PACIFIC JOURNAL OF MATHEMATICS, ISSN 0030-8730, 11/2019, Volume 303, Issue 1, pp. 325 - 335

The sum formula for finite and symmetric multiple zeta values, established by Wakabayashi and the authors, implies that if the weight and depth are fixed and the specified component is required...

MATHEMATICS | restricted sum formula | finite real multiple zeta values | sum formula | symmetric multiple zeta values | finite multiple zeta values | symmetrised multiple zeta values

MATHEMATICS | restricted sum formula | finite real multiple zeta values | sum formula | symmetric multiple zeta values | finite multiple zeta values | symmetrised multiple zeta values

Journal Article

Journal of approximation theory, ISSN 0021-9045, 2015, Volume 197, pp. 49 - 61

For n≥1 let An≔{P:P(z)=∑j=1nzkj:0≤k1 Large sieve inequalities | [formula omitted] norm | Newman polynomials | Littlewood polynomials | Sums of monomials | Mahler measure | Fekete polynomials | Constrained coefficients | L 1 norm | INEQUALITIES | L-1 norm | REMEZ-TYPE | RESTRICTED COEFFICIENTS | SUBARCS | MATHEMATICS | BOUNDS | NORM | LARGE SIEVE | ZEROS

Journal Article

PACIFIC JOURNAL OF MATHEMATICS, ISSN 0030-8730, 06/2014, Volume 269, Issue 2, pp. 371 - 384

.... This corresponds to the Jantzen sum formula of a baby Verma module over a modular Lie algebra. This also implies a new proof of the linkage principle which was already derived...

LIE-ALGEBRAS | MATHEMATICS | Jantzen sum formula | critical representations of affine Kac-Moody algebras | REPRESENTATIONS | Jantzen filtration | category O | affine Kac-Moody algebras | restricted Verma modules at the critical level | Mathematics - Representation Theory

LIE-ALGEBRAS | MATHEMATICS | Jantzen sum formula | critical representations of affine Kac-Moody algebras | REPRESENTATIONS | Jantzen filtration | category O | affine Kac-Moody algebras | restricted Verma modules at the critical level | Mathematics - Representation Theory

Journal Article

Journal of Number Theory, ISSN 0022-314X, 09/2016, Volume 166, pp. 452 - 472

.... The restricted sum formula is a generalization of the celebrated sum formula which is the special case p=0...

Secondary | Restricted sum formula | Duality theorem | Multiple zeta value | Primary | Sum formula

Secondary | Restricted sum formula | Duality theorem | Multiple zeta value | Primary | Sum formula

Journal Article

Archiv der Mathematik, ISSN 0003-889X, 7/2016, Volume 107, Issue 1, pp. 9 - 22

The main goal of this paper is the presentation of an elementary analytic technique which enables the evaluation of the so-called restricted sum formulas...

Restricted sum formulas | Infinite series and products | Primary 11M32 | 11M35 | Generating functions | Riemann zeta function | Mathematics, general | Mathematics | Secondary 11B68 | MATHEMATICS

Restricted sum formulas | Infinite series and products | Primary 11M32 | 11M35 | Generating functions | Riemann zeta function | Mathematics, general | Mathematics | Secondary 11B68 | MATHEMATICS

Journal Article

Rocky Mountain Journal of Mathematics, ISSN 0035-7596, 2017, Volume 47, Issue 7, pp. 2107 - 2131

...)(-alpha r) (k(r) - 1)(-1) = zeta(r), which may be compared to the well-known sum formula...

Restricted sum formula | Duality theorem | Multiple zeta value | Sum formula | MATHEMATICS | restricted sum formula | sum formula | duality theorem

Restricted sum formula | Duality theorem | Multiple zeta value | Sum formula | MATHEMATICS | restricted sum formula | sum formula | duality theorem

Journal Article

Results in Mathematics, ISSN 1422-6383, 3/2018, Volume 73, Issue 1, pp. 1 - 22

... integral representations of series. By applying the formulas obtained, we prove that the multiple zeta star values whose indices are the sequences $$(\bar{1},\{1\}_m,\bar{1})$$ (1¯,{1}m,1¯) and $$(2,\{1\}_m,\bar{1})$$ (2,{1}m,1...

11M40 | restricted sum formula | 33E20 | multiple zeta star value | 40B05 | Mathematics, general | Mathematics | 11M06 | Multiple zeta value | MATHEMATICS | MATHEMATICS, APPLIED | ANALOGS | HARMONIC SUMS | DUALITY | FORMULAS

11M40 | restricted sum formula | 33E20 | multiple zeta star value | 40B05 | Mathematics, general | Mathematics | 11M06 | Multiple zeta value | MATHEMATICS | MATHEMATICS, APPLIED | ANALOGS | HARMONIC SUMS | DUALITY | FORMULAS

Journal Article

Electronic Journal of Combinatorics, ISSN 1077-8926, 11/2015, Volume 22, Issue 4, pp. 1 - 9

...] with rn inversions is given by broken vertical bar S-n,S-m(321)broken vertical bar = Sigma b proves m (n - Delta(b)/2 l(b)) where the sum runs over all compositions b...

