Mapping Properties Research Articles

Mirror Symmetry as an Algebra of Operators in Noncommutative Geometry of Space-Time

The analysis of the geometric and algebraic properties of mirror mappings allowed the latter to be used as the operator algebra of a noncommutative geometry. The coordinates of the noncommutative geometry are auto- or cross-correlation coordinates in the mirror-mapped spaces. A particular case of the six-dimensional Kahler manifold which is mapped on the noncommutative geometry with the vector Clifford algebra Cl4 has been considered. This mapping corresponds to a tetraquark composed from two quark–anti-quark pairs with the charges ±2/3q taken from different generations.

Acquiring new memories in neocortex of hippocampal-lesioned mice

The hippocampus interacts with the neocortical network for memory retrieval and consolidation. Here, we found the lateral entorhinal cortex (LEC) modulates learning-induced cortical long-range gamma synchrony (20–40 Hz) in a hippocampal-dependent manner. The long-range gamma synchrony, which was coupled to the theta (7–10 Hz) rhythm and enhanced upon learning and recall, was mediated by inter-cortical projections from layer 5 neurons of the LEC to layer 2 neurons of the sensory and association cortices. Artificially induced cortical gamma synchrony across cortical areas improved memory encoding in hippocampal lesioned mice for originally hippocampal-dependent tasks. Mechanistically, we found that activities of cortical c-Fos labeled neurons, which showed egocentric map properties, were modulated by LEC-mediated gamma synchrony during memory recall, implicating a role of cortical synchrony to generate an integrative memory representation from disperse features. Our findings reveal the hippocampal mediated organization of cortical memories and suggest brain-machine interface approaches to improve cognitive function.

Grand Lebesgue space for p = ∞ and its application to Sobolev–Adams embedding theorems in borderline cases

AbstractWe define the grand Lebesgue space corresponding to the case and similar grand spaces for Morrey and Morrey type spaces, also for , on open sets in . We show that such spaces are useful in the study of mapping properties of the Riesz potential operator in the borderline cases for Lebesgue spaces and for Morrey and Morrey type spaces, providing the target space more narrow than BMO. While for Lebesgue spaces there are known results on the description of the target space in terms better than BMO, the results obtained for Morrey and Morrey type spaces are entirely new. We also show that the obtained results are sharp in a certain sense.

Boundary Integral Operators for the Heat Equation in Time-Dependent Domains

This article provides a functional analytical framework for boundary integral equations of the heat equation in time-dependent domains. More specifically, we consider a non-cylindrical domain in space-time that is the C^2-diffeomorphic image of a cylinder, i.e., the tensor product of a time interval and a fixed domain in space. On the non-cylindrical domain, we introduce Sobolev spaces, trace lemmata and provide the mapping properties of the layer operators. Here it is critical that the Neumann trace requires a correction term for the normal velocity of the moving boundary. Therefore, one has to analyze the situation carefully.

Analysis of shape grammars: Continuity of rules

The rules in a shape grammar apply in terms of embedding to take advantage of the parts that emerge visually in the appearance of shapes. While the shapes are kept unanalyzed as a computation moves forward, part-structures for shapes can be defined retrospectively by analyzing how the rules were applied. An important outcome of this is that rule continuity is not built-in but it is “fabricated” retrospectively to analyze the computation as a continuous process. An aspect of continuity analysis that has not been addressed in the literature is how to decide which mapping forms to use to study the continuity of rule applications. This is addressed in this paper using recent results on shape topology and continuous mappings. A characterization is provided that distinguishes the suitable mapping forms from those that are inherently discontinuous or practically inconsequential for continuity analysis. It is also shown that certain intrinsic properties of shape topologies and continuous mappings provide an effective method of computing topologies algorithmically.

A Fuzzy Approach to the Synthesis of Cognitive Maps for Modeling Decision Making in Complex Systems

The object of this study is fuzzy cognitive modeling as a means of studying semistructured socio-economic systems. The features of constructing cognitive maps, providing the ability to choose management decisions in complex semistructured socio-economic systems, are described. It is shown that further improvement of technologies necessary for developing decision support systems and their practical use is still relevant. This work aimed to improve the accuracy of cognitive modeling of semistructured systems based on a fuzzy cognitive map of structuring nonformalized situations (MSNS) with the evaluation of root-mean-square error (RMSE) and mean average squared error (MASE) coefficients. In order to achieve the goal, the following main methods were used: systems analysis methods, fuzzy logic and fuzzy sets theory postulates, theory of integral wavelet transform, correlation and autocorrelation analyses. As a result, a new methodology for constructing MSNS was proposed—a map of structuring nonformalized situations that combines the positive properties of previous fuzzy cognitive maps. The solution of modeling problems based on this methodology should increase the reliability and quality of analysis and modeling of semistructured systems and processes under uncertainty. The analysis using open datasets proved that compared to the classical ARIMA, SVR, MLP, and Fuzzy time series models, our proposed model provides better performance in terms of MASE and RMSE metrics, which confirms its advantage. Thus, it is advisable to use our proposed algorithm in the future as a mathematical basis for developing software tools for the analysis and modeling of problems in semistructured systems and processes. Doi: 10.28991/ESJ-2022-06-02-012 Full Text: PDF

