Dichotomy for the Hausdorff dimension of the set of nonergodic directions

Given an irrational 0<λ<1, we consider billiards in the table P _λ formed by a \(\tfrac{1}{2}\times1\) rectangle with a horizontal barrier of length \(\frac{1-\lambda}{2}\) with one end touching at the midpoint of a vertical side. Let NE?(P _λ ) be the set of θ such that the flow on P _λ in direction θ is not ergodic. We show that the Hausdorff dimension of NE?(P _λ ) can only take on the values 0 and \(\tfrac{1}{2}\), depending on the summability of the series \(\sum_{k}\frac{\log\log q_{k+1}}{q_{k}}\) where {q _k } is the sequence of denominators of the continued fraction expansion of λ. More specifically, we prove that the Hausdorff dimension is \(\frac{1}{2}\) if this series converges, and 0 otherwise. This extends earlier results of Boshernitzan and Cheung.     

Extendability of Contractible Configurations for Nowhere-Zero Flows and Modulo Orientations

Let H be a connected graph and G be a supergraph of H. It is trivial that for any k-flow (D, f) of G, the restriction of (D, f) on the edge subset E(G / H) is a k-flow of the contracted graph G / H. However, the other direction of the question is neither trivial nor straightforward at all: for any k-flow \((D',f')\) of the contracted graph G / H, whether or not the supergraph G admits a k-flow (D, f) that is consistent with \((D',f')\) in the edge subset E(G / H). In this paper, we will investigate contractible configurations and their extendability for integer flows, group flows, and modulo orientations. We show that no integer flow contractible graphs are extension consistent while some group flow contractible graphs are also extension consistent. We also show that every modulo \((2k+1)\)-orientation contractible configuration is also extension consistent and there are no modulo (2k)-orientation contractible graphs.     

Stability and convergence of a fully discrete local discontinuous Galerkin method for multi-term time fractional diffusion equations

In this paper, a fully discrete local discontinuous Galerkin method for a class of multi-term time fractional diffusion equations is proposed and analyzed. Using local discontinuous Galerkin method in spatial direction and classical L1 approximation in temporal direction, a fully discrete scheme is established. By choosing the numerical flux carefully, we prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^2?α ), where k, h, and Δt are the degree of piecewise polynomial, the space, and time step sizes, respectively. Numerical examples are carried out to illustrate the effectiveness of the numerical scheme.     

Simulation and study of stratified flows around finite bodies

The flows past a sphere and a square cylinder of diameter d moving horizontally at the velocity U in a linearly density-stratified viscous incompressible fluid are studied. The flows are described by the Navier–Stokes equations in the Boussinesq approximation. Variations in the spatial vortex structure of the flows are analyzed in detail in a wide range of dimensionless parameters (such as the Reynolds number Re = Ud/ν and the internal Froude number Fr = U/(Nd), where ν is the kinematic viscosity and N is the buoyancy frequency) by applying mathematical simulation (on supercomputers of Joint Supercomputer Center of the Russian Academy of Sciences) and three-dimensional flow visualization. At 0.005 < Fr < 100, the classification of flow regimes for the sphere (for 1 < Re < 500) and for the cylinder (for 1 < Re < 200) is improved. At Fr = 0 (i.e., at U = 0), the problem of diffusion-induced flow past a sphere leading to the formation of horizontal density layers near the sphere’s upper and lower poles is considered. At Fr = 0.1 and Re = 50, the formation of a steady flow past a square cylinder with wavy hanging density layers in the wake is studied in detail.     

The Shape of Orthogonal Cycles in Three Dimensions

Let σ be a directed cycle whose edges have each been assigned a desired direction in 3D (East, West, North, South, Up, or Down) but no length. We say that σ is a shape cycle. We consider the following problem. Does there exist an orthogonal representation Γ of σ in 3D space such that no two edges of Γ intersect except at common endpoints and such that each edge of Γ has the direction specified in σ? If the answer is positive, we say that σ is simple. This problem arises in the context of extending orthogonal graph drawing techniques from 2D to 3D. We give a combinatorial characterization of simple shape cycles that yields linear time recognition and drawing algorithms.     

