systems of diagonal diophantine inequalities we treat systems of real diagonal forms f 1 x f 2 x f r x of degree k in s variables we give a lower bound s 0 r k which depends only on r and k such that if s s 0 r k holds then under certain conditions on the forms and for any positive real number e there is a nonzero integral simultaneous solution x z s of the system of diophantine inequalities f i x e for i i r in particular our result is one of the first to treat systems of inequalities of even degree the result is an extension of earlier work by the author on quadratic forms also a restriction in that work is removed which enables us to now treat combined systems of diophantine equations and inequalities

tensor network states and geometry tensor network states are used to approximate ground states of local hamiltonians on a lattice in d spatial dimensions different types of tensor network states can be seen to generate different geometries matrix product states mps in d 1 dimensions as well as projected entangled pair states peps in d 1 dimensions reproduce the d dimensional physical geometry of the lattice model in contrast the multi scale entanglement renormalization ansatz mera generates a d 1 dimensional holographic geometry here we focus on homogeneous tensor networks where all the tensors in the network are copies of the same tensor and argue that certain structural properties of the resulting many body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters indeed the asymptotic decay of correlations in homogeneous mps and mera for d 1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries respectively whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law that is again in the relevant geometry this geometrical interpretation offers a simple and unifying framework to understand the structural properties of and helps clarify the relation between different tensor network states in addition it has recently motivated the branching mera a generalization of the mera capable of reproducing violations of the entropic boundary law in d 1 dimensions

commutators and images of noncommutative polynomials abstract let a be an algebra and let f be a nonconstant noncommutative polynomial in the first part of the paper we consider the relationship between a a the linear span of commutators in a and span f a the linear span of the image of f in a in particular we show that a a a implies span f a a in the second part we establish some waring type results for images of polynomials for example we show that if c is a commutative unital algebra over a field f of characteristic 0 a is the matrix algebra m n c and the polynomial f is neither an identity nor a central polynomial of m n f then every commutator in a can be written as a difference of two elements each of which is a sum of 7788 elements from f a if c f is an algebraically closed field then 4 elements suffice similar results are obtained for some other algebras in particular for the algebra b h of all bounded linear operators on a hilbert space h

the representation of numbers as sums of unlike powers ii n zx i 1 the principal tool in the proof is the hardy littlewood circle method incorporating results and techniques of a powerful new iterative method developed by vaughan and wooley va3 va4 vw1 wol vw2 in section 2 an algorithm developed in fo for optimizing the parameters in mixed power mean value theorems is generalized and analyzed section 3 details a more sophisticated method of generating mixed power mean value theorems by a limited adaptation of the iterative method itself the form of these estimates offers many advantages over those of section 2 and provides the key to the elimination of the 16th power from 1 1 these mean value theorems are then applied to the proof of theorem 1 in section 4 the tools developed here are applicable to a wide range of mixed power representation problems and we briefly illustrate in section 5 the application to the problem of determining the number of terms required to represent all large n when the lowest power used is a kth power instead of a square throughout n is a large natural number whose representation as a sum of mixed powers is at issue and e is an arbitrarily small positive real number constants implied by the landau oand vinogradov symbols may depend on e or k for a real number x write e x for e27ix x for the greatest integer not exceeding x and llxll for the distance from x to the nearest integer unless otherwise specified lowercase latin letters denote natural numbers and greek letters denote real numbers let a p r denote the set of natural numbers not exceeding p with no

maximum principle satisfying second order discontinuous galerkin schemes for convection diffusion equations on triangular meshes we propose second order accurate discontinuous galerkin dg schemes which satisfy a strict maximum principle for general nonlinear convection diffusion equations on unstructured triangular meshes motivated by genuinely high order maximum principle satisfying dg schemes for hyperbolic conservation laws perthame 1996 and zhang 2010 14 26 we prove that under suitable time step restriction for forward euler time stepping for general nonlinear convection diffusion equations the same scaling limiter coupled with second order dg methods preserves the physical bounds indicated by the initial condition while maintaining uniform second order accuracy similar to the purely convection cases the limiters are mass conservative and easy to implement strong stability preserving ssp high order time discretizations will keep the maximum principle following the idea in zhang 2012 30 we extend the schemes to two dimensional convection diffusion equations on triangular meshes there are no geometric constraints on the mesh such as angle acuteness numerical results including incompressible navier stokes equations are presented to validate and demonstrate the effectiveness of the numerical methods

