a read once lower bound and a 1 p k hierarchy for branching programs branching programs b p s or decision diagrams are a general graph based model of sequential computation the b p s of polynomial size are a nonuniform counterpart of log lower bounds for different kinds of restricted b p s are intensively investigated an important restriction are the so called k b p s where each computation reads each input variable at most k times although exponential lower bounds have been proven for syntactic k b p s this is not true for general nonsyntactic k b p s even for k 2 therefore the so called 1 k b p s are investigated for some explicit functions exponential lower bounds for 1 k b p s are known we prove that the hierarchy of 1 k b p s w r t k is strict more exactly we present multipointer functions fn k which are polynomially easy for 1 k b p s but exponentially hard for 1 k 1 b p s for k 12n1 6 log1 3n this is a generalization of a similar result of sieling 20 for syntactic 1 k branching programs as a by product we prove a lower bound of 2n 3n for an explicit pointer function in p

the local langlands conjecture for gsp 4 we prove the local langlands conjecture for gsp4 f where f is a nonarchimedean local eld of characteristic zero

average and worst case number of nodes in decision diagrams of symmetric multiple valued functions we derive the average and worst case number of nodes in decision diagrams of r valued symmetric functions of n variables we show that for large n both numbers approach n sup r rl for binary decision diagrams r 2 we compute the distribution of the number of functions on n variables with a specified number of nodes subclasses of symmetric functions appear as features in this distribution for example voting functions are noted as having an average of n sup 2 6 nodes for large n compared to n sup 2 2 for general binary symmetric functions

hierarchy theorems for k obdds and k ibdds almost the same types of restricted branching programs or binary decision diagrams bdds are considered in complexity theory and in applications like hardware verification these models are read once branching programs free bdds and certain types of oblivious branching programs ordered and indexed bdds with k layers the complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed bdds of polynomial size

a lower bound on branching programs reading some bits twice by 1 k n branching programs b p s we mean those b p s which during each of their computations are allowed to test at most k n input bits repeatedly for a boolean function computable within polynomial time a trade off is presented between the size and the number of repeatedly tested input bits of any b p p computing the function namely if at most k n repeated tests are allowed where log2 n k n n 1000 log2 n then the size of p is at least exp n k n log2 n 12 this is exponential whenever k n n for a fixed 1 and superpolynomial whenever k n o nlog32 n the presented result is a step towards a superpolynomial lower bound for 2 b p s which is an open problem since 1984 when the first superpolynomial lower bounds for 1 b p s were proven wegener 1988 ak 1984 the present result is an improvement on ak 1995

on special zeros of p adic l functions of hilbert modular forms let e be a modular elliptic curve over a totally real number field f we prove the weak exceptional zero conjecture which links a higher derivative of the p adic l function attached to e to certain p adic periods attached to the corresponding hilbert modular form at the places above p where e has split multiplicative reduction under some mild restrictions on p and the conductor of e we deduce the exceptional zero conjecture in the strong form i e where the automorphic p adic periods are replaced by the mathcal l invariants of e defined in terms of tate periods from a special case proved earlier by mok crucial for our method is a new construction of the p adic l function of e in terms of local data

p adic l functions and p adic periods of modular forms let e be an elliptic curve which is defined over q and has stable reduction modulo a given prime p assuming that e is modular one can associate to e a p adic l function lp e s see mz swd a v vi mz t t for its construction in various cases this function is defined by a certain interpolation property and is analytic for sezp in this paper we will assume that e has split multiplicative reduction at p under this assumption the interpolation property implies that lp e 1 0 we will prove a formula for ep e 1 which was discovered experimentally by mazur tate and teitelbaum mz t t by tate s p adic urfiformization theory there is a p adic integer q epzp which we refer to as the tate period for e and a p adic analytic isomorphism

