Pure Mathematics

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This is the collection for the University of Waterloo's Department of Pure Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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2-Semilattices: Residual Properties and Applications to the Constraint Satisfaction Problem
(University of Waterloo, 2017-08-22) Payne, Ian
Semilattices are algebras known to have an important connection to partially ordered sets. In particular, if a partially ordered set $(A,\leq)$ has greatest lower bounds, a semilattice $(A;\wedge)$ can be associated to the order where $a\wedge b$ is the greatest lower bound of $a$ and $b$. In this thesis, we study a class of algebras known as 2-semilattices, which is a generalization of the class of semilattices. Similar to the correspondence between partial orders and semilattices, there is a correspondence between certain digraphs and 2-semilattices. That is, to every 2-semilattice, there is an associated digraph which holds information about the 2-semilattice. Making frequent use of this correspondence, we explore the class of 2-semilattices from three perspectives: (i) Tame Congruence Theory, (ii) the ``residual character" of the class of 2-semilattices, and (iii), the constraint satisfaction problem. Tame Congruence Theory, developed in [29], is a structure theory on finite algebras driven by understanding their prime congruence quotients. The theory assigns to each such quotient a type from 1 to 5. We show that types 3, 4, and 5 can occur in the class of 2-semilattices, but type 4 can not occur in a finite simple 2-semilattice. Classes of algebras contain ``subdirectly irreducible" members which hold information about the class. Specifically, the size of these members has been of interest to many authors. We show for certain subclasses of the class of 2-semilattices that there is no cardinal bound on the size of the irreducible members in that subclass. The ``fixed template constraint satisfaction problem" can be identified with the decision problem hom$(\mathbb{A})$ where $\mathbb{A}$ is a fixed finite relational structure. The input to hom$(\mathbb{A})$ is a finite structure $\mathbb{B}$ similar to $\mathbb{A}$. The question asked is ``does there exist a homomorphism from $\mathbb{B}$ to $\mathbb{A}$?" Feder and Vardi [22] conjectured that for fixed $\mathbb{A}$, this decision problem is either NP-complete or solvable in polynomial time. Bulatov [15] confirmed this conjecture in the case that $\mathbb{A}$ is invariant under a 2-semilattice operation. We extend this result.
Abelian, amenable operator algebras are similar to C∗ -algebras
(Duke University Press, 2016-12) Marcoux, Laurent W.; Popov, Alexey I.
Suppose that H is a complex Hilbert space and that ℬ(H) denotes the bounded linear operators on H. We show that every abelian, amenable operator algebra is similar to a C∗-algebra. We do this by showing that if 𝒜⊆ℬ(H) is an abelian algebra with the property that given any bounded representation ϱ:𝒜→ℬ(Hϱ) of 𝒜 on a Hilbert space Hϱ, every invariant subspace of ϱ(𝒜) is topologically complemented by another invariant subspace of ϱ(𝒜), then 𝒜 is similar to an abelian C∗-algebra.
Abstract and Explicit Constructions of Jacobian Varieties
(University of Waterloo, 2018-08-10) Urbanik, David
Abelian varieties, in particular Jacobian varieties, have long attracted interest in mathematics. Their influence pervades arithmetic geometry and number theory, and understanding their construction was a primary motivator for Weil in his work on developing new foundations for algebraic geometry in the 1930s and 1940s. Today, these exotic mathematical objects find applications in cryptography and computer science, where they can be used to secure confidential communications and factor integers in subexponential time. Although in many respects well-studied, working in concrete, explicit ways with abelian varieties continues to be difficult. The issue is that, aside from the case of elliptic curves, it is often difficult to find ways of modelling and understanding these objects in ways amenable to computation. Often, the approach taken is to work ``indirectly'' with abelian varieties, in particular with Jacobians, by working instead with divisors on their associated curves to simplify computations. However, properly understanding the mathematics underlying the direct approach --- why, for instance, one can view the degree zero divisor classes on a curve as being points of a variety --- requires sophisticated mathematics beyond what is usually understood by algorithms designers and even experts in computational number theory. A direct approach, where explicit polynomial and rational functions are given that define both the abelian variety and its group law, cannot be found in the literature for dimensions greater than two. In this thesis, we make two principal contributions. In the first, we survey the mathematics necessary to understand the construction of the Jacobian of a smooth algebraic curve as a group variety. In the second, we present original work with gives the first instance of explicit rational functions defining the group law of an abelian variety of dimension greater than two. In particular, we derive explicit formulas for the group addition on the Jacobians of hyperelliptic curves of every genus g, and so give examples of explicit rational formulas for the group law in every positive dimension.
