In this note, using Calabi’s method, we construct rotationally symmetric KählerRicci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kähler-Einstein. These examples generalize the construction of Koiso, Cao and Feldman-Ilmanen-Knopf. 1 A little motivation In [1], the authors constructed...

The molecular basis of the processes that control two closely related traits, the absorption of cholesterol from the intestines and plasma plant sterol levels, are only partially understood. The discovery that mutations in two novel hemitransporters, ATP binding cassette transporter G5 (ABCG5) and ABCG8, underlie a rare inborn error in plant sterol metabolism, beta-sitosterolemia, represents a ...

The prime spectra of two families of algebras, Sw and Šw, w ∈ W, indexed by the Weyl group W of a semisimple finitely dimensional Lie algebra g, are studied in the spirit of [J3]. The algebras Sw have been introduced by A. Joseph (see [J4], Sect. 3). They are q-analogues of the algebras of regular functions on w-translates of the open Bruhat cell of a semisimple Lie group G corresponding to the...

We determine the order of magnitude of H(k+1)(x,y, 2y), the number of integers n ≤ x that are divisible by a product d1 · · · dk with yi < di ≤ 2yi, when the numbers log y1, . . . , log yk have the same order of magnitude and k ≥ 2. This generalizes a result by Kevin Ford when k = 1. As a corollary of these bounds, we determine the number of elements up to multiplicative constants that appear i...

(2) If K ∈ B(X) is compact, then for all λ ∈ C \ {0}, K − λ1 is Fredholm with index zero. (3) The shift operator S± ∈ B(`p) for 1 ≤ p ≤ ∞ defined by (S±x)n = xn±1 is Fredholm with index ±1. (4) If X,Y are finite dimensional and T ∈ B(X,Y ), then by the Rank-Nullity Theorem, ind(T ) = dim(X)− dim(Y ). Lemma 3. Suppose E,F ⊆ X are closed subspaces with F finite dimensional. (1) The subspace E + F...

The proofs that these maps are continuous are simple estimates that you probably remember from calculus. Since the continuity of all the maps we’ll look at in these notes is proved by expressing them in terms of the maps a and m, we include the proofs of continuity of a and m for completeness. Proof. To prove that the addition map a is continuous, suppose x = (x1, x2) ∈ R and > 0 are given. We ...

Example 1.1 (Cosets in R). Consider the vector space X = R. Let M be any onedimensional subspace of R, i.e., M is a line in R through the origin. A coset of M is a rigid translate of M by a vector in R. For concreteness, let us consider the case where M is the x1-axis in R , i.e., M = {(x1, 0) : x1 ∈ R}. Then given a vector y = (y1, y2) ∈ R , the coset y +M is the set y +M = {y +m : m ∈ M} = {(...

Let JG denote the binomial edge ideal of a connected undirected graph on n vertices. This is the ideal generated by the binomials xiyj − xjyi, 1 ≤ i < j ≤ n, in the polynomial ring S = K[x1, . . . , xn, y1, . . . , yn] where {i, j} is an edge of G. We study the arithmetic properties of S/JG for G, the complete bipartite graph. In particular we compute dimensions, depths, Castelnuovo-Mumford reg...

(a linear unimodular change of variables). Then the problem reduces to studying values of the standard form Sm,n of signature (m,n) applied to the collection of vectors of the form {gx | x ∈ Z}. And the dynamical approach consists of studying the action of the stabilizer of the form Sm,n on such collections. Problem 2. Given m vectors y1, . . . ,ym ∈ R n (viewed as linear forms x 7→ yi ·x, x ∈ ...