on the size of binary decision diagrams representing boolean functions we consider the size of the representation of boolean functions by several classes of binary decision diagrams bdds also called branching programs namely the classes of arbitrary bdds of real time bdd rbdd i e bdds where each computation path is limited to the number of variables of free bdds fbdds also called read once only branching programs of ordered bdds obdds i e fbdds where variables are tested in the same order along all paths and binary decision trees bdts using well known techniques we first establish asymptotically sharp bounds as a function of n on the minimum size of arbitrary bdds representing almost all boolean functions of n variables and provide asymptotic lower and upper bounds differing only by a factor of two on the minimum size obdds representing almost all boolean functions of n variables we then using a method to obtain exponential lower bounds on complexity of computation of boolean functions by rbdd fbdd and obdd that originated in breitbart 1968 present the highest such bounds to date and also present improved bounds on the relative economy of description of particular boolean functions by the above classes of bdds for each nontrivial pair of bdd classes considered we exhibit infinite families of boolean functions representable much more concisely by bdds in one class than by bdds in the other

arbitrary lagrangian eulerian finite element analysis of free surface flow an arbitrary lagrangian eulerian ale description of newtonian fluid flow is presented it allows the simulation of free surfaces with the finite element method fem we develop a robust node repositioning algorithm which consists of two parts an energy minimization principle controls the positions of the internal nodes a good approximation of the boundaries is ensured by explicitly enforcing the free surface conditions numerical examples are done employing the mini element

a level set approach for computing solutions to incompressible two phase flow a level set approach for computing solutions to incompressible two phase flow is presented the interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function a new treatment of the level set method allows us to efficiently maintain the level set function as the signed distance from the interface we never have to explicitly reconstruct or find the zero level set consequently we are able to handle arbitrarily complex topologies large density and viscosity ratios and surface tension on relatively coarse grids we use a second order projection method along with a second order upwinded procedure for advecting the momentum and level set equations we consider the motion of air bubbles and water drops we also compute flows with multiple fluids such as air oil and water

approximation of boolean functions by combinatorial rectangles this paper deals with the number of monochromatic combinatorial rectangles required to approximate a boolean function on a constant fraction of all inputs where each rectangle may use its own partition of the input variables the main result of the paper is that the number of rectangles required for the approximation of boolean functions in this model is very sensitive to the allowed error there is an explicitly defined sequence of boolean functions fn on n variables such that fn has rectangle approximations with a constant number of rectangles and one sided error 13 o 1 or two sided error 14 o 1 but on the other hand fn requires exponentially many rectangles if the error bounds are decreased by an arbitrarily small constant as applications of this result the following separation results for read once branching programs are obtained the functions from the main result require only linear size for nondeterministic read once branching programs and randomized read once branching programs with two sided error 13 o 1 while randomized read once branching programs with constant two sided error smaller than 13 and unambiguous nondeterministic read once branching programs require exponential size

more on the complexity of slice functions abstract one of the hardest problems in complexity theory seems to be to find a proof that the circuit complexity of some explicitly defined boolean function is nonlinear in the number of inputs and outputs an interesting approach for such a bound is the investigation of slice functions since for these functions circuits over a complete basis monotone circuits and the so called set circuits are nearly equivalent in this paper the theory of slice functions is extended a monotone representation of each boolean function whose monotone complexity is at most a factor n larger than its circuit complexity is presented for some slice functions and some other functions new pseudocomplements with small monotone complexity are presented these results lead to new and intuitive problems whose solution would lead to nonlinear lower bounds on circuit complexity some connections between the complexity of n output and 1 output slices and between the complexity of the k slice and the k 1 slice of the same function are discussed

