Polynomial Action Research Articles

Mixing and rigidity along asymptotically linearly independent sequences

AbstractWe use Gaussian measure-preserving systems to prove the existence and genericity of Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let $\unicode{x3bb} _1,\ldots ,\unicode{x3bb} _N\in [0,1]$ and let $\phi _1,\ldots ,\phi _N:\mathbb N\rightarrow \mathbb Z$ be asymptotically linearly independent (that is, for any $(a_1,\ldots ,a_N)\in \mathbb Z^N\setminus \{\vec 0\}$ , $\lim _{k\rightarrow \infty }|\sum _{j=1}^Na_j\phi _j(k)|=\infty $ ). Then the class of invertible Lebesgue measure-preserving transformations $T:[0,1]\rightarrow [0,1]$ for which there exists a sequence $(n_k)_{k\in \mathbb {N}}$ in $\mathbb {N}$ with for any measurable $A,B\subseteq [0,1]$ and any $j\in \{1,\ldots ,N\}$ , is generic. This result is a refinement of a result due to Stëpin (Theorem 2 in [Spectral properties of generic dynamical systems. Math. USSR-Izv.29(1) (1987), 159–192]) and a generalization of a result due to Bergelson, Kasjan, and Lemańczyk (Corollary F in [Polynomial actions of unitary operators and idempotent ultrafilters. Preprint, 2014, arXiv:1401.7869]).

Maximum force for black holes and Buchdahl stars

Black hole and Buchdahl star are identified respectively by $\Phi(R)=1/2, 4/9$ where $g_{tt}=1-2\Phi(R)$ for a spherically symmetric static metric. We investigate the maximum force for black hole and Buchdahl star when one of the participating objects is charged and/or rotating while the other is neutral and non-rotating. It turns out that the maximum force between two Schwarzschild objects is universal, given in terms of the fundamental constant velocity of light and the gravitational constant in general relativity (GR) in the usual four dimensional spacetime. In general this feature uniquely picks out the pure Lovelock gravity (having only one $N$th order term in action which includes GR in the linear order $N=1$) and the dimensional spectrum, $D=3N+1$, where $N$ is degree of the Lovelock polynomial action.

LINEARIZATION OF HOLOMORPHIC FAMILIES OF ALGEBRAIC AUTOMORPHISMS OF THE AFFINE PLANE

Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.

On covariant actions for chiral p-forms

We construct a Lorentz and generally covariant, polynomial action for free chiral p-forms, classically equivalent to the Pasti-Sorokin-Tonin (PST) formulation. The minimal set up requires introducing an auxiliary p-form on top of the physical gauge p-form and the PST scalar. The action enjoys multiple duality symmetries, including those that exchange the roles of physical and auxiliary p-form fields. Same type of actions are available for duality-symmetric formulations, which is demonstrated on the example of electromagnetic field in four dimensions. There, the degrees of freedom of a single Maxwell field are described employing four distinct vector gauge fields and a scalar field.

Pure Lovelock black holes in dimensions d=3N+1 are stable

In this paper we show that pure Lovelock static Schwarzschild's analogue black hole in dimensions $d>3N+1$, where $N$ is the degree of Lovelock polynomial action, is stable even though pure Gauss-Bonnet $N=2$ black hole is unstable in dimension $d<7$. We also discuss and compare quasinormal modes for pure Lovelock and the corresponding Einstein black hole in the same dimension. We find that perturbations decay with characteristic time which is weakly dimensional dependent as it depends only on the gravitational potential of the background solution, while frequency of oscillations however depend on the dimension. Also we show that spectrum of perturbations is not isospectral except in $d=4$.

Endomorphisms of regular rooted trees induced by the action of polynomials on the ring ℤd of d-adic integers

We show that every polynomial in [Formula: see text] defines an endomorphism of the [Formula: see text]-ary rooted tree induced by its action on the ring [Formula: see text] of [Formula: see text]-adic integers. The sections of this endomorphism also turn out to be induced by polynomials in [Formula: see text] of the same degree. In the case of permutational polynomials acting on [Formula: see text] by bijections, the induced endomorphisms are automorphisms of the tree. In the case of [Formula: see text], such polynomials were completely characterized by Rivest in [Permutation polynomials modulo [Formula: see text], Finite Fields Appl. 7(2) (2001) 287–292]. As our main application, we utilize the result of Rivest to derive the condition on the coefficients of a permutational polynomial [Formula: see text] that is necessary and sufficient for [Formula: see text] to induce a level transitive automorphism of the binary tree, which is equivalent to the ergodicity of the action of [Formula: see text] on [Formula: see text] with respect to the normalized Haar measure.

