Abstract In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms …
Abstract In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms of comaximal graphs and the rings in question. In addition, we investigate the relation between the comaximal graph of a ring and its subrings of a certain type.
Abstract We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a …
Abstract We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a commutative ring R and that of the matrix ring M n (R). Key Words: Commutative ringsMatrix ringsZero-divisor graph2000 Mathematics Subject Classification: Primary 16S50Secondary 13A99, 05C99 ACKNOWLEDGMENTS The authors would like to thank the anonymous referee for his/her helpful comments which have significantly improved the presentation of the results in this article. This work was partially supported by the Ministry of Science and Environmental Protection of the Republic of Serbia Project #144020. Notes Communicated by I. Swanson.
We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, …
We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with.
Mod 2 cohomology of the Grassmann manifold G 2,n is a polynomial algebra modulo a certain well-known ideal.A Groebner basis for this ideal is obtained.Using this basis, some new immersion …
Mod 2 cohomology of the Grassmann manifold G 2,n is a polynomial algebra modulo a certain well-known ideal.A Groebner basis for this ideal is obtained.Using this basis, some new immersion results for Grassmannians G 2,n are established.
Any square matrix over an algebraically closed field has a Jordan normal form. In this paper, we prove that every infinite upper triangular matrix over an arbitrary field has a …
Any square matrix over an algebraically closed field has a Jordan normal form. In this paper, we prove that every infinite upper triangular matrix over an arbitrary field has a generalized infinite Jordan normal form.
The canonical circle action on complex Stiefel manifolds is considered. The corresponding mod $p$ index for all primes $p$ is computed and some Borsuk--Ulam type theorems are proved.
The canonical circle action on complex Stiefel manifolds is considered. The corresponding mod $p$ index for all primes $p$ is computed and some Borsuk--Ulam type theorems are proved.
Abstract Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def …
Abstract Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ cohomology of the real flag manifolds $F(1,1,n)$ and $F(1,2,n)$ are obtained. These are used to compute appropriate Stiefel–Whitney classes in order to establish some new nonembedding and nonimmersion results for the manifolds $F(1,2,n)$ .
Coefficients of the cyclotomic polynomial can be interpreted topologically, as the torsion in the homology of a certain simplicial complex associated with the degree of the cyclotomic polynomial, which was …
Coefficients of the cyclotomic polynomial can be interpreted topologically, as the torsion in the homology of a certain simplicial complex associated with the degree of the cyclotomic polynomial, which was studied by Musiker and Reiner. We answer a question posed by the two authors regarding homotopy type of certain subcomplexes of the associated simplicial complex when the degree of the cyclotomic polynomial is a product of three distinct primes.
Abstract Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases …
Abstract Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases for such spaces of maximum dimension can be chosen in such a way to consist of matrices of the minimal rank. An estimate of the ranks of matrices in particular types of bases for maximal such spaces is also given. This extends previously known results which were valid only in the case of spaces consisting of matrices of rank not equal to one. Keywords: spaces of matricesrank conditionnon-singular bilinear mapsAMS subject classifications:: 15A0315A69 Acknowledgements This work was partially supported by Ministry of Science and Environmental Protection of Republic of Serbia Project #144020. The author would like to thank the anonymous referee for his/her valuable comments which have not only improved the presentation but have also helped in extending the results of this article.
The knowledge of cohomology of a manifold has shown to be quite relevant in various investigations: the question of vector fields, immersion and embedding dimension, and recently even in topological …
The knowledge of cohomology of a manifold has shown to be quite relevant in various investigations: the question of vector fields, immersion and embedding dimension, and recently even in topological robotics. The method of Gröbner bases is applicable when the cohomology of the manifold is a quotient of a polynomial algebra. The mod 2 cohomology of the real flag manifold F(n 1 , n 2 , …, n r ) is known to be isomorphic to a polynomial algebra modulo a certain ideal. Reduced Gröbner bases for these ideals are obtained in the case of manifolds F(1, 1, …, 1, n) including the complete flag manifolds (n = 1).
