Finite Singularities Research Articles

End of the world brane networks for infinite distance limits in CY moduli space

Dynamical Cobordism provides a powerful method to probe infinite distance limits in moduli/field spaces parameterized by scalars constrained by generic potentials, employing configurations of codimension-1 end of the world (ETW) branes. These branes, characterized in terms of critical exponents, mark codimension-1 boundaries in the spacetime in correspondence of finite spacetime distance singularities at which the scalars diverge. Using these tools, we explore the network of infinite distance singularities in the complex structure moduli space of Calabi-Yau fourfolds compactifications in M-theory with a four-form flux turned on, which is described in terms of normal intersecting divisors classified by asymptotic Hodge theory. We provide spacetime realizations for these loci in terms of networks of intersecting codimension-1 ETW branes classified by specific critical exponents which encapsulate the relevant information of the asymptotic Hodge structure characterizing the corresponding divisors.

Conformal structure of singularities in some varying fundamental constants bimetric cosmologies

In this paper, we explore the conformal structure of singularities arising from varying fundamental constants using the method of Penrose diagrams. We employ a specific type of bimetric model featuring two different metrics. One metric describes the causal structure for matter, while the other characterizes the causal structure for gravitational interactions, which is related to variations in fundamental constants such as the gravitational constant and the speed of light. For this reason, we focused on the gravitational metric to calculate the conformal transformation and compose Penrose diagrams for the singularities arising from the varying fundamental constants. We have shown that, in one case, the parameter such as the scale factor, the density and the pressure resemble those of the finite scale factor singularity (FSF). Despite singularity appears in constant conformal time in our case and in the case of FSF the Misner-Sharp horizon looks different. Our another case is similar to sudden future singularity (SFS), but there are differences in the conformal structures. We have also shown that in our cases the behavior of Misner-Sharp horizon strongly depends on initial conditions. The last analytical solution which we introduced is identical to conformal structure of the standard exotic singularity for the matter.

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

We demonstrate that the linear quotient singularity for the exceptional subgroup G G in S p ( 4 , C ) \mathrm {Sp}(4,\mathbb {C}) of order 32 is isomorphic to an affine quiver variety for a 5-pointed star-shaped quiver. This allows us to construct uniformly all 81 projective crepant resolutions of C 4 / G \mathbb {C}^4/G as hyperpolygon spaces by variation of GIT quotient, and we describe both the movable cone and the Namikawa Weyl group action via an explicit hyperplane arrangement. More generally, for the n n -pointed star shaped quiver, we describe completely the birational geometry for the corresponding hyperpolygon spaces in dimension 2 n − 6 2n-6 ; for example, we show that there are 1684 projective crepant resolutions when n = 6 n=6 . We also prove that the resulting affine cones are not quotient singularities for n ≥ 6 n \geq 6 .

Oriented and Standard Shadowing Properties for Topological Flows

We prove that oriented and standard shadowing properties are equivalent for topological flows with finite singularities that are Lyapunov stable or backward Lyapunov stable. Moreover, we prove that the direct product $\phi_1 \times \phi_2$ of two topological flows has the oriented shadowing property if $\phi_1$ with finite singularities has the oriented shadowing property, while $\phi_2$ has the limit set consisting of finite singularities that are Lyapunov stable or backward Lyapunov stable.

Existence and regularity for global weak solutions to the 𝜆-family water wave equations

In this paper, we prove the global existence of Hölder continuous solutions for the Cauchy problem of a family of partial differential equations, named as λ \lambda -family equations, where λ \lambda is the power of nonlinear wave speed. The λ \lambda -family equations include Camassa-Holm equation ( λ = 1 \lambda =1 ) and Novikov equation ( λ = 2 \lambda =2 ) modelling water waves, where solutions generically form finite time cusp singularities, or, in other words, show wave breaking phenomenon. The global energy conservative solution we construct is Hölder continuous with exponent 1 − 1 2 λ 1- \frac {1}{2\lambda } . The existence result also paves the way for the future study on uniqueness and Lipschitz continuous dependence.

A Finsler type Lipschitz optimal transport metric for a quasilinear wave equation

We consider the global well-posedness of weak energy conservative solution to a general quasilinear wave equation through variational principle, where the solution may form finite time cusp singularity, when energy concentrates. As a main result in this paper, we construct a Finsler type optimal transport metric, then prove that the solution flow is Lipschitz under this metric. We also prove a generic regularity result by applying Thom's transversality theorem, then find piecewise smooth transportation paths among a dense set of solutions. The results in this paper are for large data solutions, without restriction on the size of solutions.

