作者机构:
[Liang Cheng; Liang Cheng Liang Cheng Liang Cheng] School of Mathematics and Statistics, Central China Normal University, Wuhan, Hebei, China;[Yongjia Zhang; Yongjia Zhang Yongjia Zhang Yongjia Zhang] School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China
通讯机构:
[Yongjia Zhang; Yongjia Zhang Yongjia Zhang Yongjia Zhang] S;School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China
关键词:
noncollapsing;Ricci flow;Type I ancient solution
摘要:
<jats:title>Abstract</jats:title><jats:p>In this paper, we study complete Type I ancient Ricci flows with positive sectional curvature. Our main results are as follows: in the complete and noncompact case, all such ancient solutions must be noncollapsed on all scales; in the closed case, if the dimension is even, then all such ancient solutions must be noncollapsed on all scales. This furthermore gives a complete classification for three‐dimensional noncompact Type I ancient solutions without assuming the noncollapsingcondition.</jats:p>
作者机构:
[Deng, Qingquan] Cent China Normal Univ, Sch Math & Stat, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.;[Fan, Dashan] Univ Wisconsin Milwaukee, Dept Math Sci, Milwaukee, WI 53201 USA.;[Zhao, Ruizhi] Cent China Normal Univ, Sch Math & Stat, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.
通讯机构:
[Deng, QQ ] C;Cent China Normal Univ, Sch Math & Stat, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.
摘要:
In this paper we study the Hp–Hq estimates of the solutions for a class of dispersive equations
$$\left\{ {\matrix{{i{\partial _t}u(t,x) = - P(|\nabla |)u(t,x),} \hfill & {(t,x) \in \mathbb{R} \times {\mathbb{R}^n},} \hfill \cr {u(0,x) = {u_0}(x),} \hfill & {x \in {\mathbb{R}^n},} \hfill \cr } } \right.$$
where P: ℝ+ → ℝ is smooth away from the origin and enjoy different scalings. As applications, we obtain the decay estimates for the solutions of higher order homogeneous and inhomogeneous Schrödinger equations.
摘要:
In coding theory, it is important to construct codes with optimal parameters. The codes meeting the Griesmer bound are called Griesmer codes. In this paper, we prove nonexistence of some Griesmer codes of dimension four over finite fields of cardinality eight and nine using projective geometry. Moreover, we improve 24 cases on code lengths for given minimal distance and dimension four. Two precise lower bounds of code lengths have also been determined.
作者机构:
[Zhang, Yiping; Zhang, YP] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Zhang, Yiping; Zhang, YP] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;[Zhang, Yiping; Zhang, YP] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.
通讯机构:
[Zhang, YP ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.
关键词:
Compressible Navier-Stokes equations;finite time blow-up behaviors;blow-up rate
摘要:
In this paper, we are interested in the finite time blow-up behaviors of strong solutions to the compressible Navier-Stokes equations with vacuum as far field density. Via the blow-up criterion obtained by [H. Wen and C. Zhu, Adv. Math., 2013], we are able to obtain a lower bound of the blow-up rate for
$$||\nabla u||_{L^2}$$
and
$$||P||_{L^2}$$
provided that
$$\gamma >3$$
and the viscous coefficients satisfy the additional condition
$$\frac{29}{3}\mu >\lambda $$
.
期刊:
Communications in Statistics - Theory and Methods,2024年:1-20 ISSN:0361-0926
通讯作者:
Jing, L
作者机构:
[Ouyang, Yang] Natl Inst Cultural Dev, Wuhan, Peoples R China.;[Jing, Luo; Zhimeng, Xu; Jing, L] South Cent Minzu Univ, Dept Math & Stat, Wuhan 430074, Peoples R China.;[Jing, Luo; Jing, L] Cent China Normal Univ, Sch Math & Stat, Minist Educ, Wuhan 430079, Peoples R China.;[Jing, Luo; Jing, L] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.;[Qiuping, Wang] Zhaoqing Univ, Sch Math & Stat, Zhaoqing, Peoples R China.
通讯机构:
[Jing, L ] S;South Cent Minzu Univ, Dept Math & Stat, Wuhan 430074, Peoples R China.;Cent China Normal Univ, Sch Math & Stat, Minist Educ, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.
摘要:
The Bradley-Terry model is a common model for analyzing paired comparison data. Under differential private mechanism, there is a lack of asymptotic properties for the parameter estimator of parameters in this model. In this article, we show that the moment estimators of the parameters based on the differential private degree sequence with Laplace noise is uniformly consistent and asymptotically normal. Simulations are provided to illustrate asymptotic results.
