In this paper, we study vanishing and splitting results on a complete smooth metric measure space \((M^n,g,\mathrm {e}^{-f}\mathrm {d}v)\) with various negative m-Bakry-Émery Ricci curvature lower bounds in terms of the first eigenvalue \(\lambda _1(\Delta _f)\) of the weighted Laplacian \(\Delta _f\), i.e., \(\mathrm {Ric}_{m,n}\ge -a\lambda _1(\Delta _f)-b\) for \(0<a\le \dfrac{m}{m-1}, b\ge 0\). In particular, we consider three main cases for different a and b with or without conditions on \(\lambda _1(\Delta _f)\). These results are extensions of Dung and Vieira, and weighted generalizations of Li-Wang, Dung-Sung, and Vieira.
相似文献We are concerned with the following \(p(x)\)-Laplacian equations in \(\mathbb{R}^{N}\)
$$ -\triangle _{p(x)} u+|u|^{p(x)-2}u= f(x,u)\quad \mbox{in } \mathbb{R} ^{N}. $$The nonlinearity is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of this problem. To overcome this difficulty, by adding potential term and using mountain pass theorem, we get the weak solution \(u_{\lambda }\) of perturbation equations. First, we prove that \(u_{\lambda }\rightharpoonup u\) as \(\lambda \rightarrow 0\). Second, by using vanishing lemma, we get that \(u\) is a nontrivial solution of the original problem.
相似文献We study the existence, nonexistence and multiplicity of solutions to Chern-Simons-Schrödinger system
$$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l} -\Delta u+u+\lambda (\frac{h^{2}(|x|)}{|x|^{2}}+\int _{|x|}^{+ \infty }\frac{h(s)}{s}u^{2}(s)ds )u=|u|^{p-2}u,\quad x\in \mathbb{R}^{2}, \\ u\in H^{1}_{r}(\mathbb{R}^{2}), \end{array}\displaystyle \right . \end{aligned}$$where \(\lambda >0\) is a parameter, \(p\in (2,4)\) and
$$ h(s)=\frac{1}{2} \int _{0}^{s}ru^{2}(r)dr. $$We prove that the system has no solutions for \(\lambda \) large and has two radial solutions for \(\lambda \) small by studying the decomposition of the Nehari manifold and adapting the fibering method. We also give the qualitative properties about the energy of the solutions and a variational characterization of these extremals values of \(\lambda \). Our results improve some results in Pomponio and Ruiz (J. Eur. Math. Soc. 17:1463–1486, 2015).
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 \(L_n\).
相似文献We study the problem of recovering an unknown signal \({\varvec{x}}\) given measurements obtained from a generalized linear model with a Gaussian sensing matrix. Two popular solutions are based on a linear estimator \(\hat{\varvec{x}}^\mathrm{L}\) and a spectral estimator \(\hat{\varvec{x}}^\mathrm{s}\). The former is a data-dependent linear combination of the columns of the measurement matrix, and its analysis is quite simple. The latter is the principal eigenvector of a data-dependent matrix, and a recent line of work has studied its performance. In this paper, we show how to optimally combine \(\hat{\varvec{x}}^\mathrm{L}\) and \(\hat{\varvec{x}}^\mathrm{s}\). At the heart of our analysis is the exact characterization of the empirical joint distribution of \(({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})\) in the high-dimensional limit. This allows us to compute the Bayes-optimal combination of \(\hat{\varvec{x}}^\mathrm{L}\) and \(\hat{\varvec{x}}^\mathrm{s}\), given the limiting distribution of the signal \({\varvec{x}}\). When the distribution of the signal is Gaussian, then the Bayes-optimal combination has the form \(\theta \hat{\varvec{x}}^\mathrm{L}+\hat{\varvec{x}}^\mathrm{s}\) and we derive the optimal combination coefficient. In order to establish the limiting distribution of \(({\varvec{x}}, \hat{\varvec{x}}^\mathrm{L}, \hat{\varvec{x}}^\mathrm{s})\), we design and analyze an approximate message passing algorithm whose iterates give \(\hat{\varvec{x}}^\mathrm{L}\) and approach \(\hat{\varvec{x}}^\mathrm{s}\). Numerical simulations demonstrate the improvement of the proposed combination with respect to the two methods considered separately.
