stochastic embedding of dynamical systems most physical systems are modeled by an ordinary or a partial differential equation like the n body problem in celestial mechanics in some cases for example when studying the long term behavior of the solar system or for complex systems there exist elements which can influence the dynamics of the system which are not well modeled or even known one way to take these problems into account consists of looking at the dynamics of the system on a larger class of objects that are eventually stochastic in this paper we develop a theory for the stochastic embedding of ordinary differential equations we apply this method to lagrangian systems in this particular case we extend many results of classical mechanics namely the least action principle the euler lagrange equations and noether s theorem we also obtain a hamiltonian formulation for our stochastic lagrangian systems many applications are discussed at the end of the paper

splittable ideals and the resolutions of monomial ideals abstract we provide a new combinatorial approach to study the minimal free resolutions of edge ideals that is quadratic square free monomial ideals with this method we can recover most of the known results on resolutions of edge ideals with fuller generality and at the same time obtain new results past investigations on the resolutions of edge ideals usually reduced the problem to computing the dimensions of reduced homology or koszul homology groups our approach circumvents the highly nontrivial problem of computing the dimensions of these groups and turns the problem into combinatorial questions about the associated simple graph we also show that our technique extends successfully to the study of graded betti numbers of arbitrary square free monomial ideals viewed as facet ideals of simplicial complexes

localized states in the conserved swift hohenberg equation with cubic nonlinearity the conserved swift hohenberg equation with cubic nonlinearity provides the simplest microscopic description of the thermodynamic transition from a fluid state to a crystalline state the resulting phase field crystal model describes a variety of spatially localized structures in addition to different spatially extended periodic structures the location of these structures in the temperature versus mean order parameter plane is determined using a combination of numerical continuation in one dimension and direct numerical simulation in two and three dimensions localized states are found in the region of thermodynamic coexistence between the homogeneous and structured phases and may lie outside of the binodal for these states the results are related to the phenomenon of slanted snaking but take the form of standard homoclinic snaking when the mean order parameter is plotted as a function of the chemical potential and are expected to carry over to related models with a conserved order parameter 2013 american physical society

ideals with stable betti numbers componentwise linear ideals were introduced earlier to generalize the result that the stanley reisner ideal i of a simplicial complex has a linear resolution if and only if its alexander dual is cohen macaulay it turns out that i is componentwise linear if and only if is sequentially cohen macaulay in this paper we discuss betti number properties of componentwise linear ideals let i be a graded ideal of a polynomial ring s and gin i the generic initial ideal of i with respect to the reverse lexicographic term order on s our main result is that i and gin i admit the same graded betti numbers if and only if i is componentwise linear for the proof of this fact we describe some properties of the betti diagram of a generic initial ideal combinatorial implications for shifted complexes will also be discussed

maximal minimal resolutions that is exact everywhere except at f0 where the cokernel is m such a resolution always exists if r is a noetherian n graded ring with r0 a field and m is a finitely generated z graded r module then a resolution may be constructed in a minimal way in which each fi is graded and each differential is homogeneous of degree 0 the rank of the ith free module in a minimal free resolution of m is an invariant of m called the ith betti number of m and denoted i m likewise r ij m the number of elements of degree j in a minimal set of homogeneous generators of fi is also an invariant of m called the i j th graded betti number of m in this paper we study modules with maximal graded betti numbers more precisely consider a set consisting of pairs r m where r is a noetherian n graded ring with r0 a field and m is a finitely generated graded r module is there some r m such that ij m ij m for every i and j and every r m there are trivial examples in which the answer is yes e g has only one element or no e g consists of all such pairs theorems 1 and 2 below give affirmative answers for certain sets defined by conditions on the hilbert series and the depths of r and m theorem 3 allows one to compute the maximal graded betti numbers for the pairs in the sets considered in theorems 1 and 2 throughout this paper k denotes a field and q the polynomial ring k x1 xn with the usual n grading given by degxi 1 the hilbert series of a finitely generated graded q module m is hm s dimk mis we define d lexicographic ideals and submodules in section 2 the poincare series of an r module m is p m s t ij m s t we write 4 for coefficient wise inequality of laurent series with coefficients in z

betti splitting via componentwise linear ideals abstract a monomial ideal i admits a betti splitting i j k if the betti numbers of i can be determined in terms of the betti numbers of the ideals j k and j k given a monomial ideal i we prove that i j k is a betti splitting of i provided j and k are componentwise linear generalizing a result of francisco ha and van tuyl if i has a linear resolution the converse also holds we apply this result recursively to the alexander dual of vertex decomposable shellable and constructible simplicial complexes moreover we determine the graded betti numbers of the defining ideal of three general fat points in the projective space