Algebraic Approaches to State Complexity of Regular Operations
(University of Waterloo, 2019-10-15) Davies, Sylvie
The state complexity of operations on regular languages is an active area of research in theoretical computer science. Through connections with algebra, particularly the theory of semigroups and monoids, many problems in this area can be simplified or completely reduced to combinatorial problems. We describe various algebraic techniques for attacking state complexity problems. We present a general method for constructing witness languages for operations -- languages that attain the worst-case state complexity when used as the argument(s) of the operation. Our construction is based on full transformation monoids, which contain all functions from a finite set into itself. When a witness for an operation is known, determining the state complexity essentially becomes a counting problem. These counting problems, however, are not necessarily easy, and the witness languages produced by this method are not ideal in the sense that they have extremely large alphabets. We thus investigate some commonly used operations in detail, and look for algebraic techniques to simplify the combinatorial side of state complexity problems and to simplify the search for small-alphabet witnesses. For boolean operations (e.g., union, intersection, difference) we show that these combinatorial problems can be solved easily in special cases by studying the subgroup of permutations in the syntactic monoid of a witness candidate. If the subgroup of permutations is known to have some strong transitivity property, such as primitivity or 2-transitivity, we can draw conclusions about the worst-case state complexity when this language is used in a boolean operation. For the operations of concatenation and Kleene star (an iterated version of concatenation), we describe a “construction set” method to simplify state complexity lower-bound proofs, and determine some algebraic conditions under which this method can be applied. For the reversal operation, we show that the state complexity of the reverse of a language is closely related to the syntactic monoid of the language, and use this fact to investigate a generalized version of the reversal state complexity problem. After describing our techniques, we demonstrate them by applying them to some classical state complexity problems. We obtain complex generalizations of the classical results that would be difficult to prove without the machinery we develop.
Algebraic characterization of multivariable dynamics
(University of Waterloo, 2009-03-26T15:37:16Z) Ramsey, Christopher
Let X be a locally compact Hausdorﬀ space along with n proper continuous maps σ = (σ1 , · · · , σn ). Then the pair (X, σ) is called a dynamical system. To each system one can associate a universal operator algebra called the tensor algebra A(X, σ). The central question in this theory is whether these algebras characterize dynamical systems up to some form of natural conjugacy. In the n = 1 case, when there is only one self-map, we will show how this question has been completely determined. For n ≥ 2, isomorphism of two tensor algebras implies that the two dynamical systems are piecewise conjugate. The converse was only established for n = 2 and 3. We introduce a new construction of the unitary group U (n) that allows us to prove the algebraic characterization question in n = 2, 3 and 4 as well as translating this conjecture into a conjecture purely about the structure of the unitary group.
ALGEBRAIC DEGREE IN SPATIAL MATRICIAL NUMERICAL RANGES OF LINEAR OPERATORS
(American Mathematical Society, 2021-07-20) Bernik, Janez; Livshits, Leo; MacDonald, Gordon W.; Marcoux, Laurent W.; Mastnak, Mitja; Radjavi, Heydar
We study the maximal algebraic degree of principal ortho-compressions of linear operators that constitute spatial matricial numerical ranges of higher order. We demonstrate (amongst other things) that for a (possibly unbounded) operator L on a Hilbert space, every principal m-dimensional ortho-compression of L has algebraic degree less than m if and only if rank(L − λI) ≤ m − 2 for some λ ∈ C
The Algebraic Kirchberg--Phillips Conjecture for Leavitt Path Algebras
(University of Waterloo, 2015-09-18) Hossain, Ehsaan
This essay is meant to be an exposition of the theory of Leavitt path algebras and graph C*-algebras, with an aim to discuss some current classification questions. These two classes of algebras sit on opposite sides of a mirror, each reflecting aspects of the other. The majority of these notes is taken to describe the basic properties of Leavitt path algebras and graph C*-algebras, the main theme being the translation of graph-theoretic properties into exclusively (C*-)algebraic properties.