A Gröbner basis for the ideal determining mod 2 cohomology of Grassmannian G_{3,n} is obtained. This is used, along with the method of obstruction theory, to establish some new immersion …
A Gröbner basis for the ideal determining mod 2 cohomology of Grassmannian G_{3,n} is obtained. This is used, along with the method of obstruction theory, to establish some new immersion results for these manifolds.
Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a …
Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper is to understand this cohomology via Gröbner bases. Reduced Gröbner bases for the ideals $I_{k,n}$ are determined. An application of these bases is given by proving an immersion theorem for Grassmann manifolds $G_{5,n}$, which establishes new immersions for an infinite family of these manifolds.
A Grobner basis for the ideal determining mod 2 cohomology of Grassmannian G3,n, obtained in a previous paper, is used, along with the method of obstruction theory, to establish some …
A Grobner basis for the ideal determining mod 2 cohomology of Grassmannian G3,n, obtained in a previous paper, is used, along with the method of obstruction theory, to establish some new immersion results for these manifolds.
Using discrete Morse theory for simplicial complexes we determine the homotopy type of ideal zerodivisor complex for finite rings and for rings with infinitely many maximal ideals.
Using discrete Morse theory for simplicial complexes we determine the homotopy type of ideal zerodivisor complex for finite rings and for rings with infinitely many maximal ideals.
The conditions that allow an element of an associative, unital, not necessarily commutative ring R, to be represented as a sum of (commuting) idempotents and one nilpotent element are analyzed.Some …
The conditions that allow an element of an associative, unital, not necessarily commutative ring R, to be represented as a sum of (commuting) idempotents and one nilpotent element are analyzed.Some applications to group rings are also presented.
Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove …
Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.
Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] …
Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] is expressible as a sum [Formula: see text] of (commuting) idempotent matrices [Formula: see text] and a nilpotent matrix [Formula: see text].
Let R be a Pr¨ufer domain of Krull dimension one. We prove the existence of Gr¨obner bases for finitely generated submodules of finitely generated free modules over R[X], where the …
Let R be a Pr¨ufer domain of Krull dimension one. We prove the existence of Gr¨obner bases for finitely generated submodules of finitely generated free modules over R[X], where the term order is POT, or, “position over term”. In order to do this, we first prove that there is a Gr¨obner basis for finitely generated ideals in R[X], which is a special case of the main result. The proof is based on the results from [3]. In addition to this we show, in the case of valuation domains, that every Gr¨obner basis is actually a strong Gr¨obner basis
<p>We introduce the general notion of a rank on a vector space, which includes both tensor rank and conventional matrix rank, but incorporates other examples as well. Extending this concept, …
<p>We introduce the general notion of a rank on a vector space, which includes both tensor rank and conventional matrix rank, but incorporates other examples as well. Extending this concept, we investigate vector spaces consisting of vectors with a lower bound on their rank. Our main result shows that bases for such spaces of maximum dimension can be chosen to consist exclusively of vectors of minimal rank. This generalization extends the results of <sup>[<xref ref-type="bibr" rid="b15">15</xref>,<xref ref-type="bibr" rid="b36">36</xref>]</sup>, with potential applications in different areas.</p>
Graphs associated to the set of irreducible character degrees of a finite group [Formula: see text] have been extensively studied as a way of understanding structure of the underlying group. …
Graphs associated to the set of irreducible character degrees of a finite group [Formula: see text] have been extensively studied as a way of understanding structure of the underlying group. Another approach, proposed by Isaacs, is to study associated simplicial complexes, namely, the common divisor simplicial complex [Formula: see text] and the prime divisor simplicial complex [Formula: see text]. These complexes can be associated to any set of positive integers and this paper shows that they are homotopy equivalent. Further, considering these complexes associated to the set of irreducible character degrees, we give a bound on the rank of the fundamental group.
Graphs associated to the set of irreducible character degrees of a finite group [Formula: see text] have been extensively studied as a way of understanding structure of the underlying group. …
Graphs associated to the set of irreducible character degrees of a finite group [Formula: see text] have been extensively studied as a way of understanding structure of the underlying group. Another approach, proposed by Isaacs, is to study associated simplicial complexes, namely, the common divisor simplicial complex [Formula: see text] and the prime divisor simplicial complex [Formula: see text]. These complexes can be associated to any set of positive integers and this paper shows that they are homotopy equivalent. Further, considering these complexes associated to the set of irreducible character degrees, we give a bound on the rank of the fundamental group.
