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We study the problem of guarding the boundary of a simple polygon with a minimum number of guards such that each guard covers a contiguous portion of the boundary. First, we present a simple greedy algorithm for this problem that returns a guard set of size at most OPT + 1, where OPT is the number of guards in an optimal solution. Then, we present a polynomial-time exact algorithm. While the algorithm is not complicated, its correctness proof is rather involved. This result is interesting in the sense that guarding problems are typically NP-hard and, in particular, it is NP-hard to minimize the number of guards to see the boundary of a simple polygon, without the contiguous boundary guarding constraint. From the combinatorial point of view, we show that any $n$-vertex polygon can be guarded by at most $\lfloor \frac{n-2}{2}\rfloor$ guards. This bound is tight because there are polygons that require this many guards., Comment: 16 pages of body text; 3 pages of references; 13 figures

Modeling the motion of ultracold neutrons (UCNs) is crucial for assessing their losses, accurately measuring their lifetime, and describing other experiments. In material traps, it is necessary to account not only for specular but also for diffuse elastic reflection of UCNs from the trap walls. Typically, the Lambert cosine law is used to describe the angular distribution of diffusely scattered neutrons. However, this law lacks a rigorous theoretical derivation and is often violated. In our work, we propose an experiment to measure the deviation of the angular distribution of UCNs during diffuse scattering from the Lambert law. This deviation can be determined by the difference in the number of neutrons exiting through the central and end windows of a long narrow UCN trap. Monte Carlo simulations corresponding to a possible experiment have been performed, demonstrating a significant effect for different trap geometries., Comment: 7 pages, 3 figures, accepted in Pisma v ZhETF vol. 120, issue 12

Chirality in tellurium derives from a Peierls distortion driven by strong electron-phonon coupling, making this material a unique candidate for observing a light-induced topological phase transition. By using time- and angle-resolved photoelectron spectroscopy (trARPES), we reveal that upon near-infrared photoexcitation the Peierls gap is modulated by displacively excited coherent phonons with $\mathrm{A_{1g}}$ symmetry as well as chiral-symmetry-breaking $\mathrm{E'_{LO}}$ modes. By comparison with state-of-the-art TDDFT+U calculations, we reveal the microscopic origin of the in-phase oscillations of band edges, due to phonon-induced modulation of the effective Hubbard $U$ term., Comment: 6 pages, 3 figures

We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves in terms of the combinatorics of the corresponding graphs., Comment: 7 pages

Let $C$ be a curve over a non-archimedean local field of characteristic zero. We formulate algebro-geometric statements that imply boundedness of functions on the moduli space of stable bundles of rank $2$ and fixed odd degree determinant over $C$, coming from the Schwartz space of $\kappa$-densities on the corresponding stack of bundles (earlier we proved that these functions are locally constant on the locus of very stable bundles). We prove the relevant algebro-geometric statements for curves of genus $2$ and for non-hyperelliptic curves of genus $3$., Comment: 37 pages

We study the behavior of the superperiod map near the boundary of the moduli space of stable supercurves and prove that it is similar to the behavior of periods of classical curves. We consider two applications to the geometry of this moduli space in genus $2$, denoted as $\bar{\mathcal S}_2$. First, we characterize the canonical projection of $\bar{\mathcal S}_2$ in terms of its behavior near the boundary, proving in particular that $\bar{\mathcal S}_2$ is not projected. Secondly, we combine the information on superperiods with the explicit calculation of genus $2$ Mumford isomorphism, due to Witten, to study the expansion of the superstring measure for genus $2$ near the boundary. We also present the proof, due to Deligne, of regularity of the superstring measure on $\bar{\mathcal S}_g$ for any genus., Comment: 66 pages

We prove that for every relatively prime pair of integers $(d,r)$ with $r>0$, there exists an exceptional pair $({\mathcal O},V)$ on any del Pezzo surface of degree 4, such that $V$ is a bundle of rank $r$ and degree $d$. As an application, we prove that every Feigin-Odesskii Poisson bracket on a projective space can be included into a 5-dimensional linear space of compatible Poisson brackets. We also construct new examples of linear spaces of compatible Feigin-Odesskii Poisson brackets of dimension $>5$, coming from del Pezzo surfaces of degree $>4$., Comment: 16 pages

