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

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

numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

multiple factorizations of words and defect effect we prove that if x is a finite prefix set and w is a non periodic bi infinite word possessing 3 disjoint x factorizations then the combinatorial rank of x is at most card x gamma 2 this is one of the rare cases when a cumulative defect effect is known to hold finally connections to the critical factorization theorem are discussed

the ehrenfeucht conjecture a algebra framework for its proof abstract we describe an algebraic framework including in particular the affine monoid of a given ring within this framework it is a simple matter to deduce ehrenfeucht s conjecture from hilbert s basis theorem

prinjective modules reflection functors quadratic forms and auslander reiten sequences let a b be artinian rings and let a m b be an a b bimodule which is a finitely generated left a module and a finitely generated right b module a right a m b prinjective module is a finitely genrated module x r x a x b x a a m b x b over the triangular matrix ring r a o a m b b such that x a is a projective a module x b is an injective b module and is a b homomorphism we study the category prin r b a of right a m b prinjective modules it is an additive krull schmidt subcategory of mod r closed under extensions

tame wild dichotomy of birkhoff type problems for nilpotent linear operators abstract inspired by recent results of ringel schmidmeier kussin lenzing meltzer xiao wu chen and pu zhang given a field k m 1 and a finite poset i i with a unique maximal element we study the category m o n i f m of mono representations of i over the frobenius k algebra f m k t t m of k dimension m viewed as k vector spaces u with an m nilpotent k linear operator t u u together with t invariant subspaces u i u j u for all i j in i the problem of when the krull schmidt k category m o n i f m is of wild resp tame representation type is called a wild resp tame birkhoff type problem for m nilpotent operators in case when k is algebraically closed we give a complete solution of the problem by describing all minimal pairs i m resp all pairs with m 1 such that category m o n i f m is of wild resp tame representation type we reduce the problem to a birkhoff type problem for the category fspr i f m m o n i f m of subprojective representations over f m of a larger poset i i the tame wild dichotomy for the category m o n i f m is also proved surprisingly in case when i i a b is the union of two incomparable chains i and i of length i a 1 1 and i b 1 1 with i i the problem is equivalent with the wildness resp tameness of the category coh x p of coherent sheaves over the weighted projective line x p for the weight triple p a b m with a b m 2 studied by kussin lenzing and meltzer 19 in relation with the hypersurface singularity f x 1 a x 2 b x 3 m

a note on decidability questions on presentations of word semigroups we apply automata theoretic tools and some recently established compactness properties in the study of f semigroups that is subsemigroups of free semigroups with each f semigroup we associate an f presentation which turns out to be finite for all finitely generated f semigroups connections between f presentations and ordinary presentations of semigroups are also pointed out it is also shown that it is undecidable whether two finitely generated f semigroups satisfy a common relation in their f presentation

a compactness property of the k abelian monoids abstract the k abelian equivalence of words counting the numbers of occurrences of factors of length at most k has been analyzed in recent years from several different directions we continue this analysis the k abelian equivalence classes are known to constitute a monoid hence equations over these monoids are well defined we show that these monoids satisfy a compactness property each system of equations with a finite number of unknowns is equivalent to some of its finite subsystems we give two proofs for this compactness result one is based the fact that the monoid can be embedded into the multiplicative monoid of matrices and the other directly on linear algebra the former method allows the application of hilbert s basis theorem the latter one in turn allows to conclude an upper bound for the size of the finite subsystem

on invariant analysis of some time fractional nonlinear systems of partial differential equations i an investigation of lie point symmetries for systems of time fractional partial differential equations including ito system coupled burgers equations coupled korteweg de vries equations hirota satsuma coupled kdv equations and coupled nonlinear hirota equations has been done using the obtained symmetries each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving erdelyi kober fractional differential operator depending on a parameter

