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The classification and the geometry of corank one map germs from (C², 0) → (C³, 0) have been studied by Mond [1, 2]. In this paper we characterize the classification of map germs of corank at most 1, in terms of certain invariants. Moreover, by using this characterization, we develop an algorithm to compute the type of map germs with out computing the normal form. Also, we give its implementation in the computer algebra system SINGULAR [15]. [ABSTRACT FROM AUTHOR]

We give an explicit formula for the log-canonical threshold of a reduced germ of plane curve. The formula depends only on the first two maximal contact values of the branches and their intersection multiplicities. We also improve the two branches formula given in [27]. [ABSTRACT FROM PUBLISHER]

In 7.1 we prove the iso-definability of Cˆ when C is a curve. This is done using Riemann-Roch. In 7.2 we explain how definable types on C correspond to germs of paths on Cˆ. The remainder of the chapter is devoted to the construction of the retraction on skeleta for curves. A key result is the finiteness of forwardbranching points proved in Proposition 7.4.5. [ABSTRACT FROM AUTHOR]

We study in this paper a class of smoothly finite determined germs of diffeomorphisms which have infinitely many resonant relations near a nonstrongly 1-resonant fixed point on and derive the simplest normal forms of such germs with nondegenerated nonlinear parts. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to show how some results about codimension one foliations in dimension three can be generalized to dimension two foliations in dimension n ≥ 4. [ABSTRACT FROM AUTHOR]

Here we state a conjecture concerning a local version of Brunella's alternative: any codimension one foliation in (C³, 0) without germ of invariant surface has a neighborhood of the origin formed by leaves containing a germ of analytic curve at the origin. We prove the conjecture for the class of codimension one foliations whose reduction of singularities is obtained by blowing-up points and curves of equireduction and such that the final singularities are free of saddle-nodes. The concept of "partial separatrix" for a given reduction of singularities has a central role in our argumentations, as well as the quantitative control of the generic Camacho-Sad index in dimension three. The "nodal components" are the only possible obstructions to get such germs of analytic curves. We use the partial separatrices to push the leaves near a nodal component towards compact dicritical divisors, finding in this way the desired analytic curves. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying the structure of codimension 1 singular holomorphic foliations on ℂ³, 0 without invariant germs of analytic surface. We focus on the socalled CH-foliations, that is, foliations without saddle nodes in 2D sections. Considering a reduction of singularities, we detect the possible existence of "nodal components", which are a higher-dimensional version of the nodal separators in dimension 2. If the foliation is without nodal components, we prove that all the leaves in a neighborhood of the origin contain at least one germ of analytic curve at the origin. We also study the structure of nodal components for the case of "Relatively Isolated CH-foliations" and we show that they cut the dicritical components or they exit the origin through a non-compact invariant curve. This allows us to give a precise statement of a local version of Brunella's alternative: if we do not have an invariant surface, all the leaves contain a germ of analytic curve or it is possible to detect the nodal components in the generic points of the singular curves before doing the reduction of singularities. [ABSTRACT FROM AUTHOR]

ABSTRACT The Quermass-interaction model allows to generalize the classical germ-grain Boolean model in adding a morphological interaction between the grains. It enables to model random structures with specific morphologies, which are unlikely to be generated from a Boolean model. The Quermass-interaction model depends in particular on an intensity parameter, which is impossible to estimate from classical likelihood or pseudo-likelihood approaches because the number of points is not observable from a germ-grain set. In this paper, we present a procedure based on the Takacs-Fiksel method, which is able to estimate all parameters of the Quermass-interaction model, including the intensity. An intensive simulation study is conducted to assess the efficiency of the procedure and to provide practical recommendations. It also illustrates that the estimation of the intensity parameter is crucial in order to identify the model. The Quermass-interaction model is finally fitted by our method to P. Diggle's heather data set. [ABSTRACT FROM AUTHOR]

We describe a natural decomposition of a normal complex surface singularity ( X, 0) into its 'thick' and 'thin' parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of ( X, 0) in terms of its topology and a finite list of numerical bilipschitz invariants. [ABSTRACT FROM AUTHOR]

