On the sectional invariants of polarized manifolds

Let (X,L) be a polarized manifold of dimension n. In this paper, for any integer i with 0≤i≤n we introduce the notion of the ith sectional invariant of (X,L). We define the ith sectional Euler number e_i(X,L), the ith sectional Betti number b_i(X,L), and the ith sectional Hodge number of type (j,i−j) of (X,L) and we will study some properties of these.     

The minimum distance of sets of points and the minimum socle degree

Let K be a field of characteristic 0. Let be a reduced finite set of points, not all contained in a hyperplane. Let be the maximum number of points of Γ contained in any hyperplane, and let . If I⊂R=K[x₀,…,x_n] is the ideal of Γ, then in Tohaˇneanu (2009) [12] it is shown that for n=2,3, d(Γ) has a lower bound expressed in terms of some shift in the graded minimal free resolution of R/I. In these notes we show that this behavior holds true in general, for any n≥2: d(Γ)≥A_n, where A_n=min{a_i−n} and ⊕_iR(−a_i) is the last module in the graded minimal free resolution of R/I. In the end we also prove that this bound is sharp for a whole class of examples due to Juan Migliore (2010) [10].     

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function f∈C(Y,Z) with Y⊆Y^′ and Z⊆Z^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=κ(XI)=_∏i∈IXi in the κ-box topology (that is, with basic open sets of the form _∏i∈IUi with Ui open in Xi and with Ui≠Xi for <κ-many i∈I). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this.

Theorem. Letω?κ?α (that means: κ<α, and[β<αandλ<κ]⇒βλ<α) with α regular,be a set of non-empty spaces with eachd(Xi)<α,π[Y]=XJfor each non-emptyJ⊆Isuch that|J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets ofZ×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for everyf∈C(Y,Z)there isJ∈[I]^<αsuch thatf(x)=f(y)wheneverxJ=yJ, and (c) every such f extends to.     

Classification of systems of Dyson-Schwinger equations in the Hopf algebra of decorated rooted trees

We consider systems of combinatorial Dyson-Schwinger equations (briefly, SDSE) , … , in the Connes-Kreimer Hopf algebra HI of rooted trees decorated by I={1,…,N}, where is the operator of grafting on a root decorated by i, and F₁,…,FN are non-constant formal series. The unique solution X=(X₁,…,XN) of this equation generates a graded subalgebra H(S) of HI. We characterise here all the families of formal series (F₁,…,FN) such that H(S) is a Hopf subalgebra. More precisely, we define three operations on SDSE (change of variables, dilatation and extension) and give two families of SDSE (cyclic and fundamental systems), and prove that any SDSE (S) such that H(S) is Hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables, a dilatation and iterated extensions.     

Harmonic measures for a point may form a square

Let X be a Green domain in Rd, d?2, x∈X, and let Mx(P(X)) denote the compact convex set of all representing measures for x. Recently it has been proven that the set of harmonic measures , U open in X, x∈U, which is contained in the set of extreme points of Mx(P(X)), is dense in Mx(P(X)). In this paper, it is shown that Mx(P(X)) is not a simplex (and hence not a Poulsen simplex). This is achieved by constructing open neighborhoods U₀, U₁, U₂, U₃ of x such that the harmonic measures are pairwise different and . In fact, these measures form a square with respect to a natural L²-structure. Since the construction is mainly based on having certain symmetries, it can be carried out just as well for Riesz potentials, the Heisenberg group (or any stratified Lie algebra), and the heat equation (or more general parabolic situations).     

Unbiased invariant minimum norm estimation in generalized growth curve model

This paper considers the generalized growth curve model subject to R(X_m)⊆R(X_m-1)⊆?⊆R(X₁), where B_i are the matrices of unknown regression coefficients, X_i,Z_i and U are known covariate matrices, i=1,2,…,m, and E splits into a number of independently and identically distributed subvectors with mean zero and unknown covariance matrix Σ. An unbiased invariant minimum norm quadratic estimator (MINQE(U,I)) of tr(CΣ) is derived and the conditions for its optimality under the minimum variance criterion are investigated. The necessary and sufficient conditions for MINQE(U,I) of tr(CΣ) to be a uniformly minimum variance invariant quadratic unbiased estimator (UMVIQUE) are obtained. An unbiased invariant minimum norm quadratic plus linear estimator (MINQLE(U,I)) of is also given. To compare with the existing maximum likelihood estimator (MLE) of tr(CΣ), we conduct some simulation studies which show that our proposed estimator performs very well.     

Modular metric spaces, I: Basic concepts

The notion of a modular is introduced as follows. A (metric) modular on a set X is a function w:(0,∞)×X×X→[0,∞] satisfying, for all x,y,z∈X, the following three properties: x=y if and only if w(λ,x,y)=0 for all λ>0; w(λ,x,y)=w(λ,y,x) for all λ>0; w(λ+μ,x,y)≤w(λ,x,z)+w(μ,y,z) for all λ,μ>0. We show that, given x₀∈X, the set X_w={x∈X:lim_λ→∞w(λ,x,x₀)=0} is a metric space with metric , called a modular space. The modular w is said to be convex if (λ,x,y)?λw(λ,x,y) is also a modular on X. In this case X_w coincides with the set of all x∈X such that w(λ,x,x₀)<∞ for some λ=λ(x)>0 and is metrizable by . Moreover, if or , then ; otherwise, the reverse inequalities hold. We develop the theory of metric spaces, generated by modulars, and extend the results by H. Nakano, J. Musielak, W. Orlicz, Ph. Turpin and others for modulars on linear spaces.     

