new reductions of the kadomtsev petviashvili and two dimensional toda lattice hierarchies via symmetry constraints new types of reductions of the kadomtsev petviashvili kp hierarchy and the two dimensional toda lattice 2dtl hierarchy are considered on the basis of sato s approach within this approach these hierarchies are represented by infinite sets of equations for potentials u1 u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp and the 2dtl hierarchies are studied under constraints of the following type n 1n nsn u1 u2 u3 x where sn are symmetries for these hierarchies n are arbitrary constants and is an arbitrary linear functional of the quantity it is shown that for the kp hierarchy these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 u3 un many known systems and several new ones are among them symmetry reductions for the 2dtl hierarchy give rise both to finite dimensional dynamical systems and 1 1 dimensional discrete systems some few results for the modified kp hierarc

an analytic description of the vector constrained kp hierarchy in this paper we give a geometric description in terms of the grassmann manifold of segal and wilson of the reduction of the kp hierarchy known as the vector k constrained kp hierarchy we also show in a geometric way that these hierarchies are equivalent to krichever s general rational reductions of the kp hierarchy

on solutions of constrained kp equations we derive solutions of general wronskian form for the vector constrained kp hierarchies as one explicit example we discuss rational solutions in order to introduce our method we give a direct elementary proof of the existence of wronskian solutions for the l modified kp hierarchies l 0 1

method of squared eigenfunction potentials in integrable hierarchies of kp type the method of squared eigenfunction potentials sep is developed systematically to describe and gain new information about the kadomtsev petviashvili kp hierarchy and its reductions interrelation to the function method is discussed in detail the principal result which forms the basis of our sep method is the proof that any eigenfunction of the general kp hierarchy can be represented as a spectral integral over the baker akhiezer ba wave function with a spectral density expressed in terms of sep in fact the spectral representations of the adjoint ba functions can in turn be considered as defining equations for the kp hierarchy the sep method is subsequently used to show how the reduction of the full kp hierarchy to the constrained kp ckp rm hierarchies can be given entirely in terms of linear constraint equations on the pertinent functions the concept of sep turns out to be crucial in providing a description of ckp rm hierarchies in the language of the universal sato grassmannian and finding the non isospectral virasoro symmetry generators acting on the underlying functions the sep method is used to write down generalized binary darboux backlund transformations for constrained kp hierarchies whose orbits are shown to correspond to a new toda model on a square lattice as a result we obtain a series of new determinant solutions for the functions generalizing the known wronskian multi soliton solutions finally applications to random matrix models in condensed matter physics are briefly discussed

inverse eigenvalue problems and lists of multiplicities of eigenvalues for matrices whose graph is a tree the case of generalized stars and double generalized stars we characterize the possible lists of ordered multiplicities among matrices whose graph is a generalized star a tree in which at most one vertex has degree greater than 2 or a double generalized star here the inverse eigenvalue problem iep for symmetric matrices whose graph is a generalized star is settled the answer is consistent with a conjecture that determination of the possible ordered multiplicities is equivalent to the iep for a given tree moreover a key spectral feature of the iep in the case of generalized stars is shown to characterize them among trees

on a reduction of the multi component kp hierarchy we show that the constrained kp hierarchies and their generalizations are natural reductions of the multi component kp hierarchy and that particular solutions of these hierarchies are obtained in a straightforward way from that of the multi component kp hierarchy

analytic bilinear approach to integrable hierarchies i generalized kp hierarchy an analytic bilinear approach for construction and study of integrable hierarchies in particular the kp hierarchy is proposed it starts with the generalized hirota identity for the cauchy baker akhiezer cba function and leads to a generalized kp hierarchy in the form of compact functional equations containing a special shift operator a generalized kp hierarchy incorporates the basic kp hierarchy modified kp hierarchy kp singularity manifold equation hierarchy and corresponding hierarchies of linear problems different vertical levels of the generalized kp hierarchy are connected via invariants of the combescure symmetry group the resolution of functional equations also gives rise to the function and the addition formulas for it

nonlinear stability criteria for tree like structures composed of branched elastic rods new necessary conditions for the nonlinear stability of branched tree like structures composed of elastic rods are presented the conditions which are found by examining the second variation of an energy functional are established by extending legendre9s classical work on this topic for the branched tree like structures of interest in this paper one of the new necessary conditions is the existence of a bounded solution to a set of riccati equations which are coupled at branching points in addition a set of new necessary conditions are established in the event that the abscissa of the branching point is not fixed the results are illustrated using a range of examples featuring planar configurations of branched elastic rods where each of the individual rods is modelled using euler9s theory of the elastica

multicomponent integrable reductions in the kadomtsev petviashvilli hierarchy new types of reductions of the kadomtsev petviashvili kp hierarchy are considered on the basis of sato s approach within this approach the kp hierarchy is represented by infinite sets of equations for potentials u2 u3 of pseudodifferential operators and their eigenfunctions and adjoint eigenfunctions the kp hierarchy was studied under constraints of the following type ni 1 i i x s x where s x are symmetries for the kp equation and i i i i are eigenfunctions with eigenvalue i it is shown that for the first three cases 2 3 4 these constraints give rise to hierarchies of 1 1 dimensional commuting flows for the variables u2 1 n 1 n bi hamiltonian structures for the new hierarchies are presented

