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The purpose of this paper is to introduce pedal curves of spacelike frontals in the hyperbolic 2-space and investigate the singularities of these hyperbolic pedal curves. We classify the singularities of hyperbolic pedal curves for non-singular and singular dual curve germs. To do this we make use of several important results from the Legendrian singularity theory. We explore how the curvatures of the original spacelike frontal and the location of the pedal point affect on the singularities of the corresponding pedal curve. More specifically, we show that for non-singular dual curve germs, the singularities of a pedal curve depend upon the singularities of the first hyperbolic Legendrian curvature germ and the location of the pedal point, whereas for singular dual curve germs, the singularities depend upon the singularities of both hyperbolic Legendrian curvature germs and also the location of the pedal point. Several examples with figures are provided to support our results. [ABSTRACT FROM AUTHOR]

A well-established AII amacrine cell model linked to blinding retinal diseases demonstrates rhythmic bursting activity. This behavior potentially impedes the retinal and cortical responses essential for visual restoration. Consequently, understanding the dynamics of AII amacrine cell firing is critical. We expanded the study by incorporating a calcium ion channel into the original model, creating a refined eight-dimensional AII model. This enhancement allowed us to conduct an in-depth exploration of system behavior via bifurcation analysis and an extensive examination of multi-timescale singularities. We identified the parameter ranges that control the firing activities of the model through codimension-one and codimension-two bifurcation analyses of key parameters. Furthermore, our research revealed that introducing a balance equation determines a transition from regular spiking to bursting. Importantly, the orchestration of three-timescale spiking and bursting attractors is enabled by the universal unfolding of the winged cusp singularity. This study provides a comprehensive analysis of firing dynamics in a high-dimensional neuron model. The insights derived significantly advance our understanding the dynamical behaviors of AII amacrine cells and are promising for improving strategies for retinal pathologies. [ABSTRACT FROM AUTHOR]

Typical singularities of solutions to the equations of one-dimensional isentropic gas dynamics in the Chaplygin and Bechert–Stanyukovich cases are studied. These two cases violate a condition even stronger than the condition of strong nonlinearity and serve as an approximation to real gases. The singularities of these cases have not been studied before. The section of the cusp singularity () for the Chaplygin case and the section of the hyperbolic umbilic singularity () for both cases are described. The complete inheritance of the canonical form from the section of the cusp singularity of the solution to the hodograph image of the wave equation in the Chaplygin case is noted. The simplification of the canonical form in the Bechert–Stanyukovich case is noted. The work uses standard methods and results from the theory of singularities of differentiable mappings. [ABSTRACT FROM AUTHOR]

Formal asymptotics are substantiated that describe a typical dropping cusp singularity in the semiclassical approximations to solutions of two cases of the integrable nonlinear Schrödinger equation , where is a small parameter. The substantiation uses the ideas and facts of the mathematical catastrophe theory and the part of Yu.F. Korobeinik's theorem concerning analytical, as , solutions of the mixed type linear equation to which the hodograph images of both cases of the systems of equations of these semiclassical approximations are equivalent. [ABSTRACT FROM AUTHOR]

Let f:(Cn,S)→(Cp,0)$f: (\mathbb {C}^n,S)\rightarrow (\mathbb {C}^p,0)$ be a finite map‐germ with n

Suppose V is a singular complex analytic curve inside C 2 . We investigate when a singular or non-singular complex analytic curve W inside C 2 with sufficiently small Hausdorff distance d H (V , W) from V must intersect V. We obtain a sufficient condition on W which when satisfied gives an affirmative answer to our question. More precisely, we show the intersection is non-empty for any such W that admits at most one non-normal crossing type discriminant point associated with some proper projection. As an application, we prove a special case of the higher dimensional analog and also a holomorphic multifunction analog of a result by Lyubich and Peters (Geom. Funct. Anal. 24, 887–915 (2014)). [ABSTRACT FROM AUTHOR]

In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding.

