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The behavior and properties of solutions of two-dimensional quadratic polynomial dynamical system on the phase plane of variables and time are considered. A complete qualitative theory is constructed which includes the analysis of all singular points and the features of solutions depending on all parameters of the problem. A main result is that, with the discriminant criteria created on the basis of the growth model constructed in the present study, it is possible to formulate practical recommendations for regulating and monitoring the process of waterflooding and the development of an oil field. [ABSTRACT FROM AUTHOR]

A novel optimization algorithm for the calibration of redundantly actuated parallel manipulator (RA-PKM) is presented. The method uses an analytical indicator function (IF) which takes a specific value at the motion reversal points (MRP). MRP are configurations where the speed of one passive actuator vanishes during a continuous motion. IF depends on the passive actuator, the locked actuators, the geometrical parameters. The main idea is to perform an optmization that makes the IF equal to the specific value at the MRP whenever the geometrical parameters match the real values. Simulations of the calibration of a planar 4-RPR RA-PKM are presented. Communicated by S. Velinsky. [ABSTRACT FROM AUTHOR]

The homological conjectures, which date back to Peskine, Szpiro and Hochster in the late 60's, make fundamental predictions about syzygies and intersection problems in commutative algebra. They were settled long ago in the presence of a base field and led to tight closure theory, a powerful tool to investigate singularities in characteristic p. Recently, perfectoid techniques coming from p-adic Hodge theory have allowed us to get rid of any base field; this solves the direct summand conjecture and establishes the existence and weak functoriality of big Cohen-Macaulay algebras, which solve in turn the homological conjectures in general. This also opens the way to the study of singularities in mixed characteristic. We sketch a broad outline of this story, taking lastly a glimpse at ongoing work by L. Ma and K. Schwede, which shows how such a study could build a bridge between singularity theory in characteristic p and in characteristic 0. [ABSTRACT FROM AUTHOR]

Abstract Let P (t) ∈ P 2 (K (t)) be a rational projective parametrization of a plane curve C. In this paper, we introduce the notion of limit point , P L , of P (t) , and some remarkable properties of P L are obtained. In particular, if the singularities of C are P 1 , ... , P n and P L (all of them ordinary) and their respective multiplicities are m 1 , ... , m n and m L , we show that T (s) = ∏ i = 1 n H P i (s) m i − 1 H P L (s) m L − 1 , where T (s) is the univariate resultant of two polynomials obtained from P (t) , and H P 1 (s) , ... , H P n (s) , H P L (s) are the fibre functions of the singularities. The fibre function of a point P is a polynomial H P (s) whose roots are the fibre of P. Thus, a complete classification of the singularities of a given plane curve, via the factorization of a resultant, is obtained. [ABSTRACT FROM AUTHOR]

Abstract In this article, we discuss formal invariants of singularly-perturbed linear differential systems in neighborhood of turning points and give algorithms which allow their computation. The algorithms proposed are implemented in the computer algebra system Maple. [ABSTRACT FROM AUTHOR]

Given a ruled surface V defined in the standard parametric form P (t 1 , t 2) , we present an algorithm that determines the singularities (and their multiplicities) of V from the parametrization P. More precisely, from P we construct an auxiliary parametric curve and we show how the problem can be simplified to determine the singularities of this auxiliary curve. Only one univariate resultant has to be computed and no elimination theory techniques are necessary. These results improve some previous algorithms for detecting singularities for the special case of parametric ruled surfaces. [ABSTRACT FROM AUTHOR]

In the present paper, we study the following singular Kirchhoff problem M ∬ R 2 N | u (x) − u (y) | 2 | x − y | N + 2 s d x d y (− Δ) s u = λ f (x) u − γ + g (x) u 2 s ∗ − 1 in Ω , u > 0 in Ω , u = 0 in R N ∖ Ω , where Ω ⊂ R N is an open bounded domain, dimension N > 2 s with s ∈ (0 , 1) , 2 s ∗ = 2 N ∕ (N − 2 s) is the fractional critical Sobolev exponent, parameter λ > 0 , exponent γ ∈ (0 , 1) , M models a Kirchhoff coefficient, f ∈ L 2 s ∗ 2 s ∗ + γ − 1 (Ω) is a positive weight, while g ∈ L ∞ (Ω) is a sign-changing function. Using the idea of Nehari manifold technique, we prove the existence of at least two positive solutions for a sufficiently small choice of λ. This approach allows us to avoid any restriction on the boundary of Ω. [ABSTRACT FROM AUTHOR]