A polynomial action on colored $\mathfrak {sl}_2$ link homology

We construct an action of a polynomial ring on the colored sl2 link homology of Cooper–Krushkal, over which this homology is finitely generated. We define a new, related link homology which is finite dimensional, extends to tangles, and categorifies a scalar multiple of the \mathfrak {sl}_2 Reshetikhin–Turaev invariant. We expect this homology to be functorial under 4-dimensional cobordisms. The polynomial action is related to a conjecture of Gorsky–Oblomkov–Rasmussen–Shende on the stable Khovanov homology of torus knots, and as an application we obtain a weak version of this conjecture. A key new ingredient is the construction of a bounded chain complex which categorifies a scalar multiple of the Jones–Wenzl projector, in which the denominators have been cleared.

Norm convergence of multiple ergodic averages on amenable groups

We apply Walsh's method for proving norm convergence of multiple ergodic averages to arbitrary amenable groups. We obtain convergence in the uniform Ces\`aro sense for their polynomial actions and for ``triangular'' averages associated to commuting homomorphic actions. The latter generalizes a result due to Bergelson, McCutcheon, and Zhang in the case of two actions.

Further evidence for asymptotic safety of quantum gravity

The asymptotic safety conjecture is examined for quantum gravity in four dimensions. Using the renormalisation group, we find evidence for an interacting UV fixed point for polynomial actions up to the 34th power in the Ricci scalar. The extrapolation to infinite polynomial order is given, and the self-consistency of the fixed point is established using a bootstrap test. All details of our analysis are provided. We also clarify further aspects such as stability, convergence, the role of boundary conditions, and a partial degeneracy of eigenvalues. Within this setting we find strong support for the conjecture.

Universality of isothermal fluid spheres in Lovelock gravity

We show universality of isothermal fluid spheres in pure Lovelock gravity where the equation of motion has only one $N$th order term coming from the corresponding Lovelock polynomial action of degree $N$. Isothermality is characterized by the equation of state, $p = \alpha \rho$ and the property, $\rho \sim 1/r^{2N}$. Then the solution describing isothermal spheres, which exist only for the pure Lovelock equation, is of the same form for the general Lovelock degree $N$ in all dimenions $d \geq 2N+2$. We further prove that the necessary and sufficient condition for the isothermal sphere is that its metric is conformal to the massless global monopole or the solid angle deficit metric, and this feature is also universal.

Quadratic gravity from BF theory in two and three dimensions

A polynomial action in the gauge fields is proposed for two and three dimensions, reproducing quadratic gravity. Such action is further amenable for quantization and based upon the topological BF theory.

Study of motion around a static black hole in Einstein and Lovelock gravity

We study motion around a static Einstein and pure Lovelock black hole in higher dimensions. It is known that in higher dimensions, bound orbits exist only for pure Lovelock black hole in all even dimensions, D=2N+2, where N is degree of Lovelock polynomial action. In particular, we compute periastron shift and light bending and the latter is given by one of transverse spatial components of Riemann curvature tensor. We also consider pseudo-Newtonian potentials and Kruskal coordinates.

Antioxidative Effects and Inhibition of Human Low Density Lipoprotein Oxidation In Vitro of Polyphenolic Compounds in Flammulina velutipes (Golden Needle Mushroom).

Dietary polyphenolic compounds mediate polynomial actions in guarding against multiple diseases. Atherosclerosis is an oxidative stress driven pathophysiological complication where free radical induced oxidative modification of low density lipoprotein (LDL) plays the ground breaking role. Mushrooms have been highly regarded for possessing an antioxidant arsenal. Polyphenolic compounds present in dietary mushrooms seem pertinent in withstanding LDL oxidation en route to controlling atherosclerosis. In this study, the antioxidative effect of five solvent fractions consisting of methanol : dichloromethane (M : DCM), hexane (HEX), dichloromethane (DCM), ethyl acetate (EA), and aqueous residue (AQ) of Flammulina velutipes was evaluated. M : DCM fraction showed the most potent 2,2-diphenyl-1-picrylhydrazyl radical scavenging effect with IC50 of 0.86 mg/mL and total phenolic content of 56.36 gallic acid equivalent/g fraction. In LDL oxidation inhibitory tests, M : DCM fraction at 1 µg/mL concentration mostly lengthened the lag time (125 mins) of conjugated diene formation and inhibited the formation of thiobarbituric acid reactive substances (48.71%, at 1 mg/mL concentration). LC-MS/MS analyses of M : DCM fraction identified the presence of polyphenolic substances protocatechuic acid, p-coumaric, and ellagic acid. These chain-breaking polyphenolics might impart the antioxidative effects of F. velutipes. Thus, mushroom-based dietary polyphenolic compounds might be implicated in slowing down the progression of atherosclerosis.

