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Articles 1 - 30 of 42
Full-Text Articles in Number Theory
Noncrystallographic Tail-Triangle C-Groups Of Rank 4 And Interlacing Number 2, Mark L. Loyola, Nonie Elvin S. Leyrita, Ma. Louise Antonette N. De Las Peñas
Noncrystallographic Tail-Triangle C-Groups Of Rank 4 And Interlacing Number 2, Mark L. Loyola, Nonie Elvin S. Leyrita, Ma. Louise Antonette N. De Las Peñas
Mathematics Faculty Publications
This work applies the modular reduction technique to the Coxeter group of rank 4 having a star diagram with labels 5, 3, and k = 3,4,5, or 6. As moduli, we use the primes in the quadratic integer ring Z[τ], where τ = (1+√5)/2, the golden ratio. We prove that each reduced group is a C-group, regardless of the prime used in the reduction. We also classify each reduced group as a reflection group over a finite field, whenever applicable.
The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
The Spectrum Of Nim-Values For Achievement Games For Generating Finite Groups, Bret J. Benesh, Dana C. Ernst, Nándor Sieben
Mathematics Faculty Publications
We study an impartial achievement game introduced by Anderson and Harary. The game is played by two players who alternately select previously unselected elements of a finite group. The game ends when the jointly selected elements generate the group. The last player able to make a move is the winner of the game. We prove that the spectrum of nim-values of these games is {0, 1, 2, 3, 4}. This positively answers two conjectures from a previous paper by the last two authors.
On Generating Functions In Additive Number Theory, Ii: Lower-Order Terms And Applications To Pdes, J. Brandes, Scott T. Parsell, C. Poulias, G. Shakan, R. C. Vaughn
On Generating Functions In Additive Number Theory, Ii: Lower-Order Terms And Applications To Pdes, J. Brandes, Scott T. Parsell, C. Poulias, G. Shakan, R. C. Vaughn
Mathematics Faculty Publications
We obtain asymptotics for sums of the form
Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n),
involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has
sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon,
and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.
Applications Of The Heine And Bauer-Muir Transformations To Rogers-Ramanujan Type Continued Fractions, Jongsil Lee, James Mclaughlin, Jaebum Sohn
Applications Of The Heine And Bauer-Muir Transformations To Rogers-Ramanujan Type Continued Fractions, Jongsil Lee, James Mclaughlin, Jaebum Sohn
Mathematics Faculty Publications
In this paper we show that various continued fractions for the quotient of general Ramanujan functions G(aq, b, λq)/G(a, b, λ) may be derived from each other via Bauer-Muir transformations. The separate convergence of numerators and denominators play a key part in showing that the continued fractions and their Bauer-Muir transformations converge to the same limit. We also show that these continued fractions may be derived from either Heine’s continued fraction for a ratio of 2φ1 functions, or other similar continued fraction expansions of ratios of 2φ1 functions. Further, by employing essentially the same methods, a new continued fraction for …
Mock Theta Function Identities Deriving From Bilateral Basic Hypergeometric Series, James Mclaughlin
Mock Theta Function Identities Deriving From Bilateral Basic Hypergeometric Series, James Mclaughlin
Mathematics Faculty Publications
The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of 2ψ2 series ∞ ∑n=−∞ (a, c;q)n (b,d;q)n z n . Three transformation formulae for this series due to Bailey are used to derive various transformation and summation formulae for both these mock theta functions and the corresponding bilateral series. New and existing summation formulae for these bilateral series are also used to make explicit in a number of cases the fact that for a mock theta function, say χ(q), and a root of unity in a …
Comparing Local Constants Of Ordinary Elliptic Curves In Dihedral Extensions, Sunil Chetty
Comparing Local Constants Of Ordinary Elliptic Curves In Dihedral Extensions, Sunil Chetty
Mathematics Faculty Publications
We establish, for a substantial class of elliptic curves, that the arithmetic local constants introduced by Mazur and Rubin agree with quotients of analytic root numbers.