Polynomial numerical indices of Banach spaces with absolute norm

We study when a Banach space with absolute norm may have polynomial numerical indices equal to one. In the real case, we show that a Banach space X with absolute norm, which has the Radon-Nikodým property or is Asplund, satisfies n⁽²⁾(X)<1 unless it is one-dimensional. In the complex case, we show that the only Banach spaces X with absolute norm and the Radon-Nikodým property which satisfy n⁽²⁾(X)=1 are the spaces . Also, the only Asplund complex space X with absolute norm which satisfies n⁽²⁾(X)=1 is c₀(Λ).     

Singular values, norms, and commutators

Let and X_i, i=1,…,n, be bounded linear operators on a separable Hilbert space such that X_i is compact for i=1,…,n. It is shown that the singular values of are dominated by those of , where ‖·‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a₁?A?a₂ and b₁?B?b₂ for some real numbers and b₂, and if X is compact, then the singular values of the generalized commutator AX-XB are dominated by those of max(b₂-a₁,a₂-b₁)(X⊕X). This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.     

A numerical characterization of the S2-ification of a Rees algebra

Let A be a local ring with maximal ideal . For an arbitrary ideal I of A, we define the generalized Hilbert coefficients . When the ideal I is -primary, j_k(I)=(0,…,0,(−1)^ke_k(I)), where e_k(I) is the classical kth Hilbert coefficient of I. Using these coefficients we give a numerical characterization of the homogeneous components of the S₂-ification of S=A[It,t⁻¹], extending previous results obtained by the author to not necessarily -primary ideals.     

Filtrations and local syzygies of multiplier ideals II

Let a be a non-zero ideal sheaf on a smooth affine variety X of dimension d and let c be a positive rational number. Let x be a closed point of X and let m_x be the maximal ideal sheaf at x. In [Robert Lazarsfeld, Kyungyong Lee, Local syzygies of multiplier ideals, Invent. Math. 167 (2007) 409-418] the authors studied the local syzygies of the multiplier ideal J(a^c). Motivated by their result, the asymptotic behavior of the local syzygies of the multiplier ideal at x for k≥d−2 was studied in [Seunghun Lee, Filtrations and local syzygies of multiplier ideals, J. Algebra (2007) 629-639]. In this note, we study the local syzygies of at x for 1≤k≤d−3. As a by-product we give a different proof of the main theorem in the former reference cited above.     

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

We consider a bipartite distance-regular graph Γ with diameter D?4, valency k?3, intersection numbers b_i,c_i, distance matrices A_i, and eigenvalues θ₀>θ₁>?>θ_D. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of Mat_X(C) generated by , where A=A₁ and denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever for 0?i?D. By the endpoint of W we mean . Assume W is thin with endpoint 2. Observe is a one-dimensional eigenspace for ; let η denote the corresponding eigenvalue. It is known where , and d=⌊D/2⌋. To describe the structure of W we distinguish four cases: (i) ; (ii) D is odd and ; (iii) D is even and ; (iv) . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694-1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D-1-e where e=1 in case (iii) and e=0 in case (iv). Let v denote a nonzero vector in . We show W has a basis , where E_i denotes the primitive idempotent of A associated with θ_i and where the set S is {1,2,…,d-1}∪{d+1,d+2,…,D-1} in case (iii) and {1,2,…,D-1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis , and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W.     

Stability of associated primes of integral closures of monomial ideals

Let I be a monomial ideal of a polynomial ring R=K[X₁,…,Xr] and d(I) the maximal degree of minimal generators of I. In this paper, we explicitly determine a number n₀ in terms of r and d(I) such that for all n?n₀. Furthermore, our n₀ is almost sharp.     

A note on the Jacobian Conjecture

In this note, we show that, if the Druzkowski mappings F(X)=X+(AX)^∗3, i.e. F(X)=(x₁+(a₁₁x₁+?+a_1nx_n)³,…,x_n+(a_n1x₁+?+a_nnx_n)³), satisfies TrJ((AX)^∗3)=0, then where δ is the number of diagonal elements of A which are equal to zero. Furthermore, we show the Jacobian Conjecture is true for the Druzkowski mappings in dimension ?9 in the case .     

The Betti polynomials of powers of an ideal

For an ideal I in a regular local ring or a graded ideal I in the polynomial ring we study the limiting behavior of as k goes to infinity. By Kodiyalam’s result it is known that β_i(S/I^k) is a polynomial for large k. We call these polynomials the Kodiyalam polynomials and encode the limiting behavior in their generating polynomial. It is shown that the limiting behavior depends only on the coefficients on the Kodiyalam polynomials in the highest possible degree. For these we exhibit lower bounds in special cases and conjecture that the bounds are valid in general. We also show that the Kodiyalam polynomials have weakly descending degrees and identify a situation where the polynomials all have the highest possible degree.     

The topology of continua that are approximated by disjoint subcontinua

Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:

(1)

X is non-Suslinean.

(2)

If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.

(3)

If X is G-like, then X is indecomposable.

(4)

If all lie in the same ray and X is finitely cyclic, then X is indecomposable.