Deformations of arcs and comparison of formal neighborhoods for a curve singularity

Abstract Let $ {\mathcal C} $ be an algebraic curve and c be an analytically irreducible singular point of ${\mathcal C}$ . The set ${\mathscr {L}_{\infty }}({\mathcal C})^c$ of arcs with origin c is an irreducible closed subset of the space of arcs on ${\mathcal C}$ . We obtain a presentation of the formal neighborhood of the generic point of this set which can be interpreted in terms of deformations of the generic arc defined by this point. This allows us to deduce a strong connection between the aforementioned formal neighborhood and the formal neighborhood in the arc space of any primitive parametrization of the singularity c. This may be interpreted as the fact that analytically along ${\mathscr {L}_{\infty }}({\mathcal C})^c$ the arc space is a product of a finite dimensional singularity and an infinite dimensional affine space.

New time-dependent solutions of viable Horndeski gravity

We generate new spherical and time-dependent solutions of viable Horndeski gravity by disforming a solution of the Einstein equations with scalar field source and positive cosmological constant. They describe dynamical objects embedded in asymptotically FLRW spacetimes and contain apparent horizons and a finite radius singularity that evolve in time in peculiar ways apparently not encountered before in Einstein and “old” scalar-tensor gravity.

Factors of $E$-operators with an $\eta$-apparent singularity at zero

In 1929, Siegel defined $E$-functions as power series in $\overline{\mathbb{Q}}[[z]]$, with Taylor coefficients satisfying certain growth conditions, and solutions of linear differential equations with coefficients in $\overline{\mathbb{Q}}(z)$. The Siegel–Shidlovskii theorem (1956) generalized to $E$-functions the Diophantine properties of the exponential function. In 2000, André proved that the finite singularities of a differential operator in $\overline{\mathbb{Q}}(z)[d/dz] \setminus \{0\}$ of minimal order for some non-zero $E$-function are apparent, except possibly 0 which is always regular singular. We pursue the classification of such operators and consider those for which 0 is $\eta$-apparent, in the sense that there exists $\eta \in \mathbb C$ such that $L$ has a local basis of solutions at 0 in $z^{\eta} \mathbb C[[z]]$. We prove that they have a $\mathbb{C}$-basis of solutions of the form $Q_{j}(z)z^{\eta} e^{\beta_{j} z}$, where $\eta \in \mathbb{Q}$, the $\beta_{j} \in \overline{\mathbb{Q}}$ are pairwise distinct and the $Q_{j}(z) \in \overline{\mathbb{Q}}[z] \setminus \{0\}$. This generalizes a previous result by Roques and the author concerning $E$-operators with an apparent singularity or no singularity at the origin, of which certain consequences are also given here.

Multiple fluid theory of cosmic evolution and its thermodynamic analysis

In this paper we have discussed the modified gravity and scalar field DE model specifically DBI essence model with the analysis of thermodynamics during cosmological evolution. We have used the modified gravity with the form f(R,T)=R+2\Lambda T in our calculations. The basic aim behind this paper is to discuss a theory that unifies modified gravity with DE models including the solutions of some cosmological problems like thermodynamics energy conditions violation problems, finite time future singularity problems, initial singularity problem, Cosmic inflation problem, decelerated expansion problem, graceful exit problem, reheating problem, bouncing nature problem, phase transformation-spontaneous symmetry breaking problem, negative heat capacity paradox problem and obviously the present day universe problems (Continuously decreasing temperature problem, present day DE dominated expansion problem). We have established the viscous effects (both positive and negative viscosity) in our calculation and discussed the negative-positive viscosity by introducing two special type of energy cycles. Finally, we have discussed the stability conditions for universe evolution through cosmic perturbation and resolved the instability problems. We have also shown the state finder trajectories for both with and without viscous fluid on the basis of our calculations to compare our research with the results of other independent DE models

Singularity Formation for the General Poiseuille Flow of Nematic Liquid Crystals

We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model. The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation. In this paper, we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation, extended from an earlier result on a special case.

New paradigm for configurational entropy in glass-forming systems

We show that on cooling towards glass transition configurational entropy exhibits more significant changes than predicted by classic relation. A universal formula according to Kauzmann temperature {T}_{K} is given: S={S}_{0}{t}^{n}, where t=left(T-{T}_{K}right)/T. The exponent n is hypothetically linked to dominated local symmetry. Such a behaviour is coupled to previtreous evolution of heat capacity Delta {C}_{P}^{config.}left(Tright)=left(nC/Tright){left(1-{T}_{K}/Tright)}^{n-1} associated with finite temperature singularity. These lead to generalised VFT relation, for which the basic equation is retrieved. For many glass-formers, basic VFT equation may have only an effective meaning. A universal-like reliability of the Stickel operator analysis for detecting dynamic crossover phenomenon is also questioned. Notably, distortions-sensitive and derivative-based analysis focused on previtreous changes of configurational entropy and heat capacity for glycerol, ethanol and liquid crystal is applied.

