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28860 results in Differential and integral equations, dynamical systems and control theory

Page 18 of 1443

  • Discrete-to-continuum limits of optimal transport with linear growth on periodic graphs

  • Lorenzo Portinale, Filippo Quattrocchi
  • Journal:
    European Journal of Applied Mathematics , First View
    Published online by Cambridge University Press:
    20 December 2024, pp. 1-29
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  • We prove discrete-to-continuum convergence for dynamical optimal transport on $\mathbb{Z}^d$-periodic graphs with cost functional having linear growth at infinity. This result provides an answer to a problem left open by Gladbach, Kopfer, Maas, and Portinale (Calc Var Partial Differential Equations 62(5), 2023), where the convergence behaviour of discrete boundary-value dynamical transport problems is proved under the stronger assumption of superlinear growth. Our result extends the known literature to some important classes of examples, such as scaling limits of $1$-Wasserstein transport problems. Similarly to what happens in the quadratic case, the geometry of the graph plays a crucial role in the structure of the limit cost function, as we discuss in the final part of this work, which includes some visual representations.

  • RANKS OF SOFT OPERATORS IN NOWHERE SCATTERED $\mathrm {C}^*$-ALGEBRAS

  • Part of
  • M. Ali Asadi-Vasfi, Hannes Thiel, Eduard Vilalta
  • Journal:
    Journal of the Institute of Mathematics of Jussieu / Volume 24 / Issue 2 / March 2025
    Published online by Cambridge University Press:
    20 December 2024, pp. 371-410
    Print publication:
    March 2025
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  • We show that for $\mathrm {C}^*$-algebras with the global Glimm property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-$\mathrm {C}^*$-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a $\mathrm {C}^*$-algebra is determined by the soft part of its Cuntz semigroup.

    Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

Page 18 of 1443