On Power-Associative Nilalgebras of Dimension N and Nilindex N-1

Whether or not a finite-dimensional, commutative, power-associative nilalgebra is solvable is a well-known open problem. In this paper, we describe commutative, power-associative nilalgebras of dimension n ≥ 6 and nilindex n ? 1 based on the condition that n ? 4 ≤ dim 𝔄³ ≤ n ? 3. This paper is a continuation of [10 Fernadez , J. C. G. , Garcia , C. I. , Montoya , M. L. R. ( 2013 ). On power-associative nilalgebras of nilindex and dimension n . Revista Colombiana de Matemáticas 47 : 1 – 11 . [Google Scholar]], where we describe commutative power-associative nilalgebras of dimension and nilindex n. We observe that the Jordan case was obtained by L. Elgueta and A. Suazo in [2 Elgueta , L. , Suazo , A. ( 2002 ). Jordan nilalgebras of nilindex n and dimension n + 1 . Comm. Algebra 30 : 5547 – 5561 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

Regularity Index of S + 2 Fat Points not on a Linear (S − 1)-Space

We will give a formula to compute the regularity index of s + 2 fat points not lying on a linear (s ? 1)-space in ? ⁿ , s ≤ n (Theorem 3.4). Our result generalizes a formula to compute the regularity index of fat points in general position in ? ⁿ ([3 Catalisano , M. V. , Trung , N. V. , Valla , G. ( 1993 ). A sharp bound for the regularity index of fat points in general position . Proc. Amer. Math. Soc. 118 : 717 – 724 .[Crossref], [Web of Science ®] , [Google Scholar]], Corollary 8). Our result also shows that the Segre bound is attained by s + 2 points not lying on a linear (s ? 1)-space.     

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

It is known that the semigroup Sing _n of all singular self-maps of X _n  = {1,2,…, n} has rank n(n ? 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing _n , has the same value as the rank. (See Gomes and Howie, 1987 Gomes , G. M. S. , Howie , J. M. ( 1987 ). On the rank of certain finite semigroups of transformations . Math. Proc. Cambridge Phil. Soc. 101 : 395 – 303 .[Crossref], [Web of Science ®] , [Google Scholar].) Idempotents generating Sing _n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005 Ay?k , G. , Ay?k , H. , Howie , J. M. ( 2005 ). On factorisations and generators in transformation semigroups . Semigroup Forum 70 : 225 – 237 .[Crossref], [Web of Science ®] , [Google Scholar]). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing _n , defined as the smallest number of (m, r)-path-cycles generating Sing _n , is once again n(n ? 1)/2.     

Unimodular Regularity,Stable Range,and Exchange Rings

Let R be an exchange ring. In this article, we show that the following conditions are equivalent: (1) R has stable range not more than n; (2) whenever x ∈ R ⁿ is regular, there exists some unimodular regular w ∈  ⁿ R such that x = xwx; (3) whenever x ∈ R ⁿ is regular, there exist some idempotent e ∈ R and some unimodular regular w ∈ R ⁿ such that x = ew; (4) whenever x ∈ R ⁿ is regular, there exist some idempotent e ∈ M _n (R) and some unimodular regular w ∈ R ⁿ such that x = we; (5) whenever a( ⁿ R) + bR = dR with a ∈ R ⁿ and b,d ∈ R, there exist some z ∈ R ⁿ and some unimodular regular w ∈ R ⁿ such that a + bz = dw; (6) whenever x = xyx with x ∈ R ⁿ and y ∈  ⁿ R, there exist some u ∈ R ⁿ and v ∈  ⁿ R such that vxyu = yx and uv = 1. These, by replacing unimodularity with unimodular regularity, generalize the corresponding results of Canfell (1995 Canfell , M. J. ( 1995 ). Completions of diagrams by automorphisms and Bas' first stable range condition . J. Algebra 176 : 480 – 513 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.9), Chen (Chen 2000 Chen , H. ( 2000 ). On stable range conditions . Comm. Algebra 28 : 3913 – 3924 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 4.2 and Proposition 4.6, Chen 2001 Chen , H. ( 2001 ). Regular rings with finite stable range . Comm. Algebra 29 : 157 – 166 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 10), and Wu and Xu (1997 Wu , T. , Xu , Y. ( 1997 ). On the stable range condition of exchagne rings . Comm. Algebra 25 : 2355 – 2363 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 9), etc.     

