A sufficient condition for the equisingularity of inverse images of plane curve singularities, as well as a formula for their Milnor numbers, are given.

Filtrations of modules over wreath products of algebras are studied and corresponding multiplicity formulas are given in terms of Littlewood–Richardson coefficients. An example relevant to Jantzen filtrations in Schur algebras is presented.

We identify all non-elementary Kleinian groups which can be generated by two elliptic elements whose commutator is also elliptic. Among those, we identify the finite sets which have finite covolume.

For any $x \in (0,1]$, let the series $ \sum_{n=1}^{\infty}1/d_n(x)$ be the Sylvester expansion of $x$. In this paper, we consider the Hausdorff dimension of the set $$B(\alpha, \beta)= \bigg\{x \in (0,1]: \lim\limits _{n \to \infty} \frac{d_{n+1}(x)}{d^{\beta}_n(x)}=\alpha\bigg\}$$ for any $\alpha \geq 0$ and $\beta \geq 2$. As a corollary, we answer the question posed by Goldie and Smith in [6].

We prove that a point $x$ in R is out of the support of a pseudomeasure $T$ on R if and only if the Fourier integral of $T$ at $x$ is summable with Cesàro means of order 1 to zero.

We study the pointwise behavior of Birkhoff sums $S_n\phi(x)$ on subshifts of finite type for Hölder continuous functions $\phi$. In particular, we show that for a given equilibrium state $\mu$ associated to a Hölder continuous potential, there are points $x$ such that $S_n\phi(x) - n \mathbb{E}_\mu \phi \sim a n^\beta$ for any $a>0$ and $0< \beta <1$. Actually the Hausdorff dimension of the set of such points is bounded from below by the dimension of $\mu$ and it is attained by some maximizing equilibrium state $\nu$ such that $\mathbb{E}_\nu \phi = \mathbb{E}_\mu \phi$. On such points the ergodic average $n^{-1}S_n\phi(x)$ converges more rapidly than predicted by the Birkhoff Theorem, the Law of the Iterated Logarithm and the Central Limit Theorem. All these sets, for different choices $(a, \beta)$, are distinct but have the same dimension. This reveals a rich multifractal structure of the symbolic dynamics. As a consequence, we prove that the set of uniform recurrent points, which are close to periodic points, has full dimension. Applications are also given to the study of syndetic numbers, Hardy–Weierstraß functions and lacunary Taylor series.

The concept of ‘varieties of Banach algebras’ was introduced by the author (Quart. J. Math. Oxford (2), 27 (1976), 481–487). Here we develop the theory, producing analogues of the ‘relatively free’ objects in varieties of universal algebras. We take as a test question the problem of describing all varieties which are closed under bicontinuous isomorphisms (i.e. varieties which are also semivarieties), and show that this may be answered easily with the aid of these new concepts. (The answer is, as expected, just those varieties which are ‘algebraically defined’.)

Let $M$ be the Hardy–Littlewood maximal function and let $ M_{(\alpha)}$ be the $\ell^\alpha$-maximal operator defined by $$M_{(\alpha)} \bar{f}(x)=\|{M\bar {f}\|_{\ell^\alpha}=\Bigg(\sum_{i=1}^\infty Mf_i(x)^\alpha\Bigg)^{1/\alpha},$$ where $\bar f=(f_i)_i$, $M\bar f=(Mf_i)_i$ and $\alpha >1$. The purpose of this work is to study modular inequalities of the form $$\int_{{\mathbb R}^n} P \big(\big|\overline M_{(\alpha)}f(x)\big| \big)\,dx \le \sum_j\int_{{\mathbb R}^n } Q(|f_j(x)|)\, dx,$$ where $P$ and $Q$ are modular functions. Our results apply to operators of the form $$\overline T_{(\alpha)} \bar f(x)= \Bigg(\sum_{i=1}^\infty |T_i f_i(x)|^\alpha\Bigg)^{1/\alpha},$$ where $T_i$ satisfies similar properties to $M$.

We obtain new stability conditions for $C_0$-semigroups on Banach spaces having nontrivial Fourier type. On Hilbert spaces these conditions are sharp. For $C_0$-semigroups on general Banach spaces, we prove a new individual stability criterion. We also show that stronger versions of the range stability condition from [4] are not necessary for stability. This answers an open problem from [4].

The notion of a completely hyperexpansive operator has been introduced in [2] by Athavale. In this paper bounded and unbounded hyperexpansive composition operators are investigated. A unique semispectral measure is associated with a bounded completely hyperexpansive composition operator. Examples of bounded and unbounded densely defined $2$-isometric (completely hyperexpansive, $2$-hyperexpansive) composition operators with invariant domains are given.

We consider an analytic function-germ $f{:}\,(\mathbb{R}^n,0) \rightarrow (\mathbb{R},0)$ with an isolated critical point at $0$. For a sufficiently small ball $B_\varepsilon$ of radius $\varepsilon$ and a sufficiently small regular value $\delta$ of $f$, we give degree formulas for the following Euler–Poincaré characteristics: $$\chi( f^{-1}(\delta) \cap B_\varepsilon \cap \{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $k \in \{1,\ldots,n\}$ and $\epsilon_i \in \{0,1\}$. This leads to degree formulas for $$ \chi (\{f * 0\} \cap S_\varepsilon \cap \{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $* \in \{\le,=,\ge\}$. Combining this with the Eisenbud–Levine–Khimshiashvili's formula, we obtain signature formulas for $$\chi( W \cap\{(-1)^{\epsilon_1} x_1\ge 0,\ldots, (-1)^{\epsilon_k} x_k \ge 0\}),$$ where $W$ is either the link of a weighted-homogeneous singularity or any compact algebraic set of $\mathbb{R}^n$.

We provide a new proof of the following results of H. Schubert: if $K$ is a satellite knot with companion $J$ and pattern $(\skew1\hat{V}, L)$ with index $k$, then the bridge numbers satisfy the following: $b(K) \geq k \cdot (b(J))$. In addition, if $K$ is a composite knot with summands $J$ and $L$, then $b(K) = b(J) + b(L) - 1.$

In this paper, we consider the energy behaviour of Rayleigh surface waves in a homogeneous isotropic 3-dimensional half space with free boundary condition. It is known that Rayleigh waves with finite energy are asymptotically concentrated along the boundary. We clarify the domain where the energy of the Rayleigh waves concentrates and obtain the decay order of the energy distributed in the complement of the domain.