102 research outputs found
Long Time Behavior of First Order Mean Field Games on Euclidean Space
The aim of this paper is to study the long time behavior of solutions to
deterministic mean field games systems on Euclidean space. This problem was
addressed on the torus in [P. Cardaliaguet, {\it Long time
average of first order mean field games and weak KAM theory}, Dyn. Games Appl.
3 (2013), 473-488], where solutions are shown to converge to the solution of a
certain ergodic mean field games system on . By adapting the
approach in [A. Fathi, E. Maderna, {\it Weak KAM theorem on non compact
manifolds}, NoDEA Nonlinear Differential Equations Appl. 14 (2007), 1-27], we
identify structural conditions on the Lagrangian, under which the corresponding
ergodic system can be solved in . Then we show that time
dependent solutions converge to the solution of such a stationary system on all
compact subsets of the whole space
Time periodic and almost periodic viscosity solutions of contact Hamilton-Jacobi equations on
This paper concerns with the time periodic viscosity solution problem for a
class of evolutionary contact Hamilton-Jacobi equations with time independent
Hamiltonians on the torus . Under certain suitable assumptions we
show that the equation has a non-trivial -periodic viscosity solution if and
only if , where is a dense subset of . Moreover, we
clarify the structure of . As a consequence, we also study the existence of
Bohr almost periodic viscosity solutions
MiniGPT-5: Interleaved Vision-and-Language Generation via Generative Vokens
Large Language Models (LLMs) have garnered significant attention for their
advancements in natural language processing, demonstrating unparalleled prowess
in text comprehension and generation. Yet, the simultaneous generation of
images with coherent textual narratives remains an evolving frontier. In
response, we introduce an innovative interleaved vision-and-language generation
technique anchored by the concept of "generative vokens," acting as the bridge
for harmonized image-text outputs. Our approach is characterized by a
distinctive two-staged training strategy focusing on description-free
multimodal generation, where the training requires no comprehensive
descriptions of images. To bolster model integrity, classifier-free guidance is
incorporated, enhancing the effectiveness of vokens on image generation. Our
model, MiniGPT-5, exhibits substantial improvement over the baseline Divter
model on the MMDialog dataset and consistently delivers superior or comparable
multimodal outputs in human evaluations on the VIST dataset, highlighting its
efficacy across diverse benchmarks.Comment: 20 pages, 9 figure
Parametric SAR Image Formation - A Promising Approach to Resolution-Unlimited Imaging
Publication in the conference proceedings of EUSIPCO, Bucharest, Romania, 201
Weak KAM approach to first-order Mean Field Games with state constraints
We study the asymptotic behavior of solutions to the constrained MFG system
as the time horizon goes to infinity. For this purpose, we analyze first
Hamilton-Jacobi equations with state constraints from the viewpoint of weak KAM
theory, constructing a Mather measure for the associated variational problem.
Using these results, we show that a solution to the constrained ergodic mean
field games system exists and the ergodic constant is unique. Finally, we prove
that any solution of the first-order constrained MFG problem on
converges to the solution of the ergodic system as
Mathematical Mechanism on Dynamical System Algorithms of the Ising Model
Various combinatorial optimization NP-hard problems can be reduced to finding
the minimizer of an Ising model, which is a discrete mathematical model. It is
an intellectual challenge to develop some mathematical tools or algorithms for
solving the Ising model. Over the past decades, some continuous approaches or
algorithms have been proposed from physical, mathematical or computational
views for optimizing the Ising model such as quantum annealing, the coherent
Ising machine, simulated annealing, adiabatic Hamiltonian systems, etc..
However, the mathematical principle of these algorithms is far from being
understood. In this paper, we reveal the mathematical mechanism of dynamical
system algorithms for the Ising model by Morse theory and variational methods.
We prove that the dynamical system algorithms can be designed to minimize a
continuous function whose local minimum points give all the candidates of the
Ising model and the global minimum gives the minimizer of Ising problem. Using
this mathematical mechanism, we can easily understand several dynamical system
algorithms of the Ising model such as the coherent Ising machine, the
Kerr-nonlinear parametric oscillators and the simulated bifurcation algorithm.
Furthermore, motivated by the works of C. Conley, we study transit and capture
properties of the simulated bifurcation algorithm to explain its convergence by
the low energy transit and capture in celestial mechanics. A detailed
discussion on -spin and -spin Ising models is presented as application.Comment: 39 pages, 2 figures(including 8 sub-figures
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