The Ramanujan Journal, ISSN 1382-4090, 4/2019, Volume 48, Issue 3, pp. 639 - 654

Let T(a, b, c, d; n) denote the number of representations of n as $$a\frac{x(x+1)}{2}+ b\frac{y(y+1)}{2}+c\frac{z(z+1) }{2}+d\frac{w(w+1)}{2}$$ a x ( x + 1 ) 2...

11E25 | Fourier Analysis | Functions of a Complex Variable | Sum of squares | Field Theory and Polynomials | 11D85 | Mathematics | Theta function identities | Number Theory | Combinatorics | Sum of triangular numbers | MATHEMATICS

11E25 | Fourier Analysis | Functions of a Complex Variable | Sum of squares | Field Theory and Polynomials | 11D85 | Mathematics | Theta function identities | Number Theory | Combinatorics | Sum of triangular numbers | MATHEMATICS

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 5/2018, Volume 46, Issue 1, pp. 1 - 18

...Ramanujan J (2018) 46:1–18 https://doi.org/10.1007/s11139-016-9841-x Congruences modulo 64 and 1024 for overpartitions Olivia X. M. Yao 1 Received: 29 May 2016...

Overpartitions | 11P83 | Fourier Analysis | Congruences | Functions of a Complex Variable | Sum of squares | Field Theory and Polynomials | Mathematics | Number Theory | Combinatorics | 05A17 | MATHEMATICS | RAMANUJAN-TYPE CONGRUENCES | POWERS | SQUARES

Overpartitions | 11P83 | Fourier Analysis | Congruences | Functions of a Complex Variable | Sum of squares | Field Theory and Polynomials | Mathematics | Number Theory | Combinatorics | 05A17 | MATHEMATICS | RAMANUJAN-TYPE CONGRUENCES | POWERS | SQUARES

Journal Article

Nature Genetics, ISSN 1061-4036, 01/2017, Volume 49, Issue 1, pp. 125 - 130

... (ATXN1, UBE2E2, EBF1, RREB1, GSDMB, GRAMD3 and ENSA; P < 5 x 10(-8); false discovery rate < 1...

ALL-CAUSE MORTALITY | CARDIOMETABOLIC RISK | QUALITY | SUSCEPTIBILITY | BIOLOGY | GENETICS & HEREDITY | VISCERAL ADIPOSITY | METABOLIC RISK-FACTORS | INCIDENT CARDIOVASCULAR-DISEASE | HYPERTENSION | SUBCUTANEOUS FAT | Genetic Predisposition to Disease | Genome-Wide Association Study | Humans | Mice, Inbred C57BL | Genetic Loci - genetics | Adipocytes - cytology | Male | Obesity - genetics | Phenotype | Animals | Ethnic Groups - genetics | Adipocytes - metabolism | Genetic Markers - genetics | Polymorphism, Single Nucleotide - genetics | Female | Cell Differentiation | Mice | Body Fat Distribution | Cohort Studies | Adipose tissues | Genome-wide association studies | Usage | Magnetic resonance imaging | Analysis | Heart | Women | Genealogy | Medical imaging | Nuclear magnetic resonance--NMR | Body fat | Genomes | Adipocytes | Metabolism | Body mass index | Mens health | Rodents | Quality | adipocyte development | GWAS | differentiation | ectopic fat | obesity

ALL-CAUSE MORTALITY | CARDIOMETABOLIC RISK | QUALITY | SUSCEPTIBILITY | BIOLOGY | GENETICS & HEREDITY | VISCERAL ADIPOSITY | METABOLIC RISK-FACTORS | INCIDENT CARDIOVASCULAR-DISEASE | HYPERTENSION | SUBCUTANEOUS FAT | Genetic Predisposition to Disease | Genome-Wide Association Study | Humans | Mice, Inbred C57BL | Genetic Loci - genetics | Adipocytes - cytology | Male | Obesity - genetics | Phenotype | Animals | Ethnic Groups - genetics | Adipocytes - metabolism | Genetic Markers - genetics | Polymorphism, Single Nucleotide - genetics | Female | Cell Differentiation | Mice | Body Fat Distribution | Cohort Studies | Adipose tissues | Genome-wide association studies | Usage | Magnetic resonance imaging | Analysis | Heart | Women | Genealogy | Medical imaging | Nuclear magnetic resonance--NMR | Body fat | Genomes | Adipocytes | Metabolism | Body mass index | Mens health | Rodents | Quality | adipocyte development | GWAS | differentiation | ectopic fat | obesity

