dodgson condensation the historical and mathematical development of an experimental method abstract dodgson s condensation method has become a powerful tool in the automation of determinant evaluations in this expository paper i describe its 19 th century roots and the major steps on the path that began in the 20 th century when the iteration of an identity derived by dodgson first was studied including its role in the discovery of the alternating sign matrix conjecture the evaluation of an important 19 th century determinant in partition theory as well as a combinatorial proof of it i then discuss additional developments that have led the way to its use in modern experimental mathematics

minor identities for quasi determinants and quantum determinants we present several identities involving quasi minors of noncommutative generic matrices these identities are specialized to quantum matrices yieldingq analogues of various classical determinantal formulas

nonlinear lie derivations of triangular algebras in this paper we prove that every nonlinear lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero

linear preservers on upper triangular operator matrix algebras abstract in this paper we obtain several characterizations of rank preserving linear maps and completely rank nonincreasing linear maps on upper triangular hilbert space operator matrix algebras and apply them to get some algebraic results we show that every automorphism of an upper triangular operator matrix algebra is inner and every weakly continuous surjective local automorphism is in fact an automorphism a weakly continuous linear bijection on an upper triangular operator matrix algebra is idempotent preserving if and only if it is a jordan homomorphism and in turn if and only if it is an automorphism or an anti automorphism as an application we also obtain a result concerning the asymptotic joint similarity of matrix tuples

jordan mappings of semiprime rings an additive mapping 8 of a ring r into a 2 torsion free ring r is called a jordan homomorphism if 6 ab z a 8 b 6 b a for all a be r a well known result of i n herstein 4 states that every jordan homomorphism onto a prime ring is either a homomorphism or an antihomomorphism suppose that r contains ideals u and v with null intersection let cp r u be a homomorphism and ii r v be an antihomomorphism a mapping 6 40 is a so called direct sum of mappings rp and 11 obviously b is a jordan homomorphism according to this construction we see that herstein s result 4 does not hold in semip me rings moreover w e baxter and w s martindale el showed by an example that a jordan homomorphism 0 of a ring r onto a semiprime ring r is not necessarily a direct sum of a homomorphism and an antihomomorphism but they proved that there always exists an essential ideal e of r such that the restriction of 8 to e is a direct sum of a homomorphism cp e r and an antihomomorphism y5 e r 1 theorem 2 71 in this paper theorem 2 3 we extend this result by showing that e can be choosen so that it is a sum of ideals u and v of r such that cp vanishes on v and tc vanishes on u even more for each x e r we have b ux e u e x for all u e u and b ux f x e o for all u e v this result also contains an affirmative answer to the question of baxter and martindale whether there is a way to choose the ideal e so that 8 e is an associative subring of r as we shall see in our case 6 e is the essential associative ideal of r an additive mapping 0 of a ring r into a ring r which satisfies b aba a 6 b 6 a for all a b e r will be called a jordan triple homomorphism an easy computation shows that every jordan homomorphism is also a

the capelli identity the double commutant theorem and multiplicity free actions 0 the capelli identity cal 3 w p 39 is one of the most celebrated and useful formulas of classical invariant theory w d cl z the double commutant theorem w p 91 is likewise a basic result in the general theory of associative algebras both play key roles in weyl s book the classical groups the main purpose of this paper is to demonstrate a close connection between the two in the context of multiplicity free actions ekj of groups on vector spaces the focus of our discussion will be the structure of the differential operators which commute with a multiplicity free action applications include a new derivation of some formulas of shimura shl and rubenthaler and schiffmann rs for b functions associated to hermitian symmetric spaces and a construction of interesting sets of generators for the center of the universal enveloping algebra of gin we also give a detailed discussion of certain aspects of multiplicity free representations here is an overview of the contents of the paper in sect 1 we review the classical capelli identities we observe how their existence is predicted by a double commutant result 1 6 and 1 8 and we show how they can be used to compute b functions we remark that capelli understood that his operators generate the center of the algebra generated by the polarization operators see b1 p 77 also capelli s motivation in introducing his operators was to explain a formula of cayley essentially the computation of a b function capelli s point of view was remarkably modern and structuralist in certain ways more modern even than that of weyl sections 2 through 9 provide a conceptual context for understanding the capelli identities as a feature of multiplicity free actions they contain a general discussion of the structure of g the algebra of polynomial coefficient differential operators which commute with a given group g of linear transformations the results of the discussion are summarized in theorem 9 1 which says in particular that the polynomial coefficient differential operators commuting with a

