relative entropy optimization and its applications in this expository article we study optimization problems specified via linear and relative entropy inequalities such relative entropy programs reps are convex optimization problems as the relative entropy function is jointly convex with respect to both its arguments prominent families of convex programs such as geometric programs gps second order cone programs and entropy maximization problems are special cases of reps although reps are more general than these classes of problems we provide solutions based on reps to a range of problems such as permanent maximization robust optimization formulations of gps and hitting time estimation in dynamical systems we survey previous approaches to some of these problems and the limitations of those methods and we highlight the more powerful generalizations afforded by reps we conclude with a discussion of quantum analogs of the relative entropy function including a review of the similarities and distinctions with respect to the classical case we also describe a stylized application of quantum relative entropy optimization that exploits the joint convexity of the quantum relative entropy function

convex inner approximations of nonconvex semialgebraic sets applied to fixed order controller design we describe an elementary algorithm to build convex inner approximations of nonconvex sets both input and output sets are basic semialgebraic sets given as lists of defining multivariate polynomials even though no optimality guarantees can be given e g in terms of volume maximisation for bounded sets the algorithm is designed to preserve convex boundaries as much as possible while removing regions with concave boundaries in particular the algorithm leaves invariant a given convex set the algorithm is based on gloptipoly 3 a public domain matlab package solving nonconvex polynomial optimisation problems with the help of convex semidefinite programming optimisation over linear matrix inequalities or lmis we illustrate how the algorithm can be used to design fixed order controllers for linear systems following a polynomial approach

small and strong formulations for unions of convex sets from the cayley embedding there is often a significant trade off between formulation strength and size in mixed integer programming when modeling convex disjunctive constraints e g unions of convex sets adding auxiliary continuous variables can sometimes help resolve this trade off however standard formulations that use such auxiliary continuous variables can have a worse than expected computational effectiveness which is often attributed precisely to these auxiliary continuous variables for this reason there has been considerable interest in constructing strong formulations that do not use continuous auxiliary variables we introduce a technique to construct formulations without these detrimental continuous auxiliary variables to develop this technique we introduce a natural non polyhedral generalization of the cayley embedding of a family of polytopes and show it inherits many geometric properties of the original embedding we then show how the associated formulation technique can be used to construct small and strong formulation for a wide range of disjunctive constraints in particular we show it can recover and generalize all known strong formulations without continuous auxiliary variables

sufficient and necessary conditions for semidefinite representability of convex hulls and sets a set s subseteq mathbb r n is called semidefinite programming sdp representable or semidefinite representable if s equals the projection of a set in higher dimensional space which is describable by some linear matrix inequality lmi clearly if s is sdp representable then s must be convex and semialgebraic it is describable by conjunctions and disjunctions of polynomial equalities or inequalities this paper proves sufficient conditions and necessary conditions for sdp representability of convex sets and convex hulls by proposing a new approach to constructing sdp representations the contributions of this paper are i for bounded sdp representable sets w 1 dots w m we give an explicit construction of an sdp representation for mathrm conv cup k 1 mw k this provides a technique for building global sdp representations from the local ones ii for the sdp representability of a compact convex semialgebraic set s we prove sufficient the boundary partial s is nonsingular and positively curved while necessary is partial s has nonnegative curvature at each nonsingular point in terms of defining polynomials for s nonsingular boundary amounts to them having nonvanishing gradient at each point on partial s and the curvature condition can be expressed as their strict versus nonstrict quasi concavity at those points on partial s where they vanish the gaps between them are partial s having or not having singular points either of the gradient or of the curvature s positivity a sufficient condition bypassing the gaps is when some defining polynomials of s satisfy an algebraic condition called sos concavity iii for the sdp representability of the convex hull of a compact nonconvex semialgebraic set t we find that the critical object is partial c t the maximum subset of partial t contained in partial mathrm conv t we prove sufficient for sdp representability partial c t is nonsingular and positively curved and necessary is partial c t has nonnegative curvature at nonsingular points the gaps between our sufficient and necessary conditions are similar to case ii the positive definite lagrange hessian pdlh condition which meshes well with constructions is also discussed

