Conformal Invariance

In this paper, we present p + q + m and p + q + 2m parametric S-functions, exploring the relationships between these two types of functions. We derive their fundamental properties, including generating functions, recurrence relations, differential formulas and integral representations. From these results, we establish several notable corollaries. Furthermore, we present a distribution function involving the S-function and compute the Laplace transform of its density function. Additionally, we explore the connections between the S-function and other significant transcendental functions.

The complete set of concomitants is given of the metric, a scalar function, and their derivatives in six dimensions without imposing conditions on the order of the derivatives and their properties are studied under conformal transformations. These results are useful in several physical problems, like finding (9')"" and ( F")"' in any geometry.

In this article we present a study of the subspaces of the manifold OscM, the total space of the osculator bundle of a real manifold M. We obtain the induced connections of the canonical metrical N-linear connection determined by the homogeneous prolongation of a Finsler metric to the manifold OscM. We present the relation between the induced and intrinsic geometric objects of the associated osculator submanifold.

This paper provides the ultraviolet (UV) completion of the Rupture–Containment Cosmology (RCC) framework. Building on the RCC Companion (technical Hilbert space foundations) and the RCC Cosmology paper (phenomenology and predictions), this work focuses on the most difficult corner of the Bronstein cube: (ℏ,c,G)
We present renormalization procedures, counterterms, and fixed-point structures, demonstrating that RCC is UV complete and consistent across all energy scales.

We consider a two-dimensional integrable and conformally invariant field theory possessing two Dirac spinors and three scalar fields. The interaction couples bilinear terms in the spinors to exponentials of the scalars. Its integrability properties are based on the sl(2) affine Kac-Moody algebra, and it is a simple example of the so-called conformal affine Toda theories coupled to matter fields. We show, using bosonization techniques, that the classical equivalence between a U (1) Noether current and the topological current holds true at the quantum level, and then leads to a bag model like mechanism for the confinement of the spinor fields inside the solitons. By bosonizing the spinors we show that the theory decouples into a sine-Gordon model and free scalars. We construct the two-soliton solutions and show that their interactions lead to the same time delays as those for the sine-Gordon solitons. The model provides a good laboratory to test duality ideas in the context of the equivalence between the sine-Gordon and Thirring theories.

This paper advances a theological-physical model in which so-called "lost civilizations"-Lemuria, Mu, and Atlantis-are interpreted as archetypal nations designed by God at primordial creation. We formalize the passage from immortal, unbounded humanity to finite, constrained humanity using (i) a hyperspace Lagrangian with Lagrange constraints and (ii) the Polyakov worldsheet action of string theory with boundary conditions, background-field couplings, and worldsheet renormalization-group (RG) flow. In this framework, divine wrath and protective love are encoded, respectively, as relevant deformations that drive flows away from conformal fixed points and as dilaton/Liouville mechanisms that regularize Weyl anomalies. We show how freedom (degrees of freedom → ∞), wisdom/strength, and timelessness contract under constraints (M < ∞; c UV > c IR), while topological/global invariants preserve a mnemonic core, analogous to a "background radiation" of collective memory. The archetypal civilizations reappear as fixed points (or neighborhoods thereof) in theory space, whose echoes persist in myth, vision, and unconscious symbolism.

This work presents a novel theoretical framework proposing a deep connection between two major relational approaches to quantum gravity: Shape Dynamics (SD) and Emergent Gravity (EG). Despite both eliminating absolute spacetime structures, SD (a global Hamiltonian theory of conformal geometry) and EG (a local thermodynamic theory deriving spacetime from entanglement) have fundamentally different mathematical structures. This paper argues that this disparity is only apparent and that their core principles—SD's "best-matching" and EG's "entanglement equilibrium"—may share a common origin in the quantum thermodynamics of shape degrees of freedom.

