blocks of small defect abstract let g be a finite solvable group let p be a prime such that p 5 and o p g 1 and we denote g p p n then g contains a block of defect less than or equal to 3 n 5 let g be a finite solvable group and let p a be the largest power of p dividing 1 for an irreducible character of g we show that g f g p p 3 a for p 5

linear preservers of minimal rank abstract it is proved that a linear transformation on the vector space of upper triangular matrices that maps the set of matrices of minimal rank 1 into itself and either has the analogous property with respect to matrices of full minimal rank or is bijective is a triangular equivalence or a flip about the south west north east diagonal followed by a triangular equivalence the result can be regarded as an analogue of marcus moyls theorem in the context of triangular matrices

maps preserving the joint numerical radius distance of operators abstract denote the joint numerical radius of an m tuple of bounded operators a a 1 a m by w a we give a complete description of maps f satisfying w a b w f a f b for any two m tuples of operators a a 1 a m and b b 1 b m we also characterize linear isometries for the joint numerical radius and maps preserving the joint numerical range of a

linear transformations on symmetric matrices that preserve commutativity it is shown that if a nonsingular linear transformation t on the space of n square real symmetric matrices preserves the commutativity where n 3 then t a qaqt q a in for all symmetric matricesa for some scalar orthogonal matrix q and linear functional q

brauer characters and normal sylow p subgroups abstract in this paper we study some variations of the well known ito michler theorem for p brauer characters using various inequalities involving the p brauer character degrees of finite groups several new criteria for the existence of a normal sylow p subgroup of finite groups are obtained using the p parts and p parts of the p brauer character degrees

ring isomorphisms and linear or additive maps preserving zero products on nest algebras it is shown that every ring isomorphism between norm closed standard subalgebras of nest algebras acting on infinite dimensional banach spaces is either a spatial isomorphism or a spatial conjugate isomorphism this is then used to characterize the linear or additive maps between nest algebras which preserve zero products

additive preservers of rank additivity on matrix spaces abstract let f be a field let v denote the vector space of all m n matrices over f or the vector space of all n n symmetric matrices over f of characteristic not 2 or 3 for each fixed positive integer s 2 let q s denote the set of all matrix pairs a b in v such that rank a b rank a rank b s we characterize additive mappings on v such that a b q s whenever a b q s for a fixed s we also describe the structure of linear mappings from the space of n n matrices over f to the space of p q matrices over f that preserve rank additivity where char f 2

additive rank one preservers on block triangular matrix spaces in this note we address a description of the general form of additive rank one preservers on block triangular matrix spaces over an arbitrary field as an application we characterize additive mappings preserving rank additivity on block triangular matrix algebras

on heights of characters of finite groups abstract in this paper we first state a conjecture on the lower bound of the maximal height of characters in a p block of a finite group then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group for all blocks of a quasi simple group g with g z g isomorphic to a 6 a 7 or a simple group of lie type with an exceptional covering group for all blocks of a symmetric group as well as for all blocks of finite general linear or unitary groups for the proof of the case with symmetric groups it relates to an open question of olsson on the existence of t core partitions where t 3 is an integer as a byproduct our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the isaacs moreto navarro tiep conjecture claiming that the number of distinct irreducible character degrees of a sylow p subgroup p of an arbitrary finite group g is at most one more than the number of irreducible character degrees of g that are multiples of p

non vanishing elements of finite groups let g be a finite group and let irr g denote the set of irreducible complex characters of g an element x of g is non vanishing if for every in irr g we have x 0 we prove that if x is a non vanishing element of g and the order of x is coprime to 6 then x lies in the fitting subgroup of g

defect zero p blocks for finite simple groups we classify those finite simple groups whose brauer graph or decomposition matrix has a p block with defect 0 completing an investigation of many authors the only finite simple groups whose defect zero p blocks remained unclassified were the alternating groups an here we show that these all have a p block with defect 0 for every prime p 5 this follows from proving the same result for every symmetric group sn which in turn follows as a consequence of the t core partition conjecture that every non negative integer possesses at least one t core partition for any t 4 for t 17 we reduce this problem to lagrange s theorem that every non negative integer can be written as the sum of four squares the only case with t 17 that was not covered in previous work was the case t 13 this we prove with a very different argument by interpreting the generating function for t core partitions in terms of modular forms and then controlling the size of the coefficients using deligne s theorem nee the weil conjectures we also consider congruences for the number of p blocks of sn proving a conjecture of garvan that establishes certain multiplicative congruences when 5 p 23 by using a result of serre concerning the divisibility of coefficients of modular forms we show that for any given prime p and positive integer m the number of p blocks with defect 0 in sn is a multiple of m for almost all n we also establish that any given prime p divides the number of p modularly irreducible representations of sn for almost all n 1996 american mathematical society