convex matrix inequalities versus linear matrix inequalities most linear control problems lead directly to matrix inequalities mis many of these are badly behaved but a classical core of problems are expressible as linear matrix inequalities lmis in many engineering systems problems convexity has all of the advantages of a lmi since lmis have a structure which is seemingly much more rigid than convex mis there is the hope that a convexity based theory will be less restrictive than lmis how much more restrictive are lmis than convex mis there are two fundamentally different classes of linear systems problems dimension free and dimension dependent a dimension free mi is a mi where the unknowns are g tuples of matrices and appear in the formulas in a manner which respects matrix multiplication most of the classic mis of control theory are dimension free dimension dependent mis have unknowns which are tuples of numbers the two classes behave very differently and this survey describes what is known in each case about the relation between convex mis and lmis the proof techniques involve and necessitate new developments in the field of semialgebraic geometry

dynamic and spatial behavior of a corrugated interface in the driven lattice gas model abstract the spatiotemporal behavior of an initially corrugated interface in the two dimensional driven lattice gas dlg model with attractive nearest neighbors interactions is investigated via monte carlo simulations by setting the system in the ordered phase with periodic boundary conditions along the external field axis i e horizontal and open along the vertical directions respectively an initial interface was imposed that consists in a series of sinusoidal profiles with amplitude a 0 and wavelength set parallel to the applied driving field axis we studied the dynamic behavior of its statistical width or roughness w t defined as the root mean square of the interface position we found that w t decays exponentially for all and lattice longitudinal sizes l x i e the lattice side that runs along the axis of the external field we determined its relaxation time and found that depends on as a power law p where p depends on the temperature and l x at low t s t t c e and large l x p approaches to p 3 2 at intermediate t s t t c e p decreases up to p 1 and is free of finite effects this indicates that the interface stabilizes faster than in the equilibrium model i e the ising lattice gas e 0 where p 3 at higher t s p increases for t t c e and the finite size dependence is recovered also if t is fixed p increases with l x until it saturates at large values of it while this regime is vanishing at t t c e in this way the dynamic relaxation process of a sinusoidal interface is improved by the external driving field with respect to its equilibrium counterpart if the system is set in an intermediate temperature stage far from t c e and in a lattice with a sufficiently large longitudinal side the behavior of was also investigated as a function of e and in the intermediate stage t t c e it was found that decreases exponentially with e in the interval 0 e 1 while for higher fields it remains constant the exponential decay depends on the wavelength of the initial profile in order to study the spatial evolution of the profiles we evaluated the structure factor of the interface and the fourier coefficients corresponding to the same wave vector of the initial profile the obtained results allowed us to conclude that the spatial evolution of the profile maintains its initial wavelength does not travel along the external field axis and its shape is preserved over all the relaxation process

quantum learning of classical stochastic processes the completely positive realization problem among several tasks in machine learning a specially important one is the problem of inferring the latent variables of a system and their causal relations with the observed behavior a paradigmatic instance of this is the task of inferring the hidden markov model underlying a given stochastic process this is known as the positive realization problem prp l benvenuti and l farina ieee trans autom control 49 5 651 664 2004 and constitutes a central problem in machine learning the prp and its solutions have far reaching consequences in many areas of systems and control theory and is nowadays an important piece in the broad field of positive systems theory we consider the scenario where the latent variables are quantum i e quantum states of a finite dimensional system and the system dynamics is constrained only by physical transformations on the quantum system the observable dynamics is then described by a quantum instrument and the task is to determine which quantum instrument if any yields the process at hand by iterative application we take as a starting point the theory of quasi realizations whence a description of the dynamics of the process is given in terms of linear maps on state vectors and probabilities are given by linear functionals on the state vectors this description despite its remarkable resemblance with the hidden markov model or the iterated quantum instrument is however devoid of any stochastic or quantum mechanical interpretation as said maps fail to satisfy any positivity conditions the completely positive realization problem then consists in determining whether an equivalent quantum mechanical description of the same process exists we generalize some key results of stochastic realization theory and show that the problem has deep connections with operator systems theory giving possible insight to the lifting problem in quotient operator systems our results have potential applications in quantum machine learning device independent characterization and reverse engineering of stochastic processes and quantum processors and more generally of dynamical processes with quantum memory m gu phys rev a 83 6 062324 2011 m gu and n yamamoto e print arxiv 1303 3771 2013

