Journal of the London Mathematical Society-Second Series最新文献

This paper is concerned with the analytical construction of piecewise smooth solutions containing a single shock wave for the radially symmetric relativistic Euler equations with polytropic gases. We derive meticulously the a priori $�C^1$-estimates on the Riemann invariants of the governing system under some assumptions on the piecewise initial data. Based on these estimates, we show that the long time of existence of smooth solutions in the angular region bounded by a characteristic curve and a shock curve. The piecewise smooth initial conditions ensured the existence of smooth solutions in the angular region are discussed. Moreover, it is verified that the existence time is proportional to the initial discontinuous position.

We construct a canonical family of elements in the reduced exterior powers of unit groups of global fields and investigate their detailed arithmetic properties. We then show that these elements specialise to recover the classical theory of cyclotomic elements in real abelian fields and also have connections to the theory of non-commutative Euler systems for $��Z_p��\left(�1�\right)��$ over general number fields.

We describe surjective linear isometries and linear isometry groups of a large class of Lipschitz-free spaces that includes, for example, Lipschitz-free spaces over any graph. We define the notion of a Lipschitz-free rigid metric space whose Lipschitz-free space only admits surjective linear isometries coming from surjective dilations (i.e., rescaled isometries) of the metric space itself. We show that this class of metric spaces is surprisingly rich and contains all 3-connected graphs as well as geometric examples such as nonabelian Carnot groups with horizontally strictly convex norms. We prove that every metric space isometrically embeds into a Lipschitz-free rigid space that has only three more points.

The aim of this paper is to revisit propagation of chaos for a Langevin-type interacting particle system recently proposed as a method to sample probability measures. The interacting particle system we consider coincides, in the setting of a log-quadratic target distribution, with the ensemble Kalman sampler [SIAM J. Appl. Dyn. Syst. 19 (2020), no. 1, 412–441], for which propagation of chaos was first proved by Ding and Li in [SIAM J. Math. Anal. 53 (2021), no. 2, 1546–1578]. Like these authors, we prove propagation of chaos with an approach based on a synchronous coupling, as in Sznitman's classical argument. Instead of relying on a boostrapping argument, however, we use a technique based on stopping times in order to handle the presence of the empirical covariance in the coefficients of the dynamics. The use of stopping times to handle the lack of global Lipschitz continuity in the coefficients of stochastic dynamics originates from numerical analysis [SIAM J. Numer. Anal. 40 (2002), no. 3, 1041–1063] and was recently employed to prove mean-field limits for consensus-based optimization and related interacting particle systems [arXiv:2312.07373, 2023; Math. Models Methods Appl. Sci. 33 (2023), no. 2, 289–339]. In the context of ensemble Langevin sampling, this technique enables proving pathwise propagation of chaos with optimal rate, whereas previous results were optimal only up to a positive $�\epsilon$.

We give series solutions to single insertion place propagator-type systems of Dyson–Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson–Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and noninteger values of the insertion parameter $�s$. We remark on interesting combinatorial connections and properties.

This work offers a new prospective on asymptotic perturbation theory for varying self-adjoint extensions of symmetric operators. Employing symplectic formulation of self-adjointness, we use a version of resolvent difference identity for two arbitrary self-adjoint extensions that facilitates asymptotic analysis of resolvent operators via first-order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati-type differential equation and the first-order asymptotic expansion for resolvents of self-adjoint extensions determined by smooth one-parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich-type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self-adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second-order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.

We consider the graphical mean curvature flow of maps $��f�:�R^m�\to �R^n�$, $��m�⩾�2�$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed submanifolds that extends the well-known maximum principle of Ecker and Huisken derived in their seminal paper [Ann. of Math. (2) 130:3(1989), 453–471]. In the case of uniformly area decreasing maps $��f�:�R^m�\to �R^2�$, $��m�⩾�2�$, we use this maximum principle to show that the graphicality and the area decreasing property are preserved. Moreover, if the initial graph is asymptotically conical at infinity, we prove that the normalized mean curvature flow smoothly converges to a self-expander.