Multidimensional Mappings Research Articles

Magnetization properties, super-stable points, and cycles of antiferromagnetic spin-1 diamond chains with nodal-nodal interactions

This study is devoted to the discovery of super-stable points and cycles in antiferromagnetic Ising and Ising-Heisenberg models with spin 1 on diamond chains with nodal-nodal interactions. These phenomena are important for understanding the complex behavior of magnetic systems. We specifically investigate their connection with magnetization plateaus, which serve as critical indicators of the model’s characteristics. Employing the recurrence relations technique, we derive multidimensional rational mappings that give insights about the statistical properties of the models. Carefully examining the stability properties of these mappings, in particular, by analyzing the maximum Lyapunov exponent, we have revealed the complex relationship between the magnetization plateau and dynamic properties. Throughout our extensive research, we have comprehensively studied the existence and behavior of super-stable points and cycles for various parameter configurations in spin-1 models on the diamond chains. By highlighting the basic properties of dynamics and stability, our research advances a fundamental understanding of complex magnetic systems and their fascinating properties.

Synchrotron-based X-ray fluorescence microscopy mapping the ionome of a toxic freshwater cyanobacterium

Harmful algal blooms (HABs) pose a major environmental concern across the globe. In abundance, cyanobacteria, or so-called green-blue algae can produce extremely dangerous cyanotoxins that harm humans and animals. This study focused on the mapping and distribution of intracellular macro-and micronutrients of the wide-spread freshwater cyanobacteria Microcystis aeruginosa (M. aeruginosa). Towards a better understanding of trace metal uptake and homeostasis throughout the cell cycle, we quantitatively mapped the spatial distribution of the elements P, K, Fe, Ca, Zn, Mn, and Cu across the ultrastructure of frozen-hydrated single cells using state-of-the-art X-ray nanofluorescence imaging at the Advanced Photon Source (APS) at Argonne National Laboratory. Bulk cellular nutrient and trace metal content correlated well with the total intracellular elemental content in individual cells obtained by quantitative synchrotron X-ray fluorescence measurements. Multi-dimensional mappings showed P and K atoms colocalized as discrete semicircular hotspots that were analyzed with respect to their stoichiometry. Elevated Cu and Ca concentrations were detected along division plane of cells. P and K were found to have similar spatial elemental distribution with about 65% and 69% of the total cellular P and K, respectively, located at the hotspots. The P and K colocalization were refined further using nanotomography, showing a K envelope surrounding the P core. Inorganic P and organic P compounds were specified using solution-state 31P nuclear magnetic resonance (NMR) spectroscopy from M. aeruginosa. Of the total extracted P determined by 31P NMR spectroscopy, 47% were found to be nucleotides while only 11% were polyphosphates. Multimodal X-ray imaging provides a better understanding of intracellular biochemical processes in cyanobacteria, helping us monitor and combat an emerging environmental threat.

Asymptotic Rules of Equilibrium Desingularization

A local bifurcation analysis of a high-dimensional dynamical system dxdt=f(x) is performed using a good deformation of the polynomial mapping P:Cn→Cn. This theory is used to construct geometric aspects of the resolution of multiple zeros of the polynomial vector field P(x). Asymptotic bifurcation rules are derived from Grothendieck’s theory of residuals. Following the Coxeter–Dynkin classification, the singularity graph is constructed. A detailed study of three types of multidimensional mappings with a large symmetry group has been carried out, namely: 1. A linear singularity (behaves similarly to a one-dimensional complex analysis theory); 2. The lattice singularity (generalized the linear and resembling regular crystal growth models); 3. The fan-shaped singularity (can be split radially like nuclear fission and fusion models).

On derivatives of fuzzy multi-dimensional mappings and applications under generalized differentiability

This article identifies and presents the generalized difference (g-difference) of fuzzy numbers, Fréchet and Gâteaux generalized differentiability (g-differentiability) for fuzzy multi-dimensional mapping which consists of a new concept, fuzzy g-(continuous linear) function; Moreover, the relationship between Fréchet and Gâteaux g-differentiability is studied and shown. The concepts of directional and partial g-differentiability are further framed and the relationship of which will the aforementioned concepts are also explored. Furthermore, characterization is pointed out for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability based on level-set and through employing level-wise gH-differentiability (LgH-differentiability) is expressed. Finally, to illustrate the ability and reliability of the aforementioned concepts we have solved some application examples.

