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

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

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

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

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

Journal of the Mathematical Society of Japan, ISSN 0025-5645, 2015, Volume 67, Issue 3, pp. 1069 - 1076

Journal Article

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

International Journal of Number Theory, ISSN 1793-0421, 08/2019, Volume 15, Issue 7, pp. 1469 - 1472

Let s ( a , b ) denote the classical Dedekind sum and S ( a , b ) = 1 2 s ( a , b ) . Recently, Du and Zhang proved the following reciprocity formula...

MATHEMATICS | Dedekind sums | reciprocity laws for Dedekind sums

MATHEMATICS | Dedekind sums | reciprocity laws for Dedekind sums

Journal Article

Optics Express, ISSN 1094-4087, 02/2014, Volume 22, Issue 3, pp. 3458 - 3467

.... Using the Standardized Residual Sum of Squares (STRESS) index, the results of our visual experiment were tested against predictions made by 12 modern color-difference formulas...

STANDARDIZED RESIDUAL SUM | OPTICS | Image Processing | Computer Science

STANDARDIZED RESIDUAL SUM | OPTICS | Image Processing | Computer Science

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

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

Journal of Physics A: Mathematical and Theoretical, ISSN 1751-8121, 06/2011, Volume 44, Issue 22, pp. 225302 - 11

We present a short alternative proof of the Voronoi summation formula which plays an important role in Dirichlet's divisor problem and has recently found an application in physics as a trace formula...

EXPONENTIAL-SUMS | PHYSICS, MULTIDISCIPLINARY | STRONGLY CHAOTIC SYSTEMS | RIEMANN ZETA-FUNCTION | PERIODIC-ORBIT THEORY | PHYSICS, MATHEMATICAL | SPECTRAL STATISTICS | QUANTUM GRAPHS | Dirichlet problem | Graphs | Operators | Schroedinger equation | Mathematical analysis | Proving

EXPONENTIAL-SUMS | PHYSICS, MULTIDISCIPLINARY | STRONGLY CHAOTIC SYSTEMS | RIEMANN ZETA-FUNCTION | PERIODIC-ORBIT THEORY | PHYSICS, MATHEMATICAL | SPECTRAL STATISTICS | QUANTUM GRAPHS | Dirichlet problem | Graphs | Operators | Schroedinger equation | Mathematical analysis | Proving

Journal Article

American journal of mathematics, ISSN 0002-9327, 4/2008, Volume 130, Issue 2, pp. 359 - 383

We give a combinatorial formula for the nonsymmetric Macdonald polynomials Eµ(x; q, t). The formula generalizes our previous combinatorial interpretation of the integral form symmetric Macdonald polynomials...

Research fellowships | Algebra | Root systems | Partial sums | Polynomials | Mathematical functions | Combinatorics | Coefficients | Symmetry | MATHEMATICS | Mathematics | Formulae | Properties | Geometry | Theorems | Numerical analysis

Research fellowships | Algebra | Root systems | Partial sums | Polynomials | Mathematical functions | Combinatorics | Coefficients | Symmetry | MATHEMATICS | Mathematics | Formulae | Properties | Geometry | Theorems | Numerical analysis

Journal Article

The European Physical Journal C, ISSN 1434-6044, 8/2015, Volume 75, Issue 8, pp. 1 - 28

The difference between the electromagnetic self-energies of proton and neutron can be calculated with the Cottingham formula, which expresses the self-energies as an integral over the electroproduction cross sections...

Nuclear Physics, Heavy Ions, Hadrons | Measurement Science and Instrumentation | Nuclear Energy | Quantum Field Theories, String Theory | Physics | Elementary Particles, Quantum Field Theory | Astronomy, Astrophysics and Cosmology | EFFECTIVE-FIELD THEORY | HYDROGEN LAMB SHIFT | VIRTUAL COMPTON-SCATTERING | PARTICLES | LOW-ENERGY | MUONIC-HYDROGEN | INVARIANT AMPLITUDES | CHIRAL PERTURBATION-THEORY | PROTON | GENERALIZED VECTOR DOMINANCE | PHYSICS, PARTICLES & FIELDS | Electromagnetism | Poles | Forward scattering | Mathematical analysis | Elastic scattering | Nucleons | Representations | Sum rules | Cross sections | Physics and Astronomy (miscellaneous) | Engineering (miscellaneous) | Nuclear Experiment | Nuclear Theory

Nuclear Physics, Heavy Ions, Hadrons | Measurement Science and Instrumentation | Nuclear Energy | Quantum Field Theories, String Theory | Physics | Elementary Particles, Quantum Field Theory | Astronomy, Astrophysics and Cosmology | EFFECTIVE-FIELD THEORY | HYDROGEN LAMB SHIFT | VIRTUAL COMPTON-SCATTERING | PARTICLES | LOW-ENERGY | MUONIC-HYDROGEN | INVARIANT AMPLITUDES | CHIRAL PERTURBATION-THEORY | PROTON | GENERALIZED VECTOR DOMINANCE | PHYSICS, PARTICLES & FIELDS | Electromagnetism | Poles | Forward scattering | Mathematical analysis | Elastic scattering | Nucleons | Representations | Sum rules | Cross sections | Physics and Astronomy (miscellaneous) | Engineering (miscellaneous) | Nuclear Experiment | Nuclear Theory

Journal Article

Advances in Difference Equations, ISSN 1687-1839, 12/2017, Volume 2017, Issue 1, pp. 1 - 5