the sharp weighted bound for general calderon zygmund operators for a general calderon zygmund operator t on r n it is shown that ktfkl2 w c t sup q a q w q w 1 a k fkl2 w for all muckenhoupt weights w 2 a2 this optimal estimate was known as the a2 conjecture a recent result of perez treil volberg reduced the problem to a testing condition on indicator functions which is verified in this paper the proof consists of the following elements i a variant of the nazarov treil volberg method of random dyadic systems with just one random system and completely without bad parts ii a resulting representation of a general calderon zygmund operator as an average of dyadic shifts and iii improvements of the lacey petermichl reguera estimates for these dyadic shifts which allow summing up the series in the obtained representation

linear matrix inequality representation of sets this article concerns the question which subsets of m can be represented with linear matrix inequalities lmis this gives some perspective on the scope and limitations of one of the most powerful techniques commonly used in control theory also before having much hope of representing engineering problems as lmis by automatic methods one needs a good idea of which problems can and cannot be represented by lmis little is currently known about such problems in this article we give a necessary condition that we call rigid convexity which must hold for a set m in order for to have an lmi representation rigid convexity is proved to be necessary and sufficient when m 2 this settles a question formally stated by pablo parrilo and berndt sturmfels in 15 as shown by lewis parillo and ramana 11 our main result also establishes in the case of three variables a 1958 conjecture by peter lax on hyperbolic polynomials 2006 wiley periodicals inc

first order conditions for semidefinite representations of convex sets defined by rational or singular polynomials a set is called semidefinite representable or semidefinite programming sdp representable if it equals the projection of a higher dimensional set which is defined by some linear matrix inequality lmi this paper discusses the semidefinite representability conditions for convex sets of the form s mathcal d f x in mathcal d f x geq 0 here mathcal d x in mathbb r n g 1 x geq 0 ldots g m x geq 0 is a convex domain defined by some nice concave polynomials g i x they satisfy certain concavity certificates and f x is a polynomial or rational function when f x is concave over mathcal d we prove that s mathcal d f has some explicit semidefinite representations under certain conditions called preordering concavity or q module concavity which are based on the positivstellensatz certificates for the first order concavity criteria f u nabla f u t x u f x geq 0 quad forall x u in mathcal d when f x is a polynomial or rational function having singularities on the boundary of s mathcal d f a perspective transformation is introduced to find some explicit semidefinite representations for s mathcal d f under certain conditions in the special case n 2 if the laurent expansion of f x around one singular point has only two consecutive homogeneous parts we show that s mathcal d f always admits an explicitly constructible semidefinite representation

mixed weak estimates of sawyer type for fractional integrals and some related operators abstract we prove mixed weak estimates of sawyer type for fractional operators more precisely let t be either the maximal fractional function m or the fractional integral operator i 0 n 1 p n and 1 q 1 p n if u v q p a 1 or if u v q p a 1 and v q a u v q p then we obtain that the estimate u v q p x r n t f v x v x t 1 q c t r n f x p u x p q v x d x 1 p holds for every positive t and every bounded function with compact support as an important application of the results above we furthermore exhibit mixed weak estimates for commutators of calderon zygmund singular integral and fractional integral operators when the symbol b is in the lipschitz class 0 1

