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# Articles

#### 2020

• J. Bréhin, F. Trier, L. M. Vicente Arche, P. Hemme, P. Noël, M. Cosset-Chéneau, J. - P. Attané, L. Vila, A. Sander, Y. Gallais, A. Sacuto, B. Dkhil, V. Garcia, S. Fusil, A. Barthélémy, M. Cazayous, M. Bibes, Switchable two-dimensional electron gas based on ferroelectric Ca:SrTiO3. Physical Review Materials. 4, 041002 (2020).
Résumé : Ferroelectric materials possess electric dipoles adding up to a macroscopic polarization that is switchable by an electric field. Most ferroelectrics are insulators but some are wide bandgap semiconductors that by doping can be turned into metallic conductors. If doping is restricted to a thin slab near the material surface, the conducting region may harbor a two-dimensional electron gas (2DEG). The 2DEG can then be affected by ferroelectric polarization switching, and may even retain ferroelectric properties coexisting with the conducting behavior. The paper by Br\'ehin $e\phantom{\rule{0}{0ex}}t$ $a\phantom{\rule{0}{0ex}}l$ reports indications of this behavior in a 2DEG at the surface of ferroelectric Ca-SrTiO${}_{3}$.

• J. - Y. Chauleau, T. Chirac, S. Fusil, V. Garcia, W. Akhtar, J. Tranchida, P. Thibaudeau, I. Gross, C. Blouzon, A. Finco, M. Bibes, B. Dkhil, D. D. Khalyavin, P. Manuel, V. Jacques, N. Jaouen, M. Viret, Electric and antiferromagnetic chiral textures at multiferroic domain walls. Nature Materials. 19, 386-390 (2020).
Résumé : Chiral electric and magnetic structures are observed at domain walls in thin films of the room-temperature multiferroic BiFeO3.

• M. W. Daniels, A. Madhavan, P. Talatchian, A. Mizrahi, M. - D. Stiles, Energy-Efficient Stochastic Computing with Superparamagnetic Tunnel Junctions. Physical Review Applied. 13, 034016 (2020).
Résumé : Most computing schemes that employ superparamagnetic tunnel junctions control them with analog currents, leading to substantial Ohmic losses and requiring digital-to-analog converters. Here the authors forego current control and embed these junctions in digital circuits to produce programmable randomness, which leads to a neural network that can recognize handwritten digits at only 150 nJ per inference. This energy efficiency is made possible by the general insight that, while nanodevices provide useful dynamics for innovative computing, understanding their integration with digital logic systems is crucial to developing viable applications.

• V. E. Demidov, S. Urazhdin, A. Anane, V. Cros, S. O. Demokritov, Spin–orbit-torque magnonics. Journal of Applied Physics. 127, 170901 (2020).
Résumé : The field of magnonics, which utilizes propagating spin waves for nanoscale transmission and processing of information, has been significantly advanced by the advent of the spin–orbit torque. The l...
Mots-clés : Magnonics.

• D. Di Nuzzo, R. Mizuta, K. Nakanishi, M. - B. Martin, A. I. Aria, R. Weatherup, R. H. Friend, S. Hofmann, J. Alexander-Webber, Graphene-passivated nickel as an efficient hole-injecting electrode for large area organic semiconductor devices. Applied Physics Letters. 116, 163301 (2020).
Résumé : Efficient injection of charge from metal electrodes into semiconductors is of paramount importance to obtain high performance optoelectronic devices. The quality of the interface between the electr...

• A. Dyrdał, J. Barnaś, A. Fert, Spin-Momentum-Locking Inhomogeneities as a Source of Bilinear Magnetoresistance in Topological Insulators. Physical Review Letters. 124, 046802 (2020).
Résumé : A new model for bilinear magnetoresistance, so called because it is linear in both the electric and the magnetic field, proposes another origin mechanism that accounts for recent contradictory experimental observations.

• K. Fallon, S. Hughes, K. Zeissler, W. Legrand, F. Ajejas, D. Maccariello, S. McFadzean, W. Smith, D. McGrouther, S. Collin, N. Reyren, V. Cros, C. H. Marrows, S. McVitie, Controlled Individual Skyrmion Nucleation at Artificial Defects Formed by Ion Irradiation. Small. 16, 1907450 (2020).
Résumé : Nanoscale artificial defects in magnetic multilayer films with perpendicular magnetic anisotropy (PMA) and Dzyaloshinskii–Moriya interaction (DMI), made in a controllable manner using a focused ion b...

• O. Gladii, M. Collet, Y. Henry, J. - V. Kim, A. Anane, M. Bailleul, Determining Key Spin-Orbitronic Parameters via Propagating Spin Waves. Physical Review Applied. 13, 014016 (2020).
Résumé : We characterize spin-wave propagation and its modification by an electrical current in permalloy($\mathrm{Py}$)/$\mathrm{Pt}$ bilayers with $\mathrm{Py}$ thickness between 4 and 20 nm. First, we analyze the frequency nonreciprocity of surface spin waves and extract from it the interfacial Dzyaloshinskii-Moriya interaction constant ${D}_{s}$ accounting for an additional contribution due to asymmetric surface anisotropies. Second, we measure the spin-wave relaxation rate and deduce from it the $\mathrm{Py}$/$\mathrm{Pt}$ spin-mixing conductance ${g}_{\mathrm{eff}}^{\ensuremath{\uparrow}\ensuremath{\downarrow}}$. Last, applying a dc electrical current, we extract the spin Hall conductivity ${\ensuremath{\sigma}}_{\mathrm{SH}}$ from the change of spin-wave relaxation rate due to the spin Hall spin-transfer torque. We obtain a consistent picture of the spin-wave propagation data for different film thicknesses using a single set of parameters ${D}_{s}=0.25$ pJ/m, ${g}_{\mathrm{eff}}^{\ensuremath{\uparrow}\ensuremath{\downarrow}}=3.2\ifmmode\times\else\texttimes\fi{}{10}^{19}\phantom{\rule{0.2em}{0ex}}{\mathrm{m}}^{\ensuremath{-}2}$ and ${\ensuremath{\sigma}}_{\mathrm{SH}}=4\ifmmode\times\else\texttimes\fi{}{10}^{5}$ S/m.
Mots-clés : Magnonics.

• T. Guillet, C. Zucchetti, Q. Barbedienne, A. Marty, G. Isella, L. Cagnon, C. Vergnaud, H. Jaffrès, N. Reyren, J. - M. George, A. Fert, M. Jamet, Observation of Large Unidirectional Rashba Magnetoresistance in Ge(111). Physical Review Letters. 124, 027201 (2020).
Résumé : A new experiment shows that the semiconductor germanium exhibits unidirectional magnetoresistance, which had previously only been seen in more exotic materials

• A. Haykal, J. Fischer, W. Akhtar, J. - Y. Chauleau, D. Sando, A. Finco, F. Godel, Y. A. Birkhölzer, C. Carrétéro, N. Jaouen, M. Bibes, M. Viret, S. Fusil, V. Jacques, V. Garcia, Antiferromagnetic textures in BiFeO3 controlled by strain and electric field. Nature Communications. 11, 1704 (2020).
Résumé : Tailoring antiferromagnetic domains is critical for the development of low-dissipative spintronic and magnonic devices. Here the authors demonstrate the control of antiferromagnetic spin textures in multiferroic bismuth ferrite thin films using strain and electric fields.

• M. - A. Husanu, L. Vistoli, C. Verdi, A. Sander, V. Garcia, J. Rault, F. Bisti, L. L. Lev, T. Schmitt, F. Giustino, A. S. Mishchenko, M. Bibes, V. N. Strocov, Electron-polaron dichotomy of charge carriers in perovskite oxides. Communications Physics. 3, 62 (2020).
Résumé : The underlying mechanisms of the metal-insulator transition in correlated oxides are a rich source of interesting physics and a topic of long-standing investigation. Here, the authors use angle-resolved photoelectron spectroscopy to investigate changes in charge carrier properties and electron-phonon interactions as a function of Ce-doping across the metal-insulator transition in CaMnO3.

• W. Legrand, D. Maccariello, F. Ajejas, S. Collin, A. Vecchiola, K. Bouzehouane, N. Reyren, V. Cros, A. Fert, Room-temperature stabilization of antiferromagnetic skyrmions in synthetic antiferromagnets. Nature Materials. 19, 34-42 (2020).
Résumé : Antiferromagnetic skyrmions—which have distinct advantages over skyrmions found in other magnetic systems—are observed at room temperature in synthetic antiferromagnets. These results hold promise for low-power spintronic devices.

• J. Liang, W. Wang, H. Du, A. Hallal, K. Garcia, M. Chshiev, A. Fert, H. Yang, Very large Dzyaloshinskii-Moriya interaction in two-dimensional Janus manganese dichalcogenides and its application to realize skyrmion states. Physical Review B. 101, 184401 (2020).
Résumé : The intense recent research on skyrmions has focused on multilayers of classical magnetic materials, such as Co, CoFeB, or Fe. Here, the authors explore skyrmions in van der Waals bi-dimensional magnets, a new type of magnetic material in the broad family of 2D materials that contains graphene. Using $a\phantom{\rule{0}{0ex}}b$ $i\phantom{\rule{0}{0ex}}n\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}t\phantom{\rule{0}{0ex}}i\phantom{\rule{0}{0ex}}o$ and Monte Carlo calculations, they demonstrate that skyrmions should exist in the so-called Janus transition metal dichalcogenides. The temperature and magnetic field range for observation of skyrmions are also predicted, as a guide for experimental exploration.

• D. Marković, N. Leroux, A. Mizrahi, J. Trastoy, V. Cros, P. Bortolotti, L. Martins, A. Jenkins, R. Ferreira, J. Grollier, Detection of the Microwave Emission from a Spin-Torque Oscillator by a Spin Diode. Physical Review Applied. 13, 044050 (2020).
Résumé : Magnetic tunnel junctions are nanoscale spintronic devices with microwave-generation and -detection capabilities. Here, we use the rectification effect called a spin diode'' in a magnetic tunnel junction to wirelessly detect the microwave emission of another junction in the auto-oscillatory regime. We show that the rectified spin-diode voltage measured at the receiving junction end can be reconstructed from the independently measured auto-oscillation and spin-diode spectra in each junction. Finally, we adapt the auto-oscillator model to the case of the spin-torque oscillator and the spin diode and we show that it accurately reproduces the experimentally observed features. These results will be useful for the design of circuits and chips based on spintronic nanodevices communicating through microwaves.

• H. Naganuma, V. Zatko, M. Galbiati, F. Godel, A. Sander, C. Carrétéro, O. Bezencenet, N. Reyren, M. - B. Martin, B. Dlubak, P. Seneor, A perpendicular graphene/ferromagnet electrode for spintronics. Applied Physics Letters. 116, 173101 (2020).
Résumé : We report on the large-scale integration of graphene layers over a FePd perpendicular magnetic anisotropy (PMA) platform, targeting further downscaling of spin circuits. An L10 FePd ordered alloy s...

• Y. Nahas, S. Prokhorenko, J. Fischer, B. Xu, C. Carrétéro, S. Prosandeev, M. Bibes, S. Fusil, B. Dkhil, V. Garcia, L. Bellaiche, Inverse transition of labyrinthine domain patterns in ferroelectric thin films. Nature. 577, 47-51 (2020).
Résumé : The labyrinthine domain patterns formed in ultrathin films of ferroelectric oxides by subcritical quenching undergo an inverse phase transition to the less-symmetric parallel-stripe domain structure upon increasing temperature.

