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https://crypto.stackexchange.com/questions/24838/is-the-salsa20-gcm-composition-secure/24846
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# Is the Salsa20-GCM composition secure?
AES-GCM seems to be used by everyone, but I have never seen even one post about Salsa20-GCM. Is it secure? Wikipedia mentions that GCM uses a block cipher, but also that it uses CTR mode.
I am seeking an authoritative citation that I can reference in my project documentation, but personal opinions and arguments are welcomed too.
• Strictly speaking GCM has been defined using a 128 bit block cipher, but it's easy to generalize it to any stream cipher that takes a nonce and it's be secure that way. But I see little reason to do so, since GCM only works well if you have specialized instructions, which are typically only available on CPUs which can also accelerate AES. – CodesInChaos Apr 7 '15 at 16:34
• You exclude the possibility of abandoning AES for different reasons. – ArekBulski Apr 7 '15 at 17:47
• The AES cipher is not just used for encryption in GCM. You probably would need to make some superficial changes before a stream cipher can be used (but I don't see any big problems with that). I doubt you will find an authoritative citation though (authoritative to whom?) – Maarten Bodewes Apr 7 '15 at 18:08
• A statement from Daniel Bernstein would be authoritative enough (to me), for example. – ArekBulski Apr 7 '15 at 18:14
• If you aren't using a block cipher, your result is not GCM. It might use GHASH, and might resemble GCM, but it isn't GCM. Also, GCM seems to assume the length of the counter is the length of the output from that counter; this is not true for Salsa20 (it has what amounts to 128 bits of nonce, but for 512 bits of output). – cpast Apr 7 '15 at 21:28
As SOJPM says in their answer, the proofs for AES-GCM assumes that AES is a PRP. I can't believe that there is anywhere in the proof that using a PRF (possibly truncated) would break things -- but I haven't looked carefully for this. Depending on how the GCM proof is structured, (using/not using) the PRP/PRF switching lemma [1] may suffice, but I don't remember well enough to say for certain.
I think that the closest reference you will find is [2], which analyses the ChaCha20-Poly1305 construction in IETF protocols [3]. As both ChaCha20 and Salsa20 can be assumed to be a PRF, this change is not significant; similarly GMAC and Poly1305 are fundamentally the same (a MAC based on a polynomial-evaluation hash). However, the scheme in [1] is not precisely ChaCha20-GCM; unless you want to dive into the GCM proofs (either to check that a PRF is ok at every point, or that you can use/not use the PRP/PRF lemma) I think it is the closest analysed scheme that you will find.
• I noticed the introduction of ChaCha20-Poly1305 and it made me thinking, but your link [2] to academic analysis pushed me through it. I will switch to that. – ArekBulski Apr 8 '15 at 23:14
• [2] quotes Bernstein: "There is nothing special about AES here. One can replace AES with an arbitrary keyed function from an arbitrary set of nonces to 16-byte strings." – ArekBulski Apr 8 '15 at 23:57
GCM is a specific mode for block ciphers that combines CTR encryption mode and GMAC authentication. Since Salsa and ChaCha are already based on CTR mode internally, that would not be a relevant mode.
However, there is no problem using GMAC. Salsa and ChaCha output larger blocks than GMAC accepts, so you would need to break them in the correct size chunks to process. This is already done in Poly1305 authentication, so that is not a problem.
• Wikipedia states that GCM security depends on "block cipher that is indistinguishable from a random permutation" which seems to imply any PRF but it still seems a bit vague to me. GMAC still uses AES to encrypt the authentication tag. Poly1305 is, even by DJB himself, called Poly1305-AES and it is stated that AES can be substituted with "identical security guarantee", whatever that is exactly. – ArekBulski Apr 7 '15 at 20:52
• @ArekBulski that means a cipher with a key strength at least as good as the security guarantee of the authentication, which is 128-bits. ChaCha with Poly1305 has been added as an RFC for the next version of TLS, and is already available in Google Chrome – Richie Frame Apr 8 '15 at 3:45
The only other issue may be the generation of the H value which is $E_K(0^{128})$. Here you may just use your stream-cipher will a well-defined IV (maybe derived from the normal IV using a hash-function?).
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2020-12-05 09:00:13
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https://physics.stackexchange.com/questions/154737/is-wave-superposition-always-equivalent-to-wave-interference
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# Is wave superposition always equivalent to wave interference?
