pages

## ‣ The Aluffi Algebra

Nejad, Abbas Nasrollah; Simis, Aron
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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We deal with the quasi-symmetric algebra introduced by Paolo Aluffi, here named (embedded) Aluffi algebra. The algebra is a sort of "intermediate" algebra between the symmetric algebra and the Rees algebra of an ideal, which serves the purpose of introducing the characteristic cycle of a hypersurface in intersection theory. The results described in the present paper have an algebraic flavor and naturally connect with various themes of commutative algebra, such as standard bases \'a la Hironaka, Artin--Rees like questions, Valabrega--Valla ideals, ideals of linear type, relation type and analytic spread. We give estimates for the dimension of the Aluffi algebra and show that, pretty generally, the latter is equidimensional whenever the base ring is a hypersurface ring. There is a converse to this under certain conditions that essentially subsume the setup in Aluffi's theory, thus suggesting that this algebra will not handle cases other than the singular locus of a hypersurface. The torsion and the structure of the minimal primes of the algebra are clarified. In the case of a projective hypersurface the results are more precise and one is naturally led to look at families of projective plane singular curves to understand how the property of being of linear type deforms/specializes for the singular locus of a member. It is fairly elementary to show that the singular locus of an irreducible curve of degree at most 3 is of linear type. This is roundly false in degree larger than 4 and the picture looks pretty wild as we point out by means of some families. Degree 4 is the intriguing case. Here we are able to show that the singular locus of the generic member of a family of rational quartics...

## ‣ The Orlik-Terao algebra and 2-formality

Schenck, Hal; Tohaneanu, Stefan
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A in C^n; it is the quotient of an exterior algebra E(V) on |A| generators. Orlik and Terao introduced a commutative analog S(V)/I of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice L(A). Motivated by topological considerations, Falk and Randell introduced the property of 2-formality; we study the relation between 2-formality and the Orlik-Terao algebra. Our main result is a necessary and sufficient condition for 2-formality in terms of the quadratic component I_2 of the Orlik-Terao ideal I: 2-formality is determined by the tangent space T_p(V(I_2)) at a generic point p.; Comment: 11 pages, 2 figures. v2 typo in Example 1.4 fixed v3 Le-Mohammadi have pointed out that one direction of Thm 2.4 (now Thm 2.5) needs a saturation hypothesis, now corrected

## ‣ On the invariants of the splitting algebra

Thorup, Anders
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 23/05/2011 Português
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For a given monic polynomial $p(t)$ of degree $n$ over a commutative ring $k$, the splitting algebra is the universal $k$-algebra in which $p(t)$ has $n$ roots, or, more precisely, over which $p(t)$ factors, $p(t)=(t-\xi_1)...(t-\xi_n)$. The symmetric group $S_r$ for $1\le r\le n$ acts on the splitting algebra by permuting the first $r$ roots $\xi_1,...,\xi_r$. We give a natural, simple condition on the polynomial $p(t)$ that holds if and only if there are only trivial invariants under the actions. In particular, if the condition on $p(t)$ holds then the elements of $k$ are the only invariants under the action of $S_n$. We show that for any $n\ge 2$ there is a polynomial $p(t)$ of degree $n$ for which the splitting algebra contains a nontrivial element invariant under $S_n$. The examples violate an assertion by A. D. Barnard from 1974.

## ‣ Algebraic structures of tropical mathematics

Izhakian, Zur; Knebusch, Manfred; Rowen, Louis
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/05/2013 Português
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Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing researchers to rely often on combinatoric techniques. In this paper we describe an alternative structure, more compatible with valuation theory, studied by the authors over the past few years, that permits fuller use of algebraic theory especially in understanding the underlying tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid $\tM$ is replaced by $R: = L\times \tM$, where $L$ is a given indexing semiring (not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When $L=\{1,\infty\}$ we recover the "standard" supertropical structure with its "ghost" layer. When $L = \NN$ we can describe multiple roots of polynomials via a "layering function" $s: R \to L$. Likewise, one can define the layering $s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called tropically dependent if each component of some nontrivial linear combination $\sum \a_i v_i$ is a ghost...

## ‣ When the orbit algebra of group is an integral domain? Proof of a conjecture of P.J. Cameron

Pouzet, Maurice
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/04/2007 Português
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P.J.Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure $R$ is an integral domain if and only if $R$ is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra.; Comment: 19 pages

## ‣ A new discriminant algebra construction

Biesel, Owen; Gioia, Alberto
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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A discriminant algebra operation sends a commutative ring $R$ and an $R$-algebra $A$ of rank $n$ to an $R$-algebra $\Delta_{A/R}$ of rank $2$ with the same discriminant bilinear form. Constructions of discriminant algebra operations have been put forward by Rost, Deligne, and Loos. We present a simpler and more explicit construction that does not break down into cases based on the parity of $n$. We then prove properties of this construction, and compute some examples explicitly.; Comment: 33 pages; the new version has been reorganized to improve readability and contains new examples, as well as a streamlined proof of the main theorem

## ‣ Elimination and nonlinear equations of Rees algebra

Busé, Laurent; Chardin, Marc; Simis, Aron
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 13/11/2009 Português
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A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additional point is that this contribution -- which of course corresponds to non linear equations of the Rees algebra -- can be described in these degrees in terms of non Koszul syzygies via certain upgrading maps in the vein of the ones introduced earlier by J. Herzog, the third named author and W. Vasconcelos. As a measure of the reach of this torsion analysis we could say that, in the case of a general everywhere defined map, half of the degrees where the torsion does not vanish are understood.

## ‣ Constructive Homological Algebra and Applications

Rubio, Julio; Sergeraert, Francis
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
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This text was written and used for a MAP Summer School at the University of Genova, August 28 to September 2, 2006. Available since then on the web site of the second author, it has been used and referenced by several colleagues working in Commutative Algebra and Algebraic Topology. To make safer such references, it was suggested to place it on the Arxiv repository. It is a relatively detailed exposition of the use of the Basic Perturbation Lemma to make constructive Homological Algebra (standard Homological Algebra is not constructive) and how this technology can be used in Commutative Algebra (Koszul complexes) and Algebraic Topology (effective versions of spectral sequences).; Comment: Version 3: Error corrected p. 111, see footnote 26