<p>We introduce the general notion of a rank on a vector space, which includes both tensor rank and conventional matrix rank, but incorporates other examples as well. Extending this concept, …
<p>We introduce the general notion of a rank on a vector space, which includes both tensor rank and conventional matrix rank, but incorporates other examples as well. Extending this concept, we investigate vector spaces consisting of vectors with a lower bound on their rank. Our main result shows that bases for such spaces of maximum dimension can be chosen to consist exclusively of vectors of minimal rank. This generalization extends the results of <sup>[<xref ref-type="bibr" rid="b15">15</xref>,<xref ref-type="bibr" rid="b36">36</xref>]</sup>, with potential applications in different areas.</p>
Let R be a Pr¨ufer domain of Krull dimension one. We prove the existence of Gr¨obner bases for finitely generated submodules of finitely generated free modules over R[X], where the …
Let R be a Pr¨ufer domain of Krull dimension one. We prove the existence of Gr¨obner bases for finitely generated submodules of finitely generated free modules over R[X], where the term order is POT, or, “position over term”. In order to do this, we first prove that there is a Gr¨obner basis for finitely generated ideals in R[X], which is a special case of the main result. The proof is based on the results from [3]. In addition to this we show, in the case of valuation domains, that every Gr¨obner basis is actually a strong Gr¨obner basis
Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] …
Let [Formula: see text] be an associative unital ring and not necessarily commutative. We analyze conditions under which every [Formula: see text] matrix [Formula: see text] over [Formula: see text] is expressible as a sum [Formula: see text] of (commuting) idempotent matrices [Formula: see text] and a nilpotent matrix [Formula: see text].
Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove …
Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.
Coefficients of the cyclotomic polynomial can be interpreted topologically, as the torsion in the homology of a certain simplicial complex associated with the degree of the cyclotomic polynomial, which was …
Coefficients of the cyclotomic polynomial can be interpreted topologically, as the torsion in the homology of a certain simplicial complex associated with the degree of the cyclotomic polynomial, which was studied by Musiker and Reiner. We answer a question posed by the two authors regarding homotopy type of certain subcomplexes of the associated simplicial complex when the degree of the cyclotomic polynomial is a product of three distinct primes.
Any square matrix over an algebraically closed field has a Jordan normal form. In this paper, we prove that every infinite upper triangular matrix over an arbitrary field has a …
Any square matrix over an algebraically closed field has a Jordan normal form. In this paper, we prove that every infinite upper triangular matrix over an arbitrary field has a generalized infinite Jordan normal form.
The conditions that allow an element of an associative, unital, not necessarily commutative ring R, to be represented as a sum of (commuting) idempotents and one nilpotent element are analyzed.Some …
The conditions that allow an element of an associative, unital, not necessarily commutative ring R, to be represented as a sum of (commuting) idempotents and one nilpotent element are analyzed.Some applications to group rings are also presented.
Using discrete Morse theory for simplicial complexes we determine the homotopy type of ideal zerodivisor complex for finite rings and for rings with infinitely many maximal ideals.
Using discrete Morse theory for simplicial complexes we determine the homotopy type of ideal zerodivisor complex for finite rings and for rings with infinitely many maximal ideals.
A Grobner basis for the ideal determining mod 2 cohomology of Grassmannian G3,n, obtained in a previous paper, is used, along with the method of obstruction theory, to establish some …
A Grobner basis for the ideal determining mod 2 cohomology of Grassmannian G3,n, obtained in a previous paper, is used, along with the method of obstruction theory, to establish some new immersion results for these manifolds.
Abstract Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def …
Abstract Groebner bases for the ideals determining mod $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}2$ cohomology of the real flag manifolds $F(1,1,n)$ and $F(1,2,n)$ are obtained. These are used to compute appropriate Stiefel–Whitney classes in order to establish some new nonembedding and nonimmersion results for the manifolds $F(1,2,n)$ .
