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Boij-Söderberg theory gives a combinatorial description of the set of Betti tables belonging to finite length modules over the polynomial ring S = k [ x 1 , ... , x n ]. We posit that a similar combinatorial description can be given for analogous numerical invariants of graded differential S-modules , which are natural generalizations of chain complexes. We prove several results that lend evidence in support of this conjecture, including a categorical pairing between the derived categories of graded differential S -modules and coherent sheaves on P n − 1 and a proof of the conjecture in the case where S = k [ t ]. [ABSTRACT FROM AUTHOR]

We work within the framework of the variational bicomplex: we define A ∞ -algebra structures on horizontal and vertical cohomologies of (formally integrable) partial differential equations with the help of Merkulov's theorem. Since higher order A ∞ -algebra operations are related to Massey products, our observation implies the existence of invariants for differential equations that go beyond conservation laws. We also propose notions of formality for PDEs, and we present examples of formal equations. [ABSTRACT FROM AUTHOR]

We apply numerical algebraic geometry to the invariant-theoretic problem of detecting symmetries between two plane algebraic curves. We describe an efficient equality test which determines, with "probability-one", whether or not two rational maps have the same image up to Zariski closure. The application to invariant theory is based on the construction of suitable signature maps associated to a group acting linearly on the respective curves. We consider two versions of this construction: differential and joint signature maps. In our examples and computational experiments, we focus on the complex Euclidean group, and introduce an algebraic joint signature that we prove determines equivalence of curves under this action and the size of a curve's symmetry group. We demonstrate that the test is efficient and use it to empirically compare the sensitivity of differential and joint signatures to different types of noise. [ABSTRACT FROM AUTHOR]

Abstract We give an expression of the irreducible invariant curves at the singular point. For analytically integrable systems, we provide an expression of its primitive first integral. This fact allows us to obtain necessary conditions of analytic integrability at degenerate singular points. [ABSTRACT FROM AUTHOR]

Abstract In this paper, mistuning of bladed-disk systems is investigated. The blades are considered as continuous Euler-Bernoulli beams with transverse and longitudinal degrees of freedom. The governing equations of motion for the tuned and mistuned bladed disk systems are derived and discretized by use of Galerkin method. The obtained equations have time-varying coefficients and cannot directly be used for calculating the system critical speeds and Campbell diagram. For the tuned case, by use of Coleman and complex transformations one can transform the time varying system to a time-invariant system. Further transformations are required for the mistuned system to change into a time-invariant system. Next, a four-bladed disk system supported by bearings was considered. The modal analysis was performed and the critical speeds and the Campbell diagram were calculated. The results indicate that both coupled modes and conjugate modes appear in the Campbell diagram which are changing with the rotating speed. Finally, a probabilistic analysis was performed on the mistuned parameters by use of Monte Carlo method. It was found that the disk dominant modes are insensitive to the various mistuning realizations. However, the blade dominant modes are sensitive to the mistuning parameters and have a coefficient of variation one order of magnitude less than the mistuning coefficient of variation. It was also seen that the probability density functions of whirl speeds for different mistuned properties are able to be considered Gaussian. [ABSTRACT FROM AUTHOR]

Abstract In [13] , initially motivated by questions about the Hodge line bundle of a Hodge-type Shimura variety, we singled out a generalization of the notion of minuscule character which we termed quasi-constant. Here we prove that the character of the Hodge line bundle is always quasi-constant. Furthermore, we classify the quasi-constant characters of an arbitrary connected, reductive group over an arbitrary field. As an application, we observe that, if μ is a quasi-constant cocharacter of an F p -group G , then our construction of group-theoretical Hasse invariants in loc. cit. applies to the stack G - Zip μ , without any restrictions on p , even if the pair (G , μ) is not of Hodge type and even if μ is not minuscule. We conclude with a more speculative discussion of some further motivation for considering quasi-constant cocharacters in the setting of our program outlined in loc. cit. [ABSTRACT FROM AUTHOR]

