Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

We consider a nematic liquid crystal occupying the exterior region in \({\mathbb {R}}^3\) outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to an external field, modeled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough, we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.     

Order and chaos in Hofstadter's Q(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2^k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2^αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/n^α, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of x_m = R(n) − R(n − m) is given by p_m(x_m) = λ_m p*(x_m/λ_m), with λ_m → √2 for large m. © 1999 John Wiley & Sons, Inc.     

Refined approximation for minimizers of a Landau-de Gennes energy functional

We study minimizers of a Landau-de Gennes energy functional in the asymptotic regime of small dimensionless elastic constant L > 0. The results on the convergence to a minimizer of the limit Oseen-Frank functional in Majumdar and Zarnescu (Arch Ration Mech Anal 196:227–280, 2010) are revisited and improved, which in effect lead to a sharp rate of convergence. The equation for the first-order correction term is derived: it has a “normal component” given by an algebraic relation and a “tangential component” given by a linear system.     

Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for L-harmonic forms

We compute the spectra of the Tanaka type Laplacians on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S²ⁿ⁻¹ in ⁿ. We prove that Szegö map is a unitary operator from a subspace of (p, q −1)-forms on the sphere defined by the operators and the normal vector field onto the space of L²-harmonic (p, q)-forms on the unit ball. Our results generalize earlier result of Folland.     

The image inH 2(Q 3;ℝ) of the set of closed 2-forms with preassigned kernel

If (P²ⁿ, ω) is a symplectic manifold and Q³ is its orientable closed submanifold such that ω/Q³≠0, then there arises a one-dimensional distribution ￡=(ω/Q³). We study the dependence of ω in a neighborhood of Q³ and of [ω] ɛH ²(Q ³,ℝ) on ￡. Bibliography: 13 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 125–136.     

Optimal lower bounds for cubature error on the sphere

We show that the worst-case cubature error E(Q_m;H^s) of an m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s=H^s(S²),s>1, has the lower bound , where the constant c_s is independent of Q_m and m. This lower bound result is optimal, since we have established in previous work that there exist sequences of cubature rules for which with a constant independent of n. The method of proof is constructive: given the cubature rule Q_m, we construct explicitly a ‘bad’ function f_mH^s, which is a function for which Q_mf_m=0 and . The construction uses results about packings of spherical caps on the sphere.     

Riesz External Field Problems on the Hypersphere and Optimal Point Separation

We consider the minimal energy problem on the unit sphere ?? ^d in the Euclidean space ? ^d+1 in the presence of an external field Q, where the energy arises from the Riesz potential 1/r ^s (where r is the Euclidean distance and s is the Riesz parameter) or the logarithmic potential log(1/r). Characterization theorems of Frostman-type for the associated extremal measure, previously obtained by the last two authors, are extended to the range d ? 2 ≤ s < d ? 1. The proof uses a maximum principle for measures supported on ?? ^d . When Q is the Riesz s-potential of a signed measure and d ? 2 ≤ s < d, our results lead to explicit point-separation estimates for (Q,s)-Fekete points, which are n-point configurations minimizing the Riesz s-energy on ?? ^d with external field Q. In the hyper-singular case s > d, the short-range pair-interaction enforces well-separation even in the presence of more general external fields. As a further application, we determine the extremal and signed equilibria when the external field is due to a negative point charge outside a positively charged isolated sphere. Moreover, we provide a rigorous analysis of the three point external field problem and numerical results for the four point problem.     

New regular embeddings of n‐cubes Qn

We investigate highly symmetrical embeddings of the n‐dimensional cube Q_n into orientable compact surfaces which we call regular embeddings of Q_n. We derive some general results and construct some new families of regular embeddings of Q_n. In particular, for n = 1, 2,3(mod 4), we classify the regular embeddings of Q_n which can be expressed as balanced Cayley maps. © 2004 Wiley Periodicals, Inc. J Graph Theory 46: 297–312, 2004     

Cubature over the sphere in Sobolev spaces of arbitrary order

This paper studies numerical integration (or cubature) over the unit sphere for functions in arbitrary Sobolev spaces H^s(S²), s>1. We discuss sequences of cubature rules, where (i) the rule Q_m(n) uses m(n) points and is assumed to integrate exactly all (spherical) polynomials of degree ≤n and (ii) the sequence (Q_m(n)) satisfies a certain local regularity property. This local regularity property is automatically satisfied if each Q_m(n) has positive weights. It is shown that for functions in the unit ball of the Sobolev space H^s(S²), s>1, the worst-case cubature error has the order of convergence O(n^-s), a result previously known only for the particular case . The crucial step in the extension to general s>1 is a novel representation of , where P_ℓ is the Legendre polynomial of degree ℓ, in which the dominant term is a polynomial of degree n, which is therefore integrated exactly by the rule Q_m(n). The order of convergence O(n^-s) is optimal for sequences (Q_m(n)) of cubature rules with properties (i) and (ii) if Q_m(n) uses m(n)=O(n²) points.     