A note on the shrinking sector problem for surfaces of variable negative curvature

Given the universal cover ? for a compact surface V of variable negative curvature and a point x? ₀ ∈ ?, we consider the set of directions \({\widetilde v_0} \in {S_{\widetilde {{x_0}}}}\widetilde V\) for which a narrow sector in the direction ?, and chosen to have unit area, contains exactly k points from the orbit of the covering group. We can consider the size of the set of such ? in terms of the induced measure on \({S_{{{\widetilde x}_0}}}\widetilde V\) by any Gibbs measure for the geodesic flow. We show that for each k the size of such sets converges as the sector grows narrower and describe these limiting values. The proof involves recasting a similar result by Marklof and Vinogradov, for the particular case of surfaces of constant curvature and the volume measure, by using the strong mixing property for the geodesic flow, relative to the Gibbs measure.     

Quantization of Donaldson’s heat flow over projective manifolds

Consider E a holomorphic vector bundle over a projective manifold X polarized by an ample line bundle L. Fix k large enough, the holomorphic sections \(H^0(E\otimes L^k)\) provide embeddings of X in a Grassmanian space. We define the balancing flow for bundles as a flow on the space of projectively equivalent embeddings of X. This flow can be seen as a flow of algebraic type hermitian metrics on E. At the quantum limit \(k\rightarrow \infty \), we prove the convergence of the balancing flow towards the Donaldson heat flow, up to a conformal change. As a by-product, we obtain a numerical scheme to approximate the Yang–Mills flow in that context.     

Optimal linear estimator of origin-destination flows with redundant data

Suppose given a network endowed with a multiflow. We want to estimate some quantities connected with this multiflow, for instance the value of an s–t flow for one of the sources–sinks pairs s–t, but only measures on some arcs are available, at least on one s–t cocycle (set of arcs having exactly one endpoint in a subset X of vertices with s∈X and t?X). These measures, supposed to be unbiased, are random variables whose variances are known. How can we combine them optimally in order to get the best estimator of the value of the s–t flow?This question arises in practical situations when the OD matrix of a transportation network must be estimated. We will give a complete answer for the case when we deal with linear combinations, not only for the value of an s–t flow but also for any quantity depending linearly from the multiflow. Interestingly, we will see that the Laplacian matrix of the network plays a central role.     

Asymptotics of the Velocity Potential of an Ideal Fluid Flowing around a Thin Body

We consider the Neumann problem outside a small neighborhood of a planar disk in the three-dimensional space. The surface of this neighborhood is assumed to be smooth, and its thickness is characterized by a small parameter ε. A uniform asymptotic expansion of the solution of this problem with respect to ε is constructed by the matching method. Since the problem turned out to be bisingular, an additional inner asymptotic expansion in the so-called stretched variables is constructed near the edge of the disk. A physical interpretation of the solution of this boundary value problem is the velocity potential of a laminar flow of an ideal fluid around a thin body, which is the neighborhood of the disk. It is assumed that this flow has unit velocity at a large distance from the disk, which is equivalent to the following condition for the potential: u(x₁, x₂, x₃, ε) = x₃+O(r^?2) as r → ∞, where r is the distance to the origin. The boundary condition of this problem is the impermeability of the surface of the body: ?u/?n = 0 at the boundary. After subtracting x₃ from the solution u(x₁, x₂, x₃, ε), we get a boundary value problem for the potential ?(x₁, x₂, x₃, ε) of the perturbed motion. Since the integral of the function ??/?n over the surface of the body is zero, we have ?(x₁, x₂, x₃, ε) = O(r^?2) as r → ∞. Hence, all the coefficients of the outer asymptotic expansion with respect to ε have the same behavior at infinity. However, these coefficients have growing singularities at the approach to the edge of the disk, which implies the bisingularity of the problem.     

A new infeasible-interior-point algorithm for linear programming over symmetric cones

In this paper we present an infeasible-interior-point algorithm, based on a new wide neighbourhood N(τ₁, τ₂, η), for linear programming over symmetric cones. We treat the classical Newton direction as the sum of two other directions. We prove that if these two directions are equipped with different and appropriate step sizes, then the new algorithm has a polynomial convergence for the commutative class of search directions. In particular, the complexity bound is O(r^1.5logε^?1) for the Nesterov-Todd (NT) direction, and O(r²logε^?1) for the xs and sx directions, where r is the rank of the associated Euclidean Jordan algebra and ε > 0 is the required precision. If starting with a feasible point (x⁰, y⁰, s⁰) in N(τ₁, τ₂, η), the complexity bound is \(O\left( {\sqrt r \log {\varepsilon ^{ - 1}}} \right)\) for the NT direction, and O(rlogε^?1) for the xs and sx directions. When the NT search direction is used, we get the best complexity bound of wide neighborhood interior-point algorithm for linear programming over symmetric cones.     

