Russian Mathematical Surveys

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class="metrics__score">2.000</span> </div> <div class="metrics__metric"> <span class="metrics__description">Citescore</span> <span class="metrics__score">2.1</span> </div> </div> </div>  <div class="cf mb-1">    <div class="tabs cf mb-2 mt-1 tabs--vertical" id="wd-jnl-hm-art-list">  <div role="tablist"> <button role="tab" aria-selected="false" aria-controls="most-read-tab" id="most-read" class="event_tabs" tabindex="-1"> Most read </button> <button role="tab" aria-selected="true" aria-controls="latest-articles-tab" id="latest-articles" class="event_tabs"> Latest articles </button> <button role="tab" aria-selected="false" aria-controls="review-articles-tab" id="review-articles" class="event_tabs" tabindex="-1"> Review articles </button> </div>   <div tabindex="0" role="tabpanel" id="most-read-tab" aria-labelledby="most-read" hidden="hidden"> <div class=" reveal-container reveal-closed reveal-enabled reveal-container--jnl-tab"> <h2 class="tabpanel__title"> <button type="button" class="reveal-trigger event_tabs-accordion" aria-expanded="false"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg>Most read</button> </h2> <div class="reveal-content tabpanel__content" style="display: none"> <p> <button data-reveal-label-alt="Close all abstracts" class="reveal-all-trigger mr-2 small" data-reveal-text="Open all abstracts" data-link-purpose-append="in this tab" data-link-purpose-append-open="in this tab"> Open all abstracts<span class="offscreen-hidden">, in this tab</span> </button> </p>  <div class="art-list"> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM1997v052n06ABEH002155" class="art-list-item-title event_main-link">Quantum computations: algorithms and error correction</a> <p class="small art-list-item-meta"> A Yu Kitaev 1997 <em>Russ. Math. Surv.</em> <b>52</b> 1191 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Quantum computations: algorithms and error correction" data-link-purpose-append-open="Quantum computations: algorithms and error correction">Open abstract</span> </button> <a href="/article/10.1070/RM1997v052n06ABEH002155/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Quantum computations: algorithms and error correction</span></a> <a href="/article/10.1070/RM1997v052n06ABEH002155/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Quantum computations: algorithms and error correction</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p>Contents §0. Introduction §1. Abelian problem on the stabilizer §2. Classical models of computations 2.1. Boolean schemes and sequences of operations 2.2. Reversible computations §3. Quantum formalism 3.1. Basic notions and notation 3.2. Transformations of mixed states 3.3. Accuracy §4. Quantum models of computations 4.1. Definitions and basic properties 4.2. Construction of various operators from the elements of a basis 4.3. Generalized quantum control and universal schemes §5. Measurement operators §6. Polynomial quantum algorithm for the stabilizer problem §7. Computations with perturbations: the choice of a model §8. Quantum codes (definitions and general properties) 8.1. Basic notions and ideas 8.2. One-to-one codes 8.3. Many-to-one codes §9. Symplectic (additive) codes 9.1. Algebraic preparation 9.2. The basic construction 9.3. Error correction procedure 9.4. Torus codes §10. Error correction in the computation process: general principles 10.1. Definitions and results 10.2. Proofs §11. Error correction: concrete procedures 11.1. The symplecto-classical case 11.2. The case of a complete basis</p><p>Bibliography </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM1997v052n06ABEH002155">https://doi.org/10.1070/RM1997v052n06ABEH002155</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM9937" class="art-list-item-title event_main-link">Newton polytopes and tropical geometry</a> <p class="small art-list-item-meta"> B. Ya. Kazarnovskii <em>et al</em> 2021 <em>Russ. Math. Surv.</em> <b>76</b> 91 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Newton polytopes and tropical geometry" data-link-purpose-append-open="Newton polytopes and tropical geometry">Open abstract</span> </button> <a href="/article/10.1070/RM9937/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Newton polytopes and tropical geometry</span></a> <a href="/article/10.1070/RM9937/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Newton polytopes and tropical geometry</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> The practice of bringing together the concepts of 'Newton polytopes', 'toric varieties', 'tropical geometry', and 'Gröbner bases' has led to the formation of stable and mutually beneficial connections between algebraic geometry and convex geometry. This survey is devoted to the current state of the area of mathematics that describes the interaction and applications of these concepts. </p><p> Bibliography: 68 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM9937">https://doi.org/10.1070/RM9937</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM9956" class="art-list-item-title event_main-link">Semantic limits of dense combinatorial objects</a> <p class="small art-list-item-meta"> L. N. Coregliano and A. A. Razborov 2020 <em>Russ. Math. Surv.