numerical radius and zero pattern of matrices abstract let a be an n n complex matrix and r be the maximum size of its principal submatrices with no off diagonal zero entries suppose a has zero main diagonal and x is a unit n vector then letting a be the frobenius norm of a we show that a x x 2 1 1 2 r 1 2 n a 2 this inequality is tight within an additive term o r n 2 if the matrix a is hermitian then a x x 2 1 1 r a 2 this inequality is sharp moreover it implies the turan theorem for graphs

a primer of perron frobenius theory for matrix polynomials abstract we present an extension of perron frobenius theory to the spectra and numerical ranges of perron polynomials namely matrix polynomials of the form l i m a m 1 m 1 a 1 a 0 where the coefficient matrices are entrywise nonnegative our approach relies on the companion matrix linearization first we recount the generalization of the perron frobenius theorem to perron polynomials and report some of its consequences subsequently we examine the role of l in multistep difference equations and provide a multistep version of the fundamental theorem of demography finally we extend issos results on the numerical range of nonnegative matrices to perron polynomials

the existence of an abelian variety over q isogenous to no jacobian we prove the existence of an abelian variety a of dimension g over q that is not isogenous to any jacobian subject to the necessary condition g 3 recently c chai and f oort gave such a proof assuming the andr eoort conjecture we modify their proof by constructing a special sequence of cm points for which we can avoid any unproven hypotheses we make use of various techniques from the recent work of klingler yafaev et al

on the number of quintic fields we show that the number of quintic number fields whose discriminant does not exceed x in absolute value is bounded by a constant times x1 e for any e 0 this may be compared with the conjecture that the number of such fields is asymptotic to a constant times x as x tends to infinity

numerical range of some doubly stochastic matrices a classification of all possible shapes is given for numerical ranges of 4 i 4 doubly stochastic matrices the tests determining the shape are also provided along with illustrating examples

an algorithm to compute relative cubic fields let k be an imaginary quadratic number field with class number 1 we describe a new essentially linear time algorithm to list all isomorphism classes of cubic extensions l k up to a bound x on the norm of the relative discriminant ideal the main tools are taniguchi s generalization of davenport heilbronn parametrisation of cubic extensions and reduction theory for binary cubic forms over imaginary quadratic fields finally we give numerical data for k q i and we compare our results with ray class field algorithm ones and with asymptotic heuristics based on a generalization of roberts conjecture

counting arithmetic lattices and surfaces we give estimates on the number al h x of conjugacy classes of arithmetic lattices of covolume at most x in a simple lie group h in particular we obtain a first concrete estimate on the number of arithmetic 3 manifolds of volume at most x our main result is for the classical case h psl 2 where we show that limx log al h x x log x 1 2 the proofs use several different techniques geometric bounding the number of generators of as a function of its covolume number theoretic bounding the number of maximal such and sharp estimates on the character values of the symmetric groups to bound the subgroup growth of

asymptotic expansions of mellin convolutions by means of analytic continuation we present a method for deriving asymptotic expansions of integrals of the form 0 f t h xt dt for small x based on analytic continuation the expansion is given in terms of two asymptotic sequences the coefficients of both sequences being mellin transforms of h and f many known and unknown asymptotic expansions of important integral transforms are derived trivially from the approach presented here this paper reconsiders earlier work of mcclure and wong explicit error terms for asymptotic expansions of stieltjes transforms j inst math appl 22 1978 129 145 exact remainders for asymptotic expansions of fractional integrals j inst math appl 24 1979 139 147 and asymptotic approximations of integrals academic press new york 1989 chaps 5 where elements of distribution theory are used and wong explicit error terms for asymptotic expansions of mellin convolutions j math anal appl 72 2 1979 740 756 where as in the present paper the asymptotic expansions are obtained without the use of distributions in this paper we re derive the expansions given in wong explicit error terms for asymptotic expansions of mellin convolutions j math anal appl 72 2 1979 740 756 by using a different approach and we obtain new results which are not present in wong explicit error terms for asymptotic expansions of mellin convolutions j math anal appl 72 2 1979 740 756 a proof of the asymptotic character of the expansions and accurate error bounds

on matrices whose numerical ranges have circular or weak circular symmetry in 18 among other equivalent conditions it is proved that a square complex matrix a is permutationally similar to a block shift matrix if and only if for any complex matrix b with the same zero pattern as a w b the numerical range of b is a circular disk centered at the origin in this paper we add a long list of further new equivalent conditions the corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2 m where m 2 is a given positive integer is also proved many interesting by products are obtained in particular on the numerical range of a square nonnegative matrix a the following unexpected results are established i when the undirected graph of a is connected if w a is a circular disk centered at the origin then so is w b for any complex matrix b with the same zero pattern as a ii when a is irreducible if is an eigenvalue in the peripheral spectrum of a that lies on the boundary of w a then is a sharp point of w a we also obtain results on the numerical range of an irreducible square nonnegative matrix which strengthen or clarify the work of issos 9 and nylen and tam 14 on this topic open questions are posed at the end

asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity we find two convergent series expansions for legendre s first incomplete elliptic integral f l k in terms of recursively computed elementary functions both expansions are valid at every point of the unit square 0 l k 1 truncated expansions yield asymptotic approximations for f l k as l and or k tend to unity including the case when logarithmic singularity l k 1 is approached from any direction explicit error bounds are given at every order of approximation for the reader s convenience we present explicit expressions for low order approximations and numerical examples to illustrate their accuracy our derivation is based on rearrangements of some known double series expansions hypergeometric summation algorithms and inequalities for hypergeometric functions

numerical ranges of companion matrices we show that an n by n companion matrix a can have at most n line segments on the boundary w a of its numerical range w a and it has exactly n line segments on w a if and only if for n odd a is unitary and for n even a is unitarily equivalent to the direct sum a1 a2 of two n 2 by n 2 companion matrices a1

an approximation for zero balanced appell function f1 near 1 1 abstract we suggest an approximation for the zero balanced appell hypergeometric function f 1 near the singular point 1 1 our approximation can be viewed as a generalization of ramanujan s approximation for zero balanced f 1 2 and is expressed in terms of f 2 3 we find an error bound and prove some basic properties of the suggested approximation which reproduce the similar properties of the appell function our approximation reduces to the approximation of carlson gustafson when the appell function reduces to the first incomplete elliptic integral

the numerical range of a nonnegative matrix abstract we offer an almost self contained development of perron frobenius type results for the numerical range of an irreducible nonnegative matrix rederiving and completing the previous work of issos nylen and tam and tam and yang on this topic we solve the open problem of characterizing nonnegative matrices whose numerical ranges are regular convex polygons with center at the origin some related results are obtained and some open problems are also posed

gau wu numbers of nonnegative matrices abstract for any n by n matrix a we consider the maximum number k k a of orthonormal vectors x j c n such that the scalar products a x j x j lie on the boundary w a of the numerical range w a this number is called the gau wu number of the matrix a if a is an n by n n 2 nonnegative matrix with the permutationally irreducible real part of the form 0 a 1 0 0 a m 1 0 0 where m 3 and the diagonal zeros are zero square matrices then k a has an upper bound m 1 in addition we also obtain necessary and sufficient conditions for k a m 1 for such a matrix a another class of nonnegative matrices we study is the doubly stochastic ones we prove that the value of k a is equal to 3 for any 3 by 3 doubly stochastic matrix a for any 4 by 4 doubly stochastic matrix we also determine its numerical range which is then applied to find its gau wu numbers furthermore a lower bound of the gau wu number k a is also found for a general n by n n 5 doubly stochastic matrix a via the possible shapes of w a

hyperbolic forms associated with cyclic weighted shift matrices abstract the singular points of the curve of a hyperbolic form associated with a cyclic weighted shift matrix are examined it is shown that the singular points of such a curve are real nodes some results related the numerical ranges of cyclic weighted shift matrices are presented in particular the existence of flat portions on the boundary of the numerical range depends on the reducibility of the hyperbolic form further an algebraic method is provided for the decomposition of reducible form which leads to a criterion for the periodicity of the weights

the boundary of higher rank numerical ranges abstract the rank k numerical range of an n n matrix a is defined as k a c pap p for some rank k orthogonal projection p this study presents a method to generate the boundary of k a and examines the flat portions of the boundary of the rank k numerical range of a matrix associated with a roulette curve a sufficient condition is given to ensure that the rank k numerical range is not attainable by a classical numerical range

numerical ranges of nilpotent operators abstract for any operator a on a hilbert space let w a w a and w 0 a denote its numerical range numerical radius and the distance from the origin to the boundary of its numerical range respectively we prove that if a n 0 then w a n 1 w 0 a and moreover if a attains its numerical radius then the following are equivalent 1 w a n 1 w 0 a 2 a is unitarily equivalent to an operator of the form aa n a where a is a scalar satisfying a 2 w 0 a a n is the n by n matrix 0 1 1 0 1 0 and a is some other operator and 3 w a bw a n for some scalar b

