Pub Date : 2026-01-14DOI: 10.1007/s11538-025-01583-2
Tyler Cassidy, Stuart T Johnston, Michael Plank, Imke Botha, Jennifer A Flegg, Ryan J Murphy, Sara Hamis
Mathematical modelling is a widely used approach to understand and interpret clinical trial data. This modelling typically involves fitting mechanistic mathematical models to data from individual trial participants. Despite the widespread adoption of this individual-based fitting, it is becoming increasingly common to take a hierarchical approach to parameter estimation, where modellers characterize the population parameter distributions, rather than considering each individual independently. This hierarchical parameter estimation is standard in pharmacometric modelling. However, many of the existing techniques for parameter identifiability do not immediately translate from the individual-based fitting to the hierarchical setting. In this work, we propose a nonparametric approach to study practical identifiability within a hierarchical parameter estimation framework. We focus on the commonly used nonlinear mixed effects framework and investigate two well-studied examples from the pharmacometrics and viral dynamics literature to illustrate the potential utility of our approach.
{"title":"A Nonparametric Approach to Practical Identifiability of Nonlinear Mixed Effects Models.","authors":"Tyler Cassidy, Stuart T Johnston, Michael Plank, Imke Botha, Jennifer A Flegg, Ryan J Murphy, Sara Hamis","doi":"10.1007/s11538-025-01583-2","DOIUrl":"10.1007/s11538-025-01583-2","url":null,"abstract":"<p><p>Mathematical modelling is a widely used approach to understand and interpret clinical trial data. This modelling typically involves fitting mechanistic mathematical models to data from individual trial participants. Despite the widespread adoption of this individual-based fitting, it is becoming increasingly common to take a hierarchical approach to parameter estimation, where modellers characterize the population parameter distributions, rather than considering each individual independently. This hierarchical parameter estimation is standard in pharmacometric modelling. However, many of the existing techniques for parameter identifiability do not immediately translate from the individual-based fitting to the hierarchical setting. In this work, we propose a nonparametric approach to study practical identifiability within a hierarchical parameter estimation framework. We focus on the commonly used nonlinear mixed effects framework and investigate two well-studied examples from the pharmacometrics and viral dynamics literature to illustrate the potential utility of our approach.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 2","pages":"19"},"PeriodicalIF":2.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12799758/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145965141","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}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2026.114106
Divyang G. Bhimani , Diksha Dhingra , Vijay Kumar Sohani
The study of low regularity Cauchy data for nonlinear dispersive PDEs has been successfully achieved using modulation spaces in recent years. In this paper, we study the inhomogeneous nonlinear Schrödinger equation (INLS) on the whole space having initial data in modulation spaces. In the subcritical regime , we establish local well-posedness in . By adapting Bourgain's high-low decomposition method, we establish global well-posedness in with and p sufficiently close to 2. This is the first global well-posedness result for INLS in modulation spaces, which contains certain Sobolev and Sobolev spaces.
