principal minor sums of a tb m abstract the question is raised whether the sum of the k k principal minors of the titled matrix is a polynomial in t with positive coefficients when a and b are positive definite this would generalize a conjecture made by bessis moussa villani as stated by e h lieb and r seiringer equivalent forms of the bessis moussa villani conjecture j stat phys 115 2004 185 190 we give a variety of evidence for this further question some of which is new

link invariant of the izergin korepin model the link invariant associated with the izergin korepin 19 vertex model is deduced using the method of statistical mechanics it is shown that the izergin korepin model leads to an invariant which is precisely the 3 state akutsu wadati polynomial previously known only for 2 and 3 braid knots we give a table of the invariant for all knots and links up to seven crossings

the constant variation formulae for singular fractional differential systems with delay abstract this paper considers the caputo singular fractional differential systems with delay and the riemann liouville singular fractional differential systems with delay a new function is defined by the d inverse matrix and function two fundamental solutions are given the constant variation formulae for singular fractional differential systems with delay are obtained

simple excitations in the nested bethe ansatz model the scaling dimensions of the operators in the nested bethe ansatz model are studied they explicitly show the structure of the underlying lie algebras they can be written in terms of the cartan matrix the author infers that they are the natural extension of those of su 2 or equivalently of the coulomb gas

trace functions as laplace transforms we study trace functions on the form t trf a tb where f is a real function defined on the positive half line and a and b are matrices such that a is positive definite and b is positive semidefinite if f is non negative and operator monotone decreasing then such a trace function can be written as the laplace transform of a positive measure the question is related to the bessis moussa villani conjecture

advances on the bessis moussa villani trace conjecture abstract a long standing conjecture asserts that the polynomial p t tr a tb m has nonnegative coefficients whenever m is a positive integer and a and b are any two n n positive semidefinite hermitian matrices the conjecture arises from a question raised by bessis et al d bessis p moussa m villani monotonic converging variational approximations to the functional integrals in quantum statistical mechanics j math phys 16 1975 2318 2325 in connection with a problem in theoretical physics their conjecture as shown recently by lieb and seiringer is equivalent to the trace positivity statement above in this paper we derive a fundamental set of equations satisfied by a and b that minimize or maximize a coefficient of p t applied to the bessis moussa villani bmv conjecture these equations provide several reductions in particular we prove that it is enough to show that 1 it is true for infinitely many m 2 a nonzero matrix coefficient of a tb m always has at least one positive eigenvalue or 3 the result holds for singular positive semidefinite matrices moreover we prove that if the conjecture is false for some m then it is false for all larger m finally we outline a general program to settle the bmv conjecture that has had some recent success

eigenvalue and stability of singular differential delay systems the eigenvalue and the stability of singular differential systems with delay are considered firstly we investigate some properties of the eigenvalue then give the exact exponential estimation for the fundamental solution and finally discuss the necessary and sufficient condition of uniform asymptotic stability

low lying excitations in the square triangle random tiling model we consider the sub dominant eigenstates of the transfer matrix for the square triangle random tiling model on an infinite strip of width l a numerical algorithm for generation of the corresponding solution of the bethe ansatz is developed numerical finite size scaling analysis of the associated eigenvalues reveals the presence of both integer and non integer critical exponents the analytical value of one of the non integer exponents is found it is also shown numerically that along with the leading correction to the free energy density for some excitations there is a term proportional to

real zero polynomials and a horn s problem abstract a horn s problem concerns finding two self adjoint matrices so that a b and a b have prescribed spectrum in this paper we show how it connects to an interpolation problem for two variable real zero polynomials and a tracial moment problem in addition we outline an algorithm to construct a pair a b

proof of the cases p 7 of the lieb seiringer formulation of the bessis moussa villani conjecture it is shown that the polynomial t tr a tb p has nonnegative coefficients when p 7 and a and b are any two complex positive semidefinite n n matrices with arbitrary n this proves a general nontrivial case of the lieb seiringer formulation of the bessis moussa villani conjecture which is a long standing problem in theoretical physics

on the existence and uniqueness of solutions of a class of fractional differential equations abstract in this paper we investigate the existence and uniqueness of solutions for the following class of multi order fractional differential equations d 1 1 1 d n n n u t i 1 n d i i i u t d i n i i u t f t u t t 0 1 u 0 0 i 1 n i 1 i 0 i 0 1 i n where d i n i i denotes the generalized erdelyi kober operator of fractional derivative of order i moreover some properties concerning the positive maximal minimal and continuation of solutions are obtained