</em> <b>75</b> 627 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Semantic limits of dense combinatorial objects" data-link-purpose-append-open="Semantic limits of dense combinatorial objects">Open abstract</span> </button> <a href="/article/10.1070/RM9956/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Semantic limits of dense combinatorial objects</span></a> <a href="/article/10.1070/RM9956/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Semantic limits of dense combinatorial objects</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> The theory of limits of discrete combinatorial objects has been thriving for the last decade or so. The syntactic, algebraic approach to the subject is popularly known as 'flag algebras', while the semantic, geometric approach is often associated with the name 'graph limits'. The language of graph limits is generally more intuitive and expressible, but a price that one has to pay for it is that it is better suited for the case of ordinary graphs than for more general combinatorial objects. Accordingly, there have been several attempts in the literature, of varying degree of generality, to define limit objects for more complicated combinatorial structures. This paper is another attempt at a workable general theory of dense limit objects. Unlike previous efforts in this direction (with the notable exception of [5] by Aroskar and Cummings), our account is based on the same concepts from first-order logic and model theory as in the theory of flag algebras. It is shown how our definitions naturally encompass a host of previously considered cases (graphons, hypergraphons, digraphons, permutons, posetons, coloured graphs, and so on), and the fundamental properties of existence and uniqueness are extended to this more general case. Also given is an intuitive general proof of the continuous version of the Induced Removal Lemma based on the compactness theorem for propositional calculus. Use is made of the notion of open interpretation that often allows one to transfer methods and results from one situation to another. Again, it is shown that some previous arguments can be quite naturally framed using this language. </p><p> Bibliography: 68 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM9956">https://doi.org/10.1070/RM9956</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM9990" class="art-list-item-title event_main-link">Equivariant minimal model program</a> <p class="small art-list-item-meta"> Yu. G. Prokhorov 2021 <em>Russ. Math. Surv.</em> <b>76</b> 461 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Equivariant minimal model program" data-link-purpose-append-open="Equivariant minimal model program">Open abstract</span> </button> <a href="/article/10.1070/RM9990/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Equivariant minimal model program</span></a> <a href="/article/10.1070/RM9990/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Equivariant minimal model program</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> The purpose of the survey is to systematize a vast amount of information about the minimal model program for varieties with group actions. We discuss the basic methods of the theory and give sketches of the proofs of some principal results. </p><p> Bibliography: 243 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM9990">https://doi.org/10.1070/RM9990</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10023" class="art-list-item-title event_main-link">Dynamical phenomena connected with stability loss of equilibria and periodic trajectories</a> <p class="small art-list-item-meta"> A. I. Neishtadt and D. V. Treschev 2021 <em>Russ. Math. Surv.</em> <b>76</b> 883 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Dynamical phenomena connected with stability loss of equilibria and periodic trajectories" data-link-purpose-append-open="Dynamical phenomena connected with stability loss of equilibria and periodic trajectories">Open abstract</span> </button> <a href="/article/10.1070/RM10023/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Dynamical phenomena connected with stability loss of equilibria and periodic trajectories</span></a> <a href="/article/10.1070/RM10023/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Dynamical phenomena connected with stability loss of equilibria and periodic trajectories</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> This is a study of a dynamical system depending on a parameter <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/5/883/revision1/RMS_76_5_883ieqn1.gif" style="max-width: 100%;" alt="$\kappa$" align="top"></img></span><script type="math/tex">\kappa</script></span></span>. Under the assumption that the system has a family of equilibrium positions or periodic trajectories smoothly depending on <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/5/883/revision1/RMS_76_5_883ieqn1.gif" style="max-width: 100%;" alt="$\kappa$" align="top"></img></span><script type="math/tex">\kappa</script></span></span>, the focus is on details of stability loss through various bifurcations (Poincaré–Andronov– Hopf, period-doubling, and so on). Two basic formulations of the problem are considered. In the first, <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/5/883/revision1/RMS_76_5_883ieqn1.gif" style="max-width: 100%;" alt="$\kappa$" align="top"></img></span><script type="math/tex">\kappa</script></span></span> is constant and the subject of the analysis is the phenomenon of a soft or hard loss of stability. In the second, <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/5/883/revision1/RMS_76_5_883ieqn1.gif" style="max-width: 100%;" alt="$\kappa$" align="top"></img></span><script type="math/tex">\kappa</script></span></span> varies slowly with time (the case of a dynamic bifurcation). In the simplest situation <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/5/883/revision1/RMS_76_5_883ieqn2.gif" style="max-width: 100%;" alt="$\kappa=\varepsilon t$" align="top"></img></span><script type="math/tex">\kappa=\varepsilon t</script></span></span>, where <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/5/883/revision1/RMS_76_5_883ieqn3.gif" style="max-width: 100%;" alt="$\varepsilon$" align="top"></img></span><script type="math/tex">\varepsilon</script></span></span> is a small parameter. More generally, <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/5/883/revision1/RMS_76_5_883ieqn4.gif" style="max-width: 100%;" alt="$\kappa(t)$" align="top"></img></span><script type="math/tex">\kappa(t)</script></span></span> may be a solution of a slow differential equation. In the case of a dynamic bifurcation the analysis is mainly focused around the phenomenon of stability loss delay. </p><p> Bibliography: 88 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10023">https://doi.org/10.1070/RM10023</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10001" class="art-list-item-title event_main-link">Analytic moduli for parabolic Dulac germs</a> <p class="small art-list-item-meta"> P. Mardešić and M. Resman 2021 <em>Russ. Math. Surv.