Showing papers on "Singularity published in 2019"
TL;DR: In this paper, the authors provided universal expressions for the classical piece of the amplitude given by the graviton/photon exchange between massive particles of arbitrary spin, at both tree and one loop level.
Abstract: We provide universal expressions for the classical piece of the amplitude given by the graviton/photon exchange between massive particles of arbitrary spin, at both tree and one loop level. In the gravitational case this leads to higher order terms in the post-Newtonian expansion, which have been previously used in the binary inspiral problem. The expressions are obtained in terms of a contour integral that computes the Leading Singularity, which was recently shown to encode the relevant information up to one loop. The classical limit is performed along a holomorphic trajectory in the space of kinematics, such that the leading order is enough to extract arbitrarily high multipole corrections. These multipole interactions are given in terms of a recently proposed representation for massive particles of any spin by Arkani-Hamed et al. This explicitly shows universality of the multipole interactions in the effective potential with respect to the spin of the scattered particles. We perform the explicit match to standard EFT operators for S = $$ \frac{1}{2} $$
and S = 1. As a natural byproduct we obtain the classical pieces up to one loop for the bending of light.
182 citations
30 May 2019
TL;DR: The Cargese summer school celebrated the 100th anniversary of the Painleve property, the property that was introduced byPainleve and subsequently by Gambier and their school to classify ordinary differential equations (ODEs) according to the singularity behavior of their solutions as mentioned in this paper.
Abstract: The Cargese summer school celebrated the 100th anniversary of the Painleve property, the property that was introduced by Painleve and subsequently by Gambier and their school to classify ordinary differential equations (ODEs) according to the singularity behavior of their solutions, [1–4]. In the other contributions to this volume the various implications of the Painleve property as well as of the particular differential equations in which this property is manifested are explained in detail.
108 citations
TL;DR: A Newton linearized Galerkin finite element method is proposed to solve nonlinear time fractional parabolic problems with non-smooth solutions in time direction and the optimal error estimate in the $$L^2$$L2-norm is obtained without any time step restrictions dependent on the spatial mesh size.
Abstract: A Newton linearized Galerkin finite element method is proposed to solve nonlinear time fractional parabolic problems with non-smooth solutions in time direction. Iterative processes or corrected schemes become dispensable by the use of the Newton linearized method and graded meshes in the temporal direction. The optimal error estimate in the $$L^2$$
-norm is obtained without any time step restrictions dependent on the spatial mesh size. Such unconditional convergence results are proved by including the initial time singularity into concern, while previous unconditional convergent results always require continuity and boundedness of the temporal derivative of the exact solution. Numerical experiments are conducted to confirm the theoretical results.
94 citations
TL;DR: In this paper, the authors derived the universal total twist of two particles at a large angular momentum from the pole structure of the fusion matrix and applied it to the light cone bootstrap.
Abstract: The light cone OPE limit provides a significant amount of information regarding the conformal field theory (CFT), like the high-low temperature limit of the partition function. We started with the light cone bootstrap in the general CFT 2 with c > 1. For this purpose, we needed an explicit asymptotic form of the Virasoro conformal blocks in the limit z → 1, which was unknown until now. In this study, we computed it in general by studying the pole structure of the fusion matrix (or the crossing kernel). Applying this result to the light cone bootstrap, we obtained the universal total twist (or equivalently, the universal binding energy) of two particles at a large angular momentum. In particular, we found that the total twist is saturated by the value $$ \frac{c-1}{12} $$
if the total Liouville momentum exceeds beyond the BTZ threshold. This might be interpreted as a black hole formation in AdS3. As another application of our light cone singularity, we studied the dynamics of entanglement after a global quench and found a Renyi phase transition as the replica number was varied. We also investigated the dynamics of the 2nd Renyi entropy after a local quench. We also provide a universal form of the Regge limit of the Virasoro conformal blocks from the analysis of the light cone singularity. This Regge limit is related to the general n-th Renyi entropy after a local quench and out of time ordered correlators.
93 citations
TL;DR: Some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors are introduced and the index of introduced systems is obtained and discussed the solvability of these systems.
