In this paper, an algorithm is presented that calculates the module of generalized invariants. Generalized invariants are useful in the study of modular invariant theory, where the characteristic of the base field divides the order of the group. The structure theorems of generalized invariants were given by the author in 2016, and computational aspects of these results are studied in this note. Implementation codes are also provided for MAGMA, computational algebra system.
{"title":"The Computation of Generalized Invariants","authors":"Uğur Madran","doi":"10.1155/2023/2175379","DOIUrl":"https://doi.org/10.1155/2023/2175379","url":null,"abstract":"In this paper, an algorithm is presented that calculates the module of generalized invariants. Generalized invariants are useful in the study of modular invariant theory, where the characteristic of the base field divides the order of the group. The structure theorems of generalized invariants were given by the author in 2016, and computational aspects of these results are studied in this note. Implementation codes are also provided for MAGMA, computational algebra system.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"51 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88615827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Convolutional neural networks (CNNs) are often used in tasks involving vision processing, and unclear images can hinder the performance of convolutional neural networks and increase its computational time. Furthermore, artificial intelligence (AI) and machine learning (ML) are related technologies, which are considered a branch of computer science, which are used to simulate and enhance human intelligence. In e-healthcare, AI and ML can be used to optimize the workflow, automatically process large amounts of medical data, and provide effective medical decision support. In this paper, the authors take several mainstream artificial intelligence models currently open on the market for reference. In this paper, the optimized model (AL-CNN) is tested for noise image recognition, and the AL-CNN model is established by using activation functions, matrix operations, and feature recognition methods, and the noisy images are processed after custom configuration. Not only does this model require no prior preparation when processing images, but it also improves the accuracy of dealing with noise in convolutional neural networks. In the AL-CNN in this paper, the architecture of the convolutional neural network includes a noise layer and a layer that can be automatically resized. After the comparison of the recognition experiments, the accuracy rate of AL-CNN is 20% higher than that of MatConvNet-moderate, and the accuracy rate is 40% higher than that of MatConvNet-chronic. In the second set of experiments, the accuracy exceeds MXNet and TensorFlow by 50% and 70%, respectively. In addition, the authors optimized the convolutional layer, pooling layer, and loss function of AL-CNN in different parameters, which improved the stability of noise processing, respectively. After customizing the two configuration optimizations, the authors found that the second optimized AL-CNN has higher recognition accuracy, and after the optimization test, the error rate can be continuously decreased as the number of recognition increases in a very short number of times.
{"title":"Establishment and Test Effect of Artificial Intelligence Optimization Model Based on Convolutional Neural Network","authors":"Chunrong Zhou, Zhenghong Jiang","doi":"10.1155/2023/4216012","DOIUrl":"https://doi.org/10.1155/2023/4216012","url":null,"abstract":"Convolutional neural networks (CNNs) are often used in tasks involving vision processing, and unclear images can hinder the performance of convolutional neural networks and increase its computational time. Furthermore, artificial intelligence (AI) and machine learning (ML) are related technologies, which are considered a branch of computer science, which are used to simulate and enhance human intelligence. In e-healthcare, AI and ML can be used to optimize the workflow, automatically process large amounts of medical data, and provide effective medical decision support. In this paper, the authors take several mainstream artificial intelligence models currently open on the market for reference. In this paper, the optimized model (AL-CNN) is tested for noise image recognition, and the AL-CNN model is established by using activation functions, matrix operations, and feature recognition methods, and the noisy images are processed after custom configuration. Not only does this model require no prior preparation when processing images, but it also improves the accuracy of dealing with noise in convolutional neural networks. In the AL-CNN in this paper, the architecture of the convolutional neural network includes a noise layer and a layer that can be automatically resized. After the comparison of the recognition experiments, the accuracy rate of AL-CNN is 20% higher than that of MatConvNet-moderate, and the accuracy rate is 40% higher than that of MatConvNet-chronic. In the second set of experiments, the accuracy exceeds MXNet and TensorFlow by 50% and 70%, respectively. In addition, the authors optimized the convolutional layer, pooling layer, and loss function of AL-CNN in different parameters, which improved the stability of noise processing, respectively. After customizing the two configuration optimizations, the authors found that the second optimized AL-CNN has higher recognition accuracy, and after the optimization test, the error rate can be continuously decreased as the number of recognition increases in a very short number of times.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"5 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88910787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of research, invention, and production is to effectively supply demand, which requires not only financial support but also time to complete. First, the differential dynamic model among funds, demand, and research (including invention and production) with time delay is constructed by using the predator-prey theory. Second, the sufficiency of local asymptotic stability of the positive equilibrium point of the model is obtained by using the Hopf bifurcation theory and stability theory. Then, the formulas for determining the direction of Hopf bifurcation and the stability of the periodic solution are calculated by using the central manifold theory and normative theory. Finally, the evolution process of the differential dynamic model with controlled time delay is numerically simulated so as to verify the correctness of the relevant analytical conclusions.
