double phase problems with variable growth abstract we consider a class of double phase variational integrals driven by nonhomogeneous potentials we study the associated euler equation and we highlight the existence of two different rayleigh quotients one of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues the notion of eigenvalue is understood in the sense of pairs of nonlinear operators as introduced by fucik necas soucek and soucek the analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the p x laplace operator the generalized mean curvature operator or the capillarity differential operator with variable exponent the results contained in this paper complement the pioneering contributions of marcellini mingione et al in the field of variational integrals with unbalanced growth

cyclic self dual codes it is shown that if the automorphism group of a binary self dual code satisfies a certain condition then the code contains words of weight congruent to 2 modulo 4 in particular no cyclic binary self dual code can have all its weights divisible by four the number of cyclic binary self dual codes of length n is determined and the shortest nontrivial code in this class is shown to have length 14

an improved 2d mof method by using high order derivatives abstract the mof moment of fluid method is one of the most accurate approaches among various interface reconstruction algorithms alike other second order methods the mof method needs to solve an implicit optimization problem to obtain the optimal approximate interface so an iteration process is inevitable under most circumstances in order to solve the optimization efficiently the properties of the objective function are worthy of studying in 2d problems the first order derivative has been deduced and applied in the previous researches in this paper the high order derivatives of the objective function are deduced on the convex polygon we show that the n th n 2 order derivatives are discontinuous and the number of the discontinuous points is two times the number of the polygon edge a rotation algorithm is proposed to successively calculate these discontinuous points thus the target interval where the optimal solution is located can be determined since the high order derivatives of the objective function are continuous in the target interval the iteration schemes based on high order derivatives can be used to improve the convergence rate moreover when iterating in the target interval the value of objective function and its derivatives can be directly updated without explicitly solving the volume conservation equation the direct update makes a further improvement of the efficiency especially when the number of edges of the polygon is increasing the halley s method which is based on the first three order derivatives is applied as the iteration scheme in this paper and the numerical results indicate that the cpu time is about half of the previous method on the quadrilateral cell and is about one sixth on the decagon cell

moment of fluid interface reconstruction method in multi material arbitrary lagrangian eulerian mmale algorithms a new lagrangian interface evolution method to use in a multi material ale mmale algorithm is proposed the input data set for the method consists of the volume fractions after the lagrangian step and the orientation position of the interfaces before this step in the present work the moment of fluid mof method is used at the end of the ale cycle to provide required interface information for this method compared to traditional reconstruction methods in addition to the volume fractions the proposed method incorporates the configuration of the interfaces in the calculations the effect of compressibility is also considered in prioritizing the materials the capability of the method is assessed by testing its potential to track interfaces through large topology change in the domains which contain two or more materials numerical results for these tough test problems provide enough evidence for the accuracy and robustness of the proposed method

a pocket guide to nonlinear differential equations in musielak orlicz spaces abstract the musielak orlicz setting unifies variable exponent orlicz weighted sobolev and double phase spaces they inherit technical difficulties resulting from general growth and inhomogeneity in this survey we present an overview of developments of the theory of existence of pdes in the setting including reflexive and non reflexive cases as well as isotropic and anisotropic ones particular attention is paid to problems with data below natural duality in absence of lavrentiev s phenomenon

regularity for double phase variational problems we prove sharp regularity theorems for minimisers of a class of variational integrals whose integrand switches between two different types of degenerate elliptic phases according to the zero set of a modulating coefficient a cdot the model case is given by the functional w mapsto int dw p a x dw q rm d x where q p and a cdot geqq 0

growth conditions and regularity for weak solutions to nonlinear elliptic pdes abstract we describe some aspects of the process approach to interior regularity of weak solutions to a class of nonlinear elliptic equations in divergence form as well as of minimizers of integrals of the calculus of variations in particular with respect to the growth conditions we emphasize some differences between solving equations and minimizing energy integrals and also between x dependence or x independence