{"title":"Low-regularity global solution of the inhomogeneous nonlinear Schrödinger equations in modulation spaces","authors":"Divyang G. Bhimani , Diksha Dhingra , Vijay Kumar Sohani","doi":"10.1016/j.jde.2026.114106","DOIUrl":"10.1016/j.jde.2026.114106","url":null,"abstract":"<div><div>The study of low regularity Cauchy data for nonlinear dispersive PDEs has been successfully achieved using modulation spaces in recent years. In this paper, we study the inhomogeneous nonlinear Schrödinger equation (INLS)<span><span><span><math><mi>i</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>Δ</mi><mi>u</mi><mo>±</mo><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mo>−</mo><mi>b</mi></mrow></msup><mo>|</mo><mi>u</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>α</mi></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mspace></mspace><mspace></mspace><mo>(</mo><mi>b</mi><mo>,</mo><mi>α</mi><mo>></mo><mn>0</mn><mo>)</mo></math></span></span></span> on the whole space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> having initial data in modulation spaces. In the subcritical regime <span><math><mo>(</mo><mn>0</mn><mo><</mo><mi>α</mi><mo><</mo><mfrac><mrow><mn>4</mn><mo>−</mo><mn>2</mn><mi>b</mi></mrow><mrow><mi>n</mi></mrow></mfrac><mo>)</mo></math></span>, we establish local well-posedness in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>2</mn><mo>,</mo><mfrac><mrow><mi>α</mi><mo>+</mo><mn>2</mn></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></msup><mo>(</mo><mo>⊃</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mspace></mspace><mtext>for</mtext><mspace></mspace><mi>s</mi><mo>></mo><mfrac><mrow><mi>n</mi><mi>α</mi></mrow><mrow><mn>2</mn><mo>(</mo><mi>α</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></mfrac><mo>)</mo></math></span>. By adapting Bourgain's high-low decomposition method, we establish global well-posedness in <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>p</mi><mo>,</mo><mfrac><mrow><mi>p</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mfrac></mrow></msup></math></span> with <span><math><mn>2</mn><mo><</mo><mi>p</mi></math></span> and <em>p</em> sufficiently close to 2. This is the first global well-posedness result for INLS in modulation spaces, which contains certain Sobolev <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span> <span><math><mo>(</mo><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><msubsup><mrow><mi>L</mi></mrow><mrow><mi>s</mi></mrow><mrow><mi>p</mi></mrow></msubsup><mo>−</mo></math></span>Sobolev spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114106"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.dam.2025.12.068
Bernard De Baets , Emilio De Santis
A set of candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For candidates with , we identify a voting profile with voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large.
{"title":"Voting profiles admitting all candidates as knockout winners","authors":"Bernard De Baets , Emilio De Santis","doi":"10.1016/j.dam.2025.12.068","DOIUrl":"10.1016/j.dam.2025.12.068","url":null,"abstract":"<div><div>A set of <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> candidates is presented to a commission. At every round, each member of this commission votes by pairwise comparison, and one-half of the candidates is deleted from the tournament, the remaining ones proceeding to the next round until the <span><math><mi>n</mi></math></span>th round (the final one) in which the final winner is declared. The candidates are arranged on a board in a given order, which is maintained among the remaining candidates at all rounds. A study of the size of the commission is carried out in order to obtain the desired result of any candidate being a possible winner. For <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup></math></span> candidates with <span><math><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, we identify a voting profile with <span><math><mrow><mn>4</mn><mi>n</mi><mo>−</mo><mn>3</mn></mrow></math></span> voters such that any candidate could win simply by choosing a proper initial order of the candidates. Moreover, in the setting of a random number of voters, we obtain the same results, with high probability, when the expected number of voters is large.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 340-351"},"PeriodicalIF":1.0,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980019","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.jde.2026.114108
Zijin Li , Ning Liu , Taoran Zhou
We consider the generalized Leray's problem with the Navier-slip boundary condition in an infinite pipe . We show that if the flux Φ of the solution is no larger than a critical value that is independent with the friction ratio of the Navier-slip boundary condition, the solution to the problem must be the parallel Poiseuille flow with the given flux. Compared with known related 3D results, this seems to be the first conclusion with the size of critical flux being uniform with the friction ratio , and it is surprising since the prescribed uniqueness breaks down immediately when , even if .
Our proof relies primarily on a refined gradient estimate of the Poiseuille flow with the Navier-slip boundary condition. Additionally, we prove the critical flux provided that Σ is a unit disk.