Amenability for the Fourier Algebra
(University of Waterloo, 2007-08-29T16:56:45Z) Tikuisis, Aaron Peter
The Fourier algebra A(G) can be viewed as a dual object for the group G and, in turn, for the group algebra L1(G). It is a commutative Banach algebra constructed using the representation theory of the group, and from which the group G may be recovered as its spectrum. When G is abelian, A(G) coincides with L1(G^); for non-abelian groups, it is viewed as a generalization of this object. B. Johnson has shown that G is amenable as a group if and only if L1(G) is amenable as a Banach algebra. Hence, it is natural to expect that the cohomology of A(G) will reflect the amenability of G. The initial hypothesis to this effect is that G is amenable if and only if A(G) is amenable as a Banach algebra. Interestingly, it turns out that A(G) is amenable only when G has an abelian group of finite index, leaving a large class of amenable groups with non-amenable Fourier algebras. The dual of A(G) is a von Neumann algebra (denoted VN(G)); as such, A(G) inherits a natural operator space structure. With this operator space structure, A(G) is a completely contractive Banach algebra, which is the natural operator space analogue of a Banach algebra. By taking this additional structure into account, one recovers the intuition behind the first conjecture: Z.-J. Ruan showed that G is amenable if and only if A(G) is operator amenable. This thesis concerns both the non-amenability of the Fourier algebra in the category of Banach spaces and why Ruan's Theorem is actually the proper analogue of Johnson's Theorem for A(G). We will see that the operator space projective tensor product behaves well with respect to the Fourier algebra, while the Banach space projective tensor product generally does not. This is crucial to explaining why operator amenability is the right sort of amenability in this context, and more generally, why A(G) should be viewed as a completely contractive Banach algebra and not merely a Banach algebra.
Applications of Operator Systems in Dynamics, Correlation Sets, and Quantum Graphs
(University of Waterloo, 2020-07-24) Kim, Se Jin
The recent works of Kalantar-Kennedy, Katsoulis-Ramsey, Ozawa, and Dykema-Paulsen have demonstrated that many problems in the theory of operator algebras and quantum information can be approached by looking at various subspaces of bounded operators on a Hilbert space. This thesis is a compilation of papers written by the author with various coauthors that apply the theory of operator systems to expand on some of these results. This thesis is split into two parts. In Part I, we start by expanding on the theory of crossed product of operator algebras of Katsoulis and Ramsey. We first develop an analogous crossed product of operator systems. We then reduce two open problems on the uniqueness of universal crossed product operator algebras into one of operator systems and show that it has answers in the negative. In the final chapter of Part I, we generalize results of Kakariadis, Dor On-Salmon, and Katsoulis- Ramsey to characterize which tensor algebras of C*-correspondences admit hyperrigidity. In Part II, we look at synchronous correlation sets, introduced by Dykema-Paulsen as a symmetric form of Tsirelson’s quantum correlation sets. These sets have the distinct advantage that there is a nice C*-algebraic characterization that we present in Chapter 6. We show that the correlation sets coming from the tensor models on finite and infinite dimensional Hilbert spaces cannot be distinguished by synchronous correlation sets and that one can distinguish this set from the correlation sets which arise as limits of correlation sets arising from finite dimensional tensor models. Beyond this, we show that Tsirelson’s problem is equivalent its synchronous analogue, expanding on a result of Dykema-Paulsen. We end the thesis by looking at generalizations of graphs by the ways of operator subspaces of the space of matrices. We construct an analogue of the graph complement and show its robustness by deriving various generalizations of known graph inequalities.
