LEADER 06948nam 2200589 450 001 996495167103316 005 20230809170104.0 010 $a9783031142055$b(electronic bk.) 010 $z9783031142048 035 $a(MiAaPQ)EBC7127772 035 $a(Au-PeEL)EBL7127772 035 $a(CKB)25219376900041 035 $a(PPN)26585640X 035 $a(EXLCZ)9925219376900041 100 $a20230317d2022 uy 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aMeasure theory, probability, and stochastic processes /$fJean-Franc?ois Le Gall 210 1$aCham, Switzerland :$cSpringer,$d[2022] 210 4$d©2022 215 $a1 online resource (409 pages) 225 1 $aGraduate texts in mathematics ;$vVolume 295 311 08$aPrint version: 9783031142048 320 $aIncludes bibliographical references (pages 401-402) and index. 327 $aIntro -- Preface -- Contents -- List of Symbols -- Part I Measure Theory -- 1 Measurable Spaces -- 1.1 Measurable Sets -- 1.2 Positive Measures -- 1.3 Measurable Functions -- Operations on Measurable Functions -- 1.4 Monotone Class -- 1.5 Exercises -- 2 Integration of Measurable Functions -- 2.1 Integration of Nonnegative Functions -- 2.2 Integrable Functions -- 2.3 Integrals Depending on a Parameter -- 2.4 Exercises -- 3 Construction of Measures -- 3.1 Outer Measures -- 3.2 Lebesgue Measure -- 3.3 Relation with Riemann Integrals -- 3.4 A Subset of ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark Which Is Not Measurable -- 3.5 Finite Measures on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper R) /StPNE pdfmark [/StBMC pdfmarkRps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark and the Stieltjes Integral -- 3.6 The Riesz-Markov-Kakutani Representation Theorem -- 3.7 Exercises -- 4 Lp Spaces -- 4.1 Definitions and the Hölder Inequality -- 4.2 The Banach Space ps: [/EMC pdfmark [/Subtype /Span /ActualText (upper L Superscript p Baseline left parenthesis upper E comma script upper A comma mu right parenthesis) /StPNE pdfmark [/StBMC pdfmarkLp(E,A,?)ps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 4.3 Density Theorems in Lp Spaces -- 4.4 The Radon-Nikodym Theorem -- 4.5 Exercises -- 5 Product Measures -- 5.1 Product ?-Fields -- 5.2 Product Measures -- 5.3 The Fubini Theorems -- 5.4 Applications -- 5.4.1 Integration by Parts -- 5.4.2 Convolution -- 5.4.3 The Volume of the Unit Ball -- 5.5 Exercises -- 6 Signed Measures -- 6.1 Definition and Total Variation -- 6.2 The Jordan Decomposition -- 6.3 The Duality Between Lp and Lq -- 6.4 The Riesz-Markov-Kakutani Representation Theorem for Signed Measures -- 6.5 Exercises. 327 $a7 Change of Variables -- 7.1 The Change of Variables Formula -- 7.2 The Gamma Function -- 7.3 Lebesgue Measure on the Unit Sphere -- 7.4 Exercises -- Part II Probability Theory -- 8 Foundations of Probability Theory -- 8.1 General Definitions -- 8.1.1 Probability Spaces -- 8.1.2 Random Variables -- 8.1.3 Mathematical Expectation -- 8.1.4 An Example: Bertrand's Paradox -- 8.1.5 Classical Laws -- 8.1.6 Distribution Function of a Real Random Variable -- 8.1.7 The ?-Field Generated by a Random Variable -- 8.2 Moments of Random Variables -- 8.2.1 Moments and Variance -- 8.2.2 Linear Regression -- 8.2.3 Characteristic Functions -- 8.2.4 Laplace Transform and Generating Functions -- 8.3 Exercises -- 9 Independence -- 9.1 Independent Events -- 9.2 Independence for ?-Fields and Random Variables -- 9.3 The Borel-Cantelli Lemma -- 9.4 Construction of Independent Sequences -- 9.5 Sums of Independent Random Variables -- 9.6 Convolution Semigroups -- 9.7 The Poisson Process -- 9.8 Exercises -- 10 Convergence of Random Variables -- 10.1 The Different Notions of Convergence -- 10.2 The Strong Law of Large Numbers -- 10.3 Convergence in Distribution -- 10.4 Two Applications -- 10.4.1 The Convergence of Empirical Measures -- 10.4.2 The Central Limit Theorem -- 10.4.3 The Multidimensional Central Limit Theorem -- 10.5 Exercises -- 11 Conditioning -- 11.1 Discrete Conditioning -- 11.2 The Definition of Conditional Expectation -- 11.2.1 Integrable Random Variables -- 11.2.2 Nonnegative Random Variables -- 11.2.3 The Special Case of Square Integrable Variables -- 11.3 Specific Properties of the Conditional Expectation -- 11.4 Evaluation of Conditional Expectation -- 11.4.1 Discrete Conditioning -- 11.4.2 Random Variables with a Density -- 11.4.3 Gaussian Conditioning -- 11.5 Transition Probabilities and Conditional Distributions -- 11.6 Exercises. 