algebraic bethe ansatz for the gl 1 2 generalized model ii the three gradings the algebraic bethe ansatz can be performed rather abstractly for whole classes of models sharing the same r matrix the only prerequisite being the existence of an appropriate pseudo vacuum state here we perform the algebraic bethe ansatz for all models with 9 9 rational gl 1 2 invariant r matrix and all three possibilities of choosing the grading our bethe ansatz solution applies for instance to the supersymmetric t j model the supersymmetric u model and a number of interesting impurity models it may be extended to obtain the quantum transfer matrix spectrum for this class of models the properties of a specific model enter the bethe ansatz solution i e the expression for the transfer matrix eigenvalue and the bethe ansatz equations through the three pseudo vacuum eigenvalues of the diagonal elements of the monodromy matrix which in this context are called the parameters of the model

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

the controllability of fractional control systems with control delay this paper discusses the controllability of fractional control systems with control delay we firstly give the solution expression for fractional control systems with control delay then give the necessary and sufficient conditions for the controllability of fractional control systems with control delay

new exact exponents for two dimensional self avoiding walks the author establishes using standard coulomb gas methods the values of a new set of geometrical exponents that have been previously conjectured these exponents give for any p the number of configurations of p two dimensional self avoiding walks of the same length l which are attached by their ends

equivalent forms of the bessis moussa villani conjecture the bmv conjecture for traces which states that text tr text exp left a lambda b right is the laplace transform of a positive measure is shown to be equivalent to two other statements i the polynomial lambda mapsto text tr left a lambda b right p has only non negative coefficients for all a b geqslant 0 p in mathbb n and ii lambda mapsto text tr left a lambda b right p is the laplace transform of a positive measure for a b geqslant 0 p 0

loop erased self avoiding walks in two dimensions exact critical exponents and winding numbers abstract loop erased self avoiding walks lesaws are defined as walks resulting from sequential erasing of the loops of random walks the critical properties of lesaws are those of a c 2 conformal theory in two dimensions and geometrically those of subgraphs of spanning trees in this paper we derive the exact asymptotic behavior of the set of probabilities p k n that k k 2 lesaws of length n starting at neighbouring points do not intersect as well as the exact asymptotic winding angle distribution of a single two dimensional lesaw

surface critical behaviour of an o n loop model related to two manhattan lattice walk problems we find and discuss the scaling dimensions of the branch 0 manifold of the nienhuis o n loop model on the square lattice concentrating on the surface dimensions the results are extracted from a bethe ansatz calculation of the finite size corrections to the eigenspectrum of the six vertex model with free boundary conditions these results are especially interesting for polymer physics at two values of the crossing parameter lambda interacting self avoiding walks on the manhattan lattice at the collapse temperature lambda pi 3 and hamiltonian walks on the manhattan lattice lambda pi 2 are discussed in detail our calculations illustrate the importance of examining both odd and even strip widths when performing finite size correction calculations to obtain scaling dimensions

two dimensional o n model in a staggered field nienhuis truncated o n model gives rise to a model of self avoiding loops on the hexagonal lattice each loop having a fugacity of n we study such loops subjected to a particular kind of staggered field w which for n has the geometrical effect of breaking the three phase coexistence linked to the three colourability of the lattice faces we show that at t 0 for w 1 the model flows to the ferromagnetic potts model with q n2 states with an associated fragmentation of the target space of the coulomb gas for t 0 there is a competition between t and w which gives rise to multicritical versions of the dense and dilute loop universality classes via an exact mapping and numerical results we establish that the latter two critical branches coincide with those found earlier in the o n model on the triangular lattice using transfer matrix studies we have found the renormalization group flows in the full phase diagram in the t w plane with fixed n superposing three copies of such hexagonal lattice loop models with staggered fields produces a variety of one or three species fully packed loop models on the triangular lattice with certain geometrical constraints possessing integer central charges 0 c 6 in particular we show that benjamini and schramm s rgb loops have fractal dimension df 3 2