• P. Noël, M. Cosset-Cheneau, V. Haspot, V. Maurel, C. Lombard, M. Bibes, A. Barthelemy, L. Vila, J. - P. Attané, Negligible thermal contributions to the spin pumping signal in ferromagnetic metal–platinum bilayers. Journal of Applied Physics. 127, 163907 (2020).
Résumé : Spin pumping by ferromagnetic resonance is one of the most common techniques to determine spin Hall angles, Edelstein lengths, or spin diffusion lengths of a large variety of materials. In recent y...

• P. Noël, F. Trier, L. M. Vicente Arche, J. Bréhin, D. - C. Vaz, V. Garcia, S. Fusil, A. Barthélémy, L. Vila, M. Bibes, J. - P. Attané, Non-volatile electric control of spin–charge conversion in a SrTiO3 Rashba system. Nature. 580, 483 (2020).
Résumé : The polarization direction of a ferroelectric-like state can be used to control the conversion of spin currents into charge currents at the surface of strontium titanate, a non-magnetic oxide.

• X. Palermo, N. Reyren, S. Mesoraca, A. V. Samokhvalov, S. Collin, F. Godel, A. Sander, K. Bouzehouane, J. Santamaria, V. Cros, A. I. Buzdin, J. - E. Villegas, Tailored Flux Pinning in Superconductor-Ferromagnet Multilayers with Engineered Magnetic Domain Morphology From Stripes to Skyrmions. Physical Review Applied. 13, 014043 (2020).
Résumé : Superconductor-ferromagnet (S/F) hybrid systems show interesting magnetotransport behaviors that result from the transfer of properties between both constituents. For instance, magnetic memory can be transferred from the F into the S through the pinning of superconducting vortices by the ferromagnetic textures. The ability to tailor this type of induced behavior is important to broaden its range of application. Here we show that engineering the F magnetization reversal allows the tuning of the strength of the vortex pinning (and memory) effects, as well as the field range in which they appear. This is done by using magnetic multilayers in which Co thin films are combined with different heavy metals ($\mathrm{Ru}$, $\mathrm{Ir}$, $\mathrm{Pt}$). By choosing the materials, thicknesses, and stacking order of the layers, we can design the characteristic domain size and morphology, from out-of-plane magnetized stripe domains to much smaller magnetic skyrmions. These changes strongly affect the magnetotransport properties. The underlying mechanisms are identified by comparing the experimental results to a magnetic pinning model.

• V. Rouco, R. El Hage, A. Sander, J. Grandal, K. Seurre, X. Palermo, J. Briatico, S. Collin, J. Trastoy, K. Bouzehouane, A. I. Buzdin, G. Singh, N. Bergeal, C. Feuillet-Palma, J. Lesueur, C. Leon, M. Varela, J. Santamaría, J. - E. Villegas, Quasiparticle tunnel electroresistance in superconducting junctions. Nature Communications. 11, 658 (2020).
Résumé : The non-volatile switching of tunnel electroresistance in ferroelectric junctions provides the basis for memory and neuromorphic computing devices. Rouco et al. show tunnel electroresistance in superconductor-based junctions that arises from a redox rather than ferroelectric mechanism and is enhanced by superconductivity.

• P. Talatchian, M. Romera, F. Abreu Araujo, P. Bortolotti, V. Cros, D. Vodenicarevic, N. Locatelli, D. Querlioz, J. Grollier, Designing Large Arrays of Interacting Spin-Torque Nano-Oscillators for Microwave Information Processing. Physical Review Applied. 13, 024073 (2020).
Résumé : Arrays of spin-torque nano-oscillators are promising for broadband microwave signal detection and processing, as well as for neuromorphic computing. In many of these applications, the oscillators should be engineered to have equally spaced frequencies and equal sensitivity to microwave inputs. Here we design spin-torque nano-oscillator arrays with these rules and estimate their optimum size for a given sensitivity, as well as the frequency range that they cover. For this purpose, we explore analytically and numerically conditions to obtain vortex spin-torque nano-oscillators with equally spaced gyrotropic oscillation frequencies and having all similar synchronization bandwidths to input microwave signals. We show that arrays of hundreds of oscillators covering ranges of several hundred MHz can be built taking into account nanofabrication constraints.

• L. Thomas, D. Guérin, B. Quinard, E. Jacquet, R. Mattana, P. Seneor, D. Vuillaume, T. Mélin, S. Lenfant, Conductance switching at the nanoscale of diarylethene derivative self-assembled monolayers on La0.7Sr0.3MnO3. Nanoscale. 12, 8268 (2020).
Résumé : We report on the phosphonic acid route for the grafting of functional molecules, optical switch (dithienylethene diphosphonic acid, DDA), on La0.7Sr0.3MnO3 (LSMO). Compact self-assembled monolayers (SAMs) of DDA are formed on LSMO as studied by topographic atomic force microscopy (AFM), ellipsometry, water contact angle measurements and X-ray photoemission spectroscopy (XPS). The conducting AFM measurements show that the electrical conductance of LSMO/DDA is about 3 decades below that of a bare LSMO substrate. Moreover, the presence of the DDA SAM suppresses the known conductance switching of the LSMO substrate that is induced by mechanical and/or bias constraints during C-AFM measurements. A partial light-induced conductance switching between the open and closed forms of the DDA is observed for the LSMO/DDA/C-AFM tip molecular junctions (closed/open conductance ratio of about 8). We show that, in the case of long-time exposure to UV light, this feature can be masked by a non-reversible decrease (a factor of about 15) of the conductance of the LSMO electrode.

• F. Trier, D. - C. Vaz, P. Bruneel, P. Noël, A. Fert, L. Vila, J. - P. Attané, A. Barthélémy, M. Gabay, H. Jaffrès, M. Bibes, Electric-Field Control of Spin Current Generation and Detection in Ferromagnet-Free SrTiO3-Based Nanodevices. Nano Letters. 20, 395 (2020).
Résumé : Spintronics entails the generation, transport, manipulation and detection of spin currents, usually in hybrid architectures comprising interfaces whose impact on performance is detrimental. In addition, how spins are generated and detected is generally material specific and determined by the electronic structure. Here, we demonstrate spin current generation, transport and electrical detection, all within a single non-magnetic material system: a SrTiO3 two-dimensional electron gas (2DEG) with Rashba spin-orbit coupling. We show that the spin current is generated from a charge current by the 2D spin Hall effect, transported through a channel and reconverted into a charge current by the inverse 2D spin Hall effect. Furthermore, by adjusting the Fermi energy with a gate voltage we tune the generated and detected spin polarization and relate it to the complex multiorbital band structure of the 2DEG. We discuss the leading mechanisms of the spin-charge interconversion processes and argue for the potential of quantum oxide materials for future all-electrical low-power spin-based logic.

• M. - W. Yoo, D. Rontani, J. Létang, S. Petit-Watelot, T. Devolder, M. Sciamanna, K. Bouzehouane, V. Cros, J. - V. Kim, Pattern generation and symbolic dynamics in a nanocontact vortex oscillator. Nature Communications. 11, 601 (2020).
Résumé : Controlling chaotic behavior in spintronic devices is promising for signal-processing applications. Here, the authors unveil the symbolic patterns hidden in the magnetization dynamics of a nanocontact vortex oscillator and detail how to control chaos complexity with a single experimental parameter.

#### 2019

• A. Arora, L. C. Phillips, P. Nukala, M. B. Hassine, A. A. Ünal, B. Dkhil, L. Balcells, O. Iglesias, A. Barthélémy, F. Kronast, M. Bibes, S. Valencia, Switching on superferromagnetism. Physical Review Materials. 3, 024403 (2019).
Résumé : Electric-field control of magnetism has emerged as a potential approach for low-power consumption spintronics. Although recent results in multiferroic systems have shown the possibility to manipulate the magnetic state of magnetic materials deposited on ferroelectric substrates by means of electric fields, the resulting magnetic state is multidomain in nature. For real multiferroic devices to become a true single magnetic domain state, lower dimensionality is required. In this work, the authors show, for an assemble of iron nanograins deposited on top of a BaTiO${}_{3}$ substrate, that an electric field-induced strain is capable of switching on a collective long-range ferromagnetic order (superferromagnetism) on an otherwise zero-dimensional superparamagnetic nanoparticle system. The effect, observed slightly above room temperature, holds promise for the implementation of nanoscale multiferroic systems in spin-based storage and logic architectures operating at ambient conditions.

• C. Cerqueira, J. Y. Qin, H. Dang, A. Djeffal, J. - C. Le Breton, M. Hehn, J. - C. Rojas-Sanchez, X. Devaux, S. Suire, S. Migot, P. Schieffer, J. G. Mussot, P. Łaczkowski, A. Anane, S. Petit-Watelot, M. Stoffel, S. Mangin, Z. Liu, B. W. Cheng, X. F. Han, H. Jaffrès, J. - M. George, Y. Lu, Evidence of Pure Spin-Current Generated by Spin Pumping in Interface-Localized States in Hybrid Metal–Silicon–Metal Vertical Structures. Nano Letters. 19, 90 (2019).
Résumé : Evidence of Pure Spin-Current Generated by Spin Pumping in Interface-Localized States in Hybrid Metal–Silicon–Metal Vertical Structures

• M. Coll, J. Fontcuberta, M. Althammer, M. Bibes, H. Boschker, A. Calleja, G. Cheng, M. Cuoco, R. Dittmann, B. Dkhil, I. El Baggari, M. Fanciulli, I. Fina, E. Fortunato, C. Frontera, S. Fujita, V. Garcia, S. T. B. Goennenwein, C. - G. Granqvist, J. Grollier, R. Gross, A. Hagfeldt, G. Herranz, K. Hono, E. Houwman, M. Huijben, A. Kalaboukhov, D. J. Keeble, G. Koster, L. F. Kourkoutis, J. Levy, M. Lira-Cantu, J. L. MacManus-Driscoll, J. Mannhart, R. Martins, S. Menzel, T. Mikolajick, M. Napari, M. D. Nguyen, G. Niklasson, C. Paillard, S. Panigrahi, G. Rijnders, F. Sánchez, P. Sanchis, S. Sanna, D. G. Schlom, U. Schroeder, K. M. Shen, A. Siemon, M. Spreitzer, H. Sukegawa, R. Tamayo, J. van den Brink, N. Pryds, F. M. Granozio, Towards Oxide Electronics: a Roadmap. Applied Surface Science. 482, 1-93 (2019).

• F. Couëdo, E. Recoba Pawlowski, J. Kermorvant, J. Trastoy, D. Crété, Y. Lemaître, B. Marcilhac, C. Ulysse, C. Feuillet-Palma, N. Bergeal, J. Lesueur, High-Tc superconducting detector for highly-sensitive microwave magnetometry. Applied Physics Letters. 114, 192602 (2019).
Résumé : We have fabricated arrays of High-Tc Superconducting Quantum Interference Devices (SQUIDs) with randomly distributed loop sizes as sensitive detectors for Radio Frequency (RF) waves. These subwavelength size devices known as Superconducting Quantum Interference Filters (SQIFs) detect the magnetic component of the electromagnetic field. We used a scalable ion irradiation technique to pattern the circuits and engineer the Josephson junctions needed to make SQUIDs. Here, we report on a 300 SQUID series array with the loop area ranging from 6 to 60 μm2, folded in a meander line covering a 3.5 mm × 120 μm substrate area, made out of a 150 nm thick YBa2Cu3O7 film. Operating at a temperature of T = 66 K in an unshielded magnetic environment under low DC bias current (I = 60 μA) and a DC magnetic field (B = 3 μT), this SQIF can detect a magnetic field of a few picoteslas at a frequency of 1.125 GHz, which corresponds to a sensitivity of a few hundreds of fT / Hz and shows a linear response over 7 decades in RF power. This work is a promising approach for the realization of low dissipative subwavelength gigahertz magnetometers.We have fabricated arrays of High-Tc Superconducting Quantum Interference Devices (SQUIDs) with randomly distributed loop sizes as sensitive detectors for Radio Frequency (RF) waves. These subwavelength size devices known as Superconducting Quantum Interference Filters (SQIFs) detect the magnetic component of the electromagnetic field. We used a scalable ion irradiation technique to pattern the circuits and engineer the Josephson junctions needed to make SQUIDs. Here, we report on a 300 SQUID series array with the loop area ranging from 6 to 60 μm2, folded in a meander line covering a 3.5 mm × 120 μm substrate area, made out of a 150 nm thick YBa2Cu3O7 film. Operating at a temperature of T = 66 K in an unshielded magnetic environment under low DC bias current (I = 60 μA) and a DC magnetic field (B = 3 μT), this SQIF can detect a magnetic field of a few picoteslas at a frequency of 1.125 GHz, which corresponds to a sensitivity of a few hundreds of fT / Hz and shows a linear response over 7 decades in ...
Mots-clés : Supra.

