数学最新文献

{"title":"Rainbow directed version of Dirac's theorem","authors":"Hao Li ,&nbsp;Luyi Li ,&nbsp;Ping Li ,&nbsp;Xueliang Li","doi":"10.1016/j.disc.2025.114506","DOIUrl":"10.1016/j.disc.2025.114506","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>s</mi><mo>]</mo><mo>}</mo></math></span> be a collection of not necessarily distinct graphs on the same vertex set <em>V</em>. A graph <em>H</em> is called <em>rainbow</em> in <span><math><mi>G</mi></math></span> if any two edges of <em>H</em> belong to different graphs of <span><math><mi>G</mi></math></span>. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem. In this paper, we prove a rainbow directed version of Dirac's theorem asymptotically: For each <span><math><mn>0</mn><mo>&lt;</mo><mi>ε</mi><mo>&lt;</mo><mn>1</mn></math></span>, there exists an integer <em>N</em> such that when <span><math><mi>n</mi><mo>≥</mo><mi>N</mi></math></span> the following holds. Let <span><math><mi>D</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>:</mo><mi>i</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo><mo>}</mo></math></span> be a collection of <em>n</em>-vertex digraphs on the same vertex set <em>V</em>. If both the out-degree and the in-degree of <em>v</em> are at least <span><math><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>+</mo><mi>ε</mi><mo>)</mo></mrow><mi>n</mi></math></span> for each vertex <span><math><mi>v</mi><mo>∈</mo><mi>V</mi></math></span> and each integer <span><math><mi>i</mi><mo>∈</mo><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, then <span><math><mi>D</mi></math></span> contains a rainbow Hamiltonian cycle. Furthermore, we provide a sufficient condition for the existence of arbitrary rainbow tournaments in a collection of <em>n</em>-vertex digraphs, where a <em>tournament</em> is an oriented graph of a complete graph.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114506"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761335","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}

{"title":"A central limit theorem for a card shuffling problem","authors":"Shane Chern ,&nbsp;Lin Jiu ,&nbsp;Italo Simonelli","doi":"10.1016/j.jcta.2025.106048","DOIUrl":"10.1016/j.jcta.2025.106048","url":null,"abstract":"<div><div>Given a positive integer <em>n</em>, consider a permutation of <em>n</em> objects chosen uniformly at random. In this permutation, we collect maximal subsequences consisting of consecutive numbers arranged in ascending order called blocks. Each block is then merged, and after all merges, the elements of this new set are relabeled from 1 to the current number of elements. We continue to permute and merge this new set uniformly at random until only one object is left. In this paper, we investigate the distribution of <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, the number of permutations needed for this process to end. In particular, we find explicit asymptotic expressions for the mean value <span><math><mi>E</mi><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, the variance <span><math><mrow><mi>Var</mi></mrow><mo>[</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></math></span>, and higher central moments, and show that <span><math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> satisfies a central limit theorem.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"214 ","pages":"Article 106048"},"PeriodicalIF":0.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759116","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}

{"title":"Every even cycle of order at least 8 has a mirror labeling","authors":"Jonathan Calzadillas ,&nbsp;Dan McQuillan ,&nbsp;James M. McQuillan","doi":"10.1016/j.disc.2025.114503","DOIUrl":"10.1016/j.disc.2025.114503","url":null,"abstract":"<div><div>A mirror labeling of the cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is a vertex-magic total labeling (VMTL) for the cycle <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with the property that if <em>x</em> is a vertex label, then <span><math><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>x</mi></math></span> is an edge label, for each <span><math><mn>1</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>2</mn><mi>n</mi></math></span>. (Note that any mirror labeling for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> can be easily converted into an edge-magic total labeling for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with the same property, and vice versa.) It has been known for decades that every odd cycle has a mirror labeling. Mirror labelings for <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> with <em>n</em> even are considerably more difficult to construct generally, with only the case <span><math><mi>n</mi><mo>≡</mo><mn>2</mn></math></span> mod 8 having been provided. In this paper, we obtain mirror labelings for all remaining cases, namely <span><math><mi>n</mi><mo>≡</mo><mn>6</mn></math></span> mod 8, <span><math><mi>n</mi><mo>≥</mo><mn>14</mn></math></span> and <span><math><mi>n</mi><mo>≡</mo><mn>0</mn></math></span> mod 4, <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>.</div><div>This result has significant ramifications for the study of vertex-magic total labelings of graphs generally. A quarter century ago, James MacDougall provided his guiding conjecture positing that every regular graph of degree at least 2 has a VMTL, except for the disjoint union <span><math><mn>2</mn><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. Ian Gray showed that every Hamiltonian regular graph of odd order possesses a VMTL, and introduced mirror vertex-magic total labelings as a tool to obtain a similar, general result for even order regular graphs. However, a key technical part of his program was missing, namely, the existence of mirror VMTLs for even order cycles. A mirror labeling is a particular kind of mirror VMTL. Thus, the results of this work provide the missing piece required for Gray's program. It now follows, that any Hamiltonian <span><math><mo>(</mo><mn>4</mn><mi>t</mi><mo>+</mo><mn>2</mn><mo>)</mo></math></span>-regular graph of any even order (<span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>, <span><math><mi>t</mi><mo>≥</mo><mn>0</mn></math></span>) must have a VMTL. This provides substantial new progress towards resolving MacDougall's Conjecture.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114503"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761184","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}

