Fixed Point Results for F-Contractions in Cone Metric Spaces over Topological Modules and Applications to Integral Equations

In this paper, the concept of F-contraction was generalized for cone metric spaces over topological left modules and some fixed point results were obtained for self-mappings satisfying a contractive condition of this type. Some applications of the main result to the study of the existence and uniqueness of the solutions for certain types of integral equations were presented in the last part of the article, one of them being a fractional integral equation.

The Density of Polynomials in a Special Space of Entire Functions of Exponential Type

The traditional approach to solving a specific problem of spectral synthesis in a complex domain involves reducing it to the problem of local description of closed submodules in a certain space of entire functions. The last problem is split into checking the stability and saturation of the submodule under study. This approach turned out to be very effective, for example, in the study of submodules of local rank 1 and in the study of submodules in topological modules associated with unbounded convex domains. Recent studies on spectral synthesis in the complex domain are based on a different scheme. This scheme involves reducing the problem of local description to checking the density of polynomials in a special module of entire functions of exponential type. Moreover, the space under study is a separable locally convex space of type (LN)*. Polynomial approximation in such a space is understood by us as sequential approximation, that is, we are talking about the approximation of space elements by ordinary (not generalized) sequences of polynomials. In this article, we study a special locally convex module of entire vector functions over the ring of polynomials in the degree of the independent variable. The theorem proved in the article can serve as a source of new results on spectral synthesis in the complex domain.

Metabolic response of dolphins to short-term fasting reveals physiological changes that differ from the traditional fasting model

Bottlenose dolphins (Tursiops truncatus) typically feed on prey that are high in lipid and protein content and nearly devoid of carbohydrate, a dietary feature shared with other marine mammals. However, unlike fasted-adapted marine mammals that predictably incorporate fasting into their life history, dolphins feed intermittently throughout the day and are not believed to be fasting-adapted. To assess whether the physiological response to fasting in the dolphin shares features with or distinguishes them from those of fasting-adapted marine mammals, the plasma metabolomes of eight bottlenose dolphins were compared between post-absorptive and 24-hour fasted states. Increases in most identified free fatty acids and lipid metabolites and reductions in most amino acids and their metabolites were consistent with the upregulation of lipolysis and lipid oxidation and the downregulation of protein catabolism and synthesis. Consistent with a previously hypothesized diabetic-like fasting state, fasting was associated with elevated glucose and patterns of certain metabolites (e.g. citrate, cis-aconitate, myristoleic acid) indicative of lipid synthesis and glucose cycling to protect endogenous glucose from oxidative disposal. Pathway analysis predicted an upregulation of cytokines, decreased cell growth, and increased apoptosis including apoptosis of insulin-secreting β-cells. Metabolomic conditional mutual information networks were estimated for the post-absorptive and fasted states and ‘topological modules’ were estimated for each using the eigenvector approach to modularity network division. A Dynamic Network Marker indicative of a physiological shift toward a negative energy state was subsequently identified that has the potential conservation application of assessing energy state balance in at-risk wild dolphins.

On the relative K-group in the ETNC, Part II

In a previous paper we showed that, under some assumptions, the relative K-group in the Burns–Flach formulation of the equivariant Tamagawa number conjecture (ETNC) is canonically isomorphic to a K-group of locally compact equivariant modules. Our approach as well as the standard one both involve presentations: One due to Bass–Swan, applied to categories of finitely generated projective modules; and one due to Nenashev, applied to our topological modules without finite generation assumptions. In this paper we provide an explicit isomorphism.

Regularity in Topological Modules

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and m∗∈M∗ is a continuous linear functional on M which is open as a map between topological spaces, then m∗−1(int(B)) is regular open and m∗−1(B) is regular closed, for B any closed-unit zero-neighborhood in R.

Automated design of a convolutional neural network with multi-scale filters for cost-efficient seismic data classification

Geoscientists mainly identify subsurface geologic features using exploration-derived seismic data. Classification or segmentation of 2D/3D seismic images commonly relies on conventional deep learning methods for image recognition. However, complex reflections of seismic waves tend to form high-dimensional and multi-scale signals, making traditional convolutional neural networks (CNNs) computationally costly. Here we propose a highly efficient and resource-saving CNN architecture (SeismicPatchNet) with topological modules and multi-scale-feature fusion units for classifying seismic data, which was discovered by an automated data-driven search strategy. The storage volume of the architecture parameters (0.73 M) is only ~2.7 MB, ~0.5% of the well-known VGG-16 architecture. SeismicPatchNet predicts nearly 18 times faster than ResNet-50 and shows an overwhelming advantage in identifying Bottom Simulating Reflection (BSR), an indicator of marine gas-hydrate resources. Saliency mapping demonstrated that our architecture captured key features well. These results suggest the prospect of end-to-end interpretation of multiple seismic datasets at extremely low computational cost.

Cone Metric Spaces over Topological Modules and Fixed Point Theorems for Lipschitz Mappings

In this paper, we introduce the concept of cone metric space over a topological left module and we establish some coincidence and common fixed point theorems for self-mappings satisfying a condition of Lipschitz type. The main results of this paper provide extensions as well as substantial generalizations and improvements of several well known results in the recent literature. In addition, the paper contains an example which shows that our main results are applicable on a non-metrizable cone metric space over a topological left module. The article proves that fixed point theorems in the framework of cone metric spaces over a topological left module are more effective and more fertile than standard results presented in cone metric spaces over a Banach algebra.

Some Results on Fuzzy Topological Modules and Fuzzy Topological Submodules

In this paper, we bring together the structure of fuzzy topological space ,fuzzy module and that of fuzzy ring to form a combined structure, that of a fuzzy topological R-module. Properties of fuzzy topological R-module ,topological R-submodule and its Properties are also briefly examined. we proved many theorems and corollaries as results shown in this paper.

Emergence of Network Motifs in Deep Neural Networks

Network science can offer fundamental insights into the structural and functional properties of complex systems. For example, it is widely known that neuronal circuits tend to organize into basic functional topological modules, called network motifs. In this article, we show that network science tools can be successfully applied also to the study of artificial neural networks operating according to self-organizing (learning) principles. In particular, we study the emergence of network motifs in multi-layer perceptrons, whose initial connectivity is defined as a stack of fully-connected, bipartite graphs. Simulations show that the final network topology is shaped by learning dynamics, but can be strongly biased by choosing appropriate weight initialization schemes. Overall, our results suggest that non-trivial initialization strategies can make learning more effective by promoting the development of useful network motifs, which are often surprisingly consistent with those observed in general transduction networks.