counting primitive points of bounded height let k be a number field and k a finite extension of k we count points of bounded height in projective space over the field k generating the extension k k as the height gets large we derive asymptotic estimates with a particularly good error term respecting the extension k k in a future paper we will use these results to get asymptotic estimates for the number of points of fixed degree over k we also introduce the notion of an adelic lipschitz height generalizing that of masser and vaaler this will lead to further applications involving points of fixed degree on linear varieties and algebraic numbers of fixed degree satisfying certain subfield conditions

tabulation of cubic function fields via polynomial binary cubic forms we present a method for tabulating all cubic function fields over mathbb f q t whose discriminant d has either odd degree or even degree and the leading coefficient of 3d is a non square in mathbb q up to a given bound x on d q deg d our method is based on a generalization of belabas method for tabulating cubic number fields the main theoretical ingredient is a generalization of a theorem of davenport and heilbronn to cubic function fields along with a reduction theory for binary cubic forms that provides an efficient way to compute equivalence classes of binary cubic forms the algorithm requires o q 4 x 1 epsilon field operations when d has odd degree and o q 5 x 1 epsilon when d has even degree it performs quite well in practice the algorithm examples and numerical data for q 5 7 11 13 are included

diagonals and numerical ranges of direct sums of matrices abstract for any n by n matrix a we consider the maximum number k k a for which there is a k by k compression of a with all its diagonal entries in the boundary w a of the numerical range w a of a if a is a normal or a quadratic matrix then the exact value of k a can be computed for a matrix a of the form b c we show that k a 2 if and only if the numerical range of one summand say b is contained in the interior of the numerical range of the other summand c and k c 2 for an irreducible matrix a we can determine exactly when the value of k a equals the size of a these are then applied to determine k a for a reducible matrix a of size 4 in terms of the shape of w a

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

small isospectral and nonisometric orbifolds of dimension 2 and 3 revisiting a construction due to vigneras we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic fuchsian and kleinian groups that are laplace isospectral in fact representation equivalent but nonisometric

numerical ranges and gersgorin discs abstract for a complex matrix a a ij i j 1 n let w a be its numerical range and let g a be the convex hull of i 1 n z c z a ii i j a ij a ji 2 and g a g u au u n by n unitary it is known that w a is always contained in g a and hence in g a in this paper we consider conditions for w a to be equal to g a or g a we show that if w a g a then the boundary of w a consists only of circular arcs and line segments if moreover a is unitarily irreducible then w a is a circular disc almost complete characterizations of 2 by 2 and 3 by 3 matrices a for which w a g a are obtained we also give criteria for the equality of w a and g a in particular such a s among the permutationally irreducible ones must have even sizes we also characterize those a s with size 2 or 4 which satisfy w a g a

on the arithmetic determination of the trace abstract let k be a number field which is tame and non totally real in this article we give a numerical criterion depending only on the ramification behavior of ramified primes in k to decide whether or not the integral trace of k is isometric to the integral trace of another number field l as a byproduct of our proofs here and in contrast with our previous results for cubic fields of positive discriminant we show that for cubic fields of negative discriminant isometry between integral traces is equivalent to equality of discriminants

series expansions for the third incomplete elliptic integral via partial fraction decompositions we find convergent double series expansions for legendre s third incomplete elliptic integral valid in overlapping subdomains of the unit square truncated expansions provide asymptotic approximations in the neighborhood of the logarithmic singularity 1 1 if one of the variables approaches this point faster than the other each approximation is accompanied by an error bound

zero dilation index of a finite matrix abstract for an n by n complex matrix a we define its zero dilation index d a as the largest size of a zero matrix which can be dilated to a this is the same as the maximum k 1 for which 0 is in the rank k numerical range of a using a result of li and sze we show that if d a 2 n 3 then under unitary similarity a has the zero matrix of size 3 d a 2 n as a direct summand it complements the known fact that if d a n 2 then 0 is an eigenvalue of a we then use it to give a complete characterization of n by n matrices a with d a n 1 namely a satisfies this condition if and only if it is unitarily similar to b 0 n 3 where b is a 3 by 3 matrix whose numerical range w b is an elliptic disc and whose eigenvalue other than the two foci of w b is 0 we also determine the value of d a for any normal matrix a and any weighted permutation matrix a with zero diagonals

tight bounds on the spectral radius of asymmetric nonnegative matrices abstract for an arbitrary asymmetric nonnegative n n matrix a we identify a pair of symmetric matrices whose largest eigenvalues bound the spectral radius of a furthermore we show that these bounding matrices are best possible by characterizing matrices a which attain equality with either the upper or the lower bounding matrix the lower bound may be extended to some matrices with negative entries provided they have no negative cycles

