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For any nonempty set U ⊂ R + , we consider the maximal operator H U defined as H U f = sup u ∈ U | H (u) f | , where H (u) represents the Hilbert transform along the monomial curve u γ (s) . We focus on the L p (R d) operator norm of H U for p ∈ (p ∘ (d) , ∞) , where p ∘ (d) is the optimal exponent known for the L p boundedness of the maximal averaging operator obtained by Ko–Lee–Oh (Invent Math 228:991–1035, 2022, Forum Math Pi 11:Paper No. e4, 33, 2023) and Beltran–Guo–Hickman–Seeger (Am J Math, ). To achieve this goal, we employ a novel bootstrapping argument to establish a maximal estimate for the Mihlin–Hörmander-type multiplier, along with utilizing the local smoothing estimate for the averaging operator and its vector-valued extension to obtain crucial decay estimates. Furthermore, our approach offers an alternative means for deriving the upper bound established in Guo–Roos–Seeger–Yung (Math Ann 377:69–114, 2020). [ABSTRACT FROM AUTHOR]

Boson stars are self-gravitating solutions made entirely of fundamental massive scalar fields. Here we investigate mini boson stars in D non-compact spacetime dimensions and we show that they are dynamically unstable for D > 4 . [ABSTRACT FROM AUTHOR]

In this paper, we have developed a class of new solutions for relativistic compact stars in higher-dimensional (D ≥ 4) space-time assuming pseudo-spheroidal geometry of the grr metric component. To study the physical properties, we consider equation of state of interior strange matter p = 1 3(ρ − 4B), as proposed in MIT bag model in the presence of a density-dependent B parameter. The interior matter of a quark star may consist of three flavor quarks. We observe some interesting results. The parameter B depends on the anisotropy (α) in the interior, and at the stellar surface, B attains a constant value independent of α. At the interior, B increases with the increase of α. We also note that the value of B approaches a constant value with increasing spheroidal parameter λ. B(ρ) also depends on space-time dimensions and it is interesting to note that B(ρ) picks up negative values near core region which limits the number of space-time dimensions available for a stellar model. All the stability criterion and energy conditions hold good for a physically realistic stellar configuration. It is observed that strange quark matter may be stable or metastable or unstable depending upon the value of energy per baryon (EB). Strange quark matter will be stable if energy per baryon EB < 930.4MeV/fm3. It is noted that the maximum mass obtainable in this model is 1.99M⊙ considering stable strange quark matter when dimension D = 4. In addition, we observe that the dimensions have some effect on the gross properties of strange stars (SS). [ABSTRACT FROM AUTHOR]

This study investigates the Lie symmetry properties of matter variables, spacetime dimension, and kinematic quantities in spherically symmetric radiating stars undergoing gravitational collapse in higher dimensional spacetimes. Our analysis explores how variations in functions and dependent variables influence the Lie algebra dimensionality and Lie symmetries of the boundary condition. Our study reveals the explicit dependence of charge and spacetime dimension in the Lie symmetry generators for the first time. We examine models with various kinematical features, such as shear, shear-free, geodesic, and non-expanding conditions with different matter variables. Using the Lie symmetries we obtain group invariant solutions to the models and observe how the kinematical features and matter variables affect the Lie symmetries. Additionally, we demonstrate the appearance of a linear equation of state. We recover known four-dimensional models from our results.

The analysis of the continual gravitational contraction of a spherically symmetric shell of charged radiation is extended to higher dimensions in Einstein–Gauss–Bonnet gravity. The spacetime metric, which is of Boulware–Deser type, is real only up to a maximum electric charge and thus collapse terminates with the formation of a branch singularity. This branch singularity divides the higher dimensional spacetime into two regions, a real and physical one, and a complex region. This is not the case in neutral Einstein–Gauss–Bonnet gravity as well as general relativity. The charged gravitational collapse process is also similar for all dimensions N ≥ 5 unlike in the neutral scenario where there is a marked difference between the N = 5 and N > 5 cases. In the case where N = 5 uncharged collapse ceases with the formation of a weaker, conical singularity which remains naked for a time depending on the Gauss–Bonnet invariant, before succumbing to an event horizon. The similarity of charged collapse for all higher dimensions is a unique feature in the theory. The sufficient conditions for the formation of a naked singularity are studied for the higher dimensional charged Boulware–Deser spacetime. For particular choices of the mass and charge functions, naked branch singularities are guaranteed and indeed inevitable in higher dimensional Einstein–Gauss–Bonnet gravity. The strength of the naked branch singularities is also tested and it is found that these singularities become stronger with increasing dimension, and no extension of spacetime through them is possible. [ABSTRACT FROM AUTHOR]