I'm confused when using these 2 words "wave superposition" and "wave interference" since their definition is very similar.
So, are these 2 term the same?
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2019-12-15 17:40:41
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http://nonequilibrium-turbulence.org.uk/aggregator/categories/5?page=7
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Latest papers in fluid mechanics
Retarding spreading of surfactant drops on solid surfaces: Interplay between the Marangoni effect and capillary flows
Physical Review Fluids - Thu, 08/27/2020 - 11:00
Author(s): Parisa Bazazi and S. Hossein Hejazi
Early time spreading of a water drop on a hydrophilic surface is characterized by the wetted radius which grows linearly in time. We report the unexpected result that the initial spread of surfactant-laden drops is impeded by Marangoni stresses, leading to a large increase in total spreading time. The nonuniform distribution of surfactants at the interface generates Marangoni stresses before the drop-solid contact suppresses film drainage and droplet expansion. Our experiments show that, remarkably, surfactants delay the initial fast motion of the three-phase contact lines.
[Phys. Rev. Fluids 5, 084006] Published Thu Aug 27, 2020
Dynamics of retracting surfactant-laden ligaments at intermediate Ohnesorge number
Physical Review Fluids - Thu, 08/27/2020 - 11:00
Author(s): Cristian R. Constante-Amores, Lyes Kahouadji, Assen Batchvarov, Seungwon Shin, Jalel Chergui, Damir Juric, and Omar K. Matar
Three-dimensional direct numerical simulations of the ligaments retraction process over a range of system parameters that account for surfactant solubility, sorption kinetics, and Marangoni stresses are presented. The presence of surfactant inhibits the “end-pinching” mechanism and promotes neck reopening through Marangoni flow induced by the formation of surfactant concentration gradients that drive flow reversal toward the neck.
[Phys. Rev. Fluids 5, 084007] Published Thu Aug 27, 2020
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Physical Review Fluids - Thu, 08/27/2020 - 11:00
Author(s): Laurence Girolami and Frédéric Risso
This paper develops a physical model able to describe the dam-break flow of particulate suspensions that sediment progressively during propagation at a constant velocity that solely depends on the mixture properties. The model considers the suspension as an equivalent fluid of constant density and negligible viscosity and leads to a good prediction of the flow duration and deposits shape. These findings allow the formulation of consistent shallow-water equations that can be used to compute the dense basal layer of small-volume pyroclastic flows.
[Phys. Rev. Fluids 5, 084306] Published Thu Aug 27, 2020
Dynamics of nonisothermal two-thin-fluid-layer systems subjected to harmonic tangential forcing under Rayleigh–Taylor instability conditions
Physics of Fluids - Wed, 08/26/2020 - 11:02
Physics of Fluids, Volume 32, Issue 8, August 2020.
The stability of a nonisothermal system consisting of two superimposed fluid layers: a thin liquid film layer and a gas layer sandwiched between differentially heated horizontal solid plates in the gravity field, is investigated. The system is assumed to be subjected to the Rayleigh–Taylor instability (RTI) with the Marangoni effect that either enhances the RTI or opposes it and to the tangential harmonic vibration of the upper substrate. A set of reduced evolution equations is derived based on the weighted-residual integral boundary layer approach, and the investigation is carried out in the framework of this set. The base state of the system represents a time-periodic flow, and its linear stability analysis is carried out using the Floquet theory in the large-time limit. The nonlinear dynamics of the system is investigated numerically in the case of either a static or vibrating substrate. Among the possible outcomes of the nonlinear dynamics, there is the emergence of ruptured states of the liquid film with rupture taking place at either the upper or lower substrate and also the emergence of saturated continuous flows of the liquid film. We also find that the nonlinear dynamics of the system is consistent with the results of the linear stability analysis in terms of enhancement or attenuation of interfacial distortion.
Interplay of Kelvin–Helmholtz instability with acoustics in a viscous potential flow
Physics of Fluids - Wed, 08/26/2020 - 11:02
Physics of Fluids, Volume 32, Issue 8, August 2020.