arbitrary lagrangian eulerian finite element method for computation of two phase flows with soluble surfactants a finite element scheme based on a coupled arbitrary lagrangian eulerian and lagrangian approach is developed for the computation of interface flows with soluble surfactants the numerical scheme is designed to solve the time dependent navier stokes equations and an evolution equation for the surfactant concentration in the bulk phase and simultaneously an evolution equation for the surfactant concentration on the interface second order isoparametric finite elements on moving meshes and second order isoparametric surface finite elements are used to solve these equations the interface resolved moving meshes allow the accurate incorporation of surface forces marangoni forces and jumps in the material parameters the lower dimensional finite element meshes for solving the surface evolution equation are part of the interface resolved moving meshes the numerical scheme is validated for problems with known analytical solutions a number of computations to study the influence of the surfactants in 3d axisymmetric rising bubbles have been performed the proposed scheme shows excellent conservation of fluid mass and of the total mass of the surfactant

shintani cocycles and the order of vanishing of p adic hecke l series at s 0 let chi be a hecke character of finite order of a totally real number field f by using hill s shintani cocycle we provide a cohomological construction of the p adic l series l p chi s associated to chi this is used to show that l p chi s has a trivial zero at s 0 of order at least equal to the number of places of f above p where the local component of chi is trivial

the complexity of central slice functions let f be a monotone boolean function over x x1 xn the k slice of f is the function fk f tkn vtk 1n where tkn is the kth threshold function berkowitz has shown that sufficiently large superlinear lower bounds on the monotone network complexity of kk imply lower bounds of the same order on the combinational complexity of f and that if f has large combinational complexity then some slice of f must have large monotone complexity however this latter result does not specify any particular slice and it is known that some nontrivial slice functions of np complete predicates have linear complexity in this paper we consider the 12n slice functions and show that for three basic monotone boolean np complete functions this slice is also np complete in addition the 12n slice of some boolean matrix functions f is studied and is proved to be no easier to compute than f

front tracking with moving least squares surfaces the representation of interfaces by means of the algebraic moving least squares amls technique is addressed this technique in which the interface is represented by an unconnected set of points is interesting for evolving fluid interfaces since there is no surface connectivity the position of the surface points can thus be updated without concerns about the quality of any surface triangulation we introduce a novel amls technique especially designed for evolving interfaces applications that we denote ramls for robust amls the main advantages with respect to previous amls techniques are increased robustness computational efficiency and being free of user tuned parameters further we propose a new front tracking method based on the lagrangian advection of the unconnected point set that defines the ramls surface we assume that a background eulerian grid is defined with some grid spacing h the advection of the point set makes the surface evolve in time the point cloud can be regenerated at any time in particular we regenerate it each time step by intersecting the gridlines with the evolved surface which guarantees that the density of points on the surface is always well balanced the intersection algorithm is essentially a ray tracing algorithm well studied in computer graphics in which a line ray is traced so as to detect all intersections with a surface also the tracing of each gridline is independent and can thus be performed in parallel several tests are reported assessing first the accuracy of the proposed ramls technique and then of the front tracking method based on it comparison with previous eulerian lagrangian and hybrid techniques encourage further development of the proposed method for fluid mechanics applications

integration on cal h _p times cal h and arithmetic applications this article describes a conjectural p adic analytic construction of global points on modular elliptic curves points which are defined over the ring class fields of real quadratic fields the resulting conjectures suggest that the classical heegner point construction and the theory of complex multiplication on which it is based should extend to a variety of contexts in which the underlying field is not a cm field

adaptive tetrahedral mesh generation by constrained delaunay refinement this paper presents a tetrahedral mesh generation method for numerically solving partial differential equations using finite element or finite volume methods in three dimensional space the main issues are the mesh quality and mesh size which directly affect the accuracy of the numerical solution and the computational cost two basic problems need to be resolved namely boundary conformity and field points distribution the proposed method utilizes a special three dimensional triangulation so called constrained delaunay tetrahedraliwtion to conform the domain boundary and create field points simultaneously good quality tetrahedra and graded mesh size can be theoretically guaranteed for a large class of mesh domains in addition an isotropic size field associated with the numerical solution can be supplied the field points will then be distributed according to it good mesh size conformity can be achieved for smooth sizing informations the proposed method has been implemented various examples are provided to illustrate its theoretical aspects as well as practical performance

