Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

This study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ X = Q + ∑ i = 1 k A i ∗ G ( X ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive-definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices, and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ X = Q + A 1 ∗ X 1 / 3 A 1 + A 2 ∗ X 1 / 3 A 2 + A 3 ∗ X 1 / 3 A 3 , and visualize this through convergence analysis and a solution graph.

Number of autonomous agents in a neural network

This paper considers several aspects of the relationship between size, structure, speed of propagation and the number of autonomous cognitive agents in a neural network. Whereas, memory and function generation capacities of neural networks with scale invariant structure have been investigated extensively, the number of autonomous agents has not received prior attention. We propose the emergence of the dichotomy of causal and noncausal regions that is related to speed of propagation, in which the autonomous cognitive agents are not bound in a causal relationship with other agents. Arguments are presented for why the count of autonomous agents is best estimated with respect to the dimensionality of the underlying space. The number of autonomous agents obtained for the human brain equals twenty-five, and it is significant that the number in the sub-system modules also turns out to be close to the same value. It is possible that this near equality across layers provides a special uniqueness to the human brain. We argue that the findings of this study will be useful in the design of neural-network based AI systems that are designed to emulate human cognitive capacity. <br><br><br><br>

Number of autonomous agents in a neural network

This paper considers several aspects of the relationship between size, structure, speed of propagation and the number of autonomous cognitive agents in a neural network. Whereas, memory and function generation capacities of neural networks with scale invariant structure have been investigated extensively, the number of autonomous agents has not received prior attention. We propose the emergence of the dichotomy of causal and noncausal regions that is related to speed of propagation, in which the autonomous cognitive agents are not bound in a causal relationship with other agents. Arguments are presented for why the count of autonomous agents is best estimated with respect to the dimensionality of the underlying space. The number of autonomous agents obtained for the human brain equals twenty-five, and it is significant that the number in the sub-system modules also turns out to be close to the same value. It is possible that this near equality across layers provides a special uniqueness to the human brain. We argue that the findings of this study will be useful in the design of neural-network based AI systems that are designed to emulate human cognitive capacity. <br><br><br><br>

Hierarchical Bayesian Spatio-Temporal Modeling for PM10 Prediction

Over the past few years, hierarchical Bayesian models have been extensively used for modeling the joint spatial and temporal dependence of big spatio-temporal data which commonly involves a large number of missing observations. This article represented, assessed, and compared some recently proposed Bayesian and non-Bayesian models for predicting the daily average particulate matter with a diameter of less than 10 (PM10) measured in Qatar during the years 2016–2019. The disaggregating technique with a Markov chain Monte Carlo method with Gibbs sampler are used to handle the missing data. Based on the obtained results, we conclude that the Gaussian predictive processes with autoregressive terms of the latent underlying space-time process model is the best, compared with the Bayesian Gaussian processes and non-Bayesian generalized additive models.

The Almost Optimal Multilinear Restriction Estimate for Hypersurfaces with Curvature: The Case of n-1 Hypersurfaces in ℝn

In this paper, we establish the optimal multilinear restriction estimate for $n-1$ hypersurfaces with some curvature, where $n$ is the dimension of the underlying space. The result is sharp up to the end point and the role of curvature is made precise in terms of the shape operator.

Uniform Regularity of Set-Valued Mappings and Stability of Implicit Multifunctions

We propose a unifying general (i.e. not assuming the mapping to have any particular structure) view on the theory of regularity and clarify the relationships between the existing primal and dual quantitative sufficient and necessary conditions including their hierarchy. We expose the typical sequence of regularity assertions, often hidden in the proofs, and the roles of the assumptions involved in the assertions, in particular, on the underlying space: general metric, normed, Banach or Asplund. As a consequence, we formulate primal and dual conditions for the stability properties of solution mappings to inclusions Comment: 24 pages

Numerical Accuracy of Ladder Schemes for Parallel Transport on Manifolds

Parallel transport is a fundamental tool to perform statistics on Riemannian manifolds. Since closed formulae do not exist in general, practitioners often have to resort to numerical schemes. Ladder methods are a popular class of algorithms that rely on iterative constructions of geodesic parallelograms. And yet, the literature lacks a clear analysis of their convergence performance. In this work, we give Taylor approximations of the elementary constructions of Schild’s ladder and the pole ladder with respect to the Riemann curvature of the underlying space. We then prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. We also contribute a new link between Schild’s ladder and the Fanning scheme which explains why the latter naturally converges only linearly. The extra computational cost of ladder methods is thus easily compensated by a drastic reduction of the number of steps needed to achieve the requested accuracy. Illustrations on the 2-sphere, the space of symmetric positive definite matrices and the special Euclidean group show that the theoretical errors we have established are measured with a high accuracy in practice. The special Euclidean group with an anisotropic left-invariant metric is of particular interest as it is a tractable example of a non-symmetric space in general, which reduces to a Riemannian symmetric space in a particular case. As a secondary contribution, we compute the covariant derivative of the curvature in this space.

A note on generalized invertibility and invariants of infinite matrices

The classes of band-dominated operators and the subclass of operators in the Wiener algebra $${\mathcal {W}}$$ W are known to be inverse closed. This paper studies and extends partially known results of that type for one-sided and generalized invertibility. Furthermore, for the operators in the Wiener algebra $${\mathcal {W}}$$ W invertibility, the Fredholm property and the Fredholm index are known to be independent of the underlying space $$l^p$$ l p , $$1\le p\le \infty $$ 1 ≤ p ≤ ∞ . Here this is completed by the observation that even the kernel and a suitable direct complement of the range as well as generalized inverses of operators in $${\mathcal {W}}$$ W are invariant w.r.t. p.

On Extended Branciari b -Distance Spaces and Applications to Fractional Differential Equations

In this work, we define new α − λ -rational contractive conditions and establish fixed-points results based on aforesaid contractive conditions for a mapping in extended Branciari b -distance spaces. We furnish two examples to justify the work. Further, we discuss results on weak well-posed property, weak limit shadowing property, and generalized w -Ulam-Hyers stability in the underlying space. Finally, as an application of our main result, we obtain sufficient conditions for the existence of solutions of a nonlinear fractional differential equation with integral boundary conditions.