proof of an extension of e sawyer s conjecture about weighted mixed weak type estimates we show that if v in a infty and u in a 1 then there is a constant c depending on the a 1 constant of u and the a infty constant of v such that begin aligned left frac t fv v right l 1 infty uv le c vert f vert l 1 uv end aligned where t can be the hardy littlewood maximal function or any calderon zygmund operator this result was conjectured in cruz uribe et al int math res not 30 1849 1871 2005 and constitutes the most singular case of some extensions of several problems proposed by sawyer and muckenhoupt and wheeden we also improve and extends several quantitative estimates

semidefinite representation of convex sets let s x in mathbb r n g 1 x geq 0 ldots g m x geq 0 be a semialgebraic set defined by multivariate polynomials g i x assume s is convex compact and has nonempty interior let s i x in mathbb r n g i x geq 0 and s resp s i be the boundary of s resp s i this paper as does the subject of semidefinite programming sdp concerns linear matrix inequalities lmis the set s is said to have an lmi representation if it equals the set of solutions to some lmi and it is known that some convex s may not be lmi representable helton and vinnikov in commun pure appl math 60 5 654 674 2007 a question arising from nesterov and nemirovski siam studies in applied mathematics society for industrial and applied mathematics siam philadelphia 1994 see helton and vinnikov in commun pure appl math 60 5 654 674 2007 and nemirovski in plenary lecture international congress of mathematicians icm madrid spain 2006 is given a subset s of mathbb r n does there exist an lmi representable set in some higher dimensional space mathbb r n n whose projection down onto mathbb r n equals s such s is called semidefinite representable or sdp representable this paper addresses the sdp representability problem the following are the main contributions of this paper i assume g i x are all concave on s if the positive definite lagrange hessian condition holds i e the hessian of the lagrange function for optimization problem of minimizing any nonzero linear function l t x on s is positive definite at the minimizer then s is sdp representable ii if each g i x is either sos concave 2 g i x w x t w x for some possibly nonsquare matrix polynomial w x or strictly quasi concave on s then s is sdp representable iii if each s i is either sos convex or poscurv convex s i is compact convex whose boundary has positive curvature and is nonsingular i e g i x 0 on s i s then s is sdp representable this also holds for s i for which s i s extends smoothly to the boundary of a poscurv convex set containing s iv we give the complexity of schmudgen and putinar s matrix positivstellensatz which are critical to the proofs of i iii

determinantal representations and bezoutians a major open question in convex algebraic geometry is whether all hyperbolicity cones are spectrahedral i e the solution sets of linear matrix inequalities we will use sum of squares decompositions of certain bilinear forms called bezoutians to approach this problem more precisely we show that for every smooth hyperbolic polynomial h there is another hyperbolic polynomial q such that q cdot h has a definite determinantal representation besides commutative algebra the proof relies on results from real algebraic geometry

on semidefinite representations of non closed sets abstract spectrahedra are sets defined by linear matrix inequalities projections of spectrahedra are called semidefinitely representable sets both kinds of sets are of practical use in polynomial optimization since they occur as feasible sets in semidefinite programming there are several recent results on the question which sets are semidefinitely representable so far all results focus on the case of closed sets in this work we develop a new method to prove semidefinite representability of sets which are not closed for example the interior of a semidefinitely representable set is shown to be semidefinitely representable more general one can remove faces of a semidefinitely representable set and preserve semidefinite representability as long as the faces are parametrized in a suitable way

exposed faces of semidefinitely representable sets a linear matrix inequality lmi is a condition stating that a symmetric matrix whose entries are affine linear combinations of variables is positive semidefinite motivated by the fact that diagonal lmis define polyhedra the solution set of an lmi is called a spectrahedron linear images of spectrahedra are called semidefinitely representable sets part of the interest in spectrahedra and semidefinitely representable sets arises from the fact that one can efficiently optimize linear functions on them by semidefinite programming such as one can do on polyhedra by linear programming it is known that every face of a spectrahedron is exposed this is also true in the general context of rigidly convex sets we study the same question for semidefinitely representable sets lasserre proposed a moment matrix method to construct semidefinite representations for certain sets our main result is that this method can work only if all faces of the considered set are exposed this necessary condition complements sufficient conditions recently proved by lasserre helton and nie

