SIAM journal on optimization, ISSN 1095-7189, 2019, Volume 29, Issue 2, pp. 1106 - 1130

.... The main contribution of this paper consists of providing formulas for such a subdifferential under weak continuity assumptions...

MATHEMATICS, APPLIED | CONVEX | convex functions | CONSTRAINT QUALIFICATIONS | qualification conditions | subdifferential calculus rules | CALCULUS RULES | SUM | supremum function | OPTIMALITY CONDITIONS | SEMIINFINITE

MATHEMATICS, APPLIED | CONVEX | convex functions | CONSTRAINT QUALIFICATIONS | qualification conditions | subdifferential calculus rules | CALCULUS RULES | SUM | supremum function | OPTIMALITY CONDITIONS | SEMIINFINITE

Journal Article

Mathematische Zeitschrift, ISSN 0025-5874, 4/2019, Volume 291, Issue 3, pp. 1337 - 1356

We prove a weighted sum formula of the zeta values at even arguments, and a weighted sum formula of the multiple zeta values with even arguments and its zeta-star analogue...

Weighted sum formulas | Bernoulli numbers | 11M32 | Multiple zeta values | Mathematics, general | Mathematics | Multiple zeta-star values | 11B68 | MATHEMATICS

Weighted sum formulas | Bernoulli numbers | 11M32 | Multiple zeta values | Mathematics, general | Mathematics | Multiple zeta-star values | 11B68 | MATHEMATICS

Journal Article

Inventiones mathematicae, ISSN 0020-9910, 12/2013, Volume 194, Issue 3, pp. 673 - 729

We develop a fairly explicit Kuznetsov formula on GL(3) and discuss the analytic behavior of the test functions on both sides...

Moments of L -functions | 11F72 | Whittaker functions | Mathematics | Kloosterman sums | 11F66 | Kuznetsov formula | Poincaré series | Large sieve | Mathematics, general | Spectral decomposition | Weyl’s law | Exceptional eigenvalues | Moments of L-functions | Weyl's law | EXISTENCE | POINCARE-SERIES | LINNIK | ASYMPTOTIC FORMULA | Poincare series | MATHEMATICS | FOURIER COEFFICIENTS | CUSP FORMS | Eigenvalues | Mathematics - Number Theory

Moments of L -functions | 11F72 | Whittaker functions | Mathematics | Kloosterman sums | 11F66 | Kuznetsov formula | Poincaré series | Large sieve | Mathematics, general | Spectral decomposition | Weyl’s law | Exceptional eigenvalues | Moments of L-functions | Weyl's law | EXISTENCE | POINCARE-SERIES | LINNIK | ASYMPTOTIC FORMULA | Poincare series | MATHEMATICS | FOURIER COEFFICIENTS | CUSP FORMS | Eigenvalues | Mathematics - Number Theory

Journal Article

Journal of Number Theory, ISSN 0022-314X, 05/2016, Volume 162, pp. 483 - 495

We discover new Voronoi formulae for automorphic forms on GL(n) for n≥4. There are [n/2] different Voronoi formulae on GL...

Kloosterman sum | Maass form | Voronoi formula | Automorphic form | Functional equation | MATHEMATICS | Mathematics - Number Theory

Kloosterman sum | Maass form | Voronoi formula | Automorphic form | Functional equation | MATHEMATICS | Mathematics - Number Theory

Journal Article

Odontology, ISSN 1618-1247, 2019, Volume 107, Issue 1, pp. 72 - 79

...) formulas and the gingival color space using the Bland and Altman limits of agreement, to use this relationship to establish the equivalences between the gingival color thresholds of perceptibility...