On The Dimension Of Algebraic-Geometric Trace Codes, Phong Le, Sunil Chetty
On The Dimension Of Algebraic-Geometric Trace Codes, Phong Le, Sunil Chetty
Mathematics Faculty Publications
We study trace codes induced from codes defined by an algebraic curve X. We determine conditions on X which admit a formula for the dimension of such a trace code. Central to our work are several dimension reducing methods for the underlying functions spaces associated to X.
General Multi-Sum Transformations And Some Implications, James Mclaughlin
General Multi-Sum Transformations And Some Implications, James Mclaughlin
Mathematics Faculty Publications
We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence {g(k)}), to be reduced to an infinite q-product times a single basic hypergeometric sum. Various applications are given, including summation formulae for some q orthogonal polynomials, and various multisums that are expressible as infinite products.
Arithmetic Local Constants For Abelian Varieties With Extra Endomorphisms, Sunil Chetty
Arithmetic Local Constants For Abelian Varieties With Extra Endomorphisms, Sunil Chetty
Mathematics Faculty Publications
This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than ℤ. We then study the growth of the p∞- Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers k ⊂ K ⊂ F in which [F : K] is not a p-power extension.
Refinements Of Some Partition Inequalities, James Mclaughlin
Refinements Of Some Partition Inequalities, James Mclaughlin
Mathematics Faculty Publications
In the present paper we initiate the study of a certain kind of partition inequality, by showing, for example, that if M ≥ 5 is an integer and the integers a and b are relatively prime to M and satisfy 1 ≤ a < b < M/2, and the c(m, n) are defined by 1 (sqa, sqM−a; qM)∞ − 1 (sqb , sqM−b ; qM)∞ := X m,n≥0 c(m, n)s mq n , then c(m, Mn) ≥ 0 for all integers m ≥ 0, n ≥ 0. A similar result is proved for the integers d(m, n) defined by (−sqa , −sqM−a ; q M)∞ − (−sqb , −sqM−b ; q M)∞ := X m,n≥0 d(m, n)s mq n . In each case there are obvious interpretations in terms of integer partitions. For example, if p1,5(m, n) (respectively p2,5(m, n)) denotes the number of partitions of n into exactly m parts ≡ ±1( mod 5) (respectively ≡ ±2( mod 5)), then for each integer n ≥ 1, p1,5(m, 5n) ≥ p2,5(m, 5n), 1 ≤ m ≤ 5n.
Further Results On Vanishing Coefficients In Infinite Product Expansions, James Mclaughlin
Further Results On Vanishing Coefficients In Infinite Product Expansions, James Mclaughlin
Mathematics Faculty Publications
We extend results of Andrews and Bressoud on the vanishing of coefficients in the series expansions of certain infinite products. These results have the form that if (q r−tk, qmk−(r−tk) ; q mk)∞ (q r, qmk−r; qmk)∞ =: X∞ n=0 cnq n , for certain integers k, m s and t, where r = sm+t, then ckn−rs is always zero. Our theorems also partly give a simpler reformulation of results of Alladi and Gordon, but also give results for cases not covered by the theorems of Alladi and Gordon. We also give some interpretations of the analytic results in terms …
Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li
Computing Local Constants For Cm Elliptic Curves, Sunil Chetty, Lung Li
Mathematics Faculty Publications
Let E/k be an elliptic curve with CM by O. We determine a formula for (a generalization of) the arithmetic local constant of Mazur-Rubin at almost all primes of good reduction. We apply this formula to the CM curves defined over Q and are able to describe extensions F/Q over which the O-rank of E grows.