nonlinear elliptic systems with general growth we prove local lipschitz continuity and as a consequence c k and c regularity of weak solutions u for a class of nonlinear elliptic differential systems of the formn 1 xi a i du 0 1 2 m the growth conditions on the dependence of functions a i on the gradient du are so mild to allow us to embrace growths between the linear and the exponential cases and they are more general than those known in the literature

regularity results for a class of functionals with non standard growth we consider the integral functional under non standard growth assumptions that we call p x type namely we assume that a relevant model case being the functional under sharp assumptions on the continuous function p x 1 we prove regularity of minimizers energies exhibiting this growth appear in several models from mathematical physics

on the minimal distance of binary self dual cyclic codes in this paper an explicit construction of binary self dual cyclic codes of length n going to infinity with a minimal distance at least half the square root of n is presented the same idea is also used to construct more general binary cyclic codes with a large minimal distance finally in the special case of self dual cyclic codes a simplified version of a proof by conway and sloane is given showing an upper bound for the distance of binary self dual codes

nonlocal harnack inequalities abstract we state and prove a general harnack inequality for minimizers of nonlocal possibly degenerate integro differential operators whose model is the fractional p laplacian

nonlinear wave equations as limits of convex minimization problems proof of a conjecture by de giorgi we prove a conjecture by de giorgi which states that global weak solutions of nonlinear wave equations such as w jwj p 2 w 0 can be obtained as limits of functions that minimize suitable functionals of the calculus of variations these functionals which are integrals in space time of a convex lagrangian contain an exponential weight with a parameter and the initial data of the wave equation serve as boundary conditions as tends to zero the minimizers v converge up to subsequences to a solution of the nonlinear wave equation there is no restriction on the nonlinearity exponent and the method is easily extended to more general equations

some plane curvature approximations second order accurate approximations to the curvature function along a sufficiently smooth plane curve are presented the curve being given in finite form and thus approximately by n 2 points taken along its full length the curvature estimates are continuous and invariant under translation and rotation and they are based on local information so are easy to implement computationally in particular second order accurate estimates of surface tension forces halfway between immediate neighbors in the curve s mesh can thereby be made for hydrodynamic simulations

the second order sharpening of blurred smooth borders the problem is to approximate with local second order accuracy the smooth boundary separating a black and a white region in the plane given discretely located gray values associated with a blurring of that border second order here is with respect to the size ital h of the scale of the prescribed blurring the locally determined approximations are line segments the algorithms discussed here can result in second order accuracy but they may not in certain geometric circumstances typical local curvature estimates based on adjacent line segments do not converge but atypical one does consideration of a class of scaled blurrings leads to a type of blurring of borders which is particularly easy for a computer to undo locally yielding a line which is locally second order accurate some extensions to three and more dimensions are appended

construction of an extremal self dual code of length 62 we construct an extremal singly even self dual 62 31 12 code the existence of such a self dual code was not previously known combined with known results the above code also determines all lengths for which there are self dual codes with minimum weight 12

lattice boltzmann model for free surface flow and its application to filling process in casting a generalized lattice boltzmann model to simulate free surface is constructed in both two and three dimensions the proposed model satisfies the interfacial boundary conditions accurately a distinctive feature of the model is that the collision processes is carried out only on the points occupied partially or fully by the fluid to maintain a sharp interfacial front the method includes an anti diffusion algorithm the unknown distribution functions at the interfacial region are constructed according to the first order chapman enskog analysis the interfacial boundary conditions are satisfied exactly by the coefficients in the chapman enskog expansion the distribution functions are naturally expressed in the local interfacial coordinates the macroscopic quantities al the interface are extracted from the least square solutions of a locally linearized system obtained from the known distribution functions the proposed method does not require any geometric front construction and is robust for any interfacial topology simulation results of realistic filling process are presented rectangular cavity in two dimensions and hammer box campbell box sheffield box and motorblock in three dimensions to enhance the stability at high reynolds numbers various upwind type schemes are developed free slip and no slip boundary conditions are also discussed

