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We study the problem of finding algebraically stable models for non-invertible holomorphic fixed point germs

We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let

The γp → pπ0η reaction has been studied with the CBELSA/TAPS detector system to search for experimental evidence for triangle singularities. Kinematically these singularities exist in the decay chain Δ* → ηΔ(1232) → ηπ0 p but do not show an experimental effect because of the weakness of the π0 − η interaction at low near threshold far below the a0(980) resonance.

This paper presents the novel concept of a singularity-free tube (SFT) in the constant orientation workspace of a spatial parallel manipulator. The concept is developed and demonstrated in the context of a $6$ - $6$ spatial parallel manipulator, namely, the semi-regular Stewart platform manipulator. Given two points in the said workspace, the SFT is a tubular volume which contains these points and is free of gain-type or forward-kinematic singularities. The purpose of identifying such regions in space is to allow abundant freedom to the path-planner to connect the said points by a path, which can be free of gain-type singularities simply by remaining inside the SFT at all times. To demonstrate the concept, two smooth paths obtained by formulating two different optimisation problems have been presented as examples. The SFT can be of great help in singularity-free path-planning in many similar manipulators.

For each n, let Mn be an n × n random matrix with independent ±1 entries. We show that ℙ{Mn is singular} = (1/2 + on (1)) n , which settles an old problem. Some generalizations are considered.

A bstract Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and the particle content of the theory. For example, four-particle amplitudes are analytic for contact interactions and have simple poles with appropriately positive residues for tree-level exchange. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlation functions are controlled by an approximate conformal symmetry on the boundary of the spacetime. After systematically classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities (and the correct normalization of physical ones) completely fixes this correlator. Moreover, a “spin-raising” operator relates it to the correlators associated with the exchange of particles with spin, while “weight-shifting” operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. Using our formalism, we reproduce many classic results in the literature, such as the three-point function of slow-roll inflation, and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. Remarkably, the inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta, which can also be directly connected with the signals for particle production seen in the squeezed limit.

A kagome lattice naturally features Dirac fermions, flat bands and van Hove singularities in its electronic structure. The Dirac fermions encode topology, flat bands favour correlated phenomena such as magnetism, and van Hove singularities can lead to instabilities towards long-range many-body orders, altogether allowing for the realization and discovery of a series of topological kagome magnets and superconductors with exotic properties. Recent progress in exploring kagome materials has revealed rich emergent phenomena resulting from the quantum interactions between geometry, topology, spin and correlation. Here we review these key developments in this field, starting from the fundamental concepts of a kagome lattice, to the realizations of Chern and Weyl topological magnetism, to various flat-band many-body correlations, and then to the puzzles of unconventional charge-density waves and superconductivity. We highlight the connection between theoretical ideas and experimental observations, and the bond between quantum interactions within kagome magnets and kagome superconductors, as well as their relation to the concepts in topological insulators, topological superconductors, Weyl semimetals and high-temperature superconductors. These developments broadly bridge topological quantum physics and correlated many-body physics in a wide range of bulk materials and substantially advance the frontier of topological quantum matter. Recent key developments in the exploration of kagome materials are reviewed, including fundamental concepts of a kagome lattice, realizations of Chern and Weyl topological magnetism, flat-band many-body correlations, and unconventional charge-density waves and superconductivity.

The Control Moment Gyroscope (CMG) is superior to other attitude actuators in terms of torque amplification capabilities. Its operating margin is hampered by the singularity problem, which needs the proper selection of a steering law. This study analyses five steering laws for avoiding or escaping different types of singularity; pseudoinverse, pseudoinverse with the null space approach, singularity robust (SR) inverse, generalized SR inverse, and new generalized SR inverse. This analysis measures the effectiveness of these laws in terms of attitude performance and command generation. Simulation results are provided to show the performance of each steering law, with an emphasis on the benefits and drawbacks of each steering law.

In this paper, the author considers semilinear elliptic equations of the form $-\\Delta u- \\frac{\\lambda}{|x|^2}u +b(x)\\,h(u)=0$ in $\\Omega\\setminus\\{0\\}$, where $\\lambda$ is a parameter with $-\\infty0$. The author completely classifies the behaviour near zero of all positive solutions of equation (0.1) when $h$ is regularly varying at $\\infty$ with index $q$ greater than $1$ (that is, $\\lim_{t\\to \\infty} h(\\xi t)/h(t)=\\xi^q$ for every $\\xi>0$). In particular, the author's results apply to equation (0.1) with $h(t)=t^q (\\log t)^{\\alpha_1}$ as $t\\to \\infty$ and $b(x)=|x|^\\theta (-\\log |x|)^{\\alpha_2}$ as $|x|\\to 0$, where $\\alpha_1$ and $\\alpha_2$ are any real numbers.