A partition of the class of all Hermite–Biehler entire functions of finite order into subclasses ${\cal P}_\kappa$ is introduced. A given function $E(z)$ belongs to ${\cal P}_\kappa$ if and only if $-z^{-1}\log E(z)\in{\mc N}_\kappa$, where the class ${\mc N}_\kappa$ is a well studied family of meromorphic functions on the upper half plane, which originates from operator theoretic problems. It is also proved that the subclasses ${\cal P}_\kappa$ are stable under bounded type perturbation.

The aim of this paper is to determine which (finite) 3-dimensional classical groups are completions of the Goldschmidt G3-amalgam. We recall, first, that an amalgam (of rank 2) consists of three groups P1, P2, B and two group monomorphisms ϕ1, ϕ2 such that

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Usually, when ϕ1 and ϕ2 are understood, this amalgam is denoted by [Ascr ](P1, P2, B). Now a group G is a completion of [Ascr ](P1, P2, B) if there exist group homomorphisms ψi[ratio ]Pi → G (i = 1, 2) satisfying G = 〈im ψ1, im ψ2〉 and ψ1ϕ1 = ψ2ϕ2[ratio ]B→G. The Goldschmidt G3-amalgam, which appears in [6], is defined as follows: P1 ≅ Sym(4) ≅ P2, B ≅ Dih(8) (Dih(n) denotes the dihedral group of order n) with ϕ−11(O2(P1)) ≠ ϕ−12(O2(P2)).

It is shown that, if Q is a finitely generated abelian group, a finitely generated Q-group A is m-tame if and only if the mth tensor power of the augmentation ideal of ℤA is finitely generated over Am [rtimes ] Q, where Q acts diagonally on both Am and the tensor power. It is proved that quotients of metabelian groups of type FP3 are again of type FP3, and a necessary condition is found for a split extension of abelian-by-(nilpotent of class two) groups to be of type FP2. A conjecture is formulated that generalises the FPm-Conjecture for metabelian groups, and it is shown that one of the implications holds in the prime characteristic case.

A characterization is given of the ‘special linear groups’ T(Ψ, W) as linear groups generated by a non-degenerate class Σ of abstract root groups such that the elements of A ∈ Σ are transvections.

The right-invariant Riemannian metric on simplex shape spaces in fact makes them particular Riemannian symmetric spaces of non-compact type. In the paper, the general properties of such symmetric spaces are made explicit for simplex shape spaces. In particular, a global matrix coordinate representation is suggested, with respect to which several geometric features, important for shape analysis, have simple and easily computable expressions. As a typical application, it is shown how to locate the Fréchet means of a class of probability measures on the simplex shape spaces, a result analogous to that for Kendall's shape spaces.

Let $G$ be a metacyclic group of order $pq$ , where $p$ and $q$ are distinct odd prime numbers, let $N| k$ be a Galois extension whose Galois group $G(N| k)$ is isomorphic to $G$ . Let $R_N, R_k$ be the rings of integers of $N$ and $k$ . As $R_k$ -module $R_N$ is completely determined by $[N:k]$ and by its class in the class group of $R_k$ . The paper determines the classes realized by tame Galois extensions $N|k$ with $G(N|k)\cong G$ and proves that they form a group.

In 1868 Zolotarev determined the polynomial which deviates least from zero with respect to the maximum norm on $[-1,1]$ among all polynomials of the form $x^n + \sigma x^{n-1} + a_{n-2}x^{n-2} + \ldots +a_1x + a_0$, where $\sigma \in {\mathbb R}$ is given. The polynomial was given explicitly in terms of elliptic functions by Zolotarev. It is now called the Zolotarev polynomial. Zolotarev also gave an explicit expression for the minimum deviation. In the sequel attempts have been made to replace the elliptic functions and to express the Zolotarev polynomial and the minimum deviation in terms of elementary functions, at least asymptotically. In 1913 Bernstein succeeded in finding an asymptotic formula for the minimum deviation, which has been improved several times since then. Here we give the first asymptotic representation of the Zolotarev polynomials. For the asymptotic representation we use the rational functions introduced by Bernstein. Furthermore, we obtain asymptotic representations of minimal polynomials with interpolation constraints which are of interest in the theory of the iterative solution of inconsistent linear systems of equations.

Here we develop estimates for Fourier coefficients of Siegel cusp forms. First we consider the case of Siegel modular forms for the full modular group $\Gamma_g$ having small weight (that is, weight k = g + 1). Afterwards the case of a certain subgroup $\Gamma_{g,0}(N)$ of $\Gamma_g$ is considered.