Inverse Problems for Differential Forms on Riemannian Manifolds with Boundary

Consider a real-analytic orientable connected complete Riemannian manifold M with boundary of dimension n ≥ 2 and let k be an integer 1 ≤ k ≤ n. In the case when M is compact of dimension n ≥ 3, we show that the manifold and the metric on it can be reconstructed, up to an isometry, from the set of the Cauchy data for harmonic k-forms, given on an open subset of the boundary. This extends a result of [14 Lassas , M. , Uhlmann , G. ( 2001 ). On determining a Riemannian manifold from the Dirichlet-to-Neumann map . Ann. Sci. École Norm. 34 : 771 – 787 .[Web of Science ®] , [Google Scholar]] when k = 0. In the two-dimensional case, the same conclusion is obtained when considering the set of the Cauchy data for harmonic 1-forms. Under additional assumptions on the curvature of the manifold, we carry out the same program when M is complete non-compact. In the case n ≥ 3, this generalizes the results of [13 Lassas , M. , Taylor , M. , Uhlmann , G. ( 2003 ). The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary . Comm. Anal. Geom. 11 : 207 – 221 .[Crossref], [Web of Science ®] , [Google Scholar]] when k = 0. In the two-dimensional case, we are able to reconstruct the manifold from the set of the Cauchy data for harmonic 1-forms.     

A Simple Proof of Tyurin's Babylonian Tower Theorem

Using the method of Coand? and Trautmann [4 Coand? , I. , Trautmann , G. ( 2006 ). The splitting criterion of Kempf and the Babylonian tower theorem . Comm. Algebra 34 : 2485 – 2488 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we give a simple proof of a theorem due, in the smooth case, to Tyurin [9 Tyurin , A. N. ( 1976 ). Finite dimensional vector bundles over infinite varieties . Math. USSR Izv. 10 : 1187 – 1204 .[Crossref], [Web of Science ®] , [Google Scholar]]: if a vector bundle E on a c-codimensional locally Cohen–Macaulay closed subscheme X of ? ⁿ extends to a vector bundle F on a similar closed subscheme Y of ? ^N , for every N > n, then E is the restriction to X of a direct sum of line bundles on ? ⁿ . Using the same method, we also provide a proof of the Babylonian tower theorem for locally complete intersection subschemes of projective spaces.     

Bounds for the Number of Conjugacy Classes of Non-Normal Subgroups in a Finite p-Group

Let ν(G) be the number of conjugacy classes of non-normal subgroups of a finite group G. We obtain two new lower bounds for ν(G) when G is a non-abelian finite p-group and p is odd. More precisely, if |G| =p ⁿ , exp Z(G) = p ^e , and exp G/G′ =p ^f , let us define λ(G) = n ? e and κ(G) = n ? f. Then we prove that ν(G) ≥ p(λ(G) ?3) +2 and ν(G) ≥ p(κ(G) ?3) +2. The first bound improves the bound ν(G) ≥ λ(G) ?1 given by [10 La Haye , R. , Rhemtulla , A. ( 1999 ). Groups with a bounded number of conjugacy classes of non-normal subgroups . J. Algebra 214 : 41 – 63 .[Crossref], [Web of Science ®] , [Google Scholar]], and almost in every case, the second one improves the bound ν(G) ≥ p(k ? 1) +1 obtained by [6 Fernández-Alcober , G. A. , Legarreta , L. ( 2008 ). Conjugacy classes of non-normal subgroups in finite nilpotent groups . J. Group Theory 11 ( 3 ): 381 – 397 .[Crossref] , [Google Scholar]], where k is defined by the condition that |G′| =p ^k .     