a monotone finite element scheme for convection diffusion equations a simple technique is given in this paper for the construction and analysis of a class of finite element discretizations for convection diffusion problems in any spatial dimension by properly averaging the pde coefficients on element edges the resulting finite element stiffness matrix is an m matrix under some mild assumption for the underlying generally unstructured finite element grids as a consequence the proposed edge averaged finite element scheme is particularly interesting for the discretization of convection dominated problems this scheme admits a simple variational formulation it is easy to analyze and it is also suitable for problems with a relatively smooth flux variable some simple numerical examples are given to demonstrate its effectiveness for convection dominated problems

a new iterative method in waring s problem on introduit une nouvelle methode iterative dans le probleme de waring on ameliore toutes les bornes superieures anterieures pour g k lorsque k est superieur ou egal a 5

renormalization and tensor product states in spin chains and lattices we review different descriptions of many body quantum systems in terms of tensor product states we introduce several families of such states in terms of the known renormalization procedures and show that they naturally arise in that context we concentrate on matrix product states tree tensor states multiscale entanglement renormalization ansatz and projected entangled pair states we highlight some of their properties and show how they can be used to describe a variety of systems

equal sums of three powers in a region maxxi b here d will be a fixed positive integer there are 6b o b trivial solutions in which x4 x5 x6 are a permutation of x1 x2 x3 we write n 0 d b for the number of non trivial solutions and our task is to estimate this quantity when d 2 it is a simple exercise with the circle method to show that nd b cb for an appropriate positive constant c so that the non trivial solutions dominate the trivial ones for larger values of d it follows from hua s inequality 11 that

the sato tate distribution in thin parametric families of elliptic curves we obtain new results concerning the sato tate conjecture on the distribution of frobenius traces over single and double parametric families of elliptic curves we consider these curves for values of parameters having prescribed arithmetic structure product sets geometric progressions and most significantly prime numbers in particular some families are much thinner than the ones previously studied

exponentiated gradient versus gradient descent for linear predictors we consider two algorithm for on line prediction based on a linear model the algorithms are the well known gradient descent gd algorithm and a new algorithm which we call eg they both maintain a weight vector using simple updates for the gd algorithm the update is based on subtracting the gradient of the squared error made on a prediction the eg algorithm uses the components of the gradient in the exponents of factors that are used in updating the weight vector multiplicatively we present worst case loss bounds for eg and compare them to previously known bounds for the gd algorithm the bounds suggest that the losses of the algorithms are in general incomparable but eg has a much smaller loss if only a few components of the input are relevant for the predictions we have performed experiments which show that our worst case upper bounds are quite tight already on simple artificial data

the hasse principle for systems of diagonal cubic forms we establish the hasse principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r times r submatrix thereby achieving the theoretical limit of the circle method for such systems

the cubic case of the main conjecture in vinogradov s mean value theorem abstract we apply a variant of the multigrade efficient congruencing method to estimate vinogradov s integral of degree 3 for moments of order 2 s establishing strongly diagonal behaviour for 1 s 6 consequently the main conjecture is now known to hold for the first time in a case of degree exceeding 2

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

on the waring goldbach problem for fourth and fifth powers it is shown that every sufficiently large integer congruent to 14 modulo 240 may be written as the sum of 14 fourth powers of prime numbers and that every sufficiently large odd integer may be written as the sum of 21 fifth powers of prime numbers the respective implicit bounds 14 and 21 improve on the previous bounds 15 following from work of davenport and 23 due to thanigasalam these conclusions are established through the medium of the hardy littlewood method the proofs being somewhat novel in their use of estimates stemming directly from exponential sums over prime numbers in combination with the linear sieve rather than the conventional methods which waste a variable or two by throwing minor arc estimates down to an auxiliary mean value estimate based on variables not restricted to be prime numbers in the work on fifth powers a switching principle is applied to a cognate problem involving almost primes in order to obtain the desired conclusion involving prime numbers alone 2000 mathematics subject classification 11p05 11n36 11l15 11p55