Positive Solutions for Anisotropic Singular Dirichlet Problems

We consider a Dirichlet problem driven by a (p(z), q(z))-Laplacian and a reaction involving the sum of a parametric singular term plus a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter lambda >0 varies. Also we show that for every admissible parameter the problem has a smallest positive solution and obtain the monotonicity and continuity properties of the minimal solution map.

Functional connectome of human cerebellum

Background Neural connectome theory has been widely used in system neuroscience, and prompted our comprehension for the topological organizations of human cerebral cortex. However, how functional connectome is organized topologically in cerebellums remains unclear.Method Resting-state functional connectivity (rs-fcMRI) data were acquired from 1416 healthy adults in two independent samples. In Sample 1 (n = 976), both voxel-wise and node-wise topological properties for functional cerebellar connectome were estimated. Moreover, the network-based topological properties of cerebellum and cerebro-cerebellar topological mapping were investigated, respectively. Given the temporal natures in the neural population, a hidden Markov model (HMM) was further capitalized to uncover the dynamic pattern of cerebellar functional connectome. In order to test the robustness of our findings, we ran all of the analyses in an independent dataset (Sample 2; n = 440).Results We found that Crus I and II exhibited prominently high degree centrality (DC) for cerebellar functional connectome. Further, the cerebellar functional connectome and even the nested network-wise cerebellar connectome was found to be organized by small-world, modular and hierarchical manners significantly. Also, three intrinsic modules were found in cerebellar functional connectome, including attention/executive network, default mode network and task-positive network. In addition, the significant cerebro-cerebellar correlations for small-world organization and hierarchical architecture were found as well. By building cerebro-cerebellar topological mapping, both frontoparietal and subcortical networks were found to be overrepresented into cerebellums than cerebral cortex (3-fold). As for temporal natures, cerebellar functional connectome was observed to be highly flexible and modular, but showed high individual-specific variances in temporal dynamic pattern.Conclusion This study identified the topological architectures and temporal hierarchy of functional cerebellar connectome, and further demonstrated prominent functional cerebro-cerebellar couplings of small-world organization and hierarchical architectures.

Mutliresolutional ensemble PartialNet for Alzheimer detection using magnetic resonance imaging data

Alzheimer's disease (AD) is an irreversible and progressive disorder where a large number of brain cells and their connections degenerate and die, eventually destroy the memory and other important mental functions that affect memory, thinking, language, judgment, and behavior. Not a single test can effectively determine AD; however, CT and magnetic resonance imaging (MRI) can be used to observe the decrease in size of different areas (mainly temporal and parietal lobes). This paper proposes an integrative deep ensemble learning framework to obtain better predictive performance for AD diagnosis. Unlike DenseNet, we present a multiresolutional ensemble PartialNet tailored to Alzheimer detection using brain MRIs. PartialNet incorporates the properties of identity mappings, diversified depth as well as deep supervision, thus, considers feature reuse that in turn results in better learning. Additionally, the proposed ensemble PartialNet demonstrates better characteristics in terms of vanishing gradient, diminishing forward flow with better training time, and a low number of parameters compared with DenseNet. Experiments performed on benchmark AD neuroimaging initiative data set that showed considerable performance gain (2 + % ↑ $\uparrow $ ) and (1.2 + % ↑ $\uparrow $ ) for multiclass and binary class in AD detection in comparison to state-of-the-art methods.

Harmonization of design-based mapping for spatial populations

The mapping of a survey variable throughout a continuum or for finite populations of units is usually performed from a model-dependent perspective. Nevertheless, when a sample of locations/units is selected by a probabilistic sampling scheme, the complex task of modelling can be avoided by using the inverse distance weighting interpolator and deriving the properties of maps in a design-based perspective. Conditions ensuring consistency of maps can be derived mainly based on some obvious assumptions about the pattern of the survey variable throughout the study region as well from the feature of the sampling scheme adopted to select locations/units. Nevertheless, in a design-based setting the totals of the survey variable for a set of domains partitioning the study region are commonly estimated by traditional estimators such as the Horvitz–Thompson estimator in the case of finite populations or the Monte-Carlo estimator in the case of continuous populations or by related estimators exploiting the information of auxiliary variables. That necessarily gives rise to different total estimates with respect to those achieved from the resulting maps as the sum of the interpolated values within domains. To obtain non-discrepant results, a harmonization of maps is here suggested, in such a way that the resulting totals arising from maps coincide with those achieved by traditional estimation. The capacity of the harmonization procedure to maintain consistency is argued theoretically and checked by a simulation study performed on some real populations.