On a Group Graded Version of BGG

A major result in Algebraic Geometry is the theorem of Bernstein–Gelfand–Gelfand that states the existence of an equivalence of triangulated categories: gr _Λ ? 𝒟^b(Coh ?ⁿ), where gr _Λ denotes the stable category of finitely generated graded modules over the n + 1 exterior algebra and 𝒟^b(Coh ?ⁿ) is the derived category of bounded complexes of coherent sheaves on projective space ?ⁿ.

Generalizations of this result were obtained in Martínez-Villa and Saorín (2004 Martínez-Villa , R. , Saorín , M. ( 2004 ). Koszul equivalences and dualities . Pacific J. Math. 214 ( 2 ): 359 – 378 . [Google Scholar]) and from a different point of view, the theorem has been extended by Yanagawa (2004 Yanagawa , K. ( 2004 ). Derived category of squarefree modules and local cohomology with monomial ideal support . J. Math. Soc. Japan 56 : 289 – 308 . [Google Scholar]) to ?ⁿ-graded modules over the polynomial algebra. This generalization has important applications in combinatorial commutative algebra.

The aim of the article is to extend the results of Martínez-Villa and Saorín (2004 Martínez-Villa , R. , Saorín , M. ( 2004 ). Koszul equivalences and dualities . Pacific J. Math. 214 ( 2 ): 359 – 378 . [Google Scholar]) to group graded algebras in order to obtain a generalization of Yanagawa's results having in mind the application to other settings (Geigle and Lenzing, 1987 Geigle , W. , Lenzing , H. ( 1987 ). A class of weghted projective curves in representation theory of finite dimensional algebras, in singularities, representations of algebras, and vector bundles . Lecture Notes in Math. 1273 : 265 – 297 .[Crossref], [Web of Science ®] , [Google Scholar]).     

On L p Resolvent Estimates for Elliptic Operators on Compact Manifolds

We prove uniform L^p estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3 Dos Santos Ferreira, D., Kenig, C., and Salo, M., 2014. On L^p resolvent estimates for Laplace-Beltrami operators on compact manifolds, Forum Math. 26 (2014), pp. 815–849.[Crossref], [Web of Science ®] , [Google Scholar]] in the case of Laplace-Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1 Bourgain, J., Shao, P., Sogge, C., and Yao, X., On L^p-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys., to appear.[Web of Science ®] , [Google Scholar]] very closely. We also show that spectral regions in our L^p resolvent estimates are optimal.     

Determining an Unbounded Potential from Cauchy Data in Admissible Geometries

In [4 Dos Santos Ferreira , D. , Kenig , C.E. , Salo , M. , Uhlmann , G. ( 2009 ). Limiting Carleman weights and anisotropic inverse problems . Invent. Math. 178 : 119 – 171 .[Crossref], [Web of Science ®] , [Google Scholar]] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n ≥ 3. In this article we extend this result to the case of unbounded potentials, namely those in L ^n/2. In the process, we derive L ^p Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison and Kenig [8 Jerison , D. , Kenig , C.E. ( 1985 ). Unique continuation and absence of positive eigenvalues for Schrödinger operators . Ann. Math. 121 : 463 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]] and Kenig et al. [9 Kenig , C.E. , Ruiz , A. , Sogge , C.D. ( 1987 ). Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators . Duke Math. J. 55 : 329 – 347 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Symmetric Group Decomposition Numbers for Some Three-Part Partitions

The decomposition numbers d _λμ for Specht modules S ^λ of partitions λ with three parts and whose third part is at most p ? 1 are obtained by induction and by using “node removal rules” developed in James and Williams (2000 James , G. D. , Williams , A. L. (2000). Decomposition numbers of symmetric groups by induction. J. Algebra 228:119–142. [CSA] [CROSSREF]  [Google Scholar]).     