{"title":"Dynamic Analysis of the Project Investment Model with Time Delay and Density Constraints","authors":"Debao Gao","doi":"10.1155/2023/3791676","DOIUrl":"https://doi.org/10.1155/2023/3791676","url":null,"abstract":"The purpose of research, invention, and production is to effectively supply demand, which requires not only financial support but also time to complete. First, the differential dynamic model among funds, demand, and research (including invention and production) with time delay is constructed by using the predator-prey theory. Second, the sufficiency of local asymptotic stability of the positive equilibrium point of the model is obtained by using the Hopf bifurcation theory and stability theory. Then, the formulas for determining the direction of Hopf bifurcation and the stability of the periodic solution are calculated by using the central manifold theory and normative theory. Finally, the evolution process of the differential dynamic model with controlled time delay is numerically simulated so as to verify the correctness of the relevant analytical conclusions.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82010618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<jats:p>In this paper, generalized <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </math> </jats:inline-formula>-implications are introduced and studied. Although they are not always fuzzy implications, the sufficient and necessary condition such that a generalized <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </math> </jats:inline-formula>-operation is a fuzzy implication is discovered and proved. The satisfaction or violation of some basic properties of fuzzy implications, such as the left neutrality property, the identity principle, and the left ordering property, is also investigated and presented. The intersections among the classes of generalized <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </math> </jats:inline-formula>-implications (respectively, operations), R-implications, and <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </math> </jats:inline-formula>-implications (respectively, operations) are also studied. It is demonstrated that the generalized <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M5"> <msup> <mrow> <mi mathvariant="normal">R</mi> </mrow> <mrow> <mo>′</mo> </mrow> </msup> </math>
{"title":"Generalized R′-Implications: A Hyper Class of R- and R′-Implications","authors":"Dimitrios S. Grammatikopoulos, Basil Papadopoulos","doi":"10.1155/2023/7111888","DOIUrl":"https://doi.org/10.1155/2023/7111888","url":null,"abstract":"<jats:p>In this paper, generalized <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msup>\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">R</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>′</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula>-implications are introduced and studied. Although they are not always fuzzy implications, the sufficient and necessary condition such that a generalized <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <msup>\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">R</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>′</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula>-operation is a fuzzy implication is discovered and proved. The satisfaction or violation of some basic properties of fuzzy implications, such as the left neutrality property, the identity principle, and the left ordering property, is also investigated and presented. The intersections among the classes of generalized <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <msup>\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">R</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>′</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula>-implications (respectively, operations), R-implications, and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <msup>\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">R</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>′</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula>-implications (respectively, operations) are also studied. It is demonstrated that the generalized <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\u0000 <msup>\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">R</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>′</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 ","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"26 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75183213","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sheeba Afridi, Muhammad Yasin Khan, G. Ali, Murtaza Ali, Irfan Nurhidayat, M. A. Arefin