operator spectrum and exact exponents of the fully packed loop model we develop a coulomb gas description of the critical fluctuations in the fully packed loop model on the honeycomb lattice we identify the complete operator spectrum of this model in terms of electric and magnetic vector charges and we calculate the scaling dimensions of these operators exactly we also study the geometrical properties of loops in this model and we derive exact results for the fractal dimension and the loop size distribution function a review of the many different representations of this model that have recently appeared in the literature is given

a new exactly solvable case of an o n model on a hexagonal lattice a new solvable case of o n model on a hexagonal lattice is presented it is shown that the model has a phase transition at n 2 this phase transition can be interpreted as phase transition between percolating and non percolating phases

extended scaling relations for the chiral and cubic crossover exponents the critical dimensions xch q of the chiral and xcb q of the cubic operator in the two dimensional q state potts model satisfy the extended scaling relations xch n2 y2 4x x and xcb m2 y2 4x x with x y 2 2 cos pi y 2 square root q n an odd and m an even integer n 1 and m 0 give the leading exponents at q 3 xch is relevant but takes the special value xt 1 9 5 the crossover exponent at the percolation point in the bond diluted random ising model determined by xcb at q 1 is equal to one along the baxter line in the ashkin teller eight vertex and annni model xch xt 4 1 xt

constrained controllability of fractional linear systems with delays in control abstract the paper presents finite dimensional dynamical control systems described by linear fractional order state equations with multiple delays in control the constrained controls are considered the set of admissible control values is assumed to be a compact set containing 0 a convex and compact set containing 0 in its interior a cone with vertex at zero and a nonempty interior or a convex and close cone with nonempty interior and vertex at zero the definitions of various types of constrained controllability of the linear fractional systems with delays in control are formulated new necessary and sufficient conditions for constrained relative controllability of fractional order control systems with delayed controls are established and proved numerical examples are provided to illustrate the obtained theoretical results

from dilute to dense self avoiding walks on hypercubic lattices a field theoretic representation is presented to count the number of configurations of a single self avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions we evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain the meanfield approximation is exact in the limit of infinite dimensions and corrections to it in powers ofd 1 can be systematically evaluated the connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions we also find that the entropy per site develops a maximum atf 1 2d 1 ford 2 d 3 this maximum occurs atf 0 80 f 0 86 and its value is about 50 30 higher than the entropy per site of a hamiltonian walk f 1

critical exponents of manhattan hamiltonian walks in two dimensions from potts and o n models we consider a set of hamiltonian circuits filling a manhattan lattice i e a square lattice with alternating traffic regulation we show that the generating function with fugacityz of this set is identical to the critical partition function of aq state potts model on an unoriented square lattice withq1 2 z the set of critical exponents governing correlations of hamiltonian circuits is derived using a coulomb gas technique these exponents are also found to be those of an o n vector model in the low temperature phase withn q1 2 z the critical exponents in the limitz 0 are then those of spanning trees q 0 and of dense polymers n 0 t tc corresponding to a conformal theory with central chargec 2 this shows that the manhattan orientation and the hamiltonian constraint of filling all the lattice are irrelevant for the infrared critical properties of hamiltonian walks

higher conformal multifractality we derive from conformal invariance and quantum gravity the multifractal spectrum f of the harmonic measure i e electrostatic potential or diffusion field near any conformally invariant fractal in two dimensions it gives the hausdorff dimension of the set of points where the potential varies with distance r to the fractal frontier as r first examples are a random walk i e a brownian motion a self avoiding walk or a critical percolation cluster the generalized dimensions d n as well as the multifractal functions f are derived and are all identical for these three cases the external frontiers of a brownian motion and of a percolation cluster are thus identical to a self avoiding walk in the scaling limit the multifractal mf function f c of the electrostatic potential near any conformally invariant fractal boundary like a critical o n loop or a q state potts cluster is given as a function of the central charge c of the associated conformal field theory the dimensions dep of the external perimeter and dh of the hull of a critical scaling curve or cluster obey the superuniversal duality equation d text ep 1 d text d 1 tfrac 1 4 finally for a conformally invariant scaling curve which is simple i e without double points we derive higher multifractal functions like the universal function f2 which gives the hausdorff dimension of the points where the potential varies jointly with distance r as r on one side of the curve and as r on the other the general case of the potential distribution between the branches of a star made of an arbitrary number of scaling paths is also treated the results apply to critical o n loops potts clusters and to the sle process we present a duality between external perimeters of potts clusters and o n loops at their critical point as well as the corresponding duality in the sle process for 16

