graphical representation of the partition function of a one dimensional δ function bose gas the one dimensional repulsive function bose system is studied by only using the bethe ansatz equation n particle partition functions are exactly calculated from this expression for the n particle partition function the n particle cluster integral is derived the results completely agree with those of the thermal bethe ansatz tba this directly proves the validity of the tba the theory of partitions and graphs is used to simplify the discussion

efficient computation of modal sensitivities for systems with repeated frequencies the physical origin and ambiguity of repeated frequency modes is discussed followed by a mathematical derivation of the governing equations for the partial derivatives of repeated eigenvalues with respect to changes in the system parameters this is followed by a derivation for the associated eigenvector derivative equations physical interpretations are included to help explain the analytical results an efficient computation procedure which preserves the bandwidth for very large systems is also proposed for systems with repeated or closely spaced eigenvalues two small examples to illustrate the solution procedure are presented 8 references

on finite regular and holomorphic mappings we show that for every hypersurface v yv y and every k nk n there are only a finite number of non equivalent finite regular mappings f x yf x y such that the discriminant d f d f equals v and f k f k in particular if knr x cn i 1rxi 0 and x is a smooth and simply connected algebraic manifold then every finite regular mapping f x cnf x cn with d f knr is equivalent to one of the mappings fd1 dr cn x1 xn x1d1 xrdr xr 1 xn cn moreover we obtain generalizations of the lamy theorem we prove the same statement in the local and sometimes global holomorphic situation in particular we show that if f cn 0 cn 0 f cn 0 cn 0 is a proper and holomorphic mapping of topological degree two then there exist biholomorphisms cn 0 cn 0 cn 0 cn 0 such that f x1 x2 xn x12 x2 xn moreover for every proper holomorphic mapping f cn 0 cn 0 f cn 0 cn 0 which has a discriminant with only simple normal crossings there exist biholomorphisms cn 0 cn 0 cn 0 cn 0 such that f x1 x2 xn x1d1 x2d2 xrdr xr 1 xn where r is the number of irreducible components of the discriminant at 0

a hypersurace which has the abhyankar moh property let x be an algebraic subset of cn we say that x has the abhyankarmoh property a m p if for any polynomial embedding f x cn there exists a polynomial automorphism f of cn such that f is a restriction of this automorphism to the set x in 6 10 11 20 it was shown that if dim x is sufficiently small relatively to n and if x has nice singularities then x has the a m p and the problem of extending the polynomial embedding of x into cn has in general many solutions unfortunately if dim x is big enough in comparison with n the situation is more difficult in particular if x is a generic hypersurface of degree n in cn n 2 then the problem of extending the polynomial embedding of x into cn has no more than one solution see 8 abhyankar and moh proved see 1 2 that a line in c2 has the a m p it can be easily inferred from this that the cross k2 x c2 x1 x2 0 has the a m p the analogous questions for a hyperplane and the n cross kn x cn x1 xn 0 in the n space n 2 has been open cf 1 and conjecture 12 in 12 the question for a hyperplane is still open moreover in the case n 2 no example of a hypersurface which had the a m p has been known

eigenvalue and eigenvector approximate analysis for repeated eigenvalue problems the method developed for approximate analysis involves a reparamaterization of the multivariable structural eigenvalue problems in terms of a single positive valued parameter the resulting equations yield first order approximations of changes in both the eigenvalues and eigenvectors associated with the repeated eigenvalue problem

mean first passage times of processes driven by white shot noise we consider mean first passage times in systems driven by white shot noise with exponentially distributed jump heights simple interpretable results are obtained and the linkage between those results and the steady state probability density function of the process is presented the virtual waiting time or takacs process constant losses and the shot noise process with linear losses are analyzed in depth along with a more complex process with useful implications for the modeling of the soil moisture dynamics in hydrology