schur product of matrices and numerical radius range preserving maps abstract let f a be the numerical range or the numerical radius of a square matrix a denote by a b the schur product of two matrices a and b characterizations are given for mappings on square matrices satisfying f a b f a b for all matrices a and b analogous results are obtained for mappings on hermitian matrices

isometries of a generalized numerical radius on compact operators abstract we describe all isometries of the q numerical radius on the space k h of compact operators on an infinite dimensional hilbert space h

formulas for partition k tuples with t cores abstract let a t k n denote the number of partition k tuples of n where each partition is t core in this paper we establish formulas of a t k n for some values of t and k by employing the method of modular forms which extends wang s result for t 3 and k 2 3

isometries of the toeplitz matrix algebra abstract we study the structure of isometries defined on the algebra a of upper triangular toeplitz matrices our first result is that a continuous multiplicative norm preserving map a m n c must be of the form either a u a u or a u a u where a is the complex conjugation and u is a unitary matrix in our second result we use a range of ideas in operator theory and linear algebra to show that every linear isometry a m n c is of the form a u a v where u and v are two unitary matrices this implies in particular that every such an isometry is a complete isometry and that a unital linear isometry a m n c is necessarily an algebra homomorphism

the heights of irreducible brauer characters in 2 blocks of the symmetric groups abstract we prove that there exists a unique irreducible brauer character of height zero in any 2 blocks of the symmetric group this generalizes the theorem of p fong and g d james that the dimension of every non trivial 2 modular simple module of the symmetric group is even

rate of decay of stable periodic solutions of duffing equations abstract in this paper we consider the second order equations of duffing type bounds for the derivative of the restoring force are given that ensure the existence and uniqueness of a periodic solution furthermore the stability of the unique periodic solution is analyzed the sharp rate of exponential decay is determined for a solution that is near to the unique periodic solution

minimal rank completions for block matrices abstract this paper deals with the following problem of completion of block matrices let aij i j be given matrices find additional matrices aij such that the completion a aij ni j 1 has lowest possible rank the structure of the set of all minimal rank completions is studied special attention is paid to the toeplitz case together with its connection to the partial realization problem

self conjugate t core partitions sums of squares and p blocks of an we prove that if t is an integer with t 8 or t 10 then every integer n 2 has a self conjugate t core partition this result has consequences in the representation theory of alternating groups and has a version as a theorem about the representation of integers by sums of squares we also give an infinite sequence of integers that have no self conjugate 9 core partition

some congruences for partitions that are p cores a number of linear congruences modulo r are proved for the number of partitions that are p cores where p is prime 5 p 23 and r is any prime divisor of 1 2 p 1 analogous results are derived for the number of irreducible p modular representations of the symmetric group s n the congruences are proved using the theory of modular forms

the duffing equation with the potential landesman lazer condition abstract in this paper we investigate the nonlinear second order ordinary differential equation at resonance u x c u m 2 c 2 4 u g x u f x x 0 x 0 x 0 we suppose that the nonlinearity g satisfies a potential landesman lazer condition and we prove the existence of a solution to this problem using a saddle point theorem

additive preservers of non zero decomposable tensors let t be an additive mapping from a tensor product of vector spaces over a field into itself we describe t for the following two cases i t is surjective and sends non zero decomposable elements to non zero decomposable elements and ii t a is a non zero decomposable element if and only if a is a non zero decomposable element

some linear preserver problems on block triangular matrix algebras in this article we classify rank one nonincreasing maps on n square block triangular matrix algebras with the assumption that in is of rank n as applications we obtain complete classifications of adjugate commuting maps and compound commuting maps on block triangular matrix algebras

a variation on theorems of jordan and gluck abstract gluck proved that any finite group g has an abelian subgroup a such that g a is bounded by a polynomial function of the largest degree of the complex irreducible characters of g this improved on a previous bound of isaacs and passman in this paper we present a variation of this result that looks at the number of prime factors all these results in turn may be seen as variations on the classical theorem of jordan on linear groups

characterizing normal sylow p subgroups by character degrees abstract suppose that g is a finite group let p be a prime and let p syl p g we prove that p is normal in g if and only if all the irreducible constituents of the permutation character 1 p g have degree not divisible by p

linear transformations which preserve or decrease rank abstract we describe some results concerning a linear transformation on a space v of matrices which is rank preserving or rank nonincreasing on a certain subset of v

non linear numerical radius isometries on atomic nest algebras and diagonal algebras abstract a nonlinear map between operator algebras is said to be a numerical radius isometry if w t s w t s for all t s in its domain algebra where w t stands for the numerical radius of t let n and m be two atomic nests on complex hilbert spaces h and k respectively denote alg n the nest algebra associated with n and d n alg n alg n the diagonal algebra we give a thorough classification of weakly continuous numerical radius isometries from alg n onto alg m and a thorough classification of numerical radius isometries from d n onto d m