on repeated root cyclic codes a parity check matrix for a q ary repeated root cyclic code is derived using the hasse derivative then the minimum distance of a q ary repeated root cyclic code is expressed in terms of the minimum distance of a certain simple root cyclic code with the help of this result several binary repeated root cyclic codes of lengths up to n 62 are shown to contain the largest known number of codewords for their given length and minimum distance the relative minimum distance d sub min n of q ary repeated root cyclic codes of rate r or r is proven to tend to zero as the largest multiplicity of a root of the generator g x increases to infinity it is further shown that repeated root cycle codes cannot be asymptotically better than simple root cyclic codes

existence of parabolic minimizers on metric measure spaces abstract the objects of our studies are vector valued parabolic minimizers u associated to a convex caratheodory integrand f obeying a p growth assumption from below and a certain monotonicity condition in the gradient variable here the functions being considered are defined on a metric measure space x d for such parabolic minimizers that coincide with a time independent cauchy dirichlet datum u 0 on the parabolic boundary of a space time cylinder 0 t with an open subset x and t 0 we prove existence in the parabolic newtonian space l p 0 t n 1 p r n in this paper we generalize results from bogelein et al 2014 2015 to the metric setting and argue completely on a variational level

a volume of fluid method based on multidimensional advection and spline interface reconstruction a new volume of fluid method for tracking two dimensional interfaces is presented the method involves a multidimensional advection algorithm based on the use of edge matched flux polygons to integrate the volume fraction evolution equation and a spline based reconstruction algorithm the accuracy and efficiency of the proposed method are analyzed using different tests and the results are compared with those obtained recently by other authors despite its simplicity the proposed method represents a significant improvement and compares favorably with other volume of fluid methods as regards the accuracy and efficiency of both the advection and reconstruction steps

regularity results for vectorial minimizers of a class of degenerate convex integrals abstract we establish the higher differentiability and the higher integrability for the gradient of vectorial minimizers of integral functionals with p q growth conditions we assume that the non homogeneous densities are uniformly convex and have a radial structure with respect to the gradient variable only at infinity the results are obtained under a possibly discontinuous dependence on the spatial variable of the integrand

repeated root cyclic codes in the theory of cyclic codes it is common practice to require that n q 1 where n is the word length and f sub q is the alphabet it is shown that the even weight subcodes of the shortened binary hamming codes form a sequence of repeated root cyclic codes that are optimal in nearly all other cases one does not find good cyclic codes by dropping the usual restriction that n and q must be relatively prime this statement is based on an analysis for lengths up to 100 a theorem shows why this was to be expected but it also leads to low complexity decoding methods this is an advantage especially for the codes that are not much worse than corresponding codes of odd length it is demonstrated that a binary cyclic code of length 2n n odd can be obtained from two cyclic codes of length n by the well known mod u mod u v mod construction this leads to an infinite sequence of optimal cyclic codes with distance 4 furthermore it is shown that low complexity decoding methods can be used for these codes the structure theorem generalizes to other characteristics and to other lengths some comparisons of the methods using earlier examples are given

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

higher integrability for minimizers of integral functionals with p q growth abstract we prove higher integrability for minimizers u r n of integral functionals f du a x u dx where f satisfies a non standard growth condition of p q type z p f z l 1 z q p

pointwise bounds for minimizers of some anisotropic functionals abstract we consider variational integral functionals g x u x d u x d x where is a bounded open subset in r n and the integrand g x s f x b x s is not subjected to any growth condition from above neither satisfies any regularity assumption growing linearly with respect to the s variable their interest relies on the fact that they are strictly connected with optimal transport problems with congestion effects the aim of this paper is to find sufficient conditions on the boundary datum u in order to obtain global and explicit estimates for the solutions of the following minimization problem min g x u x d u x d x u u w 0 1 1

