Periodicity of Adams operations on the Green ring of a finite group

The Adams operations and on the Green ring of a group G over a field K arise from the study of the exterior powers and symmetric powers of KG-modules. When G is finite and K has prime characteristic p we show that and are periodic in n if and only if the Sylow p-subgroups of G are cyclic. In the case where G is a cyclic p-group we find the minimum periods and use recent work of Symonds to express in terms of .     

On certain cohomological invariants of groups

Let R be a left and right ℵ₀-Noetherian ring. We show that if all projective left and all projective right R-modules have finite injective dimension, then all injective left and all injective right R-modules have finite projective dimension. Using this result, we prove that the invariants and , which were introduced by Gedrich and Gruenberg (1987) [15], are equal for any group G. As an application of the latter equality, we show that a group G is finite if and only if , where is the generalized cohomological dimension of groups introduced by Ikenaga (1984) [21].     

Homology of artinian and Matlis reflexive modules, I

Let R be a commutative local noetherian ring, and let L and L^′ be R-modules. We investigate the properties of the functors and . For instance, we show the following:

(a)

if L and L^′ are artinian, then is artinian, and is noetherian over the completion ;

(b)

if L is artinian and L^′ is Matlis reflexive, then , , and are Matlis reflexive.

Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.     

On the number of graphs with a given endomorphism monoid

For a given finite monoid , let be the number of graphs on n vertices with endomorphism monoid isomorphic to . For any nontrivial monoid we prove that where and are constants depending only on with .For every k there exists a monoid of size k with , on the other hand if a group of unity of has a size k>2 then .     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .     

Higher Kurtz randomness

A real x is -Kurtz random (-Kurtz random) if it is in no closed null set ( set). We show that there is a cone of -Kurtz random hyperdegrees. We characterize lowness for -Kurtz randomness as being -dominated and -semi-traceable.     

On the Wiener integral with respect to a sub-fractional Brownian motion on an interval

The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , k≠0, is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions.     

Derivative of power series attached to Γ-transform of p-adic measures

We extend the result of Anglès (2007) [1], namely for the Iwasawa power series . For the derivative , a numerical polynomial Q on Zp, and a prime π in over p, we show that if and only if i.e. for all x∈Zp. This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in .     

Polar syzygies in characteristic zero: The monomial case

Given a set of forms , where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module , whose elements are called differential syzygies of . There is a distinct submodule P⊂Z coming from the polynomial relations of through its transposed Jacobian matrix, the elements of which are called polar syzygies of . We say that is polarizable if equality P=Z holds. This paper is concerned with the situation where are monomials of degree 2, in which case one can naturally associate to them a graph with loops and translate the problem into a combinatorial one. The main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra and that the converse holds provided the graph is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of has diameter at most 2 then is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.     

On the computation of the jdeg of blowup algebras

For a graded algebra , its is a global degree that can be used to study issues of complexity of the normalization . Here some techniques grounded on Rees algebra theory are used to estimate . A closely related notion, of divisorial generation, is introduced to count numbers of generators of .     

On wreathed lexicographic products of graphs

This paper proves a necessary and sufficient condition for the endomorphism monoid of a lexicographic product G[H] of graphs G,H to be the wreath product of the monoids and . The paper also gives respective necessary and sufficient conditions for specialized cases such as for unretractive or triangle-free graphs G.     

Extension of Fourier algebra homomorphisms to duals of algebras of uniformly continuous functionals

For a locally compact group G, let XG be one of the following introverted subspaces of VN(G): , the C^∗-algebra of uniformly continuous functionals on A(G); , the space of weakly almost periodic functionals on A(G); or , the C^∗-algebra generated by the left regular representation on the measure algebra of G. We discuss the extension of homomorphisms of (reduced) Fourier-Stieltjes algebras on G and H to cb-norm preserving, weak^∗-weak^∗-continuous homomorphisms of into , where (XG,XH) is one of the pairs , , or . When G is amenable, these extensions are characterized in terms of piecewise affine maps.     

Basic sets in defining characteristic for general linear groups of small rank

Let q be a power of some prime number p. Let be a connected reductive group defined over the field with q elements and let F be the corresponding Frobenius map. In this note, we give methods to find relations between the restrictions on semisimple elements of the irreducible characters of . As illustration, we explicitly determine a p-basic set for , and .     

Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let be the space of solutions to the parabolic equation having finite norm. We characterize nonnegative Radon measures μ on having the property , 1≤p≤q<∞, whenever . Meanwhile, denoting by v(t,x) the solution of the above equation with Cauchy data v₀(x), we characterize nonnegative Radon measures μ on satisfying , β∈(0,n), p∈[1,n/β], q∈(0,∞). Moreover, we obtain the decay of v(t,x), an isocapacitary inequality and a trace inequality.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form

     

Posets of annular non-crossing partitions of types B and D

We study the set of annular non-crossing permutations of type B, and we introduce a corresponding set of annular non-crossing partitions of type B, where p and q are two positive integers. We prove that the natural bijection between and is a poset isomorphism, where the partial order on is induced from the hyperoctahedral group B_p+q, while is partially ordered by reverse refinement. In the case when q=1, we prove that is a lattice with respect to reverse refinement order.We point out that an analogous development can be pursued in type D, where one gets a canonical isomorphism between and . For q=1, the poset coincides with a poset “NC^(D)(p+1)” constructed in a paper by Athanasiadis and Reiner [C.A. Athanasiadis, V. Reiner, Noncrossing partitions for the group D_n, SIAM Journal of Discrete Mathematics 18 (2004) 397-417], and is a lattice by the results of that paper.     

Fixed points of dual quantum operations

Let ?A be a normal completely positive map on B(H) with Kraus operators . Denote M the subset of normal completely positive maps by . In this note, the relations between the fixed points of ?A and are investigated. We obtain that , where K(H) is the set of all compact operators on H and is the dual of ?A∈M. In addition, we show that the map is a bijection on M.     

A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees

We introduce a property of forcing notions, called the anti-, which comes from Aronszajn trees. This property canonically defines a new chain condition stronger than the countable chain condition, which is called the property .In this paper, we investigate the property . For example, we show that a forcing notion with the property does not add random reals. We prove that it is consistent that every forcing notion with the property has precaliber ℵ₁ and for forcing notions with the property fails. This negatively answers a part of one of the classical problems about implications between fragments of .