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Recent documents in Digital Commons @ West Chester Universityen-usMon, 25 Jan 2021 01:47:43 PST36002C-B-FLY in Platinic Chloride (H₂PtCl₆)
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_ptbr/2
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_ptbr/2Thu, 14 Jan 2021 18:00:10 PSTMonica Joshi2C-B-FLY in Platinic Bromide (H₂PtBr₆)
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_ptbr/1
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_ptbr/1Thu, 14 Jan 2021 18:00:06 PSTMonica Joshi2C-B-FLY in Mercuric Chloride (HgCl₂)
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_hgcl/2
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_hgcl/2Thu, 14 Jan 2021 17:55:08 PSTMonica Joshi2C-B-FLY in Mercuric Chloride (HgCl₂)
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_hgcl/1
https://digitalcommons.wcupa.edu/chem_lmtnpa_2cbfly_hgcl/1Thu, 14 Jan 2021 17:55:04 PSTMonica JoshiOn generating functions in additive number theory, II: lower-order terms and applications to PDEs
https://digitalcommons.wcupa.edu/math_facpub/78
https://digitalcommons.wcupa.edu/math_facpub/78Thu, 14 Jan 2021 11:06:10 PST
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.
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J. Brandes et al.Empathy and Community in the Age of Refugees: Petzold’s Radical Translation of Seghers’ Transit
https://digitalcommons.wcupa.edu/langcult_facpub/24
https://digitalcommons.wcupa.edu/langcult_facpub/24Thu, 14 Jan 2021 10:53:17 PST
Petzold’s film constitutes a radical translation of Seghers’ novel by transforming her tale of political refugees in Vichy France into an existential allegory depicting the fluidity of identities and relationships in a globalized world. The transitory existence of Petzold’s war refugee serves as an extreme example of the instability of modern life, which allows spectators to identify and empathize with migrants’ unpredictable journeys. Moreover, the director conveys the universality of his protagonist’s story by portraying him as an Everyman bereft of distinctive personality traits, by intermingling the past (Seghers’ plot) with the present (contemporary settings), and by situating his experiences in non-descript, liminal “non-places.” Both thematically and aesthetically, narrative is portrayed as establishing a community in an unstable contemporary world. Like the anti-hero of many modern Bildungsromane, Petzold’s protagonist fails to develop a stable identity and enduring friendships that anchor him in a community, but he creates his own family of listeners through his storytelling. In a similar vein, the film’s voice-over/narrator that bridges the fictional world with that of the audience underscores the film’s (and the novel’s) central theme: in a world of rapid change and mobility, the individual who may not be able to establish a stable identity or relationships, can create, as a narrator, a community of empathic listeners.
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Margarete LandwehrLibrary of Microcrystalline Tests for Novel Psychoactive Substances
https://digitalcommons.wcupa.edu/chem_facpub/35
https://digitalcommons.wcupa.edu/chem_facpub/35Wed, 13 Jan 2021 09:28:42 PST
A microcrystalline test is a precipitation reaction between a drug and a reagent, forming an insoluble drug-reagent complex that is unique to that specific test. These tests are quick, requiring minimal sample preparation and can be non-destructive. Therefore, they can be used as preliminary and confirmatory tests with expertise. Microcrystalline tests are one of the oldest analytical chemistry practices and their use for classic drugs such as cocaine, heroin and amphetamines is well-documented. However, there is very limited research on microcrystalline tests for the novel compounds encountered by law enforcement today. This research is an effort to increase understanding and promote use of microcrystalline tests for novel psychoactive substances.
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Matthew Quinn et al.Ramanujan and the Regular Continued Fraction Expansion of Real Numbers
https://digitalcommons.wcupa.edu/math_facpub/77
https://digitalcommons.wcupa.edu/math_facpub/77Mon, 11 Jan 2021 10:20:38 PST
In some recent papers, the authors considered regular continued fractions of the form [a0; a, · · · , a | {z } m , a2 , · · · , a2 | {z } m , a3 , · · · , a3 | {z } m , · · · ], where a0 ≥ 0, a ≥ 2 and m ≥ 1 are integers. The limits of such continued fractions, for general a and in the cases m = 1 and m = 2, were given as ratios of certain infinite series. However, these formulae can be derived from known facts about two continued fractions of Ramanujan. Motivated by these observations, we give alternative proofs of the results of the previous authors for the cases m = 1 and m = 2 and also use known results about other q-continued fractions investigated by Ramanujan to derive the limits of other infinite families of regular continued fractions
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James McLaughlin et al.Some Applications of a Bailey-type Transformation
https://digitalcommons.wcupa.edu/math_facpub/76
https://digitalcommons.wcupa.edu/math_facpub/76Mon, 11 Jan 2021 10:20:32 PST
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.