existence of evolutionary variational solutions via the calculus of variations abstract in this paper we introduce a purely variational approach to time dependent problems yielding the existence of global parabolic minimizers that is 0 t u t f x d u d x d t 0 t f x d u d d x d t whenever t 0 and c 0 0 t r n for the integrand f r n n 0 we merely assume convexity with respect to the gradient variable and coercivity these evolutionary variational solutions are obtained as limits of maps depending on space and time minimizing certain convex variational functionals in the simplest situation with some growth conditions on f the method provides the existence of global weak solutions to cauchy dirichlet problems of parabolic systems of the type t u div d f x d u 0 in 0

harnack inequalities for double phase functionals we prove a harnack inequality for minimisers of a class of non autonomous functionals with non standard growth conditions they are characterised by the fact that their energy density switches between two types of different degenerate phases

on a degenerate nonlocal parabolic equation with variable source abstract we study the dirichlet problem for the degenerate nonlocal parabolic equation u t div a u l 2 2 u b u l 2 2 u q x t 2 u f in q t where q t 0 t t 0 r n n 2 is a bounded domain with a sufficiently smooth boundary a b are real valued functions defined on r and q x t is a measurable function in q t with values in an interval q q 1 we show that under appropriate conditions on a b q the problem has a global in time solution and find conditions of uniqueness sufficient conditions for extinction in finite time or strict positivity of u t l 2 for all t 0 are found in the case f 0 the decay rates for u t l 2 are derived

transitive and self dual codes attaining the tsfasman vla spl breve dut 80 zink bound a major problem in coding theory is the question of whether the class of cyclic codes is asymptotically good in this correspondence as a generalization of cyclic codes the notion of transitive codes is introduced see definition 1 4 in section i and it is shown that the class of transitive codes is asymptotically good even more transitive codes attain the tsfasman vlabrevedut 80 zink bound over fq for all squares q l2 it is also shown that self orthogonal and self dual codes attain the tsfasman vlabrevedut 80 zink bound thus improving previous results about self dual codes attaining the gilbert varshamov bound the main tool is a new asymptotically optimal tower e0subee1 subee2sube middotmiddotmiddot of function fields over fq with q lscr2 where all extensions en e0 are galois

numerical methods for tracking interfaces an overview is given of special numerical methods for tracking discontinuous fronts and interfaces these methods include surface tracking methods based on connected marker points along the interface volume tracking methods that track the volume occupied by the solution regions bounded by the interfaces and moving mesh methods where the underlying mesh is aligned and moved with the interface the pros and cons of the current methods are discussed and a new method is proposed that overcomes some of the difficulties encountered in approximating equations with multiple interacting interfaces

a newton krylov solver for remapping based volume of fluid methods a new conservative remapping method is presented here for generating multi dimensional fluxes used for the time integration of hyperbolic equations through single stage unsplit advection a solution to the lagrangian mesh location is generated that converges the volumes of all remapped flux definitions to the volumes of equivalent one dimensional flux definitions at their corresponding cell faces the method is most feasible within localized subdomains of an eulerian solution domain the main utility of the method is the generation of fluxes in the vicinity of discontinuities and sharp gradients that does not introduce physically spuriously extrema while locally conserving mass in generating solutions to lagrangian mesh location by solving a nonlinear equation system the new remap requires the anchoring of the lagrangian mesh removal of degrees of freedom to make the problem definite the method is based on characteristic tracing along trajectories assumed to be linear this paper investigates the convergence and sensitivity of the lagrangian mesh solver to mesh and timestep size in assessing its merit for application in transient cfd codes for single stage unsplit advection volume of fluid methods the new remap suppresses the introduction of undershoots overshoots and wisps in combination with a conservative remapping phase the new method for fixing the location of the lagrangian mesh eliminates the need for heuristic redistribution or clipping procedures for spurious extrema removal

higher integrability for constrained minimizers of integral functionals with p q growth in low dimension abstract we prove higher summability for the gradient of minimizers of strongly convex integral functionals of the calculus of variations f x d u x d x u r n s n 1 with growth conditions of p q type p f x c q 1 p q in low dimension our procedure is set in the framework of fractional sobolev spaces and renders the desired regularity as the result of an approximation technique relying on estimates obtained through a careful use of difference quotients