Abstract: One of the important classes of coupled systems of algebraic, differential and fractional differential equations (CSADFDEs) is fractional differential algebraic equations (FDAEs). The main difference of such systems with other class of CSADFDEs is that their singularity remains constant in an interval. However, complete classifying and analyzing of these systems relay mainly to the concept of the index which we introduce in this paper. For a system of linear differential algebraic equations (DAEs) with constant coefficients, we observe that the solvability depends on the regularity of the corresponding pencils. However, we show that in general, similar properties of DAEs do not hold for FDAEs. In this paper, we introduce some practical applications of systems of FDAEs in physics such as a simple pendulum in a Newtonian fluid and electrical circuit containing a new practical element namely fractors. We obtain the index of introduced systems and discuss the solvability of these systems. We numerically solve the FDAEs of a pendulum in a fluid with three different fractional derivatives (Liouville–Caputo’s definition, Caputo–Fabrizio’s definition and with a definition with Mittag–Leffler kernel) and compare the effect of different fractional derivatives in this modeling. Finally, we solved some existing examples in research and showed the effectiveness and efficiency of the proposed numerical method.
93 citations
TL;DR: In this paper, it was shown that flat bands can be categorized into two distinct classes, singular and nonsingular flat bands, by exploiting the singular behavior of their Bloch wave functions in momentum space.
Abstract: We show that flat bands can be categorized into two distinct classes, that is, singular and nonsingular flat bands, by exploiting the singular behavior of their Bloch wave functions in momentum space. In the case of a singular flat band, its Bloch wave function possesses immovable discontinuities generated by the band-crossing with other bands, and thus the vector bundle associated with the flat band cannot be defined. This singularity precludes the compact localized states from forming a complete set spanning the flat band. Once the degeneracy at the band crossing point is lifted, the singular flat band becomes dispersive and can acquire a finite Chern number in general, suggesting a new route for obtaining a nearly flat Chern band. On the other hand, the Bloch wave function of a nonsingular flat band has no singularity, and thus forms a vector bundle. A nonsingular flat band can be completely isolated from other bands while preserving the perfect flatness. All one-dimensional flat bands belong to the nonsingular class. We show that a singular flat band displays a novel bulk-boundary correspondence such that the presence of the robust boundary mode is guaranteed by the singularity of the Bloch wave function. Moreover, we develop a general scheme to construct a flat band model Hamiltonian in which one can freely design its singular or nonsingular nature. Finally, we propose a general formula for the compact localized state spanning the flat band, which can be easily implemented in numerics and offer a basis set useful in analyzing correlation effects in flat bands.
92 citations
TL;DR: In this paper, it was shown that the solution of a time-fractional partial differential equation lacks regularity at the initial time t = 0, which is often inconsistent with the physical problem being modeled.
88 citations
TL;DR: A new class of parity-time symmetric systems that combine EPs with another type of mathematical singularity associated with the poles of complex functions are introduced that can exhibit an unprecedentedly large eigenvalue bifurcation beyond those obtained by standard EPs.
Abstract: Standard exceptional points (EPs) are non-Hermitian degeneracies that occur in open systems. At an EP, the Taylor series expansion becomes singular and fails to converge-a feature that was exploited for several applications. Here, we theoretically introduce and experimentally demonstrate a new class of parity-time symmetric systems [implemented using radio frequency (rf) circuits] that combine EPs with another type of mathematical singularity associated with the poles of complex functions. These nearly divergent exceptional points can exhibit an unprecedentedly large eigenvalue bifurcation beyond those obtained by standard EPs. Our results pave the way for building a new generation of telemetering and sensing devices with superior performance.
82 citations
TL;DR: This work provides a formal derivation for a novel multiloop LTD expression and studies its threshold singularity structure, and applies its findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed.
Abstract: Loop-tree duality (LTD) offers a promising avenue to numerically integrate multiloop integrals directly in momentum space. It is well established at one loop, but there have been only sparse numerical results at two loops. We provide a formal derivation for a novel multiloop LTD expression and study its threshold singularity structure. We apply our findings numerically to a diverse set of up to four-loop finite topologies with kinematics for which no contour deformation is needed. We also lay down the ground work for constructing such a deformation. Our results serve as an important stepping stone towards a generalized and efficient numerical implementation of LTD, which is applicable to the computation of virtual corrections.
