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2. Book Review: Collected papers of C. S. Seshadri. Volume 1. Vector bundles and invariant theory; Collected papers of C. S. Seshadri. Volume 2. Schubert geometry and representation Theory
- Author
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Usha N. Bhosle
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Vector bundle ,Representation theory ,Invariant theory ,Mathematics ,Volume (compression) - Published
- 2013
3. Erratum to the paper 'L∞(L∞)-boundedness and convergence of DG(p)-solutions for nonlinear conservation laws with boundary conditions'
- Author
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Christian Henke and Lutz Angermann
- Subjects
Conservation law ,Pure mathematics ,Lemma (mathematics) ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Lebesgue integration ,Computational Mathematics ,Nonlinear system ,symbols.namesake ,Convergence (routing) ,symbols ,Boundary value problem ,Affine transformation ,Constant (mathematics) ,Mathematics - Abstract
In the paper (HA14), unfortunately, a computational error occurred in one estimate. Although the wrong estimate does not affect the main results, we want to present the necessary corrections. Essentially, Lemma 5.2 has to be corrected and, since it is used in the proof of Theorem 5.1, the proof of this theorem also requires an adaptation. (i) The corrected formulation of Lemma 5.2 is as follows. Lemma 5.2 For Lagrange finite elements with a shape-regular family of affine meshes { T n h } h>0 there is a constant C > 0 independent of q and h such that for all w ∈ Wh and q = 2m, m ∈N: CΛq−2 p (∇w,∇Ip h (wq−1))T ∫ T ‖∇w‖l2‖w‖ q−2 0,∞,T dx, ∀T ∈ T n h , (5.1) where Λp = ‖ ∑ndof i=1 |φi|‖0,∞,T is the Lebesgue constant.
- Published
- 2015
4. Book Review: Selected papers on the classification of varieties and moduli spaces
- Author
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János Kollár
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Moduli space ,Mathematics - Published
- 2005
5. On the Paper 'Asymptotics for the Moments of Singular Distributions'
- Author
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H.-J. Fischer
- Subjects
Pure mathematics ,Mathematics(all) ,Numerical Analysis ,General Mathematics ,Applied Mathematics ,Mathematical analysis ,Dimension (graph theory) ,Contrast (statistics) ,Singular measure ,Absolute continuity ,Measure (mathematics) ,WIMP ,Analysis ,Mathematics - Abstract
In their 1993 paper, W. Goh and J. Wimp derive interesting asymptotics for the moments cn(?) ? cn = ?10tnd?(t), n = 0, 1, 2, ..., of some singular distributions ? (with support ? 0, 1]), which contain oscillatory terms. They suspect, that this is a general feature of singular distributions and that this behavior provides a striking contrast with what happens for absolutely continuous distributions. In the present note, however, we give an example of an absolutely continuous measure with asymptotics of moments containing oscillatory terms, and an example of a singular measure having very regular asymptotic behavior of its moments. Finally, we give a short proof of the fact that the drop-off rate of the moments is exactly the local measure dimension about 1 (if it exists).
- Published
- 1995
- Full Text
- View/download PDF
6. Addendum to the paper: 'Existence of weak solutions for the Navier-Stokes equations with initial data in 𝐿^{𝑝}' [Trans. Amer. Math. Soc. 318 (1990), no. 1, 179–200; MR0968416 (90k:35199)]
- Author
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Calixto P. Calderón
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Addendum ,Navier–Stokes equations ,Mathematics - Abstract
This paper considers the existence of global weak solutions for the Navier-Stokes equations in the infinite cylinder R n × R + {{\mathbf {R}}^n} \times {{\mathbf {R}}_ + } with initial data in L r {L^r} , n ≥ 3 n \geq 3 , 1 > r > ∞ 1 > r > \infty . An imbedding theorem as well as related initial value problems are also studied, thus completing results in [2].
- Published
- 1990
7. Erratum to the paper: F. J. Gallego, B. P. Purnaprajna Projective normality and syzygies of algebraic surfaces
- Author
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Francisco Javier Gallego and Bangere P. Purnaprajna
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Algebraic surface ,Calculus ,Projective space ,Projective test ,Normality ,Mathematics ,media_common - Published
- 2000
8. Noncomplete linear systems on elliptic curves and Abelian varieties: Addendum to a paper by Ch. Birkenhake
- Author
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E. Ballico
- Subjects
Pure mathematics ,Elliptic curve ,Applied Mathematics ,General Mathematics ,Linear system ,Addendum ,Abelian group ,Algorithm ,Mathematics - Abstract
Here we give a result on the postulation (i.e. the 2-normality) of nonlinearly normal embeddings of Abelian varieties. This result improves some of the results proved in a recent paper by Ch. Birkenhake.
- Published
- 1998
9. REMARKS ON THE PAPER 'A NOTE ON EVERITT TYPE INTEGRAL INEQUALITY' OF B. G. PACHPATTE
- Author
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Dragoslav S. Mitrinović and Josip Pečarić
- Subjects
Pure mathematics ,Inequality ,Applied Mathematics ,General Mathematics ,media_common.quotation_subject ,Type (model theory) ,media_common ,Mathematics - Abstract
n the present note we give an interpolating inequality for Pachpatte's inequality from [1]. A discrete analogouos is also given.
