Bulletin of Mathematical Biology最新文献

{"title":"Topological Structure of Epigenetic Forests in Flower Morphogenesis.","authors":"Yuriria Cortés-Poza, J Rogelio Pérez-Buendía","doi":"10.1007/s11538-026-01688-2","DOIUrl":"10.1007/s11538-026-01688-2","url":null,"abstract":"<p><p>Understanding how stable developmental patterns emerge from gene regulatory networks remains a central problem in developmental biology. Here, we study how classical homeotic mutations reshape the epigenetic landscape of the floral gene regulatory network of Arabidopsis thaliana. We represent this landscape as an Epigenetic Forest: a collection of rooted in-arborescences induced by the state transition graph of a Boolean gene regulatory network, where each tree is the basin of attraction of a stable gene expression pattern associated with a floral or meristematic identity. We apply this framework to the wild-type network, three single homeotic mutants (ap1, pi, and ag), and three double mutants (ap3-pi, ag-pi, and ap1-ag). For each genotype, we quantify landscape organization using complementary descriptors of basin structure, convergence depth, fate diversity, dominance, inequality, and Jensen-Shannon divergence from wild type. The resulting landscapes reveal distinct modes of mutant-induced deformation. The ap1 mutant restricts fate accessibility and concentrates trajectories into dominant basins, whereas pi eliminates B-function-dependent identities while largely preserving global basin organization. In contrast, ag increases effective fate diversity despite the loss of reproductive identity, reflecting defective meristem termination and convergence to WUS-associated states. Double mutants exhibit non-additive deformation: ap3-pi is indistinguishable from pi across all reported descriptors, consistent with logical saturation of the AND-like B-function module, whereas ag-pi and ap1-ag produce distinct redistributions of fate accessibility. The framework recovers canonical floral identities and experimentally observed mutant phenotypes while treating the epigenetic landscape as a finite, computable object determined by regulatory logic. The framework thus provides a topology-based description of developmental robustness, epistasis, and mutant-induced landscape deformation.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13294335/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148307029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Uncertainty and Sensitivity Analysis of Hair Growth Duration in Human Scalp Follicles Under Normal and Alopecic Conditions.","authors":"Atanaska Dobreva, Damon Comer, N G Cogan, Ralf Paus","doi":"10.1007/s11538-026-01680-w","DOIUrl":"10.1007/s11538-026-01680-w","url":null,"abstract":"<p><p>Hair follicles, the organs that produce hair, go through a constant cycle composed of phases of growth, regression, and rest. During this cycle, matrix keratinocytes (MKs), the cells responsible for hair fiber synthesis, proliferate for several years and then undergo spontaneous apoptosis. Damage to MKs and perturbations in their normal dynamics result in a shortened growth phase of the hair cycle, leading to hair loss. The most common factors causing such disruption are hormonal imbalance and attacks by the immune system. Androgenetic alopecia (AGA) is a form of hair loss caused by high sensitivity to androgens, and alopecia areata (AA) is a condition where hair loss is caused by an autoimmune reaction against MKs. In this study, we inform a mathematical model for the human hair cycle with experimental data for the lengths of hair cycle phases available from male control subjects and subjects with AGA. We also connect a mathematical model for AA with estimates for the duration of hair cycle phases obtained from the literature. Subsequently, with each model we perform parameter screening, uncertainty quantification, and global sensitivity analysis, and we compare the results across control, AGA, and AA conditions. The findings reveal that, in AGA subjects, there is greater uncertainty associated with the duration of hair growth than in control subjects. Additionally, compared to control and AGA conditions, in AA it is more certain that longer hair growth phase could not be expected. The global sensitivity analysis results show that, in AGA conditions, synthesis of regulatory molecules in the dermal papilla and stem cell input to the MK population have high impact on hair growth duration, which agrees with physiological understanding for the effect of androgens on hair follicles in AGA.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148307048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modeling, Analysis, and Optimal Control of Leukemic Cell Population Dynamics Under Therapy.","authors":"Pauline Mazel, Frédéric Grognard, Thomas Stiehl, Alain Rapaport, Walid Djema","doi":"10.1007/s11538-026-01692-6","DOIUrl":"10.1007/s11538-026-01692-6","url":null,"abstract":"<p><p>Building upon the ODE model describing the dynamics of healthy and leukemic cells introduced in Kumar et al. (2024); Stiehl and Marciniak-Czochra (2012), we propose an extended framework that incorporates a control variable representing the effects of chemotherapy. This extension aims to provide a more refined mathematical basis for investigating anti-cancer strategies. First, we perform a stability analysis of the equilibria associated with healthy and leukemic states, partly estimated from clinical data. This analysis reveals a complex structure, including the emergence of a continuum of coexistence states and bifurcation thresholds that play a key role in the