the number of extensions of a number field with fixed degree and bounded discriminant we give an upper bound on the number of extensions of a fixed number field of prescribed degree and discriminant x these bounds improve on work of schmidt we also prove various related results such as lower bounds for the number of extensions and upper bounds for galois extensions

circularity of the numerical range abstract an equivalent condition on a 3 square complex or a 4 square real upper triangular matrix is found for its numerical range to be a circular disk centered at the origin sufficient conditions for the circularity of the numerical range of certain sparse matrices are also given in terms of graphs

numerical ranges of weighted shift matrices abstract we show that if a is an n by n n 3 matrix of the form 0 a 1 0 a n 1 a n 0 then the boundary of its numerical range contains a line segment if and only if the a j s are nonzero and the numerical ranges of the n 1 by n 1 principal submatrices of a are all equal for n 3 this is the case if and only if a 1 a 2 a 3 0 in which case w a the numerical range of a is the equilateral triangular region with vertices the three cubic roots of a 1 a 2 a 3 for n 4 the condition becomes a 1 a 3 0 and a 2 a 4 0 in which case w a is the convex hull of two degenerate or otherwise ellipses

homological stability for hurwitz spaces and the cohen lenstra conjecture over function fields we prove a homological stabilization theorem for hurwitz spaces moduli spaces of branched covers of the complex projective line this has the following arithmetic consequence let 2 be prime and a a nite abelian group then there exists q q a such that for q greater than q a positive fraction of quadratic extensions of fq t have the part of their class group isomorphic to a

numerical ranges of row stochastic matrices abstract in this paper we consider properties of the numerical range of an n by n row stochastic matrix a it is shown that the numerical radius of a satisfies 1 w a 1 n 2 and moreover w a 1 resp w a 1 n 2 if and only if a is doubly stochastic resp a 0 1 1 0 j th for some j 1 j n a complete characterization of the a s for which the zero matrix of size n 1 can be dilated to a is also given finally for each n 2 we determine the smallest rectangular region in the complex plane whose sides are parallel to the x and y axis and which contains the numerical ranges of all n by n row stochastic matrices

numerical range of a doubly stochastic matrix abstract we discuss the numerical range of a doubly stochastic matrix and relate it to the structure of the matrix also we characterize those subsets of which are the numerical range of a 3 3 doubly stochastic matrix

an effective chebotarev density theorem for families of number fields with an application to formula see text torsion in class groups we prove a new effective chebotarev density theorem for galois extensions l mathbb q that allows one to count small primes even as small as an arbitrarily small power of the discriminant of l this theorem holds for the galois closures of almost all number fields that lie in an appropriate family of field extensions previously applying chebotarev in such small ranges required assuming the generalized riemann hypothesis the error term in this new chebotarev density theorem also avoids the effect of an exceptional zero of the dedekind zeta function of l without assuming grh we give many different appropriate families including families of arbitrarily large degree to do this we first prove a new effective chebotarev density theorem that requires a zero free region of the dedekind zeta function then we prove that almost all number fields in our families yield such a zero free region the innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non cuspidall functions this builds on and greatly generalizes the applicability of work of kowalski and michel on the average density of zeroes of a family of cuspidall functions a surprising feature of this new method which we expect will have independent interest is that we control the number of zeroes in the family of l functions by bounding the number of certain associated fields with fixed discriminant as an application of the new chebotarev density theorem we prove the first nontrivial upper bounds for ell torsion in class groups for all integers ell ge 1 applicable to infinite families of fields of arbitrarily large degree

3 3 matrices with a flat portion on the boundary of the numerical range the set of unitarily irreducible complex 3 3 matrices whose numerical range has a flat portion is explicitly described it is proved that the set is connected and real analytic properties of the set are studied connectedness of the sets of unitarily irreducible 3 3 matrices having two other possible shapes of their numerical ranges is proved as well

perron frobenius type results on the numerical range we present results connecting the shape of the numerical range to intrinsic properties of a matrix a w hena is a nonnegative matrix these results are to a large extent analogous to the perron frobenius theory especially as it pertains to irreducibility and cyclicity in the combinatorial sense special attention is given to polygonal circular and elliptic numerical ranges the main vehicles for obtaining these results are the hermitian and skew hermitian parts of a as well as levinger s transformation aa 1 a a 2002 elsevier science

on the gau wu number for some classes of matrices abstract for a given n n matrix a let k a stand for the maximal number of orthonormal vectors x j such that the scalar products a x j x j lie on the boundary of the numerical range w a this number was recently introduced by gau and wu and we therefore call it the gau wu number of the matrix a we compute k a for two classes of n n matrices a a simple and explicit expression for k a for tridiagonal toeplitz matrices a is derived furthermore we prove that k a 2 for every pure almost normal matrix a note that for every matrix a we have k a 2 and for normal matrices a we have k a n so our results show that pure almost normal matrices are in fact as far from normal as possible with respect to the gau wu number finally matrices with maximal gau wu number k a n are considered

