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Quartz Crystal Microbalance with Dissipation monitoring (QCM-D) has become a major tool in the analysis of adsorption of nanometric objects, such as proteins, viruses, liposomes and inorganic particles from the solution. While in vacuum extremely accurate mass measurements are possible, in a liquid phase the quantitative analysis is intricate due to the complex interplay of hydrodynamic and adhesion forces, varying with the physicochemical properties of adsorbent and the quartz resonator surfaces. In the present paper we dissect the role of hydrodynamics for the analytically tractable scenario of a stiff contact, whereas the adsorbed particles oscillate with the resonator as a whole without rotation. Under the assumption of the low surface coverage, we theoretically study the excess shear force exerted on the resonator due to presence of a single adsorbed particle. The excess shear force has two contributions: (i) the fluid-mediated force due to flow disturbance created by the particle and (ii) the viscous force exerted on the particle by the fluid and transmitted to the resonator via contact. We found that for small adsorbates, there is mutual cancellation of the above contributions to the excess shear force at the leading order approximation, reducing the overall effect of the hydrodynamics to the order-of-magnitude of the inertial force. However, accurate numerical solution shows that for small particles the viscous force dominates over the inertia force, rendering the standard Sauerbrey model inapplicable. These findings indicate that the accurate account of hydrodynamics in the analysis of QCM-D response is critical. The resulting dimensionless frequency and dissipation shifts and the corresponding acoustic ratio computed numerically, showing a fair agreement with previously published experimental results at low oscillation frequencies.

A double partition problem asks for a number of nonnegative integer solutions to a system of two linear Diophantine equations with integer coefficients. Artur Cayley suggested a reduction of a double partition to a sum of scalar partitions with an algorithm subject to a set of conditions. We show that when these conditions are not satisfied and the original algorithm fails its modification solves the reduction problem., Comment: 18 pages

In the present paper we theoretically study the shear force exerted on an infinite horizontal plane undergoing fast lateral oscillations in presence of a rigid particle suspended in the viscous liquid above the plate. The study is largely motivated by Quartz Crystal Microbalance (QCM-D) technique which relies on analyzing response (complex impedance) of fast oscillating (in MHz range) quartz crystal disk in the liquid medium due to small substances adsorbed at its surface. In fact, small substances suspended in the liquid medium in the vicinity of the oscillating crystal may also contribute to impedance, as they modify the local shear force the suspending liquid exerts on the quartz crystal. For a dilute suspension the contributions of individual particles are additive and, therefore, our analysis is restricted to the excess shear force due a single spherical particle located at arbitrary distance above the plane. Three distinct cases are considered: (i) limiting case of high solid inertia, whereas the heavy particle can be considered as stationary; (ii) a freely suspended particle of arbitrary mass, undergoing fluid-mediated time-periodic rotation and translation and (iii) an adsorbed particle moving with the plate as a whole without rotation. For small-amplitude plane oscillations the unsteady Stokes flow equations apply. We construct the series solution of these equations using the method of reflections, whereas its terms are written explicitly. Due to the exponential decay of the flow away from the oscillating plate, the truncated series containing only few low-order terms shows an excellent agreement with the rigorous numerical results for a wide range of particle sizes and separation distances. The present results support the notion that the hydrodynamic contribution of the suspended small substances to the measured impedance is non-negligible or even dominant., Comment: 23 pages, 8 figures

We present the results of a study of the Poynting vector field generic singularities at the resonant light scattering of a plane monochromatic linearly polarized electromagnetic wave by a subwavelength particle. We reveal the impact of the problem symmetry, the spatial dimension, and the energy conservation law on the properties of the singularities. We show that, in the cases when the problem symmetry results in the existence of an invariant plane for the Poynting vector field lines, a formation of a standing wave in the immediate vicinity of a singularity gives rise to a saddle-type singular point. All other types of singularities are associated with vanishing at the singular points, either (i) magnetic field, for the polarization plane parallel to the invariant plane, or (ii) electric field, at the perpendicular orientation of the polarization plane. We also show that in the case of two-dimensional problems (scattering by a cylinder), the energy conservation law restricts the types of possible singularities only to saddles and centers in the non-dissipative media and to saddles, foci, and nodes in dissipative. Finally, we show that dissipation affects the (i)-type singularities much stronger than the (ii)-type. The same conclusions are valid for the imaginary part of the Poynting vector in problems where the latter is regarded as a complex quantity. The singular points associated with the formation of standing waves are different for real and imaginary parts of this complex vector field, while all other singularities are common. We illustrate the general discussion by analyzing singularities at light scattering by a subwavelength Germanium cylinder with the actual dispersion of its refractive index., Comment: 20 pages, 3 figures, 2 tables