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results. [ABSTRACT FROM AUTHOR]

The rotations of rigid bodies in Euclidean space are characterized by their instantaneous angular velocity and angular momentum. In an arbitrary number of spatial dimensions, these quantities are represented by bivectors (antisymmetric rank-2 tensors), and they are related by a rank-4 inertia tensor. Remarkably, this inertia tensor belongs to a well-studied class of algebraic curvature tensors that have the same index symmetries as the Riemann curvature tensor field used in general relativity. Any algebraic curvature tensor can be decomposed into irreducible representations of the orthogonal group via the Ricci decomposition. We calculate the Ricci decomposition of the inertia tensor for a rigid body in any number of dimensions, and we find that (unlike for the Riemann curvature tensor field) its traceless Weyl tensor is always zero, so the inertia tensor is completely characterized by its (rank-2) Ricci contraction. So unlike in general relativity, the traceless Weyl tensor does not cause any qualitatively new phenomenology for rigid-body dynamics in n ≥ 4 dimensions. [ABSTRACT FROM AUTHOR]

Static equilibrium configurations of continua supported by surface tension are given by constant mean curvature (CMC) surfaces which are critical points of a variational problem to extremize the area while keeping the volume fixed. CMC surfaces are used as mathematical models of a variety of continua, such as tiny liquid drops, stars, and nuclei, to play important roles in both mathematics and physics. Therefore, the geometry of CMC surfaces and their properties such as stability are of special importance in differential geometry and in a variety of physical sciences. In this paper we examine the stability of CMC hypersurfaces in arbitrary dimensions, possibly having boundaries on two parallel hyperplanes, by investigating the second variation of the area. We determine the stability of non-uniform liquid bridges or unduloids for the first time in all dimensions and all parameter (the ratio of the neck radius to bulge radius) regimes. The analysis is assisted by numerical computations. [ABSTRACT FROM AUTHOR]

It has been claimed that wormholes are just as good a prediction of Einstein’s theory as black holes, but they are subject to severe restrictions from quantum field theory. The purpose of this paper is to show that the claim can be substantiated in spite of these restrictions.

Studies of scattering amplitudes for electric and magnetic charges have identified previously overlooked multiparticle representations of the Poincaré group in four dimensions. Such representations associate nontrivial quantum numbers (known as pairwise helicities) with asymptotically separated pairs of particles, and thus cannot be described as tensor products of one-particle states. We extend this construction to sources and spacetimes of higher dimension. We first establish the dynamical origin of pairwise helicity in p -form electrodynamics coupled to mutually nonlocal branes. We then interpret this pairwise helicity as a quantum number under an SO(2) pairwise little group associated with pairs of distinct branes. We further characterize the "higher" little groups that could in principle be used to induce multiparticle or multi-brane representations of the Lorentz group. [ABSTRACT FROM AUTHOR]

Using an open thermodynamic systems theory, the effect of particle creation on the evolution and dynamics of the standard cosmological FLRW model in a higher-dimensional spacetime with functionally dependent cosmological and gravitational constants Λ and G is investigated. The gravitational field equations have been transformed into a dimensionless system of non-linear, first-order, coupled differential equations (DEs) as functions of the universe's density parameters Ω i and rate of particle creation Ψ in redshift space, which can be numerically casted. Two cosmological models are obtained, depending on the choice of particle creation rate— Ψ ∼ H 2 and Ψ ∼ n 2 for dust-, radiation- and dark-energy-dominated universes, respectively. The dynamic behaviour of each model is discussed. [ABSTRACT FROM AUTHOR]

First, the solution uniqueness, existence and regularity for stationary anisotropic (linear) Stokes and generalised Oseen systems with constant viscosity coefficients in a compressible framework are analysed in a range of periodic Sobolev (Bessel‐potential) spaces in ℝn$$ {\mathrm{\mathbb{R}}}^n $$. By the Galerkin algorithm and the Brower fixed point theorem, the existence of solution to the stationary anisotropic (nonlinear) Navier–Stokes incompressible system is shown in a periodic Sobolev space for any n≥2$$ n\ge 2 $$. Then the solution uniqueness and regularity results for stationary anisotropic periodic Navier–Stokes system are established for n∈{2,3,4}$$ n\in \left\{2,3,4\right\} $$. [ABSTRACT FROM AUTHOR]