327 $aPart III Stochastic Processes -- 12 Theory of Martingales -- 12.1 Definitions and Examples -- 12.2 Stopping Times -- 12.3 Almost Sure Convergence of Martingales -- 12.4 Convergence in Lp When p> -- 1 -- 12.5 Uniform Integrability and Martingales -- 12.6 Optional Stopping Theorems -- 12.7 Backward Martingales -- 12.8 Exercises -- 13 Markov Chains -- 13.1 Definitions and First Properties -- 13.2 A Few Examples -- 13.2.1 Independent Random Variables -- 13.2.2 Random Walks on ps: [/EMC pdfmark [/Subtype /Span /ActualText (double struck upper Z Superscript d) /StPNE pdfmark [/StBMC pdfmarkZdps: [/EMC pdfmark [/StPop pdfmark [/StBMC pdfmark -- 13.2.3 Simple Random Walk on a Graph -- 13.2.4 Galton-Watson Branching Processes -- 13.3 The Canonical Markov Chain -- 13.4 The Classification of States -- 13.5 Invariant Measures -- 13.6 Ergodic Theorems -- 13.7 Martingales and Markov Chains -- 13.8 Exercises -- 14 Brownian Motion -- 14.1 Brownian Motion as a Limit of Random Walks -- 14.2 The Construction of Brownian Motion -- 14.3 The Wiener Measure -- 14.4 First Properties of Brownian Motion -- 14.5 The Strong Markov Property -- 14.6 Harmonic Functions and the Dirichlet Problem -- 14.7 Harmonic Functions and Brownian Motion -- 14.8 Exercises -- A A Few Facts from Functional Analysis -- Normed Linear Spaces and Banach Spaces -- Hilbert Spaces -- Notes and Suggestions for Further Reading -- References -- Index. 410 0$aGraduate texts in mathematics ;$vVolume 295. 606 $aMeasure theory 606 $aProbabilities 606 $aStochastic processes 606 $aTeoria de la mesura$2thub 606 $aProbabilitats$2thub 606 $aProcessos estocàstics$2thub 608 $aLlibres electrònics$2thub 615 0$aMeasure theory. 615 0$aProbabilities. 615 0$aStochastic processes. 615 7$aTeoria de la mesura 615 7$aProbabilitats 615 7$aProcessos estocàstics 676 $a515.42 700 $aLe Gall$b J. F$g(Jean-Franc?ois),$0348889 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a996495167103316 996 $aMeasure Theory, Probability, and Stochastic Processes$92963436 997 $aUNISA LEADER 05794nam 22007215 450 001 9910254601803321 005 20200705132837.0 010 $a3-319-64316-9 024 7 $a10.1007/978-3-319-64316-8 035 $a(CKB)3710000001631595 035 $a(DE-He213)978-3-319-64316-8 035 $a(MiAaPQ)EBC6312780 035 $a(MiAaPQ)EBC5588959 035 $a(Au-PeEL)EBL5588959 035 $a(OCoLC)1001463317 035 $a(PPN)203853857 035 $a(EXLCZ)993710000001631595 100 $a20170807d2017 u| 0 101 0 $aeng 135 $aurnn|008mamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 14$aThe Story of Light Science $eFrom Early Theories to Today's Extraordinary Applications /$fby Dennis F. Vanderwerf 205 $a1st ed. 2017. 210 1$aCham :$cSpringer International Publishing :$cImprint: Springer,$d2017. 215 $a1 online resource (XIV, 332 p. 218 illus., 5 illus. in color.) 311 $a3-319-64315-0 327 $aEmergent Theories of Light and Measurements of Light Speed -- Light as an Electromagnetic Wave -- Light and its Application to Relativity -- The Quantum Nature of Light -- Natural and Artificial Sources of Light -- Laser Light -- Variation and Control of Light Propagation Properties -- Quantum Mechanics of the Photon -- Quantum Applications of the Photon -- Light and the Cosmos -- Lorentz contraction of a modern spacecraft -- A derivation of E = mc2 -- Time dilation and muon lifetime calculations -- Derivation of Wien?s displacement law from Planck?s law -- Combustion-based light sources -- Multiple-laser white light illumination -- Circuit model calculations for Deutsch?s algorithm -- Timeline of some notable Achievements in Light Science -- A Selection of Additional Readings. 330 $aThis book traces the evolution of our understanding and utilization of light from classical antiquity and the early thoughts of Pythagoras to the present time. From the earliest recorded theories and experiments to the latest applications in photonic communication and computation, the ways in which light has been put to use are numerous and astounding. Indeed, some of the latest advances in light science are in fields that until recently belonged to the realm of science fiction. The author, writing for an audience of both students and other scientifically interested readers, describes fundamental investigations of the nature of light and ongoing methods to measure its speed as well as the emergence of the wave theory of light and the complementary photon theory. The importance of light in the theory of relativity is discussed as is the development of electrically-driven light sources and lasers. The information here covers the range of weak single-photon light sources to super-high power lasers and synchrotron light sources. Many cutting-edge topics are also introduced, including entanglement-based quantum communication through optical fibers and free space, quantum teleportation, and quantum computing. The nature and use of "squeezed light" - e.g. for gravitational wave detection - is another fascinating excursion, as is the topic of fabricated metamaterials, as used to create invisibility cloaks. Here the reader also learns about the realization of extremely slow speed and time-reversed light. The theories, experiments, and applications described in this book are, whenever possible, derived from original references. The many annotated drawings and level of detail make clear the goals, procedures, and conclusions of the original investigators. Where they are required, all specialist terms and mathematical symbols are defined and explained. The final part of the book covers light experi ments in the free space of the cosmos, and also speculates about scenarios for the cosmological origins of light and the expected fate of the photon in a dying universe. The final part of the book covers light experiments in the free space of the cosmos, and also speculates about scenarios for the cosmological origins of light and the expected fate of the photon in a dying universe. 606 $aLasers 606 $aPhotonics 606 $aQuantum computers 606 $aSpintronics 606 $aPhysics 606 $aMicrowaves 606 $aOptical engineering 606 $aOptics, Lasers, Photonics, Optical Devices$3https://scigraph.springernature.com/ontologies/product-market-codes/P31030 606 $aQuantum Information Technology, Spintronics$3https://scigraph.springernature.com/ontologies/product-market-codes/P31070 606 $aPopular Science in Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/Q29000 606 $aHistory and Philosophical Foundations of Physics$3https://scigraph.springernature.com/ontologies/product-market-codes/P29000 606 $aMicrowaves, RF and Optical Engineering$3https://scigraph.springernature.com/ontologies/product-market-codes/T24019 615 0$aLasers. 615 0$aPhotonics. 615 0$aQuantum computers. 615 0$aSpintronics. 615 0$aPhysics. 615 0$aMicrowaves. 615 0$aOptical engineering. 615 14$aOptics, Lasers, Photonics, Optical Devices. 615 24$aQuantum Information Technology, Spintronics. 615 24$aPopular Science in Physics. 615 24$aHistory and Philosophical Foundations of Physics. 615 24$aMicrowaves, RF and Optical Engineering. 676 $a535 700 $aVanderwerf$b Dennis F$4aut$4http://id.loc.gov/vocabulary/relators/aut$0825060 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910254601803321 996 $aThe Story of Light Science$91974906 997 $aUNINA