The proposed synthesis is built on several key pillars:

1. Relational Thermodynamic Variables: The definition of fundamental thermodynamic quantities (entropy, energy flux, temperature) within the configuration space of Shape Dynamics, making them invariant under diffeomorphisms and conformal transformations.
2. A Connecting Principle: A detailed argument, based on information geometry, linking the minimization of kinematic energy in SD (best-matching) to the maximization of entanglement entropy in EG.
3. A Novel Lagrangian: The introduction of a new action principle that incorporates a kinetic term from SD, a potential term from General Relativity, and a new entropic term based on a conformally invariant definition of relational entropy.
4. An Entropic Clock: A proposed solution to the "problem of time" in quantum gravity by defining a relational time parameter based on the rate of change of total entanglement entropy.

This work is primarily a research proposal and a theoretical framework. It includes detailed mathematical derivations of foundational results from both SD and EG to ensure clarity and provide a common starting point. A comprehensive research program is outlined to develop this framework into a testable theory, including steps for formal derivation, quantization, and the investigation of phenomenological implications in cosmology and black hole physics.

In this article, Shailesh Shirali begins with a seemingly simple question but develops the answer into not one, but four different proofs! While the content focuses on mathematics that has many applications, some of which are mentioned here, the multiplicity of proofs is an added draw, helping the teacher to illustrate innovative ways of thinking and connections across approaches

We investigate conjugacy classes of germs of hyperbolic 1-dimensional vector fields at the origin in low regularity. We show that the classical linearization theorem of Sternberg strongly fails in this setting by providing explicit uncountable families of mutually non-conjugate flows with the same multipliers, where conjugacy is considered in the bi-Lipschitz, C 1 and C 1+ac settings.

We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycle introduced by the second author. Hence we obtain a polynomial-time algorithm for computing these special values. Second, we show how to use our techniques to compute certain special values of the Witten zetafunction, and compute some explicit examples.

In this paper the geometric entanglement (GE) of systems in one spatial dimension (1D) and in the thermodynamic limit is analyzed focusing on two aspects. First, we reexamine the calculation of the GE for translation-invariant matrix product states (MPSs) in the limit of infinite system size. We obtain a lower bound to the GE which collapses to an equality under certain sufficient conditions that are fulfilled by many physical systems, such as those having unbroken space (P) or space-time (PT) inversion symmetry. Our analysis justifies the validity of several derivations carried out in previous works. Second, we derive scaling laws for the GE per site of infinite-size 1D systems with correlation length ξ 1. In the case of MPSs, we combine this with the theory of finite-entanglement scaling, allowing to understand the scaling of the GE per site with the MPS bond dimension at conformally invariant quantum critical points.

The Einstein static universe has played a central role in a number of emergent scenarios recently put forward to deal with the singular origin of the standard cosmological model. Here we study the existence and stability of the Einstein static solution in the presence of vacuum energy corresponding to conformally invariant fields. We show that the presence of vacuum energy stabilizes this solution by changing it to a center equilibrium point, which is cyclically stable. This allows nonsingular emergent cosmological models to be constructed in which initially the Universe oscillates indefinitely about an initial Einstein static solution and is thus past eternal.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature. We show that a complete noncompact manifold with asymptoticaly nonnegative Ricci curvature and sectional curvature decay at most quadratically is diffeomorphic to a Euclidean n-space R^n under some conditions on the density of rays starting from the base point p or on the volume growth of geodesic balls in M.

We present a gravity dual of local operator quench in a two-dimensional CFT with conformal boundaries. This is given by a massive excitation in a three-dimensional AdS space with the end of the world brane (EOW brane). Due to the gravitational backreaction, the EOW brane gets deformed in a nontrivial way. We show that the energy-momentum tensor and entanglement entropy computed from the gravity dual and from the BCFT in the large c limit match perfectly. Interestingly, this comparison avoids the folding of the EOW brane in an elegant way.

We present a gravity dual of local operator quench in a two-dimensional CFT with conformal boundaries. This is given by a massive excitation in a three-dimensional AdS space with the end of the world brane (EOW brane). Due to the gravitational backreaction, the EOW brane gets deformed in a nontrivial way. We show that the energy-momentum tensor and entanglement entropy computed from the gravity dual and from the BCFT in the large c limit match perfectly. Interestingly, this comparison avoids the folding of the EOW brane in an elegant way.