We investigate the statistical properties of piecewise expanding maps on the unit interval, whose inverse Jacobian may have low regularity near singularities. The method is new yet simple: instead of directly working with the 1-d map, we first lift the 1-d expanding map to a hyperbolic map on the unit square, and then take advantage of the functional analytic method developed by Demers and Zhang in [21,22,23] for hyperbolic systems with singularities. By projecting back to the 1-d map, we are able to prove that it inherits nice statistical properties, including the large deviation principle, the exponential decay of correlations, as well as the almost sure invariance principle for the expanding map on a large class of observables. Moreover, we are able to prove that the projected SRB measure has a piecewise continuous density function. Our results apply to rather general 1-d expanding maps, including some C1 perturbations of the Lorenz-like map and the Gauss map whose statistical properties are still unknown as they fail all other available methods. [ABSTRACT FROM AUTHOR]

Although some authors highlight the benefits of journal rankings, previous research is often highly critical of them, insinuating that they can lead to desingularization of academic journals (i.e. their impoverishment and standardization) and dequalification of researchers (i.e. a weakening of researchers' ability to evaluate academic research). However, as very few authors have empirically assessed these presumptions, we aim to address this gap in the literature. Based on Lucien Karpik's notions of singularities, judgment devices, forms of involvement, and emulation and rivalry, we assess whether the processes surrounding the production and use of journal rankings might lead to desingularization and dequalification. Our findings support previous research by highlighting that processes where passivity and heteronomy (i.e. lack of autonomy) prevail are conducive to desingularization, rivalry and dequalification. Our findings, however, introduce some nuances into the debate by underscoring instances where emulation logic is employed instead of mere rivalry logic, and where substantial judgment devices and active involvement are mobilized in the production and use of rankings, thereby somewhat alleviating the spread of desingularization and dequalification. Ultimately, our study raises questions that point to a need for serious collective reflection within the academic community on the processes by which published research is evaluated. [ABSTRACT FROM AUTHOR]

We consider the reconstruction of spike train signals of the form F (x) = ∑ i = 1 d a i δ (x - x i) , from their moments measurements m k (F) = ∫ x k F (x) d x = ∑ i = 1 d a i x k . When some of the nodes x i near collide the inversion becomes unstable. Given noisy moments measurements, a typical consequence is that reconstruction algorithms estimate the signal F with a signal having fewer nodes, F ~ . We derive lower bounds for the moments difference between a signal F with d nodes and a signal F ~ with strictly less nodes, l. Next we consider the geometry of the non generic case of d nodes signals F, for which there exists an l < d nodes signal F ~ , with moments m 0 (F ~) = m 0 (F) , ... , m p (F ~) = m p (F) , p > 2 l - 1. We give a complete description for the case of a general d, l = 1 and p = 2 . We give a reference for the case p = 2 l - 1 which can be inferred from earlier work. [ABSTRACT FROM AUTHOR]

For mechanical Hamiltonian systems on the torus, we study the dynamical properties of the generalized characteristic semiflows associated with the Hamilton-Jacobi equations, and build the relation between the ω -limit sets of the semiflows and the projected Aubry sets. [ABSTRACT FROM AUTHOR]

We report on recent developments toward a relativistic quantum-mechanical theory of motion for a fixed, finite number of electrons, photons and their anti-particles, as well as its possible generalizations to other particles and interactions. [ABSTRACT FROM AUTHOR]

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant Gaussian curvature surfaces correspond to harmonic maps, and we examine the relationship between the two types of maps and their singularities. Finally, we determine which finitely A -determined map-germs from the plane to the plane can be represented by harmonic maps. [ABSTRACT FROM AUTHOR]

Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no longer well-defined and has to be replaced by new concepts. This article discusses the nature of two such concepts, recently introduced by Bowman and Valdez: linear approaches and rotational components. We show that there is a large flexibility in the spaces of rotational components and even more in the spaces of linear approaches. In particular, we prove that every finite topological space arises as space of rotational components. However, this space will still not contain enough information to describe an infinite translation surface. We showcase this through an uncountable family with the same space of rotational components but different spaces of linear approaches. Additionally, we study several known and new examples to illustrate the concept of linear approaches and rotational components. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to characterize ideals I in the power series ring R = K [ [ x 1 , ... , x s ] ] that are finitely determined up to contact equivalence by proving that this is the case if and only if I is an isolated complete intersection singularity, provided dim (R / I) > 0 and K is an infinite field (of arbitrary characteristic). Here two ideals I and J are contact equivalent if the local K –algebras R / I and R / J are isomorphic. If I is minimally generated by a 1 , ... , a m , we call I finitely contact determined if it is contact equivalent to any ideal J that can be generated by b 1 , ... , b m with a i − b i ∈ 〈 x 1 , ... , x s 〉 k for some integer k. We give also computable and semicontinuous determinacy bounds. The above result is proved by considering left–right equivalence on the ring M m , n of m × n matrices A with entries in R and we show that the Fitting ideals of a finitely determined matrix in M m , n have maximal height, a result of independent interest. The case of ideals is treated by considering 1-column matrices. Fitting ideals together with a special construction are used to prove the characterization of finite determinacy for ideals in R. Some results of this paper are known in characteristic 0, but they need new (and more sophisticated) arguments in positive characteristic partly because the tangent space to the orbit of the left-right group cannot be described in the classical way. In addition we point out several other oddities, including the concept of specialization for power series, where the classical approach (due to Krull) does not work anymore. We include some open problems and a conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we study the formation of finite-time singularities of smooth solutions to the compressible Euler equations with damping in arbitrary dimensions for isentropic, polytropic fluid flows. The radial component of initial momentum is assumed to be large enough and the smooth solutions are assumed to be decay at far fields, but it is not required that the initial velocity field has a compact support. It is shown that the smooth solutions to the Cauchy problem will break down in a finite time. Our result partially complements the work for the compactly supported initial velocity field (Sideris et al., 2003). [ABSTRACT FROM AUTHOR]

The paper describes the configuring and analysis of a class of generalized reconfigurable 2-axis parallel kinematic machine (R2PKM). A generalized model which is used for solving inverse and direct kinematic problem is presented. Generalized equations that present the solutions of kinematic problems are derived and they are valid for any configuration of R2PKM. Characteristic configurations of the mechanism are given as an example of the concept realization. For selected configurations, workspaces are defined and possible singular positions of the mechanism are analyzed. Optimization of the mechanism legs length was performed using the condition number. Determination of optimal legs length has been shown for several different configurations of the parallel mechanism. Control and programming system is configured and experimental prototype of R2PKM is realized. Verification of the configured machine, configured control and programming system was performed on the virtual machine and also on the prototype of the machine by drawing several different contours. [ABSTRACT FROM AUTHOR]

SUMMARY: This paper deals with the kinematic analysis and enumeration of singularities of the six degree-of-freedom 3-RPS-3-SPR series–parallel manipulator (S–PM). The characteristic tetrahedron of the S–PM is established, whose degeneracy is bijectively mapped to the serial singularities of the S–PM. Study parametrization is used to determine six independent parameters that characterize the S–PM and the direct kinematics problem is solved by mapping the transformation matrix between the base and the end-effector to a point in ℙ7. The inverse kinematics problem of the 3-RPS-3-SPR S–PM amounts to find the location of three points on three lines. This problem leads to a minimal octic univariate polynomial with four quadratic factors. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of mappings of two metric spaces that distort the modulus of families of paths by analogy with the Poletskiî inequality.We deal with the situation when the mapping acts in a space that admits weak sphericalization, while the corresponding extended metric space is uniform. For such mappings, the possibility of continuous extension to an isolated boundary point is proved and, as a consequence, an analogue of the Sokhotski-Casorati-Weierstrass theorem is obtained. [ABSTRACT FROM AUTHOR]

We study the local symplectic algebra of curves with semigroups (4, 5, 6, 7), (4, 5, 6) and (4, 5, 7). We use the method of algebraic restrictions to parameterized curves as in Domitrz (Fund Math 204(1):57–86, 2009). A new discrete invariant for algebraic restrictions to parameterized quasi-homogeneous curves is introduced. This invariant together with the method of algebraic restriction can distinguish different symplectic orbits of quasi-homogeneous curves. [ABSTRACT FROM AUTHOR]

This paper discusses two different problems. The first is the classic system of the Euler equations for the rotation of a rigid body about a fixed point. We study the Euler system using singularity analysis for the zero torque and nonzero torque cases. Secondly, we implement an alternate approach to study a system consisting of three or more first-order odes. An existing dependent variable is considered to be independent and the old or original system is rewritten as a new system with a fewer number of first-order odes. We study the analysis of the new system using the Lie symmetry approach. The main reason behind following this approach is to overcome the tedious calculation of the Lie point symmetries for the original system. The method is explained using the Ramanujan system, which arises in modular theory, for which we introduce a new independent variable to reduce the system to two first-order equations, from which we obtain a single second-order equation with one Lie-Point symmetry. The reduced equation is an Abel's equation of the first kind which is the same result as can be obtained by the usual procedure. [ABSTRACT FROM AUTHOR]

We obtain a list of all simple classes of singularities of curves (irreducible and reducible) in real spaces of any dimension with respect to the quasi equivalence relation. [ABSTRACT FROM AUTHOR]

Highlights • Algebraic description of the manipulator's workspace. • Detailed analysis of the input and output singularities. • Description of singularities in Study space and joint space. • Possible transition between different operation modes. • Investigation of a self motion. Abstract The aim of this paper is to give a detailed examination of the input and output singularities of a 3- R UU parallel manipulator in the translational operation mode. This task is achieved by using algebraic constraint equations. For this type of manipulator a complete workspace representation in Study coordinates is presented after elimination of the input parameters. Both, input and output singularities are mapped into a Study subspace as well as into the joint space. Therewith a detailed singularity investigation of the translational operation mode of a 3- R UU parallel manipulator is provided. This paper is an extended version of a previous publication. The addendum comprises the discovery of a possible transition between two operation modes as well as a self motion and an examination of another component of the output singularity surface, most of them for arbitrary design parameters. [ABSTRACT FROM AUTHOR]

We study frames in ℝ3 and mapping from a surface M in ℝ3 to the space of frames. We consider in detail mapping frames determined by a unit tangent principal or asymptotic direction field U and the normal field N. We obtain their generic local singularities as well as the generic singularities of the direction field itself. We show, for instance, that the cross-cap singularities of the principal frame map occur precisely at the intersection points of the parabolic and subparabilic curves of different colours. We study the images of the asymptotic and principal foliations on the unit sphere by their associated unit direction fields. We show that these curves are solutions of certain first order differential equations and point out a duality in the unit sphere between some of their configurations. [ABSTRACT FROM AUTHOR]

The main objective of our work is to optimize the positional and orientation error of a platform using the direct geometric model of a parallel plane manipulator robot. It is well known that the main disadvantage of parallel manipulators is the existence of singularities within its workspace, the adaptive neuro-fuzzy solution is proposed in this study. Intermediate methods have been used to determine the optimal solution. The first method is a graphical method which determines all possible positions of the platform based on the intersection of circles. The second method is the polynomial method used to calculate the coordinates of the center of gravity and the orientation of the platform. Matlab programming simulation of these methods makes it possible to find all the solutions deduced from these methods. The analysis shows that the polynomial method is the one that provides the optimal solution. [ABSTRACT FROM AUTHOR]

We find exact static solutions of the Einstein equations in the spacetime with plane symmetry, where an infinite slab with finite thickness and homogeneous energy (mass) density is present. In the first solution the pressure is isotropic, while in the second solution the tangential components of the pressure are equal to zero. In both cases the pressure vanishes at the boundaries of the slab. Outside the slab these solutions are matched with the Rindler spacetime and with the Weyl-Levi-Civita spacetime, which represent special cases of the Kasner solution. [ABSTRACT FROM AUTHOR]

Abstract Numerical analysis of physical/mathematical problems based on generalized/extended isogeometric analysis suffers from the major drawbacks of sub optimal convergence rates and ill conditioning of system matrices. Blending elements and linear dependency of basis functions are some of the causes attributed to these drawbacks. The presence of blending elements reduces the overall convergence rates and the ill conditioning of system matrices results in either increasing computational time when iterative solvers are used or erroneous results when direct solvers are employed. In order to alleviate these drawbacks, three different Stable Generalized IsoGeometric Analysis (SGIGA) methods are proposed in this paper. In SGIGA, the enrichment functions are modified by shifting the enrichment function using linear or least square interpolant of the enrichment function. Problems with weak and strong discontinuities, singularities and combination of both discontinuities and singularities are considered as case studies to illustrate the performance of the proposed SGIGA methods. From the results, it is observed that SGIGA yields optimal convergence rates as well as better conditioning of system matrices. The results obtained from the proposed SGIGA methods are also compared with the results from the established methods, Stable Generalized Finite Element Method (SGFEM) and eXtended IsoGeometric Analysis (XIGA), to study the relative performances with respect to accuracy and conditioning. Highlights • A novel method, Stable Generalized IsoGeometric Analysis, is proposed. • Three different types of SGIGA are employed to overcome the issues arising in XIGA. • The ill conditioning is studied using the angle measure between the spaces. • The proposed SGIGA-LS-C0 showed better results compared to SGFEM and XIGA. [ABSTRACT FROM AUTHOR]