This paper describes the connections between the notions of Hamiltonian system, contact Hamiltonian system and nonholonomic system from the perspective of differential equations and dynamical systems. It shows that action minimizing curves of nonholonomic system satisfy the dissipative Lagrange system, which is equivalent to the contact Hamiltonian system under some generic conditions. As the initial research of contact Hamiltonian dynamics in this direction, we investigate the dynamics of contact Hamiltonian systems in some special cases including invariants, completeness of phase flows and periodic behavior. [ABSTRACT FROM AUTHOR]

A well known result of Forstnerić [15] states that most real-analytic strictly pseudoconvex hypersurfaces in complex space are not holomorphically embeddable into spheres of higher dimension. A more recent result by Forstnerić [16] states even more: most real-analytic hypersurfaces do not admit a holomorphic embedding even into a merely algebraic hypersurface of higher dimension, in particular, a hyperquadric. We emphasize that both cited theorems are proved by showing that the set of embeddable hypersurfaces is a set of first Baire category. At the same time, the classical theorem of Webster [30] asserts that every real-algebraic Levi-nondegenerate hypersurface admits a transverse holomorphic embedding into a nondegenerate real hyperquadric in complex space. In this paper, we provide effective results on the non-embeddability of real-analytic hypersurfaces into a hyperquadric. We show that, under the codimension restriction N ≤ 2 n , the defining functions φ ( z , z ¯ , u ) of all real-analytic hypersurfaces M = { v = φ ( z , z ¯ , u ) } ⊂ C n + 1 containing Levi-nondegenerate points and locally transversally holomorphically embeddable into some hyperquadric Q ⊂ C N + 1 satisfy an universal algebraic partial differential equation D ( φ ) = 0 , where the algebraic-differential operator D = D ( n , N ) depends on n ≥ 1 , n < N ≤ 2 n only. To the best of our knowledge, this is the first effective result characterizing real-analytic hypersurfaces embeddable into a hyperquadric of higher dimension. As an application, we show that for every n , N as above there exists μ = μ ( n , N ) such that a Zariski generic real-analytic hypersurface M ⊂ C n + 1 of degree ≥ μ is not transversally holomorphically embeddable into any hyperquadric Q ⊂ C N + 1 . We also provide an explicit upper bound for μ in terms of n , N . To the best of our knowledge, this gives the first effective lower bound for the CR-complexity of a Zariski generic real-algebraic hypersurface in complex space of a fixed degree. [ABSTRACT FROM AUTHOR]

This paper contains a weak lifting result for differential graded (DG) modules that is a generalization of Auslander, Ding, and Solberg's weak lifting for modules and Yoshino's weak lifting for complexes. [ABSTRACT FROM AUTHOR]

In the present paper a generalized Kählerian space is considered, as a generalized Riemannian space GR N with almost complex structure F i h that is covariantly constant with respect to the first and the second kind of covariant derivative. In the general case of a holomorphically projective mapping f of two non-symmetric generalized Kählerian spaces GK N and G K ‾ N it is impossible to obtain a generalization of the holomorphically projective curvature tensor. In the present paper we study the case when GK N and G K ‾ N have the same torsion in corresponding points. Such a mapping we call “equitorsion mapping”. We obtain quantities H P W θ ( θ = 1 , ⋯ , 5 ) , that are generalizations of the holomorphically projective tensor, i.e. they are invariants based on f . Among H P W θ only H P W 5 is a tensor. Using another five linearly independent curvature tensors, we can prove that there exist three holomorphically projective tensors. [ABSTRACT FROM AUTHOR]

