The Landscape of the Spiked Tensor Model

We consider the problem of estimating a large rank-one tensor u ^⊗k ∈ (ℝⁿ)^⊗k , k ≥ 3 , in Gaussian noise. Earlier work characterized a critical signal-to-noise ratio λ  _Bayes = O(1) above which an ideal estimator achieves strictly positive correlation with the unknown vector of interest. Remarkably, no polynomial-time algorithm is known that achieved this goal unless λ ≥ Cn^{(k − 2)/4} , and even powerful semidefinite programming relaxations appear to fail for 1 ≪ λ ≪ n^{(k − 2)/4} . In order to elucidate this behavior, we consider the maximum likelihood estimator, which requires maximizing a degree-k homogeneous polynomial over the unit sphere in n dimensions. We compute the expected number of critical points and local maxima of this objective function and show that it is exponential in the dimensions n , and give exact formulas for the exponential growth rate. We show that (for λ larger than a constant) critical points are either very close to the unknown vector u or are confined in a band of width Θ(λ^{−1/(k − 1)}) around the maximum circle that is orthogonal to u . For local maxima, this band shrinks to be of size Θ(λ^{−1/(k − 2)}) . These “uninformative” local maxima are likely to cause the failure of optimization algorithms. © 2019 Wiley Periodicals, Inc.     

Self-similar asymptotic behavior for the solutions of a linear coagulation equation

In this paper we consider the long-time asymptotics of a linear version of the Smoluchowski equation which describes the evolution of a tagged particle moving in a random distribution of fixed particles. The volumes v of these particles are independently distributed according to a probability distribution which decays asymptotically as a power law $v^{?\sigma }$. The validity of the equation has been rigorously proved in [22] taking as a starting point a particle model and for values of the exponent $\sigma >3$, but the model can be expected to be valid, on heuristic grounds, for $\sigma >\frac{5}{3}$. The resulting equation is a non-local linear degenerate parabolic equation. The solutions of this equation display a rich structure of different asymptotic behaviors according to the different values of the exponent σ. Here we show that for $\frac{5}{3}<\sigma <2$ the linear Smoluchowski equation is well-posed and that there exists a unique self-similar profile which is asymptotically stable.     

Coordinate Bond Breaking Induced by Collapse of Poly(\begin{document}$N$\end{document}-isopropyl acrylamide) as Ligands of a Rare Earth Complex

A novel water-soluble luminescent complex consisting of Eu(ally-dbm)\begin{document}$_3$\end{document}-2Tppo and poly(N-isopropyl acrylamide) (PNIPAM) is synthesized through a series of chemical reactions. The structure of the complex is characterized by TGA, GPC, HNMR, and the thermal-responsive fluorescence of the complex in aqueous solution is investigated. It is found that PNIPAM collapse above the lower critical solution temperature causes the coordination bond breaking, leading to weakening of the fluorescence from Eu\begin{document}$^{3+}$\end{document} and enhancing of the fluorescence from the ligands. When temperature decreases, the fluorescence from Eu\begin{document}$^{3+}$\end{document} is found to boost up and the fluorescence from ligands weakens accordingly. It is deduced from this phenomenon that the ligands re-coordinate with europium ions again along with the temperature decreasing, which is further confirmed by IR measurements. This thermal-responsive fluorescence is of reversibility, which can be used as molecular probes for biological imaging and collapse studying of PNIPAM.     

Three-term polynomial progressions in subsets of finite fields

We prove that the group of diffeomorphisms of the interval [0, 1] contains surface groups whose action on (0, 1) has no global fix point and such that only countably many points of the interval (0, 1) have non-trivial stabiliser.     

Application of ion-imprinted polymer synthesized by precipitation polymerization as an efficient and selective sorbent for separation and pre-concentration of chromium ions from some real samples

In this study, a new Cr(III)-imprinted polymer (Cr(III)-IIP) is prepared from CrCl₃·6H₂O, methacrylic acid functional monomer, ethyleneglycoldimethacrylate cross-linking agent, 2,2?-azobisisobutyronitrile radical initiator and 2,2-(azanediylbis (ethane-2,1-diyl))bis(isoindoline-1,3-dione) ligand. To obtain the maximum adsorption capacity, the optimum condition was studied through pH, type and concentration of eluent, IIP weight, sample volume as well as the adsorption and desorption times. The Cr(III) ion content was determined via flame atomic absorption spectrometer. In optimum conditions, the adsorption capacity of the IIP for Cr(III) was obtained to be 74.65 mg g^?1, using 50 mg of IIP and the initial pH solution of 3.0. Both the adsorption and desorption times for quantitative analyses of Cr(III) ions were 15 and 5 min; respectively. After elution of the adsorbed ions by 3 mL of 4 mol L^?1 HNO₃ aqueous solution, the established IIP-based SPE procedure provides a reasonable pre-concentration factor of 100. The IIP-based pre-concentration method provides a low detection limit of 1.7 µg L^?1 with good repeatability (RSD?=?3.22%). Reusability studies confirmed that synthesis IIP is reusable and recoverable up to six cycles. According to the selectivity experiments, it was concluded that the prepared sorbent possesses more affinity toward Cr(III) ions than other ions such as Al³⁺, Pb²⁺, Cu²⁺, Mn²⁺, Fe²⁺, Zn²⁺, and Ni²⁺ ions. To evaluate the potential applicability of the proposed separation method, the pre-concentration and determination of trace amounts of Cr(III) were performed successfully in food samples with complex matrices, a bestial sample (i.e. cow liver) and an herbal product (i.e., broccoli) as real samples.     

On the output function in a Ginsburg’s machine

We give the form of the output function in Ginsburg’s machine in which the input and output dictionaries are abelian groups and the transition function is of a special form.     

Optical vortices in the Ginzburg–Landau equation with cubic–quintic nonlinearity

In a dissipative system with cubic–quintic nonlinearity, the curious evolution of optical vortex beams characterized by different topological charges (TCs) is simulated numerically and presented their evolution profiles. We find that new vortices will be induced during propagation, and the behavior of vortices, as affected by the TC and the number of beads of the incident beam, as well as its size, is also discussed. Common rules associated with the initial conditions coming from various incident beams are developed to determine the number of induced vortices and the corresponding rotation direction. Attributed to the nonlinearity, during propagation we see the beams slowly expand to induce new vortices, which commonly appear in oppositely charged pairs, while the net topological charge of the vortex is conserved. Our results not only deepen the understanding of optical vortices, but also widen their potential applications.