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

embeddings of hypersurfaces in affine spaces abstract in this paper we address the following two general problems given two algebraic varieties in cn find out whether or not they are 1 isomorphic and 2 equivalent under an automorphism of cn although a complete solution of either of those problems is out of the question at this time we give here some handy and useful invariants of isomorphic as well as of equivalent varieties furthermore and more importantly we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one and use it to construct numerous examples of isomorphic but inequivalent algebraic varieties in cn among other things we establish the following interesting fact for isomorphic hypersurfaces p x1 xn 0 and q x1 xn 0 the number of zeros of grad p might be different from that of grad q this implies in particular that although the fibers p 0 and q 0 are isomorphic there are some other fibers p c and q c which are not we construct examples like this for any n 2

on the russell problem we give a partial answer to the russell conjecture about characterization of the affine space we also characterize testing sets for properness and non properness sets of polynomial mappings of k uniruled varieties where k is an algebraically closed field

the jacobian conjecture and the extensions of polynomial embeddings let x be an algebraic subset of c we say that x has the abhyanka r moh property amp if for any polynomial embedding 4 x of there exists a polynomial au tomorphism b of c such that b is a restriction of this au tomorphism to the set x in 3 9 it was shown that if dim x is sufficiently small relatively to n and if x has nice singularities then x has the am p and the problem of extending the polynomial embedding of x into c has in general many solutions unfortunately if dim x is big enough in compar ison with n the situation is more difficult in particular if x is a generic hypersurface of degree n in n 2 then the problem of extending the polynomial embedding of x into c has no more than one solution see 5 abhyankar and mob proved see 1 that a line in c 2 has the amp it can be easily inferred from this that the c r o s s k 2 x g 2 xl x2 0 has the amp the analogous questions for a hyperplane and the n cross k x c x l x 0 in the n space n 3 are open cf 1 and conjecture 12 in 6 moreover in the case n 3 no example of a hypersurface which has the a m p is known on the other hand in 4 we showed that for every n 2 the smooth hypersurface x x c xl x 2 2 is any non zero complex constant does not have the amp we recall an important conjecture connected with this subject

a direct calculation of the free energy from the bethe ansatz equation for the heisenberg model thermodynamics of the xxx heisenberg model is studied the trace of the boltzmann weight with respect to the hilbert space is taken in the thermodynamic limit with the number of up spins being fixed the expression of the trace gives an explanation why the correct thermodynamic quantities are derived from the string hypothesis a combination of this with the previous result leads to a conclusion that the free energy can be calculated only from the bethe ansatz equation the method is more direct than other known methods which were used to derive the free energy

eigenvector derivatives with repeated eigenvalues in this paper an algorithm is derived for computing the derivatives of eigenvalues and eigenvectors for real symmetric matrices in the case of repeated eigenvalues where the matrices are functions of real parameters such as mass density or moment of inertia the algorithm is an extension of recent work by i u ojalvo the key step in this extended derivation is to differentiate the eigenvalue equation twice the algorithm preserves the symmetry and band structure of the matrices allowing efficient computer storage and solution techniques applications include sensitivity analysis and optimization of the normal modes of finite element modeled structures such as large space structures a cantilever beam finite element example is included

simplified calculation of eigenvector derivatives with repeated eigenvalues an efficient and practical procedure based on the generalized schmidt orthogonalization is presented to compute the eigenvector derivatives when repeated eigenvalues exist

calculation of eigenvector derivatives for structures with repeated eigenvalues in structural optimization and system identification eigenvector derivatives provide important information for updating design model parameters when the current parameter values yield repeated eigenvalues it has not been possible previously to calculate unique eigenvector derivatives recent work has provided a method for determining unique eigenvalue derivatives for this case but methods for calculating the eigenvector sensitivities have been incomplete in this work a complete method for calculation of repeated root eigenvector derivatives is shown for the real symmetric structural eigenproblem the derivation is completed by using information from the second derivative of the eigen problem and is limited to the case of distinct eigenvalue sensitivities as an example the repeated root eigenvector sensitivities are calculated for a simple three degree of freedom beam grillage comparisons of linear approximations using these derivatives to the calculated eigenvectors support the accuracy of the formulation

statistical mechanics of an exactly integrable system the equilibrium thermodynamics of the nonlinear schrodinger model at finite temperature is calculated by means of the quantum inverse method working directly in an infinite volume we derive the equation of state and the integral equation which determines the excitation spectrum this integral equation is found to be closely related to the gelfand levitan expression for the charge density operator

testing sets for properness of polynomial mappings in this paper we characterize testing sets for properness of polynomial mappings f mathbb c n rightarrow mathbb c m we also study the set of points at which such mappings are not proper as the first application we give a proof of the complementary conjecture of mckay and wang conjecture 12 in 16 the second application is an answer to the problem of kraft russell about a characterization of mathbb c n the third application is given in 13 a solution of the problem of van den essen and shpilrain about endomorphism of the ring mathbb c x 1 x n the fourth application is a theorem about extensions of affine varieties

