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Content of PHYSICS OF GASES, PLASMAS, AND ELECTRIC DISCHARGES in our journal
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 Select Generation of Ultrahigh-Velocity Collisionless Electrostatic Shocks Using an Ultra-Intense Laser Pulse Interacting with Foil-Gas Target Shu-Kai He, Jin-Long Jiao, Zhi-Gang Deng, Feng Lu, Lei Yang, Fa-Qiang Zhang, Ke-Gong Dong, Wei Hong, Zhi-Meng Zhang, Bo Zhang, Jian Teng, Wei-Min Zhou, Yu-Qiu Gu Chin. Phys. Lett.    2019, 36 (10): .   DOI: 10.1088/0256-307X/36/10/105201 Abstract   HTML   PDF (949KB) Ultra high-velocity collisionless shocks are generated using an ultra-intense laser interacting with foil-gas target, which consists of copper foil and helium gas. The energy of helium ions accelerated by shock and the proton probing image of the shock electrostatic field show that the shock velocity is 0.02$c$, where $c$ is the light speed. The numerical and theory studies indicate that the collisionless shock velocity exceeding 0.1$c$ can be generated by a laser pulse with picosecond duration and an intensity of 10$^{20}$ W/cm$^{2}$. This system may be relevant to the study of mildly relativistic velocity collisionless shocks in astrophysics.
 Select Comparison of ITG and TEM Microturbulence in DIII–D Tokamak Wei Hu, Hong-Ying Feng, Wen-Lu Zhang Chin. Phys. Lett.    2019, 36 (8): .   DOI: 10.1088/0256-307X/36/8/085201 Abstract   HTML   PDF (1798KB) Microturbulence excited by ion temperature gradient (ITG)-dominant and trapped electron mode (TEM)-dominant instabilities is compared in the fusion plasmas using gyrokinetic simulations based on the realistic equilibrium data from DIII–D discharges. Collisions make a difference between two plasmas and give rise to similar results to those found in previous research experiments [Chin. Phys. Lett. 35 (2018) 105201]. The mode structures and frequency spectrum of the most unstable modes characterized by the ITG-dominant and TEM-dominant instabilities are excited in the lower and higher $T_{\rm e}$ plasmas in the linear simulations. In the nonlinear simulations, contour plots of the perturbed potential are shown in the saturated stage, with the radial correlation lengths being microscopic on the order of the ion thermal gyroradius $\rho_{\rm i}$ in both the ITG and the TEM microturbulences. The dominant mode wavelengths of the perturbed potential increase when evolving from linear to nonlinear stages in both simulations, with the fluctuation energy spreading from the linearly dominant modes to the nonlinearly dominant modes. The radial correlation lengths are about 4$\rho_{\rm i}$ and the electron density fluctuation intensities are about 0.85% in the nonlinear saturated stage, which are in agreement with the experimental results.
 Select Electron-Electron Collision Term Describing the Reflections Induced Scattering in a Magnetized Plasma Chao Dong, Ding Li, Chang Jiang Chin. Phys. Lett.    2019, 36 (7): .   DOI: 10.1088/0256-307X/36/7/075201 Abstract   HTML   PDF (437KB) For an electron-electron collision with characteristic scale length larger than the relative gyro-radius of the two colliding electrons, when the initial relative parallel kinetic energy cannot surmount the Coulomb repulsive potential, reflection will occur with interchange of the parallel velocities of the two electrons after the collision. The Fokker–Planck approach is employed to derive the electron collision term $\mathcal{C}_{\rm R}$ describing parallel velocity scattering due to the reflections for a magnetized plasma where the average electron gyro-radius is much smaller than the Debye length but much larger than the Landau length. The electron parallel velocity friction and diffusion coefficients due to the reflections are evaluated, which are found not to depend on the electron perpendicular velocity. By studying the temporal evolution of the $H$ quantity due to $\mathcal{C}_{\rm R}$, it is found that $\mathcal{C}_{\rm R}$ eventually makes the system relax to a state in which the electron parallel velocity distribution is decoupled from the perpendicular velocity distribution.
 Select Impact of Sheath Boundary Conditions and Magnetic Flutter on Evolution and Distribution of Transient Particle and Heat Fluxes in the Edge-Localized Mode Burst by Experimental Advanced Superconducting Tokamak Simulation Yan-Bin Wu, Tian-Yang Xia, Fang-Chuan Zhong, Bin Gui, EAST Team Chin. Phys. Lett.    2019, 36 (4): .   DOI: 10.1088/0256-307X/36/4/045201 Abstract   HTML   PDF (1527KB) To study the evolution and distribution of the transient particle and heat fluxes during the edge-localized modes (ELMs) burst on the experimental advanced superconducting tokamak (EAST), the BOUT$^{++}$ six-field two-fluid model with sheath boundary conditions (SBCs) and magnetic flutter terms in the parallel thermal conduction is used to simulate the evolution of the profiles and growing process of the fluxes at divertor targets. Although SBCs hardly play a role in the linear phase, in the nonlinear phase both SBCs and magnetic flutter can change the dominant toroidal mode. SBCs are able to broaden the frequency distribution of the turbulence. The magnetic flutter increases the ELM size from 2.8% to 8.4%, and it doubles the amplitudes of the radial heat and particle transport coefficients at outer midplane (OMP), at around 1.0 m$^{2}$s$^{-1}$. It is then able to increase the particle and heat flux at the divertor targets and to broaden the radial distribution of the parallel heat flux towards the targets.
 Select Successive Picket Drive for Mitigating the Ablative Richtmyer–Meshkov Instability Meng Li, Wen-Hua Ye Chin. Phys. Lett.    2019, 36 (2): .   DOI: 10.1088/0256-307X/36/2/025201 Abstract   HTML   PDF (1535KB) The ablative Richtmyer–Meshkov instability (ARMI) is crucial to the successful ignition implosion of the inertial confinement fusion (ICF) because of its action as the seed of the Rayleigh–Taylor instability. In usual ICF implosions, the first shock driven by various foots of the pulses plays a central role in the ARMI growth. We propose a new scheme for refraining from ARMI with a pulse of successive pickets. With the successive-picket pulse design, a rippled capsule surface is compressed by three successive shocks with sequentially strengthening intensities and ablated stabilization, and the ablative Richtmyer–Meshkov growth is mitigated quite effectively. Our numerical simulations and theoretical analyses identify the validity of this scheme.
 Select Electron Magnetohydrodynamics Magnetic Reconnection Experiment on Keda Linear Magnetized Plasma Device Feibin Fan, Jinlin Xie, Qiaofeng Zhang, Longlong Sang, Weixing Ding Chin. Phys. Lett.    2019, 36 (1): .   DOI: 10.1088/0256-307X/36/1/015201 Abstract   HTML   PDF (894KB) We conduct an electron magnetohydrodynamics magnetic reconnection experiment with guide-field in our Keda linear magnetized plasma device, in which two pulsed currents with the same direction are conducted in parallel with the axial direction of the main chamber of the device using two long aluminum sticks. After approximately 5 μs, an X-type magnetic field line topology is formed at the center of the chamber. With the formation of the X-type topology of magnetic field lines, we can also find the rapid increase of the current and ratio of the common flux to the private flux in this area. Additionally, a reduction in the plasma density and the plasma density concentration along one pair of separatrices can also be found.
 Select Particle Growth in an Experimental Dusty Plasma System Shou-Zhi Jiang, Xue-Ni Hou, Jie Kong, Lorin S. Matthews, Truell W. Hyde, Feng Huang, Min-Juan Wang Chin. Phys. Lett.    2018, 35 (12): .   DOI: 10.1088/0256-307X/35/12/125201 Abstract   HTML   PDF (1102KB) The coagulation and growth process of dust particles is investigated through laboratory experiment in a plasma system. A large number of dust particles with different sizes and shapes are formed. The growth process is characterized by the scattering laser intensity and fractal dimension. The comparisons of dust particles and scattering laser intensity obtained at different rf powers are presented. The three-dimensional distribution of dust particles is also given. These results provide an experimental basis for dust growth investigation.
 Select Relativistic Spherical Plasma Waves in a Collisional and Warm Plasma Zhong-Kui Kuang, Li-Hong Cheng, Pan-Fei Geng, Rong-An Tang, Ju-Kui Xue Chin. Phys. Lett.    2018, 35 (12): .   DOI: 10.1088/0256-307X/35/12/125202 Abstract   HTML   PDF (611KB) Under Lagrange coordinates, the relativistic spherical plasma wave in a collisional and warm plasma is discussed theoretically. Within the Lagrange coordinates and using the Maxwell and hydrodynamics equations, a wave equation describing the relativistic spherical wave is derived. The damped oscillating spherical wave solution is obtained analytically using the perturbation theory. Because of the coupled effects of spherical geometry, thermal pressure, and collision effect, the electron damps the periodic oscillation. The oscillation frequency and the damping rate of the wave are related to not only the collision and thermal pressure effect but also the space coordinate. Near the center of the sphere, the thermal pressure significantly reduces the oscillation period and the damping rate of the wave, while the collision effect can strongly influence the damping rate. Far away from the spherical center, only the collision effect can reduce the oscillation period of the wave, while the collision effect and thermal pressure have weak influence on the damping rate.
 Select Collisional Effects on Drift Wave Microturbulence in Tokamak Plasmas Wei Hu, Hong-Ying Feng, Chao Dong Chin. Phys. Lett.    2018, 35 (10): .   DOI: 10.1088/0256-307X/35/10/105201 Abstract   HTML   PDF (563KB) Collisional effects on the microturbulence, excited by the electrostatic drift-wave instability, are investigated through first-principle large scale gyrokinetic particle simulations using the realistic discharge parameters of the DIII–D Tokamak. In the linear simulations, the growth rates of the drift waves are decreased by the collisions compared to the collisionless simulations in the lower and higher $T_{\rm e}$ plasmas. In the lower $T_{\rm e}$ plasma, the collisions can promote the transition of the drift wave regime from the TEM-dominant instability to the ITG-dominant instability. The zonal flows are excited by the microturbulence and work as a modulation mechanism for the microturbulence in the nonlinear simulations. Microturbulence can excite high frequency zonal flows in the collisionless plasmas, which is in agreement with the theoretical work. In the lower $T_{\rm e}$ plasma, the collisions decrease the microturbulence in the nonlinear saturated stage compared to the collisionless simulations, which are beneficial for the plasma confinement. In the higher $T_{\rm e}$ plasma, the final saturated microturbulence shows a slight change.