The canonical circle action on complex Stiefel manifolds is considered. The corresponding mod $p$ index for all primes $p$ is computed and some Borsuk--Ulam type theorems are proved.
The canonical circle action on complex Stiefel manifolds is considered. The corresponding mod $p$ index for all primes $p$ is computed and some Borsuk--Ulam type theorems are proved.
A Gröbner basis for the ideal determining mod 2 cohomology of Grassmannian G_{3,n} is obtained. This is used, along with the method of obstruction theory, to establish some new immersion …
A Gröbner basis for the ideal determining mod 2 cohomology of Grassmannian G_{3,n} is obtained. This is used, along with the method of obstruction theory, to establish some new immersion results for these manifolds.
Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a …
Grassmann manifolds $G_{k,n}$ are among the central objects in geometry and topology. The Borel picture of the mod 2 cohomology of $G_{k,n}$ is given as a polynomial algebra modulo a certain ideal $I_{k,n}$. The purpose of this paper is to understand this cohomology via Gröbner bases. Reduced Gröbner bases for the ideals $I_{k,n}$ are determined. An application of these bases is given by proving an immersion theorem for Grassmann manifolds $G_{5,n}$, which establishes new immersions for an infinite family of these manifolds.
The knowledge of cohomology of a manifold has shown to be quite relevant in various investigations: the question of vector fields, immersion and embedding dimension, and recently even in topological …
The knowledge of cohomology of a manifold has shown to be quite relevant in various investigations: the question of vector fields, immersion and embedding dimension, and recently even in topological robotics. The method of Gröbner bases is applicable when the cohomology of the manifold is a quotient of a polynomial algebra. The mod 2 cohomology of the real flag manifold F(n 1 , n 2 , …, n r ) is known to be isomorphic to a polynomial algebra modulo a certain ideal. Reduced Gröbner bases for these ideals are obtained in the case of manifolds F(1, 1, …, 1, n) including the complete flag manifolds (n = 1).
We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, …
We discuss the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with.
Mod 2 cohomology of the Grassmann manifold G 2,n is a polynomial algebra modulo a certain well-known ideal.A Groebner basis for this ideal is obtained.Using this basis, some new immersion …
Mod 2 cohomology of the Grassmann manifold G 2,n is a polynomial algebra modulo a certain well-known ideal.A Groebner basis for this ideal is obtained.Using this basis, some new immersion results for Grassmannians G 2,n are established.
Abstract In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms …
Abstract In this paper we investigate the center, radius and girth of comaximal graphs of commutative rings. We also provide some counterexamples to the results concerning the relation between isomorphisms of comaximal graphs and the rings in question. In addition, we investigate the relation between the comaximal graph of a ring and its subrings of a certain type.
Abstract We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a …
Abstract We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a commutative ring R and that of the matrix ring M n (R). Key Words: Commutative ringsMatrix ringsZero-divisor graph2000 Mathematics Subject Classification: Primary 16S50Secondary 13A99, 05C99 ACKNOWLEDGMENTS The authors would like to thank the anonymous referee for his/her helpful comments which have significantly improved the presentation of the results in this article. This work was partially supported by the Ministry of Science and Environmental Protection of the Republic of Serbia Project #144020. Notes Communicated by I. Swanson.
Abstract Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases …
Abstract Motivated by the problem concerning the existence of non-singular bilinear maps, vector spaces of matrices consisting of matrices with rank bounded below are investigated. It is shown that bases for such spaces of maximum dimension can be chosen in such a way to consist of matrices of the minimal rank. An estimate of the ranks of matrices in particular types of bases for maximal such spaces is also given. This extends previously known results which were valid only in the case of spaces consisting of matrices of rank not equal to one. Keywords: spaces of matricesrank conditionnon-singular bilinear mapsAMS subject classifications:: 15A0315A69 Acknowledgements This work was partially supported by Ministry of Science and Environmental Protection of Republic of Serbia Project #144020. The author would like to thank the anonymous referee for his/her valuable comments which have not only improved the presentation but have also helped in extending the results of this article.