On sait associer à certaines structures de Poisson sur ℝn, de 1-jet nul en 0, des actions de ℝ2 sur ℝn, données par le ‘rotationnel’ de leur partie quadratique et un autre champ de vecteurs. Lorsque ces actions sont ‘non résonantes’ et ‘hyperboliques’, on montre que ces structures sont ‘quadratisables’, en ce sens qu'il existe des coordonnées dans lesquelles, elles sont quadratiques. Dans le cas de la dimension 3, nos résultats mènent à la ‘non-dégénérescence’ générique des structures de Poisson quadratiques à rotationnels inversibles.

We can associate with some Poisson structures defined on ℝn with a zero 1-jet at zero, actions from ℝ2 on ℝn, given by the ‘curl’ of their quadratic part and another vector field. Assuming that those actions are ‘hyperbolics’ and without ‘resonances’, we give a normal form for those structures. On ℝ3, we prove that every quadratic Poisson structure with invertible curl, is generically ‘non degenerate’.

The positivity of the partial sums of the series of Legendre polynomials is a classical result of Fejér. The paper presents corresponding results for both even and odd Legendre polynomials with best possible leading constants. These results have applications to demonstrating positivity of Cotes numbers at Jacobi abscissas.

We prove a complex analytic analogue of the classical Rolle theorem asserting that the number of zeros of a real smooth function can exceed that of its derivative by at most 1. This result is used then to obtain upper bounds for the number of complex isolated zeros of:

(1) functions defined by linear ordinary differential equations (in terms of the magnitude of the coefficients of the equations);

(2) elements from the polynomial envelope of a linear differential equation with an irreducible monodromy group (in terms of the degree of the envelope);

(3) successive derivatives of a function defined by a linear irreducible equation (in terms of the order of the derivative).

These results generalize the bounds from [2, 5, 6] that were previously obtained for the number of real isolated zeros.

The purpose of this paper is to give characterizations of weights for which reverse inequalities of Hölder type for quasi-monotone functions are satisfied. Our inequalities with general weights and with sharp constants complement the results of [2, 8, 9] and [16, 17] for the values of parameters p and q with 1[les ]p[les ]q<∞.

Let u be a solution of the differential equation u″+Ru=0, where R is rational. Newton's method of finding the zeros of u consists of iterating the function f(z) =z−u(z)/u′(z). With suitable hypotheses on R and u, it is shown that the iterates of f converge on an open dense subset of the plane if they converge for the zeros of R. The proof is based on the iteration theory of meromorphic functions, and in particular on the result that, if the family of K-quasiconformal deformations of a meromorphic function f depends on only finitely many parameters, then every cycle of Baker domains of f contains a singularity of f−1. This result, together with classical results of Hille concerning the asymptotic behaviour of solutions of the above differential equations, is also used to study their value distribution. For example, it is shown that, if R is a rational function which satisfies R(z)∼amzm as z→∞ and has only k distinct zeros where k<(m+2)/2, then δ(0, u)[les ]k/(m+2−k)<1.

Let $\Omega \subset \R^2$ be a bounded Lipschitz domain and let \[ F: \Omega \times {\mathbb R}_{+}^{2 \times 2} \longrightarrow {\mathbb R} \] be a Carathèodory integrand such that $F\left(x, \cdot\right)$ is polyconvex for ${\mathcal L}^2$-a.e. $x \in \Omega$. Moreover assume that $F$ is bounded from below and satisfies the condition $F\left(x, \xi\right) \searrow \infty$ as $\det \xi \searrow 0$ for ${\mathcal L}^2$-a.e. $x \in \Omega$. The paper describes the effect of domain topology on the existence and multiplicity of strong local minimizers of the functional \[ {\mathbb F} \left[u\right] := \int_{\Omega } F\left(x, \nabla u\left(x\right)\right) \, dx,\] where the map $u$ lies in the Sobolev space $W_{\rm {id}}^{1,p} (\Omega, {\mathbb R}^2)$ with $p \geqslant 2$ and satisfies the pointwise condition $\det \nabla u\left(x\right) >0$ for ${\mathcal L}^2$-a.e. $x \in \Omega$. The question is settled by establishing that ${\mathbb F}\left[\cdot\right]$ admits a set of strong local minimizers on {W^{1,p}_{\rm{id}}(\Omega, {\mathbb R}^2)} that can be indexed by the group $\Pn \oplus \Z^n$, the direct sum of Artin's pure braid group on $n$ strings and $n$ copies of the infinite cyclic group. The dependence on the domain topology is through the number of holes $n$ in $\Omega$ and the different mechanisms that give rise to such local minimizers are fully exploited by this particular representation.