<Emphasis Type="Italic">W</Emphasis>-entropy formulas on super Ricci flows and Langevin deformation on Wasserstein space over Riemannian manifolds

In this survey paper, we give an overview of our recent works on the study of the W-entropy for the heat equation associated with the Witten Laplacian on super-Ricci flows and the Langevin deformation on the Wasserstein space over Riemannian manifolds. Inspired by Perelman’s seminal work on the entropy formula for the Ricci flow, we prove the W-entropy formula for the heat equation associated with the Witten Laplacian on n-dimensional complete Riemannian manifolds with the CD(K,m)-condition, and the W-entropy formula for the heat equation associated with the time-dependent Witten Laplacian on n-dimensional compact manifolds equipped with a (K,m)-super Ricci flow, where K ∈ R and m ∈ [n,∞]. Furthermore, we prove an analogue of the W-entropy formula for the geodesic flow on the Wasserstein space over Riemannian manifolds. Our result improves an important result due to Lott and Villani (2009) on the displacement convexity of the Boltzmann-Shannon entropy on Riemannian manifolds with non-negative Ricci curvature. To better understand the similarity between above two W-entropy formulas, we introduce the Langevin deformation of geometric flows on the tangent bundle over the Wasserstein space and prove an extension of the W-entropy formula for the Langevin deformation. We also make a discussion on the W-entropy for the Ricci flow from the point of view of statistical mechanics and probability theory. Finally, to make this survey more helpful for the further development of the study of the W-entropy, we give a list of problems and comments on possible progresses for future study on the topic discussed in this survey.     

On homeomorphisms of the plane which have no fixed points

Homeomorphisms of the planeR ² onto itself are studied, subject to the restriction that they should preserve the sense of orientation and have no fixed points. The results of this investigation are then applied to the problem of determining which homeomorphisms can be embedded in flows, i.e., in one-parameter subgroups of the full homeomorphism group of the plane. A “free mapping” ofR ² onto itself is defined to be a homeomorphismT, without fixed points, such thatC∩TC=0 impliesC∩T ⁿ C=0 for alln≠0 wheneverC is a compact connected subset ofR ². Free mappings turn out to be just those homeomorphisms ofR ² onto itself that preserve orientation and have no fixed points. A fundamental property of free mappingsis the fact that ifT is a free mapping andA is any compact subset ofR ² then\(\mathop U\limits_{ - \infty }^{ + \infty } T^n A\) does not meet some unbounded connected subsetB ofR ². The proof of this theorem is lengthy, and will appear elsewhere. The theorem can be weakened by adding the extra assumption thatT be embedded in a flow; the proof of this weakened version is much easier, and is included in the present article. It is found that for an arbitrary free mappingT there exists a natural partition of the plane into a collection of “fundamental regions”, with the property that ifT is embedded in a flow then each of the fundamental regions is invariant under the flow. An example is given of a free mapping whose fundamental regions are bad enough so that the mapping cannot be embedded in a flow. It is proved, on the other hand, that if a free mappingT has just one fundamental region thenT is equivalent to a translation, i.e., there is a homeomorphismU ofR ² onto itself such thatUTU ^?1 is just the translation(x, y)→(x+1, y). Indeed,T is equivalent to a translation if and only ifT has just one fundamental region.     

On the hyperbolicity of flows

Let X be a C~1 vector field on a compact boundaryless Riemannian manifold M(dim M≥2),and A a compact invariant set of X.Suppose that A has a hyperbolic splitting,i.e.,T∧M = E~sX E~u with E~s uniformly contracting and E~u uniformly expanding.We prove that if,in addition,A is chain transitive,then the hyperbolic splitting is continuous,i.e.,A is a hyperbolic set.In general,when A is not necessarily chain transitive,the chain recurrent part is a hyperbolic set.Furthermore,we show that if the whole manifold M admits a hyperbolic splitting,then X has no singularity,and the flow is Anosov.     

