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The authors consider operators of the form L=\\sum_{i=1}^{n}X_{i}^{2}+X_{0} in a bounded domain of \\mathbb{R}^{p} where X_{0},X_{1},\\ldots,X_{n} are nonsmooth Hörmander's vector fields of step r such that the highest order commutators are only Hölder continuous. Applying Levi's parametrix method the authors construct a local fundamental solution \\gamma for L and provide growth estimates for \\gamma and its first derivatives with respect to the vector fields. Requiring the existence of one more derivative of the coefficients the authors prove that \\gamma also possesses second derivatives, and they deduce the local solvability of L, constructing, by means of \\gamma, a solution to Lu=f with Hölder continuous f. The authors also prove C_{X,loc}^{2,\\alpha} estimates on this solution.

In this work we deal with linear second order partial differential operators of the following type:

Let

The authors consider the nonlinear equation -\\frac 1m=z+Sm with a parameter z in the complex upper half plane \\mathbb H , where S is a positivity preserving symmetric linear operator acting on bounded functions. The solution with values in \\mathbb H is unique and its z-dependence is conveniently described as the Stieltjes transforms of a family of measures v on \\mathbb R. In a previous paper the authors qualitatively identified the possible singular behaviors of v: under suitable conditions on S we showed that in the density of v only algebraic singularities of degree two or three may occur. In this paper the authors give a comprehensive analysis of these singularities with uniform quantitative controls. They also find a universal shape describing the transition regime between the square root and cubic root singularities. Finally, motivated by random matrix applications in the authors' companion paper they present a complete stability analysis of the equation for any z\\in \\mathbb H, including the vicinity of the singularities.

A number of new formulas are obtained for the vector field characteristics used in vector field geometry, vector analysis, and differential geometry: curvature vector, associated vector field, degree of nonholonomy, and Laplacian. Nonclassical characteristics such as the vector potential of the curvature vector field of a vector field and the sum of three curvature vectors of vector lines of the Frenet unit vector fields of a family of curves are also studied. It is shown that all the listed quantities are related to Aminov’s divergent representations for the Gaussian curvature or for the total curvature of the second kind. The obtained formulas can be considered as properties of the family of curves. Some formulas have divergence form, which makes it possible to derive differential conservation laws for the family of curves as well as for the eikonal equation and the Euler equations of hydrodynamics.

We study hyperbolic Ricci solitons on sequential warped products. The necessary conditions are obtained for a hyperbolic Ricci soliton with the structure of a sequential warped product to be an Einstein manifold when we consider the potential field as a Killing or a conformal vector field. Some physical applications are also given.

In this paper, we study the conformal vector fields on Finsler warped product manifolds. We obtain a system of equivalent equations that the conformal vector fields on Finsler warped product manifolds satisfy and completely characterize conformal vector fields on such manifolds. Further, by solving the equation, we give the classification. And we also give some examples.

Cognitive maps and spatial memory are fundamental paradigms of brain functioning. Here, we present a spiking neural network (SNN) capable of generating an internal representation of the external environment and implementing spatial memory. The SNN initially has a non-specific architecture, which is then shaped by Hebbian-type synaptic plasticity. The network receives stimuli at specific loci, while the memory retrieval operates as a functional SNN response in the form of population bursts. The SNN function is explored through its embodiment in a robot moving in an arena with safe and dangerous zones. We propose a measure of the global network memory using the synaptic vector field approach to validate results and calculate information characteristics, including learning curves. We show that after training, the SNN can effectively control the robot’s cognitive behavior, allowing it to avoid dangerous regions in the arena. However, the learning is not perfect. The robot eventually visits dangerous areas. Such behavior, also observed in animals, enables relearning in time-evolving environments. If a dangerous zone moves into another place, the SNN remaps positive and negative areas, allowing escaping the catastrophic interference phenomenon known for some AI architectures. Thus, the robot adapts to changing world.

We investigate η-Ricci-Bourguignon solitons structure on sequential warped product manifolds and prove that an η-Ricci-Bourguignon soliton sequential warped manifold whose potential vector field is Killing or conformal must be a quasi-Einstein manifold. Finally, we deduce two applications of η-Ricci-Bourguignon solitons sequential warped product namely standard static space-times and generalized Robertson-Walker space-times.