arXiv subjects
Publications and source records attributed to Dandan Chen.
In this paper, we establish a $q$-integral formula by using the orthogonality relation, and also provide a new proof of the $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials. A new $q$-beta integral with five parameters is evaluated.
It is well known that Ramanujan conjectured congruences modulo powers of $5$, $7$ and and $11$ for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of $5$ for the crank parity function. The generating function for the analogous rank parity function is $f(q)$, the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. Recently we proved congruences modulo powers of $5$ for the rank parity function, and here we extend these congruences for powers of $7$. We also show how these congruences imply congruences modulo powers of $5$ and $7$ for the coefficients of the related third order mock theta function $ω(q)$, using Atkin-Lehner involutions and transformation results of Zwegers. Finally we a prove a family of congruences modulo powers of $7$ for the crank parity function.
We find the Hecke-Rogers type series representations of generating functions of the Hurwitz class numbers which is very close to certain mock theta functions. We also prove two combinatorial interpretation of Hurwitz class numbers appeared on OEIS.
We experimentally and computationally study the flow of a quasi-two-dimensional emulsion through a constricting hopper shape. Our area fractions are above jamming such that the droplets are always in contact with one another and are in many cases highly deformed. At the lowest flow rates, the droplets exit the hopper via intermittent avalanches. At the highest flow rates, the droplets exit continuously. The transition between these two types of behaviors is a fairly smooth function of the mean strain rate. The avalanches are characterized by a power law distribution of the time interval between droplets exiting the hopper, with long intervals between the avalanches. Our computational studies reproduce the experimental observations by adding a flexible compliance to the system. The compliance results in continuous flow at high flow rates, and allows the system to clog at low flow rates leading to avalanches. The computational results suggest that the interplay of the flow rate and compliance controls the presence or absence of the avalanches.
We find and prove a class of congruences modulo 4 for Andrews' partition with certain ternary quadratic form. We also discuss distribution of $\overline{\mathcal{EO}}(n)$ and further prove that $\overline{\mathcal{EO}}(n)\equiv0\pmod4$ for almost all $n$. This study was inspired by similar congruences modulo 4 in the work by the second author and Garvan.
It is well known that Ramanujan conjectured congruences modulo powers of 5, 7 and and 11 for the partition function. These were subsequently proved by Watson (1938) and Atkin (1967). In 2009 Choi, Kang, and Lovejoy proved congruences modulo powers of 5 for the crank parity function. The generating function for rank parity function is f(q), which is the first example of a mock theta function that Ramanujan mentioned in his last letter to Hardy. We prove congruences modulo powers of 5 for the rank parity function.
Standardized large-scale testing can be a debatable topic, in which test fairness sits at its very core. This study found that two out of five recent multi-group DIF detection methods are capable of capturing both the uniform and nonuniform DIF that affects test fairness. Still, no prior research has demonstrated the relative performance of these two methods when they are compared with each other. These two methods are the improved Wald test and the generalized logistic regression procedure. This study assessed the commonalities and differences between two sets of empirical results from these two methods with the latest TIMSS math score data. The primary conclusion was that the improved Wald test is relatively more established than the generalized logistic regression procedure for multi-group DIF analysis. Empirical results from this study may inform the selection of a multi-group DIF method in the ILSA score analysis.
We construct a class of companion elliptic functions associated with the even Dirichlet characters. Using the well-known properties of the classical Weierstrass elliptic function $\wp(z|τ)$ as the blueprint, we will derive their representations in terms of $q$-series and partial fractions, explore the significance of the coefficients of their power series expansions and establish the modular properties under the actions of the arithmetic groups $Γ_0(N)$ and $Γ_1(N)$.
Based on the values of the Weierstrass elliptic function $\wp(z|τ)$ at $z=πτ/2$, $(π+πτ)/{2}, (π+πτ)/{4},(π+2πτ)/{4}$ and the theory of modular forms on the arithmetic group $Γ_0(2)$, we decompose $θ_2^{2n}(τ)$ as sum of Eisenstein series and a cusp forms. Using the recurrence relation of $\wp^{(2n)}(z|τ)$, we provide an algorithm to determine the exact form of these cusp forms.
Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.
Recently, Jelínek conjectured that there exists a bijection between certain restricted permutations and Fishburn matrices such that the bijection verifies the equidistribution of several statistics. The main objective of this paper is to establish such a bijection.
We experimentally study the jamming of quasi-two-dimensional emulsions. Our experiments consist of oil-in-water emulsion droplets confined between two parallel plates. From the droplet outlines, we can determine the forces between every droplet pair to within 8% over a wide range of area fractions $ϕ$. We study three bidisperse samples that jam at area fractions $ϕ_c \approx 0.86$. Our data show that for $ϕ> ϕ_c$, the contact numbers and pressure have power-law dependence on $ϕ-ϕ_c$ in agreement with the critical scaling found in numerical simulations. Furthermore, we see a link between the interparticle force law and the exponent for the pressure scaling, supporting prior computational observations. We also observe linear-like force chains (chains of large inter-droplet forces) that extend over 10 particle lengths, and examine the origin of their linearity. We find that the relative orientation of large force segments are random and that the tendency for force chains to be linear is not due to correlations in the direction of neighboring large forces, but instead occurs because the directions are biased towards being linear to balance the forces on each droplet.
We experimentally study the shear flow of oil-in-water emulsion droplets in a thin sample chamber with a hopper shape. In this thin chamber, the droplets are quasi-2D in shape. The sample is at an area fraction above jamming and forced to flow with a constant flux rate. Stresses applied to a droplet from its neighbors deform the droplet outline, and this deformation is quantified to provide an ad hoc measure of the stress. As the sample flows through the hopper we see large fluctuations of the stress, similar in character to what has been seen in other flows of complex fluids. Periods of time with large decreases in stress are correlated with bursts of elementary rearrangement events ("T1 events" where four droplets rearrange). More specifically, we see a local relationship between these observations: a T1 event decreases the inter-droplet forces up to 3 droplet diameters away from the event. This directly connects microscopic structural changes to macroscopic fluctuations, and confirms theoretical pictures of local rearrangements influencing nearby regions. These local rearrangements are an important means of reducing and redistributing stresses within a flowing material.
The rheology of dense amorphous materials under large shear strain is not fully understood, partly due to the difficulty of directly viewing the microscopic details of such materials. We use a colloidal suspension to simulate amorphous materials, and study the shear-induced structural relaxation with fast confocal microscopy. We quantify the plastic rearrangements of the particles in several ways. Each of these measures of plasticity reveals spatially heterogeneous dynamics, with localized regions where many particles are strongly rearranging by these measures. We examine the shapes of these regions and find them to be essentially isotropic, with no alignment in any particular direction. Furthermore, individual particles are equally likely to move in any direction, other than the overall bias imposed by the strain.