Correction to : Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

A Correction to this paper has been published: https://doi.org/10.1007/s10883-021-09575-w

Thermodynamics and energy condition analysis for Van-Der-Waals EOS without viscous cosmology

In this paper we have studied Van-Der-Waals fluid system with the generalized EOS as as discussed in the recent works of Kremer, G.M [1, 4], Vardiashvili, G [2], Jantsch, R.C [3], Capozziello, S [5]. and are functions of energy density and time that are different for the three types of Van der Waals fluid which are one parameter model, two parameters model and three parameters model. We have studied the changes in the parameters for different cosmic phases. We have also investigated the thermodynamics and the stability conditions for these three models. Finally, we have resolved the finite time future singularity problems.

Dynamical Cobordism and Swampland Distance Conjectures

We consider spacetime-dependent solutions to string theory models with tadpoles for dynamical fields, arising from non-trivial scalar potentials. The solutions have necessarily finite extent in spacetime, and are capped off by boundaries at a finite distance, in a dynamical realization of the Cobordism Conjecture. We show that as the configuration approaches these cobordism walls of nothing, the scalar fields run off to infinite distance in moduli space, allowing to explore the implications of the Swampland Distance Conjecture. We uncover new interesting scaling relations linking the moduli space distance and the SDC tower scale to spacetime geometric quantities, such as the distance to the wall and the scalar curvature. We show that walls at which scalars remain at finite distance in moduli space correspond to domain walls separating different (but cobordant) theories/vacua; this still applies even if the scalars reach finite distance singularities in moduli space, such as conifold points.We illustrate our ideas with explicit examples in massive IIA theory, M-theory on CY threefolds, and 10d non-supersymmetric strings. In 4d mathcal{N} = 1 theories, our framework reproduces a recent proposal to explore the SDC using 4d string-like solutions.

Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

Stokes Matrices of a Reducible Double Confluent Heun Equation via Monodromy Matrices of a Reducible General Huen Equation with Symmetric Finite Singularities

A Liouville-type theorem for fully nonlinear CR invariant equations on the Heisenberg group

We obtain a Liouville-type theorem for cylindrical viscosity solutions of fully nonlinear CR invariant equations on the Heisenberg group. As a by-product, we also prove a comparison principle with finite singularities for viscosity solutions to more general fully nonlinear operators on the Heisenberg group.

Liftable vector fields, unfoldings and augmentations

We study liftable vector fields of smooth map-germs. We show how to obtain the module of liftable vector fields of any map-germ of finite singularity type from the module of liftable vector fields of a stable unfolding of it. As an application, we obtain the liftable vector fields for the family H_k in Mond’s list. We then show the relation between the liftable vector fields of a stable germ and its augmentations.

Quadratic Differential Systems with a Finite Saddle-Node and an Infinite Saddle-Node (1, 1)SN - (B)

This paper presents a global study of the class [Formula: see text] of all real quadratic polynomial differential systems which have a finite semi-elemental saddle-node and an infinite saddle-node formed by the coalescence of a finite singularity and an infinite singularity. This class can be divided into two different families, namely, [Formula: see text] phase portraits possessing a finite saddle-node as the only finite singularity and [Formula: see text] phase portraits possessing a finite saddle-node and also a simple finite elemental singularity. Each one of these two families is given by a specific normal form. The study of family [Formula: see text] was reported in [Artés et al., 2020b] where the authors obtained [Formula: see text] topologically distinct phase portraits for systems in the closure [Formula: see text]. In this paper, we provide the complete study of the geometry of family [Formula: see text]. This family which modulo the action of the affine group and time homotheties is three-dimensional and we give the bifurcation diagram of its closure with respect to a specific normal form, in the three-dimensional real projective space. The respective bifurcation diagram yields 631 subsets with 226 topologically distinct phase portraits for systems in the closure [Formula: see text] within the representatives of [Formula: see text] given by a specific normal form. Some of these phase portraits are proven to have at least three limit cycles.

Correlation functions in finite temperature CFT and black hole singularities

We compute thermal 2-point correlation functions in the black brane AdS5 background dual to 4d CFT’s at finite temperature for operators of large scaling dimension. We find a formula that matches the expected structure of the OPE. It exhibits an exponentiation property, whose origin we explain. We also compute the first correction to the two-point function due to graviton emission, which encodes the proper time from the event horizon to the black hole singularity.