nonuniformly elliptic energy integrals with p q growth abstract we study the local boundedness of minimizers of a nonuniformly energy integral of the form f x d v d x under p q growth conditions of the type x p f x x 1 q for some exponents q p 1 and with nonnegative functions satisfying some summability conditions we use here the original notation introduced in 1971 by trudinger 26 where x and x had the role of the minimum and the maximum eigenvalues of an n n symmetric matrix a i j x and f x i j 1 n a i j x i j was the energy integrand associated to a linear nonuniformly elliptic equation in divergence form in this paper we consider a class of energy integrals associated to nonlinear nonuniformly elliptic equations and systems with integrands f x satisfying the general growth conditions above

a variational approach to parabolic equations under general and p q growth conditions abstract we consider variational solutions to the cauchy dirichlet problem t u div d f x u d u d u f x u d u in t u u 0 on par t where the function f f x u f r n r n r n n 0 is convex with respect to u and coercive in r n n but it not necessarily satisfies a growth condition from above a motivation to consider a class of such energy functions f can be also easily found in the stationary case where a large literature in the calculus of variations is devoted to the minimization of p q growth problems 45 and to double phase problems 23 24 4 5 6 in the parabolic context the notion of variational solution see the references from 8 to 15 is compatible with the lack of the same polynomial growth from below and from above

two phase flows on interface refined grids modeled with vof staggered finite volumes and spline interpolants abstract a two phase 2d model that combines the volume of fluid vof method with implicit staggered finite volumes discretization of the navier stokes equation is presented staggered finite volumes are developed on the basis of nonconforming crouzeix raviart finite elements where all components of the velocity lie in the middle of the element edges and the pressure degrees of freedom are found in the centers of mass of the elements staggered finite volumes extend marker and cell mac regular staggered grids to unstructured mesh a linear saddle point problem resulting from either the discretization or the newton method is solved for all unknown pressures and velocities interface is represented with spline interpolants which follow the vof distribution adaptive mesh refinement is used to obtain a high level of uniform refining at the domain of dependence of the interface the aligned grid is obtained by irregular refining of the cells which are intersected by a curve the boundaries of its elements coincide with the slope segments going through the intersections of the curve with the underlying regular elements boundary the deformable computational grids are used only to discretize the navier stokes equation the advection of volume fractions is done on the advection mesh which corresponds to highest regular refining on the computational grid approximation of the surface tension on spline interpolants offers a straightforward way to describe correctly the pressure jumps on interface fitted staggered grids this allows deletion of the anomalous currents around a statical bubble and their effective reduction in real simulations on the aligned grid the continuity of the viscous stress is modeled exactly due to the finite volume approach using the proposed numerical techniques single bubble rise is analyzed

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

a second order accurate material order independent interface reconstruction technique for multi material flow simulations a new second order accurate volume conservative material order independent interface reconstruction method for multi material flow simulations is presented first materials are located in multi material computational cells using a piecewise linear reconstruction of the volume fraction function these material locator points are then used as generators to reconstruct the interface with a weighted voronoi diagram that matches the volume fractions the interfaces are then improved by minimizing an objective function that smoothes interface normals while enforcing convexity and volume constraints for the pure material subcells convergence tests are shown demonstrating second order accuracy static and dynamic examples are shown illustrating the superior performance of the method over existing material order dependent methods

an improved 3d mof method based on analytical partial derivatives abstract mof moment of fluid method is one of the most accurate approaches among various surface reconstruction algorithms as other second order methods mof method needs to solve an implicit optimization problem to obtain the optimal approximate surface therefore the partial derivatives of the objective function have to be involved during the iteration for efficiency and accuracy however to the best of our knowledge the derivatives are currently estimated numerically by finite difference approximation because it is very difficult to obtain the analytical derivatives of the object function for an implicit optimization problem employing numerical derivatives in an iteration not only increase the computational cost but also deteriorate the convergence rate and robustness of the iteration due to their numerical error in this paper the analytical first order partial derivatives of the objective function are deduced for 3d problems the analytical derivatives can be calculated accurately so they are incorporated into the mof method to improve its accuracy efficiency and robustness numerical studies show that by using the analytical derivatives the iterations are converged in all mixed cells with the efficiency improvement of 3 to 4 times