inflation algorithm for loop free non negative edge bipartite graphs of corank at least two we continue the study of finite connected loop free edge bipartite graphs with m 3m 3 vertices a class of signed graphs we started in simson 2013 48 and m g siorek et al 2016 19 by means of the non symmetric gram matrix g mm z of its symmetric gram matrix g 12 g g tr mm 12z and the gram quadratic form q zm zq zm z in particular we study connected non negative edge bipartite graphs with n r 3n r 3 vertices of corank r 2r 2 in the sense that the symmetric gram matrix g mn r z g mn r z of is positive semi definite of rank n 1n 1 the edge bipartite graphs of corank r 2r 2 are studied up to the weak gram zz congruence z z where z z means that g btr g bg btr g b for some b mn r z b mn r z such that det b 1det b 1 our main result of the paper asserts that given a connected edge bipartite graph with n r 3n r 3 vertices of corank r 2r 2 there exists a suitably chosen sequence t of the inflation operators of the form tab such that the composite operator t reduces to a connected bigraph d n r such that zd n r and d n r is one of the canonical r vertex extensions a n r n 1n 1 d n r n 4n 4 e 6 r e 7 r and e 8 r with n rn r vertices of the simply laced dynkin diagrams an dn e6 e7 e8an dn e6 e7 e8 with n 1n 1 vertices the algorithm constructs also a matrix b mn r z b mn r z with det b 1det b 1 defining the weak gram zz congruence zd n r that is satisfying the equation gd n r btr g b

algebraic properties of word equations abstract the question about maximal size of independent system of word equations is one of the most striking problems in combinatorics on words recently aleksi saarela has introduced a new approach to the problem that is based on linear algebraic properties of polynomials encoding the equations and their solutions in this paper we develop further this approach and take into account other algebraic properties of polynomials namely their factorization this in particular allows to improve the bound for the number of independent equations with maximal rank from quadratic to linear

a coxeter type classification of one peak principal posets abstract we present a complete coxeter spectral classification of finite one peak posets i t that are principal in the sense that their symmetric gram matrix g i 1 2 c i c i t r m i q is positive semi definite of rank i 1 where c i m i z is the incidence matrix of i equivalently posets t with the quadratic tits form q t z t 1 z in the sense of drozd non negative and the kernel ker q t v z t 1 q t v 0 z t 1 infinite cyclic we continue the study started in m g siorek d simson 2012 11 we show that up to the strong gram z congruence and the weak gram z congruence every one peak principal poset is uniquely determined by its complex coxeter spectrum specc i c moreover we calculate the exact number of all such posets of a given size up to isomorphism and present their hasse quivers we finish the paper with conjecture on the exact number of principal positive posets according to our knowledge no formula is known for the number of non isomorphic posets in a general case

on systems of word equations over three unknowns with at most six occurrences of one of the unknowns in this paper we investigate the open question formulated in 1983 by culik ii and karhumaki asking whether there exist independent systems of three word equations over three unknowns admitting non periodic solutions in particular we answer negatively the above mentioned question for systems in which one of the unknowns occurs at most six times that is we show that such systems admit only periodic solutions or they are not independent

a bounded and efficient scheme for multidimensional problems with anomalous convection and diffusion abstract this work is motivated by a generalization of the well known burgers fisher and burgers huxley equations in multiple dimensions considering riesz fractional diffusion and convection initial boundary conditions which are positive and bounded are imposed on a closed and bounded rectangular domain in this manuscript we propose a finite difference method to approximate the positive and bounded solutions of the fractional model the methodology is a linear three steps crank nicolson technique which is based on the use of fractional centered differences the properties of fractional centered differences are employed to establish the existence and the uniqueness of solutions of the finite difference method as well as the capability of the technique to preserve the positivity and the boundedness of the approximations we show in this work that the method is capable of preserving some of the constant solutions of the continuous model additionally we prove that our technique is a second order consistent stable and quadratically convergent scheme suitable bounds for the numerical solutions are also derived in this work finally some illustrative simulations show that the method is able to preserve the positivity and the boundedness of the numerical approximations in agreement with the analytic results proved in this work numerical comparisons provided in this work confirm the rate of convergence of the numerical technique