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 10/2018, Volume 47, Issue 1, pp. 221 - 235

.... Xia 1 · Olivia X. M. Yao 1 Received: 12 January 2017 / Accepted: 13 February 2017 / Published online: 29 April 2017 © Springer Science+Business Media New York...

11P83 | Fourier Analysis | Congruences | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | t -Core partitions | Mathematics | Number Theory | Combinatorics | 05A17 | t-Core partitions | MATHEMATICS | CORE PARTITIONS | FACTS

11P83 | Fourier Analysis | Congruences | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | t -Core partitions | Mathematics | Number Theory | Combinatorics | 05A17 | t-Core partitions | MATHEMATICS | CORE PARTITIONS | FACTS

Journal Article

Graphs and Combinatorics, ISSN 0911-0119, 1/2015, Volume 31, Issue 1, pp. 299 - 307

...Graphs and Combinatorics (2015) 31:299–307 DOI 10.1007/s00373-013-1378-9 ORIGINAL PAPER Two Formulas of 2-Color Off-Diagonal Rado Numbers Olivia X. M. Yao...

Off-diagonal Rado number | 05D10 | 05C55 | Schur number | 11B75 | Mathematics | Engineering Design | Combinatorics | Rado number | MATHEMATICS | A(X+Y)=BZ | Graphs | Integers | Texts | Mathematical analysis | Combinatorial analysis | Color

Off-diagonal Rado number | 05D10 | 05C55 | Schur number | 11B75 | Mathematics | Engineering Design | Combinatorics | Rado number | MATHEMATICS | A(X+Y)=BZ | Graphs | Integers | Texts | Mathematical analysis | Combinatorial analysis | Color

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 8/2017, Volume 43, Issue 3, pp. 619 - 631

... Olivia X. M. Yao 1 · Ya Juan Wang 1 Received: 16 September 2015 / Accepted: 7 April 2016 / Published online: 26 July 2016 © Springer Science+Business Media New York...

k )-parametrization of theta function | 11P83 | Congruence | Functions of a Complex Variable | Field Theory and Polynomials | Mathematics | p | Fourier Analysis | Theta function | Broken k -diamond partition | Number Theory | Combinatorics | Newman’s identity | 05A17 | (p; k)-parametrization of theta function | Broken k-diamond partition | FORMS | MATHEMATICS | PARITY | ANDREWS | (p; k)-parametrization of theta function | Newman's identity | K-DIAMOND PARTITIONS

k )-parametrization of theta function | 11P83 | Congruence | Functions of a Complex Variable | Field Theory and Polynomials | Mathematics | p | Fourier Analysis | Theta function | Broken k -diamond partition | Number Theory | Combinatorics | Newman’s identity | 05A17 | (p; k)-parametrization of theta function | Broken k-diamond partition | FORMS | MATHEMATICS | PARITY | ANDREWS | (p; k)-parametrization of theta function | Newman's identity | K-DIAMOND PARTITIONS

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 05/2014, Volume 10, Issue 3, pp. 669 - 674

Let bt(n) denote the number of t-regular partitions of n. Recently, Keith discovered some congruences modulo 3 for 9-regular partitions. He also conjectured...

Regular partitions | congruences | MATHEMATICS

Regular partitions | congruences | MATHEMATICS

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 5/2014, Volume 34, Issue 1, pp. 109 - 117

...Ramanujan J (2014) 34:109–117 DOI 10.1007/s11139-013-9493-z Parity results for 9-regular partitions Ernest X.W. Xia · Olivia X.M. Yao Received: 23 October 2012...