- Published
- 1990
10. Some remarks on a paper by R. H. Bruck
- Author
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Trevor Evans
- Subjects
Loop (topology) ,Pure mathematics ,Rank (linear algebra) ,Subvariety ,Applied Mathematics ,General Mathematics ,Variety (universal algebra) ,Ring of integers ,Identity (music) ,Mathematics - Abstract
Introduction. In a recent paper [2] R. H. Bruck has introduced the concept of right neoring and discussed some properties of these systems. In particular, he has considered analogues of certain properties of the ring of integers. This paper is essentially a commentary on Bruck's paper and we generalize some of his results as follows. The construction of the universal right neoring in [2] is applied to the free monogenic $3-loop in any subvariety $3 of the variety of loops and a complete analogue of Theorem 4.1 of [2] is obtained for any one of these subvarieties. Then, using a result similar to those obtained in [5], it is shown that this construction yields uncountably many right neorings with an identity which generates the additive loop of the right neoring. Conversely, every right neoring with an identity which generates its additive loop can be obtained from a free monogenic $3-loop by the above construction. Each of these right neorings has some properties resembling those of the ring of integers. One possible answer is given to the question raised by Bruck concerning the existence of universal right neorings with free additive loop of arbitrary rank. A brief proof is given, using the results of [4; 5], of the cancellation properties of the monogenic universal right neoring. Finally, we discuss briefly the relationship between right neorings and the logarithmetics of Etherington.
- Published
- 1956
11. Direct product of division rings and a paper of Abian
- Author
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M. Chacron
- Subjects
Subdirect product ,Nilpotent ,Ring (mathematics) ,Pure mathematics ,Noncommutative ring ,Applied Mathematics ,General Mathematics ,Order (ring theory) ,Von Neumann regular ring ,Commutative property ,Direct product ,Mathematics - Abstract
It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.
- Published
- 1971
12. An operator valued function space integral: A sequel to Cameron and Storvick’s paper
- Author
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D. L. Skoug and G. W. Johnson
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Multiple integral ,Integral representation theorem for classical Wiener space ,Mathematical analysis ,Riemann integral ,Riemann–Stieltjes integral ,Singular integral ,Fourier integral operator ,Volume integral ,symbols.namesake ,symbols ,Daniell integral ,Mathematics - Abstract
Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.
- Published
- 1971
13. Remarks on the preceding paper of James A. Clarkson: 'Uniformly convex spaces' [Trans. Amer. Math. Soc. 40 (1936), no. 3; MR1501880]
- Author
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Nelson Dunford and Anthony P. Morse
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Pure mathematics ,Applied Mathematics ,General Mathematics ,Uniformly convex space ,Mathematics - Published
- 1936
14. Generalized Limits in General Analysis, First Paper
- Author
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Charles N. Moore
- Subjects
Pure mathematics ,Series (mathematics) ,Basis (linear algebra) ,Simple (abstract algebra) ,Generalization ,Applied Mathematics ,General Mathematics ,Multiple integral ,Partial derivative ,Divergent series ,Equivalence (measure theory) ,Mathematics - Abstract
The analogies that exist between infinite series and infinite integrals are well known and have frequently served to indicate the extension of a theorem or a method from one of these domains of investigation to the other. According to a principle of generalization that has been formulated by E. H. Moore, the presence of such analogies implies the existence of a general theory which incltudes the central features of both the special theories.t It is the purpose of the present paper to develop the fundamental principles of that sectioll of this general theorv which contains as particular instances the theories of Cesaro and H6lder summability of divergent series and divergent integrals. Furthermore, the usefulness of the theory will be illustrated by proving a general theorem in it which includes as special cases the Knopp-Schnee-Ford theoremt with regard to the equivalence of the Cesaro and Holder means for summing divergent series, an analogous theorem due to Landau ? concerning divergent integrals, and a further new theorem with regard to the equivalence of certain generalized derivatives. The general theorem just mentioned can be extended to the case of multiple limits so as to include other new theorems, analogous to those referred to above, with regard to multiple series, multiple integrals, and partial derivatives. This extension, however, involves formulas that are considerably more complicated than in the case of simple limits. I shall therefore reserve it for a second paper, as I wish to avoid algebraic complexity in this first presentation of the general theory. Following the terminology introduced by E. H. Moore, we indicate the basis of our general theory as follows
- Published
- 1922
15. Concerning the Arc-Curves and Basic Sets of a Continuous Curve, Second Paper
- Author
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W. Leake Ayres
- Subjects
Arc (geometry) ,Set (abstract data type) ,Pure mathematics ,Relation (database) ,Applied Mathematics ,General Mathematics ,Metric (mathematics) ,Point (geometry) ,Locally compact space ,Notation ,Separable space ,Mathematics - Abstract