Let f(z) = e2piaz + O(z²), a ∈ R, be a germ of a holomorphic diffeomorphism in C. For a rational and f of infinite order, the space of conformal conjugacy classes of germs topologically conjugate to f is parametrized by the Ecalle-Voronin invariants (and in particular is infinite-dimensional). When a is irrational and f is nonlinearizable it is not known whether f admits quasiconformal deformations. We show that if f has a sequence of repelling periodic orbits converging to the fixed point, then f embeds into an infinite-dimensional family of quasi-conformally conjugate germs, no two of which are conformally conjugate. [ABSTRACT FROM AUTHOR]

We produce new examples supporting the Mond conjecture which can be stated as follows. The number of parameters needed for a miniversal unfolding of a finitely determined map-germ from n-space to (n + 1)-space is less than (or equal to if the map-germ is weighted homogeneous) the rank of the nth homology group of the image of a stable perturbation of the map-germ. We give examples of finitely determined map-germs of corank 2 from 3-space to 4-space satisfying the conjecture. We introduce a new type of augmentations to generate series of finitely determined map-germs in dimensions (n, n + 1) from a given one in dimensions (n - 1, n). We present more examples in dimensions (4, 5) and (5, 6) based on our examples, and verify the conjecture for them. [ABSTRACT FROM AUTHOR]

In this paper, we describe the moduli space of germs of generic families of analytic diffeomorphisms which unfold a parabolic fixed point of codimension 1. In [11] (and also [14]), it was shown that the Ecalle–Voronin modulus can be unfolded to give a complete modulus for such germs. The modulus is defined on a ramified sector in the canonical perturbation parameter ϵ. As in the case of the Ecalle–Voronin modulus, the modulus is defined up to a linear scaling depending only on ϵ. Here, we characterize the moduli space for such unfoldings by finding the compatibility conditions on the modulus which are necessary and sufficient for realization as the modulus of an unfolding. The compatibility condition is obtained by considering the region of sectorial overlap in ϵ-space. This lies in the Glutsyuk sector where the two fixed points are hyperbolic and connected by the orbits of the diffeomorphism. In this region, we have two representatives of the modulus which describe the same dynamics. We identify the necessary compatibility condition between these two representatives by comparing them both with their common Glutsyuk modulus. The compatibility condition implies the existence of a linear scaling for which the modulus is -summable in ϵ, whose direction of nonsummability coincides with the direction of real multipliers at the fixed points. Conversely, we show that the compatibility condition (which implies the summability property) is sufficient to realize the modulus as coming from an analytic unfolding, thus giving a complete description of the space of moduli. The terminology “space” of moduli is justified by the fact that the moduli depend analytically on extra parameters. [ABSTRACT FROM AUTHOR]

Abstract: We focus on topological equisingularity of families of holomorphic function germs with 1-dimensional critical set. We introduce the notion of equisingularity at the critical set and prove that any family which is equisingular at the critical set is topologically equisingular. We show that if a family of germs with 1-dimensional critical set has constant generic Lê numbers then it is equisingular at the critical set, and hence topologically equisingular (answering a question of D. Massey [13]). It is worth to remark that this does not happen for higher dimensional critical set [5]. We use these topological triviality results to modify the definition of singularity stem present in the literature, introducing and characterising topological stems (being this concept closely related with Arnoldʼs series of singularities). We provide another sufficient condition for topological equisingularity for families whose reduced critical set is deformed flatly. Finally we study how the critical set can be deformed in a topologically equisingular family and provide examples of topologically equisingular families whose critical set is a non-flat deformation with singular special fibre and smooth generic fibre. [Copyright &y& Elsevier]

Abstract: In this paper we present the algorithms used by the second author in Hernandes (2011) for the implementation in MAPLE of the results of Hefez and Hernandes (2011) that allow to perform the analytic classification of irreducible germs of singular complex plane curves. [Copyright &y& Elsevier]