Bounding the sum of powers of the Laplacian eigenvalues of graphs

For a non-zero real number α, let s _α (G) denote the sum of the αth power of the non-zero Laplacian eigenvalues of a graph G. In this paper, we establish a connection between s _α (G) and the first Zagreb index in which the Hölder’s inequality plays a key role. By using this result, we present a lot of bounds of s _α (G) for a connected (molecular) graph G in terms of its number of vertices (atoms) and edges (bonds). We also present other two bounds for s _α (G) in terms of connectivity and chromatic number respectively, which generalize those results of Zhou and Trinajsti? for the Kirchhoff index [B Zhou, N Trinajsti?. A note on Kirchhoff index, Chem. Phys. Lett., 2008, 455: 120–123].     

Analysis of a new finite difference/local discontinuous Galerkin method for the fractional Cattaneo equation

In this paper, we first present a new finite difference scheme to approximate the time fractional derivatives, which is defined in the sense of Caputo, and give a semidiscrete scheme in time with the truncation error O((Δt)^3?α ), where Δt is the time step size. Then a fully discrete scheme based on the semidiscrete scheme for the fractional Cattaneo equation in which the space direction is approximated by a local discontinuous Galerkin method is presented and analyzed. We prove that the method is unconditionally stable and convergent with order O(h ^k+1 + (Δt)^3?α ), where k is the degree of piecewise polynomial. Numerical examples are also given to confirm the theoretical analysis.     

About the spectrum of Nagata rings

In this paper, we deal with the Nagata ring R(n) in case R is obtained by a (T, I, D) construction. We characterize when R(n) is a strong S-domain and catenarian. This study allows us to provide several interesting applications and examples.     

Absolutely continuous spectrum of Stark operators

We prove several new results on the absolutely continuous spectra of perturbed one-dimensional Stark operators. First, we find new classes of perturbations, characterized mainly by smoothness conditions, which preserve purely absolutely continuous spectrum. Then we establish stability of the absolutely continuous spectrum in more general situations, where imbedded singular spectrum may occur. We present two kinds of optimal conditions for the stability of absolutely continuous spectrum: decay and smoothness. In the decay direction, we show that a sufficient (in the power scale) condition is |q(x)|≤C(1+|x|)^?1/4?ε; in the smoothness direction, a sufficient condition in Hölder classes isq∈C^1/2+ε(R). On the other hand, we show that there exist potentials which both satisfy |q(x)|≤C(1+|x|)^?1/4 and belong toC^1/2(R) for which the spectrum becomes purely singular on the whole real axis, so that the above results are optimal within the scales considered.     

Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

In the paper, a formula to calculate the probability that a random segment L(ω, u) in R ⁿ with a fixed direction u and length l lies entirely in the bounded convex body D ? R ⁿ (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ? D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ? D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.     

Capacity inverse minimum cost flow problems under the weighted Hamming distance

Given a network N(V, A, u, c) and a feasible flow \(x^0\), the inverse minimum cost flow problem is to modify the capacity vector or the cost vector as little as possible to make \(x^0\) form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the capacity inverse minimum cost flow problems under the weighted Hamming distance, where we use the weighted Hamming distance to measure the modification of the arc capacities. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in \(O(n\cdot m^2)\) time.     

Polar factorization of conformal and projective maps of the sphere in the sense of optimal mass transport

Let M be a compact Riemannian manifold and let μ, d be the associated measure and distance on M. Robert McCann, generalizing results for the Euclidean case by Yann Brenier, obtained the polar factorization of Borel maps S: M → M pushing forward μ to a measure ν: each S factors uniquely a.e. into the composition S = T ? U, where U: M → M is volume preserving and T: M → M is the optimal map transporting μ to ν with respect to the cost function d²/2.In this article we study the polar factorization of conformal and projective maps of the sphere S ⁿ . For conformal maps, which may be identified with elements of O_o(1, n+1), we prove that the polar factorization in the sense of optimal mass transport coincides with the algebraic polar factorization (Cartan decomposition) of this Lie group. For the projective case, where the group GL+(n + 1) is involved, we find necessary and sufficient conditions for these two factorizations to agree.