Rings Whose Nilpotent Elements form a Levitzki Radical Ring

In this article, we first discuss the relations among JH^r = 0, JH^r?1·H = 0, and JH^r·x = 0. Then we give a counterexample to the question mentioned in the Remarks of [3 Cheng, C. C., Sakkalis, T., Wang, S. S. S. (1994). A case of the Jacobian conjecture. J. Pure and Applied Algebra 96:15–18.[Crossref], [Web of Science ®] , [Google Scholar]] and prove the equivalence among JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r)) = 0, JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r?1))·H(x^(r)) = 0, and JH(x⁽¹⁾)JH(x⁽²⁾)…JH(x^(r))·x^(r) = 0. Finally, we give partial answer to Conjecture 2 in [4 Connell, E., Zweibel, J. (1991). Exact and coexact matrices. J. Algebra 142:110–117.[Crossref], [Web of Science ®] , [Google Scholar]].     

On Commuting Automorphisms of p-Groups

Let G be a group. If the set 𝒜(G) = {α ∈Aut(G) | xα(x) = α(x)x, for all x ∈ G} forms a subgroup of Aut(G), then G is called 𝒜(G)-group. We show that the minimum order of a non-𝒜(G) p-group is p ⁵ for any prime p. We also find the smallest group order of a non-𝒜(G) group. This is related to a question introduced by Deaconescu, Silberberg, and Walls [4 Deaconescu , M. , Silberberg , Gh. , Walls , G. ( 2002 ). On commuting automorphisms of groups . Arch. Math 79 : 423 – 429 .[Crossref] , [Google Scholar]]. Moreover, we prove that for any prime p and for all integer n ≥ 5, there exists a non-𝒜(G) group of order p ⁿ .     

Hurwitz Generation of PSp 6(q)

We show that the symplectic groups PSp₆(q) are Hurwitz for all q = p ^m  ≥ 5, with p an odd prime. The result cannot be improved since, for q even and q = 3, it is known that PSp₆(q) is not Hurwitz. In particular, n = 6 turns out to be the smallest degree for which a family of classical simple groups of degree n, over 𝔽 _{p ^m} , contains Hurwitz groups for infinitely many values of m. This fact, for a given (possibly large) p, also follows from [9 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups I . J. Eur. Math. Soc. (JEMS) 16 ( 7 ): 1349 – 1375 .[Crossref], [Web of Science ®] , [Google Scholar]] and [10 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups II . Groups Geom. Dyn. 8 ( 3 ): 811 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Convergence of the Structure Function of a Multifractal Random Walk in a Mixed Asymptotic Setting

Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L ¹ convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire [19 Ossiander , M. , and Waymire , E.C. 2000 . Statistical estimation for multiplicative cascades . Ann. Stat. 28 : 1533 – 1560 .[Crossref], [Web of Science ®] , [Google Scholar]] and Bacry et al. [1 Bacry , E. , Gloter , A. , Hoffmann , M. , and Muzy , J.F. Multifractal analysis in a mixed asymptotic framework . Ann. Appl. Prob. (to appear) . [Google Scholar]] in the simpler framework of Mandelbrot cascades.     

Polynomial Identities of Algebras of Small Dimension

It is well known that given an associative algebra or a Lie algebra A, its codimension sequence c _n (A) is either polynomially bounded or grows at least as fast as 2 ⁿ . In [2 Giambruno , A. , Mishchenko , S. , Zaicev , M. ( 2006 ). Algebras with intermediate growth of the codimensions . Adv. Appl. Math. 37 ( 3 ): 360 – 377 .[Crossref], [Web of Science ®] , [Google Scholar]] we proved that for a finite dimensional (in general nonassociative) algebra A, dim A = d, the sequence c _n (A) is also polynomially bounded or c _n (A) ≥ a ⁿ asymptotically, for some real number a > 1 which might be less than 2. Nevertheless, for d = 2, we may take a = 2. Here we prove that for d = 3 the same conclusion holds. We also construct a five-dimensional algebra A with c _n (A) < 2 ⁿ .