raising and lowering operators for uq gl n a set of raising and lowering operators are constructed for uq gl n being hermitian conjugates of one another they are easily normalized and allow us to express arbitrary gelfand tzetlin basis states as a product of lowering operators acting on the maximal weight state

h c1 maps and elliptic spdes with polynomial and exponential perturbations of nelson s euclidean free field abstract elliptic stochastic partial differential equations spde with polynomial and exponential perturbation terms defined in terms of nelson s euclidean free field on rd are studied using results by s kusuoka and a s ustunel and m zakai concerning transformation of measures on abstract wiener space spdes of this type arise in particular in euclidean quantum field theory with interactions of the polynomial or exponential type the probability laws of the solutions of such spdes are given by girsanov probability measures that are non linearly transformed measures of the probability law of nelson s free field defined on subspaces of schwartz space of tempered distributions

the clebsch gordan coefficients for the two parameter quantum algebra sup q 2 in the lowdin shapiro approach the structure of the irreducible representations of the two parameter quantum algebra sup q 2 is studied the projection operators for this algebra are constructed in the lowdin shapiro form the explicit analytical expressions for the sup q 2 clebsch gordan coefficients are obtained with the help of these projection operators there are clear perspectives to elaborate the theory of tensor operators universal r matrix 6j 9j symbols etc for the sup q 2 algebra using the same tools

periodic and partially periodic representations of su n q the gelfand zetlin basis is adapted tosu n q forq a root of unit extra parameters are incorporated in the matrix elements of the generators to obtain all the invariants corresponding to the augmented center a crucial identity is derived and proved which guarantees the periodicity of the action of the generators full periodicity is relaxed by stages some raising and lowering operators remaining injective while others become nilpotent with corresponding changes in the dimension of the representation in the extreme case of highest weight representations all the raising and lowering operators are nilpotent as an alternative approach an auxiliary algebra giving all the periodic representations is presented an explicit solution of this system forn 3 while fully equivalent to the g z basis turns out to be much simpler

reduction formula for moments of stochastic integrals we show how moments of stochastic integrals can be expressed explicitly as expectation values of ordinary lebesgue integrals this result generalizes the well known relationship for the second moment

polynomial realization of the uq sl 3 gel fand weyl zetlin basis an explicit realization of the u uq sl 3 gel fand weyl zetlin gwz basis as polynomial functions in three variables real or complex is given this realization is obtained in two complementary ways first a known correspondence is used between the abstract gwz basis and explicit polynomials in the quantum subgroup u of the raising generators then an explicit construction is used of arbitrary lowest weight holomorphic representations of u in terms of three variables on which the generators of u are realized as q difference operators the application of the gwz corresponding polynomials in this realization of the lowest weight vector the function 1 produces the first realization of this gwz basis another realization of the gwz polynomial basis is found by the explicit diagonalization of the operators of isospin i2 third component of isospin iz and hypercharge in the same realization as q difference operators the result is that the eigenvectors can be written in terms of q hypergeometric

symmetries of clebsch gordan coefficients of the quantum group uq n permutation symmetries for clebsch gordan coefficients of the quantum group uq n and for uq n 1 reduced clebsch gordan coefficients of uq n are derived in particular the formula relating clebsch gordan coefficients for the tensor product tm1 tm2 with clebsch gordan coefficients for the tensor product with permuted representations tm1 and tm is obtained where tm is one of the irreducible representations in the decomposition of tm1 tm2 contrary to the classical case coefficients connecting these clebsch gordan coefficients depend on gel fand tsetlin patterns of one of irreducible representations this dependence is explicitly determined

finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple lie algebra let g be a complex simple lie algebra we show that whent is not a root of 1 all finite dimensional representations of the quantum analogu t g are completely reducible and we classify the irreducible ones in terms of highest weights in particular they can be seen as deformations of the representations of the classical u g