A concurrent history represented by a causality structure that captures the intrinsic, invariant dependencies between its actions, can be interpreted as defining a set of closely related observations (e.g., step sequences). Depending on the relationships observed in the histories of a system, the concurrency paradigm to which it adheres may be identified, with different concurrency paradigms underpinned by different kinds of causality structures. Elementary net systems with inhibitor arcs and mutex arcs (ENIM-systems) are a system model that through its process semantics and associated causality structures fits the least restrictive concurrency paradigm. One can also investigate the abstract behaviour of an ENIM-system by grouping together step sequences in equivalence classes (generalised comtraces) using the structural relations between its transitions. The resulting concurrent histories of the ENIM-system are consistent with the generalised stratified order structures underlying its processes. The paper establishes a link between ENIM-systems and trace theory allowing one to discuss different observations of concurrent behaviour in a way that is consistent with the causality semantics defined by the operationally defined processes. [ABSTRACT FROM AUTHOR]

Let α1,…,αn be irrational numbers which are linearly independent over the rationals, and f1,…,fn commuting germs of holomorphic diffeomorphisms in ℂ such that fk(0)=0,f′k(0)=e2πiαk,k=1,…,n. Moser showed that f 1,…,fn are simultaneously linearizable (i.e. conjugate by a germ of holomorphic diffeomorphism h to the rigid rotations Rαk(z)=e2πiαkz) if the vector of rotation numbers (α 1,…,αn) satisfies a Diophantine condition. Adapting Yoccoz’s renormalization to the setting of commuting germs, we show that simultaneous linearization holds in the presence of a weaker Brjuno-type condition ℬ(α1,…,αn)<+∞, where ℬ(α1,…,αn) is a multivariable analogue of the Brjuno function. If there are no periodic orbits for the action of the germs f1,…,fn in a neighbourhood of the origin, then a weaker arithmetic condition ℬ′ (α 1,…,αn)<+∞ analogous to Perez-Marco’s condition for linearization in the absence of periodic orbits is shown to suffice for linearizability. Normalizing the germs to be univalent on the unit disc, in both cases the Siegel discs are shown to contain discs of radii $C e^{-2\pi \mathcal {B}}, C e^{-2\pi \mathcal {B'}}$ for some universal constant C. [ABSTRACT FROM PUBLISHER]

We prove hypoellipticity in the sense of germs for the operator P = LqLq + Lqt2kLq + Q2, where Lq = Dt + itq-1√-Δx and Q = x1D2 - x2D1, even though it fails to be hypoelliptic in the strong sense. The primary tool is an a priori estimate. [ABSTRACT FROM AUTHOR]

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ℂ. More precisely, we are interested in the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms φ, η such that every formal conjugation $$\hat \sigma$$ (i.e. $$\eta \circ \hat \sigma = \hat \sigma \circ \phi$$) does not extend to the fixed points set Fix( φ) of φ, meaning that it is not transversally formal (or semi-convergent) along Fix( φ). We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points. [ABSTRACT FROM AUTHOR]

Abstract: We investigate a relationship between the caustics of a submanifold of general dimension and of a canal hypersurface of the submanifold in Euclidean space. As a consequence, these caustics are the same. Moreover, induced Lagrangian immersion germs are Lagrangian equivalent under a suitable condition. In order to show the results, we use the theory of Lagrangian singularity and of Legendrian singularity. [Copyright &y& Elsevier]

Abstract: We characterize finite determinacy of map germs in terms of the Milnor number of the double point curve in and we provide an explicit description of the double point scheme in terms of elementary symmetric functions. Also we prove that the Whitney equisingularity of 1-parameter families of map germs is equivalent to the constancy of both and with respect to t, where is a generic plane. [Copyright &y& Elsevier]

Abstract: Suppose that is a smoothly stable, -level preserving germ which unfolds ; then f is smoothly stable if and only if we can find a pair of smooth retractions and such that . Unfortunately, we do not know whether f will be topologically stable if we can find a pair of continuous retractions r and s. The class of extremely tame (E-tame) retractions, introduced by du Plessis and Wall, are defined by their nice geometric properties, which are sufficient to ensure that f is topologically stable. In this article, we present the E-tame retractions and their relation with topological stability, survey recent results by the author concerning their construction, and illustrate the use of our techniques by constructing E-tame retractions for certain germs belonging to the E- and Z-series of singularities. [Copyright &y& Elsevier]