 Select Electron Dynamics and Characteristics of Attosecond Electromagnetic Emissions in Relativistic Laser-Plasma Interactions Yi-Ying Wu, Quan-Li Dong, Zhao-Hua Wang, Ping Liu, Cheng-Zhen Wang, Yi-Hui Zhang, Zheng-Ming Sheng, Jie Zhang Chin. Phys. Lett.    2018, 35 (9): .   DOI: 10.1088/0256-307X/35/9/095201 Abstract   HTML   PDF (1387KB) Generation of attosecond electromagnetic (EM) pulses and the associated electron dynamics are studied using particle-in-cell simulations of relativistic laser pulses interacting with over-dense plasma foil targets. The interaction process is found to be so complicated even in the situation of utilizing driving laser pulses of only one cycle. Two electron bunches closely involved in the laser-driven wavebreaking process contribute to attosecond EM pulses through the coherent synchrotron emission process whose spectra are found to follow an exponential decay rule. Detailed investigations of electron dynamics indicate that the early part of the reflected EM emission is the high-harmonics produced through the relativistic oscillating mirror mechanism. High harmonics are also found to be generated through the Bremsstrahlung radiation by one electron bunch that participates in the wavebreaking process and decelerates when it experiences the local wavebreaking-generated high electrostatic field in the moving direction.
 Select Resistance and Reactance of Monopole Fields Induced by a Test Charge Drifting Off-Axis in a Cold and Collisional Cylindrical Plasma M. S. Bawa'aneh, A. M. Al-Khateeb, Y. -c. Ghim Chin. Phys. Lett.    2018, 35 (8): .   DOI: 10.1088/0256-307X/35/8/085101 Abstract   HTML   PDF (521KB) We study the interaction of a uniform, cold and collisional plasma with a test charged particle moving off-axis at a constant speed down a cylindrical tube with a resistive thick metallic wall. Upon matching the electromagnetic field components at all interfaces, the induced monopole electromagnetic fields in the plasma are obtained in the frequency domain. An expression for the plasma electric resistance and reactance is derived and analyzed numerically for some representative parameters. Near the plasma resonant frequency, the plasma resistance evolves with frequency like a parallel RLC resonator with peak resistance at the plasma frequency $\omega_{\rm pe}$, while the plasma reactance can be capacitive or inductive in nature depending on the frequency under consideration.
 Select Propagation and Damping of Two-Fluid Magnetohydrodynamic Waves in Stratified Solar Atmosphere Yong-Jian Jin, Hui-Nan Zheng, Zhen-Peng Su Chin. Phys. Lett.    2018, 35 (7): .   DOI: 10.1088/0256-307X/35/7/075201 Abstract   HTML   PDF (634KB) An extreme ultra-violet (EUV) wave is characterized as a bright pulse that has emanated from the solar eruption source and can propagate globally in the solar corona. According to one leading theory, the EUV wave is a fast magnetoacoustic wave, as the coronal counterpart of the Moreton wave in the chromosphere. However, previous observations have shown that the EUV wave differs significantly from the Moreton wave in both velocity and lifetime. To reconcile these differences, here we analyze the wave characteristics of a two-fluid MHD model in the stratified solar atmosphere with a height-dependent ionization rate. It is found that the collision between neutral and ionized fluids is able to attenuate the wave amplitude, while causing a slight change in its propagation velocity. Because the chromosphere has the lower ionization rate and the stronger magnetic fields than the corona, the velocity of the Moreton wave is much higher than that of the EUV wave. In contrast to the Moreton waves damped strongly by the collision between neutral and ionized fluids, the EUV wave in the fully ionized corona is able to propagate globally on a time scale of several hours. Our results support the previous theory that fast magnetoacoustic waves account for both EUV and Moreteon waves in the different layers of the solar atmosphere.
 Select Dynamics of Ring-to-Volume Discharge Transition in $H$ Mode in Inductively Coupled Plasma Torches at Atmospheric Pressure Qi-Jia Guo, Guo-Hua Ni, Lin Li, Qi-Fu Lin, Yan-Jun Zhao, Si-Yuan Sui, Hong-Bing Xie, Wen-Xue Duan, Yue-Dong Meng Chin. Phys. Lett.    2018, 35 (7): .   DOI: 10.1088/0256-307X/35/7/075202 Abstract   HTML   PDF (832KB) The transition process in ring-to-volume discharge in $H$ mode in inductively coupled plasma torches at atmospheric pressure is investigated by analyzing the time resolved image taken by a high speed camera. The effects of input power, plasma working gas flow rate, and its composition on the transition dynamics are also discussed. The results show that the discharge plasma has experienced ring discharge, and the development stage diffused from the boundary to the center in the confinement tube, and steady volume discharge after entering the $H$ mode. Increasing input power, sheath gas flow rate and hydrogen contents in plasma working gas are all able to lessen the time consumed in the transition process in ring-to-volume discharge.
 Select Effect of Hyper-Resistivity on Nonlinear Tearing Modes Wen Yang, Ding Li, Xue-qiao Xu Chin. Phys. Lett.    2018, 35 (6): .   DOI: 10.1088/0256-307X/35/6/065201 Abstract   HTML   PDF (422KB) We analytically investigate nonlinear tearing modes with the anomalous electron viscosity or, as it is normally called, hyper-resistivity. In contrast to the flux average method used by previous work, we employ the standard singular perturbation technique and a quasilinear method to obtain the time evolution equation of tearing modes. The result that the magnetic flux grows with time in a scaling as $t^{2/3}$ demonstrates that nonlinear tearing modes with the hyper-resistivity effect alone have a weaker dependence on time than that of the corresponding resistive case.
 Select Weakly Nonlinear Rayleigh–Taylor Instability in Cylindrically Convergent Geometry Hong-Yu Guo, Li-Feng Wang, Wen-Hua Ye, Jun-Feng Wu, Wei-Yan Zhang Chin. Phys. Lett.    2018, 35 (5): .   DOI: 10.1088/0256-307X/35/5/055201 Abstract   HTML   PDF (549KB) The Rayleigh–Taylor instability (RTI) in cylindrical geometry is investigated analytically through a second-order weakly nonlinear (WN) theory considering the Bell–Plesset (BP) effect. The governing equations for the combined perturbation growth are derived. The WN solutions for an exponentially convergent cylinder are obtained. It is found that the BP and RTI growths are strongly coupled, which results in the bubble-spike asymmetric structure in the WN stage. The large Atwood number leads to the large deformation of the convergent interface. The amplitude of the spike grows faster than that of the bubble especially for large mode number $m$ and large Atwood number $A$. The averaged interface radius is small for large mode number perturbation due to the mode-coupling effect.
 Select Demonstration of a Shock-Timing Experiment in a CH Layer at the ShenGuang III Laser Facility Feng WANG, Yu-Long Li, Zhe-Bin Wang, Tao Xu, Wei-Yi Zha, Dong Yang Chin. Phys. Lett.    2018, 35 (5): .   DOI: 10.1088/0256-307X/35/5/055202 Abstract   HTML   PDF (1340KB) Shock-timing experiments are indispensable to inertial confinement fusion mainly because the timing of multiple shock waves is crucial to understanding the processes of laser irradiation of targets. Investigations into shock waves driven by a two-step radiation pulse in polystyrene (CH) capsule targets are experimentally conducted at the ShenGuang III laser facility. Differing from the traditional shock-timing implementation in which one shock wave could catch up with another one in solid CH, in this experiment, the second shock front in a rarefaction CH layer is observed through velocity interferometry. This second shock could also be made to converge with rarefaction waves within only a few micrometers of the CH capsule by designing the two-shock coalescence time. A shock-timing diagnostic technique to tune the multi-shock convergence in the CH capsule can thereby be achieved. The experimental results in the CH layer are quasi-quantitatively interpreted using streamlines simulated with the Multi-1D program. The experimental results are expected to offer important information for target structure and laser pulse design, both of which are important for realizing inertial confinement fusion.
 Select Mechanism of Electromagnetic Flow Control Enhanced by Electro-Discharge in Water Yan-Liang Ji, Ben-Mou Zhou, Ya-Dong Huang Chin. Phys. Lett.    2018, 35 (5): .   DOI: 10.1088/0256-307X/35/5/055203 Abstract   HTML   PDF (1041KB) Pulsed discharge utilized to achieve large current density in the electromagnetic flow control is numerically studied. A mathematic discharge model is established to calculate the plasma channel, and an actuator is designed to generate the Lorentz force in the micro plasma channel. During the discharge process, the resistance in the channel decreases rapidly and a large current density appears between the discharge electrodes. After the actuator is applied in the leading edge of a flat plate, the separation region and downstream turbulent boundary layer on the plate disappear. Meanwhile, a skin-friction drag force reduction is achieved.
 Select Extraordinary Electromagnetic Waves in Weakly Relativistic Degenerate Spin-1/2 Magnetized Quantum Plasmas Chun-Hua Li, Shao-Wei Wang, Yun-Hao Liu, Zhen-Wei Xia, Xiao-Hui Zhang, Dan-Dan Zou Chin. Phys. Lett.    2018, 35 (4): .   DOI: 10.1088/0256-307X/35/4/045201 Abstract   HTML   PDF (564KB) By using the relativistic quantum magnetohydrodynamic model, the extraordinary electromagnetic waves in magnetized quantum plasmas are investigated with the effects of particle dispersion associated with the quantum Bohm potential effects, the electron spin-1/2 effects, and the relativistic degenerate pressure effects. The electrons are treated as a quantum and magnetized species, while the ions are classical ones. The new general dispersion relations are derived and analyzed in some interesting special cases. Quantum effects are shown to affect the dispersion relations of the extraordinary electromagnetic waves. It is also shown that the relativistic degenerate pressure effects significantly modify the dispersive properties of the extraordinary electromagnetic waves. The present investigation should be useful for understanding the collective interactions in dense astrophysical bodies, such as the atmosphere of neutron stars and the interior of massive white dwarfs.
 Select Effect of Parallel-Plate Geometry on Mode Transition Behavior in Argon Microplasmas: Two-Dimensional Simulation Xiang-Mei Liu, Yuan-Hong Song, Wei Jiang, Wen-Zhu Jia Chin. Phys. Lett.    2018, 35 (4): .   DOI: 10.1088/0256-307X/35/4/045202 Abstract   HTML   PDF (599KB) A two-dimensional self-consistent fluid model is employed to investigate radio-frequency process parameters on the plasma properties in Ar microdischarges. The neutral gas density and temperature balance equations are taken into account. We mainly investigate the effect of the electrode gap on the spatial distribution of the electron density and electron temperature profiles, due to a mode transition from the $\gamma$ regime (secondary electrons emission is responsible for the significant ionization) to the $\alpha$ regime (sheath oscillations and bulk electrons are responsible for sustaining discharge) induced by a sudden decrease of electron density and electron temperature. The pressure, radio-frequency sources frequency and voltage effects on the electron density are also elaborately investigated.
 Select Particle Trajectory Integrator in Guiding Center Phase Space Jing-Bo Lin, Wen-Lu Zhang, Peng-Fei Liu, Chao Dong, Jin-Tao Cao, Ding Li Chin. Phys. Lett.    2018, 35 (2): .   DOI: 10.1088/0256-307X/35/2/025201 Abstract   HTML   PDF (620KB) A trajectory integrator is developed based on a particle's guiding center Hamiltonian. It is verified by a series of benchmarks, which are in good accordance with theoretical prediction. This integrator can be used as the guiding center trajectory integrator of a particle-in-cell simulation platform, such as the newly developed VirtEx. It can also be used as a stand-alone tool to investigate particle dynamics in a given background field.