CIEDE2000 formula | CIELAB formula | Dentistry | Oral and Maxillofacial Surgery | Gingiva color measurement | Bland and Altman limits of agreement | Bland and Altman limits ofagreement | agreement | ZIRCONIA | PERFORMANCE | SOFT-TISSUE | RESTORATIONS | MUCOSA | DENTISTRY, ORAL SURGERY & MEDICINE | IMPLANT ABUTMENTS | CIELAB | Bland and Altman limits of | DELTA-E-ASTERISK | STANDARDIZED RESIDUAL SUM | SHADE GUIDE | Confidence intervals | Transformation | Gingiva | Color

CIEDE2000 formula | CIELAB formula | Dentistry | Oral and Maxillofacial Surgery | Gingiva color measurement | Bland and Altman limits of agreement | Bland and Altman limits ofagreement | agreement | ZIRCONIA | PERFORMANCE | SOFT-TISSUE | RESTORATIONS | MUCOSA | DENTISTRY, ORAL SURGERY & MEDICINE | IMPLANT ABUTMENTS | CIELAB | Bland and Altman limits of | DELTA-E-ASTERISK | STANDARDIZED RESIDUAL SUM | SHADE GUIDE | Confidence intervals | Transformation | Gingiva | Color

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 1/2016, Volume 39, Issue 1, pp. 31 - 47

In this paper, we obtain analogues of Jacobi’s derivative formula in terms of the theta constants with rational characteristics...

14K25 | Fourier Analysis | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | Mathematics | The sum of two squares | Number Theory | Combinatorics | Rational characteristics | Jacobi’s derivative formula | MATHEMATICS | INFINITE SERIES | Jacobi's derivative formula | IDENTITIES | PRODUCTS

14K25 | Fourier Analysis | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | Mathematics | The sum of two squares | Number Theory | Combinatorics | Rational characteristics | Jacobi’s derivative formula | MATHEMATICS | INFINITE SERIES | Jacobi's derivative formula | IDENTITIES | PRODUCTS

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 03/2016, Volume 12, Issue 2, pp. 383 - 408

In this paper, explicit formulas involving a generalized Ramanujan sum are derived...

Ramanujan sums | prime number theorem | explicit formulas | MATHEMATICS | MOBIUS FUNCTION

Ramanujan sums | prime number theorem | explicit formulas | MATHEMATICS | MOBIUS FUNCTION

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 1/2018, Volume 45, Issue 1, pp. 73 - 94

In terms of Abel’s transformation on difference operators, we establish four families of summation formulas involving generalized harmonic numbers...

Fourier Analysis | Secondary 40A25 | Functions of a Complex Variable | Difference operator | Harmonic numbers | Field Theory and Polynomials | Primary 05A19 | Abel’s transformation | Mathematics | Number Theory | Combinatorics | MATHEMATICS | IDENTITIES | Abel's transformation | SUMS

Fourier Analysis | Secondary 40A25 | Functions of a Complex Variable | Difference operator | Harmonic numbers | Field Theory and Polynomials | Primary 05A19 | Abel’s transformation | Mathematics | Number Theory | Combinatorics | MATHEMATICS | IDENTITIES | Abel's transformation | SUMS

Journal Article

Journal of Inequalities and Applications, ISSN 1025-5834, 12/2016, Volume 2016, Issue 1, pp. 1 - 7

In this paper we present two computational formulae for one kind of reciprocal sums related to the Riemann zeta-function at integer points s = 4 , 5 $s=4,5...

inequality | telescoping method | continued fraction | 11B83 | Analysis | multiple-correction method | Mathematics, general | Mathematics | 11M06 | 11J70 | Applications of Mathematics | Riemann zeta-function | PELL | MATHEMATICS | MATHEMATICS, APPLIED | RECIPROCAL FIBONACCI | FIBONACCI POLYNOMIALS | MULTIPLE-CORRECTION | INFINITE SUM | Integers | Texts | Computation | Inequalities | Sums

inequality | telescoping method | continued fraction | 11B83 | Analysis | multiple-correction method | Mathematics, general | Mathematics | 11M06 | 11J70 | Applications of Mathematics | Riemann zeta-function | PELL | MATHEMATICS | MATHEMATICS, APPLIED | RECIPROCAL FIBONACCI | FIBONACCI POLYNOMIALS | MULTIPLE-CORRECTION | INFINITE SUM | Integers | Texts | Computation | Inequalities | Sums