subsequent optimization stage. Based on these findings, we investigate an optimal control problem to minimize leukemia stem cells while limiting drug toxicity. Pontryagin's Maximum Principle provides necessary conditions for optimality, and direct numerical optimization confirms the predicted structures, motivating the study of the static problem. This static formulation reveals an unconventional feature: the cost functional becomes set-valued due to the continuum of equilibria, placing the problem outside the scope of standard methods. Simulations reveal a turnpike phenomenon, where over long time horizons the dynamic trajectories closely approximate the ideal static structure. Finally, a sensitivity analysis of the performance criterion with respect to key parameters complements the study, providing preliminary insights into which biological mechanisms may influence the optimal therapeutic outcomes. We conclude with a discussion of these findings.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148315732","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Coupled SDE-ODE Modeling of Tumor-Immune Dynamics to Infer Biomarker Release.","authors":"Pujan Shrestha, Yijia Fan, Jason T George","doi":"10.1007/s11538-026-01665-9","DOIUrl":"10.1007/s11538-026-01665-9","url":null,"abstract":"<p><p>Tumor-immune interactions are central to cancer progression and treatment response, driving cell death through immune-mediated killing and resource-limited competition. In early-stage disease or following effective treatment, cancer populations are often small and difficult to observe directly. Disease monitoring therefore relies on the detection of biomarkers such as circulating tumor DNA (ctDNA) as noisy proxies to cancer size. Existing approaches lack robust frameworks to infer tumor burden from these signals when populations approach detection thresholds. We present a coupled deterministic-stochastic framework that links tumor-immune dynamics to biomarker release. A two prey, one predator Lotka-Volterra model captures interactions between immune cells and competing tumor subpopulations under shared resource constraints. Biomarker production is modeled using stochastic differential equations driven by tumor cell death arising from immune-mediated apoptosis and intratumor competition-induced necrosis. We incorporate both square-root (CIR-type) noise, representing count-limited fluctuations near the detection limit, and multiplicative (geometric-type) noise, representing proportional variability at higher concentrations. Within this framework, we derive analytical expressions for biomarker trajectories and first-passage statistics, including mean detection times under minimal detection thresholds. Our results demonstrate how tumor heterogeneity, immune pressure, and stochastic variance structure jointly shape biomarker emergence, persistence, and detectability. Together, this work provides a mechanistically interpretable and mathematically tractable foundation for inferring tumor burden from noisy biomarker signals across clinically relevant concentration regimes.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13290969/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148300918","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Dual-age Structured Epidemiological Model with Waning Immunity and Reinfection.","authors":"Raimund M Kovacevic, Nikolaos I Stilianakis, Vladimir M Veliov","doi":"10.1007/s11538-026-01686-4","DOIUrl":"10.1007/s11538-026-01686-4","url":null,"abstract":"<p><p>Waning immunity and reinfection are critical features of many infectious diseases, but epidemiological models often fail to capture the interaction between an individual's immunity history and their current infection status, or do so only simplistically. We develop a dual-age structured model tracking immunity age (time since last recovery) and infection age (time since infection) to analyze epidemic dynamics under waning immunity and reinfection. The model is formulated as a system of age-structured partial differential equations describing susceptible and infected populations stratified by both age variables. The contact rate, mortality and recovery rates, susceptibility, and pathogen load are all treated as parameter functions depending on both immunity and infection age. We derive basic reproduction numbers and numerically solve the system using a second-order Runge-Kutta scheme along characteristic lines. We further extend the model to treat vaccination interventions, specifically booster vaccination strategies targeting individuals by immunity age - interventions that cannot be formulated in standard models. Numerical results reveal that higher contact rates produce larger oscillation amplitudes with longer inter-epidemic periods. However, long-term oscillation amplitude and cumulative infections depend non-monotonically on the initial infected population size, indicating that the relationship between initial infection levels and long-term epidemic outcomes is not straightforward. Vaccination efficiency depends critically on the pathogen load profile, with more concentrated distributions requiring higher vaccination rates for elimination. Most efficient strategies target intermediate immunity ages rather than only fully waned individuals.