The Beltrami Model of De Sitter Space: From Snyder's quantized space-time to de Sitter invariant relativity

In terms of the Beltrami model of de Sitter space we show that there is an interchangeable relation between Snyder's quantized space-time model in dS-space of momenta at the Planck length $\ell_P=(G\hbar c^{-3})^{1/2}$ and the dS-invariant special relativity in dS-spacetime of radius $R\simeq(3\Lambda^{-1})^{1/2}$, which is another fundamental length related to the cosmological constant. Here, the cosmological constant $\Lambda$ is regarded as a fundamental constant together with the speed of light $c$, Newton constant $G$ and Planck constant $\hbar$. Furthermore, the physics at two fundamental scales of length, the \dS-radius $R$ and the Planck length $\ell_P$, should be dual to each other and linked via the gravity with local dS-invariance characterized by a dimensionless coupling constant $g= \sqrt{3} \ell_P/R\simeq(G\hbar c^{-3}\Lambda)^{1/2}\sim 10^{-61}$.Comment: 15 pages. Invited talk given at `International workshop on noncommutative geometry and physics', Beijing, Nov. 7-10, 2005. To appear in the proceeding

Spin Mixing in Spinor Fermi Gases

We study a spinor fermionic system under the effect of spin-exchange interaction. We focus on the interplay between the spin-exchange interaction and the effective quadratic Zeeman shift. We examine the static and the dynamic properties of both two- and many-body system. We find that the spin-exchange interaction induces coherent Rabi oscillation in the two-body system, but the oscillation is quickly damped when the system is extended to the many-body case

Mean-variance portfolio selection under Volterra Heston model

Motivated by empirical evidence for rough volatility models, this paper investigates continuous-time mean-variance (MV) portfolio selection under the Volterra Heston model. Due to the non-Markovian and non-semimartingale nature of the model, classic stochastic optimal control frameworks are not directly applicable to the associated optimization problem. By constructing an auxiliary stochastic process, we obtain the optimal investment strategy, which depends on the solution to a Riccati-Volterra equation. The MV efficient frontier is shown to maintain a quadratic curve. Numerical studies show that both roughness and volatility of volatility materially affect the optimal strategy.Comment: Final version, 22 pages, 5 figures, to appear in Applied Mathematics & Optimizatio

A Mixed-Binary Convex Quadratic Reformulation for Box-Constrained Nonconvex Quadratic Integer Program

In this paper, we propose a mixed-binary convex quadratic programming reformulation for the box-constrained nonconvex quadratic integer program and then implement IBM ILOG CPLEX 12.6 to solve the new model. Computational results demonstrate that our approach clearly outperform the very recent state-of-the-art solvers.Comment: 8page

Conformal Triality of de Sitter, Minkowski and Anti-de Sitter Spaces

We describe how conformal Minkowski, dS- and AdS-spaces can be united into a single submanifold [N] of RP^5. It is the set of generators of the null cone in M^{2,4}. Conformal transformations on the Mink-, dS- and AdS-spaces are induced by O(2,4) linear transformations on M^{2,4}. We also describe how Weyl transformations and conformal transformations can be resulted in on [N]. In such a picture we give a description of how the conformal Mink-, dS- and AdS-spaces as well as [N] are mapped from one to another by conformal maps. This implies that a CFT in one space can be translated into a CFT in another. As a consequence, the AdS/CFT-correspondence should be extended.Comment: Talk on the XXIII International Conference of Differential Geometric Methods in Theoretical Physics, Chern Institute of Mathematics (former Nankai Institute of Mathematics), August 20--26, 2005. To appear in the proceedings, published by World Scientific. Plain LaTeX, 10 pages, no figure

