“…REFEREN CES . 156 9 Transforma.tion of a Boundary Value Problem to an Initial Value Problem . 157 9.0 Introduction . 157 9.1 Blasius Equation in Boundary Layer Flow . 157 9.2 Longitudinal Impact of Nonlinear Viscoplastic Rods . 163 9.3 Summary . 168 REFERENCES . . . . . . . . . . . . . . . . . . 168 . 10 From Nonlinear to Linear Differential Equa.tions Using Transformation Groups. . . . . . . . . . . . . . 169 . 10.1 From Nonlinear to Linear Differential Equations . 170 10.2 Application to Ordinary Differential Equations -Bernoulli's Equation . . . . . . . . . . . 173 10.3 Application to Partial Differential Equations -A Nonlinear Chemical Exchange Process . 178 10.4 Limitations of the Inspectional Group Method . 187 10.5 Summary . 188 REFERENCES . . . . 188 11 Miscellaneous Topics . 190 11.1 Reduction of Differential Equations to Algebraic Equations 190 11.2 Reduction of Order of an Ordinary Differential Equation . 191 11.3 Transformat.ion From Ordinary to Partial Differential Equations-Search for First Integrals . . . . . . " 193 . 11.4 Reduction of Number of Variables by Multiparameter Groups of Transformations . . . . . . . . .. . . . 194 11.5 Self-Similar Solutions of the First and Second Kind . . 202 11.6 Normalized Representation and Dimensional Consideration 204 REFERENCES .206 Problems . 208 .220 Index .. …”
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“…This book explains some recent applications of the theory of polynomials and algebraic geometry to combinatorics and other areas of mathematics. …”
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“…Specifically, we consider the stochastic linear quadratic optimal control problem with different forms of noise disturbances, operator differential algebraic equations arising in fluid dynamics, stationary equations and fractional versions of the equations studied – applications never covered in the extant literature. …”
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“…This monograph provides a unified, self-contained introduction to the theory and its applications, making it accessible to a student who has completed a first course in algebraic topology and manifold theory. The book can be used as a source for research projects for master's students, as a thorough introduction to the field for graduate students, and as a valuable landmark text for researchers, providing both the details of the theory of bounded cohomology and links of the theory to other closely related areas. …”
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“…In the most popular and most rapidly growing areas (the theory of random graphs, Ramsey theory, extremal graph theory, algebraic graph theory, etc.), graphs were considered as abstract binary relations rather than geometric objects. …”
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“…The book presents inference methods based on the algebraic nature of the A-hypergeometric system, and introduces the holonomic gradient methods, which numerically solve holonomic systems without combinatorial enumeration, to compute the normalizing constant. …”
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“…Applications of the contraction principle include the proofs of algebraic Weierstrass preparation theorem, a Cauchy–Kowalevsky theorem for partial differential equations and the central limit theorem. …”
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“…Substituting the obtained forces into the beam-envelope equations establishes a set of six modified equations describing the beam dynamics using simple algebraic expressions. The whole solution considers the energy spread inherent to an electron beam. …”
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“…The former are dual to the asymptotically flat Kerr-Taub-NUT family, while the latter leads to the homonymous class of algebraically special Ricci-flat spacetimes.…”
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“…It searches equation space to discover algebraic relations approximating a dataset. We use PySR (software for uncovering these expressions based on evolutionary algorithm) and extend the functionality of hls4ml (a package for machine learning inference in FPGAs) to support PySR-generated expressions for resource-constrained production environments. …”
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“…The usefulness of the constructed LMT-ANNs for such a modelled issue is demonstrated by the best promising algebraic outputs in the E−03 to E−08 range, as well as error histogram and regression analysis measures. …”
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“…As in the original study, we tested the hypothesis that infants are able to learn abstract “algebraic” rules and apply them to novel input. Our results did not replicate the original findings: infants showed no difference in looking time between test patterns consistent or inconsistent with the familiarization pattern they were exposed to.…”
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“…The proposed technology uses an innovative data acquisition and reconstruction approach by integrating developments in field cycling, parallel imaging and non-Fourier based algebraic reconstruction. The scanner uses field cycling to image in an inhomogeneous B(0)-field; in this way magnetization is maximized during the high field polarization phase, and B(0) inhomogeneity effects are minimized by using a low field during image acquisition. …”
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“…The proposed methods convert solving the original problems into solving systems of algebraic equations. RESULTS: The accuracy and convergence order of the proposed methods are examined by solving several examples. …”
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“…Utilizing the density fitting approximation, efficient implementations are presented for the genuine DH ansatz relying on the perturbative second-order correction, while an iterative analogue is also elaborated using our second-order algebraic-diagrammatic construction [ADC(2)]-based DH approach. …”
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“…Third, multiple permutation and diffusion operations are conducted on different dimensions with dynamic eight-base DNA-level encoding and algebraic operation rules determined via the hyperchaotic sequence. …”
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“…We show that, under extendability conditions, those pre-Dynkin systems equipped with probabilities can be embedded into algebras of sets. Surprisingly, the extendability conditions elaborated in a strand of work in quantum probability are equivalent to coherence from the imprecise probability literature. …”
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“…The generated S-box's statistical and algebraic resilience is demonstrated by its very low linear probability and differential probability scores of 0.125 and 0.039, respectively, and a high non-linearity score of 111.5. …”
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“…Our framework supports general energy models, partition function computations, recursive substructures and partial folding, and could pave the way for algebraic dynamic programming beyond the context-free case.…”
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“…Under this view, an algebraic binary form of the degree n corresponds to a chemical atom with valence n and each physical molecule or ion has an invariant-theoretic counterpart. …”
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