<jats:p>Molecular descriptors are a basic tool in the spectral graph, molecular chemistry, and various other fields of mathematics and chemistry. Kulli–Basava <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <mfenced open="(" close=")" separators="|"> <mrow> <mi mathvariant="double-struck">K</mi> <mi mathvariant="fraktur">B</mi> </mrow> </mfenced> </math> </jats:inline-formula> indices were initiated for chemical applications of various substances in chemistry. For simple graph <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mi>G</mi> </math> </jats:inline-formula>, <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <mi mathvariant="double-struck">K</mi> <mi mathvariant="fraktur">B</mi> </math> </jats:inline-formula> indices in generalized forms are <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi mathvariant="double-struck">K</mi> <msubsup> <mrow> <mi mathvariant="fraktur">B</mi> </mrow> <mrow> <mn>1</mn> </mrow> <mrow> <mi mathvariant="normal">ϱ</mi> </mrow> </msubsup> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> <mo>=</mo> <mrow> <msub> <mrow> <mstyle displaystyle="true"> <mo stretchy="false">∑</mo> </mstyle> </mrow> <mrow> <mi>g</mi> <mi>h</mi> <mo>∈</mo> <mi>E</mi> <mrow> <mfenced open="(" close=")" separators="|"> <mrow> <mi>G</mi> </mrow> </mfenced> </mrow>
分子描述符是光谱图、分子化学以及其他数学和化学领域的基本工具。Kulli-Basava K - B指数在化学中应用于各种物质。
{"title":"Sharp Bounds of Kulli–Basava Indices in Generalized Form for k-Generalized Quasi Trees","authors":"Sheeba Afridi, Muhammad Yasin Khan, G. Ali, Murtaza Ali, Irfan Nurhidayat, M. A. Arefin","doi":"10.1155/2023/7567411","DOIUrl":"https://doi.org/10.1155/2023/7567411","url":null,"abstract":"<jats:p>Molecular descriptors are a basic tool in the spectral graph, molecular chemistry, and various other fields of mathematics and chemistry. Kulli–Basava <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi mathvariant=\"double-struck\">K</mi>\u0000 <mi mathvariant=\"fraktur\">B</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula> indices were initiated for chemical applications of various substances in chemistry. For simple graph <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <mi>G</mi>\u0000 </math>\u0000 </jats:inline-formula>, <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mi mathvariant=\"double-struck\">K</mi>\u0000 <mi mathvariant=\"fraktur\">B</mi>\u0000 </math>\u0000 </jats:inline-formula> indices in generalized forms are <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <mi mathvariant=\"double-struck\">K</mi>\u0000 <msubsup>\u0000 <mrow>\u0000 <mi mathvariant=\"fraktur\">B</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">ϱ</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 <mo>=</mo>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mstyle displaystyle=\"true\">\u0000 <mo stretchy=\"false\">∑</mo>\u0000 </mstyle>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mi>h</mi>\u0000 <mo>∈</mo>\u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </mrow>\u0000 ","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90552796","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<jats:p>In this paper, we consider an exceptional study of differentially private stochastic gradient descent (SGD) algorithms in the stochastic convex optimization (SCO). The majority of the existing literature requires that the losses have additional assumptions, such as the loss functions with Lipschitz, smooth and strongly convex, and uniformly bounded of the model parameters, or focus on the Euclidean (i.e. <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msubsup> <mrow> <mi mathvariant="script">l</mi> </mrow> <mrow> <mn>2</mn> </mrow> <mrow> <mi>d</mi> </mrow> </msubsup> </math> </jats:inline-formula>) setting. However, these restrictive requirements exclude many popular losses, including the absolute loss and the hinge loss. By loosening the restrictions, we proposed two differentially private SGD without shuffle model and with shuffle model algorithms (in short, DP-SGD-NOS and DP-SGD-S) for the <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <mfenced open="(" close=")" separators="|"> <mrow> <mi>α</mi> <mo>,</mo> <mi>L</mi> </mrow> </mfenced> </math> </jats:inline-formula>-Hölder smooth loss by adding calibrated Laplace noise under no shuffling scheme and shuffling scheme in the <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M3"> <msubsup> <mrow> <mi mathvariant="script">l</mi> </mrow> <mrow> <mi>p</mi> </mrow> <mrow> <mi>d</mi> </mrow> </msubsup> </math> </jats:inline-formula>-setting for <jats:inline-formula> <math xmlns="http://www.w3.org/1998/Math/MathML" id="M4"> <mi>p</mi> <mo>∈</mo> <mfenced open="[" close="]" separators="|"> <mrow> <mn>1,2</mn> </mrow> </mfenced> </math> </jats:inline-formula>. We provide privacy guarantees by using advanced compositio
