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This monograph presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley–Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

Cytosolic-free calcium ions play an important role in various physical and physiological processes. A vital component of neural signaling is the free calcium ion concentration often known as the second messenger. There are many parameters that effect the cytosolic free calcium concentration like buffer, voltage-gated ion channels, Endoplasmic reticulum, Mitochondria, etc. Mitochondria are small organelles located within the nervous system that are involved in processes within cells such as calcium homeostasis management, energy generation, response to stress, and cell demise pathways. In this work, a mathematical model with fuzzy boundary values has been developed to study the effect of Mitochondria and ER fluxes on free Calcium ions. The intended findings are displayed utilizing the physiological understanding that amyloid beta plaques and tangles of neurofibrillary fibers have been identified as the two main causes of AD. The key conclusion of the work is the investigation of Ca2+ for healthy cells and cells affected by Alzheimer's disease, which may aid in the study of such processes for computational scientists and medical practitioners. Also, it has been shown that when a unique solution is found for a specific precise problem, it also successfully deals with any underlying ambiguity within the problem by utilizing a technique based on the principles of linear transformation. Furthermore, the comparison between the analytical approach and the generalized hukuhara derivative approach is shown here, which illustrates the benefits of the analytical approach. The simulation is carried out in MATLAB.

Problems with dominant advection, discontinuities, travelling features, or shape variations are widespread in computational mechanics. However, classical linear model reduction and interpolation methods typically fail to reproduce even relatively small parameter variations, making the reduced models inefficient and inaccurate. This work proposes a model order reduction approach based on the Radon Cumulative Distribution Transform (RCDT). We demonstrate numerically that this non-linear transformation can overcome some limitations of standard proper orthogonal decomposition (POD) reconstructions and is capable of interpolating accurately some advection-dominated phenomena, although it may introduce artefacts due to the discrete forward and inverse transform. The method is tested on various test cases coming from both manufactured examples and fluid dynamics problems.

This paper deals with the competing risks data with covariate measurement error. A semiparametric linear transformation model for the right-censored competing risks data when the covariates are measured with error is proposed. The parameters involved in the model are estimated using a set of estimating equations. An adaptable simulation extrapolation (SIMEX) technique is employed to handle the covariate measurement error. Simulation studies are conducted, to examine the finite sample properties of the estimators. Also, we demonstrated the proposed method using a real dataset.

Let V be an infinite-dimensional vector space over a field F and let $I(V)$ be the inverse semigroup of all injective partial linear transformations on V. Given $\\alpha \\in I(V)$ , we denote the domain and the range of $\\alpha $ by ${\\mathop {\\textrm {dom}}}\\,\\alpha $ and ${\\mathop {\\textrm {im}}}\\,\\alpha $ , and we call the cardinals $g(\\alpha )={\\mathop {\\textrm {codim}}}\\,{\\mathop {\\textrm {dom}}}\\,\\alpha $ and $d(\\alpha )={\\mathop {\\textrm {codim}}}\\,{\\mathop {\\textrm {im}}}\\,\\alpha $ the ‘gap’ and the ‘defect’ of $\\alpha $ . We study the semigroup $A(V)$ of all injective partial linear transformations with equal gap and defect and characterise Green’s relations and ideals in $A(V)$ . This is analogous to work by Sanwong and Sullivan [‘Injective transformations with equal gap and defect’, Bull. Aust. Math. Soc. 79 (2009), 327–336] on a similarly defined semigroup for the set case, but we show that these semigroups are never isomorphic.

