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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
Journal Article
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
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
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
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
Journal Article
by Xu, Ce
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
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
Journal Article
Computational Statistics and Data Analysis, ISSN 0167-9473, 05/2016, Volume 97, pp. 33 - 46
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
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
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
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
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
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
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
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
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
Journal Article
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