linear quadratic control for stochastic equations in a hilbert space with fractional brownian motions a linear quadratic control problem with a finite time horizon for some infinite dimensional controlled stochastic differential equations driven by a fractional gaussian noise is formulated and solved the feedback form of the optimal control and the optimal cost are given explicitly the optimal control is the sum of the well known linear feedback control for the associated deterministic linear quadratic control problem and a suitable prediction of the adjoint optimal system response to the future noise the covariance of the noise as well as the control operator in the system equation can in general be unbounded so the results can also be applied where the noise or the control are on the boundary of the domain or at discrete points in the domain some examples of controlled stochastic partial differential equations are given

regularity of laws and ergodicity of hypoelliptic sdes driven by rough paths we consider differential equations driven by rough paths and study the regularity of the laws and their long time behavior in particular we focus on the case when the driving noise is a rough path valued fractional brownian motion with hurst parameter h 2 1 1 our contribution in this work is twofold first when the driving vector fields satisfy hormander s c elebrated lie bracket condition we derive explicit quantitative bounds on the i nverse of the malliavin ma trix en route to this we provide a novel deterministic version of norris s lemma for differential equations driven by rough paths this result with the added assumption that the linearized equation has moments will then yield that the transition laws have a smooth density with respect to lebesgue measure our second main result states that under hormander s condi tion the solutions to rough differential equations driven by fractional brownian motion with h 2 1 1 enjoy a suitable version of the strong feller property under a standard controllability condition this implies that they admit a unique stationary solution that is physical in the sense that it does not look into the future

model spaces of regularity structures for space fractional spdes we study model spaces in the sense of hairer for stochastic partial differential equations involving the fractional laplacian we prove that the fractional laplacian is a singular kernel suitable to apply the theory of regularity structures our main contribution is to study the dependence of the model space for a regularity structure on the three parameter problem involving the spatial dimension the polynomial order of the nonlinearity and the exponent of the fractional laplacian the goal is to investigate the growth of the model space under parameter variation in particular we prove several results in the approaching subcriticality limit leading to universal growth exponents of the regularity structure a key role is played by the viewpoint that model spaces can be identified with families of rooted trees our proofs are based upon a geometrical construction similar to newton polygons for classical taylor series and various combinatorial arguments we also present several explicit examples listing all elements with negative homogeneity by implementing a new symbolic software package to work with regularity structures we use this package to illustrate our analytical results and to obtain new conjectures regarding coarse grained network measures for model spaces

long time behavior of solutions to a class of stochastic parabolic equations with homogeneous white noise stratonovitch s case in this article we prove new results concerning the long time behavior of random fields that are solutions in some sense to a class of semilinear parabolic equations subjected to a homogeneous and multiplicative white noise our main results state that these random fields eventually homogeneize with respect to the spatial variable and finally converge to a non random global attractor which consists of two spatially and temporally homogeneous asymptotic states more precisely we prove that the random fields either stabilize exponentially rapidly with probability one around one of the asymptotic states or that they set out to oscillate between them in the first case we can also determine exactly the corresponding lyapunov exponents in the second case we prove that the random fields are in fact recurrent in that they can reach every point between the two asymptotic states in a finite time with probability one in both cases we also interpret our results in terms of stability properties of the global attractor and we provide estimates for the average time that the random fields spend in small neighborhoods of the asymptotic states our methods of proof rest upon the use of a suitable regularization of the brownian motion along with a related wong zakai approximation procedure

on the conicity of eigenvalues intersections for parameter dependent self adjoint operators motivated by recent controllability results for the bilinear schrodinger equation based on the existence of conical intersections in this paper we identify two physically interesting families of parameter dependent hamiltonians that admit residual and prevalent subfamilies for which all double eigenvalues are conical in order to obtain such a result we exploit a characterization of conical intersections in terms of a transversality condition which allows to apply a suitable transversality theorem

the evolution of a random vortex filament we study an evolution problem in the space of continuous loops in a three dimensional euclidean space modeled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids we establish existence of a local solution starting from holder regular loops with index greater than 1 3 when the holder regularity of the initial condition x is smaller or equal to 1 2 we require x to be a rough path in the sense of lyons rev mat iberoamericana 14 1998 215 310 system control and rough paths 2002 oxford univ press the solution will then live in an appropriate space of rough paths in particular we can construct local solution starting from almost every brownian loop