on the positivity of the coefficients of a certain polynomial defined by two positive definite matrices it is shown that the polynomial p t tr a tb m has positive coefficients when m 6 and a and b are any two 3 by 3 complex hermitian positive definite matrices this case is the first that is not covered by prior general results this problem arises from a conjecture raised by bessis moussa and villani in connection with a long standing problem in theoretical physics the full conjecture as shown recently by lieb and seiringer is equivalent to p t having positive coefficients for any m and any two n by n positive definite matrices we show that generally the question in the real case reduces to that of singular a and b and this is a key part of our proof

critical properties of the izergin korepin and solvable o n models and their related quantum spin chains the central charge of the izergin korepin model the corresponding quantum spin chain and the o n model is calculated analytically via the bethe ansatz the calculation extends a technique recently developed for the zamolodchikov fateev model in addition critical exponents and the central charge for these models are obtained from numerical solutions of bethe ansatz equations for finite systems as a physical application the authors find the exponents v 12 23 and gamma 53 46 for the theta transition of polymers in two dimensions

o n field theory with n continuous as a model for equilibrium polymerisation a grand canonical ensemble of continuous chains and rings is introduced for describing polymerisation equilibrium like that of sulphur equivalence is shown to o n field theory with n continuous n to 1

the packing of two species of polygons on the square lattice we decorate the square lattice with two species of polygons under the constraint that every lattice edge is covered by only one polygon and every vertex is visited by both types of polygons we end up with a 24 vertex model which is known in the literature as the fully packed double loop model fpl2 in the particular case in which the fugacities of the polygons are the same the model admits an exact solution the solution is obtained using coordinate bethe ansatz and provides a closed expression for the free energy in particular we find the free energy of the four colouring model and the double hamiltonian walk and recover the known entropy of the ice model when both fugacities are set equal to 2 the model undergoes an infinite order phase transition

exact solution and surface critical behaviour of an o n model on the honeycomb lattice the authors obtain the exact bethe ansatz solution of an o n model on the honeycomb lattice with open boundaries across a finite strip several quantities relevant to the surface critical behaviour of the o n model are derived from this solution the exact surface critical exponents are in agreement with those obtained by cardy 1987 and by duplantier and saleur 1986 based on conformal invariance arguments

exact critical exponents for two dimensional dense polymers using conformal invariance and coulomb gas results the author gives the exact value in two dimensions of the eta exponent of l dense polymers attached by their extremities eta l l2 4 8 the value in two dimensions of the gamma exponent of a dense branched polymer of fixed topology with nl l leg vertices l or 1 is then deduced to be gamma sigma l or 1 nl 2 l l 18 32 these values correspond to a conformally invariant theory with central charge c 2

universality of surface exponents of self avoiding walks on a manhattan lattice the authors use phenomenological renormalization group techniques to study the bulk and surface properties of self avoiding walks on a manhattan lattice they find that as is the case for the bulk exponents the underlying bond directionality does not change the universality class of the surface exponents relative to undirected lattices they also obtain an estimate for the position of the binding transition

scalar products of bethe vectors in models with mathfrak gl 2 1 symmetry 1 super analog of reshetikhin formula we study scalar products of bethe vectors in integrable models solvable by nested algebraic bethe ansatz and possessing mathfrak gl 2 1 symmetry using explicit formulas of the monodromy matrix entries multiple actions onto bethe vectors we obtain a representation for the scalar product in the most general case this explicit representation appears to be a sum over partitions of the bethe parameters it can be used for the analysis of scalar products involving on shell bethe vectors as a by product we obtain a determinant representation for the scalar products of generic bethe vectors in integrable models with mathfrak gl 1 1 symmetry

fast analytic solver of rational bethe equations in this note we propose an approach for a fast analytic determination of all possible sets of bethe roots corresponding to eigenstates of rational integrable spin chains of given not too large length in terms of baxter q functions we observe that all exceptional solutions if any are automatically correctly accounted the key intuition behind the approach is that the equations on the q functions are determined solely by the young diagram and not by the choice of the rank of the symmetry hence we can choose arbitrary and that accommodate the desired representation then we consider all distinguished q functions at once not only those following a certain kac dynkin path

o n matrix difference equations and a nested bethe ansatz a system of u n matrix difference equations is solved by means of a nested version of a generalized bethe ansatz the highest weight property of the solutions is proved and some examples of solutions are calculated explicitly

