Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $c<{{\aleph }_{\omega }}$, we construct a weakly tight family under the hypothesis $\mathfrak{s}\le \mathfrak{b}<{{\aleph }_{\omega }}$. The case when $\mathfrak{s}<\mathfrak{b}$ is handled in ZFC and does not require $\mathfrak{b}<{{\aleph }_{\omega }}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b}<{{\aleph }_{\omega }}$ is used to treat the case $\mathfrak{s}=\mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well-studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira [8], who applied it to the Katétov order on almost disjoint families.

In this paper, we consider a generalized Kähler–Einstein equation on a Kähler manifold $M$. Using the twisted $\mathcal{K}$–energy introduced by Song and Tian, we show that the existence of generalized Kähler–Einstein metrics with semi–positive twisting (1, 1)–form $\theta$ is also closely related to the properness of the twisted $\mathcal{K}$-energy functional. Under the condition that the twisting form $\theta$ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler–Einstein metric implies a Moser–Trudinger type inequality.

Throughout this paper, we work in the PL and pseudosimplicial categories, for which we refer to [17] and [10] respectively. For the graph theory involved see [9].

An h-coloured graph (Γ, γ) is a multigraph Γ = (V(Γ), E(Γ)) regular of degree h, endowed with an edge-coloration γ by h colours. If is the colour set, for each we set

For each set . For n ∊ Z, n ≧ 1, set

Δn will be mostly used to denote the colour set for an (n + 1)-coloured graph.

Let Σ be the projective 3-space over the field GF(q) where q = pe, p an odd prime. A spread W in ∑ is a set of q2 + 1 lines in ∑ which are such that each point of Σ lies on exactly one line of W. Thus the lines of W are all mutually skew. The notion of a spread extends to higher dimensions and also applies for arbitrary fields [1; 3; 6, p. 29; 7, p. 5]. Our concern, however, will be within the narrower but still extensive bounds indicated.

For a locally compact group G, let LP(G) be the usual Lebesgue space with respect to left Haar measure m on G. For x ϵ G define the left and right translation operators Lx and Rx by Lx f(y) = f(xy), Rx f(y) = f(yx)(f ϵ Lp(G),y ϵ G). The purpose of this paper is to prove the following theorem.

1. Introduction. After establishing a duplication principle (§2) which enables us to derive a group of order 2h with k + 1 involutory generators from any group of order h with k generators, we shall prove the following combination principle :

Let and be two groups, each having k generators, such that the rth generators of and have relatively prime periods for each r from 1 to k.

It is well known that many polynomials which solve extremal problems on a single interval as the Chebyshev or the Bernstein-Szegö polynomials can be represented by trigonometric functions and their inverses. On two intervals one has elliptic instead of trigonometric functions. In this paper we show that the counterparts of the Chebyshev and Bernstein-Szegö polynomials for several intervals can be represented with the help of automorphic functions, so-called Schottky-Burnside functions. Based on this representation and using the Schottky-Burnside automorphic functions as a tool several extremal properties of such polynomials as orthogonality properties, extremal properties with respect to the maximum norm, behaviour of zeros and recurrence coefficients etc. are derived.

It is well known that epimorphisms in the category Top (Top1, respectively) of topological spaces (T0spaces, respectively) and continuous maps are precisely onto continuous maps. Since every mono-reflective subcategory of a category is also epi-reflective and every embedding in Top (Top1, respectively) is a monomorphism, there is no nontrivial reflective subcategory of Top (Top1 respectively) such that every reflection is an embedding. However, in the category Top0 of T0-spaces and continuous maps as well as in the category Haus of Hausdorff spaces and continuous maps, there are epimorphisms which are not onto. Moreover, every reflection of a reflective subcategory of Top0, which contains a non T1-space, is an embedding [16].

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute $\mathbb{Z}/2$–Betti numbers of these spaces.

In this paper we compare the properties separability and modularity for field extensions. Let be fields of characteristic . K is separable over if K and are linearly disjoint over . K is modular over if K and are linearly disjoint over their intersection for all n > 0. The latter definition is due to Sweedler [12] and is important particularly for Galois theories of purely inseparable extensions [2; 3; 4; 7].

A class of (universal) algebras [4; 5] of the same type or species τ is called equationally complete if the variety [4] Var() generated by has exactly two subvarieties —namely Var() itself and the class of all one element algebras. I t follows that equationally complete varieties are the atoms in the complete lattice of all varieties of a given type of algebras. J. Kalicki, D. Scott [7; 8; 9; 10] and others [2; 3; 6; 12] have considered several questions about equational completeness. A good many of these results have appeared in books. (See, for instance, [4, 5], to which we also refer the reader for more extensive bibliography on subjects related to equational completeness.)