{"title":"A refined uniqueness result of Leray's problem in an infinite-long pipe with the Navier-slip boundary","authors":"Zijin Li , Ning Liu , Taoran Zhou","doi":"10.1016/j.jde.2026.114108","DOIUrl":"10.1016/j.jde.2026.114108","url":null,"abstract":"<div><div>We consider the generalized Leray's problem with the Navier-slip boundary condition in an infinite pipe <span><math><mi>D</mi><mo>=</mo><mi>Σ</mi><mo>×</mo><mi>R</mi></math></span>. We show that if the flux Φ of the solution is no larger than a critical value that is <em>independent with the friction ratio</em> of the Navier-slip boundary condition, the solution to the problem must be the parallel Poiseuille flow with the given flux. Compared with known related 3D results, this seems to be the first conclusion with the size of critical flux being uniform with the friction ratio <span><math><mi>α</mi><mo>∈</mo><mo>]</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>]</mo></math></span>, and it is surprising since the prescribed uniqueness breaks down immediately when <span><math><mi>α</mi><mo>=</mo><mn>0</mn></math></span>, even if <span><math><mi>Φ</mi><mo>=</mo><mn>0</mn></math></span>.</div><div>Our proof relies primarily on a refined gradient estimate of the Poiseuille flow with the Navier-slip boundary condition. Additionally, we prove the critical flux <span><math><msub><mrow><mi>Φ</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>≥</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>16</mn></mrow></mfrac></math></span> provided that Σ is a unit disk.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"461 ","pages":"Article 114108"},"PeriodicalIF":2.3,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1007/s11538-025-01587-y
Alexis M Hill, Kelly To, Claus O Wilke
Synonymous codon usage can influence protein expression, since codons with high numbers of corresponding tRNAs are naturally translated more rapidly than codons with fewer corresponding tRNAs. Although translation efficiency ultimately depends on the concentration of aminoacylated (charged) tRNAs, many theoretical models of translation have ignored tRNA dynamics and treated charged tRNAs as fixed resources. This simplification potentially limits these models from making accurate predictions in situations where charged tRNAs become limiting. Here, we derive a mathematical model of translation with explicit tRNA dynamics and tRNA re-charging, based on a stochastic simulation of this system that was previously applied to investigate codon usage in the context of gene overexpression. We use the mathematical model to systematically explore the relationship between codon usage and the protein expression rate, and find that in the regime where tRNA charging is a limiting reaction, it is always optimal to match codon frequencies to the tRNA pool. Conversely, when tRNA charging is not limiting, using 100% of the preferred codon is optimal for protein production. We also use the tRNA dynamics model to augment a whole-cell simulation of bacteriophage T7. Using this model, we demonstrate that the high expression rate of the T7 major capsid gene causes rare charged tRNAs to become entirely depleted, which explains the sensitivity of the major capsid gene to codon deoptimization.
{"title":"Availability of Charged tRNAs Drives Maximal Protein Synthesis at Intermediate Levels of Codon Usage Bias.","authors":"Alexis M Hill, Kelly To, Claus O Wilke","doi":"10.1007/s11538-025-01587-y","DOIUrl":"10.1007/s11538-025-01587-y","url":null,"abstract":"<p><p>Synonymous codon usage can influence protein expression, since codons with high numbers of corresponding tRNAs are naturally translated more rapidly than codons with fewer corresponding tRNAs. Although translation efficiency ultimately depends on the concentration of aminoacylated (charged) tRNAs, many theoretical models of translation have ignored tRNA dynamics and treated charged tRNAs as fixed resources. This simplification potentially limits these models from making accurate predictions in situations where charged tRNAs become limiting. Here, we derive a mathematical model of translation with explicit tRNA dynamics and tRNA re-charging, based on a stochastic simulation of this system that was previously applied to investigate codon usage in the context of gene overexpression. We use the mathematical model to systematically explore the relationship between codon usage and the protein expression rate, and find that in the regime where tRNA charging is a limiting reaction, it is always optimal to match codon frequencies to the tRNA pool. Conversely, when tRNA charging is not limiting, using 100% of the preferred codon is optimal for protein production. We also use the tRNA dynamics model to augment a whole-cell simulation of bacteriophage T7. Using this model, we demonstrate that the high expression rate of the T7 major capsid gene causes rare charged tRNAs to become entirely depleted, which explains the sensitivity of the major capsid gene to codon deoptimization.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 2","pages":"16"},"PeriodicalIF":2.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.ncbi.nlm.nih.gov/pmc/articles/PMC12799730/pdf/","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145965198","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}
Pub Date : 2026-01-14DOI: 10.1016/j.chaos.2026.117902
Xuyang Wang , Xin Ning , Zheng Wang , Zhansheng Chen
This paper proposes a fixed-time stability-based control strategy for high-speed unmanned systems (HSUSs) considering practical switching topology scenarios. First, a fixed-time disturbance observer (FixTDO) with initial condition-independent convergence time is designed to handle the system disturbances. Subsequently, an auxiliary system is constructed to address communication switching topology constraints. Furthermore, a fixed-time controller is synthesized to enhance transient performance and system robustness, with a first-order filter incorporated to eliminate differentiation-induced computational divergence. Finally, comparative simulation studies validate the effectiveness and operational feasibility of the proposed algorithm.