Applications of the minimal modelprogram in arithmetic dynamics
(University of Waterloo, 2021-09-07) Nasserden, Brett
Let F be a surjective endomorphism of a normal projective variety X defined over a number field. The dynamics of F may be studied through the dynamics of the linear action of an associated linear pull-back action on divisors. This linear action is governed by the spectral theory of pull-back. We first study eigen-divisors (that is eigenvectors of the pullback action) that have Iitaka dimension 0. We analyze the base locus of such divisors and interpret the set of small eigenvalues in terms of the canonical heights of Jordan blocks described by Kawaguchi and Silverman. We identify a linear algebraic condition on surjective morphisms that may be useful in proving instances of the Kawaguchi-Silverman conjecture. We prove the Kawaguchi-Silverman conjecture and verify the aforementioned linear algebraic condition holds for projective bundles over an elliptic curve that are direct sums of Atiyah bundles. This represents new progress in the last remaining case of the Kawaguchi-Silverman conjecture for projective bundles over curves. By a result of Silverman and Kawaguchi, the arithmetic degree of a point P on X is an eigenvalue of the associated linear-pull back mapping. We give examples of endomorphisms of abelian varieties that possess an eigenvalue which is not an arithmetic degree. On the other hand, we show using the minimal model program that if X is a simplicial toric variety then every eigenvalue of the linear pullback action is an arithmetic degree. Finally, we give a program to study the arithmetic dynamics of higher-dimensional projective varieties using the minimal model program. In particular, we describe how one might use the minimal model program to determine if certain surjective morphisms have a dense set of pre-periodic points, and how to study the Medvedev-Scanlon conjecture for certain surjective endomorphisms using the minimal model program.
Approximation Constants for Closed Subschemes of Projective Varieties
(University of Waterloo, 2019-06-19) Rollick, Nickolas
Diophantine approximation is a branch of number theory with a long history, going back at least to the work of Dirichlet and Liouville in the 1840s. The innocent-looking question of how well an arbitrary real algebraic number can be approximated by rational numbers (relative to the size of the denominator of the approximating rational number) took more than 100 years to resolve, culminating in the definitive Fields Medal-winning work of Klaus Roth in 1955. Much more recently, David McKinnon and Mike Roth have re-phrased and generalized this Diophantine approximation question to apply in the setting of approximating algebraic points on projective varieties defined over number fields. To do this, they defined an "approximation constant", depending on the point one wishes to approximate and a given line bundle. This constant measures the tradeoff between the closeness of the approximation and the arithmetic complexity of the point used to make the approximation, as measured by a height function associated to the line bundle. In particular, McKinnon and Roth succeeded in proving lower bounds on the approximation constant in terms of the "Seshadri constant" associated to the given point and line bundle, measuring local positivity of the line bundle around the point. Appropriately interpreted, these results generalize the classical work of Liouville and Roth, and the corresponding McKinnon-Roth theorems are therefore labelled "Liouville-type" and "Roth-type" results. Recent work of Grieve and of Ru-Wang have taken the Roth-type theorems even further; in contrast, we explore results of Liouville-type, which are more elementary in nature. In Chapter 2, we lay the groundwork necessary to define the approximation constant at a point, before generalizing the McKinnon-Roth definition to approximations of arbitrary closed subschemes. We also introduce the notion of an essential approximation constant, which ignores unusually good approximations along proper Zariski-closed subsets. After verifying that our new approximation constant truly does generalize the constant of McKinnon-Roth, Chapter 3 establishes a fundamental lower bound on the approximation constants of closed subschemes of projective space, depending only on the equations cutting out the subscheme. In Chapter 4, we provide a series of explicit computations of approximation constants, both for subschemes satisfying suitable geometric conditions, and for curves of low degree in projective 3-space. We will encounter difficulties computing the approximation constant exactly for general cubic curves, and we spend some time showing why some of the more evident approaches do not succeed. To conclude the chapter, we take up the question of large gaps between the ordinary and essential approximation constants, by considering approximations to a certain rational point on a diagonal quartic surface. Finally, in Chapter 5, we generalize the Liouville-type results of McKinnon-Roth.