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

controlled differential equations as young integrals a simple approach the theory of rough paths allows one to define controlled differential equations driven by a path which is irregular the most simple case is the one where the driving path has finite p variations with 1 p 2 in which case the integrals are interpreted as young integrals the prototypal example is given by stochastic differential equations driven by fractional brownian motion with hurst index greater than 1 2 using simple computations we give the main results regarding this theory existence uniqueness convergence of the euler scheme flow property which are spread out among several articles

rough paths and 1d sde with a time dependent distributional drift application to polymers motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions like the kardar parisi zhang kpz equation we reconsider the unique solvability of one dimensional stochastic differential equations the drift of which is a distribution by means of rough paths theory existence and uniqueness are established in the weak sense when the drift reads as the derivative of a holder continuous function regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed which makes the structure of the solutions more explicit than within the earlier framework of dirichlet processes

some identities for the generalized fibonacci numbers and the generalized lucas numbers in this paper we study some properties of the generalized fibonacci numbers and the generalized lucas numbers these numbers are equal to the total numbers of k independent sets in special graphs we give some identities for the generalized fibonacci numbers and the generalized lucas numbers which can be useful also in problems of counting of k independent sets in graphs

a rough path perspective on renormalization abstract we develop the algebraic theory of rough path translation particular attention is given to the case of branched rough paths whose underlying algebraic structure connes kreimer grossman larson makes it a useful model case of a regularity structure in the sense of hairer pre lie structures are seen to play a fundamental rule which allow a direct understanding of the translated i e renormalized equation under consideration this construction is also novel with regard to the algebraic renormalization theory for regularity structures due to bruned hairer zambotti 2016 the links with which are discussed in detail

eigenvalue bifurcations in kac murdock szegő matrices with a complex parameter abstract for complex the spectral properties of the toeplitz matrix k n j k j k 1 n often called the kac murdock szeg matrix have been examined in detail in two recent papers the second paper in particular introduced the concept of borderline curves these are two closed curves in the complex plane that consist of all the for which k n possesses some eigenvalue whose magnitude equals the matrix dimension n the purpose of the present paper is to examine eigenvalue bifurcations in both a qualitative and a quantitative manner and to discuss connections between bifurcations and the borderline curves

algebraic renormalisation of regularity structures we give a systematic description of a canonical renormalisation procedure of stochastic pdes containing nonlinearities involving generalised functions this theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms this subgroup is sufficiently large to be able to implement a version of the bphz renormalisation prescription in this context this is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations our construction is based on bialgebras of decorated coloured forests in cointeraction more precisely we have two hopf algebras in cointeraction coacting jointly on a vector space which represents the generalised functions of the theory two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic birkhoff factorisation that arises naturally in perturbative quantum field theory

random attractor for the 3d viscous primitive equations driven by fractional noises abstract we develop a new and general method to prove the existence of the random attractor strong attractor for the primitive equations pes of large scale ocean and atmosphere dynamics under non periodic boundary conditions and driven by infinite dimensional additive fractional wiener processes in contrast to our new method the common method compact sobolev embedding theorem is to obtain the time uniform a priori estimates in some sobolev space whose regularity is higher than the solution space but this method can not be applied to the 3d stochastic pes with the non periodic boundary conditions therefore the existence of universal attractor weak attractor was established in previous works see 15 16 the main idea of our method is that we first derive that p almost surely the solution operator of stochastic pes is compact then we construct a compact absorbing set by virtue of the compact property of the solution operator and the existence of a absorbing set we should point out that our method has some advantages over the common method of using compact sobolev embedding theorem i e using our method we only need to obtain time uniform a priori estimates in the solution space to prove the existence of random attractor for the corresponding stochastic system while the common method need to establish time uniform a priori estimates in a more regular functional space than the solution space take the stochastic pes for example as the unique strong solution to the stochastic pes belongs to c 0 t h 1 3 in view of our method we only need to obtain the time uniform a priori estimates in the solution space h 1 3 to prove the existence of random attractor for this stochastic system while the common method need to establish time uniform a priori estimates for the solution in the functional space h 2 3 however time uniform a priori estimates in h 2 3 for the solution to stochastic pes are too difficult to be established the present work provides a general way for proving the existence of random attractor for common classes of dissipative stochastic partial differential equations driven by wiener noises fractional noises and even jump noises in a forth coming paper using this new method we 46 prove the existence of random attractor for the stochastic nematic liquid crystals equations this is the first result about the long time behavior of stochastic nematic liquid crystals equations

