We obtain a new characterisation for weighted Bergman spaces Apα on the unit ball n of ℂn in terms of a double integral of the functions |f(z) − f(w)|/|z − w| and |f(z) − f(w)|/|1 − 〈 z, w〉|.

We study the set of all m-tuples (λ(1), . . ., λ(m)) of possible types of finite abelian p-groups Mλ(1), . . ., Mλ(m) for which there exists a long exact sequence Mλ(1) → ⋅⋅⋅ → Mλ(m). When m=3, we recover W. Fulton's (Eigenvalues of majorized Hermitian matrices and Littlewood-Richardson coefficients (Special Issue: Workshop on Geometric and combinatorial Methods in the Hermitian Sum Spectral Problem), Linear Algebra Appl. 319(1–3) (2000), 23–36) results on the possible eigenvalues of majorized Hermitian matrices.

Let x0 < x1 < x2 < ⋅⋅⋅ be an increasing sequence of positive integers given by the formula xn=⌊βxn−1 + γ⌋ for n=1, 2, 3, . . ., where β > 1 and γ are real numbers and x0 is a positive integer. We describe the conditions on integers bd, . . ., b0, not all zero, and on a real number β > 1 under which the sequence of integers wn=bdxn+d + ⋅⋅⋅ + b0xn, n=0, 1, 2, . . ., is bounded by a constant independent of n. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequence qxn+1 − pxn ∈ {0, 1, . . ., q−1}, n=0, 1, 2, . . ., where x0 is a positive integer, p > q > 1 are coprime integers and xn=⌈pxn−1/q⌉ for n=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 or S:x↦(3x+1)/2) in the 3x+1 problem is also given.

It is shown that every finitely generated projective module PR over a semiprime ring R has the smallest FI-extending essential module extension (called the absolute FI-extending hull of PR) in a fixed injective hull of PR. This module hull is explicitly described. It is proved that , where is the smallest right FI-extending right ring of quotients of End(PR) (in a fixed maximal right ring of quotients of End(PR). Moreover, we show that a finitely generated projective module PR over a semiprime ring R is FI-extending if and only if it is a quasi-Baer module and if and only if End(PR) is a quasi-Baer ring. An application of this result to C*-algebras is considered. Various examples which illustrate and delimit the results of this paper are provided.

An exact formula for the expected length of the minimum spanning tree of a connected graph, with independent and identical edge distribution, is given, which generalizes Steele's formula in the uniform case. For a complete graph, the difference of expected lengths between exponential distribution, with rate one, and uniform distribution on the interval (0, 1) is shown to be positive and of rate ζ(3)/n. For wheel graphs, precise values of expected lengths are given via calculations of the associated Tutte polynomials.

A d-simplex is a collection of d + 1 sets such that every d of them has non-empty intersection and the intersection of all of them is empty. Fix k ≥ d + 2 ≥ 3 and let be a family of k-element subsets of an n-element set that contains no d-simplex. We prove that if , then there is a vertex x of such that the number of sets in omitting x is o(nk−1) (here o(1)→ 0 and n → ∞). A similar result when n/k is bounded from above was recently proved in [10].

Our main result is actually stronger, and implies that if for any ϵ < 0 and n sufficiently large, then contains d + 2 sets A, A1, . . . ,Ad+1 such that the Ais form a d-simplex, and A contains an element of ∩j≠iAj for each i. This generalizes, in asymptotic form, a recent result of Vestraëte and the first author [18], who proved it for d = 1, ϵ = 0 and n ≥ 2k.

In this paper we describe a special class of self-adjoint operators associated with the singular self-adjoint second-order differential expression ℓ. This class is defined by the requirement that the sesquilinear form q(u, v) obtained from ℓ by integration by parts once agrees with the inner product 〈ℓu, v〉. We call this class Type I operators. The Friedrichs Extension is a special case of these operators. A complete characterization of these operators is given, for the various values of the deficiency index, in terms of their domains and the boundary conditions they satisfy (separated or coupled).

An instance of a size-n stable marriage problem involves n men and n women, each individually ranking all members of opposite sex in order of preference as a potential marriage partner. A complete matching, a set of n marriages, is called stable if no unmatched man and woman prefer each other to their partners in the matching. It is known that, for every instance of marriage partner preferences, there exists at least one stable matching, and that there are instances with exponentially many stable matchings. Our focus is on a random instance chosen uniformly from among all (n!)2n possible instances. The second author had proved that the expected number of stable marriages is of order nlnn, while its likely value is of order n1/2−o(1) at least. In this paper the second moment of that number is shown to be of order (nlnn)2. The combination of the two moment estimates implies that the fraction of problem instances with roughly cnlnn solutions is at least 0.84. Whether this fraction is asymptotic to 1 remains an open question.

Let M be an n-dimensional closed hypersurface with constant mean curvature H satisfying |H| ≤ ϵ(n) in a unit sphere Sn+1, n ≤ 7, and S the square of the length of the second fundamental form of M. There exists a constant δ(n, H) > 0, which depends only on n and H, such that if S0 ≤ S ≤ S0 + δ(n, H), then S ≡ S0 and M is isometric to a Clifford hypersurface, where ϵ(n) is a sufficiently small constant depending on n and .

