Abstract:
This paper proposes the Lagrange multiplier test for the null hypothesis thatthe bivariate time series has only a single common stochastic volatility factor and noidiosyncratic volatility factor. The test statistic is derived by representing the model in alinear state-space form under the assumption that the log of squared measurement error isnormally distributed. The empirical size and power of the test are examined in Monte Carloexperiments. We apply the test to the Asian stock market indices.

Abstract:
We point out that the internal spin symmetry of the order parameter manifests itself at the core of a fractional vortex in real space without spin-orbit coupling. Such symmetry breaking arises from a topological constraint and the commensurability between spin symmetries of the order parameters inside and outside the core. Our prediction can be applied to probe the cyclic order parameter in a rotating spin-2 $^{87}$Rb condensate as a non-circular vortex core in a biaxial nematic state.

Abstract:
We develop Bogoliubov theory of spin-1 and spin-2 Bose-Einstein condensates (BECs) in the presence of a quadratic Zeeman effect, and derive the Lee-Huang-Yang (LHY) corrections to the ground-state energy, sound velocity, and quantum depletion. We investigate all the phases of spin-1 and spin-2 BECs that can be realized experimentally. We also examine the stability of each phase against quantum fluctuations and the quadratic Zeeman effect. Furthermore, we discuss a relationship between the number of symmetry generators that are spontaneously broken and that of Nambu-Goldstone (NG) modes. It is found that in the spin-2 nematic phase there are special Bogoliubov modes that have gapless linear dispersion relations but do not belong to the NG modes.

Abstract:
One of the central issues in the hidden subgroup problem is to bound the sample complexity, i.e., the number of identical samples of coset states sufficient and necessary to solve the problem. In this paper, we present general bounds for the sample complexity of the identification and decision versions of the hidden subgroup problem. As a consequence of the bounds, we show that the sample complexity for both of the decision and identification versions is $\Theta(\log|\HH|/\log p)$ for a candidate set $\HH$ of hidden subgroups in the case that the candidate subgroups have the same prime order $p$, which implies that the decision version is at least as hard as the identification version in this case. In particular, it does so for the important instances such as the dihedral and the symmetric hidden subgroup problems. Moreover, the upper bound of the identification is attained by the pretty good measurement. This shows that the pretty good measurement can identify any hidden subgroup of an arbitrary group with at most $O(\log|\HH|)$ samples.

Abstract:
Topological objects can influence each other if the underlying homotopy groups are non-Abelian. Under such circumstances, the topological charge of each individual object is no longer a conserved quantity and can be transformed to each other. Yet, we can identify the conservation law by considering the back-action of topological influence. We develop a general theory of topological influence and back-action based on the commutators of the underlying homotopy groups. We illustrate the case of the topological influence of a half-quantum vortex on the sign change of a point defect and point out that the topological back-action from the point defect is such twisting of the vortex that the total twist of the vortex line carries the change in the point-defect charge to conserve the total charge. We use this theory to classify charge transfers in condensed matter systems and show that a non-Abelian charge transfer can be realized in a spin-2 Bose-Einstein condensate.

Abstract:
Grand unified theories of fundamental forces predict that magnetic monopoles are inevitable in the Universe because the second homotopy group of the order parameter manifold is $\mathbb{Z}$. We point out that monopoles can annihilate in pairs due to an influence of Alice strings. As a consequence, a monopole charge is charactarized by $\mathbb{Z}_2$ rather than $\mathbb{Z}$ if the Universe can accommodate Alice strings, which is the case of certain grand unified theories.

Abstract:
Topological excitations are usually classified by the $n$th homotopy group $\pi_n$. However, for topological excitations that coexist with vortices, there are case in which an element of $\pi_n$ cannot properly describe the charge of a topological excitation due to the influence of the vortices. This is because an element of $\pi_n$ corresponding to the charge of a topological excitation may change when the topological excitation circumnavigates a vortex. This phenomenon is referred to as the action of $\pi_1$ on $\pi_n$. In this paper, we show that topological excitations coexisting with vortices are classified by the Abe homotopy group $\kappa_n$. The $n$th Abe homotopy group $\kappa_n$ is defined as a semi-direct product of $\pi_1$ and $\pi_n$. In this framework, the action of $\pi_1$ on $\pi_n$ is understood as originating from noncommutativity between $\pi_1$ and $\pi_n$. We show that a physical charge of a topological excitation can be described in terms of the conjugacy class of the Abe homotopy group. Moreover, the Abe homotopy group naturally describes vortex-pair creation and annihilation processes, which also influence topological excitations. We calculate the influence of vortices on topological excitations for the case in which the order parameter manifold is $S^n/K$, where $S^n$ is an $n$-dimensional sphere and $K$ is a discrete subgroup of $SO(n+1)$. We show that the influence of vortices on a topological excitation exists only if $n$ is even and $K$ includes a nontrivial element of $O(n)/SO(n)$.

Abstract:
We investigate the collision dynamics of two non-Abelian vortices and find that, unlike Abelian vortices, they neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. Our predictions are verified using the model of the cyclic phase of a spin-2 spinor Bose-Einstein condensate.

Abstract:
We investigate the collision dynamics of two non-Abelian vortices and find that, unlike Abelian vortices, they neither reconnect themselves nor pass through each other, but create a rung between them in a topologically stable manner. Our predictions are verified using the model of the cyclic phase of a spin-2 spinor Bose-Einstein condensate.

Abstract:
We classify vortex-core structures according to the topology of the order parameter space. By developing a method to characterize how the order parameter changes inside the vortex core. We apply this method to the spin-1 Bose-Einstein condensates and show that the vortex-core structures are classified by winding numbers that are locally defined in the core region. We also show that a vortex-core structure with a nontrivial winding number can be stabilized under a negative quadratic Zeeman effect.