arXiv subjects
Publications and source records attributed to M. Zyskin.
We obtain and investigate explicit analytic solution via universal transform of the diffusion equation in a spherical particles which appears in the so-called single particle model, a popular simple model of an electric battery.
The Doyle-Fuller-Newman framework is the most popular physics-based continuum-level description of the chemical and dynamical internal processes within operating lithium-ion-battery cells. With sufficient flexibility to model a wide range of battery designs and chemistries, the framework provides an effective balance between detail, needed to capture key microscopic mechanisms, and simplicity, needed to solve the governing equations at a relatively modest computational expense. Nevertheless, implementation requires values of numerous model parameters, whose ranges of applicability, estimation, and validation pose challenges. This article provides a critical review of the methods to measure or infer parameters for use within the isothermal DFN framework, discusses their advantages or disadvantages, and clarifies limitations attached to their practical application. Accompanying this discussion we provide a searchable database, available at www.liiondb.com, which aggregates many parameters and state functions for the standard Doyle-Fuller-Newman model that have been reported in the literature.
The compressive strength of unidirectional fiber composites in the form of Kevlar yarn with a thin outer layer of graphene was investigated and modeled. Such fiber structure may be fabricated by using a strong chemical bond between Kevlar yarn and graphene sheets. Chemical functionalization of graphene and Kevlar may achieved by modification of appropriate surface-bound functional (e.g., carboxylic acid) groups on their surfaces. In this report we studied elastic response to unidirectional in-plane applied load with load peaks along the diameter. The 2D linear elasticity model predicts that significant strengthening occurs when graphene outer layer radius is about 4 % of kevlar yarn radius. The polymer chains of Kevlar are linked into locally planar structure by hydrogen bonds across the chains, with transversal strength considerably weaker than longitudinal one. This suggests that introducing outer enveloping layer of graphene, linked to polymer chains by strong chemical bonds may significantly strengthen Kevlar fiber with respect to transversal deformations.
For nice functions, invariant means over integral currents (certain generalized surfaces), can be uniquely defined.
Let O be a closed geodesic polygon in S^2. Maps from O into S^2 are said to satisfy tangent boundary conditions if the edges of O are mapped into the geodesics which contain them. Taking O to be an octant of S^2, we compute the infimum Dirichlet energy, Ecal(H), for continuous maps satisfying tangent boundary conditions of arbitrary homotopy type H. The expression for Ecal(H) involves a topological invariant - the spelling length - associated with the (nonabelian) fundamental group of the n-times punctured two-sphere, pi_1(S^2 - {s_1,..., s_n},*). The lower bound for Ecal(H) is obtained from combinatorial group theory arguments, while the upper bound is obtained by constructing explicit representatives which, on all but an arbitrarily small subset of O, are alternatively locally conformal or anticonformal. For conformal and anticonformal classes (classes containing wholly conformal and anticonformal representatives respectively), the expression for Ecal(H) reduces to a previous result involving the degrees of a set of regular values s_1,..., s_n in the target S^2 space. These degrees may be viewed as invariants associated with the abelianization of π_1(S^2 - {s_1,..., s_n}, *). For nonconformal classes, however, Ecal(H) may be strictly greater than the abelian bound. This stems from the fact that, for nonconformal maps, the number of preimages of certain regular values may necessarily be strictly greater than the absolute value of their degrees. This work is motivated by the theoretical modelling of nematic liquid crystals in confined polyhedral geometries. The results imply new lower and upper bounds for the Dirichlet energy (one-constant Oseen-Frank energy) of reflection-symmetric tangent unit-vector fields in a rectangular prism.
We obtain complete topological classification of states of nematic liquid crystal in the geometry of periodic array of rectangular posts between two parallel slabs, with tangent or normal boundary conditions. This classification has applications in bi-stable pos-aligned liquid crystal display design and have technological significance. Methods used in classification are those of algebraic topology and go beyond relative homotopy groups.
We study nematic liquid crystal configurations in a prototype bistable device - the Post Aligned Bistable Nematic (PABN) cell. Working within the Oseen-Frank continuum model, we describe the liquid crystal configuration by a unit-vector field, in a model version of the PABN cell. Firstly, we identify four distinct topologies in this geometry. We explicitly construct trial configurations with these topologies which are used as initial conditions for a numerical solver, based on the finite-element method. The morphologies and energetics of the corresponding numerical solutions qualitatively agree with experimental observations and suggest a topological mechanism for bistability in the PABN cell geometry.
Nematic liquid crystals in a polyhedral domain, a prototype for bistable displays, may be described by a unit-vector field subject to tangent boundary conditions. Here we consider the case of a rectangular prism. For configurations with reflection-symmetric topologies, we derive a new lower bound for the one-constant elastic energy. For certain topologies, called conformal and anticonformal, the lower bound agrees with a previous result. For the remaining topologies, called nonconformal, the new bound is an improvement. For nonconformal topologies we derive an upper bound, which differs from the lower bound by a factor depending only on the aspect ratios of the prism.