The most accurate neutron lifetime measurements now use the material or magnetic traps of ultracold neutrons (UCN). The precision of these experiments is determined by the accuracy of estimating the neutron loss rate. In material UCN traps the main source of neutron losses is the absorption by trap walls. In this paper we analyze the standard methods and their approximations for the calculation of UCN absorption rate by the walls of material traps. We emphasize the approximations used both in the standard analytical formulas and in the numerical Monte-Carlo simulations. For the two simplest trap geometries, rectangular and cylindrical, we obtain analytical formulas for this absorption rate provided the UCN velocity distribution is isotropic at trap bottom. Then we perform numerical calculations of UCN velocity distribution and absorption rate taking into account the diffuse elastic UCN reflections by trap walls obeying two different laws: Lambert`s cosine law and isotropic reflection. We compare the results with the standard estimation methods and discuss the differences. We indicate the difference between the UCN number and density velocity distribution. Our results may be useful to resolve the puzzling four-second discrepancy between the magnetic and material-trap measurements of neutron lifetime., Comment: 16 pages, 10 figures

We consider the ${\mathbb Z}^n$-graded algebra of global sections of line bundles generated by the standard line bundles $L_1,\ldots,L_n$ on $\bar{M}_{0,n}$. We find a simple presentation of this algebra by generators and quadratic relations. As an application we prove that the moduli space $\bar{M}_{0,n}[\psi]$ of $\psi$-stable curves of genus $0$ is Cohen-Macaulay and normal, and the natural map $\bar{M}_{0,n}\to \bar{M}_{0,n}[\psi]$ is a rational resolution., Comment: 10 pages

Magnetic quantum oscillations (MQO) are traditionally applied to investigate the electronic structure of metals. In layered quasi-two-dimensional (Q2D) materials the MQO have several qualitative features giving additional useful information, provided their theoretical description is developed. Within the framework of the Kubo formula and the self-consistent Born approximation, we reconsider the phase of beats in the amplitude of Shubnikov oscillations of interlayer conductivity in Q2D metals. We show that the phase shift of beats of the Shubnikov (conductivity) oscillations relative to the de Haas - van Alphen (magnetization) oscillations is larger than expected previously and, under certain conditions, can reach the value of $\pi/2$, as observed experimentally. We explain the phase inversion of MQO during the 3D - 2D crossover and predict the decrease of relative MQO amplitude of interlayer magnetoresistance in a strong magnetic field, larger than the beat frequency., Comment: 13 pages

The Spin Physics Detector collaboration proposes to install a universal detector in the second interaction point of the NICA collider under construction (JINR, Dubna) to study the spin structure of the proton and deuteron and other spin-related phenomena using a unique possibility to operate with polarized proton and deuteron beams at a collision energy up to 27 GeV and a luminosity up to $10^{32}$ cm$^{-2}$ s$^{-1}$. As the main goal, the experiment aims to provide access to the gluon TMD PDFs in the proton and deuteron, as well as the gluon transversity distribution and tensor PDFs in the deuteron, via the measurement of specific single and double spin asymmetries using different complementary probes such as charmonia, open charm, and prompt photon production processes. Other polarized and unpolarized physics is possible, especially at the first stage of NICA operation with reduced luminosity and collision energy of the proton and ion beams. This document is dedicated exclusively to technical issues of the SPD setup construction.

We establish a link between open positroid varieties in the Grassmannians $G(k,n)$ and certain moduli spaces of complexes of vector bundles over Kodaira cycle $C^n$, using the shifted Poisson structure on the latter moduli spaces and relating them to the standard Poisson structure on $G(k,n)$. This link allows us to solve a classification problem for extensions of vector bundles over $C^n$. Based on this solution we further classify the symplectic leaves of all positroid varieties in $G(k,n)$ with respect to the standard Poisson structure. Moreover, we get an explicit description of the moduli stack of symplectic leaves of $G(k,n)$ with the standard Poisson structure as an open substack of the stack of vector bundles on $C^n$.

