We consider the functions $T_n(x)$ defined as the $n$th partial derivative of Lebesgue's singular function $L_a(x)$ with respect to $a$ at $a=\frac{1}{2}$. This sequence includes a multiple of the Takagi function as the case $n=1$. We show that $T_n$ is continuous but nowhere differentiable for each $n$, and determine the Hölder order of $T_n$. From this, we derive that the Hausdorff dimension of the graph of $T_n$ is one. Using a formula of Lomnicki and Ulam, we obtain an arithmetic expression for $T_n(x)$ using the binary expansion of $x$, and use this to find the sets of points where $T_2$ and $T_3$ take on their absolute maximum and minimum values. We show that these sets are topological Cantor sets. In addition, we characterize the sets of local maximum and minimum points of $T_2$ and $T_3$.

We define equivariant Chern–Schwartz–MacPherson classes of a possibly singular algebraic $G$-variety over the base field $\mathbb{C}$, or more generally over a field of characteristic 0. In fact, we construct a natural transformation $C^G_*$ from the $G$-equivariant constructible function functor $\cal{F}^G$ to the $G$-equivariant homology functor $H^G_*$ or $A^G_*$ (in the sense of Totaro–Edidin–Graham). This $C^G_*$ may be regarded as MacPherson's transformation for (certain) quotient stacks. The Verdier–Riemann–Roch formula takes a key role throughout.

We derive universal Diophantine properties for the Patterson measure $\mu_{_{\Gamma}}$ associated with a convex cocompact Kleinian group $\Gamma$ acting on $(n+1)$-dimensional hyperbolic space. We show that $\mu_{_{\Gamma}}$ is always an ${\cal S}$-friendly measure, for every $(\Gamma,\mu_{_{\Gamma }})$-neglectable set ${\cal S}$, and deduce that if $\Gamma$ is of non-Fuchsian type then $\mu_{_{\Gamma}}$ is an absolutely friendly measure in the sense of Pollington and Velani. Consequently, by a result of Kleinbock, Lindenstrauss and Weiss, $\mu_{_{\Gamma}}$ is strongly extremal which means that $\mu_{_{\Gamma}}$-almost every point is not very well multiplicatively approximable. This is remarkable, since by a well-known result in classical metric Diophantine analysis the set of very well multiplicatively approximable points is of $n$-dimensional Lebesgue measure zero but has Hausdorff dimension equal to $n$.

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier–Stieltjes transform of a discrete measure on the dual group.

We show that if $\varepsilon\,{<}\,\sqrt2$ then an $\varepsilon$-Kronecker set is $I_0$, but this is not true for at least one $\sqrt 2$-Kronecker set. $\varepsilon$-Kronecker sets in ${\mathbb Z}$ need not be finite unions of Hadamard sets. As with Sidon sets, $\varepsilon$-Kronecker sets with $\varepsilon\,{<}\,2$ do not contain arbitrarily long arithmetic progressions or large squares. When $\varepsilon\,{<}\,\sqrt 2$ they can contain only a bounded number of pairs with common differences and their step length tends to infinity. Related results and examples are given to show the sharpness of these results.

A subset $E$ of the locally compact abelian group $\Gamma$ is “$\varepsilon$-Kronecker” if every continuous function from $E$ to the unit circle can be uniformly approximated on $E$ by a character with error less than $\varepsilon$. The set $E\subset \Gamma$ is $I_0$ if every bounded function on $E$ can be interpolated by the Fourier Stieltjes transform of a discrete measure on the dual group.

We show that products (sums) of $\varepsilon$-Kronecker sets can be all of the group if the number of terms is sufficiently large, but are shown to be $U_0$ sets (sets of uniqueness in the weak sense) if the number is small. Results about cluster points of products are extended from Hadamard to $\varepsilon$-Kronecker sets. One consequence of that is that finite unions of translates of a fixed $\varepsilon$-Kronecker set are $I_0$.

We consider the class $S_{p}$ of functions \[ f(z) = z + a_{p+1}z^{p+1} +\cdots \] univalent in the unit disk $\triangle$. We show that, if $f\in S_{p}$ and $p$ is large, \[ \alpha (f) = \lim\limits_{n{\to}\infty}\frac{|a_{n}|}{n} \le \alpha (p) = \frac{2(\log p\log\log p)^2}{p^4}\]. We also obtain estimates for $|a_{n}|$ when $n\in S_p$.

In a companion paper [1] it will be shown that there exists $f$ in $S_{p}$ for $p=1,2,\ldots$ such that \[ \alpha (f) \ge \frac{C_{0}}{p^4},\] where $C_{0}$ is a positive absolute constant.