Catalan number | Dyck path | Pattern avoidance | Generating function | POLYNOMIALS | MATHEMATICS | INVOLUTIONS | MATHEMATICS, APPLIED | DYCK PATHS | generating function | PATTERNS | pattern avoidance | RESTRICTED PERMUTATIONS

Catalan number | Dyck path | Pattern avoidance | Generating function | POLYNOMIALS | MATHEMATICS | INVOLUTIONS | MATHEMATICS, APPLIED | DYCK PATHS | generating function | PATTERNS | pattern avoidance | RESTRICTED PERMUTATIONS

Journal Article

Computational Statistics and Data Analysis, ISSN 0167-9473, 05/2016, Volume 97, pp. 33 - 46

.... They have nice Markov properties for the exact and efficient recursion formulae for calculating the p-value...

Maximal contrast test | Markov property | Goodness-of-fit test | Cumulative sum statistic | Recursion formula | Restricted alternative | HOMOGENEITY | STATISTICS & PROBABILITY | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PATTERN | SEQUENCE | ORDERED-ALTERNATIVES | ASSOCIATION | Slopes | Hypotheses | Inflection points | Data processing | Statistical tests | Recursion | Statistics | Monitoring

Maximal contrast test | Markov property | Goodness-of-fit test | Cumulative sum statistic | Recursion formula | Restricted alternative | HOMOGENEITY | STATISTICS & PROBABILITY | COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS | PATTERN | SEQUENCE | ORDERED-ALTERNATIVES | ASSOCIATION | Slopes | Hypotheses | Inflection points | Data processing | Statistical tests | Recursion | Statistics | Monitoring

Journal Article

Frontiers of Mathematics in China, ISSN 1673-3452, 2/2015, Volume 10, Issue 1, pp. 69 - 89

... ⩽ ... ⩽ x r , and got some properties and a neat formula of the solutions. Due to the lack of a simple computational method for calculating the number of the solution...

11P81 | 11P83 | Congruence | multiset congruence solution | Mathematics, general | restricted integer partition | Mathematics | 05A15 | 05A17 | MATHEMATICS | Studies | Mathematical analysis | Recursive algorithms | Algorithms | Enumeration | Congruences | China | Texts | Mathematical models | Recursive

11P81 | 11P83 | Congruence | multiset congruence solution | Mathematics, general | restricted integer partition | Mathematics | 05A15 | 05A17 | MATHEMATICS | Studies | Mathematical analysis | Recursive algorithms | Algorithms | Enumeration | Congruences | China | Texts | Mathematical models | Recursive

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 12/2016, Volume 12, Issue 8, pp. 2167 - 2171

..., by combining ideas from the finite Fourier transform of arithmetic functions and Ramanujan sums, we give a short proof for the following result...

Ramanujan sum | Restricted linear congruence | finite Fourier transform | MATHEMATICS | CYCLIC GROUPS

Ramanujan sum | Restricted linear congruence | finite Fourier transform | MATHEMATICS | CYCLIC GROUPS

Journal Article

Journal of Number Theory, ISSN 0022-314X, 02/2017, Volume 171, pp. 128 - 144

In this paper, using properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, we give an explicit formula for the number...

Discrete Fourier transform | Ramanujan sum | Restricted linear congruence | MATHEMATICS | NUMBER | MODULUS | REPRESENTATIONS | MAPS | CYCLIC GROUPS | Computer science

Discrete Fourier transform | Ramanujan sum | Restricted linear congruence | MATHEMATICS | NUMBER | MODULUS | REPRESENTATIONS | MAPS | CYCLIC GROUPS | Computer science

Journal Article

IEEE transactions on signal processing, ISSN 1053-587X, 2020, Volume 68, pp. 3169 - 3178

Restricted Isometry Property (RIP) is of fundamental importance in the theory of compressed sensing and forms the base of many exact and robust recovery...

random matrix | Lower bound | Upper bound | order statistics | restricted isometry property | Signal processing algorithms | Matching pursuit algorithms | Robustness | Random variables | Sparse matrices | Compressed sensing | STABILITY | SPARSE SIGNALS | ENGINEERING, ELECTRICAL & ELECTRONIC | RECOVERY | MATRICES

random matrix | Lower bound | Upper bound | order statistics | restricted isometry property | Signal processing algorithms | Matching pursuit algorithms | Robustness | Random variables | Sparse matrices | Compressed sensing | STABILITY | SPARSE SIGNALS | ENGINEERING, ELECTRICAL & ELECTRONIC | RECOVERY | MATRICES

Journal Article

Duke mathematical journal, ISSN 0012-7094, 2011, Volume 159, Issue 1, pp. 145 - 185

.... Key ingredients in our proof are new estimates for sumsets in product sets and for exponential sums with the products of sets possessing special additive structure...