In this work, spherically symmetric thin-shell wormholes with a conformally invariant Maxwell field for N -dimensional F (R) gravity and constant scalar curvature R are built. Two cases are considered: wormholes symmetric across the throat and asymmetric ones having different values of the scalar curvature across the throat. Their stability under radial perturbations is analyzed, finding that unstable and stable solutions are possible for suitable values of the parameters, always made of exotic matter. The stable solutions are found for a short range, slightly over a large critical value of charge. [ABSTRACT FROM AUTHOR]

First, the solution uniqueness, existence and regularity for stationary anisotropic (linear) Stokes and generalised Oseen systems with constant viscosity coefficients in a compressible framework are analysed in a range of periodic Sobolev (Bessel‐potential) spaces in ℝn$$ {\mathrm{\mathbb{R}}}^n $$. By the Galerkin algorithm and the Brower fixed point theorem, the existence of solution to the stationary anisotropic (nonlinear) Navier–Stokes incompressible system is shown in a periodic Sobolev space for any n≥2$$ n\ge 2 $$. Then the solution uniqueness and regularity results for stationary anisotropic periodic Navier–Stokes system are established for n∈{2,3,4}$$ n\in \left\{2,3,4\right\} $$. [ABSTRACT FROM AUTHOR]

This article deals with the study of Hyers–Ulam stability (HU stability) and Hyers–Ulam–Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results.

The desingularization of a class of black hole space-times arising as solutions to the string equations is considered in connection with the consistency of the quantum theory and the description of nonperturbative states with quantum numbers from the particle spectrum. The geometry arising in an extreme limit of one of the singular solutions to the gravitational field equations is demonstrated to be a background of N = 2 string theory. The positivity of the masses in the particle spectrum is proven through quasilocal integrals near the resolved singularities in these limits of black hole space-times. [ABSTRACT FROM AUTHOR]

In this study, we investigate the thermodynamic properties and quasinormal modes of Dymnikova black holes within the context of higher dimensions in Einstein's general theory of relativity. We calculate the thermodynamic parameters, including the Hawking temperature and heat capacity, which allowed us to investigate the black hole's stability. Lastly the quasinormal modes with the WKB formula were calculated. [ABSTRACT FROM AUTHOR]

Assuming that the geometry of spacetime is uniquely determined by the energy–momentum tensor of matter alone, i.e. without any interactions, enables us to construct the Lagrangian from which the metric of higher dimensional spacetime follows. From the geodesic equations that follow, it becomes clear that the incorrect mass of elementary particles predicted by Kaluza–Klein theories arises from the assumption that in the absence of gravity the solution to the Einstein field equations reduces to the Minkowski metric. From construction of a consistent theory of 4 D electromagnetism, we find that this assumption does not only result in the incorrect mass of elementary particles, but also the incorrect value of the cosmological constant. This suggests that these incorrect predictions, which are often regarded as major flaws of Kaluza–Klein theories, just reflect the inconsistency of the assumption that the solution to Einstein field equations reduces to Minkowski metric in the absence of gravity and Weyl invariance which is the symmetry of gauge theories in 4 D spacetime. Abandoning this assumption results in modification of general relativity. We show that the unified description of fundamental interactions naturally incorporates the Higgs mechanism. For non-Abelian gauge fields, we find that the manifold comprising the extra dimensions has to be a group manifold and show that the standard model is realized in 16 D spacetime. We show that charge and spin are the same concept, but what makes them different is that the former follows from symmetry of 4 D spacetime while the latter follows from symmetry of the internal space. • Kaluza–Klein metric is generally not the solution to vacuum Einstein equations. • Solution to unrealistic mass problem is possible for modified general relativity. • Compactification does not result in infinite tower of massive modes. • Unified description of particles is possible in higher dimensions. [ABSTRACT FROM AUTHOR]

We present a simple gedanken experiment in which a compact object traverses a spacetime with three macroscopic spatial dimensions and n compact dimensions. The compactification radius is allowed to vary, as a function of the object’s position in the four-dimensional space, and we show that the conservation of gravitational self-energy implies the dimensional dependence of the mass-radius relation. In spacetimes with extra dimensions that are compactified at the Planck scale, no deviation from the four-dimensional result is found, but, in spacetimes with extra dimensions that are much larger than the Planck length, energy conservation implies a deviation from the normal Compton wavelength formula. The new relation restores the symmetry between the Compton wavelength and Schwarzschild radius lines on the mass-radius diagram and precludes the formation of black holes at TeV scales, even if large extra dimensions exist. We show how this follows, intuitively, as a direct consequence of the increased gravitational field strength at distances below the compactification scale. Combining these results with the heuristic identification between the Compton wavelength and the minimum value of the position uncertainty, due to the Heisenberg uncertainty principle, suggests the existence of generalised, higher-dimensional uncertainty relations. These relations may be expected to hold for self-gravitating quantum wave packets, in higher-dimensional spacetimes, with interesting implications for particle physics and cosmology in extra-dimensional scenarios.