finite average lengths in critical loop models a relation between the average length of loops and their free energy is obtained for a variety of o n type models on two dimensional lattices by extending to finite temperatures a calculation due to kast we show that the number averaged loop length stays finite for all non zero fugacities n and in particular it does not diverge upon entering the critical regime n2 fully packed loop fpl models with n 2 seem to obey the simple relation 3lmin where lmin is the smallest loop length allowed by the underlying lattice we demonstrate this analytically for the fpl model on the honeycomb lattice and for the 4 state potts model on the square lattice and based on numerical estimates obtained from a transfer matrix method we conjecture that this is also true for the two flavour fpl model on the square lattice we present in addition numerical results for the average loop length on the three critical branches compact dense and dilute of the o n model on the honeycomb lattice and discuss the limit n0 contact is made with the predictions for the distribution of loop lengths obtained by conformal invariance methods

almost automorphic mild solutions to fractional differential equations abstract we introduce the concept of resolvent families to prove the existence of almost automorphic mild solutions to the differential equation d t u t a u t t n f t 1 2 n z considered in a banach space x where f r x is almost automorphic we also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t 1 2 assuming f t x is almost automorphic in t for each x x satisfies a global lipschitz condition and takes values on x finally we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation d t u t a u t f t u t u t 1 2 under analogous conditions as in the previous case

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

critical behavior and associated conformal algebra of the z 3 potts model the conformal algebra for operators of thez 3 model at the phase transition point is built critical exponents are found in this approach as solutions of simple algebraic equations which are consistency conditions of the algebra multipoint correlation functions obey linear differential equations some solutions are given for the four point correlation functions of thez 3 model at the phase transition point

two dimensional polymers and conformal invariance abstract the critical properties of self avoiding walks percolation clusters and critical exponents associated with the planar brownian motion are reviewed in two dimensions 2d the conformal spectrum of these geometrical critical systems is made of an infinite descrete series of scaling dimensions the latter are calculated in the framework of coulomb gas methods and conformal invariance the statistical mechanics of the same critical models on a random 2d lattice i e on a random surface namely in presence of a critically fluctuating metric in the so called 2d quantum gravity is also addressed and the above critical geometrical systems are exactly solvable in this case the new gravitational conformal spectrum is related to the standard conformal spectrum in the plane as predicted recently

proof of the bmv conjecture we prove the bmv bessis moussa villani 1 conjecture which states that the function t mapsto mathop rm tr exp a tb t geqslant 0 is the laplace transform of a positive measure on 0 if a and b are n times n hermitian matrices and b is positive semidefinite a semi explicit representation for this measure is given

hamiltonian walks on the sierpinski gasket we derive the exact number of hamiltonian walks h n on the two dimensional sierpinski gasket sg n at stage n whose asymptotic behavior is given by 3 23 3n 13 52 72 172212 35 13 16 n we also obtain the number of hamiltonian walks with one end at a specific outmost vertex of sg n with asymptotic behavior 3 23 3n 13 7 1724 33 4n the distribution of hamiltonian walks on sg n with one end at a specific outmost vertex and the other at an arbitrary vertex of sg n is investigated we rigorously prove that the exponent for the mean l displacement between the two end vertices of such hamiltonian walks on sg n is lln 2 ln 3 for l 0

traveling salesman problem conformal invariance and dense polymers we propose that the statistics of the optimal tour in the planar random euclidean traveling salesman problem is conformally invariant on large scales this is exhibited in the power law behavior of the probabilities for the tour to zigzag repeatedly between two regions and in subleading corrections to the length of the tour the universality class should be the same as for dense polymers and minimal spanning trees the conjectures for the length of the tour on a cylinder are tested numerically

off shell bethe ansatz equations and n point correlators in the su 2 wznw theory we prove that the wavevectors of the off shell bethe ansatz equation for the inhomogeneous su 2 lattice vertex model render in the quasiclassical limit the solutions of the knizhnik zamolodchikov equation