RGB-D cameras have been attracting increasing researches for solving traditional problems in the domain of computer vision and robotics. Among the existing local features, most are proposed for the color channel or depth channel separately, while little attention has been paid to designing new composite features based on the physical characteristics. In this work, we propose a novel perspective invariant feature transform (PIFT) for RGB-D images. We integrate the color and depth information together making full use of the intrinsic characteristics of the two types of information to enhance the robustness and adaptability to large spatial variations of local appearance. The depth information is used to project the feature patch to its tangent plane to make it consistent with different views. It also helps to filter out the “fake keypoints” which are unstable in 3D space. Binary descriptors are then generated in the feature patches using a color coding method. Experiments on publicly available RGB-D datasets show that the proposed method has the best precision and the second best recall rate comparing against state-of-the-art local features, when applied to feature matching with large spatial variations. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamical behavior of the differential equation x ″ + a x + − b x − = f ( t ) , where x + = max ⁡ { x , 0 } , x − = max ⁡ { − x , 0 } , a and b are two different positive constants, f ( t ) is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see [11] ). Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation. [ABSTRACT FROM AUTHOR]

Let G be a graph of order n and λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n be the eigenvalues of G . The energy of G is defined as E ( G ) = ∑ k = 1 n | λ k | . A well-known result regarding the energy of graphs is the Coulson integral formula, which defines the relationship between the energy and the characteristic polynomial of graphs. Let μ 1 ≥ μ 2 ≥ ⋯ ≥ μ n = 0 be the Laplacian eigenvalues of G . The general Laplacian energy-like invariant of G , denoted by L E L α ( G ) , is defined as ∑ μ k ≠ 0 μ k α when μ 1 ≠ 0 , and 0 when μ 1 = 0 , where α is a real number. In this study, we give some Coulson-type integral formulas for the general Laplacian energy-like invariant of graphs in the case where α is a rational number. Based on this result, we also give some Coulson-type integral formulas for the general energy and general Laplacian energy of graphs in the case where α is a rational number. We also show that our formulas hold when α is an irrational number where 0 < | α | < 1 , whereas they do not hold when | α | > 1 . [ABSTRACT FROM AUTHOR]

Motivated by the challenge of defining twisted quantum field theories in the context of higher categories, we develop a general framework for lax and oplax transformations and their higher analogs between strong ( ∞ , n ) -functors. We construct a double ( ∞ , n ) -category built out of the target ( ∞ , n ) -category governing the desired diagrammatics. We define (op)lax transformations as functors into parts thereof, and an (op)lax twisted field theory to be a symmetric monoidal (op)lax natural transformation between field theories. We verify that lax trivially-twisted relative field theories are the same as absolute field theories. As a second application, we extend the higher Morita category of E d -algebras in a symmetric monoidal ( ∞ , n ) -category C to an ( ∞ , n + d ) -category using the higher morphisms in C . [ABSTRACT FROM AUTHOR]

We show that the local equivalence problem of second-order ordinary differential equations under point transformations is completely characterized by differential invariants of order at most 10 and that this upper bound is sharp. We also demonstrate that, modulo Cartan duality and point transformations, the Painlevé-I equation can be characterized as the simplest second-order ordinary differential equation belonging to the class of equations requiring 10th order jets for their classification. [ABSTRACT FROM AUTHOR]

Bernhard Riemann's habilitation lecture of 1854 on the foundations of geometry contains a stunningly precise concept of curvature without any supporting calculations. Another memoir of 1861 contains formulas in which we may recognize our Riemann tensor, though in a different context and without much geometrical interpretation. The first text is mysterious by the lack of formulas, the second by the excess of formulas. The purpose of this essay is to investigate this double mystery and the stimulating effect it had on some of Riemann's early readers, from Richard Dedekind to Tullio Levi-Civita. Use is made of some heretofore unexploited manuscript sheets by Riemann. [ABSTRACT FROM AUTHOR]

Abstract: We consider systems of Euler–Lagrange equations with two degrees of freedom and with Lagrangian being quadratic in velocities. For this class of equations the generic case of the equivalence problem is solved with respect to point transformations. Using Lieʼs infinitesimal method we construct a basis of differential invariants and invariant differentiation operators for such systems. We describe certain types of Lagrangian systems in terms of their invariants. The results are illustrated by several examples. [Copyright &y& Elsevier]