complementarity of suq 3 and uq 2 and q boson realization of the suq 3 irreducible representations the complementarity relationship also termed duality that arises between the irreps of the su 3 and u 2 lie algebras when their direct sum su 3 u 2 is embedded into a larger u 6 algebra and single row irreps of the latter are considered is extended to the corresponding q algebras suq 3 and uq 2 it is demonstrated by explicitly constructing the unique q boson state that is simultaneously maximal in suq 3 and uq 2 for a given u q 2 weight in addition the relations between the suq 3 and uq 2 casimir operators resulting from their complementarity are explicitly found together with the q bargmann representation of q boson operators the complementarity relationship is then used to construct a gel fand tseitlin basis for arbitrary suq 3 irreps in terms of q boson operators

q deformation of the lorentz group we describe a q deformation of the lorentz group in terms of a q deformation of the van der waerden spinor algebra

matrix elements and wigner coefficients for uq gl n in this series of papers the fundamental wigner coefficients and the matrix elements of the generators for the quantum group uq gl n are derived the uq gl n uq gl n 1 reduced wigner coefficients and reduced matrix elements are determined algebraically as eigenvalues of certain uq gl n 1 invariants in the quantum group uq gl n the matrix elements of the elementary quantum group generators in the gel fand tsetlin basis are derived below while all fundamental wigner coefficients are derived in the second paper of the series

toroidal and level 0 uq sln 1 actions on uq gln 1 modules 1 utilizing a braid group action on a completion of uq sln 1 an algebra homomorphism from the toroidal algebra uq sln 1 tor n 2 to a completion of uq gln 1 is obtained 2 the toroidal actions by saito induces a level 0 uq sln 1 action on level 1 integrable highest weight modules of uq sln 1 another level 0 uq sln 1 action was defined by jimbo et al in the case n 1 using the fact that the intertwiners of uq sln 1 modules are intertwiners of toroidal modules for an appropriate comultiplication the relation between these two level 0 uq sln 1 actions is clarified

biorthogonal coupling coefficients of uq n the coupling wigner clebsch gordan coefficients and their isofactors for the unitary quantum algebras uq n with the repeating irreducible representations in the coproduct decomposition are considered generalizing the u n case the biorthogonal systems of the uq n isofactors with the dual multiplicity labels are constructed by means of the recoupling technique in terms of the isofactors with simpler multiplicity structure a first construction correlated with the inverted littlewood richardson rules gives the bilinear combinations of isofactors after applying the proportionality of the q recoupling racah coefficients to the boundary q isofactors an alternative recursive construction gives the nonorthogonal q isofactors satisfying the most elementary boundary conditions and proportional to the uq n 1 recoupling coefficients for some less restricted values of parameters some multiplicity free and more general uq n recoupling coefficients are found the blocks bilinear combinations of which equal to the resubducing coefficients of the complementary chains of q algebras are proposed to use for the orthonormalization of some uq n biorthogonal isofactors including the general uq 3 case

multiplicity free uq n coupling coefficients the coupling wigner clebsch gordan coefficients of the unitary quantum algebras uq n are considered the tensorial properties of the generator powers and their ordered products used in the explicit projectors and weight lowering procedures are established different expressions for the multiplicity free isofactors of uq n coupling coefficients those coupling an arbitrary and symmetric representations are derived explicit expression of the arbitrary isofactors in terms of their boundary values are proposed proportionality of the semistretched isofactors to the stretched q recoupling coefficients of uq n 1 q analogues of 9j symbols is demonstrated the stretched isofactors of uq 3 are expressed in terms of clebsch gordan coefficients of uq 2

representations of uq sl n at roots of unity the gelfand zetlin basis for representations of uq sl n is improved to better fit the case when q is a root of unity the usual q deformed representations as well as the nilpotent periodic cyclic semi periodic semi cyclic and also some atypical representations are now described with the same formalism

quantum r matrix for g2 and a solvable 175 vertex model the quantum r matrix for cartan s exceptional simple lie algebra g2 in its seven dimensional minimal representation is presented the r matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra uq g2 this defines a new solvable seven state 175 vertex model on the planar square lattice whose boltzmann weights satisfy the yang baxter equation for the equivalent face model the local state probability is obtained in terms of a relative of a level 1 string function related to the affine lie algebra pair b3 1 contains implies g2 1