在本文中，我们考虑了随机凸优化(SCO)中微分私有随机梯度下降(SGD)算法的特殊研究。现有文献大多要求损失具有附加假设，如损失函数具有Lipschitz、光滑且强凸、模型参数均匀有界等;或专注于欧几里得(即二维)设置。然而，这些限制性要求排除了许多常见的损耗，包括绝对损耗和铰链损耗。通过放宽限制，我们提出了两种不同的私有SGD算法(即DP-SGD-NOS和DP-SGD-S)，用于α，L -Hölder在无洗牌方案和在L p中加入校准拉普拉斯噪声的平滑损失D -设为p∈1,2。我们通过使用先进的组合和隐私放大技术提供隐私保证。我们还分析了DP-SGD-NOS和DP-SGD-S的收敛界，得到了最优超额群体风险O 1 / n +D log1 / δ/ nO (1 / n + dlog1 / δ logN / δ 在本文中，我们考虑了随机凸优化(SCO)中微分私有随机梯度下降(SGD)算法的特殊研究。现有文献大多要求损失具有附加假设，如损失函数具有Lipschitz、光滑且强凸、模型参数均匀有界等;或专注于欧几里得(即二维)设置。然而，这些限制性要求排除了许多常见的损耗，包括绝对损耗和铰链损耗。通过放宽限制，我们提出了两种不同的私有SGD算法(即DP-SGD-NOS和DP-SGD-S)，用于α，L -Hölder在无洗牌方案和在L p中加入校准拉普拉斯噪声的平滑损失D -设为p∈1,2。我们通过使用先进的组合和隐私放大技术提供隐私保证。
{"title":"Privacy-Preserving SGD on Shuffle Model","authors":"Lingjie Zhang, Hai Zhang","doi":"10.1155/2023/4055950","DOIUrl":"https://doi.org/10.1155/2023/4055950","url":null,"abstract":"<jats:p>In this paper, we consider an exceptional study of differentially private stochastic gradient descent (SGD) algorithms in the stochastic convex optimization (SCO). The majority of the existing literature requires that the losses have additional assumptions, such as the loss functions with Lipschitz, smooth and strongly convex, and uniformly bounded of the model parameters, or focus on the Euclidean (i.e. <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msubsup>\u0000 <mrow>\u0000 <mi mathvariant=\"script\">l</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 </math>\u0000 </jats:inline-formula>) setting. However, these restrictive requirements exclude many popular losses, including the absolute loss and the hinge loss. By loosening the restrictions, we proposed two differentially private SGD without shuffle model and with shuffle model algorithms (in short, DP-SGD-NOS and DP-SGD-S) for the <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>,</mo>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula>-Hölder smooth loss by adding calibrated Laplace noise under no shuffling scheme and shuffling scheme in the <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <msubsup>\u0000 <mrow>\u0000 <mi mathvariant=\"script\">l</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 </msubsup>\u0000 </math>\u0000 </jats:inline-formula>-setting for <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <mi>p</mi>\u0000 <mo>∈</mo>\u0000 <mfenced open=\"[\" close=\"]\" separators=\"|\">\u0000 <mrow>\u0000 <mn>1,2</mn>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula>. We provide privacy guarantees by using advanced compositio","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83992221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The plant-pollinator model is a common model widely researched by scholars in population dynamics. In fact, its complex dynamical behaviors are universally and simply expressed as a class of delay differential-difference equations. In this paper, based on several early plant-pollinator models, we consider a plant-pollinator model with two combined delays to further describe the mutual constraints between the two populations under different time delays and qualitatively analyze its stability and Hopf bifurcation. Specifically, by selecting different combinations of two delays as branch parameters and analyzing in detail the distribution of roots of the corresponding characteristic transcendental equation, we investigate the local stability of the positive equilibrium point of equations, derive the sufficient conditions of asymptotic stability, and demonstrate the Hopf bifurcation for the system. Under the condition that two delays are not equal, some explicit formulas for determining the direction of Hopf bifurcation and some conditions for the stability of periodic solutions of bifurcation are obtained for delay differential equations by using the theory of norm form and the theorem of center manifold. In the end, some examples are presented and corresponding computer numerical simulations are taken to demonstrate and support effectiveness of our theoretical predictions.
