complexity of matrix problems in representation theory the classification problem is called wild if it contains the problem of classifying pairs of matrices up to simultaneous similarity we show in an explicit form that the last problem contains all classification matrix problems given by quivers or posets then we prove that this problem does not contain but is contained in the problem of classifying three valent tensors hence every wild classification problem given by a quiver or poset has the same complexity moreover a solution of one of them implies a solution of each of the remaining problems the problem of classifying three valent tensors is more complicated

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of n n complex matrices a b under the simultaneous similarity tat 1 tbt 1 roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set m 0n 2 r mn mn mn the set of n n complex valued matrices here r and are two positive integers then we construct a finite number of rational functions o1 os in the entries of a and b whose values are constant on all pairs similar in m n 2 r to a b the values of the functions oi a b i 1 s determine a finite number at most n 2 r of similarity classes in m n 2 r let sn be the subspace of complex symmetric matrices in mn for a b sn sn we consider the similarity class tatt tbtt where t ranges over all complex orthogonal matrices then the characteristic polynomial i a xb determines a finite number of similarity classes for almost all pairs a b sn sn

nature of complex singularities for the 2d euler equation abstract a detailed study of complex space singularities of the two dimensional incompressible euler equation is performed in the short time asymptotic regime when such singularities are very far from the real domain this allows an exact recursive determination of arbitrarily many spatial fourier coefficients using high precision arithmetic we find that the fourier coefficients of the stream function are given over more than two decades of wavenumbers by f k c k e k where k k cos sin the prefactor exponent typically between 5 2 and 8 3 is determined with an accuracy better than 0 01 it depends on the initial condition but not on the vorticity diverges as s where 7 2 and s is the distance to the complex singular manifold this new type of non universal singularity is permitted by the strong reduction of nonlinearity depletion which is associated to incompressibility spectral calculations show that the scaling reported above persists well beyond the time of validity of the short time asymptotics a simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent 5 2

integral bilinear forms coxeter transformations and coxeter polynomials of finite posets abstract linear algebra technique in the study of linear representations of finite posets is developed in the paper a concept of a quadratic wandering on a class of posets i is introduced and finite posets i are studied by means of the four integral bilinear forms b i b i b i b i z i z i z 1 1 the associated coxeter transformations and the coxeter polynomials in connection with bilinear forms of dynkin diagrams extended dynkin diagrams and irreducible root systems are also studied bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of dynkin diagrams and extended dynkin diagrams are discussed a homological interpretation of the bilinear forms 1 1 is given and z bilinear equivalences between them are discussed by applying well known results of bongartz loupias and zavadskij shkabara we give several characterisations of posets i with the euler form q i x b i x x weakly positive resp with the reduced euler form q i x b i x x weakly positive and posets i with the tits form q i x b i x x weakly positive

varieties with too many rational points we investigate fano varieties defined over a number field that contain subvarieties whose number of rational points of bounded height is comparable to the total number on the variety

lipschitz property for systems of linear mappings and bilinear forms abstract let g be a graph with undirected and directed edges its representation is given by assigning a vector space to each vertex a bilinear form on the corresponding vector spaces to each directed edge and a linear map to each directed edge two representations a and a of g are called isomorphic if there is a system of linear bijections between the vector spaces corresponding to the same vertices that transforms a to a we prove that if two representations are isomorphic and close to each other then their isomorphism can be chosen close to the identity

generalized solutions and hydrostatic approximation of the euler equations abstract solutions to the euler equations on a 3d domain d 3 typically the unit cube or the periodic unit cube can be formally obtained by minimizing the action of an incompressible fluid moving inside d 3 between two given configurations when these two configurations are very close to each other classical solutions do exist as shown by ebin and marsden however shnirelman found a class of data essentially 2d in the sense that they trivially depend on the vertical coordinate for which there cannot be any classical minimizer for such data generalized solutions can be shown to exist as a substitute for classical solutions these generalized solutions have unusual features that look highly unphysical in particular different fluid parcels can cross at the same point and at the same time but the pressure field which does not depend on the vertical coordinate is well and uniquely defined in the present paper we show that these generalized solutions are actually quite conventional in the sense they obey up to a suitable change of variable a well known variant widely used for geophysical flows of the 3d euler equations for which the vertical acceleration is neglected according to the so called hydrostatic approximation

