Bounded gaps between primes with a given primitive root, II

Let $m$ be a natural number, and let $\mathcal{Q}$ be a set containing at least $\exp(C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\in\mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\mathcal{Q}}(\exp(C'm))$. This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin's conjecture. Let $E/\mathbb{Q}$ be an elliptic curve with an irrational $2$-torsion point. Assume GRH. Then for every $m$, there are infinitely many strings of $m$ consecutive primes $p$ for which $E(\mathbb{F}_p)$ is cyclic, all lying an interval of length $O_E(\exp(C'' m))$. If $E$ has CM, then the GRH assumption can be removed. Here $C$, $C'$, and $C''$ are absolute constants

Sequences that omit a box (modulo 1)

AbstractLet S=(aj)j=1∞ be a strictly increasing sequence of real numbers satisfying(0.1)aj+1−aj⩾σ>0. For an open box I in [0,1)d, we writeEI(d)(S)={x∈Rd:ajx∉I(mod1)forj⩾1}. It is shown that the Hausdorff dimension of EI(d)(S) is d−1 wheneverlimj→∞aj+1aj=1. The case d=1 is due to Boshernitzan. The proof builds on his approach.Now let S1,…,Sd be strictly increasing in N. Define E1′=EI′(S1,…,Sd) to be the set of x in [0, 1) for whichx(n1,…,nr)∉I(mod1)fornj∈Sj,n1<⋯<nd. A sequence S is said to fulfill condition D(C) if it containsBr=[ur,vr]∩S for which vr−ur→∞ and1+vr−ur⩽C#(Br). Kaufman has shown that EI′ is countable whenever S1,…,Sd fulfill condition D(C). Here it is shown that EI′ is finite under this hypothesis. An upper bound for #(EI′) is provided

Gaps of Smallest Possible Order between Primes in an Arithmetic Progression

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists). Suppose further that, $$\max \{p : p | q \} < \exp (\frac{\log x}{C \log \log x}) \; \; {and} \; \; \prod_{p | q} p < x^{\delta},$$ where $C$ and $\delta$ are suitable positive constants depending on $t$ and $\eta$. Let $a \in \mathbb{Z}$, $(a,q)=1$ and $$\mathcal{A} = \{n \in (x/2, x]: n \equiv a \pmod{q} \} .$$ We prove that there are primes $p_1 < p_2 < ... < p_t$ in $\mathcal{A}$ with $$p_t - p_1 \ll qt \exp (\frac{40 t}{9-20 \theta}) .$$ Here $\theta = (\log q) / \log x$.Comment: 18 page

Diophantine problems in variables restricted to the values 0 and 1

AbstractLet Fx1,…,xs be a form of degree d with integer coefficients. How large must s be to ensure that the congruence F(x1,…,xs) ≡ 0 (mod m) has a nontrivial solution in integers 0 or 1? More generally, if F has coefficients in a finite additive group G, how large must s be in order that the equation F(x1,…,xs) = 0 has a solution of this type? We deal with these questions as well as related problems in the group of integers modulo 1 and in the group of reals

Demonstration of a Nano-Enabled Space Power System

The Nano-Enabled Space Power System will demonstrate power systems with nanomaterial-enhanced components as are placement for CubeSat power generation, transmission, and storage. Successful flights of these nano-power systems will accelerate the use of this revolutionary technology in the aerospace industry. The use of nano materials in solar cells, wire harnesses,and lithium ion batteries can increase the device performance without significantly altering the devices physical dimensions or the devices operating range (temperature,voltage, current). In many cases, the use of nanomaterials widens the viable range of operating conditions, such as increased depth of discharge of lithium ion batteries, tunable bandgaps in solar cells, and increased flexure tolerance of wire harnesses

Distributed Timing and Localization (DiGiTaL)

The Distributed Timing and Localization (DiGiTaL) system provides nano satellite formations with unprecedented,centimeter-level navigation accuracy in real time and nanosecond-level time synchronization. This is achieved through the integration of a multi-constellation Global Navigation Satellite System (GNSS) receiver, a Chip-Scale Atomic Clock (CSAC), and a dedicated Inter-Satellite Link (ISL). In comparison, traditional single spacecraft GNSS navigation solutions are accurate only to the meter-level due to the sole usage of coarse pseudo-range measurements. To meet the strict requirements of future miniaturized distributed space systems, DiGiTaL uses powerful error-cancelling combinations of raw carrier-phase measurements which are exchanged between the swarming nano satellites through a decentralized network. A reduced-dynamics estimation architecture on board each individual nano satellite processes the resulting millimeter-level noise measurements to reconstruct the fullformation state with high accuracy

Development of New Research-Quality Low-Resource Magnetometers for Small Satellites

Researchers from the University of Michigan (UM) and NASA Goddard Spaceflight Center (GSFC) are partnering to develop new types of magnetometers for use on future small satellites. These new instruments not only fulfill stringent requirements for low-amplitude and high-precision measurements, they are also enabling the team to develop a new approach to achieve high-quality magnetic measurements from space, without the need for a boom. Typically, space-based magnetometers are deployed on a boom that extends from the space vehicle to reduce exposure of magnetic noise emanating from the spacecraft, which could potentially contaminate measurements. The UMNASA team has developed algorithms to identify and eliminate spacecraft magnetic noise, which will allow placement of these economical, science-grade instrument magnetometers on and inside the satellite bus, instead of on a boom

Highly Integrated THz Receiver Systems for Small Satellite Remote Sensing Applications

We are developing miniaturized, highly integrated Schottky receiver systems suitable for use in CubeSats or other small spacecraft platforms, where state-of-the-art performance and ultra-low mass, power, and volume are required. Current traditional Schottky receivers are too large to employ on a CubeSat. We will develop highly integrated receivers operating from 520-600 GHz and 1040-1200 GHz that are based on state-of-the-art receivers already developed at Jet Propulsion Laboratory (JPL) by using novel 3D multi layer packaging. This process will reduce both mass and volume by more than an order of magnitude, while preserving state-of-the-art noise performance. The resulting receiver systems will have a volume of approximately 25 x 25 x 40 millimeters (mm), a mass of 250 grams (g), and power consumption on the order of of 7 watts (W). Using these techniques, we will also integrate both receivers into a single frame, further reducing mass and volume for applications where dual band operation is advantageous. Additionally, as Schottky receivers offer significant gains in noise performance when cooled to 100 K, we will investigate the improvement gained by passively cooling these receivers. Work by Sierra Lobo Inc., with their Cryo Cube technology development program, offers the possibility of passive cooling to 100 K on CubeSat platforms for 1-unit (1U) sized instruments