Cutting a Convex Polyhedron Out of a Sphere

Given a convex polyhedron P of n vertices inside a sphere Q, we give an O(n ³)-time algorithm that cuts P out of Q by using guillotine cuts and has cutting cost O(log² n) times the optimal.     

On quadrilaterals in layers of the cube and extremal problems for directed and oriented graphs

Erdős has conjectured that every subgraph of the n‐cube Q_n having more than (1/2 + o(1))e(Q_n) edges will contain a 4‐cycle. In this note we consider ‘layer’ graphs, namely, subgraphs of the cube spanned by the subsets of sizes k − 1, k and k + 1, where we are thinking of the vertices of Q_n as being the power set of {1,…, n}. Observe that every 4‐cycle in Q_n lies in some layer graph. We investigate the maximum density of 4‐cycle free subgraphs of layer graphs, principally the case k = 2. The questions that arise in this case are equivalent to natural questions in the extremal theory of directed and undirected graphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 33: 66–82, 2000     

Approximation Numbers of Operators on Normed Linear Spaces

In [1], B?ttcher et. al. showed that if T is a bounded linear operator on a separable Hilbert space H, {e_j}_j=1^￥H, \{e_{j}\}_{j=1}^{\infty} is an orthonormal basis of H and P_n is the orthogonal projection onto the span of {e_j}_j=1ⁿ\{e_{j}\}_{j=1}^{n}, then for each k ? \mathbbNk \in {\mathbb{N}}, the sequence {s_k(P_nTP_n)}\{s_{k}(P_{n}TP_{n})\} converges to s_k(T), where for a bounded operator A on H, s_k(A) denotes the kth approximation number of A, that is, s_k(A) is the distance from A to the set of all bounded linear operators of rank at most k − 1. In this paper we extend the above result to more general cases. In particular, we prove that if T is a bounded linear operator from a separable normed linear space X to a reflexive Banach space Y and if {P_n} and {Q_n} are sequences of bounded linear operators on X and Y, respectively, such that ||P_n|| ||Q_n|| ￡ 1\|P_n\| \|Q_n\| \leq 1 for all n ? \mathbbNn \in {\mathbb{N}} and {Q_nTP_n} converges to T under the weak operator topology, then {s_k(Q_nTP_n)}\{s_{k}(Q_{n}TP_{n})\} converges to s_k(T). We also obtain a similar result for the case of any normed linear space Y which is the dual of some separable normed linear space. For compact operators, we give this convergence of s_k(Q_nTP_n)s_{k}(Q_{n}TP_{n}) to s_k(T) with separability assumptions on X and the dual of Y. Counter examples are given to show that the results do not hold if additional assumptions on the space Y are removed. Under separability assumptions on X and Y, we also show that if there exist sequences of bounded linear operators {P_n} and {Q_n} on X and Y respectively such that (i) Q_nTP_nQ_{n}TP_{n} is compact, (ii) ||P_n|| ||Q_n|| ￡ 1\|P_{n}\| \|Q_{n}\| \leq 1 and (iii) {Q_nTP_n}\{Q_{n}TP_{n}\} converges to T in the weak operator topology, then {s_k(Q_nTP_n)}\{s_k(Q_{n}TP_{n})\} converges to s_k(T) if and only if s_k(T) = s_k(T￠)s_{k}(T) = s_{k}(T^\prime). This leads to a generalization of a result of Hutton [3], proved for compact operators between normed linear spaces.     

On the conductance of order Markov chains

Let Q be a convex solid in ⁿ , partitioned into two volumes u and v by an area s. We show that s>min(u,v)/diam Q, and use this inequality to obtain the lower bound n ^-5/2 on the conductance of order Markov chains, which describe nearly uniform generators of linear extensions for posets of size n. We also discuss an application of the above results to the problem of sorting of posets.Computing Center of the USSR Academy of Sciences USSR     

Approximation power of RBFs and their associated SBFs: a connection

Error estimates for scattered data interpolation by “shifts” of a conditionally positive definite function (CPD) for target functions in its native space, which is its associated reproducing kernel Hilbert space (RKHS), have been known for a long time. Regardless of the underlying manifold, for example ℝⁿ or S ⁿ, these error estimates are determined by the rate of decay of the Fourier transform (or Fourier series) of the CPD. This paper deals with the restriction of radial basis functions (RBFs), which are radial CPD functions on ℝⁿ⁺¹, to the unit sphere S ⁿ. In the paper, we first strengthen a result derived by two of us concerning an explicit representation of the Fourier–Legendre coefficients of the restriction in terms of the Fourier transform of the RBF. In addition, for RBFs that are related to completely monotonic functions, we derive a new integral representation for these coefficients in terms of the measure generating the completely monotonic function. These representations are then utilized to show that if an RBF has a native space equivalent to a Sobolev space H _s(ℝⁿ⁺¹), then the restriction to S ⁿ has a native space equivalent to H _s−1/2(S ⁿ). In addition, they are used to recover the asymptotic behavior of such coefficients for a wide variety of RBFs. Some of these were known earlier. Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation.     