In this paper, we mainly investigate (contra)pedals and (anti)orthotomics of frontals in the de Sitter 2‐space from the viewpoint of singularity theory and differential geometry. We utilize the de Sitter Legendrian Frenet frames to provide parametric representations of (contra)pedal curves of spacelike and timelike frontals in the de Sitter 2‐space and to investigate the geometric and singularity properties of these (contra)pedal curves. We then introduce orthotomics of frontals in the de Sitter 2‐space and explain these orthotomics as wavefronts from the viewpoint of Legendrian singularity theory. Furthermore, we generalize these methods to study antiorthotomics of frontals in the de Sitter 2‐space. [ABSTRACT FROM AUTHOR]

In this paper, we study the singularities on a non-developable ruled surface according to Blaschke's frame in the Euclidean 3-space. Additionally, we prove that singular points occur on this kind of ruled surface and use the singularity theory technique to examine these singularities. Finally, we construct an example to confirm and demonstrate our primary finding. [ABSTRACT FROM AUTHOR]

We propose a method of investigating the structural stability of phenomenological models of the phase transition by means of the singularity theory (catastrophe theory) combined with the Poincare normal form theory. Also we find the structurally-stable thermodynamic potential models invariant under the order parameter symmetry group L = 3 m and with n = 1∼5 varying parameters, which are defined in the ranges of modules satisfying the conditions of global minimality and thermodynamic stability. And various phase diagrams of the potential models are constructed, and based on these, the theoretical T-x-P phase diagram for Ni 1-x Cu x Cr 2 O 4 solid solutions is constructed, which is in good agreement with the experimental data. [ABSTRACT FROM AUTHOR]

In the paper, the main parameters affecting the stability of a rotating pretwisted cylindrical shell model with a presetting angle are found by the extended singularity theory. Based on the nonlinear second-order ordinary differential equations of rotating pretwisted cylindrical shell, the universal unfoldings of these equations are given, such that the derivative terms of these equations are treated as periodic perturbation parameters. Analysis results show that these equations have: one form of the universal unfolding codimension 4; one form of the universal unfolding codimension 2; one form of the universal unfolding codimension 2. Calculating the transition set of the universal unfolding codimension 2, the transition set covers all parameters which affect the stability of the rotating pretwisted cylindrical shell. The numerical results show that the amplitudes of periodic excitation and part of the functions of the geometric and physical parameters have strong effect on bifurcation and hysteresis of the rotating pretwisted cylindrical shell subjected to small perturbations near the singularity; however, small external disturbances have weak effect on bifurcation and hysteresis. Due to the periodicity of perturbation parameters, settled in different persistent regions by change in other parameters, the governing equations produce chaotic solutions. The present results are validated by comparison with those in the existing literature. [ABSTRACT FROM AUTHOR]

In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and focal surfaces of timelike Sabban curves in de Sitter space. We find that de Sitter focal surfaces can be regarded as caustics and de Sitter evolutes corresponding to the locus of the polar vectors of osculating de Sitter subspaces. By using singularity theory, we classify the singularities of the de Sitter focal surfaces and de Sitter evolutes and show that there is a close relationship between a new geometric invariant and the types of singularities. Moreover, the Legendrian dual relationships between the hyperbolic tangent indicatrix of timelike Sabban curves and the focal surfaces are given. Finally, we provide an example to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this study, a uniformly distributed mathematical model for the self-heating process within compost piles is formulated and investigated. It consists of mass balance equations for oxygen and energy as well as the heat-generation processes due to both biological and chemical activities. This model is an extension of the study of Luangwilai et al. [2013] which consists of only biological activity. The singularity and degenerate Hopf bifurcation theories are used to determine the loci of different singularities: the isola, cusp, double-limit points and boundary limit set as well as the double-Hopf, generalized Hopf (Bautin) and Bogdanov–Takens bifurcations. In this investigation it is found that these loci separate the secondary parameter plane into twenty-two regions of different steady-state solution behaviors, whereas the model in the earlier study reported only eight regions. With more topological detail in the parameter space, a clearer understanding of the thermal behavior of a compost pile can be obtained, thus assisting compost operators in controlling the temperature within the pile more effectively: for example, achieving desirable elevated temperature range for ideal composting conditions via periodic solutions and S-shaped solution branch, or alternatively, understanding conditions, if not monitored carefully, that can also increase the likelihood of spontaneous ignition within the pile. [ABSTRACT FROM AUTHOR]