• T. Devolder, D. Rontani, S. Petit-Watelot, K. Bouzehouane, S. Andrieu, J. Létang, M. - W. Yoo, J. - P. Adam, C. Chappert, S. Girod, V. Cros, M. Sciamanna, J. - V. Kim, Chaos in Magnetic Nanocontact Vortex Oscillators. Physical Review Letters. 123, 147701 (2019).
Résumé : We present an experimental study of spin-torque driven vortex self-oscillations in magnetic nanocontacts. We find that, above a certain threshold in applied currents, the vortex gyration around the nanocontact is modulated by relaxation oscillations, which involve periodic reversals of the vortex core. This modulation leads to the appearance of commensurate but also, more interestingly here, incommensurate states, which are characterized by devil's staircases in the modulation frequency. We use frequency- and time-domain measurements together with advanced time-series analyses to provide experimental evidence of chaos in incommensurate states of vortex oscillations, in agreement with theoretical predictions.

• P. Domenichini, C. P. Quinteros, M. Granada, S. Collin, J. - M. George, J. Curiale, S. Bustingorry, M. G. Capeluto, G. Pasquini, Transient magnetic-domain-wall ac dynamics by means of magneto-optical Kerr effect microscopy. Physical Review B. 99, 214401 (2019).
Résumé : Magnetic domain walls can be described within the general framework of disordered elastic systems, which accounts well for most of their dynamical and geometrical properties. Beyond the theoretical predictions, the authors report here unexpected features observed under the application of alternating square pulses. In particular, magneto-optical images show that domain walls evolve toward strongly distorted shapes concomitantly with a reduction of domain area. The evolution of the morphology is characterized, and the underlying physics is discussed.

• J. - Y. Duquesne, P. Rovillain, C. Hepburn, M. Eddrief, P. Atkinson, A. Anane, R. Ranchal, M. Marangolo, Surface-Acoustic-Wave Induced Ferromagnetic Resonance in Fe Thin Films and Magnetic Field Sensing. Physical Review Applied. 12, 024042 (2019).
Résumé : Surface acoustic waves (SAWs) are ubiquitous in sensor technology, their main benefits being sensitivity, amenability to remote control, and no need for an embedded energy source. The authors discuss magnetic sensors based on the resonant interaction between a surface acoustic wave and the magnetization of a thin ferromagnetic layer. This interaction can be switched on and off by tiny angular variation of the magnetic field, which could be useful to remotely monitor periodic motion, such as the rotation of a shaft. Using a simple model of magnetization dynamics accounting for the properties of the layer, they describe the salient properties of the SAW versus in-plane magnetic field.

• M. Ernoult, J. Grollier, D. Querlioz, Using Memristors for Robust Local Learning of Hardware Restricted Boltzmann Machines. Scientific Reports. 9, 1851 (2019).
Résumé : One of the biggest stakes in nanoelectronics today is to meet the needs of Artificial Intelligence by designing hardware neural networks which, by fusing computation and memory, process and learn from data with limited energy. For this purpose, memristive devices are excellent candidates to emulate synapses. A challenge, however, is to map existing learning algorithms onto a chip: for a physical implementation, a learning rule should ideally be tolerant to the typical intrinsic imperfections of such memristive devices, and local. Restricted Boltzmann Machines (RBM), for their local learning rule and inherent tolerance to stochasticity, comply with both of these constraints and constitute a highly attractive algorithm towards achieving memristor-based Deep Learning. On simulation grounds, this work gives insights into designing simple memristive devices programming protocols to train on chip Boltzmann Machines. Among other RBM-based neural networks, we advocate using a Discriminative RBM, with two hardware-oriented adaptations. We propose a pulse width selection scheme based on the sign of two successive weight updates, and show that it removes the constraint to precisely tune the initial programming pulse width as a hyperparameter. We also propose to evaluate the weight update requested by the algorithm across several samples and stochastic realizations. We show that this strategy brings a partial immunity against the most severe memristive device imperfections such as the non-linearity and the stochasticity of the conductance updates, as well as device-to-device variability.

• K. Fallon, S. McVitie, W. Legrand, F. Ajejas, D. Maccariello, S. Collin, V. Cros, N. Reyren, Quantitative imaging of hybrid chiral spin textures in magnetic multilayer systems by Lorentz microscopy. Physical Review B. 100, 214431 (2019).
Résumé : Chiral magnetic textures in ultrathin perpendicularly magnetized multilayer film stacks with an interfacial Dzyaloshinskii-Moriya interaction have been the focus of much research recently. The chirality associated with the broken inversion symmetry at the interface between an ultrathin ferromagnetic layer and a heavy metal with large spin-orbit coupling supports homochiral N\'eel domain walls and hedgehog (N\'eel) skyrmions. Under spin-orbit torques these N\'eel-type magnetic structures are predicted, and have been measured, to move at high velocities. However recent studies have indicated that some multilayered systems may possess a more complex hybrid domain wall configuration, due to the competition between interfacial DMI and interlayer dipolar fields. These twisted textures are expected to have thickness dependent N\'eel and Bloch contributions to the domain or skyrmion walls. In this work, we use the methods of Lorentz microscopy to determine quantitatively experimentally both (i) the contributions of the N\'eel and Bloch components and (ii) their spatial spin variation at high resolution. These are compared with modeled and simulated structures that are in excellent agreement with our experimental results. Our quantitative analysis provides powerful direct evidence of the Bloch wall component which exists in these hybrid walls and will be significant when exploiting chirality in spintronic applications.

• A. Fert, F. Nguyen Van Dau, Spintronics, from giant magnetoresistance to magnetic skyrmions and topological insulators. Comptes Rendus Physique. 20, 817-831 (2019).
Résumé : This article aims at giving a general presentation of spintronics, an important field of research developing today along many new directions in physic…

• N. Figueiredo-Prestes, R. C. Oliveira, M. A. B. Tavares, D. S. Costa, I. Mazzaro, H. F. Jurca, J. Zarpellon, M. D. Martins, C. Deranlot, J. - M. George, D. H. Mosca, Stabilization and tuning of perpendicular magnetic anisotropy in room-temperature ferromagnetic transparent CeO2 films. Journal of Applied Physics. 126, 183903 (2019).
Résumé : The development of multifunctional materials that combine optical transparency to room-temperature magnetism is still a great challenge. Here, we present an investigation on the room-temperature ferromagnetism of ceria (CeO 2) nanocrystalline films integrated to cobalt and platinum multilayered films (Co/Pt-ML) with perpendicular magnetic anisotropy (PMA). The magnetic coupling between Co/Pt-ML and CeO 2 films spaced by Pt interlayers with different thicknesses (from 3 nm to 20 nm) enable stabilization and tuning of PMA along with the stacking at remanence. CeO 2 films on the Co/Pt-ML films exhibit domain structures consisting of nonconnected labyrinthine patterns dominated by protruding fingers similar to that of individual Co/Pt-ML films. Our present results demonstrate that the PMA of Co/Pt-ML films can be used to exploit the room-temperature ferromagnetism of transparent CeO 2 films and their multifunctionalities for emerging transparent electronics and spintronics such as high-density magnetic devices.The development of multifunctional materials that combine optical transparency to room-temperature magnetism is still a great challenge. Here, we present an investigation on the room-temperature ferromagnetism of ceria (CeO 2) nanocrystalline films integrated to cobalt and platinum multilayered films (Co/Pt-ML) with perpendicular magnetic anisotropy (PMA). The magnetic coupling between Co/Pt-ML and CeO 2 films spaced by Pt interlayers with different thicknesses (from 3 nm to 20 nm) enable stabilization and tuning of PMA along with the stacking at remanence. CeO 2 films on the Co/Pt-ML films exhibit domain structures consisting of nonconnected labyrinthine patterns dominated by protruding fingers similar to that of individual Co/Pt-ML films. Our present results demonstrate that the PMA of Co/Pt-ML films can be used to exploit the room-temperature ferromagnetism of transparent CeO 2 films and their multifunctionalities for emerging transparent electronics and spintronics such as high-density mag...

• M. Galbiati, S. Tatay, S. M. - M. Dubois, F. Godel, R. Galceran, S. Mañas-Valero, M. Piquemal-Banci, A. Vecchiola, J. - C. Charlier, A. Forment-Aliaga, E. Coronado, B. Dlubak, P. Seneor, Path to Overcome Material and Fundamental Obstacles in Spin Valves Based on MoS2 and Other Transition-Metal Dichalcogenides. Physical Review Applied. 12, 044022 (2019).
Résumé : The recent introduction of two-dimensional materials into magnetic tunnel junctions (2D MTJs) offers very promising properties for spintronics, such as atomically defined interfaces, spin filtering, perpendicular anisotropy, and modulation of spin-orbit torque. Nevertheless, the difficulty of integrating exfoliated 2D materials into spintronic devices has limited exploration. Here the authors find a fabrication process leading to superior performance in MTJs based on transition-metal dichalcogenides, and further suggest a path to alleviate basic issues of technology and physics for 2D MTJs.

• S. Hurand, A. Jouan, E. Lesne, G. Singh, C. Feuillet-Palma, M. Bibes, A. Barthélémy, J. Lesueur, N. Bergeal, Josephson-like dynamics of the superconducting LaAlO3/SrTiO3 interface. Physical Review B. 99, 104515 (2019).
Résumé : In this article, we show that the two-dimensional electron gas formed at the ${\mathrm{LaAlO}}_{3}/{\mathrm{SrTiO}}_{3}$ interface behaves as a Josephson junction array. In particular, it exhibits a stochastic switching of the superconducting critical current, which qualitatively follows the dynamics of the resistively and capacitively shunted Josephson junction model. The switching current distribution (SCD) has been measured as a function of temperature and back-gate voltage. At low temperatures a clear saturation of the standard deviation of the SCD is observed, possibly indicating the presence of a macroscopic quantum tunneling regime with phase diffusion. Through the gate voltage we modify the damping of the array and compare it to artificial arrays of junctions.

• L. - M. Kern, R. Galceran, V. Zatko, M. Galbiati, F. Godel, D. Perconte, F. Bouamrane, E. Gaufrès, A. Loiseau, P. Brus, O. Bezencenet, M. - B. Martin, B. Servet, F. Petroff, B. Dlubak, P. Seneor, Atomic layer deposition of a MgO barrier for a passivated black phosphorus spintronics platform. Applied Physics Letters. 114, 053107 (2019).
Résumé : We demonstrate a stabilized black phosphorus (BP) 2D platform thanks to an ultrathin MgO barrier, as required for spintronic device integration. The in-situ MgO layer deposition is achieved by using a large-scale atomic layer deposition process with high nucleation density. Raman spectroscopy studies show that this layer protects the BP from degradation in ambient conditions, unlocking in particular the possibility to carry out usual lithographic fabrication steps. The resulting MgO/BP stack is then integrated in a device and probed electrically, confirming the tunnel properties of the ultrathin MgO contacts. We believe that this demonstration of a BP material platform passivated with a functional MgO tunnel barrier provides a promising perspective for BP spin transport devices.