In an earlier paper t with the same title, we have defined and studied the properties of certain subsets of a continuous curve? which we call the arc-curves of the continuous curve. In a recent paper, G. T. Whyburnil has defined the cyclic elements of a continuous curve, and he has considered a continuous curve as composed of its cyclic elements and has given a large number of the properties of connected collections of cyclic elements. On examining the two papers it is found that arc-curves and connected collections of cyclic elements have many properties in common; and, in fact, in part II of the present paper we shall show that, although these two sets were defined very differently, every connected collection of cyclic elements of a continuous curve is an arc-curve of the continuous curve, and conversely, every arc-curve that contains more than one point is a collection of cyclic elements of the continuous curve. In part III we will develop some new theory concerning the basic sets of a continuous curve, which were defined in Arc-curves, first paper, and shall show the relation between the basic sets and the nodes of a continuous curve. In part IV we shall show that an irreducible basic set of a continuous curve resembles 'in its properties the set of all end points of the continuous curve. All point sets considered in this paper are assumed to lie in a metric, separable, locally compact space. Notation. We shall use the common notation of the theory of sets, such as A +B, A-B, A B, etc., in its usual meaning. If H is a point set, the symbol H denotes the point set consisting of the points of H together
- Published
- 1929
16. A Note on Quillen's Paper 'Projective Modules Over Polynomial Rings'
- Author
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Moshe Roitman
- Subjects
Discrete mathematics ,Pure mathematics ,Collineation ,Applied Mathematics ,General Mathematics ,Polynomial ring ,Complex projective space ,Projective cover ,Projective line over a ring ,Projective space ,Projective module ,Quaternionic projective space ,Mathematics - Published
- 1977
17. On the Zeros of Dirichlet L-Functions.II (With Corrections to Ön the Zeros of Dirichlet L-Functions.I' and the Subsequent Papers)
- Author
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Akio Fujii
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Dirichlet L-function ,Dirichlet's energy ,Dirichlet eta function ,Class number formula ,symbols.namesake ,Dirichlet kernel ,Dirichlet's principle ,symbols ,General Dirichlet series ,Dirichlet series ,Mathematics - Published
- 1981
18. Algebraic Surfaces Invariant Under An Infinite Discontinuos Group of Birational Transformations: (Second Paper)
- Author
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Virgil Snyder
- Subjects
Algebraic cycle ,Pure mathematics ,Function field of an algebraic variety ,Applied Mathematics ,General Mathematics ,Algebraic group ,Algebraic surface ,Dimension of an algebraic variety ,Geometric invariant theory ,Invariant (mathematics) ,Algebraic closure ,Mathematics - Published
- 1913
19. Application of the Theory of Relative Cyclic Fields to both Cases of Fermat's Last Theorem (Second Paper)
- Author
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H. S. Vandiver
- Subjects
Pure mathematics ,Fermat's little theorem ,Proofs of Fermat's little theorem ,Applied Mathematics ,General Mathematics ,Regular prime ,Fermat's theorem on sums of two squares ,Wieferich prime ,Fermat's factorization method ,symbols.namesake ,Fermat's theorem ,symbols ,Mathematics ,Fermat number - Published
- 1927
20. Note on a paper by Mandelbrojt and MacLane
- Author
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Jacqueline Ferrand
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,Domain (ring theory) ,Mathematics::Differential Geometry ,Function (mathematics) ,Mathematical proof ,Mathematics - Abstract
Let A8 be the domain in the s-plane (s =o+it) defined by -gl(a) -Ah. The proofs of Theorems 1, I1, and III are the same, with the new function S(oI5.
- Published
- 1947
21. Comment on a Paper of C. Ulucay
- Author
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W. T. Scott
- Subjects
Pure mathematics ,Argument ,Applied Mathematics ,General Mathematics ,Relation (history of concept) ,Mathematics - Abstract
must hold. By an elementary theorem of geometry, C3 /t2s3M3(s) = 1, and a similar argument shows that tc satisfies this same relation in the case where wx/2 1. Because of the restriction on tc only those values of s may be used for which sM(s) oo as s->1, it is not permissible to let s-*1. Added in proof. See review by E. Reich, Math. Rev. vol. 19 (1958) p. 736.
- Published
- 1959
22. Correction to the Paper On the Zeros of Polynomials over Division Rings
- Author
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B. Gordon and T. S. Motzkin
- Subjects
Classical orthogonal polynomials ,Algebra ,Pure mathematics ,Difference polynomials ,Gegenbauer polynomials ,Macdonald polynomials ,Discrete orthogonal polynomials ,Applied Mathematics ,General Mathematics ,Orthogonal polynomials ,Hahn polynomials ,Koornwinder polynomials ,Mathematics - Published
- 1966
23. A note on the Hitchin-Thorpe inequality and Ricci flow on 4-manifolds
- Author
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Yuguang Zhang and Zhenlei Zhang
- Subjects
Mathematics - Differential Geometry ,Hitchin–Thorpe inequality ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Short paper ,Ricci flow ,Type inequality ,Differential Geometry (math.DG) ,Bounded function ,FOS: Mathematics ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,53C20, 53C44 ,Yamabe invariant ,Mathematics ,Scalar curvature - Abstract
In this short paper, we prove a Hitchin-Thorpe type inequality for closed 4-manifolds with non-positive Yamabe invariant, and admitting long time solutions of the normalized Ricci flow equation with bounded scalar curvature.
- Published
- 2012
24. On the classification of solutions of -Δ𝑢=𝑒^{𝑢} on ℝ^{ℕ}: Stability outside a compact set and applications
- Author
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Alberto Farina and E. N. Dancer
- Subjects
Pure mathematics ,Compact space ,Euclidean space ,Applied Mathematics ,General Mathematics ,Bounded function ,Short paper ,Topology ,Stability (probability) ,Domain (mathematical analysis) ,Mathematics - Abstract
In this short paper we prove that, for3≤N≤93 \le N \le 9, the problem−Δu=eu-\Delta u = e^uon the entire Euclidean spaceRN\mathbb {R}^Ndoes not admit any solution stable outside a compact set ofRN\mathbb {R}^N. This result is obtained without making any assumption about the boundedness of solutions. Furthermore, as a consequence of our analysis, we also prove the non-existence of finite Morse Index solutions for the considered problem. We then use our results to give some applications to bounded domain problems.