82 citations
TL;DR: In this paper, images of thin accretion disks around black holes and two classes of naked singularity spacetimes are compared and compared with the images obtained from those models significantly differ from those of black holes.
Abstract: We study here images of thin accretion disks around black holes and two classes of naked singularity spacetimes and compare these scenarios. The naked singularity models which have photon spheres have single accretion disk with its inner edge lying outside the photon sphere. The images and shadows created by these models mimic those of black holes. It follows, therefore, that further and more detailed analysis of the images and shadows structure in such case is needed to confirm or otherwise the existence of an event horizon for the compact objects such as the galactic centers. However, naked singularity models which do not have any photon spheres can have either double disks or a single disk extending up to the singularity. The images obtained from such models significantly differ from those of black holes. Moreover, the images of the two classes of naked singularities in this latter case, differ also from one another, thereby allowing them to be distinguish from one another through the observation of the images.
79 citations
TL;DR: In this article, the effects of f (G, T) terms on the study of structure scalars and their influence on the formulation of the Raychaudhuri, shear and Weyl scalar equations are explored.
Abstract: This work is devoted to exploring the effects of f (G, T) terms on the study of structure scalars and their influence on the formulation of the Raychaudhuri, shear and Weyl scalar equations. For this purpose, we have assumed non-static spherically symmetric geometry coupled with shearing viscous locally anisotropic dissipative matter content. We have developed relations among Misner-Sharp mass, Weyl scalar, matter and structure variables. We have also formulated a set of f (G, T) structure scalars after orthogonally breaking down the Riemann curvature tensor. The influences of these scalar functions on the modeling of relativistic radiating spheres are also studied. The factor involved in the emergence of inhomogeneities is also explored for the constant and varying modified curvature corrections. We inferred that f (G, T) structure scalars could provide an effective tool to study the Penrose-Hawking singularity theorems and the Newman-Penrose formalism.
TL;DR: In this paper, the authors show that the topology of monochromatic light singularities can be modeled as a torus knot, which reflects the subgroup generated by the torus-knot angular momentum Jγ.
Abstract: The fundamental polarization singularities of monochromatic light are normally associated with invariance under coordinated rotations: symmetry operations that rotate the spatial dependence of an electromagnetic field by an angle θ and its polarization by a multiple γθ of that angle. These symmetries are generated by mixed angular momenta of the form Jγ = L + γS, and they generally induce Mobius-strip topologies, with the coordination parameter γ restricted to integer and half-integer values. In this work we construct beams of light that are invariant under coordinated rotations for arbitrary rational γ, by exploiting the higher internal symmetry of ‘bicircular’ superpositions of counter-rotating circularly polarized beams at different frequencies. We show that these beams have the topology of a torus knot, which reflects the subgroup generated by the torus-knot angular momentum Jγ, and we characterize the resulting optical polarization singularity using third- and higher-order field moment tensors, which we experimentally observe using nonlinear polarization tomography. The polarization structure around polarization singularities can exhibit arbitrary fractional rotations when tracing around the singularity, due to an underlying topology of a torus knot imprinted by the chosen ratio of frequencies contained in the light beam.
TL;DR: In this paper, a singularity ring (SR) representation of the real pseudospin has been proposed to capture the structure of generic spectral exceptional degeneracies, which emerge only if the real pseudoospin vector actually falls on the SR.
Abstract: This work unveils how geometric features of two-band non-Hermitian Hamiltonians can classify the topology of their eigenstates and energy manifolds. Our approach generalizes the Bloch sphere visualization of Hermitian systems to a ``Bloch torus'' picture for non-Hermitian systems, by extending the origin of the Bloch sphere to a singularity ring (SR) in the vector space of the real pseudospin. The SR captures the structure of generic spectral exceptional degeneracies, which emerge only if the real pseudospin vector actually falls on the SR. Applicable to non-Hermitian systems that may or may not have exceptional degeneracies, this SR picture affords convenient visualization of various symmetry constraints and reduces their topological characterization to the classification of simple intersection or winding behavior, as detailed by our explicit study of chiral, sublattice, particle-hole, and conjugated particle-hole symmetries. In 1D, the winding number about the SR corresponds to the band vorticity measurable through the Berry phase. In 2D, more complicated winding behavior leads to a variety of phases that illustrates the richness of the interplay between SR topology and geometry beyond mere Chern number classification. Through a normalization procedure that puts generic two-band non-Hermitian Hamiltonians on equal footing, our SR approach also allows for vivid visualization of the non-Hermitian skin effect.
