Exponential growth of the vorticity gradient for the Euler equation on the torus

We prove that there are solutions to the Euler equation on the torus with C^1,α$C^{1,\alpha }$ vorticity and smooth except at one point such that the vorticity gradient grows in L^∞$L^{\infty }$ at least exponentially as t→∞$t\to \infty$. The same result is shown to hold for the vorticity Hessian and smooth solutions. Our proofs use a version of a recent result by Kiselev and Šverák [6].     

Stream-vorticity NS equation ADI scheme to investigate properties of impulsively started translatory flows around the rotational circular cylinder without wall vorticity conditions

The properties of flow around a circular cylinder impulsively started into translatory and, rotatory motion with rotational parameter a less than or equal to 8.0 and Reynolds number Re=100 and 200 are investigated in the present paper. The vorticity and stream function N-S equations are adopted here, with a 2nd-order spatial and temporal accuracy ADI (alternating direction implicit) scheme. Moreover the wall vorticity obtain through the principle of conservation of the total computational domain vorticity is determined by domain vorticity and stream function, therefore, through the wall vorticity iteration, the wall vorticity condition is not fixed during the time step. And the present model results indicate: (1) when α>4.0, vortex street suppression is obvious for the computational period (t<60) for all the Re numbers here studied; (2) the higher the αnumber for the same Reynolds number, the slower the upper main vortex proceeds; (3) the maximum instantaneous transverse coefficient exceeds the limitation 4π.     

Unsteady response of non-Newtonian blood flow through a stenosed artery in magnetic field

Current theoretical investigation of atherosclerotic arteries deals with mathematical models that represent non-Newtonian flow of blood through a stenosed artery in the presence of a transverse magnetic field. Here, the rheology of the flowing blood is characterised by a generalised Power law model. The distensibility of an arterial wall has been accounted for based on local fluid mechanics. A radial coordinate transformation is initiated to map cosine geometry of the stenosis into a rectangular grid. An appropriate finite difference scheme has been adopted to solve the unsteady non-Newtonian momentum equations in cylindrical coordinate system. Exploiting suitably prescribed conditions based on the assumption of an axial symmetry under laminar flow condition rendered the problem effectively to two dimensions. An extensive quantitative analysis has been performed based on numerical computations in order to estimate the effects of Hartmann number (M$M$), Power law index (n$n$), generalised Reynolds number (R_eG)$\left(R{e}_G\right)$, severity of the stenosis (δ)$\left(\delta \right)$ on various parameters such as flow velocity, flux and wall shear stress by means of their graphical representations so as to validate the applicability of the proposed mathematical model. The present results agree with some of the existing findings in the literature.     

High resolution of 2D natural convection in a cavity by the DQ method

In this paper, the differential quadrature method (DQ) is applied to solve the benchmark problem of 2D natural convection in a cavity by utilizing the velocity–vorticity form of the Navier–Stokes equations, which is governed by the velocity Poisson equation, continuity equation and vorticity transport equation as well as energy equation. The coupled equations are simultaneously solved by imposing the vorticity definition at boundary without any iterative procedure. The present model is properly utilized to obtain results in the range of Rayleigh number (10³${10}^3$–10⁷${10}^7$) and H/L$H/L$ aspect ratios varying from 1 to 3. Nusselt numbers computed for 10³?Ra?10⁷${10}^3?\text{<italic>Ra</italic>}?{10}^7$ in a cavity show excellent agreement with the results available in the literature. Additionally, the detailed features of flow phenomena such as velocity, temperature, vorticity, and streamline plots are also delineated in this work. Thus, it is convinced that the DQ method is capable of solving coupled differential equations efficiently and accurately.     

On testing Hamiltonicity of graphs

Let us fix a function f(n)=o(nlnn)$f\left(n\right)=o(nlnn)$ and real numbers 0≤α<β≤1$0\le \alpha <\beta \le 1$. We present a polynomial time algorithm which, given a directed graph G$G$ with n$n$ vertices, decides either that one can add at most βn$\beta n$ new edges to G$G$ so that G$G$ acquires a Hamiltonian circuit or that one cannot add αn$\alpha n$ or fewer new edges to G$G$ so that G$G$ acquires at least e^−f(n)n!$e^{-f\left(n\right)}n!$ Hamiltonian circuits, or both.     