We investigate the existence and branching patterns of wave trains (also called periodic traveling waves) in a 2D face-centered square lattice consisting of alternating light and heavy atoms, with linear coupling between nearest particles and a nonlinear substrate potential. In contrast to monatomic chains, we consider two different periodic waveform functions corresponding to light and heavy particles, respectively. As a result, we have to solve coupled advanced-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction. Then we obtain the small-amplitude solutions in the Hamiltonian system near equilibria in nonresonance and p : q resonance, respectively. In particular, the results can be applied to some one-dimensional diatomic lattices. [ABSTRACT FROM AUTHOR]

We analyze the generating mechanisms for heteroclinic cycles in Z³2 -equivariant ODEs, not involving Hopf bifurcations. Such cycles have been observed in different areas of physics see [1, 8, 12], as well in modelling the geomagnetic field of Earth [15], and as far as we know, there is no available theoretical data explaining these phenomena. We use singularity theory to study the equivalence in the group-symmetric context, as well as the recognition problem for the simplest bifurcation problems with this symmetry group. Singularity results highlight different mechanisms for the appearance of heteroclinic cycles, based on the transition between the bifurcating branches. [ABSTRACT FROM AUTHOR]

This paper presents a methodology to study non-twist invariant circles and their bifurcations for area preserving maps, which is supported on the theoretical framework developed in Gonzalez-Enriquez et al. (Mem. Amer. Math. Soc. 227:vi+115, 2014). We recall that non-twist invariant circles are characterized not only by being invariant, but also by having some specified normal behavior. The normal behavior may endow them with extra stability properties (e.g., against external noise), and hence, they appear as design goals in some applications, e.g., in plasma physics, astrodynamics and oceanography. The methodology leads to efficient algorithms to compute and continue, with respect to parameters, non-twist invariant circles. The algorithms are quadratically convergent and, when implemented using FFT, have low storage requirement and low operations count per step. Furthermore, the algorithms are backed up by rigorous a posteriori theorems, proved and discussed in detail in Gonzalez-Enriquez et al. (Mem. Amer. Math. Soc. 227:vi+115, 2014), which give sufficient conditions guaranteeing the existence of a true non-twist invariant circle, provided an approximate invariant circle is known. Hence, one can compute confidently even very close to breakdown. With some extra effort, the calculations could be turned into computer-assisted proofs, see Figueras et al. (Found. Comput. Math. 17:1123–1193, 2017) for examples of the latter. The algorithms are also guaranteed to converge up to the breakdown of the invariant circles, and then, they are suitable to compute regions of parameters where the non-twist invariant circles exist. The calculations involved in the computation of the boundary of these regions are very robust, and they do not require symmetries and can run without continuous manual adjustments, largely improving methods based on the computation of very long period periodic orbits to approximate invariant circles. This paper contains a detailed description of our algorithms, the corresponding implementation and some numerical results, obtained by running the computer programs. In particular, we include calculations for two-dimensional parameter regions where non-twist invariant circles (with a prescribed frequency) exist. Indeed, we present systematic results in systems that do not contain symmetry lines, which seem to be unaccessible for previous methods. These numerical explorations lead to some open questions, also included here. [ABSTRACT FROM AUTHOR]

The Ribaucour transformation is a generalization of parallel surfaces and Darboux transformations. It has been studied by many researchers. In this paper, first we introduce the construction of Ribaucour transforms via Cauchy–Kovalevskaya theorem, including an explicit integral formula of Ribaucour transformations for surfaces of revolution. In the latter half of this paper, we also study the singularities appearing on Ribaucour transforms and give the criteria for cuspidal edges, swallowtails, cuspidal cross caps, D 4 ± -singularities. [ABSTRACT FROM AUTHOR]

This study provides semi-analytical solutions to the Selkov-Schnakenberg reaction-diffusion system. The Galerkin method is applied to approximate the system of partial differential equations by a system of ordinary differential equations. The steady states of the system and the limit cycle solutions are delineated using bifurcation diagram analysis. The influence of the diffusion coefficients as they change, on the system stability is examined. Near the Hopf bifurcation point, the asymptotic analysis is developed for the oscillatory solution. The semi-analytical model solutions agree accurately with the numerical results.