Recent research demonstrate that prediction of time series by predictive recurrent neural networks based on the noisy input generates a smooth anticipated trajectory. We examine influence of the noise component in both the training data sets and the input sequences on network prediction quality. We propose and discuss an explanation of the observed noise compression in the predictive process. We also discuss importance of this property of recurrent networks in the neuroscience context for the evolution of living organisms., Comment: 13 pages, 8 figures

We consider a model of mRNA production governed by the dynamics of a gene that exists in three possible states -- inactive, poised and active. The transitions between the adjacent states are controlled by stochastic processes characterized by corresponding on/off rates. mRNA is produced only when the gene is in active state and we also consider mRNA denaturation leading to its death., Comment: 7 pages, 3 figures

Singularities of the Poynting vector field at resonant light scattering by nanoparticles are discussed and classified. It is shown that there are two generic types of them, namely (i) the singularities related to the vanishing of the magnetic (and/or electric) field at the singular points and (ii) the singularities related to the formation of standing waves in the proximity of the singular points. The connection of these types of singularities to the topology of the singular points and the space dimension (3D vs. 2D) is revealed., Comment: 9 pages, 4 figures

We extend an algorithm suggested in 1858 by Sylvester and implemented in 1860 by Cayley for a problem of double partitions and apply it to derivation of explicit expressions for coefficients of the Gaussian polynomials through convolution of restricted partition functions., Comment: 14 pages

We derive the general solution of the unsteady Stokes equations for an unbounded fluid in spherical polar coordinates, in both time and frequency domains. The solution is an expansion in vector spherical harmonics and given as a sum of a particular solution, proportional to pressure gradient exhibiting power-law spatial dependence, and a solution of vector Helmholtz equation decaying exponentially in far field, the decomposition originally introduced by Lamb. The solution can be applied to construct the transient exterior flow induced by an arbitrary velocity distribution at the spherical boundary, such as arising in the squirmer model of a microswimmer. It can be used to construct solutions for transient flows driven by initial conditions, unbounded flows driven by volume forces or disturbance to the unsteady flow due to a stationary spherical particle. The long-time behavior of solution is controlled by the flow component corresponding to average (or collective) motion of the boundary. This conclusion is illustrated by the study of decay of transversal wave in the presence of a fixed sphere. We further show that the general representation reduces to the well-known solutions for unsteady flow around a sphere undergoing oscillatory rigid-body (translation and rotation) motion. The proposed solution representation provides an explicit form of the velocity potential far from an oscillating body ("generalized" Darcy's law) and high- and low-frequency expansions. The leading-order high-frequency expansion yields the well-known ideal (inviscid) flow approximation, and the leading-order low-frequency expansion yields the steady Stokes equations. We derive the higher-order corrections to these approximations and discuss d'Alembert paradox. Continuation of the general solution to imaginary frequency yields the general solution of the Brinkman equations describing viscous flow in porous medium., Comment: 30 pages, revised version

Neural networks mapping sequences to sequences (seq2seq) lead to significant progress in machine translation and speech recognition. Their traditional architecture includes two recurrent networks (RNs) followed by a linear predictor. In this manuscript we perform analysis of a corresponding algorithm and show that the parameters of the RNs of the well trained predictive network are not independent of each other. Their dependence can be used to significantly improve the network effectiveness. The traditional seq2seq algorithms require short term memory of a size proportional to the predicted sequence length. This requirement is quite difficult to implement in a neuroscience context. We present a novel memoryless algorithm for seq2seq predictive networks and compare it to the traditional one in the context of time series prediction. We show that the new algorithm is more robust and makes predictions with higher accuracy than the traditional one., Comment: 8 pages, 4 figures

In the recent publication (arxiv:2007.08063v2 [cs.LG]) a fast prediction algorithm for a single recurrent network (RN) was suggested. In this manuscript we generalize this approach to a chain of RNs and show that it can be implemented in natural neural systems. When the network is used recursively to predict sequence of values the proposed algorithm does not require to store the original input sequence. It increases robustness of the new approach compared to the standard moving/expanding window predictive procedure. We consider requirements on trained networks that allow to implement the proposed algorithm and discuss them in the neuroscience context., Comment: 10 pages, 8 figures

Recent research demonstrate that prediction of time series by recurrent neural networks (RNNs) based on the noisy input generates a smooth anticipated trajectory. We examine the internal dynamics of RNNs and establish a set of conditions required for such behavior. Based on this analysis we propose a new approximate algorithm and show that it significantly speeds up the predictive process without loss of accuracy., Comment: 15 pages, 10 figures; typos corrected; the notation table removed; an appendix added

