| 1. |
- Alsaody, Seidon
(author)
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Albert algebras over rings and related torsors
- 2020
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - : Canadian Mathematical Society. - 0008-414X .- 1496-4279. ; 73:3, s. 875-898
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Journal article (peer-reviewed)abstract
- We study exceptional Jordan algebras and related exceptional group schemes over commutative rings from a geometric point of view, using appropriate torsors to parametrize and explain classical and new constructions, and proving that over rings, they give rise to nonisomorphic structures.We begin by showing that isotopes of Albert algebras are obtained as twists by a certain F4-torsor with total space a group of type E6 and, using this, that Albert algebras over rings in general admit nonisomorphic isotopes even in the split case, as opposed to the situation over fields. We then consider certain D4 -torsors constructed from reduced Albert algebras, and show how these give rise to a class of generalised reduced Albert algebras constructed from compositions of quadratic forms. Showing that this torsor is nontrivial, we conclude that the Albert algebra does not uniquely determine the underlying composition, even in the split case. In a similar vein, we show that a given reduced Albert algebra can admit two coordinate algebras which are nonisomorphic and have nonisometric quadratic forms, contrary, in a strong sense, to the case over fields, established by Albert and Jacobson.
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| 2. |
- Arone, Gregory, et al.
(author)
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Spaces of knotted circles and exotic smooth structures
- 2022
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - 0008-414X .- 1496-4279. ; 74:1, s. 1-23
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Journal article (peer-reviewed)abstract
- Suppose that N1 and N2 are closed smooth manifolds of dimension n that are homeomorphic. We prove that the spaces of smooth knots, Emb(S1,N1) and Emb(S1,N2), have the same homotopy (2n−7) -type. In the four-dimensional case, this means that the spaces of smooth knots in homeomorphic 4 -manifolds have sets π0 of components that are in bijection, and the corresponding path components have the same fundamental groups π1. The result about π0 is well-known and elementary, but the result about π1 appears to be new. The result gives a negative partial answer to a question of Oleg Viro. Our proof uses the Goodwillie–Weiss embedding tower. We give a new model for the quadratic stage of the Goodwillie–Weiss tower, and prove that the homotopy type of the quadratic approximation of the space of knots in N does not depend on the smooth structure on N. Our results also give a lower bound on π2Emb(S1,N) . We use our model to show that for every choice of basepoint, each of the homotopy groups, π1 and π2, of Emb(S1,S1×S3) contains an infinitely generated free abelian group.
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| 3. |
- Astashkin, Sergey, et al.
(author)
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Isomorphic structure of Cesàro and Tandori spaces
- 2019
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - : Cambridge University Press. - 0008-414X .- 1496-4279. ; 71:3, s. 501-532
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Journal article (peer-reviewed)abstract
- We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space Ces∞ and its sequence counterpart ces∞ are isomorphic. This is rather surprising since Ces∞ (like Talagrand’s example) has no natural lattice predual. We prove that ces∞ is not isomorphic to ℓ∞ nor is Ces∞ isomorphic to the Tandori space L1 with the norm ∥f∥L1 = ∥f∥L1, where f(t) = esssups≥tf(s). Our investigation also involves an examination of the Schur and Dunford–Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro–Marcinkiewicz and Cesàro–Lorentz spaces have the latter property.
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| 4. |
- Berezhoi, Evgenii I., et al.
(author)
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Representation of Banach ideal spaces and factorization of operators
- 2005
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - 0008-414X .- 1496-4279. ; 57:5, s. 897-940
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Journal article (peer-reviewed)abstract
- Representation theorems are proved for Banach ideal spaces with the Fatou property which are built by the Calderón-Lozanovskiǐ construction. Factorization theorems for operators in spaces more general than the Lebesgue Lp spaces are investigated. It is natural to extend the Gagliardo theorem on the Schur test and the Rubio de Francia theorem on factorization of the Muckenhoupt Ap weights to reflexive Orlicz spaces. However, it turns out that for the scales far from Lp-spaces this is impossible. For the concrete integral operators it is shown that factorization theorems and the Schur test in some reflexive Orlicz spaces are not valid. Representation theorems for the Calderón-Lozanovskiǐ construction are involved in the proofs
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| 5. |
- Björn, Anders, 1966-, et al.