on matrices with cyclic structure abstract the relations between several necessary conditions for a square complex matrix a to be m cyclic are examined these conditions are known to be equivalent if a is an irreducible nonnegative matrix whose index of imprimitivity is m in particular we find that if the digraph of a contains at least one cycle with nonzero signed length then the following conditions are each equivalent to the m cyclicity of a i a is diagonally similar to e 2 i m a ii all cycles in the digraph of a have signed length an integral multiple of m in the course of our investigations we lay down the groundwork of the theory of cyclically m partite or linearly partite digraphs and characterize these digraphs in terms of the signed lengths of their cycles the characterization of diagonal similarity between matrices in terms of matrix cycle products due to saunders and schneider plays a key role in our development our investigations also lead to a new illuminating conceptual proof for the second part of the frobenius theorem on an irreducible nonnegative matrix the connection between the m cyclicity of a square complex matrix and that of its associated collection of elementary jordan blocks is studied in the second half of this paper in particular for an m cyclic collection u it is found that the problem of determining all m tuples k 1 k m of positive integers for which there exists an m cyclic matrix a in the superdiagonal k 1 k m block form such that u a u is equivalent to determine all row sum vectors of 0 1 matrices that have a prescribed column sum vector and each of whose column vectors has cyclically consecutive equal components as a by product we also obtain an equivalent condition on m given square complex matrices b 1 b m so that there exist complex rectangular matrices a 1 a m that satisfy b j a j a m a 1 a j 1 for j 1 m thus completing the work of flanders for the case m 2

maximizing numerical radii of weighted shifts under weight permutations abstract let w i c 1 i n and l s n the symmetric group of all permutations of 1 2 n suppose a l is the weighted cyclic matrix a l 0 w l 1 0 w l n 1 w l n 0 and w a l denotes its numerical radius we characterize those s n which satisfy w a max l s n w a l the characterizations for unilateral and bilateral weighted backward shifts are also obtained

diagonals and numerical ranges of weighted shift matrices for any n by n matrix a we consider the maximum number k k a forwhichthereisak by kcompressionofawithallitsdiagonal entries in the boundary w a of the numerical range w a of a for any such compression we give a standard model under unitary equivalence fora this is then applied to determine the value ofk a for a of size 3 in terms of the shape of w a w hena is a matrix of the form

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

condition for the numerical range to contain an elliptic disc for an n by n matrix a and an elliptic disc e in the plane we show that the sum of the number of common supporting lines and the number of common intersection points to e and the numerical range w a of a should be at least 2n 1 in order to guarantee that e be contained in w a this generalizes previous results of anderson and thompson as an application our result is used to verify a special case of the poncelet property conjecture 2003 elsevier science inc all rights reserved

the envelope of tridiagonal toeplitz matrices and block shift matrices abstract the envelope of a square complex matrix is a spectrum encompassing region in the complex plane it is contained in and is akin to the numerical range in the sense that the envelope is obtained as an infinite intersection of unbounded regions contiguous to cubic curves rather than half planes in this article the geometry and properties of the envelopes of special matrices are examined in particular symmetries of the envelope of a tridiagonal toeplitz matrix are obtained and the envelopes of block shift matrices jordan blocks and 2 2 matrices are explicitly characterized

on the numerical radius and its applications abstract we give a brief account of the numerical radius of a linear bounded operator on a hilbert space and some of its better known properties both finite and infinite dimensional aspects are discussed as well as applications to stability theory of finite difference approximations for hyperbolic initial value problems

numerical ranges of weighted shifts abstract let a be a unilateral resp bilateral weighted shift with weights w n n 0 resp n eckstein and racz showed before that a has its numerical range w a contained in the closed unit disc if and only if there is a sequence a n n 0 resp a n n in 1 1 such that w n 2 1 a n 1 a n 1 for all n in terms of such a n s we obtain a necessary and sufficient condition for w a to be open if the w n s are periodic we show that the a n s can also be chosen to be periodic as a result we give an alternative proof for the openness of w a for an a with periodic weights which was first proven by stout more generally a conjecture of his on the openness of w a for a with split periodic weights is also confirmed

counting maximal arithmetic subgroups we study the growth rate of the number of maximal arithmetic subgroups of bounded covolumes in a semisimple lie group using an extension of the method developed by borel and prasad