Set Of Critical Points Research Articles

Universal stratifications and a Bertini-type theorem

A stratification of the set of critical points of a map is universal in the class of stratifications satisfying the classical Thom and Whitney-a conditions if it is the coarsest among all such stratifications. We show that a universal stratification exists if and only if the ‘canonical subbundle’ of the cotangent bundle of the source of the map (constructed via operations introduced by Glaeser) is Lagrangian. The proof relies on a new Bertini-type theorem for singular varieties proved via an intriguing use of resolution of singularities. Many examples are provided, including those of maps without universal stratifications.

Asymptotic stabilization of coupled oscillators with dry friction by feedback control

In this paper we design a feedback control u=u(x,ẋ) so that each solution x(⋅) of the closed-loop system ẍ(t)+∂Φ(ẋ)+∇f(x)+u(x,ẋ)∋0 approaches the set of critical points of f(x) with |ẋ(t)|→0 as t→+∞. The robustness of the control is also discussed in the case where f(x) has only a finite number of critical values. The approach is mainly based on LaSalle’s invariance principles and Morse decomposition theory of attractors for differential inclusions.

Skeleton growing: an algorithm to extract a curve skeleton from a pseudonormal vector field

A curve skeleton is used to represent a 3D object in many different applications. It is a 1D curve that captures topology of the 3D object. The proposed method extracts a curve skeleton from the vector field inside the 3D object. A vector at each voxel of the 3D object is calculated using a pseudonormal vector. By using such a calculation, the computation time is significantly reduced compared with using a typical potential field. A curve skeleton is then extracted from the pseudonormal vector field by using a skeleton-growing algorithm. The proposed algorithm uses high-curvature boundary voxels to search for a set of critical points and skeleton branches near high-curvature areas. The set of detected critical points is then used to grow a curve skeleton in the next step. All parameters of our algorithms are calculated from the 3D object itself, without user intervention. The effectiveness of our method is demonstrated in our experiments.

Upper bounds for the number of zeroes for some Abelian integrals

Consider the vector field x′=−yG(x,y),y′=xG(x,y), where the set of critical points {G(x,y)=0} is formed by K straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree n and study the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of K and n. Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and on a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for K≤4 we recover or improve some results obtained in several previous works.

Critical points of solutions to quasilinear elliptic problems

We describe the set of critical points (points of vanishing gradient) associated to solutions of an important class of quasilinear elliptic problems with zero Dirichlet condition in planar domains. We show that the critical set is made up of finitely many isolated points and finitely many (regular) analytic Jordan curves. Further, we generalize the well-known result of Makar-Limanov, according to which the solution to the Poisson equation Δu=1, with zero Dirichlet condition in a convex domain, has a unique critical point.

Computing parameter ranges in constructive geometric constraint solving: Implementation and correctness proof

In parametric design, changing values of parameters to get different solution instances to the problem at hand is a paramount operation. One of the main issues when generating the solution instance for the actual set of parameters is that the user does not know in general which is the set of parameter values for which the parametric solution is feasible. Similarly, in constraint-based dynamic geometry, knowing the set of critical points where construction feasibility changes would allow to avoid unexpected and unwanted behaviors.We consider parametric models in the Euclidean space with one internal degree of freedom. In this scenario, in general, the set of values of the variant parameter for which the parametric model is realizable and defines a valid shape is a set of intervals on the real line.In this work we report on our experiments implementing the van der Meiden Approach to compute the set of parameter values that bound intervals for which the parametric object is realizable. The implementation is developed on top of a constructive, ruler-and-compass geometric constraint solver. We formalize the underlying concepts and prove that our implementation is correct, that is, the approach exactly computes all the feasible interval bounds.

Extrapolation of the boundary of uni- and bimodality of a two-component Gaussian mixture

A two-component Gaussian mixture with equal variances and various mathematical expectations has been studied. It has been found that a set of singular critical points of its probability density is a boundary of uni- and bimodality. The approximation equation of this boundary has been obtained on a line segment as a second-degree polynomial. The bifurcation area of the function being investigated is presented in the probabilistic form. The extrapolation algorithm of this boundary has been developed on a small line, and the experimental results have been presented.

