Annals of mathematics, ISSN 0003-486X, 2014, Volume 179, Issue 3, pp. 1121 - 1174

It is proved that $\lim_{n\rightarrow \infty }inf(p_{n+1}-p_n)\textless 7\times 10^7$, where pn is the n-th prime. Our method is a refinement of the recent...

Integers | Cauchy Schwarz inequality | Prime numbers | Arithmetic progressions | Mathematical theorems | Partial sums | Mathematical functions | Factorization | Estimation methods | MATHEMATICS | ARITHMETIC PROGRESSIONS | LARGE MODULI | NUMBERS

Integers | Cauchy Schwarz inequality | Prime numbers | Arithmetic progressions | Mathematical theorems | Partial sums | Mathematical functions | Factorization | Estimation methods | MATHEMATICS | ARITHMETIC PROGRESSIONS | LARGE MODULI | NUMBERS

Journal Article

Georgian Mathematical Journal, ISSN 1072-947X, 06/2018, Volume 25, Issue 2, pp. 313 - 316

A trigonometric sine series with special properties is constructed.
One of the properties of this series shows that Bary’s conjecture on subsequences
of...

40A30 | 40A05 | partial sums | Trigonometric series | subsequences of partial sums | MATHEMATICS

40A30 | 40A05 | partial sums | Trigonometric series | subsequences of partial sums | MATHEMATICS

Journal Article

The Annals of probability, ISSN 0091-1798, 11/2011, Volume 39, Issue 6, pp. 2441 - 2473

The results of Komlós, Major and Tusnády give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in...

Integers | Epidemics | Approximation | Statistical theories | Stochastic processes | Stationary processes | Partial sums | Mathematics | Random variables | Probabilities | KMT approximation | Strong invariance principle | Increments of partial sums | Dependence | STRONG LAWS | AUGMENTED GARCH SEQUENCES | STATISTICS & PROBABILITY | EMPIRICAL PROCESS | STRONG APPROXIMATION | MOMENT INEQUALITIES | ITERATED RANDOM FUNCTIONS | increments of partial sums | CONDITIONAL HETEROSCEDASTICITY | CENTRAL LIMIT THEOREM | PARTIAL-SUMS | dependence | strong invariance principle | INDEPENDENT RANDOM-VARIABLES | Mathematics - Probability | 60G10 | 60G17 | 60F17

Integers | Epidemics | Approximation | Statistical theories | Stochastic processes | Stationary processes | Partial sums | Mathematics | Random variables | Probabilities | KMT approximation | Strong invariance principle | Increments of partial sums | Dependence | STRONG LAWS | AUGMENTED GARCH SEQUENCES | STATISTICS & PROBABILITY | EMPIRICAL PROCESS | STRONG APPROXIMATION | MOMENT INEQUALITIES | ITERATED RANDOM FUNCTIONS | increments of partial sums | CONDITIONAL HETEROSCEDASTICITY | CENTRAL LIMIT THEOREM | PARTIAL-SUMS | dependence | strong invariance principle | INDEPENDENT RANDOM-VARIABLES | Mathematics - Probability | 60G10 | 60G17 | 60F17

Journal Article

The Annals of probability, ISSN 0091-1798, 3/2014, Volume 42, Issue 2, pp. 794 - 817

The celebrated results of Komlós, Major and Tusnády [
32 (1975) 111–131;
34 (1976) 33–58] give optimal Wiener approximation for the partial sums of i.i.d....

Brownian motion | Ergodic theory | Approximation | Central limit theorem | Articles | Time series | Stationary processes | Partial sums | Random variables | Dynamical systems | Probabilities | KMT approximation | Strong invariance principle | Ergodic sums | Nonlinear time series | Weak dependence | STATISTICS & PROBABILITY | ergodic sums | STRICTLY STATIONARY-PROCESSES | RANDOM-VARIABLES | ITERATED RANDOM FUNCTIONS | weak dependence | INVARIANCE-PRINCIPLES | nonlinear time series | PARTIAL-SUMS | TIME-SERIES | SYSTEMS | strong invariance principle | CENTRAL-LIMIT-THEOREM | 60G10 | 60G17 | 60F17