• P. Laczkowski, M. Cosset-Cheneau, W. Savero-Torres, V. T. Pham, G. Zahnd, H. Jaffrès, N. Reyren, J. - C. Rojas-Sànchez, A. Marty, L. Vila, J. - M. George, J. - P. Attané, Spin-dependent transport characterization in metallic lateral spin valves using one-dimensional and three-dimensional modeling. Physical Review B. 99, 134436 (2019).
Résumé : We present the analysis of the spin signals obtained in NiFe based metallic lateral spin valves. We exploit the spin-dependent diffusive equations in both the conventional one-dimensional (1D) analytic modeling and in 3D finite element method simulations. These approaches are used for extracting the spin diffusion length and the effective spin polarization in Py/Al, Py/Cu, and Py/Au based lateral nanostructures at both 300 K and 77 K. Both the analytic modeling and 3D finite element method simulations give consistent results. The combination of these models provides a powerful tool for reliable spin transport characterization in all metallic spin valves and gives an insight into the spin/charge current and spin accumulations 3D distributions in these devices. We provide the necessary ingredients to develop the 3D finite element modeling of diffusive spin transport.

• J. H. Lee, F. Trier, T. Cornelissen, D. Preziosi, K. Bouzehouane, S. Fusil, S. Valencia, M. Bibes, Imaging and Harnessing Percolation at the Metal–Insulator Transition of NdNiO3 Nanogaps. Nano Letters. 11, 7801 (2019).
Résumé : Competition between coexisting electronic phases in first-order phase transitions can lead to a sharp change in the resistivity as the material is subjected to small variations in the driving parameter, for example, the temperature. One example of this phenomenon is the metal–insulator transition (MIT) in perovskite rare-earth nickelates. In such systems, reducing the transport measurement area to dimensions comparable to the domain size of insulating and metallic phases around the MIT should strongly influence the shape of the resistance–temperature curve. Here we measure the temperature dependence of the local resistance and the nanoscale domain distribution of NdNiO3 areas between Au contacts gapped by 40–260 nm. We find that a sharp resistance drop appears below the bulk MIT temperature at ∼105 K, with an amplitude inversely scaling with the nanogap width. By using X-ray photoemission electron microscopy, we directly correlate the resistance drop to the emergence and distribution of individual metallic domains at the nanogap. Our observation provides useful insight into percolation at the MIT of rare-earth nickelates.

• Y. Li, V. V. Naletov, O. Klein, J. L. Prieto, M. Muñoz, V. Cros, P. Bortolotti, A. Anane, C. Serpico, G. de Loubens, Nutation Spectroscopy of a Nanomagnet Driven into Deeply Nonlinear Ferromagnetic Resonance. Physical Review X. 9, 041036 (2019).
Résumé : Magnetization precession in ferromagnetic media remains coherent at remarkably high precession angles, an important insight for reliable control of nanomagnetic devices.

• J. Létang, S. Petit-Watelot, M. - Y. Yoo, T. Devolder, K. Bouzehouane, V. Cros, J. - V. Kim, Modulation and phase-locking in nanocontact vortex oscillators. Physical Review B. 100, 144414 (2019).
Résumé : We have conducted experiments to probe how the dynamics of nanocontact vortex oscillators can be modulated by an external signal. We explore the phase-locking properties in both the commensurate and chaotic regimes, where chaos appears to impede phase-locking while a more standard behavior is seen in the commensurate phase. These different regimes correspond to how the periodicity of the vortex core reversal relates to the frequency of core gyration around the nanocontact; a commensurate phase appears when the reversal rate is an integer fraction of the gyration frequency, while a chaotic state appears when this ratio is irrational. External modulation where the power spectral density exhibits rich features appears due to the modulation between the external source frequency, gyration frequency, and core-reversal frequency. We explain these features with first- or second-order modulation between the three frequencies. Phase-locking is also visible between the external source frequency and internal vortex modes (gyration and core-reversal modes).

• D. Marković, N. Leroux, M. Riou, F. Abreu Araujo, J. Torrejon, D. Querlioz, A. Fukushima, S. Yuasa, J. Trastoy, P. Bortolotti, J. Grollier, Reservoir computing with the frequency, phase, and amplitude of spin-torque nano-oscillators. Applied Physics Letters. 114, 012409 (2019).
Résumé : Spin-torque nano-oscillators can emulate neurons at the nanoscale. Recent works show that the non-linearity of their oscillation amplitude can be leveraged to achieve waveform classification for an input signal encoded in the amplitude of the input voltage. Here, we show that the frequency and phase of the oscillator can also be used to recognize waveforms. For this purpose, we phase-lock the oscillator to the input waveform, which carries information in its modulated frequency. In this way, we considerably decrease the amplitude, phase, and frequency noise. We show that this method allows classifying sine and square waveforms with an accuracy above 99% when decoding the output from the oscillator amplitude, phase, or frequency. We find that recognition rates are directly related to the noise and non-linearity of each variable. These results prove that spin-torque nano-oscillators offer an interesting platform to implement different computing schemes leveraging their rich dynamical features.

• R. Matsumoto, S. Lequeux, H. Imamura, J. Grollier, Chaos and Relaxation Oscillations in Spin-Torque Windmill Spiking Oscillators. Physical Review Applied. 11, 044093 (2019).
Résumé : Spintronic neurons that emit sharp voltage spikes are needed for neural-network hardware, for fast data processing with low power consumption. Conventional spintronic neurons, $i.e.$ spin-torque nano-oscillators (STNOs), emit sinusoidal waveforms, though. The authors suggest how to imitate neuron spiking in STNOs in which both magnetic layers are free and thin enough to be switched by current. Simulations show that the chaotic behavior of these frequency-tunable devices can be adjusted by the magnetic stack and current. The proposed spintronic relaxation oscillator is a promising building block for hardware neuromorphic chips, leveraging nonlinear dynamics for computing.

• H. Meng, Y. Ren, J. - E. Villegas, A. I. Buzdin, Josephson current through a ferromagnetic bilayer: Beyond the quasiclassical approximation. Physical Review B. 100, 224514 (2019).
Résumé : Based on the Bogoliubov--de Gennes equations, we provide an exact numerical solution for the critical current of Josephson junctions with a composite ferromagnetic bilayer. We demonstrate that for the antiparallel orientation of the magnetic moments of the bilayer, the presence of a potential barrier at the bilayer interface results in large oscillations of the critical current as a function of ferromagnet thickness and/or exchange field. Because of this, and remarkably, in the range of small exchange field and thicknesses, the magnetism leads to the increase of the critical current. This effect is well pronounced at low temperature but disappears near ${T}_{c}$. If the potential barrier is replaced by a spin-active barrier at the bilayer interface the conventional 0-$\ensuremath{\pi}$ transition, similar to the case of an uniform ferromagnetic Josephson junction, is observed. Strikingly, for a parallel orientation of the magnetic moments of the bilayer, the presence of the spin-active barrier restores the anomalous behavior---potential barrier in the antiparallel case. These behaviors result from the resonant tunneling of Cooper pairs across the composite barrier---an effect related to the spin-dependent Fermi vector in the presence of the ferromagnets' exchange field.

• M. Riou, J. Torrejon, B. Garitaine, F. A. Araujo, P. Bortolotti, V. Cros, S. Tsunegi, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, D. Querlioz, M. D. Stiles, J. Grollier, Temporal Pattern Recognition with Delayed-Feedback Spin-Torque Nano-Oscillators. Physical Review Applied. 12, 024049 (2019).
Résumé : The recent demonstration of neuromorphic computing with spin-torque nano-oscillators points to substantial energy-saving in data analysis. However, the limited intrinsic memory of these devices (much less than a microsecond) limits their utility for analyzing temporal sequences. Here the authors overcome the short memory of a spin-torque oscillator by using a feedback loop with a delay of 1 $\ensuremath{\mu}$s, reducing the error rate during temporal-pattern classification by 99%. In addition, they determine optimal operating point of the oscillator, in terms of the current and magnetic field, to take advantage of this memory for recognition tasks.

• J. - C. Rojas-Sánchez, A. Fert, Compared Efficiencies of Conversions between Charge and Spin Current by Spin-Orbit Interactions in Two- and Three-Dimensional Systems. Physical Review Applied. 11, 054049 (2019).
Résumé : Two-dimensional (2D) materials such as topological insulators (TIs) are promising components for the design of spintronic devices, and this work offers a way to quantify their advantage. This article presents a study of the conversions between charge and spin currents based on the regular and inverse spin Hall effects in heavy metals, and based on the regular and inverse Edelstein effects in a 2D electron gas. The authors propose an approach for comparing the efficiency of conversion in 3D and 2D systems; this comparison is not straightforward, as those different physical effects are responsible at different dimensionalities. Spoiler: TIs turn out to be far more effective converters.

• V. Rubio-Giménez, C. Bartual-Murgui, M. Galbiati, A. Núñez-López, J. Castells-Gil, B. Quinard, P. Seneor, E. Otero, P. Ohresser, A. Cantarero, E. Coronado, J. Antonio Real, R. Mattana, S. Tatay, C. Martí-Gastaldo, Effect of nanostructuration on the spin crossover transition in crystalline ultrathin films. Chemical Science. 10, 4038-4047 (2019).

• D. Sando, F. Appert, S. R. Burns, Q. Zhang, Y. Gallais, A. Sacuto, M. Cazayous, V. Garcia, S. Fusil, C. Carrétéro, J. M. L. Breton, A. Barthélémy, M. Bibes, J. Juraszek, V. Nagarajan, Influence of flexoelectricity on the spin cycloid in (110)-oriented BiFeO3 films. Physical Review Materials. 3, 104404 (2019).
Résumé : The influence of film orientation, strain relaxation, and flexoelectric fields on the stability of the spin cycloid in (110)-oriented $\mathrm{BiFe}{\mathrm{O}}_{3}$ epitaxial films grown on $\mathrm{LaAl}{\mathrm{O}}_{3}$ substrates is investigated. By means of advanced x-ray-diffraction techniques, we show that thinner films have very large strain gradients which give rise to high flexoelectric fields. Using low-energy Raman spectroscopy and conversion electron M\"ossbauer spectroscopy (CEMS) we show that films up to 53 nm thick possess collinear antiferromagnetic order, with no cycloidal modulation. This suppression of the cycloid is proposed to be from strain and strain-gradient-induced flexoelectric fields. On the other hand, films thicker than 90 nm show a complex spin texture consistent with two separate cycloids, likely with different propagation directions. Interestingly, CEMS analysis suggests that the two cycloids have the same spin rotation plane. The multiple cycloids are suggested to arise from different ferroelastic domains (in turn influenced by twinning in the substrate) with different strain relaxation behaviors. These results offer insight into the factors that influence cycloid stability in the less common (110) film orientation and have implications for future magnonic devices.