non cuspidal hida theory for siegel modular forms and trivial zeros of p adic l functions we study the derivative of the standard p adic l function associated with a p ordinary siegel modular form for p a parabolic subgroup of mathrm gl n when it presents a semi stable trivial zero this implies part of greenberg s conjecture on the order and leading coefficient of p adic l functions at such trivial zero we use the method of greenberg stevens for the construction of the improvedp adic l function we develop hida theory for non cuspidal siegel modular forms

transfinite mappings and their application to grid generation the two essential ingredients of any boundary value problem are the field equations which describe the physics of the problem and a set of relations which specify the geometry of the problem domain mesh generators or grid generators are preprocessors whicd decompose the problem domain into a large number of interconnected finite elements or curvilinear finite difference stencils a number of such techiques have been developed over the past decade to alleviate the frustation and reduce the time involved in the tedious manual subdividing of a complex shaped region or 3 d structure into finite elements our purpose here is to describe how the techniques of bivariate and trivariate blending function interpolation which were originally developed for and applied to geometric problems of computer aided design of sculptured surface and 3 d solids can be adapted and applied to the geometric problems of grid generation in contrast to other techniques which require the numerical solution of complex partial differential equations and hence a great deal of computing the transfinite methods proposed herein are computationally inexpensive

a lower bound for integer multiplication with read once branching programs we prove that read once branching programs computing integer multiplication require size 2 omega sqrt n this is the first nontrivial lower bound for multiplication on branching programs that are not oblivious by the appropriate problem reductions we obtain the same lower bound for other arithmetic functions

a 3d moving mesh finite element method for two phase flows a 3d ale finite element method is developed to study two phase flow phenomena using a new discretization method to compute the surface tension forces the computational method is based on the arbitrary lagrangian eulerian formulation ale and the finite element method fem creating a two phase method with an improved model for the liquid gas interface an adaptive mesh update procedure is also proposed for effective management of the mesh to remove add and repair elements since the computational mesh nodes move according to the flow the ale description explicitly defines the two phase interface position by a set of interconnected nodes which ensures a sharp representation of the boundary including the role of the surface tension the proposed methodology for computing the curvature leads to accurate results with moderate programming effort and computational cost static and dynamic tests have been carried out to validate the method and the results have compared well to analytical solutions and experimental results found in the literature demonstrating that the new proposed methodology provides good accuracy to describe the interfacial forces and bubble dynamics this paper focuses on the description of the proposed methodology with particular emphasis on the discretization of the surface tension force the new remeshing technique and the validation results additionally a microchannel simulation in complex geometry is presented for two elongated bubbles c 2014 elsevier inc all rights reserved

optimal decision trees and one time only branching programs for symmetric boolean functions combinational complexity and depth are the most important complexity measures for boolean functions it has turned out to be very hard to prove good lower bounds on the combinational complexity or the depth of explicitly defined boolean functions therefore one has restricted oneself to models where nontrivial lower bounds are easier to prove here decision trees branching programs and one time only branching programs are considered where each variable may be tested on each path of computation only once efficient algorithms for the construction of optimal decision trees and optimal one time only branching programs for symmetric boolean functions are presented furthermore the following trade off results are proved an exponential lower bound on the decision tree complexity of some boolean function is shown having linear formula size and linear one time only branching program complexity furthermore a quadratic lower bound on the one time only branching program complexity of some boolean function is shown having linear combinational complexity

separating complexity classes related to ω decision trees abstract by proving exponential lower and polynomial upper bounds for parity decision trees and collecting similar bounds for nondeterministic and co nondeterministic decision trees the complexity classes related to polynomial size deterministic nondeterministic co nondeterministic parity and alternating decision trees are completely separated considering alternating decision trees it is shown that the number of alternations between say nodes and nodes strongly influences their computational power