A comparative analysis of structural and flow approaches to analysis of vulnerability of complex network systems (NS) from targeted attacks and non-target lesions of various types was carried out. Typical structural and functional scenarios of successive targeted attacks on the most important by certain characteristics system elements were considered, and scenarios of simultaneous group attacks on the most significant NS's components were proposed. The problem of system lesions scale from heterogeneous negative impacts was investigated and it was confirmed that the flow approach allows us to obtain a much more realistic picture of consequences of such lesions. It is shown that scenarios of group targeted attacks built on the basis of the NC flow model are more optimal from the point of view of choosing attack targets than structural ones., Comment: 11 pages, 4 figures, in Ukrainian

We study the standard family of supercurves of genus 1 with an underlying odd spin structures. We give a simple algebraic description of this family and of the compactified family of stable supercurves with one Neveu-Schwarz puncture. We also describe the Gauss-Manin connection on the first de Rham cohomology of this family and compute the superperiods of global differentials., Comment: 13 pages

Given a geometric domain $P$, visibility-based search problems seek routes for one or more mobile agents ("watchmen") to move within $P$ in order to be able to see a portion (or all) of $P$, while optimizing objectives, such as the length(s) of the route(s), the size (e.g., area or volume) of the portion seen, the probability of detecting a target distributed within $P$ according to a prior distribution, etc. The classic watchman route problem seeks a shortest route for an observer, with omnidirectional vision, to see all of $P$. In this paper we study bicriteria optimization problems for a single mobile agent within a polygonal domain $P$ in the plane, with the criteria of route length and area seen. Specifically, we address the problem of computing a minimum length route that sees at least a specified area of $P$ (minimum length, for a given area quota). We also study the problem of computing a length-constrained route that sees as much area as possible. We provide hardness results and approximation algorithms. In particular, for a simple polygon $P$ we provide the first fully polynomial-time approximation scheme for the problem of computing a shortest route seeing an area quota, as well as a (slightly more efficient) polynomial dual approximation. We also consider polygonal domains $P$ (with holes) and the special case of a planar domain consisting of a union of lines. Our results yield the first approximation algorithms for computing a time-optimal search route in $P$ to guarantee some specified probability of detection of a static target within $P$, randomly distributed in $P$ according to a given prior distribution.

Let $\rm{Bun}$ be the moduli stack of rank $2$ bundles with fixed determinant on a smooth proper curve $C$ over a local field $F$. We show how to associate with a Schwartz $\kappa$-density, for $\rm{Re}(\kappa)\ge 1/2$, a smooth function on the corresponding coarse moduli space of very stable bundles. In the non-archimedean case we also prove that the stack $\rm{Bun}$ is $\kappa$-bounded in the sense of Definition 2.10 of [arXiv:2112.08139] for any $\kappa\in\mathbb{C}$., Comment: 31 pages; v2: corrected definition of $\kappa$-nice

We prove that a pair of Feigin-Odesskii Poisson brackets on ${\mathbb P}^4$ associated with elliptic curves given as linear sections of the Grassmannian $G(2,5)$ are compatible if and only if this pair of elliptic curves is contained in a del Pezzo surface obtained as a linear section of $G(2,5)$.

A flow approach to community detection in complex network and multilayer network systems is proposed. Two methods have been developed to search for communities in a network system (NS). The first of them is based on the calculation of flow influence parameters of NS's subsystems, selected according to the principle of nesting hierarchy. The second method uses the concept of flow core of network system. Two methods are also proposed for community detection in multilayer network system (MLNS). The first of them is based on the concept of MLNS aggregate-network and subsequent allocation of its flow core. The second method uses the concept of flow core of the process of intersystem interactions in general. All developed methods are based on the use of flow criterion that the selected group of nodes really forms a community. The results of application of developed approaches are illustrated by examples for which known methods are ineffective., Comment: 14 page, 8 figures, in Ukrainian

We study the relation between the Hodge filtration of the de Rham cohomology of a proper smooth supervariety $X$ and the usual Hodge filtration of the corresponding reduced variety $X_0$., Comment: 7 pages

