replica treatment of non hermitian disordered hamiltonians we employ the fermionic and bosonic replicated nonlinear models to treat ginibre unitary symplectic and orthogonal ensembles of non hermitian random matrix hamiltonians using the saddle point approach combined with the borel resummation procedure we derive the exact large n results for microscopic density of states in all three ensembles we also obtain tails of the density of states as well as the two point function for the unitary ensemble

on multilinear square function and its applications to multilinear littlewood paley operators with non convolution type kernels abstract let m 2 n 1 and x r n define the multilinear square function t by t f x 0 r n m k t x y 1 y m j 1 m f j y j d y 1 d y m 2 d t t 1 2 where the kernel k satisfies a class of integral smooth conditions which is much weaker than the standard calderon zygmund kernel conditions in this paper we first established the l p 1 w 1 l p m w m l p estimate of t when each p i 1 and weak type l p 1 w 1 l p m w m l p estimate of t when there is a p i 1 where i 1 m i p p i and each w i is a nonnegative function on r n as applications of the above results we obtained the boundedness of multilinear littlewood paley operators with non convolution type kernels including multilinear g function marcinkiewicz integral and g function

application of the τ function theory of painleve equations to random matrices piv pii and the gue tracy and widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite gue in terms of a piv and pii transcendent respectively we generalise these results to the evaluation of where for and otherwise and the average is with respect to the joint eigenvalue distribution of the gue as well as to the evaluation of of particular interest are and f n 2 and their scaled limits which give the distribution of the largest eigenvalue and the density respectively our results are obtained by applying the okamoto function theory of piv and pii for which we give a self contained presentation based on the recent work of noumi and yamada we point out that the same approach can be used to study the quantities and f n a for the other classical matrix ensembles

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

on multilinear littlewood paley operators abstract in this paper we established the weighted estimates for the multilinear littlewood paley operators including the multilinear marcinkiewicz integral and the multilinear area integral of lusin these operators are generalizations of the linear operators for higher dimension which were first defined and studied by e m stein in 1958 strong l p 1 1 l p m m l p estimates when each p i 1 and weak type l p 1 1 l p m m l p estimates if there is a p i 1 are obtained where i 1 m i p p i and each w i i 1 m is a nonnegative function on r n

the solution of the kato problem for divergence form elliptic operators with gaussian heat kernel bounds which in turnhas applications to the perturbation theory for certain classes of hyperbolicequations see 16 we remark that 1 3 is equivalent to the oppositeinequality for the square root of the adjoint operator l in 13 14 kato conjectured that an abstract version of 1 3 might hold for regularly accretive operators see 14 18 for the details a counter example to this abstract conjecture was obtained by mcintosh 17 whothen reformulated the conjecture in the following form bearing in mind thatkato s interest in the problem had been motivated by the special case ofelliptic di erential operators conjecture 1 4 the estimate 1 3 holds for lde ned as in 1 2 for anyl

studies on the painlev equations iii second and fourth painlev equations p ii andp iv etude des transformations birationnelles canoniques du systeme de painleve associe a chacune des equations

replica field theories painleve transcendents and exact correlation functions exact solvability is claimed for nonlinear replica ensuremath sigma models derived in the context of random matrix theories contrary to other approaches reported in the literature the framework outlined does not rely on traditional replica symmetry breaking but rests on a previously unnoticed exact relation between replica partition functions and painlev e transcendents while expected to be applicable to matrix models of arbitrary symmetries the method is used to treat fermionic replicas for the gaussian unitary ensemble gue chiral gue symmetry classes a and aiii in cartan classification and ginibre s ensemble of complex non hermitian random matrices further applications are briefly discussed

multilinear littlewood paley estimates with applications to partial differential equations abstract we obtain a collection of multilinear littlewood paley estimates which we then apply to two problems in partial differential equations the first problem is the estimation of the square root of an elliptic operator in divergence form and the second is the estimation of solutions to the cauchy problem for nondivergence form parabolic equations