Mod 2 cohomology of the Grassmann manifold G 2,n is a polynomial algebra modulo a certain well-known ideal.A Groebner basis for this ideal is obtained.Using this basis, some new immersion …
Mod 2 cohomology of the Grassmann manifold G 2,n is a polynomial algebra modulo a certain well-known ideal.A Groebner basis for this ideal is obtained.Using this basis, some new immersion results for Grassmannians G 2,n are established.
Using fiberings, we determine the cup-length and the Lyusternik–Shnirel'man category for some infinite families of real flag manifolds $O(n_1+\dots+n_q)/ O(n_1)\times\dots\times O(n_q)$, $q\geq 3$. We also give, or describe ways to …
Using fiberings, we determine the cup-length and the Lyusternik–Shnirel'man category for some infinite families of real flag manifolds $O(n_1+\dots+n_q)/ O(n_1)\times\dots\times O(n_q)$, $q\geq 3$. We also give, or describe ways to obtain, interesti
For n = 2 m+1 -4, m 2, we determine the cup-length of H * ( G n,3 ; Z/2) by finding a Gröbner basis associated with a certain subring, …
For n = 2 m+1 -4, m 2, we determine the cup-length of H * ( G n,3 ; Z/2) by finding a Gröbner basis associated with a certain subring, where G n,3 is the oriented Grassmann manifold SO(n + 3)/SO(n) × SO(3).As an application, we provide not only a lower but also an upper bound for the LS-category of G n,3 .We also study the immersion problem of G n,3 .
Let Gk;n be the Grassmann manifold of k-planes in R n+k : Borel showed that H ¤ (Gk;n;Z2 )= Z2 (w1;:::;wk)=Ik;n where Ik;n is the ideal generated by the dual …
Let Gk;n be the Grassmann manifold of k-planes in R n+k : Borel showed that H ¤ (Gk;n;Z2 )= Z2 (w1;:::;wk)=Ik;n where Ik;n is the ideal generated by the dual Stiefel-Whitney classes wn+1;:::;wn+k: We compute Groebner bases for the ideals I2;2ii3 and I2;2ii4 and use these results along with the theory of modi- …ed Postnikov towers to prove immersion results, namely that G2;2ii3 immerses in R 2 i+2 i15 : As a bene…t of the Groebner basis theory we also obtain a simple descrip- tion of H ¤ i G2;2ii3;Z2 ¢ and H ¤ i G2;2ii4;Z2 ¢ and use these results to give a simple proof of some non-immersion results of Oproui.
Let M n (R) and S n (R) be the spaces of n × n real matrices and real symmetric matrices respectively.We continue to study d(n, n -2, R): The …
Let M n (R) and S n (R) be the spaces of n × n real matrices and real symmetric matrices respectively.We continue to study d(n, n -2, R): The minimal number such that every -dimensional subspace of S n (R) contains a nonzero matrix of rank n-2 or less.We show that d(4, 2, R) = 5 and obtain some upper bounds and monotonicity properties of d(n, n -2, R).We give upper bounds for the dimensions of n -1 subspaces (subspaces where every nonzero matrix has rank n -1) of M n (R) and S n (R), which are sharp in many cases.We study the subspaces of M n (R) and S n (R) where each nonzero matrix has rank n or n -1.For a fixed integer q > 1 we find an infinite sequence of n such that any q+1
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript k Baseline left-parenthesis bold upper R Superscript n plus k Baseline right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>G</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext mathvariant="bold">R</mml:mtext></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">{G_k}({\textbf {R}^{n + k}})</mml:annotation></mml:semantics></mml:math></inline-formula>denote the …
Let<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G Subscript k Baseline left-parenthesis bold upper R Superscript n plus k Baseline right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>G</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext mathvariant="bold">R</mml:mtext></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">{G_k}({\textbf {R}^{n + k}})</mml:annotation></mml:semantics></mml:math></inline-formula>denote the grassman manifold of<italic>k</italic>-planes in real<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis n plus k right-parenthesis"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mspace width="thinmathspace" /><mml:mo>+</mml:mo><mml:mspace width="thinmathspace" /><mml:mi>k</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">(n\, +\, k)</mml:annotation></mml:semantics></mml:math></inline-formula>-space and<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w 1 element-of upper H Superscript 1 Baseline left-parenthesis upper G Subscript k Baseline left-parenthesis bold upper R Superscript n plus k Baseline right-parenthesis semicolon bold upper Z Subscript 2 Baseline right-parenthesis"><mml:semantics><mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:mspace width="thinmathspace" /><mml:mo>∈<!