Among the hydrodynamic instabilities influencing the evolution, stabilization, and control of flows, the Kelvin–Helmholtz (KH) instability mode is a profound trigger to induce unsteadiness and turbulence—either within a single fluid, by means of a velocity shear, or along the interface of multiple fluids. This mechanism has been analytically studied by Funada and Joseph [“Viscous potential flow analysis of Kelvin–Helmholtz instability in a channel,” J. Fluid Mech. 445, 263 (2001)], for the surface separating two fluids within the approximation of inviscid and viscous potential flows. The present investigation extends the Funada–Joseph formulation to incorporate the effect of imposed acoustic waves on the system under consideration. Specifically, the KH–acoustic interaction is studied by employing a modification of the Bychkov approach [V. Bychkov, “Analytical scalings for flame interaction with sound waves,” Phys. Fluids 11, 3168 (1999)], which has been originally derived for the acoustic coupling to the combustion instability. The analytic formulae for the dispersion relations, growth rates, and neutral curves describing the perturbed interface of the KH instability/acoustic region are derived. Specifically, the limits for stable/unstable regimes as a function of hydrodynamic and acoustic parameters are identified. Two interacting modes are of particular interest: resonant and parametric modes, characterized by acoustic fields having the same frequency (resonant) and twice the frequency (parametric) of the instability oscillations. It is shown that while relatively weak acoustics provide a promising contribution to stabilize the KH instability, those of higher strength can excite the parametric instability. Overall, a comprehensive parametric study of the KH–acoustic coupling and stability limits shows that a global stability region may exist between that of the resonant and parametrically unstable regimes.
Global eigenmodes of thin liquid sheets by means of Volume-of-Fluid simulations
Physics of Fluids - Wed, 08/26/2020 - 11:02
Physics of Fluids, Volume 32, Issue 8, August 2020.
The unsteady dynamics of planar liquid sheet flows, interacting with unconfined gaseous environments located on both sides of the liquid phase, is numerically investigated by means of the Volume-of-Fluid (VOF) technique for supercritical regimes. The global behavior of the non-parallel flow is analyzed by perturbing the initial steady configuration by means of a Gaussian bump in the transverse velocity component of relatively small amplitude, thereby exciting sinuous modes. To gain more physical insights into the fluid system, a theoretical linear one-dimensional model is also developed. A physical interpretation of this model relates the sheet dynamics to transverse vibrations of tensional string forced by terms containing the lateral velocity and subjected to a total damping coefficient, which can assume negative values. The VOF simulation satisfactorily confirms that the velocity impulse perturbation splits into two wave fronts traveling downstream with the theoretical wave velocities. A good agreement is found in comparing the crossing times over the entire domain length of such waves with the almost constant spacing between the frequencies of the eigenvalue spectrum. Surface tension plays a stabilizing role, and for relatively high values of density ratio rρ of gaseous-to-liquid phases, the sheet becomes unstable. It is argued that the distribution of transverse velocity component of the gaseous phase represents the forcing term, which leads the system toward the instability when, for relatively high rρ, the total damping becomes negative. An analogy seems to exist between the global unstable behavior exhibited by the liquid sheet as rρ increases and the shear-induced global instability found by Tammisola et al. [Surface tension-induced global instability of planar jets and wakes,” J. Fluid Mech. 713, 632–658 (2012)] in the presence of surface tension. However, for the gravitational sheet, the surface tension is stabilizing.
Modeling drug delivery from multiple emulsions
Physical Review E - Wed, 08/26/2020 - 11:00
Author(s): G. Pontrelli, E. J. Carr, A. Tiribocchi, and S. Succi
We present a mechanistic model of drug release from a multiple emulsion into an external surrounding fluid. We consider a single multilayer droplet where the drug kinetics are described by a pure diffusive process through different liquid shells. The multilayer problem is described by a system of di...
[Phys. Rev. E 102, 023114] Published Wed Aug 26, 2020
Interaction of wave-driven particles with slit structures
Physical Review E - Wed, 08/26/2020 - 11:00
Author(s): Clive Ellegaard and Mogens T. Levinsen
Just over a decade ago Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)] published a provocative paper suggesting that a classical system might be able to simulate the truly fundamental quantum mechanical single- and double-slit experiment. The system they investigated was that of an oil droplet ...
[Phys. Rev. E 102, 023115] Published Wed Aug 26, 2020
Specific features of the gas-dynamic structure of supersonic axisymmetric microjets of a nonequilibrium $\mathrm{S}{\mathrm{F}}_{6}$ gas
Physical Review Fluids - Wed, 08/26/2020 - 11:00
Author(s): Vladimir Aniskin, Nikolay Maslov, Sergey Mironov, Elena Tsybulskaya, and Ivan Tsyryulnikov
The effect of vibrational nonequilibrium of SF6 molecules on the gas dynamic structure of microjets is revealed, manifesting in a decrease in the longitudinal cell size of the wave structure and weakening of variations in flow parameters along the axis of the microjet compared to the equilibrium flow. The physical justification is found for the diameter of the nozzle, which is the boundary value between the macro- and microjets of vibrationally relaxing gases.