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

heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible hamiltonian hopft bifurcation this paper considers an unfolding of a degenerate reversible 1 1 resonance or hamiltonian hopf bifurcation for four dimensional systems of time reversible ordinary differential equations odes this bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs the degeneracy occurs via the vanishing of a normal form coefficient q2 0 that determines whether the bifurcation is super or subcritical of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits a partial unfolding of such solutions already occurs in the work of dias and iooss eur j mech b fluids 15 1996 367 393 given a sign of the coefficient of a higher order term q4 0 and 1 q2 0 the normal form then shows a small amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit a genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies this is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies at each limit point the homoclinic orbit generates another bump close to the periodic orbit as q2 is further decreased from zero the snake widens until at a critical value a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium a kink for q2 less than this value the kinks heteroclinic tangencies and snake like curves no longer occur

integration by parts and time reversal for diffusion processes in this paper we obtain necessary and sufficient conditions for the reversibility of the diffusion property assuming the existence of a density at every time t the proofs are based on techniques of the stochastic calculus of variations

snaking of radial solutions of the multi dimensional swift hohenberg equation a numerical study abstract the bifurcation structure of localized stationary radial patterns of the swift hohenberg equation is explored when a continuous parameter n is varied that corresponds to the underlying space dimension whenever n is an integer in particular we investigate how 1d pulses and 2 pulses are connected to planar spots and rings when n is increased from 1 to 2 we also elucidate changes in the snaking diagrams of spots when the dimension is switched from 2 to 3

artinian level modules and cancellable sequences in order to use dualization to study hilbert functions of artinian level algebras we extend the notion of level sequences and cancellable sequences introduced by geramita and lorenzini to include

ideals containing the squares of the variables we study the betti numbers of graded ideals containing the squares of the variables in a polynomial ring we prove the lex plus powers conjecture for such ideals

initial ideals veronese subrings and rates of algebras abstract let s be a polynomial ring over an infinite field and let i be a homogeneous ideal of s let t d be a polynomial ring whose variables correspond to the monomials of degree d in s we study the initial ideals of the ideals v d i t d that define the veronese subrings of s i in suitable orders they are easily deduced from the initial ideal of i we show that in v d i is generated in degree max reg i d 2 where reg i is the regularity of the ideal i in other words the d th veronese subrings of any commutative graded ring s i has a grobner basis of degree max i d 2 we also give bounds on the regularity of i in terms of the degrees of the generators of in i and some combinatorial data this implies a version of backelin s theorem that high veronese subrings of any ring are homogeneous koszul algebras in the sense of priddy trans amer math soc 152 1970 39 60 we also give a general obstruction for a homogeneous ideal i s to have an initial ideal in i that is generated by quadrics beyond the obvious requirement that i itself should be generated by quadrics and the stronger statement that s i is koszul we use the obstruction to show that in certain dimensions a generic complete intersection of quadrics cannot have an initial ideal that is generated by quadrics for the application to backelin s theorem we require a result of backelin whose proof has never appeared we give a simple proof of a sharpened version bounding the rate of growth of the degrees of generators for syzygies of any multihomogenous module over a polynomial ring modulo an ideal generated by monomials following a method of bruns and herzog

algebra structures on some canonical resolutions the study of algebra structures on finite free resolutions of cyclic modules begins with buchsbaum and eisenbud b el the classes of cyclic modules whose minimal resolutions are known to admit an algebra structure include the residue field gu complete intersections koszul complex modules of homological dimension atmost 3 b el gorenstein of codimension four k m and herzog algebras k m23 avramov al gave examples to show that there are cyclic modules whose minimal resolutions do not admit an algebra structure let r be a noetherian local ring with maximal ideal m in this paper we construct algebra structures on the minimal resolutions of two classes of cyclic modules r i namely when z is of the form jk where j is an ideal generated by a regular sequence and when z is the ideal of maximal minors of a generic n x m matrix provided r contains the rationals in al avramov defined certain obstructions to the existance of an algebra structure on the minimal resolution of a module and then produced modules with non zero obstructions the general question is whether the vanishing of these obstructions is also sufficient for the existence of a minimal algebra resolution of a cyclic module when z is an ideal generated by a regular sequence in r and m r z for any positive integer k then these obstructions are all zero therefore avramov and schlessinger asked whether the minimal resolutions of r z admit an algebra structure corollary 3 6 of this paper provides an affirmative answer to this question

