Let \( \pi_{x} \) be the set of primes greater than \( x \). We prove that for all \( x\in{??} \) the classes of finite groups \( D_{\pi_{x}} \) and \( E_{\pi_{x}} \) coincide; i.e., a finite group \( G \) possesses a \( \pi_{x} \)-Hall subgroup if and only if \( G \) satisfies the complete analog of the Sylow Theorems for a \( \pi_{x} \)-subgroup.
相似文献\({{\mathfrak{L}}_{II}}\) operator is introduced by Xin (2015), which is an important extrinsic elliptic differential operator of divergence type and has profound geometric meaning. In this paper, we extend \({{\mathfrak{L}}_{II}}\) operator to a more general elliptic differential operator \({{\mathfrak{L}}_\nu}\), and investigate the clamped plate problem of bi-\({{\mathfrak{L}}_\nu}\) operator, which is denoted by \({\mathfrak{L}}_\nu ^2\) on the complete Riemannian manifolds. A general formula of eigenvalues for the \({\mathfrak{L}}_\nu ^2\) operator is established. Applying this formula, we estimate the eigenvalues on the Riemannian manifolds. As some further applications, we establish some eigenvalue inequalities for this operator on the translating solitons with respect to the mean curvature flows, submanifolds of the Euclidean spaces, unit spheres and projective spaces. In particular, for the case of translating solitons, all of the eigenvalue inequalities are universal.
相似文献Bounds are obtained for the \(L^p\) norm of the torsion function \(v_{\varOmega }\), i.e. the solution of \(-\varDelta v=1,\, v\in H_0^1(\varOmega ),\) in terms of the Lebesgue measure of \(\varOmega \) and the principal eigenvalue \(\lambda _1(\varOmega )\) of the Dirichlet Laplacian acting in \(L^2(\varOmega )\). We show that these bounds are sharp for \(1\le p\le 2\).
相似文献In this paper, our aim is to revisit the nonparametric estimation of a square integrable density f on \({\mathbb {R}}\), by using projection estimators on a Hermite basis. These estimators are studied from the point of view of their mean integrated squared error on \({\mathbb {R}}\). A model selection method is described and proved to perform an automatic bias variance compromise. Then, we present another collection of estimators, of deconvolution type, for which we define another model selection strategy. Although the minimax asymptotic rates of these two types of estimators are mainly equivalent, the complexity of the Hermite estimators is usually much lower than the complexity of their deconvolution (or kernel) counterparts. These results are illustrated through a small simulation study.
相似文献We prove that an overcomplete Gabor frame in \({\ell }^2({\mathbb {Z}})\) generated by a finitely supported sequence is always linearly dependent. This is a particular case of a general result about linear dependence versus independence for Gabor systems in \({\ell }^2({\mathbb {Z}})\) with modulation parameter 1 / M and translation parameter N for some \(M,N\in {\mathbb {N}},\) and generated by a finite sequence g in \({\ell }^2({\mathbb {Z}})\) with K nonzero entries.
相似文献We characterize the completeness and frame/basis property of a union of under-sampled windowed exponentials of the form
$$ {\mathcal{F}}(g): =\bigl\{ e^{2\pi i n x}: n\ge 0\bigr\} \cup \bigl\{ g(x)e^{2\pi i nx}: n< 0\bigr\} $$for \(L^{2}[-1/2,1/2]\) by the spectra of the Toeplitz operators with the symbol \(g\). Using this characterization, we classify all real-valued functions \(g\) such that \({\mathcal{F}}(g)\) is complete or forms a frame/basis. Conversely, we use the classical non-harmonic Fourier series theory to determine all \(\xi \) such that the Toeplitz operators with the symbol \(e^{2\pi i \xi x}\) is injective or invertible. These results demonstrate an elegant interaction between frame theory of windowed exponentials and Toeplitz operators. Finally, we use our results to answer some open questions in dynamical sampling, and derivative samplings on Paley-Wiener spaces of bandlimited functions.
相似文献We study non reflexive Orlicz spaces \(L^\varPsi \) and their Morse subspace \(M^\varPsi \), i.e. the closure of \(L^\infty \) in \(M^\varPsi \) to determine when \((M^\varPsi ,L^\varPsi )\) can be described as having an o–O type structure with respect to an equivalent norm on \(L^\varPsi \). Examples of classes of Young functions for which the answer is affirmative are provided, but also examples are given to show that this is not possible for all non-reflexive Orlicz spaces. An equivalent expression of the distance in \(L^\varPsi \) to \(M^\varPsi \), induced by the new norm, is also provided.
相似文献In this paper, we derive the explicit expressions of the Markov semi-groups constructed by Biane (ESAIM Probab Stat 15:S2–S10, 2011) from the restriction of a particular positive definite function on the complex unimodular group \(SL(2,{\mathbb {C}})\) to two commutative subalgebras of its universal \(C^{\star }\)-algebra. Our computations use Euclidean Fourier analysis together with the generating function of Laguerre polynomials with index \(-\,1\), and yield absolutely-convergent double series representations of the semi-group densities. We also supply some arguments supporting the coincidence, noticed by Biane as well, occurring between the heat kernel on the Heisenberg group and the semi-group corresponding to the intersection of the principal and the complementary series. To this end, we appeal to the metaplectic representation \(Mp(4,{\mathbb {R}})\) and to the Landau operator in the complex plane.
相似文献In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial \(L_n\). In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for \(\pi \): the first can be seen as a generalization of the known formula
$$\begin{aligned} \pi =\lim _{n\rightarrow \infty } 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{n}}, \end{aligned}$$related to the smallest positive zero of \(L_n\); the second is an exact formula for \(\pi \) achieved thanks to some identities valid for \(L_n\).
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