[Phys. Rev. Fluids 5, 083401] Published Wed Aug 26, 2020
Behavior of the square-back Ahmed body global modes at low ground clearance
Physical Review Fluids - Wed, 08/26/2020 - 11:00
Author(s): Baptiste Plumejeau, Laurent Keirsbulck, Sébastien Delprat, Marc Lippert, and Wafik Abassi
A study of the evolution of the wake flow of a square-back Ahmed body is presented. Various ground clearance configurations around the critical case associated with the onset of the lateral bistability are investigated. The oscillation modes vary between stable and bistable states, the corresponding Strouhal numbers for the horizontal (respectively, vertical) evolve from 0.16 (respectively, 0.27) to 0.13 (respectively, 0.18).
[Phys. Rev. Fluids 5, 084701] Published Wed Aug 26, 2020
3-dimensional particle image velocimetry based evaluation of turbulent skin-friction reduction by spanwise wall oscillation
Physics of Fluids - Wed, 08/26/2020 - 02:56
Physics of Fluids, Volume 32, Issue 8, August 2020.
The reduction of turbulent skin-friction drag and the response of vortical structures in a zero-pressure gradient, turbulent boundary layer subjected to spanwise wall oscillation is investigated using planar and tomographic particle image velocimetry (PIV). The experiments are conducted at a momentum based Reynolds number of 1000, while the range of spanwise oscillation amplitude and frequency is chosen around the optimum reported in previous studies. A high-resolution planar PIV measurement is employed to determine the drag reduction directly from wall shear measurements and to analyze the accompanying modifications in the turbulent vortical structures. Drag reduction of up to 15% is quantified, with variations following the trends reported in the literature. The analysis of the turbulence structure of the flow is made in terms of Reynolds shear stresses, turbulence production, and vortex visualization. A pronounced drop of turbulence production is observed up to a height of 100 wall units from the wall. The vorticity analysis, both in the streamwise wall-normal plane and in the volumetric results, indicates a reduction of vorticity fluctuations in the near-wall domain. A distortion of the hairpin-packet arrangement is hypothesized, suggesting that the drag-reduction mechanism lies in the inhibition of the hairpin auto-generation by the spanwise wall oscillations.
Elliptic supersonic jet morphology manipulation using sharp-tipped lobes
Physics of Fluids - Tue, 08/25/2020 - 12:45
Physics of Fluids, Volume 32, Issue 8, August 2020.
Elliptic nozzle geometry is attractive for mixing enhancement of supersonic jets. However, jet dynamics, such as flapping, gives rise to high-intensity tonal sound. We experimentally manipulate the supersonic elliptic jet morphology by using two sharp-tipped lobes. The lobes are placed on either end of the minor axis in an elliptic nozzle. The design Mach number and the aspect ratio of the elliptic nozzle and the lobed nozzle are 2.0 and 1.65. The supersonic jet is exhausted into ambient under almost perfectly expanded conditions. Time-resolved schlieren imaging, longitudinal and cross-sectional planar laser Mie scattering imaging, planar Particle Image Velocimetry (PIV), and near-field microphone measurements are performed to assess the fluidic behavior of the two nozzles. Dynamic Mode Decomposition (DMD) and proper orthogonal decomposition analyses are carried out on the schlieren and the Mie scattering images. Mixing characteristics are extracted from the Mie scattering images through the image processing routines. The flapping elliptic jet consists of two dominant DMD modes, while the lobed nozzle has only one dominant mode, and the flapping is suppressed. Microphone measurements show the associated noise reduction. The jet column bifurcates in the lobed nozzle enabling a larger surface contact area with the ambient fluid and higher mixing rates in the near-field of the nozzle exit. The jet width growth rate of the two-lobed nozzle is about twice that of the elliptic jet in the near-field, and there is a 40% reduction in the potential core length. PIV contours substantiate the results.
Non-autoregressive time-series methods for stable parametric reduced-order models
Physics of Fluids - Tue, 08/25/2020 - 11:37
Physics of Fluids, Volume 32, Issue 8, August 2020.