The seminal Batchelor-Green's (BG) theory on the hydrodynamic interaction of two spherical particles of radii a suspended in a viscous shear flow neglects the effect of the boundaries. In the present paper we study how a plane wall modifies this interaction. Using an integral equation for the surface traction we derive the expression for the particles' relative velocity as a sum of the BG's velocity and the term due to the presence of a wall at finite distance, z_0. Our calculation is not the perturbation theory of the BG solution, so the contribution due to the wall is not necessarily small. The distance at which the wall significantly alters the particles interaction scales as z_0^{3/5}. The phase portrait of the particles' relative motion is different from the BG theory, where there are two singly-connected regions of open and closed trajectories both of infinite volume. For finite z_0, there is a new domain of closed (dancing) and open (swapping) trajectories. The width of this region behaves as 1/z_0. Along the swapping trajectories, that have been previously observed numerically, the incoming particle is turning back after the encounter with the reference particle, rather than passing it by, as in the BG theory. The region of dancing trajectories has infinite volume. We found a one-parameter family of equilibrium states, overlooked previously, whereas the pair of spheres flows as a whole without changing its configuration. These states are marginally stable and their perturbation yields a two-parameter family of the dancing trajectories, where the particle is orbiting around a fixed point in a frame co-moving with the reference particle. We suggest that the phase portrait obtained at z_0>>a is topologically stable and can be extended down to rather small z_0 of several particle diameters. We confirm this by direct numerical simulations of the Navier-Stokes equations with z_0=5a., Comment: 24 pages, 13 figures

Steering of magnetic nano-/microhelices by a rotating magnetic field is considered as a promising technique for controlled navigation of tiny objects through viscous fluidic environments. It has been recently demonstrated that simple geometrically achiral planar structures can also be steered efficiently. Such planar propellers are interesting for practical reasons, as they can be mass-fabricated using standard micro/nanolithography techniques. While planar magnetic structures are prone to in-plane magnetization, under the effect of an in-plane rotating magnetic field, they exhibit, at most, propulsion due to spontaneous symmetry breaking, i.e., they can propel either parallel or anti-parallel to the rotation axis of the field depending on their initial orientation. Here we demonstrate that actuation by a conically rotating magnetic field (i.e., superposition of in-plane rotating field and constant field orthogonal to it) can yield efficient unidirectional propulsion of planar and magnetized in-plane structures. In particular, we found that a highly symmetrical V-shape magnetized along its symmetry axis which exhibits no net propulsion in in-plane rotating field, exhibits unidirectional in-sync propulsion with a constant (frequency-independent) velocity when actuated by the conically rotating field.

We study stability of axisymmetric liquid bridges between two axisymmetric solid bodies in the absence of gravity under arbitrary asymmetric perturbations which are expanded into a set of angular Fourier modes. We determine the stability region boundary for every angular mode in case of both fixed and free contact lines. Application of this approach allows us to demonstrate existence of stable convex nodoid menisci between two spheres., Comment: 32 pages, 13 figures

We present English translation of the classical article of Hermann Amadeus Schwarz (1843--1921) "Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume" which was published in 1884, in Proceedings of the K"onigliche Gesellschaft der Wissenschaften and the Georg-Augusts-Universit"at, G"ottingen. We preserved the author notations throughout the text and tried to follow his grammar construction of the sentences., Comment: 10 pages

We consider a coexistence of two axisymmetric liquid bridges LB_i and LB_m of two immiscible liquids i and m which are immersed in a third liquid (or gas) e and trapped between two smooth solid bodies with axisymmetric surfaces S_1,S_2 and free contact lines. Evolution of liquid bridges allows two different configurations of LB_i and LB_m with multiple (five or three) interfaces of non-smooth shape. We formulate a variational problem with volume constraints and present its governing equations supplemented by boundary conditions. We find a universal relationship between curvature of the interfaces and discuss the Young relation at the singular curve where all liquids meet together., Comment: 14 pages, 4 Figured

Based on the Weierstrass representation of second variation we develop a non-spectral theory of stability for isoperimetric problem with minimized and constrained two-dimensional functionals of general type and free endpoints allowed to move along two given planar curves. We apply this theory to the axisymmetric pendular ring between two solid bodies without gravity to determine the stability of menisci with free contact lines. For catenoid and cylinder menisci and different solid shapes we determine the stability domain. The other menisci (unduloid, nodoid and sphere) are considered in a simple setup between two plates. We find the existence conditions of stable unduloid menisci with and without inflection points., Comment: 28 pages, 10 Figures