new exact solutions of nonlinear fractional acoustic wave equations in ultrasound in this paper new exact solutions of fractional nonlinear acoustic wave equations have been devised the travelling periodic wave solutions of fractional burgers hopf equation and khokhlov zabolotskaya kuznetsov kzk equation have obtained by first integral method nonlinear ultrasound modelling is found to have an increasing number of applications in both medical and industrial areas where due to high pressure amplitudes the effects of nonlinear propagation are no longer negligible taking nonlinear effects into account the ultrasound beam analysis makes more accurate in these applications the burgers hopf equation is one of the extensively studied models in mathematical physics in addition the kzk parabolic nonlinear wave equation is one of the most widely employed nonlinear models for propagation of 3d diffraction sound beams in dissipative media in the present analysis these nonlinear equations have solved by first integral method as a result new exact analytical solutions have been obtained first time ever for these fractional order acoustic wave equations the obtained results are presented graphically to demonstrate the efficiency of this proposed method

kleiner s theorem for unitary representations of posets abstract a subspace representation of a poset s s 1 s t is given by a system v v 1 v t consisting of a vector space v and its subspaces v i such that v i v j if s i s j for each real valued vector 1 t with positive components we define a unitary representation of s as a system u u 1 u t that consists of a unitary space u and its subspaces u i such that u i u j if s i s j and satisfies 1 p 1 t p t 1 in which p i is the orthogonal projection onto u i we prove that s has a finite number of unitarily nonequivalent indecomposable representations for each weight if and only if s has a finite number of nonequivalent indecomposable subspace representations that is if and only if s contains any of kleiner s critical posets

on dimension of poset variety abstract for a finite partially ordered set we calculate the dimension of the variety of its subspace representations having fixed dimension vector the dimension is given in terms of the euler quadratic form associated with a partially ordered set which gives a geometric interpretation of this form

symbolic computation of strong gram congruences for cox regular positive edge bipartite graphs with loops abstract the paper can be viewed as a second part of the author s paper simson 2018 43 our main aim is to construct symbolic algorithms for the coxeter spectral classification of a class of signed graphs called edge bipartite graphs with n 2 vertices we started in simson 2013 37 and bocian et al 2014 6 more precisely we construct algorithms for the classification up to the strong gram z congruence z of all finite connected cox regular edge bipartite graphs with at least one loop bigraphs for short that are positive in the sense that the associated symmetric gram matrix g 1 2 g g t r m n 1 2 z is positive definite where g m n z is the non symmetric gram matrix of defined in section 1 and z means that there is a z invertible matrix b m n z such that g b t r g b we recall from 43 that every such a bigraph with a loop is strongly gram z congruent with a bigraph d that is one of the positive cox regular bigraphs b n n 2 c n n 3 f 4 m 4 g 2 presented in section 1 here by applying the geometry of mesh root system technique we construct symbolic algorithms that compute the correspondence d and the set gl n z d of all z invertible matrices b m n z defining the strong gram z congruence z d for any connected positive bigraph with n 2 vertices and at least one loop

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

explicit test sets for iterated morphisms in free monoids and metabelian groups for checking equivalence of any two word morphisms restricted on a subset language it suffices to do this for a finite subset of the language the so called finite test set a way of effectively obtaining a finite test set for a special class of languages the so called iterated morphisms is presented here together with an explicit upper bound for its size the method can be extended to free metabelian groups and to certain other metabelian groups

