“…To achieve the Vector Decomposition, linear algebra (matrix inversion) is used with the numeric algorithm for the Singular Value Decomposition. …”
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“…It assumes knowledge only of basic calculus, matrix algebra and elementary statistics. This third edition contains detailed instructions for the use of the professional version of the Windows-based computer package ITSM2000, now available as a free download from the Springer Extras website. …”
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“…Optimization without Calculus Chapter Summary The Arithmetic Mean-Geometric Mean Inequality An Application of the AGM Inequality: the Number e Extending the AGM Inequality Optimization Using the AGM Inequality The Holder and Minkowski Inequalities Cauchy's Inequality Optimizing using Cauchy's Inequality An Inner Product for Square Matrices Discrete Allocation Problems Geometric Programming Chapter Summary An Example of a GP Problem Posynomials and the GP Problem The Dual GP Problem Solving the GP Problem Solving the DGP Problem Constrained Geometric Programming Basic Analysis Chapter Summary Minima and Infima Limits Completeness Continuity Limsup and Liminf Another View Semi-Continuity Convex Sets Chapter SummaryThe Geometry of Real Euclidean Space A Bit of Topology Convex Sets in RJ More on Projections Linear and Affine Operators on RJ The Fundamental Theorems Block-Matrix Notation Theorems of the Alternative Another Proof of Farkas' Lemma Gordan's Theorem Revisited Vector Spaces and Matrices Chapter Summary Vector Spaces Basic Linear Algebra LU and QR Factorization The LU Factorization Linear Programming Chapter Summary Primal and Dual Converting a Problem to PS Form Duality Theorems The Basic Strong Duality Theorem Another Proof Proof of Gale's Strong Duality Theorem Some Examples The Simplex Method Yet Another Proof The Sherman-Morrison-Woodbury Identity An Example of the Simplex Method Another Example Some Possible Difficulties Topics for Projects Matrix Games and Optimization Chapter Summary Two-Person Zero-Sum Games Deterministic Solutions Randomized Solutions Symmetric Games Positive Games Example: The "Bluffing" Game Learning the Game Non-Constant-Sum Games Differentiation Chapter Summary Directional Derivative Partial Derivatives Some Examples Gâteaux Derivative Fréchet Derivative The Chain Rule Convex Functions Chapter Summary Functions of a Single Real Variable Functions of Several Real Variables Sub-Differentials and Sub-Gradients Sub-Gradients and Directional Derivatives Functions and Operators Convex Sets and Convex Functions Convex Programming Chapter Summary The Primal Problem From Constrained to Unconstrained Saddle Points The Karush-Kuhn-Tucker TheoremOn Existence of Lagrange Multipliers The Problem of Equality Constraints Two Examples The Dual Problem Nonnegative Least-Squares Solutions An Example in Image Reconstruction Solving the Dual Problem Minimum One-Norm Solutions Iterative OptimizationChapter Summary The Need for Iterative Methods Optimizing Functions of a Single Real Variable The Newton-Raphson Approach Approximate Newton-Raphson Methods Derivative-Free MethodsRates of Convergence Descent Methods Optimizing Functions of Several Real Variables Auxiliary-Function Methods Projected Gradient-Descent Methods Feasible-Point Methods Quadratic Programming Simulated Annealing Solving Systems of Linear Equations Chapter Summary Arbitrary Systems of Linear Equations Regularization Nonnegative Systems of Linear Equations Regularized SMART and EMML Block-Iterative Methods Conjugate-Direction MethodsChapter Summary Iterative Minimization Quadratic Optimization Conjugate Bases for RJ The Conjugate Gradient Method Krylov Subspaces Extensions of the CGM Operators Chapter Summary Operators Contraction Operators Orthogonal-Projection Operators Two Useful Identities Averaged OperatorsGradient Operators Affine-Linear OperatorsParacontractive Operators Matrix Norms Looking Ahead Chapter Summary Sequential Unconstrained Minimization Examples of SUM Auxiliary-Function Methods The SUMMA Class of AF Methods Bibliography Index Exercises appear at the end of each chapter.…”
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“…HISTORICAL EXPERIMENTS AND THEORIESDates of Important Discoveries and Events Blackbody RadiationPhotoelectrice Effect Quantum Theory of Spectra TheComptone Effect Matterwaves, the de Broglie HypothesisThe Davisson -Germer Experiment Heisenberg's Uncertainity PrincipleDifference Between Particles and Waves Interpretation of the Wavefunction AXIOMATIC STRUCTURE OF QUANTUM MECHANICSThe Necessity of Quantum TheoryFunction Spaces Postulates of Quantum Mechanics The Kronecker Delta and the Dirac Delta Function Dirac Notation OBSERVABLES AND SUPERPOSITIONFree Particle Particle In A Box Ensemble Average Hilbert -Space Interpretation The Initial Square Wave Particle Beam Superposition and Uncertainty Degeneracy of States Commutators and Uncertainty TIME DEVELOPMENT AND CONSERVATION THEOREMSTime Development of State Functions, The Discrete Case The Continuous Case, Wave Packets Particle Beam Gaussian Wave Packet Free Particle Propagator The Limiting Cases of the Gaussian Wave Packets Time Development of Expectation Values Conservation of Energy and MomentumConservation of Parity BOUND AND UNBOUND STATES IN ONE-DIMENSIONOne-Dimensional Schrödinger Equation The Simple Harmonic Oscillator Unbound States One-Dimensional Barrier Problems The Finite Potential Well N-PARTICLE SYSTEMSThe Schrödinger Equation for N-Particle Systems Identical Particles The Pauli Principle; Fermions and Bosons THE SCHRÖDINGER EQUATION IN THREE-DIMENSIONSThe Two-Body Systems Separation of Variables in the Two-Body Systems Rotational Invariance The Schrödinger Equation for Non-Central Potentials ANGULAR MOMENTUMCommutation Relations Raising and Lowering Operators Eigen Solutions of Angular Momentum Operators Kinetic Energy and Angular Momentum THE RADIAL EQUATION FOR FREE AND BOUND PARTICLESThe Radial Schrödinger