The focal surface of a generic space curve in Euclidean 3-space is a classical subject which is a two dimensional caustic and has Lagrangian singularities. In this paper, we define the first de Sitter focal surface and the second de Sitter focal surface of de Sitter spacelike curve and consider their singular points as an application of the theory of caustics and Legendrian dualities. The main results state that de Sitter focal surfaces can be seen as two dimensional caustics which have Lagrangian singularities. To characterize these singularities, a useful new geometric invariant ρ(s) is discovered and two dual relationships between focal surfaces and spacelike curve are given. Three examples are used to demonstrate the main results.

Modern cosmology has two competing theories of the origin of the universe: the "Singularity theory" and the "Superstring theory". Four Dilemmas of the "Superstring theory" are presented: the incompleteness of the eleven space–time dimensions, the inextricable dependence on the "Space–Time Background", the "Zero-Brane theory" admitting stuff smaller than the Planck scale, and the pure mathematical theory that cannot be falsified by experiments. Although the "Singularity theory" is faced with many critiques from the "Superstring theory", from the perspective of Information Ontology, treating the "Singularity" as "Origin Information" can dissolve these troubles well. For the "Singularity theory", the new philosophical thinking framework effectively explains the parameter problems of the universe, and gives satisfied response to the challenges from the "Superstring theory". As a result, the "Singularity theory" has a more competitive advantage on the origin of the universe. [ABSTRACT FROM AUTHOR]

We study Landau–Ginzburg orbifolds with and , where and is either the maximal group of scalar symmetries of or the intersection of the maximal diagonal symmetries of with . We construct a mirror map between the corresponding phase spaces and prove that it is an isomorphism restricted to a certain subspace of the phase space when is a prime number. When satisfies the parity condition of Ebeling–Gusein-Zade, this subspace coincides with the full space. We also show that two phase spaces are isomorphic for . [ABSTRACT FROM AUTHOR]

In the theory of cosmology, de Sitter space is the symmetrical model of accelerated expansions of the universe. It is derived from the solution of the Einstein field equation, which has a positive cosmological constant. In this paper, we define the evolutes and focal surfaces of timelike Sabban curves in de Sitter space. We find that de Sitter focal surfaces can be regarded as caustics and de Sitter evolutes corresponding to the locus of the polar vectors of osculating de Sitter subspaces. By using singularity theory, we classify the singularities of the de Sitter focal surfaces and de Sitter evolutes and show that there is a close relationship between a new geometric invariant and the types of singularities. Moreover, the Legendrian dual relationships between the hyperbolic tangent indicatrix of timelike Sabban curves and the focal surfaces are given. Finally, we provide an example to illustrate our main results.

Bifurcation characteristics of a fractional non-smooth oscillator containing clearance constraints under sinusoidal excitation are investigated. First, the bifurcation response equation of the fractional non-smooth system is obtained via the K–B method. Second, the stability of the bifurcation response equation is analyzed, and parametric conditions for stability are acquired. The bifurcation characteristics of the fractional non-smooth system are then studied using singularity theory, and the transition set and bifurcation diagram under six different constrained parameters are acquired. Finally, the analysis of the influence of fractional terms on the dynamic characteristics of the system is emphasized through numerical simulation. Local bifurcation diagrams of the system under different fractional coefficients and orders verify that the system will present various motions, such as periodic motion, multiple periodic motion, and chaos, with the change in fractional coefficient and order. This manifestation indicates that fractional parameters have a direct effect on the motion form of this non-smooth system. Thus, these results provide a theoretical reference for investigating and repressing oscillation problems of similar systems. [ABSTRACT FROM AUTHOR]