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148307096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A Mathematical Model to Predict Growth and Treatment for UPS Cancer.","authors":"Sumit Roy","doi":"10.1007/s11538-026-01694-4","DOIUrl":"10.1007/s11538-026-01694-4","url":null,"abstract":"<p><p>We propose a mathematical model for the growth and treatment of Undifferentiated Pleomorphic Sarcoma (UPS) using a system of nonlinear differential equations. The model combines Gompertz-type tumor growth with surface-dependent necrotic loss, surgical resection with residual disease, postoperative recovery, tumor-immune interaction, and radiation treatment scheduling. We study the mathematical properties of the model and obtain several results. The growth equation shows the existence of a threshold below which the tumor cannot survive and may disappear. The postoperative phase exhibits an early inflammatory stage followed by proliferative recovery. For the tumor-immune subsystem, equilibrium states and local stability conditions are identified. The radiation treatment problem is formulated as an optimal control problem, and the optimal strategy is shown to be of bang-bang type. The model suggests that tumor recurrence depends not only on tumor growth itself but also on residual disease, postoperative dynamics, immune response, and treatment timing.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148300976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Dynamical Behavior and Control Optimization of a Zoonotic Epidemic Model Incorporating Temperature Effects: Analysis and Simulations.","authors":"Kedeng Cheng, Yuanhua Qiao","doi":"10.1007/s11538-026-01681-9","DOIUrl":"10.1007/s11538-026-01681-9","url":null,"abstract":"<p><p>The transmission of numerous zoonotic diseases hinges on the tripartite interactions among humans, animals and the environment. The One Health paradigm underscores the inextricable interconnection between human health, animal health and ecosystem health. This study innovatively develops a multi-host zoonotic disease transmission model incorporating both pathogen compartment and temperature effects. Through rigorous mathematical analysis, we prove the non-negativity and boundedness of solutions and systematically investigate the existence conditions and global asymptotic stability characteristics of equilibrium points. For intervention strategies, we establish an optimal control framework, demonstrating not only the existence of optimal solutions but also deriving explicit analytical expressions. Numerical simulations validate the model's dynamic characteristics and enable quantitative evaluation of different interventions. Empirical analysis using China's brucellosis surveillance data reveal: there is a potential correlation between temperature changes and the cases number and the model achieves a goodness-of-fit of 0.98 for cumulative cases, strongly validating its reliability; temperature influence quantification showed each <math><msup><mn>10</mn> <mo>∘</mo></msup> </math> C increase in mean annual temperature reduced peak incidence by 3888 cases (17.91%) and 20-year cumulative cases by 49556 (20.37%). Multi-strategy comparative studies identified pharmaceutical intervention with 40% enhanced treatment efficacy as the optimal strategy, providing scientific evidence for zoonotic disease control.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148270680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An Idealized Model of Extrusome Ejection Through the Collapse of a Capsule.","authors":"Addie Harrison, Wanda Strychalski, Christina Hamlet, Laura Miller","doi":"10.1007/s11538-026-01684-6","DOIUrl":"10.1007/s11538-026-01684-6","url":null,"abstract":"<p><p>Extrusomes are extrusive organelles found in a variety of organisms, including cnidarians, dinoflagellates, and protists. These organelles typically contain a fluid-filled capsule that houses a structure, such as a coiled tubule or barb, which is rapidly ejected toward a target. Although the ejection mechanisms and morphology vary widely, a common feature is the rapid acceleration of the ejected structure through a fluid. In this paper, we develop an idealized model to simulate the collapse of an extrusome capsule, which enables the ejection of a barb or other internal structure. Specifically, we used the immersed boundary method to numerically simulate the collapse of a simplified capsule, modeled as the two sides of an elliptical shell with a flat plate along the bottom. As the capsule collapses, the elliptical sides straighten, ejecting both the internal fluid and the enclosed structure towards either free fluid or a flexible target. We investigate the effects of key model parameters, such as the size of the capsule opening. We also explore the role of the Reynolds number (Re), to consider the fluid dynamics across a range of regimes, from inertial-dominated flows relevant to some extrusome firings to viscous-dominated flows characteristic of cellular-scale processes. Our results demonstrate that decreasing the capsule opening gap size leads to increased firing velocity and shorter ejection times. Similarly, increasing the capsule's minor axis reduces the time it takes the barb to reach the target as a larger volume of fluid is moved. The relationship between Re and the time to contact is nonmonotonic, but higher Re values generally result in faster target contact, even at longer initial distances. Furthermore, we observe that higher Re values enhance the robustness of the target contact in different configurations. Finally, we quantify how the stiffness of the barb affects its ability to reach the target. We find that the large deformations of the flexible barbs slow their trajectories and that the stiffer barbs hit their targets sooner. These findings provide a foundational understanding of the biomechanics and fluid dynamics of extrusome ejection through the collapse of a capsule. The insights gained may contribute to the development of microinjectors in drug delivery, where precise and rapid mechanical movements are critical.