{"title":"Bounded compact and dual compact approximation properties of Hardy spaces: New results and open problems","authors":"Oleksiy Karlovych , Eugene Shargorodsky","doi":"10.1016/j.indag.2023.10.004","DOIUrl":"10.1016/j.indag.2023.10.004","url":null,"abstract":"<div><p>The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such spaces, to provide background for the open problems. Namely, we consider abstract Hardy spaces <span><math><mrow><mi>H</mi><mrow><mo>[</mo><mi>X</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>]</mo></mrow></mrow></math></span> built upon translation-invariant Banach function spaces <span><math><mi>X</mi></math></span> with weights <span><math><mi>w</mi></math></span> such that <span><math><mrow><mi>w</mi><mo>∈</mo><mi>X</mi></mrow></math></span> and <span><math><mrow><msup><mrow><mi>w</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>∈</mo><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></math></span>, where <span><math><msup><mrow><mi>X</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is the associate space of <span><math><mi>X</mi></math></span>. We prove that if <span><math><mi>X</mi></math></span> is separable, then <span><math><mrow><mi>H</mi><mrow><mo>[</mo><mi>X</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>]</mo></mrow></mrow></math></span> has the BCAP with the approximation constant <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>H</mi><mrow><mo>[</mo><mi>X</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span>. Moreover, if <span><math><mi>X</mi></math></span> is reflexive, then <span><math><mrow><mi>H</mi><mrow><mo>[</mo><mi>X</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>]</mo></mrow></mrow></math></span> has the BCAP and the DCAP with the approximation constants <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>H</mi><mrow><mo>[</mo><mi>X</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span> and <span><math><mrow><msup><mrow><mi>M</mi></mrow><mrow><mo>∗</mo></mrow></msup><mrow><mo>(</mo><mi>H</mi><mrow><mo>[</mo><mi>X</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>]</mo></mrow><mo>)</mo></mrow><mo>≤</mo><mn>2</mn></mrow></math></span>, respectively. In the case of classical weighted Hardy space <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><mi>H</mi><mrow><mo>[</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>]</mo></mrow></mrow></math></span> with <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></math></span>, one has a sharper result: <span><math><mrow><mi>M</mi><mrow><mo>(</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>≤</mo><msup><mrow><mn>2</mn></mrow><mrow><mrow><mo>|</mo><mn>1</mn><mo>−</mo><mn>2</mn><mo>/</mo><mi>p</mi><mo>|</mo></mrow></mrow></msup","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000964/pdfft?md5=a439055dfb56920bebd7105cab40d8a0&pid=1-s2.0-S0019357723000964-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136009247","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Non-stationary α-fractal functions and their dimensions in various function spaces","authors":"Anarul Islam Mondal, Sangita Jha","doi":"10.1016/j.indag.2023.10.006","DOIUrl":"10.1016/j.indag.2023.10.006","url":null,"abstract":"<div><p><span>In this article, we study the novel concept of non-stationary iterated function systems (IFSs) introduced by Massopust in 2019. At first, using a sequence of different contractive operators, we construct non-stationary </span><span><math><mi>α</mi></math></span>-fractal functions on the space of all continuous functions. Next, we provide some elementary properties of the fractal operator associated with the non-stationary <span><math><mi>α</mi></math></span>-fractal functions. Further, we show that the proposed interpolant generalizes the existing stationary interpolant in the sense of IFS. For a class of functions defined on an interval, we derive conditions on the IFS parameters so that the corresponding non-stationary <span><math><mi>α</mi></math></span><span>-fractal functions are elements of some standard spaces like bounded variation space, convex Lipschitz space, and other function spaces. Finally, we discuss the dimensional analysis of the corresponding non-stationary </span><span><math><mi>α</mi></math></span>-fractal functions on these spaces.