In this paper, we perform a fine blow-up analysis for a boundary value elliptic equation involving the critical trace Sobolev exponent related to the conformal deformation of the metrics on the standard ball, namely the problem of prescribing the boundary mean curvature. From this analysis some a priori estimates in low dimension are obtained. With these estimates, we prove the existence of at least one solution when an index-counting formula associated to the prescribed mean curvature is different from zero.

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

In the presence of consistent regulators, the standard procedure of BRST gauge xing (or moving from one gauge to another) can require non-trivial modications. These modications occur at the quantum level, and gauges exist which are only well-dened when quantum mechanical modications are correctly taken into account. We illustrate how this phenomenon manifests itself in the solvable case of two-dimensional bosonization in the path-integral formalism. As a by-product, we show h o w to derive smooth bosonization in Batalin-Vilkovisky Lagrangian BRST quantization. UUITP-8/95 NIKHEF-H 95-017 hep-th/9505040 On leave of absence from the Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen, Denmark.

It is proved that the arbitrary nondegenerate system in a linear complete topological space has a correspondence complete topological space of coefficients with canonical basis. Basicity criterion for systems in such spaces is given in terms of coefficient operator.

Quantum gravity in the region very near the horizon of an extreme Kerr black hole (whose angular momentum and mass are related by J = GM 2 ) is considered. It is shown that consistent boundary conditions exist, for which the asymptotic symmetry generators form one copy of the Virasoro algebra with central charge c L = 12J . This implies that the near-horizon quantum states can be identified with those of (a chiral half of) a two-dimensional conformal field theory (CFT). Moreover, in the extreme limit, the Frolov-Thorne vacuum state reduces to a thermal density matrix with dimensionless temperature T L = 1 2π and conjugate energy given by the zero mode generator, L 0 , of the Virasoro algebra. Assuming unitarity, the Cardy formula then gives a microscopic entropy S micro = 2πJ for the CFT, which reproduces the macroscopic Bekenstein-Hawking entropy S macro = Area 4 G . The results apply to any consistent unitary quantum theory of gravity with a Kerr solution. We accordingly conjecture that extreme Kerr black holes are holographically dual to a chiral two-dimensional conformal field theory with central charge c L = 12J , and in particular that the near-extreme black hole GRS 1915+105 is approximately dual to a CFT with c L ∼ 2 × 10 79 .

We consider the conformally-invariant coupling of topologically massive gravity to a dynamical massless scalar field theory on a three-manifold with boundary. We show that, in the phase of spontaneously broken Lorentz and Weyl symmetries, this theory induces the target space zero mode of the vertex operator for the string dilaton field on the boundary of the three-dimensional manifold. By a further coupling to topologically massive gauge fields in the bulk, we demonstrate directly from the three-dimensional theory that this dilaton field transforms in the expected way under duality transformations so as to preserve the mass gaps in the spectra of the gauge and gravitational sectors of the quantum field theory. We show that this implies an intimate dynamical relationship between T -duality and S-duality transformations of the quantum string theory. The dilaton in this model couples bulk and worldsheet degrees of freedom to each other and generates a dynamical string coupling.

We study self-consistent D=4 gravity-matter systems coupled to a new class of Weyl-conformally invariant lightlike branes (WILL}-branes). The latter serve as material and charged source for gravity and electromagnetism. Further, due to the natural coupling to a 3-index antisymmetric tensor gauge field, the WILL-brane dynamically produces a space-varying bulk cosmological constant. We find static spherically-symmetric solutions where the space-time consists of two regions with black-hole-type geometries separated by the WILL-brane which &quot;straddles&quot; their common event horizon and, therefore, provides an explicit dynamical realization of the &quot;membrane paradigm&quot; in black hole physics. Finally, by matching via WILL-brane of internal Schwarzschild-de-Sitter with external Reissner-Nordstrom-de-Sitter (or external Schwarzschild-de-Sitter)geometries we discover the emergence of a potential &quot;well&quot; for infalling test particles in the vicinity of the WILL-brane (th...