Abstract We present a notion of a random toric surface modeled on a notion of a random graph. We then study some threshold phenomena related to the smoothness of the resulting surfaces. [ABSTRACT FROM AUTHOR]

Abstract Let k be a possibly non-perfect field of characteristic p > 0. In this work we prove the local existence of absolute p -bases for regular algebras of finite type over k. Namely, consider a regular variety Z over k. Kimura and Niitsuma proved that, for every ξ ∈ Z , the local ring O Z , ξ has a p -basis over O Z , ξ p. Here we show that, for every ξ ∈ Z , there exists an open affine neighborhood of ξ , say ξ ∈ Spec (A) ⊂ Z , so that A admits a p -basis over A p. This passage from the local ring to an affine neighborhood of ξ has geometrical consequences, some of which will be discussed in the second part of the article. As we will see, given a p -basis B of the algebra A over A p , there is a family of differential operators on A naturally associated to B. These differential operators will enable us to give a Jacobian criterion for regularity for varieties defined over k , as well as a method to compute the order of an ideal I ⊂ A. [ABSTRACT FROM AUTHOR]

It has long been suspected that flows of incompressible fluids at large or infinite Reynolds number (namely at small or zero viscosity) may present finite time singularities. We review briefly the theoretical situation on this point. We discuss the effect of a small viscosity on the self-similar solution to the Euler equations for inviscid fluids. Then we show that single-point records of velocity fluctuations in the Modane wind tunnel display correlations between large velocities and large accelerations in full agreement with scaling laws derived from Leray's equations (1934) for self-similar singular solutions to the fluid equations. Conversely, those experimental velocity–acceleration correlations are contradictory to the Kolmogorov scaling laws. [ABSTRACT FROM AUTHOR]

Highlights • Derivation of design conditions related to the base and moving-platform and 3-dof operation modes. • Parametrization of orientation workspace and determination of actuation singularity loci in both Polar and Cartesian coordinates. • Performance index named Maximum Inscribed Circle Diameter (MICD) and its distribution in the design parameter space. • Dimension synthesis based on MICD for five distinct moving-platform heights. • Reconfiguration strategy to ensure the moving-platform working above the minimum required orientation for any moving-platform height. Abstract This paper deals with the constraint-based design, optimum design and reconfiguration strategy of a 3-RPS parallel manipulator. Some design conditions related to the base and moving-platform design parameters and the three degree-of-freedom operation modes are derived. A 3-RPS parallel manipulator with two types of operation modes is generated by following those conditions. Due to its potential advantages, this manipulator is used as an ankle rehabilitation device which can cover the ankle joint motion. To derive the optimum parameters, kinematic optimization is conducted by initially parametrizing the orientation workspace and it turns out that its orientation workspace is not symmetrical. The singularity loci are traced in its orientation workspace. A performance index, named Maximum Inscribed Circle Diameter (MICD) is presented to assess the maximum tilt of the moving-platform for any azimuth angle. The distributions of MICD are plotted in the design space for different moving-platform heights. The optimum region with regard to MICD is obtained. It is noteworthy that the evolution of MICD as a function of moving-platform height in both operation modes is the opposite. Therefore, a reconfiguration strategy is proposed to ensure the moving-platform working above the minimum required orientation for any moving-platform height. [ABSTRACT FROM AUTHOR]

This paper studies the rate of convergence for conditional quasi-Monte Carlo (QMC), which is a counterpart of conditional Monte Carlo. We focus on discontinuous integrands defined on the whole of Rd, which can be unbounded. Under suitable conditions, we show that conditional QMC not only has the smoothing effect (up to infinitely times differentiable) but also can bring orders of magnitude reduction in integration error compared to plain QMC. Particularly, for some typical problems in options pricing and Greeks estimation, conditional randomized QMC that uses n samples yields a mean error of O(n 1+ϵ ) for arbitrarily small ϵ > 0. As a byproduct, we find that this rate also applies to randomized QMC integration with all terms of the analysis of variance decomposition of the discontinuous integrand, except the one of highest order. [ABSTRACT FROM AUTHOR]