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

addendum and corrigendum to the representation of numbers as sums of unlike powers ii professor j6rg briidern has kindly pointed out some errors in table 1 and table 2 of my paper 1 specifically the values of a 4 4 and a 4 5 in table 1 are better than can actually be deduced from the literature and they lead to errors in the first three lines of table 2 the main mistake was an erroneous combination of results from 3 and 4 to determine a 4 4 4 5627 which incidentally has also appeared in another paper 2 the correct values for table 1 rounded up in the last decimal place are

vinogradov s mean value theorem via efficient congruencing ii we apply the efficient congruencing method to estimate vino gradov s integral for moments of order 2s with 1 6 s 6 k 2 1 thereby we show that quasi diagonal behaviour holds when s o k 2 we obtain near optimal estimates for 1 6 s 6 1 k 2 k and optimal estimates for s k 2 1 in this way we come half way to proving the main conjecture in two different directions there are consequences for estimates of weyl type and in several allied applications thus for example the anticipated asymptotic formula in waring s problem is established for sums of s kth powers of natural numbers whenever s 2k 2 2k 8 k 6 estimates stemming from vinogradov s mean value theorem deliver bounds for exponential sums of large degree both in mean and pointwise beyond the competence of alternate approaches the ubiquity of such exponential sums in analytic number theory in the analysis for example of the riemann zeta function in waring s problem and beyond accounts for the high pro file of vinogradov s methods in the associated literature in recent work we established a version of vinogradov s mean value theorem which achieves an essentially optimal upper bound with a number of variables only twice the number conjectured to be best possible see 19 for systems of degree k previous estimates missed such a bound by a factor of order logk our earlier approach provides no upper bounds when the number of variables is smaller precluding the possibility of applications involving the finer features of these mean values our goal in this paper is to remedy this deficiency at the same time strengthening our previous conclusions it transpires that we are able to come within a hair s breadth of proving the main conjecture concerning vino gradov s mean value theorem in half of the basic interval of relevant moments such developments illustrate the flexibility of the new efficient congruencing method introduced in 19 we now introduce some notation when k n and r k define

on integer solutions of parsell vinogradov systems we prove a sharp upper bound on the number of integer solutions of the parsell vinogradov system in every dimension d ge 2

solitary wave solutions of the generalised burgers huxley equation exact solitary wave solutions of the generalised burgers huxley equation delta u delta t alpha udelta delta u delta chi delta 2u delta chi 2 beta u 1 udelta udelta gamma are obtained by using the relevant nonlinear transformations the results obtained are the generalisation of former work the method in this paper can also be applied to the burgers fisher equation

nonnegativity of exact and numerical solutions of some chemotactic models we investigate nonnegativity of exact and numerical solutions to a generalized keller segel model this model includes the so called minimal keller segel model but can cover more general chemistry we use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative we then derive numerical methods using finite element techniques for the generalized keller segel model adapting the ideas in our proof of nonnegativity of exact solutions to the discrete setting we are able to show nonnegativity of discrete solutions from the numerical methods under certain standard assumptions one of the numerical methods is then applied to the minimal keller segel model recalling known results on the qualitative behavior of this model we are able to choose parameters that yield convergence to a nonhomogeneous stationary solution while proceeding to exhibit these stationary patterns we also demonstrate how naive choices of numerical methods can give physically unrealistic solutions thereby justifying the need to study positivity preserving methods

on the hasse principle for quartic hypersurfaces we establish the hasse principle for smooth projective quartic hypersurfaces of dimension greater than or equal to 28 defined over q

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

a generalization of vinogradov s mean value theorem we obtain new upper bounds for the number of integral solutions of a complete system of symmetric equations which may be viewed as a multi dimensional version of the system considered in vinogradov s mean value theorem we then use these bounds to obtain weyl type estimates for an associated exponential sum in several variables finally we apply the hardy littlewood method to obtain asymptotic formulas for the number of solutions of the vinogradov type system and also of a related system connected to the problem of finding linear spaces on hypersurfaces

subconvexity for additive equations pairs of undenary cubic forms we investigate pairs of diagonal cubic equations with integral coefficients for a class of such diophantine systems with 11 or more variables we are able to establish that the number of integral solutions in a large box is at least as large as the expected order of magnitude