Abstract: Acceleration methods are commonly used for computing precisely the effects of loops in the reachability analysis of counter machine models. Applying these methods on synchronous data-flow programs, e.g. Lustre programs, requires to deal with the non-deterministic transformations due to numerical input variables. In this article, we address this problem by extending the concept of abstract acceleration of Gonnord et al. to numerical input variables. Moreover, we describe the dual analysis for co-reachability. We compare our method with some alternative techniques based on abstract interpretation pointing out its advantages and limitations. At last, we give some experimental results. [Copyright &y& Elsevier]

Abstract: We consider the space of polydifferential operators on n functions on symplectic manifolds invariant under symplectic automorphisms, whose study was initiated by Mathieu in 1995. Permutations of inputs yield an action of , which extends to an action of . We study this structure viewing n as a parameter, in the sense of Deligneʼs category. For manifolds of dimension 2d, we show that the isotypic part of this space of -th tensor powers of the reflection representation of is spanned by Poisson polynomials. We also prove a partial converse, and compute explicitly the isotypic part of ⩽4-th tensor powers of the reflection representation. We give generating functions for the isotypic parts corresponding to Young diagrams which only differ in the length of the top row, and prove that they are rational fractions whose denominators are related to hook lengths of the diagrams obtained by removing the top row. This also gives such a formula for the same isotypic parts of induced representations from to where n is viewed as a parameter. We show that the space of invariant operators of order 2m has polynomial dimension in n of degree equal to 2m, while the part not coming from Poisson polynomials has polynomial dimension of degree . We use this to compute asymptotics of the dimension of invariant operators. We also give new bounds on the order of invariant operators for a fixed n. We apply this to the Poisson and Hochschild homology associated to the singularity . Namely, the canonical surjection from to (the Brylinski spectral sequence in degree zero) restricts to an isomorphism in the aforementioned isotypic part , and also in . We prove a partial converse. Finally, the kernel of the entire surjection has dimension on the order of times the dimension of the homology group. [Copyright &y& Elsevier]

Abstract: In this paper, we provide an algorithm to compute explicit rational solutions of a rational system of autonomous ordinary differential equations (ODEs) from its rational invariant algebraic curves. The method is based on the proper rational parametrization of these curves and the fact that by linear reparametrizations, we can find the rational solutions of the given system of ODEs. Moreover, if the system has a rational first integral, we can decide whether it has a rational general solution and compute it in the affirmative case. [Copyright &y& Elsevier]

Abstract: This is the fourth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern–Gauss–Bonnet integrand. The present paper lays out the second half of this entire work: The second half proves certain purely algebraic statements regarding local Riemannian invariants; these were used extensively in the first two papers in this series, see Alexakis (2007, 2009) . These results may be of independent interest, applicable to related problems. [Copyright &y& Elsevier]

Abstract: In this paper we consider a general class of systems of two linear hyperbolic equations. Motivated by the existence of the Laplace invariants for the single linear hyperbolic equation, we adopt the problem of finding differential invariants for the system. We derive the equivalence group of transformations for this class of systems. The infinitesimal method, which makes use of the equivalence group, is employed for determining the desired differential invariants. We show that there exist four differential invariants and five semi-invariants of first order. Applications of systems that can be transformed by local mappings to simple forms are provided. [Copyright &y& Elsevier]

Abstract: The aim of this paper is to describe, in as much detail as possible and constructively, the structure of the algebra of differential invariants of a Lie pseudo-group acting on the submanifolds of an analytic manifold. Under the assumption of local freeness of a suitably high order prolongation of the pseudo-group action, we develop computational algorithms for locating a finite generating set of differential invariants, a complete system of recurrence relations for the differentiated invariants, and a finite system of generating differential syzygies among the generating differential invariants. In particular, if the pseudo-group acts transitively on the base manifold, then the algebra of differential invariants is shown to form a rational differential algebra with non-commuting derivations. The essential features of the differential invariant algebra are prescribed by a pair of commutative algebraic modules: the usual symbol module associated with the infinitesimal determining system of the pseudo-group, and a new “prolonged symbol module” constructed from the symbols of the annihilators of the prolonged pseudo-group generators. Modulo low order complications, the generating differential invariants and differential syzygies are in one-to-one correspondence with the algebraic generators and syzygies of an invariantized version of the prolonged symbol module. Our algorithms and proofs are all constructive, and rely on combining the moving frame approach developed in earlier papers with Gröbner basis algorithms from commutative algebra. [Copyright &y& Elsevier]