摘要:
Literature reviews reveal that there is a very close connection between experimental design and coding theory. Based on a code mapping transformation, this paper provides a new method to construct a class of mixed designs with two- and four -level. A general construction method is described and some theoretical results of obtained designs are given. Analytic connections are established between the generated and the initial designs in terms of aberration criteria and discrepancies. Sharp lower bounds of the wrap -around L-2 - and Lee discrepancies are obtained and used as the benchmarks to measure the uniformity of the generated designs. Examples are provided to illustrate the effectiveness of the construction and lend our results further support.
期刊:
ARCHIV DER MATHEMATIK,2024年:1-5 ISSN:0003-889X
通讯作者:
Huang, X
作者机构:
[Huang, Xin; Huang, X] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Huang, Xin; Huang, X] Southern Univ Sci & Technol, Shenzhen Int Ctr Math, Shenzhen 518055, Peoples R China.
通讯机构:
[Huang, X ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Southern Univ Sci & Technol, Shenzhen Int Ctr Math, Shenzhen 518055, Peoples R China.
摘要:
We extend a theorem of Kessar and Linckelmann concerning Morita equivalences and Galois compatible bijections between Brauer characters to virtual Morita equivalences. As a corollary, we obtain that the block version of Navarro’s refinement of Alperin’s weight conjecture holds for blocks with cyclic and Klein four defect groups, blocks of symmetric and alternating groups with abelian defect groups, and p-Blocks of
$$\textrm{SL}_2(q)$$
and
$$\textrm{GL}_2(q)$$
, where p|q.
摘要:
It is well known that the Hausdorff dimension and box dimension of a BedfordMcMullen carpet coincide if and only if the carpet has uniform horizontal fibers. Yang et al. [17] gave a complete Lipschitz classification of such carpets which are totally disconnected and possessing vacant rows. In this paper, we show that if we replace the uniform horizontal fibers condition by a locally uniform horizontal fibers condition, the conclusions of Yang et al. still hold. (c) 2024 Elsevier B.V. All rights reserved.
作者机构:
[Qu, Lianqiang] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Sun, LQ; Sun, Liuquan] Chinese Acad Sci, Inst Appl Math, Acad Math & Syst Sci, Beijing 100190, Peoples R China.;[Sun, Yanqing] Univ North Carolina Charlotte, Dept Math & Stat, 9201 Univ City Blvd, Charlotte, NC 28223 USA.
通讯机构:
[Sun, LQ ] C;Chinese Acad Sci, Inst Appl Math, Acad Math & Syst Sci, Beijing 100190, Peoples R China.
摘要:
Quantile regression has become a widely used tool for analysing competing risk data. However, quantile regression for competing risk data with a continuous mark is still scarce. The mark variable is an extension of cause of failure in a classical competing risk model where cause of failure is replaced by a continuous mark only observed at uncensored failure times. An example of the continuous mark variable is the genetic distance that measures dissimilarity between the infecting virus and the virus contained in the vaccine construct. In this article, we propose a novel mark-specific quantile regression model. The proposed estimation method borrows strength from data in a neighbourhood of a mark and is based on an induced smoothed estimation equation, which is very different from the existing methods for competing risk data with discrete causes. The asymptotic properties of the resulting estimators are established across mark and quantile continuums. In addition, a mark-specific quantile-type vaccine efficacy is proposed and its statistical inference procedures are developed. Simulation studies are conducted to evaluate the finite sample performances of the proposed estimation and hypothesis testing procedures. An application to the first HIV vaccine efficacy trial is provided.
摘要:
The proliferation of single-cell RNA-seq data has greatly enhanced our ability to comprehend the intricate nature of diverse tissues. However, accurately annotating cell types in such data, especially when handling multiple reference datasets and identifying novel cell types, remains a significant challenge. To address these issues, we introduce Single Cell annotation based on Distance metric learning and Optimal Transport (scDOT), an innovative cell-type annotation method adept at integrating multiple reference datasets and uncovering previously unseen cell types. scDOT introduces two key innovations. First, by incorporating distance metric learning and optimal transport, it presents a novel optimization framework. This framework effectively learns the predictive power of each reference dataset for new query data and simultaneously establishes a probabilistic mapping between cells in the query data and reference-defined cell types. Secondly, scDOT develops an interpretable scoring system based on the acquired probabilistic mapping, enabling the precise identification of previously unseen cell types within the data. To rigorously assess scDOT's capabilities, we systematically evaluate its performance using two diverse collections of benchmark datasets encompassing various tissues, sequencing technologies and diverse cell types. Our experimental results consistently affirm the superior performance of scDOT in cell-type annotation and the identification of previously unseen cell types. These advancements provide researchers with a potent tool for precise cell-type annotation, ultimately enriching our understanding of complex biological tissues.
