Simplicial Ricci Flow

We construct a discrete form of Hamilton’s Ricci flow (RF) equations for a d-dimensional piecewise flat simplicial geometry, ${{\mathcal S}}$ . These new algebraic equations are derived using the discrete formulation of Einstein’s theory of general relativity known as Regge calculus. A Regge–Ricci flow (RRF) equation can be associated to each edge, ?, of a simplicial lattice. In defining this equation, we find it convenient to utilize both the simplicial lattice ${{\mathcal S}}$ and its circumcentric dual lattice, ${{\mathcal S}^*}$ . In particular, the RRF equation associated to ? is naturally defined on a d-dimensional hybrid block connecting ? with its (d?1)-dimensional circumcentric dual cell, ? ^*. We show that this equation is expressed as the proportionality between (1) the simplicial Ricci tensor, Rc _? , associated with the edge ${\ell\in{\mathcal S}}$ , and (2) a certain volume weighted average of the fractional rate of change of the edges, ${\lambda\in \ell^*}$ , of the circumcentric dual lattice, ${{\mathcal S}^*}$ , that are in the dual of ?. The inherent orthogonality between elements of ${\mathcal S}$ and their duals in ${{\mathcal S}^*}$ provide a simple geometric representation of Hamilton’s RF equations. In this paper we utilize the well established theories of Regge calculus, or equivalently discrete exterior calculus, to construct these equations. We solve these equations for a few illustrative examples.     

Detection of α‐Methylacyl‐CoA Racemase in Serum and Urine Using a Highly Sensitive Electrochemical Immunodetector

The detection of α‐methylacyl‐CoA racemase (AMACR), a novel biomarker for prostate cancer, is demonstrated in serum and urine using a novel immuno‐detection method. The detection system consists of a three‐electrode conventional electrochemical cell modified with a gating electrode for applying a gating voltage V_G to the immune complex immobilized on the working electrode to provide signal amplification. The detection system is realized by integrating gating electrodes with screen‐printed electrodes. This detection method does not require involved sample preparation procedures. The detection was demonstrated in serum and urine samples on the nanogram/mL level with V_G equal to 0.6 V. Detection in serum was also performed on the picogram/mL level with a limit of 100 picogram/mL with V_G=0.6 V being a necessary condition.     

Cytotoxic Activity of Capnellene‐8β, 10α‐Diol Derivatives from a Taiwanese Soft Coral Capnella sp.

The sesquiterpene capnellene‐8β, 10α‐diol ( 1 ) was isolated from non‐polar extract of the soft coral Capnella sp. Ten acylation products of capnellene‐8β, 10α‐diol were prepared: 10α‐hydroxy‐8β‐O‐benzoylcapnellene ( 2 ), 10α‐hydroxy‐8β‐O‐p‐toluoylcapnellene ( 3 ), 10α‐hydroxy‐8β‐O‐4‐chlorobenzoyl‐capnellene ( 4 ), 10α‐hydroxy‐8β‐O‐2‐furoylcapnellene ( 5 ), 10α‐hydroxy‐8β‐O‐2‐thiophenoylcapnellene ( 6 ), 10α‐hydroxy‐8β‐O‐4‐fluorobenzoylcapnellene ( 7 ), 10α‐hydroxy‐8β‐O‐4‐propylbenzoylcapnellene ( 8 ), 10α‐hydroxy‐8β‐O‐cinnamoylcapnellene ( 9 ), 10α‐hydroxy‐8β‐O‐4‐nitrobenzoylcapnellene ( 10 ), and 10α‐hydroxy‐8β‐O‐4‐anisoylcapnellene ( 11 ). The structures of capnellene‐8β, 10α‐diol as well as its derivatives were established through standard spectroscopic analysis. The in vitro cytotoxic activities of the eleven compounds were evaluated against Hela, KB, Daoy, and WiDr human tumor cell lines.     

Determination of pharmaceuticals classified as emerging pollutants using capillary electrophoresis with capacitively coupled contactless conductivity detection

A study on the simultaneous separation of 13 pharmaceutical products by capillary electrophoresis with capacitively coupled contactless conductivity detection was presented. The parameters of the background electrolyte, such as pH, organic additives as well as types and concentrations of cyclodextrins (CD) were studied. The optimal separation conditions were achieved with a background electrolyte consisting of 9 mM Tris/5 mM lactic acid at pH 8.0, containing 5% n-propanol, 0.025% gamma-CD, 0.075% hydroxyl-beta-CD and 0.15% dimethyl-beta-CD. Limits of detections ranged from 61 to 1676 microg/L (S/N=3) and the relative standard deviations for migration time and peak area were below 2 and 6%, respectively. This demonstrated the potential of the capillary electrophoresis-capacitively coupled contactless conductivity detection method for biomedical and environmental analysis, as shown in the determination of pharmaceuticals identified as emerging pollutants in water samples.     