spectrum and commutativity preserving mappings on triangular matrices we give the description of continuous mappings on the algebra of all upper triangular complex matrices that preserve spectrum and commutativity

explicit formulas for partition pairs and triples with 3 cores abstract let a 3 n resp b 3 n denote the number of partition pairs resp triples of n where each partition is 3 core by applying ramanujan s 1 1 formula and bailey s 6 6 formula we find the explicit formulas for a 3 n and b 3 n using these formulas we confirm a conjecture of xia and establish many arithmetic identities satisfied by a 3 n and b 3 n

on the orders of zeros of irreducible characters abstract let g be a finite group and p a prime number we say that an element g in g is a vanishing element of g if there exists an irreducible character of g such that g 0 the main result of this paper shows that if g does not have any vanishing element of p power order then g has a normal sylow p subgroup also we prove that this result is a generalization of some classical theorems in character theory of finite groups

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

a note on additive mappings decreasing rank one kuzma characterized additive mappings on the space of all finite rank bounded linear operators on a real or complex banach space that decreases operators of rank one in this note we give a short proof of his result in a slightly more general setting of the tensor product of a right and a left vector space over a division ring

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

rank k preservers and preservers of sets of ranks abstract let t be a linear transformation on the set of m n matrices with entries in an algebraically closed field if t maps the set of all matrices whose rank is k into itself and if n 3 k 2 then the rank of a is the rank of t a for every m n matrix

commutativity preserving operators on symmetric matrices abstract if is a nonsingular linear operator on n n symmetric matrices over a formally real field n 3 and if preserves commutativity then a cu 1 au f a i where u t u and c are scalar and is a linear functional this is an extension of a result of chan and lim

additive rank one preserving surjections on symmetric matrix spaces we characterize additive rank one preserving surjections on symmetric matrix spaces over a field of characteristic not 2 or 3 as applications the invertibility preservers the determinant preservers and characteristic polynomial preservers have been characterized

linear transformations on matrices the invariance of sets of ranks abstract let r e denote the set of all m n matrices over an algebraically closed field f whose ranks lie in the set e where e is a subset of 1 2 m let t be a linear transformation which maps r e into itself under some restrictions on e or when t is nonsingular there are nonsingular matrices u and v such that t a uav for every m n matrix a

additive preservers of numerical range abstract we characterize all additive mappings preserving classical numerical range on various subalgebras of m n the subalgebra of all diagonal matrices d n the subalgebra of all upper triangular matrices t n and subalgebras of block diagonal and block triangular matrices characterizations obtained show that in some cases m n for example all such preservers are linear on the other hand we show that the group of all additive preservers on t n contains some additional mappings

duffing equation and action functional abstract in this paper we investigate the periodic nonlinear second order ordinary differential equation with friction x t c x t g t x t f t t 0 t x 0 x t x 0 x t where c r g is a caratheodory function f l 1 0 t a quotient g t s s lies between 0 and c 2 4 2 t 2 and a nonlinearity g satisfies the potential landesman lazer type condition we introduce a corresponding energy functional and prove that its critical point is a classical solution to this problem

on operators preserving commutativity abstract let l x be the algebra of all bounded operators on a non trivial complex banach space x and f l x l x a bijective linear operator such that f and f 1 both send commuting pairs of operators into commuting pairs then either f a uau 1 p a i or f a ua u 1 p a i where p is a linear functional on l x u is a bounded linear bijective operator between the appropriate two spaces is a complex constant and a is the adjoint of a the form of an operator f for which f and f 1 both send projections of rank one into projections of rank one is also determined

coherence invariant mappings on block triangular matrix spaces abstract in this paper we classify bijective mappings on the space of block upper triangular matrices such that both and 1 preserve pairs of matrices whose differences have rank one applications to rank one preservers and semi isomorphisms are considered

c numerical radius isometries on matrix algebras and triangular matrix algebras abstract let c c 1 c n t r n and m n be the set of n n complex matrices for any a m n define the c numerical range and the c numerical radius of a by w c a i 1 n c i a x i x i x 1 x n is an orthonormal set in c n and w c a max z z w c a respectively let t n be the set of n n upper triangular matrices when w c is a norm on m n mappings t on m n or t n satisfying w c t a t b w c a b for all a b are characterized as an intermediate step we also characterize additive c numerical range preservers on m n or t n

cranks and t cores new statistics on partitions calledcranks are defined which combinatorially prove ramanujan s congruences for the partition function modulo 5 7 11 and 25 explicit bijections are given for the equinumerous crank classes the cranks are closely related to thet core of a partition usingq series some explicit formulas are given for the number of partitions which aret cores some related questions for self conjugate and distinct partitions are discussed