期刊:
Journal of Mathematical Analysis and Applications,2024年537(1):128270 ISSN:0022-247X
通讯作者:
Huang, JC
作者机构:
[Chen, Shuang; Huang, Jicai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Chen, Shuang; Huang, Jicai] Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
通讯机构:
[Huang, JC ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;Cent China Normal Univ, Hubei Key Lab Math Sci, Wuhan 430079, Hubei, Peoples R China.
摘要:
In this paper, we study the destabilization of synchronous periodic solutions for patch models. By applying perturbation theory for matrices, we derive asymptotic expressions of the Floquet spectra and provide a destabilization criterion for synchronous periodic solutions arising from closed orbits or degenerate Hopf bifurcations in terms of period functions. Finally, we apply the main results to the well-known two -patch Holling-Tanner model. (c) 2024 Elsevier Inc. All rights reserved.
期刊:
Journal of Mathematical Biology,2024年88(4):1-30 ISSN:0303-6812
通讯作者:
Lu, M
作者机构:
[Lu, Min; Huang, Jicai; Pan, Qin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;[Lu, Min; Huang, Jicai; Pan, Qin] Cent China Normal Univ, Key Lab NAA, MOE, Wuhan 430079, Peoples R China.;[Ruan, Shigui] Univ Miami, Dept Math, Coral Gables, FL 33146 USA.
通讯机构:
[Lu, M ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.;Cent China Normal Univ, Key Lab NAA, MOE, Wuhan 430079, Peoples R China.
关键词:
Bogdanov–Takens bifurcation;Harvesting;Holling I functional response;Hopf bifurcation;Krill–whale interaction;Predator–prey model
摘要:
In the Antarctic, the whale population had been reduced dramatically due to the unregulated whaling. It was expected that Antarctic krill, the main prey of whales, would grow significantly as a consequence and exploratory krill fishing was practiced in some areas. However, it was found that there has been a substantial decline in abundance of krill since the end of whaling, which is the phenomenon of krill paradox. In this paper, to study the krill–whale interaction we revisit a harvested predator–prey model with Holling I functional response. We find that the model admits at most two positive equilibria. When the two positive equilibria are located in the region
$$\big \{(N,P)|0\le N< 2N_c,\ P\ge 0\big \}$$
, the model exhibits degenerate Bogdanov–Takens bifurcation with codimension up to 3 and Hopf bifurcation with codimension up to 2 by rigorous bifurcation analysis. When the two positive equilibria are located in the region
$$\big \{(N,P)|N>2N_c,\ P\ge 0\big \}$$
, the model has no complex bifurcation phenomenon. When there is one positive equilibrium on each side of
$$N=2N_c$$
, the model undergoes Hopf bifurcation with codimension up to 2. Moreover, numerical simulation reveals that the model not only can exhibit the krill paradox phenomenon but also has three limit cycles, with the outmost one crosses the line
$$N=2N_c$$
under some specific parameter conditions.
作者:
Banerjee, Malay;Huang, Jicai;Pan, Qin;Zou, Lan
期刊:
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B,2024年0:0-0 ISSN:1531-3492
通讯作者:
Zou, L
作者机构:
[Banerjee, Malay] IIT Kanpur, Dept Math & Stat, Kanpur 208016, India.;[Huang, Jicai; Pan, Qin] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Zou, Lan; Zou, L] Capital Normal Univ, Sch Math Sci, Beijing 100089, Peoples R China.
通讯机构:
[Zou, L ] C;Capital Normal Univ, Sch Math Sci, Beijing 100089, Peoples R China.
关键词:
Key words and phrases. Generalist predator;stability;limit cycle;Hopf bifurcation;Bogdanov- Takens bifurcation
摘要:
. Consideration of generalist predators leads to relatively complex dynamics due to alternative food sources. Here, we propose and analyze a prey-predator model with a generalist predator. The availability of alternative food sources for the predator and a density-dependent growth rate induces not only bistability and tristability, but also more complicated dynamical behaviors. We have studied the possible number and geometric configurations of positive equilibria in detail. A systematic bifurcation analysis has revealed the existence of the degenerate Bogdanov-Takens bifurcation of codimension four and degenerate Hopf bifurcation of codimension three. We found that degenerate local bifurcations with a higher codimension are responsible for three limit cycles. Derivation of the analytical conditions for three limit cycles for a suitable range of parameters is a crucial finding of this work.