Scale effects on the estimation of erosion thresholds through a distributed and physically-based hydrological model

Slope incision and subsequent development of rills, gullies and channels are responsible for significant soil losses and are often irreversible with very high restoration costs particularly in semiarid environments. The location of potential areas of erosion where these processes occur is vital to land management and conservation. The study of the relationship between the local slope (S) and the drainage area (A) through the geomorphological relationship S=αAb, combined with hydrologic simulation models, has proved to be appropriate for the identification and characterisation of potential areas of incision, especially when it takes into account the spatial distribution of soil properties and the evolution of hydrological processes. However, physical properties of soil, digital elevation models (DEMs) and flow algorithms used may affect the results. This study employs a distributed, physically-based hydrological model to evaluate the infiltration–runoff relationships and their influence on selecting critical area from three DEMs with different resolutions. The results show a significant scale effect on flow distribution and the location of threshold points on slopes. The results obtained from a 30-m DEM significantly differ from those obtained from 10 and 5-m DEMs because the former was unable to capture the spatial variability of geomorphic processes. The selected set of critical points shows high S–A correlations for different values of critical shear stress. The physical model confirmed the dominance of surface runoff in the study site and was validated from field identification of erosion risk areas, although for incision areas <2ha, an appreciable error remains in relation to the calculation algorithm used for the drainage network and DEM resolution.

Modeling the risk factors in ergonomic processes in Brick kilns workers using Fuzzy Logic

The paper presents a part of my research work for investigating ergonomic approach for modeling the risk factors for better productivity without compromising or minimize the risk for transforming measured body data between various working postures in Brick kiln .The measured human body is substituted by a proper set of critical body points under stress using fuzzy logic. The critical points data are used as a basis of transforming the data, the obtained data are used to describe specific working body postures.

Phase Diagram of XY Model with Nematic Coupling on the Simple Cubic Lattice

Using cluster Monte Carlo method, we numerically investigate the criticality in the XY model with nematic coupling on the simple cubic lattice. We determine critical lines belong to the three-dimensional XY universality class in variable of θ (2θ) between the XY-ferromagnetic (nematic) and disordered states. Furthermore, the phase transition between the XY-ferromagnetic and the nematic states is found to be in the three-dimensional Ising universality class. The critical points are determined from the intersections of Binder ratios for various system sizes. With two sets of critical points obtained, we finally construct the phase diagram on the Δ-J plane.

From Self-Organized to Extended Criticality

We address the issue of criticality that is attracting the attention of an increasing number of neurophysiologists. Our main purpose is to establish the specific nature of some dynamical processes that although physically different, are usually termed as “critical,” and we focus on those characterized by the cooperative interaction of many units. We notice that the term “criticality” has been adopted to denote both noise-induced phase transitions and Self-Organized Criticality (SOC) with no clear connection with the traditional phase transitions, namely the transformation of a thermodynamic system from one state of matter to another. We notice the recent attractive proposal of extended criticality advocated by Bailly and Longo, which is realized through a wide set of critical points rather than emerging as a singularity from a unique value of the control parameter. We study a set of cooperatively firing neurons and we show that for an extended set of interaction couplings the system exhibits a form of temporal complexity similar to that emerging at criticality from ordinary phase transitions. This extended criticality regime is characterized by three main properties: (i) In the ideal limiting case of infinitely large time period, temporal complexity corresponds to Mittag-Leffler complexity; (ii) For large values of the interaction coupling the periodic nature of the process becomes predominant while maintaining to some extent, in the intermediate time asymptotic region, the signature of complexity; (iii) Focusing our attention on firing neuron avalanches, we find two of the popular SOC properties, namely the power indexes 2 and 1.5 respectively for time length and for the intensity of the avalanches. We derive the main conclusion that SOC emerges from extended criticality, thereby explaining the experimental observation of Plenz and Beggs: avalanches occur in time with surprisingly regularity, in apparent conflict with the temporal complexity of physical critical points.

Critical points of maximal D = 8 gauged supergravities

We study the general deformations of maximal eight-dimensional supergravity by using the embedding tensor approach. The scalar potential induced by these gaugings is determined. Subsequently, by combining duality covariance arguments and algebraic geometry techniques, we find the complete set of critical points of the scalar potential. Remarkably, up to SO(2) × SO(3) rotations there turns out to be a unique theory admitting extrema. The gauge group of the theory is CSO(2, 0, 1).

Critical points of Green's function and geometric function theory

We study questions related to critical points of the Green's func- tion of a bounded multiply connected domain in the complex plane. The motion of critical points, their limiting positions as the pole approaches the boundary and the differential geometry of the level lines of the Green's func- tion are main themes in the paper. A unifying role is played by various affine and projective connections and corresponding Mobius invariant differential op- erators. In the doubly connected case the three Eisenstein series E2, E4, E6 are used. A specific result is that a doubly connected domain is the disjoint union of the set of critical points of the Green's function, the set of zeros of the Bergman kernel and the separating boundary limit positions for these. At the end we consider the projective properties of the prepotential associated to a second order differential operator depending canonically on the domain.

