Abstract : We study some K3 surfaces obtained as minimal resolutions of quotients of subgroups of special reflection groups. Some of these were already studied in a previous paper by W.~Barth and the second author. We give here an easy proof that these are K3 surfaces, give equations in weighted projective space and describe their geometry.
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 : For an arbitrary integer $x$, an integer of the form $T(x)\!=\!\frac{x^2+x}{2}$ is called a triangular number. Let $\alpha_1,\dots,\alpha_k$ be positive integers. A sum $\Delta_{\alpha_1,\dots,\alpha_k}(x_1,\dots,x_k)=\alpha_1 T(x_1)+\cdots+\alpha_k T(x_k)$ of triangular numbers is said to be {\it almost universal with one exception} if the Diophantine equation $\Delta_{\alpha_1,\dots,\alpha_k}(x_1,\dots,x_k)=n$ has an integer solution $(x_1,\dots,x_k)\in\mathbb{Z}^k$ for any nonnegative integer $n$ except a single one. In this article, we classify all almost universal sums of triangular numbers with one exception. Furthermore, we provide an effective criterion on almost universality with one exception of an arbitrary sum of triangular numbers, which is a generalization of ``15-theorem" of Conway, Miller, and Schneeberger.
Abstract : In this paper, we present a method of characterizing minimal polynomials on the ring ${\mathbf Z}_p$ of $p$-adic integers in terms of their coefficients for an arbitrary prime $p$. We first revisit and provide alternative proofs of the known minimality criteria given by Larin [11] for $p=2$ and Durand and Paccaut [6] for $p=3$, and then we show that for any prime $p\geq 5,$ the proposed method enables us to classify all possible minimal polynomials on ${\mathbf Z}_p$ in terms of their coefficients, provided that two prescribed prerequisites for minimality are satisfied.
Abstract : For two dimensional lattices, a Gaussian basis achieves all two successive minima. For dimension larger than two, constructing a pairwise Gaussian basis is useful to compute short vectors of the lattice. For three dimensional lattices, Semaev showed that one can convert a pairwise Gaussian basis to a basis achieving all three successive minima by one simple reduction. A pairwise Gaussian basis can be obtained from a given basis by executing Gauss algorithm for each pair of basis vectors repeatedly until it returns a pairwise Gaussian basis. In this article, we prove a necessary and sufficient condition for a pairwise Gaussian basis to achieve the first $k$ successive minima for three dimensional lattices for each $k\in\{1,2,3\}$ by modifying Semaev's condition. Our condition directly checks whether a pairwise Gaussian basis contains the first $k$ shortest independent vectors for three dimensional lattices. LLL is the most basic lattice basis reduction algorithm and we study how to use LLL to compute a pairwise Gaussian basis. For $\delta\ge 0.9$, we prove that LLL($\delta$) with an additional simple reduction turns any basis for a three dimensional lattice into a pairwise SV-reduced basis. By using this, we convert an LLL reduced basis to a pairwise Gaussian basis in a few simple reductions. Our result suggests that the LLL algorithm is quite effective to compute a basis with all three successive minima for three dimensional lattices.
Abstract : This paper considers a parabolic-hyperbolic-hyperbolic type chemotaxis system in $\mathbb{R}^{d}$, $d\ge3$, describing tumor-induced angiogenesis. The global existence result and temporal decay estimate for a unique mild solution are established under the assumption that some Sobolev norms of initial data are sufficiently small.
Abstract : Let $\alpha\in(0,\infty)$, $p\in(0,\infty)$ and $q(\cdot): {{\mathbb R}}^{n}\rightarrow[1,\infty)$ satisfy the globally log-H\"{o}lder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator $M$ and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley $g$-function and the Littlewood-Paley $g^{\ast}_{\lambda}$-function.
Abstract : Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$. We give some descriptions of the $\mathfrak{a}$-depth of $\mathfrak{a}$-relative Cohen-Macaulay modules by cohomological dimensions, and study how relative Cohen-Macaul-\\ayness behaves under flat extensions. As applications, the perseverance of relative Cohen-Macaulayness in a polynomial ring, formal power series ring and completion are given.
Abstract : In this paper, using the theory of majorization, we discuss the Schur $m$ power convexity for $L$-conjugate means of $n$ variables and the Schur convexity for weighted $L$-conjugate means of $n$ variables. As applications, we get several inequalities of general mean satisfying Schur convexity, and a few comparative inequalities about $n$ variables Gini mean are established.
Abstract : We study the Dirichlet problem for the degenerate nonlocal parabolic equation \[ u_{t}-a\left(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2}\right)\Delta u=C_b\left\Vert u\right\Vert _{L^2(\Omega)}^{\beta}\left\vert u \right\vert^{q\left(x,t\right)-2}u\log|u|+f \quad \text{in $Q_T$}, \] where $Q_{T}:=\Omega \times (0,T)$, $T>0$, $\Omega \subset \mathbb{R}^{N}$, $N\geq 2$, is a bounded domain with a sufficiently smooth boundary, $q(x,t)$ is a measurable function in $Q_{T}$ with values in an interval $[q^{-},q^{+}]\subset(1,\infty)$ and the diffusion coefficient $a(\cdot)$ is a continuous function defined on $\mathbb{R}_+$. It is assumed that $a(s)\to 0$ or $a(s)\to \infty$ as $s\to 0^+$, therefore the equation degenerates or becomes singular as $\|\nabla u(t)\|_{2}\to 0$. For both cases, we show that under appropriate conditions on $a$, $\beta$, $q$, $f$ the problem has a global in time strong solution which possesses the following global regularity property: $\Delta u\in L^2(Q_T)$ and $a(\left\Vert \nabla u\right\Vert _{L^2(\Omega)}^{2})\Delta u\in L^2(Q_T)$.
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014
Hyungbin Park, Junsu Park
J. Korean Math. Soc. 2024; 61(5): 875-898
https://doi.org/10.4134/JKMS.j230053
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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