nineteenth century roots of quasideterminants abstract in this expository paper i provide a complete record of the nineteenth century publications that bear on the development of quasideterminants in the twentieth century two important recursive feasible algorithms sylvester s from 1851 and dodgson s from 1866 are discussed and the antecedents of both are traced back to work by jacobi

unfoldings in knot theory in in r the link at infinity of a hypersurface v cie r defined by a good polynomial map f ty ii2 was used as an extended example f was called good if it had only isolated singularities and it was claimed that the link at infinity then always has a milnor fibration this is incorrect although we have found it to be a common misconception to correct it the definition of good must be modified as follows

nonlinear mappings on upper triangular matrices derivable at zero point abstract let t n f be the set of all n n upper triangular matrices over a field f recently by using a technique from graph theory l wang in 8 proved that a bijective mapping on t n f with f a finite field preserves zero product i e x y 0 whenever x y 0 if and only if can be decomposed into the product of an inner automorphism an automorphism of t n f induced by an automorphism of f and a regular automorphism of the zero divisor graph of t n f algebraic group theory shows that derivations of an algebra often go parallel with automorphisms of the algebra this viewpoint encourages us to determine nonlinear mappings on t n f which are derivable at zero point i e x y x y 0 whenever x y 0 it is proved that a mapping on t n f is derivable at zero point if and only if is the sum of an inner derivation an additive derivation of t n f induced by an additive derivation of f and a regular mapping on t n f

with a cole hopf transformation to solutions of the noncommutative kp hierarchy in terms of wronski matrices in the case of the kp hierarchy where the dependent variable takes values in an arbitrary associative algebra it is known that there are solutions which can be expressed in terms of quasideterminants of a wronski matrix which solves the linear heat hierarchy we obtain these solutions without the help of quasideterminants in a simple way via solutions of matrix kp hierarchies over and by use of a cole hopf transformation for this class of exact solutions we work out a correspondence with weakly nonassociative algebras

on the computation of minimal polynomials let k be a field and let f and g be non constant elements of k t assume thatgcd deg f deg g is not divisible by the characteristic of k this paper describes an algorithm to compute the polynomial p x y of minimal degree such that p f g 0 using ideas needed to justify the algorithm a new proof is given of the fact that if k f g k t then either deg g dividesdeg f or deg f divides deg g

jordan isomorphisms of triangular matrix algebras over a connected commutative ring abstract let c be a 2 torsionfree commutative ring with identity 1 and let t r c r 2 be the algebra of all upper triangular r r r 2 matrices over c then c contains no idempotents except 0 and 1 if and only if every jordan isomorphism of t r c onto an arbitrary algebra over c is either an isomorphism or an anti isomorphism

maps determined by action on identity product elements abstract let m n r be the algebra of all n n matrices over a unital commutative ring r with 6 invertible for a given element z m n r a map on m n r is called preserving z product if x y z whenever xy z a map on m n r is called derivable at the given point z if x y x y z whenever xy z using elementary matrix technique we show that if an invertible linear map on m n r preserves identity product then it is a jordan automorphism and a linear map on m n r is derivable at the identity matrix if and only if it is an inner derivation

operators on triangular algebras abstract we study the algebra of differential operators on the triangular algebras and the upper triangular algebras we further identify all the ideals of the algebra of differential operators on the upper triangular algebras