For any k ≥ 2 and fixed b = (b 1 , ⋯ , b k) ∈ N k , this paper concerns the number of integer lattice points in Z k which are b -visible from a set of N watch-points. Moreover, for N = 2 and b ˜ = ( b ˜ 1 , ⋯ , b ˜ k) ∈ N k with b ˜ ≠ b , we consider the mixed visibility, that is, the number of integer lattice points which are b -visible and b ˜ -visible from two watch-points respectively. We prove asymptotic formulas for the number of corresponding visible lattice points. Part of our results improves a classical result of Rearick. [ABSTRACT FROM AUTHOR]

This study investigates the Lie symmetry properties of matter variables, spacetime dimension, and kinematic quantities in spherically symmetric radiating stars undergoing gravitational collapse in higher dimensional spacetimes. Our analysis explores how variations in functions and dependent variables influence the Lie algebra dimensionality and Lie symmetries of the boundary condition. Our study reveals the explicit dependence of charge and spacetime dimension in the Lie symmetry generators for the first time. We examine models with various kinematical features, such as shear, shear-free, geodesic, and non-expanding conditions with different matter variables. Using the Lie symmetries we obtain group invariant solutions to the models and observe how the kinematical features and matter variables affect the Lie symmetries. Additionally, we demonstrate the appearance of a linear equation of state. We recover known four-dimensional models from our results., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Authors.)

Abstract We formulate a set of general rules for computing d-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism [1]. With these rules, the procedure for determining any conformal block of interest is reduced to (1) identifying the relevant projection operators and tensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate the bookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We present several concrete examples to illustrate the general procedure as well as to demonstrate and test the explicit application of the rules. In particular, we consider four-point functions involving scalars S and some specific irreducible representations R, namely 〈SSSS〉, 〈SSSR〉, 〈SRSR〉 and 〈SSRR〉 (where, when allowed, R is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations.

Black holes play an important role in linking microphysics with macrophysics, with those of the Planck mass (MP ∼ 10−5 g) featuring in any theory of quantum gravity. In particular, the Compton-Schwarzschild correspondence posits a smooth transition between the Compton wavelength (RC ∝ 1/M) below the Planck mass and the Schwarzschild radius (RS ∝ M) above it. The duality between RC and RS implies a form of the Generalized Uncertainty Principle (GUP) and suggests that elementary particles may be sub-Planckian black holes. The simplest possibility is that the ADM mass has the form M+βMP2/M for some constant β and this model can be extended to charged and rotating black holes, clearly relevant to elementary particles. Another possibility is that sub-Planckian black holes may arise in loop quantum gravity and this explicitly links black holes and elementary particles. Higher dimensions may modify both proposals. If there are n extra dimensions, all with the same compactification scale, one expects RS ∝ M1/(1+n) below this scale but RC depends on the form of the higher-dimensional wave-function. If it is spherically symmetric, then RC ∝ M−1, so duality is broken and the Planck mass is reduced, allowing the possibility of TeV quantum gravity. If the wave-function is pancaked in the extra dimensions, RC ∝ M−1/(1+n) and so duality is preserved but the Planck mass is unchanged.

Neutrino physics is one of the most active fields of research with important implications for particle physics, cosmology and astrophysics. On the other hand, motivated by some theories including string theory, formulation of physical theories in more than four spacetime dimensions has been the subject of increasing attention in recent years. Interaction of neutrinos with gravitational fields is one of the interesting phenomena which can lead to transition between different helicity states (spin oscillations). We study neutrino spin oscillations in Schwarzschild and RN backgrounds in higher-dimensional gravitational fields. We calculate the transition probability as a function of time and also study the dependence of the oscillation frequency on the orbital radius. The results help us to better understand the behavior of gravity and neutrinos in higher dimensions. [ABSTRACT FROM AUTHOR]