Метод анализа экономических процессов в условиях нестабильности (на примере анализа динамики выручки российских компаний)

New methods are proposed that allow using the properties of nonlinear dynamics models to analyze the dynamics of real socio-economic processes. For this, a complex of models has been developed in the form of connected nonlinear one-dimensional and multidimensional map pings that meet the general laws of socio-economic processes - development in a rapidly changing environment under resource constraints. The proposed set of models allows you to analyze the dynamics of processes under the assumptions of both their independence and interconnection with each other. The definition of interrelated processes is introduced as processes developing under conditions of general constraints. Three criteria are proposed for identifying interrelated processes based on empirical data. In the case of interrelation of processes, the parameters of one-dimensional and multidimensional models are also interrelated, which allows a deeper analysis of the situations under consideration. The use of the proposed method is demonstrated by the example of analyzing the dynamics of revenue of Russian companies in the pe­riod from 2006 to 2015 (38 thousand companies in total). All companies were divided into four groups, depending on the rate of revenue growth from 2010 to 2015. The hypothesis of a possible relationship between the decline of one and the parallel strengthening of another group of Russian firms after the crisis of 2008-2009 is investigated and rejected. Innovative firms that boosted their revenue growth potential after the crisis were targeting markets other than con ser­vative firms whose revenue growth potential was exhausted after the crisis. The peculiarities of the dynamics of revenue of groups of companies are visually presented in the form of normalized values, when the median estimates of each of the groups in 2007 are taken as a unit.

Thinking Camera – Powered by the CAOS Camera Platform

Introduced is the system level design of a Thinking Camera powered by the CAOS camera platform. The proposed camera provides features such as extreme linear dynamic range and full spectrum operations with selectable space-time-frequency CAOS pixel modes using active and passive light. The imager has triple output ports design and is controlled via application-specific classical image processing, machine learning techniques, and human/user feedback. Imaging capabilities cover multi-dimensional mappings allowing diverse full spectrum (350 nm to 2700 nm) applications from industrial metrology to quantitative medical imaging to artefact preserving colour photography.

Novel Method for Multi-Dimensional Mapping of Higher Order Modulations for BICM-ID Over Rayleigh Fading Channels

Multi-dimensional (MD) mapping offers more flexibility in mapping design for bit-interleaved coded modulation with iterative decoding (BICM-ID) and potentially improves the bandwidth efficiency. However, for higher order signal constellations, finding suitable MD mappings is a very complicated task due to the large number of possible mappings. In this paper, a novel mapping method is introduced to construct efficient MD mappings to improve the error performance of BICM-ID over Rayleigh fading channels. We propose to break the MD mapping design problem into four distinct 2-D mapping functions. The 2-D mappings are designed such that the resulting MD mapping improves the BICM-ID error performance at low signal-to-noise ratios (SNRs). We also develop cost functions that can be optimized to improve the error performance at high SNRs. The proposed mapping method is very simple compared to well-known mapping methods, and it can achieve suitable MD mappings for different modulations including higher order modulations for BICM-ID. The simulation results show that our mappings significantly outperform the previously known mappings at a target bit error rate (BER) of 10−6. Our mappings also offer a lower error-floor compared to their well-known counterparts.

Efficient Multi-Dimensional Mapping Using QAM Constellations for BICM-ID

Bit-interleaved coded modulation with iterative decoding (BICM-ID) offers improved error performance over additive white Gaussian noise and fading channels if it uses a wisely designed signal mapping. Furthermore, error performance of BICM-ID can be significantly improved by employing multi-dimensional (MD) modulation. However, suitable MD mappings are obtained by computer search techniques, except for MD modulations that use small constellations, e.g., binary phase shift keying, quadrature phase shift keying, and 8-ary phase shift keying as basic modulation. The alphabet size of MD modulations increases exponentially as the order of the basic modulation increases and computer searching becomes intractable. In this paper, we propose a systematic mapping method for MD modulations. The innovation of our proposed method is that it generates MD mappings using 16- and 64-quadrature amplitude modulation very efficiently. The presented numerical results show that the proposed method improves the bit error rate of BICM-ID.

