Dimensional Theorem Research Articles

A global diffeomorphism theorem and a unique weak solution of Dirichlet problem

ABSTRACTUsing an improved version of the infinite dimensional theorem on diffeomorphisms, we obtain conditions for the existence of the unique weak solution of Dirichlet problem, where is a bounded domain with a -boundary and is a -Carathéodory function. It is also obtained that the solution operator, which assigns to each the unique solution of the equation, is a diffeomorphism.

Nonlocal curvature and topology of locally conformally flat manifolds

In this paper, we focus on the geometry of compact conformally flat manifolds (Mn,g) with positive scalar curvature. Schoen–Yau proved that its universal cover (Mn˜,g˜) is conformally embedded in Sn such that Mn is a Kleinian manifold. Moreover, the limit set of the Kleinian group has Hausdorff dimension <n−22. If additionally we assume that the non-local curvature Q2γ≥0 for some 1<γ<2, the Hausdorff dimension of the limit set is less than or equal to n−2γ2. If Q2γ>0, then the above inequality is strict. Moreover, the above upper bound is sharp. As applications, we obtain some topological rigidity and classification theorems in lower dimensions.

KAM for the nonlinear wave equation on the circle: A normal form theorem

In this paper we prove a KAM theorem in infinite dimension which treats the case of multiple eigenvalues (or frequencies) of finite order. More precisely, we consider a Hamiltonian normal form in infinite dimension: $$\begin{aligned} h(\rho )=\omega (\rho )\cdot r + \frac{1}{2} \langle \zeta ,A(\rho )\zeta \rangle , \end{aligned}$$ where $$ r \in \mathbb {R}^n $$ , $$\zeta =((p_s,q_s)_{s \in \mathcal {L}})$$ and $$ \mathcal {L}$$ is a subset of $$\mathbb {Z}$$ and $$\rho $$ is a parameter that belongs to a compact. We assume that the infinite matrix $$A(\rho )$$ is of the form $$A(\rho )= D(\rho )+N(\rho )$$ , where $$D(\rho )$$ is a diagonal matrix with finite eigenvalue multiplicity and N is a bloc diagonal matrix. We assume that the size of each bloc of N is the multiplicity of the corresponding eigenvalue in D. In this context, if we start from a torus, then the solution of the associated Hamiltonian system remains on that torus. Under certain conditions emitted on the frequencies, we can affirm that the trajectory of the solution fills the torus. In this context, the starting torus is an invariant torus. Then, we perturb this integrable Hamiltonian and we want to prove that the starting torus is a persistent torus. We prove that, if the perturbation is small and under certain conditions of non-resonance of the frequencies, then the starting torus is a persistent torus. We use this result to prove the existence of quasi-periodic solutions for the nonlinear wave equation with convolutive potential on the circle. In Bouthelja (KAM for the nonlinear wave equation on the circle: small amplitude solutions (preprint), arXiv:1712.01597 , 2017) we use this result to prove the existence of such solutions for the nonlinear wave equation on the circle without an external parameter.

A -theorem for Horndeski gravity at the critical point

We study holographic conformal anomalies and the corresponding $a$-theorem for Einstein gravity extended with Horndeski terms that involve up to and including linear curvature tensors. We focus on our discussion in $D=5$ bulk dimensions. For the generic Horndeski coupling, the $a$-charge is the same as that in Einstein gravity, but the inclusion of the Horndeski term violates the $a$-theorem. However, there exists a critical point of the Horndeski coupling, for which the theory admits nearly AdS spacetimes with non-vanishing Horndeski scalar. The full AdS isometry is broken down by the logarithmic scalar hair to the Poincar\'e group plus the scale invariance. We find that in this case the $a$-charge depends on the AdS radius $\ell$ and the integration constant $\chi_s$ of the Horndeski scalar. In addition, we find that two new central charges emerge, that are absent in gravities with minimally-coupled matter. We call them $b$-charges. These $b$-charges also depend on $\ell$ and $\chi_s$. We construct an $a$-function for fixed $\ell$ but with the running Horndeski scalar $\chi$ replacing the constant $\chi_s$, and establish the holographic $a$-theorem using the null energy condition in the bulk. Furthermore, we find that there exist analogous monotonous $b$-functions as well. We also obtain the $a$-charge and the $a$-theorem in general odd bulk dimensions.

