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

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

geometric invariants for liaison of space curves let k be an algebraically closed field and let s k x x x2 x3 for a curve cc pi the hartshorne rao module m c bneie h p3 xc n is a graded s module of finite length and up to duals and shifts is a complete invariant of liaison cf 12 1 given a graded s module m z m of finite length the action of s p p 6 1 between two consecutive components i e ql s hom m m gives rise to a degeneracy locus which can be thought of as lying in psi p namely let i p les irk4 l r and let v v s where s max r i v 5 p3 or else v 0 these loci are isomo hism invariants and are preserved under duals and shifts for m n c these loci are related to the geometry of c itself the main philosophy that emerges section 2 is that the degeneracy locus generally corresponds to those planes h in p3 which meet c nongenerically either containing a component of c or having cn n impose an unusually small number of conditions on some plane curves on h we conclude section 2 by applying these ideas to derive some necessary conditions for m c to have components in negative degrees the remaining sections give various apphcations of the techniques introduced in section 2 first in section 3 we classify sets of skew lines up to liaison that is if c and c consist of t 2 3 and t skew lines respectively then we ask when c can be linked to c the answer in general is never but the situation changes signi cantly if c lies on a quadric surface cf theorem 3 1 a large part of the answer can be read from the degeneracy locus v

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

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

green s function for an infinite slot printed between two homogeneous dielectrics i magnetic currents this sequence of papers presents an analytical closed form for the green s function of an infinite slot printed between two different homogeneous dielectrics this first part is devoted to the derivation of the slot magnetic current and to the discussion of the relevant physical implications the fourier spectrum of the magnetic current is derived by solving in analytical form under small width approximation the integral equation ie representing the continuity of the magnetic field trough the slot axis the accuracy of the result is validated trough a fine meshing method of moments from the consequent spectral expression a closed form approximation of the leaky wave propagation and attenuation constants is derived an asymptotic expression for the current is also obtained by steepest descent path evaluation of the pertinent spectral integral analytical expressions of the quasi static reactive contribution is given for both elementary dipole and delta gap excitations

cycles of linear and semilinear mappings abstract we give a canonical form of matrices of a cycle of linear or semilinear mapping v 1 v 2 v t v 1 in which all v i are complex vector spaces each line is an arrow or and each arrow denotes a linear or semilinear mapping

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

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

green s function of an infinite slot printed between two homogeneous dielectrics part ii uniform asymptotic solution this second part of a two paper sequence deals with the uniform asymptotic description of the green s function of an infinite slot printed between two different homogeneous dielectric media starting from the magnetic current derived in part i the dyadic green s function is first formulated in integral form in both spectral and spatial domains next the asymptotic solution for the vector potential is evaluated asymptotically the asymptotic ray field is structured in three contributions a spherical wave radiated by the source space wave a conical leaky wave and a lateral wave these contributions are first introduced by the stationary phase method applied to the space domain radiation integral from this approach it is seen that the lateral wave contribution is negligible in actual configurations next a rigorous uniform asymptotic evaluation of the radiated field is formulated in the spectral domain by using a steepest descent path deformation which accounts for the vicinity of the pole to the saddle point through this rigorous asymptotics the domain of existence of the leaky wave is found to be limited by a conical shadow boundary which deviates from that defined by the stationary phase regime along this conical shadow boundary phase matching occurs between space and leaky wave which facilitates the transition mechanism between the two wave types this transition occurs inside a conical transition region with elliptical cross section the interference between space and leaky waves from the near to the far zone are discussed by means of illustrative examples which also confirm the accuracy of the asymptotics

on the componentwise linearity and the minimal free resolution of a tetrahedral curve abstract a tetrahedral curve is an unmixed usually nonreduced one dimensional subscheme of projective 3 space whose homogeneous ideal is the intersection of powers of the ideals of the six coordinate lines the second and third authors have shown that these curves have very nice combinatorial properties and they have made a careful study of the even liaison classes of these curves we build on this work by showing that they are almost always componentwise linear i e their homogeneous ideals have the property that for any d the degree d component of the ideal generates a new ideal whose minimal free resolution is linear the one type of exception is clearly spelled out and studied as well the main technique is a careful study of the way that basic double linkage behaves on tetrahedral curves and the connection to the tetrahedral curves that are minimal in their even liaison classes with this preparation we also describe the minimal free resolution of a tetrahedral curve and in particular we show that in any fixed even liaison class there are only finitely many tetrahedral curves with linear resolution finally we begin the study of the generic initial ideal gin of a tetrahedral curve we produce the gin for arithmetically cohen macaulay tetrahedral curves and for minimal arithmetically buchsbaum tetrahedral curves and we show how to obtain it for any nonminimal tetrahedral curve in terms of the gin of the minimal curve in that even liaison class