Abstract: This work focuses on the existence of quasi-periodic solutions for linear autonomous delay differential equation under quasi-periodic time-dependent perturbation near an elliptic–hyperbolic equilibrium point. Using the time-1 map of the solution operator, Newton iteration scheme, space splitting and KAM techniques, it is shown that under appropriate hypothesis, there exist quasi-periodic solutions with the same frequencies as the perturbation for most parameters. We show that if the delay differential equation is analytic, we obtain analytic parameterizations of the solutions. [Copyright &y& Elsevier]

Abstract: Certain nonlinear evolution PDEs in variables (time and space) are identified, featuring a positive parameter ω and evolving, for a large class of initial data, periodically with the fixed period (or perhaps with p a small integer). They are autonomous (i.e., they do not feature any explicit dependence on the time variable), but they generally (although not quite all of them) depend explicitly on the space variable hence are not translation-invariant. They are integrable, having been obtained by applying an appropriate change of dependent and independent variables to certain nonlinear evolution PDEs whose integrable character has been recently ascertained. Solutions of some of these PDEs are exhibited. [Copyright &y& Elsevier]

Abstract: We describe the historical and ideological context that brought to the fore the study of a centro-affine invariant that subsequently received much attention. The invariant was introduced by Ţiţeica in 1907, and this discovery has been viewed by many as a consequence of Klein''s Erlangen program. We thus present the starting point of affine differential geometry, as it was discovered by Ţiţeica after his years in the Ph.D. program in Paris (1896–1899) under the guidance of Gaston Darboux. [Copyright &y& Elsevier]

Abstract: In this paper we are going to apply the invariant theory to give invariant conditions on the coefficients of any non-degenerate quadratic system in order to determine if it has or not a polynomial first integral without using any normal form. We obtain that the existence of polynomial first integral is directly related with the fact that all the roots of a convenient cubic polynomial are rational and negative. The coefficients of this cubic polynomial are invariants related with some geometric properties of the system. [Copyright &y& Elsevier]

Abstract: In this paper, we consider one-dimensional nonlinear Schrödinger equation on under the boundary conditions , , , for . It is proved that for a prescribed and analytic positive potential , the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system. [Copyright &y& Elsevier]

Abstract: The Camassa–Holm equation can be viewed as the geodesic equation on some diffeomorphism group with respect to the invariant metric. We derive the geodesic equations on that group with respect to the invariant metric, which we call the modified Camassa–Holm equation, and then study the well-posedness and dynamics of a modified Camassa–Holm equation on the unit circle S, which has some significant difference from that of Camassa–Holm equation, e.g., it does not admit finite time blowup solutions. [Copyright &y& Elsevier]

Abstract: Given a group action, known by its infinitesimal generators, we exhibit a complete set of syzygies on a generating set of differential invariants. For that we elaborate on the reinterpretation of Cartan’s moving frame by Fels and Olver [Fels, M., Olver, P.J., 1999. Moving coframes. II. Regularization and theoretical foundations. Acta Appl. Math. 55 (2), 127–208]. This provides constructive tools for exploring algebras of differential invariants. [Copyright &y& Elsevier]

Abstract: We generalize to some PDEs a theorem by Eliasson and Nekhoroshev on the persistence of invariant tori in Hamiltonian systems with r integrals of motion and n degrees of freedom, . The result we get ensures the persistence of an r-parameter family of r-dimensional invariant tori. The parameters belong to a Cantor-like set. The proof is based on the Lyapunov–Schmidt decomposition and on the standard implicit function theorem. Some of the persistent tori are resonant. We also give an application to the nonlinear wave equation with periodic boundary conditions on a segment and to a system of coupled beam equations. In the first case we construct 2-dimensional tori, while in the second case we construct 3-dimensional tori. [Copyright &y& Elsevier]