 Select Formation of Nonlinear Solitary Vortical Structures by Coupled Electrostatic Drift and Ion-Acoustic Waves L. Z. Kahlon, I. Javaid Chin. Phys. Lett.    2017, 34 (12): .   DOI: 10.1088/0256-307X/34/12/125201 Abstract   HTML   PDF (419KB) Propagation of coupled electrostatic drift and ion-acoustic waves (DIAWs) is presented. It is shown that nonlinear solitary vortical structures can be formed by low-frequency coupled electrostatic DIAWs. Primary waves of distinct (small, intermediate and large) scales are considered. Appropriate set of 3D equations consisting of the generalized Hasegawa–Mima equation for the electrostatic potential (involving both vector and scalar nonlinearities) and the equation of motion of ions parallel to magnetic field are obtained. According to experiments of laboratory plasma mainly focused to large scale DIAWs, the possibility of self-organization of DIAWs into the nonlinear solitary vortical structures is shown analytically. Peculiarities of scalar nonlinearities in the formation of solitary vortical structures are widely discussed.
 Select Effects of Three Typical Resistivity Models on Pulsed Inductive Plasma Acceleration Modeling Xin-Feng Sun, Yan-Hui Jia, Tian-Ping Zhang, Chen-Chen Wu, Xiao-Dong Wen, Ning Guo, Hai Jin, Yu-Jun Ke, Wei-Long Guo Chin. Phys. Lett.    2017, 34 (12): .   DOI: 10.1088/0256-307X/34/12/125202 Abstract   HTML   PDF (661KB) The effects of three different typical resistivity models (Spitzer, Z&L and M&G) on the performance of pulsed inductive acceleration plasma are studied. Numerical results show that their influences decrease with the increase of the plasma temperature. The significant discriminations among them appear at the plasma temperature lower than 2.5 eV, and the maximum gap of the pulsed inductive plasma accelerated efficiency is approximately 2.5%. Moreover, the pulsed inductive plasma accelerated efficiency is absolutely related to the dynamic impedance parameters, such as voltage, inductance, capacitance and flow rate. However, the distribution of the efficiency as a function of plasma temperature with three resistivity models has nothing to do with the dynamic impedance parameter.
 Select Microstructure and Deuterium Retention of Tungsten Deposited by Hollow Cathode Discharge in Deuterium Plasma Zhong-Chao Sun, Zi-Wei Lian, Wei-Na Qiao, Jian-Gang Yu, Wen-Jia Han, Qing-Wei Fu, Kai-Gui Zhu Chin. Phys. Lett.    2017, 34 (12): .   DOI: 10.1088/0256-307X/34/12/125203 Abstract   HTML   PDF (1799KB) Tungsten has been chosen as one of the most promising candidates as the plasma-facing material in future fusion reactors. Although tungsten has numerous advantages compared with other materials, issues including dust are rather difficult to deal with. Dust is produced in fusion devices by energetic plasma-surface interaction. The re-deposition of dust particles could cause the retention of fuel atoms. In this work, tungsten is deposited with deuterium plasma by hollow cathode discharge to simulate the dust production in a tokamak. The morphology of the deposited tungsten can be described as a film with spherical particles on it. Thermal desorption spectra of the deposited tungsten show extremely high desorption of the peak positions. It is also found that there is a maximum retention of deuterium in the deposited tungsten samples due to the dynamic equilibrium of the deposition and sputtering process on the substrates.
 Select Enhancement of Heat-Resistance of Carbonyl Iron Particles by Coating with Silica and Consequent Changes in Electromagnetic Properties Zhao-Wen Ren, Hui Xie, Ying-Ying Zhou Chin. Phys. Lett.    2017, 34 (10): .   DOI: 10.1088/0256-307X/34/10/105201 Abstract   HTML   PDF (3242KB) Silica-coated carbonyl iron particles (CIPs) are fabricated with the Stober method to improve their heat-resistance and wave-absorption properties. The morphology, heat-resistance, electromagnetic properties and microwave absorption of raw-CIPs and silica-coated CIPs are investigated using a scanning electron microscope, an energy dispersive spectrometer, a thermal-gravimetric analyzer, and a network analyzer. The results show that the heat-resistance of silica-coated CIPs is better than that of raw CIPs. The reflection losses exceeding $-$10 dB of silica-coated CIPs are obtained in the frequency range 9.3–12.4 GHz for the absorber thickness of 2.3 mm, and the same reflection losses of uncoated CIPs reach the data in the lower frequency range for the same thickness. The enhanced microwave absorption of silica-coated CIPs can be ascribed to the combination of proper electromagnetic impedance match and the decrease of dielectric permittivity.
 Select Heat Flux on EAST Divertor Plate in H-mode with LHCD/LHCD+NBI Bo Shi, Zhen-Dong Yang, Bin Zhang, Cheng Yang, Kai-Fu Gan, Mei-Wen Chen, Jin-Hong Yang, Hui Zhang, Jun-Li Qi, Xian-Zu Gong, Xiao-Dong Zhang, Wei-Hua Wang Chin. Phys. Lett.    2017, 34 (9): .   DOI: 10.1088/0256-307X/34/9/095201 Abstract   HTML   PDF (776KB) Based on the surface temperature measured by the infrared camera on the experimental advanced superconducting tokamak (EAST), the heat fluxes on the lower outer divertor target plate during H-mode with the lower-hybrid wave current drive (LHCD) only and with the LHCD combined with the neutral beam injection (NBI) are calculated by the DFLUX code and compared. The analyzed discharges are lower single null divertor configuration discharges. In the case with the LHCD only ($I_{\rm p}\sim 400$ kA, $P_{\rm LHCD}\sim2$ MW), ELM-free appears after L-H transition with the peak heat flux on the lower outer target plate less than 1 MW/m$^{2}$. However, there is no ELM-free appearing after the L-H transition in the case with the LHCD+NBI ($I_{\rm p}\sim300$ kA, $P_{\rm LHCD}+P_{\rm NBI}\sim2$ MW). The results show that the peak heat fluxes on the lower outer target plate in the LHCD+NBI H-mode cases are larger than those in the LHCD H-mode under the similar auxiliary heating power. This is because the heat flux profiles of the lower outer target plate as a function of plate location in ELMing with the LHCD+NBI are narrower than those with the LHCD only. The results are consistent with the results in terms of the scrape-off layer width observed in the EAST.
 Select High-Yield High-Efficiency Positron Generation in High-$Z$ Metal Targets Irradiated by Laser Produced Electrons from Near-Critical Density Plasmas Wei Song, Rong-Hao Hu, Yin-Ren Shou, Zheng Gong, Jin-Qing Yu, Chen Lin, Wen-Jun Ma, Yan-Yin Zhao, Hai-Yang Lu, Xue-Qing Yan Chin. Phys. Lett.    2017, 34 (8): .   DOI: 10.1088/0256-307X/34/8/085201 Abstract   HTML   PDF (978KB) An improved indirect scheme for laser positron generation is proposed. The positron yields in high-$Z$ metal targets irradiated by laser produced electrons from near-critical density plasmas and underdense plasma are investigated numerically. It is found that the positron yield is mainly affected by the number of electrons of energies up to several hundreds of MeV. Using near-critical density targets for electron acceleration, the number of high energy electrons can be increased dramatically. Through start-to-end simulations, it is shown that up to $6.78\times10^{10}$ positrons can be generated with state-of-the-art Joule-class femtosecond laser systems.
 Select The Diffusion Coefficient Using Sawtooth Oscillation in IR-T1 Tokamak N. Hasanvand, S. Meshkani, M. Ghoranneviss Chin. Phys. Lett.    2017, 34 (8): .   DOI: 10.1088/0256-307X/34/8/085202 Abstract   HTML   PDF (596KB) The investigation of runaway electrons is expanded by different methods. The aim of this study is to show sawtooth oscillations of hard x-ray emission and with the help of sawtooth oscillations to obtain radial diffusion coefficient and magnetic fluctuations. In the same way, the hard x-ray spectral evaluation is compared in several time intervals and it is shown that during discharge, the energy of the runaway electrons is less than 200 keV. Also, for typical plasmas, population of runaway electrons is measured at seven time intervals of 5 ms and temporal evaluation of runaway electron mean energy. The sawtooth-like shape is observed in the hard x-ray range (10–1000 keV). By the sawtooth oscillation method, the RE diffusion coefficient in radial transport in the IR-T1 plasma is $D_{\rm r}\sim 0.5$ m$^2$s$^{-1}$. The magnetic field fluctuation due to magnetic diffusion $D_{\rm m}$ is given as $\frac{b_{\rm r}}{B_{\rm t}}\sim 10^{-4}$.
 Select A Method of Using a Carbon Fiber Spiral-Contact Electrode to Achieve Atmospheric Pressure Glow Discharge in Air Wen-Zheng Liu, Shuai Zhao, Mao-Lin Chai, Jiang-Qi Niu Chin. Phys. Lett.    2017, 34 (8): .   DOI: 10.1088/0256-307X/34/8/085203 Abstract   HTML   PDF (1037KB) During discharge, appropriately changing the development paths of electron avalanches and increasing the number of initial electrons can effectively inhibit the formation of filamentary discharge. Based on the aforementioned phenomenon, we propose a method of using microdischarge electrodes to produce a macroscopic discharge phenomenon. In the form of an asymmetric structure composed of a carbon fiber electrode, an electrode structure of carbon fiber spiral-contact type is designed to achieve an atmospheric pressure glow discharge in air, which is characterized by low discharge voltage, low energy consumption, good diffusion and less ozone generation.
 Select Jeans Gravitational Instability with $\kappa$-Deformed Kaniadakis Distribution Hui Chen, Shi-Xuan Zhang, San-Qiu Liu Chin. Phys. Lett.    2017, 34 (7): .   DOI: 10.1088/0256-307X/34/7/075101 Abstract   HTML   PDF (513KB) The Jeans instabilities in an unmagnetized, collisionless, isotropic self-gravitating matter system are investigated in the context of $\kappa$-deformed Kaniadakis distribution based on kinetic theory. The result shows that both the growth rates and critical wave numbers of Jeans instability are lower in the $\kappa$-deformed Kaniadakis distributed self-gravitating matter systems than the Maxwellian case. The standard Jeans instability for a Maxwellian case is recovered under the limitation $\kappa=0$.
 Select Linear Growth of Rayleigh–Taylor Instability of Two Finite-Thickness Fluid Layers Hong-Yu Guo, Li-Feng Wang, Wen-Hua Ye, Jun-Feng Wu, Wei-Yan Zhang Chin. Phys. Lett.    2017, 34 (7): .   DOI: 10.1088/0256-307X/34/7/075201 Abstract   HTML   PDF (577KB) The linear growth of Rayleigh–Taylor instability (RTI) of two superimposed finite-thickness fluids in a gravitational field is investigated analytically. Coupling evolution equations for perturbation on the upper, middle and lower interfaces of the two stratified fluids are derived. The growth rate of the RTI and the evolution of the amplitudes of perturbation on the three interfaces are obtained by solving the coupling equations. It is found that the finite-thickness fluids reduce the growth rate of perturbation on the middle interface. However, the finite-thickness effect plays an important role in perturbation growth even for the thin layers which will cause more severe RTI growth. Finally, the dependence of the interface position under different initial conditions are discussed in some detail.
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I have been asking more than a few questions here. I have never – that I know of – marked any of them as "answered". How can I bring up a screen of questions I have asked so I can mark them – individually – as answered as appropriate?