• D. Sando, F. Appert, B. Xu, O. Paull, S. R. Burns, C. Carrétéro, B. Dupé, V. Garcia, Y. Gallais, A. Sacuto, M. Cazayous, B. Dkhil, J. - M. Le Breton, A. Barthélémy, M. Bibes, L. Bellaiche, V. Nagarajan, J. Juraszek, A magnetic phase diagram for nanoscale epitaxial BiFeO3 films. Applied Physics Reviews. 6, 041404 (2019).
Résumé : BiFeO3 thin films have attracted considerable attention by virtue of their potential application in low-energy spintronic and magnonic devices. BiFeO3 possesses an intricate magnetic structure, characterized by a spin cycloid with period ∼62 nm that governs the functional magnonic response, and which can be modulated or even destroyed by strain, magnetic and electric fields, or chemical doping. The literature on (110)-oriented BiFeO3 films is not explicit in defining the conditions under which this cycloid persists, as its presence depends on synthesis method and thin-film boundary conditions, especially in the sub-100 nm thickness regime. This report aims to end “trial and error” approaches in determining the conditions under which this cycloid and its associated functional magnonic response exist. We show that in specific crystallographic orientations of epitaxial BiFeO3, an unexplored strain parameter—the distortion in the ab plane of the monoclinic unit cell—significantly influences the spin structure. Combining Mössbauer spectroscopy and low-energy Raman spectroscopy with first-principles-based effective Hamiltonian calculations, we show that both average strain and this distortion destroy the cycloid. For films grown on (110)-oriented SrTiO3 substrates, if the BiFeO3 lattice parameters a and b differ by more than about 1.2%, the cycloid is destabilized, resulting in a pseudocollinear magnetic order ground state. We are thereby able to construct a phase diagram of the spin structure for nanoscale epitaxial BiFeO3 films, which aims to resolve long-standing literature inconsistencies and provide powerful guidelines for the design of future magnonic and spintronic devices.BiFeO3 thin films have attracted considerable attention by virtue of their potential application in low-energy spintronic and magnonic devices. BiFeO3 possesses an intricate magnetic structure, characterized by a spin cycloid with period ∼62 nm that governs the functional magnonic response, and which can be modulated or even destroyed by strain, magnetic and electric fields, or chemical doping. The literature on (110)-oriented BiFeO3 films is not explicit in defining the conditions under which this cycloid persists, as its presence depends on synthesis method and thin-film boundary conditions, especially in the sub-100 nm thickness regime. This report aims to end “trial and error” approaches in determining the conditions under which this cycloid and its associated functional magnonic response exist. We show that in specific crystallographic orientations of epitaxial BiFeO3, an unexplored strain parameter—the distortion in the ab plane of the monoclinic unit cell—significantly influences the spin structure...

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--- Exporter la sélection au format


text:
# linear relation W=mA+c

#### thatsnotalion

Hi i need help with this question.

i need to show that the data fit closely with the linear relation W=mA+c, and determine the constants m and c?

the data in the table is as follows:

A: 20, 22, 30, 34, 42, 43, 46, 54, 56, 68, 70
W:0.08, 0.10, 0.15, 0.20, 0.27, 0.26, 0.31, 0.36, 0.40, 0.49, 0.49

#### pickslides

MHF Helper
Do you have a copy of MS excel, a graphical calculator or any other software that can do this for you?

M guess is... $$\displaystyle W = 0.0085A - 0.0905$$

Last edited:

#### thatsnotalion

how id you guess that? and no, i don't have any programs to help.

do i use simultaneous eqns?
for example
0.08=20m+c
0.10=22m+c

#### pickslides

MHF Helper
do i use simultaneous eqns?
for example
0.08=20m+c
0.10=22m+c
You can do that. But don't pick to points so close to each other. You really should plot the points first then draw a line through the points for a best fit (make sure there is an even amount of points above and below the your line. Then pick points that are close to and at either end of the line.

I have made an edit above. (Happy)(Rofl)

#### thatsnotalion

hmm thanks! i'm still a little unsure though =/

#### pickslides

MHF Helper
Try solving.

$$\displaystyle 0.08=20m+c$$
$$\displaystyle 0.49=70m+c$$

Do you know how?

#### thatsnotalion

i used elimintation and got c=-0.084, m=0.0082?

#### pickslides

MHF Helper
i used elimintation and got c=-0.084, m=0.0082?
So did I. Well done!

#### thatsnotalion

awesomesauce. thanks for that!

format: plain


text:
# How to compute gradient of partial log-likelihood function in Cox proportional hazards model?

The partial log-likelihood function in Cox proportional hazards is given with such formula $${}_{p}\ell(\beta) = \sum\limits_{i=1}^{K}X_i'\beta - \sum\limits_{i=1}^{K}\log\Big(\sum\limits_{l\in \mathscr{R}(t_i)}^{}e^{X_l'\beta}\Big),$$ where $K$ is the number of observations for which we have observed an event (where generaly there were $n$ observations, so $K-n$ observations were censored) and $\mathscr{R}(t_i)$ is a risk set for time $t_i$ defined as: $\mathscr{R}(t_i) = : \{X_j: t_j >= t_i, j = 1, \dots, n \}$ (as described here How to compute partial log-likelihood function in Cox proportional hazards model?).

Then I am wondering on how to implement and compute gradient of partial log-likelihood function in Cox proportional hazards model, which is given with formula ($k$-th coordinate for $p$-dimenational vector of partial derivatives, for $k=1,\dots,p$) $$U_k(\beta)=\dfrac{\partial{}_{p}\ell_k(\beta)}{\partial\beta_k}=\sum\limits_{i=1}^{K}\Big(X_{ik}-A_{ik}\Big),$$ where $$A_{ik} = \dfrac{\sum\limits_{l\in \mathscr{R}(t_i)}^{} X_{lk} e^{X_l'\beta}}{\sum\limits_{l\in \mathscr{R}(t_i)}^{} e^{X_l'\beta}}$$ is a mean of $X_{.k}$ ($k$-th variables) over finite population $\mathscr{R}(t_i)$, with the use of \textit{exponentially weightened} form of sampling.

I've tried myself to implement gradient with this code:

library(foreach)
library(dplyr)
coxphGD_step <- function(formula, data, learningRate, beta){
# collect times, status, variables and reorder samples
# to make the algorithm more clear to read and track
preparedData <- prepareData(formula = formula, data = data)
# calculate the log-likelihood for this batch sample
partial_sum <- list()
foreach(k = 1:nrow(preparedData)) %do% {
# risk set for current time/observation
risk_set <- preparedData %>% filter(times >= preparedData$times[k]) nominator <- apply(risk_set[, -c(1,2)], MARGIN = 1, function(element){ element * exp(element * beta) }) %>% rowSums() denominator <- apply(risk_set[, -c(1,2)], MARGIN = 1, function(element){ exp(element * beta) }) %>% rowSums() partial_sum[[k]] <- preparedData[k, "event"] * (preparedData[k, -c(1,2)] - nominator/denominator) } do.call(rbind, partial_sum) %>% colSums() -> U_batch return(beta + learningRate * U_batch) }  which transforms data to the input format as in survival::coxph function (and also sorts observations at the end to facilitate the implantation) with below function prepareData <- function(formula, data) { # Parameter identification as in survival::coxph(). Call <- match.call() indx <- match(c("formula", "data"), names(Call), nomatch = 0) if (indx[1] == 0) stop("A formula argument is required") temp <- Call[c(1, indx)] temp[[1]] <- as.name("model.frame") mf <- eval(temp, parent.frame()) Y <- model.extract(mf, "response") if (!inherits(Y, "Surv")) stop("Response must be a survival object") type <- attr(Y, "type") if (type != "right" && type != "counting") stop(paste("Cox model doesn't support \"", type, "\" survival data", sep = "")) # collect times, status, variables and reorder samples # to make the algorithm more clear to read and track cbind(event = unclass(Y)[,2], # 1 indicates event, 0 indicates cens times = unclass(Y)[,1], mf[, -1]) %>% arrange(times) }  which both functions are used in the above function that implements the gradient descent (order I) algorithm to estimate the coefficients in the cox proportional hazards model (this is different method from ?coxph.fit which uses cholesky decomposition of second derivatives to perform Newton-Raphson gradient descent order II algorithm) library(assertthat) library(survival) coxphGD <- function(formula, data, learningRates = function(x){1/x}, beta_0 = 0, epsilon = 1e-5, max.iter = 500 ) { checkArguments(formula, data, learningRates, beta_0, epsilon) -> beta_start # check arguments n <- length(data) diff <- epsilon + 1 i <- 1 beta_new <- list() # steps are saved in a list so that they can beta_old <- beta_start # be tracked in the future # estimate while(i <= max.iter & diff > epsilon) { beta_new[[i]] <- coxphGD_step(formula = formula, beta = beta_old, learningRate = learningRates(i), data = data) %>% unlist # unlist as this might be the result of foreach diff <- sqrt(sum((beta_new[[i]] - beta_old)^2)) beta_old <- beta_new[[i]] i <- i + 1 ; cat("\r iteration: ", i, "\r") } # return results list(Call = match.call(), epsilon = epsilon, learningRates = learningRates, steps = i, coefficients = c(list(beta_start), beta_new)) } checkArguments <- function(formula, data, learningRates, beta_0, epsilon) { # + check names and types for every variables assert_that(is.function(learningRates)) assert_that(is.numeric(epsilon)) assert_that(is.numeric(beta_0)) # check length of the start parameter if (length(beta_0) == 1) { beta_0 <- rep(beta_0, as.character(formula)[3] %>% strsplit("\\+") %>% unlist %>% length) } return(beta_0) }  but I think this implementation is somehow wrong because when I tried to perform gradient descent order I algorithm to estimate coefficients in the Cox proportional model for artificial data (generation based on How to create a toy survival (time to event) data with right censoring) I received results that are far from real coefficients beta = c(2,2) from which I have generated data set.seed(456) dataCox <- function(N, lambda, rho, x, beta, censRate){ # real Weibull times u <- runif(N) Treal <- (- log(u) / (lambda * exp(x %*% beta)))^(1 / rho) # censoring times Censoring <- rexp(N, censRate) # follow-up times and event indicators time <- pmin(Treal, Censoring) status <- as.numeric(Treal <= Censoring) # data set data.frame(id=1:N, time=time, status=status, x=x) } x <- matrix(sample(0:1, size = 2000, replace = TRUE), ncol = 2) dataCox(10^3, lambda = 5, rho = 1.5, x, beta = c(2,2), censRate = 0.2) -> dCox library(assertthat) library(magrittr) library(survival) library(dplyr) library(foreach) coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/x}, beta_0 = 0, epsilon = 1e-5, max.iter = 50 ) -> results coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/(1000*x)}, beta_0 = 0, epsilon = 1e-5, max.iter = 50 ) -> results2 coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/(1000*sqrt(x))}, beta_0 = 0, epsilon = 1e-5, max.iter = 50 ) -> results3 coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/(10000*x)}, beta_0 = 0, epsilon = 1e-5, max.iter = 50 ) -> results4 coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/(100*x)}, beta_0 = 0, epsilon = 1e-5, max.iter = 50 ) -> results5 coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/(100*sqrt(x))}, beta_0 = 0, epsilon = 1e-5, max.iter = 50 ) -> results6 coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/(100*sqrt(x))}, beta_0 = 0, epsilon = 1e-4, max.iter = 50 ) -> results7 coxphGD(formula = Surv(time, status)~x.1+x.2, data = dCox, learningRates = function(x){1/(50*sqrt(x))}, beta_0 = 0, epsilon = 1e-5, max.iter = 150 ) -> results8 library(ggplot2) png('results1.png') as.data.frame(t(simplify2array(results$coefficients))) %>%
ggplot(aes(V1, V2)) + geom_path() +
coord_cartesian(xlim=c(0,3), ylim=c(0,3))
dev.off()

png('results2.png')
as.data.frame(t(simplify2array(results2$coefficients))) %>% ggplot(aes(V1, V2)) + geom_path() + coord_cartesian(xlim=c(0,3), ylim=c(0,3)) dev.off() png('results3.png') as.data.frame(t(simplify2array(results3$coefficients))) %>%
ggplot(aes(V1, V2)) + geom_path() +
coord_cartesian(xlim=c(0,3), ylim=c(0,3))
dev.off()

png('results4.png')
as.data.frame(t(simplify2array(results4$coefficients))) %>% ggplot(aes(V1, V2)) + geom_path() + coord_cartesian(xlim=c(0,3), ylim=c(0,3)) dev.off() png('results5.png') as.data.frame(t(simplify2array(results5$coefficients))) %>%
ggplot(aes(V1, V2)) + geom_path() +
coord_cartesian(xlim=c(0,3), ylim=c(0,3))
dev.off()

png('results6.png')
as.data.frame(t(simplify2array(results6$coefficients))) %>% ggplot(aes(V1, V2)) + geom_path() + coord_cartesian(xlim=c(0,3), ylim=c(0,3)) dev.off() png('results7.png') as.data.frame(t(simplify2array(results7$coefficients))) %>%
ggplot(aes(V1, V2)) + geom_path() +
coord_cartesian(xlim=c(0,3), ylim=c(0,3))
dev.off()

png('results8.png')
as.data.frame(t(simplify2array(results8$coefficients))) %>% ggplot(aes(V1, V2)) + geom_path() + coord_cartesian(xlim=c(0,3), ylim=c(0,3)) dev.off()  I mean estimation process stops for$\beta\$'s that are not even close to beta=c(2,2). My question is where is a mistake in the philosophy or implementation or is there a simpler way to compute gradient for partial log-likelihood in Cox proportional hazards model?