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135455026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Remarks on weak convergence of complex Monge–Ampère measures","authors":"Mohamed El Kadiri","doi":"10.1016/j.indag.2023.08.001","DOIUrl":"10.1016/j.indag.2023.08.001","url":null,"abstract":"<div><p>Let <span><math><mrow><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></math></span> be a decreasing sequence of psh functions in the domain of definition <span><math><mi>D</mi></math></span> of the Monge–Ampère operator on a domain <span><math><mi>Ω</mi></math></span> of <span><math><msup><mrow><mi>ℂ</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> such that <span><math><mrow><mi>u</mi><mo>=</mo><msub><mrow><mo>inf</mo></mrow><mrow><mi>j</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></math></span> is plurisubharmonic on <span><math><mi>Ω</mi></math></span>. In this paper we are interested in the problem of finding conditions insuring that <span><span><span><math><mrow><munder><mrow><mo>lim</mo></mrow><mrow><mi>j</mi><mo>→</mo><mo>+</mo><mi>∞</mi></mrow></munder><mo>∫</mo><mi>φ</mi><msup><mrow><mrow><mo>(</mo><mi>d</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><mo>∫</mo><mi>φ</mi><mo>NP</mo><msup><mrow><mrow><mo>(</mo><mi>d</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span></span></span>for any continuous function on <span><math><mi>Ω</mi></math></span> with compact support, where <span><math><mrow><mo>NP</mo><msup><mrow><mrow><mo>(</mo><mi>d</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> is the nonpolar part of <span><math><msup><mrow><mrow><mo>(</mo><mi>d</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></math></span>, and conditions implying that <span><math><mrow><mi>u</mi><mo>∈</mo><mi>D</mi></mrow></math></span>. For <span><math><mrow><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mo>max</mo><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>j</mi><mo>)</mo></mrow></mrow></math></span> these conditions imply also that <span><span><span><math><mrow><munder><mrow><mo>lim</mo></mrow><mrow><mi>j</mi><mo>→</mo><mo>+</mo><mi>∞</mi></mrow></munder><msub><mrow><mo>∫</mo></mrow><mrow><mi>K</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>d</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msub><mrow><mo>∫</mo></mrow><mrow><mi>K</mi></mrow></msub><mo>NP</mo><msup><mrow><mrow><mo>(</mo><mi>d</mi><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><mi>u</mi><mo>)</mo></mrow></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span></span></span>for any compact set <span><math><mrow><mi>K</mi><mo>⊂</mo><mrow><mo>{</mo><mi>u</mi><mo>></mo><mo>−</mo><mi>∞</mi><mo>}</mo></mrow></mrow></math","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45857362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Helena Ferreira , Marta Ferreira , Luís A. Alexandre
{"title":"A crossinggram for random fields on lattices","authors":"Helena Ferreira , Marta Ferreira , Luís A. Alexandre","doi":"10.1016/j.indag.2023.10.003","DOIUrl":"10.1016/j.indag.2023.10.003","url":null,"abstract":"<div><p>The modeling of risk situations that occur in a space framework can be done using max-stable random fields on lattices. Although the summary coefficients for the spatial behavior do not characterize the finite-dimensional distributions of the random field, they have the advantage of being immediate to interpret and easier to estimate. The coefficients that we propose give us information about the tendency of a random field for local oscillations of its values in relation to real valued high levels. It is not the magnitude of the oscillations that is being evaluated, but rather the greater or lesser number of oscillations, that is, the tendency of the trajectories to oscillate. We can observe surface trajectories more smooth over a region according to higher crossinggram value. It takes value in <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></math></span> and increases with the concordance of the variables of the random field.