On a theorem of Bishop and commutants of Toeplitz operators in $$\mathbb {C}^n$$

We prove an approximation theorem on a class of domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-problem is solvable in $L^{\infty}$. Furthermore, as a corollary, we obtain a version of the Axler-\v{C}u\v{c}kovi\'c-Rao Theorem in higher dimensions.

On the mass aspect function and positive energy theorems for asymptotically hyperbolic manifolds

We prove positivity of energy for a class of asymptotically locally hyperbolic manifolds in dimensions . The result is established by first proving deformation-of-mass-aspect theorems in dimensions . Our positivity results extend to the case n = 3 when more stringent conditions are imposed.

Ricci flow on asymptotically Euclidean manifolds

In this paper, we prove that if an asymptotically Euclidean manifold $(M^n,g)$ under the condition that $R \ge 0$ has long time existence of Ricci flow, the mass of $(M^n,g)$ is nonnegative. In addition, we give an independent proof of positive mass theorem in dimension $3$.

Ricci flow with surgery in higher dimensions

We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's neck-like curvature pinching estimate. Using this estimate, we are able to prove a version of Perelman's Canonical Neighborhood Theorem in higher dimensions. This makes it possible to extend the flow beyond singularities by a surgery procedure in the spirit of Hamilton and Perelman. As a corollary, we obtain a classification of all diffeomorphism types of such manifolds in terms of a connected sum decomposition. In particular, the underlying manifold cannot be an exotic sphere. Our result is sharp in many interesting situations. For example, the curvature tensors of $\mathbb{CP}^{n/2}$, $\mathbb{HP}^{n/4}$, $S^{n-k} \times S^k$ ($2 \leq k \leq n-2$), $S^{n-2} \times \mathbb{H}^2$, $S^{n-2} \times \mathbb{R}^2$ all lie on the boundary of our curvature cone. Another borderline case is the pseudo-cylinder: this is a rotationally symmetric hypersurface which is weakly, but not strictly, two-convex. Finally, the curvature tensor of $S^{n-1} \times \mathbb{R}$ lies in the interior of our curvature cone.

The Non-homogeneous Groshev Convergence theorem for Diophantine Approximation on Manifolds

This paper is based on Khintchine theorem, Groshev theorem and measure and dimension theorems for non-degenerate manifolds. The inhomogeneous Diophantine approximation of Groshev type on manifolds is studied. Major work is to discuss the inhomogeneous convergent theory of Diophantine approximation restricted to non-degenerate manifold in , based on the proof of Barker-Sprindzuk conjecture, the homogeneous theory of Diophantine approximation and inhomogeneous Groshev type theory for Diophantine approximation, by the decomposition of the set in manifold, with the aid of Borel Cantell lemma and transformation of lemma and its properties and the main inhomogeneous conversion principle, we know these two types of set in sense of Lebesgue measure is zero provided that the convergent sum condition is satisfied, from which several conclusions about the inhomogeneous convergent theory of Diophantine approximation is obtained. The main result is that Lebesgue measure is inhomogeneous strongly extremal. At last we use the fact that friendly measure is strongly contracting measure to develop an inhomogeneous strong extreme measure which is restricted to matrices with dependent quantities

Markov Property of the Conformal Field Theory Vacuum and the a Theorem.

We use strong subadditivity of entanglement entropy, Lorentz invariance, and the Markov property of the vacuum state of a conformal field theory to give new proof of the irreversibility of the renormalization group in d=4 space-time dimensions-the a theorem. This extends the proofs of the c and F theorems in dimensions d=2 and d=3 based on vacuum entanglement entropy, and gives a unified picture of all known irreversibility theorems in relativistic quantum field theory.