generalized chiral potts models and minimal cyclic representations of u_q widehat mathfrak g mathfrak l n c finite dimensional cyclic representations of the quantized universal enveloping algebra u q g exist only when the deformation parameter q is a root of unity q n 1 in such representations the n th power of each chevalley generator acts as a non zero scalar these values of e i n f i n and t i n are part of a continuous family of parameters upon which the cyclic representations depend

non integrable representations of the restricted quantum analogue of sl 3 the structure of irreducible representations of restricted at roots of unity is understood within the gelfand zetlin basis the latter needs a weakened definition for non integrable representations where the quadratic casimir operator of the quantum subalgebra is not completely diagonalized this is necessary in order to take into account the indecomposable modules that appear the set of redefined mixed states has a teepee shape inside the pyramid made with the whole representation

phaseequal convex phase retrieval via alternating direction method of multipliers the problem of estimating an unknown complex signal from its magnitude only measurements is a classical problem known as phase retrieval this problem arises naturally because in some applications it is difficult or costly to measure the phase of the measurements in this paper we propose a new convex formulation referred to as phaseequal for solving the phase retrieval problem similar to phasemax t goldstein and c studer 2018 by goldstein and studer iee tit 2018 phaseequal works in the natural signal space and thus is computationally efficient we then extend phaseequal to the sparse signal case with the proposed convex formulation termed as compressed phaseequal different from existing convex compressed phase retrieval methods the proposed compressed phaseequal formulation does not involve any regularization parameter and thus is free of the parameter tuning issue which is always tricky in practice order wise recovery conditions for phaseequal and its sparse version i e compressed phaseequal are analyzed our theoretical results show that phaseequal resp compressed phaseequal achieves perfect recovery with mathcal o n resp mathcal o k text log frac n k magnitude measurements provided that a well correlated reference vector is available where k and n denote the number of nonzero entries in the complex sparse signal and the dimension of the signal respectively simulation results are provided to illustrate the effectiveness of the proposed methods

a quantum group structure in integrable conformal field theories we discuss a quantization prescription of the conformal algebras of a class ofd 2 conformal field theories which are integrable we first give a geometrical construction of certain extensions of the classical virasoro algebra known as classicalw algebras in which these algebras arise as the lie algebra of the second hamiltonian structure of a generalized korteweg de vries hierarchy this fact implies that thew algebras obtained as a reduction with respect to the nilpotent subalgebras of the kac moody algebra describe the integrability of a toda field theory subsequently we determine the coadjoint operators of thew algebras and relate these to classical yang baxter matrices the quantization of these algebras can be carried out using the concept of a so called quantum group we derive the condition under which the representations of these quantum groups admit a hilbert space completion by exploring the relation with the braid group then we consider a modification of the miura transformation which we use to define a quantumw algebra this leads to an alternative interpretation of the coset construction for kac moody algebras in terms of nonlinear integrable hierarchies subsequently we use the connection between the induced braid group representations and the representations of the mapping class group of riemann surfaces to identify an action of thew algebras on the moduli space of stable curves and we give the invariants of this action this provides a generalization of the situation for the virasoro algebra where such as invariant is given by the so called mumford form which describes the partition function of the bosonic string

exponential loss of memory for the 2 dimensional allen cahn equation with small noise we prove an asymptotic coupling theorem for the 2 dimensional allen cahn equation perturbed by a small space time white noise we show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology in the 1 dimensional case a similar result was shown in martinelli et al commun math phys 120 1 25 69 1988 j stat phys 55 3 4 477 504 1989 it is well known that in two or more dimensions solutions of this equation are distribution valued and the equation has to be interpreted in a renormalised sense formally this renormalisation corresponds to moving the minima of the potential infinitely far apart and making them infinitely deep we show that despite this renormalisation solutions behave like perturbations of the deterministic system without renormalisation they spend large stretches of time close to the minimisers of the un renormalised potential and the exponential contraction rate of different profiles is given by the second derivative of the potential in these points as an application we prove an eyring kramers law for the transition times between the stable solutions of the deterministic system for fixed initial conditions