Let \$C\$ be an irreducible, smooth, projective curve of genus \$g$\backslash$geqslant 3\$ over the complex field \$$\backslash$mathbb\{C\}\$. The curve \$C\$ is called \{$\backslash$em bielliptic\} if it admits a degree-two morphism \$$\backslash$pi$\backslash$colon C$\backslash$longrightarrow E\$ onto an elliptic curve \$E\$; such a morphism is called a \{$\backslash$em bielliptic structure\} on \$C\$. If \$C\$ is bielliptic and \$g$\backslash$geqslant 6\$, then the bielliptic structure on \$C\$ is unique, but if \$g=3,4,5\$, then this holds true only generically and there are curves carrying \$n>1\$ bielliptic structures. The sharp bounds \$n$\backslash$leqslant 21,10,5\$ exist for \$g=3,4,5\$ respectively. Let \$$\backslash$mathfrak\{M\}\_g\$ be the coarse moduli space of irreducible, smooth, projective curves of genus \$g=3,4,5\$. Denote by \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ the locus of points in \$$\backslash$mathfrak\{M\}\_g\$ representing curves carrying at least \$n\$ bielliptic structures. It is then natural to ask the following questions. Clearly \$$\backslash$mathfrak\{B\}\_g{\^{}}n$\backslash$subseteq $\backslash$mathfrak\{B\}\_g{\^{}}\{n-1\}\$; does \$$\backslash$mathfrak\{B\}\_g{\^{}}n$\backslash$neq$\backslash$mathfrak\{B\}\_g{\^{}}\{n-1\}\$ hold? What are the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$? Are the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ rational? How do the irreducible components of \$$\backslash$mathfrak\{B\}\_g{\^{}}n\$ intersect each other? Let \$C$\backslash$in$\backslash$mathfrak\{B\}\_g{\^{}}2\$; how many non-isomorphic elliptic quotients can it have? Complete answers are given to the above questions in the case \$g=4\$.

Let f(z) and g(z) be transcendental entire functions. Fuchs and Song proved that if (f(z); g(z)) parametrizes some complex algebraic curve, then f and g must have a transcendental common right factor. The paper proves this result by a different method that also allows a similar result to be proved for some transcendental curves. This result is then used to solve some factorization problems of entire functions.

Upper and lower bounds are established for the entropy numbers of certain diagonal operators between Banach sequence spaces. These diagonal operators are isomorphisms between the spaces considered in the paper and weighted sequence spaces considered by Leopold so that the entropy numbers in question coincide with those considered by Leopold. The results in the paper improve the previous results in at least two ways. The estimates in the paper are ‘almost’ sharp in the sense that the upper and lower estimates differ only by logarithmic factors for a much wider range of parameters. Moreover, all the upper estimates are improvements on the previous ones, the improvement being quite significant in some cases.

The paper examines blow-up phenomena for the inequality \renewcommand{\theequation}{*} \begin{equation} u_t - Lu - \vert u \vert^{q-1} u \leqslant v_t - Lv - \vert v \vert^{q-1}v \end{equation} in the half-space ${\bb R}^1_+ \times {\bb R}^n,\; n \geqslant 1$ , where $L$ is a linear second-order partial differential operator in divergence form.

The paper studies weak solutions of (*) that belong only locally to the corresponding Sobolev spaces in the half-space ${\bb R}^1_+ \times {\bb R}^n$ . It also requires no conditions for the behavior of solutions of (*) on the hyperplane $t = 0$ .

The existence of critical blow-up exponents is obtained for solutions of (*) as a special case of a comparison principle for the corresponding solutions of (*). For example, the well-known Fujita result is a consequence of the comparison principle.

The approach developed in the paper is directly applicable to the study of analogous problems involving nonlinear differential operators. Its elliptic analogue has been recently developed by the authors.

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [lscr ]1(Γ) and [lscr ]∞(Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

An n-affine manifold is a differentiable manifold endowed with an atlas whose transition functions are locally affine transformations of ℝn . The paper studies affine dynamic, that is, affine endomorphisms of affine manifolds. The main goal is to relate the dynamical viewpoint to the completeness problem. In particular, it is shown that the Markus conjecture is true when dim Aff(M, [dtri ]) > dim M − 2.