positive polynomials and projections of spectrahedra this article is concerned with different aspects of spectrahedra and their projections sets that are important in semidefinite optimization we prove results on the limitations of so called lasserre and theta body relaxation methods for semialgebraic sets and varieties as a special case we obtain the main result of netzer plaumann and schweighofer siam j optim 20 2010 pp 1944 1955 on nonexposed faces we also solve the open problems from that work we further give a unified account of several results on convex hulls of curves and images of polynomial maps we finally prove a positivstellensatz for projections of spectrahedra which exceeds the known results that only work for basic closed semialgebraic sets

convexity in semialgebraic geometry and polynomial optimization we review several and provide new results on the theory of moments sums of squares and basic semialgebraic sets when convexity is present in particular we show that under convexity the hierarchy of semidefinite relaxations for polynomial optimization simplifies and has finite convergence a highly desirable feature as convex problems are in principle easier to solve in addition if a basic semialgebraic set mathbf k is convex but its defining polynomials are not we provide two algebraic certificates of convexity which can be checked numerically the second is simpler and holds if a sufficient and almost necessary condition is satisfied it also provides a new condition for mathbf k to have semidefinite representation for this we use and extend some of the recent results from the author and helton and nie math program to appear finally we show that when restricting to a certain class of convex polynomials the celebrated jensen s inequality in convex analysis can be extended to linear functionals that are not necessarily probability measures

field theory of critical behavior in a driven diffusive system we use a field theoretic renormalization group method to study the critical properties of a diffusive system with a single conserved density subject to a constant uniform external field a fixed point stable belowd c 5 is found to govern the critical behavior scaling forms of density correlation functions are derived and critical exponents are obtained to all orders in 5 d spatial correlations are found to be very anisotropic with elongated correlations along the external field long wavelength transverse fluctuations are suppressed completely to yield mean field transverse exponents

linear optimization with cones of moments and nonnegative polynomials let mathcal a a be a finite subset of mathbb n n nn and mathbb r x mathcal a r x a be the space spanned by monomials x alpha x with alpha in mathcal a a let k k be a compact semialgebraic set of mathbb r n rn such that a polynomial in mathbb r x mathcal a r x a is positive on k k denote by fancyscript p mathcal a k pa k the cone of polynomials in mathbb r x mathcal a r x a that are nonnegative on k k the dual cone of fancyscript p mathcal a k pa k is fancyscript r mathcal a k ra k the set of all truncated moment sequences in mathbb r mathcal a ra that admit representing measures supported in k k first we study geometric properties of the cones fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k like interiors closeness duality memberships and construct a convergent hierarchy of semidefinite relaxations for each of them second we propose a semidefinite algorithm for solving linear optimization problems with the cones fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k and prove its asymptotic and finite convergence third we show how to check whether fancyscript p mathcal a k pa k and fancyscript r mathcal a k ra k intersect affine subspaces if they do we show how to get a point in the intersections if they do not we prove certificates for the empty intersection

endpoint estimates for commutators of singular integral operators abstract we prove endpoint estimates for commutatotrs of singular integrals with bmo functions we first show that they satisfy l log l type inequalities and then that there is a hardy space type estimate where the usual atomic hardy space is replaced by an appropriate subspace

convex hulls of curves of genus one abstract let c be a real nonsingular affine curve of genus one embedded in affine n space whose set of real points is compact for any polynomial f which is nonnegative on c r we prove that there exist polynomials f i with f i f i 2 mod i c and such that the degrees deg f i are bounded in terms of deg f only using lasserre s relaxation method we deduce an explicit representation of the convex hull of c r in r n by a lifted linear matrix inequality this is the first instance in the literature where such a representation is given for the convex hull of a nonrational variety the same works for convex hulls of singular curves whose normalization is c we then make a detailed study of the associated degree bounds these bounds are directly related to size and dimension of the projected matrix pencils in particular we prove that these bounds tend to infinity when the curve c degenerates suitably into a singular curve and we provide explicit lower bounds as well