We derive a lower bound for energies of harmonic maps of convex polyhedra in $ \R^3 $ to the unit sphere $S^2,$ with tangent boundary conditions on the faces. We also establish that $C^\infty$ maps, satisfying tangent boundary conditions, are dense with respect to the Sobolev norm, in the space of continuous tangent maps of finite energy.
It was suggested in hep-th/0002106, that semiclassically, a partition function of a string theory in the 5 dimensional constant negative curvature space with a boundary condition at the absolute satisfy the loop equation with respect to varying the boundary condition, and thus the partition function of the string gives the expectation value of a Wilson loop in the 4 dimensional QCD. In the paper, we present the geometrical framework, which reveals that the equations of motion of such string theory are integrable, in the sense that they can be written via a Lax pair with a spectral parameter. We also show, that the issue of the loop equation rests solely on the properly posing the boundary condition.
We constructed canonical non-highest weight unitary irreducible representation of $\hat{so}(1,n)$ current algebra as well as canonical non-highest weight non-unitary representations, We constructed certain Laplacian operators as elements of the universal enveloping algebra, acting in representation space. We speculated about a possible relation of those Laplacians with the loop operator for the Yang-Mills.
A conjectured duality between supergravity and $N=\infty$ gauge theories gives predictions for the glueball masses as eigenvalues for a supergravity wave equations in a black hole geometry, and describes a physics, most relevant to a high-temeperature expansion of a lattice QCD. We present an analytical solution for eigenvalues and eigenfunctions, with eigenvalues given by zeroes of a certain well-computable function $r(p)$, which signify that the two solutions with desired behaviour at two singular points become linearly dependent. Our computation shows corrections to the WKB formula $m^2= 6n(n+1)$ for eigenvalues corresponding to glueball masses QCD-3, and gives the first states with masses $m^2=$ 11.58766; 34.52698; 68.974962; 114.91044; 172.33171; 241.236607; 321.626549, ... . In $QCD_4$, our computation gives squares of masses 37.169908; 81.354363; 138.473573; 208.859215; 292.583628; 389.671368; 500.132850; 623.97315 ... for $O++$. In both cases, we have a powerful method which allows to compute eigenvalues with an arbitrary precision, if needed so, which may provide quantative tests for the duality conjecture. Our results matches with the numerical computation of [5] well withing precision reported there in both $QCD_3$ and $QCD_4$ cases. As an additional curiosity, we report that for eigenvalues of about 7000, the power series, although convergent, has coefficients of orders ${10}^{34}$; tricks we used to get reliably the function $r(p)$, as also the final answer gets small, of order ${10}^{-6}$ in $QCD_4$. In principle we can go to infinitely high eigenavalues, but such computations maybe impractical due to corrections.
We consider d-dimensional Riemanian manifolds which admit d-2 commuting space-like Killing vector fields, orthogonal to a surface, containing two one-parametric families of light-like curves. The condition of the Ricci tensor to be zero gives Ernst equations for the metric. We write explicitly a family of local solutions of this equations corresponding to arbitrary initial data on two characteristics in terms of a series. These metrics describe scattering of 2 gravitational waves, and thus we expect they are very interesting. Ernst equations can be written as equations of motion for some 2D Lagrangian, which governs fluctuations of the metric, constant in the Killing directions. This Lagrangian looks essentially as a 2D chiral field model, and thus is possibly treatable in the quantum case by standart methods. It is conceivable that it may describe physics of some specially arranged scattering experiment, thus giving an insight for 4D gravity, not treatable by standart quantum field theory methods. The renormalization flow for our Lagrangian is different from the flow for the unitary chiral field model, the difference is essentially due to the fact that here the field is taking values in a non-compact space of symmetric matrices. We investigate the model and derive the renormalized action in one loop.
We consider a real manifold of dimension 3 or 4 with Minkovsky metric, and with a connection for a trivial GL(n,C) bundle over that manifold. To each light ray on the manifold we assign the data of paralel transport along that light ray. It turns out that these data are not enough to reconstruct the connection, but we can add more data, which depend now not from lines but from 2-planes, and which in some sence are the data of parallel transport in the complex light-like directions, then we can reconstruct the connection up to a gauge transformation. There are some interesting applications of the construction: 1) in 4 dimensions, the self-dual Yang Mills equations can be written as the zero curvature condition for a pair of certain first order differential operators; one of the operators in the pair is the covariant derivative in complex light-like direction we studied. 2) there is a relation of this Radon transform with the supersymmetry. 3)using our Radon transform, we can get a measure on the space of 2 dimensional planes in 4 dimensional real space. Any such measure give rise to a Crofton 2-density. The integrals of this 2-density over surfaces in R^4 give rise to the Lagrangian for maps of real surfaces into R^4, and therefore to some string theory. 4) there are relations with the representation theory. In particular, a closely related transform in 3 dimensions can be used to get the Plancerel formula for representations of SL(2,R).
We construct quantum operators solving the quantum versions of the Sturm-Liouville equation and the resolvent equation, and show the existence of conserved currents. The construction depends on the following input data: the basic quantum field $O(k)$ and the regularization .