</em> <b>76</b> 389 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Analytic moduli for parabolic Dulac germs" data-link-purpose-append-open="Analytic moduli for parabolic Dulac germs">Open abstract</span> </button> <a href="/article/10.1070/RM10001/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Analytic moduli for parabolic Dulac germs</span></a> <a href="/article/10.1070/RM10001/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Analytic moduli for parabolic Dulac germs</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> This paper gives moduli of analytic classification for parabolic <i>Dulac</i> germs (that is, <i>almost regular</i> germs). Dulac germs appear as first return maps of hyperbolic polycycles. Their moduli are given by a sequence of <i>Écalle–Voronin</i>-type germs of analytic diffeomorphisms. The result is stated in a broader class of <i>parabolic generalized Dulac germs</i> having power- logarithmic asymptotic expansions. </p><p> Bibliography: 23 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10001">https://doi.org/10.1070/RM10001</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM9963" class="art-list-item-title event_main-link">Adjunction in 2-categories</a> <p class="small art-list-item-meta"> D. B. Kaledin 2020 <em>Russ. Math. Surv.</em> <b>75</b> 883 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Adjunction in 2-categories" data-link-purpose-append-open="Adjunction in 2-categories">Open abstract</span> </button> <a href="/article/10.1070/RM9963/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Adjunction in 2-categories</span></a> <a href="/article/10.1070/RM9963/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Adjunction in 2-categories</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> The aim of the paper is to introduce an approach to the theory of 2-categories which is based on systematic use of the Grothendieck construction and the Segal Machine and to show how adjunction questions can be investigated by means of this approach and what its connections are with more traditional approaches. As an application, the derived Morita 2-category and the Fourier–Mukai 2-category over a Noetherian ring are constructed and the embedding of the latter in the former is demonstrated. </p><p> Bibliography: 15 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM9963">https://doi.org/10.1070/RM9963</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM9972" class="art-list-item-title event_main-link">Fenchel–Nielsen coordinates and Goldman brackets</a> <p class="small art-list-item-meta"> L. O. Chekhov 2020 <em>Russ. Math. Surv.</em> <b>75</b> 929 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Fenchel–Nielsen coordinates and Goldman brackets" data-link-purpose-append-open="Fenchel–Nielsen coordinates and Goldman brackets">Open abstract</span> </button> <a href="/article/10.1070/RM9972/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Fenchel–Nielsen coordinates and Goldman brackets</span></a> <a href="/article/10.1070/RM9972/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Fenchel–Nielsen coordinates and Goldman brackets</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> It is explicitly shown that the Poisson bracket on the set of shear coordinates defined by V. V. Fock in 1997 induces the Fenchel–Nielsen bracket on the set of gluing parameters (length and twist parameters) for pair-of-pants decompositions of Riemann surfaces <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/75/5/929/revision3/RMS_75_5_929ieqn1.gif" style="max-width: 100%;" alt="$\Sigma_{g,s}$" align="top"></img></span><script type="math/tex">\Sigma_{g,s}</script></span></span> with holes. These structures are generalized to the case of Riemann surfaces <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/75/5/929/revision3/RMS_75_5_929ieqn2.gif" style="max-width: 100%;" alt="$\Sigma_{g,s,n}$" align="top"></img></span><script type="math/tex">\Sigma_{g,s,n}</script></span></span> with holes and bordered cusps. </p><p> Bibliography: 49 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM9972">https://doi.org/10.1070/RM9972</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10009" class="art-list-item-title event_main-link">Tetrahedron equation: algebra, topology, and integrability</a> <p class="small art-list-item-meta"> D. V. Talalaev 2021 <em>Russ. Math. Surv.</em> <b>76</b> 685 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Tetrahedron equation: algebra, topology, and integrability" data-link-purpose-append-open="Tetrahedron equation: algebra, topology, and integrability">Open abstract</span> </button> <a href="/article/10.1070/RM10009/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Tetrahedron equation: algebra, topology, and integrability</span></a> <a href="/article/10.1070/RM10009/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Tetrahedron equation: algebra, topology, and integrability</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> The Zamolodchikov tetrahedron equation inherits almost all the richness of structures and topics in which the Yang–Baxter equation is involved. At the same time, this transition symbolizes the growth of the order of the problem, the step from the Yang–Baxter equation to the local Yang–Baxter equation, from the Lie algebra to the 2-Lie algebra, from ordinary knots in <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/685/revision1/RMS_76_4_685ieqn1.gif" style="max-width: 100%;" alt="$\mathbb{R}^3$" align="top"></img></span><script