Abstract : We classify toric log del Pezzo surfaces of Picard number one by introducing the notion, cascades. As an application, we show that if such a surface admits a K"ahler--Einstein metric, then it should admit a special cascade and it satisfies the equality of the orbifold Bogomolov--Miyaoka--Yau inequality, i.e., $K^2 = 3e_{orb}.$ Moreover, we provide an algorithm to compute a toric log del Pezzo surfaces of Picard number one for a given input of singularity types.
Abstract : Let $\{X,X_n;n\geq1\}$ be a $\beta$-mixing sequence of identical non-negative random variables with super-heavy tailed distributions and $S_n=X_1+X_2+\cdots+X_n$. For $\varepsilon>0$, $b>1$ and appropriate values of $x$, we obtain the logarithmic asymptotics behaviors for the tail probabilities $\mathbb{P}(S_n>e^{\varepsilon n^{x}})$ and $\mathbb{P}(S_n>e^{\varepsilon b^{n}})$. Moreover, our results are applied to the log-Pareto distribution and the distribution for the super-Petersburg game.
Abstract : This study estimates the true price of an asset and finds the optimal bid/ask prices for market makers. We provide a novel state-space model based on the exponential Ornstein--Uhlenbeck volatility and the Heston models with Gaussian noise, where the traded price and volume are available, but the true price is not observable. An objective of this study is to use Bayesian filtering to estimate the posterior distribution of the true price, given the traded price and volume. Because the posterior density is intractable, we employ the guided particle filtering algorithm, with which adaptive rejection metropolis sampling is used to generate samples from the density function of an unknown distribution. Given a simulated sample path, the posterior expectation of the true price outperforms the traded price in estimating the true price in terms of both the mean absolute error and root-mean-square error metrics. Another objective is to determine the optimal bid/ask prices for a market maker. The profit-and-loss of the market maker is the difference between the true price and its bid/ask prices multiplied by the traded volume or bid/ask size of the market maker. The market maker maximizes the expected utility of the PnL under the posterior distribution. We numerically calculate the optimal bid/ask prices using the Monte Carlo method, finding that its spread widens as the market maker becomes more risk-averse, and the bid/ask size and the level of uncertainty increase.
Abstract : Let $f(z)$ be a primitive holomorphic cusp form and $g(z)$ be a Maass cusp form. In this paper, we give quantitative results for the sign changes of coefficients of triple product $L$-functions $L(f\times f\times f, s)$ and $L(f\times f\times g, s)$.
Abstract : We consider a vector bundle map $F\colon E_{1}\longrightarrow E_{2}$ between Lie algebroids $E_{1}$ and $E_{2}$ over arbitrary bases $M_{1}$ and $M_{2}$. We associate to it different notions of curvature which we call A-curvature, Q-curvature, P-curvature, and S-curvature using the different characterizations of Lie algebroid structure, namely Lie algebroid, Q-manifold, Poisson and Schouten structures. We will see that these curvatures generalize the ordinary notion of curvature defined for a vector bundle, and we will prove that these curvatures are equivalent, in the sense that $F$ is a morphism of Lie algebroids if and only if one (and hence all) of these curvatures is null. In particular we get as a corollary that $F$ is a morphism of Lie algebroids if and only if the corresponding map is a morphism of Poisson manifolds (resp. Schouten supermanifolds).
Abstract : Let $R=\bigoplus_{\alpha \in \Gamma} R_{\alpha}$ be a commutative ring graded by an arbitrary torsionless monoid $\Gamma$. A homogeneous prime ideal $P$ of $R$ is said to be strongly homogeneous prime if $aP$ and $bR$ are comparable for any homogeneous elements $a,b $ of $R$. We will say that $R$ is a graded pseudo-valuation ring (gr-PVR for short) if every homogeneous prime ideal of $R$ is strongly homogeneous prime. In this paper, we introduce and study the graded version of the pseudo-valuation rings which is a generalization of the gr-pseudo-valuation domains in the context of arbitrary $\Gamma$-graded rings (with zero-divisors). We then study the possible transfer of this property to the graded trivial ring extension and the graded amalgamation. Our goal is to provide examples of new classes of $\Gamma$-graded rings that satisfy the above mentioned property.
Abstract : In this paper, we utilize Nevanlinna theory to study the existence and forms of solutions for quadratic trinomial complex partial differential-difference equations of the form $ aF^2+2\omega FG+bG^2=\exp(g) $, where $ ab\neq 0, \omega\in\mathbb{C} $ with $ \omega^2\neq 0, ab $ and $g$ is a polynomial in $\mathbb{C}^n$. In order to achieve a comprehensive and thorough analysis, we study the characteristics of solutions in two specific cases: one when $ \omega^2\neq 0, ab $ and the other when $ \omega=0 $. Because polynomials in several complex variables may exhibit periodic behavior, a property that differs from polynomials in single complex variables, our study of finding solutions of equations in $\mathbb{C}^n$ is significant. The main results of the paper improved several known results in $\mathbb{C}^n$ for $n\geq 2$. Additionally, the corollaries generalize results of Xu \emph{et al.} [Rocky Mountain J. Math. \textbf{52}(6) (2022), 2169--2187] for trinomial equations with arbitrary coefficients in $\mathbb{C}^n$. Finally, we provide examples that endorse the validity of the conclusions drawn from the main results and their related remarks.
Abstract : We use Borcherds products to give a new construction of the weight~$3$paramodular nonlift eigenform~$f_N$ for levels~$N=61,73,79$.We classify the congruences of~$f_N$ to Gritsenko lifts.We provide techniques that compute eigenvalues to supportfuture modularity applications.Our method does not compute Hecke eigenvalues from Fouriercoefficients but instead uses elliptic modular forms, specificallythe restrictions of Gritsenko lifts and their images under the slashoperator to modular curves.
Abstract : The Cameron--Martin translation theorem describes how Wiener measure changes under translation by elements of the Cameron--Martin space in an abstract Wiener space (AWS). Translation theorems for the analytic Feynman integrals also have been established in the literature. In this article, we derive a more general translation theorem for the analytic Feynman integral associated with bounded linear operators (B.L.OP.) on AWSs. To do this, we use a certain behavior which exists between the analytic Fourier--Feynman transform (FFT) and the convolution product (CP) of functionals on AWS. As an interesting application, we apply this translation theorem to evaluate the analytic Feynman integral of the functional \[ F(x)=\exp\bigg(-iq \int_0^Tx(t)y(t)d t \bigg),\quad y\in C_0[0,T], \, q\in\mathbb R\setminus\{0\} \] defined on the classical Wiener space $C_0[0,T]$.
Hyungbin Park, Junsu Park
J. Korean Math. Soc. 2024; 61(5): 875-898
https://doi.org/10.4134/JKMS.j230053
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231
Sangtae Jeong
J. Korean Math. Soc. 2023; 60(1): 1-32
https://doi.org/10.4134/JKMS.j210494
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231
Sangtae Jeong
J. Korean Math. Soc. 2023; 60(1): 1-32
https://doi.org/10.4134/JKMS.j210494
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Xing Yu Song, Ling Wu
J. Korean Math. Soc. 2023; 60(5): 1023-1041
https://doi.org/10.4134/JKMS.j220589
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