wigner dyson statistics from the replica method we compute the correlation functions of the eigenvalues in the gaussian unitary ensemble using the fermionic replica method we show that non trivial saddle points which break replica symmetry must be included in the calculation in order to correctly reproduce the exact asymptotic behaviour of the correlation functions at a large distance

addendum to on the variation in the cohomology of the symplectic form of the reduced phase space here we have used the notation rlx for the inner product of the vector field x with the form a and ix m x j m for the x component of a in an earlier paper 4 it was shown that the push forward j dm of the liouville measure dm on m under the momentum mapping j is a piecewise polynomial measure on moreover in case x has isolated isolated zeros on m an explicit formula for the integral

the laplacian on a random one dimensional lattice the authors study the spectrum of minus the laplacian on a random one dimensional lattice it extends monotonically throughout the positive real axis with a continuum limit behaviour for small values the authors find as expected localisation effects throughout the whole range they use a variety of techniques which are shown to be consistent and in agreement with a monte carlo simulation they include small and large disorder expansions direct integral equations for probability distributions as well as the replica method the latter is also used to investigate green functions

application of the τ function theory of painleve equations to random matrices pv piii the lue jue and cue with denoting an average with respect to the eigenvalue pdf for the laguerre unitary ensemble the object of our study is e n i a x l i l 0 i l for i 0 s and i s where x i l 1 for l i and x i l 0 other wise using okamoto s development of the theory of the painleve v equation it is shown that e n i a is a function associated with the hamiltonian therein and so can be characterized as the solution of a certain second order second degree differential equation or in terms of the solution of certain difference equations the cases 0 and 2 are of particular interest because they correspond to the cumulative distribution and density function respectively for the smallest and largest eigenvalue in the case i s e n i a is simply related to an average in the jacobi unitary ensemble and this in turn is simply related to certain averages over the orthogonal group the unitary symplectic group and the circular unitary ensemble the latter integrals are of interest for their combinatorial content also considered are the hard edge and soft edge scaled limits of e n i a in particular in the hard edge scaled limit it is shown that the limiting quantity e hard 0 s a can be evaluated as a function associated with the hamiltonian in okamoto s theory of the painleve iii equation

on the variation in the cohomology of the symplectic form of the reduced phase space is called the momentum mapping of the hamiltonian t action given 1 1 the condition 1 2 just means that t acts along the fibers of j for the basic definitions and properties of non commutative hamiltonian group actions see am the results of this paper can easily be extended to hamiltonian actions of arbitrary compact connected lie groups by applying our results to the action of its maximal toms and using the equivariance of the momentum mapping for some more details see the remarks at the end of sect 2 we will assume throughout this paper that the momentum map is proper that is j i u is compact for each compact subset u of t now let r be a regular value of j that is t j tmm t is surjective for all meye j 1 4 then ye is a smooth submanifold of m compact because

topological optimization of compliant adaptive wing structure load path based topology optimization is used to synthesize a compliant adaptive aircraft wing leading edge which deforms in a prescribed way when subject to a single point internal actuation the load path based optimization method requires the specification of a parent lattice increasing the complexity of this lattice means the number of parameters required for a complete representation of the structure in the topology optimization becomes prohibitive although it is desirable to enable a full exploration of the design space a new method based on graph theory and network analysis is proposed which enables a substantial reduction in the required number of parameters to represent the parent lattice the results from this load path based approach are compared with those obtained from the better known density based topology optimization method

the connections between dirichlet regularity and neumann problems for second order elliptic operators with complex bounded measurable coefficients abstract the present paper discusses relations between regularity dirichlet and neumann problems we investigate the boundary problems for block operators and prove in particular that the solvability of the regularity problem does not imply the solvability of the dual dirichlet problem for general elliptic operators with complex bounded measurable coefficients this is strikingly different from the case of real operators for which such an implication was established in 1993 by c kenig j pipher invent math 113 3 1993 447 509 and since then has served as an integral part of many results