作者机构:
[Lu, Min; Huang, Jicai] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Lu, Min; Huang, Jicai] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Hubei, Peoples R China.;[Xiang, Chuang; Xiang, C] Nanjing Normal Univ, Sch Math Sci, Nanjing 210023, Peoples R China.;[Xiang, Chuang; Xiang, C] Nanjing Normal Univ, Key Lab NSLSCS, Minist Educ, Nanjing 210023, Peoples R China.;[Ruan, Shigui] Univ Miami, Dept Math, Coral Gables, FL 33146 USA.
通讯机构:
[Xiang, C ] N;Nanjing Normal Univ, Sch Math Sci, Nanjing 210023, Peoples R China.;Nanjing Normal Univ, Key Lab NSLSCS, Minist Educ, Nanjing 210023, Peoples R China.
关键词:
Two-patch model;Allee effects in predators;Environment change;Bifurcations;Extinction;Regime shifts
摘要:
In this paper, we propose a generalized Rosenzweig-MacArthur predator-prey model with two patches and Allee effects in predators. In a constant environment, for the single -patch model we show the existence of degenerate Bogdanov-Takens bifurcation with codimension up to 4 and degenerate Hopf bifurcation with codimension up to 3, which indicate that the nilpotent cusp of codimension 4 is the organizing center of the bifurcation set. Our results show that Allee effects in predators can not only induce much more complex bifurcations and dynamics (such as three limit cycles, multitype tristability), but also serve as a destabilizing factor to make predators extinction (deterministically) when the environment is sufficiently enriched, which provides strong support for the so-called "paradox of enrichment". For the two -patch model, we numerically reveal the existence of eight positive steady states including two new coexisting attractors, which means that population dispersal can help species coexist or increase the population abundance. In a changing environment, for the single -patch model we find that the population can track unstable states, fast positive or slow negative environmental change can be beneficial to predators by delaying or even avoiding extinction; while for the twopatch model we observe some interesting phenomena: fast negative (or slow positive) environmental change can make predators in different patches trend to extinction synchronously, and rate -induced different regime shifts.
摘要:
The Erdos-Gyaras conjecture asserts that every graph with minimum degree at least three has a cycle whose length is a power of 2. Let G be a graph with minimum degree at least 3. We show that if G contains no induced path of order 10, then G contains a cycle of length 4 or 8, and hence the conjecture holds in this case. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
期刊:
CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS,2024年63(7):1-39 ISSN:0944-2669
通讯作者:
Pan, KF
作者机构:
[Luo, Peng; Peng, Shuangjie] Cent China Normal Univ, Sch Math & Stat, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.;[Pan, Kefan] Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
通讯机构:
[Pan, KF ] C;Cent China Normal Univ, Sch Math & Stat, Wuhan 430079, Peoples R China.
摘要:
We revisit the following nonlinear Schrödinger equation
$$\begin{aligned} -\varepsilon ^2\Delta u+ V(x)u=u^{p},\quad u>0,\;\; u\in H^1({\mathbb {R}}^N), \end{aligned}$$
where
$$\varepsilon >0$$
is a small parameter,
$$N\ge 2$$
and
$$1<p<2^*-1$$
. It is known that the Morse index gives a strong qualitative information on the solutions, such as non-degeneracy, uniqueness, symmetries, singularities as well as classifying solutions. Here we compute the Morse index of positive k-peak solutions to above problem when the critical points of V(x) are non-isolated and degenerate. We also give a specific formula for the Morse index of k-peak solutions when the critical point set of V(x) is a low-dimensional ellipsoid. Our main difficulty comes from the non-uniform degeneracy of potential V(x). Our results generalize Grossi and Servadei’s work (Ann Math Pura Appl 186: 433–453, (2007)) to very degenerate (non-admissible) potentials and show that the structure of potentials highly affects the properties of concentrated solutions.
期刊:
Journal of Functional Analysis,2024年286(2):110206 ISSN:0022-1236
通讯作者:
Xie, HF
作者机构:
[Luo, Peng] Cent China Normal Univ, Sch Math & Stat, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan 430079, Peoples R China.;[Tang, Zhongwei] Beijing Normal Univ, Sch Math Sci, Beijing 100875, Peoples R China.;[Xie, Huafei] Nanyang Normal Univ, Sch Math & Stat, Nanyang 473061, Peoples R China.