on the complexity of branching programs and decision trees for clique functions exponential lower bounds on the complexity of computing the clique functions in the boolean decision tree model are proved for one time only branching programs large polynomial lower bounds are proved for k clique functions if k is fixed and exponential lower bounds if k increases with n finally the hierarchy of the classes bp d p of all sequences of boolean functions that may be computed by d times only branching programs of polynomial size is introduced it is shown constructively that bp 1 p is a proper subset of bp 2 p

polynomial size ω branching programs and their computational power abstract in the following new types of branching programs so called branching programs b 2 are introduced the complexity classes related to polynomial size branching programs will be completely classified in addition to identifying a new class p bp l poly between l poly and p poly new characterizations of such fundamental space complexity classes like nl poly co nl poly and p poly are obtained using these characterizations we relate the complexity of the mentioned classes to those of some extremely restricted problems resembling the graph accessibility problem

efficient mesh deformation based on radial basis function interpolation by means of the inverse fast multipole method abstract radial basis function interpolation is often employed in mesh deformation algorithms for unstructured meshes for example in fluid structure interaction or design optimization problems this is known to be a robust methodology that results in high quality deformed meshes the applicability of this method to large problems is currently hampered by its prohibitive computational cost however which is caused by the need to solve a dense system of equations the computation time grows as o n b 3 if a direct solver is employed for the solution where n b denotes the number of boundary nodes in the mesh while alternative iterative solvers often suffer from an unfavorable convergence behavior in this paper we present the inverse fast multipole method as a novel fast approximate direct solver with a computational cost scaling as o n b the linear complexity is achieved by transforming the dense system into an extended sparse system along with the compression of certain matrix blocks into low rank factorizations the solver is inexact although the error can be controlled and made as small as needed a low accuracy solver can hence be used as an efficient preconditioner in an iterative scheme numerical benchmarks are presented demonstrating that the proposed approach enhances the computational efficiency of mesh deformation algorithms based on radial basis function interpolation significantly without jeopardizing their robustness and quality

properties of complexity measures for prams and wrams abstract the computation of boolean functions by parallel computers with shared memory prams and wrams is considered in particular complexity measures for parallel computers like critical and sensitive complexity are compared with other complexity measures for boolean functions like branching program depth and length of prime implicants and clauses the relations between these complexity measures and their asymptotic behaviour are investigated for the classes of boolean functions monotone functions and symmetric functions

a fictitious domain approach for a mixed finite element method solving the two phase stokes problem with surface tension forces abstract in this article we study a mixed finite element formulation for solving the stokes problem with general surface forces that induce a jump of the normal trace of the stress tensor on an interface that splits the domain into two subdomains equality of velocities is assumed at the interface the interface conditions are taken into account with multipliers a suitable lagrangian functional is introduced for deriving a weak formulation of the problem a specificity of this work is the consideration of the interface with a fictitious domain approach the latter is inspired by the xfem approach in the sense that cut off functions are used but it is simpler to implement since no enrichment by singular functions is provided in that context getting convergence for the dual variables defined on the interface is non trivial for that purpose an augmented lagrangian technique stabilizes the convergence of the multipliers which is important because their value would determine the dynamics of the interface in an unsteady framework theoretical analysis is provided where we show that a discrete inf sup condition independent of the mesh size is satisfied for the stabilized formulation this guarantees optimal convergence rates that we observe with numerical tests the capacity of the method is demonstrated with robustness tests and with an unsteady model tested for deformations of the interface that correspond to ellipsoidal shapes in dimension 2