canonical matrices of isometric operators on indefinite inner product spaces abstract we give canonical matrices of a pair a b consisting of a nondegenerate form b and a linear operator a satisfying b ax ay b x y on a vector space over f in the following cases f is an algebraically closed field of characteristic different from 2 or a real closed field and b is symmetric or skew symmetric f is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field and b is hermitian or skew hermitian with respect to any nonidentity involution on f these classification problems are wild if b may be degenerate we use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver this method was described in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 3 1988 481 501

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

on the distribution of points of bounded height on equivariant compactifications of vector groups we prove asymptotic formulas for the number of rational points of bounded height on smooth equivariant compactifications of the affine space

singularities of euler flow not out of the blue does three dimensional incompressible euler flow with smooth initial conditions develop a singularity with infinite vorticity after a finite time this blowup problem is still open after briefly reviewing what is known and pointing out some of the difficulties we propose to tackle this issue for the class of flows having analytic initial data for which hypothetical real singularities are preceded by singularities at complex locations we present some results concerning the nature of complex space singularities in two dimensions and propose a new strategy for the numerical investigation of blowup

a finite tame wild trichotomy theorem for tensor diagrams abstract in this paper we consider the problem of determining when two tensor networks are equivalent under a heterogeneous change of basis in particular to a tensor or string diagram in a certain monoidal category which we call tensor diagrams we formulate an associated abelian category of representations each representation corresponds to a tensor network on that diagram we then classify which tensor diagrams give rise to categories that are finite tame or wild in the traditional sense of representation theory for those tensor diagrams of finite and tame type we classify the indecomposable representations our main result is that a tensor diagram is wild if and only if it contains a vertex of degree at least three otherwise it is of tame or finite type

operators on positive semidefinite inner product spaces abstract let u be a semiunitary space i e a complex vector space with scalar product given by a positive semidefinite hermitian form if a linear operator a u u is bounded i e a u c u for some c r and all u u then the subspace u 0 u u u u 0 is invariant and so a defines the linear operators a 0 u 0 u 0 and a 1 u u 0 u u 0 let a be an indecomposable bounded operator on u such that 0 u 0 u let be an eigenvalue of a 0 we prove that the algebraic multiplicity of in a 1 is not less than the geometric multiplicity of in a 0 and the geometric multiplicity of in a 1 is not less than the number of jordan blocks j t of each fixed size t t in the jordan canonical form of a 0 we give canonical forms of selfadjoint and isometric operators on u and of hermitian forms on u for an arbitrary system of semiunitary spaces and linear mappings on between them we give an algorithm that reduces their matrices to canonical form its special cases are belitskii s and littlewood s algorithms for systems of linear operators on vector spaces and unitary spaces respectively

rigid systems of second order linear differential equations abstract we say that a system of differential equations x t a x t bx t cu t a b c m m c c m n is rigid if it can be reduced by substitutions x t sy t u t u y t vy t pv t with nonsingular s and p to each system obtained from it by a small enough perturbation of its matrices a b c we prove that there exists a rigid system for given m and n if and only if m n 1 5 2 and describe all rigid systems

wildness of the problem of classifying nilpotent lie algebras of vector fields in four variables abstract let f be a field of characteristic zero let w n f be the lie algebra of all f derivations with the lie bracket d 1 d 2 d 1 d 2 d 2 d 1 on the polynomial ring f x 1 x n the problem of classifying finite dimensional subalgebras of w n f was solved if n 2 and f c or f r we prove that this problem is wild if n 4 which means that it contains the classical unsolved problem of classifying matrix pairs up to similarity the structure of finite dimensional subalgebras of w n f is interesting since each derivation in case f r can be considered as a vector field with polynomial coefficients on the manifold r n