We investigate the interlayer coupling between two thin ferromagnetic (F) films mediated by an antiferromagnetic (AF) spacer in F*/AF/F trilayers and show how it transitions between different regimes on changing the AF thickness. Employing layer-selective Kerr magnetometry and ferromagnetic-resonance techniques in a complementary manner enables us to distinguish between three functionally distinct regimes of such ferromagnetic interlayer coupling. The F layers are found to be individually and independently exchange-biased for thick FeMn spacers - the first regime of no interlayer F-F* coupling. F-F* coupling appears on decreasing the FeMn thickness below 9 nm. In this second regime found in structures with 6.0-9.0 nm thick FeMn spacers, the interlayer coupling exists only in a finite temperature interval just below the effective N\'eel temperature of the spacer, which is due to magnon-mediated exchange through the thermally softened antiferromagnetic spacer, vanishing at lower temperatures. The third regime, with FeMn thinner than 4 nm, is characterized by a much stronger interlayer coupling in the entire temperature interval, which is attributed to a magnetic-proximity induced ferromagnetic exchange. These experimental results, spanning the key geometrical parameters and thermal regimes of the F*/AF/F nanostructure, complemented by a comprehensive theoretical analysis, should broaden the understanding of the interlayer exchange in magnetic multilayers and potentially be useful for applications in spin-thermionics., Comment: 14 pages, 9 figures

The derived moduli stack of complexes of vector bundles on a Gorenstein Calabi-Yau curve admits a 0-shifted Poisson structure. Feigin-Odesskii Poisson varieties are examples of such moduli spaces over complex elliptic curves. Using moduli stack of chains we construct an auxiliary Poisson varieties with a Poisson morphism from it to a Feigin-Odesskii variety. We call it the \emph{bosonization} of Feigin-Odesskii variety. As an application, we show that the Feigin-Odesskii Poisson brackets on projective spaces (associated with stable bundles of arbitrary rank on elliptic curves) admit no infinitesimal symmetries.

We study Hecke operators associated with curves over a non-archimedean local field $K$ and over the rings $O/{\mathfrak m}^N$, where $O\subset K$ is the ring of integers. Our main result is commutativity of a certain "small" local Hecke algebra over $O/{\mathfrak m}^N$, associated with a connected split reductive group $G$ such that $[G,G]$ is simple and simpy connected. The proof uses a Hecke algebra associated with $G(K(\!(t)\!))$ and a global argument involving $G$-bundles on curves., Comment: 34 pages; v2: Proposition 4.8 corrected

We study the de Rham cohomology and the Hodge to de Rham spectral sequence for supervarieties., Comment: 14 pages; v2: 15 pages, added Prop. 2.3, edited proof of Thm. 2.15

We present all-electrical operation of a Fe$_x$Cr$_{1-x}$-based Curie switch at room temperature. More specifically, we study the current-induced thermally-driven transition from ferromagnetic to antiferromagnetic Ruderman-Kittel-Kasuya-Yosida (RKKY) indirect coupling in a Fe/Cr/Fe$_{17.5}$Cr$_{82.5}$/Cr/Fe multilayer. Magnetometry measurements at different temperatures show that the transition from the ferromagnetic to the antiferromagnetic coupling at zero field is observed at $\sim$325K. Analytical modelling confirms that the observed temperature-dependent transition from indirect ferromagnetic to indirect antiferromangetic interlayer exchange coupling originates from the modification of the effective interlayer exchange constant through the ferromagnetic-to-paramagnetic transition in the Fe$_{17.5}$Cr$_{82.5}$ spacer with minor contributions from the thermally-driven variations of the magnetization and magnetic anisotropy of the Fe layers. Room-temperature current-in-plane magnetotransport measurements on the patterned Fe/Cr/Fe$_{17.5}$Cr$_{82.5}$/Cr/Fe strips show the transition from the 'low-resistance' parallel to the 'high-resistance' antiparallel remanent magnetization configuration, upon increased probing current density. Quantitative comparison of the switching fields, obtained by magnetometry and magnetotransport, confirms that the Joule heating is the main mechanism responsible for the observed current-induced resistive switching., Comment: 8 pages, 4 figures

We define and study the subspace of cuspidal functions for $G$-bundles on a class of nilpotent extensions $C$ of curves over a finite field. We show that this subspace is preserved by the action of a certain noncommutative Hecke algebra $\mathcal{H}_{G,C}$. In the case $G=\rm{GL}_2$, we construct a commutative subalgebra in $\mathcal{H}_{G,C}$ of Hecke operators associated with simple divisors. In the case of length 2 extensions and of $G=\rm{GL}_2$, we prove that the space of cuspidal functions (for bundles with a fixed determinant) is finite-dimensional and provide bounds on its dimension. In this case we also construct some Hecke eigenfunctions using the relation to Higgs bundles over the corresponding reduced curve., Comment: 62 pages