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

on vinogradov s mean value theorem strongly diagonal behaviour via efficient congruencing we enhance the efficient congruencing method for estimating vinogradov s integral for moments of order 2s with 1 leqslant s leqslant k 2 1 in this way we prove the main conjecture for such even moments when 1 leqslant s leqslant tfrac 1 4 k 1 2 showing that the moments exhibit strongly diagonal behaviour in this range there are improvements also for larger values of s these finding application to the asymptotic formula in waring s problem

on waring s problem for cubes and smooth weyl sums non trivial estimates for fractional moments of smooth cubic weyl sums are developed complemented by bounds for such sums of use on both the major and minor arcs in a hardy littlewood dissection these estimates are applied to derive an upper bound for the s th moment of the smooth cubic weyl sum of the expected order of magnitude as soon as s ge 7 691 related arguments demonstrate that all large integers n are represented as the sum of eight cubes of natural numbers all of whose prime divisors are at most exp c log n log log n 1 2 for a suitable positive number c this conclusion improves a previous result of g harcos in which nine cubes are required 1991 mathematics subject classification 11p05 11l15 11p55

new integer representations as the sum of three cubes we describe a new algorithm for finding integer solutions to x 3 y 3 z 3 k for specific values of k we use this to find representations for values of k for which no solution was previously known including k 30 and k 52

random diophantine equations i abstract we consider additive diophantine equations of degree k in s variables and establish that whenever s 3 k 2 then almost all such equations satisfy the hasse principle the equations that are soluble form a set of positive density and among the soluble ones almost all equations admit a small solution our bound for the smallest solution is nearly best possible

travelling wave solutions for the generalized burgers huxley equation abstract in this paper travelling wave solutions for the generalized burgers huxley equation are studied by using the first integral method which is based on the ring theory of commutative algebra we obtain a class of travelling solitary wave solutions for the generalized burgers huxley equation a minor error in the literature is clarified

discrete maximum principle and adequate discretizations of linear parabolic problems in this paper we analyze the connections between the different qualitative properties of numerical solutions of linear parabolic problems with dirichlet type boundary condition first we formulate the qualitative properties for the differential equations and shed light on their relations then we show how the well known discretization schemes can be written in the form of a one step iterative process we give necessary and sufficient conditions of the main qualitative properties of these iterations we apply the results to the finite difference and galerkin finite element solutions of linear parabolic problems in our main result we show that the nonnegativity preservation property is equivalent to the maximum minimum principle and they imply the maximum norm contractivity in one two and three dimensions we list sufficient a priori conditions that ensure the required qualitative properties finally we demonstrate the above results on numerical examples

asymptotic estimates for rational linear spaces on hypersurfaces we develop a repeated efficient differencing procedure for estimating mean values of certain multidimensional exponential sums over smooth numbers as a consequence we obtain asymptotic lower bounds for the number of linear spaces of fixed dimension up to a given height lying on the hypersurface defined by an additive equation

the numerical solution of a generalized burgers huxley equation through a conditionally bounded and symmetry preserving method in this article we propose a non standard finite difference scheme to approximate the solutions of a generalized burgers huxley equation from fluid dynamics our numerical method preserves the skew symmetry of the partial differential equation under study and under some analytical constraints of the model constants and the computational parameters involved it is capable of preserving the boundedness and the positivity of the solutions in the linear regime the scheme is consistent to first order in time due partially to the inclusion of a tuning parameter in the approximation of a temporal derivative and to second order in space we compare the results of our computational technique against the exact solutions of some particular initial boundary value problems our simulations indicate that the method presented in this work approximates well the theoretical solutions and moreover that the method preserves the boundedness of solutions within the analytical constraints derived here in the problem of approximating solitary wave solutions of the model under consideration we present numerical evidence on the existence of an optimum value of the tuning parameter of our technique for which a minimum relative error is achieved finally we linearly perturb a steady state solution of the partial differential equation under investigation and show that our simulations still converge to the same constant solution establishing thus robustness of our method in this sense

the hasse principle for pairs of diagonal cubic forms by means of the hardy littlewood method we apply a new mean value theorem for exponential sums to confirm the truth over the rational numbers of the hasse principle for pairs of diagonal cubic forms in thirteen or more variables

on sums of seven cubes we show that every integer between 1290741 and 3 375 10 12 is a sum of 5 non negative cubes from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 non negative cubes

the asymptotic formulae in waring s problem for cubes write rs n for the number of representations of a natural number as the sum of s positive integral cubes we investigate the higher moments of rs n establishing asymptotic formulae for n n rs n h when s 5 and h 3 when s 6 and h 28 5 and when s 7 and h 12 hitherto such an asymptotic formula was available only when s 4 and h 2