We consider generic degenerate subvarieties Xi ⊂ Pn. We determine an integer N, depending on the varieties, and for n ≥ N we compute dimension and degree formulas for the Hadamard product of the varieties Xi. Moreover, if the varieties Xi are smooth, their Hadamard product is smooth too. For n < N, if the Xi are generically di-parameterized, the dimension and degree formulas still hold. However, the Hadamard product can be singular and we give a lower bound for the dimension of the singular locus. [ABSTRACT FROM AUTHOR]

Let R be a polynomial ring over a field k with irrelevant ideal m and dimension d. Let I be a homogeneous ideal in R. We study the asymptotic behavior of the length of the modules Hmi(R/In) for n >> 0. We show that for a fixed number α ∈ ℤ, lim supn→∞ λ(Hmi(R/In)≥−αn) / nd < ∞. It follows that lim supn→∞ λ(Hmi(R/In)) / nd < ∞ when X = Proj R/I is locally a complete intersection (lci) and i ≤ dimX. We also establish that the actual limit exists and is rational for certain classes of monomial ideals I such that the lengths of local cohomology of In are eventually finite. Our proofs use Gr¨obner deformation and Presburger arithmetic. Finally, we utilize more traditional commutative algebra techniques to show that lim infn→∞ λ(Hmi(R/In) / nd > 0 under certain conditions when R/I is either F-pure or lci. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of normal forms and singularities of non-holonomic robotic systems represented by control-affine systems. By means of the concept of the end-point map of the system, and of the system's Jacobian, the configuration singularities have been defined as the control functions for which the Jacobian is not surjective. The presence of these singularities impairs performance of Jacobian motion planning algorithms. Being the singular optimal controls, the configuration singularities can be examined using the tools from the optimal control theory. The main idea of this paper is to rely the analysis of configuration singularities on normal forms of robotic systems. This idea has been applied to the dynamics of a space manipulator. Normal forms of this manipulator under the feedback equivalence have been obtained, and exploited in the analysis of its configuration singularities. [ABSTRACT FROM AUTHOR]

Moser proved in 1965 in his seminal paper [15] that two volume forms on a compact manifold can be conjugated by a diffeomorphism, that is to say they are equivalent, if and only if their associated cohomology classes in the top cohomology group of a manifold coincide. In particular, this yields a classification of compact symplectic surfaces in terms of De Rham cohomology. In this paper we generalize these results for volume forms admitting transversal zeroes. In this case there is also a cohomology capturing the classification: the relative cohomology with respect to the critical hypersurface. We compare this classification scheme with the classification of Poisson structures on surfaces which are symplectic away from a hypersurface where they fulfill a transversality assumption (b-Poisson structures). We do this using the desingularization technique introduced in [10] and extend it to bm-Nambu structures. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we prove the existence of an initial trace T u for any positive solution u to the semilinear fractional diffusion equation (H) ∂ t u + (− Δ) s u + f (t , x , u) = 0 in (0 , + ∞) × R N , where N ≥ 1 , the operator (− Δ) s with s ∈ (0 , 1) is the fractional Laplacian, f : R + × R N × R + → R is a Caratheodory function satisfying f (t , x , u) u ≥ 0 for all (t , x , u) ∈ R + × R N × R + and R + = [ 0 , + ∞). We define the regular set of the trace T u as an open subset of R u ⊂ R N carrying a nonnegative Radon measure ν u such that lim t → 0 ⁡ ∫ R u u (t , x) ζ (x) d x = ∫ R u ζ d ν u , ∀ ζ ∈ C 0 2 (R u) , and the singular set S u = R N ∖ R u as the set points a such that lim sup t → 0 ∫ B ρ (a) u (t , x) d x = + ∞ for any ρ > 0. We also study the reverse problem of constructing a positive solution to (H) with a given initial trace (S , ν) , where S ⊂ R N is a closed set and ν is a positive Radon measure on R = R N ∖ S and develop the case f (t , x , u) = t β u p with β > − 1 and p > 1. [ABSTRACT FROM AUTHOR]

In this paper, we continue to solve as accurately as possible singular eigenvalues problems attached to the Schrödinger equation. We use the conventional ChC and SiC as well as Chebfun. In order to quantify the accuracy of our outcomes, we use the drift with respect to some parameters, i.e., the order of approximation N, the length of integration interval X, or a small parameter ε, of a set of eigenvalues of interest. The deficiency of orthogonality of eigenvectors, which approximate eigenfunctions, is also an indication of the accuracy of the computations. The drift of eigenvalues provides an error estimation and, from that, one can achieve an error control. In both situations, conventional spectral collocation or Chebfun, the computing codes are simple and very efficient. An example for each such code is displayed so that it can be used. An extension to a 2D problem is also considered.