analytic bilinear approach to integrable hierarchies ii multicomponent kp and 2d toda lattice hierarchies an analytic bilinear approach for the construction and study of integrable hierarchies is discussed generalized multicomponent kp and 2d toda lattice hierarchies are considered this approach allows us to represent generalized hierarchies of integrable equations in a condensed form of finite functional equations a generalized hierarchy incorporates basic hierarchy modified hierarchy singularity manifold equation hierarchy and corresponding linear problems different levels of generalized hierarchy are connected via invariants of combescure symmetry transformation the resolution of functional equations also leads to the function and addition formulas to it

the number of distinct eigenvalues for which an index decreases multiplicity abstract for an hermitian matrix a whose graph is a tree t we study the number of eigenvalues of a whose multiplicity decreases when a particular vertex is deleted from t explicit results are given when that number of eigenvalues is less than 4 and an inductive result thereafter the work is based in part on classical results about multiplicities but also on some new facts including a useful identity this allows us to give strong bounds based on simple facts about the location of the vertex in the tree some facts about matrices whose graphs are not trees are included and the classical diameter bound about the number of distinct eigenvalues for a tree follows

darboux theorems and wronskian formulas for integrable systems i constrained kp flows abstract generalizations of the classical darboux theorem are established for pseudo differential scattering operators of the form l i 0 n u i i i 1 m i 1 i i iteration of the darboux transformations leads to a gauge transformed operator with coefficients given by wronskian formulas involving a set of eigenfunctions of l nonlinear integrable partial differential equations are associated with the scattering operator l which arise as a symmetry reduction of the multicomponent kp hierarchy with a suitable linear time evolution for the eigenfunctions the darboux transformation is used to obtain solutions of the integrable equations in terms of wronskian determinants

kp symmetry reductions and a generalized constraint we study the link between symmetry reductions and constraints of the kadomtsev petviashvili equations in terms of the tau function we propose a generalization adapted to non zero boundary conditions of the standard constraints and show a particular class of solutions solitons

kp constraints from reduced multi component hierarchies the m vector k constrained kadomtsev petviashvili kp hierarchy is shown to be a pseudo reduction of the m 1 component kp hierarchy to facilitate the implementation of this reduction on the level of the solutions the typical multi component kp solutions are mapped onto solutions of a toda molecule type equation from which wronskian and grammian solutions for the constrained kp hierarchy follow the reduction of the associated linear systems is discussed and its importance for the choice of bilinear representation of the reduced systems is explained

on the evolution of intrinsic curvature in rod based models of growth in long slender plant stems in many applications of rod theories as models for plant stem growth and development it is necessary to allow the intrinsic curvature and flexural stiffness of the rod to evolve in the present paper the application of evolution equations for these quantities is examined and a new evolution equation for the intrinsic curvature is proposed to illustrate the new evolution equation several examples of the evolution of rods in the presence of external forces and tip growth are presented growth in plants frequently results in residual stresses and the compatibility of rod models with these fields is also discussed

constraints of the kadomtsev petviashvili hierarchy for the kadomtsev petviashvili kp hierarchy constructed in terms of the famous sato theory a k constraint is proposed that leads the hierarchy to the nonlinear system involving a finite number of dynamical coordinates the eigenvalue problem of the k constrained system is naturally obtained from the linear system of the kp hierarchy which takes the form of kth order polynomial coupled with a first order one thus we are able to derive the correspondent lax pair recursion operator bi hamiltonian structures and conserved quantities the constraints for the bkp hierarchy are also sketched

the structure of matrices with a maximum multiplicity eigenvalue there is remarkable and distinctive structure among hermitian matrices whose graph is a given tree t and that have an eigenvalue of multiplicity that is a maximum for t among such structure we give several new results 1 no vertex of t may be neutral 2 neutral vertices may occur if the largest multiplicity is less than the maximum 3 every parter vertex has at least two downer branches 4 removal of a parter vertex changes the status of no other vertex and 5 every set of parter vertices forms a parter set statements 3 4 and 5 are also not generally true when the multiplicity is less than the maximum some of our results are used to give further insights into prior results and both the review of necessary background and the development of new structural lemmas may be of independent interest

undirected graphs of hermitian matrices that admit only two distinct eigenvalues abstract we consider the problem of determining those undirected n vertex graphs with a corresponding hermitian matrix that admits only two distinct eigenvalues with multiplicities k and n k after giving some general algebraic characterizations of these dual multiplicity graphs we then prove two major graph theoretic necessary conditions on such graphs construction techniques are then developed and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two

constraints of the kp hierarchy and multilinear forms we consider the trilinear form of the kaup broer system which gives rise to solutions in wronskian form the kaup broer system is connected with akns system through a gauge transformation the akns hierarchy can be understood as a generalized 1 constraint of the kp hierarchy imposing that constraint on sato s equation we obtain the basic trilinear form and moreover a hierarchy of trilinear equations governing the akns flows similary hierarchies of multilinear forms are derived in the case of generalized k constraints

an algebraic description of generalized k constraints the generalized k constrained kp hierarchy is shown to correspond to a so called pseudo reduction of the two dimensional toda lattice hierarchy described in a free fermion approach which is adapted to the case of two singularities in the spectral parameter range wronskian solutions are discussed and in particular soliton solutions are recovered through a reduction of the toda solitons