Canadian Mathematical Bulletin最新文献

We discuss the class of functions, which are well approximated on compacta by the geometric mean of the eigenvalues of a unital (completely) positive map into a matrix algebra or more generally a type $II_1$ factor, using the notion of a Fuglede–Kadison determinant. In two variables, the two classes are the same, but in three or more noncommuting variables, there are generally functions arising from type $II_1$ von Neumann algebras, due to the recently established failure of the Connes embedding conjecture. The question of whether or not approximability holds for scalar inputs is shown to be equivalent to a restricted form of the Connes embedding conjecture, the so-called shuffle-word-embedding conjecture.

In this work, we study the existence of solutions of nonlinear fractional coupled system of $varphi $-Hilfer type in the frame of Banach spaces. We improve a property of a measure of noncompactness in a suitably selected Banach space. Darbo’s fixed point theorem is applied to obtain a new existence result. Finally, the validity of our result is illustrated through an example.

In this article, we construct some examples of noncommutative projective Calabi–Yau schemes by using noncommutative Segre products and quantum weighted hypersurfaces. We also compare our constructions with commutative Calabi–Yau varieties and examples constructed in Kanazawa (2015, Journal of Pure and Applied Algebra 219, 2771–2780). In particular, we show that some of our constructions are essentially new examples of noncommutative projective Calabi–Yau schemes.

If X is a topological space and Y is any set, then we call a family $mathcal {F}$ of maps from X to Y nowhere constant if for every non-empty open set U in X there is $f in mathcal {F}$ with $|f[U]|> 1$, i.e., f is not constant on U. We prove the following result that improves several earlier results in the literature.

If X is a topological space for which $C(X)$, the family of all continuous maps of X to $mathbb {R}$, is nowhere constant and X has a $pi $-base consisting of connected sets then X is $mathfrak {c}$-resolvable.

Following Faber–Pandharipande, we use the virtual localization formula for the moduli space of stable maps to $mathbb {P}^{1}$ to compute relations between Hodge integrals. We prove that certain generating series of these integrals are polynomials.

We characterize the finite codimension sub-${mathbf {k}}$-algebras of ${mathbf {k}}[![t]!]$ as the solutions of a computable finite family of higher differential operators. For this end, we establish a duality between such a sub-algebras and the finite codimension ${mathbf {k}}$-vector spaces of ${mathbf {k}}[u]$, this ring acts on ${mathbf {k}}[![t]!]$ by differentiation.

Let K be a non-Archimedean valued field with valuation ring R. Let $C_eta $ be a K-curve with compact-type reduction, so its Jacobian $J_eta $ extends to an abelian R-scheme J. We prove that an Abel–Jacobi map $iota colon C_eta to J_eta $ extends to a morphism $Cto J$, where C is a compact-type R-model of J, and we show this is a closed immersion when the special fiber of C has no rational components. To do so, we apply a rigid-analytic “fiberwise” criterion for a morphism to extend to integral models, and geometric results of Bosch and Lütkebohmert on the analytic structure of $J_eta $.

We prove a conjectural formula for the Brumer–Stark units. Dasgupta and Kakde have shown the formula is correct up to a bounded root of unity. In this paper, we resolve the ambiguity in their result. We also remove an assumption from Dasgupta–Kakde’s result on the formula.