In this paper, we give an overview of new results that define and explicitly construct Ramanujan Cayley biregular bipartite graphs. We also study the extremal combinatorial properties of these graphs. This parallels the work of Lubotzky, Phillips and Sarnak on regular Ramanujan Cayley graphs, with several interesting differences. Furthermore, this work also proposes a stronger definition of Ramanujan graphs than has been used in the past, which opens the door to future studies.
We introduce conditions weaker than Gauduchon K & auml;hler-like and use them to derive rigidity results. These results extend those obtained in [15,16] under the stronger Gauduchon K & auml;hler-like assumption.
Llarull's scalar curvature rigidity theorem states that a 1-Lipschitz map f : M -> Sn from a closed connected Riemannian spin manifold M with scalar curvature scal >= n(n-1) to the standard sphere Sn is an isometry if the degree off is nonzero. We investigate if one can replace the condition deg(f) /= 0 by the weaker condition that f is surjective. The answer turns out to be "no" for n >= 3 but "yes" for n = 2. If we replace the scalar curvature by Ricci curvature, the answer is "yes"in all dimensions.
In this article we classify all simple modules over a noncommutative and noncocommutative bialgebra M ( p , q ) assuming q is a root of unity.
This paper contributes to the historical understanding of the developments surrounding the Levi-Civita parallel transport problem, exploring its connections with the local problem of isometric immersions and alternative proposals. Additionally, it highlights one of its remarkable applications: the geometric interpretation of Foucault’s pendulum precession. It also recalls how other geometric explanations of this phenomenon emerged in the context of Berry and Hannay phases.
We study threshold phenomena in weighted Q2-spaces. Our main result is a summable Baire category version of K & ouml;rner's topological Ivashev-Musatov Theorem, which we show is optimal in several respects.
We study existence and nonexistence of diagonal and separating coordinates for Riemannian symmetric spaces of rank 1. We generalize the results of Gauduchon and Moroianu (2020) by showing that a symmetric space of rank 1 has diagonal coordinates if and only if it has constant sectional curvature. This implies that orthogonal separation of variables on a symmetric space of rank 1 is possible only in the constant sectional curvature case. We show that on the complex projective space ℂ P n and on complex hyperbolic space ℂ H n , with n ≥ 2 , separating coordinates necessarily have precisely n ignorable coordinates. In view of results of Boyer et al. (1983, 1985) and later results of Winternitz et al. (1994), this completes the description of separation of variables on ℂ P n for all n and on ℂ H n for n = 2 , 3 .
Given a symplectic manifold, can one pack uncountably many Lagrangian submanifolds in a given Hamiltonian isotopy class of this symplectic manifold? We address C^∞ and C^0 versions of this question.
Let triangle N be the multidimensional discrete Laplacian on ZN (N >= 1). In this note, we prove that, when N = 1, the right-hand derivative of (-triangle 1)s at 0 is an exotic discrete Riesz potential (namely, the endpoint case: the order is 0) in Stein-Wainger sense (J. Anal. Math., 2000), and when N >= 2, the corresponding derivative is also an exotic discrete Riesz potential with an additional corrector. A similar conclusion for the left-hand derivative case is also considered. All results obtained in this note extend the logarithmic Laplacian of Chen-Weth (Commun. Partial Differ. Equations, 2019) to the discrete setting.
We review the notion and the properties of the generalized principal eigenvalue for elliptic operators in unbounded domains, and we relate it to the criticality theory. We focus on operators with almost periodic coefficients. We present a Liouville-type result in dimension N <= 2. Next, we show with a counterexample that criticality is not equivalent to the existence of an almost periodic principal eigenfunction, even for self-adjoint operators. Finally, we exhibit an almost periodic operator which is subcritical but which admits a critical limit operator. This is a manifestation of the instability character of the criticality property in the almost periodic setting.
We are interested in how regular a transport velocity field must be in order to control Riesz-type commutators. Estimates for these commutators play a central role in the analysis of the mean-field limit and fluctuations for systems of particles with pairwise Riesz interactions, which we start by reviewing. Our first new result shows that the usual L^∞ assumption on the gradient of the velocity field cannot, in general, be relaxed to a BMO assumption. We construct counterexamples in all dimensions and all Riesz singularities -2< s<d, except for the one-dimensional logarithmic endpoint s=0. At this exceptional endpoint, such a relaxation is possible, a fact related to the classical Coifman-Rochberg-Weiss commutator bound for the Hilbert transform. Our second result identifies a trade-off between the singularity of the interaction potential and the required regularity of the velocity field. Roughly speaking, smoother (less singular) interactions require stronger velocity control if one wants a commutator estimate in the natural energy seminorm determined by the potential. We formulate this principle for a broad class of potentials and show that, in the sub-Coulomb Riesz regime, the velocity regularity appearing in the known commutator inequality is sharp. Despite these negative findings, we show as our third result that a defective commutator estimate holds for almost-Lipschitz transport fields. Such a defective estimate, which is a consequence of the celebrated Brezis-Wainger-Hansson inequality, allows us to prove rates of convergence when the mean-field density belongs to the scaling-critical Sobolev space.
The Jacobian conjecture is thought to have been proposed by O. H. Keller in 1939. However, we have found that the statement of the conjecture is precisely the main result of a paper published by L. Kraus in 1884. Although the final step of Kraus's proof is flawed, the ideas he introduced anticipated approaches to the problem that would only emerge more than a century later. Interestingly, the root of Kraus's error remains the principal obstacle to algebro-geometric approaches: controlling the ramification at infinity.
We show the following feature of the relation between Brownian loop-soups on cable-graphs and their total occupation time-field Lambda: when conditioned on Lambda, the conditional law of individual loops becomes singular with respect to that of unconditioned loops. The idea of the proof is to see that some type of fast points on the curve Lambda impose an exceptional behaviour of all the loops when they go through these points.
We introduce the notion of directed scheme of ideals to characterize peculiar ideals on the reals, which comes from a formalization of the framework of Yorioka ideals for strong measure zero sets. We prove general theorems for directed schemes and propose a directed scheme (M) over right arrow = {MI : I is an element of I} for the ideal MA of meager-additive sets of reals. This directed scheme does not only helps us to understand more the combinatorics of MA and its cardinal characteristics, but provides us new characterizations of the additivity and cofinality numbers of the meager ideal of the reals. In addition, we display connections between the characteristics associated with MI and other classical characteristics. Furthermore, we demonstrate the consistency of cov(NA) 14].
We construct algebraic surfaces with a large number of type A singularities. Bivariate polynomials presented in previous works for the construction of nodal surfaces and certain families of Belyi polynomials are used. In some cases explicit expressions in terms of classical Jacobi polynomials are obtained.
The stability of optimal transport maps with respect to perturbations of the marginals is a question of interest for several reasons, ranging from numerical analysis and statistics to the justification of the linearized optimal transport framework. Under various assumptions on the source measure, it is known that optimal transport maps are stable with respect to variations of the target measure. In this note, we focus on the mechanisms that can, on the contrary, lead to instability . We identify two of them. We first show that instability may arise from the unboundedness of the density: we exhibit a source density on the unit ball of ℝ d which blows up at two points of the boundary and for which optimal transport maps are highly unstable. Then we prove that even for uniform densities on bounded open sets, optimal transport maps can be rather unstable sufficiently close to configurations where uniqueness of optimal plans is lost.
In this short note, we answer a question raised by E. De Giorgi, showing that a Mumford-Shah minimizer in dimension 2 can admit at most three limit values as approaching the singular set. This result actually stems from tools developed in the early 2000s by G. David, A. Bonnet, and J.-C. L & eacute;ger.