We split the generic conformal mechanical system into a "radial" and an "angular" part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N =4 supersymmetric "angular" Hamiltonian system one may construct a new system with full N =4 superconformal D(1, 2; α) symmetry.

The global symmetry implied bythe fact that one can multiply all masses with a common constant is made into a local, gauge symmetry. The matter action then becomes conformally invariant and it seems natural to choose for the corresponding scalar gauge field the action for a conformally invariant (massless) scalar field. The resulting conformally invariant theory turns out to be equivalent to general relativity. Since this means that the usual Einstein-Hilbert action is not, in fact, a true gauge action for the space-time geometry, the full theory ought to be supplied with such a term. Gauge-theoretic arguments and conformal invariance requirements dictate its form. But a truly infinitesimal geometry must recognize only the transference of length from one point to another point infinitely near the first.

Exact solutions of traversable wormholes are found under the assumption of spherical symmetry and the existence of a non-static conformal symmetry, which presents a more systematic approach in searching for exact wormhole solutions. In this work, a wide variety of solutions are deduced by considering choices for the form function, a specific linear equation of state relating the energy density and the pressure anisotropy, and various phantom wormhole geometries are explored. A large class of solutions impose that the spatial distribution of the exotic matter is restricted to the throat neighborhood, with a cut-off of the stress-energy tensor at a finite junction interface, although asymptotically flat exact solutions are also found. Using the "volume integral quantifier," it is found that the conformally symmetric phantom wormhole geometries may, in principle, be constructed by infinitesimally small amounts of averaged null energy condition violating matter. Considering the tidal acceleration traversability conditions for the phantom wormhole geometry, specific wormhole dimensions and the traversal velocity are also deduced.

We give a new sufficient condition for existence and completeness of wave operators in abstract scattering theory. This condition generalises both trace class and smooth approaches to scattering theory. Our construction is based on estimates for the Cauchy transforms of operator valued measures. 2000 Mathematics Subject Classification. 47A40.

In the first part of this series of papers we developed the invariant differentiation with respect to a Cartan connection, we described this procedure in the terms of the underlying principal connections, and we discussed applications of this theory to the construction of natural operators. In this part we will extend the results of [Ochiai, 70] on the existence and the uniqueness of the so called normal Cartan connections on manifolds with almost Hermitian symmetric structures to first order structures which do not admit a torsion free linear connection. Moreover, for each of these structures we obtain explicit (universal) formulae for these canonical connections in the terms of the curvatures of the underlying linear connections. In the sequel, we shall use the notation and results from the preceding part of this series of papers, see [ Čap, Slovák, Souček]. In particular, the citations like I.2.3 mean the corresponding items in that part. In the first part we defined almost Hermitian symmetric structures as 'second order structures', see I.3.4. In this part we will first show that any first order structure with the 'right' structure group gives rise to an almost Hermitian symmetric structure in this sense. Basically, the construction is just the standard first prolongation of G-structures, see [Kobayashi] or [Sternberg]. Due to the special situation there is a canonical prolongation in this case which admits the structure of a principal bundle with the structure group B. Moreover, it turns out that for the almost Hermitian structures in question, there exists a unique normal Cartan connection. We shall prove this in section 2. Thus, the calculus developed in part I, will yield natural operators in all these cases. Our approach is quite different to that of Ochiai whose vanishing torsion assumption restricts in fact the considerations to the locally flat structures in many cases, cf. [Baston, 91] or [ Čap, Slovák, 95]. Another approach to the construction of the canonical Cartan connections on certain auxiliary vector bundles, thus avoiding the construction of the prolongation, can be found in [Baston, 91]. We shall discuss the applications to the individual almost Hermitian symmetric structures in the next part of this series.