通讯机构:
[Xie, HF ] N;Nanyang Normal Univ, Sch Math & Stat, Nanyang 473061, Peoples R China.
摘要:
In this paper we study the following eigenvalue problem {-Delta v = lambda C(alpha)(p(alpha) - epsilon)vertical bar x vertical bar(alpha)u(epsilon)(p alpha-epsilon-1)v in Omega, u=0 on partial derivative Omega, where Omega subset of R-N is a smooth bounded domain containing the origin, C(alpha) = (N + alpha)(N - 2), N >= 3, p(alpha) = N+2+2 alpha/N-2, alpha > 0, epsilon > 0 is a small parameter and u(epsilon) is a single peaked solution of Henon equation {-Delta u = C(alpha)vertical bar x vertical bar(alpha)u(p alpha-epsilon) in Omega, u > 0 in Omega, u=0 on partial derivative Omega, which established by Gladiali and Grossi (2012) [21]. By using various local Pohozaev identities and blow-up analysis, we prove some asymptotic behavior of the eigen values lambda(epsilon,i) and corresponding eigen functions v(epsilon,i), i = 2, ... , Sigma(1 <= k <= 2+alpha/2) (N + 2k - 2) (N + k - 3)!/(N - 2)!k! + 2 when alpha is not an even integer. As a consequence, if 0 < alpha < 2, we have that the Morse index of the single peaked solutions is N + 1, which gives an affirmative answer to a conjecture raised by Gladiali and Grossi. (c) 2023 Elsevier Inc. All rights reserved.
期刊:
ELECTRONIC JOURNAL OF COMBINATORICS,2024年31(1) ISSN:1077-8926
通讯作者:
Li, SC
作者机构:
[Lei, Xingyu; Li, Shuchao; Li, SC] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构:
[Li, SC ] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
摘要:
<jats:p>For a given graph $F$, we say that a graph $G$ is $F$-free if it does not contain $F$ as a subgraph. A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. Let $K_r^{+}(a_1, a_2, \ldots, a_r)$ be the graph obtained from complete $r$-partite graph with parts of sizes $a_1\geqslant 2, a_2, \ldots, a_r$, by adding an edge to the first part. In this paper, we focus on the spectral extrema of disjoint color-critical graphs. For fixed $t, a_1,\ldots, a_r\ (r\geqslant 2)$ and large enough $n$, we characterize the unique $n$-vertex $tK_r^+(a_1,\ldots,a_r)$-free graph having the largest spectral radius. Moreover, let $F_1,\ldots,F_t$ be $t$ disjoint color-critical graphs with the same chromatic number. We identify the unique $n$-vertex $\bigcup_{i=1}^tF_i$-free graph having the largest spectral radius for sufficiently large $n$. Consequently, we generalize the main results obtained by Ni, Wang and Kang [Electron. J. Combin. 30 (1) (2023), #P1.20] and by Fang, Zhai and Lin [arXiv:2302.03229v2].</jats:p>
期刊:
INVERSE PROBLEMS AND IMAGING,2024年0:0-0 ISSN:1930-8337
通讯作者:
Li, ZY
作者机构:
[Jiang, Daijun] Cent China Normal Univ, Sch Math & Stat, Wuhan, Peoples R China.;[Jiang, Daijun] Cent China Normal Univ, Key Lab Nonlinear Anal & Applicat, Minist Educ, Wuhan, Peoples R China.;[Li, ZY; Li, Zhiyuan] Ningbo Univ, Sch Math & Stat, Ningbo, Peoples R China.
通讯机构:
[Li, ZY ] N;Ningbo Univ, Sch Math & Stat, Ningbo, Peoples R China.
关键词:
Fractional diffusion equation;Hopf's lemma;inverse source problem;inverse Robin problem;Tikhonov regularization
摘要:
. This paper is devoted to an inverse problem of simultaneously determining the spatially dependent source term and the Robin boundary coefficient in a time fractional diffusion equation, with the aid of extra measurement data at a subdomain near the accessible boundary. Firstly, the spatially varying source is uniquely determined in view of the unique continuation principle and Duhamel principle for the fractional diffusion equation. The Hopf lemma for a homogeneous time-fractional diffusion equation is proved and then used to prove the uniqueness of recovering the Robin boundary coefficient. Numerically, based on the theoretical uniqueness result, we apply the classical Tikhonov regularization method to transform the inverse problem into a minimization problem, which is solved by an iterative thresholding algorithm. Finally, several numerical examples are presented to show the accuracy and effectiveness of the proposed algorithm.