The aim of this paper is to characterize the finite simple group PSp4(3) by the structure of the centralizer of an element of order three contained in the center of its Sylow 3-subgroup. More precisely, we shall prove the following results.

We study the effect of two types of degeneration of a Riemannian metric on the first eigenvalue of the Laplace operator on surfaces. In both cases we prove that the first eigenvalue of the round sphere is an optimal asymptotic upper bound. The first type of degeneration is concentration of the density to a point within a conformal class. The second is degeneration of the conformal class to the boundary of the moduli space on the torus and on the Klein bottle. In the latter, we follow the outline proposed by N. Nadirashvili in 1996.

In (5) Hammer and Sobczyk defined an outwardly simple line family in the plane as a family of lines in the plane having the property that each point outside some given circle lies on exactly one line of the family, and they characterized planar outwardly simple line families as follows (5): the extended diameters of a convex body, whose boundary has no pair of parallel line segments in it, form an outwardly simple line family; moreover, each outwardly simple line family is the family of extended diameters of a convex body having constant width.

We introduce a new conjecture concerning the construction of elements in the annihilator ideal associated to a Galois action on the higher-dimensional algebraic $K$-groups of rings of integers in number fields. Our conjecture ismotivic in the sense that it involves the (transcendental) Borel regulator as well as being related to $l$–adic étale cohomology. In addition, the conjecture generalises the wellknown Coates–Sinnott conjecture. For example, for a totally real extension when $r\,=\,-2,\,-4,\,-6,\,\ldots $ the Coates–Sinnott conjecture merely predicts that zero annihilates ${{K}_{-2r}}$ of the ring of $S$–integers while our conjecture predicts a non-trivial annihilator. By way of supporting evidence, we prove the corresponding (conjecturally equivalent) conjecture for the Galois action on the étale cohomology of the cyclotomic extensions of the rationals.

A function ƒ(z) is said to be p-valent in a region if it is regular in if the equation

1

has p distinct roots in for some particular w0 , and if for each complex w0 , equation (1) does not have more than p roots in . The function ƒ(z) is also said to have valence p in . In the case when p = 1, the function is said to be univalent in .

Let A1 denote the class of all complex functions on the unit circle which are sums of absolutely convergent trigonometric series; that is to say, the class of all f of the form

Similarly, if 1 < p < ∞, let Ap be the class of all complex functions f on the circle whose Fourier coefficients

satisfy the condition

In his paper (3) on finite H-planes, Kleinfeld has defined invariants s and t for an H-plane π as follows: let P and k be any point and line of π such that P is incident with k, then let s be the number of non-neighbour points to P on k and let t be the number of neighbour points to P on k. He has shown that s and t are independent of the choice of P and k, t divides s, π has s2 + st + t2 points (lines), π* has order s/t, and, if t ≠ 1, s ≤ t2. In the case s = t2, π is called uniform and has the property that each pair of neighbour lines (points) has exactly t points (lines) in common.

In 1972, Lelek introduced the notion of Class (W) in his seminar at the University of Houston [see below for definitions of concepts mentioned here]. Since then there has been much interest in classifying and characterizing continua in Class (W). For example, Cook has a result [5, Theorem 4] which implies that any hereditarily indecomposible continuum is in Class (W) Read [21, Theorem 4] showed that all chainable continua are in Class (W), and Feuerbacher proved the following result:

(1.1) THEOREM [7, Theorem 7]. A non-chainable circle-like continuum is in Class (W) if and only if it is not weakly chainable

In [14, 4.2 and section 6], a covering property (denoted here and in [18] by CP) was defined and studied primarily for the purpose of proving that indecomposability is a Whitney property for the class of chainable continua [14, 4.3].

Let $A$ be an amenable separable ${{C}^{*}}$-algebra and $B$ be a non-unital but $\sigma $-unital simple ${{C}^{*}}$- algebra with continuous scale. We show that two essential extensions ${{\tau }_{1}}$ and ${{\tau }_{2}}$ of $A$ by $B$ are approximately unitarily equivalent if and only if

$$\left[ {{\tau }_{1}} \right]\,=\,\left[ {{\tau }_{2}} \right]\,\text{in}\,KL\left( A,\,M\left( B \right)/B \right).$$

If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL\left( A,\,M\left( B \right)/B \right)$. Using $KL\left( A,\,M\left( B \right)/B \right)$, we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.