waring s problem in function fields let fq t denote the ring of polynomials over the nite eld fq of characteristic p and write j k t for the additive closure of the set of kth powers of polynomials in fq t dene gq k to be the least integer s satisfying the property that every polynomial in j k t of suciently large degree admits a strict representation as a sum of s kth powers we employ a version of the hardy littlewood method involving the use of smooth polynomials in order to establish a bound of the shape gq k ck logk o k log logk here the coecient c is equal to 1 when k p but in any case satises c 4 3 there are associated conclusions for the solubility of diagonal equations over fq t and for exceptional set estimates in waring s problem z to fq t therefore is the derivation of results uniform in the characteristic in this paper we investigate the analogue of waring s problem over fq t our aim being to establish conclusions that are relatively robust to changes in the characteristic of fq we concentrate in particular on methods having the potential to impact questions that concern the density of rational points on algebraic varieties in function elds a topic to which we intend to return on a future occasion some preparation is required before we can announce our principal conclusions let k be an integer with k 2 let s2 n and consider a polynomial m in fq t we seek to

further improvements in waring s problem 1 i n t r o d u c t i o n 147 2 i t e ra t ive schemes 149 3 e s t i m a t e s for the n u m b e r of so lu t ions of auxi l ia ry equa t ions 155 4 m a j o r and mino r arc e s t ima tes 168 5 t he i t e ra t ive scheme for fifth powers i 182 6 t h e i t e ra t ive scheme for fifth powers ii 185 7 t he proof of t h e o r e m 1 1 for fifth powers 191 8 t he i te ra t ive scheme for k exceeding 5 second differences 197 9 t he i t e ra t ive scheme for k exceeding 5 t h i r d differences 202 10 t he i t e ra t ive scheme for k exceeding 6 four th differences 206 11 t h e i t e ra t ive scheme for k exceeding 7 fifth and s ix th differences 212 12 t h e proof of t h e o r e m 1 1 for s ix th powers 216 13 t h e h a r d y l i t t l e w o o d dissec t ion for larger k 230 14 t h e p roof of t h e o r e m 1 1 for s even th powers 232 15 t he proof of t h e o r e m 1 1 for e igh th powers 234 16 t he proof of t h e o r e m 1 1 for n i n t h powers 234 append ix numer ica l values of p a r a m e t e r s 236 references 240

the asymptotic estimates and hasse principle for multidimensional waring s problem abstract motivated by the asymptotic estimates and hasse principle for multidimensional waring s problem via the circle method we prove for the first time that the corresponding singular series is bounded below by an absolute positive constant without any nonsingular local solubility assumption the number of variables we need is near optimal by proving a more general uniform density result over certain complete discrete valuation rings with finite residue fields we also establish uniform lower bounds for both singular series and singular integral in f q t we thus obtain asymptotic formulas and the hasse principle for multidimensional waring s problem in f q t via a variant of the circle method

maximum principle in linear finite element approximations of anisotropic diffusion convection reaction problems a mesh condition is developed for linear finite element approximations of anisotropic diffusion convection reaction problems to satisfy a discrete maximum principle loosely speaking the condition requires that the mesh be simplicial and mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 nonobtuse when the dihedral angles are measured in the metric specified by the inverse of the diffusion matrix where h h denotes the mesh size and varvec b b and c c are the coefficients of the convection and reaction terms in two dimensions the condition can be replaced by a weaker mesh condition an mathcal o vert varvec b vert infty h vert c vert infty h 2 o b h c h 2 perturbation of a generalized delaunay condition these results include many existing mesh conditions as special cases numerical results are presented to verify the theoretical findings

preserving nonnegativity of an affine finite element approximation for a convection diffusion reaction problem an affine finite element scheme approximation of a time dependent linear convection diffusion reaction problem in 2d and 3d is presented for these equations which do not satisfy an underlying maximum principle sufficient conditions are given in terms of the coefficient functions the computational grid and the discretization parameters to ensure that the nonnegativity property of the true solution is also satisfied by its approximation numerical examples are given which confirm the necessity and sufficiency of the discretization conditions to guarantee the nonnegativity of the approximation we present a monotone fem scheme for a convection diffusion reaction problem in 2 3d the considered equation does not possess an underlying maximum principle sufficient conditions are given to ensure the nonnegativity of the approximations numerical examples confirm the necessity and sufficiency of the conditions

