Dimensional Theorem Research Articles

Direct and Inverse Embedding Theorems for Different Dimensions for a Class of Multianisotropic Sobolev Spaces

Direct and Inverse Embedding Theorems for Different Dimensions for a Class of Multianisotropic Sobolev Spaces

Restoring the Fluctuation-Dissipation Theorem in Kardar-Parisi-Zhang Universality Class through a New Emergent Fractal Dimension.

The Kardar-Parisi-Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation-dissipation theorem (FDT) for spatial dimension d>1. Notably, these questions were answered exactly only for 1+1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d+1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension dn. We present relations between the KPZ exponents and two emergent fractal dimensions, namely df, of the rough interface, and dn. Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α, the surface fractal dimension df and, through our relations, the noise fractal dimension dn. Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class.

Spectral flow, Llarull’s rigidity theorem in odd dimensions and its generalization

Spectral flow, Llarull’s rigidity theorem in odd dimensions and its generalization

Dynamic Crack Growth in Viscoelastic Materials with Memory

AbstractIn this paper we introduce a model of dynamic crack growth in viscoelastic materials, where the damping term depends on the history of the deformation. The model is based on a dynamic energy dissipation balance and on a maximal dissipation condition. Our main result is an existence theorem in dimension two under some a priori regularity constraints on the cracks.

Irreversibility, QNEC, and defects

We first present an analysis of infinitesimal null deformations for the entanglement entropy, which leads to a major simplification of the proof of the C, F and A-theorems in quantum field theory. Next, we study the quantum null energy condition (QNEC) on the light-cone for a CFT. Finally, we combine these tools in order to establish the irreversibility of renormalization group flows on planar d-dimensional defects, embedded in D-dimensional conformal field theories. This proof completes and unifies all known defect irreversibility theorems for defect dimensions d ≤ 4. The F-theorem on defects (d = 3) is a new result using information-theoretic methods. For d ≥ 4 we also establish the monotonicity of the relative entropy coefficient proportional to Rd−4. The geometric construction connects the proof of irreversibility with and without defects through the QNEC inequality in the bulk, and makes contact with the proof of strong subadditivity of holographic entropy taking into account quantum corrections.

Nearly optimal central limit theorem and bootstrap approximations in high dimensions

In this paper, we derive new, nearly optimal bounds for the Gaussian approximation to scaled averages of n independent high-dimensional centered random vectors X1,…,Xn over the class of rectangles in the case when the covariance matrix of the scaled average is nondegenerate. In the case of bounded Xi’s, the implied bound for the Kolmogorov distance between the distribution of the scaled average and the Gaussian vector takes the form C(Bn2log3d/n)1/2logn, where d is the dimension of the vectors and Bn is a uniform envelope constant on components of Xi’s. This bound is sharp in terms of d and Bn, and is nearly (up to logn) sharp in terms of the sample size n. In addition, we show that similar bounds hold for the multiplier and empirical bootstrap approximations. Moreover, we establish bounds that allow for unbounded Xi’s, formulated solely in terms of moments of Xi’s. Finally, we demonstrate that the bounds can be further improved in some special smooth and moment-constrained cases.

Scalar curvature volume comparison theorems for almost rigid spheres

Bray’s football theorem gives a sharp volume upper bound for a three dimensional manifold with scalar curvature no less than n ( n − 1 ) n(n-1) and Ricci curvature at least ε 0 g ¯ \varepsilon _0 \bar {g} . This paper extends Bray’s football theorem in higher dimensions, assuming the manifold is axisymmetric or the Ricci curvature has a uniform upper bound. Effectively, we show that if the Ricci curvature of an n n -manifold is close to that of a round n-sphere, a lower bound on scalar curvature gives an upper bound on the total volume.

Entropic g Theorem in General Spacetime Dimensions.

We establish the irreversibility of renormalization group flows on a pointlike defect inserted in a d-dimensional Lorentzian conformal field theory. We identify the impurity entropy g with the quantum relative entropy in two equivalent ways. One involves a null deformation of the Cauchy surface, and the other is given in terms of a local quench protocol. Positivity and monotonicity of the relative entropy imply that g decreases monotonically along renormalization group flows, and provides a clear information-theoretic meaning for this irreversibility.

On the local systolic optimality of Zoll contact forms

We prove a normal form for contact forms close to a Zoll one and deduce that Zoll contact forms on any closed manifold are local maximizers of the systolic ratio. Corollaries of this result are: (1) sharp local systolic inequalities for Riemannian and Finsler metrics close to Zoll ones, (2) the perturbative case of a conjecture of Viterbo on the symplectic capacity of convex bodies, (3) a generalization of Gromov’s non-squeezing theorem in the intermediate dimensions for symplectomorphisms that are close to linear ones.

Scattering on singular Yamabe spaces

We apply scattering theory on asymptotically hyperbolic manifolds to singular Yamabe metrics, applying the results to the study of the conformal geometry of compact manifolds with boundary. In particular, we define extrinsic versions of the conformally invariant powers of the Laplacian, or GJMS operators, on the boundary of any such manifold, along with associated extrinsic Q -curvatures. We use the existence and uniqueness of a singular Yamabe metric in any conformal class to define also nonlocal extrinsic fractional GJMS operators on the boundary, and draw other global conclusions about the scattering operator, including a Gauss–Bonnet theorem in dimension four.

Cultural Phenomenon is the measure problem of Ethnic Students in learning mathematics

The aim of the study was to identify the measure language difficulties in mathematics of the students in basic level class room and relation between language and learning mathematics. The finding of the study indicates that the student’s language difficulties in two categories the first one are sematic difficulties. These difficulties include that the student suffering mathematics due to specific terms. The terms having different meaning in other Description in the theorem properties and the use of notation. Another difficulty are synthetic difficulties; this difficulty is use of language; mathematics need to be explained with the help of language. Every language has its specific formal structure and pattern which is called the synthetic language. The study has a hypothesis that the ethnic students have great discontinuity between their everyday life and school activities as the get practical Knowledge in home and theoretical knowledge at school. This study tries to find out the problem faced by the ethnic student in learning mathematics to be making co-operative class environment and to be serious thought while making decision about curriculum and teaching learning process.

A nonlinear projection theorem for Assouad dimension and applications

We prove a general nonlinear projection theorem for Assouad dimension. This theorem has several applications including to distance sets, radial projections, and sum-product phenomena. In the setting of distance sets we are able to completely resolve the planar distance set problem for Assouad dimension, both dealing with the awkward `critical case' and providing sharp estimates for sets with Assouad dimension less than 1. In the higher dimensional setting we connect the problem to the dimension of the set of exceptions in a related (orthogonal) projection theorem. We also obtain results on pinned distance sets and our results still hold when the distances are taken with respect to a sufficiently curved norm. As another application we prove a radial projection theorem for Assouad dimension with sharp estimates on the Hausdorff dimension of the exceptional set.

Conformally Prescribed Scalar Curvature on Orbifolds

We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension 4, and an existence theorem which holds in dimensions n ge 4. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the textrm{U}(2)-invariant Leray–Schauder degree for a family of negative-mass orbifolds found by LeBrun.

Positivity of Mass in Higher Dimensions

The positive mass theorem in higher dimensions is proved using causality arguments inspired by those of Penrose et al. (a positive mass theorem based on the focusing and retardation of null geodesics, 1993) in 3+1 dimensions.

Central limit theorem and bootstrap approximation in high dimensions: Near 1/n rates via implicit smoothing

Nonasymptotic bounds for Gaussian and bootstrap approximation have recently attracted significant interest in high-dimensional statistics. This paper studies Berry–Esseen bounds for such approximations with respect to the multivariate Kolmogorov distance, in the context of a sum of n random vectors that are p-dimensional and i.i.d. Up to now, a growing line of work has established bounds with mild logarithmic dependence on p. However, the problem of developing corresponding bounds with near n−1/2 dependence on n has remained largely unresolved. Within the setting of random vectors that have sub-Gaussian or subexponential entries, this paper establishes bounds with near n−1/2 dependence, for both Gaussian and bootstrap approximation. In addition, the proofs are considerably distinct from other recent approaches, and make use of an “implicit smoothing” operation in the Lindeberg interpolation.

Improved central limit theorem and bootstrap approximations in high dimensions

This paper deals with the Gaussian and bootstrap approximations to the distribution of the max statistic in high dimensions. This statistic takes the form of the maximum over components of the sum of independent random vectors and its distribution plays a key role in many high-dimensional estimation and testing problems. Using a novel iterative randomized Lindeberg method, the paper derives new bounds for the distributional approximation errors. These new bounds substantially improve upon existing ones and simultaneously allow for a larger class of bootstrap methods.

Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

Abstract The classical Khintchine–Groshev theorem is a generalization of Khintchine’s theorem on simultaneous Diophantine approximation, from approximation of points in ${\mathbb {R}}^m$ to approximation of systems of linear forms in ${\mathbb {R}}^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine–Groshev theorem that does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani [11] that monotonicity is not required when $nm>1$. That result resolved a conjecture of Beresneich et al. [5], and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin–Schaeffer conjecture. When $nm=1$, it is known by work of Duffin and Schaeffer [16] that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\times m)$-dimensional Khintchine–Groshev theorem ($k\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\times m)$-dimensional theorem. Hence, it is shown that Khintchine’s theorem itself underpins the Khintchine–Groshev theory.

Dilaton in scalar QFT: a no-go theorem in 4-epsilon and 3-epsilon dimensions

Spontaneous scale invariance breaking and the associated Goldstone boson, the dilaton, is investigated in renormalizable, unitary, interacting non-supersymmetric scalar field theories in 4-\varepsilon4−ε dimensions. At leading order it is possible to construct models which give rise to spontaneous scale invariance breaking classically and indeed a massless dilaton can be identified. Beyond leading order, in order to have no anomalous scale symmetry breaking in QFT, the models need to be defined at a Wilson-Fisher fixed point with exact conformal symmetry. It is shown that this requirement on the couplings is incompatible with having the type of flat direction which would be necessary for an exactly massless dilaton. As a result spontaneous scale symmetry breaking and an exactly massless dilaton can not occur in renormalizable, unitary 4-\varepsilon4−ε dimensional scalar QFT. The arguments apply to \phi^6ϕ6 theory in 3-\varepsilon3−ε dimensions as well.

Fenchel–Rockafellar theorem in infinite dimensions via generalized relative interiors

In this paper we provide further studies of Fenchel duality theory in the general framework of locally convex topological vector (LCTV) spaces. We prove the validity of the Fenchel strong duality under some qualification conditions via generalized relative interiors imposed on the epigraphs and the domains of the functions involved. Our results directly generalize the classical Fenchel–Rockafellar theorem on strong duality from finite dimensions to LCTV spaces.

"On the maximum modulus principle and the identity theorem in arbitrary dimension"

"We prove an identity theorem for Gˆateaux holomorphic functions on polygonally connected 2- open sets, which yields a very general maximum norm principle and a sublinear “max-min” principle. All results apply in particular to vector-valued functions which are holomorphic (in any sense that implies Gˆateaux holomorphy) on domains in Hausdorff locally convex spaces."