Advection-dominated dynamical systems, characterized by partial differential equations, are found in applications ranging from weather forecasting to engineering design where accuracy and robustness are crucial. There has been significant interest in the use of techniques borrowed from machine learning to reduce the computational expense and/or improve the accuracy of predictions for these systems. These rely on the identification of a basis that reduces the dimensionality of the problem and the subsequent use of time series and sequential learning methods to forecast the evolution of the reduced state. Often, however, machine-learned predictions after reduced-basis projection are plagued by issues of stability stemming from incomplete capture of multiscale processes as well as due to error growth for long forecast durations. To address these issues, we have developed a non-autoregressive time series approach for predicting linear reduced-basis time histories of forward models. In particular, we demonstrate that non-autoregressive counterparts of sequential learning methods such as long short-term memory (LSTM) considerably improve the stability of machine-learned reduced-order models. We evaluate our approach on the inviscid shallow water equations and show that a non-autoregressive variant of the standard LSTM approach that is bidirectional in the principal component directions obtains the best accuracy for recreating the nonlinear dynamics of partial observations. Moreover—and critical for many applications of these surrogates—inference times are reduced by three orders of magnitude using our approach, compared with both the equation-based Galerkin projection method and the standard LSTM approach.
On the role of surface grooves in the reduction of pressure losses in heated channels
Physics of Fluids - Tue, 08/25/2020 - 11:36
Physics of Fluids, Volume 32, Issue 8, August 2020.
Pressure-gradient-driven flows in grooved horizontal channels were investigated. The results show that a significant reduction in pressure losses can be achieved by exposing such channels to spatially distributed heating. The system response strongly depends on the characterization of both patterns and on their relative position, leading to a pattern interaction problem. Mismatch and misplacement of both patterns may result in a significant increase in pressure losses or may have no effect on such losses. The reduction in pressure loss is associated with the formation of convection rolls on the bounding surfaces due to spatially distributed buoyancy along the streamwise direction. The pressure-gradient-reducing effect is active only in small Reynolds number flows. Explicit results are given for fluids with the Prandtl number Pr = 0.71, representing air.
Statistical transition to turbulence in plane channel flow
Physical Review Fluids - Tue, 08/25/2020 - 11:00
Author(s): Sébastien Gomé, Laurette S. Tuckerman, and Dwight Barkley
The subcritical route to turbulence in shear flows is characterized by metastable localized turbulent-laminar patterns. In plane channel flow, these take the form of intermittent oblique turbulent bands, which either proliferate or decay on timescales that depend on the Reynolds number. A statistical study via direct numerical simulations in a narrow tilted domain leads to the determination of a crossing Reynolds number of around 965, above which the probability for a band to split outpaces its probability to disappear.
[Phys. Rev. Fluids 5, 083905] Published Tue Aug 25, 2020
Universal trends in human cough airflows at large distances
Physics of Fluids - Tue, 08/25/2020 - 03:45
Physics of Fluids, Volume 32, Issue 8, August 2020.
Coughs are one of the primary means of transmission of diseases such as influenza and SARS-CoV-2 (COVID-19). Disease spreading occurs by the expulsion of pathogen containing aerosol droplets. Fine droplets can pass through layers of masks and are carried away by the exhaled airflow unlike larger droplets that settle down due to gravity. Hence, it is important to quantitatively assess the maximum distance of travel of typical human coughs with and without different types of masks. Even though near field data are available near the mouth, far field data are scarce. In this study, the schlieren method that is a highly sensitive, non-intrusive flow visualization technique is used. It can directly image weak density gradients produced by coughs. An assessment of different methods of covering the mouth while coughing is arrived at by using observations from high speed schlieren images. The effectiveness of coughing into the elbow is examined. The velocity of propagation of coughs and the distance of propagation with and without masks are quantified. It is also found that normalizing the distance–velocity profiles causes all the data to collapse onto a universal non-dimensional curve irrespective of the usage of different types of masks or test subjects. Visualization of cough flow fields and analysis of experimental data reveal that the flow physics is governed by the propagation of viscous vortex rings.
Reopening dentistry after COVID-19: Complete suppression of aerosolization in dental procedures by viscoelastic Medusa Gorgo
Physics of Fluids - Tue, 08/25/2020 - 03:45
Physics of Fluids, Volume 32, Issue 8, August 2020.