Elucidating multi-input processing 3-node gene regulatory network topologies capable of generating striped gene expression patterns.

A central problem in developmental and synthetic biology is understanding the mechanisms by which cells in a tissue or a Petri dish process external cues and transform such information into a coherent response, e.g., a terminal differentiation state. It was long believed that this type of positional information could be entirely attributed to a gradient of concentration of a specific signaling molecule (i.e., a morphogen). However, advances in experimental methodologies and computer modeling have demonstrated the crucial role of the dynamics of a cell’s gene regulatory network (GRN) in decoding the information carried by the morphogen, which is eventually translated into a spatial pattern. This morphogen interpretation mechanism has gained much attention in systems biology as a tractable system to investigate the emergent properties of complex genotype-phenotype maps. In this study, we apply a Markov chain Monte Carlo (MCMC)-like algorithm to probe the design space of three-node GRNs with the ability to generate a band-like expression pattern (target phenotype) in the middle of an arrangement of 30 cells, which resemble a simple (1-D) morphogenetic field in a developing embryo. Unlike most modeling studies published so far, here we explore the space of GRN topologies with nodes having the potential to perceive the same input signal differently. This allows for a lot more flexibility during the search space process, and thus enables us to identify a larger set of potentially interesting and realizable morphogen interpretation mechanisms. Out of 2061 GRNs selected using the search space algorithm, we found 714 classes of network topologies that could correctly interpret the morphogen. Notably, the main network motif that generated the target phenotype in response to the input signal was the type 3 Incoherent Feed-Forward Loop (I3-FFL), which agrees with previous theoretical expectations and experimental observations. Particularly, compared to a previously reported pattern forming GRN topologies, we have uncovered a great variety of novel network designs, some of which might be worth inquiring through synthetic biology methodologies to test for the ability of network design with minimal regulatory complexity to interpret a developmental cue robustly.

FPGA simulation of chaotic tent map‐based S‐Box design

AbstractThe chaotic system has a characteristically random behavior by nature, and these systems have their own characteristics in a completely deterministic structure. This feature of a chaotic system makes it difficult to predict encryptions designed based on such a system. Thanks to this unpredictable and strong feature, maps produced from chaotic systems are an important alternative in the field of encryption. One of the structures obtained by employing chaotic maps is the substitution box. S‐Box, which provides the confusion principle used in block ciphers, is the main block that dynamically replaces unencrypted data with confidential data and makes a significant contribution to ensuring high security in the encryption system. Therefore, S‐Boxes hold a critical role in block ciphers. Speed and reliability are important parameters in the creation of this main block. Especially, applications performed on hardware are more reliable and high performance. Therefore, in this study, an S‐Box was designed using field‐programmable gate arrays (FPGA) simulation from a chaotic tent map to create a fast and reliable S‐Box because FPGAs offer solutions that may be important in this field considering their fast and customizable architecture. In the proposed method, the S‐Box was created in 0.16 s. In addition, the dynamic properties of the chaotic tent map were analyzed with Lyapunov exponents, and the NIST SP 800‐22 test was applied for the information encryption suitability of the proposed chaotic system. Also, to test the reliability of the produced S‐Box structures, SAC, non‐linearity, bit independence criteria, and input/output XOR distribution table metrics were implemented. The results showed that the proposed chaotic map was dynamic and passed the reliability tests successfully.

An extension problem and Hardy's inequality for the fractional Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type

In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.

Inference of cosmic-ray source properties by conditional invertible neural networks

The inference of physical parameters from measured distributions constitutes a core task in physics data analyses. Among recent deep learning methods, so-called conditional invertible neural networks provide an elegant approach owing to their probability-preserving bijective mapping properties. They enable training the parameter-observation correspondence in one mapping direction and evaluating the parameter posterior distributions in the reverse direction. Here, we study the inference of cosmic-ray source properties from cosmic-ray observations on Earth using extensive astrophysical simulations. We compare the performance of conditional invertible neural networks (cINNs) with the frequently used Markov Chain Monte Carlo (MCMC) method. While cINNs are trained to directly predict the parameters’ posterior distributions, the MCMC method extracts the posterior distributions through a likelihood function that matches simulations with observations. Overall, we find good agreement between the physics parameters derived by the two different methods. As a result of its computational efficiency, the cINN method allows for a swift assessment of inference quality.