{"title":"Constructing balanced 2p-variable rotation symmetric Boolean functions with optimal algebraic immunity, high nonlinearity and high algebraic degree","authors":"Jiao Du ,&nbsp;Xiaoting Chen ,&nbsp;Yongxia Mao ,&nbsp;Qiang Gao ,&nbsp;Tianyin Wang","doi":"10.1016/j.disc.2025.114513","DOIUrl":"10.1016/j.disc.2025.114513","url":null,"abstract":"<div><div>How to design cryptographic Boolean functions is a challenge work in the design of stream and block ciphers. Cryptographic criteria of Boolean functions are connected with some known cryptanalytic attacks. To resist these known attacks, it is important to search Boolean functions with some properties, including balancedness, optimal algebraic immunity, high algebraic degree, good nonlinearity, high correlation immunity, etc. Rotation symmetric Boolean functions (RSBFs) can have these properties simultaneously. In this paper, we propose a new class of balanced 2<em>p</em>-variable RSBFs based on the compositions of an integer, where <em>p</em> is an odd prime. It is found that the functions of this class have optimal algebraic immunity, and their nonlinearity reaches <span><math><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>p</mi><mo>−</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mi>p</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><mn>2</mn><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>3</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>i</mi><mo>−</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>p</mi><mo>−</mo><mi>i</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>+</mo><mn>1</mn></math></span> (where <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>η</mi></mrow></msub><mo>=</mo><mfrac><mrow><mi>p</mi><mo>−</mo><mn>2</mn><mo>−</mo><mrow><mo>(</mo><mi>p</mi><mspace></mspace><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> and <em>p</em> is an odd prime), which is higher than the previously constructed balanced even-variable RSBFs with optimal algebraic immunity. At the same time, the algebraic degree of the constructed functions are studied, and the results show that they can be optimal under certain conditions.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 8","pages":"Article 114513"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761336","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}

{"title":"Two-grid mixed finite element analysis of semi-linear second order hyperbolic problem","authors":"Jiansong Zhang,&nbsp;Yanyu Liu","doi":"10.1016/j.camwa.2025.03.035","DOIUrl":"10.1016/j.camwa.2025.03.035","url":null,"abstract":"<div><div>A novel two-grid symmetric mixed finite element analysis is considered for semi-linear second order hyperbolic problem. To overcome the saddle-point problem resulted by the traditional mixed element methods, a new symmetric and positive definite mixed procedure is first introduced to solve semi-linear hyperbolic problem. Then the a priori error estimates both in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> and <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-norm senses are derived. Meanwhile, the two-grid technique proposed by Xu is applied to improve the resulting nonlinear numerical algorithm. Theoretical analysis is considered and the corresponding error estimate is derived under the relation <span><math><mi>h</mi><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. Finally, numerical examples are provided to test theoretical results and the efficiency of the proposed two-grid mixed element method.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"189 ","pages":"Pages 70-85"},"PeriodicalIF":2.9,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759981","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}

{"title":"Complete weight enumerators and weight hierarchies of two classes of linear codes","authors":"Jiawei He,&nbsp;Yinjin Liao","doi":"10.1016/j.disc.2025.114510","DOIUrl":"10.1016/j.disc.2025.114510","url":null,"abstract":"<div><div>The study of generalized Hamming weights for linear coding is an important area of research in coding theory as it provides valuable structural information about coding and plays a crucial role in determining the performance of coding in various applications. In this paper, two distinct classes of linear codes are devised through the selection of two particular defining sets. Initially, the weight distributions of these codes are ascertained. Subsequently, by conducting a detailed analysis of the intersections between the defining sets and the duals of all <em>r</em>-dimensional subspaces, the complete weight hierarchies of the two classes of linear codes are successfully determined.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 9","pages":"Article 114510"},"PeriodicalIF":0.7,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759598","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}

{"title":"Taut foliations, braid positivity, and unknot detection","authors":"Siddhi Krishna","doi":"10.1016/j.aim.2025.110233","DOIUrl":"10.1016/j.aim.2025.110233","url":null,"abstract":"<div><div>We study <em>positive braid knots</em> (the knots in the three–sphere realized as positive braid closures) through the lens of the L-space conjecture. This conjecture predicts that if <em>K</em> is a non-trivial positive braid knot, then for all <span><math><mi>r</mi><mo>&lt;</mo><mn>2</mn><mi>g</mi><mo>(</mo><mi>K</mi><mo>)</mo><mo>−</mo><mn>1</mn></math></span>, the 3-manifold obtained via <em>r</em>-framed Dehn surgery along <em>K</em> admits a taut foliation. Our main result provides some positive evidence towards this conjecture: we construct taut foliations in such manifolds whenever <span><math><mi>r</mi><mo>&lt;</mo><mi>g</mi><mo>(</mo><mi>K</mi><mo>)</mo><mo>+</mo><mn>1</mn></math></span>. As an application, we produce a novel braid positivity obstruction for cable knots by proving that <em>the</em> <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mo>±</mo><mn>1</mn><mo>)</mo></math></span><em>–cable of a knot K is braid positive if and only if K is the unknot</em>. We also present some curious examples demonstrating the limitations of our construction; these examples can also be viewed as providing some negative evidence towards the L-space conjecture. Finally, we apply our main result to produce taut foliations in some splicings of knot exteriors.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"470 ","pages":"Article 110233"},"PeriodicalIF":1.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759383","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}