bounds on eigenvalues of real and complex interval matrices we present a cheap and tight formula for bounding real and imaginary parts of eigenvalues of real or complex interval matrices it outperforms the classical formulae not only for the complex case but also for the real case in particular it generalizes and improves the results by rohn 1998 5 and hertz 2009 19 the main idea behind is to reduce the problem to enclosing eigenvalues of symmetric interval matrices for which diverse methods can be utilized the result helps in analysing stability of uncertain dynamical systems since the formula gives sufficient conditions for testing schur and hurwitz stability of interval matrices it may also serve as a starting point for some iteration methods

on the existence of sle trace finite energy drivers and non constant kappa existence of loewner trace is revisited we identify finite energy paths the skeleton of wiener measure as natural class of regular drivers for which we find simple and natural estimates in terms of their cameron martin norm secondly now dealing with potentially rough drivers a representation of the derivative of the inverse of the loewner flow is given in terms of a rough and then pathwise follmer integral assuming the driver within a class of ito processes an exponential martingale argument implies existence of trace in contrast to classical exact sle computations our arguments are well adapted to perturbations such as non constant kappa assuming 2 for technical reasons and additional finite energy drift terms

evolution equations driven by a fractional brownian motion abstract in this paper we study nonlinear stochastic evolution equations in a hilbert space driven by a cylindrical fractional brownian motion with hurst parameter h 1 2 and nuclear covariance operator we establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter h this result is applied to stochastic parabolic equation perturbed by a fractional white noise in this case if the coefficients are lipschitz continuous and bounded the existence and uniqueness of a solution holds if h d 4 the proofs of our results combine techniques of fractional calculus with semigroup estimates

variational solutions for partial differential equations driven by a fractional noise abstract in this article we develop an existence and uniqueness theory of variational solutions for a class of nonautonomous stochastic partial differential equations of parabolic type defined on a bounded open subset d r d and driven by an infinite dimensional multiplicative fractional noise we introduce two notions of such solutions for them and prove their existence and their indistinguishability by assuming that the noise is derived from an l 2 d valued fractional wiener process w h with hurst parameter h 1 1 1 whose covariance operator satisfies appropriate integrability conditions and where 0 1 denotes the holder exponent of the derivative of the nonlinearity in the stochastic term of the equations we also prove the uniqueness of solutions when the stochastic term is an affine function of the unknown random field our existence and uniqueness proofs rest upon the construction and the convergence of a suitable sequence of faedo galerkin approximations while our proof of indistinguishability is based on certain density arguments as well as on new continuity properties of the stochastic integral we define with respect to w h

beyond the limitations of perturbation methods for real random eigenvalue problems using exceptional points and analytic continuation a numerical method is proposed to approximate the solution of parametric eigenvalue problem when the variability of the parameters exceed the radius of convergence of low order perturbation methods the radius of convergence of eigenvalue perturbation methods based on taylor series is known to decrease when eigenvalues are getting closer to each other this phenomenon knwon as veering in structural dynamics is a direct consequence of the existence of branch point singularity in the complex plane of the varying parameters where some eigenvalues are defective when this degeneracy referred to as exceptional point ep is close to the real axis the veering becomes stronger the main idea of the proposed approach is to combined a pair of eigenvalues to remove this singularity to do so two analytic auxiliary functions are introduced and are computed through high order derivatives of the eigenvalue pair with respect to the parameter this yields a new robust eigenvalue reconstruction scheme which is compared to taylor and puiseux series in all cases theoretical bounds are established and all approximations are compared numerically on a three degrees of freedom toy model this system illustrate the ability of the method to handle the vibrations of a structure with a randomly varying parameter computationally efficient the proposed algorithm could also be relevant for actual numerical models of large size arising from other applications involving parametric eigenvalue problems e g waveguides rotating machinery or instability problems such as squeal or flutter

