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In today’s fast-paced and technology-driven world, the gap between the skills taught in educational institutions and the requirements of the industry has become a significant challenge. Industries constantly evolve, demanding new competencies and up-to-date knowledge from the workforce. This research aims to develop a machine learning-driven framework to dynamically match industry skill requirements with educational curricula and generate adaptive course recommendations. Data collection from diverse sources, including job portals, industry reports, company career pages, and labor market analysis, captures the latest skill demands across multiple sectors. The framework utilizes an advanced machine learning Dynamic Termite Life Cycle Optimizer-driven Euclidean-Support Vector Machine (DTLC-Euclidean SVM) to predict both current and emerging skill demands in the job market. By analyzing patterns in graduate skillsets and employer requirements, the system identifies gaps and aligns educational offerings accordingly. The dynamic matching algorithm employs similarity metrics and k-Means Clustering techniques map industry needs to existing course content, while an automated course generation module suggests new or updated courses to address identified skill shortages. The overall performance accuracy (96.8%), precision (97.6%), recall (96.2%), and f1-score (97.3%), further establish its robustness and reliability in bridging the skill gap. This data-driven approach enables continuous curriculum adaptation, fostering stronger alignment between academia and industry. Finally, the framework supports educators and policymakers in developing responsive, targeted educational programs that prepare students for real-world career opportunities, enhancing employability and addressing the demands of a rapidly evolving labor market.

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

Big data has many divergent types of sources, from physical (sensor/IoT) to social and cyber (web) types, rendering it messy and, imprecise, and incomplete. Due to its quantitative (volume and velocity) and qualitative (variety) challenges, big data to the users resembles something like “the elephant to the blind men”. It is imperative to enact a major paradigm shift in data mining and learning tools so that information from diversified sources must be integrated together to unravel information hidden in the massive and messy big data, so that, metaphorically speaking, it would let the blind men “see” the elephant. This talk will address yet another vital “V”-paradigm: “Visualization”. Visualization tools are meant to supplement (instead of replace) the domain expertise (e.g. a cardiologist) and provide a big picture to help users formulate critical questions and subsequently postulate heuristic and insightful answers. For big data, the curse of high feature dimensionality is causing grave concerns on computational complexity and over-training. In this talk, we shall explore various projection methods for dimension reduction - a prelude to visualization of vectorial and non-vectorial data. A popular visualization tool for unsupervised learning is Principal Component Analysis (PCA). PCA aims at the best recoverability of the original data in the Euclidean Vector Space (EVS). However, PCA is not effective for supervised and collaborative learning environment. Discriminant Component Analysis (DCA), basically a supervised PCA, can be derived via a notion of Canonical Vector Space (CVS). The signal subspace components of DCA are associated with the discriminant distance/power (related to the classification effectiveness) while the noise subspace components of DCA are tightly coupled with the recoverability and/or privacy protection. DCA enjoys two major merits: First, because the rank of the signal subspace is limited by the number of classes, DCA can effectively support classification using a relatively small dimensionality (i.e. high compression). Second, in DCA, the eigenvalues of the noise-space are ordered according to their corresponding reconstruction errors and can thus be used to control recoverability or anti-recoverability by applying respectively an negative or positive ridge. Via DCA, individual data can be highly compressed before being uploaded to the cloud, and thus better enabling privacy protection. In many practical scenarios, additional privacy protection can be incorporated by allowing individual participants to selectively hide some personal features. The classification of masked data calls for a kernel approach to Incomplete Data Analysis (KAIDA). More specifically, we extend PCA/DCA to their kernel variants. The success of kernel machines hinges upon the kernel function adopted to characterize the similarity of pairs of partially-specified vectors. Simulations on the HAR dataset confirm that DCA far outperforms PCA, both in their conventional or kernelized variants. For the latter, the visualization/classification results suggest favorable performance by the proposed partial correlation kernels over the imputed RBF kernel. In addition, the visualization results further points to a potentially promising approach via multiple kernels such as combining an imputed Gaussian RBF kernel and a non-imputed partial correlation kernel.

