Turán problems for hypergraphs:avoiding path or star forests

This thesis contains a number of new contributions regarding Turán problems for hypergraphs. These new results involve the Turán number of Berge linear forests, Turán numbers of star forests (in three different variants), linear Turán numbers of generalized crowns, and connected Turán numbers for Berge paths.A classic result due to Erdos and Gallai about paths states that the number of edges e(G) of an n-vertex graph G containing no copy of a path Pk+1 of length k as its subgraph satisfies: e(G) ≤ (k−1)n/2 , with equality holding if and only if k divides n and G consists of vertex-disjoint copies of the complete graph K_k on k vertices.It is natural to consider determining the Turán number of forests, since a forest consists of one or more trees. Motivated by the previous results, in this thesis we focus on the Turán numbers of linear forests and star forests in hypergraphs. In Chapter 2, by considering avoiding Berge linear forests in uniform hypergraphs, we determine the value of ex_r (n, Berge-L_{n,k}) for the cases 3 ≤ r ≤(k+1)/2−3 and r ≥ k+1, when k is odd, and for the cases 3 ≤ r ≤ k/2−1−\sqrt{(k+2)/2} and r ≥ k+1, when k is even, and we characterize the extremal hypergraphs. We establish an upper bound on ex_r (n, Berge-L_{n,k}) for several other cases.In Chapter 3, we determine the Turán numbers for analogues of star forests in the hypergraph setting. In Chapter 4, we generalize the notion of a crown to r-uniform hypergraph and obtain upper and lower bounds on the Turán number for it. In Chapter 5, we consider the Turán numbers of Berge paths in connected hypergraphs and determine them asymptotically

Quantitative Strongest Post: A Calculus for Reasoning about the Flow of Quantitative Information

We present a novel strongest-postcondition-style calculus for quantitative reasoning about non-deterministic programs with loops. Whereas existing quantitative weakest pre allows reasoning about the value of a quantity after a program terminates on a given initial state, quantitative strongest post allows reasoning about the value that a quantity had before the program was executed and reached a given final state. We show how strongest post enables reasoning about the flow of quantitative information through programs. Similarly to weakest liberal preconditions, we also develop a quantitative strongest liberal post. As a byproduct, we obtain the entirely unexplored notion of strongest liberal postconditions and show how these foreshadow a potential new program logic - partial incorrectness logic - which would be a more liberal version of O'Hearn's recent incorrectness logic

Dynamic Gust Load Analysis for Rotors

Dynamic load of helicopter rotors due to gust directly affects the structural stress and flight performance for helicopters. Based on a large deflection beam theory, an aeroelastic model for isolated helicopter rotors in the time domain is constructed. The dynamic response and structural load for a rotor under the impulse gust and slope-shape gust are calculated, respectively. First, a nonlinear Euler beam model with 36 degrees-of-freedoms per element is applied to depict the structural dynamics for an isolated rotor. The generalized dynamic wake model and Leishman-Beddoes dynamic stall model are applied to calculate the nonlinear unsteady aerodynamic forces on rotors. Then, we transformed the differential aeroelastic governing equation to an algebraic one. Hence, the widely used Newton-Raphson iteration algorithm is employed to simulate the dynamic gust load. An isolated helicopter rotor with four blades is studied to validate the structural model and the aeroelastic model. The modal frequencies based on the Euler beam model agree well with published ones by CAMRAD. The flap deflection due to impulse gust with the speed of 2m/s increases twice to the one without gust. In this numerical example, results indicate that the bending moment at the blade root is alleviated due to elastic effect

Quantitative Weakest Hyper Pre: Unifying Correctness and Incorrectness Hyperproperties via Predicate Transformers

We present a novel \emph{weakest pre calculus} for \emph{reasoning about quantitative hyperproperties} over \emph{nondeterministic and probabilistic} programs. Whereas existing calculi allow reasoning about the expected value that a quantity assumes after program termination from a \emph{single initial state}, we do so for \emph{initial sets of states} or \emph{initial probability distributions}. We thus (i)~obtain a weakest pre calculus for hyper Hoare logic and (ii)~enable reasoning about so-called \emph{hyperquantities} which include expected values but also quantities (e.g. variance) out of scope of previous work. As a byproduct, we obtain a novel strongest post for weighted programs that extends both existing strongest and strongest liberal post calculi. Our framework reveals novel dualities between forward and backward transformers, correctness and incorrectness, as well as nontermination and unreachability

Compact on-chip power splitter based on topological photonic crystal

We propose and demonstrate an on-chip 1*N power splitter based on topological photonic crystal (TPC) on a monolithic silicon photonic platform. Benefiting from the valley-locked propagation mode at the interface of TPCs with different topological phases, the proposed power splitter has negligible backscattering around the sharp bendings and good robustness to fabrication defects, which therefore enable lower insertion loss, better uniformity, and more compact footprint than the conventional designs. For the fabricated 1*2 (8) power splitter, the uniformity among the output ports is below 0.35 (0.65) dB and the maximum insertion loss is 0.38 (0.58) dB with compact footprint of 5*5 um2 (10*12 um2) within a bandwidth of 70 nm. In addition, the topological power splitter only requires simple configurations of TPCs with different topological phases, which is more reliable in design and fabrication compared with the conventional designs.Comment: 8 pages,4 figure

Quantitative Phase Imaging Camera With a Weak Diffuser

We introduce the quantitative phase imaging camera with a weak diffuser (QPICWD) as an effective scheme of quantitative phase imaging (QPI) based on normal microscope platforms. The QPICWD is an independent compact camera measuring object induced phase delay under low-coherence quasi-monochromatic illumination by examining the deformation of the speckle intensity pattern. By interpreting the speckle deformation with an ensemble average of the geometric flow, we can obtain the high-resolution distortion field via the transport of intensity equation (TIE). Since the phase measured by TIE is the generalized phase of the partially coherent image, rather than the phase of the measured object, we analyze the effect of illumination coherence and imaging numerical aperture (NA) on the accuracy of phase retrieval, revealing that the sample's phase can be reliably reconstructed under the conditions that the coherence parameter (the ratio of illumination NA to objective NA) of the Köhler illumination is between 0.3 and 0.5. We present some applications for the proposed design involving nondestructive optical testing of microlens array with nanometric thickness and imaging of fixed and live unstained HeLa cells. Since the designed QPI camera does not require any modification of the widely available bright-field microscope or additional accessories for its use, it is expected to be applied by the broader communities of biology and medicine

Fano-like resonance in an all-in-fiber structure

We achieve Fano-like resonances in an all-in-fiber structure embedded with an in-line Mach-Zehnder interferometer (MZI). A fiber Bragg grating is inserted into MZI's one arm to form a resonance, which functions as the discrete state of the Fano-like resonance to couple with the continuum propagating mode of MZI in the fiber core. A theoretical model predicts the controllable resonance lineshape by changing the phase difference between the MZI's two interference pathways. Fano-like resonances with an extinction ratio over 20 dB are experimentally observed, which are reliably tuned into Lorentzian and electromagnetically induced transparency-like resonances by versatile methods. The realization of Fano-like resonances with broad tunability in this all-in-fiber structure holds potentials in fiber-based applications of sensing, signal processing and nonlinear optics