Brownian motion | Ergodic theory | Approximation | Central limit theorem | Articles | Time series | Stationary processes | Partial sums | Random variables | Dynamical systems | Probabilities | KMT approximation | Strong invariance principle | Ergodic sums | Nonlinear time series | Weak dependence | STATISTICS & PROBABILITY | ergodic sums | STRICTLY STATIONARY-PROCESSES | RANDOM-VARIABLES | ITERATED RANDOM FUNCTIONS | weak dependence | INVARIANCE-PRINCIPLES | nonlinear time series | PARTIAL-SUMS | TIME-SERIES | SYSTEMS | strong invariance principle | CENTRAL-LIMIT-THEOREM | 60G10 | 60G17 | 60F17

Journal Article

1974, Probability and mathematical statistics., ISBN 9780126727500, x, 381 p. --

Book

Journal of mathematical analysis and applications, ISSN 0022-247X, 06/2018, Volume 462, Issue 2, pp. 1073 - 1086

In this article we prove a general result which in particular suggests that, on a simply connected domain Ω in C, all the derivatives and anti-derivatives of...

Generic property | Universal Taylor series | Differentiation and integration operators | Partial sums | Taylor expansion | Baire's Theorem | MATHEMATICS | MATHEMATICS, APPLIED | UNIVERSAL TAYLOR-SERIES | BOUNDARY-BEHAVIOR

Generic property | Universal Taylor series | Differentiation and integration operators | Partial sums | Taylor expansion | Baire's Theorem | MATHEMATICS | MATHEMATICS, APPLIED | UNIVERSAL TAYLOR-SERIES | BOUNDARY-BEHAVIOR

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2013, Volume 2013, Issue 1, pp. 1 - 9

The aim of the present paper is to consider geometric properties such as starlikeness and convexity of the Cesáro partial sums of certain analytic functions in...

unit disk | partial sums | Analysis | analytic function | starlike functions | Cesáro partial sums | Mathematics, general | Mathematics | univalent functions | Applications of Mathematics | Starlike functions | Partial sums | Analytic function | Unit disk | Univalent functions | MATHEMATICS | MATHEMATICS, APPLIED | Cesaro partial sums | Disks | Convexity | Analytic functions | Classification | Inequalities | Sums

unit disk | partial sums | Analysis | analytic function | starlike functions | Cesáro partial sums | Mathematics, general | Mathematics | univalent functions | Applications of Mathematics | Starlike functions | Partial sums | Analytic function | Unit disk | Univalent functions | MATHEMATICS | MATHEMATICS, APPLIED | Cesaro partial sums | Disks | Convexity | Analytic functions | Classification | Inequalities | Sums

Journal Article

Journal of mathematical analysis and applications, ISSN 0022-247X, 05/2016, Volume 437, Issue 1, pp. 513 - 525

This paper proves that the real projection of each zero of any function P(z) in a large class of exponential polynomials is an interior point of the closure of...

Exponential polynomials | Partial sums of the Riemann zeta function | Kronecker theorem | Zeros of entire functions | MATHEMATICS | MATHEMATICS, APPLIED

Exponential polynomials | Partial sums of the Riemann zeta function | Kronecker theorem | Zeros of entire functions | MATHEMATICS | MATHEMATICS, APPLIED

Journal Article

The Annals of probability, ISSN 0091-1798, 3/2014, Volume 42, Issue 2, pp. 760 - 793

In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in 𝓗 (a real...

