invariants of 3 manifolds via link polynomials and quantum groups the aim of this paper is to construct new topological invariants of compact oriented 3 manifolds and of framed links in such manifolds our invariant of a link in a closed oriented 3 manifold is a sequence of complex numbers parametrized by complex roots of 1 for a framed link in s 3 the terms of the sequence are equale to the values of the suitably parametrized jones polynomial of the link in the corresponding roots of 1 in the case of manifolds with boundary our invariant is a sequence of finite dimensional complex linear operators this produces from each root of unity q a 3 dimensional topological quantum field theory

idempotency of linear combinations of an idempotent matrix and a tripotent matrix abstract the problem of characterizing situations in which a linear combination c c 1 a c 2 b of an idempotent matrix a and a tripotent matrix b is an idempotent matrix is thoroughly studied in two particular cases of this problem when either b or b is an idempotent matrix a complete solution follows from the main result in linear algebra appl 321 2000 3 in the present paper a complete solution is established in all the remaining cases when b is an essentially tripotent matrix in the sense that both idempotent matrices b 1 and b 2 constituting its unique decomposition b b 1 b 2 are nonzero the problem is considered also under the additional assumption that the differences a b 1 and a b 2 are hermitian matrices this obviously covers the case when a b 1 and b 2 are hermitian themselves when the problem can be interpreted from a statistical point of view

convexity and matrix means abstract in this article we present some mean inequalities for convex functions that lead to some generalized inequalities treating the arithmetic geometric and harmonic means for numbers and matrices our first main inequality will be 1 f 0 f 1 f 1 f 0 f 1 f 1 1 for the convex function f when 1 and 0 1 moreover when 1 the inequality will be valid for operator convex functions then by selecting an appropriate convex function we obtain certain matrix inequalities in particular we obtain several mixed mean inequalities for operators using real and operator convexity our discussion will lead to new multiplicative refinements and reverses of the heinz and holder inequalities for matrices new and refined trace and determinant inequalities the significance of this work is its general treatment where convexity is the only needed property

numerical radius of hadamard product of matrices it is known that the numerical radius of the hadamard product a b of two n by n matrices a and b is related to those of a and b by a w a b 2w a w b b w a b w a w b if one of a and b is normal and c w a b maxi aii w b if a aij i j 1n is positive semidefinite in this paper we give complete characterizations of a and b for which the equality is attained the matrices involved can be considered as elaborate generalizations of the equality attaining a 00a0 and b 00b0 for a a a100a2 a1 a2 and b w b for b and a a1a3a2a4 0 a1 a4 and b w b for c

characterizations and linear combinations of k generalized projectors abstract we extend generalized projectors introduced by gros and trenkler generalized and hypergeneralized projectors linear algebra appl 264 1997 463 474 to k generalized projectors and we characterize them obtaining results in the aforesaid paper as a consequence moreover we list all situations when a linear combination of commuting k generalized projectors is a k generalized projector the method for obtaining this result permits to give a revisited version of the main result by baksalary and baksalary on linear combinations of generalized projectors linear algebra appl 388 2004 17 24 in addition the case of orthogonal projectors is also analyzed

integrality for tqfts we discuss ways that the ring of coefficients for a tqft can be reduced if one restricts somewhat the allowed cobordisms when we apply these methods to a tqft associated to so 3 at an odd prime p we obtain a functor from a somewhat restricted cobordism category to the category of free finitely generated modules over a ring of cyclotomic integers z 2p if p 1 mod 4 and z 4p if p 1 mod 4 where k is a primitive kth root of unity we study the quantum invariants of prime power order simple cyclic covers of 3 manifolds we define new invariants arising from strong shift equivalence and integrality similar results are obtained for some other tqfts but the modules are only guaranteed to be projective

triangular truncation and finding the norm of a hadamard multiplier for b a fixed matrix we study the problem of using a criterion of haagerup to find the norm of the map a a b where is the hadamard or entrywise product of matrices the techniques developed are applied to triangular truncation and it is proved that if kn is the norm of triangular truncation of n n matrices then knlogn 1