p critical integral quadratic forms and positive unit forms an algorithmic approach abstract we introduce the class of p critical integral unit forms q z n z containing the critical forms in the sense of ovsienko 13 several characterisations of p critical forms are given in particular it is proved that q is p critical if and only if there is a uniquely determined extended dynkin diagram a n d n e 6 e 7 e 8 and a special group isomorphism t z n z n such that q t is the quadratic form q z n z n 0 of the diagram a correspondence between positive forms p z n 1 z with a sincere root and p critical forms q z n z is described and efficient linear algebra algorithms for computing p critical unit forms and positive forms are constructed

many aspects of defect theorems we give a survey and a unified presentation of the defect theorem its generalizations and recent aspects of interest in its basic form the defect theorem states that if a set of n words satisfies a nontrivial relation then these words can be expressed simultaneously as products of at most n 1 words in other words dependency of words causes a defect effect there does not exist just one defect theorem but several ones depending on the restrictions that are put to the n 1 words the defect theorem is closely related to equations of words and in this way to the compactness theorem for systems of word equations

equations over finite sets of words and equivalence problems in automata theory abstract several observations concerning equations over monoids of finite sets of words are made monoids considered are the monoid of finite prefix codes the monoid of all finite sets and the monoid of all finite multisets as well as some others as an application some known decidability results in automata theory are extended finally several open questions are raised

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

the ehrenfeucht conjecture a compactness claim for finitely generated free monoids abstract we survey recent results on the so called ehrenfeucht conjecture which states for each language l over a finite alphabet there exists a finite subset f of l such that for each pair g h of morphisms on the equation g x h x holds for all x in l if and only if holds for all x in f we point out that the conjecture is closely related to the theory of equations in free monoids we also state a suprising consequence of the conjecture if it holds even noneffectively for all dol languages then the hdol sequence equivalence problem is decidable furthermore we give examples of when the conjecture is known to hold in particular we establish it for all binary languages as well as for all languages when attention is restricted to bounded delay morphisms of some fixed delay

test sets for finite substitutions abstract ehrenfeucht s conjecture that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s has recently been verified in this paper we extend this result from endomorphisms to certain types of finite substitutions by applying konig s lemma and embeddings of the monoid into a free algebra

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

exceptional modules are tree modules abstract we consider representations of a quiver over an arbitrary field recall that an indecomposable representation m without self extensions is said to be exceptional we are going to show that exceptional representations can be exhibited using matrices involving as coefficients just 0 and 1 actually if d is the dimension of m there exists such a matrix presentation with precisely d 1 non zero coefficients the corresponding coefficient quiver is a tree

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

the expressibility of languages and relations by word equations classically several properties and relations of words such as being a power of the same word can be expressed by using word equations this paper is devoted to a general study of the expressive power of word equations as main results we prove theorems which allow us to show that certain properties of words are not expressible as components of solutions of word equations in particular the primitiveness and the equal length are such properties as well as being any word over a proper subalphabet

on systems of word equations with simple loop sets consider the infinite system s of word equations x 0u 1 ix 1u 2 ix 2 u m ix m y 0v 1 iy 1v 2 iy 2 v n iy n email protected n for each email protected n let t k be the subsystem of s given by email protected k k 1 k 2 we prove two properties of the above system 1 let k 1 if f is a solution of t k such that primitive roots of f u 1 f u 2 f u m are of equal length as well as primitive roots of f v 1 f v 2 f v n then f is a solution of the whole s 2 if n 1 then for any k 2 a solution f of t k is also a solution of s

multiple flag varieties of finite type abstract we classify all products of flag varieties with finitely many orbits under the diagonal action of the general linear group we also classify the orbits in each case and construct explicit representatives

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

multiplicities a deterministic view of nondeterminism abstract we discuss four important problems in automata theory each of which has three natural variants a deterministic one a nondeterministic one and as an in between case a nondeterministic one with multiplicities our goal is to support a general view that the third variant is much closer to the first one than to the second one

local and global cyclicity in free semigroups it is shown that the system of equations x1r xnr s x1s xns r r s n is equivalent to its two element subset x1a xna b x1b xnb a x1a xna c x1c xnc a whenever a b c are integers such that 1 1

