It is proved that every solution of the Neumann initial-boundary problem

[formula here]

converges to some equilibrium, if the system satisfies (i) ∂Fi/∂uj [ges ] 0 for all 1 [les ] i ≠ j [les ] n, (ii) F(u * g(s)) [ges ] h(s) [midast ] F(u) whenever u ∈ ℝn+ and 0 [les ] s [les ] 1, where x * y = (x1y1, …, xnyn) and g, h : [0, 1] → [0, 1]n are continuous functions satisfying gi(0) = hi(0) = 0, gi(1) = hi(1) = 1, 0 < gi(s); hi(s) < 1 for all s ∈ (0, 1) and i = 1, 2, …, n, and (iii) the solution of the corresponding ordinary differential equation system is bounded in ℝn+. We also study the convergence of the solution of the Lotka–Volterra system

[formula here]

where ri > 0, α [ges ] 0, and aij [ges ] 0 for i ≠ j.

Let $\tau$ be a conjugation, alias a conjugate linear isometry of order 2, on a complex Banach space $X$ and let $X^{\tau}$ be the real form of $X$ of $\tau$-fixed points. In contrast to the Dunford–Pettis property, the alternative Dunford–Pettis property need not lift from $X^{\tau}$ to $X$. If $X$ is a C*-algebra it is shown that $X^{\tau}$ has the alternative Dunford–Pettis property if and only if $X$ does and an analogous result is shown when $X$ is the dual space of a C*-algebra. One consequence is that both Dunford–Pettis properties coincide on all real forms of C*-algebras.

It is proved that the simultaneous lattice packing and lattice covering constant of every three-dimensional centrally symmetric convex body is less than or equal to 7/4. Consequently, for any three-dimensional convex body K there is a corresponding lattice packing in which no extra translate of K can be added.

It is shown that the composition operator w[map ]w∘u generated by an inner function u is a self-map of the Muckenhoupt class A2, but that it need not preserve the Muckenhoupt classes Ap for p∈(1, 2)∪(2, ∞).

According to a 1904 conjecture of Dickson [2] unless prevented by congruence conditions, any finite collection of linear forms in $\Z[x]$ with positive leading coefficients infinitely often simultaneously represent primes. For the forms $x,x+2$ this includes the conjectured infinitude of prime-pairs.

A correspondence between the equivariant degree introduced by Ize, Massabó, and Vignoli and an unstable version of the equivariant fixed point index defined by Prieto and Ulrich is shown. With the help of conormal maps and properties of the unstable index, a sum decomposition formula is proved for the index and consequently also for the degree. As an application, equivariant homotopy groups are decomposed as direct sums of smaller groups of fixed orbit types, and a geometric interpretation of each summand is given in terms of conormal maps.

The theory of order $p$ linear differential equations with univalued solutions can be presented in terms of the Schubert calculus in the Grassmannian of $p$-dimensional subspaces of the vector space of complex polynomials in one variable. The regular singular points and the exponents of an equation determine an intersection of Schubert varieties.

Heine and Stieltjes in their studies of order $p = 2$ linear differential equations with polynomial coefficients having a polynomial solution of a prescribed degree discovered that such a solution was given by a critical point of a certain remarkable symmetric function. In terms of the Schubert calculus, the critical points determine the elements of the intersection of Schubert varieties.

The case $p > 2$ is studied. A function whose critical points determine the non-degenerate elements in the intersection of Schubert varieties is presented. The number of Fuchsian equations with univalued solutions and prescribed exponents at singular points is estimated from above by the intersection number of the corresponding Schubert classes. Conjecturally, for generic disposition of singular points, these numbers coincide, that is, the intersection of Schubert varieties is transversal and consists of non-degenerate elements only. For $p = 2$ the conjecture was proved earlier by the author.

The spectral gaps and thus the exponential rates of convergence to equilibrium are compared for ergodic one-dimensional diffusions on an interval. One of the results may be thought of as the diffusion analogue of a recent result for the spectral gap of one-dimensional Schrödinger operators. The similarities and differences between spectral gap results for diffusions and for Schrödinger operators are also discussed.

For each permutation f of the set of positive integers, all triples s, t, u are determined such that t and u are the lower and upper limits of the sequence of partial sums of the ‘f-rearrangement’ [sum ]af(n) of some real series [sum ]an with sum s.

Cameron–Praeger designs with parameters $t-(v,k,\lambda)$ are studied. Cameron and Praeger showed that in such designs, $t=2$ or 3. In 1989, Delandtsheer and Doyen proved that if $t=2$ then \[ v \leqslant \left(\left({k \atop 2}\right)-1\right)^2. \] In 2000, Mann and Tuan improved this equality and showed that if $t=3$ then \[ v \leqslant \left({k \atop 2}\right)+1. \] Three infinite families of Cameron–Praeger 3-designs for which this bound is met have been constructed by Mann and Tuan and by Sebille. The paper constructs infinitely many infinite families of such Cameron–Praeger 3-designs via a study of a divisibility problem for polynomials. Further, the construction generalizes the previous constructions.

In a previous article [11] we studied the central limit theorem for infinitesimal triangular arrays of probability measures on a Lie group. Since the work of Berg [2] and more recently of Bendikov [1] similar studies appeared to be urgent for the infinite-dimensional torus group and beyond that for the class of Lie projective groups which among others contains all compact groups. Consequently we started extending the existing theory within the enlarged framework of Lie projective groups. In the present contribution continuous convolution hemigroups (μ(s, t)) of probability measures on a Lie projective group G are investigated with respect to generation, representation and occurrence as limits of non-commutative infinitesimal triangular arrays.