periodic and flat irreducible representations ofsu 3 q we construct all the periodic irreducible representations ofu su 3 q forq am root of unity their dimensions arek 2m 2 fork 1 m onlyk 1 m 2 for evenm their interest is that they could be a tool to generalize the chiral potts model by truncation of these representations we construct flat representations ofu su 3 q in which all the multiplicities of the weights are set to 1

roots of unity representations of quantum groups representations of quantum groupsu e g n g n any semi simple lie algebra of rankn are constructed from arbitrary representations of rankn 1 quantum groups for e a root of unity representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases

singular vectors of representations of quantum groups the author gives explicit formulae for singular vectors of verma modules over uq g they give the general formula for g an for some cases when g dn and for some rank two subalgebras of g not an dn for this he uses a special basis of uq g where g is the negative root subalgebra of g which was introduced in his earlier work on the case q 1 this basis seems more economical than the poincare birkhoff witt type of basis used by malikov feigin and fuchs 1986 for the construction of singular vectors of verma modules in the case q 1 furthermore this basis turns out to be part of a general basis introduced recently for other reasons by lusztig 1988 for uq b where b is a borel subalgebra of g

multiplicity free wigner coefficients for semisimple lie groups i the u n pattern calculus this is the first paper in a series of two dedicated to a new algebraic determination of the multiplicity free reduced wigner coefficients for the lie groups u n and o n the approach employed enables a direct nonrecursive derivation of reduced wigner coefficients the absolute value squared of the reduced wigner coefficients is expressed as a rational polynomial function numerator polynomial divided by denominator polynomial whose weyl group symmetries are seen to fall out naturally in our approach from the transformation properties of polynomial functions determined by casimir invariants a unified treatment of the composition laws of reduced wigner coefficients and the evaluation of their associated denominator polynomials is presented which applies to both u n and o n an explicit formula for the numerator polynomials of u n is also derived the numerator polynomials for the orthogonal groups will be given in the second paper of the series

quantum line bundles on s2 and method of orbits for suq 2 the quantum sphere is covered by two quantum complex planes quantum line bundle on s2 is introduced by specifying a transition function the structure of the quantum complex planes enables one to construct representations of suq 2 acting in spaces of holomorphic sections in the quantum case dressing transformation replaces coadjoint action

generalized gel fand invariants and characteristic identities for quantum groups the generalized gel fand invariants for an arbitrary quantum group are explicitly constructed and their eigenvalues in any irreducible representation are computed these invariants enable one to develop characteristic identities for the quantum group and as a natural application these identities are used to construct projection operators for tensor products of representations to illustrate the general theory the quantum group uq gl m is studied in detail

irreducible highest weight representations of quantum groupsu q gl n c explicit recurrence formulas of canonical realization boson representation for quantum enveloping algebrasu q gl n c are given using them irreducible highest weight representations ofu q gl n c are obtained as restriction of representation of fock space to invariant subspace generated by vacuum as a cyclic vector

quantum lie algebras associated to and quantum lie algebras are non associative algebras which are embedded into the quantized enveloping algebras of drinfeld and jimbo in the same way as ordinary lie algebras are embedded into their enveloping algebras the quantum lie product on is induced by the quantum adjoint action of we construct the quantum lie algebras associated to and we determine the structure constants and the quantum root systems which are now functions of the quantum parameter q they exhibit an interesting duality symmetry under

the pattern calculus for tensor operators in quantum groups an explicit algebraic evaluation is given for all q tensor operators with unit norm belonging to the quantum group uq u n and having extremal operator shift patterns acting on arbitrary uq u n irreps these rather complicated results are shown to be easily comprehensible in terms of a diagrammatic calculus of patterns a more conceptual derivation of these results is discussed using fundamental properties of the q 6j coefficients

