edge decompositions of graphs with high minimum degree abstract a fundamental theorem of wilson states that for every graph f every sufficiently large f divisible clique has an f decomposition here a graph g is f divisible if e f divides e g and the greatest common divisor of the degrees of f divides the greatest common divisor of the degrees of g and g has an f decomposition if the edges of g can be covered by edge disjoint copies of f we extend this result to graphs g which are allowed to be far from complete in particular together with a result of dross our results imply that every sufficiently large k 3 divisible graph of minimum degree at least 9 n 10 o n has a k 3 decomposition this significantly improves previous results towards the long standing conjecture of nash williams that every sufficiently large k 3 divisible graph with minimum degree at least 3 n 4 has a k 3 decomposition we also obtain the asymptotically correct minimum degree thresholds of 2 n 3 o n for the existence of a c 4 decomposition and of n 2 o n for the existence of a c 2 l decomposition where l 3 our main contribution is a general iterative absorption method which turns an approximate or fractional decomposition into an exact one in particular our results imply that in order to prove an asymptotic version of nash williams conjecture it suffices to show that every k 3 divisible graph with minimum degree at least 3 n 4 o n has an approximate k 3 decomposition

global regularity for the 2d mhd equations with mixed partial dissipation and magnetic diffusion abstract whether or not classical solutions of the 2d incompressible mhd equations without full dissipation and magnetic diffusion can develop finite time singularities is a difficult issue a major result of this paper establishes the global regularity of classical solutions for the mhd equations with mixed partial dissipation and magnetic diffusion in addition the global existence conditional regularity and uniqueness of a weak solution is obtained for the 2d mhd equations with only magnetic diffusion

proof of the bandwidth conjecture of bollobas and komlos in this paper we prove the following conjecture by bollobas and komlos for every 0 and integers r 1 and there exists 0 with the following property if g is a sufficiently large graph with n vertices and minimum degree at least r 1 r n and h is an r chromatic graph with n vertices bandwidth at most n and maximum degree at most then g contains a copy of h

global existence and decay of smooth solution for the 2 d mhd equations without magnetic diffusion abstract we prove the global existence and the decay estimates of small smooth solution for the 2 d mhd equations without magnetic diffusion this confirms the numerical observation that the energy of the mhd equations is dissipated at a rate independent of the ohmic resistivity

correcting for unknown errors in sparse high dimensional function approximation we consider sparsity based techniques for the approximation of high dimensional functions from random pointwise evaluations to date almost all the works published in this field contain some a priori assumptions about the error corrupting the samples that are hard to verify in practice in this paper we instead focus on the scenario where the error is unknown we study the performance of four sparsity promoting optimization problems weighted quadratically constrained basis pursuit weighted lasso weighted square root lasso and weighted lad lasso from the theoretical perspective we prove uniform recovery guarantees for these decoders deriving recipes for the optimal choice of the respective tuning parameters on the numerical side we compare them in the pure function approximation case and in applications to uncertainty quantification of odes and pdes with random inputs our main conclusion is that the lesser known square root lasso is better suited for high dimensional approximation than the other procedures in the case of bounded noise since it avoids both theoretically and numerically the need for parameter tuning

well posedness and dynamics of the stochastic fractional magneto hydrodynamic equations abstract the current paper is devoted to the well posedness and dynamics of the stochastic 2d incompressible fractional magneto hydrodynamic mhd equations driven by gaussian multiplicative noise the nonlocal fractional diffusion leads to a new difficulty in the convergence since higher order estimates cannot be obtained the commutator estimates are introduced to overcome these difficulties using the stopping time technique and monotonicity arguments the global existence and uniqueness of the weak solution are obtained in a fixed probability space finally the existence of a random attractor for the random dynamical systems generated by the solution of stochastic mhd equation is presented

global regularity for the 2d magnetic benard fluid system with mixed partial viscosity abstract inspired by the previous works the main goal of this paper is to establish the global well posedness and conditional regularity of the incompressible magnetic benard fluid system with mixed partial viscosity in two dimensions as a result our results generalize and extend the previous results

cycle decompositions v complete graphs into cycles of arbitrary lengths we show that the complete graph on n vertices can be decomposed into t cycles of specified lengths m1 mt if and only if n is odd 3 mi n for i 1 t and we also show that the complete graph on n vertices can be decomposed into a perfect matching and t cycles of specified lengths m1 mt if and only if n is even 3 mi n for i 1 t and m1 mt n2 n 2

hamilton decompositions of regular expanders a proof of kelly s conjecture for large tournaments abstract a long standing conjecture of kelly states that every regular tournament on n vertices can be decomposed into n 1 2 edge disjoint hamilton cycles we prove this conjecture for large n in fact we prove a far more general result based on our recent concept of robust expansion and a new method for decomposing graphs we show that every sufficiently large regular digraph g on n vertices whose degree is linear in n and which is a robust outexpander has a decomposition into edge disjoint hamilton cycles this enables us to obtain numerous further results e g as a special case we confirm a conjecture of erd s on packing hamilton cycles in random tournaments as corollaries to the main result we also obtain several results on packing hamilton cycles in undirected graphs giving e g the best known result on a conjecture of nash williams we also apply our result to solve a problem on the domination ratio of the asymmetric travelling salesman problem which was raised e g by glover and punnen as well as alon gutin and krivelevich

