Dimensional Theorem Research Articles

The Parabola Theorem on Continued Fractions

Using geometric methods borrowed from the theory of Kleinian groups, we interpret the parabola theorem on continued fractions in terms of sequences of Mobius transformations. This geometric approach allows us to relate the Stern–Stolz series, which features in the parabola theorem, to the dynamics of certain sequences of Mobius transformations acting on three-dimensional hyperbolic space. We also obtain a version of the parabola theorem in several dimensions.

Marstrand-type Theorems for the Counting and Mass Dimensions in ℤ

The counting and (upper) mass dimensions of a set A ⊆ $\mathbb{R}^d$ are $$D(A) = \limsup_{\|C\| \to \infty} \frac{\log | \lfloor A \rfloor \cap C |}{\log \|C\|}, \quad \smash{\overline{D}}\vphantom{D}(A) = \limsup_{\ell \to \infty} \frac{\log | \lfloor A \rfloor \cap [-\ell,\ell)^d |}{\log (2 \ell)},$$ where ⌊A⌋ denotes the set of elements of A rounded down in each coordinate and where the limit supremum in the counting dimension is taken over cubes C ⊆ $\mathbb{R}^d$ with side length ‖C‖ → ∞. We give a characterization of the counting dimension via coverings: $$D(A) = \text{inf} \{ \alpha \geq 0 \mid {d_{H}^{\alpha}}(A) = 0 \},$$ where $${d_{H}^{\alpha}}(A) = \lim_{r \rightarrow 0} \limsup_{\|C\| \rightarrow \infty} \inf \biggl\{ \sum_i \biggl(\frac{\|C_i\|}{\|C\|} \biggr)^\alpha \ \bigg| \ 1 \leq \|C_i\| \leq r \|C\| \biggr\}$$ in which the infimum is taken over cubic coverings {Ci} of A ∩ C. Then we prove Marstrand-type theorems for both dimensions. For example, almost all images of A ⊆ $\mathbb{R}^d$ under orthogonal projections with range of dimension k have counting dimension at least min(k, D(A)); if we assume D(A) = D(A), then the mass dimension of A under the typical orthogonal projection is equal to min(k, D(A)). This work extends recent work of Y. Lima and C. G. Moreira.

A noncommutative Borsuk-Ulam theorem for Natsume-Olsen spheres

Natsume-Olsen noncommutative spheres are C*-algebras which generalize C(S^k) when k is odd. These algebras admit natural actions by finite cyclic groups, and if one of these actions is fixed, any equivariant homomorphism between two Natsume-Olsen spheres of the same dimension induces a nontrivial map on odd K-theory. This result is an extended, noncommutative Borsuk-Ulam theorem in odd dimension, and just as in the topological case, this theorem has many (almost) equivalent formulations in terms of theta-deformed spheres of arbitrary dimension. In addition, we present theorems on graded Banach algebras, motivated by algebraic Borsuk-Ulam results of A. Taghavi.

Method for Identifying Vickers Hardness by Instrumented Indentation Curves with Berkovich/Vickers Indenter

Applying dimensional theorem and finite element method, Vickers indentation diagonal length and the load–displacement curves of instrumented indentation test on elastoplastic materials with Berkovich triangular and Vickers quadrangular pyramid diamond indenters were systematically analyzed. Consequently, the approximate functional relationships between instrumented indentation hardness and traditional Vickers hardness were established, and a method for identifying traditional Vickers hardness by instrumented Berkovich or Vickers indentation curves was proposed. Through the error evaluation on the proposed method determined by finite element numerical simulation and the results of the experimental tests on representative metal and ceramic materials, it was demonstrated that the Vickers hardness of the indented materials could be effectively identified only by using the instrumented indentation curves without observing and measuring the Vickers indentation diagonal length.

