A new sum-product estimate in prime fields

In this paper we obtain a new sum-product estimate in prime fields. In particular, we show that if $A\subseteq \mathbb{F}_p$ satisfies $|A|\le p^{64/117}$ then $$\max\{|A\pm A|, |AA|\} \gtrsim |A|^{39/32}.$$ Our argument builds on and improves some recent results of Shakan and Shkredov which use the eigenvalue method to reduce to estimating a fourth moment energy and the additive energy $E^+(P)$ of some subset $P\subseteq A+A$. Our main novelty comes from reducing the estimation of $E^+(P)$ to a point-plane incidence bound of Rudnev rather than a point line incidence bound of Stevens and de Zeeuw as done by Shakan and Shkredov.Comment: 16 page

Dual Frequency Microwave Resonator for Non-invasive detection of Aqueous Glucose

A novel dual-band microwave sensor for noninvasive detection of glucose concentration is presented. The proposed sensor consists of an open-loop resonator coupled to the input and the output of the structure. The resonator is loaded with modified split-ring resonators (SRRs) for dual-band operation as a sensing area. The open-loop resonator with electric coupling operating at low band functions as a host. The SRRs embedded into the open-loop resonator operate at a high band. In the proposed sensor, the overall size is miniaturized using the embedded resonator structure. This configuration has two transmission poles (TPs) and one transmission zero (TZ) in transmission coefficients, which are all sensitive to glucose-level (GL) variation. A dielectric container made with 3-D printer is used for dropping the aqueous glucose samples on the sensing section of the sensor. The experimental results obtained from the prototype having a dielectric container shows two resonance frequencies at 1.8 and 2.67 GHz as well as a TZ at 2.32 GHz. A glucose solution with deionized water in the range from 89 to 456 mg/dL is used in the measurements. For this range of glucose concentrations, the experimental frequency resolutions are 0.78 and 0.95 MHz/(mg/dL) based on the TP and the TZ, respectively.</p

T-Junction Loaded with Interdigital Capacitor for Differential Measurement of Permittivity

The microwave sensors have been successfully used for permittivity measurement. These sensors suffer from limited sensitivity and environmental effects. This article presents a novel T-junction highly sensitive microwave sensor for permittivity measurement of low-loss solid materials. The proposed sensor operation principle is based on downshifting the transmission zero (TZ) of the outputs of T-junction with the coupling of the material under test (MUT). The sensing section consists of an interdigital capacitor (IDC) located in between the lines of the T-junction. IDC is directly connected to output arms of T-junction, so that it could disturb the outputs strongly. Any change in electric field concentration in IDC directly is transmitted to the outputs and is translated as TZ change. Design steps including T-junction and IDC effects on outputs are presented in detail. The sensor operation principle is described through an equivalent circuit model, which is validated by simulation and experimental results. Two outputs of the proposed sensor show the same electrical performances, which allow differential operation mode. Hence, cross sensitivity due to environmental factors can be tolerated by the sensor. Measurement results of the fabricated prototype show 112-MHz frequency shift per unit permittivity change and a normalized sensitivity of 3.9%, which are larger than available similar sensors. The proposed sensor is implemented on a 22.22 × 18.76 × 1.6 mm3 printed circuit board.</p