Ergodic theory | Integers | Approximation | Central limit theorem | Mathematical theorems | Articles | Stationary processes | Partial sums | Markov chains | Hilbert spaces | Martingales | Ergodic theorems | Moderate deviations | Stationary process | Wasserstein distances | Martingale approximation | Quenched invariance principle | ergodic theorems | INEQUALITIES | MARKOV-CHAINS | MIXING SEQUENCES | POINTWISE ERGODIC-THEOREMS | STATISTICS & PROBABILITY | moderate deviations | DEPENDENT RANDOM-VARIABLES | PROJECTIVE CRITERIA | STATIONARY-PROCESSES | stationary process | PARTIAL-SUMS | quenched invariance principle | CENTRAL-LIMIT-THEOREM | Mathematics - Probability | 60F05 | 60F15

Ergodic theory | Integers | Approximation | Central limit theorem | Mathematical theorems | Articles | Stationary processes | Partial sums | Markov chains | Hilbert spaces | Martingales | Ergodic theorems | Moderate deviations | Stationary process | Wasserstein distances | Martingale approximation | Quenched invariance principle | ergodic theorems | INEQUALITIES | MARKOV-CHAINS | MIXING SEQUENCES | POINTWISE ERGODIC-THEOREMS | STATISTICS & PROBABILITY | moderate deviations | DEPENDENT RANDOM-VARIABLES | PROJECTIVE CRITERIA | STATIONARY-PROCESSES | stationary process | PARTIAL-SUMS | quenched invariance principle | CENTRAL-LIMIT-THEOREM | Mathematics - Probability | 60F05 | 60F15

Journal Article

Algorithmica, ISSN 1432-0541, 2012, Volume 68, Issue 3, pp. 610 - 625

We present new results on Cartesian trees with applications in range minimum queries and bottleneck edge queries. We introduce a cache-oblivious Cartesian tree...

Range minimum query | Theory of Computation | Bottleneck paths | Partial sums | Computer Systems Organization and Communication Networks | Data Structures, Cryptology and Information Theory | Cache-oblivious | Minimum spanning tree verification | Algorithms | Mathematics of Computing | Computer Science | Lowest common ancestor | Algorithm Analysis and Problem Complexity | Cartesian tree | Partial sums | MATHEMATICS, APPLIED | ALGORITHM | LOWEST COMMON ANCESTORS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | CONNECTIVITY | MAXIMUM CAPACITY | SPANNING TREE | VERIFICATION | Computer science

Range minimum query | Theory of Computation | Bottleneck paths | Partial sums | Computer Systems Organization and Communication Networks | Data Structures, Cryptology and Information Theory | Cache-oblivious | Minimum spanning tree verification | Algorithms | Mathematics of Computing | Computer Science | Lowest common ancestor | Algorithm Analysis and Problem Complexity | Cartesian tree | Partial sums | MATHEMATICS, APPLIED | ALGORITHM | LOWEST COMMON ANCESTORS | COMPUTER SCIENCE, SOFTWARE ENGINEERING | CONNECTIVITY | MAXIMUM CAPACITY | SPANNING TREE | VERIFICATION | Computer science

Journal Article

IZVESTIYA MATHEMATICS, ISSN 1064-5632, 04/2019, Volume 83, Issue 2, pp. 273 - 286

We prove an exponential integral estimate for the cubic partial sums of multiple Fourier series on sets of large measure. This estimate yields some new...

MATHEMATICS | multiple Fourier series | exponential integral estimates | SUMMABILITY | cubic partial sums | Fourier series | Sums

MATHEMATICS | multiple Fourier series | exponential integral estimates | SUMMABILITY | cubic partial sums | Fourier series | Sums

Journal Article

Proceedings of the National Academy of Sciences - PNAS, ISSN 0027-8424, 10/2005, Volume 102, Issue 40, pp. 14150 - 14154

Based on the nonlinear system theory, we introduce previously undescribed dependence measures for stationary causal processes. Our physical and predictive...

Physical Sciences | Mathematical theorems | Asymptotic properties | Stochastic processes | Probability theory | Time series | Stationary processes | Partial sums | Systems theory | Random variables | Distribution functions | Weak convergence | Nonlinear time series | Kernel estimation | Limit theory | STATIONARY-PROCESSES | MULTIDISCIPLINARY SCIENCES | nonlinear time series | limit theory | VARIANCE | weak convergence | kernel estimation | FUNCTIONALS | CENTRAL-LIMIT-THEOREM | Dynamical systems | Time-series analysis | Research