generalized lie derivations on triangular algebras let a be a unital algebra and let m be a unitary a bimodule we consider generalized lie derivations mapping from a to m our results are applied to triangular algebras in particular to nest algebras and block upper triangular matrix algebras we prove that under certain conditions each generalized lie derivation of a triangular algebra a is the sum of a generalized derivation and a central map which vanishes on all commutators of a

derivations of upper triangular matrix rings we give a description of the derivations in tn r the ringof upper triangular matrices over a ring r assuming only the existence of an identity element we show that every derivation is the sum of an inner derivation and another one induced from r

automorphisms and derivations of upper triangular matrix rings abstract kezlan proved that for a commutative ring c every c automorphism of the ring of upper triangular matrices over c is inner we generalize this result to rings in which all idempotents are central moreover we show that for a semiprime ring a and central subring c every c automorphism of the ring of upper triangular matrices over c is the composite of an inner automorphism and an automorphism induced from a c automorphism of a by the method of proof we re prove results of s p coelho and c p milies and of mathis stating that a derivation of a ring of upper triangular matrices of a c algebra n n matrices over a is a sum of an inner derivation and a derivation induced from a c derivation of a by an example we show that an extra assumption is needed for proving the above result of automorphisms of upper triangular matrices finally we consider automorphisms of subrings of n n matrices over a commutative ring c where entries over the diagonal are from c and below the diagonal are taken from a nil ideal we prove that all such automorphisms are inner

binomial determinants paths and hook length formulae abstract we give a combinatorial interpretation for any minor or binomial determinant of the matrix of binomial coefficients this interpretation involves configurations of nonintersecting paths and is related to young tableaux and hook length formulae

quantum hyperdeterminants and hyper pfaffians the notion of generalized quantum monoids is introduced it is proved that the quantum coordinate ring of the monoid can be lifted to a quantum hyper algebra in which the quantum determinant and quantum pfaffian are sent to the quantum hyperdeterminant and quantum hyper pfaffian respectively the quantum hyperdeterminant in even dimension is shown to be a q analog of cayley s first hyperdeterminant

n derivations of triangular algebras abstract let a be a triangular algebra let n 2 be an integer a map a a a a is said to be a n derivation if it is a derivation in each argument in this paper we investigate n derivations n 3 for a certain class of triangular algebras the main result is then applied to upper triangular matrix algebras and nest algebras

on identities satisfied by minors of a matrix abstract the algebraic relations between minors of a generic matrix are of great interest in many domains invariant theory representations of the symmetric group algebraic geometry schur functions etc we show that most of these relations result easily from a particular identity which was stated by h w turnbull in 1909 but has remained little known since the introduction of cayley or peano algebras by g c rota throws light on this identity indeed turnbull s formula turns out to be nothing but the expression of associativity and anticommutativity for the second product the meet operation of these algebras a constant problem when dealing with minors identities is the choice of appropriate notations the difficulty of grasping the cumbersome symbols used by many authors of the last century is no doubt responsible for the present neglect of this field we propose to code these identities by permutations of letters in tableaux as an illustration of these principles we derive explicitly a great number of classical theorems due to sylvester bazin reiss picquet muir etc and we give also several new formulae which generalize some of these theorems however we do not try to give an exhaustive list of identities it seems to us more interesting to show that these tools provide an effective calculus on minors which enables anyone to find the formulae old or new suitable for a particular application

quasideterminant solutions of the generalized heisenberg magnet model in this paper we present the darboux transformation for the generalized heisenberg magnet ghm model based on the general linear lie group gl n and construct multi soliton solutions in terms of quasideterminants further we relate the quasideterminant multi soliton solutions obtained by means of darboux transformation with those obtained by the dressing method we also discuss the model based on the lie group su n and obtain explicit soliton solutions of the model based on su 2

isomorphism between the r matrix and drinfeld presentations of yangian in types b c and d it is well known that the gauss decomposition of the generator matrix in the r matrix presentation of the yangian in type a yields generators of its drinfeld presentation defining relations between these generators are known in an explicit form thus providing an isomorphism between the presentations it has been an open problem since the pioneering work of drinfeld to extend this result to the remaining types we give a solution for the classical types b c and d by constructing an explicit isomorphism between the r matrix and drinfeld presentations of the yangian it is based on an embedding theorem which allows us to consider the yangian of rank n 1 as a subalgebra of the yangian of rank n of the same type