sums of hermitian squares and the bmv conjecture we show that all the coefficients of the polynomial mathop mathrm tr a tb m in mathbb r t are nonnegative whenever m 13 is a nonnegative integer and a and b are positive semidefinite matrices of the same size this has previously been known only for m 7 the validity of the statement for arbitrary m has recently been shown to be equivalent to the bessis moussa villani conjecture from theoretical physics in our proof we establish a connection to sums of hermitian squares of polynomials in noncommuting variables and to semidefinite programming as a by product we obtain an example of a real polynomial in two noncommuting variables having nonnegative trace on all symmetric matrices of the same size yet not being a sum of hermitian squares and commutators

algebraic bethe ansatz for the fpl2 model an exact solution of the model of fully packed loops of two colours on a square lattice has recently been proposed by dei cont and nienhuis using the coordinate bethe ansatz approach we point out here a simpler alternative in which the transfer matrix is directly identified as a product of r matrices this allows us to apply the nested algebraic bethe ansatz which leads to the same bethe equations we comment on some of the applications of this result

compact self avoiding circuits on two dimensional lattices close packed self avoiding walks and circuits as models for condensed polymer phases are studied on the square planar and honeycomb lattices exact solutions for strips from these lattices are obtained via transfer matrix methods extrapolations are made for the leading asymptotic terms in the count of compact conformations on the square planar lattice the leading asymptotic term for each lattice is bounded from below and it is noted that boundary effects can be important

long range properties of spanning trees we compute some large scale properties of the uniform spanning tree process on bounded regions in z2 in particular we compute the distribution of the meeting point of the branches of the tree issued from three boundary points we also compute the crossing probabilities of branches of the tree on rectangular and annular regions as well as the winding number of the branches of the tree

on multivariate trace inequalities of sutter berta and tomamichel we consider a family of multivariate trace inequalities recently derived by sutter berta and tomamichel these inequalities generalize the golden thompson inequality and lieb s triple matrix inequality to an arbitrary number of matrices in a way that features complex matrix powers i e certain unitaries we show that their inequalities can be rewritten as an n matrix generalization of lieb s original triple matrix inequality the complex matrix powers are replaced by resolvents and appropriate maximally entangled states we expect that the technically advantageous properties of resolvents in particular for perturbation theory can be of use in applications of the n matrix inequalities e g for analyzing the performance of the rotated petz recovery map in quantum information theory and for removing the unitaries altogether

diagonalisation of gl n invariant transfer matrices and quantum n wave system lee model the algebraic bethe ansatz is constructed for the gl n invariant transfer matrices with arbitrary gl n spin for this purpose the notion of vacuum subspace is introduced it is shown that the gl n magnet can be interpreted as an integrable discrete approximation of vector or matrix nonlinear schrodinger models or of the quantum n wave system

tiles and colors tiling models are classical statistical models in which different geometric shapes the tiles are packed together such that they cover space completely in this paper we discuss a class of two dimensional tiling models in which the tiles are rectangles and isosceles triangles some of these models have been solved recently by means of bethe ansatz we discuss the question why only these models in a larger family are solvable and we search for the yang baxter structure behind their integrability in this quest we find the bethe ansatz solution of the problem of coloring the edges of the square lattice in four colors such that edges of the same color never meet in the same vertex

on systems of linear fractional differential equations with constant coefficients abstract this paper deals with the study of linear systems of fractional differential equations such as the following system 1 y a x y b x where y is the riemann liouville or the caputo fractional derivative of order 0 2 a x a 11 x a 1 n x a n 1 x a nn x b x b 1 x b n x are matrices of known real functions in a way analogous to the usual case we show how a generalized matrix exponential function and certain fractional green function in connection with the mittag leffler type functions would allow us to obtain an explicit representation of the general solution to the system 1 when a is a constant matrix

sum of squares results for polynomials related to the bessis moussa villani conjecture we show that the polynomial sm k a b that is the sum of all words in noncommuting variables a and b having length m and exactly k letters equal to b is not equal to a sum of commutators and hermitian squares in the algebra r x y where x2 a and y2 b for all even values of m and k with 6 k m 10 and also for m k 12 6 this leaves only the case m k 16 8 open this topic is of interest in connection with the lieb seiringer formulation of the bessis moussa villani conjecture which asks whether tr sm k a b 0 holds for all positive semidefinite matrices a and b these results eliminate the possibility of using descent sum of squares to prove the bmv conjecture