least upper bounds for the size of obdds using symmetry properties this paper investigates reduced ordered binary decision diagrams obdd of partially symmetric boolean functions when using variable orders where symmetric variables are adjacent we prove upper bounds for the size of such symmetry ordered obdds symobdd they generalize the upper bounds for the size of obdds of totally symmetric boolean functions and nonsymmetric boolean functions proven by m a heap and m r mercer 1994 and i wegener 1984 experimental results based on these upper bounds show that the nontrivial symmetry sets of a boolean function should be located either right up at the beginning or right up at the end of the variable order in order to obtain best upper bounds

boolean functions whose monotone complexity is of size n2 log n we construct a sequence of monotone boolean functions hn 0 1 n 0 1 n such that the monotone complexity of hn is of order n2 log n this result includes the largest known lower bound of this kind previously there were an n3 2 bound for the boolean matrix product an n5 3 bound for boolean sums and an n2 log2n bound of the author for the same functions hn this new lower bound is proved by new methods which probably will turn out to be useful also for other problems

on the complexity of vlsi implementations and graph representations of boolean functions with application to integer multiplication lower bound results on boolean function complexity under two different models are discussed the first is an abstraction of tradeoffs between chip area and speed in very large scale integrated vlsi circuits the second is the ordered binary decision diagram obdd representation used as a data structure for symbolically representing and manipulating boolean functions the lower bounds demonstrate the fundamental limitations of vlsi as an implementation medium and that of the obdd as a data structure it is shown that the same technique used to prove that any vlsi implementation of a single output boolean function has area time complexity at sup 2 omega n sup 2 also proves that any obdd representation of the function has omega c sup n vertices for some c 1 but that the converse is not true an integer multiplier for word size n with outputs numbered 0 least significant through 2n 1 most significant is described for the boolean function representing either output i 1 or output 2n i 1 where 1

stark heegner points and the cohomology of quaternionic shimura varieties let f be a totally real field of narrow class number one and let e f be a modular semistable elliptic curve of conductor n 1 l etk f be a non cm quadratic extension with disc k n 1 such that the sign in the functional equation of l e k s is 1 suppose further that there is a prime p n that is inert in k w e describe a p adic construction of points on e which we conjecture to be rational over ring class fields of k f and satisfy a shimura reciprocity law these points are expected to behave like classical heegner points and can be viewed as new instances of the stark heegner point construction of 5 the key idea in our construction is a reinterpretation of darmon s theory of modular symbols and mixed period integrals in terms of group cohomology

numerical simulation of 3d bubbles rising in viscous liquids using a front tracking method the rise of bubbles in viscous liquids is not only a very common process in many industrial applications but also an important fundamental problem in fluid physics an improved numerical algorithm based on the front tracking method originally proposed by tryggvason and his co workers has been validated against experiments over a wide range of intermediate reynolds and bond numbers using an axisymmetric model j hua j lou numerical simulation of bubble rising in viscous liquid j comput phys 22 2007 769 795 in the current paper this numerical algorithm is further extended to simulate 3d bubbles rising in viscous liquids with high reynolds and bond numbers and with large density and viscosity ratios representative of the common air water two phase flow system to facilitate the 3d front tracking simulation mesh adaptation is implemented for both the front mesh on the bubble surface and the background mesh on the latter mesh the governing navier stokes equations for incompressible newtonian flow are solved in a moving reference frame attached to the rising bubble specifically the equations are solved using a finite volume scheme based on the semi implicit method for pressure linked equations simple algorithm and it appears to be robust even for high reynolds numbers and high density and viscosity ratios the 3d bubble surface is tracked explicitly using an adaptive unstructured triangular mesh the numerical model is integrated with the software package paramesh a block based adaptive mesh refinement amr tool developed for parallel computing paramesh allows background mesh adaptation as well as the solution of the governing equations in parallel on a supercomputer further peskin distribution function is applied to interpolate the variable values between the front and the background meshes detailed sensitivity analysis about the numerical modeling algorithm has been performed the current model has also been applied to simulate a number of cases of 3d gas bubbles rising in viscous liquids e g air bubbles rising in water simulation results are compared with experimental observations both in aspect of terminal bubble shapes and terminal bubble velocities in addition we applied this model to simulate the interaction between two bubbles rising in a liquid which illustrated the model s capability in predicting the interaction dynamics of rising bubbles