Physical Sciences | Mathematical theorems | Asymptotic properties | Stochastic processes | Probability theory | Time series | Stationary processes | Partial sums | Systems theory | Random variables | Distribution functions | Weak convergence | Nonlinear time series | Kernel estimation | Limit theory | STATIONARY-PROCESSES | MULTIDISCIPLINARY SCIENCES | nonlinear time series | limit theory | VARIANCE | weak convergence | kernel estimation | FUNCTIONALS | CENTRAL-LIMIT-THEOREM | Dynamical systems | Time-series analysis | Research

Journal Article

Journal of inequalities and applications, ISSN 1025-5834, 2018, Volume 2018, Issue 1, pp. 1 - 14

Let {X,Xn}n∈N\(\{X, X_{n}\}_{n\in N}\) be a strictly stationary ρ−\(\rho^{-}\)-mixing sequence of positive random variables, under the suitable conditions, we...

Almost sure central limit theorem | Mixing sequence | Products of the some partial sums | Self-normalized | Random variables

Almost sure central limit theorem | Mixing sequence | Products of the some partial sums | Self-normalized | Random variables

Journal Article

Integral transforms and special functions, ISSN 1476-8291, 2019, Volume 31, Issue 3, pp. 221 - 231

We establish some supercongruences related to a supercongruence of Van Hamme, such as
where p is an odd prime and
is the
th Euler number. Our proof uses some...

Whipple's | Euler numbers | WZ-pair | transformation | Supercongruence | gamma function | Q-ANALOG | MATHEMATICS | MATHEMATICS, APPLIED | PARTIAL-SUMS | PROOF | Whipple's F-7 transformation | Congruences

Whipple's | Euler numbers | WZ-pair | transformation | Supercongruence | gamma function | Q-ANALOG | MATHEMATICS | MATHEMATICS, APPLIED | PARTIAL-SUMS | PROOF | Whipple's F-7 transformation | Congruences

Journal Article

The Annals of statistics, ISSN 0090-5364, 2010, Volume 38, Issue 2, pp. 1034 - 1070

Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte...

Ergodic theory | Integers | Statistical variance | Mathematical theorems | Markov chains | Standard error | Interval estimators | Estimators | Consistent estimators | Estimation methods | Markov chain | Batch means | Monte Carlo | Standard errors | Spectral methods | batch means | spectral methods | APPROXIMATION | STATISTICS & PROBABILITY | WIDTH OUTPUT ANALYSIS | SIMULATION | standard errors | STRONG CONSISTENCY | PARTIAL SUMS | GIBBS SAMPLERS | GEOMETRIC ERGODICITY | CONVERGENCE-RATES | 62M15 | 60J22

Ergodic theory | Integers | Statistical variance | Mathematical theorems | Markov chains | Standard error | Interval estimators | Estimators | Consistent estimators | Estimation methods | Markov chain | Batch means | Monte Carlo | Standard errors | Spectral methods | batch means | spectral methods | APPROXIMATION | STATISTICS & PROBABILITY | WIDTH OUTPUT ANALYSIS | SIMULATION | standard errors | STRONG CONSISTENCY | PARTIAL SUMS | GIBBS SAMPLERS | GEOMETRIC ERGODICITY | CONVERGENCE-RATES | 62M15 | 60J22

Journal Article

Chaos, solitons and fractals, ISSN 0960-0779, 01/2018, Volume 106, pp. 233 - 242

In this paper we propose an alternative to the coupling of Berkes, Liu and Wu [1] to obtain strong approximations for partial sums of dependent sequences. The...

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | DYNAMICAL-SYSTEMS | SURE INVARIANCE-PRINCIPLE | PARTIAL-SUMS | PHYSICS, MATHEMATICAL | CENTRAL-LIMIT-THEOREM | DEPENDENCE | Mathematics - Probability

MATHEMATICS, INTERDISCIPLINARY APPLICATIONS | PHYSICS, MULTIDISCIPLINARY | DYNAMICAL-SYSTEMS | SURE INVARIANCE-PRINCIPLE | PARTIAL-SUMS | PHYSICS, MATHEMATICAL | CENTRAL-LIMIT-THEOREM | DEPENDENCE | Mathematics - Probability

Journal Article

Statistica Sinica, ISSN 1017-0405, 7/2011, Volume 21, Issue 3, pp. 1397 - 1413

We obtain an invariance principle for non-stationary vector-valued stochastic processes. It is shown that, under mild conditions, the partial sums of...