gaussian type lower bounds for the density of solutions of sdes driven by fractional brownian motions in this paper we obtain gaussian type lower bounds for the density of solutions to stochastic differential equations sdes driven by a fractional brownian motion with hurst parameter h in the one dimensional case with additive noise our study encompasses all parameters h 0 1 while the multidimensional case is restricted to the case h 1 2 we rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools

exceptional points of non hermitian operators exceptional points associated with non hermitian operators i e operators being non hermitian for real parameter values are investigated the specific characteristics of the eigenfunctions at the exceptional point are worked out within the domain of real parameters the exceptional points are the points where eigenvalues switch from real to complex values these and other results are exemplified by a classical problem leading to exceptional points of a non hermitian matrix

the reconstruction theorem in besov spaces abstract the theory of regularity structures 9 sets up an abstract framework of modelled distributions generalising the usual holder functions and allowing one to give a meaning to several ill posed stochastic pdes a key result in that theory is the so called reconstruction theorem it defines a continuous linear operator that maps spaces of modelled distributions into the usual space of distributions in the present paper we extend the scope of this theorem to analogues to the whole class of besov spaces b p q with non integer regularity indices we then show that these spaces behave very much like their classical counterparts by obtaining the corresponding embedding theorems and schauder type estimates

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

stochastic analysis rough path analysis and fractional brownian motions in this paper we show by using dyadic approximations the existence of a geometric rough path associated with a fractional brownian motion with hurst parameter greater than 1 4 using the integral representation of fractional brownian motions we furthermore obtain a skohorod integral representation of the geometric rough path we constructed by the results in ly1 a stochastic integration theory may be established for fractional brownian motions and strong solutions and a wong zakai type limit theorem for stochastic differential equations driven by fractional brownian motions can be deduced accordingly the method can actually be applied to a larger class of gaussian processes with covariance functions satisfying a simple decay condition

stochastic calculus with respect to gaussian processes in this paper we develop a stochastic calculus with respect to a gaussian process of the form b t int t 0 k t s dw s where w is a wiener process and k t s is a square integrable kernel using the techniques of the stochastic calculus of variations we deduce change of variable formulas for the indefinite integrals and we study the approximation by riemann sums the particular case of the fractional brownian motion is discussed

stability in mean of partial variables for stochastic reaction diffusion systems abstract it is with the aid of the lyapunov method that stochastic ordinary differential equations and stochastic functional differential equations have been investigated intensively in the past decades however for stochastic reaction diffusion equations this useful technique seems to feel helpless on account of the emptiness of its own ito s formula to get over this difficulty we will manage to regard the integral of the considered trajectory with respect to spatial variables as the solution of the corresponding stochastic ordinary differential equations and then employ the ito formula to study stability in mean of partial variables for ito stochastic reaction diffusion equations some sufficient conditions for uniform stability in mean asymptotic stability in mean uniformly asymptotic stability in mean exponential stability in mean of partial variables are presented

the optical singularities of birefringent dichroic chiral crystals using a new formalism involving projection from the sphere of directions to the stereographic plane and associated complex variables explicit formulae are obtained for the two refractive indices and polarizations in optically anisotropic crystals that are both dichroic absorbing and chiral optically active this enables three types of polarization singularity to be classified and explored singular axes which are degeneracies where the two refractive indices are equal and which for a transparent non chiral crystal condense pairwise onto the optic axes c points where the polarization is purely circular right or left handed with topological index 1 or and whose positions are independent of the chirality and l lines where the polarization is purely linear dividing direction space into regions with right and left handedness a local model captures essential features of the general theory interference figures generated by slabs of crystal viewed directly or through a polarizer and or analyser enable the singularities to be displayed directly

rough differential equations driven by signals in besov spaces rough differential equations are solved for signals in general besov spaces unifying in particular the known results in holder and p variation topology to this end the paracontrolled distribution approach which has been introduced by gubinelli imkeller and perkowski 24 to analyze singular stochastic pdes is extended from holder to besov spaces as an application we solve stochastic differential equations driven by random functions in besov spaces and gaussian processes in a pathwise sense