type="math/tex">\mathbb{R}^3</script></span></span> to 2-knots in <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/685/revision1/RMS_76_4_685ieqn2.gif" style="max-width: 100%;" alt="$\mathbb{R}^4$" align="top"></img></span><script type="math/tex">\mathbb{R}^4</script></span></span>. These transitions are followed in several examples, and there are also discussions of the manifestation of the tetrahedron equation in the long-standing question of integrability of the three-dimensional Ising model and a related model of neural network theory: the Hopfield model on a two-dimensional lattice. </p><p> Bibliography: 82 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10009">https://doi.org/10.1070/RM10009</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10008" class="art-list-item-title event_main-link">Chaos and integrability in <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn1.gif" alt="$\operatorname{SL}(2,\mathbb R)$" align="top"></img></span><script type="math/tex">\operatorname{SL}(2,\mathbb R)</script></span></span>-geometry</a> <p class="small art-list-item-meta"> A. V. Bolsinov <em>et al</em> 2021 <em>Russ. Math. Surv.</em> <b>76</b> 557 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Chaos and integrability in -geometry" data-link-purpose-append-open="Chaos and integrability in -geometry">Open abstract</span> </button> <a href="/article/10.1070/RM10008/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Chaos and integrability in -geometry</span></a> <a href="/article/10.1070/RM10008/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Chaos and integrability in -geometry</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> We review the integrability of the geodesic flow on a threefold <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn2.gif" style="max-width: 100%;" alt="$\mathcal M^3$" align="top"></img></span><script type="math/tex">\mathcal M^3</script></span></span> admitting one of the three group geometries in Thurston's sense. We focus on the <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn1.gif" style="max-width: 100%;" alt="$\operatorname{SL}(2,\mathbb R)$" align="top"></img></span><script type="math/tex">\operatorname{SL}(2,\mathbb R)</script></span></span> case. The main examples are the quotients <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn3.gif" style="max-width: 100%;" alt="$\mathcal M^3_\Gamma=\Gamma\backslash \operatorname{PSL}(2,\mathbb R)$" align="top"></img></span><script type="math/tex">\mathcal M^3_\Gamma=\Gamma\backslash \operatorname{PSL}(2,\mathbb R)</script></span></span>, where <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn4.gif" style="max-width: 100%;" alt="$\Gamma \subset \operatorname{PSL}(2,\mathbb R)$" align="top"></img></span><script type="math/tex">\Gamma \subset \operatorname{PSL}(2,\mathbb R)</script></span></span> is a cofinite Fuchsian group. We show that the corresponding phase space <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn5.gif" style="max-width: 100%;" alt="$T^*\mathcal M_\Gamma^3$" align="top"></img></span><script type="math/tex">T^*\mathcal M_\Gamma^3</script></span></span> contains two open regions with integrable and chaotic behaviour, with zero and positive topological entropy, respectively. </p><p> As a concrete example we consider the case of the modular threefold with the modular group <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn6.gif" style="max-width: 100%;" alt="$\Gamma=\operatorname{PSL}(2,\mathbb Z)$" align="top"></img></span><script type="math/tex">\Gamma=\operatorname{PSL}(2,\mathbb Z)</script></span></span>. In this case <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn7.gif" style="max-width: 100%;" alt="$\mathcal M^3_\Gamma$" align="top"></img></span><script type="math/tex">\mathcal M^3_\Gamma</script></span></span> is known to be homeomorphic to the complement of a trefoil knot <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn8.gif" style="max-width: 100%;" alt="$\mathcal K$" align="top"></img></span><script type="math/tex">\mathcal K</script></span></span> in a 3-sphere. Ghys proved the remarkable fact that the lift of a periodic geodesic on the modular surface to <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn7.gif" style="max-width: 100%;" alt="$\mathcal M^3_\Gamma$" align="top"></img></span><script type="math/tex">\mathcal M^3_\Gamma</script></span></span> produces the same isotopy class of knots as that which appears in the chaotic version of the celebrated Lorenz system and was studied in detail by Birman and Williams. We show that these knots are replaced by trefoil knot cables in the integrable limit of the geodesic system on <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/76/4/557/revision1/RMS_76_4_557ieqn7.gif" style="max-width: 100%;" alt="$\mathcal M^3_\Gamma$" align="top"></img></span><script type="math/tex">\mathcal M^3_\Gamma</script></span></span>. </p><p> Bibliography: 60 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10008">https://doi.org/10.1070/RM10008</a> </div> </div> </div> </div> </div>  </div> </div> </div>   <div tabindex="0" role="tabpanel" id="latest-articles-tab" aria-labelledby="latest-articles"> <div class=" reveal-container reveal-closed reveal-enabled reveal-container--jnl-tab"> <h2 class="tabpanel__title"> <button type="button" class="reveal-trigger event_tabs-accordion" aria-expanded="false"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg>Latest articles</button> </h2> <div class="reveal-content tabpanel__content" style="display: none"> <p> <button data-reveal-label-alt="Close all abstracts" class="reveal-all-trigger mr-2 small" data-reveal-text="Open all abstracts" data-link-purpose-append="in this tab" data-link-purpose-append-open="in this tab"> Open all abstracts<span class="offscreen-hidden">, in this tab</span> </button> </p>  <div class="art-list"> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10011" class="art-list-item-title event_main-link">On the Dirichlet problem for not strongly elliptic second-order equations</a> <p class="small art-list-item-meta"> A. O. Bagapsh <em>et al</em> 2022 <em>Russ. Math. Surv.