analytic structure of solutions of the one dimensional burgers equation with modified dissipation we use the one dimensional burgers equation to illustrate the effect of replacing the standard laplacian dissipation term by a more general function of the laplacian of which hyperviscosity is the best known example in equations of hydrodynamics we analyze the asymptotic structure of solutions in the fourier space at very high wave numbers by introducing an approach applicable to a wide class of hydrodynamical equations whose solutions are calculated in the limit of vanishing reynolds numbers from algebraic recursion relations involving iterated integrations we give a detailed analysis of their analytic structure for two different types of dissipation a hyperviscous and an exponentially growing dissipation term our results obtained in the limit of vanishing reynolds numbers are validated by high precision numerical simulations at non zero reynolds numbers we then study the bottleneck problem an intermediate asymptotics phenomenon which in the case of the burgers equation arises when ones uses dissipation terms such as hyperviscosity growing faster at high wave numbers than the standard laplacian dissipation term a linearized solution of the well known boundary layer limit of the burgers equation involving two numerically determined parameters gives a good description of the bottleneck region

tracing complex singularities with spectral methods abstract a numerical method for investigating the possibility of blow up after a finite time is introduced for a large class of nonlinear evolution problems with initial data analytic in the space variable s the solutions have for any t 0 complex space singularities at the edge of an analyticity strip of width t loss of regularity corresponds to the vanishing of t numerical integration by high resolution spectral methods reveals the large wavenumber behavior of the fourier transform of the solutions from which t is readily obtained its time evolution can be traced down to about one mesh length by extrapolation of t such numerical experiments provide evidence suggesting finite time blow up or all time regularity the method is tested on the inviscid and viscous burgers equations and is applied to the one dimensional nonlinear schrodinger equation with quartic potential and to the two dimensional incompressible euler equation all with periodic boundary conditions in the latter case evidence is found suggesting that existing all time regularity results can be substantially sharpened

on the euler equations of incompressible fluids euler equations of incompressible fluids use and enrich many branches of mathematics from integrable systems to geometric analysis they present important open physical and mathematical problems examples include the stable statistical behavior of ill posed free surface problems such as the rayleightaylor and kelvin helmholtz instabilities the paper describes some of the open problems related to the incompressible euler equations with emphasis on the blowup problem the inviscid limit and anomalous dissipation some of the recent results on the quasigeostrophic model are also mentioned

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

pairs of mutually annihilating operators abstract pairs a b of mutually annihilating operators ab ba 0 on a finite dimensional vector space over an algebraically closed field were classified by gelfand and ponomarev russian math surveys 23 1968 1 58 by method of linear relations the classification of a b over any field was derived by nazarova roiter sergeichuk and bondarenko j soviet math 3 1975 636 654 from the classification of finitely generated modules over a dyad of two local dedekind rings we give canonical matrices of a b over any field in an explicit form and our proof is constructive the matrices of a b are sequentially reduced to their canonical form by similarity transformations a b s 1 as s 1 bs

remarks on the classification of a pair of commuting semilinear operators abstract gelfand and ponomarev i m gelfand v a ponomarev remarks on the classification of a pair of commuting linear transformations in a finite dimensional vector space funct anal appl 3 1969 325 326 proved that the problem of classifying pairs of commuting linear operators contains the problem of classifying k tuples of linear operators for any k we prove an analogous statement for semilinear operators

on the geometry of thin exceptional sets in manin s conjecture manin s conjecture predicts the rate of growth of rational points of a bounded height after removing those lying on an exceptional set we study whether the exceptional set in manin s conjecture is a thin set using the minimal model program and boundedness of log fano varieties

on apolarity and generic canonical forms abstract the notion of s algebra is introduced the theory of apolarity and generic canonical forms for polynomials is generalized to s algebras over the complex field c we apply this theory to the problem of finding the essential rank of general symmetric and skew symmetric tensors upper bounds for the essential ranks are found by different combinatorial coverings