The layout of our exposition is as follows. In Section 2 some facts on Lie projective groups G with Lie algebra [Lscr ](G), Lie system [Hfr ], projective basis (Xi)i∈I and corresponding projective weak coordinate system (xi)i∈I are collected to make the reader familiar with the setting. Section 3 contains the basic methodical result yielding the intended generalisations. It is shown in Proposition 3.3 that a hemigroup (μ(s, t)) in the set [Mfr ]1(G) of probability measures on G corresponds to a triplet (a, B, η) in the set ℙbv(ℝ+, G) of characteristics of bounded variation if and only if for each H∈[Hfr ] the projection (pH(μ(s, t))) in [Mfr ]1(G/H) corresponds to a certain triplet (aH, BH, ηH) in the set ℙbv(ℝ+, G/H). Here the correspondence between hemigroups and triplets is achieved via weak backward evolution equations with respect to (Xi)i∈I and (xi)i∈I (see Definition 3.2). Applying the reduction method based on this result, the convergence of infinitesimal arrays {μn[lscr ][ratio ](n,[lscr ])∈ℕ2} in [Mfr ]1(G) towards hemigroups (μ(s, t)) of continuous weak bounded variation is proved in Section 4. In fact, sufficient conditions involving the triplet (a, B, η) are given in order that the sequence (μn (s, t))n[ges ]1 of ‘scaled’ convolution products μn, kn(s)+1[midast ]…[midast ] μn, kn(t) converges weakly to (μ(s, t)). Moreover, necessary and sufficient conditions are given for the limiting hemigroup to be a diffusion hemigroup. Section 5 is devoted to the existence and uniqueness results leading to the one-to-one correspondence between the set ℙbv(ℝ+, G) and the set of hemigroups of continuous weak bounded variation on G. In Section 6 we illustrate the technique of Lie projectivity of the infinite-dimensional torus group and of the p-adic solenoidal group. Moreover, we sketch a little-known example of a hemigroup appearing in atomic physics and derive from it some thoughts towards a perturbation theory for hemigroups.

Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. These results are unified and extended to the much wider class of rings with Auslander dualizing complexes.

In this paper we study sequence spaces that arise from the concept of strong weighted mean summability. Let q = (qn) be a sequence of positive terms and set Qn = [sum ]nk=1qk. Then the weighted mean matrix Mq = (ank) is defined by

formula here

It is well known that Mq defines a regular summability method if and only if Qn→∞. Passing to strong summability, we let 0<p<∞. Then

formula here

are the spaces of all sequences that are strongly Mq-summable with index p to 0, strongly Mq-summable with index p and strongly Mq-bounded with index p, respectively. The most important special case is obtained by taking Mq = C1, the Cesàro matrix, which leads to the familiar sequence spaces

formula here

respectively, see [4, 21]. We remark that strong summability was first studied by Hardy and Littlewood [8] in 1913 when they applied strong Cesàro summability of index 1 and 2 to Fourier series; orthogonal series have remained the main area of application for strong summability. See [32, §6] for further references.

When we abstract from the needs of summability theory certain features of the above sequence spaces become irrelevant; for instance, the qk simply constitute a diagonal transform. Hence, from a sequence space theoretic point of view we are led to study the spaces

The paper gives new examples of [Ascr ]-structures on certain SO(3)-principal fibre bundles.

Let ϕ be a hyperbolic map. Cocycle equations of the form f = uϕ·g·αu−1 are considered, with f, g, u taking values in a compact connected Lie group G, α being an automorphism of G and f, g being Hölder continuous. When the eigenvalues of the derivative of α have modulus 1, it is proved that any measurable solution u has a Hölder continuous version. This condition on α is optimal. When f, g are Ck then u may be taken to be Ck−1+ε for any ε ∈ (0, 1).

Let Σ be a compact Riemann surface of genus g, X={x1, …, xn}⊆Σ a finite set of points, and Ω1Σ(log X) be the sheaf of 1-forms, holomorphic over Σ[setmn ]X and generated near xj by dzj/zj for a coordinate zj centred at xj.

The asymptotic behaviour of entropy numbers of Trudinger–Strichartz embeddings of radial Besov spaces on ${\mathbb R}^n$ into exponential Orlicz spaces is calculated. Estimates of the entropy numbers as well as estimates of entropy numbers of Sobolev embeddings of radial Besov spaces are applied to spectral theory of certain pseudo-differential operators.

The paper studies a generalized spherical function, or a generalized Whittaker model for generalized principal series representations of $G={\rm Sp}(2,{\bf R})$ induced from the Jacobi maximal parabolic subgroup $P_{\rm J}$ , which is called the Fourier–Jacobi type. In particular, a multiplicity theorem and an explicit formula via the Meijer $G$ -functions for this function are given.

Remez-type inequalities provide upper bounds for the uniform norms of polynomials $p$ on given compact sets $K$, provided that $|p(x)|\leq1$ for every $x\in K\setminus E$, where $E$ is a subset of $K$ of small measure. In this paper we prove sharp Remez-type inequalities for homogeneous polynomials on star-like surfaces in $\mathbb{R}^d$. In particular, this covers the case of spherical polynomials (when $d=2$ we deduce a result of Erdélyi for univariate trigonometric polynomials).

The problem of completing a partially prescribed Toeplitz matrix to a positive definite Toeplitz matrix is considered. The main result is a characterization of the patterns of specified entries that guarantee the existence of a positive definite Toeplitz completion of a partial positive definite Toeplitz matrix. This characterization resolves affirmatively a conjecture of C. R. Johnson and L. Rodman.