gauge symmetry and non abelian topological sectors in a geometrically constrained model on the honeycomb lattice we study a constrained statistical mechanical model in two dimensions that has three useful descriptions they are i the ising model on the honeycomb lattice constrained to have three up spins and three down spins on every hexagon ii the three color and fully packed loop model on the links of the honeycomb lattice with loops around a single hexagon forbidden and iii three ising models on interleaved triangular lattices with domain walls of the different ising models not allowed to cross unlike the three color model the configuration space on the sphere or plane is connected under local moves on higher genus surfaces there are infinitely many dynamical sectors labeled by a noncontractible set of nonintersecting loops we demonstrate that at infinite temperature the transfer matrix admits an unusual structure related to a gauge symmetry for the same model on an anisotropic lattice this enables us to diagonalize the original transfer matrix for up to 36 sites finding an entropy per plaquette s k b ensuremath approx 0 3661 ensuremath cdots and substantial evidence that the model is not critical we also find the striking property that the eigenvalues of the transfer matrix on an anisotropic lattice are given in terms of fibonacci numbers we comment on the possibility of a topological phase with infinite topological degeneracy in an associated two dimensional quantum model

fractional order nonlinear systems with delay in iterative learning control in this paper we discuss a p type iterative learning control ilc scheme for a class of fractional order nonlinear systems with delay by introducing the norm and using gronwall inequality the sufficient condition for the robust convergence of the tracking errors is obtained based on this convergence the p type ilc updating laws can be determined

fractals in two dimensions and conformal invariance abstract the exact fractal properties of critical geometrical systems in two dimensions are briefly reviewed they are generically described by an infinite discrete spectrum of scaling dimensions associated with underlying conformal operators applications involve polymers percolation brownian motion and ising clusters a seemingly essential distinction between critical phenomena spectra and multifractal ones is given

on d hagele s approach to the bessis moussa villani conjecture abstract the reformulation of the bessis moussa villani bmv conjecture given by lieb and seiringer asserts that the coefficient m k a b of t k in the polynomial tr a tb m with a b positive semidefinite matrices is nonnegative for all m k we propose a natural extension of a method of attack on this problem due to hagele and investigate for what values of m k the method is successful obtaining a complete determination when either m or k is odd

asymptotic positivity of hurwitz product traces two proofs abstract consider the polynomial tr a tb m in t for positive hermitian matrices a and b with m n the bessis moussa villani conjecture in the equivalent form of lieb and seiringer states that this polynomial has nonnegative coefficients only we prove that they are at least asymptotically positive for the nontrivial case of ab 0 more precisely we show once complex analytically once combinatorially that the k th coefficient is positive for all integer m m 0 where m 0 depends on a b and k

self avoiding walk on a three dimensional manhattan lattice we have extended the definition of the manhattan lattice from two dimensional to three dimensional 3d spaces the number of self avoiding walks on the 3d manhattan lattice cn and their mean square end to end distances rn2 were counted exactly up to 31 and 30 steps respectively analysis using the method of the dlog pad approximant gave the exponents 1 1615 0 0002 and 0 5870 0 0025 which are in good agreement with corresponding values for self avoiding walks on the ordinary 3d lattice this result suggests that self avoiding walks on the 3d manhattan lattice belong to the same universality class as self avoiding walks on the ordinary 3d lattice

more on operator monotone and operator convex functions of several variables abstract let c 1 c 2 c k be positive matrices in m n and f be a continuous real valued function on 0 in addition consider as a positive linear functional on m n and define t 1 t 2 t 3 t k f t 1 c 1 t 2 c 2 t 3 c 3 t k c k as a k variables continuous function on 0 0 in this paper we show that if f is an operator convex function of order mn then is a k variables operator convex function of order n 1 n k such that m n 1 n 2 n k also if f is an operator monotone function of order n k 1 then is a k variables operator monotone function of order n in particular if f is a non negative operator decreasing function on 0 then the function t f a t b is an operator decreasing and can be written as a laplace transform of a positive measure

littlewood richardson coefficients for grothendieck polynomials from integrability abstract we study the littlewood richardson coefficients of double grothendieck polynomials indexed by grassmannian permutations geometrically these are the structure constants of the equivariant k theory ring of grassmannians representing the double grothendieck polynomials as partition functions of an integrable vertex model we use its yang baxter equation to derive a series of product rules for the former polynomials and their duals the littlewood richardson coefficients that arise can all be expressed in terms of puzzles without gashes which generalize previous puzzles obtained by knutson tao and vakil