We study the composition calculus of Fourier integral operators F, with Morin singularities. This composition calculus falls outside of the classical transverse or clean intersection calculus of FIOs and it is motivated by the inverse problems studied in Felea and Greenleaf (Math Res Lett 17(5):867–886, 2010) and Felea and Nolan (J Fourier Anal Appl 21(4):799–821, 2015). In this case, the normal operator F ∗ F is an operator with the wave front set in Δ ∪ Λ where Λ is a singular canonical relation called an open umbrella. We show that the operator F ∗ F has the same order on Δ \ Λ and on Λ \ Δ . This result is a generalization of the results obtained in Felea and Greenleaf (2010) and Felea and Nolan (2015) where the canonical relations have the fold/cusp and cusp/cusp singularities. [ABSTRACT FROM AUTHOR]

We consider the Curie–Weiss Potts model in zero external field under independent symmetric spin-flip dynamics. We investigate dynamical Gibbs–non-Gibbs transitions for a range of initial inverse temperatures β < 3 , which covers the phase transition point β = 4 log 2 (Ellis and Wang in Stoch Process Appl 35(1):59–79, 1990). We show that finitely many types of trajectories of bad empirical measures appear, depending on the parameter β , with a possibility of re-entrance into the Gibbsian regime, of which we provide a full description. [ABSTRACT FROM AUTHOR]

The direct numerical simulation of the acoustic wave scattering created by very small obstacles is very expensive, especially in three dimensions and even more so in time domain. The use of asymptotic models is very efficient and the purpose of this work is to provide a rigorous justification of a new asymptotic model for low-cost numerical simulations. This model is based on asymptotic near-field and far-field developments that are then matched by a key procedure that we describe and demonstrate. We show that it is enough to focus on the regular part of the wave field to rigorously establish the complete asymptotic expansion. For that purpose, we provide an error estimate which is set in the whole space, including the transition region separating the near-field from the far-field area. The proof of convergence is established through Kondratiev's seminal work on the Laplace equation and involves the Mellin transform. Numerical experiments including multiple scattering illustrate the efficiency of the resulting numerical method by delivering some comparisons with solutions computed with a finite element software. [ABSTRACT FROM AUTHOR]

Tête-à-tête graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed tête-à-tête graphs provide a generalization which define mixed tête-à-tête twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed tête-à-tête twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of tête-à-tête twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities. [ABSTRACT FROM AUTHOR]

Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal projections in a number of directions. We discuss relations between (1) a space curve and the projected curves from several distinct directions, and (2) a surface and the apparent contours of projections from several distinct directions, in terms of differential geometry and singularity theory. In particular, formulae for recovering certain information on the original curves or surfaces from their projected images are given.

This paper discusses how a semi-analytical approach can be used alongside the reversible Schnakenberg model in a reaction-diffusion cell. It is worth noting that semi-analytical techniques are aligned to the Galerkin method, which offers an approximation of the governing partial differential equations through a system of ordinary differential equations. The bifurcation diagrams, steady-state curves and regions of the parameter space where bifurcations take place are part of the semi-analytical solutions and the numerical solution.

In this paper, we investigate the effects of Type IV singularity through f (T) gravity description of inflationary Universe, where T denotes the torsion scalar. With the Friedmann equations of the theory, we reconstruct a f (T) model according to a given Hubble rate susceptible to describe the inflationary era near the Type IV singularity. One obtains an interesting well-known f (T) model but with additional constant parameter c 0 staying as the Type IV singularity contribution. Moreover, we calculate the Hubble flow parameters in order to determine the dynamical evolution of the cosmological system. The results show that some of the Hubble flow parameters are small near the Type IV singularity and become singular at Type IV singularity, indicating that a dynamical instability of the cosmological system occurs at that point. This means that the dynamical cosmological evolution up to that point ceases to be the final attractor since the system is abruptly interrupted. Furthermore, by considering the f (T) trace anomaly equation, the previous result on the Type IV singularity is consolidated by the conditional instability coming from the de Sitter inflationary description of the reconstructed f (T) model. The model leads to instability strongly governed by the Type IV singularity parameter c 0 is viewed as the graceful exit from inflation. Our theoretical f (T) description based on slow-roll parameters not only confirms some observational data on spectral index and the scalar-to-tensor ratio from Planck data and BICEP 2 /Keck-Array data, but also shows the property of f (T) gravity in describing the early and late-time evolution of our Universe. [ABSTRACT FROM AUTHOR]