A structure theorem for small sumsets in nonabelian groups

Let G$G$ be an arbitrary finite group and let S$S$ and T$T$ be two subsets such that |S|≥2$\left|S\right|\ge 2$, |T|≥2$\left|T\right|\ge 2$, and |TS|≤|T|+|S|−1≤|G|−2$\left|TS\right|\le \left|T\right|+\left|S\right|-1\le \left|G\right|-2$. We show that if |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ then either S$S$ is a geometric progression or there exists a non-trivial subgroup H$H$ such that either |HS|≤|S|+|H|−1$\left|HS\right|\le \left|S\right|+\left|H\right|-1$ or |SH|≤|S|+|H|−1$\left|SH\right|\le \left|S\right|+\left|H\right|-1$. This extends to the nonabelian case classical results for abelian groups. When we remove the hypothesis |S|≤|G|−4|G|^1/2$\left|S\right|\le \left|G\right|-4{\left|G\right|}^{1/2}$ we show the existence of counterexamples to the above characterization whose structure is described precisely.     

[1, 2]-sets in graphs

A subset S⊆V$S\subseteq V$ in a graph G=(V,E)$G=(V,E)$ is a [j,k]$[j,k]$-set if, for every vertex v∈V?S$v\in V?S$, j≤|N(v)∩S|≤k$j\le |N\left(v\right)\cap S|\le k$ for non-negative integers j$j$ and k$k$, that is, every vertex v∈V?S$v\in V?S$ is adjacent to at least j$j$ but not more than k$k$ vertices in S$S$. In this paper, we focus on small j$j$ and k$k$, and relate the concept of [j,k]$[j,k]$-sets to a host of other concepts in domination theory, including perfect domination, efficient domination, nearly perfect sets, 2-packings, and k$k$-dependent sets. We also determine bounds on the cardinality of minimum [1, 2]-sets, and investigate extremal graphs achieving these bounds. This study has implications for restrained domination as well. Using a result for [1, 3]-sets, we show that, for any grid graph G$G$, the restrained domination number is equal to the domination number of G$G$.     

Isocapacity estimates for Hessian operators

Through a new powerful potential-theoretic analysis, this paper is devoted to discovering the geometrically equivalent isocapacity forms of Chou–Wang's Sobolev type inequality and Tian–Wang's Moser–Trudinger type inequality for the fully nonlinear 1≤k≤n/2$1\le k\le n/2$ Hessian operators.     

Potential operators associated with Jacobi and Fourier–Bessel expansions

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier–Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those 1≤p,q≤∞$1\le p,q\le \infty$, for which the potential operators are of strong type (p,q)$(p,q)$, of weak type (p,q)$(p,q)$ and of restricted weak type (p,q)$(p,q)$. These results may be thought of as analogues of the celebrated Hardy–Littlewood–Sobolev fractional integration theorem in the Jacobi and Fourier–Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier–Bessel expansions.     

The boundary element solution of the magnetohydrodynamic flow in an infinite region

We consider the magnetohydrodynamic (MHD) flow which is laminar, steady and incompressible, of a viscous and electrically conducting fluid on the half plane (y≥0)$(y\ge 0)$. The boundary y=0$y=0$ is partly insulated and partly perfectly conducting. An external circuit is connected so that current enters the fluid at discontinuity points through external circuits and moves up on the plane. The flow is driven by the interaction of imposed electric currents and a uniform, transverse magnetic field applied perpendicular to the wall, y=0$y=0$. The MHD equations are coupled in terms of the velocity and the induced magnetic field. The boundary element method (BEM) is applied here by using a fundamental solution which enables treating the MHD equations in coupled form with general wall conditions. Constant elements are used for the discretization of the boundary y=0$y=0$ only since the boundary integral equation is restricted to this boundary due to the regularity conditions at infinity. The solution is presented for the values of the Hartmann number up to M=700$M=700$ in terms of equivelocity and induced magnetic field contours which show the well-known characteristics of the MHD flow. Also, the thickness of the parabolic boundary layer propagating in the field from the discontinuity points in the boundary conditions, is calculated.     

Optimal decay rates of solutions for a multidimensional generalized Benjamin–Bona–Mahony equation

In this paper, we study the optimal decay rates of solutions for the generalized Benjamin–Bona–Mahony equation in multi-dimensional space (n≥3$n\ge 3$). By using Fourier transform and the energy method, we obtain the L^q(2≤q≤∞)$L^q(2\le q\le \infty )$ convergence rates of the solutions under the condition that the initial data is small. The optimal decay rates obtained in this paper are found to be the same as the decay rate for the Heat equation.     

A new upper bound on the acyclic chromatic indices of planar graphs

An acyclic edge coloring of a graph G$G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a^′(G)$a^{\prime }\left(G\right)$ of G$G$ is the smallest integer k$k$ such that G$G$ has an acyclic edge coloring using k$k$ colors. It was conjectured that a^′(G)≤Δ+2$a^{\prime }\left(G\right)\le \Delta +2$ for any simple graph G$G$ with maximum degree Δ$\Delta$. In this paper, we prove that if G$G$ is a planar graph, then a^′(G)≤Δ+7$a^{\prime }\left(G\right)\le \Delta +7$. This improves a result by Basavaraju et al. [M. Basavaraju, L.S. Chandran, N. Cohen, F. Havet, T. Müller, Acyclic edge-coloring of planar graphs, SIAM J. Discrete Math. 25 (2011) 463–478], which says that every planar graph G$G$ satisfies a^′(G)≤Δ+12$a^{\prime }\left(G\right)\le \Delta +12$.     