prime and almost prime integral points on principal homogeneous spaces we develop the affine sieve in the context of orbits of congruence subgroups of semisimple groups acting linearly on affine space in particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors in many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable where the orbit consists of the integers when the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number and thus establish the zariski density of matrices all of whose entries are prime numbers among the key tools used are explicit approximations to the generalized ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms

computation of canonical matrices for chains and cycles of linear mappings van dooren linear algebra appl 27 1979 103 constructed an algorithm for the computation of all irregular summands in kronecker s canonical form of a matrix pencil the algorithm is numerically stable since it uses only unitary transformations we construct a unitary algorithm for computation of the canonical form of the matrices of a chain of linear mappings v1 v2 vt andextend van dooren s algorithm to the matrices of a cycle of linear mappings whereall vi are complex vector spaces and each line denotes or

on the representation of integers by quadratic forms let n 4 and let q 2 z x1 xn be a non singular quadratic form when q is indefinite we provide new upper bounds for the least non trivial integral solution to the equation q 0 and when q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation q k is insoluble in integers despite being soluble modulo every prime power

sieving rational points on varieties an upper bound sieve for rational points on suitable varieties isdeveloped together with applications tocounting rational points in thin sets to local solubility in families and to the notion of friable rational pointswith respect to divisors in the special case of quadrics sharper estimates areobtained by developing a version of the selberg sieve for rational points

normal form of m by n by 2 matrices for equivalence abstract we study m n 2 matrices up to equivalence and give a canonical form of m 2 2 matrices over any field

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

the problems of classifying pairs of forms and local algebras with zero cube radical are wild we prove that over an algebraically closed field of characteristic not two the problems of classifying pairs of sesquilinear forms in which the second is hermitian pairs of bilinear forms in which the second is symmetric skew symmetric and local algebras with zero cube radical and square radical of dimension 2 are hopeless since each of them reduces to the problem of classifying pairs of n by n matrices up to simultaneous similarity

wildness of the problems of classifying two dimensional spaces of commuting linear operators and certain lie algebras abstract for each two dimensional vector space v of commuting n n matrices over a field f with at least 3 elements we denote by v the vector space of all n 1 n 1 matrices of the form a 0 0 with a v we prove the wildness of the problem of classifying lie algebras v with the bracket operation u v u v v u we also prove the wildness of the problem of classifying two dimensional vector spaces consisting of commuting linear operators on a vector space over a field

length realizability for pairs of quasi commuting matrices abstract for the pairs of quasi commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators

problems of classifying associative or lie algebras and triples of symmetric or skew symmetric matrices are wild we prove that the problems of classifying triples of symmetric or skew symmetric matrices up to congruence local commutative associative algebras with zero cube radical and square radical of dimension 3 and lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n by n matrices up to simultaneous similarity

isometric and selfadjoint operators on a vector space with nondegenerate diagonalizable form abstract let v be a vector space over a field f with scalar product given by a nondegenerate sesquilinear form whose matrix is diagonal in some basis if f c then we give canonical matrices of isometric and selfadjoint operators on v using known classifications of isometric and selfadjoint operators on a complex vector space with nondegenerate hermitian form if f is a field of characteristic different from 2 then we give canonical matrices of isometric selfadjoint and skewadjoint operators on v up to classification of symmetric and hermitian forms over finite extensions of f

a canonical form for nonderogatory matrices under unitary similarity abstract a square matrix is nonderogatory if its jordan blocks have distinct eigenvalues we give canonical forms for nonderogatory complex matrices up to unitary similarity and pairs of complex matrices up to similarity in which one matrix has distinct eigenvalues the types of these canonical forms are given by undirected and respectively directed graphs with no undirected cycles