## migrated from superuser.comJul 23 '15 at 12:19

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• This lists all your questions and indicates if they have answers or not. – Ramhound Jul 23 '15 at 12:32
• You have 10 accepted answers, by the way. Not sure how you could be unaware of that. Wouldn't doing the same thing 10 times give you at least a vague memory of having done it before? – allquixotic Jul 25 '15 at 12:01
• I don't hang out on SuperUser and have never read the manual/instructions: as the kids so annoyingly say -- "my bad". I come in, ask my question and leave. Wait: it looks like Sathya has answered me,c, completely ... – greenber Jul 26 '15 at 10:09

Whilst this is overkill since you currently only have 38 questions some of which are closed, it is possible to query StackExchange sites for this kind of information using the Stack Exchange Data Explorer.

The following query will show questions with answers but none that are marked as 'accepted':

select q.Title, 'http://superuser.com/questions/' + cast(q.ID as varchar) as [Link]
from Posts q
where q.PostTypeId = 1 and q.OwnerUserId = 73534 and q.AcceptedAnswerId is null and ClosedDate is null and AnswerCount > 0
order by CreationDate desc

Query results can be found here.

Note that it seems to take a little while for the results to be synchronised with the live data.

• Check my answer for an easier approach – Sathyajith Bhat Jul 24 '15 at 13:27
• +1 for the approach, most simple and straightforward way to do it. I didn't know superuser has db until I read this answer, this is much more powerful than only searching answer which is un-answered, it can slot any status of the question/user/answer. – Bilo Jul 26 '15 at 11:00

An easier solution is to make use of user:me hasaccepted:no search operator. This will give you all questions that you have not accepted.

Check the Search page and click on "Advanced Search Tips" for more such tips

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Home > Standard Error > What Is The Formula To Find Standard Error

# What Is The Formula To Find Standard Error

## Contents

This estimate may be compared with the formula for the true standard deviation of the sample mean: SD x ¯   = σ n {\displaystyle {\text{SD}}_{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} you can't ask millions of people, so instead you ask maybe 1,000 people. mathtutordvd 128,799 views 8:53 95% Confidence Interval - Duration: 9:03. The "population" is all 20 rose bushes, and the "sample" is the 6 bushes that Sam counted the flowers of. check over here

Loading... The ages in that sample were 23, 27, 28, 29, 31, 31, 32, 33, 34, 38, 40, 40, 48, 53, 54, and 55. Andale Post authorAugust 6, 2014 at 10:45 am Thanks for pointing that out Kim. Brandon Foltz 70,478 views 32:03 Standard Error - Duration: 7:05. see this here

## Standard Error Formula Statistics

The margin of error and the confidence interval are based on a quantitative measure of uncertainty: the standard error. How to cite this article: Siddharth Kalla (Sep 21, 2009). Similarly, the sample standard deviation will very rarely be equal to the population standard deviation. What is the Standard Error Formula?

This often leads to confusion about their interchangeability. National Center for Health Statistics typically does not report an estimated mean if its relative standard error exceeds 30%. (NCHS also typically requires at least 30 observations – if not more OK. Standard Error Of Estimate Formula If values of the measured quantity A are not statistically independent but have been obtained from known locations in parameter space x, an unbiased estimate of the true standard error of

The sample standard deviation s = 10.23 is greater than the true population standard deviation σ = 9.27 years. Loading... Take it with you wherever you go. https://en.wikipedia.org/wiki/Standard_error Recall that the regression line is the line that minimizes the sum of squared deviations of prediction (also called the sum of squares error).

For any random sample from a population, the sample mean will very rarely be equal to the population mean. Standard Error Of Proportion Next, consider all possible samples of 16 runners from the population of 9,732 runners. Leave a Reply Cancel reply Your email address will not be published. For the age at first marriage, the population mean age is 23.44, and the population standard deviation is 4.72.

## Standard Error Formula Excel

Rating is available when the video has been rented. their explanation The service is unavailable. Standard Error Formula Statistics Please answer the questions: feedback Standard Error Calculator Using a sample to estimate the standard error In the examples so far, the population standard deviation σ was assumed to be known.

Search Statistics How To Statistics for the rest of us! check my blog Probability and Statistics > Statistics Definitions > What is the standard error? If σ is known, the standard error is calculated using the formula σ x ¯   = σ n {\displaystyle \sigma _{\bar {x}}\ ={\frac {\sigma }{\sqrt {n}}}} where σ is the The following expressions can be used to calculate the upper and lower 95% confidence limits, where x ¯ {\displaystyle {\bar {x}}} is equal to the sample mean, S E {\displaystyle SE} Standard Error Of The Mean Definition

Richard Frederick 5,799 views 9:30 How to Determine Standard Deviation and Standard Error of the Mean - Duration: 6:39. If people are interested in managing an existing finite population that will not change over time, then it is necessary to adjust for the population size; this is called an enumerative But how do we say "add them all up" in mathematics? http://3cq.org/standard-error/what-is-the-formula-for-standard-error-of-the-mean.php The standard error is the standard deviation of the Student t-distribution.

The data set is ageAtMar, also from the R package openintro from the textbook by Dietz et al.[4] For the purpose of this example, the 5,534 women are the entire population Standard Error Definition It is rare that the true population standard deviation is known. The standard error (SE) is the standard deviation of the sampling distribution of a statistic,[1] most commonly of the mean.

## Spider Phobia Course More Self-Help Courses Self-Help Section .

Therefore, the standard error of the estimate is There is a version of the formula for the standard error in terms of Pearson's correlation: where ρ is the population value of Work out the Mean (the simple average of the numbers) 2. Howcast 480,467 views 1:29 Standard Deviation vs Standard Error - Duration: 3:57. Standard Error Vs Standard Deviation p = Proportion of successes.


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# Variational problems with several volume constraints on the level sets

created on 04 Dec 2000
modified on 17 Jun 2002

[BibTeX]

Published Paper

Inserted: 4 dec 2000
Last Updated: 17 jun 2002

Journal: Calculus of Variations
Year: 2001

Abstract:

We consider functionals of the kind $I(u)= \int F(x,u,\nabla u)$ on $W^{1,p}$, and we study the problem $\min_{K} I$, where $K\subset W^{1,p}$ consists of those functions $u$ whose level sets satisfy certain volume constraints $meas(\{u=l_i\})=\alpha_i>0$, where $meas(\cdot)$ denotes Lebesgue measure, and $\{l_i\}$, $\{\alpha_i\}$ are given numbers. Examples show that this problem may have no solution, even for simple smooth $F$. As a consequence, we relax the constraint $u\in K$ to $u\in K_+$, i.e. $meas(\{u=l_i\})>= \alpha_i$, and we show that the minimizers over $K_+$ exist and are Hölder continuous. Then we prove several existence theorems for the original problem, showing that, under suitable assumptions on the integrand function $F$, every minimizer over $K_+$ actually belongs to $K$.