We show that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that non-symplectic integrators do not. We provide a universal description of the breaking of umbilic bifurcations by nonysmplectic integrators. We discover extra structure induced from certain types of boundary value problems, including classical Dirichlet problems, that is useful to locate bifurcations. Geodesics connecting two points are an example of a Hamiltonian boundary value problem, and we introduce the jet-RATTLE method, a symplectic integrator that easily computes geodesics and their bifurcations. Finally, we study the periodic pitchfork bifurcation, a codimension-1 bifurcation arising in integrable Hamiltonian systems. It is not preserved by either symplectic or non-symplectic integrators, but in some circumstances symplecticity greatly reduces the error. [ABSTRACT FROM AUTHOR]

We develop the contact singularity theory for singularly perturbed (or 'slow–fast') vector fields of the general form z ′ = H (z , ε) , z ∈ R n and 0 < ε ≪ 1 . Our main result is the derivation of computable, coordinate-independent defining equations for contact singularities under an assumption that the leading-order term of the vector field admits a suitable factorization. This factorization can in turn be computed explicitly in a wide variety of applications. We demonstrate these computable criteria by locating contact folds and, for the first time, contact cusps in general slow–fast models of biochemical oscillators and the Yamada model for self-pulsating lasers. [ABSTRACT FROM AUTHOR]

We study the metastable minima of the Curie–Weiss Potts model with three states, as a function of the inverse temperature, and for arbitrary vector-valued external fields. Extending the classic work of Ellis and Wang (Stoch Process Appl 35(1):59–79, 1990) and Wang (Stoch Process Appl 50(2):245–252, 1994) we use singularity theory to provide the global structure of metastable (or local) minima. In particular, we show that the free energy has up to four local minimizers (some of which may at the same time be global) and describe the bifurcation geometry of their transitions under variation of the parameters. [ABSTRACT FROM AUTHOR]

A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter ℤ 2 -equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression. [ABSTRACT FROM AUTHOR]

In this paper, we study cylindrical helices, Mannheim curves and Darboux developable surfaces associated to Mannheim curves. We give a method for constructing Mannheim curves from spherical curves and demonstrate that all Mannheim curves can be constructed by such a way. We also introduce the spherical Darboux images of Mannheim curves, and we give a classification of singularities of Darboux developable surfaces and spherical Darboux images. Moreover, we study the geometric properties of singularities as applications of singularity theory for spherical curves. [ABSTRACT FROM AUTHOR]

Quasi-periodic high order degenerate bifurcation theories have been well established, but works which are related to almost-periodic bifurcations seem to be very few. In this paper, we consider the almost-periodic time-dependent perturbations of one-dimensional degenerate vector field x ˙ = x l. With the KAM theory and singularity theory, we show that the universal unfolding of the vector field can persist under small almost-periodic perturbation if some appropriate non-resonant conditions are satisfied, which implies strongly non-resonant invariant tori in the integrable part and all bifurcation scenario can survive under any small almost-periodic perturbation. [ABSTRACT FROM AUTHOR]

We give an explicit simple construction for classifying spaces of maps obtained as hyperplane projections of immersions. We prove structure theorems for these classifying spaces. [ABSTRACT FROM AUTHOR]