profinite completions of burnside type quotients of surface groups using quantum representations of mapping class groups we prove that profinite completions of burnside type surface group quotients are not virtually prosolvable in general further we construct infinitely many finite simple characteristic quotients of surface groups

idempotency of linear combinations of an idempotent matrix and a t potent matrix that commute abstract this paper deals with idempotent matrices i e a 2 a and t potent matrices i e b t b when both matrices commute we derive a list of all complex numbers c 1 and c 2 such that c 1 a c 2 b is an idempotent matrix in addition the real case is also analyzed

idempotency of linear combinations of three idempotent matrices two of which are commuting abstract the considerations of the present paper were inspired by baksalary o m baksalary idempotency of linear combinations of three idempotent matrices two of which are disjoint linear algebra appl 388 2004 67 78 who characterized all situations in which a linear combination p c 1 p 1 c 2 p 2 c 3 p 3 with c i i 1 2 3 being nonzero complex scalars and p i i 1 2 3 being nonzero complex idempotent matrices such that two of them p 1 and p 2 say are disjoint i e satisfy condition p 1 p 2 0 p 2 p 1 is an idempotent matrix in the present paper by utilizing different formalism than the one used by baksalary the results given in the above mentioned paper are generalized by weakening the assumption expressing the disjointness of p 1 and p 2 to the commutativity condition p 1 p 2 p 2 p 1

schur multiplier norms for loewner matrices abstract we study upper bounds on the schur multiplier norm of loewner matrices for concave and convex functions these bounds then immediately lead to upper bounds on the ratio of schatten q norms of commutators a f b q a b q we also consider operator monotone functions for which sharper bounds are obtained

numerical radius inequalities for certain 2 2 operator matrices ii abstract we give several sharp numerical radius inequalities for certain 2 2 operator matrices among other inequalities it is shown that if a and b be operators in b h then w a b 0 0 1 2 a a a b b 1 2 and 2 w 0 a b 0 max a b 1 2 a t b 1 t b t a 1 t for all t 0 1 where w and denote the numerical radius and the usual operator norm respectively the second inequality refines and generalizes earlier inequalities

matrix young inequalities for the hilbert schmidt norm abstract let a b and x be n n complex matrices such that a and b are positive semidefinite if p q 1 with 1 p 1 q 1 it is shown that 1 p a p x 1 q xb q 2 2 1 r 2 a p x xb q 2 2 axb 2 2 where r max p q and 2 is the hilbert schmidt norm generalizations and applications of this inequality are also considered

holder type norm inequalities for schur products of matrices abstract for a unitarily invariant norm on m n and p 1 we define a p by a p 1 p then p is again a unitarily invariant norm we give holder type inequalities for schur products of the form a b 0 p0 a 1 p1 b 2 p2 as a corollary we settle in a stronger form a conjecture of marcus et al on submultiplicativity of a unitarily invariant norm with respect to schur multiplication we prove also holder type inequalities for operator radii w of the form w a b w a w b

induced norms of the schur multiplier operator abstract fix an n by n complex matrix a and consider the operator x sa x a x on n by n complex matrices x where a x denotes the schur product of a and x we show that the induced norm of sa with respect to the numerical radius norm is at most one if and only if the matrix a admits a factorization a b wb where w is a contractive matrix and the euclidean norms of the columns of b are at most one we give other equivalent characterizations and derive as a consequence a formally similar result about the induced norm of sa with respect to the spectral norm

on linear combinations of two tripotent idempotent and involutive matrices abstract let a c 1 a 1 c 2 a 2 where c 1 c 2 are nonzero complex numbers and a 1 a 2 is a pair of two n n nonzero matrices we consider the problem of characterizing all situations where a linear combination of the form a c 1 a 1 c 2 a 2 is i a tripotent or an involutive matrix when a 1 and a 2 are commuting involutive or commuting tripotent matrices respectively ii an idempotent matrix when a 1 and a 2 are involutive matrices and iii an involutive matrix when a 1 and a 2 are involutive or idempotent matrices