- Published
- 2008
25. A new proof of Mok’s generalized Frankel conjecture theorem
- Author
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Hui-Ling Gu
- Subjects
Pure mathematics ,Conjecture ,Maximum principle ,Simple (abstract algebra) ,Applied Mathematics ,General Mathematics ,Short paper ,Calculus ,Maximal principle ,Mathematics::Differential Geometry ,Transcendental number ,Mathematics - Abstract
In this short paper, we will give a simple and transcendental proof for Mok's theorem of the generalized Frankel conjecture. This work is based on the maximum principle proposed by Brendle and Schoen.
- Published
- 2008
26. Homological stability of non-orientable mapping class groups with marked points
- Author
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Elizabeth Hanbury
- Subjects
Discrete mathematics ,Pure mathematics ,Mathematics::Algebraic Geometry ,Applied Mathematics ,General Mathematics ,Short paper ,Homology (mathematics) ,Mathematics::Geometric Topology ,Mapping class group ,Mathematics - Abstract
Wahl recently proved that the homology of the non-orientable mapping class group stabilizes as the genus increases. In this short paper we analyse the situation where the underlying non-orientable surfaces have marked points.
- Published
- 2008
27. Iterates of Generic Polynomials and Generic Rational Functions
- Author
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Jamie Juul
- Subjects
Pure mathematics ,Degree (graph theory) ,Mathematics - Number Theory ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,MathematicsofComputing_GENERAL ,Galois group ,37P05, 11G50, 14G25 ,Rational function ,01 natural sciences ,Unpublished paper ,Generic polynomial ,Number theory ,Symmetric group ,Iterated function ,0103 physical sciences ,FOS: Mathematics ,Number Theory (math.NT) ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
In 1985, Odoni showed that in characteristic 0 0 the Galois group of the n n -th iterate of the generic polynomial with degree d d is as large as possible. That is, he showed that this Galois group is the n n -th wreath power of the symmetric group S d S_d . We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.
- Published
- 2014
28. AN ALMOST SCHUR THEOREM ON 4-DIMENSIONAL MANIFOLDS
- Author
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Guofang Wang, Yuxin Ge, Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), and Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Short paper ,01 natural sciences ,Schur's theorem ,Computer Science::Computers and Society ,[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph] ,Ricci-flat manifold ,0103 physical sciences ,Sectional curvature ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Schur product theorem ,Mathematics ,Scalar curvature - Abstract
International audience; In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.
- Published
- 2012
29. Non-negative Ricci curvature on closed manifolds under Ricci flow
- Author
-
Davi Maximo
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Short paper ,Ricci flow ,01 natural sciences ,Mathematics::Geometric Topology ,Mathematics - Analysis of PDEs ,Differential Geometry (math.DG) ,Bounded curvature ,0103 physical sciences ,FOS: Mathematics ,010307 mathematical physics ,Mathematics::Differential Geometry ,0101 mathematics ,10. No inequality ,Mathematics::Symplectic Geometry ,Ricci curvature ,Mathematics ,Analysis of PDEs (math.AP) - Abstract
In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of bounded curvature. This brings down to four dimensions a similar result B\"ohm and Wilking have for dimensions twelve and above, \cite{BW}. Moreover, the manifolds constructed here are \Kahler manifolds and relate to a question raised by Xiuxiong Chen in \cite{XC}, \cite{XCL}., Comment: New version with added references and corrected typos
- Published
- 2009
- Full Text
- View/download PDF
30. The Gottlieb group of finite linear quotients of odd-dimensional spheres
- Author
-
S. Allen Broughton
- Subjects
Pure mathematics ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Homotopy ,Short paper ,Geometry ,SPHERES ,Isomorphism ,Geometric proof ,Quotient ,Homeomorphism ,Mathematics - Abstract
Let G be a finite, freely acting group of homeomorphisms of the odd-dimensional sphere S 2n−1 . John Oprea has proven that the Gottlieb group of S 2n−1 /G equals Z(G), the centre of G. The purpose of this short paper is to give a considerably shorter, more geometric proof of Oprea's theorem in the important case where G is a linear group
- Published
- 1991
31. On the geometry of irreversible metric-measure spaces: Convergence, stability and analytic aspects
- Author
-
Wei Zhao and Alexandru Kristály
- Subjects
Pure mathematics ,Class (set theory) ,Applied Mathematics ,General Mathematics ,Stability (learning theory) ,Function (mathematics) ,Stability result ,Measure (mathematics) ,Metric (mathematics) ,Convergence (routing) ,Mathematics::Metric Geometry ,Mathematics::Differential Geometry ,Topology (chemistry) ,Mathematics - Abstract
The paper is devoted to the study of Gromov-Hausdorff convergence and stability of irreversible metric-measure spaces, both in the compact and noncompact cases. While the compact setting is mostly similar to the reversible case developed by J. Lott, K.-T. Sturm and C. Villani, the noncompact case provides various surprising phenomena. Since the reversibility of noncompact irreversible spaces might be infinite, it is motivated to introduce a suitable nondecreasing function that bounds the reversibility of larger and larger balls. By this approach, we are able to prove satisfactory convergence/stability results in a suitable – reversibility depending – Gromov-Hausdorff topology. A wide class of irreversible spaces is provided by Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the reversible and irreversible settings. We conclude the paper by proving various geometric and functional inequalities (as Brunn-Minkowski, Bishop-Gromov, log-Sobolev and Lichnerowicz inequalities) on irreversible structures.