MATHEMATICS | RESTRICTED ISOMETRY PROPERTY | RECONSTRUCTION | SIGNAL RECOVERY | FOURIER | 41A46 | 94B60 | 94A12 | 11B13 | 11B30 | 11T23

MATHEMATICS | RESTRICTED ISOMETRY PROPERTY | RECONSTRUCTION | SIGNAL RECOVERY | FOURIER | 41A46 | 94B60 | 94A12 | 11B13 | 11B30 | 11T23

Journal Article

京都産業大学総合学術研究所所報, ISSN 1348-8465, 07/2014, Volume 9, pp. 193 - 196

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 11/2018, Volume 47, Issue 2, pp. 339 - 381

.... In the nineteenth century, Sylvester described these partitions as a sum of waves. We give detailed descriptions of these waves and, for the first time, show the asymptotics of the initial waves as N and n both go to infinity at about the same rate...

Fourier Analysis | Sylvester waves | 11P82 | Functions of a Complex Variable | Saddle-point method | Field Theory and Polynomials | Mathematics | Restricted partitions | 41A60 | Number Theory | Combinatorics | Asymptotics | MATHEMATICS | DILOGARITHM | FORMULAS

Fourier Analysis | Sylvester waves | 11P82 | Functions of a Complex Variable | Saddle-point method | Field Theory and Polynomials | Mathematics | Restricted partitions | 41A60 | Number Theory | Combinatorics | Asymptotics | MATHEMATICS | DILOGARITHM | FORMULAS

Journal Article

京都産業大学総合学術研究所所報, ISSN 1348-8465, 07/2013, Volume 8, pp. 177 - 180

Journal Article

Nutrients, ISSN 2072-6643, 2019, Volume 11, Issue 7, p. 1653

...) and the German Interview and Examination Survey for Children (KiGGS) (6–17 years) to evaluate BMI trajectories of infants receiving either lower protein (LP) or higher protein (HP) content formula; and (b...

cost-effectiveness | childhood | RISK | early nutrition | CHILDHOOD OVERWEIGHT | NUTRITION & DIETETICS | GERMANY | markov model | GROWTH | formula | ADIPOSITY | obesity prevention | BMI | Body Mass Index | Costs and Cost Analysis | Europe | Humans | Child, Preschool | Infant | Cost-Benefit Analysis - statistics & numerical data | Infant Formula - economics | Diet, Protein-Restricted - economics | Pediatric Obesity - prevention & control | Child Nutritional Physiological Phenomena - physiology | Adolescent | Child | Germany | Infant, Newborn | Intervention | Pediatrics | Economic analysis | Breastfeeding & lactation | Systematic review | Infants | Education policy | Prevention | Proteins | Body mass index | Randomization | Child development | Fat body | Children | Age | Obesity | Nutrition | Baby foods | Preventive medicine | Primary care | Babies | Health care expenditures | Body mass | Body size | Strategy | Cost analysis

cost-effectiveness | childhood | RISK | early nutrition | CHILDHOOD OVERWEIGHT | NUTRITION & DIETETICS | GERMANY | markov model | GROWTH | formula | ADIPOSITY | obesity prevention | BMI | Body Mass Index | Costs and Cost Analysis | Europe | Humans | Child, Preschool | Infant | Cost-Benefit Analysis - statistics & numerical data | Infant Formula - economics | Diet, Protein-Restricted - economics | Pediatric Obesity - prevention & control | Child Nutritional Physiological Phenomena - physiology | Adolescent | Child | Germany | Infant, Newborn | Intervention | Pediatrics | Economic analysis | Breastfeeding & lactation | Systematic review | Infants | Education policy | Prevention | Proteins | Body mass index | Randomization | Child development | Fat body | Children | Age | Obesity | Nutrition | Baby foods | Preventive medicine | Primary care | Babies | Health care expenditures | Body mass | Body size | Strategy | Cost analysis

Journal Article

Discrete Mathematics, ISSN 0012-365X, 02/2020, Volume 343, Issue 2, p. 111690

...), and using a result from the theory of partitions and also properties of Ramanujan sums and of the discrete Fourier transform of arithmetic functions, give an explicit formula for the number...

Discrete Fourier transform | Ramanujan sum | Order-restricted linear congruence | MATHEMATICS

Discrete Fourier transform | Ramanujan sum | Order-restricted linear congruence | MATHEMATICS

Journal Article

Journal of number theory, ISSN 0022-314X, 2018, Volume 188, pp. 324 - 334

...] considered the above linear congruence with s=1 and gave a formula for the number of solutions in terms of the Ramanujan sums...

Generalized Ramanujan sum | Restricted linear congruence | Discrete Fourier transforms | Generalized gcd | MATHEMATICS | Mathematics - Number Theory

Generalized Ramanujan sum | Restricted linear congruence | Discrete Fourier transforms | Generalized gcd | MATHEMATICS | Mathematics - Number Theory

Journal Article

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