Abstract: We study the geometry of an important class of generic curves in the Grassmann manifolds of n-dimensional subspaces and Lagrangian subspaces of under the action of the linear and linear symplectic groups. [Copyright &y& Elsevier]

Abstract: We consider linear hyperbolic equations of the form We derive equivalence transformations which are used to obtain differential invariants for the cases and . Motivated by these results, we present the general results for the n-dimensional case. It appears (at least for ) that this class of hyperbolic equations admits differential invariants of order one, but not of order two. We employ the derived invariants to construct interesting mappings between equivalent equations. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the semilinear elliptic problem in Ω, on ∂Ω, where Ω is a smooth bounded domain in , , , is the critical Sobolev exponent, is a continuous function. When Ω and are invariant under a group of orthogonal transformations, we prove the existence of nodal and positive solutions for , where is the first Dirichlet eigenvalue of on Ω. [Copyright &y& Elsevier]

Abstract: The topological Tverberg theorem claims that for any continuous map of the -simplex to there are q disjoint faces of such that their images have a non-empty intersection. This has been proved for affine maps, and if q is a prime power, but not in general. We extend the topological Tverberg theorem in the following way: Pairs of vertices are forced to end up in different faces. This leads to the concept of constraint graphs. In Tverberg''s theorem with constraints, we come up with a list of constraints graphs for the topological Tverberg theorem. The proof is based on connectivity results of chessboard-type complexes. Moreover, Tverberg''s theorem with constraints implies new lower bounds for the number of Tverberg partitions. As a consequence, we prove Sierksma''s conjecture for and . [Copyright &y& Elsevier]

Abstract: The concept of rigid spherical -designs was introduced by Eiichi Bannai. We want to find examples of rigid but not tight spherical designs. Sali investigated the case when is an orbit of a finite reflection group and proved that is rigid if and only if tight for the groups , , , , , , , . There are two cases left open, namely the group and the isometry group of the four-dimensional regular polytope, the 600-cell. In this paper, we study the rigidity of spherical -designs that are orbits of a finite reflection groups and , and prove that is rigid if and only if tight or the 600-cell. [Copyright &y& Elsevier]

Abstract: We introduce various families of irreducible homaloidal hypersurfaces in projective space , for all . Some of these are families of homaloidal hypersurfaces whose degrees are arbitrarily large as compared to the dimension of the ambient projective space. The existence of such a family solves a question that has naturally arisen from the consideration of the classes of homaloidal hypersurfaces known so far. The result relies on a fine analysis of hypersurfaces that are dual to certain scroll surfaces. We also introduce an infinite family of determinantal homaloidal hypersurfaces based on a certain degeneration of a generic Hankel matrix. The latter family fit non-classical versions of de Jonquières transformations. As a natural counterpoint, we broaden up aspects of the theory of Gordan–Noether hypersurfaces with vanishing Hessian determinant, bringing over some more precision into the present knowledge. [Copyright &y& Elsevier]

Abstract: We give an algebraic description of regular holonomic -modules whose characteristic variety is the union of conormal bundles to the orbits of the action of the general linear group on square matrices. [Copyright &y& Elsevier]

Abstract: Darboux-deformations of short range one-dimensional potentials are constructed by means of Gamow–Siegert functions (resonance states). Results include both Hermitian and non-Hermitian short range potentials which are exactly solvable. As illustration, the method is applied to square wells and barriers for which the transmission coefficient is a superposition of Fock–Breit–Wigner distributions. Resonance levels are calculated in the long lifetime limit by means of analytical and numerical approaches. The new complex potentials behave as an optical device which both refracts and absorbs light waves. [Copyright &y& Elsevier]

Abstract: We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose “natural” values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox–Ingersoll–Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation. [Copyright &y& Elsevier]

Abstract: Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs. [Copyright &y& Elsevier]