Artin's Conjecture: Unconditional Approach and Elliptic Analogue
(University of Waterloo, 2008-08-11T17:46:04Z) Sen Gupta, Sourav
In this thesis, I have explored the different approaches towards proving Artin's `primitive root' conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by E. Artin in the year 1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on the assumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid of the assumption and it seemed quite a daunting task. In 1983, the pioneering attempt was made by R. Gupta and M. Ram Murty, who proved unconditionally that there exists a specific set of 13 distinct numbers such that for at least one of them, the conjecture is true. Along the same line, using sieve theory, D. R. Heath-Brown reduced this set down to 3 distinct primes in the year 1986. This is the best unconditional result we have so far. In the first part of this thesis, we will review the sieve theoretic approach taken by Gupta-Murty and Heath-Brown. The second half of the thesis will deal with the elliptic curve analogue of the Artin's conjecture, which is also known as the Lang-Trotter conjecture. Lang and Trotter proposed the elliptic curve analogue in 1977, including the higher rank version, and also proceeded to set up the mathematical formulation to prove the same. The analogue conjecture was proved by Gupta and Murty in the year 1986, assuming the generalized Riemann hypothesis, for curves with complex multiplication. They also proved the higher rank version of the same. We will discuss their proof in details, involving the sieve theoretic approach in the elliptic curve setup. Finally, I will conclude the thesis with a refinement proposed by Gupta and Murty to find out a finite set of points on the curve such that at least one satisfies the conjecture.
Artin's Primitive Root Conjecture and its Extension to Compositie Moduli
(University of Waterloo, 2008-08-11T17:42:21Z) Camire, Patrice
If we fix an integer a not equal to -1 and which is not a perfect square, we are interested in estimating the quantity N_{a}(x) representing the number of prime integers p up to x such that a is a generator of the cyclic group (Z/pZ)*. We will first show how to obtain an aymptotic formula for N_{a}(x) under the assumption of the generalized Riemann hypothesis. We then investigate the average behaviour of N_{a}(x) and we provide an unconditional result. Finally, we discuss how to generalize the problem over (Z/mZ)*, where m > 0 is not necessarily a prime integer. We present an average result in this setting and prove the existence of oscillation.
Asymptotic Distributions for Block Statistics on Non-crossing Partitions
(University of Waterloo, 2014-01-23) Li, Boyu
The set of non-crossing partitions was first studied by Kreweras in 1972 and was known to play an important role in combinatorics, geometric group theory, and free probability. In particular, it has a natural embedding into the symmetric group, and there is an extensive literature on the asymptotic cycle structures of random permutations. This motivates our study on analogous results regarding the asymptotic block structure of random non-crossing partitions. We first investigate an analogous result of the asymptotic distribution for the total number of cycles of random permutations due to Goncharov in 1940's: Goncharov showed that the total number of cycles in a random permutation is asymptotically normally distributed with mean log(n) and variance log(n). As a analog of this result, we show that the total number of blocks in a random non-crossing partition is asymptotically normally distributed with mean n/2 and variance n/8. We also investigate the outer blocks, which arise naturally from non-crossing partitions and has many connections in combinatorics and free probability. It is a surprising result that among many blocks of non-crossing partitions, the expected number of outer blocks is asymptotically 3. We further computed the asymptotic distribution for the total number of blocks, which is a shifted negative binomial distribution.
Asymptotic Estimates for Rational Spaces on Hypersurfaces in Function Fields
(University of Waterloo, 2010-06-29T18:31:41Z) Zhao, Xiaomei
The ring of polynomials over a finite field has many arithmetic properties similar to those of the ring of rational integers. In this thesis, we apply the Hardy-Littlewood circle method to investigate the density of rational points on certain algebraic varieties in function fields. The aim is to establish asymptotic relations that are relatively robust to changes in the characteristic of the base finite field. More notably, in the case when the characteristic is "small", the results are sharper than their integer analogues.
Branched Covering Constructions and the Symplectic Geography Problem
(University of Waterloo, 2008-08-15T18:09:05Z) Hughes, Mark Clifford
We apply branched covering techniques to construct minimal simply-connected symplectic 4-manifolds with small χ_h values. We also use these constructions to provide an alternate proof that for each s ≥ 0, there exists a positive integer λ(s) such that each pair (j,8j+s) with j ≥ λ(s) is realized as (χ_h(M),c_1^2(M)) for some minimal simply-connected symplectic M. The smallest values of λ(s) currently known to the author are also explicitly computed for 0 ≤ s ≤ 99. Our computations in these cases populate 19 952 points in the (χ,c)-plane not previously realized in the existing literature.