Journal Article

Taiwanese journal of mathematics, ISSN 1027-5487, 4/2016, Volume 20, Issue 2, pp. 243 - 261

In this paper, we introduce the vectorization of shuffle products of two sums of multiple zeta values, which generalizes the weighted sum formula obtained by Ohno and Zudilin...

Integers | Mathematics | Weighted sum formula | Multiple zeta value | Shuffle relation | Sum formula | MATHEMATICS

Integers | Mathematics | Weighted sum formula | Multiple zeta value | Shuffle relation | Sum formula | MATHEMATICS

Journal Article

Journal of Algebraic Combinatorics, ISSN 0925-9899, 8/2019, Volume 50, Issue 1, pp. 73 - 98

.... Closed formulas for exponential sums of symmetric Boolean functions were found by Cai, Green and Thierauf in the late 1990s...

Convex and Discrete Geometry | Symmetric functions | 05E05 | Mathematics | Order, Lattices, Ordered Algebraic Structures | Group Theory and Generalizations | Linear recurrences | Combinatorics | Computer Science, general | Exponential sums | 11T23 | 11B37 | MATHEMATICS | PERTURBATIONS | Cryptography

Convex and Discrete Geometry | Symmetric functions | 05E05 | Mathematics | Order, Lattices, Ordered Algebraic Structures | Group Theory and Generalizations | Linear recurrences | Combinatorics | Computer Science, general | Exponential sums | 11T23 | 11B37 | MATHEMATICS | PERTURBATIONS | Cryptography

Journal Article

Frontiers of Mathematics in China, ISSN 1673-3452, 2/2014, Volume 9, Issue 1, pp. 101 - 109

...Front. Math. China 2014, 9(1): 101–109 DOI 10.1007/s11464-013-0348-0 Multiplication formulas for Kubert functions Hailong LI 1 , Jing MA 2 , Yuichi URAMATSU 3...

multiplication formula | integral formula | Mathematics, general | Kubert function | Mathematics | Bernoulli polynomial | mean value | 11A25 | 11B68 | MATHEMATICS | DEDEKIND SUMS | IDENTITIES | HECKE OPERATORS | HURWITZ ZETA-FUNCTION | Studies | Polynomials | Multiplication & division | Integrals | Mathematical analysis | Dirichlet problem

multiplication formula | integral formula | Mathematics, general | Kubert function | Mathematics | Bernoulli polynomial | mean value | 11A25 | 11B68 | MATHEMATICS | DEDEKIND SUMS | IDENTITIES | HECKE OPERATORS | HURWITZ ZETA-FUNCTION | Studies | Polynomials | Multiplication & division | Integrals | Mathematical analysis | Dirichlet problem

Journal Article

Journal of Functional Analysis, ISSN 0022-1236, 2012, Volume 262, Issue 4, pp. 1515 - 1528

Let X and Y be two n × n Hermitian matrices. In the article Proof of a conjectured exponential formula...

Functional calculus | Operator identity | Unitary operators | MATHEMATICS | EIGENVALUES | INEQUALITIES | HONEYCOMB MODEL | HERMITIAN MATRICES | CONJECTURE | SUMS

Functional calculus | Operator identity | Unitary operators | MATHEMATICS | EIGENVALUES | INEQUALITIES | HONEYCOMB MODEL | HERMITIAN MATRICES | CONJECTURE | SUMS

Journal Article

Entropy (Basel, Switzerland), ISSN 1099-4300, 2014, Volume 16, Issue 9, pp. 4892 - 4910

...) Hartley did put forth his rule twenty years before Shannon; (2) Shannon's formula as a fundamental tradeoff between transmission rate, bandwidth, and signal-to-noise ratio came out unexpected in 1948; (3...