- Published
- 2022
32. Unique Continuation at the Boundary for Harmonic Functions in C 1 Domains and Lipschitz Domains with Small Constant
- Author
-
Xavier Tolsa
- Subjects
Surface (mathematics) ,Pure mathematics ,Applied Mathematics ,General Mathematics ,Mathematics::Analysis of PDEs ,Boundary (topology) ,Lipschitz continuity ,Measure (mathematics) ,Domain (mathematical analysis) ,Mathematics - Analysis of PDEs ,Harmonic function ,Lipschitz domain ,Mathematics - Classical Analysis and ODEs ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,Constant (mathematics) ,31B05 31B20 ,Analysis of PDEs (math.AP) ,Mathematics - Abstract
Let $\Omega\subset\mathbb R^n$ be a $C^1$ domain, or more generally, a Lipschitz domain with small local Lipschitz constant. In this paper it is shown that if $u$ is a function harmonic in $\Omega$ and continuous in $\overline \Omega$ which vanishes in a relatively open subset $\Sigma\subset\partial\Omega$ and moreover the normal derivative $\partial_\nu u$ vanishes in a subset of $\Sigma$ with positive surface measure, then $u$ is identically $0$., Comment: More detailed explanation in some argument involving integration by parts and in Remark 3.3. An additional appendix with a self-contained proof of Lemma 4.3, whose proof was not included in the paper previously
- Published
- 2021
33. Aspherical manifolds, Mellin transformation and a question of Bobadilla–Kollár
- Author
-
Yongqiang Liu, Botong Wang, and Laurenţiu G. Maxim
- Subjects
Mathematics - Algebraic Geometry ,Pure mathematics ,Transformation (function) ,Applied Mathematics ,General Mathematics ,14F05, 14F35, 14F45, 32S60, 32L05, 58K15 ,Mathematics - Algebraic Topology ,Mathematics - Abstract
In their 2012 paper, Bobadilla and Koll\'ar studied topological conditions which guarantee that a proper map of complex algebraic varieties is a topological or differentiable fibration. They also asked whether a certain finiteness property on the relative covering space can imply that a proper map is a fibration. In this paper, we answer positively the integral homology version of their question in the case of abelian varieties, and the rational homology version in the case of compact ball quotients. We also propose several conjectures in relation to the Singer-Hopf conjecture in the complex projective setting., Comment: published/final version
- Published
- 2021
34. Chaotic behavior of the p-adic Potts–Bethe mapping II
- Author
-
Otabek Khakimov and Farrukh Mukhamedov
- Subjects
Pure mathematics ,Applied Mathematics ,General Mathematics ,ComputingMethodologies_DOCUMENTANDTEXTPROCESSING ,Chaotic ,Mathematics - Abstract
The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst.38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$ ). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs).
- Published
- 2021
35. On the Baum–Connes conjecture for discrete quantum groups with torsion and the quantum Rosenberg conjecture
- Author
-
Adam Skalski and Yuki Arano
- Subjects
Pure mathematics ,Conjecture ,Mathematics::Operator Algebras ,Quantum group ,Applied Mathematics ,General Mathematics ,Mathematics - Operator Algebras ,Crossed product ,Unimodular matrix ,Mathematics::K-Theory and Homology ,Primary 46L67, Secondary 46L80 ,FOS: Mathematics ,Baum–Connes conjecture ,Countable set ,Equivariant map ,Operator Algebras (math.OA) ,Quantum ,Mathematics - Abstract
We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition., Comment: 15 pages, v2 corrects a few minor points. The final version of the paper will appear in the Proceedings of the American Mathematical Society
- Published
- 2021
36. The nilpotent cone for classical Lie superalgebras
- Author
-
Daniel K. Nakano and L. Jenkins
- Subjects
Pure mathematics ,Nilpotent cone ,17B20, 17B10 ,Applied Mathematics ,General Mathematics ,Group Theory (math.GR) ,Representation theory ,Mathematics::Quantum Algebra ,FOS: Mathematics ,Representation Theory (math.RT) ,Mathematics::Representation Theory ,Mathematics - Group Theory ,Mathematics - Representation Theory ,Mathematics - Abstract
In this paper the authors introduce an analogue of the nilpotent cone, N {\mathcal N} , for a classical Lie superalgebra, g {\mathfrak g} , that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g 0 ¯ ⊕ g 1 ¯ {\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1} with Lie G 0 ¯ = g 0 ¯ \text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0} , it is shown that there are finitely many G 0 ¯ G_{\bar 0} -orbits on N {\mathcal N} . Later the authors prove that the Duflo-Serganova commuting variety, X {\mathcal X} , is contained in N {\mathcal N} for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.
- Published
- 2021
37. Multiplicative constants and maximal measurable cocycles in bounded cohomology
- Author
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Marco Moraschini, Alessio Savini, Moraschini M., and Savini A.