Approximation | Logical proofs | Time series | Stationary processes | Partial sums | General | Mathematical vectors | Random variables | Covariance matrices | Martingales | Probabilities | Non-stationary nonlinear multiple time series | Functional linear models | Central limit theorem | Gaussian approximation | Local stationarity | SEQUENCES | non-stationary nonlinear multiple time series | STATISTICS & PROBABILITY | STRONG INVARIANCE-PRINCIPLES | INFERENCE | RANDOM-VARIABLES | MODELS | THEOREMS | PARTIAL-SUMS | local stationarity | functional linear models

Approximation | Logical proofs | Time series | Stationary processes | Partial sums | General | Mathematical vectors | Random variables | Covariance matrices | Martingales | Probabilities | Non-stationary nonlinear multiple time series | Functional linear models | Central limit theorem | Gaussian approximation | Local stationarity | SEQUENCES | non-stationary nonlinear multiple time series | STATISTICS & PROBABILITY | STRONG INVARIANCE-PRINCIPLES | INFERENCE | RANDOM-VARIABLES | MODELS | THEOREMS | PARTIAL-SUMS | local stationarity | functional linear models

Journal Article

Issues of analysis, ISSN 2306-3432, 12/2018, Volume 25, Issue 2, pp. 131 - 143

In this article, we prove a two-points distortion theorem and obtain sharp coefficient estimates for the families of close-toconvex harmonic mappings whose...

sectionsб | Univalent Harmonic | closeto-convex | partial sums | convex in one direction | sections

sectionsб | Univalent Harmonic | closeto-convex | partial sums | convex in one direction | sections

Journal Article

Monatshefte für Mathematik, ISSN 0026-9255, 7/2018, Volume 186, Issue 3, pp. 453 - 470

Let $$\mathcal {S}_H^0$$
SH0
denote the class of all functions $$f(z)=h(z)+\overline{g(z)}=z+\sum ^\infty _{n=2} a_nz^n +\overline{\sum ^\infty _{n=2}...

Harmonic functions | Harmonic univalent functions | 30C55 | 30C50 | Mathematics, general | Coefficient bounds | Partial sums | Mathematics | Secondary 30C45 | Primary 31A05 | Affine invariant family | Linear invariant family | MATHEMATICS

Harmonic functions | Harmonic univalent functions | 30C55 | 30C50 | Mathematics, general | Coefficient bounds | Partial sums | Mathematics | Secondary 30C45 | Primary 31A05 | Affine invariant family | Linear invariant family | MATHEMATICS

Journal Article

Bioinformatics, ISSN 1367-4803, 2017, Volume 33, Issue 5, pp. 654 - 660

Motivation: The local score of a biological sequence analysis is a mathematical tool largely used to analyse biological sequences. Consequently, determining an...

DISTRIBUTIONS | APPROXIMATION | ALIGNMENT | BIOTECHNOLOGY & APPLIED MICROBIOLOGY | BIOCHEMICAL RESEARCH METHODS | MATHEMATICAL & COMPUTATIONAL BIOLOGY | PARTIAL-SUMS | GOODNESS-OF-FIT | VARIABLES | STRONG LIMIT-THEOREMS | EXCEEDANCES | Sequence Analysis, Protein | Statistics, Nonparametric | Models, Statistical | Statistics | Mathematics

DISTRIBUTIONS | APPROXIMATION | ALIGNMENT | BIOTECHNOLOGY & APPLIED MICROBIOLOGY | BIOCHEMICAL RESEARCH METHODS | MATHEMATICAL & COMPUTATIONAL BIOLOGY | PARTIAL-SUMS | GOODNESS-OF-FIT | VARIABLES | STRONG LIMIT-THEOREMS | EXCEEDANCES | Sequence Analysis, Protein | Statistics, Nonparametric | Models, Statistical | Statistics | Mathematics

Journal Article

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