local calderon zygmund estimates for parabolic minimizers abstract we establish local calderon zygmund estimates for parabolic minimizers u to certain variational inequalities the formal corresponding parabolic partial differential equation would be t u div f z u d u u f z u d u div f p 2 f h with p 2 n n 2 however since we do not impose any condition on the existence of the derivative u f the pde above may not have any meaning at all for proving the calderon zygmund estimates we thus remain solely on the level of parabolic minimizers precisely for q p we prove that d u l loc q if u is locally holder continuous and f l loc q and h l loc q for a q depending on q

nonlocal diffusion equations abstract in this paper we present a variational approach to establish the existence and uniqueness of variational solutions to nonlocal evolutionary problems the model we have in mind is the following nonlocal version of the parabolic p laplacian u t div a d u t l p p d u p 2 d u h which has recently attracted many authors we emphasize that our approach also applies in situations where the considered functionals do not allow the derivation of the associated euler lagrange equation

parabolic systems with p q growth a variational approach we consider the evolution problem associated with a convex integrand f mathbb r nn to 0 infty satisfying a non standard p q growth assumption to establish the existence of solutions we introduce the concept of variational solutions in contrast to weak solutions that is mappings u colon omega t to mathbb r n which solve partial tu rm div df du 0 weakly in omega t variational solutions exist under a much weaker assumption on the gap q p here we prove the existence of variational solutions provided the integrand f is strictly convex and frac 2n n 2 p le q p 1 these variational solutions turn out to be unique under certain mild additional assumptions on the data moreover if the gap satisfies the natural stronger assumption 2 le p le q p rm min big 1 frac 4 n big we show that variational solutions are actually weak solutions this means that solutions u admit the necessary higher integrability of the spatial derivative du to satisfy the parabolic system in the weak sense that is we prove that u in l q rm loc big 0 t w 1 q rm loc omega mathbb r n big

a predicted newton s method for solving the interface positioning equation in the mof method on general polyhedrons abstract solving the interface positioning equation is the key procedure of the plic vof methods most of previous research only focused on the planar constant calculation and paid less attention to the relationship between the planar constant and the approximate interface orientation the latter issue is important especially for the second order iteration based plic vof method such as the mof and lvira method in these methods the most accurate interface orientation is calculated through an iterative procedure so the interface positioning equation has to be solved multiple times for the given volume fraction with different interface orientations in this situation if the incremental relation between the planar constant and the interface orientation is known a predicted planar constant can be estimated in this paper we deduce the analytical partial derivatives of the planar constant with respect to the interface orientation and use them to predict the planar constant a predicted newton s method is proposed to solve the interface positioning equation which takes the predicted planar constant as the initial guess a great deal of numerical tests are also presented in this paper to verify the robustness of the new scheme the efficiency of the proposed predicted newton s method is compared with the commonly used secant bisection method by ahn and shashkov and the numerical results indicate that the new method can reduce the iteration steps by 60 66 in solving the interface positioning equation and reduce the cpu time by 32 39 when implemented in the mof method

a new upper bound on the minimal distance of self dual codes it is shown that the minimal distance d of a binary self dual code of length n or 74 is at most 2 n 6 10 this bound is a consequence of some new conditions on the weight enumerator of a self dual code obtained by considering a particular translate of the code called its shadow these conditions also enable one to find the highest possible minimal distance of a self dual code for all n or 60 to show that self dual codes with d or 22 with d or 8 exist precisely for n 24 32 and n or 26 and with d or 10 exist precisely for n or 46 and to show that there are exactly eight self dual codes of length 32 with d 8 several of the self dual codes of length 34 have trivial group this appears to be the smallest length where this can happen