on universal quadratic identities for minors of quantum matrices abstract we give a complete combinatorial characterization of homogeneous quadratic relations of universal character valid for minors of quantum matrices more precisely for minors in the quantized coordinate ring o q m m n k of m n matrices over a field k where q k this is obtained as a consequence of a study of quantized minors of matrices generated by paths in certain planar graphs called se graphs generalizing the ones associated with cauchon diagrams our efficient method of verifying universal quadratic identities for minors of quantum matrices is illustrated with many appealing examples

multiply refined enumeration of alternating sign matrices four natural boundary statistics and two natural bulk statistics are considered for alternating sign matrices asms specifically these statistics are the positions of the 1 s in the first and last row and column of an asm and the numbers of generalized inversions and 1 s in an asm previously known results for the exact enumeration of asms with prescribed values of some of these statistics are reviewed in detail a quadratic relation which fully determines the generating function associated with all six statistics is then obtained the derivation of the relation involves combining the desnanot jacobi determinant identity with the izergin korepin formula for the partition function of the six vertex model with domain wall boundary conditions

quasideterminant multisoliton solutions of a supersymmetric chiral field model in two dimensions the darboux transformation of a supersymmetric principal chiral field model in two dimensions is studied we obtain exact superfield multisoliton solutions of the model by means of iterated darboux transformation and express them in terms of quasideterminants

quasi commuting families of quantum minors in b leclerc a zelevinsky quasicommuting families of quantum plucker coordinates in kirillov s seminar on representation theory in amer math soc transl 2 vol 181 amer math soc providence ri 1998 pp 85 108 a combinatorial criterion for quasi commutativity is established for pairs of quantum plucker coordinates in the quantized coordinate algebra cq f of the flag variety of type a this paper attempts to generalize these results by producing necessary and sufficient conditions for pairs of quantum minors in the quantized coordinate algebra cq matk m to quasi commute in addition we study the combinatorics of maximal by inclusion families of pairwise quasi commuting quantum minors and pose relevant conjectures

jordan derivations and antiderivations on triangular matrices abstract we define an antiderivation from an algebra a into an a bimodule m as a linear map a m such that ab b a b a for all a b a the main result states that every jordan derivation from the algebra of all upper triangular matrices into its bimodule is the sum of a derivation and an antiderivation

multivariate orthogonal polynomials and integrable systems multivariate orthogonal polynomials in d real dimensions are considered from the perspective of the cholesky factorization of a moment matrix the approach allows for the construction of corresponding multivariate orthogonal polynomials associated second kind functions jacobi type matrices and associated three term relations and also christoffel darboux formulae the multivariate orthogonal polynomials their second kind functions and the corresponding christoffel darboux kernels are shown to be quasi determinants as well as schur complements of bordered truncations of the moment matrix quasi tau functions are introduced it is proven that the second kind functions are multivariate cauchy transforms of the multivariate orthogonal polynomials discrete and continuous deformations of the measure lead to toda type integrable hierarchy being the corresponding flows described through lax and zakharov shabat equations bilinear equations are found varying size matrix nonlinear partial difference and differential equations of the 2d toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials the discrete flows which are shown to be connected with a gauss borel factorization of the jacobi type matrices and its quasi determinants lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi tau matrices which generalize to the multidimensional realm those that relate the baker and adjoint baker functions to ratios of miwa shifted tau functions in the 1d scenario in this context the multivariate extension of the elementary darboux transformation is given in terms of quasi determinants of matrices built up by the evaluation at a poised set of nodes lying in an appropriate hyperplane in r d of the multivariate orthogonal polynomials the multivariate christoffel formula for the iteration of m elementary darboux transformations is given as a quasi determinant it is shown using congruences in the space of semi infinite matrices that the discrete and continuous flows are intimately connected and determine nonlinear partial difference differential equations that involve only one site in the integrable lattice behaving as a kadomstev petviashvili type system finally a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given in particular it is shown that the toda times that preserve the invariance condition lay in a secant variety of the veronese variety of the fixed point set of the linear isometry

