Journal of the
Korean Mathematical Society
JKMS
ISSN(Print) 0304-9914
ISSN(Online) 2234-3008
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2023-09-01
Almost universal sums of triangular numbers with one exception
Jangwon JuAbstract : 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.
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2023-01-01
Minimal polynomial dynamics on the $p$-adic integers
Sangtae JeongAbstract : 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.
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2023-07-01
Complex reflection groups and K3 surfaces II. The groups ${\boldsymbol{G_{29}}}$, ${\boldsymbol{G_{30}}}$ and ${\boldsymbol{G_{31}}}$
Cédric Bonnafé, Alessandra SartiAbstract : 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.
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2024-09-01
Price Estimation via Bayesian Filtering and Optimal Bid--Ask Prices for Market Makers
Hyungbin Park, Junsu ParkAbstract : 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.
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2023-09-01
Li-Yau gradient estimates on closed manifolds under Bakry-\'Emery Ricci curvature conditions
Xing Yu Song, Ling WuAbstract : In this paper, motivated by the work of Q.~S.~Zhang in [25], we derive optimal Li-Yau gradient bounds for positive solutions of the $f$-heat equation on closed manifolds with Bakry-\'Emery Ricci curvature bounded below.
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2023-01-01
Minimal surface system in Euclidean four-space
Hojoo LeeAbstract : We construct generalized Cauchy-Riemann equations of the first order for a pair of two $\mathbb{R}$-valued functions to deform a minimal graph in ${\mathbb{R}}^{3}$ to the one parameter family of the two dimensional minimal graphs in ${\mathbb{R}}^{4}$. We construct the two parameter family of minimal graphs in ${\mathbb{R}}^{4}$, which include catenoids, helicoids, planes in ${\mathbb{R}}^{3}$, and complex logarithmic graphs in ${\mathbb{C}}^{2}$. We present higher codimensional generalizations of Scherk's periodic minimal surfaces.
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2023-01-01
Simple zeros of $L$-functions and the Weyl-type subconvexity
Peter Jaehyun Cho
, Gyeongwon OhAbstract : Let $f$ be a self-dual primitive Maass or modular forms for level $4$. For such a form $f$, we define \begin{align*} N_f^s(T)\!:=\!|\{\rho \in \mathbb{C} : |\Im(\rho)| \leq T, \text{ $\rho$ is a non-trivial simple zero of $L_f(s)$} \}|. \end{align*} We establish an omega result for $N_f^s(T)$, which is $N_f^s(T)=\Omega \big( T^{\frac{1}{6}-\epsilon} \big)$ for any $\epsilon>0$. For this purpose, we need to establish the Weyl-type subconvexity for $L$-functions attached to primitive Maass forms by following a recent work of Aggarwal, Holowinsky, Lin, and Qi.
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2023-09-01
Forbidden theta graph, bounded spectral radius and size of non-bipartite graphs
Shuchao Li, Wanting Sun, Wei WeiAbstract : Zhai and Lin recently proved that if $G$ is an $n$-vertex connected $\theta(1, 2, r+1)$-free graph, then for odd $r$ and $n \geqslant 10r$, or for even $r$ and $n \geqslant 7r$, one has $\rho(G) \le \sqrt{\lfloor\frac{n^2}{4}\rfloor}$, and equality holds if and only if $G$ is $K_{\lceil\frac{n}{2}\rceil, \lfloor\frac{n}{2}\rfloor}$. In this paper, for large enough $n$, we prove a sharp upper bound for the spectral radius in an $n$-vertex $H$-free non-bipartite graph, where $H$ is $\theta(1, 2, 3)$ or $\theta(1, 2, 4)$, and we characterize all the extremal graphs. Furthermore, for $n \geqslant 137$, we determine the maximum number of edges in an $n$-vertex $\theta(1, 2, 4)$-free non-bipartite graph and characterize the unique extremal graph.
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2023-07-01
Generalized hexagons embedded in metasymplectic spaces
Sebastian Petit, Hendrik Van MaldeghemAbstract : We consider thick generalized hexagons fully embedded in metasymplectic spaces, and we show that such an embedding either happens in a point residue (giving rise to a full embedding inside a dual polar space of rank 3), or happens inside a symplecton (giving rise to a full embedding in a polar space of rank 3), or is isometric (that is, point pairs of the hexagon have the same mutual position whether viewed in the hexagon or in the metasymplectic space--these mutual positions are \emph{equality, collinearity, being special, opposition}). In the isometric case, we show that the hexagon is always a Moufang hexagon, its little projective group is induced by the collineation group of the metasymplectic space, and the metasymplectic space itself admits central collineations (hence, in symbols, it is of type $\mathsf{F_{4,1}}$). We allow non-thick metasymplectic spaces without non-thick lines and obtain a full classification of the isometric embeddings in this case.
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2023-09-01
Topologically stable points and uniform limits
Namjip Koo, Hyunhee LeeAbstract : In this paper we study a pointwise version of Walters topological stability in the class of homeomorphisms on a compact metric space. We also show that if a sequence of homeomorphisms on a compact metric space is uniformly expansive with the uniform shadowing property, then the limit is expansive with the shadowing property and so topologically stable. Furthermore, we give examples to illustrate our results.
Most Read
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Price Estimation via Bayesian Filtering and Optimal Bid--Ask Prices for Market Makers
Hyungbin Park, Junsu Park
J. Korean Math. Soc. 2024; 61(5): 875-898
https://doi.org/10.4134/JKMS.j230053 -
Complex reflection groups and K3 surfaces II. The groups ${\boldsymbol{G_{29}}}$, ${\boldsymbol{G_{30}}}$ and ${\boldsymbol{G_{31}}}$
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014 -
Almost universal sums of triangular numbers with one exception
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231 -
Minimal polynomial dynamics on the $p$-adic integers
Sangtae Jeong
J. Korean Math. Soc. 2023; 60(1): 1-32
https://doi.org/10.4134/JKMS.j210494
Most Downloaded
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Almost universal sums of triangular numbers with one exception
Jangwon Ju
J. Korean Math. Soc. 2023; 60(5): 931-957
https://doi.org/10.4134/JKMS.j220231 -
Minimal polynomial dynamics on the $p$-adic integers
Sangtae Jeong
J. Korean Math. Soc. 2023; 60(1): 1-32
https://doi.org/10.4134/JKMS.j210494 -
Complex reflection groups and K3 surfaces II. The groups ${\boldsymbol{G_{29}}}$, ${\boldsymbol{G_{30}}}$ and ${\boldsymbol{G_{31}}}$
Cédric Bonnafé, Alessandra Sarti
J. Korean Math. Soc. 2023; 60(4): 695-743
https://doi.org/10.4134/JKMS.j220014 -
Price Estimation via Bayesian Filtering and Optimal Bid--Ask Prices for Market Makers
Hyungbin Park, Junsu Park
J. Korean Math. Soc. 2024; 61(5): 875-898
https://doi.org/10.4134/JKMS.j230053
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