Computation of blowing up centers

Given a birational projective morphism of quasi-projective varieties . We want to find the ideal sheaf over X such that the blowing up of X along corresponds to f. In this paper we approach the problem from two directions, solving two subcases. First we present a method that determines when f is the composition of blowing ups along known centers, then by another method, we compute directly from .     

Mixed Artin-Tate motives over number rings

This paper studies Artin-Tate motives over bases , for a number field F. As a subcategory of motives over S, the triangulated category of Artin-Tate motives is generated by motives , where ? is any finite map. After establishing the stability of these subcategories under pullback and pushforward along open and closed immersions, a motivic t-structure is constructed. Exactness properties of these functors familiar from perverse sheaves are shown to hold in this context. The cohomological dimension of mixed Artin-Tate motives () is two, and there is an equivalence .     

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 .     

The coarse shape groups

The (pointed) coarse shape category Sh^* (), having (pointed) topological spaces as objects and having the (pointed) shape category as a subcategory, was recently constructed. Its isomorphisms classify (pointed) topological spaces strictly coarser than the (pointed) shape type classification. In this paper we introduce a new algebraic coarse shape invariant which is an invariant of shape and homotopy, as well. For every pointed space (X,?) and for every k∈N₀, the coarse shape group , having the standard shape group for its subgroup, is defined. Furthermore, a functor is constructed. The coarse shape and shape groups already differ on the class of polyhedra. An explicit formula for computing coarse shape groups of polyhedra is given. The coarse shape groups give us more information than the shape groups. Generally, does not imply (e.g. for solenoids), but from pro-πk(X,?)=0 follows . Moreover, for pointed metric compacta (X,?), the n-shape connectedness is characterized by , for every k?n.     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.     

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.     

Homological finiteness properties of monoids, their ideals and maximal subgroups

We consider the general question of how the homological finiteness property left-(resp. right-) holding in a monoid influences, and conversely depends on, the property holding in the substructures of that monoid. This is done by giving methods for constructing free resolutions of substructures from free resolutions of their containing monoids, and vice versa. In particular, we show that left- is inherited by the maximal subgroups in a completely simple minimal ideal, in the case that the minimal ideal has finitely many left ideals. For completely simple semigroups, we prove the converse, and as a corollary, show that a completely simple semigroup is of type left- and right- if and only if it has finitely many left and right ideals and all of its maximal subgroups are of type . Also, given an ideal of a monoid, we show that if the ideal has a two-sided identity element then the containing monoid is of type left- if and only if the ideal is of type left-. Applying this result, we obtain necessary and sufficient conditions for a Clifford monoid (and more generally a strong semilattice of monoids) to be of type left-. Examples are provided showing that for each of the results all of the hypotheses are necessary.     

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 an additive decomposition of the BLUE in a multiple-partitioned linear model

Necessary and sufficient conditions are derived for the BLUE in a general multiple-partitioned linear model to be the sum of the BLUEs under the k small models , …, . Some consequences and further research topics are also given.     

New applications of Besov-type and Triebel-Lizorkin-type spaces

In this paper, the authors prove that Besov-Morrey spaces are proper subspaces of Besov-type spaces and that Triebel-Lizorkin-Morrey spaces are special cases of Triebel-Lizorkin-type spaces . The authors also establish an equivalent characterization of when τ∈[0,1/p). These Besov-type spaces and Triebel-Lizorkin-type spaces were recently introduced to connect Besov spaces and Triebel-Lizorkin spaces with Q spaces. Moreover, for the spaces and , the authors investigate their trace properties and the boundedness of the pseudo-differential operators with homogeneous symbols in these spaces, which generalize the corresponding classical results of Jawerth and Grafakos-Torres by taking τ=0.     

Kirillov-Reshetikhin crystals for nonexceptional types

We provide combinatorial models for all Kirillov-Reshetikhin crystals of nonexceptional type, which were recently shown to exist. For types , , we rely on a previous construction using the Dynkin diagram automorphism which interchanges nodes 0 and 1. For type we use a Dynkin diagram folding and for types , a similarity construction. We also show that for types and the analog of the Dynkin diagram automorphism exists on the level of crystals.     

How to play the Majority game with a liar

The Majority game is played by a questioner () and an answerer (). holds n elements, each of which can be labeled as 0 or 1. is trying to identify some element holds as having the Majority label or, in the case of a tie, claim there is none. To do this asks questions comparing whether two elements have the same or different label. ’s goal is to ask as few questions as possible while ’s goal is to delay as much as possible. Let q^∗ denote the minimal number of questions needed for to identify a Majority element regardless of ’s answers.In this paper we investigate upper and lower bounds for q^∗ in a variation of the Majority game, where is allowed to lie up to t times. We consider two versions of the game, the adaptive (where questions are asked sequentially) and the oblivious (where questions are asked in one batch).     

The least eigenvalue of the complements of trees

Let be the set of the complements of trees of order n. In this paper, we characterize the unique graph whose least eigenvalue attains the minimum among all graphs in .     

Exact rates in log law for positively associated random variables

Let be a strictly stationary sequence of positively associated random variables with mean zero and finite variance. Set , Mn=maxk?n|Sk|, n?1. Suppose . In this paper, we study the exact convergence rates of a kind of weighted infinite series of , and as ε↘0, respectively.     

Cut elimination and strong separation for substructural logics: An algebraic approach

We develop a general algebraic and proof-theoretic study of substructural logics that may lack associativity, along with other structural rules. Our study extends existing work on (associative) substructural logics over the full Lambek Calculus (see, for example, Ono (2003) [34], Galatos and Ono (2006) [18], Galatos et al. (2007) [17]). We present a Gentzen-style sequent system that lacks the structural rules of contraction, weakening, exchange and associativity, and can be considered a non-associative formulation of . Moreover, we introduce an equivalent Hilbert-style system and show that the logic associated with and is algebraizable, with the variety of residuated lattice-ordered groupoids with unit serving as its equivalent algebraic semantics.Overcoming technical complications arising from the lack of associativity, we introduce a generalized version of a logical matrix and apply the method of quasicompletions to obtain an algebra and a quasiembedding from the matrix to the algebra. By applying the general result to specific cases, we obtain important logical and algebraic properties, including the cut elimination of and various extensions, the strong separation of , and the finite generation of the variety of residuated lattice-ordered groupoids with unit.     

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 .     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .