bifix codes and sturmian words we prove new results concerning the relation between bifix codes episturmian words and subgroups of free groups we study bifix codes in factorial sets of words we generalize most properties of ordinary maximal bifix codes to bifix codes maximal in a recurrent set f of words f maximal bifix codes in the case of bifix codes contained in sturmian sets of words we obtain several new results let f be a sturmian set of words defined as the set of factors of a strict episturmian word our results express the fact that an f maximal bifix code of degree d behaves just as the set of words of f of length d an f maximal bifix code of degree d in a sturmian set of words on an alphabet with k letters has k 1 d 1 elements this generalizes the fact that a sturmian set contains k 1 d 1 words of length d moreover given an infinite word x if there is a finite maximal bifix code x of degree d such that x has at most d factors of length d in x then x is ultimately periodic our main result states that any f maximal bifix code of degree d on the alphabet a is the basis of a subgroup of index d of the free group on a

a system of matrix equations and a linear matrix equation over arbitrary regular rings with identity abstract in this paper we consider the classical system of matrix equations a 1 xb 1 c 1 a 2 xb 2 c 2 over r an arbitrary regular ring with identity necessary and sufficient conditions for the existence and the expression of the general solution to the system are derived as an application the linear matrix equation axb cyd e over r is considered

levels of knotting of spatial handlebodies if h is a spatial handlebody i e a handlebody embedded in the 3 sphere a spine of h is a graph s3 such that h is a regular neighbourhood of usually h is said to be unknotted if it admits a planar spine this suggests that a handlebody should be considered not very knotted if it admits spines that enjoy suitable special properties building on this remark we define several levels of knotting of spatial handlebodies and we provide a complete description of the relationships between these levels focusing our attention on the case of genus 2 we also relate the knotting level of a spatial handlebody h to classical topological properties of its complement m s3 h such as its cut number more precisely we show that if h is not highly knotted then m admits special cut systems for m and we discuss the extent to which the converse implication holds along the way we construct obstructions that allow us to determine the knotting level of several families of spatial handlebodies these obstructions are based on recent quandle coloring invariants for spatial handlebodies on the extension to the context of spatial handlebodies of tools coming from the theory of homology boundary links on the analysis of appropriate coverings of handlebody complements and on the study of the classical alexander elementary ideals of their fundamental groups

there are no unexpected tunnel number one knots of genus one author s scharlemann martin abstract we show that the only knots that are tunnel number one and genus one are those that are already known 2 bridge knots obtained by plumbing together two unknotted annuli and the satellite examples classified by eudave munoz and by morimoto and sakuma this confirms a conjecture first made by goda and teragaito

on the absence of mcshane type identities for the outer space abstract a remarkable result of mcshane states that for a punctured torus with a complete finite volume hyperbolic metric we have 1 e l 1 1 2 where varies over the homotopy classes of essential simple closed curves and l is the length of the geodesic representative of we prove that there is no reasonable analogue of mcshane s identity for the culler vogtmann outer space of a free group

quivers and matrix equations abstract we discuss methods for solving some familiar matrix equations the methods were derived from the theory of quivers a powerful tool in the representation theory of algebras it is not necessary that the reader understand quivers but we include a brief explanation for those who may be interested

solvability conditions and general solution for mixed sylvester equations in this paper we give some necessary and sufficient solvability conditions for the mixed sylvester matrix equations and parameterize general solution when it is solvable moreover we investigate the maximal and minimal ranks of the general solution and maximal and minimal ranks and inertias of hermitian part of solution respectively

balance 0 1 matrices bicoloring and total dual integrality a 0 1 matrixa is balanced if in every submatrix with two nonzero entries per row and column the sum of the entries is a multiple of four this definition was introduced by truemper 1978 and generalizes the notion of a balanced 0 1 matrix introduced by berge 1970 in this paper we extend a bicoloring theorem of berge 1970 and total dual integrality results of fulkerson hoffman and oppenheim 1974 to balanced 0 1 matrices

equations in free groups and coincidence of mappings on surfaces let f colon s h to s g be a continuous map and c colon s h to s g a constant map between closed orientable surfaces of genus h g respectively by definition the pair f c has the wecken property if f can be deformed into a map f such that every coincidence classes of f c is essential and consists of exactly one point the main result is that f c has the wecken property if and only if deg f leq frac 2h 2 ell 2g 1 where ell pi 1 s g f pi 1 s h certain quadratic equations in free groups closely related to the coincidence problem are solved

automorphic orbits in free groups abstract let fn be the free group of a finite rank n we study orbits orb u where u is an element of the group fn under the action of an automorphism if an orbit like that is finite we determine precisely what its cardinality can be if u runs through the whole group fn and runs through the whole group aut fn another problem that we address here is related to whitehead s algorithm that determines whether or not a given element of a free group of finite rank is an automorphic image of another given element it is known that the first part of this algorithm reducing a given free word to a free word of minimum possible length by elementary whitehead automorphisms is fast of quadratic time with respect to the length of the word on the other hand the second part of the algorithm applied to two words of the same minimum length was always considered very slow we give here an improved algorithm for the second part and we believe this algorithm always terminates in polynomial time with respect to the length of the words we prove that this is indeed the case if the free group has rank 2

the mapping class groups of reducible heegaard splittings of genus two the first named author was supported by the basic science research program through the national research foundation of korea nrf funded by the ministry of science ict and future planning nrf 2015r1a1a1a05001071 and by the ministry of education nrf 201800000001768 the second author was supported by jsps kakenhi grant numbers 15h03620 17k05254 17h06463 and jst crest grant number jpmjcr17j4

a system of real quaternion matrix equations with applications abstract let h be the real quaternion algebra and h n m denote the set of all n m matrices over h let p h n n and q h m m be involutions i e p 2 i q 2 i a matrix a h n m is said to be p q symmetric if a paq this paper studies the system of linear real quaternion matrix equations a 1 x 1 c 1 x 1 b 1 c 2 a 2 x 2 c 3 x 2 b 2 c 4 a 3 x 1 b 3 a 4 x 2 b 4 c c we present some necessary and sufficient conditions for the existence of a solution to this system and give an expression of the general solution to the system when the solvability conditions are satisfied as applications we discuss the necessary and sufficient conditions for the system a a x c a xb b c b a c xb c c c to have a p q symmetric solution we also show an expression of the p q symmetric solution to the system when the solvability conditions are met moreover we provide an algorithm and a numerical example to illustrate our results the findings of this paper extend some known results in the literature

a system of quaternary coupled sylvester type real quaternion matrix equations abstract in this paper we establish some necessary and sufficient solvability conditions for a system of quaternary coupled sylvester type real quaternion matrix equations in terms of ranks and generalized inverses of matrices an expression of the general solution to this system is given when it is solvable also a numerical example is presented to illustrate the main result of this paper the findings of this paper widely generalize the known results in the literature the main results are also valid over the real number field and the complex number field

simultaneous solutions of matrix equations and simultaneous equivalence of matrices given matrices ai bi and ci i i of corresponding dimensions over a field f we prove that i if aiciobi are simultaneously similar to aioobi then there exists a simultaneous solution x to the matrix sylvester equations aix xbi ci and ii if aiciobi are simultaneously equivalent to aioobi then there exist simultaneous solutions x y to the matrix equations aix ybi ci we also show that analogous results hold for mixed pairs of matrix sylvester equations a1x1 yb1 c1 a2x2 yb2 c2 and for generalized stein equations x ayb c

perfect and ideal 0 1 matrices a 0 1 matrix a is said to be perfect resp ideal if the corresponding generalized packing resp covering polytope is integral given a 0 1 matrix a we construct a 0 1 matrix that is perfect if and only if a is perfect a similar result is obtained for the generalized covering problem we also extend some known results on perfect 0 1 matrices to the 0 1 case

the quaternion lms algorithm for adaptive filtering of hypercomplex processes the quaternion least mean square qlms algorithm is introduced for adaptive filtering of three and four dimensional processes such as those observed in atmospheric modeling wind vector fields these processes exhibit complex nonlinear dynamics and coupling between the dimensions which make their component wise processing by multiple univariate lms bivariate complex lms clms or multichannel lms mlms algorithms inadequate the qlms accounts for these problems naturally as it is derived directly in the quaternion domain the analysis shows that qlms operates inherently based on the so called ldquoaugmentedrdquo statistics that is both the covariance e xx h and pseudocovariance e xx t of the tap input vector x are taken into account in addition the operation in the quaternion domain facilitates fusion of heterogeneous data sources for instance the three vector dimensions of the wind field and air temperature simulations on both benchmark and real world data support the approach

right eigenvalue equation in quaternionic quantum mechanics we study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n dimensional quaternionic vector spaces for quaternionic linear operators the eigenvalue spectrum consists of n complex values for these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations our discussion is also extended to complex linear operators whose spectrum is characterized by 2n complex eigenvalues we show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products finally we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution

perfect 0 pl matrices and perfect bidirected graphs abstract perfect 0 1 matrices and perfect bidirected graphs are independently defined but are closely related in this paper we discuss these relations and give several characterizations of perfectness the generalized stable set problem associated with bidirected graphs is a generalization of the maximum weighted stable set problem we also give a brief proof that if a given bidirected graph is perfect then the problem can be solved in polynomial time

quaternions and matrices of quaternions abstract we give a brief survey on quaternions and matrices of quaternions present new proofs for certain known results and discuss the quaternionic analogues of complex matrices the methods of converting a quaternion matrix to a pair of complex matrices and homotopy theory are emphasized

generalized schur methods with condition estimators for solving the generalized sylvester equation stable algorithms are presented for solving the generalized sylvester equation they are based on orthogonal equivalence transformations of the original problem perturbation theory and rounding error analysis are included condition estimators dif sup 1 estimators are developed which when substituted into derived error bounds give accuracy estimates of a computed solution results from numerical experiments on well conditioned and ill conditioned problems are reported