The classical divergence theorem for an $n$-dimensional domain $A$ and a smooth vector field $F$ in $n$-space \[\int_{\partial A} F \cdot n = \int_A div F\] requires that a normal vector field $n(p)$ be defined a.e. $p \in \partial A$. In this paper we give a new proof and extension of this theorem by replacing $n$ with a limit $\star \partial A$ of 1-dimensional polyhedral chains taken with respect to a norm. The operator $\star$ is a geometric dual to the Hodge star operator and is defined on a large class of $k$-dimensional domains of integration $A$ in $n$-space called chainlets. Chainlets include a broad range of domains, from smooth manifolds to soap bubbles and fractals. We prove as our main result the Star theorem \[ \displaystyle\int_{\star A} \omega = (-1)^{k(n-k)}\int_A \star \omega \] where $\omega$ is a $k$-form and $A$ is an $(n-k)$-chainlet. When combined with the general Stokes' theorem ([H1, H2]) \[\int_{\partial A} \omega = \int_A d \omega\] this result yields optimal and concise forms of Gauss' divergence theorem \[\int_{\star \partial A}\omega = (-1)^{(k)(n-k)} \int_A d\star \omega\] and Green's curl theorem \[\int_{\partial A} \omega = \int_{\star A} \star d\omega.\]

This paper provides a fairly general approach to summability questions for multi-dimensional Fourier transforms. It is based on the use of Wiener amalgam spaces $W(L_p,\ell_q)({\mathbb R}^d)$, Herz spaces and weighted versions of Feichtinger's algebra and covers a wide range of concrete special cases (20 of them are listed at the end of the paper). It is proved that under some conditions the maximal operator of the $\theta$-means $\sigma_T^\theta f$ can be estimated pointwise by the Hardy–Littlewood maximal function. From this it follows that $\sigma_T^\theta f \,{\to}\, f$ a.e. for all $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$, hence $f\in L_p({\mathbb R}^d)$ for any $1\leq p\leq \infty$. Moreover, $\sigma_T^\theta f(x)$ converges to $f(x)$ at each Lebesgue point of $f\in L_1({\mathbb R}^d)$ (resp. $f\in W(L_1,\ell_\infty)({\mathbb R}^d)$) if and only if the Fourier transform of $\theta$ is in a suitable Herz space. In case $\theta$ is in a Besov space or in a weighted Feichtinger's algebra or in a Sobolev-type space then the a.e. convergence is obtained. Some sufficient conditions are given for $\theta$ to be in the weighted Feichtinger's algebra. The same results are presented for multi-dimensional Fourier series.

We investigate the iterative behaviour of continuous order preserving subhomogeneous maps $f: K\,{\rightarrow}\, K$, where $K$ is a polyhedral cone in a finite dimensional vector space. We show that each bounded orbit of $f$ converges to a periodic orbit and, moreover, the period of each periodic point of $f$ is bounded by \[ \beta_N = \max_{q+r+s=N}\frac{N!}{q!r!s!}= \frac{N!}{\big\lfloor\frac{N}{3}\big\rfloor!\big\lfloor\frac{N\,{+}\,1}{3}\big\rfloor! \big\lfloor\frac{N\,{+}\,2}{3}\big\rfloor!}\sim \frac{3^{N+1}\sqrt{3}}{2\pi N}, \] where $N$ is the number of facets of the polyhedral cone. By constructing examples on the standard positive cone in $\mathbb{R}^n$, we show that the upper bound is asymptotically sharp.

These results are an extension of work by Lemmens and Scheutzow concerning periodic orbits in the interior of the standard positive cone in $\mathbb{R}^n$.

We give a simple constructive proof of a geometric form of Hahn-Banach theorem, simplifying a proof in [5].

The classical result of Gohberg, Markus and Feldman states that, when $E$ is one of the classical Banach sequence spaces $E=l^p$ for $1\leq p<\infty$ or $E=c_0$, the only closed, two-sided, non-trivial ideal in $\cal B(E)$, the Banach algebra of operators on a Banach space $E$, is $\cal K(E)$, the ideal of compact operators. Gramsch and Luft completely classified the closed, two-sided ideals in $\cal B(H)$ for an arbitrary Hilbert space $H$ through the idea of $\kappa$-compact operators, for infinite cardinals $\kappa$. This paper presents an extension of this result to the non-separable versions of $l^p$ and $c_0$.

We investigate spherically symmetric equilibrium states of the Vlasov–Poisson system, relevant in galactic dynamics. We recast the equations into a regular three-dimensional system of autonomous first order ordinary differential equations on a region with compact closure. Based on a dynamical systems analysis we derive theorems that guarantee that the steady state solutions have finite mass and compact support.

We prove monadic second order limit laws for classes of logarithmic adequate relational structures by using probabilistic methods.

Using coupling techniques, we prove uniqueness in $G$-measures under a weak regularity condition and give estimates of the associated rates of convergence. We also show how to generate a random variable distributed according to the unique $G$-measure on cylinder sets for any fixed level of precision.

We develop the theory of contractive Markov systems initiated in [8]. We construct a coding map for such systems and investigate some of its properties.

A wide class of scalar first order linear autonomous neutral delay differential equations with distributed type delays is considered. By the use of two distinct real roots of the corresponding characteristic equation, a new result on the behavior of the solutions is established.

We develop new criteria for the stability of a periodic solution of a given newtonian equation based on the $L^1$-norm of the coefficients of the third order approximation, by proving that the twist coefficient is different form zero.