The Existence of ϒ-fixed Point for the Multidimensional Nonlinear Mappings Satisfying (ψ,θ,φ)-weak Contractive Conditions

In this paper we prove the existence of ϒ-fixed point for a multidimensional nonlinear mappings F : Xk → X defined on the partially ordered metric spaces and satisfying (ψ, θ, ϕ)-weak contractive conditions. Moreover, we prove the uniqueness of that fixed point under extra conditions to (ψ, θ, ϕ)-weak contractive conditions.

The Study of Discrete Mappings in TQ-Space. Basic principles

For multidimensional discrete mappings of the form s k+1 = g(s k , p) we develop the method of symbolic CTQ-analysis and study the main properties of the T-alphabet. We formulate basic principles for the analytic study of discrete mappings in terms of the proposed formalism. Bibliography: 15 titles. Illustrations: 5 figures.

A Fault Detection and Isolation Software Framework for Repeatable and Comparable Experimentation

There is an extensive literature available about condition monitoring relying on multi-dimensional data-driven system models and mappings, including proposal of new methods and algorithms,comparison of state-of-the-art methods, and stateof- the-art revisions. But, when practitioners start to implement their own software to carry out their research, there is a lack of articles in the literature with detailed documentation about how to design a framework for repeatable and comparable experimentation. We propose a design for repeatable and comparable experimentation on the field of Data-Driven Residual-Based Fault Detection and Isolation. The framework has already been used for several experiments, with successful results, eliciting features such as (i) decreasing of developing times, (ii) facilitating of configuration management, and (iii) facilitating of collection and comparison of results.

Estimation of the TQ-complexity of chaotic sequences

A new approach is proposed to the quantitative estimation of the complexity of multidimensional discrete sequences in terms of the shapes of their trajectories in the extended space of states. This approach is based on the study of the structural properties of sequences and is suitable for estimating the complexity of both chaotic and stochastic sequences. It is constructed on the method, proposed earlier by the author, of symbolic CTQ-analysis of multidimensional discrete sequences and mappings. The algorithm proposed manipulates not only the frequency of occurrence of symbols, but also takes into account their sequence order. An example (financial time series) is given that demonstrates the application of the tools developed.

A Transient Diesel EMS Strategy for Online Implementation

A recently developed strategy for diesel engine management systems is modified to reduce the implementation complexity. The strategy calculates set points for engine management system controllable quantities with an aim to minimize fuel consumption for a given engine speed and requested torque profile, while keeping accumulated emissions below given limits. The strategy is based on the methodology for steady-state engine operation, but extended to handle transient effects in the engine caused by dynamics in the air system. The strategy leads to the parametrization of mappings with two, three and four input dimensions respectively. In this paper, a modification of the strategy is proposed such that the memory demanding multidimensional mappings can be approximated in an engine management system using only two-dimensional grid maps. The modified strategy has been evaluated using a complete diesel engine vehicle system model simulating the NEDC driving cycle. The performance of the modified strategy has been compared with the original performance of the strategy. It is demonstrated that the modification of the strategy has very little impact on resulting performance of a vehicle but requires considerably less memory for implementation.

Multidimensional dynamic processes studied by symbolic analysis in velocity-curvature space

A new computer-aided method for the symbolic analysis of discrete mappings and sequences is proposed that is based on a finite discretization of the velocity-curvature space. A minimum alphabet is introduced in a natural way. A number of initial analytical measures are defined that make it possible to study the dynamics structure of multidimensional discrete mappings, continuous systems, and dynamic processes. The proposed analytical method is tested by applying it to a logistic oscillator in the domain to the right of the period-doubling limit point, and the method is shown to be informative. It is revealed that the oscillation structure of the logistic map is highly asymmetric. The critical parameter values are found at which the geometric structure of the map trajectories changes qualitatively.