Additive noise channel | Channel capacity | Pulse-amplitude modulation (PAM) | Additive white Gaussian noise (AWGN) channel | Signal-to-noise ratio | Shannon's formula | Differential entropy | Uniform sum distribution | Characteristic function | Central limit theorem | Uniform noise channel | Uniform B-spline function | Hartley's rule | uniform B-spline function | PHYSICS, MULTIDISCIPLINARY | channel capacity | additive noise channel | pulse-amplitude modulation (PAM) | differential entropy | central limit theorem | TRANSMISSION | uniform noise channel | signal-to-noise ratio | INFORMATION-THEORY | uniform sum distribution | WORK | additive white Gaussian noise (AWGN) channel | characteristic function | COMMUNICATION | ENTROPY | Shannon’s formula | Hartley’s rule

Additive noise channel | Channel capacity | Pulse-amplitude modulation (PAM) | Additive white Gaussian noise (AWGN) channel | Signal-to-noise ratio | Shannon's formula | Differential entropy | Uniform sum distribution | Characteristic function | Central limit theorem | Uniform noise channel | Uniform B-spline function | Hartley's rule | uniform B-spline function | PHYSICS, MULTIDISCIPLINARY | channel capacity | additive noise channel | pulse-amplitude modulation (PAM) | differential entropy | central limit theorem | TRANSMISSION | uniform noise channel | signal-to-noise ratio | INFORMATION-THEORY | uniform sum distribution | WORK | additive white Gaussian noise (AWGN) channel | characteristic function | COMMUNICATION | ENTROPY | Shannon’s formula | Hartley’s rule

Journal Article

Mathematica Slovaca, ISSN 0139-9918, 8/2013, Volume 63, Issue 4, pp. 733 - 758

... ). And we obtain four infinite families of recursive formulas for the power moments of Kloosterman sums and four those of 2-dimensional Kloosterman sums in terms of the frequencies of weights in the codes...

special orthogonal group | Kloosterman sum | Algebra | weight distribution | Mathematics, general | Primary 11T23, 20G40, 94B05 | Mathematics | 2-dimensional Kloosterman sum | double cosets | Pless power moment identity | maximal parabolic subgroup | orthogonal group | MATHEMATICS | CODES | FINITE-FIELD | SYMPLECTIC GROUPS | GAUSS SUMS

special orthogonal group | Kloosterman sum | Algebra | weight distribution | Mathematics, general | Primary 11T23, 20G40, 94B05 | Mathematics | 2-dimensional Kloosterman sum | double cosets | Pless power moment identity | maximal parabolic subgroup | orthogonal group | MATHEMATICS | CODES | FINITE-FIELD | SYMPLECTIC GROUPS | GAUSS SUMS

Journal Article

16.
Full Text
Transformation formulas of a character analogue of $$\log \theta _{2}(z)$$ log θ 2 ( z )

The Ramanujan Journal, ISSN 1382-4090, 2/2019, Volume 48, Issue 2, pp. 323 - 349

In this paper, transformation formulas for the function $$\begin{aligned} A_{1}\left( z,s:\chi \right) =\sum \limits _{n=1}^{\infty }\sum \limits _{m=1...

Bernoulli and Euler polynomials | Fourier Analysis | Functions of a Complex Variable | Hardy–Berndt sums | Dedekind sums | Field Theory and Polynomials | Mathematics | Number Theory | Combinatorics | 11F20 | 11B68

Bernoulli and Euler polynomials | Fourier Analysis | Functions of a Complex Variable | Hardy–Berndt sums | Dedekind sums | Field Theory and Polynomials | Mathematics | Number Theory | Combinatorics | 11F20 | 11B68

Journal Article

Annals of the Institute of Statistical Mathematics, ISSN 0020-3157, 10/2014, Volume 66, Issue 5, pp. 833 - 864

...–Pitman sampling formula: One is the prediction of the number of new species if the catch is continued, and the other is how the number of species will decrease in random subsamples...

