We provide moment bounds for expressions of the type
Stochastic networks for the clock were identified by ensemble methods using genetic algorithms that captured the amplitude and period variation in single cell oscillators of
- Award ID(s):
- 1713746
- NSF-PAR ID:
- 10192589
- Publisher / Repository:
- Nature Publishing Group
- Date Published:
- Journal Name:
- Scientific Reports
- Volume:
- 10
- Issue:
- 1
- ISSN:
- 2045-2322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
Abstract where$$(X^{(1)} \otimes \cdots \otimes X^{(d)})^T A (X^{(1)} \otimes \cdots \otimes X^{(d)})$$ denotes the Kronecker product and$$\otimes $$ are random vectors with independent, mean 0, variance 1, subgaussian entries. The bounds are tight up to constants depending on$$X^{(1)}, \ldots , X^{(d)}$$ d for the case of Gaussian random vectors. Our proof also provides a decoupling inequality for expressions of this type. Using these bounds, we obtain new, improved concentration inequalities for expressions of the form .$$\Vert B (X^{(1)} \otimes \cdots \otimes X^{(d)})\Vert _2$$ -
Abstract We perform path-integral molecular dynamics (PIMD), ring-polymer MD (RPMD), and classical MD simulations of H
O and D$$_2$$ O using the q-TIP4P/F water model over a wide range of temperatures and pressures. The density$$_2$$ , isothermal compressibility$$\rho (T)$$ , and self-diffusion coefficients$$\kappa _T(T)$$ D (T ) of H O and D$$_2$$ O are in excellent agreement with available experimental data; the isobaric heat capacity$$_2$$ obtained from PIMD and MD simulations agree qualitatively well with the experiments. Some of these thermodynamic properties exhibit anomalous maxima upon isobaric cooling, consistent with recent experiments and with the possibility that H$$C_P(T)$$ O and D$$_2$$ O exhibit a liquid-liquid critical point (LLCP) at low temperatures and positive pressures. The data from PIMD/MD for H$$_2$$ O and D$$_2$$ O can be fitted remarkably well using the Two-State-Equation-of-State (TSEOS). Using the TSEOS, we estimate that the LLCP for q-TIP4P/F H$$_2$$ O, from PIMD simulations, is located at$$_2$$ MPa,$$P_c = 167 \pm 9$$ K, and$$T_c = 159 \pm 6$$ g/cm$$\rho _c = 1.02 \pm 0.01$$ . Isotope substitution effects are important; the LLCP location in q-TIP4P/F D$$^3$$ O is estimated to be$$_2$$ MPa,$$P_c = 176 \pm 4$$ K, and$$T_c = 177 \pm 2$$ g/cm$$\rho _c = 1.13 \pm 0.01$$ . Interestingly, for the water model studied, differences in the LLCP location from PIMD and MD simulations suggest that nuclear quantum effects (i.e., atoms delocalization) play an important role in the thermodynamics of water around the LLCP (from the MD simulations of q-TIP4P/F water,$$^3$$ MPa,$$P_c = 203 \pm 4$$ K, and$$T_c = 175 \pm 2$$ g/cm$$\rho _c = 1.03 \pm 0.01$$ ). Overall, our results strongly support the LLPT scenario to explain water anomalous behavior, independently of the fundamental differences between classical MD and PIMD techniques. The reported values of$$^3$$ for D$$T_c$$ O and, particularly, H$$_2$$ O suggest that improved water models are needed for the study of supercooled water.$$_2$$ -
Abstract In this paper, we study multistage stochastic mixed-integer nonlinear programs (MS-MINLP). This general class of problems encompasses, as important special cases, multistage stochastic convex optimization with
non-Lipschitzian value functions and multistage stochastic mixed-integer linear optimization. We develop stochastic dual dynamic programming (SDDP) type algorithms with nested decomposition, deterministic sampling, and stochastic sampling. The key ingredient is a new type of cuts based on generalized conjugacy. Several interesting classes of MS-MINLP are identified, where the new algorithms are guaranteed to obtain the global optimum without the assumption of complete recourse. This significantly generalizes the classic SDDP algorithms. We also characterize the iteration complexity of the proposed algorithms. In particular, for a -stage stochastic MINLP satisfying$$(T+1)$$ L -exact Lipschitz regularization withd -dimensional state spaces, to obtain an -optimal root node solution, we prove that the number of iterations of the proposed deterministic sampling algorithm is upper bounded by$$\varepsilon $$ , and is lower bounded by$${\mathcal {O}}((\frac{2LT}{\varepsilon })^d)$$ for the general case or by$${\mathcal {O}}((\frac{LT}{4\varepsilon })^d)$$ for the convex case. This shows that the obtained complexity bounds are rather sharp. It also reveals that the iteration complexity depends$${\mathcal {O}}((\frac{LT}{8\varepsilon })^{d/2-1})$$ polynomially on the number of stages. We further show that the iteration complexity dependslinearly onT , if all the state spaces are finite sets, or if we seek a -optimal solution when the state spaces are infinite sets, i.e. allowing the optimality gap to scale with$$(T\varepsilon )$$ T . To the best of our knowledge, this is the first work that reports global optimization algorithms as well as iteration complexity results for solving such a large class of multistage stochastic programs. The iteration complexity study resolves a conjecture by the late Prof. Shabbir Ahmed in the general setting of multistage stochastic mixed-integer optimization. -
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ of dimension$$\Gamma $$ , the now usual distance to the boundary$$d < n-1$$ given by$$D = D_\beta $$ for$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$ , where$$X \in \Omega $$ and$$\beta >0$$ . In this paper we show that the Green function$$\gamma \in (-1,1)$$ G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ , in the sense that the function$$D^{1-\gamma }$$ satisfies a Carleson measure estimate on$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$ . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).$$\Omega $$ -
Abstract Let us fix a prime
p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ with coefficients$$j=1,\dots ,m$$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ , that$$k\ge 3m$$ for$$a_{j,1}+\dots +a_{j,k}=0$$ and that every$$j=1,\dots ,m$$ minor of the$$m\times m$$ matrix$$m\times k$$ is non-singular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ contains a solution$$|A|> C\cdot \Gamma ^n$$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ C and are constants only depending on$$\Gamma $$ p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma in the solution$$x_1,\dots ,x_k$$ to be distinct. If we relax this condition and only demand that$$(x_1,\dots ,x_k)\in A^k$$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.$$x_1,\dots ,x_k$$