Cycle-maximal triangle-free graphs

We conjecture that the balanced complete bipartite graph _{K⌊n/2⌋,⌈n/2⌉}$K_{\lfloor n/2\rfloor ,\lceil n/2\rceil }$ contains more cycles than any other n$n$-vertex triangle-free graph, and we make some progress toward proving this. We give equivalent conditions for cycle-maximal triangle-free graphs; show bounds on the numbers of cycles in graphs depending on numbers of vertices and edges, girth, and homomorphisms to small fixed graphs; and use the bounds to show that among regular graphs, the conjecture holds. We also consider graphs that are close to being regular, with the minimum and maximum degrees differing by at most a positive integer k$k$. For k=1$k=1$, we show that any such counterexamples have n≤91$n\le 91$ and are not homomorphic to _C5$C_5$; and for any fixed k$k$ there exists a finite upper bound on the number of vertices in a counterexample. Finally, we describe an algorithm for efficiently computing the matrix permanent (a #P$\#P$-complete problem in general) in a special case used by our bounds.     

Second main theorem and uniqueness theorem with moving targets on parabolic manifolds

In this paper, with the motivation from Diophantine approximation, a truncated second main theorem is established for meromorphic maps from M   into P(V)$\mathbb{P}(V)$ with moving targets _gj:M→P(V^?)$g_j:M\to \mathbb{P}(V^{?})$, 1≤j≤q$1\le j\le q$, where M is a parabolic manifold and V is a Hermitian vector space. As an application of this second main theorem, a uniqueness theorem without counting multiplicities is given.     

On first-fit coloring of ladder-free posets

Bosek and Krawczyk exhibited an on-line algorithm for partitioning an on-line poset of width w$w$ into w^14lgw$w^{14lgw}$ chains. They also observed that the problem of on-line chain partitioning of general posets of width w$w$ could be reduced to First-Fit chain partitioning of 2w²+1$2w^2+1$-ladder-free posets of width w$w$, where an m$m$-ladder is the transitive closure of the union of two incomparable chains _x1≤?≤_xm$x_1\le ?\le x_m$, _y1≤?≤_ym$y_1\le ?\le y_m$ and the set of comparabilities {_x1≤_y1,…,_xm≤_ym}$\{x_1\le y_1,\dots ,x_m\le y_m\}$. Here, we provide a subexponential upper bound (in terms of w$w$ with m$m$ fixed) for the performance of First-Fit chain partitioning on m$m$-ladder-free posets, as well as an exact quadratic bound when m=2$m=2$, and an upper bound linear in m$m$ when w=2$w=2$. Using the Bosek–Krawczyk observation, this yields an on-line chain partitioning algorithm with a somewhat improved performance bound. More importantly, the algorithm and the proof of its performance bound are much simpler.     

Disjointly homogeneous rearrangement invariant spaces via interpolation

A Banach lattice E is called p-disjointly homogeneous  , 1≤p≤∞$1\le p\le \infty$, when every sequence of pairwise disjoint normalized elements in E   has a subsequence equivalent to the unit vector basis of _?p${?}_p$. Employing methods from interpolation theory, we clarify which r.i. spaces on [0,1]$[0,1]$ are p  -disjointly homogeneous. In particular, for every 1<p<∞$1<p<\infty$ and any increasing concave function φ   on [0,1]$[0,1]$, which is not equivalent to neither 1 nor t, there exists a p-disjointly homogeneous r.i. space with the fundamental function φ  . Moreover, it is shown that given 1<p<∞$1<p<\infty$ and an increasing concave function φ with non-trivial dilation indices, there is a unique p-disjointly homogeneous space among all interpolation spaces between the Lorentz and Marcinkiewicz spaces associated with φ.     

Quadrature rule for Abel’s equations: Uniformly approximating fractional derivatives

An automatic quadrature method is presented for approximating fractional derivative D^qf(x)$D^{q}f\left(x\right)$ of a given function f(x)$f\left(x\right)$, which is defined by an indefinite integral involving f(x)$f\left(x\right)$. The present method interpolates f(x)$f\left(x\right)$ in terms of the Chebyshev polynomials in the range [0, 1] to approximate the fractional derivative D^qf(x)$D^{q}f\left(x\right)$ uniformly for 0≤x≤1$0\le x\le 1$, namely the error is bounded independently of x$x$. Some numerical examples demonstrate the performance of the present automatic method.