Existence of Compactly Supported Global Minimisers for the Interaction Energy

The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "gaps" it may have. The class of potentials for which we prove existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. We also show that the support of local minimisers is compact under suitable assumptions.Comment: Final version after referee reports taken into accoun

Numerical Study of a Particle Method for Gradient Flows

We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level, and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.Comment: 27 pages, 21 figure

On a Class of Nonlocal Continuity Equations on Graphs

Motivated by applications in data science, we study partial differential equations on graphs. By a classical fixed-point argument, we show existence and uniqueness of solutions to a class of nonlocal continuity equations on graphs. We consider general interpolation functions, which give rise to a variety of different dynamics, e.g., the nonlocal interaction dynamics coming from a solution-dependent velocity field. Our analysis reveals structural differences with the more standard Euclidean space, as some analogous properties rely on the interpolation chosen

PP272—Migraine and parthenolide inhibition of transient receptor potential ankyrin 1

2013 e103 emerged as a major complication of bortezomib therapy, which usually appears in the first courses of therapy with a number of sensory and painful symptoms, including reduced threshold to mechanical and cold stimuli. No satisfactory explanation or effective treatment exists for bortezomib-evoked CIPN. Patients (or Materials) and Methods: In this study, we evaluated whether TRPA1 acted as a critical mediator of CIPN by bortezomib or oxaliplatin in a mouse model system. Results: Our data demonstrated that CIPN hypersensitivity phenotype that was stably established by bortezomib could be transiently reverted by systemic or local treatment with the TRPA1 antagonist HC-030031. A similar effect was produced by the oxidative stress scavenger α -lipoic acid. Notably, the CIPN phenotype was abolished completely in mice that were genetically deficient in TRPA1, highlighting its essential role. Administration of bortezomib or oxaliplatin, which also elicits TRPA1-dependent hypersensitivity, produced a rapid, transient increase in plasma of carboxy-methyllysine, a byproduct of oxidative stress. Short-term systemic treatment with either HC-030031 or α -lipoic acid could completely prevent hypersensitivity if administered before the cytotoxic drug. Conclusion: Our findings highlight a key role for early activation/ sensitization of TRPA1 by oxidative stress by-products in producing CIPN. Furthermore, they suggest prevention strategies for CIPN in patients through the use of early, short-term treatments with TRPA1 antagonists. Disclosure of Interest: None declared