topological classification of chains of linear mappings abstract we consider systems of linear mappings a 1 a t 1 of the form a u 1 a 1 2 a 2 u 3 a 3 a t 1 u t in which u 1 u t are unitary or euclidean spaces and each line is either the arrow or the arrow let a be transformed to b v 1 b 1 v 2 b 2 v 3 b 3 b t 1 v t by a system i u i v i i 1 t of bijections we say that a and b are linearly isomorphic if all i are linear considering all u i and v i as metric spaces we say that a and b are topologically isomorphic if all i and i 1 are continuous we prove that a and b are topologically isomorphic if and only if they are linearly isomorphic

canonical forms of two by two by two matrices we determine the canonical forms of two by two by two matrices in order to do so we present a collection of covariants that separates the canonical forms the covariants also fulfill a number of identities moreover we show that these canonical forms and covariants are related to those of the binary cubic and skew symmetric tensors of step 3 in the exterior algebra ext u where dim u 6

poincare series of semi invariants of 2 2 matrices abstract the poincare series of the algebra of sl 2 c sl 2 c invariants of m tuples of 2 2 matrices is presented both as a rational function and as a series of schur functions we show that this algebra of invariants is generated by the determinants the mixed discriminants and the discriminants of 2 2 matrices consequences on invariants of three dimensional matrices of the shape 2 2 m are discussed for arbitrary n 2 we prove an explicit functional equation for the poincare series of the sl n c sl n c invariants of m tuples of n n matrices

canonical form of m by 2 by 2 matrices over a field of characteristic other than two we give a canonical form of m 2 2 matrices for equivalence over any field of characteristic not two

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

cubic hypersurfaces and a version of the circle method for number fields a version of the hardy littlewood circle method is developed for number fields k q and is used to show that non singular projective cubic hypersurfaces over k always have a k rational point when they have dimension at least 8

systems of subspaces of a unitary space abstract for a finite poset p p 1 p t we study systems u 1 u t u of subspaces of a unitary space u such that u i u j if p i p j two systems u 1 u t u and v 1 v t v are said to be isometric if there exists an isometry u v such that u i v i we classify such systems up to isometry if p is a semichain we prove that the problem of their classification is unitarily wild if p is not a semichain a classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space which is hopeless in a certain sense

unitary and euclidean representations of a quiver abstract a unitary euclidean representation of a quiver is given by assigning to each vertex a unitary euclidean vector space and to each arrow a linear mapping of the corresponding vector spaces we recall an algorithm for reducing the matrices of a unitary representation to canonical form give a certain description of the representations of canonical form and reduce the problem of classifying euclidean representations to the problem of classifying unitary representations we also describe the set of dimensions of all indecomposable unitary euclidean representations of a quiver and establish the number of parameters in an indecomposable unitary representation of a given dimension

rational points on quartic hypersurfaces let x be a projective non singular quartic hypersurface of dimension 39 or more which is defined over we show that x is non empty provided that x is non empty and x has p adic points for every prime p 2009 walter de gruyter

some remarks on li sinai type solutions of the burgers equation for a class of complex valued initial conditions we construct solutions for the inviscid burgers equations of the same type as those recently introduced by li and sinai obtained by very simple considerations these solutions correspond to stable distributions with 1 2 we furthermore show that solutions of non gaussian type can be constructed for the viscous burgers equation

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

wildness for tensors abstract in representation theory a classification problem is called wild if it contains the problem of classifying matrix pairs up to simultaneous similarity the latter problem is considered hopeless it contains the problem of classifying an arbitrary finite system of vector spaces and linear mappings between them we prove that an analogous universal problem in the theory of tensors of order at most 3 over an arbitrary field is the problem of classifying three dimensional arrays up to equivalence transformations a i j k i 1 m j 1 n k 1 t i j k a i j k u i i v j j w k k i 1 m j 1 n k 1 t in which u i i v j j w k k are nonsingular matrices this problem contains the problem of classifying an arbitrary system of tensors of order at most three

three equipped posets and their representations and corepresentations inseparable case abstract we define three equipped posets i e partially ordered sets equipped with three kinds of binary relations and their representations and corepresentations over an inseparable cubic field extension f g in characteristic 3 these representations and corepresentations lead to some matrix problems of mixed type over the pair of fields f g through these problems of linear algebra we completely describe in evident matrix form the indecomposables for some critical three equipped posets reducing the task for the rest of the critical sets to some matrix problems containing the pseudolinear pencil problem as a subproblem