pruned cellular free resolutions of monomial ideals abstract using discrete morse theory we give an algorithm that prunes the excess of information in the taylor resolution and constructs a new cellular free resolution for an arbitrary monomial ideal the pruned resolution is not simplicial in general but we can slightly modify our algorithm in order to obtain a simplicial resolution we also show that the lyubeznik resolution fits into our pruning strategy the pruned resolution is not always minimal but it is a lot closer to the minimal resolution than the taylor and the lyubeznik resolutions as we will see in some examples we finally use our methods to give a different approach to the theory of splitting of monomial ideals we deduce from this splitting strategy that the pruned resolution is always minimal in the case of edge ideals of paths and cycles

non degenerate curves with maximal hartshorne rao module extending results for space curves we establish bounds for the cohomology of a non degenerate curve in projective n space as a consequence for any given n we determine all possible pairs d g where d is the degree and g is the arithmetic genus of the curve furthermore we show that curves attaining our bounds always exist and describe properties of these extremal curves in particular we determine the hartshorne rao module the generic initial ideal and the graded betti numbers of an extremal curve

reverse lex ideals we study reverse lex ideals in a polynomial ring and compare their properties to those of lex ideals in particular we provide an analogue of green s theorem for reverse lex ideals we also compare the betti numbers of strongly stable and square free strongly stable monomial ideals to those of reverse lex ideals

free resolutions of lex ideals over a koszul toric ring in this paper we study the minimal free resolution of lex ideals over a koszul toric ring in particular we study in which toric ring r all lex ideals are componentwise linear we give a certain necessity and sufficiency condition for this property and show that lex ideals in a strongly koszul toric ring are componentwise linear in addition it is shown that in the toric ring arising from the segre product p p every hilbert function of a graded ideal is attained by a lex ideal and that lex ideals have the greatest graded betti numbers among all ideals having the same hilbert function

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

hilbert schemes and maximal betti numbers over veronese rings macaulay s theorem macaulay in proc lond math soc 26 531 555 1927 characterizes the hilbert functions of graded ideals in a polynomial ring over a field we characterize the hilbert functions of graded ideals in a veronese ring r the coordinate ring of a veronese embedding of pr 1 we also prove that the hilbert scheme which parametrizes all graded ideals in r with a fixed hilbert function is connected this is an analogue of hartshorne s theorem hartshorne in math ihes 29 5 48 1966 that hilbert schemes over a polynomial ring are connected furthermore we prove that each lex ideal in r has the greatest betti numbers among all graded ideals with the same hilbert function

non linear behaviour of castelnuovo mumford regularity abstract it is shown that the castelnuovo mumford regularity of lex segment ideals of ordinary powers as well as of frobenius powers of a homogeneous polynomial ideal is a function of polynomial type of huge degree

on the resolution of ideals of fat points abstract in this paper we determine the hilbert function and the minimal system of generators of r 1 n 1 general fat points of p n furthermore by looking at the minimal system of generators we can show that the ideal associated to two fat points in p n or the ideal associated to r 1 n general fat points all with the same multiplicities is a splittable ideal and this is the first step in constructing a minimal resolution

betti numbers of chordal graphs and f vectors of simplicial complexes abstract let g be a chordal graph and i g its edge ideal let i g 0 1 p denote the betti sequence of i g where i stands for the ith total betti number of i g and where p is the projective dimension of i g it will be shown that there exists a simplicial complex of dimension p whose f vector f f 0 f 1 f p coincides with i g

generic sequences of polynomials abstract let i be an ideal in a polynomial ring s over an infinite field such that i is generated by a generic sequence of homogeneous polynomials of specified degrees froberg has conjectured a formula for the hilbert series of s i moreno socias has conjectured a combinatorial property for the initial ideal of i with respect to degree reverse lexicographic order i show that moreno socias conjecture implies froberg s conjecture i also give a criterion for a hilbert series to admit an ideal with the property proposed by moreno socias and show that the hilbert series proposed by froberg does have this property