global regularity for the 2d incompressible mhd α system without full dissipation and magnetic diffusions abstract in this paper we consider the global regularity for the 2d incompressible mhd system without full dissipation and magnetic diffusions we establish the global regularity of classical solutions for the mhd system with mixed partial dissipation and magnetic diffusion 1 0 2 0 1 0 2 0 or horizontal dissipation and horizontal magnetic diffusion 1 0 2 0 1 0 2 0 as 0 the solution converges to that of the corresponding mhd equations we also obtain the convergence rate of the limit as 0 in l 2 norm

a fast approximation algorithm for computing thefrequencies of subgraphs in a given graph in this paper we give an algorithm which given a labeled graph on n vertices and a list of all labeled graphs on k vertices provides for each graph h of this list an approximation to the number of induced copies of h in g with total error small this algorithm has running time o n 1 over log log n cdot m n where m n is the time needed to square a n by n matrix with 0 1 entries over the integers the main tool in designing this algorithm is a variant of the regularity lemma of szemeredi

global regularity of the two dimensional incompressible generalized magnetohydrodynamics system abstract in this paper we consider the global regularity of the two dimensional generalized incompressible magnetohydrodynamics system with velocity dissipation exponent 1 4 and magnetic diffusion exponent 1 based on the improved integrability in terms of vorticity this result is a further improvement of very recent works jiu and zhao 2014 and yamazaki 2013 which require 1 3 and 1 therefore our result is more closer to the resolution of the well known global regularity issue on the 2d magnetohydrodynamics system 0 1

packing degenerate graphs abstract given d and 0 whenever c 0 is sufficiently small and n sufficiently large if g is a family of d degenerate graphs of individual orders at most n maximum degrees at most c n log n and total number of edges at most 1 n 2 then g packs into the complete graph k n our proof proceeds by analysing a natural random greedy packing algorithm

on the local well posedness and a prodi serrin type regularity criterion of the three dimensional mhd boussinesq system without thermal diffusion abstract we prove a prodi serrin type global regularity condition for the three dimensional magnetohydrodynamic boussinesq system 3d mhd boussinesq without thermal diffusion in terms of only two velocity and two magnetic components to the best of our knowledge this is the first prodi serrin type criterion for such a 3d hydrodynamic system which is not fully dissipative and indicates that such an approach may be successful on other systems in addition we provide a constructive proof of the local well posedness of solutions to the fully dissipative 3d mhd boussinesq system and also the fully inviscid irresistive non diffusive mhd boussinesq equations we note that as a special case these results include the 3d non diffusive boussinesq system and the 3d mhd equations moreover they can be extended without difficulty to include the case of a coriolis rotational term

remarks on pressure regularity criterion for the 3d boussinesq equations abstract this paper establishes two logarithmically improved regularity criteria for the cauchy problem of the 3d boussinesq equations in termsof the pressure on framework of the anisotropic lebesgue spaces more precisely it is proved that if 0 t p l x 1 p l x 2 q l x 3 r 1 ln 1 t l 4 d t where 2 1 p 1 q 1 r 2 and 2 p q r 1 1 p 1 q 1 r 0 or 0 t p l x 1 p l x 2 q l x 3 r 1 ln 1 t l 4 d t where 2 1 p 1 q 1 r 3 and 1 p q r 1 1 2 p 1 2 q 1 2 r 0 for some t 0 then the corresponding weak solution u to the 3d boussinesq equations is regular on 0 t

global weak solutions and regularity for two dimensional boussinesq equations with partial dissipation and thermal diffusivity abstract in this paper we establish the global weak solutions for two dimensional boussinesq equations with mixed partial dissipation and thermal diffusivity moreover we give global regularity criterion for weak solutions to the two dimensional boussinesq equations with mixed partial dissipation and thermal diffusivity

global existence and uniqueness theorem to 2 d incompressible non resistive mhd system with non equilibrium background magnetic field abstract in this article we consider the global existence and uniqueness of the solution to the 2d incompressible non resistive mhd system with non equilibrium background magnetic field our result implies that a strong enough non equilibrium background magnetic field will guarantee the stability of the nonlinear mhd system beside the classical energy method the interpolation inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments

the 2d mhd equations with horizontal dissipation and horizontal magnetic diffusion this paper studies the global regularity of classical solutions to the 2d incompressible magnetohydrodynamic mhd equations with horizontal dissipation and horizontal magnetic diffusion it is shown here that the horizontal component of any solution admits a global in time bound in any lebesgue space l2r with 1 r and the bound grows no faster than the order of rlogr as r increases in addition we establish a conditional global regularity in terms of the lt2lx norm of the horizontal component and the global regularity of a slightly regularized version of the aforementioned mhd equations

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references