We present a brief description of noncompactified higher-dimensional theories from the perspective of general relativity. More concretely, the Space–Time–Matter theory, or Induced Matter theory, and the reduction procedure used to construct the modified Brans–Dicke theory and the modified Sáez–Ballester theory are briefly explained. Finally, we apply the latter to the Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological models in arbitrary dimensions and analyze the corresponding solutions. [ABSTRACT FROM AUTHOR]

Using an open thermodynamic systems theory, the effect of particle creation on the evolution and dynamics of the standard cosmological FLRW model in a higher-dimensional spacetime with functionally dependent cosmological and gravitational constants Λ and G is investigated. The gravitational field equations have been transformed into a dimensionless system of non-linear, first-order, coupled differential equations (DEs) as functions of the universe’s density parameters Ωi and rate of particle creation Ψ in redshift space, which can be numerically casted. Two cosmological models are obtained, depending on the choice of particle creation rate—Ψ∼H2 and Ψ∼n2 for dust-, radiation- and dark-energy-dominated universes, respectively. The dynamic behaviour of each model is discussed.

We establish a new family of hairy black hole solutions of quartic quasi-topological gravity sourced by logarithmic electrodynamics and conformal scalar field. Here, we derive the metric function defining black hole structures and investigate the effects of electric charge, scalar hair, and dimensionality parameters on the location of the event horizon. The corresponding hairy black hole solution of the quartic quasi-topological gravity sourced by Maxwell field is also deduced in this setup. In addition, we also compute thermodynamic quantities for these objects such as temperature, entropy and heat capacity, and check the first law of thermodynamics. Furthermore, the impacts of scalar hair and electric charge on the divergences of heat capacity and extreme black hole's horizon radius are also investigated. Finally, we also analyze thermodynamic stability and critical behavior of the resultant hairy black holes at the event horizon. [ABSTRACT FROM AUTHOR]

Rényi Holographic Dark Energy (RHDE) Models have been studied in different dimensions considering a flat FRW metric and subsequently exploring the nature of evolution of different cosmological parameters. It is seen that the number of dimensions affect the evolution of the cosmological parameters and the transition of the universe from decelerating to accelerating phase. The statefinder diagnostics have been performed and it is seen that the model corresponds to the Λ CDM model for the present and future universe. [ABSTRACT FROM AUTHOR]

Kaluza was the first to realize that the four-dimensional gravitational field of general relativity and the classical electromagnetic field behave as if they were components of a five-dimensional gravitational field. We present a novel experimental test of the macroscopic classical interpretation of the Kaluza fifth dimension. Our experiment design probes a key feature of Kaluza unification—that electric charge is identified with motion in the fifth dimension. Therefore, we tested for a time dilation effect on an electrically charged clock. This test can also be understood as a constraint on time dilation from a constant electric potential of any origin. This is only the second such test of time dilation under electric charge reported in the literature, and a null result was obtained here. We introduce the concept of a charged clock in the Kaluza context, and discuss some ambiguities in its interpretation. We conclude that a classical, macroscopic interpretation of the Kaluza fifth dimension may require a timelike signature in the five-dimensional metric, and the associated absence of a rest frame along the fifth coordinate.

We formulate a set of general rules for computing d-dimensional four-point global conformal blocks of operators in arbitrary Lorentz representations in the context of the embedding space operator product expansion formalism [1]. With these rules, the procedure for determining any conformal block of interest is reduced to (1) identifying the relevant projection operators and tensor structures and (2) applying the conformal rules to obtain the blocks. To facilitate the bookkeeping of contributing terms, we introduce a convenient diagrammatic notation. We present several concrete examples to illustrate the general procedure as well as to demonstrate and test the explicit application of the rules. In particular, we consider four-point functions involving scalars S and some specific irreducible representations R, namely 〈SSSS〉, 〈SSSR〉, 〈SRSR〉 and 〈SSRR〉 (where, when allowed, R is a vector or a fermion), and determine the corresponding blocks for all possible exchanged representations. [ABSTRACT FROM AUTHOR]