Partition | 11P83 | Fourier Analysis | Congruence | Functions of a Complex Variable | Field Theory and Polynomials | Regular partition | Mathematics | Number Theory | Combinatorics | 05A17 | MATHEMATICS | IDENTITIES | PARTS | DISTINCT

Partition | 11P83 | Fourier Analysis | Congruence | Functions of a Complex Variable | Field Theory and Polynomials | Regular partition | Mathematics | Number Theory | Combinatorics | 05A17 | MATHEMATICS | IDENTITIES | PARTS | DISTINCT

Journal Article

Journal of applied mathematics, ISSN 1110-757X, 9/2012, Volume 2012, pp. 1 - 14

.... In their paper, J. M. Borwein and P. B. Borwein [1] introduced the following three functions: (1.1)a(q)=∑m,n=-∞∞qm2+mn+n2,b(q)=∑m,n=-∞∞ωm-nqm2+mn+n2,c(q)=∑m,n=-∞∞q(m+1/3)2...

RAMANUJAN | MATHEMATICS, APPLIED | MODULAR IDENTITIES | CONVOLUTION SUMS | Series | Theorems (Mathematics) | Functions | Research | Functional equations | Studies | Mathematical models | Convolution | Elliptic functions | Mathematical analysis | Sums

RAMANUJAN | MATHEMATICS, APPLIED | MODULAR IDENTITIES | CONVOLUTION SUMS | Series | Theorems (Mathematics) | Functions | Research | Functional equations | Studies | Mathematical models | Convolution | Elliptic functions | Mathematical analysis | Sums

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 8/2013, Volume 31, Issue 3, pp. 373 - 396

... $p_{[c^{l}d^{m}]}(n)$ which is defined by $$\sum_{n=0}^\infty p_{[c^ld^m]}(n)q^n= \frac{1}{\prod_{j=1}^\infty (1-q^{cj})^{l}(1-q^{dj})^m}. $$ They discovered some analogues...

Partitions | 11P83 | Fourier Analysis | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | Partition congruences | Mathematics | Number Theory | Combinatorics | 2-Dissections | 05A17 | MATHEMATICS | PROOFS | BEAUTIFUL IDENTITY | EQUATIONS | MODULAR RELATIONS | GOLLNITZ-GORDON FUNCTIONS | Heterocyclic compounds

Partitions | 11P83 | Fourier Analysis | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | Partition congruences | Mathematics | Number Theory | Combinatorics | 2-Dissections | 05A17 | MATHEMATICS | PROOFS | BEAUTIFUL IDENTITY | EQUATIONS | MODULAR RELATIONS | GOLLNITZ-GORDON FUNCTIONS | Heterocyclic compounds

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 06/2015, Volume 11, Issue 4, pp. 1063 - 1072

The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let k(n) denote the number of Fu's k dots bracelet...

partitions | k dots bracelet partitions | Congruence | MATHEMATICS | CONGRUENCES MODULO POWERS | Jewelry

partitions | k dots bracelet partitions | Congruence | MATHEMATICS | CONGRUENCES MODULO POWERS | Jewelry

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 02/2014, Volume 10, Issue 1, pp. 115 - 123

In this paper, the convolution sums ∑i+3j=nσ(i)σ3(j) and ∑3i+j=nσ(i)σ3(j) are evaluated for all n ∈ ℕ, which answers an open problem given by Huard, Ou,...

sum of divisors function | Eisenstein series | convolution sums

sum of divisors function | Eisenstein series | convolution sums

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 09/2014, Volume 10, Issue 6, pp. 1421 - 1430

In this paper, the convolution sum ∑i+25j=n σ(i)σ(j) is evaluated for all n ∈ ℕ based on some identities due to Berndt, Lemire and Williams.

Journal Article

Nature (London), ISSN 0028-0836, 11/2018, Volume 563, Issue 7729, pp. 72 - 78

....1] [sup.2], Jeremy Miller [sup.1], Olivia Fong [sup.1], Karla E. Hirokawa [sup.1], Kanan Lathia [sup.1], Christine Rimorin [sup.1], Michael Tieu [sup.1...