Equation The Free Particle Three-Dimensional Square Well Potential The Hydrogenatom The Spectra of Hydrogenic Atoms The Virial Theorem INTERACTION OF ELECTRONS WITH ELECTROMAGNETIC FIELDMaxwell 's Equations and Gauge Transformations Motion of a Free Electron in a Uniform Magnetic Field Motion of a Bound Electron in a Uniform Magnetic Field The Principal of Gauge Invariance and Flux Quantization MATRIX REPRESENTATIONSMatrix Representations of Wave Functions and OperatorsMatrix Algebra Types of Matrix Representations Harmonic Oscillator in Matrix Representations Matrix Representations of Angular Momentum Operators SPIN AND THE ADDITION OF ANGULAR MOMENTASystems with Spin One-Half The Addition of Angular Momenta TIME-INDEPENDENT PERTURBATION THEORYNondegenerate Perturbation Theory Degenerate Perturbation Theory THE VARIATIONAL METHODThe Variational Principle Linear Variation Functions THE WKB APPROXIMATION Turning Points The Connection Formulas The WKB Approximation to a Potential Well The WKB Approximation to a Potential Barrier TIME-DEPENDENT PERTURBATION THEORYTime -Dependent Schrödinger Equation Time -Dependent Perturbation Approximations Sinusoidal Perturbations Emission and Absorption of Radiation Incoherent Perturbations Selection RulesTHE ADIABATIC APPROXIMATIONThe Adiabatic Processes The Adiabatic TheoremNonholonomic Processes Experimental Evidences of Nonholonomic Processes PATH-INTEGRATION METHODAn Approximation to Time -Evolution for a Free Particle Path Integral Evaluation of the Free Particle Propagator Equivalence to the Schrödinger Equation SCATTERING THEORYClassical Scattering TheoryCenter-of-Mass and Laboratory Frames Quantum Scattering Theory The Method of Partial waves (Low -Energy Case)Expansion of a Plane Wave into Spherical Waves Expansion of the Scattering Amplitude Scattering from a Delta Potential Scattering from a Square -Well Potential Scattering from a Hardsphere Scattering of Identical Particles Energy Dependence and Resonance Scattering The Lippman-Schwinger Equation (High-Energy Case)The Greens Function for the Scattering Problem Born Approximation INELASTIC SCATTERING Bibliography and References 409Appendix 413Index 417.…”
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“…The latter are pitched at a more formal, abstract level than the applications and, therefore, encompass a number of concepts drawn from the abstract theory of sets and relations, network theory, automata theory, Boolean and n-valued logics, abstract algebra, topology and category theory. The purpose of these theoretical sections is to provide the ans for approaching a number of basic biological questions: (1) What are the essential characteristics of a biological organism as opposed to an automaton? …”
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“…Thus, the foundation is based on linear algebra and reasonable physical assumptions, which provide numerous opportunities for experimental testing and further elaboration to deal with complex multicellular organisms that are currently beyond its scope. …”
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“…Genomic DNA from both parents, pedigree genotypes, unrelated potato varieties lacking the wart resistance traits and pools of resistant and susceptible siblings were sequenced. Set algebra and depth filtering of subsequences (k-mers) were used to delete unlinked and common SNPs and to enrich for SNPs from the haplotype(s) harboring the resistance gene(s). …”
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“…Our method—JacSketch—is motivated by novel developments in randomized numerical linear algebra, and operates by maintaining a stochastic estimate of a Jacobian matrix composed of the gradients of individual functions. …”
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“…The manuscript provides new information about the use of configurational analysis, which uses Boolean algebra to analyze pathways to implementation success considering each variable, within and across diverse clinical sites across the USA. …”
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“…In vitro, the effects of PI on bacterial growth were analyzed. 2.5%, 1.25%, 0.5%, 0.25%, 0.05%, 0.025%, 0.005%, 0.0025% and 0% PI was added into the bacterial suspension, besides, the bacterial algebra and growth rate were tested. Meanwhile, the fluorescence intensity of viable bacteria was also tested to evaluate the effects of PI on bacterial survival. …”
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“…Configurational analysis, based on Boolean algebra and set theory, represents a mathematical approach to data analysis well-suited for discerning how conditions interact and identifying multiple pathways leading to the same outcome. …”
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“…Based on the pseudophakic eye model, OOM and MOM were extracted using 4 × 4 matrix algebra for the corrected eye (with P(IOL)eq/P(IOL)cyl (scenario 1) or with P(IOL)eq and spectacle correction of the residual refractive cylinder (scenario 2) or with P(IOL)eq remaining the residual uncorrected refractive cylinder (blurry image) (scenario 3)). …”
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“…Image processing and analyses were conducted using Statistical Parametric Mapping (SPM8) software with the diffeomorphic anatomical registration through exponentiated Lie algebra (DARTEL) algorithm. Group effects on regional gray matter (GM) volumes were investigated through whole-brain voxel-wise comparisons using General Linear Model Analysis of Co-variance (ANCOVA), always including total GM volume, scan protocol, age and gender as nuisance variables. …”
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“…Following graduation in 1969, he was hired as a teacher Hamden Hall Country Day School in Connecticut where he taught high school chemistry, algebra, and social sciences for 2 years. After his first year of teaching, he decided to give research another try. …”
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