We develop new multilevel Monte Carlo (MLMC) methods to estimate the expectation of the smallest eigenvalue of a stochastic convection-diffusion operator with random coefficients. The MLMC method is based on a sequence of finite element (FE) discretizations of the eigenvalue problem on a hierarchy of increasingly finer meshes. For the discretized, algebraic eigenproblems we use both the Rayleigh quotient (RQ) iteration and implicitly restarted Arnoldi (IRA), providing an analysis of the cost in each case. By studying the variance on each level and adapting classical FE error bounds to the stochastic setting, we are able to bound the total error of our MLMC estimator and provide a complexity analysis. As expected, the complexity bound for our MLMC estimator is superior to plain Monte Carlo. To improve the efficiency of the MLMC further, we exploit the hierarchy of meshes and use coarser approximations as starting values for the eigensolvers on finer ones. To improve the stability of the MLMC method for convection-dominated problems, we employ two additional strategies. First, we consider the streamline upwind Petrov--Galerkin formulation of the discrete eigenvalue problem, which allows us to start the MLMC method on coarser meshes than is possible with standard FEs. Second, we apply a homotopy method to add stability to the eigensolver for each sample. Finally, we present a multilevel quasi-Monte Carlo method that replaces Monte Carlo with a quasi-Monte Carlo (QMC) rule on each level. Due to the faster convergence of QMC, this improves the overall complexity. We provide detailed numerical results comparing our different strategies to demonstrate the practical feasibility of the MLMC method in different use cases. The results support our complexity analysis and further demonstrate the superiority over plain Monte Carlo in all cases.

The study concerns dynamics of standing spin waves in arrays of sub-100 nm elliptic synthetic-antiferromagnet (SAF) nanodisks. We performed a detailed ferromagnetic resonance analysis in conjunction with micromagnetic modeling to find out several prominent traits of such systems. One broad line is shown to be the sole resonant response for a SAF of the considered sizes. We demonstrate that this mode is degenerated, and its excitation map resembles a superposition of in-center and edge-type oscillations. We also show how this hybrid excitation leads to almost twofold enhancement in the shape-induced anisotropy of the mode., Comment: 3 pages, 1 figure, submitted to 2022 IEEE 12th International Conference Nanomaterials: Applications & Properties (NAP)

The approach to decomposition of multidimensional network systems on monoflow multilayer network systems (MLNS) is proposed, which significantly simplifies the analysis of intersystem interactions in multipurpose and multifunctional supersystem formations. A flow model of monoflow partially overlapped MLNS is built, which reflects the process of intersystem interactions between network systems that ensure the movement of flows of a certain type. The main local and global flow characteristics of MLNS elements are determined, which allow us more adequately from a functional point of view to establish the importance of nodes and edges of a multilayer system compared to their structural characteristics. The constructed model is proposed to be used for the development of methods of protecting complex network and multilayer network systems from targeted attacks and non-targeted lesions of various types and solving other practically important problems of system analysis., Comment: 18 pages, in Ukrainian language

We give explicit formulas for ten compatible Poisson brackets on $\mathbb P^5$ found in arXiv:2007.12351.

Methods for interactive evaluation of the functioning efficiency of the motor transport system for a large city based on the use of U-statistics methods has been formalized. To optimize this technique, effective algorithmic constructions for parallel execution of local, aggregated and forecasting evaluation of system components on modern computing means - multi-core computers, clusters, hybrid architectures and high-performence computing environments are proposed. The obtained scientific results allow for real-time evaluation of the efficiency for the city's motor transport system., Comment: 12 pages, in Ukrainian language

We consider complete convergence and closely related Hsu-Robbins-Erdos-Spitzer-Baum-Katz series for sums whose terms are elements of a linear 2-nd order autoregressive sequences of random variables and prove sufficient conditions for the convergence of this series.

Surface acoustic waves (SAWs) are a versatile tool for coherently interfacing with a variety of solid-state quantum systems spanning microwave to optical frequencies, including superconducting qubits, spins, and quantum emitters. Here, we demonstrate SAW cavity optomechanics with quantum emitters in 2D materials, specifically monolayer WSe$_2$, on a planar lithium niobate SAW resonator driven by superconducting electronics. Using steady-state photoluminescence spectroscopy and time-resolved single-photon counting, we map the temporal dynamics of modulated 2D emitters under coupling to different SAW cavity modes, showing energy-level splitting consistent with deformation potential coupling of 30 meV/%. We leverage the large anisotropic strain from the SAW to modulate the excitonic fine-structure splitting on a nanosecond timescale, which may find applications for on-demand entangled photon-pair generation from 2D materials. Cavity optomechanics with SAWs and 2D quantum emitters provides opportunities for compact sensors and quantum electro-optomechanics in a multi-functional integrated platform that combines phononic, optical, and superconducting electronic quantum systems.

