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Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…
A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of…
In this paper, the dispersive revival and fractalization phenomena for bidirectional dispersive equations on a bounded interval subject to periodic boundary conditions and discontinuous initial profiles are investigated. Firstly, we study…
Neurons are often connected, spatially and temporally, in phenomenal ways that promote wave propagation. Therefore, it is essential to analyze the emergent spatiotemporal patterns to understand the working mechanism of brain activity,…
In this article, a notion of viscosity solutions is introduced for first order path-dependent Hamilton-Jacobi-Bellman (PHJB) equations associated with optimal control problems for path-dependent evolution equations in Hilbert space. We…
In this paper, we propose an equivariant degree based method to study bifurcation of periodic solutions (of constant period) in symmetric networks of reversible FDEs. Such a bifurcation occurs when eigenvalues of linearization move along…
The 3D-primitive equations with only horizontal viscosity are considered on a cylindrical domain $\Omega=(-h,h) \times G$, $G\subset \mathbb{R}^2$ smooth, with the physical Dirichlet boundary conditions on the sides. Instead of considering…
We study neural field equations, which are prototypical models of large-scale cortical activity, subject to random data. We view this spatially-extended, nonlocal evolution equation as a Cauchy problem on abstract Banach spaces, with…
We study the planar front solution for a class of reaction diffusion equations in multidimensional space in the case when the essential spectrum of the linearization in the direction of the front touches the imaginary axis. At the linear…
Spatial cognition is essential for human intelligence, enabling problem-solving through visual simulations rather than solely relying on verbal reasoning. However, existing AI benchmarks primarily assess verbal reasoning, neglecting the…
We prove an exponential separation in sample complexity between Euclidean and hyperbolic representations for learning on hierarchical data under standard Lipschitz regularization. For depth-$R$ hierarchies with branching factor $m$, we…
In this paper well-posedness is proved for an initial and boundary value problem (IBVP) relative to a large class of quasilinear hyperbolic systems, in $p+q$ equations, on a strip, arising from a model of $H_2O$-phase transitions in the…
Neural collapse describes the geometry of activation in the final layer of a deep neural network when it is trained beyond performance plateaus. Open questions include whether neural collapse leads to better generalization and, if so, why…
This paper proposes a new approach to address the problem of unmeasured confounding in spatial designs. Spatial confounding occurs when some confounding variables are unobserved and not included in the model, leading to distorted…
In this paper, we study the stochastic heat equation in the spatial domain $\mathbb{R}^d$ subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and…
Given any elliptic system with $t$-independent coefficients in the upper-half space, we obtain representation and trace for the conormal gradient of solutions in the natural classes for the boundary value problems of Dirichlet and Neumann…
For long it has been known that specific patterns of folding are necessary for an optimally functioning brain. For instance, lissencephaly and polymicrogyria can lead to severe mental retardation, short life expectancy, epileptic seizures,…
Recently, data-driven deep saliency models have achieved high performance and have outperformed classical saliency models, as demonstrated by results on datasets such as the MIT300 and SALICON. Yet, there remains a large gap between the…
We introduce a probabilistic model of early visual processing, beginning with the interaction between a light wavefront and the retina. We argue that perception originates not with deterministic transduction, but with probabilistic…