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Applied Mathematics ,General Mathematics ,Multiplicative function ,Lattice ,Geometric Topology (math.GT) ,Cohomology ,Mathematics - Geometric Topology ,Maximal cocycle ,Mathematics::Quantum Algebra ,Bounded function ,FOS: Mathematics ,Bounded cohomology ,Boundary map ,Invariant (mathematics) ,Zimmer cocycle ,Mathematics - Abstract
Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $\textup{PU}(m,1)$-cocycles of complex hyperbolic lattices., Comment: 35 pages; Major corrections along the paper following the referee's suggestions. To appear in Ergod. Theory Dyn. Syst
- Published
- 2021
38. Hodge theory of the Turaev cobracket and the Kashiwara–Vergne problem
- Author
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Richard Hain
- Subjects
Surface (mathematics) ,Pure mathematics ,Group (mathematics) ,Applied Mathematics ,General Mathematics ,Hodge theory ,Mathematics::Geometric Topology ,Mathematics::Algebraic Geometry ,Morphism ,Mathematics::Quantum Algebra ,Torsor ,Algebraic curve ,Algebraic number ,Hodge structure ,Mathematics - Abstract
In this paper we show that, after completing in the $I$-adic topology, the Turaev cobracket on the vector space freely generated by the closed geodesics on a smooth, complex algebraic curve $X$ with an algebraic framing is a morphism of mixed Hodge structure. We combine this with results of a previous paper (arXiv:1710.06053) on the Goldman bracket to construct torsors of solutions of the Kashiwara--Vergne problem in all genera. The solutions so constructed form a torsor under a prounipotent group that depends only on the topology of the framed surface. We give a partial presentation of these groups. Along the way, we give a homological description of the Turaev cobracket.
- Published
- 2021
39. On curves with circles as their isoptics
- Author
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Waldemar Cieślak and Witold Mozgawa
- Subjects
Pure mathematics ,Class (set theory) ,Plane curve ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Regular polygon ,02 engineering and technology ,Characterization (mathematics) ,Ellipse ,01 natural sciences ,Discrete Mathematics and Combinatorics ,0101 mathematics ,021101 geological & geomatics engineering ,Mathematics - Abstract
In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.
- Published
- 2021
40. Stability and collapse of the Lyapunov spectrum for Perron–Frobenius operator cocycles
- Author
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Anthony Quas and Cecilia González-Tokman
- Subjects
Pure mathematics ,Mathematics::Dynamical Systems ,Dense set ,Applied Mathematics ,General Mathematics ,Blaschke product ,Banach space ,Lyapunov exponent ,Fixed point ,symbols.namesake ,Unit circle ,symbols ,Invariant measure ,Mathematics ,Analytic function - Abstract
In this paper, we study random Blaschke products, acting on the unit circle, and consider the cocycle of Perron-Frobenius operators acting on Banach spaces of analytic functions on an annulus. We completely describe the Lyapunov spectrum of these cocycles. As a corollary, we obtain a simple random Blaschke product system where the Perron-Frobenius cocycle has infinitely many distinct Lyapunov exponents, but where arbitrarily small natural perturbations cause a complete collapse of the Lyapunov spectrum, except for the exponent 0 associated with the absolutely continuous invariant measure. That is, under perturbations, the Lyapunov exponents become 0 with multiplicity 1, and $-\infty$ with infinite multiplicity. This is superficially similar to the finite-dimensional phenomenon, discovered by Bochi \cite{Bochi-thesis}, that away from the uniformly hyperbolic setting, small perturbations can lead to a collapse of the Lyapunov spectrum to zero. In this paper, however, the cocycle and its perturbation are explicitly described; and further, the mechanism for collapse is quite different. We study stability of the Perron-Frobenius cocycles arising from general random Blaschke products. We give a necessary and sufficient criterion for stability of the Lyapunov spectrum in terms of the derivative of the random Blaschke product at its random fixed point, and use this to show that an open dense set of Blaschke product cocycles have hyperbolic Perron-Frobenius cocycles. In the final part, we prove a relationship between the Lyapunov spectrum of a single cocycle acting on two different Banach spaces, allowing us to draw conclusions for the same cocycles acting on $C^r$ functions spaces.
- Published
- 2021
41. Volume preserving flow and Alexandrov–Fenchel type inequalities in hyperbolic space
- Author
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Ben Andrews, Xuzhong Chen, and Yong Wei
- Subjects
Pure mathematics ,Geodesic dome ,Applied Mathematics ,General Mathematics ,Hyperbolic space ,010102 general mathematics ,Type (model theory) ,Curvature ,01 natural sciences ,law.invention ,Hypersurface ,Flow (mathematics) ,Principal curvature ,law ,Mathematics::Differential Geometry ,Sectional curvature ,0101 mathematics ,Mathematics - Abstract
In this paper, we study flows of hypersurfaces in hyperbolic space, and apply them to prove geometric inequalities. In the first part of the paper, we consider volume preserving flows by a family of curvature functions including positive powers of $k$-th mean curvatures with $k=1,\cdots,n$, and positive powers of $p$-th power sums $S_p$ with $p>0$. We prove that if the initial hypersurface $M_0$ is smooth and closed and has positive sectional curvatures, then the solution $M_t$ of the flow has positive sectional curvature for any time $t>0$, exists for all time and converges to a geodesic sphere exponentially in the smooth topology. The convergence result can be used to show that certain Alexandrov-Fenchel quermassintegral inequalities, known previously for horospherically convex hypersurfaces, also hold under the weaker condition of positive sectional curvature. In the second part of this paper, we study curvature flows for strictly horospherically convex hypersurfaces in hyperbolic space with speed given by a smooth, symmetric, increasing and homogeneous degree one function $f$ of the shifted principal curvatures $\lambda_i=\kappa_i-1$, plus a global term chosen to impose a constraint on the quermassintegrals of the enclosed domain, where $f$ is assumed to satisfy a certain condition on the second derivatives. We prove that if the initial hypersurface is smooth, closed and strictly horospherically convex, then the solution of the flow exists for all time and converges to a geodesic sphere exponentially in the smooth topology. As applications of the convergence result, we prove a new rigidity theorem on smooth closed Weingarten hypersurfaces in hyperbolic space, and a new class of Alexandrov-Fenchel type inequalities for smooth horospherically convex hypersurfaces in hyperbolic space.