Dimensions of fibers of generic continuous maps

In an earlier paper Buczolich, Elekes, and the author described the Hausdorff dimension of the level sets of a generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space K by introducing the notion of topological Hausdorff dimension. Later on, the author extended the theory for maps from K to $${\mathbb {R}}^n$$ . The main goal of this paper is to generalize the relevant results for topological and packing dimensions and to obtain new results for sufficiently homogeneous spaces K even in the case case of Hausdorff dimension. Let K be a compact metric space and let us denote by $$C(K,{\mathbb {R}}^n)$$ the set of continuous maps from K to $${\mathbb {R}}^n$$ endowed with the maximum norm. Let $$\dim _{*}$$ be one of the topological dimension $$\dim _T$$ , the Hausdorff dimension $$\dim _H$$ , or the packing dimension $$\dim _P$$ . Define $$\begin{aligned} d_{*}^n(K)=\inf \left\{ \dim _{*}(K{\setminus } F): F\subset K \text { is } \sigma \text {-compact with } \dim _T F<n\right\} . \end{aligned}$$ We prove that $$d^n_{*}(K)$$ is the right notion to describe the dimensions of the fibers of a generic continuous map $$f\in C(K,{\mathbb {R}}^n)$$ . In particular, we show that $$\sup \{\dim _{*}f^{-1}(y): y\in {\mathbb {R}}^n\} =d^n_{*}(K)$$ provided that $$\dim _T K\ge n$$ , otherwise every fiber is finite. Proving the above theorem for packing dimension requires entirely new ideas. Moreover, we show that the supremum is attained on the left hand side of the above equation. Assume $$\dim _T K\ge n$$ . If K is sufficiently homogeneous, then we can say much more. For example, we prove that $$\dim _{*}f^{-1}(y)=d^n_{*}(K)$$ for a generic $$f\in C(K,{\mathbb {R}}^n)$$ for all $$y\in {{\mathrm{int}}}f(K)$$ if and only if $$d^n_{*}(U)=d^n_{*}(K)$$ or $$\dim _T U<n$$ for all open sets $$U\subset K$$ . This is new even if $$n=1$$ and $$\dim _{*}=\dim _H$$ . It is known that for a generic $$f\in C(K,{\mathbb {R}}^n)$$ the interior of f(K) is not empty. We augment the above characterization by showing that $$\dim _T \partial f(K)=\dim _H \partial f(K)=n-1$$ for a generic $$f\in C(K,{\mathbb {R}}^n)$$ . In particular, almost every point of f(K) is an interior point. In order to obtain more precise results, we use the concept of generalized Hausdorff and packing measures, too.

The Whitney extension theorem in high dimensions

We prove a variant of the standard Whitney extension theorem for \mathcal C^m(\mathbb R^n) , in which the norm of the extension operator has polynomial growth in n for fixed m .

Rigidity of Free Boundary Surfaces in Compact 3-Manifolds with Strictly Convex Boundary

In this paper we obtain an analogue of Toponogov theorem in dimension 3 for compact manifolds $M^3$ with nonnegative Ricci curvature and strictly convex boundary $\partial M$. Here we obtain a sharp upper bound for the length $L(\partial\Sigma)$ of the boundary $\partial\Sigma$ of a free boundary minimal surface $\Sigma^2$ in $M^3$ in terms of the genus of $\Sigma$ and the number of connected components of $\partial\Sigma$, assuming $\Sigma$ has index one. After, under a natural hypothesis on the geometry of $M$ along $\partial M$, we prove that if $L(\partial\Sigma)$ saturates the respective upper bound, then $M^3$ is isometric to the Euclidean 3-ball and $\Sigma^2$ is isometric to the Euclidean disk. In particular, we get a sharp upper bound for the area of $\Sigma$, when $M^3$ is a strictly convex body in $\mathbb R^3$, which is saturated only on the Euclidean 3-balls (by the Euclidean disks). We also consider similar results for stationary stable surfaces.

Methodology for measuring fracture toughness of ceramic materials by instrumented indentation test with vickers indenter

AbstractVirtual crack closure technique and elastoplastic finite element method were employed to calculate the stress intensity factors (SIF) of ceramic materials on the tip of both half‐penny crack (HPC) and radial crack (RC) induced by Vickers indenter and the value of fracture toughness (KIC) was extracted by the design of equi‐SIF contour of HPC and RC crack front. Through dimensional theorem and regressive analysis, a functional relationship between instrumented indentation parameters, crack length of Vickers impression and fracture toughness of ceramic materials was established, thus a novel methodology has been presented for measuring fracture toughness of ceramic materials by instrumented Vickers indentation. Both numerical analysis and experiments have indicated that this methodology enjoys higher measurement precision compared with other available indentation methods. The methodology is universally suitable for HPC, RC as well as transition cracks and capable of determining fracture toughness and elastic modulus in a single indentation test. In addition, it saves the effort of measuring the diagonal length of Vickers impression in case that the impression remains unclear.