-- ∈ --></mml:mo><mml:mspace width="thinmathspace" /><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mi>H</mml:mi><mml:mn>1</mml:mn></mml:msup></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>G</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow class="MJX-TeXAtom-ORD"><mml:msup><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext mathvariant="bold">R</mml:mtext></mml:mrow><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mi>k</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:mo stretchy="false">)</mml:mo><mml:mo>;</mml:mo><mml:mspace width="thinmathspace" /><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mrow class="MJX-TeXAtom-ORD"><mml:mtext mathvariant="bold">Z</mml:mtext></mml:mrow><mml:mn>2</mml:mn></mml:msub></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">{w_1}\, \in \, {H^1}({G_k}({\textbf {R}^{n + k}});\,{\textbf {Z}_2})</mml:annotation></mml:semantics></mml:math></inline-formula>the first Stiefel-Whitney class of the universal bundle. Using Schubert calculus techniques and the cohomology of flag manifolds we estimate the height of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="w 1"><mml:semantics><mml:mrow class="MJX-TeXAtom-ORD"><mml:msub><mml:mi>w</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">{w_1}</mml:annotation></mml:semantics></mml:math></inline-formula>in the cohomology ring. We then apply this to improve earlier lower bounds on the Lusternik-Schnirelmann category of real grassmanians.
The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. …
The question of vector fields on spheres arises in homotopy theory and in the theory of fibre bundles, and it presents a classical problem, which may be explained as follows. For each n, let Sn-I be the unit sphere in euclidean n-space Rn. A vector field on Sn-1 is a continuous function v assigning to each point x of Sn-1 a vector v(x) tangent to Sn-1 at x. Given r such fields v1, v2, ..., Vr, we say that they are linearly independent if the vectors v1(x), v2(x), *--, vr(x) are linearly independent for all x. The problem, then, is the following: for each n, what is the maximum number r of linearly independent vector fields on Sn-i? For previous work and background material on this problem, we refer the reader to [1, 10, 11, 12, 13, 14, 15, 16]. In particular, we recall that if we are given r linearly independent vector fields vi(x), then by orthogonalisation it is easy to construct r fields wi(x) such that w1(x), w2(x), *I * , wr(x) are orthonormal for each x. These r fields constitute a cross-section of the appropriate Stiefel fibering. The strongest known positive result about the problem derives from the Hurwitz-Radon-Eckmann theorem in linear algebra [8]. It may be stated as follows (cf. James [13]). Let us write n = (2a + 1)2b and b = c + 4d, where a, b, c and d are integers and 0 < c < 3; let us define p(n) = 2c + 8d. Then there exist p(n) 1 linearly independent vector fields on Sn-1. It is the object of the present paper to prove that the positive result stated above is best possible.
We derive an inequality for the $\mathbb Z_2$-cup-length of any smooth closed connected manifold unorientedly cobordant to zero. In relation to this, we introduce a new numerical invariant of a …
We derive an inequality for the $\mathbb Z_2$-cup-length of any smooth closed connected manifold unorientedly cobordant to zero. In relation to this, we introduce a new numerical invariant of a smooth closed connected manifold, called the characteristic rank. In particular, our inequality yields strong upper bounds for the cup-length of the oriented Grassmann manifolds $\tilde G_{n,k}\cong SO(n)/SO(k)\times SO(n-k)$ $(6\leq 2k\leq n)$ if $n$ is odd; if $n$ is even, we obtain new upper bounds in a different way. We also derive lower bounds for the cup-length of $\tilde G_{n,k}$. For $\tilde G_{2^t-1,3}$ $(t\geq 3)$ our upper and lower bounds coincide, giving that the $\mathbb Z_2$-cup-length is $2^t-3$ and the characteristic rank equals $2^t-5$. Some applications to the Lyusternik-Shnirel'man category are also presented.