</em> <b>77</b> 372 </p> <div class="art-list-item-tools small wd-abstr-upper"> <a href="/article/10.1070/RM10011/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, On the Dirichlet problem for not strongly elliptic second-order equations</span></a> <a href="/article/10.1070/RM10011/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, On the Dirichlet problem for not strongly elliptic second-order equations</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10011">https://doi.org/10.1070/RM10011</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10035" class="art-list-item-title event_main-link">On a canonical basis of a pair of compatible Poisson brackets on a symplectic Lie algebra</a> <p class="small art-list-item-meta"> A. A. Garazha 2022 <em>Russ. Math. Surv.</em> <b>77</b> 375 </p> <div class="art-list-item-tools small wd-abstr-upper"> <a href="/article/10.1070/RM10035/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, On a canonical basis of a pair of compatible Poisson brackets on a symplectic Lie algebra</span></a> <a href="/article/10.1070/RM10035/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, On a canonical basis of a pair of compatible Poisson brackets on a symplectic Lie algebra</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10035">https://doi.org/10.1070/RM10035</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10039" class="art-list-item-title event_main-link">Effective results in the theory of birational rigidity</a> <p class="small art-list-item-meta"> A. V. Pukhlikov 2022 <em>Russ. Math. Surv.</em> <b>77</b> 301 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Effective results in the theory of birational rigidity" data-link-purpose-append-open="Effective results in the theory of birational rigidity">Open abstract</span> </button> <a href="/article/10.1070/RM10039/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Effective results in the theory of birational rigidity</span></a> <a href="/article/10.1070/RM10039/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Effective results in the theory of birational rigidity</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> This paper is a survey of recent effective results in the theory of birational rigidity of higher-dimensional Fano varieties and Fano–Mori fibre spaces. </p><p> Bibliography: 59 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10039">https://doi.org/10.1070/RM10039</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10040" class="art-list-item-title event_main-link">R. Thompson's group <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="https://content.cld.iop.org/journals/0036-0279/77/2/251/revision2/RMS_77_2_251ieqn1.gif" alt="$F$" align="top"></img></span><script type="math/tex">F</script></span></span> and the amenability problem</a> <p class="small art-list-item-meta"> V. S. Guba 2022 <em>Russ. Math. Surv.</em> <b>77</b> 251 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="R. Thompson’s group and the amenability problem" data-link-purpose-append-open="R. Thompson’s group and the amenability problem">Open abstract</span> </button> <a href="/article/10.1070/RM10040/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, R. Thompson's group and the amenability problem</span></a> <a href="/article/10.1070/RM10040/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, R. Thompson's group and the amenability problem</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> This paper focuses on Richard Thompson's group <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/77/2/251/revision2/RMS_77_2_251ieqn1.gif" style="max-width: 100%;" alt="$F$" align="top"></img></span><script type="math/tex">F</script></span></span>, which was discovered in the 1960s. Many papers have been devoted to this group. We are interested primarily in the famous problem of amenability of this group, which was posed by Geoghegan in 1979. Numerous attempts have been made to solve this problem in one way or the other, but it remains open. </p><p> In this survey we describe the most important known properties of this group related to the word problem and representations of elements of the group by piecewise linear functions as well as by diagrams and other geometric objects. We describe the classical results of Brin and Squier concerning free subgroups and laws. We include a description of more modern important results relating to the properties of the Cayley graphs (the Belk–Brown construction) as well as Bartholdi's theorem about the properties of equations in group rings. We consider separately the criteria for (non-)amenability of groups that are useful in the work on the main problem. At the end we describe a number of our own results about the structure of the Cayley graphs and a new algorithm for solving the word problem. </p><p> Bibliography: 69 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10040">https://doi.org/10.1070/RM10040</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM10049" class="art-list-item-title event_main-link">The normal derivative lemma and surrounding issues</a> <p class="small art-list-item-meta"> D. E. Apushkinskaya and A. I. Nazarov 2022 <em>Russ. Math. Surv.