TL;DR: In this article, the existence and uniqueness of solutions and Hyers-Ulam stability for ABC-fractional DEs with p-Laplacian operator involving spatial singularity are derived with the help of the well-known Guo-Krasnoselskii theorem.
Abstract: In this article, we have focused on the existence and uniqueness of solutions and Hyers–Ulam stability for ABC-fractional DEs with p-Laplacian operator involving spatial singularity. The existence and uniqueness of solutions are derived with the help of the well-known Guo-Krasnoselskii theorem. Our work is a continuation of the study carried out in the recently published article ” Chaos Solitons & Fractals. 2018;117:16-20.” To manifest the results, we include an example with specific parameters and assumptions.
TL;DR: In this paper, the existence, uniqueness, and structural stability of solutions to nonlinear tempered fractional differential equations involving the Caputo tempered fractions derivative with generalized boundary conditions were studied and a singularity preserving spectral-collocation method for numerical solution of such equations was developed.
TL;DR: In this article, the authors consider a quantum scalar field in RNdS spacetime and show that quantum theory comes to the rescue of strong cosmic censorship and shows that for any state that is nonsingular (i.e., Hadamard) within the domain of dependence, the expected stress-tensor blows up with affine parameter, $V$, along a radial null geodesic transverse to the Cauchy horizon as $T_{VV} \sim C/V^2$ with $C$ independent of the state and $C \
Abstract: In classical General Relativity, the values of fields on spacetime are uniquely determined by their values at an initial time within the domain of dependence of this initial data surface. However, it may occur that the spacetime under consideration extends beyond this domain of dependence, and fields, therefore, are not entirely determined by their initial data. This occurs, for example, in the well-known (maximally) extended Reissner-Nordstrom or Reissner-Nordstrom-deSitter (RNdS) spacetimes. The boundary of the region determined by the initial data is called the "Cauchy horizon." It is located inside the black hole in these spacetimes. The strong cosmic censorship conjecture asserts that the Cauchy horizon does not, in fact, exist in practice because the slightest perturbation (of the metric itself or the matter fields) will become singular there in a sufficiently catastrophic way that solutions cannot be extended beyond the Cauchy horizon. Thus, if strong cosmic censorship holds, the Cauchy horizon will be converted into a "final singularity," and determinism will hold. Recently, however, it has been found that, classically this is not the case in RNdS spacetimes in a certain range of mass, charge, and cosmological constant. In this paper, we consider a quantum scalar field in RNdS spacetime and show that quantum theory comes to the rescue of strong cosmic censorship. We find that for any state that is nonsingular (i.e., Hadamard) within the domain of dependence, the expected stress-tensor blows up with affine parameter, $V$, along a radial null geodesic transverse to the Cauchy horizon as $T_{VV} \sim C/V^2$ with $C$ independent of the state and $C
eq 0$ generically in RNdS spacetimes. This divergence is stronger than in the classical theory and should be sufficient to convert the Cauchy horizon into a strong curvature singularity.
TL;DR: In this paper, images of thin accretion disks around black holes and two classes of naked singularity spacetimes are compared and compared with the images obtained from those models significantly differ from those of black holes.
Abstract: We study here images of thin accretion disks around black holes and two classes of naked singularity spacetimes and compare these scenarios. The naked singularity models which have photon spheres have single accretion disk with its inner edge lying outside the photon sphere. The images and shadows created by these models mimic those of black holes. It follows, therefore, that further and more detailed analysis of the images and shadows structure in such case is needed to confirm or otherwise the existence of an event horizon for the compact objects such as the galactic centers. However, naked singularity models which do not have any photon spheres can have either double disks or a single disk extending up to the singularity. The images obtained from such models significantly differ from those of black holes. Moreover, the images of the two classes of naked singularities in this latter case, differ also from one another, thereby allowing them to be distinguish from one another through the observation of the images.