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

unfolding a diabolic point a generalized crossing scenario the typical avoided crossings for hermitian quantum systems depending on parameters the diabolic crossing scenario are generalized to the non hermitian case e g for resonances two types of crossings appear for type i the real parts show an avoided and the imaginary parts a true crossing of the eigenenergies and for type ii the opposite is found a simple symmetric non hermitian two state matrix hamiltonian is analysed in detail the diabolic point bifurcates into two exceptional ones on exceptional lines where the matrices are defective the adiabatic transport of eigenvectors and eigenstates in parameter space is discussed in this generalized diabolic crossing scenario in particular the geometric berry phases for a cyclic variation of system parameters depending on the topology of the closed curves with respect to the exceptional lines

stochastic calculus for fractional brownian motion i theory in this paper a stochastic calculus is given for the fractional brownian motions that have the hurst parameter in 1 2 1 a stochastic integral of ito type is defined for a family of integrands so that the integral has zero mean and an explicit expression for the second moment this integral uses the wick product and a derivative in the path space some ito formulae or change of variables formulae are given for smooth functions of a fractional brownian motion or some processes related to a fractional brownian motion a stochastic integral of stratonovich type is defined and the two types of stochastic integrals are explicitly related a square integrable functional of a fractional brownian motion is expressed as an infinite series of orthogonal multiple integrals

on a modelled rough heat equation we use the formalism of hairer s regularity structures theory to study a heat equation model with non linear perturbation driven by a space time fractional noise different regimes are observed depending on the global pathwise roughness of the noise to this end and following the procedure exhibited in hairer invent math 198 2 269 504 2014 the equation is first lifted into some abstract regularity structure and therein solved through a fixed point argument then we construct a consistent model above the fractional noise by relying on a smooth fourier type approximation of the process

unfolding of eigenvalue surfaces near a diabolic point due to a complex perturbation the paper presents a new theory of unfolding of eigenvalue surfaces of real symmetric and hermitian matrices due to an arbitrary complex perturbation near a diabolic point general asymptotic formulae describing deformations of a conical surface for different kinds of perturbing matrices are derived as a physical application singularities of the surfaces of refractive indices in crystal optics are studied

diabolical points in the spectra of triangles accidental degeneracies between energy levels e j and e j 1 of a real hamiltonian can occur generically in a family of hamiltonians labelled by at least two parameters x y energy level surfaces in e x y space have locally a double cone diabolo connection and we refer to the degeneracies themselves as diabolical points we studied the family of systems in which a particle moves freely within hard walled triangles vibrations of triangular membranes with x and y labelling two of the angles using an efficient green function technique to compute the levels we found several diabolical points for low lying levels as well as some symmetry degeneracies the lowest diabolical point occurred for levels 5 and 6 of the triangle 130 57 30 73 18 70 the conical structure was confirmed by noting that the normal derivative u of the wavefunction at a boundary point changed sign during a small circuit of the diabolical point the form of the variation of u around a circuit and the changing pattern of nodal lines of agreed with theoretical expectations an estimate of the total number of degeneracies n d j involving levels 1 through j based on the energy scaling of cone angles and the level spacing distribution gave n d j j 2 5 and our limited data support this prediction analytical theory confirmed that for thin triangles where our computational method is slow there are no degeneracies in the energy range studied

collapse of the keldysh chains and stability of continuous nonconservative systems in the present paper eigenvalue problems for non self adjoint linear differential operators smoothly dependent on a vector of real parameters are considered bifurcation of eigenvalues along smooth curves in the parameter space is studied the case of a multiple eigenvalue with the keldysh chain of arbitrary length is investigated explicit expressions describing bifurcation of eigenvalues are found the obtained formulas use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space these results are important for the stability problems and sensitivity analysis of nonconservative systems as a mechanical application the extended beck problem of stability of an elastic column subjected to a partially tangential follower force is considered and discussed in detail

a version of hormander s theorem for the fractional brownian motion it is shown that the law of an sde driven by fractional brownian motion with hurst parameter greater than 1 2 has a smooth density with respect to lebesgue measure provided that the driving vector fields satisfy hormander s condition the main new ingredient of the proof is an extension of norris lemma to this situation

