Distribution of the k-regular partition function modulo composite integers M

Let $b_k(n)$ denote the $k-$regular partitons of a natural number $n$. In this paper, we study the behavior of $b_k(n)$ modulo composite integers $M$ which are coprime to $6$. Specially, we prove that for arbitrary $k-$regular partiton function $b_k(n)$ and integer $M$ coprime to $6$, there are infinitely many Ramanujan-type congruences of $b_k(n)$ modulo $M$

Fully Printed High-Frequency Phased-Array Antenna on Flexible Substrate

To address the issues of flexible electronics needed for surface-to-surface, surface-to-orbit, and back-to-Earth communications necessary for manned exploration of the Moon, Mars, and beyond, a room-temperature printing process has been developed to create active, phased-array antennas (PAAs) on a flexible Kapton substrate. Field effect transistors (FETs) based on carbon nanotubes (CNTs), with many unique physical properties, were successfully proven feasible for phased-array antenna systems. The carrier mobility of an individual CNT is estimated to be at least 100,000 sq cm/V(dot)s. The CNT network in solution has carrier mobility as high as 46,770 sq cm/V(dot)s, and has a large current-density carrying capacity of approx. 1,000 mA/sq cm , which corresponds to a high carrying power of over 2,000 mW/ sq cm. Such high carrier mobility, and large current carrying capacity, allows the achievement of high-speed (>100 GHz), high-power, flexible electronic circuits that can be monolithically integrated on NASA s active phasedarray antennas for various applications, such as pressurized rovers, pressurized habitats, and spacesuits, as well as for locating beacon towers for lunar surface navigation, which will likely be performed at S-band and attached to a mobile astronaut. A fully printed 2-bit 2-element phasedarray antenna (PAA) working at 5.6 GHz, incorporating the CNT FETs as phase shifters, is demonstrated. The PAA is printed out at room temperature on 100-mm thick Kapton substrate. Four CNT FETs are printed together with microstrip time delay lines to function as a 2-bit phase shifter. The FET switch exhibits a switching speed of 0.2 ns, and works well for a 5.6-GHz RF signal. The operating frequency is measured to be 5.6 GHz, versus the state-of-the-art flexible FET operating frequency of 52 MHz. The source-drain current density is measured to be over 1,000 mA/sq cm, while the conventional organic FETs, and single carbon nanotube-based FETs, are typically in the mA to mA/sq cm range. The switching voltage used is 1.8 V, while the state-of-the-art flexible FET has a gate voltage around 50 V. The gate voltage can effectively control the source-drain current with an ON-OFF ratio of over 1,000 obtained at a low Vds bias of 1.8 V. The azimuth steering angles of PAA are measured at 0deg, -14.5deg, -30deg, and 48.6deg. The measured far-field patterns agree well with simulation results. The efficiency of the 2-bit 2-element PAA is measured to be 39 percent, including the loss of transmission line, FET switch, and coupling loss of RF probes. With further optimization, the efficiency is expected to be around 50-60 percent

An improvement on the parity of Schur's partition function

We improve S.-C. Chen's result on the parity of Schur's partition function. Let $A(n)$ be the number of Schur's partitions of $n$, i.e., the number of partitions of $n$ into distinct parts congruent to $1, 2 \mod{3}$. S.-C. Chen \cite{MR3959837} shows $\small \frac{x}{(\log{x})^{\frac{47}{48}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}$. In this paper, we improve Chen's result to $\frac{x}{(\log{x})^{\frac{11}{12}}} \ll \sharp \{0\le n\le x:A(2n+1)\; \text{is odd}\}\ll \frac{x}{(\log{x})^{\frac{1}{2}}}.

Surface Plasmonics and Its Applications in Infrared Sensing

Surface plasmonic waves have been extensively researched due to their strong surface confinement. The strong surface confinement allows high absorption in an infrared (IR) detector with a thin active absorption region. The excitation of surface plasmonic resonance (SPR) depends on the metallic structures and the interface materials. This enables engineering of plasmonic-enhanced IR detector properties (e.g. detection wavelength, polarization and angular dependence) by properly designing the plasmonic structures. This chapter first gives a brief review of the surface plasmonic waves, followed by the description of SPR excitation in a metallic two-dimensional (2D) sub-wavelength hole array (2DSHA) structure. The applications of the 2DSHA SPR in IR detector enhancement are then presented with a discussion of the polarization and angular dependence

The Effect of FRAX on the Prediction of Osteoporotic Fractures in Urban Middle-aged and Elderly Healthy Chinese Adults

OBJECTIVE: We aimed to analyze the applicability of a fracture risk assessment tool for the prediction of osteoporotic fractures in middle-aged and elderly healthy Chinese adults. METHODS: A standard questionnaire was administered, and bone mineral density was measured in residents visiting the Dongliu Street Community Health Service Center. Paired t-tests were used to compare the FRAX-based probabilities of fractures estimated with and without consideration of bone mineral density. Risk stratification and partial correlation analyses were applied to analyze the associations between FRAX-based probabilities and body mass index or bone mineral density at different sites. RESULTS: A total of 444 subjects were included in this study. Of these subjects, 175 (39.59%) were diagnosed as osteoporotic, and 208 (47.06%) were diagnosed as osteopenic. The Kappa value for the detection of osteoporosis at the L1-L4 lumbar spine and femoral neck was 0.314. The FRAX-based 10-year major osteoporotic fracture probability and hip osteoporotic fracture probability estimated without considering bone mineral density were 4.93% and 1.64%, respectively; when estimated while considering bone mineral density, these probabilities were 4.97% and 1.54%, respectively. A significant positive association was observed between the FRAX-based fracture probabilities estimated with and without consideration of bone mineral density, while significant negative associations between body mass index and the estimated FRAX-based fracture probabilities after adjustment for age and the estimated FRAX-based fracture probabilities and femoral neck bone mineral density were identified. These results remained the same after controlling for lumbar spine bone mineral density. CONCLUSIONS: The Chinese FRAX model could predict osteoporotic fracture risk regardless of whether bone mineral density was considered and was especially appropriate for predicting osteoporotic fractures of the femoral neck