{"title":"Fixed-time control strategy for high-speed unmanned system under switching topology","authors":"Xuyang Wang , Xin Ning , Zheng Wang , Zhansheng Chen","doi":"10.1016/j.chaos.2026.117902","DOIUrl":"10.1016/j.chaos.2026.117902","url":null,"abstract":"<div><div>This paper proposes a fixed-time stability-based control strategy for high-speed unmanned systems (HSUSs) considering practical switching topology scenarios. First, a fixed-time disturbance observer (FixTDO) with initial condition-independent convergence time is designed to handle the system disturbances. Subsequently, an auxiliary system is constructed to address communication switching topology constraints. Furthermore, a fixed-time controller is synthesized to enhance transient performance and system robustness, with a first-order filter incorporated to eliminate differentiation-induced computational divergence. Finally, comparative simulation studies validate the effectiveness and operational feasibility of the proposed algorithm.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"206 ","pages":"Article 117902"},"PeriodicalIF":5.6,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145976411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1007/s11538-025-01582-3
Gabriel K Kosmacher, Dillon Max, Zoi Rapti, Carla E Cáceres, Tara E Stewart Merrill
In both human and wildlife disease systems, temporal shifts in host immunity may shape the timing and severity of epidemics. Yet, immune responses, as well as seasonal patterns in their expression, are difficult to measure. Rather, field studies collect phenomenological data on infection outcomes. Pairing epidemic data of multiple outbreaks with models that directly parameterize immune metrics can be a powerful approach for exploring the role of time-varying immunity on disease. Field data can be used to determine how well a parameterized model can reproduce trends and differences observed among outbreaks.Previous work in the Daphnia dentifera-Metschnikowia bicuspidata focal host-fungal pathogen disease system has not taken full advantage of coupling patterns in nature with mechanisms predicted by theory. Here, we study a mathematical model accounting for host immunity in the form of resistance to and recovery from M. bicuspidata infections and temporal variation in key aspects of the system's epidemiology and ecology. Specifically, host population birth, predation and transmission rates, the fraction of recovering hosts, as well as the fungal spore yield were allowed to vary within the epidemic season. Modifying the system's carrying capacity produces good correspondence between observed and model-estimated densities. Adjusting the transmission rate, spore yield, and the fraction of recovering hosts, captures the timing of disease outbreaks, as well as other qualitative features of outbreaks, such as the disparity between the prevalence of early- and late-stage infections. Our findings suggest that host immunological parameters are an important within-host constraint on disease dynamics.