{"title":"Bifurcations in a Plant-Pollinator Model with Multiple Delays","authors":"Long Li, Yanxia Zhang, Jianfei Yao, Xiuxing Wu","doi":"10.1155/2023/9950187","DOIUrl":"https://doi.org/10.1155/2023/9950187","url":null,"abstract":"The plant-pollinator model is a common model widely researched by scholars in population dynamics. In fact, its complex dynamical behaviors are universally and simply expressed as a class of delay differential-difference equations. In this paper, based on several early plant-pollinator models, we consider a plant-pollinator model with two combined delays to further describe the mutual constraints between the two populations under different time delays and qualitatively analyze its stability and Hopf bifurcation. Specifically, by selecting different combinations of two delays as branch parameters and analyzing in detail the distribution of roots of the corresponding characteristic transcendental equation, we investigate the local stability of the positive equilibrium point of equations, derive the sufficient conditions of asymptotic stability, and demonstrate the Hopf bifurcation for the system. Under the condition that two delays are not equal, some explicit formulas for determining the direction of Hopf bifurcation and some conditions for the stability of periodic solutions of bifurcation are obtained for delay differential equations by using the theory of norm form and the theorem of center manifold. In the end, some examples are presented and corresponding computer numerical simulations are taken to demonstrate and support effectiveness of our theoretical predictions.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"10 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77763003","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the stability of various forms for the general octic functional equation We first find a special way of representing a given mapping as the sum of eight mappings. And by using the above representation, we will investigate the hyperstability of the general octic functional equation. Furthermore, we will discuss the Hyers–Ulam–Rassias stability of the general octic functional equation.
在本文中，本文研究了广义octic泛函方程∑I = 09 - 9的各种形式的稳定性C i−1 9−1 f x +I y = 0。我们首先找到一种将给定映射表示为八个映射的和的特殊方法。利用上述表述，我们将研究一般octic泛函方程的超稳定性。此外，我们将讨论一般octic泛函方程的Hyers-Ulam-Rassias稳定性。
{"title":"Some Remarks Concerning the General Octic Functional Equation","authors":"Yang-Hi Lee, Jaiok Roh","doi":"10.1155/2023/2930056","DOIUrl":"https://doi.org/10.1155/2023/2930056","url":null,"abstract":"<jats:p>In this article, we study the stability of various forms for the general octic functional equation <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msubsup>\u0000 <mrow>\u0000 <mstyle displaystyle=\"true\">\u0000 <mo stretchy=\"false\">∑</mo>\u0000 </mstyle>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 <mo>=</mo>\u0000 <mn>09</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>9</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow>\u0000 </msub>\u0000 <msup>\u0000 <mrow>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </mfenced>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>9</mn>\u0000 <mo>−</mo>\u0000 <mi>i</mi>\u0000 </mrow>\u0000 </msup>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>+</mo>\u0000 <mi>i</mi>\u0000 <mi>y</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </mrow>\u0000 </mrow>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 <mtext>.</mtext>\u0000 </math>\u0000 </jats:inline-formula> We first find a special way of representing a given mapping as the sum of eight mappings. And by using the above representation, we will investigate the hyperstability of the general octic functional equation. Furthermore, we will discuss the Hyers–Ulam–Rassias stability of the general octic functional equation.</jats:p>","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"5 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72960105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yun-Ling Cui, L. Ceng, Fang-Fei Zhang, Liang He, Jie Yin, Cong-Shan Wang, Huiying Hu
In real Hilbert spaces, let the CFPP indicate a common fixed-point problem of asymptotically nonexpansive operator and countably many nonexpansive operators, and suppose that the HVI and VIP represent a hierarchical variational inequality and a variational inequality problem, respectively. We put forward Mann hybrid deepest-descent extragradient approach for solving the HVI with the CFPP and VIP constraints. The proposed algorithms are on the basis of Mann’s iterative technique, viscosity approximation method, subgradient extragradient rule with linear-search process, and hybrid deepest-descent rule. Under suitable restrictions, it is shown that the sequences constructed by the algorithms converge strongly to a solution of the HVI with the CFPP and VIP constraints.
