{"title":"Uniform bounds for the density in Artin's conjecture on primitive roots","authors":"Antonella Perucca, Igor E. Shparlinski","doi":"10.1112/blms.70011","DOIUrl":"https://doi.org/10.1112/blms.70011","url":null,"abstract":"<p>We consider Artin's conjecture on primitive roots over a number field <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math>, reducing an algebraic number <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mi>K</mi>\u0000 <mo>×</mo>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$alpha in K^times$</annotation>\u0000 </semantics></math>. Under the Generalised Riemann Hypothesis, there is a density <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dens</mo>\u0000 <mo>(</mo>\u0000 <mi>α</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{dens}(alpha)$</annotation>\u0000 </semantics></math> counting the proportion of the primes of <span></span><math>\u0000 <semantics>\u0000 <mi>K</mi>\u0000 <annotation>$K$</annotation>\u0000 </semantics></math> for which <span></span><math>\u0000 <semantics>\u0000 <mi>α</mi>\u0000 <annotation>$alpha$</annotation>\u0000 </semantics></math> is a primitive root. This density <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>dens</mo>\u0000 <mo>(</mo>\u0000 <mi>α</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$operatorname{dens}(alpha)$</annotation>\u0000 </semantics></math> is a rational multiple of an Artin constant <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>(</mo>\u0000 <mi>τ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$A(tau)$</annotation>\u0000 </semantics></math> that depends on the largest integer <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>τ</mi>\u0000 <mo>⩾</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$tau geqslant 1$</annotation>\u0000 </semantics></math> such that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>α</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mfenced>\u0000 <msup>\u0000 <mi>K</mi>\u0000 <mo>×</mo>\u0000 </msup>\u0000 </mfenced>\u0000 <mi>τ</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$alpha in {left(K^ti","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"978-991"},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The Shi variety corresponding to an affine Weyl group","authors":"Nathan Chapelier-Laget","doi":"10.1112/blms.70007","DOIUrl":"https://doi.org/10.1112/blms.70007","url":null,"abstract":"<p>Let <span></span><math>\u0000 <semantics>\u0000 <mi>W</mi>\u0000 <annotation>$W$</annotation>\u0000 </semantics></math> be an irreducible Weyl group and <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math> its affine Weyl group. In this article we show that there exists a bijection between <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math> and the integral points of an affine variety, denoted <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>X</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$widehat{X}_{W_a}$</annotation>\u0000 </semantics></math>, which we call the Shi variety of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math>. In order to do so, we use Jian-Yi Shi's characterization of alcoves in affine Weyl groups. We then study this variety further. We introduce a new representation of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 <annotation>$W_a$</annotation>\u0000 </semantics></math>, called the <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Φ</mi>\u0000 <mo>+</mo>\u0000 </msup>\u0000 <annotation>$Phi ^+$</annotation>\u0000 </semantics></math>-representation, and we highlight combinatorial and geometrical properties of the irreducible components of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mover>\u0000 <mi>X</mi>\u0000 <mo>̂</mo>\u0000 </mover>\u0000 <msub>\u0000 <mi>W</mi>\u0000 <mi>a</mi>\u0000 </msub>\u0000 </msub>\u0000 <annotation>$widehat{X}_{W_a}$</annotation>\u0000 </semantics></math> via this representation. We also show how the components are related to a fundamental parallelepiped <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>P</mi>\u0000 ","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"913-940"},"PeriodicalIF":0.8,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Conformal classes of Lorentzian surfaces with Killing fields","authors":"Pierre Mounoud","doi":"10.1112/blms.70010","DOIUrl":"https://doi.org/10.1112/blms.70010","url":null,"abstract":"<p>We study the conformal classes of two-dimensional Lorentzian tori with (nonzero) Killing fields. We define a map that associate to such a class a vector field on the circle (up to a scalar factor). This map is not injective but has finite-dimensional fiber. It allows us to characterize the conformal classes of tori with Killing field satisfying a condition related to the existence of conjugate points given by Mehidi.