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 : For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on each of these groups. Our study is a continuation and extension of the previous studies done in [3] for Riemannian metrics and in [1] for Lorentzian metrics on unimodular Lie groups.
Abstract : In this paper, we study backward stochastic differential equations (BSDEs shortly) with jumps that have Lipschitz generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. Under just integrability on the data we show that such equations admit a unique solution which belongs to class $\mathbb{D}$.
Abstract : In this paper, we give a general method to compute the linear independence measure of $1, \log(1-1/r),\log(1+1/s)$ for infinitely many integers $r$ and $s$. We also give improvements for the special cases when $r=s$, for example, $\nu(1, \log 3/4, \log 5/4) \leq 9.197$.
Abstract : In this paper, we consider the existence of time periodic solutions for the compressible magneto-micropolar fluids in the whole space $\mathbb{R}^3$. In particular, we first solve the problem in a sequence of bounded domains by the topological degree theory. Then we obtain the existence of time periodic solutions in $\mathbb{R}^3$ by a limiting process.
Abstract : Let $M$ be a real hypersurface in the complex hyperbolic quadric~${Q^{m}}^{*}$, $m \geq 3$. The Riemannian curvature tensor field~$R$ of~$M$ allows us to define a symmetric Jacobi operator with respect to the Reeb vector field~$\xi$, which is called the structure Jacobi operator~$R_{\xi} = R(\, \cdot \, , \xi) \xi \in \text{End}(TM)$. On the other hand, in~\cite{Semm03}, Semmelmann showed that the cyclic parallelism is equivalent to the Killing property regarding any symmetric tensor. Motivated by his result above, in this paper we consider the cyclic parallelism of the structure Jacobi operator~$R_{\xi}$ for a real hypersurface~$M$ in the complex hyperbolic quadric~${Q^{m}}^{*}$. Furthermore, we give a complete classification of Hopf real hypersurfaces in ${Q^{m}}^{*}$ with such a property.
Abstract : There have been numerous studies on the characteristics of the solutions of ordinary differential equations for optimization methods, including gradient descent methods and alternating direction methods of multipliers. To investigate computer simulation of ODE solutions, we need to trace pseudo-orbits by real orbits and it is called shadowing property in dynamics. In this paper, we demonstrate that the flow induced by the alternating direction methods of multipliers (ADMM) for a $C^2$ strongly convex objective function has the eventual shadowing property. For the converse, we partially answer that convexity with the eventual shadowing property guarantees a unique minimizer. In contrast, we show that the flow generated by a second-order ODE, which is related to the accelerated version of ADMM, does not have the eventual shadowing property.
Abstract : Feller introduced an unfair-fair-game in his famous book \cite{Feller-1968}. In this game, at each trial, player will win $2^k$ yuan with probability $p_k=1/2^kk(k+1)$, $k\in \mathbb{N}$, and zero yuan with probability $p_0=1-\sum_{k=1}^{\infty}p_k$. Because the expected gain is 1, player must pay one yuan as the entrance fee for each trial. Although this game seemed ``fair", Feller \cite{Feller-1945} proved that when the total trial number $n$ is large enough, player will loss $n$ yuan with its probability approximate 1. So it's an ``unfair" game. In this paper, we study in depth its convergence in probability, almost sure convergence and convergence in distribution. Furthermore, we try to take $2^k=m$ to reduce the values of random variables and their corresponding probabilities at the same time, thus a new probability model is introduced, which is called as the related model of Feller's unfair-fair-game. We find out that this new model follows a long-tailed distribution. We obtain its weak law of large numbers, strong law of large numbers and central limit theorem. These results show that their probability limit behaviours of these two models are quite different.
Abstract : Given a dimension function $\omega$, we introduce the notion of an $\omega$-vector weighted digraph and an $\omega$-equivalence between them. Then we establish a bijection between the weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of simplices $\Delta^{\omega(1)}\times\cdots \times \Delta^{\omega(m)}$ and the set of $\omega$-equivalence classes of $\omega$-vector weighted digraphs with $m$-labeled vertices, where $n$ is the sum of the dimensions of the simplicies. Using this bijection, we obtain a formula for the number of weakly $(\mathbb{Z}/2)^n$-equivariant homeomorphism classes of small covers over a product of three simplices.
Abstract : In this paper, we investigate a few strategies to construct Ulrich bundles of small ranks over smooth fourfolds in $\mathbb{P}^5$, with a focus on the case of special quartic fourfolds. First, we give a necessary condition for Ulrich bundles over a very general quartic fourfold in terms of the rank and the Chern classes. Second, we give an equivalent condition for Pfaffian fourfolds in every degree in terms of arithmetically Gorenstein surfaces therein. Finally, we design a computer-based experiment to look for Ulrich bundles of small rank over special quartic fourfolds via deformation theory. This experiment gives a construction of numerically Ulrich sheaf of rank $4$ over a random quartic fourfold containing a del Pezzo surface of degree $5$.
Daewoong Cheong, Jinbeom Kim
J. Korean Math. Soc. 2023; 60(4): 799-822
https://doi.org/10.4134/JKMS.j220333
Aslı Güçlükan İlhan, Sabri Kaan Gürbüzer
J. Korean Math. Soc. 2022; 59(5): 963-986
https://doi.org/10.4134/JKMS.j220104
In-Jee Jeong, Sun-Chul Kim
J. Korean Math. Soc. 2023; 60(2): 465-477
https://doi.org/10.4134/JKMS.j220317
Alexandru Chirvasitu, S. Paul Smith
J. Korean Math. Soc. 2023; 60(4): 745-777
https://doi.org/10.4134/JKMS.j220200
Chong-Kyu Han, Taejung Kim
J. Korean Math. Soc. 2022; 59(6): 1171-1184
https://doi.org/10.4134/JKMS.j220043
Rasul Mohammadi, Ahmad Moussavi, masoome zahiri
J. Korean Math. Soc. 2023; 60(6): 1233-1254
https://doi.org/10.4134/JKMS.j220625
Helena Jonsson, Volodymyr Mazorchuk, Elin Persson Westin, Shraddha Srivastava, Mateusz Stroinski, Xiaoyu Zhu
J. Korean Math. Soc. 2023; 60(6): 1255-1302
https://doi.org/10.4134/JKMS.j230020
Tatyana Barron, Manimugdha Saikia
J. Korean Math. Soc. 2024; 61(1): 91-107
https://doi.org/10.4134/JKMS.j230152
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