Abstract:
We prove that for each positive integer $N$ the set of smooth, zero degree maps $\psi\colon\mathbb{S}^2\to \mathbb{S}^2$ which have the following three properties: (1) there is a unique minimizing harmonic map $u\colon \mathbb{B}^3\to \mathbb{S}^2$ which agrees with $\psi$ on the boundary of the unit ball; (2) this map $u$ has at least $N$ singular points in $\mathbb{B}^3$; (3) the Lavrentiev gap phenomenon holds for $\psi$, i.e., the infimum of the Dirichlet energies $E(w)$ of all smooth extensions $w\colon \mathbb{B}^3\to\mathbb{S}^2$ of $\psi$ is strictly larger than the Dirichlet energy $\int_{\mathbb{B}^3} |\nabla u|^2$ of the (irregular) minimizer $u$, is dense in the set of all smooth zero degree maps $\phi\colon \mathbb{S}^2\to\mathbb{S}^2$ endowed with the $W^{1,p}$-topology, where $1\le p < 2$. This result is sharp: it fails in the $W^{1,2}$ topology on the set of all smooth boundary data.

Abstract:
We consider a class of fourth order elliptic systems which include the Euler-Lagrange equations of biharmonic mappings in dimension 4 and we prove that weak limit of weak solutions to such systems is again a weak solution to a limit system.

Abstract:
Aim. The objective of this study was to analyze the changes in depressive and extrapyramidal symptomatology during glycine augmentation of antipsychotic treatment in patients with schizophrenia.Materials and methods. Twenty-nine schizophrenic patients (ICD-10) with predominant negative symptoms in stable mental state participated in a 10-week open-label prospective study. Patients received stable doses of antipsychotic drugs for at least 3 months before glycine application. During the next 6 weeks patients received augmentation of antipsychotic treatment with glycine (up to 60 g per day). The first and last two weeks of observation were used to assess stability of mental state. Symptom severity was assessed using the Hamilton Depression Rating Scale (HDRS), the Positive and Negative Syndrome Scale (PANSS), and the Simpson-Angus Extrapyramidal Symptom Rating Scale (SAS)Results. In the studied group after 6 weeks of administration of glycine a significant improvement in depressive symptoms (reduced scores by 25.8% in HDRS, p <0.001) and reduced scoring in mood symptoms of PANSS were observed. In SAS a reduction of extrapyramidal symptoms’ severity (p <0.05) was also noted. Two weeks after the glycine augmentation the symptom severity in the HDRS, PANSS, and SAS remained at similar levels.Conclusions. Glycine augmentation of antipsychotic treatment may reduce the severity of depressive and extrapyramidal symptoms. Glycine use was safe and well tolerated.

Abstract:
We investigate knot-theoretic properties of geometrically defined curvature energies such as integral Menger curvature. Elementary radii-functions, such as the circumradius of three points, generate a family of knot energies guaranteeing self-avoidance and a varying degree of higher regularity of finite energy curves. All of these energies turn out to be charge, minimizable in given isotopy classes, tight and strong. Almost all distinguish between knots and unknots, and some of them can be shown to be uniquely minimized by round circles. Bounds on the stick number and the average crossing number, some non-trivial global lower bounds, and unique minimization by circles upon compaction complete the picture.

Abstract:
We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau}\cap W^{2,p}$, where $p>m$ and $\tau=1-m/p$. The results are based on a careful analysis of Morrey estimates for integral curvature--like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on $\Sigma$ or (b) the size of spheres tangent to $\Sigma$ at one point and passing through another point of $\Sigma$. Appropriately defined \emph{maximal functions} of such integrands turn out to be of class $L^p(\Sigma)$ for $p>m$ if and only if the local graph representations of $\Sigma$ have second order derivatives in $L^p$ and $\Sigma$ is embedded. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set $\Sigma$ is a round sphere.

Abstract:
We prove an $\varepsilon$-regularity result for a wide class of parabolic systems $$ u_t-\text{div}\big(|\nabla u|^{p-2}\nabla u) = B(u, \nabla u) $$ with the right hand side $B$ growing like $|\nabla u|^p$. It is assumed that the solution $u(t,\cdot)$ is uniformly small in the space of functions of bounded mean oscillation. The crucial tool is provided by a sharp nonlinear version of the Gagliardo-Nirenberg inequality which has been used earlier in an elliptic context by T. Rivi\`ere and the last named author.

Abstract:
In this paper, we establish compactness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined a priori on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. It turns out that due to a smoothing effect any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold. This result has two applications. The first one is an isotopy finiteness theorem: there are only finitely many isotopy types of such submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. The second one is the lower semicontinuity - with respect to Hausdorff-convergence of submanifolds - of all geometric curvature energies under consideration, which can be used to minimise each of these energies within prescribed isotopy classes.

Abstract:
during the operation of turbounit its bearings displace as a result of heat elongation of bearings supports. it changes the static deflection line of rotor determined during assembly of the turbounit, causing an increase in the stresses on the bearing edges and a decrease in the dynamic state of the machine. one of possibilities to avoid the edge stresses is to apply the bearings with variable axial profile, e.g. hyperboloidal, convex profile in the axial cross-section of bearing. application of journal bearings with hyperboloidal profile allows to extend the bearing operation range without the stress concentration on the edges of bush. these bearings successfully carry the extreme load in conditions of misaligned axis of journal and the bush eliminating the necessity of using self-aligning bearings. operating characteristics of bearing include the resulting force, attitude angle, oil film pressure and temperature distributions, minimum oil film thickness, maximum oil film temperature. in literature there is a lack of data on the operating characteristics of heavy loaded hyperboloidal journal bearings operating at the conditions of adiabatic oil film and static equilibrium position of the journal. for the hyperboloidal bearing the operating characteristics have been obtained. different values of length to diameter ratio, assumed shape and inclination ratio coefficients have been assumed. iterative solution of the reynolds', energy and viscosity equations was applied. adiabatic oil film, laminar flow in the bearing gap as well as aligned and misaligned orientation of journal in the bush were considered.

Abstract:
Let $\mathcal{H}^{\mathbb{T}}$ denote the Hilbert transform on the circle. The paper contains the proofs of the sharp estimates \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq \frac{4}{\pi}\arctan\left(\exp\left(\frac{\pi}{2}\|f\|_1\right)\right) -1, \quad f\in L^{1}(\mathbb{T}), \end{equation*} and \begin{equation*} \frac{1}{2\pi}|\{ \xi\in\mathbb{T} : \mathcal{H}^{\mathbb{T}}f(\xi) \geq 1 \}| \leq \frac{\|f\|_2^2}{1+\|f\|_2^2},\quad f\in L^{2}(\mathbb{T}). \end{equation*} Related estimates for orthogonal martingales satisfying a subordination condition are also established.

Abstract:
We calculate the norms of the operators connected to the action of the Beurling-Ahlfors transform on radial function subspaces introduced by Ba\~nuelos and Janakiraman. In particular, we find the norm of the Beurling-Ahlfors transform acting on radial functions for $p>2$, extending the results obtained by Ba\~nuelos and Janakiraman, Ba\~nuelos and Os\c{e}kowski, and Volberg for $1