on the observability and asymmetry of adiabatic state flips generated by exceptional points in open quantum systems where the effective hamiltonian is not hermitian it is known that the adiabatic or instantaneous basis can be multivalued by adiabatically transporting an eigenstate along a closed loop in the parameter space of the hamiltonian it is possible to end up in an eigenstate different from the initial eigenstate this adiabatic flip effect is an outcome of the appearance of a degeneracy known as an exceptional point inside the loop we show that contrary to what is expected of the transport properties of the eigenstate basis the interplay between gain loss and non adiabatic couplings imposes fundamental limitations on the observability of this adiabatic flip effect

a class of generalized tribonacci sequences applied to counting problems abstract generalized tribonacci numbers with the third order linear recurrence with constant coefficients t k n t k n 1 t k n 2 k t k n 3 for n 2 are investigated for some sets of the initial triples t0 t1 t2 in particular generating functions the binet formula and the limit of ratio of consecutive terms t k n 1 t k n are discussed these numbers are related to numbers of path graphs colorings with k 2 colors or equivalently to counting of q ary sequences of length n for q k 2 satisfying requirements which follow the problem of degeneration in the ising model with the second neighbor interactions it is shown that the results obtained can be considered as the base for considerations of cycle graph colorings cyclic q ary sequences these are counting problems so t0 t1 t2 and k should be natural numbers but these sequences can be considered for any real numbers the special cases k 0 1 lead to the fibonacci and the usual tribonacci numbers respectively so the results can be applied to binary and ternary sequences

a priori estimates for rough pdes with application to rough conservation laws we introduce a general framework to study pdes driven by rough paths we develop new a priori estimates based on a rough gronwall lemma argument for weak solutions to rough pdes this will allow us to follow standard pde strategies to obtain existence and uniqueness results in particular our approach does not rely on any sort of transformation formula flow transformation feynman kac representation formula etc and is therefore rather flexible as an application we study conservation laws driven by rough paths establishing well posedness for the corresponding kinetic formulation

the geometry of the space of branched rough paths we construct an explicit transitive free action of a banach space of holder functions on the space of branched rough paths which yields in particular a bijection between theses two spaces this endows the space of branched rough paths the structure of a principal homogeneous space over a banach space and allows to characterize its automorphisms the construction is based on the baker campbell hausdorff formula and on the hairer kelly map which allows to describe branched rough paths in terms of anisotropic geometric rough paths

interaction of eigenvalues in multi parameter problems the paper presents a new general theory of interaction of eigenvalues of matrix operators depending on parameters both complex and real eigenvalues are considered strong and weak interactions are distinguished and their geometric interpretation on the complex plane is given mechanical examples are presented and discussed in detail

rough burgers like equations with multiplicative noise we construct solutions to vector valued burgers type equations perturbed by a multiplicative space time white noise in one space dimension due to the roughness of the driving noise solutions are not regular enough to be amenable to classical methods we use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution subject to the choice of the correct reference rough path we prove unique solvability for the equation and we show that our solutions are stable under smooth approximations of the driving noise

projective hilbert space structures at exceptional points a non hermitian complex symmetric 2 2 matrix toy model is used to study projective hilbert space structures in the vicinity of exceptional points eps the bi orthogonal eigenvectors of a diagonalizable matrix are puiseuxexpanded in terms of the root vectors at the ep it is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the ep limit can be resolved by projectively extending the original hilbert space the complementary normalization conditions correspond then to two different affine charts of this enlarged projective hilbert space geometric phase and phase jump behaviour are analysed and the usefulness of the phase rigidity as measure for the distance to ep configurations is demonstrated finally ep related aspects of pt symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated

the topology of the adiabatic passage process for molecular photodissociative dynamics we study population transfer processes for photodissociation as described within the optical potential model in the context of adiabatic or quasi adiabatic dynamics by a reformulation of the adiabatic passage theory in terms of the fibre bundle topology we extend the analysis made by yatsenko et al 2002 phys rev a 65 043407 from conservative to dissipative systems we show that this topology is associated with the problem of eigenvalue labelling and hence with the continuous eigenvalue following process in the control parameter space problems involving adiabatic passage the direct chirping process and non adiabatic transitions are studied with particular regard to the presence of resonance states we also discuss the role played by the molecular continua