We present the theory of liquid bridges between two axisymmetric solids, sphere and plane, with prescribed contact angles in a general setup, when the solids are non-touching, touching or intersecting, We give a detailed derivation of expressions for curvature, volume and surface area of pendular ring as functions of the filling angle \psi for all available types of menisci: catenoid Cat, sphere Sph, cylinder Cyl, nodoid Nod and unduloid Und (the meridional profile of the latter may have inflection points). The Young-Laplace equation with boundary conditions can be viewed as a nonlinear eigenvalue problem. Its unduloid solutions, menisci shapes z_n^s(r) and their curvatures H_n^s(\psi), exhibit a discrete spectrum and are enumerated by two indices: the number n of inflection points on the meniscus meridional profile M and the convexity index s=\pm 1 determined by the shape of a segment of M contacting the solid sphere: the shape is either convex, s=1, or concave, s=-1. For the fixed contact angles the set of the functions H_n^s(\psi) behaves in such a way that in the plane (\psi,H) there exists a bounded domain where H_n^s(\psi) do not exist for any distance between solids. The curves H_n^s(\psi) may be tangent to the boundary of domain which is a smooth closed curve. This topological representation allows to classify possible curves and introduce a saddle point notion. We observe several types of saddle points, and give their classification., Comment: 58 pages, 12 Figures

The relations between the Bernoulli and Eulerian polynomials of higher order and the complete Bell polynomials are found that lead to new identities for the Bernoulli and Eulerian polynomials and numbers of higher order. General form of these identities is considered and generating function for polynomials satisfying this general identity is found., Comment: 9 pages

We consider transformations of deterministic and random signals governed by simple dynamical mappings. It is shown that the resulting signal can be a random process described in terms of fractal distributions and fractal domain integrals. In typical cases a steady state satisfies a dilatation equation, relating an unknown function $f(x)$ to $f(\kappa x)$ (for example, ${f(x)=g(x)f(\kappa x)}$). We discuss simple linear models as well as nonlinear systems with chaotic behavior including dissipative circuits with delayed feedback., Comment: 10 pages, one figure

Following the ideas of L. Carlitz we introduce a generalization of the Bernoulli and Eulerian polynomials of higher order to vectorial index and argument. These polynomials are used for computation of the vector partition function $W({\bf s},{\bf D})$, i.e., a number of integer solutions to a linear system ${\bf x} \ge 0, {\bf D x} = {\bf s}$. It is shown that $W({\bf s},{\bf D})$ can be expressed through the vector Bernoulli polynomials of higher order., Comment: 18 pages

The explicit formulas for the sums of positive powers of the integers $s_i$ unrepresentable by the triple of integers $d_1,d_2,d_3\in {\mathbb N}, \gcd(d_1,d_2,d_3)=1$, are derived., Comment: 6 pages

Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called Sylvester waves) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli polynomials of higher order with periodic coefficients., Comment: 8 pages, submitted to The Ramanujan Journal

A new method for constructing of composite coherent states of the hydrogen atom, based on the dynamical group approach and various schemes of reduction to subgroups, is presented. A wide class of well-localized (Gaussian) hydrogenic wave packets for circular and elliptic orbits is found using the saddle-point method., Comment: 6 pages

Explicit expressions for restricted partition function $W(s,{\bf d}^m)$ and its quasiperiodic components $W_j(s,{\bf d}^m)$ (called {\em Sylvester waves}) for a set of positive integers ${\bf d}^m = \{d_1, d_2, ..., d_m\}$ are derived. The formulas are represented in a form of a finite sum over Bernoulli and Euler polynomials of higher order with periodic coefficients. A novel recursive relation for the Sylvester waves is established. Application to counting algebraically independent homogeneous polynomial invariants of the finite groups is discussed., Comment: 15 pages, 2 figures, references added, submitted to The Ramanujan Journal

A new recursive procedure for calculation of restricted partition function is suggested. An explicit formula for the restricted partition function is found based on this procedure., Comment: 11 pages, LaTeX, no figures; errors corrected, new results added

The profiles of a spreading wetting film are computed taking into account intermolecular forces and introducing a kinetic slip condition at a molecular cut-off distance. This eliminates the stress singularity, so that both "true" and "visible" contact angles are defined unequivocally. The "true" contact angle at the cut-off distance depends on the slip length as well as on the edge propagation speed, but not on gravity or asymptotic inclination angle. These macroscopic factors influence, however, the "visible" contact angle observed in the interval where the actual film profile departs from the intermediate asymptotic curve.

A new recursive procedure of the calculation of partition numbers function $W(s,{\bf d}^m)$ is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of $W(s,{\bf d}^m)$ and their periods $\tau(G)$ for the irreducible Coxeter groups and a list for the first ten symmetric group ${\cal S}_m$ are presented. A {\it least common multiple} ${\cal L}(m)$ of the series of the natural numbers 1,2,..,$m$ plays a role of the period $\tau({\cal S}_m)$ of $W(s,{\bf d}^m)$ in ${\cal S}_m$. An asymptotic behaviour of ${\cal L}(m)$ with $m \to \infty$ is found., Comment: One figure, 21 pages, submitted to Quart. J. Math