an estimate for schur multipliers in sp classes abstract an upper estimate for the norm of a matrix a as a schur multiplier in the schattern classes s p is given the estimate turns out to be sharp

a note on linear combinations of commuting tripotent matrices abstract the purpose of this note is to characterize all situations in which a linear combination of two commuting tripotent matrices is also a tripotent matrix in the case of real scalars and real symmetric matrices this problem admits an interesting statistical interpretation namely it is equivalent to the question of when a linear combination of two quadratic forms in normal variables each distributed as a difference of two independent 2 variables is also distributed as such a difference

numerical radius inequalities via convexity abstract the main goal of this article is to present numerical radius inequalities for matrices based on convexity of certain numerical radius functions this simple approach extends some unitarily invariant norms inequalities such as heinz and young inequalities to the context of numerical radius however due to weak unitary invariance of the numerical radius these extended inequalities will be weaker than the corresponding unitarily invariant norms versions

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

on idempotency and tripotency of linear combinations of two commuting tripotent matrices abstract let t 1 and t 2 be two nonzero commuting n n tripotent matrices and c 1 c 2 two nonzero complex numbers necessary and sufficient conditions for the tripotency and the idempotency of c 1 t 1 c 2 t 2 are obtained the problems considered here have also statistical importance when c 1 c 2 are real scalars and t 1 t 2 are real symmetric matrices

on the use of hadamard weightings in the partial decoupling of h based designs abstract in this paper a solution to the hadamard weighted h hwh 1 block problem is proposed the solution is based on a method by safonov and le 1988 the robust stability of designs based on the hwh method is also discussed

interpolated inequalities for unitarily invariant norms abstract in this article we interpolate the well known young and heinz inequalities for unitarily invariant norms and some of their known refinements then we prove new interpolated refinements in the end we use this interpolation idea to prove a hidden monotonicity behavior these inequalities obey

some generalized numerical radius inequalities for hilbert space operators abstract we generalize several inequalities involving powers of the numerical radius for product of two operators acting on a hilbert space for any a b x b h such that a b are positive we establish some numerical radius inequalities for a x b and a x b 1 0 1 and heinz means under mild conditions

rank equalities for idempotent matrices with applications we show by elementary block matrix operations a variety of rank equalities related to sums of two idempotent matrices and give various consequences and applications

finding norms of hadamard multipliers abstract the norm of a matrix b as a hadamard multiplier is the norm of the map x at x b where is the hadamard or entrywise product of matrices watson proposed an algorithm for finding lower bounds for the hadamard multiplier norm of a matrix it is shown how watson s algorithm can be used to give upper bounds as well which in many cases yield the hadamard multiplier norm to any desired accuracy a sharp form of wittstock s decomposition theorem is proved for the special case of hadamard multiplication

quantum representations and monodromies of fibered links abstract andersen masbaum and ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo anosov mapping classes to elements of infinite order for large enough level r in this paper we relate the amu conjecture to a question about the growth of the turaev viro invariants t v r of hyperbolic 3 manifolds we show that if the r growth of t v r m for a hyperbolic 3 manifold m that fibers over the circle is exponential then the monodromy of the fibration of m satisfies the amu conjecture building on earlier work 11 we give broad constructions of oriented hyperbolic fibered links of arbitrarily high genus whose s o 3 turaev viro invariants have exponential r growth as a result for any g n 2 we obtain infinite families of non conjugate pseudo anosov mapping classes acting on surfaces of genus g and n boundary components that satisfy the amu conjecture we also discuss integrality properties of the traces of quantum representations and we answer a question of chen and yang about turaev viro invariants of torus links

majorizations and inequalities in matrix theory abstract in matrix theory majorization plays a significant role for instance majorization relations among eigenvalues and singular values of matrices produce a lot of norm inequalities and even matrix inequalities this survey article is intended as a review of recent results in matrix theory related to majorization