相似文献In this paper we study the following fractional Hamiltonian systems
$$\begin{aligned} \left\{ \begin{array}{lllll} -_{t}D^{\alpha }_{\infty }(_{-\infty }D^{\alpha }_{t}x(t))- L(t).x(t)+\nabla W(t,x(t))=0, \\ x\in H^{\alpha }(\mathbb {R}, \mathbb {R}^{N}), \end{array} \right. \end{aligned}$$where \(\alpha \in \left( {1\over {2}}, 1\right] ,\ t\in \mathbb {R}, x\in \mathbb {R}^N,\ _{-\infty }D^{\alpha }_{t}\) and \(_{t}D^{\alpha }_{\infty }\) are the left and right Liouville–Weyl fractional derivatives of order \(\alpha \) on the whole axis \(\mathbb {R}\) respectively, \(L:\mathbb {R}\longrightarrow \mathbb {R}^{2N}\) and \(W: \mathbb {R}\times \mathbb {R}^{N}\longrightarrow \mathbb {R}\) are suitable functions. One ground state solution is obtained by applying the monotonicity trick of Jeanjean and the concentration-compactness principle in the case where the matrix L(t) is positive definite and \(W \in C^{1}(\mathbb {R}\times \mathbb {R}^{N},\mathbb {R})\) is superquadratic but does not satisfy the usual Ambrosetti–Rabinowitz condition.
相似文献We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on \(\ell ^r\)-valued extensions of linear operators. We show that for certain \(1 \le p, q_1, \dots , q_m, r \le \infty \), there is a constant \(C\ge 0\) such that for every bounded multilinear operator \(T:L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m) \rightarrow L^p(\nu )\) and functions \(\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(\mu _1), \dots , \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(\mu _m)\), the following inequality holds
$$\begin{aligned} \left\| \left( \sum _{k_1, \dots , k_m} |T(f_{k_1}^1, \dots , f_{k_m}^m)|^r\right) ^{1/r} \right\| _{L^p(\nu )} \le C \Vert T\Vert \prod _{i=1}^m \left\| \left( \sum _{k_i=1}^{n_i} |f_{k_i}^i|^r\right) ^{1/r} \right\| _{L^{q_i}(\mu _i)}. \end{aligned}$$ (1)In some cases we also calculate the best constant \(C\ge 0\) satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.
相似文献In this paper, we show that bounded weak solutions of the Cauchy problem for general degenerate parabolic equations of the form
$$ u_{t} + \operatorname{div}f(x,t,u) = \operatorname{div}\bigl( |u|^{\alpha } \nabla u\bigr), \quad x \in \mathbb{R}^{n} , \ t > 0, $$where \(\alpha > 0 \) is constant, decrease to zero, under fairly broad conditions on the advection flux \(f\). Besides that, we derive a time decay rate for these solutions.
相似文献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We prove a density lower bound for some functionals involving bulk and interfacial energies. The bulk energies are convex functions with p-power growth not subjected to any further structure conditions. The interface \(\partial E\) is the boundary of a set \(E\subset \Omega \) such that \(|E|=d\) is prescribed. Then we get \(\mathcal {H}^{n-1}((\partial E{\setminus }\partial E^*)\cup \Omega )=0\).
相似文献In this article, I explore in a unified manner the structure of uniform slash and \(\alpha \)-slash distributions which, in the continuous case, are defined to be the distributions of Y / U and \( Y_\alpha /U^{1/\alpha }\) where Y and \(Y_\alpha \) follow any distribution on \(\mathbb {R}^+\) and, independently, U is uniform on (0, 1). The parallels with the monotone and \(\alpha \)-monotone distributions of \( Y \times U\) and \(Y_\alpha \times U^{1/\alpha }\), respectively, are striking. I also introduce discrete uniform slash and \(\alpha \)-slash distributions which arise from a notion of negative binomial thinning/fattening. Their specification, although apparently rather different from the continuous case, seems to be a good one because of the close way in which their properties mimic those of the continuous case.
相似文献The problem of the minimax testing of the Poisson process intensity \({\mathbf{s}}\) is considered. For a given intensity \({\mathbf{p}}\) and a set \(\mathcal{Q}\), the minimax testing of the simple hypothesis \(H_{0}: {\mathbf{s}} = {\mathbf{p}}\) against the composite alternative \(H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}\) is investigated. The case, when the 1-st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2-nd kind error probability \(\beta ({\mathbf{p}},\mathcal{Q})\), is considered. What is the maximal set \(\mathcal{Q}\), which can be replaced by an intensity \({\mathbf{q}} \in \mathcal{Q}\) without any loss of testing performance? In the asymptotic case (\(T\rightarrow \infty \)) that maximal set \(\mathcal{Q}\) is described.