fractional embedding of differential operators and lagrangian systems this paper is a contribution to the general program of embedding theories of dynamical systems following our previous work on the stochastic embedding theory developed with darses c r acad sci ser i math 342 333 2006 preprint ihes 06 27 p 87 2006 we define the fractional embedding of differential operators and ordinary differential equations we construct an operator combining in a symmetric way the left and right riemann liouville fractional derivatives for lagrangian systems our method provides a fractional euler lagrange equation we prove developing the corresponding fractional calculus of variations that such equation can be derived via a fractional least action principle we then obtain naturally a fractional noether theorem and a fractional hamiltonian formulation of fractional lagrangian systems all these constructions are coherents i e the embedding procedure is compatible with the fractional calculus of variations we then extend our results to cover the ostrogradski

bounds for betti numbers abstract in this paper we prove parts of a conjecture of herzog giving lower bounds on the rank of the free modules appearing in the linear strand of a graded kth syzygy module over the polynomial ring if in addition the module is z n graded we show that the conjecture holds in full generality furthermore we give lower and upper bounds for the graded betti numbers of graded ideals with a linear resolution and a fixed number of generators

resolutions of a stable ideals abstract let a a 1 a 2 a n be a sequence of integers or we introduce a stable ideals in a polynomial ring and study their homological properties our results generalize results on square free monomial ideals by aramova avramov herzog hibi and srinivasan

the weak and strong lefschetz properties for artinian k algebras abstract let a i 0ai be a standard graded artinian k algebra where chark 0 then a has the weak lefschetz property if there is an element l of degree 1 such that the multiplication l a i a i 1 has maximal rank for every i and a has the strong lefschetz property if l d a i a i d has maximal rank for every i and d the main results obtained in this paper are the following 1 every height three complete intersection has the weak lefschetz property our method surprisingly uses rank two vector bundles on p 2 and the grauert mulich theorem 2 we give a complete characterization including a concrete construction of the hilbert functions that can occur for k algebras with the weak or strong lefschetz property and the characterization is the same one 3 we give a sharp bound on the graded betti numbers achieved by our construction of artinian k algebras with the weak or strong lefschetz property and fixed hilbert function this bound is again the same for both properties some hilbert functions in fact force the algebra to have the maximal betti numbers 4 every artinian ideal in k x y possesses the strong lefschetz property this is false in higher codimension

lefschetz properties and hyperplane arrangements abstract in this article we study the weak and strong lefschetz properties and the related notion of almost revlex ideal in the non artinian case proving that several results known in the artinian case hold also in this more general setting we then apply the obtained results to the study of the jacobian algebra of hyperplane arrangements

spatially complex localisation in twisted elastic rods constrained to a cylinder abstract we consider the problem of a long thin weightless rod constrained to lie on a cylinder while being held by end tension and twisting moment applications of this problem are found for instance in the buckling of drill strings inside a cylindrical hole in a previous paper the general geometrically exact formulation was given and the case of a rod of isotropic cross section analysed in detail it was shown that in that case the static equilibrium equations are completely integrable and can be reduced to those of a one degree of freedom oscillator whose non trivial fixed points correspond to helical solutions of the rod a critical load was found where the rod coils up into a helix here the anisotropic case is studied it is shown that the equations are no longer integrable and give rise to spatial chaos with infinitely many multi loop localised solutions helices become slightly modulated we study the bifurcations of the simplest single loop solution and a representative multi loop as the aspect ratio of the rod s cross section is varied it is shown how the anisotropy unfolds the coiling bifurcation the resulting post buckling behaviour is of the softening hardening softening type typically seen in the cellular buckling of long structures and can be interpreted in terms of a so called maxwell effective failure load

