Theta Maps for Combinatorial Hopf Algebras

This thesis introduces a way to generalize of peak algebra. There are several equivalent denitions for the peak algebra. Stembridge describes it via enriched P-partitions to generalize marked shifted tableaux and Schur's Q functions. Nyman shows that it is a the sum of permutations with the same peak set. Aguiar, Bergeron and Sottile show that the peak algebra is the odd Hopf sub-algebra of quasi symmetric functions using their theory of combinatorial Hopf algebras. In all these cases, there is a very natural and well-behaved Hopf algebra morphism from quasi-symmetric functions or non-commutative symmetric functions to their respective peak algebra, which we call the theta map. This thesis focuses on generalizing the peak algebra by constructing generalized theta maps for an arbitrary combinatorial Hopf algebra. The motivating example of this thesis is the Malvenuto-Reutenauer Hopf algebra of permutations. Our main result is a combinatorial description of all of the theta maps of this Hopf algebra whose images are generalizations of the peak algebra. We also give a criterion to check whether a map is a theta map, and we nd theta maps for Hopf sub-algebras of quasi-symmetric functions. We also show the existence of theta maps for any commutative and cocommutative Hopf algebras. From there, we study the diagonally symmetric functions and diagonally quasi-symmetric functions. Lastly, we describe theta maps for a Hopf algebra V on permutations

Two component dark matter with multi-Higgs portals

With the assistance of two extra groups, i.e., an extra hidden gauge group $SU(2)_D$ and a global $U(1)$ group, we propose a two component dark matter (DM) model. After the symmetry $SU(2)_D\times U(1)$ being broken, we obtain both the vector and scalar DM candidates. The two DM candidates communicate with the standard model (SM) via three Higgs as multi-Higgs portals. The three Higgs are mixing states of the SM Higgs, the Higgs of the hidden sector and real part of a supplement complex scalar singlet. We study relic density and direct detection of DM in three scenarios. The resonance behaviors and interplay between the two component DM candidates are represented through investigating of the relic density in the parameter spaces of the two DMs masses. The electroweak precision parameters constrains the two Higgs portals couplings ($\lambda_m$ and $\delta_2$). The relevant vacuum stability and naturalness problem in the parameter space of $\lambda_m$ and $\delta_2$ are studied as well. The model could alleviate these two problems in some parameter spaces under the constraints of electroweak precision observables and Higgs indirect search.Comment: 27 pages, 16 figures. Version accepted for publication in JHE

The Effects of Minimal Length, Maximal Momentum and Minimal Momentum in Entropic Force

In this paper, the modified entropic force law is studied by using a new kind of generalized uncertainty principle which contains a minimal length, a minimal momentum and a maximal momentum. Firstly, the quantum corrections to the thermodynamics of a black hole is investigated. Then, according to Verlinde's theory, the generalized uncertainty principle (GUP) corrected entropic force is obtained. The result shows that the GUP corrected entropic force is related not only to the properties of the black holes, but also to the Planck length and the dimensionless constants $\alpha _{\rm{0}}$ and $\beta _{\rm{0}}$. Moreover, based on the GUP corrected entropic force, we also derive the modified Einstein's field equation (EFE) and the modified Friedmann equation.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1604.0470

Deterministic versus probabilistic quantum information masking

We investigate quantum information masking for arbitrary dimensional quantum states. We show that mutually orthogonal quantum states can always be served for deterministic masking of quantum information. We further construct a probabilistic masking machine for linearly independent states. It is shown that a set of d dimensional states, $\{ |a_1 \rangle_A, |t a_2 \rangle_A, \dots, |a_n \rangle_A \}$, $n \leq d$, can be probabilistically masked by a general unitary-reduction operation if they are linearly independent. The maximal successful probability of probabilistic masking is analyzed and derived for the case of two initial states.Comment: 5 pages, 1 figure

Quantification of the influence of drugs on zebrafish larvae swimming kinematics and energetics

The use of zebrafish larvae has aroused wide interest in the medical field for its potential role in the development of new therapies. The larvae grow extremely quickly and the embryos are nearly transparent which allows easy examination of its internal structures using fluorescent imaging techniques. Medical treatment of zebrafish larvae can directly influence its swimming behaviours. These behaviour changes are related to functional changes of central nervous system and transformations of the zebrafish body such as muscle mechanical power and force variation, which cannot be measured directly by pure experiment observation. To quantify the influence of drugs on zebrafish larvae swimming behaviours and energetics, we have developed a novel methodology to exploit intravital changes based on observed zebrafish locomotion. Specifically, by using an in-house MATLAB code to process the recorded live zebrafish swimming video, the kinematic locomotion equation of a 3D zebrafish larvae was obtained, and a customised Computational Fluid Dynamics tool was used to solve the fluid flow around the fish model which was geometrically the same as experimentally tested zebrafish. The developed methodology was firstly verified against experiment, and further applied to quantify the fish internal body force, torque and power consumption associated with a group of normal zebrafish larvae vs. those immersed in acetic acid and two neuroactive drugs. As indicated by our results, zebrafish larvae immersed in 0.01% acetic acid display approximately 30% higher hydrodynamic power and 10% higher cost of transport than control group. In addition, 500 μM diphenylhydantoin significantly decreases the locomotion activity for approximately 50% lower hydrodynamic power, whereas 100 mg/L yohimbine has not caused any significant influences on 5 dpf zebrafish larvae locomotion. The approach has potential to evaluate the influence of drugs on the aquatic animal’s behaviour changes and thus support the development of new analgesic and neuroactive drugs

Kullback-Leibler entropy and Penrose conjecture in the Lemaitre-Tolman-Bondi model

Our universe hosts various large-scale structures from voids to galaxy clusters, so it would be interesting to find some simple and reasonable measure to describe the inhomogeneities in the universe. We explore two different methods for this purpose: the Kullback-Leibler entropy and the Weyl curvature tensor. These two quantities characterize the deviation of the actual distribution of matter from the unperturbed background. We calculate these two measures in the spherically symmetric Lemaitre-Tolman-Bondi model in the dust universe. Both exact and perturbative calculations are presented, and we observe that these two measures are in proportion up to second order.Comment: 8 page