Quantum diffusion of the random Schrödinger evolution in the scaling limit

We consider random Schrödinger equations on R ^d for d???3 with a homogeneous Anderson–Poisson type random potential. Denote by λ the coupling constant and \(\psi_t\) the solution with initial data \(\psi_0\). The space and time variables scale as \(x\sim\lambda ^{{ - 2 - \varkappa/2}} {\text{ and }}t\sim\lambda ^{{ - 2 - \varkappa}} {\text{ with }}0 < \varkappa < \varkappa_{0} {\left( d \right)}\). We prove that, in the limit λ?→?0, the expectation of the Wigner distribution of \(\psi_t\) converges weakly to the solution of a heat equation in the space variable x for arbitrary L ² initial data.The proof is based on analyzing the phase cancellations of multiple scatterings on the random potential by expanding the propagator into a sum of Feynman graphs. In this paper we consider the non-recollision graphs and prove that the amplitude of the non-ladder diagrams is smaller than their “naive size” by an extra λ ^c factor per non-(anti)ladder vertex for some c?>?0. This is the first rigorous result showing that the improvement over the naive estimates on the Feynman graphs grows as a power of the small parameter with the exponent depending linearly on the number of vertices. This estimate allows us to prove the convergence of the perturbation series.     

Bulk universality for generalized Wigner matrices

Consider N × N Hermitian or symmetric random matrices H where the distribution of the (i, j) matrix element is given by a probability measure ν _ij with a subexponential decay. Let ${\sigma_{ij}^2}$ be the variance for the probability measure ν _ij with the normalization property that ${\sum_{i} \sigma^2_{ij} = 1}$ for all j. Under essentially the only condition that ${c\le N \sigma_{ij}^2 \le c^{-1}}$ for some constant c?>?0, we prove that, in the limit N → ∞, the eigenvalue spacing statistics of H in the bulk of the spectrum coincide with those of the Gaussian unitary or orthogonal ensemble (GUE or GOE). We also show that for band matrices with bandwidth M the local semicircle law holds to the energy scale M ^?1.     

Diffeomorphic types of complements of nice point arrangements in CP<Superscript><Emphasis Type="Italic">l</Emphasis></Superscript>

We use a new method to study arrangement in CP ^l , define a class of nice point arrangements and show that if two nice point arrangements have the same combinatorics, then their complements are diffeomorphic to each other. In particular, the moduli space of nice point arrangements with same combinatorics in CP ^l is connected. It generalizes the result on point arrangements in CP ³ to point arrangements in CP ^l for any l.     

Adaptive sliding mode control of a high-precision ball-screw-driven stage

High-precision position control has been widely used in scientific instruments and semiconductor fabrication equipment. Traditionally, controllable displacements with sub-micron and nano-level resolution are usually achieved by piezoelectric actuators because of their high bandwidth and ease of control. However, the travel range of piezoelectric actuator is usually small. In this paper, the ball-screw-driven system is studied to provide long-range and high-precision performance for positioning and tracking control. In such a system, the friction dynamics are divided into the static and the dynamic regimes to describe the dynamic behavior of a conventional ball-screw-driven x-y stage. The same form of adaptive sliding mode controllers are designed in the static and dynamic regimes to obtain the precision performance for controlled stage. A proportional-integral switching surface is proposed to make it easy to assign the performance of the systems in the sliding mode motion and the controller is robust without knowing the bound of disturbance in advance. An illustrative example is included to demonstrate that the system achieves high precision (10 nm) and long-range (10 cm) positioning performance with repeatability and robustness by the proposed control approach.     

On formulas for Dedekind sums and the number of lattice points in tetrahedra

This paper explores a simple yet powerful relationship between the problem of counting lattice points and the computation of Dedekind sums. We begin by constructing and proving a sharp upper estimate for the number of lattice points in tetrahedra with some irrational coordinates for the vertices. Besides providing a sharper estimate, this upper bound (Theorem 1.1) becomes an equality (i.e. gives the exact number of lattice points) in a tetrahedron where the lengths of the edges divide each other. This equality condition can then be applied to the explicit computation of the classical Dedekind sums, a topic that is the central focus in the second half of our paper. In this half of the paper, we come up with a number of interesting results related to Dedekind sums, based on our upper estimate (Theorem 1.1). Among these findings, Theorem 1.9 and Theorem 1.10 deserve special attention, for they successfully generalize two of Apostol's formulas in [T.M. Apostol, Modular Functions and Dirichlet Series in Number Theory, Springer-Verlag, New York, 1997], and also directly imply the famous Reciprocity Law of Dedekind sums.     

Robust decentralized adaptive control for uncertain large-scale delayed systems with input nonlinearities

The paper is concerned with the problem of robust stabilization for uncertain large-scale time-varying delayed systems with input nonlinearities. Based on the sliding mode control, a memoryless decentralized adaptive sliding mode controller (DASMC) is developed. The proposed controller ensures the occurrence of the sliding manifold of the composite system even subjected to input nonlinearity. It shows that the uncertain nonlinear large-scale system also possesses the property of insensitivity to uncertainties and disturbances as a linear system does. A numerical example is given to verify the validity of the developed memoryless DASMC.