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"964-977"},"PeriodicalIF":0.8,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On volume and surface area of parallel sets. II. Surface measures and (non)differentiability of the volume","authors":"Jan Rataj, Steffen Winter","doi":"10.1112/blms.70006","DOIUrl":"https://doi.org/10.1112/blms.70006","url":null,"abstract":"<p>We prove that at differentiability points <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>r</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>></mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$r_0>0$</annotation>\u0000 </semantics></math> of the volume function of a compact set <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>A</mi>\u0000 <mo>⊂</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$Asubset mathbb {R}^d$</annotation>\u0000 </semantics></math> (associating to <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math> the volume of the <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math>-parallel set of <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math>), the surface area measures of <span></span><math>\u0000 <semantics>\u0000 <mi>r</mi>\u0000 <annotation>$r$</annotation>\u0000 </semantics></math>-parallel sets of <span></span><math>\u0000 <semantics>\u0000 <mi>A</mi>\u0000 <annotation>$A$</annotation>\u0000 </semantics></math> converge weakly to the surface area measure of the <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>r</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <annotation>$r_0$</annotation>\u0000 </semantics></math>-parallel set as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 <mo>→</mo>\u0000 <msub>\u0000 <mi>r</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$rrightarrow r_0$</annotation>\u0000 </semantics></math>. We further study the question which sets of parallel radii can occur as sets of nondifferentiability points of the volume function of some compact set. We provide a full characterization for dimensions <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$d=1$</annotation>\u0000 </semantics></math> and 2.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"895-912"},"PeriodicalIF":0.8,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581632","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Dominic Welsh, 1938–2023","authors":"Geoffrey R. Grimmett","doi":"10.1112/blms.13224","DOIUrl":"https://doi.org/10.1112/blms.13224","url":null,"abstract":"<p>Dominic Welsh was born in Port Talbot on 29 August 1938, the eldest child in a family of educators, and died in Oxford on 30 November 2023. He was the first student from his school to attend the University of Oxford, where he remained for the rest of his life as a Fellow of Merton College and a Professor of the University. He combined excellence as tutor and supervisor over nearly 40 years with a distinguished research record in probability and discrete mathematics, where he excelled in both original and expository work. With his DPhil supervisor John Hammersley, he introduced first-passage percolation, and in so doing formulated and proved the first subadditive ergodic theorem. His is the ‘W’ in the ‘RSW’ method that is now central to the theory of random planar media. He was a pioneer in matroid theory with numerous significant results and conjectures, and his monograph has been influential. He worked on computational complexity and particularly the complexity of computing the Tutte polynomial. Throughout his career, he inspired generations of undergraduates and postgraduates, and through his personal enthusiasm and warmth he helped develop a community of scholars in aspects of combinatorics who remember him with love and respect.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"992-1004"},"PeriodicalIF":0.8,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.13224","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581622","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Full Galois groups of polynomials with slowly growing coefficients","authors":"Lior Bary-Soroker, Noam Goldgraber","doi":"10.1112/blms.70008","DOIUrl":"https://doi.org/10.1112/blms.70008","url":null,"abstract":"<p>Choose a polynomial <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> uniformly at random from the set of all monic polynomials of degree <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> with integer coefficients in the box <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mo>−</mo>\u0000 <mi>L</mi>\u0000 <mo>,</mo>\u0000 <mi>L</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <mi>n</mi>\u0000 </msup>\u0000 <annotation>$[-L,L]^n$</annotation>\u0000 </semantics></math>. The main result of the paper asserts that if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo>=</mo>\u0000 <mi>L</mi>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$L=L(n)$</annotation>\u0000 </semantics></math> grows to infinity, then the Galois group of <span></span><math>\u0000 <semantics>\u0000 <mi>f</mi>\u0000 <annotation>$f$</annotation>\u0000 </semantics></math> is the full symmetric group, asymptotically almost surely, as <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>n</mi>\u0000 <mo>→</mo>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 <annotation>$nrightarrow infty$</annotation>\u0000 </semantics></math>. When <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> grows rapidly to infinity, say <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo>></mo>\u0000 <msup>\u0000 <mi>n</mi>\u0000 <mn>7</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$L>n^7$</annotation>\u0000 </semantics></math>, this theorem follows from a result of Gallagher. When <span></span><math>\u0000 <semantics>\u0000 <mi>L</mi>\u0000 <annotation>$L$</annotation>\u0000 </semantics></math> is bounded, the analog of the theorem is open, while the state-of-the-art is that the Galois group is large in the sense that it contains the alternating group (if <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo><</mo>\u0000 <mn>17</mn>\u0000 </mrow>\u0000 <annotation>$L< 17$</annotation>\u0000 </semantics></math>, it is conditional on the Extended Riemann Hypothesis). Hence the m","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"941-955"},"PeriodicalIF":0.8,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581485","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On Torelli groups and Dehn twists of smooth 4-manifolds","authors":"Manuel Krannich, Alexander Kupers","doi":"10.1112/blms.70009","DOIUrl":"https://doi.org/10.1112/blms.70009","url":null,"abstract":"<p>This note has two related but independent parts. Firstly, we prove a generalisation of a recent result of Gay on the smooth mapping class group of <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>4</mn>\u0000 </msup>\u0000 <annotation>$S^4$</annotation>\u0000 </semantics></math>. Secondly, we give an alternative proof of a consequence of work of Saeki, namely that the Dehn twist along the boundary sphere of a simply connected closed smooth 4-manifold <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> with <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>∂</mi>\u0000 <mi>X</mi>\u0000 <mo>≅</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>3</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$partial Xcong S^3$</annotation>\u0000 </semantics></math> is trivial after taking connected sums with enough copies of <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>×</mo>\u0000 <msup>\u0000 <mi>S</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$S^2times S^2$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"956-963"},"PeriodicalIF":0.8,"publicationDate":"2025-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On minimal presentations of numerical monoids","authors":"Alessio Moscariello, Alessio Sammartano","doi":"10.1112/blms.70005","DOIUrl":"https://doi.org/10.1112/blms.70005","url":null,"abstract":"<p>We consider the classical problem of determining the largest possible cardinality of a minimal presentation of a numerical monoid with given embedding dimension and multiplicity. Very few values of this cardinality are known. In addressing this problem, we apply tools from Hilbert functions and free resolutions of artinian standard graded algebras. This approach allows us to solve the problem in many cases and, at the same time, identify subtle difficulties in the remaining cases. As a by-product of our analysis, we deduce results for the corresponding problem for the type of a numerical monoid.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"878-894"},"PeriodicalIF":0.8,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/blms.70005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143582065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Sharp maximal function estimates for multilinear pseudo-differential operators of type (0,0)","authors":"Bae Jun Park, Naohito Tomita","doi":"10.1112/blms.70003","DOIUrl":"https://doi.org/10.1112/blms.70003","url":null,"abstract":"<p>In this paper, we study sharp maximal function estimates for multilinear pseudo-differential operators. Our target is operators of type (0,0) for which a differentiation does not make any decay of the associated symbol. Analogous results for operators of type <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>ρ</mi>\u0000 <mo>,</mo>\u0000 <mi>ρ</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$(rho,rho)$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>0</mn>\u0000 <mo><</mo>\u0000 <mi>ρ</mi>\u0000 <mo><</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$0<rho <1$</annotation>\u0000 </semantics></math>, appeared in an earlier work of the authors [17], but a different approach is given for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ρ</mi>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$rho =0$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":55298,"journal":{"name":"Bulletin of the London Mathematical Society","volume":"57 3","pages":"854-870"},"PeriodicalIF":0.8,"publicationDate":"2025-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143582066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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