Abstract: This article is devoted to criteria of Lipschitz equisingularity for families of real analytic map germs from to with like , where λ is a small real number. The main result of this article, Theorem 4.2 states a condition on the order of the terms of , in such a way that the family is bi-Lipschitz -trivial. Theorem 4.2 gives the conditions in terms of Newton polyhedron associated to the germ g. The tools used here are based in the construction of convenient controlled vector fields. [Copyright &y& Elsevier]

Abstract: We establish a short exact sequence to relate the germ model of invariant subspaces of a Hilbert space of vector-valued analytic functions and the sheaf model of the corresponding coinvariant subspaces. As a consequence we obtain an additive formula for Samuel multiplicities. As an application, we give a different proof for a formula relating the fibre dimension and the Samuel multiplicity which is first proved in Fang (2005) . The feature of the new proof is that the analytic arguments in Fang (2005) are now subsumed by algebraic machinery. [ABSTRACT FROM AUTHOR]

Let F( z) = Re( P( z)) + h.o.t be such that M = ( F = 0) defines a germ of real analytic Levi-flat at 0 ∈ ℂ, n ≥ 2, where P ( z) is a homogeneous polynomial of degree k with an isolated singularity at 0 ∈ ℂ and Milnor number µ. We prove that there exists a holomorphic change of coordinate ϕ such that ϕ( M) = ( Re( h) = 0), where h( z) is a polynomial of degree µ + 1 and j ( h) = P. [ABSTRACT FROM AUTHOR]

We answer a question of Vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles. We show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality. We deduce several other characterizations of the lattice property. [ABSTRACT FROM AUTHOR]

Abstract: We study the subvariety of integrable 1-forms in a finite-dimensional vector space . We prove that the irreducible components with dimension comparable with the rank of W are of minimal degree. [Copyright &y& Elsevier]

Abstract: In this Note, we study germs of holomorphic vector fields which are suitable perturbations of a quasihomogeneous vector field in a neighborhood of the origin of , fixed point of the vector fields. In particular, we define a “diophantine condition” on the quasihomogeneous initial part S which ensures that if such a perturbation of S is formally conjugate to S then it is also holomorphically conjugate to it. To cite this article: E. Lombardi, L. Stolovitch, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract. Let Δ ⊂ H be the germ of a nonnormal surface along a proper curve with smooth components such that the high index points of H are semi-log-terminal and the Gorenstein singular points are semi-log-canonical of embedding dimension at most 4. We describe the sheaf T[sup1][subqG](H) of Q-Gorenstein deformations of H and we obtain criteria for Δ ⊂ H to have Q-Gorenstein terminal smoothings. [ABSTRACT FROM AUTHOR]

Abstract: Let be tangent to the identity and with order . We study the dynamics of f near the set of his fixed points. Using some known results in this domain, we prove that if the set of fixed points of f is smooth at the origin, f is tangential to this set, and the origin is not singular, then there are no parabolic curves for f at the origin. After that, we use some adapted techniques to prove the existence of parabolic curves for f at the origin if the set of fixed points of f is smooth at the origin and this last one is a singular point of f with the pure order of f . Finally, we prove that if the origin is dicritical, then there exist infinitely many parabolic curves. [Copyright &y& Elsevier]

AbstractWe investigate in this paper the topological stability of pairs (?, X), where ? is a germ of an integrable 1-form and Xis a germ of a vector field tangent to the foliation determined by ?. [ABSTRACT FROM AUTHOR]

The completion of genome sequences and subsequent high-throughput mapping of molecular networks have allowed us to study biology from the network perspective. Experimental, statistical and mathematical modeling approaches have been employed to study the structure, function and dynamics of molecular networks, and begin to reveal important links of various network properties to the functions of the biological systems. In agreement with these functional links, evolutionary selection of a network is apparently based on the function, rather than directly on the structure of the network. Dynamic modularity is one of the prominent features of molecular networks. Taking advantage of such a feature may simplify network-based biological studies through construction of process-specific modular networks and provide functional and mechanistic insights linking genotypic variations to complex traits or diseases, which is likely to be a key approach in the next wave of understanding complex human diseases. With the development of ready-to-use network analysis and modeling tools the networks approaches will be infused into everyday biological research in the near future.Cell Research (2008) 18:224–237. doi: 10.1038/cr.2008.16; published online 29 January 2008 [ABSTRACT FROM AUTHOR]