相似文献When a measure \(\varPsi(x)\) on the real line is subjected to the modification \(d\varPsi^{(t)}(x) = e^{-tx} d \varPsi(x)\), then the coefficients of the recurrence relation of the orthogonal polynomials in \(x\) with respect to the measure \(\varPsi^{(t)}(x)\) are known to satisfy the so-called Toda lattice formulas as functions of \(t\). In this paper we consider a modification of the form \(e^{-t(\mathfrak{p}x+ \mathfrak{q}/x)}\) of measures or, more generally, of moment functionals, associated with orthogonal L-polynomials and show that the coefficients of the recurrence relation of these L-orthogonal polynomials satisfy what we call an extended relativistic Toda lattice. Most importantly, we also establish the so called Lax pair representation associated with this extended relativistic Toda lattice. These results also cover the (ordinary) relativistic Toda lattice formulations considered in the literature by assuming either \(\mathfrak{p}=0\) or \(\mathfrak{q}=0\). However, as far as Lax pair representation is concern, no complete Lax pair representations were established before for the respective relativistic Toda lattice formulations. Some explicit examples of extended relativistic Toda lattice and Langmuir lattice are also presented. As further results, the lattice formulas that follow from the three term recurrence relations associated with kernel polynomials on the unit circle are also established.
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A result of Vietoris states that if the real numbers \(a_1,\ldots ,a_n\) satisfy
$$\begin{aligned} \text{(*) } \qquad a_1\ge \frac{a_2}{2} \ge \cdots \ge \frac{a_n}{n}>0 \quad \text{ and } \quad a_{2k-1}\ge a_{2k} \quad (1\le k\le n/2), \end{aligned}$$then, for \(x_1,\ldots ,x_m>0\) with \(x_1+\cdots +x_m <\pi \),
$$\begin{aligned} \begin{aligned} \text{(**) } \qquad \sum _{k=1}^n a_k \frac{\sin (k x_1) \cdots \sin (k x_m)}{k^m}>0. \end{aligned} \end{aligned}$$We prove that \((**)\) (with “\(\ge \)” instead of “>”) holds under weaker conditions. It suffices to assume, instead of \((*)\), that
$$\begin{aligned} \sum _{k=1}^N a_k \frac{\sin (kt)}{k}>0 \quad (N=1,\ldots ,n; \, 0<t<\pi ), \end{aligned}$$and, moreover, \((**)\) is valid for a larger region, namely, \(x_1,\ldots ,x_m\in (0,\pi )\).
相似文献Discrete and continuous frames can be considered as positive operator-valued measures (POVMs) that have integral representations using rank-one operators. However, not every POVM has an integral representation. One goal of this paper is to examine the POVMs that have finite-rank integral representations. More precisely, we present a necessary and sufficient condition under which a positive operator-valued measure \(F: \varOmega \to B(H)\) has an integral representation of the form
$$ F(E) =\sum_{k=1}^{m} \int _{E} G_{k}(\omega )\otimes G_{k}(\omega )\, d \mu (\omega ) $$for some weakly measurable maps \(G_{k}\ (1\leq k\leq m) \) from a measurable space \(\varOmega \) to a Hilbert space ℋ and some positive measure \(\mu \) on \(\varOmega \). Similar characterizations are also obtained for projection-valued measures. As special consequences of our characterization we settle negatively a problem of Ehler and Okoudjou about probability frame representations of probability POVMs, and prove that an integral representable probability POVM can be dilated to a integral representable projection-valued measure if and only if the corresponding measure is purely atomic.
相似文献We prove that given any \(\epsilon >0\), a non-zero adelic Hilbert cusp form \({\mathbf {f}}\) of weight \(k=(k_1,k_2,\ldots ,k_n)\in ({\mathbb {Z}}_+)^n\) and square-free level \(\mathfrak {n}\) with Fourier coefficients \(C_{{\mathbf {f}}}(\mathfrak {m})\), there exists a square-free integral ideal \(\mathfrak {m}\) with \(N(\mathfrak {m})\ll k_0^{3n+\epsilon }N(\mathfrak {n})^{\frac{6n^2+1}{2}+\epsilon }\) such that \(C_{{\mathbf {f}}}(\mathfrak {m})\ne 0\). The implied constant depends on \(\epsilon , F\).
相似文献