For every k 1 0 < k < m ≧ n , there are linear spaces of real n × m matrices which have dimension ( m − k )( …
For every k 1 0 < k < m ≧ n , there are linear spaces of real n × m matrices which have dimension ( m − k )( n − k ) and every nonzero element has rank greater than k . Examples of such spaces are constructed and conditions are given under which they have the largest possible dimension.
Aims to cover the most important aspects of the theory of matrices whose entries come from a given commutative ring. Essential facts about commutative rings are developed throughout the book, …
Aims to cover the most important aspects of the theory of matrices whose entries come from a given commutative ring. Essential facts about commutative rings are developed throughout the book, and proofs that follow from concrete matrix calculations are also provided.
A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a …
A ring R is strongly nil clean if every element in R is the sum of an idempotent and a nilpotent that commutes. We prove, in this article, that a ring R is strongly nil clean if and only if for any a ∈ R, there exists an idempotent e ∈ ℤ[a] such that a − e ∈ N(R), if and only if R is periodic and R∕J(R) is Boolean, if and only if each prime factor ring of R is strongly nil clean. Further, we prove that R is strongly nil clean if and only if for all a ∈ R, there exist n ∈ ℕ,k ≥ 0 (depending on a) such that an−an+2k∈N(R), if and only if for fixed m,n ∈ ℕ, a−a2n+2m(n+1)∈N(R) for all a ∈ R. These also extend known theorems, e.g, [5 Diesl, A. J. (2013). Nil clean rings. J. Algebra 383:197–211.[Crossref], [Web of Science ®] , [Google Scholar], Theorem 3.21], [6 Hirano, Y., Tominaga, H., Yaqub, A. (1988). On rings in which every element is uniquely expressible as a sum of a nilpotent element and a certain potent element. Math. J. Okayama Univ. 30:33–40. [Google Scholar], Theorem 3], [7 Kosan, M. T., Wang, Z., Zhou, Y. Nil-clean and strongly nil-clean rings. J. Pure Appl. Algebra. doi:10.1016/j.jpaa.2015.07.009 [Google Scholar], Theorem 2.7] and [12 Stancu, A. (2016). A note on commutative weakly nil clean rings. J. Algebra Appl. 15. doi:http://dx.doi.org/10.1142/S0219498816200012[Crossref] , [Google Scholar], Theorem 2].
A ring [Formula: see text] is strongly 2-nil-clean if every element in [Formula: see text] is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such …
A ring [Formula: see text] is strongly 2-nil-clean if every element in [Formula: see text] is the sum of two idempotents and a nilpotent that commute. Fundamental properties of such rings are obtained. We prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if for all [Formula: see text], [Formula: see text] is nilpotent, if and only if for all [Formula: see text], [Formula: see text] is strongly nil-clean, if and only if every element in [Formula: see text] is the sum of a tripotent and a nilpotent that commute. Furthermore, we prove that a ring [Formula: see text] is strongly 2-nil-clean if and only if [Formula: see text] is tripotent and [Formula: see text] is nil, if and only if [Formula: see text] or [Formula: see text], where [Formula: see text] is a Boolean ring and [Formula: see text] is nil; [Formula: see text] is a Yaqub ring and [Formula: see text] is nil. Strongly 2-nil-clean group algebras are investigated as well.
ABSTRACT We recall several results about zero-divisor graphs of commutative rings. Then we examine the preservation of diameter and girth of the zero-divisor graph under extension to polynomial and power …
ABSTRACT We recall several results about zero-divisor graphs of commutative rings. Then we examine the preservation of diameter and girth of the zero-divisor graph under extension to polynomial and power series rings.