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Notebooks

## What Is the Right Null Model for Linear Regression?

16 Aug 2011 21:44

When social scientists do linear regressions, they commonly take as their null hypothesis the model in which all the independent variables have zero regression coefficients. An independent variable is usually taken to have a systematic influence on the dependent outcome, or, more cautiously, some systematic predictive power, if the regression coefficient is significantly different from zero, generally as assessed by a t-test.

There are a number of things wrong with this picture --- the easy slide from regression to causation, the assumption of linearity, the usual assumption of Gaussian noise, etc. --- but what I want to focus on here is taking the zero-coefficient model as the right null. The point of the null model, after all, is that it embodies a deflating explanation of an apparent pattern, that it's somehow due to a boring, uninteresting mechanism, and any appearance to the contrary is just due to chance. In other words, testing a null hypothesis should check for a particular kind of error. (See further to that theme.) A good example of this comes from evolutionary biology, where the way one detects adaptation is by first setting up a "neutral" model, of how evolutionary dynamics would change gene frequencies in the absence of adaptation, and then looking for deviations from neutrality. This only works if the neutral model is a good picture of all the non-adaptive mechanisms shaping gene frequencies. Setting up a trivial null hypothesis which you are pretty sure is wrong is not actually very informative.

So, the question here is, what is the right null model would be in the kinds of situations where economists, sociologists, etc., generally use linear regression. Typically, it seems to me, these are situations where we know there are an immense number of factors, interacting in complicated ways, but we get to measure only a very small of highly aggregated variables, few if any of which would appear in an actual causal model of the process. In this kind of situation, should we expect variables which are substantively unrelated to have zero regression coefficients? (I emphasize "substantively" because it's a tautology that an independent variable's regression coefficient will be zero if and only if it contributes nothing to the optimal linear predictor formed from the other independent variables.) My intuition is rather that, unless one is spectacularly lucky in picking the observables, the regression coefficient should only rarely be zero.

Of course I don't really trust my intuition about things like this. (If it was intuitively clear then there wouldn't be an issue!) I think however that one could make some headway by tackling the following sort of model. Suppose that the system we are really concerned with is described by a very large system of linear equations, with let us say N variables. These may be either difference equations (for time evolution) or simultaneous equations (for equilibria), I'm not sure which would be easier to pursue. Exogenous terms can also be taken to be noise. Most entries in the N*N matrix describing the system are zero, say only O(N) entries are non-zero; those we take to be random from some distribution jiggered so that solutions are sensible. In other words, imagine the distribution of coefficients to be random, with a big spike of probability mass at zero. If we could measure these variables, then linear regression would give us sensible estimates of the parameters of the system (modulo all the qualifiers filling econometrics textbooks).

The trick, however, is that we don't get to observe them. Suppose then that we get to observe K variables, each of which is a linear combination of the N underlying variables; K is much smaller than N. We form these aggregate observables in some totally dumb way, say uniform random sampling over variables, together with IID standard Gaussian weights. We designate one of them as the independent variable, again at random.

Questions:

1. What is the distribution of regression weights for aggregate quantities, holding the low-level parameters fixed but randomizing over aggregations?
2. What is the distribution if we also randomize over the low-level parameters?
3. Does either distribution have a mass spike at zero?
4. In what limits do the distributions converge to zero?
5. What changes in the special case where the K observables are averages (or sums) of K observables for each of M units (so that N=MK)?
6. What changes in the special case where the K observables are a sample of the low-level variables, rather than being linear combinations of them?

— Except in th last case, I don't think this is the same as omitted variable bias. (That is what happens when your specification doesn't include variables it should, but does include ones correlated with it, which induces a bias in the estimate of the coefficients for the included variables.) It may, however, be possible to treat all this using the same machinery.

— This idea, it occurs to me, is probably related to the Thomson sampling model (which I've discussed elsewhere).

Recommended and relevant:
• Richard A. Berk, Regression Analysis: A Constructive Critique
• Brad DeLong and Kevin Lang, "Are All Economic Hypotheses False?" [PDF]
• Trygve Haavelmo, "The Probability Approach in Econometrics", Econometrica 12 (1944): iii--115 [JSTOR]


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# Max Planck

Max Karl Ernst Ludwig Planck (April 23, 1858October 4, 1947) was one of the most important German physicists of the late 19th and early 20th century; he is considered to be the inventor of quantum theory.

## Origin and youth

Planck came from a traditional, intellectual family. His paternal great-grandfather and grandfather were both theology professors in Göttingen, his father was a law professor in Kiel and Munich, and his paternal uncle was a judge.

Max Planck was born in Kiel on April 23, 1858 to Johann Julius Wilhelm Planck and his second wife, Emma Patzig. He was the sixth child in the family, though two of his siblings were from his father's first marriage. In 1867 the family moved to Munich, where Planck attended high school, graduating early at age 16.

## Education

Planck was musically gifted: he took voice lessons in addition to playing the piano, organ and cello, and composing songs and operas. However, instead of music he chose to study physics.

The Munich physics professor Philipp von Jolly advised him against going into physics, saying, "in this field, almost everything is already discovered, and all that remains is to fill a few holes." Planck replied that he didn't wish to discover new things, only to understand the known fundamentals of the field and began his studies in 1874 in Munich. With Jolly, Planck performed the only experiment of his entire intellectual career (studying the diffusion of hydrogen through heated platinum), but soon transferred to theoretical physics.

In 1877 he went to Berlin for a year of study with the famous physicists Hermann von Helmholtz and Gustav Kirchhoff and also the mathematician Karl Weierstrass. He wrote that Helmholtz was never quite prepared, spoke slowly, miscalculated endlessly, and bored his listeners, while Kirchhoff spoke in carefully prepared lectures, which were, however, dry and monotonous. Despite this he soon became close friends with Helmholtz. While there he mostly undertook a program of self-study of Clausius's writings, which led him to choose heat theory as his field.

In October 1878 Planck passed his qualifying exams and in February of 1879 defended his dissertation, "Über den zweiten Hauptsatz der mechanischen Wärmetheorie" (On the second fundamental theorem of the mechanical heat theory). In June 1880 he presented his habilitation thesis, "Gleichgewichtszustände isotroper Körper in verschiedenen Temperaturen" (Equilibrium states of isotropic bodies at different temperatures).

With the completion of his habilitation thesis, Planck became an unpaid private lecturer in Munich, waiting until he would be offered an academic position. Although he was initially ignored by the academic community, he furthered his work on the field of heat theory and discovered one after the other the same thermodynamical formalism as Gibbs without realizing it. Clausius's ideas on entropy occupied a central role in his work.

In April 1885 the University of Kiel appointed Planck an associate professor of theoretical physics. Further work on entropy and its treatment, especially as applied in physical chemistry, followed. He proposed a thermodynamic basis for Arrhenius's theory of electrolytic dissociation.

Within four years he was named the successor to Kirchhoff's position in Berlin — presumably thanks to Helmholtz's intercession — and by 1892 became a full professor. In 1907 Planck was appointed to Boltzmann's position in Vienna, but turned it down to stay in Berlin. He retired on January 10, 1926, and the successor to his position was Erwin Schrödinger.

## Family

In March 1887 Planck married Marie Merck (1861-1909), sister of a school fellow, and moved with her into a sublet apartment in Kiel. Four children were born to the couple: Karl (1888-1916), the twins Emma (1889-1919) and Grete (1889-1917), and Erwin (1893-1945).

After the appointment to Berlin the Planck family lived in a villa in Berlin-Grunewald, Wangenheimstraße 21. In the vicinity of this address several other professors of Berlin University were living, among them the famous theologian Adolf von Harnack, who became a close friend of Planck. Soon the Planck home became a social and cultural centre; numerous well-known scientist were frequent visitors, such as Albert Einstein, Otto Hahn and Lise Meitner. The tradition of jointly playing music had already been established in the home of Helmholtz.

After several happy years the Planck family was struck by a series of disasters: In October 1909 Marie Planck died, possibly from tuberculosis. In March 1911 Max Planck married his second wife, Marga von Hoesslin (1882-1948), in December his third son, Herrmann, was born.

During the First World War Planck's oldest son, Karl, was killed in action in Verdun, Erwin had already in 1914 been taken prisoner by the French. Grete died in 1917 while giving birth to her first child; her sister lost her life two years later under the same circumstances, after marrying Grete's widower. Both granddaughters survived and were named after their mothers. Planck endured all these losses with stoic submission into fate.

Finally in January 1945 Erwin, to whom Max Planck had been particularly close, was executed by the Nazis because of his participation in the failed attempt to assassinate Hitler in July 1944.

## Professor at Berlin University

In Berlin Planck joined the local Physical Society: later he wrote about this time: "In those days I was essentially the only Theoretical physicist there, whence things were not so easy for me, because I started mentioning entropy, but this was not quite fashionable, since it was regarded as a mathematical spook". Thanks to his initiative the society merged in 1898 to form the German Physical Society (Deutsche Physikalische Gesellschaft, DPG); from 1905 to 1909 Planck was the president of the DPG.