On the Kozachenko–Leonenko entropy estimator

We study in detail the bias and variance of the entropy estimator proposed by Kozachenko and Leonenko (1987) for a large class of densities on Rd. We then use the work of Bickel and Breiman (1983) to prove a central limit theorem in dimensions 1 and 2. In higher dimensions, we provide a development of the bias in terms of powers of N−2/d. This allows us to use a Richardson extrapolation to build, in any dimension, a root-n consistent entropy estimator satisfying a central limit theorem which allows for explicit (asymptotic) confidence intervals. To our knowledge, all the previous general root-n consistency results were concerning dimension 1.

Weyl consistency conditions in non-relativistic quantum field theory

Weyl consistency conditions have been used in unitary relativistic quantum field theory to impose constraints on the renormalization group flow of certain quantities. We classify the Weyl anomalies and their renormalization scheme ambiguities for generic non-relativistic theories in 2+1 dimensions with anisotropic scaling exponent z=2; the extension to other values of z are discussed as well. We give the consistency conditions among these anomalies. As an application we find several candidates for a C-theorem. We comment on possible candidates for a $C$-theorem in higher dimensions.

On combinatorial Gauss-Bonnet Theorem for general Euclidean simplicial complexes

For a finitely triangulated closed surface M2, let αx be the sum of angles at a vertex x. By the well-known combinatorial version of the 2- dimensional Gauss-Bonnet Theorem, it holds Σx(2π - αx) = 2πχ(M2), where χ denotes the Euler characteristic of M2, αx denotes the sum of angles at the vertex x, and the sum is over all vertices of the triangulation. We give here an elementary proof of a straightforward higher-dimensional generalization to Euclidean simplicial complexes K without assuming any combinatorial manifold condition. First, we recall some facts on simplicial complexes, the Euler characteristics and its local version at a vertex. Then we define δ(τ) as the normed dihedral angle defect around a simplex τ. Our main result is Στ (-1)dim(τ)δ(τ) = χ(K), where the sum is over all simplices τ of the triangulation. Then we give a definition of curvature κ(x) at a vertex and we prove the vertex-version Σx∈K0 κ(x) = χ(K) of this result. It also possible to prove Morse-type inequalities. Moreover, we can apply this result to combinatorial (n + 1)-manifolds W with boundary B, where we prove that the difference of Euler characteristics is given by the sum of curvatures over the interior of W plus a contribution from the normal curvature along the boundary B: $$\chi \left( W \right) - \frac{1}{2}\chi \left( B \right) = \sum {_{\tau \in W - B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}} + \sum {_{\tau \in B}} {\left( { - 1} \right)^{\dim \left( \tau \right)}}\rho \left( \tau \right)$$ .

Adaptive integrand decomposition in parallel and orthogonal space

We present the integrand decomposition of multiloop scattering amplitudes in parallel and orthogonal space-time dimensions, $d=d_\parallel+d_\perp$, being $d_\parallel$ the dimension of the parallel space spanned by the legs of the diagrams. When the number $n$ of external legs is $n\le 4$, the corresponding representation of the multiloop integrals exposes a subset of integration variables which can be easily integrated away by means of Gegenbauer polynomials orthogonality condition. By decomposing the integration momenta along parallel and orthogonal directions, the polynomial division algorithm is drastically simplified. Moreover, the orthogonality conditions of Gegenbauer polynomials can be suitably applied to integrate the decomposed integrand, yielding the systematic annihilation of spurious terms. Consequently, multiloop amplitudes are expressed in terms of integrals corresponding to irreducible scalar products of loop momenta and external momenta. We revisit the one-loop decomposition, which turns out to be controlled by the maximum-cut theorem in different dimensions, and we discuss the integrand reduction of two-loop planar and non-planar integrals up to $n=8$ legs, for arbitrary external and internal kinematics. The proposed algorithm extends to all orders in perturbation theory.

Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger restriction theorem in higher dimensions

We prove the range of exponents in the general $L^2$ Fourier restriction theorem due to Mockenhaupt, Mitsis, Bak and Seeger is sharp for a large class of measures on $\mathbb{R}^d$. This extends to higher dimensions the sharpness result of Hambrook and {\L}aba.

Boundary of the range of transient random walk

We study the boundary of the range of simple random walk on $$\mathbb {Z}^d$$ in the transient case $$d\ge 3$$ . We show that volumes of the range and its boundary differ mainly by a martingale. As a consequence, we obtain an upper bound on the variance of order $$n\log n$$ in dimension three. We also establish a Central Limit Theorem in dimension four and larger.