MOUSE-BRAIN | VISUAL-CORTEX | RNA-SEQ | MULTIDISCIPLINARY SCIENCES | BRAIN-DEVELOPMENT | NEURONS | MOTOR CORTEX | MICE | CAJAL-RETZIUS CELLS | CEREBRAL-CORTEX | ADULT | Motor Cortex - anatomy & histology | Neocortex - anatomy & histology | Visual Cortex - cytology | Biomarkers - analysis | Male | Gene Expression Profiling | Organ Specificity | Neocortex - metabolism | Animals | Motor Cortex - metabolism | Sequence Analysis, RNA | Female | Neocortex - cytology | Glutamic Acid - metabolism | Mice | GABAergic Neurons - metabolism | Visual Cortex - metabolism | Single-Cell Analysis | Motor Cortex - cytology | Visual Cortex - anatomy & histology | RNA sequencing | Brain cells | Usage | Research | Neocortex | Visual cortex | Neurons | Taxonomy | Labelling | Cortex (motor) | Clustering | Ribonucleic acid--RNA | Gene sequencing | Datasets | γ-Aminobutyric acid | Glutamatergic transmission | Bioinformatics | Dissection

MOUSE-BRAIN | VISUAL-CORTEX | RNA-SEQ | MULTIDISCIPLINARY SCIENCES | BRAIN-DEVELOPMENT | NEURONS | MOTOR CORTEX | MICE | CAJAL-RETZIUS CELLS | CEREBRAL-CORTEX | ADULT | Motor Cortex - anatomy & histology | Neocortex - anatomy & histology | Visual Cortex - cytology | Biomarkers - analysis | Male | Gene Expression Profiling | Organ Specificity | Neocortex - metabolism | Animals | Motor Cortex - metabolism | Sequence Analysis, RNA | Female | Neocortex - cytology | Glutamic Acid - metabolism | Mice | GABAergic Neurons - metabolism | Visual Cortex - metabolism | Single-Cell Analysis | Motor Cortex - cytology | Visual Cortex - anatomy & histology | RNA sequencing | Brain cells | Usage | Research | Neocortex | Visual cortex | Neurons | Taxonomy | Labelling | Cortex (motor) | Clustering | Ribonucleic acid--RNA | Gene sequencing | Datasets | γ-Aminobutyric acid | Glutamatergic transmission | Bioinformatics | Dissection

Journal Article

15.
Full Text
Some modular relations for the Göllnitz-Gordon functions and Ramanujan’s modular equations

Indian Journal of Pure and Applied Mathematics, ISSN 0019-5588, 2/2014, Volume 45, Issue 1, pp. 53 - 74

Huang used the methods of Rogers, Watson and Bressoud to derive some new modular relations involving the Göllnitz-Gordon functions. In this paper, using...

theta functions | Numerical Analysis | Rogers-Ramanujan functions | Mathematics, general | Mathematics | Applications of Mathematics | Ramanujan’s modular equation | Göllnitz-Gordon functions | Ramanujan's modular equation | MATHEMATICS | PROOFS | Gollnitz-Gordon functions | IDENTITIES | Moduli spaces | Functions | Research | Functional equations | Mathematical research

theta functions | Numerical Analysis | Rogers-Ramanujan functions | Mathematics, general | Mathematics | Applications of Mathematics | Ramanujan’s modular equation | Göllnitz-Gordon functions | Ramanujan's modular equation | MATHEMATICS | PROOFS | Gollnitz-Gordon functions | IDENTITIES | Moduli spaces | Functions | Research | Functional equations | Mathematical research

Journal Article

Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, ISSN 0308-2105, 10/2019, Volume 149, Issue 5, pp. 1189 - 1205

.... For example, we prove that a(((15 X 17(3 alpha) +1)/8)) equivalent to (mod 5) for alpha > 0.

cubic partitions | congruences | θ function identities | theta function identities | MATHEMATICS | MATHEMATICS, APPLIED | CONTINUED-FRACTION | RAMANUJANS | Partitions (mathematics) | Partitions | Congruences

cubic partitions | congruences | θ function identities | theta function identities | MATHEMATICS | MATHEMATICS, APPLIED | CONTINUED-FRACTION | RAMANUJANS | Partitions (mathematics) | Partitions | Congruences

Journal Article

Ramanujan Journal, ISSN 1382-4090, 05/2017, Volume 43, Issue 1, pp. 215 - 228

In a recent work, Andrews gave a definition of combinatorial objects which he called singular overpartitions and proved that these singular overpartitions,...