The aerosol transmissibility of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has impacted the delivery of health care and essentially stopped the provision of medical and dental therapies. Dentistry uses rotary, ultrasonic, and laser-based instruments that produce water-based aerosols in the daily, routine treatment of patients. Abundant aerosols are generated, which reach health care workers and other patients. Viruses, including SARS-CoV-2 virus and related coronavirus disease (COVID-19) pandemic, continued expansion throughout the USA and the world. The virus is spread by both droplet (visible drops) and aerosol (practically invisible drops) transmission. The generation of aerosols in dentistry—an unavoidable part of most dental treatments—creates a high-risk situation. The US Centers for Disease Control and The Occupational Safety and Health Administration consider dental procedures to be of “highest risk” in the potential spreading of SARS-CoV-2 and other respiratory viruses. There are several ways to reduce or eliminate the virus: (i) cease or postpone dentistry (public and personal health risk), (ii) screen patients immediately prior to dental treatment (by appropriate testing, if any), (iii) block/remove the virus containing aerosol by engineering controls together with stringent personal protective equipment use. The present work takes a novel, fourth approach. By altering the physical response of water to the rotary or ultrasonic forces that are used in dentistry, the generation of aerosol particles and the distance any aerosol may spread beyond the point of generation can be markedly suppressed or completely eliminated in comparison to water for both the ultrasonic scaler and dental handpiece.
Parameter space mapping of the Princeton magnetorotational instability experiment
Physical Review E - Mon, 08/24/2020 - 11:00
Author(s): Himawan W. Winarto, Hantao Ji, Jeremy Goodman, Fatima Ebrahimi, Erik P. Gilson, and Yin Wang
Extensive simulations of the Princeton Magnetorotational Instability (MRI) Experiment with the Spectral/Finite Element code for Maxwell and Navier-Stokes Equations (SFEMaNS) have been performed to map the MRI-unstable region as a function of inner cylinder angular velocity and applied vertical magne...
[Phys. Rev. E 102, 023113] Published Mon Aug 24, 2020
Linear analysis of dewetting instability in multilayer planar sheets for composite nanostructures
Physical Review Fluids - Mon, 08/24/2020 - 11:00
Author(s): Bingrui Xu and Daosheng Deng
Dewetting instability of multilayer planar sheets are studied by linear analysis. Several unstable modes are identified, while the maximum growth rate depends on fluid properties. These results provide theoretical guidance to enhance or suppress the dewetting instability via material selection and structure design, enabling fabrication of sophisticated nanostructures for functional fibers and wearable textiles.
[Phys. Rev. Fluids 5, 083904] Published Mon Aug 24, 2020
Explicit algebraic relation for calculating Reynolds normal stresses in flows dominated by bubble-induced turbulence
Physical Review Fluids - Mon, 08/24/2020 - 11:00
Author(s): Tian Ma, Dirk Lucas, and Andrew D. Bragg
Two new algebraic turbulence models for flows dominated by bubble-induced turbulence (BIT) are presented. The first model, referred to as the algebraic Reynolds normal stress model, is derived from a differential Reynolds stress model for bubbly flows. The second model utilizes one two-equation turbulence model to achieve algebraic expressions for k and ϵ in the BIT dominated cases. If both models are combined, it results in a purely algebraic, explicit relation for the Reynolds normal stresses.
[Phys. Rev. Fluids 5, 084305] Published Mon Aug 24, 2020
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2020-09-22 11:14:16
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https://docs.vocitec.com/en/voice-activity-detection-controls.html
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## Voice Activity Detection Controls
The following parameters are most often used in real-time transcription scenarios using V‑Blaze.
Table 1. Voice Activity Detection controls
Name
Type
Values
Description
activitylevel
string
integer
default is 175
Specifies the volume threshold for active versus inactive audio. This value should be high enough to screen out noise, but low enough to clearly trigger on speech. Range is 0-32768, correlating to the average magnitude of a signed 16-bit LPCM frame.
insecure
boolean
true, false (default)
This option explicitly allows curl to perform "insecure" SSL connections and transfers. All SSL connections are attempted to be made secure by using the CA certificate bundle installed by default. This makes all connections considered "insecure" fail unless -k/--insecure is used.\
This option is only relevant when HTTPS URLs are provided for callback or utterance_callback.
