nonlinear wave interactions in a complex hamiltonian formalism a hamiltonian formalism is used to study the set of equations governing the changes in the amplitudes of some coherent waves due to their nonlinear interactions the main point of the present description is that it uses these complex wave amplitudes as canonical variables for a real hamiltonian thus greatly simplifying the algebra three examples are then dealt with first the hamiltonian for the four wave interactions is derived in accordance with previous studies such systems are then known to be integrable as a second example one looks at a very special case of three interacting waves where a fundamental interacts simultaneously with its second and third harmonic even though this system can be hamiltonian it seems to be nonintegrable finally attention is focused on the elliptically polarised third harmonic generation both in anisotropic and in isotropic media this set of equations can be hamiltonian under certain restrictions on the coupling coefficients but it also appears to be nonintegrable because not enough independent invariants are found

proof of integrability for five wave interactions in a case with unequal coupling constants complete integrability is proved for a system of ode describing the double three wave interaction between five waves when the coupling constant in one triplet is twice as large as in the other known results from the painleve analysis for this case are combined with the yoshida kovalevskaya approach 1983 to gain insight about the degree of the missing first integral this integral of degree six is then obtained via a direct search based on irreducible forms elementary building blocks for polynomial first integrals in involution with the manley rowe invariants

numerical methods and applications iterative methods based on linearization for nonlinear elliptic grid systems e g dyakonov how to solve stiff systems of differential equations by explicit methods v i lebedev on numerical methods of solving navier stokes equations in velocity pressurevariables g m kobelkov stiff systems of ordinary differential equations r p fedorenko convergence rate estimates of finite element methods for second order hyperbolic equations a a zlotnik fictitious domain methods and computation of homogenized properties of composites n s bakhvalov and a v knyazev

integrability of restricted multiple three wave interactions using a hamiltonian framework with complex canonical variables allows for the determination of irreducible forms which serve as building blocks for polynomial invariants all the independent invariants in involution are thus obtained for the restricted multiple three wave interactions where all triads are coupled through a common pump or daughter wave in the case of equal coupling strengths in all triads the mixed common pump daughter wave case is not integrable

second order chebyshev methods based on orthogonal polynomials stabilized methods also called chebyshev methods are explicit runge kutta methods with extended stability domains along the negative real axis these methods are intended for large mildly stiff problems originating mainly from parabolic pdes the aim of this paper is to show that with the use of orthogonal polynomials we can construct nearly optimal stability polynomials of second order with a three term recurrence relation these polynomials can be used to construct a new numerical method which is implemented in a code called rock2 this new numerical method can be seen as a combination of van der houwen sommeijer type methods and lebedev type methods

gopakumar vafa invariants do not determine flops two 3 fold flops are exhibited both of which have precisely one flopping curve one of the two flops is new and is distinct from all known algebraic d 4 flops it is shown that the two flops are neither algebraically nor analytically isomorphic yet their curve counting gopakumar vafa invariants are the same we further show that the contraction algebras associated to both are not isomorphic so the flops are distinguished at this level this shows that the contraction algebra is a finer invariant than various curve counting theories and it also provides more evidence for the proposed analytic classification of 3 fold flops via contraction algebras

double three wave interaction of four waves lax representations and exact solutions the nonlinear resonant interaction of coherent waves is a fundamental process in the study of wave phenomena which has received a great deal of attention in its many aspects in the present article a system of four interacting waves which constitute two resonant triplets is considered the system is described in a simplified model by a hamiltonian system of eight autonomous ordinary differential equations with time as the independent variable both positive and negative energy waves are allowed in the interaction two distinct lax representations for this system two new classes of exact solutions in terms of elliptic functions a solution in the form of a convergent generic laurent series expansion around a movable pole in the independent variable and a stabilization criterion for the explosive instability that may occur when waves of different energy sign interact are obtained herein

integrability of restricted multiple three wave interactions ii coupling constants with ratios 1 and 2 the system of ordinary differential equations describing the multiple three wave interactions with a common pump daughter wave is proved to be completely integrable by obtaining the necessary 2n 1 independent first integrals in involution for the case when the coupling constants and the frequency mismatches have ratio 1 and or 2 this case was deemed integrable on the basis of a painleve analysis but a direct proof has been lacking for some time the first integrals are the n 1 quadratic manley rowe relations the cubic hamiltonian n 1 quartic integrals analogous to the ones needed for complete integrability in the case of equal coupling constants and detunings in all wave triads and a new sixth order integral involving all wave quantities the form of this last invariant was deduced from the recent result for the analogous interaction between five waves n 2 and essentially made possible by the proper use of irreducible forms elementary building blocks for polynomial first integrals in involution with the manley rowe invariants

