Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{\Phi_n(q)},$$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for $a,m\geq 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to appear in Adv. Appl. Mat

Proof of two conjectures of Z.-W. Sun on congruences for Franel numbers

For all nonnegative integers n, the Franel numbers are defined as $$f_n=\sum_{k=0}^n {n\choose k}^3.$$ We confirm two conjectures of Z.-W. Sun on congruences for Franel numbers: \sum_{k=0}^{n-1}(3k+2)(-1)^k f_k &\equiv 0 \pmod{2n^2}, \sum_{k=0}^{p-1}(3k+2)(-1)^k f_k &\equiv 2p^2 (2^p-1)^2 \pmod{p^5}, where n is a positive integer and p>3 is a prime.Comment: 8 pages, minor changes, to appear in Integral Transforms Spec. Func

Factors of binomial sums from the Catalan triangle

By using the Newton interpolation formula, we generalize the recent identities on the Catalan triangle obtained by Miana and Romero as well as those of Chen and Chu. We further study divisibility properties of sums of products of binomial coefficients and an odd power of a natural number. For example, we prove that for all positive integers $n_1, ..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, the expression $$n_1^{-1}{n_1+n_{m}\choose n_1}^{-1} \sum_{k=1}^{n_1}k^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}\choose n_i+k}$$ is either an integer or a half-integer. Moreover, several related conjectures are proposed.Comment: 15 pages, final versio

Systematic Digitized Treatment of Engineering Line-Diagrams

YesIn engineering design, there are many functional relationships which are difficult to express into a simple and exact mathematical formula. Instead they are documented within a form of line graphs (or plot charts or curve diagrams) in engineering handbooks or text books. Because the information in such a form cannot be used directly in the modern computer aided design (CAD) process, it is necessary to find a way to numerically represent the information. In this paper, a data processing system for numerical representation of line graphs in mechanical design is developed, which incorporates the process cycle from the initial data acquisition to the final output of required information. As well as containing the capability for curve fitting through Cubic spline and Neural network techniques, the system also adapts a novel methodology for use in this application: Grey Models. Grey theory have been used in various applications, normally involved with time-series data, and have the characteristic of being able to handle sparse data sets and data forecasting. Two case studies were then utilized to investigate the feasibility of Grey models for curve fitting. Furthermore, comparisons with the other two established techniques show that the accuracy was better than the Cubic spline function method, but slightly less accurate than the Neural network method. These results are highly encouraging and future work to fully investigate the capability of Grey theory, as well as exploiting its sparse data handling capabilities is recommended

Projected shell model study of odd-odd f-p-g shell proton-rich nuclei

A systematic study of 2-quasiparticle bands of the proton-rich odd-odd nuclei in the mass A ~ 70-80 region is performed using the projected shell model approach. The study includes Br-, Rb-, and Y-isotopes with N = Z+2, and Z+4. We describe the energy spectra and electromagnetic transition strengths in terms of the configuration mixing of the angular-momentum projected multi-quasiparticle states. Signature splitting and signature inversion in the rotational bands are discussed and are shown to be well described. A preliminary study of the odd-odd N = Z nucleus, 74Rb using the concept of spontaneous symmetry breaking is also presented.Comment: 14 pages, 7 figures, final version accepted by Phys. Rev.

BES3 time of flight monitoring system

A Time of Flight monitoring system has been developed for BES3. The light source is a 442-443 nm laser diode, which is stable and provides a pulse width as narrow as 50 ps and a peak power as large as 2.6 W. Two optical-fiber bundles with a total of 512 optical fibers, including spares, are used to distribute the light pulses to the Time of Flight counters. The design, operation, and performance of the system are described.Comment: 8 pages 16 figures, submitted to NI

Transport across nanogaps using semiclassically consistent boundary conditions

Charge particle transport across nanogaps is studied theoretically within the Schrodinger-Poisson mean field framework and the existence of limiting current investigated. It is shown that the choice of a first order WKB wavefunction as the transmitted wave leads to self consistent boundary conditions and gives results that are significantly different in the non-classical regime from those obtained using a plane transmitted wave. At zero injection energies, the quantum limiting current density, J_c, is found to obey the local scaling law J_c ~ (V_g)^alpha/(D)^{5-2alpha} with the gap separation D and voltage V_g. The exponent alpha > 1.1 with alpha --> 3/2 in the classical regime of small de Broglie wavelengths. These results are consistent with recent experiments using nanogaps most of which are found to be in a parameter regime where classical space charge limited scaling holds away from the emission dominated regime.Comment: 4 pages, 4 ps figure

Fewer epistemological challenges for connectionism

Seventeen years ago, John McCarthy wrote the note Epistemological challenges for connectionism as a response to Paul Smolensky’s paper 'On the proper treatment of connectionism'. I will discuss the extent to which the four key challenges put forward by McCarthy have been solved, and what are the new challenges ahead. I argue that there are fewer epistemological challenges for connectionism, but progress has been slow. Nevertheless, there is now strong indication that neural-symbolic integration can provide effective systems of expressive reasoning and robust learning due to the recent developments in the field

Pooling techniques for bioassay screening

Pooling techniques commonly are used to increase the throughput of samples used for screening purposes. While the advantages of such techniques are increased analytical efficiency and cost savings, the sensitivity of measurements decreases because it is inversely proportional to the number of samples in the pools. Consequently, uncertainties in estimates of dose and risk which are based on the results of pooled samples increase as the number of samples in the pools increases in all applications. However, sensitivities may not be seriously degraded, for example, in urinalysis, if the samples in the pools are of known time duration, or if the fraction of some attribute of the grab urine samples to that in a 24-hour composite is known (e.g., mass, specific gravity, creatinine, or volume, per 24-h interval). This paper presents square and cube pooling schemes that greatly increase throughput and can considerably reduce analytical costs (on a sample basis). The benefit-cost ratios for 5{times}5 square and 5{times}5{times}5 cube pooling schemes are 2.5 and 8.3, respectively. Three-dimensional and higher arrayed pooling schemes would result in even greater economies; however, significant improvements in analytical sensitivity are required to achieve these advantages. These are various other considerations for designing a pooling scheme, where the number of dimensions and of samples in the optimum array are influenced by: (1) the minimal detectable amount (MDA) of the analytical processes, (2) the screening dose-rate requirements, (3) the maximum masses or volumes of the composite samples that can be analyzed, (4) the information already available from results of composite analysis, and (5) the ability of an analytical system to guard against both false negative and false positive results. Many of these are beyond the scope of this paper but are being evaluated