</em> <b>77</b> 189 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="The normal derivative lemma and surrounding issues" data-link-purpose-append-open="The normal derivative lemma and surrounding issues">Open abstract</span> </button> <a href="/article/10.1070/RM10049/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, The normal derivative lemma and surrounding issues</span></a> <a href="/article/10.1070/RM10049/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, The normal derivative lemma and surrounding issues</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p> In this survey we describe the history and current state of one of the key areas in the qualitative theory of elliptic partial differential equations related to the strong maximum principle and the boundary point principle (normal derivative lemma). </p><p> Bibliography: 234 titles. </p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM10049">https://doi.org/10.1070/RM10049</a> </div> </div> </div> </div> </div>  </div> </div> </div>     <div tabindex="0" role="tabpanel" id="review-articles-tab" aria-labelledby="review-articles" hidden="hidden"> <div class=" reveal-container reveal-closed reveal-enabled reveal-container--jnl-tab"> <h2 class="tabpanel__title"> <button type="button" class="reveal-trigger event_tabs-accordion" aria-expanded="false"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg>Review articles</button> </h2> <div class="reveal-content tabpanel__content" style="display: none"> <p> <button data-reveal-label-alt="Close all abstracts" class="reveal-all-trigger mr-2 small" data-reveal-text="Open all abstracts" data-link-purpose-append="in this tab" data-link-purpose-append-open="in this tab"> Open all abstracts<span class="offscreen-hidden">, in this tab</span> </button> </p>  <div class="art-list"> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM2014v069n03ABEH004896" class="art-list-item-title event_main-link">Local formulae for the hydrodynamic pressure and applications</a> <p class="small art-list-item-meta"> P. Constantin 2014 <em>Russ. Math. Surv.</em> <b>69</b> 395 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Local formulae for the hydrodynamic pressure and applications" data-link-purpose-append-open="Local formulae for the hydrodynamic pressure and applications">Open abstract</span> </button> <a href="/article/10.1070/RM2014v069n03ABEH004896/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Local formulae for the hydrodynamic pressure and applications</span></a> <a href="/article/10.1070/RM2014v069n03ABEH004896/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Local formulae for the hydrodynamic pressure and applications</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p>We provide local formulae for the pressure of incompressible fluids. The pressure can be expressed in terms of its average and averages of squares of velocity increments in arbitrarily small neighbourhoods. As an application, we give a brief proof of the fact that <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/395/revision1/rms_69_3_395ieqn1.gif" style="max-width: 100%;" alt="$C^{\alpha}$" align="top"></img></span><script type="math/tex">C^{\alpha}</script></span></span> velocities have <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/395/revision1/rms_69_3_395ieqn2.gif" style="max-width: 100%;" alt="$C^{2\alpha}$" align="top"></img></span><script type="math/tex">C^{2\alpha}</script></span></span> (or Lipschitz) pressures. We also give some regularity criteria for 3D incompressible Navier–Stokes equations.</p><p>Bibliography: 9 titles.</p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM2014v069n03ABEH004896">https://doi.org/10.1070/RM2014v069n03ABEH004896</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM2014v069n03ABEH004897" class="art-list-item-title event_main-link">An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle</a> <p class="small art-list-item-meta"> B. Fiedler <em>et al</em> 2014 <em>Russ. Math. Surv.</em> <b>69</b> 419 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle" data-link-purpose-append-open="An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle">Open abstract</span> </button> <a href="/article/10.1070/RM2014v069n03ABEH004897/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle</span></a> <a href="/article/10.1070/RM2014v069n03ABEH004897/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, An explicit Lyapunov function for reflection symmetric parabolic partial differential equations on the circle</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p>An explicit Lyapunov function is constructed for scalar parabolic reaction-advection-diffusion equations under periodic boundary conditions. The non-linearity is assumed to be even with respect to the advection term. The method followed was originally suggested by H. Matano for, and limited to, separated boundary conditions.</p><p>Bibliography: 20 titles.</p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM2014v069n03ABEH004897">https://doi.org/10.1070/RM2014v069n03ABEH004897</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM2014v069n03ABEH004898" class="art-list-item-title event_main-link">Boundary layer theory for convection-diffusion equations in a circle</a> <p class="small art-list-item-meta"> C.-Y. Jung and R. Temam 2014 <em>Russ. Math. Surv.</em> <b>69</b> 435 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Boundary layer theory for convection-diffusion equations in a circle" data-link-purpose-append-open="Boundary layer theory for convection-diffusion equations in a circle">Open abstract</span> </button> <a href="/article/10.1070/RM2014v069n03ABEH004898/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Boundary layer theory for convection-diffusion equations in a circle</span></a> <a href="/article/10.1070/RM2014v069n03ABEH004898/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Boundary layer theory for convection-diffusion equations in a circle</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p>This paper is devoted to boundary layer theory for singularly perturbed convection-diffusion equations in the unit circle. Two characteristic points appear, <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/435/revision1/rms_69_3_435ieqn1.gif" style="max-width: 100%;" alt="$(\pm 1,0)$" align="top"></img></span><script type="math/tex">(\pm 1,0)</script></span></span>, in the context of the equations considered here, and singularities may occur at these points depending on the behaviour