This paper resolves the long-standing computational spectral problem. That is to determine the existence of algorithms that can compute spectra sp(A) of classes of bounded operators A = {a ij } i,j∈N ∈ B(l 2 (N)), given the matrix elements {a ij } i,j∈N , that are sharp in the sense that they realise the boundaries of what a digital computer can achieve. Similarly, for a Schrödinger operator H = -∆ + V , determine the existence of algorithms that can compute the spectrum sp(H) given point samples of the potential function V . In order to solve the problems, we establish the Solvability Complexity Index (SCI) hierarchy, based on the SCI introduced by one of the authors in . This is a classification hierarchy for all types of problems in computational mathematics that allows for classifications determining the boundaries of what computers can achieve in scientific computing. As a consequence, the SCI hierarchy provides classifications of computational problems that can be used in computer-assisted proofs, see §1.3 and §3. The SCI hierarchy captures many key computational issues in the history of mathematics including the insolvability of the quintic, Smale's problem on the existence of iterative generally convergent algorithm for polynomial root finding, the computational spectral problem, inverse problems, optimisation etc. Given the many applications in mathematical physics, analysis, quantum chemistry, statistical mechanics, quantum mechanics, quasicrystals, optics etc., the problem of computing spectra of operators has fascinated and frustrated mathematicians since the early work by H. Goldstine, F. Murray and J. von Neumann [53] in the 1950s, yielding a vast literature (see §5). W. Arveson [8] pointed out in the early 1990s that: "Unfortunately, there is a dearth of literature on this basic problem, and so far as we have been able to tell, there are no proven techniques" (see also A. Böttcher's Problem I in [17]). Arveson considered computing spectra from matrix elements {a ij } i,j∈N ∈ B(l 2 (N)), however, the situation is not better for the Schrödinger case. In particular, despite more than 90 years of quantum mechanics, it is still unknown how to compute spectra of -∆ discrete + V on lattices and -∆ + V on L 2 (R d ) given point samples from the potential function V . We solve these problems by providing algorithms that compute spectra and approximate eigenvectors, allowing for problems that were previously out of reach. We prove lower bounds yielding sharp classification results and optimality of the algorithms. The results may be surprising and link to many areas of mathematics. Classifications and new algorithms: The SCI hierarchy induces a total ordering ≤ SCI (see Remark 1.4) on the family of computational spectral problems describing their difficulty. For example, given infinite matrices of the form A = {a ij } i,j∈N ∈ B(l 2 (N)), we prove the following: Computing sp(A), A is diagonal = SCI computing sp(-∆ + V ) with bounded V = SCI computing sp(-∆ discrete + V ) on any lattice < SCI computing sp(A), A is compact = SCI computing sp(-∆ + V ) with V blowing up at ∞ < SCI computing sp(A), A is self-adjoint. (1.1) 2010 Mathematics Subject Classification. 47A10 (primary) and 81Q10, 34L16, 46N40 (secondary).

Given a compact four dimensional smooth Riemannian manifold $(M,g)$ with smooth boundary, we consider the evolution equation by $Q$-curvature in the interior keeping the $T$-curvature and the mean curvature to be zero and the evolution equation by $T$-curvature at the boundary with the condition that the $Q$-curvature and the mean curvature vanish. Using integral method, we prove global existence and

Working in a given conformal class, we prove existence of constant Q-curvature metrics on compact manifolds of arbitrary dimension under generic assumptions. The problem is equivalent to solving a nth-order nonlinear elliptic differential (or integral) equation with variational structure, where n is the dimension of the manifold. Since the corresponding Euler functional is in general unbounded from above and below, we use critical point theory, jointly with a compactness result for the above equation.

In this paper we prove that, given a compact four-dimensional smooth Riemannian manifold (M, g) with smooth boundary, there exists a metric in the conformal class [g] of the background metric g with constant Q-curvature, zero T -curvature and zero mean curvature under generic conformally invariant assumptions. The problem is equivalent to solving a fourth-order non-linear elliptic boundary value problem (BVP) with boundary condition given by a third-order pseudodifferential operator and homogeneous Neumann condition. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from above and below, we need to use min-max methods combined with a new topological argument and a compactness result for the above BVP.