a diagrammatic derivation of convective pattern equations in complicated bifurcation problems where more than one instability can arise at onset reasonably sound derivations of the equations that govern the amplitudes of the nearly marginal modes have been developed when the spectrum of the modes is discrete the basis of these derivations lies in the center manifold theorem of dynamical systems theory but when the spectrum of the modes is continuous and we no longer have that theorem to fall back on there is nevertheless an equation the swift hohenberg equation that well describes the patterns seen in rayleigh benard convection indeed several derivations of the s h equation have been offered and here we describe how to obtain the s h equation using bogoliubov s method we suggest that this procedure clarifies and simplifies though it does not make rigorous the derivation of the s h equation looking ahead to the derivation of pattern equations for more complicated problems with continuous spectra we also describe a diagrammatic procedure that once mastered is useful in performing the complicated perturbative developments that are needed in such derivations here we illustrate the proposed combination of the ideas of bogoliubov and feynman for the standard form of the rayleigh benard convection problem the resulting pattern equation is nonlocal but it reduces without approximation to the 1 d swift hohenberg equation in the case of 2 d convection like the s h equation the nonlocal version admits a lyapunov functional and we briefly indicate its utility in pattern selection both for the swift hohenberg equation and its nonlocal extension we conclude by describing the kinds of problems for which we intend the combined method but reserve the exhibition of the required calculations for a future festschrift

snakes and ladders in an inhomogeneous neural field model continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity we analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes and ladders bifurcation structure in the case of heaviside firing rates we construct analytically symmetric and asymmetric states and hence derive closed form expressions for the corresponding bifurcation diagrams we show that the ideas proposed by beck and co workers to analyze snaking solutions to the swift hohenberg equation remain valid for the neural field model even though the corresponding spatial dynamical formulation is non autonomous we investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the heaviside limit

second order stochastic differential equations and non gaussian reciprocal diffusions we first prove that a markov diffusion satisfies a second order stochastic differential equation involving the invariants associated to its reciprocal class as a reciprocal process some properties of the noise term are given we also prove that this equation can be viewed as an euler lagrange equation in a problem of calculus of variations in the non markovian case a bernstein bridge is shown to satisfy the same equation but in a weak sense

solitary wave interaction phenomena in a strut buckling model incorporating restabilisation the archetypal model of the buckling of a compressed long elastic strut resting on a nonlinear elastic foundation is studied localised buckling is investigated when the foundation has both quadratic and cubic terms which initially destabilise but subsequently restabilise the structure the primary solution can be detected by a double scale perturbation procedure and is reminiscent of a solitary wave essentially it consists of a fast periodic oscillation which is slowly modulated and decays exponentially in both directions particular interest is paid to the process of adapting the procedure to account for the post buckling behaviour of two packet or double humped solitary waves in this model we employ the methods of beyond all orders asymptotics to reveal terms formally exponentially small in the perturbation parameter which have macroscopic effects on the post buckling behaviour of the system including the interaction phenomenon of interest the analysis is reinforced by direct numerical computations which reveal the so called snaking behaviour in the subsidiary homoclinic orbits as is observed in the case of primary solutions however additional phenomena arise for these subsidiary forms including the formation of bridges linking solution paths and the appearance of a multitude of closed isolated loops disconnected from other features of the bifurcation diagram

on localised hotspots of an urban crime model abstract we investigate stationary spatially localised crime hotspots on the real line and the plane of an urban crime model of short et al m short m dorsogna a statistical model of criminal behavior mathematical models and methods in applied sciences 18 2008 1249 1267 extending the weakly nonlinear analysis of short et al we show in one dimension that localised hotspots should bifurcate off the background spatially homogeneous state at a turing instability provided the bifurcation is subcritical using path following techniques we continue these hotspots and show that the bifurcating pulses can undergo the process of homoclinic snaking near the singular limit we analyse the singular limit to explain the existence of spike solutions and compare the analytical results with the numerical computations in two dimensions we show that localised radial spots should also bifurcate off the spatially homogeneous background state localised planar hexagon fronts and hexagon patches are found and depending on the proximity to the singular limit these solutions either undergo homoclinic snaking or act like multi spot solutions finally we discuss applications of these localised patterns in the urban crime context and the full agent based model

two dimensional localized structures in harmonically forced oscillatory systems two dimensional spatially localized structures in the complex ginzburg landau equation with 1 1 resonance are studied near the simultaneous presence of a steady front between two spatially homogeneous equilibria and a supercritical turing bifurcation on one of them the bifurcation structures of steady circular fronts and localized target patterns are computed in the turing stable and turing unstable regimes in particular localized target patterns grow along the solution branch via ring insertion at the core in a process reminiscent of defect mediated snaking in one spatial dimension stability of axisymmetric solutions on these branches with respect to axisymmetric and nonaxisymmetric perturbations is determined and parameter regimes with stable axisymmetric oscillons are identified direct numerical simulations reveal novel depinning dynamics of localized target patterns in the radial direction and of circular and planar localized hexagonal patterns in the fully two dimensional system