Brain tumor detection and classification are crucial steps in evaluating life-threatening abnormal tissues to provide appropriate treatment plans. For clinical assessment, Magnetic resonance imaging (MRI) is normally used because of its excellent quality and lack of ionizing radiation. However, as the volume of the data grows, manual processing of MRI images becomes expensive, time-taking, and error prone. Also, traditional automated detection systems struggle to handle complex image patterns, leading to reduced classification accuracy. So, this paper designs a reliable and effective brain tumor detection mechanism as a solution to these problems. The proposed \"Vision Transformer with Attention and Linear Transformation module (VITALT)\" system is a combination of modules such as Vision Transformer (ViT), Split bidirectional feature pyramid network (S-BiFPN), linear transformation module (LTM) and soft-quantization that effectively extracts features from complex brain structures. At first, to mitigate the training inaccuracies developed by dimension and quality constraints, the preprocessing steps such as resizing and normalization are executed. The preprocessed images are divided into number of patches and embedded into high-dimensional vector to provide more compact image representation. Subsequently, the global and local features in the image are captured through ViT module by learning the relationship between image patches. The multi-scale spatial features formed are then fused using S-BiFPN to increase the accuracy of prediction. By using LTM to improve the linear expression capability of the design, the characteristics that are most important for the classification of brain tumors are discovered. Also, soft quantization is used to minimize memory footprint and minimize quantization errors in detection. Finally, the head module with set of fully connected layers accurately classifies different classes of brain tumors. The experimental analysis conducted using four different benchmark brain tumor datasets shows the viability and reliability of the suggested VITALT system in predicting brain tumors, as measured by multiple evaluation metrics. The proposed system achieves classification accuracy of 99.08% for Dataset A, 98.97% for Dataset B, 98.82% for Dataset C and 99.15% for Dataset D. A high level of classification accuracy attained by the suggested system highlights its potential in medical imaging applications and its ability to contribute to improved surgical treatments.

How to find the nonlinear invariants is the core step for the nonlinear invariant attack. In this paper, to reduce the costs in finding nonlinear invariants for linear transformation, an improved algorithm is proposed to obtain nonlinear invariants with low algebraic degree not more than k , whose time complexity is O ( ∑ i = 1 k C n i 3 ) for an n -dimension linear transformations on binary field. Besides, for the m × n -dimension linear transformation on binary field which applies an n -dimension linear transformation on binary field m times in parallel, this paper make further improvement of getting its all invariants with algebraic degree not more than k ( k < m ) , whose time complexity is O ( k ∑ i = 1 k C n ( k - 1 ) i 3 + n 3 k ) . In particular, we take the lightweight block cipher Scream as example to demonstrate the efficiency of our methods, which indicates that the methods in this paper have significant advantages than before.

The global coastal seascape offers a multitude of ecosystem functions and services to the natural and human-induced ecosystems. However, the current anthropogenic global warming above pre-industrial levels is inducing the degradation of seascape health with adverse impacts on biodiversity, economy, and societies. Bathymetric knowledge empowers our scientific, financial, and ecological understanding of the associated benefits, processes, and pressures to the coastal seascape. Here we leverage two commercial high-resolution multispectral satellite images of the Pleiades and two multibeam survey datasets to measure bathymetry in two zones (0–10 m and 10–30 m) in the tropical Anguilla and British Virgin Islands, northeast Caribbean. A methodological framework featuring a combination of an empirical linear transformation, cloud masking, sun-glint correction, and pseudo-invariant features allows spatially independent calibration and test of our satellite-derived bathymetry approach. The best R2 and RMSE for training and validation vary between 0.44–0.56 and 1.39–1.76 m, respectively, while minimum vertical errors are less than 1 m in the depth ranges of 7.8–10 and 11.6–18.4 m for the two explored zones. Given available field data, the present methodology could provide simple, time-efficient, and accurate spatio-temporal satellite-derived bathymetry intelligence in scientific and commercial tasks i.e., navigation, coastal habitat mapping and resource management, and reducing natural hazards.

Subgradient methods are frequently used for optimization problems. However, subgradient techniques are characterized by slow convergence for minimizing ravine convex functions. To accelerate subgradient methods, special linear non-orthogonal transformations of the original space are used. This paper provides an overview of these transformations based on Shor’s original idea. Two one-rank linear transformations of Euclidean space are considered. These simple transformations form the basis of variable metric methods for convex minimization that have a natural geometric interpretation in the transformed space. Along with the space transformation, a search direction and a corresponding step size must be defined. Subgradient Fejer-type methods are analyzed to minimize convex functions, and Polyak step size is used for problems with a known optimal objective value. Convergence theorems are provided together with the results of numerical experiments. Directions for future research are discussed.

Let V be a vector space over a field F and let W be a subspace of V . The semigroup of partial linear transformations on V whose restriction to W belongs to an injective partial linear transformation semigroup I(W) is denoted by PI(W)(V) . In this paper, we describe Green's relations for PI(W)(V) , characterize its regular elements, and give necessary and sufficient conditions for PI(W)(V) to be regular, inverse, or completely regular. We also analyze the ideal structure of PI(W)(V) , identifying its maximal and minimal ideals.