On the basis of structural model of intersystem interactions, the main local and global structural characteristics of nodes of the multilayer network (MLN) are determined. The notions of weighted and binary aggregate-networks of MLN are introduced and integral indicators of the importance of nodes of the multilayer network are determined. The effectiveness of application of aggregate-networks for solving a number of practically important problems of the theory of complex networks is shown. To determine the most important components of intersystem interactions structure, the concepts of p-core of multilayer network and kag-core of its weighted aggregate-network are introduced. The effective scenarios of successive and simultaneous group attacks on the structure of multilayer network systems have been built based on the use of these cores. Indicators of the importance of separate layers of MLN are introduced and approaches to building scenarios of successive and simultaneous system-wide attacks on the structure of intersystem interactions are formulated., Comment: 19 pages, in Ukrainian language, 3 figures

The main types of negative internal and external influences on complex network systems (NS) and intersystem interactions in monoflow partially overlapped multilayer network systems (MLNS) are analyzed. Among such influences, targeted attacks on real complex systems and their non-targeted damages of various nature are primarily singled out. On the basis of structural and flow models of NS and MLNS, sequential and simultaneous local, group and system-wide damages to the structure and operation process of complex network systems and intersystem interactions are considered. The existing structural and functional scenarios of consecutive attacks on the most important components of the system and the need to neutralize the sources of such attacks as one of the means of system protection are analyzed. The peculiarities of application of such scenarios for the protection of NS and MLNS from system-wide non-target lesions are considered. Approaches to evaluation and overcoming the consequences of negative impacts on the structure and operation process of complex network systems and intersystem interactions are proposed., Comment: 18 pages, in Ukrainian language, 2 figures

We design and demonstrate a compact, robust and simple to assemble and tune permanent magnet suitable for NMR relaxometry measurements of microfluidic flows. Soft-magnetic stainless steel plates, incorporated inside the magnet airgap, are key for obtaining substantially improved and tunable field homogeneity. The design is scalable for different NMR probe sizes with the region of suitable field homogeneity, less than 200 ppm, achievable in a capillary length of about 50% of the total magnet length. The built physical prototype, having 3.5x3.5x8.0 cm3 in size and 5 mm high airgap, provides a field strength of 0.5 T and sufficient field homogeneity for NMR relaxometry measurements in capillaries up to 1.6 mm i.d. and 20 mm long. The magnet was used for test flow rate measurements in a wide range, from 0.001 ml/min to 20 ml/min., Comment: 10 pages, 4 figures

Ferromagnetic/antiferromagnetic bilayers are interfaced with normal metal/ferromagnetic bilayers to form F*/AF/N/F valves. The N-spacer thickness is chosen such that it mediates strong indirect exchange (RKKY) between the outer magnetic layers, which varies in strength/direction depending on the N thickness and in direction on switching F. The system exhibits a strong modulation of the F*/AF exchange bias, oscillating in strength syncronously with the oscillation in the interlayer RKKY exchange across the normal metal spacer. The effect is explained as due to a superposition taking place within the antiferromagnetic layer of the direct-exchange proximity effect from the F*/AF interface and the indirect RKKY exchange from F penetrating AF via N. The modulation, expressed via the strength of the F*/AF bias field, reaches 400% at the first RKKY peak., Comment: 5 pages, 4 figures

We study magnetic multilayers, incorporating dilute ferromagnetic spacers between strongly-ferromagnetic layers exhibiting a proximity-enhanced magnetocaloric effect. Using magnetometry and direct measurements of the adiabatic temperature change based on a nanomembrane-calorimetry, we find that the magnetocaloric effect in the studied multilayer is indeed enhanced compared to that in the bulk spacer material. We develop a phenomenological numerical model of the studied trilayer and find that a long-range exchange interaction through the weakly-ferromagnetic spacer is required to adequately describe the magnetic and magnetocaloric properties of the system.