TL;DR: In this article, the authors numerically study spherical gravitational collapse in shift symmetric Einstein dilaton Gauss Bonnet (EdGB) gravity and find evidence that there are open sets of initial data for which the character of the system of equations changes from hyperbolic to elliptic type in a compact region of the spacetime.
Abstract: We numerically study spherical gravitational collapse in shift symmetric Einstein dilaton Gauss Bonnet (EdGB) gravity. We find evidence that there are open sets of initial data for which the character of the system of equations changes from hyperbolic to elliptic type in a compact region of the spacetime. In these cases evolution of the system, treated as a hyperbolic initial boundary value problem, leads to the equations of motion becoming ill-posed when the elliptic region forms. No singularities or discontinuities are encountered on the corresponding effective ``Cauchy horizon.'' Therefore it is conceivable that a well-posed formulation of EdGB gravity (at least within spherical symmetry) may be possible if the equations are appropriately treated as mixed type.
TL;DR: Variable separation exponential-form solution of (1+1)-dimensional coupled integrable dispersionless equations in physics and mathematics is obtained via the projective Riccati equation method and the singularity structure without the physical meaning is found for the original components of the system.
TL;DR: In this paper, the authors obtained asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher-Hartwig singularities.
Abstract: We obtain asymptotics of large Hankel determinants whose weight depends on a one-cut regular potential and any number of Fisher-Hartwig singularities. This generalises two results: 1) a result of Berestycki, Webb and Wong [5] for root-type singularities, and 2) a result of Its and Krasovsky [37] for a Gaussian weight with a single jump-type singularity. We show that when we apply a piecewise constant thinning on the eigenvalues of a random Hermitian matrix drawn from a one-cut regular ensemble, the gap probability in the thinned spectrum, as well as correlations of the characteristic polynomial of the associated conditional point process, can be expressed in terms of these determinants.
TL;DR: In this article, the authors study static and rotating, uncharged and charged, AdS black holes in higher-dimensional $f(T)$ gravity, focusing on the power-law ansatz which is the most viable according to observations.
Abstract: The investigation on higher-dimensional AdS black holes is of great importance under the light of AdS/CFT correspondence. In this work we study static and rotating, uncharged and charged, AdS black holes in higher-dimensional $f(T)$ gravity, focusing on the power-law ansatz which is the most viable according to observations. We extract AdS solutions characterized by an effective cosmological constant that depends on the parameters of the $f(T)$ modification, as well as on the electric charge, even if the explicit cosmological constant is absent. These solutions do not have a general relativity or an uncharged limit, hence they correspond to a novel solution class, whose features arise solely from the torsional modification alongside the Maxwell sector incorporation. We examine the singularities of the solutions, calculating the values of various curvature and torsion invariants, finding that they do possess the central singularity, which however is softer comparing to standard general relativity case due to the $f(T)$ effect. Additionally, we investigate the horizons structure, showing that the solutions possess an inner Cauchy horizon as well as an outer event one, nevertheless for suitably large electric charge and small mass we obtain the appearance of a naked singularity. Finally, we calculate the energy of the obtained solutions, showing that the $f(T)$ modification affects the mass term.
TL;DR: In this article, the evolution towards a finite-time singularity of the Navier-Stokes equations for flow of an incompressible fluid of kinematic viscosity is considered.
Abstract: The evolution towards a finite-time singularity of the Navier–Stokes equations for flow of an incompressible fluid of kinematic viscosity ) is considered, and it is conjectured on the basis of the above dynamical system that a finite-time singularity can indeed occur in this case.
TL;DR: In this paper, the authors considered the case of Volterra integral equations with a double singularity and obtained explicit L ∞ bounds rather than L 1 bounds, which are applicable to weakly singular integral equations similar to the ones first studied by D. Henry.
TL;DR: In this article, the singularity structure of the phase space corresponding to an exponential quintessence dark energy model was investigated, and it was shown that the solutions of the dynamical system are non-singular for all cosmic times.