On A Pair Of Identities From Ramanujan's Lost Notebook, James Mclaughlin, Andrew Sills
On A Pair Of Identities From Ramanujan's Lost Notebook, James Mclaughlin, Andrew Sills
Mathematics Faculty Publications
Using a pair of two variable series-product identities recorded by Ramanujan in the lost notebook as inspiration, we find some new identities of similar type. Each identity immediately implies an infinite family of Rogers-Ramanujan type identities, some of which are well-known identities from the literature. We also use these identities to derive some general identities for integer partitions.
A Reciprocity Relation For Wp-Bailey Pairs, James Mclaughlin, Peter Zimmer
A Reciprocity Relation For Wp-Bailey Pairs, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
We derive a new general transformation for WP-Bailey pairs by considering the a certain limiting case of a WP-Bailey chain previously found by the authors, and examine several consequences of this new transformation. These consequences include new summation formulae involving WP-Bailey pairs. Other consequences include new proofs of some classical identities due to Jacobi, Ramanujan and others, and indeed extend these identities to identities involving particular specializations of arbitrary WP-Bailey pairs.
A Hardy-Ramanujan-Rademacher-Type Formula For (R,S)-Regular Partitions, James Mclaughlin, Scott Parsell
A Hardy-Ramanujan-Rademacher-Type Formula For (R,S)-Regular Partitions, James Mclaughlin, Scott Parsell
Mathematics Faculty Publications
Let pr,s(n) denote the number of partitions of a positive integer n into parts containing no multiples of r or s, where r > 1 and s > 1 are square-free, relatively prime integers. We use classical methods to derive a Hardy-Ramanujan-Rademacher-type infinite series for pr,s(n).
Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, Rüdiger Göbel, Katrin Leistner, Peter Loth, Lutz Strüngmann
Infinitary Equivalence Of Zp- Modules With Nice Decomposition Bases, Rüdiger Göbel, Katrin Leistner, Peter Loth, Lutz Strüngmann
Mathematics Faculty Publications
Warfield modules are direct summands of simply presented Zp - modules, or, alternatively, are Zp - modules possessing a nice decomposition basis with simply presented cokernel. They have been classified up to isomorphism by theor Ilm-Kaplansky and Warfield invariants. Taking a model theoretic point of view and using infinitary languages we give here a complete theoretic characterization of a large class of Zp - modules having a nice decomposition basis. As a corollary, we obtain the classical classification of countable Warfield modules. This generalizes results by Barwise and Eklof.
Semi-Direct Galois Covers Of The Affine Line, Linda Gruendken, Laura L. Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson
Semi-Direct Galois Covers Of The Affine Line, Linda Gruendken, Laura L. Hall-Seelig, Bo-Hae Im, Ekin Ozman, Rachel Pries, Katherine Stevenson
Mathematics Faculty Publications
Let k be an algebraically closed field of characteristic p > 0. Let G be a semi-direct product of the form (Z/`Z) b o Z/pZ where b is a positive integer and ` is a prime distinct from p. In this paper, we study Galois covers ψ : Z → P 1 k ramified only over ∞ with Galois group G. We find the minimal genus of a curve Z which admits a covering map of this form and we give an explicit formula for this genus in terms of ` and p. The minimal genus occurs when b equals the …
General Wp-Bailey Chains, James Mclaughlin, Peter Zimmer
General Wp-Bailey Chains, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
Motivated by a recent paper of Liu and Ma, we describe a number of general WP-Bailey chains. We show that many of the existing WP-Bailey chains (or branches of the WP-Bailey tree), including chains found by Andrews, Warnaar and Liu and Ma, arise as special cases of these general WP-Bailey chains. We exhibit three new branches of the WP-Bailey tree, branches which also follow as special cases of these general WP-Bailey chains. Finally, we describe a number of new transformation formulae for basic hypergeometric series which arise as consequences of these new WP-Bailey chains.
Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi
Continued Fraction Proofs Of M-Versions Of Some Identities Of Rogers-Ramanujan-Slater Type, Douglas Bowman, James Mclaughlin, Nancy Wyshinksi
Mathematics Faculty Publications
We derive two general transformations for certain basic hypergeometric series from the recurrence formulae for the partial numerators and denominators of two q-continued fractions previously investigated by the authors. By then specializing certain free parameters in these transformations, and employing various identities of Rogers-Ramanujan type, we derive m-versions of these identities. Some of the identities thus found are new, and some have been derived previously by other authors, using different methods. By applying certain transformations due to Watson, Heine and Ramanujan, we derive still more examples of such m-versions of Rogers Ramanujan-type identities.
An Identity Motivated By An Amazing Identity Of Ramanujan, James Mclaughlin
An Identity Motivated By An Amazing Identity Of Ramanujan, James Mclaughlin
Mathematics Faculty Publications
Ramanujan stated an identity to the effect that if three sequences {an}, {bn} and {cn} are defined by r1(x) =: ∑∞ n=0 anx n , r2(x) =: ∑∞ n=0 bnx n and r3(x) =: ∑∞ n=0 cnx n (here each ri(x) is a certain rational function in x), then a 3 n + b 3 n − c 3 n = (−1)n , ∀ n ≥ 0. Motivated by this amazing identity, we state and prove a more general identity involving eleven sequences, the new identity being ”more general” in the sense that equality holds not just for the power …
Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin
Some New Transformations For Bailey Pairs And Wp-Bailey Pairs, James Mclaughlin
Mathematics Faculty Publications
We derive several new transformations relating WP-Bailey pairs. We also consider the corresponding relations relating standard Bailey pairs, and as a consequence, derive some quite general expansions for products of theta functions which can also be expressed as certain types of Lambert series.
Some Applications Of A Bailey-Type Transformation, James Mclaughlin, Peter Zimmer
Some Applications Of A Bailey-Type Transformation, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
If k is set equal to aq in the definition of a WP Bailey pair, βn(a, k) = Xn j=0 (k/a)n−j (k)n+j (q)n−j (aq)n+j αj (a, k), this equation reduces to βn = Pn j=0 αj . This seemingly trivial relation connecting the αn’s with the βn’s has some interesting consequences, including several basic hypergeometric summation formulae, a connection to the Prouhet-Tarry-Escott problem, some new identities of the Rogers-Ramanujan-Slater type, some new expressions for false theta series as basic hypergeometric series, and new transformation formulae for poly-basic hypergeometric series.
Some Implications Of The Wp-Bailey Tree, James Mclaughlin, Peter Zimmer
Some Implications Of The Wp-Bailey Tree, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
We consider a special case of a WP-Bailey chain of George Andrews, and use it to derive a number of curious transformations of basic hypergeometric series. We also derive two new WP-Bailey pairs, and use them to derive some new transformations for basic hypergeometric series. Finally, we briefly consider the implications of WP-Bailey pairs (αn(a, k), βn(a, k)), in which αn(a, k) is independent of k, for generalizations of identities of the Rogers-Ramanujan type.
Some Identities Between Basic Hypergeometric Series Deriving From A New Bailey-Type Transformation, James Mclaughlin, Peter Zimmer
Some Identities Between Basic Hypergeometric Series Deriving From A New Bailey-Type Transformation, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
We prove a new Bailey-type transformation relating WPBailey pairs. We then use this transformation to derive a number of new 3- and 4-term transformation formulae between basic hypergeometric series.