{"title":"Dynamic analysis of the nonlinear fiber oscillator with fractional-order control in multi-filament fiber winding","authors":"Xinlei Fang ,&nbsp;Jianguo Liang ,&nbsp;Jiaquan Xie ,&nbsp;Zhanchun Chen ,&nbsp;Ting Wu ,&nbsp;Jianglin Liu","doi":"10.1016/j.chaos.2025.116385","DOIUrl":"10.1016/j.chaos.2025.116385","url":null,"abstract":"<div><div>This work aims to investigate the dynamic behavior of the carbon fiber oscillator in the winding process of the novel multi-filament fiber winding equipment for high-pressure vessels, employing an improved fractional-order controller. Specifically, a coupling suppression fractional-order PD (CS-FOPD) control strategy is proposed for the two-degree-of-freedom carbon fiber oscillator. The multiple-scales method is employed to analyze the primary resonance of the system and derive the steady-state amplitude and phase solutions. The stability of the system is assessed using the Routh-Hurwitz criterion, and the analytical solutions are validated through numerical simulations. Furthermore, numerical simulations are conducted to investigate the effects of CS-FOPD control on the amplitude-frequency characteristics, controller performance, multi-stable phenomenon, attraction domain structure, and global bifurcation behavior of the system under varying system parameters. In the absence of coupling suppression strategy, the primary-superharmonic resonance characteristics are analyzed using the multiple-scales method, and stability is evaluated via Lyapunov's first method. The nonlinear dynamics of the system are further explored through amplitude-frequency response curves and attraction domain evolution. Finally, the Melnikov method is employed to calculate the distance between stable and unstable manifolds, providing an analytical prediction of the onset of chaos. The main obtained simulation results showed the excellent performance of the proposed control strategy and the evolution of the system's dynamic characteristics. By analyzing the nonlinear dynamics of the system under the proposed control strategy, this study provides key insights for optimizing the stability and performance of the novel multi-filament carbon fiber winding equipment.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"196 ","pages":"Article 116385"},"PeriodicalIF":5.3,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759621","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}

{"title":"Conformal structures with (G_2)-symmetric twistor distribution","authors":"Pawel Nurowski,&nbsp;Katja Sagerschnig,&nbsp;Dennis The","doi":"10.1007/s13324-025-01039-9","DOIUrl":"10.1007/s13324-025-01039-9","url":null,"abstract":"<div><p>For any 4D split-signature conformal structure, there is an induced twistor distribution on the 5D space of all self-dual totally null 2-planes, which is (2, 3, 5) when the conformal structure is not anti-self-dual. Several examples where the twistor distribution achieves maximal symmetry (the split-real form of the exceptional simple Lie algebra of type <span>(textrm{G}_2)</span>) were previously known, and these include fascinating examples arising from the rolling of surfaces without twisting or slipping. We establish a complete local classification result for achieving maximal symmetry of the twistor distribution, identified among those homogeneous 4D split-conformal structures for which the conformal symmetry algebra induces a multiply-transitive action on the 5D space. Furthermore, we discuss geometric properties of these conformal structures such as their curvature, holonomy, and existence of Einstein representatives.</p></div>","PeriodicalId":48860,"journal":{"name":"Analysis and Mathematical Physics","volume":"15 3","pages":""},"PeriodicalIF":1.4,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s13324-025-01039-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143761764","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}

{"title":"Embedding spaces of split links","authors":"Rachael Boyd ,&nbsp;Corey Bregman","doi":"10.1016/j.aim.2025.110235","DOIUrl":"10.1016/j.aim.2025.110235","url":null,"abstract":"<div><div>We study the homotopy type of the space <span><math><mi>E</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span> of unparametrised embeddings of a split link <span><math><mi>L</mi><mo>=</mo><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊔</mo><mo>…</mo><mo>⊔</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. Our main result is a simple description of the fundamental group, or motion group, of <span><math><mi>E</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span>, and we extend this to a description of the motion group of embeddings in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>. The main tool we build is a semi-simplicial space of separating systems, which we show is homotopy equivalent to <span><math><mi>E</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span>. This combinatorial object provides a gateway to studying the homotopy type of <span><math><mi>E</mi><mo>(</mo><mi>L</mi><mo>)</mo></math></span> via the homotopy type of the spaces <span><math><mi>E</mi><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></math></span>.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"470 ","pages":"Article 110235"},"PeriodicalIF":1.5,"publicationDate":"2025-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143759384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}