Homeostasis occurs in a system where an output variable is approximately constant on an interval on variation of an input variable I . Homeostasis plays an important role in the regulation of biological systems, cf. Ferrell (Cell Syst 2:62–67, 2016), Tang and McMillen (J Theor Biol 408:274–289, 2016), Nijhout et al. (BMC Biol 13:79, 2015), and Nijhout et al. (Wiley Interdiscip Rev Syst Biol Med 11:e1440, 2018). A method for finding homeostasis in mathematical models is given in the control theory literature as points where the derivative of the output variable with respect to I is identically zero. Such points are called perfect homeostasis or perfect adaptation. Alternatively, Golubitsky and Stewart (J Math Biol 74:387–407, 2017) use an infinitesimal notion of homeostasis (namely, the derivative of the input–output function is zero at an isolated point) to introduce singularity theory into the study of homeostasis. Reed et al. (Bull Math Biol 79(9):1–24, 2017) give two examples of infinitesimal homeostasis in three-node chemical reaction systems: feedforward excitation and substrate inhibition. In this paper we show that there are 13 different three-node networks leading to 78 three-node input–output network configurations, under the assumption that there is one input node, one output node, and they are distinct. The different configurations are based on which node is the input node and which node is the output node. We show nonetheless that there are only three basic mechanisms for three-node input–output networks that lead to infinitesimal homeostasis and we call them structural homeostasis, Haldane homeostasis, and null-degradation homeostasis. Substantial parts of this classification are given in Ma et al. (Cell 138:760–773, 2009) and Ferrell (2016) among others. Our contributions include giving a complete classification using general admissible systems (Golubitsky and Stewart in Bull Am Math Soc 43:305–364, 2006) rather than specific biochemical models, relating the types of infinitesimal homeostasis to the graph theoretic existence of simple paths, and providing the basis to use singularity theory to study higher codimension homeostasis singularities such as the chair singularities introduced in Nijhout and Reed (Integr Comp Biol 54(2):264–275, 2014. 10.1093/icb/icu010) and Nijhout et al. (Math Biosci 257:104–110, 2014). See Golubitsky and Stewart (2017). The first two of these mechanisms are illustrated by feedforward excitation and substrate inhibition. Structural homeostasis occurs only when the network has a feedforward loop as a subnetwork; that is, when there are two distinct simple paths connecting the input node to the output node. Moreover, when the network is just the feedforward loop motif itself, one of the paths must be excitatory and one inhibitory to support infinitesimal homeostasis. Haldane homeostasis occurs when there is a single simple path from the input node to the output node and then only when one of the couplings along this path has strength 0. Null-degradation homeostasis is illustrated by a biochemical example from Ma et al. (2009); this kind of homeostasis can occur only when the degradation constant of the third node is 0. The paper ends with an analysis of Haldane homeostasis infinitesimal chair singularities. [ABSTRACT FROM AUTHOR]

The envelope of straight lines affine normal to a plane curve C is its affine evolute; the envelope of the affine lines tangent to C is the original curve, together with the entire affine tangent line at each inflexion of C. In this paper, we consider plane curves without inflexions. We use some techniques of singularity theory, such as unfoldings, discriminants and functions on discriminants, to explain how the first envelope turns into the second, as the (constant) slope between the set of lines forming the envelope and the set of affine tangents to C changes from 0 to 1. In particular, we guarantee the existence of the first slope for which singularities occur. Moreover, we explain how these singularities evolve on discriminant surface. [ABSTRACT FROM AUTHOR]

We consider a hypersurface in Cn with an isolated singular point at the origin, and study the mixed Hodge structure of the stalk of its intersection cohomology complex at the origin. In particular, we express the dimension of each graded piece of the weight filtration of this mixed Hodge structure in terms of the numbers of Jordan blocks in the Milnor monodromy. [ABSTRACT FROM AUTHOR]