Singular overpartitions | Theta functions | Arithmetic properties | MATHEMATICS | ANALOGS

Singular overpartitions | Theta functions | Arithmetic properties | MATHEMATICS | ANALOGS

Journal Article

The Ramanujan Journal, ISSN 1382-4090, 1/2013, Volume 30, Issue 1, pp. 1 - 7

...Ramanujan J (2013) 30:1–7 DOI 10.1007/s11139-012-9385-7 On the 4-dissections of certain inﬁnite products Olivia X.M. Yao · Ernest X.W. Xia Received: 30...

Fourier Analysis | Ramanujan’s continued fraction | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | 11A55 | Mathematics | Number Theory | Infinite product | 30B70 | Combinatorics | Ramanujan's continued fraction | MATHEMATICS | RAMANUJANS CONTINUED-FRACTION | Surgery

Fourier Analysis | Ramanujan’s continued fraction | Functions of a Complex Variable | Theta functions | Field Theory and Polynomials | 11A55 | Mathematics | Number Theory | Infinite product | 30B70 | Combinatorics | Ramanujan's continued fraction | MATHEMATICS | RAMANUJANS CONTINUED-FRACTION | Surgery

Journal Article

NATURE, ISSN 0028-0836, 09/2019, Volume 573, Issue 7772, pp. 61 - 61

Elucidating the cellular architecture of the human cerebral cortex is central to understanding our cognitive abilities and susceptibility to disease. Here we...

NEOCORTEX | HUMAN BRAIN | SINGLE | LONG NONCODING RNAS | MULTIDISCIPLINARY SCIENCES | TRANSCRIPTION | NEURONS | INTERNEURONS | SEQ | REVEALS | Astrocytes - cytology | Neural Inhibition | Species Specificity | Humans | Middle Aged | Male | Neurons - cytology | Transcriptome - genetics | Cerebral Cortex - cytology | Cerebral Cortex - metabolism | Astrocytes - classification | Biological Evolution | Young Adult | Neurons - classification | Animals | RNA-Seq | Adolescent | Adult | Female | Aged | Mice | Single-Cell Analysis | Principal Component Analysis | Physiological aspects | Cerebral cortex | Man | Human beings | Comparative analysis | Brain | Ontology | Architecture | Neurons | Conservation | Cognitive ability | Homology | Genomes | Ribonucleic acid--RNA | Gene expression | Temporal gyrus | Gene sequencing | Morphology | Nuclei (cytology) | Bioinformatics | Cortex (temporal) | Animal cognition

NEOCORTEX | HUMAN BRAIN | SINGLE | LONG NONCODING RNAS | MULTIDISCIPLINARY SCIENCES | TRANSCRIPTION | NEURONS | INTERNEURONS | SEQ | REVEALS | Astrocytes - cytology | Neural Inhibition | Species Specificity | Humans | Middle Aged | Male | Neurons - cytology | Transcriptome - genetics | Cerebral Cortex - cytology | Cerebral Cortex - metabolism | Astrocytes - classification | Biological Evolution | Young Adult | Neurons - classification | Animals | RNA-Seq | Adolescent | Adult | Female | Aged | Mice | Single-Cell Analysis | Principal Component Analysis | Physiological aspects | Cerebral cortex | Man | Human beings | Comparative analysis | Brain | Ontology | Architecture | Neurons | Conservation | Cognitive ability | Homology | Genomes | Ribonucleic acid--RNA | Gene expression | Temporal gyrus | Gene sequencing | Morphology | Nuclei (cytology) | Bioinformatics | Cortex (temporal) | Animal cognition

Journal Article

International Journal of Number Theory, ISSN 1793-0421, 02/2013, Volume 9, Issue 1, pp. 189 - 204

In this paper, using the (p, k)-parametrization of theta functions given by Alaca, Alaca and Williams, we establish some theta function identities. Explicit...

quadratic form | Eisenstein series | sum of squares | MATHEMATICS | IDENTITIES | SUMS

quadratic form | Eisenstein series | sum of squares | MATHEMATICS | IDENTITIES | SUMS

Journal Article

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