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Für inhomogene Poissonprozesse mit konvexen Partikeln wird eine Klasse von Maßen auf der Einheitssphäre definiert. Nach geeigneter Normierung kann jedes dieser Maße als Verteilung der äußeren Normalen der Partikel des zugrundeliegenden Prozesses interpretiert werden. Wir benutzen diese Maße nun, um Stützfunktionen von Zonoiden zu definieren, deren innere Volumina dann als Dichten spezieller Schnittprozesse interpretiert werden können.

Poisson process.

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Newton foresaw the limitations of geometry’s description of planetary behavior and developed fluxions (differentials) as the new language for celestial mechanics and as the way to implement his laws of mechanics. Two hundred years later Mandelbrot introduced the notion of fractals into the scientific lexicon of geometry, dynamics, and statistics and in so doing suggested ways to see beyond the limitations of Newton’s laws. Mandelbrot’s mathematical essays suggest how fractals may lead to the understanding of turbulence, viscoelasticity, and ultimately to end of dominance of the Newton’s macroscopic world view.Fractional Calculus and the Future of Science examines the nexus of these two game-changing contributions to our scientific understanding of the world. It addresses how non-integer differential equations replace Newton’s laws to describe the many guises of complexity, most of which lay beyond Newton’s experience, and many had even eluded Mandelbrot’s powerful intuition. The book’s authors look behind the mathematics and examine what must be true about a phenomenon’s behavior to justify the replacement of an integer-order with a noninteger-order (fractional) derivative. This window into the future of specific science disciplines using the fractional calculus lens suggests how what is seen entails a difference in scientific thinking and understanding.

fractional diffusion --- continuous time random walks --- reaction–diffusion equations --- reaction kinetics --- multidimensional scaling --- fractals --- fractional calculus --- financial indices --- entropy --- Dow Jones --- complex systems --- Skellam process --- subordination --- Lévy measure --- Poisson process of order k --- running average --- complexity --- chaos --- logistic differential equation --- liouville-caputo fractional derivative --- local discontinuous Galerkin methods --- stability estimate --- Mittag-Leffler functions --- Wright functions --- fractional relaxation --- diffusion-wave equation --- Laplace and Fourier transform --- fractional Poisson process complex systems --- distributed-order operators --- viscoelasticity --- transport processes --- control theory --- fractional order PID control --- PMSM --- frequency-domain control design --- optimal tuning --- Gaussian watermarks --- statistical assessment --- false positive rate --- semi-fragile watermarking system --- fractional dynamics --- fractional-order thinking --- heavytailedness --- big data --- machine learning --- variability --- diversity --- telegrapher’s equations --- fractional telegrapher’s equation --- continuous time random walk --- transport problems --- fractional conservations laws --- variable fractional model --- turbulent flows --- fractional PINN --- physics-informed learning --- n/a --- reaction-diffusion equations --- Lévy measure --- telegrapher's equations --- fractional telegrapher's equation

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This monograph concerns the mathematical aspects of software reliability and quality assurance and consists of 11 technical papers in this emerging area. Included are the latest research results related to formal methods and design, automatic software testing, software verification and validation, coalgebra theory, automata theory, hybrid system and software reliability modeling and assessment.

Technology: general issues --- History of engineering & technology --- fuzzy automata --- coalgebra --- fuzzy language --- bisimulation --- composition --- test data generation --- genetic algorithm --- specification-based testing --- regression testing --- mutation testing --- eventual property --- model checking --- Maude --- textual question answering --- visual question answering --- metamorphic testing --- metamorphic relations --- quality assessment --- software rejuvenation --- checkpointing --- optimal rejuvenation-trigger timing --- steady-state system availability --- phase expansion --- human-error factors --- petri net --- concurrent software systems --- model-checking --- data-flows --- software reliability model --- maximum likelihood estimation --- EM algorithm --- non-homogeneous Poisson process --- generalized failure count data --- moth flame optimization --- island-based model --- feature selection --- software defect prediction --- software reliability --- search-based test case generation --- branch coverage --- object-oriented --- deep learning --- long short-term memory --- project similarity and clustering --- cross-project prediction --- Nervos CKB --- consensus protocol --- UPPAAL --- fuzzy automata --- coalgebra --- fuzzy language --- bisimulation --- composition --- test data generation --- genetic algorithm --- specification-based testing --- regression testing --- mutation testing --- eventual property --- model checking --- Maude --- textual question answering --- visual question answering --- metamorphic testing --- metamorphic relations --- quality assessment --- software rejuvenation --- checkpointing --- optimal rejuvenation-trigger timing --- steady-state system availability --- phase expansion --- human-error factors --- petri net --- concurrent software systems --- model-checking --- data-flows --- software reliability model --- maximum likelihood estimation --- EM algorithm --- non-homogeneous Poisson process --- generalized failure count data --- moth flame optimization --- island-based model --- feature selection --- software defect prediction --- software reliability --- search-based test case generation --- branch coverage --- object-oriented --- deep learning --- long short-term memory --- project similarity and clustering --- cross-project prediction --- Nervos CKB --- consensus protocol --- UPPAAL

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This monograph concerns the mathematical aspects of software reliability and quality assurance and consists of 11 technical papers in this emerging area. Included are the latest research results related to formal methods and design, automatic software testing, software verification and validation, coalgebra theory, automata theory, hybrid system and software reliability modeling and assessment.

fuzzy automata --- coalgebra --- fuzzy language --- bisimulation --- composition --- test data generation --- genetic algorithm --- specification-based testing --- regression testing --- mutation testing --- eventual property --- model checking --- Maude --- textual question answering --- visual question answering --- metamorphic testing --- metamorphic relations --- quality assessment --- software rejuvenation --- checkpointing --- optimal rejuvenation-trigger timing --- steady-state system availability --- phase expansion --- human-error factors --- petri net --- concurrent software systems --- model-checking --- data-flows --- software reliability model --- maximum likelihood estimation --- EM algorithm --- non-homogeneous Poisson process --- generalized failure count data --- moth flame optimization --- island-based model --- feature selection --- software defect prediction --- software reliability --- search-based test case generation --- branch coverage --- object-oriented --- deep learning --- long short-term memory --- project similarity and clustering --- cross-project prediction --- Nervos CKB --- consensus protocol --- UPPAAL

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Encompassing both introductory and more advanced research material, these notes deal with the author's contributions to stochastic processes and focus on Brownian motion processes and its derivative white noise.Originally published in 1970.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Stationary processes --- Stationary processes. --- Stochastic processes --- 519.216 --- 519.216 Stochastic processes in general. Prediction theory. Stopping times. Martingales --- Stochastic processes in general. Prediction theory. Stopping times. Martingales --- Bochner integral. --- Bochner's theorem. --- Bounded operator. --- Bounded variation. --- Brownian motion. --- Characteristic exponent. --- Characteristic function (probability theory). --- Complexification. --- Compound Poisson process. --- Computation. --- Conditional expectation. --- Continuous function (set theory). --- Continuous function. --- Continuous linear operator. --- Convergence of random variables. --- Coset. --- Covariance function. --- Cyclic subspace. --- Cylinder set. --- Degrees of freedom (statistics). --- Derivative. --- Differential equation. --- Dimension (vector space). --- Dirac delta function. --- Discrete spectrum. --- Distribution function. --- Dual space. --- Eigenfunction. --- Equation. --- Existential quantification. --- Exponential distribution. --- Exponential function. --- Finite difference. --- Fourier series. --- Fourier transform. --- Function (mathematics). --- Function space. --- Gaussian measure. --- Gaussian process. --- Harmonic analysis. --- Hermite polynomials. --- Hilbert space. --- Homeomorphism. --- Independence (probability theory). --- Independent and identically distributed random variables. --- Indicator function. --- Infinitesimal generator (stochastic processes). --- Integral equation. --- Isometry. --- Joint probability distribution. --- Langevin equation. --- Lebesgue measure. --- Lie algebra. --- Limit superior and limit inferior. --- Linear combination. --- Linear function. --- Linear interpolation. --- Linear subspace. --- Mean squared error. --- Measure (mathematics). --- Monotonic function. --- Normal distribution. --- Normal subgroup. --- Nuclear space. --- One-parameter group. --- Orthogonality. --- Orthogonalization. --- Parameter. --- Poisson point process. --- Polynomial. --- Probability distribution. --- Probability measure. --- Probability space. --- Probability. --- Projective linear group. --- Radon–Nikodym theorem. --- Random function. --- Random variable. --- Reproducing kernel Hilbert space. --- Self-adjoint operator. --- Self-adjoint. --- Semigroup. --- Shift operator. --- Special case. --- Stable process. --- Stationary process. --- Stochastic differential equation. --- Stochastic process. --- Stochastic. --- Subgroup. --- Summation. --- Symmetrization. --- Theorem. --- Transformation semigroup. --- Unitary operator. --- Unitary representation. --- Unitary transformation. --- Variance. --- White noise. --- Zero element.