exact phase retrieval in principal shift invariant spaces we address the problem of phase retrieval from fourier transform magnitude spectrum for continuous time signals that lie in a shift invariant space spanned by integer shifts of a generator kernel the phase retrieval problem for such signals is formulated as one of reconstructing the combining coefficients in the shift invariant basis expansion we develop sufficient conditions on the coefficients and the bases to guarantee exact phase retrieval by which we mean reconstruction up to a global phase factor we present a new class of discrete domain signals that are not necessarily minimum phase but allow for exact phase retrieval from their fourier magnitude spectra we also establish hilbert transform relations between log magnitude and phase spectra for this class of discrete signals it turns out that the corresponding continuous domain counterparts need not satisfy a hilbert transform relation notwithstanding the continuous domain signals can be reconstructed from their fourier magnitude spectra we validate the reconstruction guarantees through simulations for some important classes of signals such as bandlimited signals and piecewise smooth signals we also present an application of the proposed phase retrieval technique for artifact free signal reconstruction in frequency domain optical coherence tomography fdoct

hidden quantum group symmetry and integrable perturbations of conformal field theories the hidden quantum group symmetry in the quantum sine gordon model is found this symmetry provides the possibility to restrict the operator algebra of the model to subalgebras it is shown that these subalgebras are massive deformations of minimal conformal field theories

sparse phase retrieval uniqueness guarantees and recovery algorithms the problem of signal recovery from its fourier transform magnitude is of paramount importance in various fields of engineering and has been around for more than 100 years due to the absence of phase information some form of additional information is required in order to be able to uniquely identify the signal of interest in this paper we focus our attention on discrete time sparse signals of length n we first show that if the discrete fourier transform dimension is greater than or equal to 2n then almost all signals with aperiodic support can be uniquely identified by their fourier transform magnitude up to time shift conjugate flip and global phase then we develop an efficient two stage sparse phase retrieval algorithm tspr which involves identifying the support i e the locations of the nonzero components of the signal using a combinatorial algorithm and identifying the signal values in the support using a convex algorithm we show that tspr can provably recover most o n 1 2 epsilon sparse signals up to a time shift conjugate flip and global phase we also show that for most o n 1 4 epsilon sparse signals the recovery is robust in the presence of measurement noise these recovery guarantees are asymptotic in nature numerical experiments complement our theoretical analysis and verify the effectiveness of tspr

induced module construction for highest weight representations of uq gl n at roots of unity a method is investigated for inducing highest weight representations for the quantum group uq gl n from the canonical subalgebra uq gl n 1 when q is a root of unity we classify the irreps into two types typical and atypical where the former is a generalization of the class of irreps with maximal dimensionality the structures of both the typical and atypical irreps are studied in particular a sufficiency condition is given for an irrep to be typical as examples we consider flat representations induced from a one dimensional representation of the canonical subalgebra and representations induced from vector representation

on the universalr matrix of u_q widehat sl _2 at roots of unityat roots of unity we show that the action of the universalr matrix of the affine u q widehat sl 2 quantum algebra whenq is a root of unity can be renormalized by some scalar factor to give a well defined nonsingular expression satisfying the yang baxter equation it can be reduced to intertwining operators of representations corresponding to chiral potts if the parameters of these representations lie on the well known algebraic curve

large deviation estimates in the stochastic quantization ofϕ 2 4 we study in the small noise limit the behaviour of field trajectories for the process constructed by the authors in connection with the stochastic quantization of 2 4 due to the presence of infinite renormalization the usual large deviation techniques do not apply immediately and a new strategy has to be developed we prove some estimates analogous to the freidlin ventzel inequalities from these it follows that the field trajectories suitably smeared in space over a scaler0 behave when the noise is small as the projection on the same scale of a field obeying a regularized stochastic equation with a large cut off however the estimates are not uniform in the cut off and an interesting feature of the problem is that the scale over which the field is smeared determines whether the noise is sufficiently small for the estimates to apply

stochastic quantization of field theory in finite and infinite volume abstract using the theory of dirichlet forms a distribution valued symmetric markov process is constructed so that it has the 4 2 measure as its invariant probability measure several properties of this process are proved and a finite to infinite volume limit theorem is established