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

efficient constructions of test sets for regular and context free languages we present a simple construction of linear size test sets for regular languages and of single exponential test sets for context free languages in the case of regular sets the size of our test set is exactly the number of transitions of the automaton this improves the best known upper bounds exponential for regular and doubly exponential for context free languages we give also an o n log n time algorithm for the morphism equivalence and an o n3log n time algorithm to test the gsm equivalence on a regular language an o n2log n time algorithm is given to test the equivalence of two deterministic gsm s as well as that of two deterministic finite transducers

simplest equation method for some time fractional partial differential equations with conformable derivative abstract the conformable fractional derivative was proposed by r khalil et al in 2014 which is natural and obeys the leibniz rule and chain rule based on the properties a class of time fractional partial differential equations can be reduced into odes using traveling wave transformation then the simplest equation method is applied to find exact solutions of some time fractional partial differential equations the exact solutions solitary wave solutions periodic function solutions rational function solutions of time fractional generalized burgers equation time fractional generalized kdv equation time fractional generalized sharma tasso olver fsto equation and time fractional fifth order kdv equation 3 1 dimensional time fractional kdv zakharov kuznetsov kdv zk equation are constructed this method presents a wide applicability to solve some nonlinear time fractional differential equations with conformable derivative

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

improved fractional sub equation method for 3 1 dimensional generalized fractional kdv zakharov kuznetsov equations in the present paper we construct the analytical exact solutions of some nonlinear evolution equations in mathematical physics namely the time fractional kdv zakharov kuznetsov kdv zk and space time fractional modified kdv zakharov kuznetsov mkdv zk equations by using improved fractional sub equation method as a result new types of exact analytical solutions are obtained the obtained results are shown graphically here the fractional derivative is described in the jumarie s modified riemann liouville sense

on the system of word equations x o u i i x 1 u i 2 x 2 u i 3 x 3 y 0 v i 1 y 1 v i 2 y 2 v i 3 y 3 i 0 1 2 l in a free monoid abstract we prove that if x 0 u 1 i x 1 u 2 i x 2 y 0 v 1 i y 1 v 2 i y 2 holds for all i 0 1 2 3 then it is true for each i 0 1 2 and that if x 0 u 1 i x 1 u 2 i x 2 u 3 i x 3 y 0 v 1 i y 1 v 2 i y 2 v 3 i y 3 holds for all i 0 1 2 3 4 5 then it is true for each i 0 1 2

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

different methods for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation in this paper the fractional derivatives in the sense of modified riemann liouville derivative and the exp function method the g g expansion method and the generalized kudryashov method are used to construct exact solutions for 3 1 dimensional space time fractional modified kdv zakharov kuznetsov equation this fractional equation can be turned into another nonlinear ordinary differential equation by fractional complex transformation and then these three methods are applied to solve it as a result some new exact solutions are obtained the three methods demonstrate power reliability and efficiency

on the independence of equations in three variables we prove that an independent system of equations in three variables with a nonperiodic solution and at least two equations consists of balanced equations only for that we show that the intersection of two different entire systems contains only balanced equations where an entire system is the set of all equations solved by a given morphism furthermore we establish that two equations which have a common nonperiodic solution have the same set of periodic solutions or are not independent

on the periodicity of coxeter transformations and the non negativity of their euler forms abstract we show that for piecewise hereditary algebras the periodicity of the coxeter transformation implies the non negativity of the euler form contrary to previous assumptions the condition of piecewise heredity cannot be omitted even for triangular algebras as demonstrated by incidence algebras of posets we also give a simple direct proof that certain products of reflections defined for any square matrix a with 2 on its main diagonal and in particular the coxeter transformation corresponding to a generalized cartan matrix can be expressed as a 1 a t where a a are closely associated with the upper and lower triangular parts of a