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James McLaughlin et al.A Hardy-Ramanujan-Rademacher-type formula for (r,s)-regular partitions
https://digitalcommons.wcupa.edu/math_facpub/75
https://digitalcommons.wcupa.edu/math_facpub/75Mon, 11 Jan 2021 10:20:26 PST
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).
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James McLaughlin et al.On a pair of identities from Ramanujan's lost notebook
https://digitalcommons.wcupa.edu/math_facpub/74
https://digitalcommons.wcupa.edu/math_facpub/74Mon, 11 Jan 2021 10:20:19 PST
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.
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James McLaughlin et al.A Reciprocity Relation for WP-Bailey Pairs
https://digitalcommons.wcupa.edu/math_facpub/73
https://digitalcommons.wcupa.edu/math_facpub/73Mon, 11 Jan 2021 10:20:13 PST
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.
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James McLaughlin et al.Further results on vanishing coefficients in infinite product expansions
https://digitalcommons.wcupa.edu/math_facpub/72
https://digitalcommons.wcupa.edu/math_facpub/72Mon, 11 Jan 2021 10:20:06 PST
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 of integer partitions.
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James McLaughlinGeneral multi-sum transformations and some implications
https://digitalcommons.wcupa.edu/math_facpub/71
https://digitalcommons.wcupa.edu/math_facpub/71Mon, 11 Jan 2021 10:19:56 PST
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.
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James McLaughlinRefinements of Some Partition Inequalities
https://digitalcommons.wcupa.edu/math_facpub/70
https://digitalcommons.wcupa.edu/math_facpub/70Mon, 11 Jan 2021 10:19:49 PST
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.
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James McLaughlinApplications of the Heine and Bauer-Muir transformations to Rogers-Ramanujan type continued fractions
https://digitalcommons.wcupa.edu/math_facpub/69
https://digitalcommons.wcupa.edu/math_facpub/69Mon, 11 Jan 2021 10:19:42 PST
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 G(aq, b, λq)/G(a, b, λ) is derived. Finally we derive a number of new versions of some beautiful continued fraction expansions of Ramanujan for certain combinations of infinite products, with the following being an example: (−a, b; q)∞ − (a, −b; q)∞ (−a, b; q)∞ + (a, −b; q)∞ = (a − b) 1 − ab − (1 − a2)(1 − b2)q 1 − abq2 − (a − bq2)(b − aq2)q 1 − abq4 − (1 − a2q2)(1 − b2q2)q3 1 − abq6 − (a − bq4)(b − aq4)q3 1 − abq8 − ··· .
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Jongsil Lee et al.Mock Theta Function Identities Deriving from Bilateral Basic Hypergeometric Series
https://digitalcommons.wcupa.edu/math_facpub/68
https://digitalcommons.wcupa.edu/math_facpub/68Mon, 11 Jan 2021 10:19:34 PST
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 certain class, say ζ , that there is a theta function θχ (q) such that lim q→ζ (χ(q)−θχ (q)) exists, as q → ζ from within the unit circle.
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James McLaughlinA Convergence Theorem for Continued Fractions of the Form K_{n=1}^{\infty}a_{n}/1
https://digitalcommons.wcupa.edu/math_facpub/67
https://digitalcommons.wcupa.edu/math_facpub/67Thu, 07 Jan 2021 12:27:56 PST
In this paper we present a convergence theorem for continued fractions of the form K∞n=1an/1. By deriving conditions on the an which ensure that the odd and even parts of K∞n=1an/1 converge, these same conditions also ensure that they converge to the same limit. Examples will be given.
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James McLaughlin et al.Powers of a matrix and combinatorial identities
https://digitalcommons.wcupa.edu/math_facpub/66
https://digitalcommons.wcupa.edu/math_facpub/66Thu, 07 Jan 2021 12:27:37 PST
In this article we obtain a general polynomial identity in k variables, where k ≥ 2 is an arbitrary positive integer. We use this identity to give a closed-form expression for the entries of the powers of a k × k matrix. Finally, we use these results to derive various combinatorial identities.
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James McLaughlin et al.The Convergence and Divergence of q-Continued Fractions outside the Unit Circle
https://digitalcommons.wcupa.edu/math_facpub/65
https://digitalcommons.wcupa.edu/math_facpub/65Thu, 07 Jan 2021 12:27:28 PST
We consider two classes of q-continued fraction whose odd and even parts are limit 1-periodic for |q| > 1, and give theorems which guarantee the convergence of the continued fraction, or of its odd- and even parts, at points outside the unit circle.
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Douglas Bowman et al.