on pointwise and weighted estimates for commutators of calderon zygmund operators abstract in recent years it has been well understood that a calderon zygmund operator t is pointwise controlled by a finite number of dyadic operators of a very simple structure called the sparse operators we obtain a similar pointwise estimate for the commutator b t with a locally integrable function b this result is applied into two directions if b b m o we improve several weighted weak type bounds for b t if b belongs to the weighted bmo we obtain a quantitative form of the two weighted bound for b t due to bloom holmes lacey wick

semidefinite descriptions of low dimensional separable matrix cones abstract let k e k e be convex cones residing in finite dimensional real vector spaces an element y in the tensor product e e is k k separable if it can be represented as finite sum y l x l x l where x l k and x l k for all l let s n h n q n be the spaces of n n real symmetric complex hermitian and quaternionic hermitian matrices respectively let further s n h n q n be the cones of positive semidefinite matrices in these spaces if a matrix a h mn h m h n is h m h n separable then it fulfills also the so called ppt condition i e it is positive semidefinite and has a positive semidefinite partial transpose the same implication holds for matrices in the spaces s m s n h m s n and for m 2 in the space q m s n we provide a complete enumeration of all pairs n m when the inverse implication is also true for each of the above spaces i e the ppt condition is sufficient for separability we also show that a matrix in q n s 2 is q n s 2 separable if and only if it is positive semidefinite

phase segregation dynamics in particle systems with long range interactions i macroscopic limits we present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter r t of a binary alloy undergoing phase segregation our model is ad dimensional lattice gas evolving via kawasaki exchange with respect to the gibbs measure for a hamiltonian which includes both short range local and long range nonlocal interactions the nonlocal part is given by a pair potential dj x y 0 x and y in d in the limit 0 the macroscopic evolution is observed on the spatial scale 1 and time scale 2 i e the density r t is the empirical average of the occupation numbers over a small macroscopic volume element centered atr x a rigorous derivation is presented in the case in which there is no local interaction in a subsequent paper part ii we discuss the phase segregation phenomena in the model in particular we argue that the phase boundary evolutions arising as sharp interface limits of the family of equations derived in this paper are the same as the ones obtained from the corresponding limits for the cahn hilliard equation

phase separation dynamics in driven diffusive systems we study phase separation dynamics in a driven diffusive system our simulations are based on the cahn hilliard equation with an additional flux term due to an external field we study the dynamical scaling parallel and perpendicular to the field a crossover is observed from isotropic domains at early times to extremely anisotropic domains at later times we find that the inverse interfacial density an isotropic measure of the domain size increases ast with 1 3 from early times independent of the field strength even though we do not observe dynamical scaling during these times our results indicate that a growth exponent 1 3 may be more universal than previously expected we analyze the dynamics in terms of surface driven instabilities and one dimensional solitary waves

a sparse approach to mixed weak type inequalities in this paper we provide some quantitative mixed weak type estimates assuming conditions that imply that uv in a infty for calderon zygmund operators rough singular integrals and commutators the main novelty of this paper lies in the fact that we rely upon sparse domination results pushing an approach to endpoint estimates that was introduced in domingo salazar et al bull lond math soc 48 1 63 73 2016 and extended in lerner et al adv math 319 153 181 2017 and li et al j geom anal 2018

classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems necessary and sufficient conditions for convexity and strong convexity respectively of connected sublevel sets that are defined by finitely many real valued c 1 1 maps are presented a novel characterization of strongly convex sets in terms of the so called local quadratic support is proved the results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous time nonlinear systems to be strongly convex an application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present

free semidefinite representation of matrix power functions abstract consider the matrix power function x p defined over the cone of positive definite matrices s n it is known that x p is convex over s n if p 1 0 1 2 and x p is concave over s n if p 0 1 we show that the hypograph of x p admits a free semidefinite representation if p 0 1 is rational and the epigraph of x p admits a free semidefinite representation if p 1 0 1 2 is rational