special representations of at the roots of unity we show how to adapt the gelfand zetlin basis for describing the special representations of when q is a root of unity the explicit construction of special representations is presented in detail for n 3

generalized chiral potts models and minimal cyclic representations of u_q widehat germ g germ l n bf c we present for odd n a construction of the n 1 state generalization of the chiral potts model proposed recently by bazhanov et al the yang baxter equation is proved

on the matrix elements of the u n generators a straightforward derivation of the matrix elements of the u n generators is presented using algebraic infinitesimal techniques an expression for the general fundamental wigner coefficients of the group is obtained as a polynomial in the group generators this enables generalized matrix elements to be defined without explicit reference to basis states such considerations are important for treating groups such as sp 2n whose basis states are not known

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

verma module of the quantum group gl n q and its q boson and heisenberg weyl relation realizations the structure and verma module of the quantum matrix element algebra a n q of the quantum group gl n q are studied the method used here is similar to that for studying the structure and verma module of semisimple lie algebras the concept of the cartan subalgebra raising and lowering operators and their pairs is defined the q boson and the corresponding heisenberg weyl relation realizations are generally studied and the cyclic representations of a n q are obtained the explicit examples a 2 q and a 3 q are discussed in detail

large deviations for the macroscopic motion of an interface we study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of ising spins with kac interaction evolving in time according to glauber non conservative dynamics such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions we derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit error terms which are valid also in the macroscopic scale furthermore using the result of a companion paper about the minimizers of this cost functional for the macroscopic motion of the interface in a fixed time we prove that the probability of such events can concentrate on nucleations should the transition happen fast enough

normalized uq sl 3 gel fand weyl zetlin basis and new summation formulas for q hypergeometric functions an explicit realization of the normalized gel fand weyl zetlin gwz basis for uq sl 3 in terms of polynomial functions in three variables real or complex is given the construction uses two different realizations of the uq sl 3 unnormalized gwz basis which were given previously and whose transformation properties were not known it turns out that finding these properties enables us to find the gwz dependent proportionality constant between these two realizations the scalar product is also fixed by this in both unnormalized realizations and then by normalization the normalized gwz states are obtained as by products new summation formulas are obtained which seem new also for q 1 the main new formula is a double sum which is given in terms of the proportionality constant mentioned above this double sum can be written as single sum over a q 3f2 hypergeometric function or as a q hypergeometric function of two variables

reduced wigner coefficients for uq gl n this is the second paper in a series concerned with the racah wigner calculus for the quantum group uq gl n in this approach all reduced wigner coefficients are determined algebraically by the eigenvalues of certain generalized casimir invariants here the method is applied to evaluate all fundamental wigner coefficients and reduced wigner coefficients for uq gl n

the fundamental invariant of the hecke algebra hn q characterizes the representations of hn q sn suq n and su n the irreducible representations irreps of the hecke algebra hn q are shown to be completely characterized by the fundamental invariant of this algebra cn this fundamental invariant is related to the quadratic casimir operator c2 of suq n and reduces to the transposition class sum 2 n of sn when q 1 the projection operators constructed in terms of cn for the various irreps of hn q are well behaved in the limit q 1 even when approaching degenerate eigenvalues of 2 n in the latter case for which the irreps of sn are not fully characterized by the corresponding eigenvalue of the transposition class sum the limiting form of the projection operator constructed in terms of cn gives rise to factors that depend on higher class sums of sn which effect the desired characterization expanding this limiting form of the projection operator into a linear combination of class sums of sn the coefficients constitute the corresponding row in the character table of sn the properties of the funda

tunneling in two dimensions tunneling is studied here as a variational problem formulated in terms of a functional which approximates the rate function for large deviations in ising systems with glauber dynamics and kac potentials 9 the spatial domain is a two dimensional square of side l with reflecting boundary conditions for l large enough the penalty for tunneling from the minus to the plus equilibrium states is determined minimizing sequences are fully characterized and shown to have approximately a planar symmetry at all times thus departing from the wulff shape in the initial and final stages of the tunneling in a final section sect 11 we extend the results to d 3 but their validity in d 3 is still open