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000952/pdfft?md5=bc26c1660b9ebd2bb412c6586f0158a0&pid=1-s2.0-S0019357723000952-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136119727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Pointwise attractors which are not strict","authors":"Magdalena Nowak","doi":"10.1016/j.indag.2023.10.002","DOIUrl":"10.1016/j.indag.2023.10.002","url":null,"abstract":"<div><p>We deal with the finite family <span><math><mi>F</mi></math></span><span> of continuous maps on the Hausdorff space </span><span><math><mi>X</mi></math></span><span>. A nonempty compact subset </span><span><math><mi>A</mi></math></span><span> of such space is called a strict attractor if it has an open neighborhood </span><span><math><mi>U</mi></math></span> such that <span><math><mrow><mi>A</mi><mo>=</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>S</mi><mo>)</mo></mrow></mrow></math></span> for every nonempty compact <span><math><mrow><mi>S</mi><mo>⊂</mo><mi>U</mi></mrow></math></span><span>. Every strict attractor is a pointwise attractor, which means that the set </span><span><math><mrow><mo>{</mo><mi>x</mi><mo>∈</mo><mi>X</mi><mo>;</mo><msub><mrow><mo>lim</mo></mrow><mrow><mi>n</mi><mo>→</mo><mi>∞</mi></mrow></msub><msup><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>A</mi><mo>}</mo></mrow></math></span> contains <span><math><mi>A</mi></math></span> in its interior.</p><p>We present a class of examples of pointwise attractors – from the finite set to the Sierpiński carpet – which are not strict when we add to the system one nonexpansive map.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135849556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Weak precompactness in projective tensor products","authors":"José Rodríguez , Abraham Rueda Zoca","doi":"10.1016/j.indag.2023.08.003","DOIUrl":"10.1016/j.indag.2023.08.003","url":null,"abstract":"<div><p>We give a sufficient condition for a pair of Banach spaces <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math></span> to have the following property: whenever <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊆</mo><mi>X</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⊆</mo><mi>Y</mi></mrow></math></span> are sets such that <span><math><mrow><mo>{</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>:</mo><mspace></mspace><mi>x</mi><mo>∈</mo><msub><mrow><mi>W</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mspace></mspace><mi>y</mi><mo>∈</mo><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>}</mo></mrow></math></span> is weakly precompact in the projective tensor product <span><math><mrow><mi>X</mi><msub><mrow><mover><mrow><mo>⊗</mo></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>π</mi></mrow></msub><mi>Y</mi></mrow></math></span>, then either <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> or <span><math><msub><mrow><mi>W</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> is relatively norm compact. For instance, such a property holds for the pair <span><math><mrow><mo>(</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ℓ</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>)</mo></mrow></math></span> if <span><math><mrow><mn>1</mn><mo><</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo><</mo><mi>∞</mi></mrow></math></span> satisfy <span><math><mrow><mn>1</mn><mo>/</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>/</mo><mi>q</mi><mo>≥</mo><mn>1</mn></mrow></math></span>. Other examples are given that allow us to provide alternative proofs to some results on multiplication operators due to Saksman and Tylli. We also revisit, with more direct proofs, some known results about the embeddability of <span><math><msub><mrow><mi>ℓ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> into <span><math><mrow><mi>X</mi><msub><mrow><mover><mrow><mo>⊗</mo></mrow><mrow><mo>̂</mo></mrow></mover></mrow><mrow><mi>π</mi></mrow></msub><mi>Y</mi></mrow></math></span> for arbitrary Banach spaces <span><math><mi>X</mi></math></span> and <span><math><mi>Y</mi></math></span>, in connection with the compactness of all operators from <span><math><mi>X</mi></math></span> to <span><math><msup><mrow><mi>Y</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0019357723000800/pdfft?md5=532c4016f038e9c133d3e9e7b6f3142c&pid=1-s2.0-S0019357723000800-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43493525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the cohomology of solvable Leibniz algebras","authors":"Jörg Feldvoss , Friedrich Wagemann","doi":"10.1016/j.indag.2023.09.002","DOIUrl":"10.1016/j.indag.2023.09.002","url":null,"abstract":"<div><p>This paper is a sequel to a previous paper of the authors in which the cohomology<span><span> of semi-simple Leibniz algebras was computed by using spectral sequences. In the present paper we generalize the vanishing theorems of Dixmier and Barnes for </span>nilpotent<span> and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we establish the Leibniz analogue of a non-vanishing theorem of Dixmier for nilpotent Leibniz algebras. In addition, we prove a Fitting lemma for Leibniz bimodules</span></span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135889547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Tangent spaces on the trianguline variety at companion points","authors":"Seginus Mowlavi","doi":"10.1016/j.indag.2023.10.007","DOIUrl":"10.1016/j.indag.2023.10.007","url":null,"abstract":"<div><p><span>Many results about the geometry of the trianguline variety have been obtained by Breuil–Hellmann–Schraen. Among them, using geometric methods, they have computed a formula for the dimension of the tangent space of the trianguline variety at dominant crystalline generic points, which has a conjectural generalisation to companion (</span><em>i.e.</em> non-dominant) points. In an earlier work, they proved a weaker form of this formula under the assumption of modularity using arithmetic methods. We prove a generalisation of a result of Bellaïche–Chenevier in <span><math><mi>p</mi></math></span>-adic Hodge theory and use it to extend the arithmetic methods of Breuil–Hellmann–Schraen to a wide class of companion points.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136127409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Amalgamation of real zero polynomials","authors":"David Sawall, Markus Schweighofer","doi":"10.1016/j.indag.2023.08.002","DOIUrl":"10.1016/j.indag.2023.08.002","url":null,"abstract":"<div><p>With this article, we hope to launch the investigation of what we call the <em>Real Zero Amalgamation Problem</em>. Whenever a polynomial arises from another polynomial by substituting zero for some of its variables, we call the second polynomial an <em>extension</em> of the first one. The <em>Real Zero Amalgamation Problem</em> asks when two (multivariate real) polynomials have a common extension (called <em>amalgam</em><span>) that is a real zero polynomial. We show that the obvious necessary conditions are not sufficient. Our counterexample is derived in several steps from a counterexample to amalgamation of matroids by Poljak and Turzík. On the positive side, we show that even a degree-preserving amalgamation is possible in three very special cases with three completely different techniques. Finally, we conjecture that amalgamation is always possible in the case of two shared variables. The analogue in matroid theory is true by another work of Poljak and Turzík. This would imply a very weak form of the Generalized Lax Conjecture.</span></p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43739370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Andrés Chirre , Markus Valås Hagen , Aleksander Simonič
{"title":"Conditional estimates for the logarithmic derivative of Dirichlet L-functions","authors":"Andrés Chirre , Markus Valås Hagen , Aleksander Simonič","doi":"10.1016/j.indag.2023.07.005","DOIUrl":"10.1016/j.indag.2023.07.005","url":null,"abstract":"<div><p><span>Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the </span><span><math><mi>q</mi></math></span><span>-aspect for the logarithmic derivative </span><span><math><mrow><mfenced><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>/</mo><mi>L</mi></mrow></mfenced><mfenced><mrow><mi>σ</mi><mo>,</mo><mi>χ</mi></mrow></mfenced></mrow></math></span> of Dirichlet <span><math><mi>L</mi></math></span>-functions, where <span><math><mi>χ</mi></math></span><span> is a primitive character modulo </span><span><math><mrow><mi>q</mi><mo>≥</mo><mn>1</mn><msup><mrow><mn>0</mn></mrow><mrow><mn>30</mn></mrow></msup></mrow></math></span> and <span><math><mrow><mn>1</mn><mo>/</mo><mn>2</mn><mo>+</mo><mn>1</mn><mo>/</mo><mo>log</mo><mo>log</mo><mi>q</mi><mo>≤</mo><mi>σ</mi><mo>≤</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo>/</mo><mo>log</mo><mo>log</mo><mi>q</mi></mrow></math></span>. In addition, for <span><math><mrow><mi>σ</mi><mo>=</mo><mn>1</mn></mrow></math></span> we improve upon the result by Ihara, Murty and Shimura (2009). Similar results for the logarithmic derivative of the Riemann zeta-function are given.</p></div>","PeriodicalId":56126,"journal":{"name":"Indagationes Mathematicae-New Series","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42154484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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