numerical radius inequalities for n n operator matrices abstract new numerical radius inequalities for general n n operator matrices are given these inequalities improve a well known inequality of j c hou and h k du a special emphasis is given to the off diagonal parts of 2 2 operator matrices

index of hadamard multiplication by positive matrices ii for each n n positive semidefinite matrix a we define the minimal index i a max 0 a b b for all b 0 and for each norm n the n index in a min n a b b 0 and n b 1 where a b aijbij is the hadamard or schur product of a aij and b bij and b 0 means that b is a positive semidefinite matrix a comparison between these indexes is done for different choices of the norm n as an application we find for each bounded invertible selfadjoint operator s on a hilbert space the best constant m s such that sts s 1ts 1 m s t for all t 0

quotients of surface groups and homology of finite covers via quantum representations we prove that for each sufficiently complicated orientable surface s there exists an infinite image linear representation rho of pi 1 s such that if gamma in pi 1 s is freely homotopic to a simple closed curve on s then rho gamma has finite order furthermore we prove that given a sufficiently complicated orientable surface s there exists a regular finite cover s rightarrow s such that h 1 s mathbb z is not generated by lifts of simple closed curves on s and we give a lower bound estimate on the index of the subgroup generated by lifts of simple closed curves we thus answer two questions posed by looijenga and independently by kent kisin marche and mcmullen the construction of these representations and covers relies on quantum text so 3 representations of mapping class groups

on idempotency of linear combinations of idempotent matrices p 1 p 2 and p 3 being any three different nonzero mutually commutative nxn idempotent matrices and c 1 c 2 and c 3 being nonzero scalars the problem of characterizing some situations where a linear combination of the form p c 1p 1 c 2p 2 or p c 1p 1 c 2p 2 c 3p 3 is also an idempotent matrix has been considered moreover two interesting results about the idempotency of linear combinations of 2x2 idempotent matrices and 3x3 idempotent matrices have been given a statistical interpretation of the idempotency problem considered in this study has also been pointed out

the nullity and rank of linear combinations of idempotent matrices abstract baksalary and baksalary j k baksalary o m baksalary nonsingularity of linear combinations of idempotent matrices linear algebra appl 388 2004 25 29 proved that the nonsingularity of p 1 p 2 where p 1 and p 2 are idempotent matrices is equivalent to the nonsingularity of any linear combinations c 1 p 1 c 2 p 2 where c 1 c 2 0 and c 1 c 2 0 in the present note this result is strengthened by showing that the nullity and rank of c 1 p 1 c 2 p 2 are constant furthermore a simple proof of the rank formula of gros and trenkler j gros g trenkler nonsingularity of the difference of two oblique projectors siam j matrix anal appl 21 1999 390 395 is obtained

on invertibility of combinations of k potent operators abstract some properties of combinations c 1 p 1 c 2 p 2 c 3 p 1 s p 2 k 1 s where p 1 and p 2 are two different nonzero k potents c 1 c 2 and c 3 are three nonzero complex numbers and positive integers k 2 and s k 1 are obtained furthermore the invertibility the group invertibility and the k potency of the linear combinations of k potents are investigated under certain commutativity properties imposed on them

idempotency of linear combinations of two idempotent matrices abstract a complete solution is established to the problem of characterizing all situations where a linear combination of two different idempotent matrices p 1 and p 2 is also an idempotent matrix including naturally three such situations known in the literature viz if the combination is either the sum p 1 p 2 or one of the differences p 1 p 2 p 2 p 1 and appropriate additional conditions are fulfilled the solution asserts that in the particular case where p 1 and p 2 are complex matrices such that p 1 p 2 is hermitian these three situations exhaust the list of all possibilities and that this list extends when the above assumption on p 1 and p 2 is violated a statistical interpretation of the idempotency problem considered in this note is also pointed out

the moore penrose inverses of products and differences of projections in a c algebra abstract for two given projections p and q in a c algebra we investigate how to express moore penrose inverses of products pq and differences p q and pq qp moreover it is shown that pq qp is moore penrose invertible if and only if pq and p q are moore penrose invertible in addition some related topics are considered