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 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.

An example is given which shows that the Denef–Loeser zeta function (usually called the topological zeta function) associated to a germ of a complex hypersurface singularity is not a topological invariant of the singularity. The idea is the following. Consider two germs of plane curves singularities with the same integral Seifert form but with different topological type and which have different topological zeta functions. Make a double suspension of these singularities (consider them in a 4-dimensional complex space). A theorem of M. Kervaire and J. Levine states that the topological type of these new hypersurface singularities is characterized by their integral Seifert form. Moreover the Seifert form of a suspension is equal (up to sign) to the original Seifert form. Hence these new singularities have the same topological type. By means of a double suspension formula the Denef–Loeser zeta functions are computed for the two 3-dimensional singularities and it is verified that they are not equal.

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.

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.

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.

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\$.

It is proved that there is a certain degree of independence between stationary reflection phenomena at different cofinalities.

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.

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.

global weierstrass representations are derived for complete minimal surfaces obtained by substituting the planar end of the costa surface by scherk ends. this gives rise to singly periodic examples, of which the fundamental piece is proved to be embedded by a simple geometric technique. this technique can be generalised for embeddedness proof of future examples of the same nature.

Let t(G, q) denote the smallest integer t such that the vertex set V of the graph G can be partitioned into t classes V(G)=[Cup ]ti=1Vi such that the number of edges in the induced subgraph 〈Vi〉 for 1[les ]i[les ]t, is divisible by q. Using an algebraic theorem due to Baker and Schmidt we prove that if q is a prime power then t(G, q) can be computed and a corresponding partition can be presented in a polynomial time.

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.

Criteria are found which classify Schatten class membership for operators on the Hardy space in terms of their action on the reproducing kernels and their derivatives. The Hilbert–Schmidt conditions are shown to be both necessary and sufficient for arbitrary operators. For other Schatten classes, the criteria are shown to be either necessary or sufficient for arbitrary operators, depending upon the exponent of the class. However, using known results of Peller and Luecking, the criteria are shown to be both necessary and sufficient for Hankel operators and Carleson embeddings.

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.

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.

A shadow of a subset A of ℝn is the image of A under a projection onto a hyperplane. Let C be a closed nonconvex set in ℝn such that the closures of all its shadows are convex. If, moreover, there are n independent directions such that the closures of the shadows of C in those directions are proper subsets of the respective hyperplanes then it is shown that C contains a copy of ℝn−2. Also for every closed convex set B ‘minimal imitations’ C of B are constructed, that is, closed subsets C of B that have the same shadows as B and that are minimal with respect to dimension.

For the nonlinear recurrence relation \[ \alpha_2 = \alpha_1(1 - \alpha_1), \quad \alpha_{k+1} = k^2 \alpha_k + \sum\limits_{m=2}^{k-1} \alpha_m \alpha_{k+1-m}, \quad k\geqslant 2, \] it is proved that the limit \[ p_{\infty}(\alpha_1) = \lim_{k \rightarrow \infty} \alpha_k /[(k-1)!]^2 \] exists and defines an entire function of $\alpha_2 = \alpha_1(1-\alpha_1)$ .

We compare the classical semidirect product V[midast ] W of pseudovarieties of semigroups with the recently found ‘regular’ semidirect product [Vscr ] [midast ]e [Wscr ] of e-pseudovarieties of regular semigroups. Applications of our findings include a substantial class of additions to the list of pseudovarieties generated by classes of regular semigroups and for which the membership problem is solved. For example, we compute the pseudovarieties generated by all locally inverse and locally orthodox semigroups, respectively, thereby answering a question raised by Almeida and Kaďourek.