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- 2021
42. Local limit theorems in relatively hyperbolic groups I: rough estimates
- Author
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Matthieu Dussaule
- Subjects
Pure mathematics ,Series (mathematics) ,010201 computation theory & mathematics ,Spectral radius ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0102 computer and information sciences ,Limit (mathematics) ,0101 mathematics ,Random walk ,01 natural sciences ,Mathematics - Abstract
This is the first of a series of two papers dealing with local limit theorems in relatively hyperbolic groups. In this first paper, we prove rough estimates for the Green function. Along the way, we introduce the notion of relative automaticity which will be useful in both papers and we show that relatively hyperbolic groups are relatively automatic. We also define the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. We then use our estimates for the Green function to prove that $p_n\asymp R^{-n}n^{-3/2}$ for spectrally positive-recurrent random walks, where $p_n$ is the probability of going back to the origin at time n and where R is the inverse of the spectral radius of the random walk.
- Published
- 2021
43. Generalized variational inequalities for maximal monotone operators
- Author
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Nguyen Quynh Nga
- Subjects
Pure mathematics ,Monotone polygon ,Fang ,Applied Mathematics ,General Mathematics ,Numerical analysis ,Variational inequality ,Banach space ,Solution set ,Structure (category theory) ,Fréchet derivative ,Mathematics - Abstract
In this paper, we present some new results on the existence of solutions of generalized variational inequalities for set-valued mappings in reflexive Banach spaces with Frechet differentiable norms. Moreover, the structure of the solution sets is investigated. The result obtained in this paper improves and extends the ones announced by Fang and Peterson [S. C. Fang and E. L. Peterson, Generalized Variational Inequalities, J. Optim. Theory Appl., 38 (1982), 363-383] and others.
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- 2021
44. Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks
- Author
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Armando Martino, Stefano Francaviglia, Francaviglia, Stefano, and Martino, Armando
- Subjects
Outer space, conjugacy problem, automorphisms of free groups, graphs ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Spectrum (functional analysis) ,Group Theory (math.GR) ,Train track map ,Automorphism ,Lipschitz continuity ,01 natural sciences ,Convexity ,Free product ,Metric (mathematics) ,FOS: Mathematics ,20E06, 20E36, 20E08 ,0101 mathematics ,Invariant (mathematics) ,Mathematics - Group Theory ,Mathematics - Abstract
This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems., 50 pages. Originally part of arXiv:1703.09945 . We decided to split that paper following the recommendations of a referee. Updated subsequent to acceptance by Transactions of the American Mathematical Society
- Published
- 2021
45. On a new class of functional equations satisfied by polynomial functions
- Author
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Chisom Prince Okeke, Timothy Nadhomi, Maciej Sablik, and Tomasz Szostok
- Subjects
Polynomial functions ,Polynomial ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Fr'echet operator ,Functional equations ,010103 numerical & computational mathematics ,Function (mathematics) ,01 natural sciences ,Continuity of monomial functions ,Monomial functions ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Linear combination ,Linear equation ,Mathematics - Abstract
The classical result of L. Székelyhidi states that (under some assumptions) every solution of a general linear equation must be a polynomial function. It is known that Székelyhidi’s result may be generalized to equations where some occurrences of the unknown functions are multiplied by a linear combination of the variables. In this paper we study the equations where two such combinations appear. The simplest nontrivial example of such a case is given by the equation$$\begin{aligned} F(x + y) - F(x) - F(y) = yf(x) + xf(y) \end{aligned}$$F(x+y)-F(x)-F(y)=yf(x)+xf(y)considered by Fechner and Gselmann (Publ Math Debrecen 80(1–2):143–154, 2012). In the present paper we prove several results concerning the systematic approach to the generalizations of this equation.