invariants and matrix elements of the quantum group u_q gl n mathbb c revisited in a previous paper the generator matrix elements and dual vector reduced wigner coefficients rwcs were evaluated via the polynomial identities satisfied by a certain matrix constructed from the r matrix r and its twisted counterpart r t t circle r here we provide an alternative evaluation utilising the r matrix r over tilde r t 1 this provides a new direct derivation of the vector rwcs obtained indirectly in earlier work via a symmetry relation this approach has the advantage that it generalises to the lie superalgebra case which will be investigated elsewhere

so 5 q and contraction chevalley basis representations for q generic and root of unity representations of so 5 q are constructed explicitly on the chevalley basis the classical representations once obtained on this basis are very simply deformed for arbitrary q generic and root of unity this is in sharp contrast to the classical canonical gelfand zetlin gz representations which lead to problems studied elsewhere in q deformations for orthogonal though not for unitary algebras our results are limited to irreducible representations labeled by three variable parameters the maximal number being four within this restriction it is shown how to construct all representations for all q s the contraction of the chevalley generators e2 f2 leads to an easily obtained complete hopf algebra the structure of the contracted algebra is quite different even at the classical level from the familiar e 4 the four dimensional euclidean algebra obtained on contracting the generator j45 in the gz formalism the contracted representations are given the so 5 q representations have their own

classification of finite dimensional unitary irreps for uq gl m n all finite dimensional irreducible unitary representations of the quantum supergroup uq gl m n are classified in terms of their highest weights for generic q 0

the q boson realizations of the quantum group uq sl n 1 c the authors give an explicit expression of the recurrency formulae of canonical realization for quantum enveloping algebras uq sl n 1 c in these formulae the generators of the algebra uq sl n 1 c are expressed by means of n canonical q boson pairs one auxiliary representation of the algebra uq gl n c

on unbiased stochastic navier stokes equations a random perturbation of a deterministic navier stokes equation is considered in the form of an spde with wick type nonlinearity the nonlinear term of the perturbation can be characterized as the highest stochastic order approximation of the original nonlinear term u nabla u this perturbation is unbiased in that the expectation of a solution of the perturbed equation solves the deterministic navier stokes equation the perturbed equation is solved in the space of generalized stochastic processes using the cameron martin version of the wiener chaos expansion it is shown that the generalized solution is a markov process and scales effectively by catalan numbers

on quantum lie algebras and quantum root systems as a natural generalization of ordinary lie algebras we introduce the concept of quantum lie algebras we define these in terms of certain adjoint submodules of quantized enveloping algebras endowed with a quantum lie bracket given by the quantum adjoint action the structure constants of these algebras depend on the quantum deformation parameter q and they go over into the usual lie algebras when q 1 the notions of q conjugation and q linearity are introduced q linear analogues of the classical antipode and cartan involution are defined and a generalized killing form q linear in the first entry and linear in the second is obtained these structures allow the derivation of symmetries between the structure constants of quantum lie algebras the explicitly worked out examples of and illustrate the results

a classification of unitary highest weight modules of the quantum analogue of the symmetric pair an an 1 abstract we study the irreducibility and unitarity of highest weight modules of the quantized enveloping algebra associated to s u n 1 we obtain a classification of these modules analogous to that of ehw in the classical context we also compute all commutation relations between root vectors in u q s u n 1

solving systems of random quadratic equations via truncated amplitude flow this paper presents a new algorithm termed truncated amplitude flow taf to recover an unknown vector x from a system of quadratic equations of the form y i langle a i x rangle 2 where a i s are given random measurement vectors this problem is known to be np hard in general we prove that as soon as the number of equations is on the order of the number of unknowns taf recovers the solution exactly up to a global unimodular constant with high probability and complexity growing linearly with both the number of unknowns and the number of equations our taf approach adapts the amplitude based empirical loss function and proceeds in two stages in the first stage we introduce an orthogonality promoting initialization that can be obtained with a few power iterations stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations which are able to handle the rather challenging nonconvex and nonsmooth amplitude based objective function in particular when vectors x and a i s are real valued our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations furthermore even under the same initialization the proposed amplitude based refinement outperforms existing wirtinger flow variants corroborating the superior performance of taf over state of the art algorithms

on a class of stochastic reaction diffusion equations in two space dimensions the authors prove for any finite time the existence of a weak dynamics for a class of reaction diffusion systems in two space dimensions perturbed by a white noise this class includes models which may exhibit complex behaviour in space and time their approach based on the methods of constructive quantum field theory extends previous results established for gradient type systems

study of stochastic differential equations by constructive methods i in this work we give an algorithm to express as a convergent series the stationary averages for a class of gradient perturbations of a nonsymmetric nongradient ornstein uhlenbeck process the method relies on a cluster expansion in time of the girsanov cameron martin formula for the density of the perturbed measure with respect to the ornstein uhlenbeck measure in the second paper of this series the approach is extended to more general perturbations

at roots of unity the quantum group at roots of unity is introduced by means of duality pairings with the quantum algebra its irreducible representations corresponding to the type of the quantum algebra are constructed through the universal t matrix the irreducible representations corresponding to the type are also found an invariant integral on this quantum group is given endowed with that some properties like unitarity and orthogonality of the irreducible representations are discussed

flat periodic representations ofu q g we give explicit expression of flat periodic representations when they exist of the quantum analogues of simple lie algebras and their affine extensions for a parameter of deformationq equal to a root of unity by flat periodic we mean that these representations have no highest weight and that all the weights have multiplicity 1

q analogs of iu n and u n 1 the starting point is the nonsemisimple inhomogeneous lie algebra un i2n denoted also as iu n where i2n represents an abelian subalgebra in semidirect product with the homogeneous part u n this is realized by explicitly giving the matrix elements of the generators on a modified gelfand zetlin basis that allows representations of infinite dimensions the enveloping algebra is q lifted by introducing q brackets in the matrix elements giving uq iu n the deformation of the abelian structure of i2n is studied for q 1 some implications are pointed out the important invariants are constructed for arbitrary n the results are compared to the corresponding ones for jimbo s construction of uq u n 1 on a gelfeld zetlin basis finally the related construction of uq u n 1 is presented and discussed here uq su 1 1 the q analog of relativistic motion in a plane is analyzed in the context of this formalism

irregular uq sl 3 representations at roots of unity via gel fand weyl zetlin basis the gel fand weyl zetlin gwz description of the uq sl 3 irregular irreps at roots of unity is explicitly given those are irreps fixed by the same parameters as the unitary irreducible representations uirs of su 3 yet having dimensions smaller than their classical counterparts the reason being that to obtain an irregular irrep one has to make factorization of an additional submodule this description is made geometrically transparent by an arrangement of the standard su 3 gwz basis in a hexagonal pyramid which is valid for any q and seems new even for q 1 the pyramid has as a base the standard hexagon which gives the weight space of the uirs of su 3 in the plane of third component of isospin iz and hypercharge y while third dimension of this pyramid is related to the isospin i algebraically this arrangement is related to a one to one correspondence between the abstract gwz states and monomials in the algebra of raising generators uq g however those monomials are not in the standard poi

phaselift exact and stable signal recovery from magnitude measurements via convex programming suppose we wish to recover a signal input amssym font abc cmmib10 def bi 1 hbox abc 1 bi x in bbb c n from m intensity measurements of the form that is from data in which phase information is missing we prove that if the vectors are sampled independently and uniformly at random on the unit sphere then the signal x can be recovered exactly up to a global phase factor by solving a convenient semidefinite program a trace norm minimization problem this holds with large probability provided that m is on the order of and without any assumption about the signal whatsoever this novel result demonstrates that in some instances the combinatorial phase retrieval problem can be solved by convex programming techniques finally we also prove that our methodology is robust vis a vis additive noise 2012 wiley periodicals inc