{"title":"The Role of Host Immunity and the Environment in Seasonal Disease Dynamics.","authors":"Gabriel K Kosmacher, Dillon Max, Zoi Rapti, Carla E Cáceres, Tara E Stewart Merrill","doi":"10.1007/s11538-025-01582-3","DOIUrl":"https://doi.org/10.1007/s11538-025-01582-3","url":null,"abstract":"<p><p>In both human and wildlife disease systems, temporal shifts in host immunity may shape the timing and severity of epidemics. Yet, immune responses, as well as seasonal patterns in their expression, are difficult to measure. Rather, field studies collect phenomenological data on infection outcomes. Pairing epidemic data of multiple outbreaks with models that directly parameterize immune metrics can be a powerful approach for exploring the role of time-varying immunity on disease. Field data can be used to determine how well a parameterized model can reproduce trends and differences observed among outbreaks.Previous work in the Daphnia dentifera-Metschnikowia bicuspidata focal host-fungal pathogen disease system has not taken full advantage of coupling patterns in nature with mechanisms predicted by theory. Here, we study a mathematical model accounting for host immunity in the form of resistance to and recovery from M. bicuspidata infections and temporal variation in key aspects of the system's epidemiology and ecology. Specifically, host population birth, predation and transmission rates, the fraction of recovering hosts, as well as the fungal spore yield were allowed to vary within the epidemic season. Modifying the system's carrying capacity produces good correspondence between observed and model-estimated densities. Adjusting the transmission rate, spore yield, and the fraction of recovering hosts, captures the timing of disease outbreaks, as well as other qualitative features of outbreaks, such as the disparity between the prevalence of early- and late-stage infections. Our findings suggest that host immunological parameters are an important within-host constraint on disease dynamics.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"88 2","pages":"18"},"PeriodicalIF":2.2,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145965302","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}
Pub Date : 2026-01-14DOI: 10.1016/j.disc.2026.114998
Reza Naserasr , Weiqiang Yu
A balanced k-coloring of a signed graph which has no negative loop is to partition its vertices into k sets each of which induces a balanced subgraph, that is a subgraph with no negative cycle. The notion, through basic graph operation, captures the classic proper coloring of graphs as special case.
Having observed the importance of balanced 2-coloring, in this work we study structural conditions which permit a signed graph to admit a 2-coloring. More precisely, slightly modifying the notion of color-critical, we say a signed graph is k-critical if it admits no balanced k-coloring but every proper subgraph of it admits such a coloring.
We show that if is a 2-critical signed graph whose underlying graph is not or an odd cycle, then where d is the maximum number of vertex disjoint digons in . As a corollary we conclude that, except for the signed graph , any signed simple graph with maximum average degree at most 4.2 admits a balanced 2-coloring.
{"title":"Density of 2-critical signed graphs","authors":"Reza Naserasr , Weiqiang Yu","doi":"10.1016/j.disc.2026.114998","DOIUrl":"10.1016/j.disc.2026.114998","url":null,"abstract":"<div><div>A balanced <em>k</em>-coloring of a signed graph <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> which has no negative loop is to partition its vertices into <em>k</em> sets each of which induces a balanced subgraph, that is a subgraph with no negative cycle. The notion, through basic graph operation, captures the classic proper coloring of graphs as special case.</div><div>Having observed the importance of balanced 2-coloring, in this work we study structural conditions which permit a signed graph to admit a 2-coloring. More precisely, slightly modifying the notion of color-critical, we say a signed graph is <em>k</em>-critical if it admits no balanced <em>k</em>-coloring but every proper subgraph of it admits such a coloring.</div><div>We show that if <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span> is a 2-critical signed graph whose underlying graph is not <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> or an odd cycle, then<span><span><span><math><mo>|</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mfrac><mrow><mn>21</mn><mo>|</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>|</mo><mo>−</mo><mn>2</mn><mi>d</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>10</mn></mrow></mfrac></math></span></span></span> where <em>d</em> is the maximum number of vertex disjoint digons in <span><math><mo>(</mo><mi>G</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>. As a corollary we conclude that, except for the signed graph <span><math><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub><mo>,</mo><mo>−</mo><mo>)</mo></math></span>, any signed simple graph with maximum average degree at most 4.2 admits a balanced 2-coloring.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 6","pages":"Article 114998"},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145980076","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-14DOI: 10.1016/j.difgeo.2026.102332
Farida Ghazawneh
Kennard, Khalili Samani, and Searle showed that for a -torus, , acting on a closed, positively curved Riemannian n-manifold, , with a non-empty fixed point set for n large enough and r approximately half the dimension of M, then is homotopy equivalent to , , , or a lens space. In this paper, we lower r to approximately and show that we still obtain the same result.