The non-linear Schrödinger equation with non-periodic potential: infinite-bump solutions and non-degeneracy

The equation $-\varepsilon^2 \Delta u+F(V(x),u)=0$ is considered in $R^n$. It is assumed that $V$ possesses a set of critical points $B$ for which the values of $V$ and $D^2V$ satisfy certain compactness and uniformity properties. Under appropriate conditions on $F$ the problem is shown to possess for each $b\in B$ and small $\varepsilon>0$ a solution that concentrates at $b$ and has detailed uniformity and decay properties. This enables the construction of solutions that concentrate at arbitrary subsets of $B$ as $\varepsilon\to 0$. Examples are given in which $B$ is infinite and $V$ non-periodic.

Interpolation and extrapolation of the boundary of uni- and bimodality of a two-component Gaussian mixture

A two-component Gaussian mixture with equal variances and different mathematical expectations is studied. It is ascertained that the set of degenerate critical points of its probability density is the boundary of the uni- and bimodality. An approximate equation of this boundary is expressed by the distribution parameters. The domain of bifurcations of the investigated function is given in the probabilistic form. An experiment on the extrapolation of the equation of this boundary to the small segment is presented.

The persistent cosmic web and its filamentary structure - I. Theory and implementation

We present DisPerSE, a novel approach to the coherent multi-scale identification of all types of astrophysical structures, and in particular the filaments, in the large scale distribution of matter in the Universe. This method and corresponding piece of software allows a genuinely scale free and parameter free identification of the voids, walls, filaments, clusters and their configuration within the cosmic web, directly from the discrete distribution of particles in N-body simulations or galaxies in sparse observational catalogues. To achieve that goal, the method works directly over the Delaunay tessellation of the discrete sample and uses the DTFE density computed at each tracer particle; no further sampling, smoothing or processing of the density field is required. The idea is based on recent advances in distinct sub-domains of computational topology, which allows a rigorous application of topological principles to astrophysical data sets, taking into account uncertainties and Poisson noise. Practically, the user can define a given persistence level in terms of robustness with respect to noise (defined as a "number of sigmas") and the algorithm returns the structures with the corresponding significance as sets of critical points, lines, surfaces and volumes corresponding to the clusters, filaments, walls and voids; filaments, connected at cluster nodes, crawling along the edges of walls bounding the voids. The method is also interesting as it allows for a robust quantification of the topological properties of a discrete distribution in terms of Betti numbers or Euler characteristics, without having to resort to smoothing or having to define a particular scale. In this paper, we introduce the necessary mathematical background and describe the method and implementation, while we address the application to 3D simulated and observed data sets to the companion paper.

The Stress Analysis of Different Circular Saw Structures during Cutting

In this paper, we analyze the stresses around the saw teeth of differently designed circular saw structures. We select a set of critical points around the saw tooth, then we calculate the equivalent stresses of these points and the maximum or minimum stresses of the different saws under different loads and rotational speeds. The saw structure factors that we examine include the diameter of circular saw blades, the depth of the circular saw, the saw tooth rake angel, the saw tooth height and the saw tooth shape. Based on our analytical results, we propose an effective approach to reduce the stress concentration around the saw tooth and thus prevent the saw tooth fracture.

What does learning to ‘draw a circle’ have to do with driving, cycling, unwinding and screwing?

What does learning to ‘draw a circle’ have to do with driving, cycling, unwinding and screwing?

A SMOOTH FUNCTION ON A MANIFOLD WITH GIVEN REEB GRAPH

We show that any finite graph without loops can be realized as the Reeb graph of a smooth function on a closed manifold with finitely many critical values, but possibly with positive dimensional critical point set. We also show that such a function can be chosen as the height function on a surface immersed in 3-space, provided that the graph has no isolated vertices.

Trace map, Cayley transform and LS category of Lie groups

The aim of this article is to use the so-called Cayley transform in order to compute the LS category of Lie groups and homogeneous spaces by giving explicit categorical open coverings. When applied to U(n), U(2n)/Sp(n) and U(n)/O(n) this method is simpler than those formerly known. We also show that the Cayley transform is related to height functions in Lie groups, allowing to give a local linear model of the set of critical points. As an application we give an explicit covering of Sp(2) by categorical open sets. The obstacles to generalize these results to Sp(n) are discussed.