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- 2021
46. Area‐Minimizing Currents mod 2 Q : Linear Regularity Theory
- Author
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Jonas Hirsch, Camillo De Lellis, Salvatore Stuvard, and Andrea Marchese
- Subjects
Pure mathematics ,multiple valued functions, Dirichlet integral, regularity theory, area minimizing currents mod(p), minimal surfaces, linearization ,Generalization ,General Mathematics ,Dimension (graph theory) ,area minimizing currents mod(p) ,linearization ,minimal surfaces ,Dirichlet integral ,01 natural sciences ,010104 statistics & probability ,symbols.namesake ,Mathematics - Analysis of PDEs ,Mod ,FOS: Mathematics ,49Q15, 49Q05, 49N60, 35B65, 35J47 ,0101 mathematics ,Mathematics ,Applied Mathematics ,010102 general mathematics ,Codimension ,regularity theory ,symbols ,multiple valued functions ,Analysis of PDEs (math.AP) - Abstract
We establish a theory of $Q$-valued functions minimizing a suitable generalization of the Dirichlet integral. In a second paper the theory will be used to approximate efficiently area minimizing currents $\mathrm{mod}(p)$ when $p=2Q$, and to establish a first general partial regularity theorem for every $p$ in any dimension and codimension., 37 pages. First part of a two-papers work aimed at establishing a first general partial regularity theory for area minimizing currents modulo p, for any p and in any dimension and codimension. v3 is the final version, to appear on Comm. Pure Appl. Math
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- 2020
47. The structure and free resolutions of the symbolic powers of star configurations of hypersurfaces
- Author
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Paolo Mantero
- Subjects
Monomial ,Pure mathematics ,Mathematics::Commutative Algebra ,Betti number ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Algebraic geometry ,Star (graph theory) ,Mathematics - Commutative Algebra ,Commutative Algebra (math.AC) ,01 natural sciences ,Representation theory ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,Young tableau ,0101 mathematics ,Commutative algebra ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Star configurations of points are configurations with known (and conjectured) extremal behaviors among all configurations of points in $\mathbb P_k^n$; additional interest come from their rich structure, which allows them to be studied using tools from algebraic geometry, combinatorics, commutative algebra and representation theory. In the present paper we investigate the more general problem of determining the structure of symbolic powers of a wide generalization of star configurations of points (introduced by Geramita, Harbourne, Migliore and Nagel) called star configurations of hypersurfaces in $\mathbb P_k^n$. Here (1) we provide explicit minimal generating sets of the symbolic powers $I^{(m)}$ of these ideals $I$, (2) we introduce a notion of $\delta$-c.i. quotients, which generalize ideals with linear quotients, and show that $I^{(m)}$ have $\delta$-c.i. quotients, (3) we show that the shape of the Betti tables of these symbolic powers is determined by certain "Koszul" strands and we prove that a little bit more than the bottom half of the Betti table has a regular, almost hypnotic, pattern, and (4) we provide a closed formula for all the graded Betti numbers in these strands. As a special case of (2) we deduce that symbolic powers of ideals of star configurations of points have linear quotients. We also improve and extend results by Galetto, Geramita, Shin and Van Tuyl, and provide explicit new general formulas for the minimal number of generators and the symbolic defects of star configurations. Finally, inspired by Young tableaux, we introduce a technical tool which may be of independent interest: it is a "canonical" way of writing any monomial in any given set of polynomials. Our methods are characteristic--free., Comment: Final revision (original paper was accepted for publication in Trans. Amer. Math. Soc.)
- Published
- 2020
48. Existence of solutions for a class of $p(x)$-curl systems arising in electromagnetism without (A-R) type conditions
- Author
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Z. Naghizadeh, Nguyen Thanh Chung, and Ghasem A. Afrouzi
- Subjects
Curl (mathematics) ,Pure mathematics ,Electromagnetism ,Applied Mathematics ,General Mathematics ,Multiplicity results ,Mountain pass theorem ,Mathematics - Abstract
In this paper, we study the existence and multiplicity of solutions for a class of of $p(x)$-curl systems arising in electromagnetism. Under suitable conditions on the nonlinearities which do not satisfy Ambrosetti-Rabinowitz type conditions, we obtain some existence and multiplicity results for the problem by using the mountain pass theorem and fountain theorem. Our main results in this paper complement and extend some earlier ones concerning the $p(x)$-curl operator in [4, 15].
- Published
- 2020
49. Multi-bump analysis for Trudinger–Moser nonlinearities. I. Quantification and location of concentration points
- Author
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Pierre-Damien Thizy and Olivier Druet
- Subjects
Work (thermodynamics) ,Pure mathematics ,Degree (graph theory) ,Liouville equation ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Strong interaction ,Dimension (graph theory) ,Mathematics::Analysis of PDEs ,01 natural sciences ,Nonlinear system ,0101 mathematics ,Mathematics - Abstract
In this paper, we investigate carefully the blow-up behaviour of sequences of solutions of some elliptic PDE in dimension two containing a nonlinearity with Trudinger-Moser growth. A quantification result had been obtained by the first author in [15] but many questions were left open. Similar questions were also explicitly asked in subsequent papers, see Del Pino-Musso-Ruf [12], Malchiodi-Martinazzi [30] or Martinazzi [34]. We answer all of them, proving in particular that blow up phenomenon is very restrictive because of the strong interaction between bubbles in this equation. This work will have a sequel, giving existence results of critical points of the associated functional at all energy levels via degree theory arguments, in the spirit of what had been done for the Liouville equation in the beautiful work of Chen-Lin [8].
- Published
- 2020
50. Tailoring a Pair of Pants: The Phase Tropical Version
- Author
-
Ilia Zharkov
- Subjects
Statistics and Probability ,Pure mathematics ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Phase (waves) ,01 natural sciences ,010305 fluids & plasmas ,Mathematics - Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Isotopy ,0101 mathematics ,Algebraic Geometry (math.AG) ,Pair of pants ,Mathematics - Abstract
We show that the phase tropical pair-of-pants is (ambient) isotopic to the complex pair-of-pants. This paper can serve as an addendum to the author's joint paper with Ruddat arXiv:2001.08267 where an isotopy between complex and ober-tropical pairs-of-pants was shown. Thus all three versions are isotopic., 10 pages, 8 figures. arXiv admin note: text overlap with arXiv:2001.08267
- Published
- 2020
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