sensitivity analysis of discrete structural systems methods for calculating sensitivity derivatives for discrete structural systems are surveyed primarily covering literature published during the past two decades methods are described for calculating derivatives of static displacements and stresses eigenvalues and eigenvectors transient structural response and derivatives of optimum structural designs with respect to problem parameters the survey is focused on publications developed in nonstructural fields such as electronics controls and physical chemistry which are directly applicable to structural problems most notable among the nonstructural based methods are the adjoint variable technique from control theory and the green s function and fast methods from physical chemistry

qr based algorithm for eigenvalue derivatives many engineering optimization problems require the calculation of eigenvalue sensitivities several straightforward methods have been developed for calculating partial derivatives of distinct complex eigenvalues recent work has expanded these methods to include systems with repeated real eigenvalues a new method is presented that embeds eigenvalue derivative computation into an established numerical eigenvalue algorithm namely qr decomposition the technique is shown to have computational advantages for high order systems

flexible varieties and automorphism groups given an irreducible affine algebraic variety x of dimension n 2 we let saut x denote the special automorphism group of x that is the subgroup of the full automorphism group aut x generated by all one parameter unipotent subgroups we show that if saut x is transitive on the smooth locus x reg then it is infinitely transitive on x reg in turn the transitivity is equivalent to the flexibility of x the latter means that for every smooth point x x reg the tangent space t x x is spanned by the velocity vectors at x of one parameter unipotent subgroups of aut x we also provide various modifications and applications

new method for eigenvector sensitivity analysis with repeated eigenvalues and eigenvalue derivatives this paper proposes a new method of eigenvector sensitivity analysis for real symmetric systems with repeated eigenvalues and eigenvalue derivatives the derivation is completed by using information from the second and third derivatives of the eigenproblem and is applicable to the case of repeated eigenvalue derivatives the extended systems with a nonsingular coefficient matrix are constructed to calculate the particular solutions an efficient approach is developed to compute the homogeneous solutions associated with the repeated eigenvalue derivatives different constraints for calculating the eigenvector sensitivities are derived and discussed four numerical examples are presented to demonstrate the validity of the proposed method

irreducible identity sets for polynomial automorphisms in this paper we study polynomial automorphisms of some algebraic affine varieties the main tool of our investigations is the notion of the uniruled set the paper is divided into five parts sections 1 5 respectively the first one is an introduction in the second one we set up notations definitions and terminology in section 3 we study the set of fixed points of a polynomial automorphism of cn the nature of this set is not well known even in the case when an automorphism has a finite order see 10 let us recall the notions which we shall use in the sequel let x cn be an algebraic set we say that x is an identity set for polynomial automorphisms of cn if every two polynomial automorphisms that coincide on x must be equal x is a set determining polynomial automorphisms of cn if the same holds under the weaker assumption that these two automorphisms transforms x onto the same set in 13 wang and mckay proved that two intersecting lines are an identity set in c2 later in 14 they generalized this result on n hyperplanes in general position in cn n 2 in our paper 8 we generalized these results and we proved that the union of every n hypersurfaces in general position in cn is an identity set in cn n 2

improved nelson s method for computing eigenvector derivatives with distinct and repeated eigenvalues t he computation of derivatives of eigenvalues and eigenvectors with respect to model parameters is crucial for many applications 1 nelson 2 proposed an approach for general real eigensystemswith distinct eigenvalueswhere only the eigenvector of interest was required ojalvo 3 mills curran 4 dailey 5 and shaw and jayasuriya 6 developed nelson s method for real symmetric eigensystems with repeated eigenvalues for computation of a particular solution themethods described in 3 5 may fail in some circumstances other methods have been suggested by millscurran 4 and song et al 7 but they are difficult to implement when a system of linear equations is solved one hopes that the calculated solution is a close representation of the true solution the computed solution for a system with a large condition number may be inexact due to extreme sensitivity of the solution to small changes in its coefficient matrix and right column vectors 8 because some diagonal elements in the coefficient matrix for particular solutions are set to units its coefficients are not all of the same order of magnitude so the coefficient matrix has a large condition number in this note nelson s method is first extended to the case of repeated eigenvalues for the symmetric real eigensystems and its improvements are then presented the condition number of the coefficient matrix of the improved method is considerably reduced