We give two extensions of the classical formula for sums of powers on arithmetic progressions...

binomial transform | Bernoulli polynomials | forward difference | sum of powers formula | binomial mixture | MATHEMATICS | MATHEMATICS, APPLIED | OPERATORS | Usage | Binomial theorem | Polynomials | Progressions | Sums | Arithmetic

binomial transform | Bernoulli polynomials | forward difference | sum of powers formula | binomial mixture | MATHEMATICS | MATHEMATICS, APPLIED | OPERATORS | Usage | Binomial theorem | Polynomials | Progressions | Sums | Arithmetic

Journal Article

Bulletin of the Korean Mathematical Society, ISSN 1015-8634, 2016, Volume 53, Issue 2, pp. 487 - 494

... the computational problem of one kind mean value function related to E(c, p) = N(c, p) - 1/2 phi(p), and give its an exact computational formula.

Computational formula | Mean value | Error term | Lehmer’s problem | MATHEMATICS | error term | mean value | DEDEKIND SUMS | Lehmer's problem | computational formula

Computational formula | Mean value | Error term | Lehmer’s problem | MATHEMATICS | error term | mean value | DEDEKIND SUMS | Lehmer's problem | computational formula

Journal Article

Annals of Physics, ISSN 0003-4916, 02/2014, Volume 341, pp. 56 - 76

.... We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving...

The perturbative chiral Potts model | Corner-to-corner resistance of a [formula omitted] resistor network | Kirchhoff index | Trigonometrical sums | The Verlinde dimension formula | Number theory | Corner-to-corner resistance of a 2 × N resistor network | PHYSICS, MULTIDISCIPLINARY | Corner-to-corner resistance of a 2 x N resistor network | 2-POINT RESISTANCE | Electronic components industry | Analysis | Resistors | Neural networks | Mathematical analysis | Mathematical models | Two dimensional | Recursion | Joints | Sums | DIAGRAMS | SPACE | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | ONE-DIMENSIONAL CALCULATIONS | RESISTORS | CHIRALITY | TWO-DIMENSIONAL CALCULATIONS

The perturbative chiral Potts model | Corner-to-corner resistance of a [formula omitted] resistor network | Kirchhoff index | Trigonometrical sums | The Verlinde dimension formula | Number theory | Corner-to-corner resistance of a 2 × N resistor network | PHYSICS, MULTIDISCIPLINARY | Corner-to-corner resistance of a 2 x N resistor network | 2-POINT RESISTANCE | Electronic components industry | Analysis | Resistors | Neural networks | Mathematical analysis | Mathematical models | Two dimensional | Recursion | Joints | Sums | DIAGRAMS | SPACE | CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS | MATRICES | ONE-DIMENSIONAL CALCULATIONS | RESISTORS | CHIRALITY | TWO-DIMENSIONAL CALCULATIONS

Journal Article

Physics Letters A, ISSN 0375-9601, 10/2011, Volume 375, Issue 42, pp. 3659 - 3663

In this Letter, a generalized Tu formula is firstly presented to construct Hamiltonian structures of fractional soliton equations...

Fractional differentiable functions | Fractional Hamiltonian system | Generalized Tu formula | FRACTAL MEDIA | PHYSICS, MULTIDISCIPLINARY | DIFFERENTIABILITY | EQUATIONS | LIE-ALGEBRAS | ORDER | NONDIFFERENTIABLE FUNCTIONS | INTEGRABLE SYSTEMS | VARIATIONAL CALCULUS | SEMIDIRECT SUMS | DERIVATIVES | Construction | Hierarchies | Mathematical analysis | Solitons | Solid state physics | Calculus | Atomic structure

Fractional differentiable functions | Fractional Hamiltonian system | Generalized Tu formula | FRACTAL MEDIA | PHYSICS, MULTIDISCIPLINARY | DIFFERENTIABILITY | EQUATIONS | LIE-ALGEBRAS | ORDER | NONDIFFERENTIABLE FUNCTIONS | INTEGRABLE SYSTEMS | VARIATIONAL CALCULUS | SEMIDIRECT SUMS | DERIVATIVES | Construction | Hierarchies | Mathematical analysis | Solitons | Solid state physics | Calculus | Atomic structure

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 08/2014, Volume 10, Issue 5, pp. 1099 - 1119

Ramanujan's formula for the number r24(n) of representations of a positive integer n as a sum of 24 squares asserts that \begin{eqnarray*} r_{24}(n) & = & \frac...

sums of squares and triangular numbers | modular forms | Ramanujan's tau function | sum of divisors function | Eisenstein series | infinite products | Dedekind eta function | Ramanujan's 24 squares formula | MATHEMATICS | Heterocyclic compounds

sums of squares and triangular numbers | modular forms | Ramanujan's tau function | sum of divisors function | Eisenstein series | infinite products | Dedekind eta function | Ramanujan's 24 squares formula | MATHEMATICS | Heterocyclic compounds

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 12/2016, Volume 12, Issue 8, pp. 2061 - 2072

Dedekind used the classical Dedekind sum D ( a , c ) to describe the transformation of log η ( z ) under the substitution z ′ = ( a z + b ) / ( c z + d ) , ( a...

Dedekind sum | function fields | Dedekind eta function | MATHEMATICS

Dedekind sum | function fields | Dedekind eta function | MATHEMATICS

Journal Article

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