We investigate the intersections of the curve $ \mathbb{R} \ni t\mapsto \zeta (\frac{1}{2} + \mathrm{i} t)$ with the real axis. We show unconditionally that the zeta function takes arbitrarily large positive and negative values on the critical line.

The first and second representation theorems for sign-indefinite, not necessarily semi-bounded quadratic forms are revisited. New and straightforward proofs of these theorems are given. A number of necessary and sufficient conditions for the second representation theorem to hold are obtained. A new simple and explicit example of a self-adjoint operator for which the second representation theorem fails to hold is also provided.

Let $\mathcal {O} \subset \mathbb {R}^d$ be a bounded domain of class $C^{1,1}$. In the Hilbert space $L_2(\mathcal {O};\mathbb {C}^n)$, we consider a matrix elliptic second order differential operator $\mathcal {A}_{D,\varepsilon }$ with the Dirichlet boundary condition. Here $\varepsilon \gt 0$ is the small parameter. The coefficients of the operator are periodic and depend on $\mathbf {x}/\varepsilon $. There are no regularity assumptions on the coefficients. A sharp order operator error estimate $\|\mathcal {A}_{D,\varepsilon }^{-1} - (\mathcal {A}_D^0)^{-1} \|_{L_2 \to L_2} \leq C \varepsilon $ is obtained. Here $\mathcal {A}^0_D$is the effective operator with constant coefficients and with the Dirichlet boundary condition.

In this paper we prove sharp estimates for generalized eigenvectors of Hermitian Jacobi matrices associated with the spectral parameter lying in a gap of their essential spectra. The estimates do not depend on the main diagonals of these matrices. The types of estimates obtained for bounded and unbounded gaps are different. These estimates extend the previous ones found in [J. Janas, S. Naboko and G. Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not.4 (2009), 736–764], for the spectral parameter being outside the whole spectrum of Jacobi matrices. Examples illustrating optimality of our results are also given.

We show that if d≥4 is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.

For an arbitrary subset $X$ of a finite-dimensional real Banach space $E$, the ball intersection with parameter $\lambda \gt 0$ is defined as the intersection of all balls of radius $\lambda $ whose centers are in $X$. On the other hand, the intersection of all balls of radius $\lambda $ that contain $X$ is said to be the respective ball hull. We present new results on these two notions and use them to get new insights into complete sets and (pairs of) sets of constant width, e.g., their representation as vector sums of suitable ball intersections and ball hulls. Also in this framework, we give partial answers to the known question, in what finite-dimensional real Banach spaces any complete set is of constant width. For polyhedral norms we obtain characterizations of monotypic balls via constant width properties of pairs formed by the ball intersection and ball hull of the same bounded and non-empty set. Finally, we present some new results on Borsuk numbers of sets of constant width in normed spaces, closely related to (unique) completions of compact sets. For example, the lower estimate on Borsuk numbers of bodies of constant width due to Lenz is extended to arbitrary normed spaces. Furthermore, we also derive the Borsuk number of the normed space with maximum norm.

We consider the convex hull ℬk of the symmetric moment curve Uk(t)=(cos t,sin t,cos 3t,sin 3t,…,cos (2k−1)t,sin (2k−1)t) in ℝ2k, where t ranges over the unit circle 𝕊=ℝ/2πℤ. The curve Uk(t) is locally neighborly: as long as t1,…,tk lie in an open arc of 𝕊 of a certain length ϕk>0 , the convex hull of the points Uk (t1),…,Uk (tk) is a face of ℬk. We characterize the maximum possible length ϕk, proving, in particular, that ϕk >π/2 for all k and that the limit of ϕk is π/2 as k grows. This allows us to construct centrally symmetric polytopes with a record number of faces.

Generalizing a result (the case $k= 1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k+ 1$ belongs to the generalized Walkup class ${ \mathcal{K} }_{k} (2k+ 1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since it is far from obvious that the vertex links of polytopal upper bound spheres should have any special combinatorial structure. It has been conjectured that for $d\not = 2k+ 1$, all $(k+ 1)$-neighborly members of the class ${ \mathcal{K} }_{k} (d)$ are tight. The result of this paper shows that the hypothesis $d\not = 2k+ 1$ is essential for every value of $k\geq 1$.

We construct dense Borel measurable subgroups of Lie groups of intermediate Hausdorff dimension. In particular, we generalize the Erdős–Volkmann construction [Additive Gruppen mit vorgegebener Hausdorffscher Dimension, J. Reine Angew. Math. 221 (1966), 203–208], showing that any nilpotent $\sigma $-compact Lie group $N$ admits dense Borel subgroups of arbitrary dimension between zero and $\dim N$. In algebraic groups defined over a finite extension of the rationals, using diophantine properties of algebraic numbers, we are also able to construct dense subgroups of arbitrary dimension, but the general case remains open. In particular, we raise the following question: does there exist a measurable proper subgroup of $ \mathbb{R} $ of positive Hausdorff dimension which is stable under multiplication by a transcendental number? Subgroups of nilpotent $p$-adic analytic groups are also discussed.

Arvanitakis [A simultaneous selection theorem. Preprint] recently established a theorem which is a common generalization of Michael’s convex selection theorem [Continuous selections I. Ann. of Math. (2) 63 (1956), 361–382] and Dugundji’s extension theorem [An extension of Tietze’s theorem, Pacific J. Math.1 (1951), 353–367]. In this note we provide a short proof of a more general version of Arvanitakis’s result.