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

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

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

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

on liaison arithmetical buchsbaum curves and monomial curves in p3 in the context of liaison by 181 and 191 for curves in ip3 the property of being the zero scheme of a section of a rank two bundle on ip3 is connected to that of being ideally the intersection of three surfaces an immediate consequence of this property is that the homogeneous ideal i c of a curve c c p3 is generated by precisely three elements if and only if c is ideally the intersection of three suraces and c is arithmetically cohen macaulay non complete intersection furthermore the characterization of the equivalence classes defined by liaison among curves in p3 yields the well known statement that a curve c is in the liaison class of a complete intersection if and only if c is arithmetically cohen macaulay see e g 2 10 13 or 141 it seems that a cayley see 7 p 1521 in 1847 was the first who posed the problem to describe this liaison class of a complete intersection now in this paper we will begin to investigate the next simple case that is the liaison classes which are characterized by a finite dimensional vector space of dimension l using the theory of buchsbaum rings see section 2 this means we will study liaison among arithmetical buchsbaum curves in p3 first from the point of view of local algebra we get the following more general result

einstein solvmanifolds attached to two step nilradicals a riemannian einstein solvmanifold possibly any noncompact homogeneous einstein space is almost completely determined by the nilradical of its lie algebra a nilpotent lie algebra which can serve as the nilradical of an einstein metric solvable lie algebra is called an einstein nilradical we give a classification of two step nilpotent einstein nilradicals with two dimensional center informally the defining matrix pencil must have no nilpotent blocks in the canonical form and no elementary divisors of a very high multiplicity we also show that the dual to a two step einstein nilradical is not in general an einstein nilradical

on the classification of degree two algebras the classification of algebras of generic degree two and of nonzero generic discriminant is presented in the case of zero discriminant a structural theorem is proven

application of the generalized weierstrass preparation theorem to the study of homogeneous ideals the system of weierstrass polynomiajs defined originally for ideals in convergent power series rings together with its sequence of degrees allows us to analyze a homogeneous ideal directly making use of it we study local cohomology modules syzygies and then graded buchsbaum rings our results give a formula which to some extent clarifies the connection among the matrices appearing in the free resolution starting from a system of weierstrass polynomials a rough classification of graded buchsbaum rings in the general case and a complete classification of graded buchsbaum integral domains of codimension two

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

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

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

on the cohen macaulay type of the general hypersurface section of a curve a theorem of strano 24 states that if c is a reduced irreducible curve in lp 3 lying on no quadric surface and if a general hyperplane section c n h of c is a complete intersection then c itself must be a complete intersection this was extended by re 22 to the case where c is a locally cohen macaulay curve in lw it was then shown by huneke and ulrich 14 that if c is reduced and connected not lying on a quadric hypersurface and if c a h is merely gorenstein then the same is true of c finally the first author 18 extended the techniques of 14 to show that if h is replaced by a general hypersurface of degree d then similar results hold recall that one definition of a gorenstein scheme is arithmetically cohenmacaulay with cohen macaulay type one in other words in a minimal free resolution

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

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

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

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

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

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

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

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

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

lie algebras of vector fields in the real plane finite dimensional real analytic lie algebras of vector fields on r 2 are completely classified up to changes of local coordinates

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

stabilizers of subspaces under similitudes of the klein quadric and automorphisms of heisenberg algebras we determine the groups of automorphisms and their orbits for nilpotent lie algebras of class 2 and small dimension over arbitrary fields including the characteristic 2 case

simultaneous unitary equivalences abstract let s 1 s 2 s 3 s 4 be given finite sets of pairs of n by n complex matrices we describe an algorithm to determine with finitely many computations whether there is a single unitary matrix u such that each pair of matrices in s 1 is unitarily similar via u each pair of matrices in s 2 is unitarily congruent via u each pair of matrices in s 3 is unitarily similar via u and each pair of matrices in s 4 is unitarily congruent via u

characteristic modes for slots in a conducting plane te case consider an infinitely long slot in a perfectly conducting plane in an unbounded medium illuminated by a uniform transverse electric te to the slot axis plane wave the theory of characteristic modes for apertures is applied to solve the problem for a narrow slot analytic expressions for the first two characteristic currents and the equivalent magnetic current are given as computed by the method of moments numerical results for the characteristic currents and fields the equivalent magnetic current and the transmitted field pattern are presented for the slot whose width is one wavelength

nilpotent lie algebras with 2 dimensional commutator ideals abstract we classify all finitely dimensional nilpotent lie k algebras h with 2 dimensional commutator ideals h extending a known result to the case where h is non central and k is an arbitrary field it turns out that while the structure of h depends on the field k if h is central it is independent of k if h is non central and is uniquely determined by the dimension of h in the case where k is algebraically or real closed we also list all nilpotent lie k algebras h with 2 dimensional central commutator ideals h and dim k h 11

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

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

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

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

characteristic modes for a slot in a conducting plane tm case consider an infinitely long slot in a conducting plane in an unbounded medium illuminated by a uniform transverse magnetic tm to the slot axis plane wave the theory of characteristic modes for apertures is applied to solve the problem for a narrow slot analytic expressions for the first two characteristic currents and the equivalent magnetic current are given as computed by the method of moments numerical results for the characteristic currents and fields the equivalent magnetic current and the transmitted field pattern are presented for the slot whose width is one wavelength

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

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

a canonical form for matrices under consimilarity abstract square complex matrices a b are said to be consimilar if a sb s 1 for some nonsingular matrix s consimilarity is an equivalence relation that is a natural matrix generalization of rotation of scalars in the complex plane we survey the known forms to which a given complex matrix may be reduced by unitary consimilarity and describe a canonical form to which it may be reduced by a general consimilarity we derive a useful criterion for two matrices to be consimilar and show that every matrix is consimilar to its own conjugate transpose and adjoint to a real matrix and to a hermitian matrix

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

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

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