stability conditions on text cy _n categories associated to a_n quivers and period maps in this paper we study the space of stability conditions on a certain n calabi yau mathrm cy n category associated to an a n quiver recently bridgeland and smith constructed stability conditions on some mathrm cy 3 categories from meromorphic quadratic differentials with simple zeros generalizing their results to higher dimensional calabi yau categories we describe the space of stability conditions as the universal cover of the space of polynomials of degree n 1 with simple zeros in particular central charges of stability conditions on mathrm cy n categories are constructed as the periods of quadratic differentials with zeros of order n 2 which are associated to polynomials

relative singularity categories i auslander resolutions abstract let r be an isolated gorenstein singularity with a non commutative resolution a end r r m in this paper we show that the relative singularity category r a of a has a number of pleasant properties such as being hom finite moreover it determines the classical singularity category d s g r of buchweitz and orlov as a certain canonical quotient category if r has finite cm type which includes for example kleinian singularities then we show the much more surprising result that d s g r determines r aus r where aus r is the corresponding auslander algebra the proofs of these results use dg algebras a koszul duality and the new concept of dg auslander algebras which may be of independent interest

quantum mechanics of complex hamiltonian systems in one dimension with a view to obtaining further insight into the nature of eigenvalues and eigenfunctions of a stationary state one dimensional schrodinger equation corresponding to a non hermitian hamiltonian h x p we investigate the ground state solutions for a variety of potentials within the framework of an extended complex phase space characterized by x x1 ip2 p p1 ix2 where x1 p1 and x2 p2 are real and considered as canonical pairs the analyticity property of the eigenfunction alone is found sufficient to throw light on the nature of eigenvalues and eigenfunctions for different systems it is noted that the imaginary part of the eigenvalue ei turns out to be zero for all potentials v x with real couplings whereas it turns out to be nonzero for the case when the couplings are complex the prescription is also extended to study the excited states the problems related to the normalization of the eigenfunction and the boundary conditions to be used within this framework are also discussed

rkc an explicit solver for parabolic pdes the fortran program rkc is intended for the time integration of parabolic partial differential equations discretized by the method of lines it is based on a family of runge kutta chebyshev formulas with a stability bound that is quadratic in the number of stages remarkable properties of the family make it possible for the program to select at each step the most efficient stable formula as well as the most efficient step size moreover they make it possible to evaluate the explicit formulas in just a few vectors of storage these characteristics of the program make it especially attractive for problems in several spatial variables rkc is compared to the bdf solver vodpk on two test problems in three spatial variables

stability conditions and quantum dilogarithm identities for dynkin quivers abstract we study the fundamental groups of the exchange graphs for the bounded derived category d q of a dynkin quiver q and the finite dimensional derived category d n q of the calabi yau n ginzburg algebra associated to q in the case of d q we prove that its space of stability conditions in the sense of bridgeland is simply connected as an application we show that the donaldson thomas type invariant associated to q can be calculated as a quantum dilogarithm function on its exchange graph in the case of d n q we show that the faithfulness of the seidel thomas braid group action which is known for q of type a or n 2 implies the simply connectedness of the space of stability conditions

silting modules and ring epimorphisms abstract there are well known constructions relating ring epimorphisms and tilting modules the new notion of silting module provides a wider framework for studying this interplay to every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the bongartz completion for hereditary rings this assignment is used to parametrise homological ring epimorphisms by silting modules we further show that homological ring epimorphisms of a hereditary ring form a lattice which completes the poset of noncrossing partitions in the case of finite dimensional algebras

