Borsuk’s Partition Problem in $$(\mathbb{R}^{n},\ell_{p})$$

Mathematical Notes - For a bounded set $$X$$ with diameter $$d_{C}(X)$$ in a finite-dimensional normed space with an origin-symmetric convex body $$C$$ as the unit ball, the Borsuk number of $$X$$...     

Erratum to: Borsuk’s Partition Problem in $$(\mathbb{R}^{n},\ell_{p})$$

Mathematical Notes - An Erratum to this paper has been published: https://doi.org/10.1134/S0001434622010321     

Hyperbolic Function Theory in the Clifford Algebra {\mathcal {C}}\ell_{n+1, 0}

The aim of this paper is to give the basic principles of hyperbolic function theory on the Clifford algebra . The structure of the theory is quite similar to the case of Clifford algebras with negative generators, but the proofs are not obvious. The (real) Clifford algebra is generated by unit vectors with positive squares e²_i = + 1. The hyperbolic Dirac operator is of the form where Q₀f is represented by the composition . If is a solution of H_kf = 0, then f is called k-hypergenic in Ω, where is an open set. We introduce some basic results of hyperbolic function theory and give some representation theorems on . Received: October, 2007. Accepted: February, 2008.     

A Few Comments on the Clifford Algebra {C\ell_2} of the Standard Euclidean Plane

This paper is a continuation of the author’s plenary lecture given at ICCA 9 which was held in Weimar at the Bauhaus University, 15–20 July, 2011. We want to study on both the mathematical and the epistemological levels the thought of the brilliant geometer W. K. Clifford by presenting a few comments on the structure of the Clifford algebra ${C\ell_2}$ associated with the standard Euclidean plane ${\mathbb{R}^2}$ . Miquel’s theorem will be given in the algebraic context of the even Clifford algebra ${C\ell^+_2}$ isomorphic to the real algebra ${\mathbb{C}}$ . The proof of this theorem will be based on the cross ratio (the anharmonic ratio) of four complex numbers. It will lead to a group of homographies of the standard projective line ${\mathbb{C}P^1 = P(\mathbb{C}^2)}$ which appeared so attractive to W. K. Clifford in his overview of a general theory of anharmonics. In conclusion it will be shown how the classical Clifford-Hopf fibration S ¹ → S ³ → S ² leads to the space of spinors ${\mathbb{C}^2}$ of the Euclidean space ${\mathbb{R}^3}$ and to the isomorphism ${{\rm {PU}(1) = \rm {SU}(2)/\{I,-I\} \simeq SO(3)}}$ .     

An Operator Theoretical Approach Towards Systems over the Signal Space \ell_{2}\mathbb{(Z)}

In this paper a system is considered as a (possibly unbounded) linear operator from l₂\mathbb(Z)^m \ell_{2}\mathbb{(Z)}^{m} to l₂\mathbb(Z)^p \ell_{2}\mathbb{(Z)}^{p} . Georgiou and Smith [6] noted that there are intrinsic difficulties in using l₂\mathbb(Z) \ell_{2}\mathbb{(Z)} as underlying signal space, since even a simple causal convolution system is not closed and an extended definition of the system is not causal. We discuss these difficulties and we develop necessary and sufficient conditions for notions such as causality, closability and causal closability.     

$$ H^{p}_{w} - L^{p}_{w} $$ Boundedness of Marcinkiewicz Integral

The Marcinkiewicz integral is essentially a Littlewood-Paley g-function, which plays a very important role in harmonic analysis. In this paper we give weaker smoothness conditions assumed on Ω to imply the boundedness of the Marcinkiewicz integral operator μ_Ω, where w belongs to the Muckenhoupt weight class.     

{L^{p}} stability for the Boltzmann equation near vacuum

The paper is concerned with the uniform time stability in the Lebesgue space \({L^{p}(\mathbb{R}^{3} \times \mathbb{R}^{3})}\) of solutions to the Boltzmann equation near vacuum. Precisely, for the soft potential case \({-2 < \gamma < 0}\), there exists \(p_{\gamma} > 1\) such that the nonnegative solution with algebraic decay rate in x, v at infinity is stable with respect to small initial data uniformly in time in \({L^{p}}\) with \({1 \leq p < p_{\gamma}}\).     

