Bounded weak solutions to superlinear Dirichlet double phase problems

In this paper we study a Dirichlet double phase problem with a parametric superlinear right-hand side that has subcritical growth. Under very general assumptions on the data, we prove the existence of at least two nontrivial bounded weak solutions to such problem by using variational methods and critical point theory. In contrast to other works we do not need to suppose the Ambrosetti-Rabinowitz condition

Fast Plasma Investigation for MMS: Simulation of the Burst Triggering System

The Magnetospheric Multiscale (MMS) mission will study small-scale reconnection structures and their rapid motions from closely spaced platforms using instruments capable of high angular, energy, and time resolution measurements. To meet these requirements, the Fast Plasma Instrument (FPI) consists of eight (8) identical half top-hat electron sensors and eight (8) identical ion sensors and an Instrument Data Processing Unit (IDPU). The sensors (electron or ion) are grouped into pairs whose 6 degree x 180 degree fields-of-view (FOV) are set 90 degrees apart. Each sensor is equipped with electrostatic aperture steering to allow the sensor to scan a 45 degree x 180 degree fan about the its nominal viewing (0 deflection) direction. Each pair of sensors, known as the Dual Electron Spectrometer (DES) and the Dual Ion Spectrometer (DIS), occupies a quadrant on the MMS spacecraft and the combination of the eight electron/ion sensors, employing aperture steering, image the full-sky every 30-ms (electrons) and 150-ms (ions), respectively. To probe the diffusion regions of reconnection, the highest temporal/spatial resolution mode of FPI results in the DES complement of a given spacecraft generating 6.5-Mb (raised dot) per second of electron data while the DIS generates 1.1-Mb (raised dot) per second of ion data yielding an FPI total data rate of 6.6-Mb (raised dot) per second. The FPI electron/ion data is collected by the IDPU then transmitted to the Central Data Instrument Processor (CIDP) on the spacecraft for science interest ranking. Only data sequences that contain the greatest amount of temporal/spatial structure will be intelligently down-linked by the spacecraft. This requires a data ranking process known as the burst trigger system. The burst trigger system uses pseudo physical quantities to approximate the local plasma environments. As each pseudo quantity will have a different value, a set of two scaling factors is employed for each pseudo term. These pseudo quantities are then combined at the instrument, spacecraft, and observatory level leading to a final ranking of data based on expected scientific interest. Here, we present simulations of the fixed point burst trigger system for the FPI. A variety of data sets based on previous mission data as well as analytical formulations are tested. Comparisons of floating point calculations versus the fixed point hardware simulation are shown. Analysis of the potential sources of error from overflows, quantization, etc. are examined and mitigation methods are presented. Finally a series of calibration curves are presented, showing the expected error in pseudo quantities based solely on the scale parameters chosen and the expected data range. We conclude with a presentation of the current base-lined FPI burst trigger approach

Elliptic Problems with Nonhomogeneous Differential Operators and Multiple Solutions

International audienceAbstract The present survey aims to report on recent advances in the study of nonlinear elliptic problems whose differential part is expressed by a general operator in divergence form. The pattern of such differential operator is the p-Laplacian Δp with 1 < p < +∞. More general operators can be considered, possibly having completely different properties, for instance not satisfying any homogeneity requirement. A major objective of our work is to provide existence theorems of multiple solutions for boundary value problems governed by such general operators. In this direction, a three nontrivial solutions theorem is presented. In the case of problems determined by the p-Laplacian, we give a theorem ensuring the existence of at leastfour nontrivial solutions. Moreover, a complete sign information is available: two positive solutions, a negative solution and a nodal (sign-changing) solution. Finally, we provide a theorem guaranteeing the existence of a positive solution for a probleminvolving the (p, q)- Laplacian operator Δp+Δq , with 1 < q < p, and a nonlinearity depending on the solution and its gradient

Fast Plasma Instrument for MMS: Data Compression Simulation Results

Magnetospheric Multiscale (MMS) mission will study small-scale reconnection structures and their rapid motions from closely spaced platforms using instruments capable of high angular, energy, and time resolution measurements. To meet these requirements, the Fast Plasma Instrument (FPI) consists of eight (8) identical half top-hat electron sensors and eights (8) identical ion sensors and an Instrument Data Processing Unit (IDPU). The sensors (electron or ion) are grouped into pairs whose 6 deg x 180 deg fields-of-view (FOV) are set 90 deg apart. Each sensor is equipped with electrostatic aperture steering to allow the sensor to scan a 45 deg x 180 deg fan about its nominal viewing (0 deg deflection) direction. Each pair of sensors, known as the Dual Electron Spectrometer (DES) and the Dual Ion Spectrometer (DIS), occupies a quadrant on the MMS spacecraft and the combination of the eight electron/ion sensors, employing aperture steering, image the full-sky every 30-ms (electrons) and 150-ms (ions), respectively. To probe the results in the DES complement of a given spacecraft generating 6.5-Mbs(exp -1) of electron data while the DIS generates 1.1-Mbs(exp -1) of ion data yielding an FPI total data rate of 6.6-MBs(exp -1). The FPI electron/ion data is collected by the IDPU then transmitted to the Central Data Instrument Processor (CIDP) on the spacecraft for science interest ranking. Only data sequences that contain the greatest amount of temporal/spatial structure will be intelligently down-linked by the spacecraft. Currently, the FPI data rate allocation to the CIDP is 1.5-Mbs(exp -1). Consequently, the FPI-IDPU must employ data/image compression to meet this CIDP telemetry allocation. Here, we present simulations of the CCSDS 122.0-B-1 algorithm-based compression of the FPI-DES electron data. Compression analysis is based upon a seed of re-processed Cluster/PEACE electron measurements. Topics to be discussed include: review of compression algorithm; data quality; data formatting/organization; and, implications for data/matrix pruning. To conclude a presentation of the base-lined FPI data compression approach is provided