Statistics for Business/Economics/Mathematical Finance/Insurance | Random sum models | Bell polynomials | Gibbs partitions | Partition data | Statistics, general | Pólya’s urn model | Trawl fishery | Statistics | Waiting time | Random number partitions | Size index | Pólya's urn model | DISTRIBUTIONS | MULTISPECIES TRAWL CATCHES | Polya's urn model | STATISTICS & PROBABILITY | Trawling | Analysis | Studies | Mathematical problems | Mathematical models | Mathematics | Polynomials | Ecology | Sampling | Statistical analysis | Samples | Tools | Ecological monitoring | Combinatorial analysis

Statistics for Business/Economics/Mathematical Finance/Insurance | Random sum models | Bell polynomials | Gibbs partitions | Partition data | Statistics, general | Pólya’s urn model | Trawl fishery | Statistics | Waiting time | Random number partitions | Size index | Pólya's urn model | DISTRIBUTIONS | MULTISPECIES TRAWL CATCHES | Polya's urn model | STATISTICS & PROBABILITY | Trawling | Analysis | Studies | Mathematical problems | Mathematical models | Mathematics | Polynomials | Ecology | Sampling | Statistical analysis | Samples | Tools | Ecological monitoring | Combinatorial analysis

Journal Article

Applied mathematics and computation, ISSN 0096-3003, 10/2020, Volume 383, p. 125380

•We provide an asymptotic formula for Vasyunin cotangent sums.•Vasyunin cotangent sums have applications in Riemann Hypothesis...

Vasyunin cotangent sums | Euler-Maclaurin summation formula | Digamma function | MATHEMATICS, APPLIED

Vasyunin cotangent sums | Euler-Maclaurin summation formula | Digamma function | MATHEMATICS, APPLIED

Journal Article

Journal of Siberian Federal University - Mathematics and Physics, ISSN 1997-1397, 2018, Volume 11, Issue 5, pp. 603 - 614

... Arising From Collection Formulas Vladimir M. Leontiev∗ Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041...

Sums of binomial coefficients | Divisibility | Collection formulas

Sums of binomial coefficients | Divisibility | Collection formulas

Journal Article

Physica A: Statistical Mechanics and its Applications, ISSN 0378-4371, 12/2017, Volume 488, pp. 46 - 55

Some new formulae of the canonical correlation functions for the one dimensional quantum transverse Ising model are found by the ST-transformation method using...

Suzuki–Trotter transformation | One-dimensional Transverse Ising model | Two-dimensional Ising model | Sum rule | Canonical correlation function | Quantum Spin models | STATISTICAL-MECHANICS | PHYSICS, MULTIDISCIPLINARY | Suzuki-Trotter transformation | DYNAMICAL CORRELATION-FUNCTIONS | ST-TRANSFORMATION APPROACH | DIAGRAMMATICAL TECHNIQUES | ANALYTIC SOLUTIONS | CHAIN | TRANSFER-MATRIX | QUANTUM-SYSTEMS | SPIN-CORRELATION

Suzuki–Trotter transformation | One-dimensional Transverse Ising model | Two-dimensional Ising model | Sum rule | Canonical correlation function | Quantum Spin models | STATISTICAL-MECHANICS | PHYSICS, MULTIDISCIPLINARY | Suzuki-Trotter transformation | DYNAMICAL CORRELATION-FUNCTIONS | ST-TRANSFORMATION APPROACH | DIAGRAMMATICAL TECHNIQUES | ANALYTIC SOLUTIONS | CHAIN | TRANSFER-MATRIX | QUANTUM-SYSTEMS | SPIN-CORRELATION

Journal Article

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