A collection of permutation classes is exhibited whose growth rates form a perfect set, thereby refuting some conjectures of Balogh, Bollobás and Morris.

Using various results from extremal set theory (interpreted in the language of additive combinatorics), we prove an asymptotically sharp version of Freiman's theorem in : if is a set for which |A + A| ≤ K|A| then A is contained in a subspace of size ; except for the error, this is best possible. If in addition we assume that A is a downset, then we can also cover A by O(K46) translates of a coordinate subspace of size at most |A|, thereby verifying the so-called polynomial Freiman–Ruzsa conjecture in this case. A common theme in the arguments is the use of compression techniques. These have long been familiar in extremal set theory, but have been used only rarely in the additive combinatorics literature.

In this paper we show existence of positive solutions for a class of quasi-linear problems with Neumann boundary conditions defined in a half-space and involving the critical exponent.

Let q be a power of a prime p, and let n, d, ℓ be integers such that 1 ≤ n, 1 ≤ ℓ < q. Consider the modulo q complete ℓ-wide family:

We describe a Gröbner basis of the vanishing ideal I() of the set of characteristic vectors of over fields of characteristic p. It turns out that this set of polynomials is a Gröbner basis for all term orderings ≺, for which the order of the variables is xn ≺ xn−1 ≺ ⋅⋅⋅ ≺ x1.

We compute the Hilbert function of I(), which yields formulae for the modulo p rank of certain inclusion matrices related to .

We apply our results to problems from extremal set theory. We prove a sharp upper bound of the cardinality of a modulo q ℓ-wide family, which shatters only small sets. This is closely related to a conjecture of Frankl [13] on certain ℓ-antichains. The formula of the Hilbert function also allows us to obtain an upper bound on the size of a set system with certain restricted intersections, generalizing a bound proposed by Babai and Frankl [6].

The paper generalizes and extends the results of [15], [16] and [17].

For any rotation-invariant positive regular Borel measure ν on the closed unit ball whose support contains the unit sphere , let L2a be the closure in L2 = L2(, dν) of all analytic polynomials. For a bounded Borel function f on , the Toeplitz operator Tf is defined by Tf(ϕ) = P(fϕ) for ϕ ∈ L2a, where P is the orthogonal projection from L2 onto L2a. We show that if f is continuous on , then Tf is compact if and only if f(z) = 0 for all z on the unit sphere. This is well known when L2a is replaced by the classical Bergman or Hardy space.

We extend the concept of umbilicity to higher order umbilicity in Riemannian manifolds saying that an isometric immersion is k-umbilical when APk−1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We determine the principal curvatures of the two-umbilical isometric immersions in terms of the mean curvatures. We give a description of the two-umbilical isometric immersions in space forms which includes the product of spheres embedded in the Euclidean sphere S2k+1 of radius 1. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1 ≡ 0 if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.

A celebrated theorem of Friedgut says that every function f : {0, 1}n → {0, 1} can be approximated by a function g : {0, 1}n → {0, 1} with , which depends only on eO(If / ε) variables, where If is the sum of the influences of the variables of f. Dinur and Friedgut later showed that this statement also holds if we replace the discrete domain {0, 1}n with the continuous domain [0, 1]n, under the extra assumption that f is increasing. They conjectured that the condition of monotonicity is unnecessary and can be removed.

We show that certain constant-depth decision trees provide counter-examples to the Dinur–Friedgut conjecture. This suggests a reformulation of the conjecture in which the function g : [0, 1]n → {0, 1}, instead of depending on a small number of variables, has a decision tree of small depth. In fact we prove this reformulation by showing that the depth of the decision tree of g can be bounded by eO(If / ε2).

Furthermore, we consider a second notion of the influence of a variable, and study the functions that have bounded total influence in this sense. We use a theorem of Bourgain to show that these functions have certain properties. We also study the relation between the two different notions of influence.

Let r ≥ 3 and (c/rr)r log n ≥ 1. If G is a graph of order n and its largest eigenvalue μ(G) satisfiesthen G contains a complete r-partite subgraph with r − 1 parts of size ⌊(c/rr)r log n⌋ and one part of size greater than n1−cr−1.

This result implies the Erdős–Stone–Bollobás theorem, the essential quantitative form of the Erdős–Stone theorem. Another easy consequence is that if F1, F2, . . . are r-chromatic graphs satisfying v(Fn) = o(log n), then

Let B be a submodule of an R-module M. The intersection of all prime (resp. weakly prime) submodules of M containing B is denoted by rad(B) (resp. wrad(B)). A generalisation of 〈E(B)〉 denoted by UE(B) of M will be introduced. The inclusions 〈E(B)〉 ⊆ UE(B) ⊆ wrad(B) ⊆ rad(B) are motivations for studying the equalities UE(B) = wrad(B) and UE(B) = rad(B) in this paper. It is proved that if R is an arithmetical ring, then UE(B) = wrad(B). In Theorem 2.5, a generalisation of the main result of [11] is given.