It always seems to be the case. No matter how hard you might work on a proposal, no matter how polished and complete the final product may be, when it is presented to a group for approval, there always seems to be some majority who wants to "improve It." Is this just an annoyance or is there a reason? The mathematical modeling provides an immediate explanation in terms of some interesting and unexpected mathematics. Even more; the mathematics describing this behavior underscores the reality that it can be surprising easy even for a group sincerely striving for excellence to make inferior decisions. Indeed, these difficulties are so pervasive and can arise in such unexpected ways that it is realistic to worry whether groups you belong to have been inadvertently victimized by these mathematical subtleties based on the orbits of symmetry groups. These problems can occur even if all decisions are reached by consensus during discus sions, such as a committee discussing the selection of a new calculus book. This paper addresses deliberations by discussing a branch of voting theory where Euclidean geometry models an "issue space." Then describing how it is possible to un intentionally make inferior choices, we will encounter mathematical behaviors remark- ably similar to "attractors" and "chaotic dynamics" from dynamical systems. Since the coexistence of chaotic and stable behavior is common in the Newtonian N -body problem and dynamical systems, it is interesting that this combination also coexists in the dynamics of discussions. Another connection arises when configurations central to the N -body problem play a suggestive role in the analysis; at another step we use singularity theory. What adds to the delight of this topic is that while the mathematics can be intricate, the issues can be described at a classroom level where some even lead to student level research projects. Keywords: spatial voting, paradoxes, singularity theory

Homeostasis is an important and common biological phenomenon wherein an output variable does not change very much as an input parameter is varied over an interval. It can be studied by restricting attention to homeostasis points — points where the output variable has a vanishing derivative with respect to the input parameter. In a feedforward network, if a node has a homeostasis point then downstream nodes will inherit it. This is the case except when the downstream node has a bifurcation point coinciding with the homeostasis point. We apply singularity theory to study the behavior of the downstream node near these homeostasis-bifurcation points. The unfoldings of low codimension homeostasis-bifurcation points are found. In the case of steady-state bifurcation, the behavior includes multiple homeostatic plateaus separated by hysteretic switches. In the case of Hopf bifurcation, the downstream node may have limit cycles with a wide range of near-constant amplitudes and periods. Homeostasis-bifurcation is therefore a mechanism by which binary, switch-like responses or stable clock rhythms could arise in biological systems. [ABSTRACT FROM AUTHOR]

Economic theory is a mathematically rich field in which there are opportunities for the formal analysis of singularities and catastrophes. This article looks at the historical context of singularities through the work of two eminent Frenchmen around the late 1960s and 1970s. Rene Thom (1923-2002) was an acclaimed mathematician having received the Fields Medal in 1958, whereas Gerard De-breu (1921-2004) would receive the Nobel Prize in economics in 1983. Both were highly influential within their fields and given the fundamental nature of their work, the potential for cross-fertilisation would seem to be quite promising. This was not to be the case: Debreu knew of Thom's work and cited it in the analysis of his own work, but despite this and other applied mathematicians taking catas-trophe theory to economics, the theory never achieved a lasting following and relatively few results were published. This article reviews Debreu's analysis of the so called regular and crtitical economies in order to draw some insights into the economic perspective of singularities before moving to how singularities arise naturally in the Nash equilibria of game theory. Finally, a modern treatment of stochastic game theory is covered through recent work on the quantal response equilibrium. In this view, the Nash equilibrium is to the quantal response equilib-rium what deterministic catastrophe theory is to stochastic catastrophe theory, with some caveats regarding when this analogy breaks down discussed at the end. [ABSTRACT FROM AUTHOR]

Symplectic integrators can be excellent for Hamiltonian initial value problems. Reasons for this include their preservation of invariant sets like tori, good energy behaviour, nonexistence of attractors, and good behaviour of statistical properties. These all refer to long-time behaviour. They are directly connected to the dynamical behaviour of symplectic maps φ: M → M on the phase space under iteration. Boundary value problems, in contrast, are posed for fixed (and often quite short) times. Symplecticity manifests as a symplectic map φ : M → M′ which is not iterated. Is there any point, therefore, for a symplectic integrator to be used on a Hamiltonian boundary value problem? In this paper, we announce results that symplectic integrators preserve bifurcations of Hamiltonian boundary value problems and that nonsymplectic integrators do not. [ABSTRACT FROM AUTHOR]

Abstract We provide a construction of Saito primitive forms for Gepner singularity by studying the relation between Saito primitive forms for Gepner singularities and primitive forms for singularities of the form F k , n = ∑ i = 1 n x i k invariant under the natural S n -action. [ABSTRACT FROM AUTHOR]