AbstractAll {\cA}-simple corank-1 germs from $\bR ^n$to $\bR ^{n+1}$, where n? 4, have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the image. [ABSTRACT FROM AUTHOR]

Abstract: Let f be a real-analytic function germ at the origin in , whose critical locus contains a given real-analytic set X, and let Y be a germ of a closed subset at the origin. We study the stability of f under perturbations u that are flat on Y and that belong to a given Denjoy–Carleman non-quasianalytic class. We obtain a condition ensuring that where Φ is a germ of diffeomorphism whose components belong to a (generally larger) Denjoy–Carleman class. Roughly speaking, this condition involves a Łojasiewicz-type separation property between Y and the complex zeros of a certain ideal associated with f and X. The relationship between the Denjoy–Carleman classes of u and Φ is controlled precisely by the inequality. This result extends, and simplifies, former work of the author on germs with isolated critical points. [Copyright &y& Elsevier]

The only stable singularities of a real map-germ $f:{\mathbb R} ^n\to{\mathbb R} ^{2n}$ are isolated transverse double-points. All ${{\cal A}}$-simple germs f have a deformation with the maximal number d(f) of real double-points (this is a partial generalization to higher n of the result of A'Campo [1] and Gusein-Zade [13] that all plane curve-germs have a deformation with δ real double points, with the extra hypothesis of ${{\cal A}}$-simplicity). The proof of this result is based on a classification of all ${{\cal A}}$-simple orbits. [ABSTRACT FROM PUBLISHER]

The article discusses the indestructibility and level by level equivalence and inequivalence. It mentions that the question of whether it is possible to have both an indestructibly supercompact cardinal and full level by level equivalence between strong compactness and supercompactness in a universe with few large cardinals is still open. It concludes by posing an analogous question for a model in which level by level equivalence between strong compactness and supercompactness fails everywhere.

Abstract: A tangential family is a 1-parameter system of regular curves emanating tangentially from another regular curve. We classify simple tangential family germs up to -equivalence. We describe perestroikas of envelopes of simple tangential family germs of small codimension under small deformations of the germ among tangential families. [Copyright &y& Elsevier]

It is shown that H(K, F) is regular for every reflexive Fréchet space F with the property (DN) where K is a balanced convex compact set in a nuclear Fréchet with the property ((...)K). Using this result we give necessary and sufficient conditions for a compact set K in a nuclear Fréchet space E with the bounded approximation property (resp. a complex space), under which every separately holomorphic function on K x F* is holomorphic, where F is a nuclear Fréchet space. [ABSTRACT FROM AUTHOR]

We give an algorithm that describes the singularity of all but finitely-many germs in a pencil generated by two arbitrary germs of plane curve. [ABSTRACT FROM AUTHOR]

We study the structure of germs of analysis function with critical sets of codimension one in the cases where the transversal singularity type is Ak. We calculated the tangent space to orbits of such germs under the group of diffeomorphisms the critical set and construct special deformations of such germs. An explicit algebraic formulation for the number of Morse point in special deformations is given and the homotopy structure of the Milnor fiber is described. The results are illustrated in the quasi-homogenous case and for certain singularities of low codimension. [ABSTRACT FROM AUTHOR]

For an integer n⩾3, a rank-n matroid is called an n-spike if it consists of n three-point lines through a common point such that, for all k in {1,2,…,n−1}, the union of every set of k of these lines has rank k+1. It is well known that there is a unique binary n-spike for each integer n⩾3. In this paper, we first prove that, for each integer n⩾3, there are exactly two distinct ternary n-spikes, and there are exactly ⌊(n2+6n+24)/12⌋ quaternary n-spikes. Then we prove that, for each integer n⩾4, there are exactly n+2+⌊n/2⌋ quinternary n-spikes and, for each integer n⩾18, the number of n-spikes representable over GF(7) is ⌊(2n2+6n+6)/3⌋. Finally, for each q>7, we find the asymptotic value of the number of distinct rank-n spikes that are representable over GF(q). [Copyright &y& Elsevier]