jordan α β derivations on triangular algebras and related mappings abstract let r be a 2 torsion free commutative ring with identity a b be unital algebras over r and m be a unital a b bimodule which is faithful as a left a module and also as a right b module let t a m 0 b be the triangular algebra consisting of a b and m and let d be an r linear mapping from t into itself suppose that a and b have only trivial idempotents then the following statements are equivalent 1 d is a jordan derivation on t 2 d is a jordan triple derivation on t 3 d is an derivation on t furthermore a generalized version of this result is also given we characterize the actions of automorphisms and skew derivations on the triangular algebra t the structure of continuous derivations of triangular banach algebras and that of generalized jordan derivations of upper triangular matrix algebras are described

spectral extension of the quantum group cotangent bundle the structure of a cotangent bundle is investigated for quantum linear groups gl q n and sl q n using a q version of the cayley hamilton theorem we construct an extension of the algebra of differential operators on sl q n otherwise called the heisenberg double by spectral values of the matrix of right invariant vector fields we consider two applications for the spectral extension first we describe the extended heisenberg double in terms of a new set of generators the weyl partners of the spectral variables calculating defining relations in terms of these generators allows us to derive sl q n type dynamical r matrices in a surprisingly simple way second we calculate an evolution operator for the model of the q deformed isotropic top introduced by a alekseev and l faddeev the evolution operator is not uniquely defined and we present two possible expressions for it the first one is a riemann theta function in the spectral variables the second one is an almost free motion evolution operator in terms of logarithms of the spectral variables the relation between the two operators is given by a modular functional equation for the riemann theta function

characterizations of derivations and jordan derivations on banach algebras abstract let a be an algebra and m be an a bimodule let x be in a and a m be a linear map which satisfies ab a b a b for all a b a with ab x it is shown that is a jordan derivation if is continuous and x is left or right invertible also it is shown that is a derivation if x is idempotent such that for m m the condition x a i x m 0 implies i x m 0 and the condition mx a i x 0 implies mx 0

a new topological proof of the abhyankar moh theorem we present a new simple proof of the famous theorem of abhyankar moh and suzuki about rational curves in a plane this proof relies on the poincare hopf theorem

manin matrices and talalaev s formula in this paper we study properties of lax and transfer matrices associated with quantum integrable systems our point of view stems from the fact that their elements satisfy special commutation properties considered by yu i manin some 20 years ago at the beginning of quantum group theory these are the commutation properties of matrix elements of linear homomorphisms between polynomial rings more explicitly these read 1 elements of the same column commute 2 commutators of the cross terms are equal mij mkl mkj mil e g m11 m22 m21 m12 the main aim of this paper is twofold on the one hand we observe and prove that such matrices which we call manin matrices in short behave almost as well as matrices with commutative elements namely the theorems of linear algebra e g a natural definition of the determinant the cayley hamilton theorem the newton identities and so on and so forth have a straightforward counterpart in the case of manin matrices on the other hand we remark that such matrices are somewhat ubiquitous in the theory of quantum integrability for instance manin matrices and their q analogs include matrices satisfying the yang baxter relation rtt ttr and the so called cartier foata matrices also they enter talalaev s remarkable formulae for the quantum spectral curve and appear in the separation of variables problem and capelli identities we show that theorems of linear algebra after being established for such matrices have various applications to quantum integrable systems and lie algebras e g in the construction of new generators in and in general in the construction of quantum conservation laws in the knizhnik zamolodchikov equation and in the problem of wick ordering we propose in the appendix a construction of quantum separated variables for the xxx heisenberg system