{"title":"Z2-torus actions on positively curved manifolds","authors":"Farida Ghazawneh","doi":"10.1016/j.difgeo.2026.102332","DOIUrl":"10.1016/j.difgeo.2026.102332","url":null,"abstract":"<div><div>Kennard, Khalili Samani, and Searle showed that for a <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-torus, <span><math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>, acting on a closed, positively curved Riemannian <em>n</em>-manifold, <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, with a non-empty fixed point set for <em>n</em> large enough and <em>r</em> approximately half the dimension of <em>M</em>, then <span><math><msup><mrow><mi>M</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is homotopy equivalent to <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>R</mi><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, <span><math><mi>C</mi><msup><mrow><mi>P</mi></mrow><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></math></span>, or a lens space. In this paper, we lower <em>r</em> to approximately <span><math><mn>2</mn><mi>n</mi><mo>/</mo><mn>5</mn></math></span> and show that we still obtain the same result.</div></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"103 ","pages":"Article 102332"},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145981244","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}
Pub Date : 2026-01-14DOI: 10.1007/s00010-026-01263-z
Péter Tóth
In this paper we investigate the functional equation
$$begin{aligned} varphi left( frac{x+y}{2} right) left( psi _1(x) - psi _2(y) right) = 0 quad left( hbox { for all } x in I_1 hbox { and } y in I_2 right) end{aligned}$$
where ( I_1 , I_2 ) are open intervals of ( mathbb {R}), ( J = frac{1}{2} left( I_1 + I_2 right) ) moreover ( psi _1: I_1 rightarrow mathbb {R}), ( psi _2: I_2 rightarrow mathbb {R}) and ( varphi : J rightarrow mathbb {R}) are unknown functions. We describe the structure of the possible solutions assuming that ( varphi ) is measurable. In the case when ( varphi ) is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations.
本文研究了泛函方程$$begin{aligned} varphi left( frac{x+y}{2} right) left( psi _1(x) - psi _2(y) right) = 0 quad left( hbox { for all } x in I_1 hbox { and } y in I_2 right) end{aligned}$$，其中( I_1 , I_2 )为( mathbb {R})、( J = frac{1}{2} left( I_1 + I_2 right) )的开区间，( psi _1: I_1 rightarrow mathbb {R})、( psi _2: I_2 rightarrow mathbb {R})、( varphi : J rightarrow mathbb {R})为未知函数。假设( varphi )是可测量的，我们描述可能解的结构。当( varphi )为导数时，给出了解的完整表征。此外，我们还给出了一个由不规则达布函数组成的解的例子。这为第59届泛函方程国际研讨会上提出的一个开放性问题提供了答案。
{"title":"Measurable solutions of an alternative functional equation","authors":"Péter Tóth","doi":"10.1007/s00010-026-01263-z","DOIUrl":"10.1007/s00010-026-01263-z","url":null,"abstract":"<div><p>In this paper we investigate the functional equation </p><div><div><span>$$begin{aligned} varphi left( frac{x+y}{2} right) left( psi _1(x) - psi _2(y) right) = 0 quad left( hbox { for all } x in I_1 hbox { and } y in I_2 right) end{aligned}$$</span></div></div><p>where <span>( I_1 , I_2 )</span> are open intervals of <span>( mathbb {R})</span>, <span>( J = frac{1}{2} left( I_1 + I_2 right) )</span> moreover <span>( psi _1: I_1 rightarrow mathbb {R})</span>, <span>( psi _2: I_2 rightarrow mathbb {R})</span> and <span>( varphi : J rightarrow mathbb {R})</span> are unknown functions. We describe the structure of the possible solutions assuming that <span>( varphi )</span> is measurable. In the case when <span>( varphi )</span> is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the <i>59th International Symposium on Functional Equations</i>.</p></div>","PeriodicalId":55611,"journal":{"name":"Aequationes Mathematicae","volume":"100 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2026-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00010-026-01263-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145982868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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