In this paper we prove that, given a compact four-dimensional smooth Riemannian manifold (M, g) with smooth boundary, there exists a metric conformal to g with constant T -curvature, zero Q-curvature and zero mean curvature under generic and conformally invariant assumptions. The problem amounts to solving a fourth-order nonlinear elliptic boundary value problem (BVP) with boundary conditions given by a third-order pseudodifferential operator and homogeneous Neumann operator. It has a variational structure, but since the corresponding Euler-Lagrange functional is in general unbounded from below, we look for saddle points. We do this by using topological arguments and min-max methods combined with a compactness result for the corresponding BVP.

The AdS/CFT correspondence between string theory in AdS space and conformal field theories in physical space-time leads to an analytic, semi-classical model for strongly-coupled QCD which has scale invariance and dimensional counting at short distances and color confinement at large distances. The AdS/CFT correspondence also provides insights into the inherently non-perturbative aspects of QCD such as the orbital and radial spectra of hadrons and the form of hadronic wavefunctions. In particular, we show that there is an exact correspondence between the fifth-dimensional coordinate of anti-de Sitter (AdS) space z and a specific light-front impact variable ζ which measures the separation of the quark and gluonic constituents within the hadron in ordinary space-time. This connection allows one to compute the analytic form of the frame-independent light-front wavefunctions of mesons and baryons, the fundamental entities which encode hadron properties and which allow the computation of decay constants, form factors and other exclusive scattering amplitudes. Relativistic light-front equations in ordinary space-time are found which reproduce the results obtained using the fifth-dimensional theory. As specific examples we compute the pion coupling constant fπ, the pion charge radius ˙r2 π ¸and examine the propagation of the electromagnetic current in AdS space, which determines the space and time-like behavior of the pion form factor and the pole of the ρ meson.

In this note we show that the roots of a polynomial are C ∞ depend of the coefficients. The main tool to show this is the Implicit Function Theorem. Resumen. En esta nota se muestra el hecho que las raíces de un polinomio son C ∞ , depende de los coeficientes. La herramienta principal para mostrar este resultado es el teorema de la función implícita.

We present a comprehensive theoretical and experimental framework for gravitational modulation using entropy-driven, prime-resonant electromagnetic fields. By embedding symbolic entropy gradients into the dynamics of prime-indexed harmonic oscillators, we demonstrate the emergence of effective curvature in spacetime. This curvature is derived from coherent resonance collapse processes and quantified via a modified gravitational field tensor. We simulate the symbolic curvature field, visualize particle geodesic trajectories, and propose experimental schematics for physical implementation. This framework suggests a novel paradigm for manipulating spacetime geometry through symbolic information dynamics.

We propose a one-dimensional nonlocal stochastic model of adsorption and desorption depending on one parameter, the adsorption rate. At a special value of this parameter, the model has some interesting features. For example, the spectrum is given by conformal field theory, and the stationary non-equilibrium probability distribution is given by the two-dimensional equilibrium distribution of the ice model with domain wall type boundary conditions. This connection is used to find exact analytic expressions for several quantities of the stochastic model. Vice versa, some understanding of the ice model with domain wall type boundary conditions can be obtained by the study of the stochastic model. At the special point we study several properties of the model, such as the height fluctuations as well as cluster and avalanche distributions. The latter has a long tail which shows that the model exhibits self organized criticality. We also find in the stationary state a special surface phase transition without enhancement and with a crossover exponent f=2/3. Furthermore, we study the phase diagram of the model as a function of the adsorption rate and find two massive phases and a scale invariant phase where conformal invariance is broken.

We propose a one-dimensional quantum Heienberg spin-2 chain, which exhibits two topologically distinct valence bond solid states in two different solvable limits. We then construct the phase diagram and study the quantum phase transition between these two states using the infinite time evolving block decimation algorithms. From the scaling relation between the entanglement entropy and correlation length, we determine that the central charge for the underlying critical conformal field theory is c = 2.