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

on oblivious branching programs of linear length abstract input oblivious decision graphs of linear length are considered among other concerns the computational complexity of three graph accessibility problems and the word problem of the free group are investigated several exponential lower bounds are proved

numerical simulation of bubble rising in viscous liquid an improved numerical algorithm for front tracking method is developed to simulate the rising of a bubble in quiescent viscous liquid due to buoyancy in the new numerical algorithm volume correction is introduced to conserve the bubble volume while tracking the bubble s rising and deforming and volume flux conservation based simple algorithm is adopted to solve the navier stokes equation for fluid flow using finite volume method the new front tracking algorithm is validated systematically by simulating single bubble rising and deforming in quiescent viscous liquid under different flow regimes the simulation results are compared with the experimental measurement in terms of terminal bubble shape and velocity the simulation results demonstrate that the new algorithm is robust in the flow regimes with larger ranges of reynolds number re 200 bond number bo 200 density ratio r l r b 1000 and viscosity ratio m l m b 500 the new front tracking algorithm is also applied to investigate bubble rising and deforming behaviour in the various flow regimes of air bubble water solution system under effects of reynolds number bond number density ratio viscosity ratio as well as the bubble initial shape which have been explored previously by experiments the predicted bubble shape and terminal velocity agree well with the experimental results hence the new modelling algorithm expands the conventional front tracking method to more realistic and wider applications

a moving mesh interface tracking method for 3d incompressible two phase flows an interface tracking method using an unstructured moving mesh has been developed for simulating three dimensional incompressible and immiscible two phase flows the interface mesh is moved in a lagrangian fashion a local mesh adaptation method is used to capture the changing interface curvature maintain good mesh quality and deal with large deformation the interface is of zero thickness so the jump and continuity conditions across the interface are implemented directly without any smoothing of the properties of the two fluids this is theoretically beneficial compared to other methods that allow the fluids properties to continuously vary in an interface region the curvature for interfacial tension calculation is geometrically computed by a least squares parabola fitting method a mesh separation scheme for interfacial flows is employed to handle topological transition the numerical technique is tested and validated by several cases which include a two layer couette flow an oscillating droplet surrounded by gas and a droplet in shear flow the results obtained from numerical simulations are found to be in excellent agreement with analytical solutions and experimental results a collapsing ligament in a quiescent gas flow and a modulated jet pinching are simulated to show the robustness of the method

almost k wise independence and hard boolean functions we construct boolean functions computable by polynomial size circuits with large lower bounds for read once branching program 1 b p s a function in p with the lower bound 2n polylog n a function in quasipolynomial time with the lower bound 2n o log n and a function in linspace with the lower bound 2n log n o 1 our constructions are simpler than those of andreev et al electronic colloq on computational complexity tr97 053 1997 as we apply the idea of almost k wise independence more directly without the use of discrepancy set generators for large affine subspaces the simplicity of our constructions also allows us to observe that there exists a boolean function in ac0 2 computable by a depth 3 polynomial size circuit over the basis 1 with the optimal lower bound 2n log n o 1 for 1 b p s

on the complexity of slice functions abstract by a result of berkowitz 1982 the monotone circuit complexity of slice functions cannot be much larger than the circuit combinational complexity of these functions for arbitrary complete bases this result strengthens the importance of the theory of monotone circuits we show in this paper that monotone circuits for slice functions can be understood as special circuits called set circuits here disjunction and conjunction are replaced by set union and set intersection all the main methods known for proving lower bounds on the monotone complexity of boolean functions fail to work in their present form for slice functions furthermore we show that the canonical slice functions of the boolean convolution the nechiporuk boolean sums and the clique function can be computed with a linear number of gates