Some New Families Of Tasoevian- And Hurwitzian Continued Fractions, James Mclaughlin
Some New Families Of Tasoevian- And Hurwitzian Continued Fractions, James Mclaughlin
Mathematics Faculty Publications
We derive closed-form expressions for several new classes of Hurwitzian- and Tasoevian continued fractions, including [0; p − 1, 1, u(a + 2nb) − 1, p − 1, 1, v(a + (2n + 1)b) − 1 ]∞ n=0, [0; c + dmn] ∞n=1 and [0; eun, fvn] ∞n=1. One of the constructions used to produce some of these continued fractions can be iterated to produce both Hurwitzian- and Tasoevian continued fractions of arbitrary long quasi-period, with arbitrarily many free parameters and whose limits can be determined as ratios of certain infinite series. We also derive expressions for arbitrarily long finite …
Some Observations On Khovanskii's Matrix Methods For Extracting Roots Of Polynomials, James Mclaughlin, B. Sury
Some Observations On Khovanskii's Matrix Methods For Extracting Roots Of Polynomials, James Mclaughlin, B. Sury
Mathematics Faculty Publications
In this article we apply a formula for the n-th power of a 3×3 matrix (found previously by the authors) to investigate a procedure of Khovanskii’s for finding the cube root of a positive integer. We show, for each positive integer α, how to construct certain families of integer sequences such that a certain rational expression, involving the ratio of successive terms in each family, tends to α 1/3 . We also show how to choose the optimal value of a free parameter to get maximum speed of convergence. We apply a similar method, also due to Khovanskii, to a …
Symmetry And Specializability In The Continued Fraction Expansions Of Some Infinite Products, James Mclaughlin
Symmetry And Specializability In The Continued Fraction Expansions Of Some Infinite Products, James Mclaughlin
Mathematics Faculty Publications
Let f(x) ∈ Z[x]. Set f0(x) = x and, for n ≥ 1, define fn(x) = f(fn−1(x)). We describe several infinite families of polynomials for which the infinite product Y∞ n=0 ( 1 + 1 fn(x) ) has a specializable continued fraction expansion of the form S∞ = [1; a1(x), a2(x), a3(x), . . . ], where ai(x) ∈ Z[x] for i ≥ 1. When the infinite product and the continued fraction are specialized by letting x take integral values, we get infinite classes of real numbers whose regular continued fraction expansion is predictable. We also show that, under some …
Some Properties Of The Distribution Of The Numbers Of Points On Elliptic Curves Over A Finite Prime Field, Saiying He, James Mclaughlin
Some Properties Of The Distribution Of The Numbers Of Points On Elliptic Curves Over A Finite Prime Field, Saiying He, James Mclaughlin
Mathematics Faculty Publications
Let p ≥ 5 be a prime and for a, b ∈ Fp, let Ea,b denote the elliptic curve over Fp with equation y 2 = x 3 + a x + b. As usual define the trace of Frobenius ap, a, b by #Ea,b(Fp) = p + 1 − ap, a, b. We use elementary facts about exponential sums and known results about binary quadratic forms over finite fields to evaluate the sums P t∈Fp ap, t, b, P t∈Fp ap, a, t, Pp−1 t=0 a 2 p, t, b, Pp−1 t=0 a 2 p, a, t and Pp−1 …
Some More Long Continued Fractions, I, James Mclaughlin, Peter Zimmer
Some More Long Continued Fractions, I, James Mclaughlin, Peter Zimmer
Mathematics Faculty Publications
In this paper we show how to construct several infinite families of polynomials D(¯x, k), such that p D(¯x, k) has a regular continued fraction expansion with arbitrarily long period, the length of this period being controlled by the positive integer parameter k. We also describe how to quickly compute the fundamental units in the corresponding real quadratic fields.
Continued Fractions With Multiple Limits, Douglas Bowman, James Mclaughlin
Continued Fractions With Multiple Limits, Douglas Bowman, James Mclaughlin
Mathematics Faculty Publications
For integers m ≥ 2, we study divergent continued fractions whose numerators and denominators in each of the m arithmetic progressions modulo m converge. Special cases give, among other things, an infinite sequence of divergence theorems, the first of which is the classical Stern-Stolz theorem. We give a theorem on a class of Poincar´e type recurrences which shows that they tend to limits when the limits are taken in residue classes and the roots of their characteristic polynomials are distinct roots of unity. We also generalize a curious q-continued fraction of Ramanujan’s with three limits to a continued fraction with …