A Novel Chaos-Based Image Encryption Using Magic Square Scrambling and Octree Diffusing

Digital chaotic maps have been widely used in the fields of cryptography owing to their dynamic characteristics, however, some unfavorable security properties arise when they operate on devices with limited precision. Thus, enhancing the properties of chaotic maps are beneficial to the improvement of chaos-based encryption algorithms. In this paper, a scheme to integrate a one-dimensional Logistic map by perturbation parameters with a delayed coupling method and feedback control is proposed and further deepens the randomness by selectively shifting the position of the chaotic sequence. Then, through a number of simulation experiments, the results demonstrate that the two-dimensional chaotic map treated by this mode exhibits better chaotic characteristics, including a larger chaos range and higher complexity. In addition, a new image encryption algorithm is designed based on these modified chaotic sequences, in which magic square theorem is incorporated to exchange pixel positions, and the octree principle is invoked to achieve pixel bit shifting. Several simulation experiments present findings that the image encryption algorithm contains a high level of security, and can compete with other encryption algorithms.

Learning orientations: a discrete geometry model

In the mammalian brain, many neuronal ensembles are involved in representing spatial structure of the environment. In particular, there exist cells that encode the animal’s location and cells that encode head direction. A number of studies have addressed properties of the spatial maps produced by these two populations of neurons, mainly by establishing correlations between their spiking parameters and geometric characteristics of the animal’s environments. The question remains however, how the brain may intrinsically combine the direction and the location information into a unified spatial framework that enables animals’ orientation. Below we propose a model of such a framework, using ideas and constructs from algebraic topology and synthetic affine geometry.

Sobolev spaces and $$\nabla $$-differential operators on manifolds I: basic properties and weighted spaces

We study covariant Sobolev spaces and $$\nabla $$ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted $$\nabla $$ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the $$\nabla $$ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection $$\nabla $$ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our $$\nabla $$ -differential operators. We also introduce and study the notion of a $$\nabla $$ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for $$\nabla $$ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

Joint Characterization of the Cryospheric Spectral Feature Space

Multispectral and hyperspectral feature spaces are useful for a variety of remote sensing applications ranging from spectral mixture modeling to discrete thematic classification. In many of these applications, models are used to project the higher dimensional continuum of reflectances (or radiances) onto lower dimensional mappings of the image target’s physical properties or categorical composition. In such cases, characterization of the feature space dimensionality, geometry and topology can provide fundamental guidance for effective model design. Utility of this characterization, however, hinges on identification of appropriate basis vectors for the feature space. The objective of this study is to compare and contrast two fundamentally different approaches for identifying feature space basis vectors via dimensionality reduction. In so doing, we illustrate how these two approaches can be combined to render a joint characterization that reveals spectral properties not apparent using either approach alone. We use a diverse collection of AVIRIS-NG reflectance spectra of ice and snow to illustrate the utility of the joint characterization to facilitate both modeling and classification of snow and ice reflectance. Joint characterization is also shown to assist with interpretation of physical properties inferred from the spectra. Spectral feature spaces combining principal components (PCs) and t-distributed Stochastic Neighbor Embeddings (t-SNEs) provide both physically interpretable dimensions representing the global structure of cryospheric reflectance properties as well as local manifold structures revealing clustering not resolved within the global continuum. The joint characterization reveals distinct continua for snow-firn gradients on different parts of the Greenland Ice Sheet and multiple clusters of ice reflectance properties common to both glacier and sea ice in different locations. The clustering revealed in the t-SNE feature spaces, and extended to the joint characterization, distinguishes subtle differences in spectral curvature specific to different spatial locations within the snow accumulation zone, as well as BRDF effects related to view geometry. The ability of the PC + t-SNE joint characterization to produce a physically interpretable spectral feature space revealing global topology while preserving local manifold structures for cryospheric hyperspectra suggests that this type of characterization might be extended to the much higher dimensional hyperspectral feature space of all terrestrial land cover.

A note on the Screaming Toes game

AbstractWe investigate properties of random mappings whose core is composed of derangements as opposed to permutations. Such mappings arise as the natural framework for studying the Screaming Toes game described, for example, by Peter Cameron. This mapping differs from the classical case primarily in the behaviour of the small components, and a number of explicit results are provided to illustrate these differences.

On Starlike Functions of Negative Order Defined by q-Fractional Derivative

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α∈(−3,1], q∈(0,1) generalizes the class Sq* of q-starlike functions and the class Tq*˜(α), α∈[−1,1], q∈(0,1) comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included.