tensor train decomposition a simple nonrecursive form of the tensor decomposition in d dimensions is presented it does not inherently suffer from the curse of dimensionality it has asymptotically the same number of parameters as the canonical decomposition but it is stable and its computation is based on low rank approximation of auxiliary unfolding matrices the new form gives a clear and convenient way to implement all basic operations efficiently a fast rounding procedure is presented as well as basic linear algebra operations examples showing the benefits of the decomposition are given and the efficiency is demonstrated by the computation of the smallest eigenvalue of a 19 dimensional operator

tangent space gradient optimization of tensor network for machine learning the gradient based optimization method for deep machine learning models suffers from gradient vanishing and exploding problems particularly when the computational graph becomes deep in this work we propose the tangent space gradient optimization tsgo for probabilistic models to keep the gradients from vanishing or exploding the central idea is to guarantee the orthogonality between variational parameters and gradients the optimization is then implemented by rotating the parameter vector towards the direction of gradient we explain and test tsgo in tensor network tn machine learning where tn describes the joint probability distribution as a normalized state in hilbert space we show that the gradient can be restricted in tangent space of 1 hypersphere instead of additional adaptive methods to control the learning rate in deep learning the learning rate of tsgo is naturally determined by rotation angle as tan our numerical results reveal better convergence of tsgo in comparison to the off the shelf adam

rational solutions of pairs of diagonal equations one cubic and one quadratic we obtain an essentially optimal estimate for the moment of order 32 3 of the exponential sum having argumentx 3 x 2 subject to modest local solubility hypotheses we thereby establish that pairs of diagonal diophantine equations one cubic and one quadratic possess non trivial integral solutions whenever the number of variables exceeds 10

multigrade efficient congruencing and vinogradov s mean value theorem we develop a multigrade enhancement of the efficient congru encing method to estimate vinogradov s integral of degree k for moments of order 2s thereby obtaining near optimal estimates for 5 k 2 1 543k 2 the asymptotic formula is also established for sums of 28 fifth powers

quasi diagonal behaviour in certain mean value theorems of additive number theory of fundamental significance in many problems of additive number theory are estimates for mean values of exponential sums over polynomial functions in this paper we shall show that the exponential sums of greatest interest in additive number theory demonstrate quasi diagonal behaviour which is to say that by taking the degree of the polynomial argument sufficiently large we can obtain upper bounds for low moments of the exponential sums arbitrarily close in a suitable sense to the diagonal estimate let us illustrate these notions by considering the situation in vinogradov s mean value theorem let s and k be positive integers and write

additive representation in short intervals ii sums of two like powers we establish that for almost all natural numbers n there is a sum of two positive integral cubes lying in the interval n n 7 18 varepsilon n here the exponent 7 18 lies half way between the trivial exponent 4 9 stemming from the greedy algorithm and the exponent 1 3 constrained by the number of integers not exceeding x that can be represented as the sum of two positive integral cubes we also provide analogous conclusions for sums of two positive integral k th powers when k ge 4

vinogradov s integral and bounds for the riemann zeta function the main result is an upper bound for the riemann zeta function in the critical strip zeta sigma it le a t b 1 sigma 3 2 log 2 3 t with a 76 2 and b 4 45 valid for frac12 le sigma le 1 and t ge 3 the previous best constant b was 18 5 tools include a variant of the korobov vinogradov method of bounding exponential sums an explicit version of t d wooley s bounds for vinogradov s integral and explicit bounds for mean values of exponential sums over numbers without small prime factors also using methods of wooley an auxiliary result is the exponential sum bound s n t le 9 463 n 1 1 133 66 lambda 2 where n is a positive integer t is a real number lambda log t log n and s n t max 0 2000 mathematical subject classification primary 11m06 11n05 11l15 secondary 11d72 11m35

a positive and bounded finite element approximation of the generalized burgers huxley equation abstract we present a finite element scheme capable of preserving the nonnegative and bounded solutions of the generalized burgers huxley equation proofs of existence and uniqueness of a solution to the continuous problem together with some results concerning the boundedness and the nonnegativity of the solution are given under appropriate conditions on the mesh and the initial and boundary data boundedness and nonnegativity of the finite element approximation are established an a priori error estimate for the approximation is also derived numerical experiments are presented which support the derived theoretical results