Arnold VJ - Academia.edu (original) (raw)

Papers by Arnold VJ

The bridge index and superbridge index of a knot are important invariants in knot theory. We defi... more The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of spherical curves as introduced by Arnold, we show that the graph of the bridge map is the union of the binormal indicatrix, its antipodal curve, and some number of great circles. Similarly, we define the inflection map of a knot conformation, interpret it in terms of the binormal indicatrix, and express its graph in terms of the tangent indicatrix. This duality relationship is also studied for another dual pair of curves, the normal and Darboux indicatrices of a knot conformation. The analogous concepts are defined and results are derived for stick knots.

Mathematics of Computation, 1997

We consider the solution of the system of linear algebraic equations which arises from the finite... more We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I − grad div. The natural setting for such problems is in the Hilbert space H (div) and the variational formulation is based on the inner product in H (div). We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.

Applied Optics, 2004

We study light propagation in biological tissue containing an absorbing obstacle. In particular, ... more We study light propagation in biological tissue containing an absorbing obstacle. In particular, we solve the infinite-domain problem in which an absorbing plate of negligible thickness prevents a portion of the light from the source from reaching the detector plane. Inasmuch as scattering in the medium is sharply peaked in the forward direction, we replace the governing radiative transport equation with the Fokker-Planck equation. The problem is solved first by application of the Kirchhoff approximation to determine the secondary source distribution over the surface of the plate. That result is propagated to the detector plane by use of Green's function. The Green's function is given as an expansion of plane-wave modes that are calculated numerically. The radiance is shown to obey Babinet's principle. Results from numerical computations that demonstrate this theory are shown.

We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimens... more We study a class of permutation-symmetric globally-coupled, phase oscillator networks on N-dimensional tori. We focus on the efiects of symmetry and of the forms of the coupling functions, derived from un- derlying Hodgkin-Huxley type neuron models, on the existence, stability, and degeneracy of phase-locked solutions in which subgroups of oscillators share common phases. We also estimate domains of attraction for

Following the Arnold-Marsden-Ebin approach, we prove local (global in 2-D) existence and uniquene... more Following the Arnold-Marsden-Ebin approach, we prove local (global in 2-D) existence and uniqueness of classical (Hölder class) solutions of stochastic Euler equation with random forcing.

International Immunology, 2005

We examined the generation and selection of the B cell antibody repertoire through crossing of mi... more We examined the generation and selection of the B cell antibody repertoire through crossing of mice bearing distinct Ig heavy (H) and light (L) chain rearranged variable region transgenes. Ig gene knockin and transgenic mice whose H and L chains pair to form a non-autoreactive, functional B cell antigen receptor (BCR) have significantly reduced pre-B cells in the bone marrow as their B cell progenitors rapidly differentiate into surface IgM 1 B cells. The presence of a pre-B cell compartment in these Ig transgenic mice, however, indicates the induction of receptor editing. Here, 18 distinct combinations of H and L chains were generated that we showed could pair in vitro to form BCRs of unknown specificities. Of these, nine induced receptor editing in vivo as evidenced by the presence of pre-B cells and endogenous L chain rearrangements in mice bearing these H and L chain transgenes. These data thus suggest that about half of the emerging antibody repertoire is negatively selected during B lymphopoiesis due to the likely encoding of autoreactive or non-functional BCRs. Fig. 4. Stromal cell cultures of bone marrow B cells derived from mice transgenic for certain H and 3-83j L chains. Figure shows the B220 versus CD43 profiles of the surface IgM-negative cells present in the stromal cell cultures. Numbers indicate the percentage of pro-B and pre-B cells as a fraction of total cultured cells recovered that are gated to exclude the ST2 stromal cells.

This is the second part of an article in two parts, which builds the foundation of a Floer-theore... more This is the second part of an article in two parts, which builds the foundation of a Floer-theoretic invariant, I_F. (See math.DG/0111313 for part I). Having constructed I_F and outlined a proof of its invariance based on bifurcation analysis in part I, in this part we prove a series of gluing theorems to confirm the bifurcation behavior predicted in part I. These gluing theorems are different from (and much harder than) the more conventional versions in that they deal with broken trajectories or broken orbits connected at degenerate rest points. The issues of orientation and signs are also settled in the last section.