Try this:

batchData <- preparedData
batchData <- batchData %>% arrange(-times)

scores <- apply(batchData[, -c(1, 2)], MARGIN = 1,
function(element) {
exp(element %*% beta)
})

nominator <- apply(batchData[, -c(1, 2)], 2, function(x) cumsum(scores*x))
denominator <- cumsum(scores)
partial_sum <- (batchData[, -c(1, 2)] - nominator/denominator)*batchData[, "event"]

U_batch <- partial_sum %>% colSums()


text:
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CHAPTER XIX.
ON FINDING AREAS BY INTEGRATING.

One use of the integral calculus is to enable us to ascertain the values of areas bounded by curves.

Let us try to get at the subject bit by bit.

Fig. 52.

Let ${\displaystyle AB}$ (Fig. 52) be a curve, the equation to which is known. That is, ${\displaystyle y}$ in this curve is some known function of ${\displaystyle x}$. Think of a piece of the curve from the point ${\displaystyle P}$ to the point ${\displaystyle Q}$.

Let a perpendicular ${\displaystyle PM}$ be dropped from ${\displaystyle P}$, and another ${\displaystyle QN}$ from the point ${\displaystyle Q}$. Then call ${\displaystyle OM=x_{1}}$ and ${\displaystyle ON=x_{2}}$, and the ordinates ${\displaystyle PM=y_{1}}$ and ${\displaystyle QN=y_{2}}$. We have thus marked out the area ${\displaystyle PQNM}$ that lies beneath the piece ${\displaystyle PQ}$. The problem is, how can we calculate the value of this area?

The secret of solving this problem is to conceive the area as being divided up into a lot of narrow strips, each of them being of the width ${\displaystyle dx}$. The smaller we take ${\displaystyle dx}$, the more of them there will be between ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$. Now, the whole area is clearly equal to the sum of the areas of all such strips. Our business will then be to discover an expression for the area of any one narrow strip, and to integrate it so as to add together all the strips. Now think of any one of the strips. It will be like this: being bounded between two vertical sides, with a flat bottom ${\displaystyle dx}$, and with a slightly curved sloping top. Suppose we take its average height as being ${\displaystyle y}$; then, as its width is ${\displaystyle dx}$, its area will be ${\displaystyle y\,dx}$. And seeing that we may take the width as narrow as we please, if we only take it narrow enough its average height will be the same as the height at the middle of it. Now let us call the unknown value of the whole area ${\displaystyle S}$, meaning surface. The area of one strip will be simply a bit of the whole area, and may therefore be called ${\displaystyle dS}$. So we may write

${\displaystyle {\text{area of 1 strip}}=dS=y\cdot dx}$.

If then we add up all the strips, we get

${\displaystyle {\text{total area S}}=\int dS=\int y\,dx}$.

So then our finding ${\displaystyle S}$ depends on whether we can integrate ${\displaystyle y\cdot dx}$ for the particular case, when we know what the value of ${\displaystyle y}$ is as a function of ${\displaystyle x}$.

For instance, if you were told that for the particular curve in question ${\displaystyle y=b+ax^{2}}$, no doubt you could put that value into the expression and say: then I must find ${\displaystyle \int (b+ax^{2})\,dx}$.

That is all very well; but a little thought will show you that something more must be done. Because the area we are trying to find is not the area under the whole length of the curve, but only the area limited on the left by ${\displaystyle PM}$, and on the right by ${\displaystyle QN}$, it follows that we must do something to define our area between those ‘limits’.

This introduces us to a new notion, namely that of integrating between limits. We suppose ${\displaystyle x}$ to vary, and for the present purpose we do not require any value of ${\displaystyle x}$ below ${\displaystyle x_{1}}$ (that is ${\displaystyle OM}$), nor any value of ${\displaystyle x}$ above ${\displaystyle x_{2}}$ (that is ${\displaystyle ON}$). When an integral is to be thus defined between two limits, we call the lower of the two values the inferior limit, and the upper value the superior limit. Any integral so limited we designate as a definite integral, by way of distinguishing it from a general integral to which no limits are assigned.

In the symbols which give instructions to integrate, the limits are marked by putting them at the top and bottom respectively of the sign of integration. Thus the instruction

${\displaystyle \int _{x=x_{1}}^{x=x_{2}}y\cdot dx}$

will be read: find the integral of ${\displaystyle y\cdot dx}$ between the inferior limit ${\displaystyle x_{1}}$ and the superior limit ${\displaystyle x_{2}}$.

Sometimes the thing is written more simply

${\displaystyle \int _{x_{1}}^{x_{2}}y\cdot dx}$.

Well, but how do you find an integral between limits, when you have got these instructions?

Look again at Fig. 52 (p. 206). Suppose we could find the area under the larger piece of curve from ${\displaystyle A}$ to ${\displaystyle Q}$, that is from ${\displaystyle x=0}$ to ${\displaystyle x=x_{2}}$, naming the area ${\displaystyle AQNO}$. Then, suppose we could find the area under the smaller piece from ${\displaystyle A}$ to ${\displaystyle P}$, that is from ${\displaystyle x=0}$ to ${\displaystyle x=x_{1}}$, namely the area ${\displaystyle APMO}$. If then we were to subtract the smaller area from the larger, we should have left as a remainder the area ${\displaystyle PQNM}$, which is what we want. Here we have the clue as to what to do; the definite integral between the two limits is the difference between the integral worked out for the superior limit and the integral worked out for the lower limit.

Let us then go ahead. First, find the general integral thus:

${\displaystyle \int y\,dx,}$.

and, as ${\displaystyle y=b+ax^{2}}$ is the equation to the curve (Fig. 52),

${\displaystyle \int (b+ax^{2})\,dx}$

is the general integral which we must find.

Doing the integration in question by the rule (p. 196), we get

${\displaystyle bx+{\frac {a}{3}}x^{3}+C}$;

and this will be the whole area from ${\displaystyle 0}$ up to any value of ${\displaystyle x}$ that we may assign.

Therefore, the larger area up to the superior limit ${\displaystyle x_{2}}$ will be

${\displaystyle bx_{2}+{\frac {a}{3}}x_{2}^{3}+C}$;

and the smaller area up to the inferior limit ${\displaystyle x_{1}}$ will be

${\displaystyle bx_{1}+{\frac {a}{3}}x_{1}^{3}+C}$.

Now, subtract the smaller from the larger, and we get for the area ${\displaystyle S}$ the value,

${\displaystyle {\text{area S}}=b(x_{2}-x_{1})+{\frac {a}{3}}(x_{2}^{3}-x_{1}^{3})}$.

This is the answer we wanted. Let us give some numerical values. Suppose ${\displaystyle b=10}$, ${\displaystyle a=0.06}$, and ${\displaystyle x_{2}=6}$ and ${\displaystyle x_{1}=6}$. Then the area ${\displaystyle S}$ is equal to

${\displaystyle 10(8-6)+{\frac {0.06}{3}}(8^{3}-6^{3})}$

{\displaystyle {\begin{aligned}&=20+0.02(512-216)\\&=20+0.02\times 296\\&=20+5.92\\&=25.92.\end{aligned}}}

Let us here put down a symbolic way of stating what we have ascertained about limits:

${\displaystyle \int _{x=x_{1}}^{x=x_{2}}y\,dx=y_{2}-y_{1}}$,

where ${\displaystyle y_{2}}$ is the integrated value of ${\displaystyle y\,dx}$ corresponding to ${\displaystyle x_{2}}$, and ${\displaystyle y_{1}}$ that corresponding to ${\displaystyle x_{1}}$.

All integration between limits requires the difference between two values to be thus found. Also note that, in making the subtraction the added constant ${\displaystyle C}$ has disappeared.

Examples

(1) To familiarize ourselves with the process, let us take a case of which we know the answer beforehand. Let us find the area of the triangle (Fig. 53), which

Fig. 53.

has base ${\displaystyle x=12}$ and height ${\displaystyle y=4}$. We know beforehand, from obvious mensuration, that the answer will come ${\displaystyle 24}$.

Now, here we have as the “curve” a sloping line for which the equation is

${\displaystyle y={\frac {x}{3}}}$.

The area in question will be

${\displaystyle \int _{x=0}^{x=12}y\cdot dx=\int _{x=0}^{x=12}{\frac {x}{3}}\cdot dx}$.

Integrating ${\displaystyle {\dfrac {x}{3}}\,dx}$ (p. 194), and putting down the value of the general integral in square brackets with the limits marked above and below, we get

{\displaystyle {\begin{aligned}{\text{area}}\;&=\left[{\frac {1}{3}}\cdot {\frac {1}{2}}x^{2}\right]_{x=0}^{x=12}+C\\&=\left[{\frac {x^{2}}{6}}\right]_{x=0}^{x=12}+C\\&=\left[{\frac {12^{2}}{6}}\right]-\left[{\frac {0^{2}}{6}}\right]\\&={\frac {144}{6}}=24.\quad Ans.\end{aligned}}}

Let us satisfy ourselves about this rather surprising dodge of calculation, by testing it on a simple example. Get some squared paper, preferably some

Fig. 54.

that is ruled in little squares of one-eighth inch or one-tenth inch each way. On this squared paper plot out the graph of the equation,

${\displaystyle y={\frac {x}{3}}}$.

The values to be plotted will be:

 ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle 0}$ ${\displaystyle 3}$ ${\displaystyle 6}$ ${\displaystyle 9}$ ${\displaystyle 12}$ ${\displaystyle 0}$ ${\displaystyle 1}$ ${\displaystyle 2}$ ${\displaystyle 3}$ ${\displaystyle 4}$

The plot is given in Fig. 54.

Now reckon out the area beneath the curve by counting the little squares below the line, from ${\displaystyle x=0}$ as far as ${\displaystyle x=12}$ on the right. There are ${\displaystyle 18}$ whole squares and four triangles, each of which has an area equal to ${\displaystyle 1{\tfrac {1}{2}}}$ squares; or, in total, ${\displaystyle 24}$ squares. Hence ${\displaystyle 24}$ is the numerical value of the integral of ${\displaystyle {\dfrac {x}{3}}\,dx}$ between the lower limit of ${\displaystyle x=0}$ and the higher limit of ${\displaystyle x=12}$.

As a further exercise, show that the value of the same integral between the limits of ${\displaystyle x=3}$ and ${\displaystyle x=15}$ is ${\displaystyle 36}$.

(2) Find the area, between limits ${\displaystyle x=x_{1}}$ and ${\displaystyle x=0}$, of the curve ${\displaystyle y={\dfrac {b}{x+a}}}$.

Fig. 55.