A comparison theorem for the law of large numbers in Banach spaces

Let $(\mathbf{B}, \|\cdot\|)$ be a real separable Banach space. Let $\{X, X_{n}; n \geq 1\}$ be a sequence of i.i.d. {\bf B}-valued random variables and set $S_{n} = \sum_{i=1}^{n}X_{i},~n \geq 1$. Let $\{a_{n}; n \geq 1\}$ and $\{b_{n}; n \geq 1\}$ be increasing sequences of positive real numbers such that $\lim_{n \rightarrow \infty} a_{n} = \infty$ and $\left\{b_{n}/a_{n};~ n \geq 1 \right\}$ is a nondecreasing sequence. In this paper, we provide a comparison theorem for the law of large numbers for i.i.d. {\bf B}-valued random variables. That is, we show that $\displaystyle \frac{S_{n}- n \mathbb{E}\left(XI\{\|X\| \leq b_{n} \} \right)}{b_{n}} \rightarrow 0$ almost surely (resp. in probability) for every {\bf B}-valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > b_{n}) < \infty$ (resp. $\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > b_{n}) = 0$) if $S_{n}/a_{n} \rightarrow 0$ almost surely (resp. in probability) for every symmetric {\bf B}-valued random variable $X$ with $\sum_{n=1}^{\infty} \mathbb{P}(\|X\| > a_{n}) < \infty$ (resp. $\lim_{n \rightarrow \infty}n\mathbb{P}(\|X\| > a_{n}) = 0$). To establish this comparison theorem for the law of large numbers, we invoke two tools: 1) a comparison theorem for sums of independent {\bf B}-valued random variables and, 2) a symmetrization procedure for the law of large numbers for sums of independent {\bf B}-valued random variables. A few consequences of our main results are provided.Comment: 19 page

Merton's portfolio problem under Volterra Heston model

This paper investigates Merton's portfolio problem in a rough stochastic environment described by Volterra Heston model. The model has a non-Markovian and non-semimartingale structure. By considering an auxiliary random process, we solve the portfolio optimization problem with the martingale optimality principle. Optimal strategies for power and exponential utilities are derived in semi-closed form solutions depending on the respective Riccati-Volterra equations. We numerically examine the relationship between investment demand and volatility roughness.Comment: 14 pages, 3 figures, exponential utility adde

Intrinsic Vertex Regularization and Renormalization in ϕ4\phi^4 Field Theory

Based upon the intrinsic relation between the divergent lower point functions and the convergent higher point ones in the renormalizable quantum field theories, we propose a new method for regularization and renormalization in QFT. As an example, we renormalize the $\phi^{4}$ theory at the one loop order by means of this method.Comment: 7p, 4figures not include

A new single-dynamical-scalar-field model of dark energy

A new single-dynamical-scalar-field model of dark energy is proposed, in which either higher derivative terms nor structures of extra dimension are needed. With the help of a fixed background vector field, the parameter for the effective equation of state of dark energy may cross $w=-1$ in the evolution of the universe. After suitable choice of the potential, the crossing $w=-1$ and transition from decelerating to accelerating occur at $z\approx 0.2$ and $z\approx 1.7$, respectively.Comment: 7 pages, 4 figure

Principle of Relativity, Dual Poincar\'e Group and Relativistic Quadruple

Based on the principle of relativity with two universal constants (c, l) and in the inertial motion group IM(1,3)\sim PGL(5,R), with Lorentz isotropy, in addition to Poincar\'e group of Einstein's SR the dual Poincar\'e group preserves the origin lightcone and its space/time-like region R_\pm appeared at common origin of intersected Minkowski/dS/AdS space. The dual Poincare kinematics is on a pair of degenerate Einstein manifolds with \Lambda_\pm=\pm3l^{-2} for R_\pm, respectively. Thus, there is a Poincar\'e double and the dS double for dS/AdS SR. Further, with other four doubles they form a relativistic quadruple for three kinds of SR on M/D_\pm, respectively. The dS SR with the dS-dual Poincare double provides new kinematics for cosmic scale physics.Comment: 16 pages. Extended versio