Planck started a six-semester course of lectures on Theoretical Physics, "dry, somewhat impersonal" according to Lise Meitner, "using no notes, never making mistakes, never faltering; the best lecturer I ever heard" according to an English participant, James R. Partington, who continues: "There were always many standing around the room. As the lecture-room was well heated and rather close, some of the listeners would from time to time drop to the floor, but this did not disturb the lecture". He did not establish an actual "school", the number of his graduate students was only about 20 all together, among them the following:

Max Abraham 1897 (1875 - 1922)
Moritz Schlick 1904 (1882 - 1936)
Walther Meißner 1906 (1882 - 1974)
Max von Laue 1906 (1879 - 1960)
Fritz Reiche 1907 (1883 - 1960)
Walter Schottky 1912 (1886 - 1976)
Walther Bothe 1914 (1891 - 1957)

In 1894 Planck turned his attention to the problem of black-body radiation. He had been commissioned by electric companies to discover how to create the most light from lightbulbs with the minimum energy. The problem had already been stated by Kirchhoff in 1859: How does the intensity of the electromagnetic radiation emitted by a black body (a perfect absorber, also known as a cavity radiator) depend on the frequency of the radiation (e.g., the colour of the light) and the temperature of the body? The question had been explored experimentally, but the Rayleigh-Jeans law, derived from classical physics, failed to explain the observed behaviour at high frequencies. Wilhelm Wien proposed Wien's law, which correctly predicted the behaviour at high frequencies, but failed at low frequencies.

Planck interpolated between the laws of Wein and Rayleigh Jeans but found that he could only derive a satisfactory equation using what he thought was merely a mathematical trick, namely that light is only emitted in "packets". He did not for many years believe that these packets, known as quanta, corresponded with reality.

His famous Planck black-body radiation law described the experimentally observed black-body spectrum very well; it was first proposed in a meeting of the DPG on October 19, 1900 and published in 1901.

A commemoration plaque for Max Planck on his discovery of Planck's constant, in front of Humboldt University, Berlin

By December 14, 1900 he was already able to present a theoretical derivation of the law, but this required him to use ideas from statistical mechanics, as introduced by Boltzmann. So far he had been holding a strong aversion against any statistical interpretation of the Second law of thermodynamics which he regarded as of axiomatic nature: "... an act of despair ... I was ready to sacrifice any of my previous convictions about physics ..." The central assumption behind his derivation was the supposition that the electromagnetic energy could be emitted only in quantized form, in other words, the energy could only be a multiple of an elementary unit ${\displaystyle E=h\nu }$, where ${\displaystyle h}$ is Planck's constant, also known as Planck's action quantum (introduced already in 1899), and ${\displaystyle \nu }$ is the frequency of the radiation.

At first Planck considered that the quantisation was only as "a purely formal assumption ... actually I did not think much about it..."; nowadays this assumption, incompatible with classical physics, is regarded as the birth of quantum physics and the greatest intellectual accomplishment of Planck's career (however, Ludwig Boltzmann had already in 1877 in a theoretical paper been discussing the possibility that the energy states of a physical system could be discrete). It was in recognition of this accomplishement that Planck was awarded the Nobel prize for physics in 1918.

The discovery of Planck's constant enabled him to define a new universal set of physical units (such as the Planck length and the Planck mass), all based on fundamental physical constants.

Subsequently, Planck tried to grasp the meaning of the energy quanta, but to no avail. "My unavailing attempts to somehow reintegrate the action quantum into classical theory extended over several years and caused me much trouble." Even several years later, other physicists like Rayleigh, Jeans, and Lorentz set Planck's constant to zero in order to align with classical physics, but Planck knew well that this constant had a precise nonzero value. "I am unable to understand Jeans' stubbornness - he is an example of a theoretician as should never be existing, the same as Hegel was for philosophy. So much the worse for the facts, if they are wrong."

Max Born wrote about Planck: "He was by nature and by the tradition of his family conservative, averse to revolutionary novelties and sceptical towards speculations. But his belief in the imperative power of logical thinking based on facts was so strong that he did not hesitate to express a claim contradicting to all tradition, because he had convinced himself that no other resort was possible."

## Einstein and the Theory of Relativity

In 1905 the three epochal papers of hitherto completely unknown Albert Einstein were published in the journal Annalen der Physik; Planck was among the few who immediately recognized the significance of the special theory of relativity. Thanks to his influence this theory was soon widely accepted in Germany. Planck also contributed considerably to extend the special theory of relativity.

However, Einstein's hypothesis of light quanta (photons), harbingered by Philipp Lenard's 1902 discovery of the photoelectric effect, was initially rejected by Planck; he was unwilling to sacrifice also Maxwell's theory of electrodynamics. "The theory of light would be thrown back not by decades, but by centuries, into the age when Christian Huygens dared to fight against the mighty emission theory of Newton ..."

In 1910 Einstein pointed at the anomalous behavior of specific heat at low temperatures as another example of a phenomenon which defies explanation by classical physics. Planck and Nernst, in order to clarify the increasing number of contradictions, organised the First Solvay Conference (Brussels 1911); at this meeting Einstein was finally able to convince Planck.

Meanwhile Planck had been appointed dean of Berlin University, whereby it was possible for him to call Einstein to Berlin and establish a new professorship for him (1914). Soon the two scientists became close friends and met frequently for jointly playing music.

## World War and Weimar Republic

At the onset of the First World War Planck was not immune to the general excitement of the public: "... besides of much horrible also much unexpectedly great and beautiful: the swift solution of the most difficult issues of domestic policy through arrangement of all parties... the higher esteem for all that is brave and truthful..." Admittedly, he refrained from the extremes of nationalism, e.g. he voted successfully for a scientific paper from Italy receiving a prize from the Prussian Academy of Sciences in 1915 (Planck was one of its four permanent presidents), although at that time Italy was about to join the Allies; nevertheless the infamous "Manifesto of the 93 intellectuals", a polemic pamphlet of war propaganda, was also signed by Planck, while Einstein retained a strictly pacifistic attitude which almost led to his imprisonment (from which he was only saved by his Swiss citizenship). But already in 1915 Planck revoked (after several meetings with Dutch physicist Lorentz) parts of the Manifesto, 1916 he signed a declaration against German annexionism.

In the turbulent post-war years Planck, by now the highest authority of German physics, issued the slogan "persevere and continue working" to his colleagues. In October 1920 Fritz Haber and he established the "Notgemeinschaft der Deutschen Wissenschaft" (emergency organisation of German science), which aimed at providing support for the destitute scientific research; the funds they could distribute were raised to a considerable part abroad. In this time Planck held leading positions also at Berlin University, the Prussian Academy of Sciences, the German Physical Society and the Kaiser-Wilhelm-Society (Kaiser-Wilhelm-Gesellschaft zur Förderung der Wissenschaften (KWG, Emperor-Wilhelm-Society for the advancement of science, which in 1948 became the Max-Planck-Gesellschaft); under such conditions he was hardly able to conduct anymore research himself.

He became a member of the Deutsche Volks-Partei (German People's Party), the party of peace Nobel prize laureate Gustav Stresemann, which aspired liberal aims for domestic policy and rather revisionistic aims for international politics. He disagreed with the introduction of universal suffrage and expressed later the view that the Nazi dictatorship was the result of "the ascent of the rule of the crowds".

## Quantum Mechanics

At the end of the Twenties Bohr, Heisenberg and Pauli had worked out the Copenhagen interpretation of quantum mechanics, but it was rejected by Planck, as also by Schrödinger and Laue; even Einstein had turned into a conservative. Heisenberg's matrix mechanics he called "disgusting", the Schrödinger equation he welcomed like salvation. He expected that wave mechanics would soon render quantum theory - his own child - unnecessary. Scientific progress ignored his concerns. He experienced the truth of his own earlier observation from his struggle with the older views in his younger years: "A new scientific truth does not establish itself by its enemies being convinced and expressing their change of opinion, but rather by its enemies gradually dying out and the younger generation being taught the truth from the beginning".

## Nazi dictatorship and Second World War

When the Nazis seized power in 1933, Planck had already reached the age of 74; he had to witness how many Jewish friends and colleagues were expelled from their positions and humiliated, and how hundreds of scientists emigrated from Germany. Again he tried "persevere and continue working" and asked scientists who were considering emigration to stay in Germany. In some cases he was successful with this request, such as in the case of Heisenberg.

Hahn asked Planck whether they should gather a number of well-known German professors in order to issue a public proclamation against the treatment of Jewish professors, but Planck replied, "If you are able to gather today 30 such gentlemen, then tomorrow 150 other will come and speak against it, because they are eager to take over the positions of the others." In the case of Fritz Haber (who was the father of German chemical warfare in WWI, but Jewish), however, Planck went so far as to try to discuss the issue with Hitler himself, naturally without any success. In the following year of 1934, Haber died in exile. One year later, Planck, being the president of the KWG since 1930, organised in a somewhat provocative style an official commemorative meeting for Haber. He also succeeded to enable a number of Jewish scientists to continue working in institutes of the KWG for several years. In 1936, his term as president of the KWG ended, and the Nazi government put pressure on him to refrain from running for another term.

As the political climate in Germany gradually became more hostile, Johannes Stark, prominent exponent of Deutsche Physik ("German Physics", also called "Aryan Physics") attacked Planck, Sommerfeld and Heisenberg for continuing to teach the theories of Einstein, calling them "white Jews." The "Hauptamt Wissenschaft" (Nazi government office for science) started an investigation of Planck's ancestry, but all they could find out was that he was "1/16 Jewish."

In 1938, Planck celebrated his 80th birthday; the DPG held an official celebration, during which the Max-Planck medal (founded as the highest medal by the DPG in 1928) was awarded to French physicist Louis de Broglie - one year before the outbreak of a new war between France and Germany. At the end of the same year the Prussian Academy lost its remaining independence and was taken over by loyal Nazis (Gleichschaltung); Planck protested by resigning his presidency. He continued to travel frequently, giving numerous public talks, such as his famous talk on "Religion and Science," and 5 years later he was sufficiently fit to climb to 3000-meter peaks in the Alps mountains.

During the Second World War, the increasing number of Allied bombing campaigns at Berlin forced Planck and his wife to leave the city temporarily and live in the countryside. In 1942 he wrote: "In me an ardent desire has grown to persevere this crisis and live long enough to be able to witness the turning point, the beginning of a new rise." In February 1944 his home in Berlin was completely destroyed by an air raid, annihilating his entire scientific records and correspondence. Finally, he got into a dangerous situation in his rural retreat due to the rapid advance of the Allied armies from both sides. After the end of the war he was brought to a relative in Göttingen.