In this paper we compute the Newton polytope M A of the Morse discriminant in the space of univariate polynomials with a given support set A. Namely, we establish a surjection between the set of all combinatorial types of Morse univariate tropical polynomials and the vertices of M A. [ABSTRACT FROM AUTHOR]

Continuum parallel robots (CPRs) are mainly constituted by flexible links arranged in parallel between a rigid platform and a rigid base, and they promise remarkable performance in human–robot collaboration applications. New CPRs modelling strategies and their experimental validation are continuously investigated due to the nonlinear phenomena complexity and the high computational effort required to solve them. This work focuses on the experimental validation of CPRs equilibrium stability prediction. We demonstrate that models based on planar displacement assumptions may fail in the equilibrium stability prediction, even though the CPR is nominally planar. A new CPR prototype for planar applications is proposed, designed, and tested for the scope. Unstable configurations that limit the robot workspace are theoretically and experimentally investigated. A singularity type, related to out-of-the-plane uncontrolled motions of the planar CPR , is experimentally identified for the first time. Experiments demonstrate that, even though the prototype is theoretically planar, a planar model neglecting out-of-the-plane phenomena is inadequate to assess equilibrium stability limits. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Yang’s list of reduced sextic plane curves to rational irreducible projective plane curves of type (6, 3, 1).

The use of large cross-section timber for structural purposes has increased in Spain, and knowledge of its properties is strategically necessary. The Spanish visual strength-grading standard UNE 56544 (2011) efficiency applied to large cross-section structural timber was analyzed using a sample of 363 specimens of radiata pine (Pinus radiata D. Don.) from the Basque Country and Catalonia, Spain. Different sizes were tested (80 × 120 × 2400 mm3, 150 × 250 × 5600 mm3, 150 × 250 × 4300 mm3, and 200 × 250 × 5000 mm3). Bending strength, modulus of elasticity, and density were obtained, and characteristic values were determined in order to assign strength class according to European standard EN 338 (2010). Knots and twists were the most relevant singularities for visual strength grading. It was concluded that large cross-section Spanish radiata pine timber was suitable for structures, and it was assigned to the C20 strength class.

Parallel robots with configurable platforms are a class of robots in which the end-effector has an inner mobility, so that its overall shape can be reconfigured: in most cases, the end-effector is thus a closed-loop kinematic chain composed of rigid links. These robots have a greater flexibility in their motion and control with respect to rigid-platform parallel architectures, but their kinematics is more challenging to analyze. In our work, we consider n-RRR planar configurable robots, in which the end-effector is a chain composed of n links and revolute joints, and is controlled by n rotary actuators located on the base of the mechanism. In particular, we study the geometrical design of such robots and their direct and inverse kinematics for n=4, n=5 and n=6; we employ the bilateration method, which can simplify the kinematic analysis and allows us to generalize the approach and the results obtained for the 3-RRR mechanism to n-RRR robots (with n>3). Then, we study the singularity configurations of these robot architectures. Finally, we present the results from experimental tests that have been performed on a 5–RRR robot prototype.

I give a historical survey of the discussions about the existence of closed timelike curves in general relativistic models of the universe, opening the physical possibility of time travel in the past, as first recognized by K. Gödel in his rotating universe model of 1949. I emphasize that journeying into the past is intimately linked to spacetime models devoid of timelike singularities. Since such singularities arise as an inevitable consequence of the equations of general relativity given physically reasonable assumptions, time travel in the past becomes possible only when one or another of these assumptions is violated. It is the case with wormhole-type solutions. S. Hawking and other authors have tried to “save” the paradoxical consequences of time travel in the past by advocating physical mechanisms of chronological protection; however, such mechanisms remain presently unknown, even when quantum fluctuations near horizons are taken into account. I close the survey by a brief and pedestrian discussion of Causal Dynamical Triangulations, an approach to quantum gravity in which causality plays a seminal role.