The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers …
The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceara, Fortaleza, BrasilWalter D. Neumann, Columbia University, New York, USAMarkus J. Pflaum, University of Colorado, Boulder, USADierk Schleicher, Jacobs University, Bremen, GermanyKatrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019)Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019)Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019)Volker Mayer, Mariusz Urbanski, and Anna Zdunik, Random and Conformal Dynamical Systems (2021)Ioannis Diamantis, Bostjan Gabrovsek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)
Let A be either the real field R, or the complex field C, or the skew field Q of quaternions. Let A1, A2, , Ak be nXn matrices with entries …
Let A be either the real field R, or the complex field C, or the skew field Q of quaternions. Let A1, A2, , Ak be nXn matrices with entries from A. Consider a typical linear combination J= X1A with real coefficients Xi; we shall say that the set { A } the property if such a linear combination is nonsingular (invertible) except when all the coefficients Xi are zero. We shall write A(n) for the maximum number of such matrices which form a set with the property P. We shall write AH(n) for the maximum number of matrices which form a set with the property P. (Here, if A = R, the word Hermitian merely means symmetric; if A= Q it is defined using the usual conjugation in Q.) Our aim is to determine the numbers A(n), AH(n). Of course, it is possible to word the problem more invariantly. Let W be a set of matrices which is a vector space of dimension k over R; we will say that W the property if every nonzero w in W is nonsingular (invertible). We now ask for the maximum possible dimension of such a space. In [1], the first named author has proved that R(n) equals the socalled Radon-Hurwitz function, defined below. In this note we determine RH(n), C(n), CH(n), Q(n) and QH(n) by deriving inequalities between them and R(n). The elementary constructions needed to prove these inequalities can also be used to give a simplified description of the Radon-Hurwitz matrices. The study of sets of real symmetric matrices {Aj} with the property P may be motivated as follows. For such a set, the system of partial differential equations
IMMERSIONS OF MANIFOLDSC) BY MORRIS W. HIRSCH Introduction Let M and N be differentiable manifolds of dimensions k and n respec- tively, k N is called an immersion if / …
IMMERSIONS OF MANIFOLDSC) BY MORRIS W. HIRSCH Introduction Let M and N be differentiable manifolds of dimensions k and n respec- tively, k N is called an immersion if / is of class C1 and the Jacobian matrix of/ has rank k at each point of M. Such a map is also called regular. Until recently, very little was known about the ex- istence and classification of immersions of one manifold in another. The present work addresses itself to this problem and reduces it to the problem of constructing and classifying cross-sections of fibre bundles. In 1944, Whitney [15] proved that every ^-dimensional manifold can be immersed in Euclidean space of 2k — 1 dimensions, P2*-1. The Whitney- Graustein theorem [13] classifies immersions of the circle S1 in- the plane E2 up to regular homotopy, which is a homotopy / En, k /') and (g, g') are regularly homotopic (in a sense to be defined later). Given two immersions/, g: £>*—»£ that agree on S*_1 and have the same first derivatives at points of S*_1, Q(f, g) is an element of a certain homotopy group, and has the following properties: (1) Q(f, g) =0 if and only if/and g are regularly homotopic rel S*-1, i.e., the homotopy agrees with / and g on Si_1 at each stage, up to the first derivative; (2) fl(/, g) enjoys the usual algebraic properties of a difference cochain. At this point we should like to be able to make the following statement: If / is an immersion of the Received by the editors September 29, 1958. (*) The material in this paper is essentially a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Chicago, 1958. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then …
It is shown that if W is a linear subspace of real n × n matrices, such that rank (A) = k for all 0 ≠ A ∈ W, then dim W≤ n. If dim W = n.5≤ n is prime, and 2 is primitive modulo n then k =1.
We define a certain type of bases of polynomial ideals whose usefulness stems from the fact that a number of computability problems in the theory of polynomial ideals (e.g. the …
We define a certain type of bases of polynomial ideals whose usefulness stems from the fact that a number of computability problems in the theory of polynomial ideals (e.g. the problem of constructing canonical forms for polynomials) is reducible to the construction of bases of this type. We prove a characterization theorem for these bases which immediately leads to an effective method for their construction.
The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K λ,μenumerates the number of column strict tableaux of shape λ and …
The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK −1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.