EMBEDDINGS OF ONE KIND OF GRAPHS

The purpose of this paper is to display a new kind of simple graphs which belong to B. inwhich any graph has its orientable genus n,n≥3. Furthermore, for any integer k,1≤k≤n,there exists a graph B^kn of B. such that the non-orientable genus of B^kn is k.     

The pauli principle,stability, and bound states in systems of identical pseudorelativistic particles

Based on analyzing the properties of the Hamiltonian of a pseudorelativistic system Z_n of n identical particles, we establish that for actual (short-range) interaction potentials, there exists an infinite sequence of integers n_s, s = 1, 2, …, such that the system is stable and that sup _s n_s+1 n_s⁻¹ < + ∞. For a stable system Z_n, we show that the Hamiltonian of relative motion of such a system has a nonempty discrete spectrum for certain fixed values of the total particle momentum. We obtain these results taking the permutation symmetry (Pauli exclusion principle) fully into account for both fermion and boson systems for any value of the particle spin. Similar results previously proved for pseudorelativistic systems did not take permutation symmetry into account and hence had no physical meaning. For nonrelativistic systems, these results (except the estimate for n_s+1 n_s⁻¹ ) were obtained taking permutation symmetry into account but under certain assumptions whose validity for actual systems has not yet been established. Our main theorem also holds for nonrelativistic systems, which is a substantial improvement of the existing result. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 157, No. 1, pp. 116–129, October, 2008.     

Local Solutions of Semilinear Wave Equations in Hs+1

The Cauchy problem for the wave equation with power type non-linearity ±∣u∣u and data in H^s+1(ℝⁿ)×H^s(ℝⁿ) is considered, where 0<s<(n/2)−1 and n≥3. Under the growth restriction σ_*⩽4/(n−2−2s) in many cases the existence of a local solution with u(t)∈H^s+1(ℝⁿ) is shown which is unique in a closely related class.     

Mean projections and finite packings of convex bodies

Consider the convex hullQ ofn non-overlapping translates of a convex bodyC inE ^d ,n be large. IfQ has minimali-dimensional projection, 1i<d then we prove thatQ is approximately a sphere.     

Reduced Landau-de Gennes functional and surface smectic state of liquid crystals

In [X.B. Pan, Landau-de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1-2) (2003) 343-382], based on the de Gennes analogy between liquid crystals and superconductivity [P.G. de Gennes, An analogy between superconductors and smectics A, Solid State Commun. 10 (1972) 753-756], the second author introduced the critical wave number Qc₃ (which is an analog of the upper critical field Hc₃ for superconductors) and predicted the existence of a surface smectic state, which was supposed to be an analogy of the surface superconducting state. In a surface smectic state, the bulk liquid crystal is in the nematic state, and a thin layer of smectic appears in a helical strip on the surface of the sample. In this paper we study an approximate form of the Landau-de Gennes model of liquid crystals, and examine the behavior of minimizers, in particular the boundary layer behavior. Our work shows the importance of the joint chirality constant qτ, which is the product of wave number q and chirality τ and also appears in the work of [P. Bauman, M. Calderer, C. Liu, D. Phillips, The phase transition between chiral nematic and smectic A^∗ liquid crystals, Arch. Rational Mech. Anal. 165 (2002) 161-186] and [X.B. Pan, Landau-de Gennes model of liquid crystals and critical wave number, Comm. Math. Phys. 239 (1-2) (2003) 343-382]. The joint chirality constant of a liquid crystal is useful to predict whether the liquid crystal is of type I or type II, and it is also useful to examine whether the liquid crystal is in a surface smectic state. The results in this paper suggest that a liquid crystal with large Ginzburg-Landau parameter κ and large joint chirality constant qτ exhibits type II behavior, and it will be in the surface smectic state if qτ∼bκ² for some β₀<b<1, where β₀ is the lowest eigenvalue of the Schrödinger operator with a unit magnetic field in the half space, and 0<β₀<1. We also show that a liquid crystal with small qτ exhibits type I behavior.     

A lower bound for the worst-case cubature error on spheres of arbitrary dimension

This paper is concerned with numerical integration on the unit sphere S^r of dimension r≥2 in the Euclidean space ℝ^r+1. We consider the worst-case cubature error, denoted by E(Q_m;H^s(S^r)), of an arbitrary m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s(S^r), where s>, and show that The positive constant c_s,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space H^s(S^r). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Q_m a `bad' function f_m, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere S^r with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.