there of a given function <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/435/revision1/rms_69_3_435ieqn2.gif" style="max-width: 100%;" alt="$f$" align="top"></img></span><script type="math/tex">f</script></span></span>, namely, the flatness or compatibility of <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/435/revision1/rms_69_3_435ieqn3.gif" style="max-width: 100%;" alt="$f$" align="top"></img></span><script type="math/tex">f</script></span></span> at these points as explained below. Two previous articles addressed two particular cases: [24] dealt with the case where the function <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/435/revision1/rms_69_3_435ieqn4.gif" style="max-width: 100%;" alt="$f$" align="top"></img></span><script type="math/tex">f</script></span></span> is sufficiently flat at the characteristic points, the so-called compatible case; [25] dealt with a generic non-compatible case (<span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/435/revision1/rms_69_3_435ieqn5.gif" style="max-width: 100%;" alt="$f$" align="top"></img></span><script type="math/tex">f</script></span></span> polynomial). This survey article recalls the essential results from those papers, and continues with the general case (<span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/435/revision1/rms_69_3_435ieqn6.gif" style="max-width: 100%;" alt="$f$" align="top"></img></span><script type="math/tex">f</script></span></span> non-flat and non-polynomial) for which new specific boundary layer functions of parabolic type are introduced in addition.</p><p>Bibliography: 49 titles.</p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM2014v069n03ABEH004898">https://doi.org/10.1070/RM2014v069n03ABEH004898</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM2014v069n03ABEH004899" class="art-list-item-title event_main-link">Non-holonomic dynamics and Poisson geometry</a> <p class="small art-list-item-meta"> A. V. Borisov <em>et al</em> 2014 <em>Russ. Math. Surv.</em> <b>69</b> 481 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="Non-holonomic dynamics and Poisson geometry" data-link-purpose-append-open="Non-holonomic dynamics and Poisson geometry">Open abstract</span> </button> <a href="/article/10.1070/RM2014v069n03ABEH004899/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, Non-holonomic dynamics and Poisson geometry</span></a> <a href="/article/10.1070/RM2014v069n03ABEH004899/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, Non-holonomic dynamics and Poisson geometry</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p>This is a survey of basic facts presently known about non-linear Poisson structures in the analysis of integrable systems in non-holonomic mechanics. It is shown that by using the theory of Poisson deformations it is possible to reduce various non-holonomic systems to dynamical systems on well-understood phase spaces equipped with linear Lie–Poisson brackets. As a result, not only can different non-holonomic systems be compared, but also fairly advanced methods of Poisson geometry and topology can be used for investigating them.</p><p>Bibliography: 95 titles.</p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM2014v069n03ABEH004899">https://doi.org/10.1070/RM2014v069n03ABEH004899</a> </div> </div> </div> </div> <div class="art-list-item reveal-container reveal-closed"> <div class="art-list-item-body"> <a href="/article/10.1070/RM2014v069n03ABEH004900" class="art-list-item-title event_main-link">A system of three quantum particles with point-like interactions</a> <p class="small art-list-item-meta"> R. A. Minlos 2014 <em>Russ. Math. Surv.</em> <b>69</b> 539 </p> <div class="art-list-item-tools small wd-abstr-upper"> <button type="button" class="reveal-trigger mr-2 nowrap"> <svg aria-hidden="true" class="fa-icon fa-icon--left fa-icon--flip" role="img" focusable="false" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path d="M137.4 374.6c12.5 12.5 32.8 12.5 45.3 0l128-128c9.2-9.2 11.9-22.9 6.9-34.9s-16.6-19.8-29.6-19.8L32 192c-12.9 0-24.6 7.8-29.6 19.8s-2.2 25.7 6.9 34.9l128 128z"/></svg><span class="reveal-trigger-label" data-reveal-text="Open abstract" data-reveal-label-alt="Close abstract" data-link-purpose-append="A system of three quantum particles with point-like interactions" data-link-purpose-append-open="A system of three quantum particles with point-like interactions">Open abstract</span> </button> <a href="/article/10.1070/RM2014v069n03ABEH004900/meta" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="View article"> <span class="icon-article"></span>View article<span class="offscreen-hidden">, A system of three quantum particles with point-like interactions</span></a> <a href="/article/10.1070/RM2014v069n03ABEH004900/pdf" class="mr-2 mb-0 nowrap event_mini-link" data-event-action="PDF"><span class="icon-file-pdf"></span>PDF<span class="offscreen-hidden">, A system of three quantum particles with point-like interactions</span></a> </div> <div class="reveal-content"> <div class="article-text view-text-small"><p>Consider a quantum three-particle system consisting of two fermions of unit mass and another particle of mass <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn1.gif" style="max-width: 100%;" alt="$m\gt 0$" align="top"></img></span><script type="math/tex">m\gt 0</script></span></span> interacting in a point-like manner with the fermions. Such systems are studied here using the theory of self-adjoint extensions of symmetric operators: the Hamiltonian of the system is constructed as an extension of the symmetric energy operator <div xmlns:xlink="http://www.w3.org/1999/xlink" class="display-eqn" id="rms_69_3_539ueq1"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" style="max-width: 100%;" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ueqn1.gif" alt=""></img></span><script type="math/tex; mode=display">\begin{equation*} H_0=-\frac{1}{2}\biggl(\frac{1}{m}\Delta_y+\Delta_{x_1}+\Delta_{x_2}\biggr), \end{equation*}</script></span></div>which