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13275610/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148263499","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A New Insight into the Threshold and Oscillatory Regimes in Plant-Pathogen Models: A Nutrient-Driven Approach.","authors":"Dhruba Pariyar Damay, Angela Peace","doi":"10.1007/s11538-026-01682-8","DOIUrl":"10.1007/s11538-026-01682-8","url":null,"abstract":"<p><p>Across ecosystems, autotroph growth and susceptibility to disease are strongly constrained by the availability of essential nutrients such as nitrogen and phosphorus. Understanding how nutrient availability influences disease transmission is important for predicting disease persistence, outbreak risk, and long-term ecosystem dynamics under changing environmental conditions. At the same time, infectious diseases in autotrophs can reshape ecosystem processes by altering elemental recycling and the nutrient supply available to hosts. Here, we formulate a five-dimensional deterministic system of nonlinear ordinary differential equations within a disease-mediated nutrient dynamic framework. We incorporate novel nutrient-driven transmission and nonlinear resource uptake kinetics to capture the bidirectional relationships linking elemental cycles with infectious disease in a natural forest ecosystem. Using a combination of qualitative mathematical analysis, including proofs of solutions boundedness and derivations of basic reproductive number, and numerical bifurcation analyses, we evaluate the system's long-term behavior. Our results show that nutrient-disease feedbacks strongly regulate the distribution of host densities and nutrients between autotrophs and the abiotic environment. Incorporating nutrient-driven transmission reveals bifurcation structures distinct from frameworks with constant transmission, highlighting high sensitivity to nutrient availability and stronger nonlinear feedbacks. Bifurcation analyses indicate that nutrient enrichment lowers the transmission threshold for disease persistence and accelerates the onset of oscillatory dynamics with greater amplitude under high nutrient levels. Similarly, higher transmission rates reduce the nutrient threshold for disease persistence and shift oscillatory dynamics to emerge at lower nutrient levels. We further show that even small differences in infected host uptake rates strongly influence dynamics: lower uptake dampens oscillations and weakens feedbacks, whereas higher uptake amplifies bottom-up nutrient effects on disease and reinforces top-down effects on nutrient cycling, producing pronounced limit cycles in hosts, nutrients, and prevalence. Overall, nutrient-driven transmission alters thresholds and oscillatory regimes in ecosystem disease models, leading to dynamics that are not captured under constant transmission assumptions. This work advances applied ecosystem and ecological disease sciences by improving our understanding of disease transmission processes in plant communities.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC13269479/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148249334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modeling the Impact of Dynamic Gastric pH on Helicobacter pylori Eradication and Antibiotic Resistance Emergence.","authors":"Alex Hermann Sockeng Koussok, Edward Richard Onyango, Koichi Fujimoto, Jean Jules Tewa","doi":"10.1007/s11538-026-01677-5","DOIUrl":"10.1007/s11538-026-01677-5","url":null,"abstract":"<p><p>Helicobacter pylori infections present a persistent global health challenge due to increasing antibiotic resistance and the bacterium's ability to survive in the acidic gastric environment. Existing within-host models of H. pylori infection neglect the gastric pH fluctuation, despite its role in modulating bacterial growth and antibiotic efficacy. To address this gap, we extend a published in-host model by explicitly incorporating gastric pH as a dynamic state variable, influenced by three key physiological processes (i) bacterial urease which neutralizes gastric acid to create a protective niche; (ii) host acid secretion response, which attempts to restore baseline acidity; and (iii) dietary perturbations, which induce temporary pH changes. Equilibrium and stability analysis reveal pH-dependent reproductive thresholds <math> <mrow><msub><mi>R</mi> <mi>s</mi></msub> <mrow><mo>(</mo> <mi>H</mi> <mo>)</mo></mrow> </mrow> </math> and <math> <mrow><msub><mi>R</mi> <mi>r</mi></msub> <mrow><mo>(</mo> <mi>H</mi> <mo>)</mo></mrow> </mrow> </math> that determine the conditions for bacterial persistence and treatment outcome. Successful eradication requires driving both thresholds below unity. Numerical simulations validate distinct clinical scenarios including complete bacterial clearance, resistant strain dominance, stable bacterial coexistence, and oscillatory persistence. These outcomes emerge from the coupled interplay between antibiotic pressure, immune response, and pH regulation. Our model provides a comprehensive theoretical framework for understanding H. pylori treatment failures and highlights how adjuvant pH-modulation strategies could enhance antibiotic efficacy against resistant infections.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 7","pages":""},"PeriodicalIF":2.3,"publicationDate":"2026-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148249392","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}