We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of minimum sum-of-squares of distances from the elements of clusters to their centers. We assume that the cardinalities of the clusters are fixed. The center of one cluster has to be optimized and is defined as the average value over all vectors in this cluster. The center of the other cluster lies at the origin. The partition satisfies the condition: the difference of the indices of the next and previous vectors in the first cluster is bounded above and below by two given constants. We propose a 2-approximation polynomial algorithm to solve this problem.

Tikhonov’s regularized method of least squares and its generalizations to non-Euclidean norms, including polyhedral, are considered. The regularized method of least squares is reduced to mathematical programming problems obtained by “instrumental” generalizations of the Tikhonov lemma on the minimal (in a certain norm) solution of a system of linear algebraic equations with respect to an unknown matrix. Further studies are needed for problems concerning the development of methods and algorithms for solving reduced mathematical programming problems in which the objective functions and admissible domains are constructed using polyhedral vector norms.

Today complex numbers have such widespread practical use--from electrical engineering to aeronautics--that few people would expect the story behind their derivation to be filled with adventure and enigma. In An Imaginary Tale, Paul Nahin tells the 2000-year-old history of one of mathematics' most elusive numbers, the square root of minus one, also known as i. He recreates the baffling mathematical problems that conjured it up, and the colorful characters who tried to solve them.In 1878, when two brothers stole a mathematical papyrus from the ancient Egyptian burial site in the Valley of Kings, they led scholars to the earliest known occurrence of the square root of a negative number. The papyrus offered a specific numerical example of how to calculate the volume of a truncated square pyramid, which implied the need for i. In the first century, the mathematician-engineer Heron of Alexandria encountered I in a separate project, but fudged the arithmetic; medieval mathematicians stumbled upon the concept while grappling with the meaning of negative numbers, but dismissed their square roots as nonsense. By the time of Descartes, a theoretical use for these elusive square roots--now called \"imaginary numbers\"--was suspected, but efforts to solve them led to intense, bitter debates. The notorious i finally won acceptance and was put to use in complex analysis and theoretical physics in Napoleonic times.Addressing readers with both a general and scholarly interest in mathematics, Nahin weaves into this narrative entertaining historical facts and mathematical discussions, including the application of complex numbers and functions to important problems, such as Kepler's laws of planetary motion and ac electrical circuits. This book can be read as an engaging history, almost a biography, of one of the most evasive and pervasive \"numbers\" in all of mathematics.Some images inside the book are unavailable due to digital copyright restrictions.

NP-completeness of two clustering (partition) problems is proved for a finite sequence of Euclidean vectors. In the optimization versions of both problems it is required to partition the elements of the sequence into a fixed number of clusters minimizing the sum of squares of the distances from the cluster elements to their centers. In the first problem the sizes of clusters are the part of input, while in the second they are unknown (they are the variables for optimization). Except for the center of one (special) cluster, the center of each cluster is the mean value of all vectors contained in it. The center of the special cluster is zero. Also, the partition must satisfy the following condition: The difference between the indices of two consecutive vectors in every nonspecial cluster is bounded below and above by two given constants.

We consider a strongly NP-hard problem of partitioning a finite sequence of vectors in Euclidean space into two clusters using the criterion of the minimal sum of the squared distances from the elements of the clusters to the centers of the clusters. The center of one of the clusters is to be optimized and is determined as the mean value over all vectors in this cluster. The center of the other cluster is fixed at the origin. Moreover, the partition is such that the difference between the indices of two successive vectors in the first cluster is bounded above and below by prescribed constants. A 2-approximation polynomial-time algorithm is proposed for this problem.

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

This chapter contains sections titled: Introduction The Simplex S D as a Compositional Space Compositional Time Series Models CTS Modelling: An Example Discussion Acknowledgements References Appendix