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

on stabilized integration for time dependent pdes an integration method is discussed which has been designed to treat parabolic and hyperbolic terms explicitly and stiff reaction terms implicitly the method is a special two step form of the one step imex implicit explicit rkc runge kutta chebyshev method the special two step form is introduced with the aim of getting a non zero imaginary stability boundary which is zero for the one step method having a non zero imaginary stability boundary allows for example the integration of pure advection equations space discretized with centered schemes the integration of damped or viscous wave equations the integration of coupled sound and heat flow equations etc for our class of methods it also simplifies the choice of temporal step sizes satisfying the von neumann stability criterion by embedding a thin long rectangle inside the stability region embedding rectangles or other tractable domains with this purpose is an idea of wesseling

categorical resolution of singularities abstract building on the concept of a smooth dg algebra we define the notion of a smooth derived category we then propose the definition of a categorical resolution of singularities our main example is the derived category d x of quasi coherent sheaves on a scheme x we prove that d x has a canonical categorical resolution if the base field is perfect and x is a separated scheme of finite type with a dualizing complex

wakamatsu tilting modules abstract we study a generalization of tilting modules to modules of possibly infinite projective dimension introduced by wakamatsu in j algebra 114 1988 106 114 in particular we characterize these modules in terms of suitable subcategories of finitely generated modules and in terms of cotorsion theories

uniqueness of enhancement for triangulated categories the paper contains general results on the uniqueness of a dg enhancement for trian gulated categories as a consequence we obtain such uniqueness for the unbounded categories of quasi coherent sheaves for the triangulated categories of perfect complexes and for the bounded de rived categories of coherent sheaves on quasi projective schemes if a scheme is projective then we also prove a strong uniqueness for the triangulated category of perfect complexes and for the bounded de rived categories of coherent sheaves these results directly imply that fully faithful functors from the bounded derived categories of coherent sheaves and the triangulated categories of perfect complexes on projective schemes can be represented by objects on the product

fourth order chebyshev methods with recurrence relation in this paper a new family of fourth order chebyshev methods also called stabilized methods is constructed these methods possess nearly optimal stability regions along the negative real axis and a three term recurrence relation the stability properties and the high order make them suitable for large stiff problems often space discretization of parabolic pdes a new code rock4 is proposed illustrated at several examples and compared to existing programs

non hermitian susy hydrogen like hamiltonians with real spectra it is shown that the radial part of the hydrogen hamiltonian factorizes as the product of two not mutually adjoint first order differential operators plus a complex constant t he 1 susy approach is used to construct non hermitian operators with hydrogen spectra other non hermitian hamiltonians are shown to admit an extra complex energy at n ew self adjoint hydrogen like hamiltonians are also derived by using a 2 susy transformation with complex conjugate pairs

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold

realisation functors in tilting theory derived equivalences and t structures are closely related we use realisation functors associated to t structures in triangulated categories to establish a derived morita theory for abelian categories with a projective generator or an injective cogenerator for this purpose we develop a theory of non compact or large tilting and cotilting objects that generalises the preceding notions in the literature within the scope of derived morita theory for rings we show that under some assumptions the realisation functor is a derived tensor product this fact allows us to approach a problem by rickard on the shape of derived equivalences finally we apply the techniques of this new derived morita theory to show that a recollement of derived categories is a derived version of a recollement of abelian categories if and only if there are tilting or cotilting t structures glueing to a tilting or a cotilting t structure as a further application we answer a question by xi on a standard form for recollements of derived module categories for finite dimensional hereditary algebras

discrete derived categories i homomorphisms autoequivalences and t structures discrete derived categories were studied initially by vossieck j algebra 243 168 176 2001 and later by bobinski et al cent eur j math 2 19 49 2004 in this article we describe the homomorphism hammocks and autoequivalences on these categories we classify silting objects and bounded t structures

eigenspectra of a complex coupled harmonic potential in three dimensions within the framework of extended complex phase space approach characterized by position and momentum coordinates we investigate the quasi exact solutions of the schrodinger equation for a coupled harmonic potential and its variants in three dimensions for this purpose ansatz method is employed and nature of the eigenvalues and eigenfunctions is determined by the analyticity property of the eigenfunctions alone the energy eigenvalue is real for the real coupling parameters and becomes complex if the coupling parameters are complex however in case of complex coupling parameters the imaginary component of energy eigenvalue reduces to zero if the p t symmetric condition is satisfied thus a non hermitian hamiltonian possesses real eigenvalue if it is p t symmetric

three kinds of mutation for a finite dimensional hereditary algebra h we consider exceptional sequences in mod h silting objects in the bounded derived category d and m cluster tilting objects in the m cluster category cm d 1 m there are mutation operations on both the set of m cluster tilting objects and the set of exceptional sequences it is also possible to define a mutation operation for silting objects we compare these three different notions of mutation