Brands of cumulants in non-commutative probability, and relations between them
(University of Waterloo, 2022-07-06) Perales, Daniel
The study of non-commutative probability revolves around the different notions of independeces, such as free, Boolean and monotone. To each type of independence one can associate a notion of cumulants that linearize the addition of independent random variables. These notions of cumulants are a clear analogue of classic cumulants that linearize the addition of independent random variables. The family of set partitions P plays a key role in the combinatorial study of probability because several formulas relating moments to a brand of cumulants can be expressed as a sum indexed by set partitions. An intriguing fact observed in the recent research literature was that non-commutative cumulants sometimes have applications to other areas of non-commutative probability than the one they were designed for. Thus one wonders if there are nice combinatorial formulas to directly transition from one brand of cumulants to another. This thesis is concerned with the study of interrelations between different brands of cumulants associated to classic, Boolean, free and monotone independences. My development is naturally divided in four topics. For the first topic I focus on free cumulants, and I use them in the study of the distribution of the anti-commutator ab + ba of two free random variables a and b. This follows up on some questions raised a while ago by Nica and Speicher in the 1990’s. I next consider the notion of convolution in the framework of a family of lattices, which goes back to work of Rota and collaborators in the 1970’s. I focus on the lattices of non-crossing partitions NC and put into evidence a certain group of semi-multiplicative functions, which encapsulate the moment-cumulant formulas and the inter-cumulant transition formulas for several known brands of cumulants in non-commutative probability. I next extend my considerations to the framework of P, which allows us to include the classical cumulants in the picture. Moreover, I do this in a more structured fashion by introducing a notion of iterative family of partitions S in P. This unifies the considerations related to NC and P, and also provides a whole gallery of new examples. Finally, it is important to make the observation that the group of semi-multiplicative functions which arise in connection to an iterative family is in fact a dual structure. Namely, it appears as the group of characters for a certain Hopf algebra over the partitions in S. In particular, the antipode promises to serve as a universal inversion tool for moment-cumulant formulas and for transition formulas between different brands of cumulants.
Classical Field Theory in the BV Formalism
(University of Waterloo, 2016-09-20) Butson, Dylan
This document is a review of the perspective on classical eld theories presented in [2] and [3].
Classification of Finitely Generated Operator Systems
(University of Waterloo, 2018-01-22) Hamzo, Chadi
For the past few decades operator systems and their C*-envelopes have provided an invaluable tool for studying the theory of C*-algebras and positive maps. They provide the natural context in which to study the theory of completely positive maps. Furthermore, many of the important open problems in quantum information theory have found equivalent formulations in terms of operator systems. The question of the classification of operator systems and computing their C*-envelopes have been the center of much interest. Borrowing from the theory of representations of commutative C*-algebras by affine maps, we construct a new tool for classifying certain types of finitely generated operator systems. Using this tool, we show that all the information regarding such operator systems is usually encoded in the joint spectra of their generating operators. Using this tool we completely classify operator systems generated by finitely many normal operators. We also provide a different proof for the classification theorem of operator systems generated by a unitary with spectrum size different that 4. Furthermore, we settle the classification problem for operator systems generated by a single unitary with four points in its spectrum. In addition, we compute the C*-envelopes of such operator systems. Furthermore, we apply this tool to the classification problem of those operator systems generated by a unilateral shift with arbitrary multiplicity or by an isometry and we compute their C*-envelopes.
Classification of Nilpotent Lie Algebras of Dimension 7 (over Algebraically Closed Field and R)
(University of Waterloo, 1998) Gong, Ming-Peng
This thesis is concerned with the classification of 7-dimensional nilpotent Lie algebras. Skjelbred and Sund have published in 1977 their method of constructing all nilpotent Lie algebras of dimension n given those algebras of dimension < n, and their automorphism groups. By using this method, we construct all nonisomorphic 7-dimensional nilpotent Lie algebras in the following two cases: (1) over an algebraically closed field of arbitrary characteristic except 2; (2) over the real field R. We have compared our lists with three of the most recent lists (those of Seeley, Ancochea-Goze, and Romdhani). While our list in case (1) over C differs greatly from that of Ancochea-Goze, which contains too many errors to be usable, it agrees with that of Seeley apart from a few corrections that should be made in his list, Our list in case (2) over R contains all the algebras on Romdhani's list, which omits many algebras.

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