biderivations of triangular algebras abstract let a be a triangular algebra a bilinear map a a a is called a biderivation if it is a derivation with respect to both arguments in this paper we define the concept of an extremal biderivation and prove that under certain conditions a biderivation of a triangular algebra a is a sum of an extremal and an inner biderivation the main result is then applied to block upper triangular matrix algebras and nest algebras we also consider the question when a derivation of a triangular algebra is an inner derivation

lie triple derivations of nest algebras on banach spaces abstract let n be a non trivial nest on x alg n be the associated nest algebra and l alg n b x be a linear mapping in this paper it is proved that l is a lie triple derivation if and only if there exist a derivation d alg n b x and a linear mapping h alg n c i with h x y z 0 for any x y z alg n such that l d h on alg n

automorphism groups of algebras of triangular matrices abstract if a is an algebra of n x n matrices over an integral domain we can associate with a a graph whose edges are pairs i j such that the i j entry of every element of a is zero the graph in turn defines a group of permutations and automorphisms of a which are conjugations by permutations matrices for a suitably restricted class of algebras a we show that the automorphism group of a is the semidirect product of this group of permutation matrices with the subgroup of inner automorphisms

chio s and dodgson s determinantal identities abstract in this expository paper we analyze and compare two determinantal identities constructed in the mid nineteenth century chio s and dodgson s from the perspective of their origins in earlier work by j j sylvester and c g j jacobi all the known proofs of chio s and dodgson s identities are cited and we present and compare them as modern algorithms for evaluating determinants restated as a recurrence we discuss the role of dodgson s identity in the development of the alternating sign matrix conjecture in the late twentieth century

quantum and quasi plucker coordinates abstract we demonstrate a passage from the quasi plucker coordinates of gelfand and retakh to the quantum plucker coordinates built from q generic matrices in the process we rediscover the defining relations of the quantum grassmannian of taft and towber and provide that algebra with more concrete geometric origins

jordan higher derivations on triangular algebras in this paper we show that any jordan higher derivation on a triangular algebra is a higher derivation this extends the main result in 13 to the case of higher derivations

darboux transformation and exact multisolitons of cpn nonlinear sigma model abstract exact multisolitons of c p n sigma model are obtained by using darboux transformation in terms of quasideterminant darboux matrix the multisoliton solutions are shown to be expressed in terms of quasideterminants of gelfand and retakh 1991 10 the method of quasideterminant darboux matrix has been shown to be related with the dressing method that has already been used to compute iteratively exact multisolitons of the c p n sigma model by means of nonlinear superposition principle of backlund transformations explicit expressions of one two and three soliton solutions of c p 1 model have been computed by using properties of quasideterminants

derivations of tensor products of nonassociative algebras abstract let r and s be nonassociative unital algebras assuming that either one of them is finite dimensional or both are finitely generated we show that every derivation of r s is the sum of derivations of the following three types a ad u where u belongs to the nucleus of r s b l z f where f is a derivation of s and z lies in the center of r and c g l w where g is a derivation of r and w lies in the center of s

modular automorphisms preserving idempotence and jordan isomorphisms of triangular matrices over commutative rings abstract suppose r is a commutative ring with 1 and 2 being the units of r let t n r be the n n upper triangular matrix modular over r and let l r be the set of all r module automorphisms on t n r that preserve idempotence the main result in this paper is that f l r if and only if there exist an invertible matrix u t n r and an idempotence e r such that f x u ex 1 e x u 1 for any x x ij t n r where x x n 1 j n 1 i as applications we determine all jordan isomomorphisms of t n r over the ring r

rank ideals and capelli identities abstract we define and study a family of completely prime rank ideals in the universal enveloping algebra u gl n a rank ideal is a noncommutative analogue of a determinantal ideal the defining ideal for the closure of the set of n n matrices of fixed rank we introduce a notion of rank for gl n modules and determine the rank of simple highest weight modules and of simple finite dimensional modules the main tools are capelli type identities and filtered algebra

jordan derivations of full matrix algebras abstract let a be a unital associative ring and m be a 2 torsion free a bimodule using an elementary and constructive method we show that every jordan derivation from m n a into m n m is a derivation