continuation of localised coherent structures in nonlocal neural field equations we study localised activity patterns in neural field equations posed on the euclidean plane such models are commonly used to describe the coarse grained activity of large ensembles of cortical neurons in a spatially continuous way we employ matrix free newton krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation this opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent pde formulation we present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking the regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function the results presented form a basis for the general study of localised cortical activity with inputs and more specifically for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input

efficient gluing of numerical continuation and a multiple solution method for elliptic pdes numerical continuation calculations for ordinary differential equations odes are by now an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences although several excellent standard software packages are available for odes there are for good reasons no standard numerical continuation toolboxes available for partial differential equations pdes which cover a broad range of different classes of pdes automatically a natural ach to this problem is to look for efficient gluing computation approaches with independent components developed by researchers in numerical analysis dynamical systems scientific computing and mathematical modeling in this paper we shall study several elliptic pdes lane emden fowler lane emden fowler with microscopic force caginalp via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the pdes in particular we solve the initialization problem of numerical continuation for pdes via a minimax algorithm to find multiple unstable solution furthermore for the caginalp system we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the fem matlab pdetoolbox even though the approach works efficiently due to the high level programming language and without developing any new algorithms we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic pdes we study in particular with respect to symmetry breaking in particular we show for a modified lane emden fowler equation with an asymmetric microscopic force how a fully connected bifurcation diagram splits up into c shaped isolas on which localized pattern deformation appears towards two different regimes we conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient time saving gluing computation for dynamical systems analysis of pdes in the near future

structural decomposition of monomial resolutions abstract we express the multigraded betti numbers of an arbitrary monomial ideal in terms of the multigraded betti numbers of two basic classes of ideals this decomposition has multiple applications in some concrete cases we use it to construct minimal resolutions of classes of monomial ideals in other cases we use it to compute projective dimensions to illustrate the effectiveness of the structural decomposition we give a new proof of a classic theorem by charalambous that states the following let k be a field and m an artinian monomial ideal in s k x 1 x n then for all i b i s m n i

stochastic derivatives for fractional diffusions in this paper we introduce some fundamental notions related to the so called stochastic derivatives with respect to a given fieldq in our framework we recall well known results about markov wiener diffusions we then focus mainly on the case where x is a fractional diffusion and where q is the past the future or the present of x we treat some crucial examples and our main result is the existence of stochastic derivatives with respect to the present of x when x solves a stochastic differential equation driven by a fractional brownian motion with hurst index h 1 2 we give explicit formulas 1 introduction there exist various ways to generalize the notion of differentiation on deterministic functions we may think of fractional derivatives or differentiation in the sense of the theory of distributions in both cases we lose a dynamical or geometric interpretation of tangent vectors velocities e g in the present work we seek to construct derivatives on stochastic processes which conserve a dynamical meaning our goal is motivated by the stochastic embedding of dynamical systems introduced in 2 this procedure aims at comprehending the following question how can we write an equation which contains the dynamical meaning of an initial ordinary differential equation and which extends this dynamical meaning to stochastic processes we refer to 3 for more details unfortunately for most of the stochastic processes used in physical models the limit zt h zt h

poset embeddings of hilbert functions and betti numbers abstract we study inequalities between graded betti numbers of ideals in a standard graded algebra over a field and their images under embedding maps defined earlier by us in caviglia and kummini 2013 5 we show that if graded betti numbers do not decrease when we replace ideals in an algebra by their embedded versions then the same behaviour is carried over to ring extensions as a corollary we give alternative inductive proofs of earlier results of bigatti hulett pardue mermin peeva stillman and murai we extend a hypersurface restriction theorem of herzog and popescu to the situation of embeddings we show that we can obtain the betti table of an ideal in the extension ring from the betti table of its embedded version by a sequence of consecutive cancellations we further show that the lex plus powers conjecture of evans reduces to the artinian situation

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude