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This book provides the first comprehensive treatment of Benford's law, the surprising logarithmic distribution of significant digits discovered in the late nineteenth century. Establishing the mathematical and statistical principles that underpin this intriguing phenomenon, the text combines up-to-date theoretical results with overviews of the law's colorful history, rapidly growing body of empirical evidence, and wide range of applications.An Introduction to Benford's Law begins with basic facts about significant digits, Benford functions, sequences, and random variables, including tools from the theory of uniform distribution. After introducing the scale-, base-, and sum-invariance characterizations of the law, the book develops the significant-digit properties of both deterministic and stochastic processes, such as iterations of functions, powers of matrices, differential equations, and products, powers, and mixtures of random variables. Two concluding chapters survey the finitely additive theory and the flourishing applications of Benford's law.Carefully selected diagrams, tables, and close to 150 examples illuminate the main concepts throughout. The text includes many open problems, in addition to dozens of new basic theorems and all the main references. A distinguishing feature is the emphasis on the surprising ubiquity and robustness of the significant-digit law. This text can serve as both a primary reference and a basis for seminars and courses.

Distribution (Probability theory) --- Probability measures. --- Measures, Normalized --- Measures, Probability --- Normalized measures --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Bayesian models. --- Benford distribution. --- Benford distributions. --- Benford functions. --- Benford sequences. --- Benford's law. --- First-digit law. --- Significant-digit law. --- base-invariance property. --- computations. --- computer science. --- diagnostics. --- differential equations. --- exponential growth. --- finitely additive probability. --- fraud detection. --- functions. --- geometric Brownian motion. --- linear processes. --- logarithmic distribution. --- mathematical hypotheses. --- mathematical theory. --- multi-dimensional models. --- natural phenomena. --- one-dimensional deterministic systems. --- one-dimensional difference. --- pedagogical tool. --- polynomial growth. --- random matrices. --- random processes. --- random variables. --- scale-invariance property. --- sequences. --- significand functions. --- significands. --- significant digits. --- statistical distribution. --- stochastic process. --- sum-invariance property. --- superexponential growth. --- surveys. --- uniform distribution.

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Precipitation is a well-recognized pillar in the global water and energy balances. The accurate and timely understanding of its characteristics at the global, regional and local scales is indispensable for a clearer insight on the mechanisms underlying the Earth’s atmosphere-ocean complex system. Precipitation is one of the elements that is documented to be greatly affected by climate change. In its various forms, precipitation comprises the primary source of freshwater, which is vital for the sustainability of almost all human activities. Its socio-economic significance is fundamental in managing this natural resource effectively, in applications ranging from irrigation to industrial and household usage. Remote sensing of precipitation is pursued through a broad spectrum of continuously enriched and upgraded instrumentation, embracing sensors which can be ground-based (e.g., weather radars), satellite-borne (e.g., passive or active space-borne sensors), underwater (e.g., hydrophones), aerial, or ship-borne. This volume hosts original research contributions on several aspects of remote sensing of precipitation, including applications which embrace the use of remote sensing in tackling issues such as precipitation estimation, seasonal characteristics of precipitation and frequency analysis, assessment of satellite precipitation products, storm prediction, rain microphysics and microstructure, and the comparison of satellite and numerical weather prediction precipitation products.

Research & information: general --- Northern China --- raindrop size distribution (DSD) --- microphysical processes --- quantitative precipitation estimation (QPE) --- satellite-based precipitation --- elevation --- extreme events --- IMERG-V05B and V06A --- MSWEP --- ERA5 --- SM2RAIN --- precipitation estimation --- soil moisture --- SM2RAIN-CCI --- SM2RAIN-ASCAT --- multi-satellite precipitation analysis (TMPA) --- error decomposition --- complex topography --- diverse climate --- gauge data --- IMERG --- TAHMO --- morphing --- field displacement --- TIGGE --- precipitation --- numerical weather prediction --- satellite --- flood --- spring 2019 --- Iran --- GPM IMERG --- satellite precipitation --- spatiotemporal analysis --- statistical distribution --- validation --- Mainland China --- GSMaP_NRT --- GSMaP_Gauge_NRT --- raindrop size distribution --- radar reflectivity --- raindrop spectrometer --- semi-arid area --- assessment --- Taiwan --- data assimilation --- WRF model --- high-impact rainfall events --- GNSS ZTD --- optimum interpolation --- geographically weighted regression --- downscaling --- Tianshan Mountains --- satellite precipitation products --- evaluation --- daily rainfall --- hourly rainfall --- GPM --- TRMM --- GNSS --- GNSS antenna --- receiver antenna calibration --- relative calibration --- Phase Center Variation --- U-blox --- goGPS --- Zenith Tropospheric Delay --- ZED-F9P --- GSMaP --- Nepal --- cloud radar --- thunderstorm --- LDR --- hydrometeor --- hydrometeor classification --- lightning --- discharge --- remote sensing --- SEVIRI --- ground radar --- precipitation interpolation --- geographically and temporally weighted regression --- time weight function --- geographically and temporally weighted regression kriging --- extreme rainfall --- polarimetric radar signatures --- quantitative precipitation estimation --- southern china --- reanalysis --- linear trends --- mainland China --- EDBF algorithm --- geospatial predictor --- spatial pattern --- weighted precipitation --- Cyprus --- bias correction --- object-based method --- storm events --- Thies --- disdrometer --- weather circulations --- convective --- stratiform --- rain spectra --- radar reflectivity–rain rate relationship --- gridded precipitation products --- abrupt changes --- trends --- statistical indicators --- agriculture --- Pakistan --- rainfall --- radar --- extreme precipitation --- spatial bootstrap --- Louisiana --- annual maxima --- n/a --- radar reflectivity-rain rate relationship

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PART I. CHILD NEUROPSYCHOLOGY: CURRENT STATUS 1. Introduction PART II. CLINICAL ISSUES 2. Behavioral Assessment 3. Communicating Results: The Interpretive Session and the Written Report PART III. DOMAINS AND TESTS 4. Preliminary Assessment and Classification Scales 5. Intelligence Testing: General Considerations 6. Executive Function 7. Attention 8. Language 9. Motor and Sensory-Perceptual Evaluation 10. Visuoperceptual, Visuospatial, and Visuoconstructional Function 11. Learning and Memory.

Statistical Distributions --- Child --- Mental Disorders --- Child Development --- Brain Diseases --- Neuropsychological Tests --- Psychiatry and Psychology --- Statistics as Topic --- Mathematical Concepts --- Age Groups --- Psychological Tests --- Human Development --- Central Nervous System Diseases --- Phenomena and Processes --- Growth and Development --- Behavioral Disciplines and Activities --- Epidemiologic Methods --- Nervous System Diseases --- Behavior and Behavior Mechanisms --- Persons --- Health Care Evaluation Mechanisms --- Investigative Techniques --- Physiological Processes --- Public Health --- Named Groups --- Quality of Health Care --- Diseases --- Analytical, Diagnostic and Therapeutic Techniques and Equipment --- Health Care Quality, Access, and Evaluation --- Environment and Public Health --- Physiological Phenomena --- Health Care --- Pediatrics --- Medicine --- Health & Biological Sciences --- Physiological Concepts --- Physiological Phenomenon --- Physiological Process --- Concept, Physiological --- Concepts, Physiological --- Phenomena, Physiological --- Phenomenas, Physiological --- Phenomenon, Physiological --- Physiological Concept --- Process, Physiological --- Processes, Physiological --- Investigative Technics --- Investigative Technic --- Investigative Technique --- Technic, Investigative --- Technics, Investigative --- Technique, Investigative --- Techniques, Investigative --- Healthcare Evaluation Mechanisms --- Evaluation Mechanism, Healthcare --- Evaluation Mechanisms, Healthcare --- Healthcare Evaluation Mechanism --- Mechanism, Healthcare Evaluation --- Mechanisms, Healthcare Evaluation --- Person --- Community-Based Distribution --- Contraceptive Distribution --- Delivery of Healthcare --- Dental Care Delivery --- Distribution, Non-Clinical --- Distribution, Nonclinical --- Distributional Activities --- Healthcare --- Healthcare Delivery --- Healthcare Systems --- Non-Clinical Distribution --- Nonclinical Distribution --- Delivery of Dental Care --- Health Care Delivery --- Health Care Systems --- Activities, Distributional --- Activity, Distributional --- Care, Health --- Community Based Distribution --- Community-Based Distributions --- Contraceptive Distributions --- Deliveries, Healthcare --- Delivery, Dental Care --- Delivery, Health Care --- Delivery, Healthcare --- Distribution, Community-Based --- Distribution, Contraceptive --- Distribution, Non Clinical --- Distributional Activity --- Distributions, Community-Based --- Distributions, Contraceptive --- Distributions, Non-Clinical --- Distributions, Nonclinical --- Health Care System --- Healthcare Deliveries --- Healthcare System --- Non Clinical Distribution --- Non-Clinical Distributions --- Nonclinical Distributions --- System, Health Care --- System, Healthcare --- Systems, Health Care --- Systems, Healthcare --- Healthcare Quality, Access, and Evaluation --- Pharmacy Audit --- Quality of Care --- Quality of Healthcare --- Audit, Pharmacy --- Care Quality --- Health Care Quality --- Healthcare Quality --- Pharmacy Audits --- Community Health --- Environment, Preventive Medicine & Public Health --- Environment, Preventive Medicine and Public Health --- Health, Community --- Health, Public --- Preventive Medicine --- Education, Public Health Professional --- Behavior And Behavior Mechanism --- Nervous System Disorders --- Neurological Disorders --- Neurologic Disorders --- Disease, Nervous System --- Diseases, Nervous System --- Disorder, Nervous System --- Disorder, Neurologic --- Disorder, Neurological --- Disorders, Nervous System --- Disorders, Neurologic --- Disorders, Neurological --- Nervous System Disease --- Nervous System Disorder --- Neurologic Disorder --- Neurological Disorder --- Neurology --- Epidemiologic Method --- Epidemiological Methods --- Methods, Epidemiologic --- Epidemiological Method --- Method, Epidemiologic --- Method, Epidemiological --- Methods, Epidemiological --- Epidemiology --- Development and Growth --- Developmental Biology --- CNS Diseases --- Central Nervous System Disorders --- CNS Disease --- Central Nervous System Disease --- Central Nervous System Disorder --- Development, Human --- Humans --- Psychology, Developmental --- Psychologic Tests --- Psychological Test --- Test, Psychological --- Tests, Psychological --- Trier Social Stress Test --- Trier Stress Test --- Psychologic Test --- Stress Test, Trier --- Stress Tests, Trier --- Test, Psychologic --- Test, Trier Stress --- Tests, Psychologic --- Tests, Trier Stress --- Trier Stress Tests --- Age Group --- Group, Age --- Groups, Age --- Concept, Mathematical --- Concepts, Mathematical --- Mathematical Concept --- Area Analysis --- Estimation Technics --- Estimation Techniques --- Indirect Estimation Technics --- Indirect Estimation Techniques --- Multiple Classification Analysis --- Service Statistics --- Statistical Study --- Statistics, Service --- Tables and Charts as Topic --- Analyses, Area --- Analyses, Multiple Classification --- Area Analyses --- Classification Analyses, Multiple --- Classification Analysis, Multiple --- Estimation Technic, Indirect --- Estimation Technics, Indirect --- Estimation Technique --- Estimation Technique, Indirect --- Estimation Techniques, Indirect --- Indirect Estimation Technic --- Indirect Estimation Technique --- Multiple Classification Analyses --- Statistical Studies --- Studies, Statistical --- Study, Statistical --- Technic, Indirect Estimation --- Technics, Estimation --- Technics, Indirect Estimation --- Technique, Estimation --- Technique, Indirect Estimation --- Techniques, Estimation --- Techniques, Indirect Estimation --- AX-CPT --- Behavioral Assessment of Dysexecutive Syndrome --- CANTAB --- Cambridge Neuropsychological Test Automated Battery --- Clock Test --- Cognitive Function Scanner --- Continuous Performance Task --- Controlled Oral Word Association Test --- Delis-Kaplan Executive Function System --- Developmental Neuropsychological Assessment --- Hooper Visual Organization Test --- NEPSY --- Neuropsychologic Tests --- Neuropsychological Test --- Paced Auditory Serial Addition Test --- Repeatable Battery for the Assessment of Neuropsychological Status --- Rey-Osterrieth Complex Figure --- Symbol Digit Modalities Test --- Test of Everyday Attention --- Test, Neuropsychological --- Tests, Neuropsychological --- Tower of London Test --- Aphasia Tests --- Cognitive Test --- Cognitive Testing --- Cognitive Tests --- Memory for Designs Test --- Neuropsychological Testing --- Neuropsychologic Test --- Test, Cognitive --- Testing, Cognitive --- Testing, Neuropsychological --- Tests, Cognitive --- Brain Disorders --- CNS Disorders, Intracranial --- Central Nervous System Disorders, Intracranial --- Central Nervous System Intracranial Disorders --- Encephalon Diseases --- Encephalopathy --- Intracranial CNS Disorders --- Intracranial Central Nervous System Disorders --- Brain Disease --- Brain Disorder --- CNS Disorder, Intracranial --- Encephalon Disease --- Encephalopathies --- Intracranial CNS Disorder --- Infant Development --- Development, Child --- Development, Infant --- Psychology, Child --- Growth --- Behavior Disorders --- Diagnosis, Psychiatric --- Mental Disorders, Severe --- Psychiatric Diagnosis --- Mental Illness --- Psychiatric Diseases --- Psychiatric Disorders --- Psychiatric Illness --- Illness, Mental --- Mental Disorder --- Mental Disorder, Severe --- Mental Illnesses --- Psychiatric Disease --- Psychiatric Disorder --- Psychiatric Illnesses --- Severe Mental Disorder --- Severe Mental Disorders --- Mentally Ill Persons --- Children --- Minors --- Distribution, Statistical --- Distributions, Statistical --- Statistical Distribution --- methods --- Psychiatry --- Names