{"title":"Mann Hybrid Deepest-Descent Extragradient Method with Line-Search Process for Hierarchical Variational Inequalities for Countable Nonexpansive Mappings","authors":"Yun-Ling Cui, L. Ceng, Fang-Fei Zhang, Liang He, Jie Yin, Cong-Shan Wang, Huiying Hu","doi":"10.1155/2023/6177912","DOIUrl":"https://doi.org/10.1155/2023/6177912","url":null,"abstract":"In real Hilbert spaces, let the CFPP indicate a common fixed-point problem of asymptotically nonexpansive operator and countably many nonexpansive operators, and suppose that the HVI and VIP represent a hierarchical variational inequality and a variational inequality problem, respectively. We put forward Mann hybrid deepest-descent extragradient approach for solving the HVI with the CFPP and VIP constraints. The proposed algorithms are on the basis of Mann’s iterative technique, viscosity approximation method, subgradient extragradient rule with linear-search process, and hybrid deepest-descent rule. Under suitable restrictions, it is shown that the sequences constructed by the algorithms converge strongly to a solution of the HVI with the CFPP and VIP constraints.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89092064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aiman Arshad, Aqsa Sattar, M. Javaid, Mamo Abebe Ashebo
The area of graph theory (GT) is rapidly expanding and playing a significant role in cheminformatics, mostly in mathematics and chemistry to develop different physicochemical, chemical structure, and their properties. The manipulation and study of chemical graphical details are made feasible by using numerical structure invariant. Investigating these chemical characteristics of topological indices (TIs) is made possible by the discipline of mathematical chemistry. In this article, we study with the Cartesian product of complete graphs, with path graphs, and find their general result of connection number (CN)-based TIs, namely, first connection- based Zagreb index (1st CBZI), second connection- based Zagreb index (2nd CBZI), and third CBZI (3rd CBZI) and then modified first connection- based Zagreb index (CBZI) and second and third modified CBZIs. We also express the general results of first multiplicative CBZI, second multiplicative CBZI, and third and fourth multiplicative CBZI, of two special types of graphs, namely, complete graphs and path graphs. More precisely, we arrange the graphical and numerical analysis of our calculated expressions for both of Cartesian product with each other.
{"title":"Connection Number- Based Topological Indices of Cartesian Product of Graphs","authors":"Aiman Arshad, Aqsa Sattar, M. Javaid, Mamo Abebe Ashebo","doi":"10.1155/2023/8272936","DOIUrl":"https://doi.org/10.1155/2023/8272936","url":null,"abstract":"The area of graph theory (GT) is rapidly expanding and playing a significant role in cheminformatics, mostly in mathematics and chemistry to develop different physicochemical, chemical structure, and their properties. The manipulation and study of chemical graphical details are made feasible by using numerical structure invariant. Investigating these chemical characteristics of topological indices (TIs) is made possible by the discipline of mathematical chemistry. In this article, we study with the Cartesian product of complete graphs, with path graphs, and find their general result of connection number (CN)-based TIs, namely, first connection- based Zagreb index (1st CBZI), second connection- based Zagreb index (2nd CBZI), and third CBZI (3rd CBZI) and then modified first connection- based Zagreb index (CBZI) and second and third modified CBZIs. We also express the general results of first multiplicative CBZI, second multiplicative CBZI, and third and fourth multiplicative CBZI, of two special types of graphs, namely, complete graphs and path graphs. More precisely, we arrange the graphical and numerical analysis of our calculated expressions for both of Cartesian product with each other.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"192 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76600586","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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