## Final years

After the war, a number of German physicists assembled in Göttingen in order to reestablish the Kaiser-Wilhelm-Gesellschaft. In July of 1945, Planck agreed to act formally as its president, again. The British occupation authorities insisted on changing the name, and therefore in February 1948 the Max-Planck-Gesellschaft was established.

Despite his deteriorating health, Planck resumed travelling in order to give public talks. In 1946, he went to London on the occasion of the 300th birthday of Isaac Newton. He was the only German invited.

On April 1st of 1946, Planck was succeded as president of the KWG by Otto Hahn.

On October 4th, 1947, he died from the consequences of a fall and several strokes.

## Honours and medals

• "Pour le Mérite" for Science and Arts 1915 (in 1930 Planck became chancellor of this order)
• Nobel prize for physics 1918 (awarded 1919)
• Adlerschild des Deutschen Reiches (1928)
• Max-Planck medal (1928, together with Einstein)
• Planck received honorary doctorates from the universities of Frankfurt, München (TH), Rostock, Berlin(TH), Graz, Athen, Cambridge, London and Glasgow
• The asteroid 1069 was given the name "Stella Planckia" (1938)


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# BrillhartTests UP¶

Author: Frederic Lehobey, James H. Davenport Date Created: 28 June 1994 Basic Operations: brillhartIrreducible? Related Domains: Also See: AMS Classifications: Keywords: factorization Examples: References: [1] John Brillhart, Note on Irreducibility Testing, Mathematics of Computation, vol. 35, num. 35, Oct. 1980, 1379-1381 [2] James Davenport, On Brillhart Irreducibility. To appear. [3] John Brillhart, On the Euler and Bernoulli polynomials, J. Reine Angew. Math., v. 234, (1969), pp. 45-64
brillhartIrreducible?(p, noLinears) returns true if p can be shown to be irreducible by a remark of Brillhart, false else. If noLinears is true, we are being told p has no linear factors false does not mean that p is reducible.
brillhartIrreducible?(p) returns true if p can be shown to be irreducible by a remark of Brillhart, false is inconclusive.
brillhartTrials() returns the number of tests in brillhartIrreducible?.
brillhartTrials(n) sets to n the number of tests in brillhartIrreducible? and returns the previous value.
noLinearFactor?(p) returns true if p can be shown to have no linear factor by a theorem of Lehmer, false else. I insist on the fact that false does not mean that p has a linear factor.


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Complex-Ion Equilibria

When a metal ion reacts with a Lewis base in solution a complex ion is formed. This reaction can be described in terms of chemical equilibria.

Introduction

A complex ion is comprised of two important parts: the central atom and its surrounding ligands. The central atom can be any metallic ion (usually a transition metal). The ligands are any combination of anions that can donate an electron pair, effectively meaning they are all Lewis bases. When combined they form coordinate covalent bonds.

Formation Constant

In general, chemical equilibrium is reached when the forward reaction rate is equal to the reverse reaction rate and can be described using an equilibrium constant, $$K$$. Complex ion equilibria are no exception to this and have their own unique equilibrium constant. This formation constant, $$K_f$$, describes the formation of a complex ion from its central ion and attached ligands. This constant may be called a stability constant or association constant; the units depend on the specific reaction it is describing. Common metal/ligand $$K_f$$ values are tabulated in Table E4.

At its most basic level, $$K_f$$ can be explained as the following, where $$M$$ is a metal ion, $$L$$ is a ligand, and $$x$$ and $$y$$ are coefficients:

$K_f = \dfrac{[M_xL_y]} {[M]^x[L]^y}$

$$K_f$$ values tend to be very large in magnitude because complex ion formation reactions heavily favor the product.

Example $$\PageIndex{1}$$

For example, observe the formation reaction of the dicyanoargentate(I) ion and the resulting formation constant expression:

$Ag^+_{(aq)} + 2CN^-\, _{(aq)} \rightleftharpoons Ag(CN)_2^- \, _{(aq)}$

$K_f = \dfrac{[Ag(CN)_2^-]}{[Ag^{+}][CN^{-}]^{2}} = 5.6 \times 10^{18}$

The larger the $$K_f$$ value of a complex ion, the more stable it is. Due to how large formation constants often are it is not uncommon to see them listed as logarithms in the form $$\log K_f$$. You may also see them in the form of a dissociation constant, $$K_d$$ (which should not be confused with pKa). This is merely the inverse value of the formation constant and as such describes the instability of a complex ion. It may help to think of $$K_f$$ and $$K_d$$ as stability and instability constants, respectively.

Using the above example, the dissociation constant expression would be:

$$K_f = \dfrac{1}{K_d}$$

$$K_{d} = \dfrac{1}{K_f} = \dfrac{[Ag^{+}][CN^{-}]^{2}}{[[Ag(CN)_2]^{-}]} = 1.8 \times 10^{-19}$$

In logarithm form $$K_f$$and $$K_d$$ would be:

$$\log K_{f} = \log(5.6 \times 10^{18}) = 18.7$$

$$\log K_{d} = \log(1.8 \times 10^{-19}) = -18.7$$

Example $$\PageIndex{2}$$: Stepwise Equilbria

One might assume that the formation of complex ions is a one step process, however this is not always the case. As an example let's look at the formation of tetraamminecuprate(II) ion in solution:

$[Cu(H_2O)_4]^{2+}_{(aq)} + 4NH_{3 \; (aq)} \rightleftharpoons [Cu(NH_3)_4]^{2+}_{(aq)} + 4H_2O_{(l)}$

The reason for using the hydrated form of the copper ion is that cations in solution usually exist in their hydrated form. It is because of this that the creation of tetraamminecuprate(II) ion is a multi-step affair, the ammine ligands have to displace the existing aqua ligands. In calculations however, the concentration of water and the number of aqua ligands are effectively constant meaning that they can be safely ignored. Each step of this process has its own formation constant, and as with other chemical equilibria, the overall reaction constant is the product of these stepwise formation constants.

Step 1:

$[Cu(H_2O)_4]^{2+}_{(aq)} + NH_{3 \; (aq)} \rightleftharpoons [Cu(H_2O)_3(NH_3)]^{2+}_{(aq)} + H_2O_{(l)}$

$K_{1} = \dfrac{[[Cu(H_2O)_3(NH_3)]^{2+}]}{[[Cu(H_2O)_4]^{2+}][NH_3]} = 1.9 \times 10^4$

Step 2:

$[Cu(H_2O)_3(NH_3)]^{2+}_{(aq)} + NH_{3 \; (aq)} \rightleftharpoons [Cu(H_2O)_2(NH_3)_2]^{2+}_{(aq)} + H_2O_{(l)}$

$K_{2} = \dfrac{[[Cu(H_2O)_2(NH_3)_2]^{2+}]}{[[Cu(H_2O)_3(NH_3)]^{2+}][NH_3]} = 3.9 \times 10^3$

Step 3:

$[Cu(H_2O)_2(NH_3)_2]^{2+}_{(aq)} + NH_{3 \; (aq)} \rightleftharpoons [Cu(H_2O)(NH_3)_3]^{2+}_{(aq)} + H_2O_{(l)}$

$K_{3} = \dfrac{[[Cu(H_2O)(NH_3)_3]^{2+}]}{[[Cu(H_2O)_2(NH_3)_2]^{2+}][NH_3]} = 1.0 \times 10^3$

Step 4:

$[Cu(H_2O)(NH_3)_3]^{2+}_{(aq)} + NH_{3 \; (aq)} \rightleftharpoons [Cu(NH_3)_4]^{2+}_{(aq)} + H_2O_{(l)}$

$K_{4} = \dfrac{[[Cu(NH_3)_4]^{2+}]}{[[Cu(H_2O)(NH_3)_3]^{2+}][NH_3]} = 1.5 \times 10^2$

Total

$K_f = \beta_4 = K_{1} \times K_{2} \times K_{3} \times K_{4} = 1.1 \times 10^{13}$

$K_f = \dfrac{[[Cu(H_2O)_3(NH_3)]^{2+}]}{[[Cu(H_2O)_4]^{2+}][NH_3]} \times \dfrac{[[Cu(H_2O)_2(NH_3)_2]^{2+}]}{[[Cu(H_2O)_3(NH_3)]^{2+}][NH_3]} \times \\ \dfrac{[[Cu(H_2O)(NH_3)_3]^{2+}]}{[[Cu(H_2O)_2(NH_3)_2]^{2+}][NH_3]} \times \dfrac{[[Cu(NH_3)_4]^{2+}]}{[[Cu(H_2O)(NH_3)_3]^{2+}][NH_3]}$

$= \dfrac{[[Cu(NH_3)_4]^{2+}]}{[[Cu(H_2O)_4]^{2+}][NH_3]^{4}} = K_f$

If you wanted to find the formation constant of one of the intermediate steps, you would simply take the product of the $$K$$ values up until that point. For example the formation constant of $$[Cu(H_2O)_2(NH_3)_2]^{2+}$$ would be $$\beta_2 = K_{1} \times K_{2} = 7.4 \times 10^7$$. As seen here and above the standard notation for these values is a lowercase beta.

You may notice that each stepwise formation constant is smaller than the one before it. This decreasing trend is due to the effects of entropy, causing each step to be progressively less likely to occur. You can think of this in the following way, continuing with the previous example:

When the first ammine ligand goes to displace an aqua ligand it has four sites from which to choose from, making it "easier" to displace one. Yet with every step the number of sites decreases making it increasingly more difficult. As always though, there are exceptions to this rule. If the values do not continually decrease then the structure of the complex ion likely changed during one of the steps.