Toeplitz Operators on Fock Spaces {F^{p}(\varphi)}

Given \({\varphi\in \verb"C"^2(\textbf{C}^n)}\) satisfying \({dd^{c}\varphi\simeq \omega_0}\) , 0 < p < ∞, let \({F^p(\varphi)}\) be the generalized Fock space of all holomorphic functions f on \({{\mathbf C}^n}\) for which the Fock norm $$\|f\|_{p, \varphi}=\left(\,\int_{{\mathbf C}^n} \left|f(z)\right|^{p}e^ {-p\varphi(z)}dv(z)\right)^{\frac{1}{p}} < \infty. $$ While \({\varphi(z)=\frac{1}{2}|z|^2}\) , \({F^{p}(\varphi)}\) is the classical Fock space F ^p . In this paper, for all possible 0 < p,q < ∞ we characterize those positive Borel measures μ on \({{\mathbf C}^n}\) for which the induced Toeplitz operators T _μ are bounded (or compact) from one generalized Fock spaces \({F^p(\varphi)}\) to another \({F^q(\varphi)}\) . With symbols \({g\in BMO}\) , we obtain Zorborska’s criterion for boundedness (or compactness) of Toeplitz operators T _g on F ^p , our work extends the known results on F ². Toeplitz operators on p-th Fock space with 0 < p < 1 have not been studied before, even in the simplest case that \({\varphi(z)=\frac{1}{2}|z|^2}\) . Our analysis shows a significant difference between Bergman spaces and Fock spaces.     

ℓ-Überlagerungen von ℙ1\\{ \mathfrak{p}_1 ,\mathfrak{p}_2 ,\mathfrak{p}_3 \}

Ohne Zusammenfassung     

Local {W^{2,m(\cdot)}_{loc}}regularity for p(.)-Laplace equations

We study ${W^{2,m(\cdot)}_{loc}}$ regularity for local weak solutions of p(·)-Laplace equations where ${p\in C^1(\Omega) \cap C(\overline{\Omega})}$ and ${\min_{x\in \overline{\Omega}} p(x) > 1}$ .     

$$L^{p}$$ L p Norms of the Lattice Point Discrepancy

Journal of Fourier Analysis and Applications - We estimate the $$L^{p}$$ norms of the discrepancy between the volume and the number of integer points in $$r\Omega -x$$ , a dilated by a factor r and...     

Large $${\varvec{p}}^{\prime }$$-orbits for $${\varvec{p}}$$p-nilpotent linear groups

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broué and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.     

Rate of convergence in L_{p} approximation

In the present paper we give a Korovkin type approximation theorem for a sequence of positive linear operators acting from \(L_{p}\left[ a,b\right] \) into itself using the concept of \(\mathcal {A}\) -summation processes. We also study the rate of convergence of these operators.     

The Distribution of {alpha}p Modulo One

In this paper the authors prove the following result. Let be an irrational number. Then for any > 0, there areinfinitely many prime numbers p such that | p | < p^–16/49+. The exponent improves on , which was obtained recently by thesecond author [Sci. China Ser. A 43 (2000) 703-721]. The resultis very close to the exponent , which can be obtained under the Generalized Riemann Hypothesis. Previous approaches to this problem have all used the same basicestimates for the trigonometric sums that arise. However, thepresent proof uses new bounds, which depend on the Kloostermansum and also on a counting problem in the geometry of numbers.In addition new techniques for the sieve method are applied.The most significant feature of the new approach is that, unlikeprevious methods, the exponential sum estimates remain non-trivialfor the exponent . This gives one hope for an unconditional result as good as that availableunder the Generalized Riemann Hypothesis. 2000 MathematicalSubject Classification: 11N36, 11J71.     

Counting domains in {p,q} tessellations

For any given regular {p,q} tessellation in the hyperbolic plane, we compute the number of vertices and tiles to be found as we distance from a given point, enabling a complete characterization of the asymptotic behavior.