A metric graph satisfying w41=1$w_4^1 = 1$ that cannot be lifted to a curve satisfying dim⁡ (W41)=1$\dim \;(W_4^1 ) = 1$

Abstract For all integers g ≥ 6 we prove the existence of a metric graph G with w 4 1 = 1 $w_4^1 = 1$ such that G has Clifford index 2 and there is no tropical modification G′ of G such that there exists a finite harmonic morphism of degree 2 from G′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

Fueter's theorem in discrete Clifford analysis

In this paper, we discretize techniques for the construction of axially monogenic functions to the setting of discrete Clifford analysis. Wherefore, we work in the discrete Hermitian Clifford setting, where each basis vector ej is split into a forward and backward basis vector: . We prove a discrete version of Fueter's theorem in odd dimension by showing that for a discrete monogenic function f(ξ0,ξ1) left‐monogenic in two variables ξ0 and ξ1 and for a left‐monogenic Pk(ξ), the m‐dimensional function is in itself left monogenic, that is, a discrete function in the kernel of the discrete Dirac operator. Closely related, we consider a Vekua‐type system for the construction of axially monogenic functions. We consider some explicit examples: the discrete axial‐exponential functions and the discrete Clifford–Hermite polynomials. Copyright © 2015 John Wiley & Sons, Ltd.

Blow-Up Rate Estimates and Liouville Type Theorems for a Semilinear Heat Equation with Weighted Source

We study the Liouville-type theorem for the semilinear parabolic equation $$u_t-\Delta u =|x|^a u^p$$ with $$p>1$$ and $$a\in {\mathbb R}$$ . Relying on the recent result of Quittner (Math Ann, doi: 10.1007/s00208-015-1219-7 , 2015), we establish the optimal Liouville-type theorem in dimension $$N=2$$ , in the class of nonnegative bounded solutions. We also provide a partial result in dimension $$N\ge 3$$ . As applications of Liouville-type theorems, we derive the blow-up rate estimates for the corresponding Cauchy problem.

Liouville-type theorems for a quasilinear elliptic equation of the Hénon-type

We consider the Henon-type quasilinear elliptic equation $${-\Delta_m u=|x|^a u^p}$$ where $${\Delta_m u={\rm div}(|\nabla u|^{m-2} \nabla u)}$$ , m > 1, p > m − 1 and $${a\geq 0}$$ . We are concerned with the Liouville property, i.e. the nonexistence of positive solutions in the whole space $${{\mathbb R}^N}$$ . We prove the optimal Liouville-type theorem for dimension N < m + 1 and give partial results for higher dimensions.

On the cohomology of finite soluble groups

Cohomological vanishing theorems in dimensions 1 and 2 are established for finite soluble groups. Applications are given to splitting and conjugacy theorems, and also to the structure of automorphism groups of finite soluble groups with nilpotent length at most 3.

The equivariant family index theorem in odd dimensions

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed’s odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.

The colored Hadwiger transversal theorem in ℝ d

Hadwiger's transversal theorem gives necessary and suffcient conditions for a family of convex sets in the plane to have a line transversal. A higher dimensional version was obtained by Goodman, Pollack and Wenger, and recently a colorful version appeared due to Arocha, Bracho and Montejano. We show that it is possible to combine both results to obtain a colored version of Hadwiger's theorem in higher dimensions. The proofs differ from the previous ones and use a variant of the Borsuk-Ulam theorem. To be precise, we prove the following. Let F be a family of convex sets in źd in bijection with a set P of points in źdź1. Assume that there is a coloring of F with suffciently many colors such that any colorful Radon partition of points in P corresponds to a colorful Radon partition of sets in F. Then some monochromatic subfamily of F has a hyperplane transversal.

A Goldberg–Sachs theorem in dimension three

We prove a Goldberg–Sachs theorem in dimension three. To be precise, given a three-dimensional Lorentzian manifold satisfying the topological massive gravity equations, we provide necessary and sufficient conditions on the tracefree Ricci tensor for the existence of a null line distribution whose orthogonal complement is integrable and totally geodetic. This includes, in particular, Kundt spacetimes that are solutions of the topological massive gravity equations.