is defined on the functions in <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn2.gif" style="max-width: 100%;" alt="$L_2(\mathbb{R}^3)\otimes L_2^{\operatorname{asym}}(\mathbb{R}^3\times\mathbb{R}^3)$" align="top"></img></span><script type="math/tex">L_2(\mathbb{R}^3)\otimes L_2^{\operatorname{asym}}(\mathbb{R}^3\times\mathbb{R}^3)</script></span></span> that vanish whenever the position of the third particle coincides with the position of a fermion. To construct a natural family of extensions of <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn3.gif" style="max-width: 100%;" alt="$H_0$" align="top"></img></span><script type="math/tex">H_0</script></span></span>, one must solve the problem of self-adjoint extensions for an auxiliary sequence <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn4.gif" style="max-width: 100%;" alt="$\{T_l,\ l=0,1,2,\dots\}$" align="top"></img></span><script type="math/tex">\{T_l,\ l=0,1,2,\dots\}</script></span></span> of symmetric operators acting in <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn5.gif" style="max-width: 100%;" alt="$L_2(\mathbb{R}^3)$" align="top"></img></span><script type="math/tex">L_2(\mathbb{R}^3)</script></span></span>. All the operators <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn6.gif" style="max-width: 100%;" alt="$T_l$" align="top"></img></span><script type="math/tex">T_l</script></span></span> with even <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn7.gif" style="max-width: 100%;" alt="$l$" align="top"></img></span><script type="math/tex">l</script></span></span> are self-adjoint, and for every odd <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn8.gif" style="max-width: 100%;" alt="$l$" align="top"></img></span><script type="math/tex">l</script></span></span> there are two numbers <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn9.gif" style="max-width: 100%;" alt="$0\lt m_l^{(1)} \lt m_l^{(2)} \lt \infty$" align="top"></img></span><script type="math/tex">0\lt m_l^{(1)} \lt m_l^{(2)} \lt \infty</script></span></span> such that <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn10.gif" style="max-width: 100%;" alt="$T_l$" align="top"></img></span><script type="math/tex">T_l</script></span></span> is self-adjoint and lower semibounded for <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn11.gif" style="max-width: 100%;" alt="$m \gt m_l^{(2)}$" align="top"></img></span><script type="math/tex">m \gt m_l^{(2)}</script></span></span>, and has deficiency indices for <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn12.gif" style="max-width: 100%;" alt="$m\leqslant m_l^{(2)}$" align="top"></img></span><script type="math/tex">m\leqslant m_l^{(2)}</script></span></span>. When <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn13.gif" style="max-width: 100%;" alt="$m\in[m_l^{(1)}, m_l^{(2)}]$" align="top"></img></span><script type="math/tex">m\in[m_l^{(1)}, m_l^{(2)}]</script></span></span>, every self-adjoint extension of <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn14.gif" style="max-width: 100%;" alt="$T_l$" align="top"></img></span><script type="math/tex">T_l</script></span></span> which is invariant under rotations of <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn15.gif" style="max-width: 100%;" alt="$\mathbb{R}^3$" align="top"></img></span><script type="math/tex">\mathbb{R}^3</script></span></span> is lower semibounded, but if <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn16.gif" style="max-width: 100%;" alt="$0 \lt m \lt m_l^{(1)}$" align="top"></img></span><script type="math/tex">0 \lt m \lt m_l^{(1)}</script></span></span>, then it has an infinite sequence of eigenvalues <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn17.gif" style="max-width: 100%;" alt="$\{\lambda_n\}$" align="top"></img></span><script type="math/tex">\{\lambda_n\}</script></span></span> of multiplicity <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn18.gif" style="max-width: 100%;" alt="$2l+1$" align="top"></img></span><script type="math/tex">2l+1</script></span></span> such that <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn19.gif" style="max-width: 100%;" alt="$\lambda_n\to-\infty$" align="top"></img></span><script type="math/tex">\lambda_n\to-\infty</script></span></span> as <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn20.gif" style="max-width: 100%;" alt="$n\to\infty$" align="top"></img></span><script type="math/tex">n\to\infty</script></span></span> (the Thomas effect). It follows from the last fact that there is a sequence of bound states of <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn21.gif" style="max-width: 100%;" alt="$H_0$" align="top"></img></span><script type="math/tex">H_0</script></span></span> with spectrum <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn22.gif" style="max-width: 100%;" alt="$P^2/(2(m+2))+z_n$" align="top"></img></span><script type="math/tex">P^2/(2(m+2))+z_n</script></span></span>, where the numbers <span xmlns:xlink="http://www.w3.org/1999/xlink" class="inline-eqn"><span class="tex"><span class="texImage"><img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mNkYAAAAAYAAjCB0C8AAAAASUVORK5CYII=" data-src="https://content.cld.iop.org/journals/0036-0279/69/3/539/revision1/rms_69_3_539ieqn23.gif" style="max-width: 100%;" alt="$z_n<0$" align="top"></img></span><script type="math/tex">z_n<0</script></span></span> cluster at 0 (Efimov's effect).</p><p>Bibliography: 19 titles.</p></div> <div class="art-list-item-tools small wd-abstr-lower"> <a class="mr-2" href="https://doi.org/10.1070/RM2014v069n03ABEH004900">https://doi.org/10.1070/RM2014v069n03ABEH004900</a> </div> </div> </div> </div> </div>  </div> </div> </div>              </div>  </div>  </div> </main> <div class="db2 tb2"> <div class="side-and-below">  <div class="sidebar-list" id="wd-jnl-links"> <h2 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