In this paper we obtain relativistic solutions of compact stars in higher dimensions assuming a pseudo-spheroidal geometry comprising of anisotropic fluid in hydrostatic equilibrium. In the Vaidya-Tikekar model the ansatz for one metric potential is parametrized by two parameter namely the spheroidal parameter (λ) and curvature parameter (R ). In this model we obtained a toy model solution for anisotropic fluid satisfying the acoustic condition. For a suitable combination of parameters λ , R and anisotropic parameter α , we note that the equation of state of interior matter content of anisotropic fluid is resemble with the MIT bag model equation of state of the form p = 1 3 (ρ − 4 B) for quark matter with the value of B lying in the range necessary bulk strange matter relative to neutron. Here B is known as Bag constant. In this model the maximum radius of a star is depend on the spheroidal parameter (λ) , dimension (D) and anisotropic parameter (α) collectively. We also note that the limiting values of λ for stellar toy model in isotropic case get modified in presence of anisotropy. In case of higher dimensional anisotropic star, we note the limiting values of λ as λ < λ 1 and λ 2 < λ < λ 1 , where λ 1 = D 2 − 6 D + 1 + 2 α D 3 + (1 − 7 α) D 2 + (2 + 15 α) D + (1 − 9 α) D 2 − 8 D − 1 + 12 α − 4 D α and λ 2 = D − 1 + 2 α + 2 α 2 + (D − 1) α + 1 D − 3 . The non-negativity of energy density (ρ) also puts another bound on spheroidal parameter (λ) with a lower limit. We also evaluated the maximum mass, surface red-shift and checked the causality and stability conditions. We also note that a large number of space-time dimensions allowed in brane-world are not permitted in the context of compact star. [ABSTRACT FROM AUTHOR]

We present a brief description of noncompactified higher-dimensional theories from the perspective of general relativity. More concretely, the Space–Time–Matter theory, or Induced Matter theory, and the reduction procedure used to construct the modified Brans–Dicke theory and the modified Sáez–Ballester theory are briefly explained. Finally, we apply the latter to the Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological models in arbitrary dimensions and analyze the corresponding solutions.

We revisit an emergent gravity scenario in (4 + 1) dimensions underlying a propagating geometric torsion ℋ 3 with a renewed interest. We show that a pair-symmetric 4 th-order curvature tensor is sourced by Neveu–Schwarz (NS) two-form in a U (1) gauge theoretic formulation. Interestingly, the new spacetime curvature governs a torsion-free geometry sourced by an NS form and shares the properties of the Riemann tensor. On the other hand, a completely anti-symmetric 4 th-order tensor in the formulation is shown to incorporate a dynamical geometric torsion and is argued to be identified with a nonperturbative correction. The four-form turns out to be U (1) gauge invariant underlying an onshell NS form. We show that an emergent gravity theory may elegantly be described with an axionic scalar presumably signifying a quintessence coupling to the Riemann-type geometries. The curvatures are appropriately worked out to obtain a d = 1 2 emergent form theory. Investigation reveals that a pair of (M M ̄) 1 0 -brane is created across an event horizon. We show that an emergent M theory in a decoupling limit identifies with the bosonic sector of N = 1 , Supergravity in d = 1 1. [ABSTRACT FROM AUTHOR]

Abstract The study of higher-dimensional black holes is a subject which has recently attracted vast interest. Perhaps one of the most surprising discoveries is a realization that the properties of higher-dimensional black holes with the spherical horizon topology and described by the Kerr–NUT–(A)dS metrics are very similar to the properties of the well known four-dimensional Kerr metric. This remarkable result stems from the existence of a single object called the principal tensor. In our review we discuss explicit and hidden symmetries of higher-dimensional Kerr–NUT–(A)dS black hole spacetimes. We start with discussion of the Killing and Killing–Yano objects representing explicit and hidden symmetries. We demonstrate that the principal tensor can be used as a “seed object” which generates all these symmetries. It determines the form of the geometry, as well as guarantees its remarkable properties, such as special algebraic type of the spacetime, complete integrability of geodesic motion, and separability of the Hamilton–Jacobi, Klein–Gordon, and Dirac equations. The review also contains a discussion of different applications of the developed formalism and its possible generalizations.

We derive new formulas for the high dimensional biharmonic potential acting on Gaussians or Gaussians times special polynomials. These formulas can be used to construct accurate cubature formulas of an arbitrary high order which are fast and effective also in very high dimensions. Numerical tests show that the formulas are accurate and provide the predicted approximation rate O (h 8) up to the dimension 10 7 . [ABSTRACT FROM AUTHOR]

In this paper we propose fast solution methods for the Cauchy problem for the multidimensional Schrödinger equation. Our approach is based on the approximation of the data by the basis functions introduced in the theory of approximate approximations. We obtain high-order approximations also in higher dimensions up to a small saturation error, which is negligible in computations, and we prove error estimates in mixed Lebesgue spaces for the inhomogeneous equation. The proposed method is very efficient in high dimensions if the densities allow separated representations. We illustrate the efficiency of the procedure on different examples, up to approximation order 6 and space dimension 200. [ABSTRACT FROM AUTHOR]