canonical form of three fermion pure states with six single particle states we construct a canonical form for pure states in 3 c6 the three fermion system with six single particle states under local unitary lu transformations i e the unitary group u 6 we also construct a minimal set of generators of the algebra of polynomial u 6 invariants on 3 c6 it turns out that this algebra is isomorphic to the algebra of polynomial lu invariants of three qubits which are additionally invariant under qubit permutations as a consequence of this surprising fact we deduce that there is a one to one correspondence between the u 6 orbits of pure three fermion states in 3 c6 and the lu orbits of pure three qubit states when qubit permutations are allowed as an important byproduct we obtain a new canonical form for pure three qubit states under lu transformations u 2 u 2 u 2 no qubit permutations allowed

on complex singularities of the 2d euler equation at short times abstract we present a study of complex singularities of a two parameter family of solutions for the two dimensional euler equation with periodic boundary conditions and initial conditions 0 z 1 z 2 f p cos p z f q cos q z in the short time asymptotic regime as has been shown numerically in pauls et al w pauls t matsumoto u frisch j bec nature of complex singularities for the 2d euler equation physica d 219 2006 40 59 the type of the singularities depends on the angle between the modes p and q thus the fourier coefficients of the solutions decrease as g k c k e k with the exponent depending on here we show for the two particular cases of going to zero and to that the type of the singularities can be determined very accurately being characterised by 5 2 and 3 respectively in these two cases we are also able to determine the subdominant corrections furthermore we find that the geometry of the singularities in these two cases is completely different the singular manifold being located over different points in the real domain

descent on fibrations over the projective line we formulate a general set up for the descent method of j l colliot thelene and j j sansuc applied to varieties fibred over the projective line over a number field this makes it possible to prove that the manin obstruction to the hasse principle and weak approximation is the only one provided there are only few degenerate fibres usually two three in some cases and that sufficiently many smooth fc fibres satisfy the hasse principle and weak approximation we introduce a new concept of what should be called a degenerate fibre the so called split fibres the property which depends only on the generic fibre and not on the choice of a particular model this yields a simpler and more general approach to the previous results of that kind 0 introduction motivated by hasse s theorem on quadratic forms one says that the hasse principle holds for a family of geometrically integral smooth and proper varieties over a number field k9 if for every member of this family the existence of rational points over all the completions of k implies the existence of a rational point the fact that the hasse principle holds for some family signif icantly simplifies the study of its arithmetic because the problem of determining whether or not there is a rational point is then reduced to a finite number of

a borel transform method for locating singularities of taylor and fourier series given a taylor series with a finite radius of convergence its borel transform defines an entire function a theorem of polya relates the large distance behavior of the borel transform in different directions to singularities of the original function with the help of the new asymptotic interpolation method of van der hoeven we show that from the knowledge of a large number of taylor coefficients we can identify precisely the location of such singularities as well as their type when they are isolated there is no risk of getting artefacts with this method which also gives us access to some of the singularities beyond the convergence disk the method can also be applied to fourier series of analytic periodic functions and is here tested on various instances constructed from solutions to the burgers equation large precision on scaling exponents up to twenty accurate digits can be achieved

specht s criterion for systems of linear mappings abstract w specht 1940 proved that two n n complex matrices a and b are unitarily similar if and only if trace w a a trace w b b for every word w x y in two noncommuting variables we extend his criterion and its generalizations by n a wiegmann 1961 and n jing 2015 to an arbitrary system a consisting of complex or real inner product spaces and linear mappings among them we represent such a system by the directed graph q a whose vertices are inner product spaces and arrows are linear mappings denote by q a the directed graph obtained by enlarging to q a the adjoint linear mappings we prove that a system a is transformed by isometries of its spaces to a system b if and only if the traces of all closed directed walks in q a and q b coincide