Exploring Cartan gravity with dynamical symmetry breaking

It has been known for some time that General Relativity can be regarded as a Yang–Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an SO(1, 4) or SO(2, 3) gauge field and Higgs field Va which acts to break the symmetry down to that of the Lorentz group SO(1, 3). This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field Φ of electroweak theory is taken as a genuine dynamical field satisfying a Klein–Gordon equation, the gauge independent norm V2 ≡ ηabVaVb of the Higgs-type field Va is typically regarded as non-dynamical. Instead, in many treatments Va does not appear explicitly in the formalism or is required to satisfy V2 = const. ≠ 0 by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection and Higgs field Va as genuine dynamical fields with no ad hoc constraints imposed. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar–tensor model of gravity. One simple choice leads to the extensively studied Peebles–Ratra rolling quintessence model. Another choice yields a scalar–tensor symmetry broken phase of the theory with positive cosmological constant and an effective mass M of the gravitational Higgs field ensuring the constancy of V2 at low energies and agreement with empirical data if M is sufficiently large. More general cases are discussed corresponding to variants of Chern–Simons modified gravity and scalar-Euler form gravity, each of which yield propagating torsion.

Bound orbits and gravitational theory

It can be easily shown that bound orbits around a static source can only exist in four dimensions for any force driven by the Laplace equation. This is true not only for Maxwell's electromagnetism and Newton's gravity, but for Einstein's theory of gravitation as well. In contrast to Maxwell's electrodynamics and Newton's gravity, general relativity has a natural and remarkable generalization in higher dimensions in Lovelock gravity. However, it is not Laplace driven and hence admits bound orbits around a static black hole in all even $D=2N+2$ dimensions, where $N$ is the degree of the Lovelock polynomial action. This is as general a result as Bertrand's theorem of classical mechanics, in which the existence of closed orbits uniquely singles out the inverse square law for a long-range central force.

The Paley–Wiener Theorem and Limits of Symmetric Spaces

We extend the Paley–Wiener theorem for Riemannian symmetric spaces to an important class of infinite-dimensional symmetric spaces. For this we define a notion of propagation of symmetric spaces and examine the direct (injective) limit symmetric spaces defined by propagation. This relies on some of our earlier work on invariant differential operators and the action of Weyl group invariant polynomials under restriction.

Polynomial actions in positive characteristic

Let q be a power of a prime p and Fq be the finite field with q elements. Denote by Fq[X] and Fq(X) the ring of polynomials with coefficients in Fq and the quotient field of Fq[X] respectively. For each P/Q ∈ Fq(X) define |P/Q| = qdeg(P )−deg(Q) where for an element g ∈ Fp[X]we have denoted its degree by deg(g). Note that with respect to the valuation function | · | the integral domain Fq[X] is a Euclidean domain and so also a principal ideal domain and a unique factorization domain. In particular Bezout’s identity is satisfied. Let Fq((X−1)) denote the field of formal Laurent series Fq((X)) = {anX + · · · + a0 + a−1X + · · · : n ∈ Z, ai ∈ Fq},

Single and Piecewise Polynomials for Modeling of Pitched Sounds

We present a compact approach to simultaneous modeling of non-stationary harmonic and transient components in pitched sound sources. The harmonic and transient components are described by separate models which are built from a common sinusoidal basis modified by a joint action of single and linear piecewise time polynomials respectively. A single polynomial accounts for slow and continuous signal time variations, while various piecewise polynomials can capture fast signal changes on smaller subintervals within the analysis window. The resulting model is linear-in-parameters and the solution to the corresponding linear system of equations provides correct model parameter estimates according to the signal content in the analysis window. The model is extended to deal with mixtures of sounds, where harmonics clustered in a small bandwidth are jointly modeled as a single harmonic. The comparative results suggest that the proposed model outperforms two reference modeling methods in terms of modeling errors and number of parameters.

Polynomial Actions in Positive Characteristic

Polynomial Actions in Positive Characteristic

Complexified path integrals and the phases of quantum field theory

The path integral by which quantum field theories are defined is a particular solution of a set of functional differential equations arising from the Schwinger action principle. In fact these equations have a multitude of additional solutions which are described by integrals over a complexified path. We discuss properties of the additional solutions which, although generally disregarded, may be physical with known examples including spontaneous symmetry breaking and theta vacua. We show that a consideration of the full set of solutions yields a description of phase transitions in quantum field theories which complements the usual description in terms of the accumulation of Lee–Yang zeroes. In particular we argue that non-analyticity due to the accumulation of Lee–Yang zeros is related to Stokes phenomena and the collapse of the solution set in various limits including but not restricted to, the thermodynamic limit. A precise demonstration of this relation is given in terms of a zero dimensional model. Finally, for zero dimensional polynomial actions, we prove that Borel resummation of perturbative expansions, with several choices of singularity avoiding contours in the complex Borel plane, yield inequivalent solutions of the action principle equations.