${\displaystyle {\text{Area}}=\int _{x=0}^{x=x_{1}}y\cdot dx=\int _{x=0}^{x=x_{1}}{\frac {b}{x+a}}\,dx}$

{\displaystyle {\begin{aligned}&=b{\bigl [}\log _{\epsilon }(x+a){\bigr ]}_{0}^{x_{1}}+C\\&=b{\bigl [}\log _{\epsilon }(x_{1}+a)-\log _{\epsilon }(0+a){\bigr ]}\\&=b\log _{\epsilon }{\frac {x_{1}+a}{a}}.\quad Ans.\end{aligned}}}

N.B–Notice that in dealing with definite integrals the constant ${\displaystyle C}$ always disappears by subtraction.

Let it be noted that this process of subtracting one part from a larger to find the difference is really a common practice. How do you find the area of a

Fig. 56.

plane ring (Fig. 56), the outer radius of which is ${\displaystyle r_{2}}$ and the inner radius is ${\displaystyle r_{1}}$? You know from mensuration that the area of the outer circle is ${\displaystyle \pi r_{2}^{2}}$; then you find the area of the inner circle, ${\displaystyle \pi r_{1}^{2}}$; then you subtract the latter from the former, and find area of ring ${\displaystyle =\pi (r_{2}^{2}-r_{1}^{2})}$; which may be written

${\displaystyle \pi (r_{2}+r_{1})(r_{2}-r_{1})}$

= mean circumference of ring × width of ring.

(3) Here’s another case–that of the die-away curve (p. 156). Find the area between ${\displaystyle x=0}$ and ${\displaystyle x=a}$, of the curve (Fig. 57) whose equation is

{\displaystyle {\begin{aligned}y&=b\epsilon ^{-x}.\\{\text{Area}}&=b\int _{x=0}^{x=a}\epsilon ^{-x}\times dx.\\\end{aligned}}}

The integration (p. 201) gives

{\displaystyle {\begin{aligned}&=b\left[-\epsilon ^{-x}\right]_{0}^{a}\\&=b{\bigl [}-\epsilon ^{-a}-(-\epsilon ^{-0}){\bigr ]}\\&=b(1-\epsilon ^{-a}).\end{aligned}}}

 Fig. 57. Fig. 58.

(4) Another example is afforded by the adiabatic curve of a perfect gas, the equation to which is ${\displaystyle pv^{n}=c}$, where ${\displaystyle p}$ stands for pressure, ${\displaystyle v}$ for volume, and ${\displaystyle n}$ is of the value ${\displaystyle 1.42}$ (Fig. 58).

Find the area under the curve (which is proportional to the work done in suddenly compressing the gas) from volume ${\displaystyle v_{2}}$ to volume ${\displaystyle v_{1}}$.

Here we have

{\displaystyle {\begin{aligned}{\text{area}}&=\int _{v=v_{1}}^{v=v_{2}}cv^{-n}\cdot dv\\&=c\left[{\frac {1}{1-n}}v^{1-n}\right]_{v_{1}}^{v_{2}}\\&=c{\frac {1}{1-n}}(v_{2}^{1-n}-v_{1}^{1-n})\\&={\frac {-c}{0.42}}\left({\frac {1}{v_{2}^{0.42}}}-{\frac {1}{v_{1}^{0.42}}}\right).\end{aligned}}}

An Exercise.

Prove the ordinary mensuration formula, that the area ${\displaystyle A}$ of a circle whose radius is ${\displaystyle R}$, is equal to ${\displaystyle \pi R^{2}}$.

Consider an elementary zone or annulus of the surface (Fig 59), of breadth ${\displaystyle dr}$, situated at a distance

Fig. 59.

${\displaystyle r}$ from the centre. We may consider the entire surface as consisting of such narrow zones, and the whole area ${\displaystyle A}$ will simply be the integral of all such elementary zones from centre to margin, that is, integrated from ${\displaystyle r=0}$ to ${\displaystyle r=R}$.

We have therefore to find an expression for the elementary area ${\displaystyle dA}$ of the narrow zone. Think of it as a strip of breadth ${\displaystyle dr}$, and of a length that is the periphery of the circle of radius ${\displaystyle r}$, that is, a length of ${\displaystyle 2\pi r}$. Then we have, as the area of the narrow zone,

${\displaystyle dA=2\pi r\,dr}$.

Hence the area of the whole circle will be:

${\displaystyle dA=2\pi r\,dr}$.

Now, the general integral of ${\displaystyle r\cdot dr}$ is ${\displaystyle {\tfrac {1}{2}}r^{2}}$. Therefore,

{\displaystyle {\begin{aligned}A&=2\pi {\bigl [}{\tfrac {1}{2}}r^{2}{\bigr ]}_{r=0}^{r=R};\\or\quad \quad \quad \quad A&=2\pi {\bigl [}{\tfrac {1}{2}}R^{2}-{\tfrac {1}{2}}(0)^{2}{\bigr ]};\\whence\quad \quad \quad \quad A&=\pi R^{2}.\end{aligned}}}.

Another Exercise.

Let us find the mean ordinate of the positive part of the curve ${\displaystyle y=x-x^{2}}$, which is shown in Fig. 60.

Fig. 60.

To find the mean ordinate, we shall have to find the area of the piece ${\displaystyle OMN}$, and then divide it by the of the base ${\displaystyle ON}$. But before we can find the area we must ascertain the length of the base, so as to know up to what limit we are to integrate. At ${\displaystyle N}$ the ordinate ${\displaystyle y}$ has zero value; therefore, we must look at the equation and see what value of ${\displaystyle x}$ will make ${\displaystyle y=0}$. Now, clearly, if ${\displaystyle x}$ is ${\displaystyle 0}$, ${\displaystyle y}$ will also be ${\displaystyle 0}$, the curve passing through the origin ${\displaystyle O}$; but also, if ${\displaystyle x=1}$, ${\displaystyle y=0}$; so that ${\displaystyle x=1}$ gives us the position of the point ${\displaystyle N}$.

Then the area wanted is

{\displaystyle {\begin{aligned}&=\int _{x=0}^{x=1}(x-x^{2})\,dx\\&=\left[{\tfrac {1}{2}}x^{2}-{\tfrac {1}{3}}x^{3}\right]_{0}^{1}\\&=\left[{\tfrac {1}{2}}-{\tfrac {1}{3}}\right]-[0-0]\\&={\tfrac {1}{6}}.\end{aligned}}}

But the base length is ${\displaystyle 1}$.

Therefore, the average ordinate of the curve ${\displaystyle ={\tfrac {1}{6}}}$.

[N.B.–It will be a pretty and simple exercise in maxima and minima to find by differentiation what is the height of the maximum ordinate. It must be greater than the average.]

The mean ordinate of any curve, over a range from ${\displaystyle x=0}$ to ${\displaystyle x=x_{1}}$, is given by the expression,

${\displaystyle {\text{mean y}}={\frac {1}{x_{1}}}\int _{x=0}^{x=x_{1}}y\cdot dx}$.

One can also find in the same way the surface area of a solid of revolution.

Example. The curve ${\displaystyle y=x^{2}-5}$ is revolving about the axis of ${\displaystyle x}$. Find the area of the surface generated by the curve between ${\displaystyle x=0}$ and ${\displaystyle x=6}$.

A point on the curve, the ordinate of which is ${\displaystyle y}$, describes a circumference of length ${\displaystyle 2\pi y}$, and a narrow belt of the surface, of width ${\displaystyle dx}$, corresponding to this point, has for area ${\displaystyle 2\pi y\,dx}$. The total area is

{\displaystyle {\begin{aligned}2\pi \int _{x=0}^{x=6}y\,dx&=2\pi \int _{x=0}^{x=6}(x^{2}-5)\,dx=2\pi \left[{\frac {x^{3}}{3}}-5x\right]_{0}^{6}\\&=6.28\times 42=263.76.\end{aligned}}}.

Areas in Polar Coordinates.

When the equation of the boundary of an area is given as a function of the distance ${\displaystyle r}$ of a point of it from a fixed point ${\displaystyle O}$ (see Fig. 61) called the pole, and

Fig. 61.

of the angle which ${\displaystyle r}$ makes with the positive horizontal direction ${\displaystyle OX}$, the process just explained can be applied just as easily, with a small modification. Instead of a strip of area, we consider a small triangle ${\displaystyle OAB}$, the angle at ${\displaystyle O}$ being ${\displaystyle d\theta }$, and we find the sum of all the little triangles making up the required area.

The area of such a small triangle is approximately ${\displaystyle {\dfrac {AB}{2}}\times r}$ or ${\displaystyle {\dfrac {r\,d\theta }{2}}\times r}$; hence the portion of the area included between the curve and two positions of r corresponding to the angles ${\displaystyle \theta _{1}}$ and ${\displaystyle \theta _{2}}$ is given by

${\displaystyle {\tfrac {1}{2}}\int _{\theta =\theta _{1}}^{\theta =\theta _{2}}r^{2}\,d\theta }$.

Examples.

(1) Find the area of the sector of ${\displaystyle 1}$ radian in a circumference of radius ${\displaystyle a}$ inches.

The polar equation of the circumference is evidently ${\displaystyle r=a}$. The area is

${\displaystyle {\tfrac {1}{2}}\int _{\theta =\theta _{1}}^{\theta =\theta _{2}}a^{2}\,d\theta ={\frac {a^{2}}{2}}\int _{\theta =0}^{\theta =1}d\theta ={\frac {a^{2}}{2}}}$.

(2) Find the area of the first quadrant of the curve (known as “Pascal’s Snail”), the polar equation of which is ${\displaystyle r=a(1+\cos \theta )}$.

{\displaystyle {\begin{aligned}{\text{Area}}&={\tfrac {1}{2}}\int _{\theta =0}^{\theta ={\frac {\pi }{2}}}a^{2}(1+\cos \theta )^{2}\,d\theta \\&={\frac {a^{2}}{2}}\int _{\theta =0}^{\theta ={\frac {\pi }{2}}}(1+2\cos \theta +\cos ^{2}\theta )\,d\theta \\&={\frac {a^{2}}{2}}\left[\theta +2\sin \theta +{\frac {\theta }{2}}+{\frac {\sin 2\theta }{4}}\right]_{0}^{\frac {\pi }{2}}\\&={\frac {a^{2}(3\pi +8)}{8}}.\end{aligned}}}

Volumes by Integration.

What we have done with the area of a little strip of a surface, we can, of course, just as easily do with the volume of a little strip of a solid. We can add up all the little strips that make up the total solid, and find its volume, just as we have added up all the small little bits that made up an area to find the final area of the figure operated upon.

Examples.

(1) Find the volume of a sphere of radius ${\displaystyle r}$.

A thin spherical shell has for volume ${\displaystyle 4\pi x^{2}dx}$ (see Fig. 59, p. 216); summing up all the concentric shells which make up the sphere, we have

volume sphere ${\displaystyle =\int _{x=0}^{x=r}4\pi x^{2}\,dx=4\pi \left[{\frac {x^{3}}{3}}\right]_{0}^{r}={\tfrac {4}{3}}\pi r^{3}}$.

Fig. 62.

We can also proceed as follows: a slice of the sphere, of thickness ${\displaystyle dx}$, has for volume ${\displaystyle \pi y^{2}dx}$ (see Fig. 62). Also ${\displaystyle x}$ and ${\displaystyle y}$ are related by the expression

${\displaystyle y^{2}=r^{2}-x^{2}}$.

{\displaystyle {\begin{aligned}{\text{Hence}}\;{\text{volume sphere}}&=2\int _{x=0}^{x=r}\pi (r^{2}-x^{2})\,dx\\&=2\pi \left[\int _{x=0}^{x=r}r^{2}\,dx-\int _{x=0}^{x=r}x^{2}\,dx\right]\\&=2\pi \left[r^{2}x-{\frac {x^{3}}{3}}\right]_{0}^{r}={\frac {4\pi }{3}}r^{3}.\end{aligned}}}

(2) Find the volume of the solid generated by the revolution of the curve ${\displaystyle y^{2}=6x}$ about the axis of ${\displaystyle x}$, between ${\displaystyle x=0}$ and ${\displaystyle x=4}$.