We describe a general procedure for computing renormalized curvature integrals on Poincaré-Einstein manifolds. In particular, we explain the connection between the Gauss-Bonnet-type formulas of Albin and Chang-Qing-Yang for the renormalized volume, and explicitly identify a scalar conformal invariant in the latter formula. Our approach constructs scalar conformal invariants of weight -n on n-manifolds, n ≥ 8, that are natural divergences; these imply that the scalar invariant in the Chang-Qing-Yang formula is not unique in dimension n ≥ 8. Our procedure also produces explicit conformally invariant Gauss-Bonnet-type formulas for compact Einstein manifolds.

Conventional wisdom says that the formation of large-scale structures in cosmology follows from the evolution of density perturbations under the combined effect of gravitation and cosmic expansion. We survey here the reasons why this framework fails to hold in primordial cosmology, due to the severe limitations placed on the Friedmann-Robertson-Walker (FRW) metric, the continuity hypothesis and the standard equations of fluid flows. We insist that understanding primordial cosmology must rely instead on a complex dynamics model of evolving dimensional fluctuations, conjectured to come into play far above the electroweak scale. A key outcome of this model is the generation of topological defects and condensates emerging from the complex Ginzburg-Landau equation.

We sketch a Weyl creation operator approach to the Riemann Hypothesis; i.e., arithmetic on the Weyl algebras with ergodic theory to transport operators. We prove that finite Hasse-Dirichlet alternating zeta functions or eta functions can be induced from a product of "creation operators". The latter idea is the result of considering complex quantum mechanics with complex time-space related concepts. Then we overview these ideas, with variations, which may provide a pathway that may settle RH; e.g., a Euler Factor analysis to study the quantum behavior of the integers as one pushes up the critical line near a macro zeta function zero.

Emphasising their connection with shell-like star-like curves, this work investigates a new subclass of star-like functions defined by q-Fibonacci numbers and q-polynomials. We study the geometric and analytic properties of this subclass, including the computation of intervals of univalence and nonunivalence for some functions. Moreover,
we define a sufficient condition for functions in this subclass to satisfy the criteria of the famous class of analytic functions with positive real components. This work improves our understanding of the link between Fibonacci-type sequences and the geometric properties of analytic functions by using subordination ideas and the features of q-Fibonacci
sequences. Emphasising the possibility for diverse research in combinatorial and analytical mathematics, the results offer fresh insights and support further study on the applications of calculus in geometric function theory

Recently, the asymptotic behaviour of three-dimensional anti-de Sitter gravity with a topological mass term was investigated. Boundary conditions were given that were asymptotically invariant under the two-dimensional conformal group and that included a fall-off of the metric sufficiently slow to consistently allow pp-wave type of solutions. Now, pp-waves can have two different chiralities. Above the chiral point and at the chiral point, however, only one chirality can be considered, namely the chirality that has the milder behaviour at infinity. The other chirality blows up faster than AdS and does not define an asymptotically AdS spacetime. By contrast, both chiralities are subdominant with respect to the asymptotic behaviour of AdS spacetime below the chiral point. Nevertheless, the boundary conditions given in the earlier treatment only included one of the two chiralities (which could be either one) at a time. We investigate in this paper whether one can generalize these boundary conditions in order to consider simultaneously both chiralities below the chiral point. We show that this is not possible if one wants to keep the two-dimensional conformal group as asymptotic symmetry group. Hence, the boundary conditions given in the earlier treatment appear to be the best possible ones compatible with conformal symmetry. In the course of our investigations, we provide general formulas controlling the asymptotic charges for all values of the topological mass (not just below the chiral point).

Let D α denote the Dirichlet-type space of functions analytic on the unit disk U and Q α the conformal invariant version of this space. Any analytic self-map φ of U induces a composition operator C φ acting on D α , respectively, Q α by C φ f = f • φ, where f ∈ D α , respectively, f ∈ Q α . The aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of φ.