Entwining Yang-Baxter maps and integrable lattices

Yang–Baxter (YB) map systems (or set-theoretic analoga of entwining YB structures) are presented. They admit zero curvature representations with spectral parameter depended Lax triples L1, L2, L3 derived from symplectic leaves of 2 � 2 binomial matrices equipped with the Sklyanin bracket. A unique factorization condition of the Lax triple implies a 3-dimensional compatibility property of these maps. In case L1 = L2 = L3 this property yields the set-theoretic quantum Yang-Baxter equation, i.e. the YB map property. By considering periodic ‘staircase’ initial value problems on quadrilateral lattices, these maps give rise to multidimensional integrable mappings which preserve the spectrum of the corresponding monodromy matrix.

Magnetization and Lyapunov exponents on a kagome chain with multi-site exchange interaction

The Ising approximation of the Heisenberg model in a strong magnetic field, with two, three and six spin exchange interactions is studied on a kagome chain. The kagome chain can be considered as an approximation of the third layer of 3He absorbed on the surface of graphite (kagome lattice). By using dynamical approach we have found one- and multi-dimensional mappings (recursion relations) for the partition function. The magnetization diagrams are plotted and they show that the kagome chain is separating into four sublattices with different magnetizations. Magnetization curves of two sublattices exhibit plateaus at zero and 2/3 of the saturation field. The maximal Lyapunov exponent for multi-dimensional mapping is considered and it is shown that near the magnetization plateaus the maximal Lyapunov exponent also exhibits plateaus.

Multidimensional mappings in antiferromagnetic models and Lyapunov exponents with the magnetization plateau

We consider the Heisenberg model with two- and three-spin exchange interactions on a recursive ladder in a strong magnetic field. Recurrent relations for branches of the partition function of the Ising model with two- and three-spin exchange interactions are deduced. As a recursive lattice the zigzag ladder is chosen. In the antiferromagnetic case magnetization plateau are observed at low temperatures. Lyapunov exponents for the three-dimensional mapping at low temperatures are calculated. It is shown that for some values of two- and three-spin exchange parameters in the antiferromagnetic case the maximum of the Lyapunov exponent approaches zero.

Neural network emulations for complex multidimensional geophysical mappings: Applications of neural network techniques to atmospheric and oceanic satellite retrievals and numerical modeling

A group of geophysical applications, which from the mathematical point of view, can be formulated as complex, multidimensional, nonlinear mappings and which in terms of the neural network (NN) technique, utilize a particular type of NN, the multilayer perceptron (MLP), is reviewed in this paper. This type of NN application covers the majority of NN applications developed in geosciences like satellite remote sensing, meteorology, oceanography, numerical weather prediction, and climate studies. The major properties of the mappings and MLP NNs are formulated and discussed. Three particular groups of NN applications are presented in this paper as illustrations: atmospheric and oceanic satellite remote sensing applications, NN emulations of model physics for developing atmospheric and oceanic hybrid numerical models, and NN emulations of the dependencies between model variables for application in data assimilation systems.

Synchronous manifold and hyperbolicity in a system of coupled identical multidimensional mappings

Two problems are solved: the obtaining of hyperbolicity conditions for families of individual multidimensional mappings and applying these hyperbolicity conditions to a system of coupled mappings and the subsequent proof of the preservation of the hyperbolicity property for the important class of coupled mappings considered.The study is performed by using one-dimensional “comparison mappings” and by introducing a metric special for this class of mappings. It is essential that the hyperbolicity conditions are formulated in terms of conditions for a concrete class of functions defining the mappings.

A Novel Multi-Dimensional Mapping of 8-PSK for BICM-ID

Employing multi-dimensional constellation and mapping to improve the error performance of the bit-interleaved coded modulation with iterative decoding (BICM-ID) has recently received a lot of attention, both in single-antenna and multiple-antenna systems. To date, except for the cases of BPSK and QPSK constellations, good multi-dimensional mappings have only been found by computer searching techniques. This paper introduces an explicit algorithm to construct a good multidimensional mapping of 8-PSK for improving the asymptotic performance of BICM-ID systems. By comparing the performance of the proposed mapping with an unachievable lower bound, it is conjectured that the proposed mapping is the global optimal mapping. The superiority of the proposed mapping over the best conventional (two-dimensional) mapping and the multidimensional mapping found previously by computer search is also thoroughly demonstrated