output feedback eigenstructure assignment using two sylvester equations the eigenstructure assignment problem with output feedback is studied for systems satisfying the condition p m n the main tool used is the concept of c a b invariance and two coupled sylvester equations the solution of which leads to the computation of an output stabilizing feedback a computationally efficient algorithm for the solution of these two coupled equations which leads to the computation of a desired output feedback is presented

primitive almost primitive test and δ primitive elements of free algebras with the nielsen schreier property abstract we study generalized primitive elements of free algebras of finite ranks with the nielsen schreier property and their automorphic orbits a primitive element of a free algebra is an element of some free generating set of this algebra almost primitive elements are not primitive elements which are primitive in any proper subalgebra primitive elements are elements whose partial derivatives generate the same one sided ideal of the universal multiplicative envelope algebra of a free algebra as the set of free generators generate we prove that an endomorphism preserving an automorphic orbit of a nonzero element of a free algebra of rank two is an automorphism an algorithm to determine test elements of free algebras of rank two is described a series of almost primitive elements is constructed and new examples of test elements are given we prove that if the rank n of the free lie algebra l is even n 2m then any primitive element of l is an automorphic image of the element w x1 x2 x2m 1 x2m there are no primitive elements of l if n is odd and the group of automorphisms of the algebra l acts transitively on the set of all primitive elements

on triangular representations of aut fr abstract if f r gl n k is a representation of a finitely generated free group f r and a m i for each basic element i e element which occurs in some basis a then we show that if f r is triangularisable it is finite this can be thought of as a generalisation of the burnside problem for these linear groups

on heegaard decompositions of torus knot exteriors and related seifert fibre spaces the heegaard decompositions of genus 2 oftorus knot exteriors are classified with respect to homeomorphisms it turns out that in general there are three different classes which are also the isotopy classes a similar result is obtained for seifert fibre spaces having a disk as base and two exceptional fibres such as the exteriors of torus knots have or having a sphere as base and three exceptional fibres

palindromic primitives and palindromic bases in the free group of rank two abstract the present paper records more details of the relationship between primitive elements and palindromes in f 2 the free group of rank two we characterize the conjugacy classes of palindromic primitive elements as those in which cyclically reduced words have odd length we identify large palindromic subwords of certain primitives in conjugacy classes which contain cyclically reduced words of even length we show that under obvious conditions on exponent sums pairs of palindromic primitives form palindromic bases for f 2 further we note that each cyclically reduced primitive element is either a palindrome or the concatenation of two palindromes

unknotting tunnels and seifert surfaces let k be a knot with an unknotting tunnel gamma and suppose that k is not a 2 bridge knot there is an invariant rho p q in mathbb q 2 mathbb z with p odd defined for the pair k gamma the invariant rho has interesting geometric properties it is often straightforward to calculate for example for k a torus knot and gamma an annulus spanning arc rho k gamma 1 although rho is defined abstractly it is naturally revealed when k cup gamma is put in thin position if rho neq 1 then there is a minimal genus seifert surface f for k such that the tunnel gamma can be slid and isotoped to lie on f one consequence is that if rho k gamma neq 1 then mathrm genus k 1 this confirms a conjecture of goda and teragaito for pairs k gamma with rho k gamma neq 1

enumerating palindromes and primitives in rank two free groups abstract let f be a free group of rank two an element of f is primitive if it along with another group element generates the group if f a b then a word w a b in a and b is a palindrome if it reads the same forwards and backwards it is known that in a rank two free group for any fixed set of two generators a primitive element will be conjugate either to a palindrome or to the product of two palindromes but known iteration schemes for all primitive words give only a representative for the conjugacy class here we derive a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresses the word as a unique product of two unique palindromes that have already appeared in the scheme we denote these words by e p q where p q is rational number expressed in lowest terms we prove that e p q is a palindrome if pq is even and the unique product of two unique palindromes if pq is odd we prove that the pair x y or x 1 y 1 generates the group if and only if x is conjugate to e p q and y is conjugate to e r s where p s r q 1 this improves a previously known result that held only for pq and rs both even the derivation of the enumeration scheme also gives a new proof of the known results about primitive words

perfect 0 1 matrices abstract perfect graphs and perfect 0 1 matrices are well studied in the literature here we introduce perfect 0 1 matrices our main result is a characterization of these matrices in terms of a family of perfect 0 1 matrices

systems of coupled generalized sylvester matrix equations this paper studies some systems of coupled generalized sylvester matrix equations we present some necessary and sufficient conditions for the solvability to these systems we give the expressions of the general solutions to the systems when their solvability conditions are satisfied

binary integer programs with two variables per inequality several recent papers have shown that some properties of the maximum weight stable set problem hold true in the more general setting of binary integer programs with two variables per inequality in this paper we show that in fact the two problems are equivalent by using the transitive closure of the binary integer program and possibly reducing the number of variables by fixing complementing or identifying them we use this equivalence to prove two conjectures made by johnson and padberg regarding the perfection of bidirected graphs