higher derivations of triangular algebras and its generalizations abstract let r be a commutative ring with identity a and b be unital algebras over r and m be a unital it a it b bimodule let t a m 0 b be the triangular algebra consisting of a it band m motivated by the work of cheung 14 we mainly consider the question whether every higher derivation on a triangular algebra is an inner higher derivation we also give some characterizations on generalized jordan triple higher derivations of triangular algebras

a quantum analogue of the dihedral action on grassmannians in recent work launois and lenagan have shown how to construct a cocycle twisting of the quantum grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the n cycle c 1 2 cdots n up to a power of q this twisting is needed because c does not naturally induce an automorphism of the quantum grassmannian as it does classically and semi classically we extend this construction to give a quantum analogue of the action on the grassmannian of the dihedral subgroup of s n generated by c and w 0 the longest element and this analogue takes the form of a groupoid we show that there is an induced action of this subgroup on the torus invariant prime ideals of the quantum grassmannian and also show that this subgroup acts on the totally nonnegative and totally positive grassmannians then we see that this dihedral subgroup action exists classically semi classically by poisson automorphisms and anti automorphisms a result of yakimov and in the quantum and nonnegative settings

jordan derivations of unital algebras with idempotents abstract we consider jordan derivations of a unital algebra a having a nontrivial idempotent it turns out that on unital algebras there exist jordan derivations that are not derivations for this purpose we introduce the term a singular jordan derivation which is a proper jordan derivation of the form that depends on peirce decomposition of the unital algebra a singular jordan derivations are usually antiderivations the main result of the paper states that under certain conditions every jordan derivation of a is the sum of a derivation and a singular jordan derivation

quasipositivity as an obstruction to sliceness for an oriented link l s 3 d 4 let s l be the greatest euler characteristic f of an oriented 2 manifold f without closed components smoothly embedded in d 4 with boundary l a knot k is slice if s k 1 realize d 4 in c 2 as z w z 2 w 2 1 it has been conjectured that if v is a nonsingular complex plane curve transverse to s 3 then s v s 3 v d 4 kronheimer and mrowka have proved this conjecture in the case that v d 4 is the milnor fiber of a singularity i explain how this seemingly special case implies both the general case and the slice bennequin inequality for braids as applications i show that various knots are not slice e g pretzel knots like p 3 5 7 all knots obtained from a positive trefoil o 2 3 by iterated untwisted positive doubling

determinantal identities gauss schur cauchy sylvester kronecker jacobi binet laplace muir and cayley abstract we give a common concise derivation of some important determinantal identities attributed to the mathematicians in the title we also give a formal treatment of determinantal identities of the minors of a matrix

determinants and alternating sign matrices let m be an n by n matrix by a connected minor of m of size k we mean a minor formed from k consecutive rows and k consecutive columns we give formulas for det m in terms of connected minors one involving minors of two consecutive sizes and one involving minors of three consecutive sizes the formulas express det m as sums indexed by sets of alternating sign matrices these matrices are described here and by w h mills d p robbins and h rumsey jr invent math 66 1982 73 87 j combin theory ser a 34 1983 340 359 the former study has led to the solution of macdonald s conjecture on cyclically symmetric plane partitions g e andrews invent math 53 1979 193 225 i g macdonald symmetric functions and hall polynomials p 53 oxford univ press clarendon oxford 1979

quasideterminant solutions of an integrable chiral model in two dimensions the darboux transformation is used to obtain multisoliton solutions of the chiral model in two dimensions the matrix solutions of the principal chiral model and its lax pair are expressed in terms of quasideterminants the iteration of the darboux transformation gives the quasideterminant multisoliton solutions of the model it has been shown that the quasideterminant multisoliton solution of the chiral model is the same as obtained by zakharov and mikhailov using the dressing method based on the matrix riemann hilbert problem

additive jordan derivations of reflexive algebras additive jordan derivations of certain reflexive algebras are investigated in particular additive jordan derivations of nest algebras on banach spaces are shown to be additive derivations