Chelation Effect

Generally, complex ions with polydentate ligands have much higher formation constants than those with monodentate ligands. This additional stability is known as the chelation effect. For example, the formation of the bis(ethylenediamine)cuprate(II) ion is identical to the tetraamminecuprate(II) ion above except the ethylenediamine ligands displace two aqua ligands at a time. Observe the differences in formation constant (Table E4):

 Complex Ion Kf $$[Cu(NH_3)_4]^{2+}$$ $$1.1 \times 10^{13}$$ $$[Cu(en)_2]^{2+}$$ $$1.0 \times 10^{20}$$

Solubility Effects

The formation of complex ions can sometimes explain the solubility of certain compounds in solution. For example, let's say we have a saturated solution of silver bromide ($$AgBr$$) which also contains additional undissolved solid silver bromide. If high enough concentrations of ammonia are added to the solution, the solid silver bromide will dissolve despite the solution previously having been saturated. This is because of the formation diammineargentate(I) ion, illustrated by the following chemical equation:

$$AgBr_{(s)} + 2NH_{3 \; (aq)} \rightarrow [Ag(NH_3)_2]^{+}_{(aq)} + Br^-_{(l)}$$

References

1. Beck, Mihály T. Chemistry of Complex Equilibria. Revised ed. New York: Halsted Press, 1990.
2. Brown, Theodore, Eugene Lemay, Bruce Bersten, and Catherine Murphey. Chemistry: The Central Science. 10th ed. Andalusia, Alabama: Pearson Publications Company, 2006.
3. Petrucci, Harwood, Herring, and Madura. General Chemistry: Principles and Modern Applications. 9th ed. Upper Saddle River, New Jersey: Pearson Education, 2007.
4. Wilkinson, Geoffrey, Robert D. Gillard, and Jon A. McCleverty, eds. Comprehensive Coordination Chemistry: The Synthesis, Reactions, Properties & Applications of Coordination Compounds. First ed. Vol. 1. Oxford, England: Pergamon Press, 1987. N. pag. 7 vols.

Problems

1. Based on the given values, rank the following complex ions from most to least stable.

$$[Cr(OH)_4]^{-}$$ $$K_f = 8.0 \times 10^{29}$$

$$[Ni(CN)_4]^{2-}$$ $$\log(K_d) = -31.3$$

$$[Co(NH_3)_6]^{3+}$$ $$K_d = 2.2 \times 10^{-34}$$

$$[Cu(CN)_4]^{3-}$$ $$\log(K_f) = 30.3$$

2. Of $$[Ni(en)_3]^{2+}$$, $$[Ni(EDTA)]^{2-}$$, and $$[Ni(NH_3)_6]^{2+}$$ which would you expect to have the largest kf value? The smallest? Explain.

3. The creation of $$[Ni(en)_3]^{2+}$$ is a three step process with the following stepwise formation constants $$K_1 = 3.3 \times 10^{7}$$, $$K_2 = 1.9 \times 10^{6}$$, and $$K_3 = 1.8 \times 10^{4}$$. Using this information find the overall formation constant and the formation constant for each of the intermediate steps.

Solutions

1. The key to solving this problem is to convert all of the given values into the same form. In this case we will convert all of them into the form Kf though any other form could be used as long as you are consistent.

$$[Cr(OH)_4]^{-}$$ $$K_f = 8.0 \times 10^{29}$$

$$[Ni(CN)_4]^{2-}$$ $$\log(K_d) = -31.3 \longrightarrow K_f = 2.0 \times 10^{31}$$

$$[Co(NH_3)_6]^{3+}$$ $$K_d = 2.2 \times 10^{-34} \longrightarrow K_f = 4.5 \times 10^{33}$$

$$[Cu(CN)_4]^{3-}$$ $$\log(K_f) = 30.3 \longrightarrow K_f = 2.0 \times 10^{30}$$

After converting it is easy to rate the ions in terms of stability. From most to least stable: $$[Co(NH_3)_6]^{3+}$$, $$[Ni(CN)_4]^{2-}$$, $$[Cu(CN)_4]^{3-}$$, $$[Cr(OH)_4]^{-}$$.

2. This is a question involving the effects of chelation. When the complex ion with EDTA is formed, the EDTA ligand displaces six aqua ligands at once. Ethylenediamine ligands displace two at a time and ammine ligands displace one at a time. Based on this we would expect $$[Ni(EDTA)]^{2-}$$ to be the most stable and $$[Ni(NH_3)_6]^{2+}$$ to be the least stable.

3. Take the product of the stepwise formation constants up until the step you are trying to find.

$$[Ni(H_2O)_4(en)]^{2+}$$ $$\beta_1 = K_1 = 3.3 \times 10^{7}$$

$$[Ni(H_2O)_2(en)_2]^{2+}$$ $$\beta_2 = K_1 \times K_2 = 6.8 \times 10^{13}$$

$$[Ni(en)_3]^{2+}$$ $$\beta_3 = K_f = K_1 \times K_2 \times K_3 = 1.1 \times 10^{18}$$

Contributors

• Katherine Barrett, Gianna Navarro, Christopher May (UCD)


text:
Eigenvalues

• April 7th 2012, 08:22 AM
renolovexoxo
Eigenvalues
Suppose A is nonsingular. Prove that the eigenvalues of A^-1 are the reciprocals of the eigenvalues of A.

I was able to come up with a proof for eigenvalues of A being eigenvalues of A^-1, but I'm unsure about how to prove the opposite way to prove the equality.
• April 7th 2012, 08:31 AM
Prove It
Re: Eigenvalues
You should know \displaystyle \begin{align*} \mathbf{A}\mathbf{x} = \lambda \mathbf{x} \end{align*}. Now, if \displaystyle \begin{align*} \mathbf{A} \end{align*} is invertible, then \displaystyle \begin{align*} \mathbf{A}^{-1} \end{align*} exists and \displaystyle \begin{align*} \mathbf{A}^{-1}\mathbf{A} = \mathbf{I} \end{align*}. Anyway...

\displaystyle \begin{align*} \mathbf{A}\mathbf{x} &= \lambda \mathbf{x} \\ \mathbf{A}^{-1}\mathbf{A}\mathbf{x} &= \mathbf{A}^{-1}\lambda \mathbf{x} \\ \mathbf{I}\mathbf{x} &= \lambda \mathbf{A}^{-1}\mathbf{x} \\ \mathbf{x} &= \lambda \mathbf{A}^{-1}\mathbf{x} \\ \frac{1}{\lambda}\mathbf{x} &= \mathbf{A}^{-1}\mathbf{x} \end{align*}

Since we have the product of two matrices (in this case \displaystyle \begin{align*} \mathbf{A}^{-1}\mathbf{x} \end{align*}) equal to a scalar multiple of the second matrix (in this case, \displaystyle \begin{align*} \frac{1}{\lambda}\mathbf{x} \end{align*}), that means that \displaystyle \begin{align*} \frac{1}{\lambda} \end{align*} must be an eigenvalue of \displaystyle \begin{align*} A^{-1} \end{align*}.
• April 7th 2012, 10:22 AM
renolovexoxo
Re: Eigenvalues
That was the direction I was able to complete. I was asking about the other way. Proving that for an eigenvalue of A^-1 there is a reciprocal in the set of eigenvalues for A.
• April 7th 2012, 11:33 AM
Haven
Re: Eigenvalues
Exact same proof, just replace A with A^{-1} and vice versa.
• April 8th 2012, 07:26 AM
renolovexoxo
Re: Eigenvalues
Okay, Thank you!


text:
Matlab label with different colors on the string using latex interpreter

129 views (last 30 days)
Oyeniyi on 14 Apr 2016
Commented: Chibuzo Nnonyelu on 18 Dec 2018
How do I specify different font colors on the same string using the TEXT, TITLE, XLABEL, YLABEL, or ZLABEL commands etc., while using the LATEX interpreter. For example, the command below works well with the TEX interpreter, how do I get it (or something similar) to work with the LATEX interpreter.
\color{red}x \color{green}+ \color{blue}y

Daniel Lyddy on 18 Sep 2017
Suppose you have a 3-vector called 'titleColor' that holds the RGB components of the color you want to assign to the title (which I will call 'titleString').
tStr = sprintf('\\color[rgb]{%f, %f, %f}%s', titleColor, titleString);
title(tStr);
I haven't tried this trick with labels yet, but I don't see why a similar approach would not work.
And yes, there are two backslashes before the color[rgb] in the code above. The sprintf command drops one of them.


text:
# Math Help - 3 numbers picked from 1 - 10 prob lowest no is 5

1. ## 3 numbers picked from 1 - 10 prob lowest no is 5

I have the question you have the numbers 1 - 10 and 3 numbers are picked at random whats the probability the lowest picked number is 5

What i guessed it would be is to use the choose functions so 10C1 * 4C1 * 3C1 / 10C3 which equals 1 which is wrong. But ive no idea how else to do it.

2. Hello, renlok!

You have the numbers 1 - 10 and three numbers are picked at random.
What's the probability the lowest picked number is 5?
Choosing three numbers from ten numbers, there are: . $_{10}C_3 \:=\:{10\choose3} \:=\:120$ outcomes.

We want the "5" and two Others.
The two Others are chosen from {6, 7, 8, 9, 10}.
.There are: . $_5C_2 \:=\:{5\choose2} \:=\:10$ ways.

Therefore: . $P(\text{lowest is 5}) \;=\;\frac{10}{120} \;=\;\frac{1}{12}$


text:
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HW Helper
Thanks
P: 26,167
Hi ╔(σ_σ)╝!

(have a theta: θ and a phi: φ )
 Quote by ╔(σ_σ)╝ … I found that P dot Q = cos($$\phi$$ - $$\theta$$) But the answer at the back of my book says cos($$\phi$$)cos( $$\theta$$) - sin($$\phi$$)sin( $$\theta$$) I guess this is the same P dot Q, so my answer is the same as the book.
If that's meant to be cos(θ - φ), then the book is wrong.
 3) … So |P - Q|/2 = 1-cos$$\varphi$$ Right ? But then my book says the answer is |sin (0.5($$\phi$$ - $$\theta$$))|
i] |A| is the square-root of A.A

ii] 1 + cos2θ = 2sin2θ