Superhydrophobcity is a well-known wetting phenomenon found in numerous plants and insects. It is achieved by the combination of the surfaces chemical properties and its surface roughness. Inspired by nature, numerous synthetic superhydrophobic surfaces have been developed for various applications. Designated surface coating is one of the fabrication routes to achieve the superhydrophobicity. Yet, many of these coatings, such as fluorine-based formulations, may pose severe health and environmental risks, limiting the applicability. Herein, we present a new family of superhydrophobic coatings comprised of natural saturated fatty acids, which are not only a part of our daily diet, but can be produced from renewable feedstock, providing a safe and sustainable alternative to existing state-of-the-art. These crystalline coatings are readily fabricated via single-step deposition routes, thermal deposition or spray-coating. The fatty acids self-assemble into highly hierarchical crystalline structures exhibiting a water contact angle of about 165 degrees and contact angle hysteresis lower than 6 degrees, while their properties and morphology depend on the specific fatty acid used as well as on the deposition technique. Moreover, the fatty acid coatings demonstrate excellent thermal stability. Importantly these new family of coatings displays excellent anti-biofouling and antimicrobial properties against Escherichia coli and Listeria innocua, used as relevant model Gram-negative and Gram-positive bacteria, respectively. We believe that these coatings have a great application potential in the fields, where other alternatives are prohibited due to safety limitations, while at the same time their usage in other regulation-free applications is not limited.

Since the first case of COVID-19 disease in Wuhan in December 2019, there is a worldwide struggle to reduce the transmission of acute respiratory syndrome coronavirus SARS-CoV-2. Many countries worldwide decided to impose local lockdowns in order to reduce person-to-person interactions, masks became obligatory especially in closed spaces, and there was a general requirement for social distance. However, the most efficient method to reduce continuing spreading of infection among the population, and in the meantime maintain a regular daily life, is early detection of infected contagious people. Up to now, the most reliable method for SARS-CoV-2 detection is reverse-transcriptase PCR test (RT-PCR). It is possible to detect the virus even if there is only one RNA strand in the sample, and run hundreds of samples simultaneously. This method has a few disadvantages, such as high cost, is time consuming, the need for medical laboratories and skilled staff to perform the test, and the major flaw: the lack of appropriate number of available tests. The latterly prominent Omicron variant (B.1.1.529) and its derivatives have caused a tremendous increase in the number of infected people due to its enhanced transmissibility. This all emphasizes the high demand for easy-to-use, cheap and available detection tests.

We prove that several Feigin-Odesskii Poisson brackets associated with normal elliptic curves in ${\mathbb P}^n$ are compatible if and only if they are contained in a scroll or in a Veronese surface in ${\mathbb P}^5$ (with an exception of one case when $n=3$). In the case $n=3$ we determine the quartic corresponding to the Schouten bracket of two (non-compatible) Poisson brackets associated with normal elliptic curves $E_1$ and $E_2$., Comment: 15 pages, v2: strengthened Corollary C for even n

We prove that for any cubic polynomial of slice rank $r$, the intersection of all linear subspaces of minimal codimension contained in the corresponding hypersurface has codimension $\le r^2+\frac{(r+1)^2}{4}+r$ in the affine space. This is deduced from the following result of independent interest. Consider the intersection $I$ of linear ideals $(P_i)$ in $k[x_1,\ldots,x_n]$, with $\dim P_i\le r$. Then the number of quadratic generators of $I$ is $\le r^2$., Comment: 8 pages

We identify Feigin-Odesskii brackets $q_{n,1}(C)$, associated with a normal elliptic curve of degree $n$, $C\subset {\mathbb P}^{n-1}$, with the skew-symmetric $n\times n$ matrix of quadratic forms introduced by Fisher in arXiv:1510.04327 in connection with some minimal free resolutions related to the secant varieties of $C$. On the other hand, we show that for odd $n$, the generators of the ideal of the secant variety of $C$ of codimension $3$ give a Cremona transformation of ${\mathbb P}^{n-1}$, generalizing the quadro-cubic Cremona transformation of ${\mathbb P}^4$. We identify this transformation with the one considered in arXiv:alg-geom/9712022 and find explict formulas for the inverse transformation. We also find polynomial formulas for Cremona transformations from arXiv:alg-geom/9712022 associated with higher rank bundles on $C$., Comment: 10 pages; v2: 12 pages, added Theorem D on Cremona transformations associated with higher rank bundles