stochastic shell models driven by a multiplicative fractional brownian motion abstract we prove existence and uniqueness of the solution of a stochastic shell model the equation is driven by an infinite dimensional fractional brownian motion with hurst parameter h 1 2 1 and contains a non trivial coefficient in front of the noise which satisfies special regularity conditions the appearing stochastic integrals are defined in a fractional sense first we prove the existence and uniqueness of variational solutions to approximating equations driven by piecewise linear continuous noise for which we are able to derive important uniform estimates in some functional spaces then thanks to a compactness argument and these estimates we prove that these variational solutions converge to a limit solution which turns out to be the unique pathwise mild solution associated to the shell model with fractional noise as driving process

a krein space related perturbation theory for mhd α2 dynamos and resonant unfolding of diabolical points the spectrum of the spherically symmetric 2 dynamo is studied in the case of idealized boundary conditions starting from the exact analytical solutions of models with constant profiles a perturbation theory and a galerkin technique are developed in a krein space approach with the help of these tools a very pronounced resonance pattern is found in the deformations of the spectral mesh as well as in the unfolding of the diabolical points located at the nodes of this mesh non oscillatory as well as oscillatory dynamo regimes are obtained an estimation technique is developed for obtaining the critical profiles at which the eigenvalues enter the right spectral half plane with non vanishing imaginary components at which overcritical oscillatory dynamo regimes form finally frechet derivative gradient based methods are developed suitable for further numerical investigations of krein space related setups such as mhd 2 dynamos or models of symmetric quantum mechanics

bivariate fibonacci like p polynomials abstract in this article we study the bivariate fibonacci and lucas p polynomials p 0 is integer from which specifying x y and p bivariate fibonacci and lucas polynomials bivariate pell and pell lucas polynomials jacobsthal and jacobsthal lucas polynomials fibonacci and lucas p polynomials fibonacci and lucas p numbers pell and pell lucas p numbers and chebyshev polynomials of the first and second kind are obtained afterwards we obtain some properties of the bivariate fibonacci and lucas p polynomials

perturbation theory of symmetric hamiltonians in the framework of perturbation theory the reality of the perturbed eigenvalues of a class of symmetric hamiltonians is proved using stability techniques we apply this method to symmetric unperturbed hamiltonians perturbed by symmetric additional interactions

sensitivity of rough differential equations an approach through the omega lemma the ito map assigns the solution of a rough differential equation a generalization of an ordinary differential equation driven by an irregular path when existence and uniqueness hold by studying how a path is transformed through the vector field which is integrated we prove that the ito map is holder or lipschitz continuous with respect to all its parameters this result unifies and weaken the hypotheses of the regularity results already established in the literature

coupling of eigenvalues of complex matrices at diabolic and exceptional points the paper presents a general theory of coupling of eigenvalues of complex matrices of an arbitrary dimension depending on real parameters the cases of weak and strong coupling are distinguished and their geometric interpretation in two and three dimensional spaces is given general asymptotic formulae for eigenvalue surfaces near diabolic and exceptional points are presented demonstrating crossing and avoided crossing scenarios two physical examples illustrate effectiveness and accuracy of the presented theory

integration questions related to fractional brownian motion let b h u u be a fractional brownian motion fbm with index h 0 1 and b h be the closure in l 2 of the span sp b h of the increments of fbm b h it is well known that when b h b 1 2 is the usual brownian motion bm an element x b 1 2 can be characterized by a unique function f x l 2 in which case one writes x in an integral form as x f x u db 1 2 u from a different though equivalent perspective the space l 2 forms a class of integrands for the integral on the real line with respect to bm b 1 2 in this work we explore whether a similar characterization of elements of b h can be obtained when h 0 1 2 or h 1 2 1 since it is natural to define the integral of an elementary function f k 1 n f k 1 uk uk 1 by k 1 n f k b h u k 1 b h u k we want the spaces of integrands to contain elementary functions these classes of integrands are inner product spaces if the space of integrands is not complete then it characterizes only a strict subset of b h when 0 h 1 2 by using the moving average representation of fbm b h we construct a complete space of integrands when 1 2 h 1 however an analogous construction leads to a space of integrands which is not complete when 0 h 1 2 or 1 2 h 1 we also consider a number of other spaces of integrands while smaller and henceincomplete they form a natural choice and are convenient to workwith we compare these spaces of integrands to the reproducing kernel hilbert space of fbm