Abstract: In this work we shall investigate the singularity structure of the phase space corresponding to an exponential quintessence dark energy model recently related to swampland models. The dynamical system corresponding to the cosmological system is an autonomous polynomial dynamical system, and by using a mathematical theorem we shall investigate whether finite-time singularities can occur in the dynamical system variables. As we demonstrate, the solutions of the dynamical system are non-singular for all cosmic times and this result is general, meaning that the initial conditions corresponding to the regular solutions, belong to a general set of initial conditions and not to a limited set of initial conditions. As we explain, a dynamical system singularity is not directly related to a physical finite-time singularity. Then, by assuming that the Hubble rate with functional form $H(t)=f_1(t)+f_2(t)(t-t_s)^{\alpha}$, is a solution of the dynamical system, we investigate the implications of the absence of finite-time singularities in the dynamical system variables. As we demonstrate, Big Rip and a Type IV singularities can always occur if $\alpha 2$ respectively. However, Type II and Type III singularities cannot occur in the cosmological system, if the Hubble rate we quoted is considered a solution of the cosmological system.
TL;DR: The extended virtual element method (X-VEM) is proposed to treat singularities and crack discontinuities that arise in the Laplace problem and results are presented that affirm the sound accuracy and demonstrate the optimal rates of convergence in the L 2 norm and energy.
TL;DR: In this article, the authors investigated the characteristics of integrability, bidirectional solitons and localized solutions for a (€ 3+1$$ )-dimensional breaking soliton (GBS) equation with general forms.
Abstract: The characteristics of integrability, bidirectional solitons and localized solutions are investigated for a (
$$3+1$$
)-dimensional breaking soliton (GBS) equation with general forms. Firstly, starting from the GBS equation, we perform the singularity manifold analysis and obtain a new integrable model in the sense of Painleve property. Secondly, taking advantage of the Bell polynomial approach, we construct the Backlund transformation, Lax pair and an infinite sequence of conservation laws. Subsequently, this new equation is also found to allow bidirectional soliton solutions, and the head-on and overtaking collisions between solitons are illustrated by some illustrative graphs. Finally, some localized excitations, such as lump solution, multi-dromions, periodic solitary waves solution, are obtained.
TL;DR: In this paper, a mathematical model of the problem of nonlinear oscillations of a viscoelastic pipeline conveying fluid is developed in the Boltzmann-Volterra integral model with weakly singular kernels of heredity.
TL;DR: In this article, it was shown that many classical results of the minimal model program do not hold over an algebraically closed field of characteristic two, and a three dimensional plt pair whose codimension one part is not normal, a klt singularity which is not rational nor Cohen-Macaulay, and Fano threefold with non-trivial intermediate cohomology.
Abstract: We show that many classical results of the minimal model programme do not hold over an algebraically closed field of characteristic two. Indeed, we construct a three dimensional plt pair whose codimension one part is not normal, a three dimensional klt singularity which is not rational nor Cohen-Macaulay, and a klt Fano threefold with non-trivial intermediate cohomology.
01 Dec 2019
TL;DR: In this article, the authors prove the existence of finite-energy strong solutions to the axi-symmetric 3D Euler equations on the domains of (x,y,z) and (1+πsilon |z|) which become singular in finite time.
Abstract: For all $$\epsilon >0$$, we prove the existence of finite-energy strong solutions to the axi-symmetric 3D Euler equations on the domains $$ \{(x,y,z)\in {\mathbb {R}}^3: (1+\epsilon |z|)^2\le x^2+y^2\}$$ which become singular in finite time. The solutions we construct have bounded vorticity before a certain time when the vorticity becomes unbounded. We further show that solutions with 0 swirl are always globally regular in the setting we consider. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a 1D system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system Elgindi and Jeong (Finite-time Singularity Formation for Strong Solutions to the Boussinesq System, 2017) though a number of modifications must be made due to anisotropy and since our domains are not scale-invariant. This seems to be the first construction of singularity formation for finite-energy strong solutions to the actual 3D Euler system.
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TL;DR: It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with Holder continuous gradient and suitable decay at infinity as mentioned in this paper.
Abstract: It has been known since work of Lichtenstein [42] and Gunther [29] in the 1920's that the $3D$ incompressible Euler equation is locally well-posed in the class of velocity fields with Holder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.