The volume of a strip of the solid is ${\displaystyle \pi y^{2}dx}$.

{\displaystyle {\begin{aligned}{\text{Hence}}\;{\text{volume}}&=\int _{x=0}^{x=4}\pi y^{2}\,dx=6\pi \int _{x=0}^{x=4}x\,dx\\&=6\pi \left[{\frac {x^{2}}{2}}\right]_{0}^{4}=48\pi =150.8.\end{aligned}}}

In certain branches of physics, particularly in the study of alternating electric currents, it is necessary to be able to calculate the quadratic mean of a variable quantity. By “quadratic mean” is denoted the square root of the mean of the squares of all the values between the limits considered. Other names for the quadratic mean of any quantity are its “virtual” value, or its “R.M.S.” (meaning root-mean-square) value. The French term is valeur efficace. If ${\displaystyle y}$ is the function under consideration, and the quadratic mean is to be taken between the limits of ${\displaystyle x=0}$ and ${\displaystyle x=l}$; then the quadratic mean is expressed as

${\displaystyle {\sqrt[{2}]{{\frac {1}{l}}\int _{0}^{l}y^{2}\,dx}}}$.

Examples.

(1) To find the quadratic mean of the function ${\displaystyle y=ax}$ (Fig. 63).

Here the integral is ${\displaystyle \int _{0}^{l}a^{2}x^{2}\,dx}$, which is ${\displaystyle {\frac {1}{3}}a^{2}l^{3}}$.

Fig. 63.

Dividing by ${\displaystyle l}$ and taking the square root, we have

quadratic mean ${\displaystyle ={\frac {1}{\sqrt {3}}}\,al}$.

Here the arithmetical mean is ${\displaystyle {\tfrac {1}{2}}al}$; and the ratio of quadratic to arithmetical mean (this ratio is called the form-factor) is ${\displaystyle {\dfrac {2}{\sqrt {3}}}=1.155}$.

(2) To find the quadratic mean of the function ${\displaystyle y=x^{a}}$.

The integral is ${\displaystyle \int _{x=0}^{x=l}x^{2a}\,dx}$, that is ${\displaystyle {\dfrac {l^{2a+1}}{2a+1}}}$.

Hence quadratic mean ${\displaystyle ={\sqrt[{2}]{\dfrac {l^{2a}}{2a+1}}}}$.

(3) To find the quadratic mean of the function ${\displaystyle y=a^{\frac {x}{2}}}$ The integral is ${\displaystyle \int _{x=0}^{x=l}(a^{\frac {x}{2}})^{2}\,dx}$, that is ${\displaystyle \int _{x=0}^{x=l}a^{x}\,dx}$, or ${\displaystyle \quad \left[{\frac {a^{x}}{\log _{\epsilon }a}}\right]_{x=0}^{x=l}}$, which is ${\displaystyle {\dfrac {a^{l}-1}{\log _{\epsilon }a}}}$.

Hence the quadratic mean is ${\displaystyle {\sqrt[{2}]{\dfrac {a^{l}-1}{l\log _{\epsilon }a}}}}$.

Exercises XVIII. (See p. 263 for Answers.)

(1) Find the area of the curve ${\displaystyle y=x^{2}+x-5}$ between ${\displaystyle x=0}$ and ${\displaystyle x=6}$, and the mean ordinates between these limits.

(2) Find the area of the parabola ${\displaystyle y=2a{\sqrt {x}}}$ between ${\displaystyle x=0}$ and ${\displaystyle x=a}$. Show that it is two-thirds of the rectangle of the limiting ordinate and of its abscissa.

(3) Find the area of the positive portion of a sine curve and the mean ordinate.

(4) Find the area of the positive portion of the curve ${\displaystyle y=\sin ^{2}x}$, and find the mean ordinate.

(5) Find the area included between the two branches of the curve ${\displaystyle y=x^{2}\pm x^{\tfrac {5}{2}}}$ from ${\displaystyle x=0}$ to ${\displaystyle x=1}$, also the area of the positive portion of the lower branch of the curve (see Fig 30, p. 108).

(6) Find the volume of a cone of radius of base ${\displaystyle r}$, and of height ${\displaystyle h}$.

(7) Find the area of the curve ${\displaystyle y=x^{3}-\log _{\epsilon }x}$ between ${\displaystyle x=0}$ and ${\displaystyle x=1}$.

(8) Find the volume generated by the curve ${\displaystyle y={\sqrt {1+x^{2}}}}$, as it revolves about the axis of ${\displaystyle x}$, between ${\displaystyle x=0}$ and ${\displaystyle x=4}$.

(9) Find the volume generated by a sine curve revolving about the axis of ${\displaystyle x}$. Find also the area of its surface.

(10) Find the area of the portion of the curve ${\displaystyle xy=a}$ included between ${\displaystyle x=1}$ and ${\displaystyle x=a}$. Find the mean ordinate between these limits.

(11) Show that the quadratic mean of the function ${\displaystyle y=\sin x}$, between the limits of ${\displaystyle 0}$ and ${\displaystyle \pi }$ radians, is ${\displaystyle {\tfrac {\sqrt {2}}{2}}}$. Find also the arithmetical mean of the same function between the same limits; and show that the form-factor is ${\displaystyle =1.11}$.

(12) Find the arithmetical and quadratic means of the function ${\displaystyle x^{2}+3x+2}$, from ${\displaystyle x=0}$ to ${\displaystyle x=3}$.

(13) Find the quadratic mean and the arithmetical mean of the function ${\displaystyle y=A_{1}\sin x+A_{1}\sin 3x}$.

(14) A certain curve has the equation ${\displaystyle y=3.42\epsilon ^{0.21x}}$. Find the area included between the curve and the axis of ${\displaystyle x}$, from the ordinate at ${\displaystyle x=2}$ to the ordinate at ${\displaystyle x=8}$. Find also the height of the mean ordinate of the curve between these points.

(15) Show that the radius of a circle, the area of which is twice the area of a polar diagram, is equal to the quadratic mean of all the values of r for that polar diagram.

(16) Find the volume generated by the curve ${\displaystyle y=\pm {\dfrac {x}{6}}{\sqrt {x(10-x)}}}$ rotating about the axis of ${\displaystyle x}$.


text:
# charge flow between a sphere (inside) a spherical shell irrespective of the charge of the shell

A small sphere of radius $r_1$ and charge $q_1$ is enclosed by a spherical shell of radius $r_2$ and charge $q_2$. If $q_1$ is non-zero and the two spheres are connected by a wire, then, charges will flow from the inner sphere to the outer one; no matter what the charge $q_2$ is. Why???

• I guess you know the fact that charges always reside on the surface of a solid pure conductor and not inside it. When you connect inner sphere to outer hollow sphere, the inner sphere becomes a part of the hollow sphere. But it's inside the hollow sphere. So, the net charges would come out on the surface of the hollow sphere. – Shubham Sep 16 '15 at 16:38
• thanks , it was so simple , may be i simply didn't put much to understand it. – shaistha Sep 17 '15 at 13:36
• when i tried to find an answer for this somewhere , i got the explanation as this, according to gauss's law , E between a sphere and a shell is determined by the charge q1 on a small sphere .Hence , potential between them is independent of charge q2.But the thing which i didn't understand in this explanation is , why E is determined by the charge q1. I TOTALLY UNDERSTAND THE CONCEPT GIVEN BY SHUBHAM BUT DON'T UNDERSTAND THE ABOVE explanation . – shaistha Sep 18 '15 at 8:58


text:
emacs-orgmode
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## Re: [O] [RFC] Change $..$ and $$...$$ into elements

 From: Rasmus Subject: Re: [O] [RFC] Change $..$ and $$...$$ into elements Date: Thu, 21 Jan 2016 16:58:56 +0100 User-agent: Gnus/5.13 (Gnus v5.13) Emacs/25.0.50 (gnu/linux)

Nicolas Goaziou <address@hidden> writes:

> Hello,
>
>
>> What's wrong with equation (potentially coupled with mathtools for
>> handling numbering automatically)?  Is it worth breaking old documents
>> (irrespective of org-lint) for this inconvenience?
>
> Actually, the idea behind this patch is to mimic the expected LaTeX
> behaviour,i.e., treat $..$ as an equivalent
> to \begin{equation*}...\end{equation*}.
>
> This was requested at least once before, so I guess it may make sense to
> follow the path of least surprise.

And I'm saying that people should just write:

aaa
$$bbb$$
ccc

Org isn't LaTeX and there’s no difference in the output from $⋯$
and \begin{equation*}⋯\end{equation*} or the above.

Nonetheless, I don't feel strongly about the (non)issue so if this
behavior is better, somehow, go for it.

Rasmus

--
This is the kind of tedious nonsense up with which I will not put


text:
# xarray.ufuncs.fmod¶

xarray.ufuncs.fmod = <xarray.ufuncs._UFuncDispatcher object>

xarray specific variant of numpy.fmod. Handles xarray.Dataset, xarray.DataArray, xarray.Variable, numpy.ndarray and dask.array.Array objects with automatic dispatching.

Documentation from numpy:

fmod(x1, x2, /, out=None, *, where=True, casting=’same_kind’, order=’K’, dtype=None, subok=True[, signature, extobj])

Return the element-wise remainder of division.

This is the NumPy implementation of the C library function fmod, the remainder has the same sign as the dividend x1. It is equivalent to the Matlab(TM) rem function and should not be confused with the Python modulus operator x1 % x2.

Parameters
x1array_like

Dividend.

x2array_like

Divisor.

outndarray, None, or tuple of ndarray and None, optional

A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned. A tuple (possible only as a keyword argument) must have length equal to the number of outputs.

wherearray_like, optional

Values of True indicate to calculate the ufunc at that position, values of False indicate to leave the value in the output alone.

**kwargs

For other keyword-only arguments, see the ufunc docs.

Returns
yarray_like

The remainder of the division of x1 by x2. This is a scalar if both x1 and x2 are scalars.

remainder

Equivalent to the Python % operator.

divide

Notes

The result of the modulo operation for negative dividend and divisors is bound by conventions. For fmod, the sign of result is the sign of the dividend, while for remainder the sign of the result is the sign of the divisor. The fmod function is equivalent to the Matlab(TM) rem function.

Examples

>>> np.fmod([-3, -2, -1, 1, 2, 3], 2)
array([-1,  0, -1,  1,  0,  1])
>>> np.remainder([-3, -2, -1, 1, 2, 3], 2)
array([1, 0, 1, 1, 0, 1])

>>> np.fmod([5, 3], [2, 2.])
array([ 1.,  1.])
>>> a = np.arange(-3, 3).reshape(3, 2)
>>> a
array([[-3, -2],
[-1,  0],
[ 1,  2]])
>>> np.fmod(a, [2,2])
array([[-1,  0],
[-1,  0],
[ 1,  0]])


text: Speaker : Dr. Yu-Chang Liang (National Sun Yat-sen University) Title : Anti-magic labeling of even regular graphs Time : 2014-11-14 (Fri) 14:00 - 15:30 Place : Seminar Room 617, Institute of Mathematics (NTU Campus) Abstract: An anti-magic labeling of a graph $G$ is a one-to-one correspondence between $E(G)$ and $\{1,2,\ldots, |E|\}$ such that the sums of the labels assigned to edges incident to distinct vertices are different. If $G$ has an anti-magic labeling, then we say $G$ is anti-magic. In this talk, we will show that any regular graph with even degree is anti-magic. This is a joint work with Xuding Zhu, Fei-Huang Chang, Zhi-Shi Pan.