additive rank one preserving mappings on triangular matrix algebras abstract we classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices

on vanishing class sizes in finite groups abstract let g be a finite group an element g of g is called a vanishing element if there exists an irreducible character of g such that g 0 in this case we say that the conjugacy class of g is a vanishing conjugacy class in this paper we discuss some arithmetical properties concerning the sizes of the vanishing conjugacy classes in a finite group

on operators preserving the numerical range abstract let f be a surjective linear mapping between the algebras l h and l k of all bounded operators on nontrivial complex hilbert spaces h and k respectively for any positive integer k let wk a denote the kth numerical range of an operator a on h if k is strictly less than one half the dimension of h and wk f a wk a for all a from l h then there is a unitary mapping u h k such that either f a uau or f a uau t for every a l h where the transposition is taken in any basis of k fixed in advance this generalizes the result of s pierce and w watkins on finite dimensional spaces the case of k greater than or equal to one half of the dimension of h is also treated using our method our proofs depend on a characterization of those linear operators preserving projections of rank one which is of independent interest

on p parts of character degrees and conjugacy class sizes of finite groups abstract let g be a finite group and irr g the set of irreducible complex characters of g let e p g be the largest integer such that p e p g divides 1 for some irr g we show that g f g p p k e p g for a constant k this settles a conjecture of a moreto 13 conjecture 4 we also study the related problems of the p parts of conjugacy class sizes of finite groups

explicit bounds for the number of p core partitions in this article we derive explicit bounds on ct n the number of t core partitions of n in the case when t p is prime we express the generating function f z as the sum

lie type like groups abstract a finite simple group of lie type in defining characteristic p has exactly two p blocks the principal block and a block of defect zero consisting of the steinberg character whose degree is the p part of the order of the group in this paper we characterize finite groups g which have exactly the principal p block and a p block of defect zero consisting of an irreducible character of degree g p

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

permanent polya problem for additive surjective maps abstract let m n f denote the set of square matrices of size n over a field f with characteristics different from two we say that the map f m n f m n f is additive if f a b f a f b for all a b m n f the main goal of this paper is to prove that for n 2 there are no additive surjective maps t m n f m n f such that per t a det a for all a m n f also we show that an arbitrary additive surjective map t m n f m n f which preserves permanent is linear and thus can be completely characterized

spectrum preserving additive maps abstract let x and y be complex banach spaces we show that a spectrum preserving surjective additive map from b x to b y is either of the form f ata 1 for a linear isomorphism a of x onto y or of the form t bt b 1 for a linear isomorphism b of x onto y

on vanishing criteria that control finite group structure abstract many results have been established that show how arithmetic conditions on conjugacy class sizes affect group structure a conjugacy class in g is called vanishing if there exists some irreducible character of g which evaluates to zero on the conjugacy class the aim of this paper is to show that for some classical results it is enough to consider the same arithmetic conditions on the vanishing conjugacy classes of the group

linear additive preservers of ranks 2 and 4 on alternate matrix spaces over fields abstract let kn f be the linear space of all n n alternate matrices over a field f an operator f kn f kn f is said to be additive if f a b f a f b for any a b kn f linear if f is additive and f aa af a for every a f and a kn f and a preserver of ranks 2 and 4 on kn f if rank f x rank x for every x kn f with rank x 2 or 4 when n 4 we characterize all linear respectively additive preservers of ranks 2 and 4 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself furthermore it is also shown that the condition f is not isomorphic to a proper subfield of itself is necessary for the obtained conclusion on additive preservers of ranks 2 and 4 on kn f

isometries of a generalized numerical radius abstract for 0 q 1 the q numerical range is defined on the algebra m n of all n n complex matrices by w q a x ay x y c n x y 1 y x q the q numerical radius is defined by r q a max w q a we characterize isometries of the metric space m n r q i e the maps m n m n that satisfy r q a b r q a b we also characterize maps on m n that preserves the q numerical range the maps are not assumed to be linear

geometry of block triangular matrices over a division ring let d be a division ring with d f2 and t ni k denote the set of k k block triangular matrices over d let be a bijective map from t ni k k n1 nk 2 to itself such that both and 1 preserve the adjacency by the method of maximal sets of rank one and affine geometry we characterize and obtain the fundamental theorem of the geometry on t ni k as a corollary weakly block additive adjacency preserving bijective maps in both directions on t ni k are characterized as applications of the fundamental theorem ring automorphisms or ring anti automorphisms of t ni k are characterized and jordan automorphisms of jordan ring j t ni k are also characterized

formulas for cubic partition with 3 cores abstract let c 3 n denote the number of cubic partitions of n with 3 cores in this paper we establish the arithmetic properties and formulas for c 3 n by employing bailey s 6 6 formula and theta function identities