(author)
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Sobolev extensions of Hölder continuous and characteristic functions on metric spaces
- 2007
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - 0008-414X .- 1496-4279. ; 59:6, s. 1135-1153
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Journal article (peer-reviewed)abstract
- We study when characteristic and Holder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions. ©Canadian Mathematical Society 2007.
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| 6. |
- Brandes, Julia, 1986, et al.
(author)
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Simultaneous Additive Equations: Repeated and Differing Degrees
- 2017
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In: Canadian Journal of Mathematics. - 1496-4279 .- 0008-414X. ; 69:2, s. 258-283
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Journal article (peer-reviewed)abstract
- We obtain bounds for the number of variables required to establish Hasse principles, both for the existence of solutions and for asymptotic formula, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning-Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems. We also obtain specialized results for systems of quadratic and cubic forms, where we are able to take advantage of some of the stronger methods available in that setting. In particular, we achieve essentially square root cancellation for systems consisting of one cubic and r quadratic equations.
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| 7. |
- Chen, Chih-Whi, et al.
(author)
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The Primitive Spectrum and Category O for the Periplectic Lie Superalgebra
- 2020
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - : Canadian Mathematical Society. - 0008-414X .- 1496-4279. ; 72:3, s. 625-655
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Journal article (peer-reviewed)abstract
- We solve two problems in representation theory for the periplectic Lie superalgebra pc (n), namely, the description of the primitive spectrum in terms of functorial realisations of the braid group and the decomposition of category O into indecomposable blocks. To solve the first problem, we establish a new type of equivalence between category O for all (not just simple or basic) classical Lie superalgebras and a category of Harish-Chandra bimodules. The latter bimodules have a left action of the Lie superalgebra but a right action of the underlying Lie algebra. To solve the second problem, we establish a BGG reciprocity result for the periplectic Lie superalgebra.
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| 8. |
- Emamizadeh, Behrouz, et al.
(author)
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Optimization Related to Some Nonlocal Problems of Kirchhoff Type
- 2016
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In: Canadian Journal of Mathematics - Journal Canadien de Mathematiques. - Toronto, ON : University of Toronto Press. - 0008-414X .- 1496-4279. ; 68:3, s. 521-540
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Journal article (peer-reviewed)abstract
- In this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton we are able to show that both problems are solvable, and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions. The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type, which is stable. Some numerical results are included to confirm the theory.
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| 9. |
- Gourevitch, Dmitry, et al.
(author)
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Fourier coefficients of minimal and next-to-minimal automorphic representations of simply-laced groups
- 2022
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In: Canadian Journal of Mathematics. - 1496-4279 .- 0008-414X. ; 74:1, s. 122-169
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Journal article (peer-reviewed)abstract
- In this paper, we analyze Fourier coefficients of automorphic forms on a finite cover G of an adelic split simply-laced group. Let ππ be a minimal or next-to-minimal automorphic representation of G. We prove that any η∈πη∈π is completely determined by its Whittaker coefficients with respect to (possibly degenerate) characters of the unipotent radical of a fixed Borel subgroup, analogously to the Piatetski-Shapiro–Shalika formula for cusp forms on GLnGLn . We also derive explicit formulas expressing the form, as well as all its maximal parabolic Fourier coefficient, in terms of these Whittaker coefficients. A consequence of our results is the nonexistence of cusp forms in the minimal and next-to-minimal automorphic spectrum. We provide detailed examples for G of type D5D5 and E8E8 with a view toward applications to scattering amplitudes in string theory.
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| 10. |
- Jin, Yongyang, et al.
(author)
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Degenerate $p$-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
- 2010
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In: Canadian Journal of Mathematics. - 0008-414X .- 1496-4279. ; 62, s. 1116-1130
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Journal article (peer-reviewed)abstract
- Let G be a step-two nilpotent group of H-type with Lie algebra G=V⊕t . We define a class of vector fields X={Xj} on G depending on a real parameter k≥1 , and we consider the corresponding p -Laplacian operator Lp,ku=divX(|∇Xu|p−2∇Xu) . For k=1 the vector fields X={Xj} are the left invariant vector fields corresponding to an orthonormal basis of V ; for G being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator Lp,k and as an application, we get a Hardy type inequality associated with X .
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