Abstract: In this paper we consider the general class of hyperbolic equations . We use equivalence transformations to derive differential invariants for this class and for two subclasses. Then we employ these invariants to construct equations that can be linearized via local mappings. Further applications of the differential invariants are given. [Copyright &y& Elsevier]

Abstract: To any finite poset we associate two graphs which we denote by and . Several standard constructions can be seen as or for suitable posets , including the comparability graph of a poset, the clique graph of a graph and the 1-skeleton of a simplicial complex. We interpret graphs and posets as simplicial complexes using complete subgraphs and chains as simplices. Then we study and compare the homotopy types of , and . As our main application we obtain a theorem, stronger than those previously known, giving sufficient conditions for a graph to be homotopy equivalent to its clique graph. We also introduce a new graph operator that preserves clique-Hellyness and dismantlability and is such that is homotopy equivalent to both its clique graph and the graph . [Copyright &y& Elsevier]

Abstract: We study the class of independence complexes of claw-free graphs. The main theorem gives good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class. Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with vertices and maximal degree is -connected, where . This can be compared with the result of Szabó and Tardos that is optimal with no restrictions on the graphs. Secondly, we calculate the connectivity of a family of complexes used in Babson and Kozlov’s proof of the Lovász conjecture. [Copyright &y& Elsevier]

Abstract: Geometric constructions applied to a rational action of an algebraic group lead to a new algorithm for computing rational invariants. A finite generating set of invariants appears as the coefficients of a reduced Gröbner basis. The algorithm comes in two variants. In the first construction the ideal of the graph of the action is considered. In the second one the ideal of a cross-section is added to the ideal of the graph. Zero-dimensionality of the resulting ideal brings a computational advantage. In both cases, reduction with respect to the computed Gröbner basis allows us to express any rational invariant in terms of the generators. [Copyright &y& Elsevier]

Abstract: We consider an elliptic problem of Ambrosetti–Prodi type involving critical Sobolev exponent. We prove that this problem has solutions blowing up near the boundary of the domain. [Copyright &y& Elsevier]

Abstract: We investigate a new lattice of generalised non-crossing partitions, constructed using the geometry of the complex reflection group . For the particular case (resp. ), our lattice coincides with the lattice of simple elements for the type (resp. ) dual braid monoid. Using this lattice, we construct a Garside structure for the braid group . As a corollary, one may solve the word and conjugacy problems in this group. [Copyright &y& Elsevier]

Abstract: The construction of a quantum groupoid out of a double groupoid satisfying a filling condition and a perturbation datum is given. Several important classes of examples of tensor categories are shown to fit into this construction. Certain invariants such as a pivotal group-like element and quantum and Frobenius–Perron dimensions of simple objects are computed. [Copyright &y& Elsevier]

Abstract: By means of time-dependent similarity transformations, we derive exact solutions of the Fokker–Planck equations with moving boundaries in the presence of: (1) a time-dependent linear force and (2) a time-dependent nonlinear force. The method of similarity transformation is simple and can be easily applied to more general Fokker–Planck equations. Furthermore, the knowledge of the exact solutions in closed form can be useful as a benchmark to test approximate numerical or analytical procedures. [Copyright &y& Elsevier]

Abstract: Selective attention systems have generally been developed as models of human attention, and they have been evaluated on that basis. Now, however, they are being used as front ends to object recognition systems, and in particular to appearance-based recognition systems. As such, they need to be evaluated by other criteria. We argue that to serve as effective front ends for object recognition, selective attention systems should (1) select fixation points in scale as well as position, and (2) be insensitive to 2D similarity transformations of the image (i.e., in-plane translations, rotations, reflections, and scales). This paper evaluates the Neuromorphic Vision Toolkit (NVT), a well-known selective attention system, and finds that it satisfies neither criterion. Further investigation, however, suggests that the sensitivity of NVT to similarity transformation is an artifact of its implementation. We develop a new system, called selective attention as a front end (SAFE), that is based on the same principles as NVT, but selects both the scale and position of fixation points and is largely invariant to 2D similarity transformations. [Copyright &y& Elsevier]