front migration in the nonlinear cahn hilliard equation the method of matched asymptotic expansions is used to describe solutions of the nonlinear cahn hilliard equation for phase separation in n 1 space dimensions the expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature on the dominant slowest timescale the interface velocity is determined by the mean curvature of the interface by a non local relation which is identical to that in a well known quasi static model of solidification which exhibits a shape instability discovered by mullins sekerka j appl phys 34 323 329 1963 on a faster timescale the cahn hilliard equation regularizes a classic two phase stefan problem similarity solutions of the two phase stefan problem should describe boundary layers existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

detecting rigid convexity of bivariate polynomials abstract given a polynomial x r n p x in n 2 variables a symbolic numerical algorithm is first described for detecting whether the connected component of the plane sublevel set p x p x 0 containing the origin is rigidly convex or equivalently whether it has a linear matrix inequality lmi representation or equivalently if polynomial p x is hyperbolic with respect to the origin the problem boils down to checking whether a univariate polynomial matrix is positive semidefinite an optimization problem that can be solved with eigenvalue decomposition when the variety c x p x 0 is an algebraic curve of genus zero a second algorithm based on bezoutians is proposed to detect whether p has an lmi representation and to build such a representation from a rational parametrization of c finally some extensions to positive genus curves and to the case n 2 are mentioned

the l log l e endpoint estimate for maximal singular integral operators abstract we prove in this paper the following estimate for the maximal operator t associated to the singular integral operator t t f l 1 w 1 r n f x m l log l w x d x w 0 0 1 this follows from the sharp l p estimate t f l p w p 1 1 p f l p m l log l p 1 w 1 p w 0 0 1 as a consequence we deduce that t f l 1 w w a 1 log e w a r n f w d x extending the endpoint results obtained in 23 and 15 to maximal singular integrals another consequence is a quantitative two weight bump estimate

smooth hyperbolicity cones are spectrahedral shadows hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials a well understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral in this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone that is a spectrahedral shadow

polynomial matrix inequality and semidefinite representation consider a convex set s x in cal d g x succeq 0 where g x is a symmetric matrix whose every entry is a polynomial or rational function d rn is a domain on which g x is defined and g x succeq 0 means g x is positive semidefinite the set s is called semidefinite representable if it equals the projection of a higher dimensional set that is defined by a linear matrix inequality lmi this paper studies sufficient conditions guaranteeing semidefinite representability of s we prove that s is semidefinite representable in the following cases i d rn g x is a matrix polynomial and matrix sos concave ii d is compact convex g x is a matrix polynomial and strictly matrix concave on d iii g x is a matrix rational function and q module matrix concave on d explicit constructions of semidefinite representations are given some examples are illustrated

interfacial growth in driven ginzburg landau models short and long time dynamics interfacial growth in driven systems is studied from the initial stage to the longtime regime numerical integrations of a ginzburg landan type equation with a new flux term introduced by an external field are presented the interfacial instabilities are induced by the external field from the numerical results we obtain the dispersion relation for the initial growth during the intermediate temporal regime fingers of a characteristic triangular shape could grow depending on the boundary conditions the final state corresponds to strips multifinger states or a one finger state the results for the initial growth are interpreted by means of surface driven and mullins sekerka instabilities the shape of the one finger state is explained in terms of the characteristic length introduced by the external field

sharp reverse holder property for a weights on spaces of homogeneous type abstract in this article we present a new proof of a sharp reverse holder inequality for a weights then we derive two applications a precise open property of muckenhoupt classes and as a consequence of this last result we obtain a simple proof of a sharp weighted bound for the hardy littlewood maximal function involving a constants m l p w c 1 p 1 w a p a 1 p where 1 p w 1 1 p and c is a dimensional constant our approach allows us to extend the result to the context of spaces of homogeneous type and prove a weak reverse holder inequality which is still sufficient to prove the open property for a p classes and the l p boundedness of the maximal function in this latter case the constant c appearing in the norm inequality for the maximal function depends only on the doubling constant of the measure and the geometric constant of the quasimetric

exact algorithms for linear matrix inequalities let a x a 0 x 1a 1 cdots x na n be a linear matrix or pencil generated by given symmetric matrices a 0 a 1 ldots a n of size m with rational entries the set of real vectors x such that the pencil is positive semidefinite is a convex semialgebraic set called spectrahedron described by a linear matrix inequality we design an exact algorithm that up to genericity assumptions on the input matrices computes an exact algebraic representation of at least one point in the spectrahedron or decides that it is empty the algorithm does not assume the existence of an interior point and the computed point minimizes the rank of the pencil on the spectrahedron the degree d of the algebraic representation of the point coincides experimentally with the algebraic degree of a generic semidefinite program associated to the pencil we provide explicit bounds for the complexity of our algorithm proving that the maximum number of arithmetic operations that are performed is essentially quadratic in a mu

tractability conditions for numeric csps abstract the computational complexity of the constraint satisfaction problem csp with semilinear relations over the reals has gained recent attraction as a result its complexity is known for all finite sets of semilinear relations containing the relation r x y z r 3 x y z we consider larger and more expressive classes of relations such as semialgebraic and o minimal relations we present a general result for characterising computationally hard fragments and under certain side conditions this result implies that polynomial time solvable fragments are only to be found within two limited families of sets of relations in the setting of semialgebraic relation our result takes on a simplified form and we provide a full complexity classification for constraint languages that consist of algebraic varieties full classifications like the one obtained here for algebraic varieties or the one for semilinear relations appear to be rare and we discuss several barriers for obtaining further such results these barriers have strong connections with well known open problems concerning the complexity of various restrictions of convex programming

dynamics of finger arrays in a diffusion limited growth model with a drift abstract we propose a phenomenological model of fingering dynamics in the presence of an external drift motivated by recent experiments on quasi two dimensional electrodeposition with hydrodynamic convection we study the dynamical transition between a laplacian growth regime and a weak finger competition regime the model proposed defines a wavelength selection problem of the finger array which is coupled in a nontrivial way to the single finger selection problem in a channel the wavelength selection mechanism proposed is associated to the crossover from a diffusive to a ballistic behavior of the aggregating particles as the coarsening spatial scale of the pattern matches the appropriate diffusion length this yields specific predictions for the selected wavelength the dynamics of the model are studied numerically with boundary integral methods in the quasistatic approximation and complementarily using a biased random walk monte carlo scheme on a lattice results are in qualitative agreement with the theoretical predictions

coordinate shadows of semidefinite and euclidean distance matrices we consider the projected semidefinite and euclidean distance cones onto a subset of the matrix entries these two sets are precisely the input data defining feasible semidefinite and euclidean distance completion problems we classify when these sets are closed and use the boundary structure of these two sets to elucidate the krislock wolkowicz facial reduction algorithm in particular we show that under a chordality assumption the minimal cones of these problems admit combinatorial characterizations as a by product we record a striking relationship between the complexity of the general facial reduction algorithm singularity degree and facial exposedness of conic images under a linear mapping

surface driven instability and enhanced relaxation in the dynamics of a nonequilibrium interface we determine the stability of a nonequilibrium interface between two coexisting solid phases in the presence of a weak external field starting at the coarsegrained cahn hilliard level we use the method of matched asymptotics to derive the macroscopic interfacial dynamics we then show that the external field leads to an instability due to flux along the interface in contrast with the more common mullins sekerka type instability which involves fluxes normal to the interface we also find that the external field produces an important modification of the gibbs thomson relation with these results we perform the linear stability analysis for an approximately flat interface if the field is tangent to the interface the modification of the gibbs thomson relation is important and the interface is stabilized if the field is normal to the interface the surface flux is important and the effect can be stabilizing or destabilizing but the orientational dependence is opposite what would be obtained if the mullins sekerka instability dominates numerical simulations are performed to study the effect of the surface current and are in agreement with our analytical results