Asymptotics of abelian group-partitions and associated Dirichlet series

We introduce a notion of a group-partition for a finite Abelian group, which is a generalized notion of the standard partition. To obtain asymptoticdistributions of group-partition, we study the Dirichlet series for group-partitions by employing the generating function of the plane partition.Comment: 9 page

Higher Selberg zeta functions for congruence subgroups

As a generalization of the results [KW3],we study the functional equation of the higher Selberg zeta function for congruence subgroups. To obtain the gamma factor of this function, we introduce a higher Dirichlet $L$-function. Then we determine the gamma factor explicitly in terms of the Barnes triple gamma function and the higher Dirichlet $L$-function.Comment: 30page

Symmetries in the third Painlev\'e equation arising from the modified Pohlmeyer-Lund-Regge hierarchy

We propose a modification of the AKNS hierarchy that includes the "modified" Pohlmeyer-Lund-Regge (mPLR) equation. Similarity reductions of this hierarchy give the second, third, and fourth Painlev\'e equations. Especially, we present a new Lax representation and a complete description of the symmetry of the third Painlev\'e equation through the similarity reduction. We also show the relation between the tau-function of the mPLR hierarchy and Painlev\'e equations.Comment: 23 page

Realized Volatility Analysis in A Spin Model of Financial Markets

We calculate the realized volatility in the spin model of financial markets and examine the returns standardized by the realized volatility. We find that moments of the standardized returns agree with the theoretical values of standard normal variables. This is the first evidence that the return dynamics of the spin financial market is consistent with the view of the mixture-of-distribution hypothesis that also holds in the real financial markets.Comment: 4 pages, 5 figure

Quest for potentials in the quintessence scenario

The time variation of the equation of state $w$ for quintessence scenario with a scalar field as dark energy is studied up to the third derivative ($d^3w/da^3$) with respect to the scale factor $a$, in order to predict the future observations and specify the scalar potential parameters with the observables. The third derivative of $w$ for general potential $V$ is derived and applied to several types of potentials. They are the inverse power-law ($V=M^{4+\alpha}/Q^{\alpha}$), the exponential ($V=M^4\exp{(\beta M/Q)}$), the cosine ($V=M^4(\cos (Q/f)+1)$) and the Gaussian types ($V=M^4\exp(-Q^2/\sigma^2)$), which are prototypical potentials for the freezing and thawing models. If the parameter number for a potential form is $n$, it is necessary to find at least for $n+2$ independent observations to identify the potential form and the evolution of the scalar field ($Q$ and $\dot{Q}$). Such observations would be the values of $\Omega_Q, w, dw/da. \cdots$, and $dw^n/da^n$. Since four of the above mentioned potentials have two parameters, it is necessary to calculate the third derivative of $w$ for them to estimate the predict values. If they are tested observationally, it will be understood whether the dark energy could be described by the scalar field with this potential. Numerical analysis for $d^3w/da^3$ are made under some specified parameters in the investigated potentials. It becomes possible to distinguish the freezing and thawing modes by the accurate observing $dw/da$ and $d^2w/da^2$ in some parameters.Comment: 6 pages, 2 figures. arXiv admin note: text overlap with arXiv:1503.0367

Rough volatility of Bitcoin

Recent studies have found that the log-volatility of asset returns exhibit roughness. This study investigates roughness or the anti-persistence of Bitcoin volatility. Using the multifractal detrended fluctuation analysis, we obtain the generalized Hurst exponent of the log-volatility increments and find that the generalized Hurst exponent is less than $1/2$, which indicates log-volatility increments that are rough. Furthermore, we find that the generalized Hurst exponent is not constant. This observation indicates that the log-volatility has multifractal property. Using shuffled time series of the log-volatility increments, we infer that the source of multifractality partly comes from the distributional property.Comment: 12 pages, 8 figure

Grand Projection State: A Single Microscopic State to Determine Free Energy

Recently, we clarify connection of spatial constraint and equilibrium macroscopic properties in disordered states of classical system under the fixed composition; namely few special microscopic states, independent of constituent elements, can describe macroscopic properties. In this study, we extend our developed approach to composition-unfixed system. Through this extension in binary system, we discover a single special microscopic state to determine not only composition but also Helmholtz free energy measured from unary system, which has not been described by a single state.Comment: 6 pages, 5 figure

On a structure of random open books and closed braids

A result of Malyutin shows that a random walk on the mapping class group gives rise to an element whose fractional Dehn twist coefficient is large or small enough. We show that this leads to several properties of random 3-manifolds and links. For example, random closed braids and open books are hyperbolic.Comment: 4 pages, no figure. Theorem 5 is adde

Statistical properties and multifractality of Bitcoin

Using 1-min returns of Bitcoin prices, we investigate statistical properties and multifractality of a Bitcoin time series. We find that the 1-min return distribution is fat-tailed, and kurtosis largely deviates from the Gaussian expectation. Although for large sampling periods, kurtosis is anticipated to approach the Gaussian expectation, we find that convergence to that is very slow. Skewness is found to be negative at time scales shorter than one day and becomes consistent with zero at time scales longer than about one week. We also investigate daily volatility-asymmetry by using GARCH, GJR, and RGARCH models, and find no evidence of it. On exploring multifractality using multifractal detrended fluctuation analysis, we find that the Bitcoin time series exhibits multifractality. The sources of multifractality are investigated, confirming that both temporal correlation and the fat-tailed distribution contribute to it. The influence of "Brexit" on June 23, 2016 to GBP--USD exchange rate and Bitcoin is examined in multifractal properties. We find that, while Brexit influenced the GBP--USD exchange rate, Bitcoin was robust to Brexit.Comment: 19 pages, 9 figure

On the center of a Coxeter group

In this paper, we show that the center of every Coxeter group is finite and isomorphic to $(\Z_2)^n$ for some $n\ge 0$. Moreover, for a Coxeter system $(W,S)$, we prove that $Z(W)=Z(W_{S\setminus\tilde{S}})$ and $Z(W_{\tilde{S}})=1$, where $Z(W)$ is the center of the Coxeter group $W$ and $\tilde{S}$ is the subset of $S$ such that the parabolic subgroup $W_{\tilde{S}}$ is the {\it essential parabolic subgroup} of $(W,S)$ (i.e.\ $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $(W,S)$). The finiteness of the center of a Coxeter group implies that a splitting theorem holds for Coxeter groups