A $$d$$ d -dimensional Extension of Christoffel Words

In this article, we extend the definition of Christoffel words to directed subgraphs of the hypercubic lattice in arbitrary dimension that we call Christoffel graphs. Christoffel graphs when $d=2$ correspond to well-known Christoffel words. Due to periodicity, the $d$-dimensional Christoffel graph can be embedded in a $(d-1)$-torus (a parallelogram when $d=3$). We show that Christoffel graphs have similar properties to those of Christoffel words: symmetry of their central part and conjugation with their reversal. Our main result extends Pirillo's theorem (characterization of Christoffel words which asserts that a word $amb$ is a Christoffel word if and only if it is conjugate to $bma$) in arbitrary dimension. In the generalization, the map $amb\mapsto bma$ is seen as a flip operation on graphs embedded in $\mathbb{Z}^d$ and the conjugation is a translation. We show that a fully periodic subgraph of the hypercubic lattice is a translate of its flip if and only if it is a Christoffel graph.

Noncommutative Tsen's theorem in dimension one

Let k be a field. In this paper, we find necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative P1-bundle. This result can be considered a noncommutative, one-dimensional version of Tsen's theorem. By specializing this theorem, we show that every arithmetic noncommutative projective line is a noncommutative curve, and conversely we characterize exactly those noncommutative curves of genus zero which are arithmetic. We then use this characterization, together with results from [9], to address some problems posed in [4].

Next to subleading soft-graviton theorem in arbitrary dimensions

We study the soft graviton theorem recently proposed by Cachazo and Strominger. We employ the Cachazo, He and Yuan formalism to show that the next to subleading order soft factor for gravity is universal at tree level in arbitrary dimensions.

Maps on real Hilbert spaces preserving the area of parallelograms and a preserver problem on self-adjoint operators

In this paper first we describe all (not necessarily linear or bijective) transformations on Rd with 2≤d<∞ which preserve the area of parallelograms spanned by any two vectors. We also characterize those (not necessarily linear) bijections on an arbitrary real Hilbert space that preserve the latter quantity. This answers a question raised by Rassias and Wagner, and it can be considered as a variant of the famous Wigner theorem on real Hilbert spaces which plays an important role in quantum mechanics. As a consequence, we solve a preserver problem of Molnár and Timmermann which has remained open stubbornly only in the two-dimensional case. Finally this two-dimensional result will be applied in order to strengthen their theorem in higher dimensions.

Sub-sub-leading soft-graviton theorem in arbitrary dimension

The CHY formula which describes an n-point tree level scattering amplitude for scalars, gluons or gravitons in arbitrary dimension [22], is used to prove the sub-sub-leading soft-graviton theorem recently proposed by Cachazo and Strominger [13].

Nonsplittability of the rational homology cobordism group of 3-manifolds

The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel. We use Heegaard-Floer methods to show that the cokernel is infinitely generated. On the level of Witt groups of linking forms and conjecturally in the topological category, the inclusion is an isomorphism. One application is the demonstration of the failure in dimension three of an algebraic and higher dimensional theorem of Stoltzfus regarding primary splittings in the knot concordance group. We also build on work of the second author with Hedden and Ruberman to prove that the homology cobordism group of rational homology three spheres that bound topological rational homology 4-balls remains infinitely generated when modded out by the homology cobordism group of integral homology 3-spheres.

On almost everywhere convergence of strong arithmetic means of Fourier series

This article establishes a real-variable argument for Zygmund’s theorem on almost everywhere convergence of strong arithmetic means of partial sums of Fourier series on T \mathbb {T} , up to passing to a subsequence. Our approach extends to, among other cases, functions that are defined on T d \mathbb {T}^d , which allows us to establish an analogue of Zygmund’s theorem in higher dimensions.

Subleading soft theorem in arbitrary dimensions from scattering equations.

We investigate the new soft graviton theorem recently proposed by Cachazo and Strominger. We use the Cachazo-He-Yuan formula to prove a universal behavior for both Yang-Mills theory and gravity scattering amplitudes at the tree level in arbitrary dimension.