In the last years, applying wavelets analysis has called the attention in a wide variety of pract... more In the last years, applying wavelets analysis has called the attention in a wide variety of practical problems, in particular for the numerical solutions of partial differential equations using different methods, as finite differences, semi-discrete techniques or the finite element method. In the construction of wavelet-based elements, instead of traditional polynomial interpolation, scaling and wavelet functions have been adopted to form the shape function to construct elements. Due to their properties, wavelets are very useful when it is necessary to approximate efficiently the solution on non-regular zones. Furthermore, in some cases it is convenient to use the Daubechies wavelet, which has properties of orthogonality and minimum compact support, and provides guaranty of convergence and accuracy of the approximation in a wide variety of situations. The aim of this research is to explore the Galerkin method using wavelets to solve plate bending problems. Some numerical examples, with B-splines and Daubechies, are presented and show the feasibility of our proposal.

Transformation Groups, 2009

We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous... more We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces SO(n)/SO(k1) ×⋯× SO(k r ), for any choice of k 1,…,k r , k 1 + ⋯ + k r ⩽ n. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented. Also, the proof of integrability of the SO(n)-invariant Einstein metrics on SO(k 1 + k 2 + k 3)/SO(k 1) × SO(k 2) × SO(k 3) and on the Stiefel manifolds V (n, k) = SO(n)/SO(k) is given.

Practically all existing approaches to structure and motion computation use only positive image c... more Practically all existing approaches to structure and motion computation use only positive image correspondences to verify the camera pose hypotheses. Incorrect epipolar geometries are solely detected by identifying outliers among the found correspondences. Ambigous patterns in the images are often incorrectly handled by these standard methods. In this work we propose two approaches to overcome such problems. First, we apply non-monotone reasoning on view triplets using a Bayesian formulation. In contrast to two-view epipolar geometry, image triplets allow the prediction of features in the third image. Absence of these features (i.e. missing correspondences) enables additional inference about the view triplet. Furthermore, we integrate these view triplet handling into an incremental procedure for structure and motion computation. Thus, our approach is able to refine the maintained 3D structure when additional image data is provided.

In this article, we prove that there exists at least one chord which is characteristic of Reeb ve... more In this article, we prove that there exists at least one chord which is characteristic of Reeb vector field connecting a given Legendre submanifold in a closed contact manifold with any contact form.

Transparent boundary conditions (TBCs) for general Schr6dinger-type equations on a bounded domain... more Transparent boundary conditions (TBCs) for general Schr6dinger-type equations on a bounded domain can be derived explicitly under the assumption that the given potential V is constant on the exterior of that domain. In 1D these boundary conditions are nonlocal in time (of memory type).

Communications in Mathematical Physics, 2003

Let (M,g) be a C ∞ compact Riemann manifold with classical Hamiltonian, HC ∞ (T * M). Assume th... more Let (M,g) be a C ∞ compact Riemann manifold with classical Hamiltonian, HC ∞ (T * M). Assume that the corresponding -quantization P 1 :=Op  (H) is quantum completely integrable. We establish an -microlocal Weyl law on short spectral intervals of size 2−ε;∀ε>0 for various families of operators P 1 u ;uI containing P 1 , both in the mean and pointwise a.e. for uI. The -microlocalization refers to a small tubular neighbourhood of a non-degenerate, stable periodic bicharacteristic γ⊂T * M−0.

We consider words over the alphabet [k] = {1, 2, . . . , k}, k ≥ 2. For a fixed nonnegative integ... more We consider words over the alphabet [k] = {1, 2, . . . , k}, k ≥ 2. For a fixed nonnegative integer p, a p-succession in a word w 1 w 2 · · · w n consists of two consecutive letters of the form (w i , w i + p), i = 1, 2, . . . , n − 1. We analyze words with respect to a given number of contained p-successions. First we find the mean and variance of the number of p-successions. We then determine the distribution of the number of p-successions in words of length n as n (and possibly k) tends to infinity; a simple instance of a phase transition (Gaussian-Poisson-degenerate) is encountered. Finally we also investigate successions in compositions of integers.

We start with a mini-survey on some problems of pseudoperiodic topology.