Refer to http://curl.haxx.se/docs/sslcerts.html for more details on this parameter.
realtime
boolean
true, false (default)
Controls whether or not the ASR engine is processing incoming audio in real-time mode or not. Real-time mode is enabled based on a license setting and cannot be enabled using this setting if it is not enabled in the license. This tag is only useful to specify that the ASR engine not process incoming audio in real-time even though real-time is enabled in the license.
utterance_callback
string
URL
Enables you to specify the URL of a callback server to which each utterance in a transcription result will be POSTed as it is transcribed. Using this option is mandatory for real-time speech processing. As used in the ASR engine, a callback is the address and (optionally) method name and parameters of a web application that can receive data via HTTP or HTTPS. In the ASR engine, callbacks are usually used to enable another application to receive and directly interact with the transcripts produced by the ASR engine.
uttmaxgap
string
integer
Specifies the maximum gap in seconds that can occur between utterances before they are combined. During text processing, each utterance is buffered for a maximum of uttmaxgap seconds for possible combination with a subsequent utterance before being released for subsequent processing.
### Tip
During real-time speech processing, uttmaxgap must be set to 0.
uttmaxsilence
string
integer
default is 800 ms
Specifies the maximum amount of silence in milliseconds that can occur between speech sounds without terminating the current utterance. Once a silence occurs that exceeds uttmaxsilence milliseconds, an utterance “cut” is made within the detected silent region.
uttmaxtime
string
integer
default is 80 seconds
Specifies the maximum amount of time in seconds that is allotted for a spoken utterance. Normally an utterance is terminated by a sufficient duration of silence, but if no such period of silence is encountered prior to reaching uttmaxtime, the utterance is terminated forcibly.
uttminactivity
string
integer
default is 500 ms
Specifies how much activity is needed (without uttpadding) to classify as an utterance. This is usually lower if activitylevel/uttpadding are high and vice-versa.
integer
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2020-04-07 18:12:25
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https://www.anl.gov/in-the-news-0?content-type=All&sort_by=created&sort_order=DESC&page=2
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# In the News
## Filter Results
• ### New Effort to Link Plant Genes to Functions in Bioenergy Crops
Quantitative Plant Science Initiative led by Brookhaven Lab aims to decode functions of genes and their impacts on productivity to guide breeding, engineering of sustainable bioenergy crops.
• ### Random Boarding May Help Airlines Reduce Covid-19 Risks
Preliminary research based on computer simulations suggests that random boarding of aircraft, rather than back-to-front boarding may have an even greater impact, reducing exposure rates by about 50 percent.
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2020-07-15 03:06:49
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http://math.stackexchange.com/questions/315868/when-is-a-map-essential-in-%c4%8cech-cohomology
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When is a map essential in Čech cohomology?
I read a nice survey of parts of game theory, Foundations of Strategic Equilibrium, by Hillas and Kohlberg. Something where I stumble is the discussion of Mertens stability. There is a definition that requires a certain map to be essential in Čech cohomology and I know nothing about cohomology. So I would like to know:
Is there a self-contained way to define essentiality of a map in Čech cohomology that can be explained to someone who knows point-set topology quite well but knows almost nothing about algebraic topology? If yes, please give the definition and maybe a bit of explanation..
-
What means "to be essential in Čech cohomology"? – Boris Novikov Mar 5 '13 at 7:45
@BorisNovikov That is the question. The whole thing appears at the end of page 45 and beginning of page 46 in the linked text. – Michael Greinecker Mar 5 '13 at 15:27
Thank you, I try to understand. – Boris Novikov Mar 5 '13 at 16:17
Hillas has an excellent working paper where you can get a nice intuition of what such an essentiality condition implies. The title is "A Game Illustrating Some Features of the Definition of Strategic Stability".
-
I found the paper where a correct definition of essentiality is given:
Srihari Govindan and Jean-Francois Mertens (1993): \An Equivalent Def- inition of Stable Equilibria"
If you want I will send it by e-mail. As to Cech cohomology, there is the classical book
Spanier, Ediwin H (1966) Algebraic Topology. McGraw Hill, New York.
However it is very thick. I can recommend, e.g.,
Morgan J. W., Lamberson P. J. Algebraic Topology
(I can send it also; though I didn't read it).
-
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2016-07-24 09:07:43
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https://zbmath.org/?q=an:0827.53018
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# zbMATH — the first resource for mathematics
On $$HB$$-recurrent hyperbolic Kaehlerian spaces. (English) Zbl 0827.53018
By a hyperbolic Kählerian space the author means a triple $$(M, F, g)$$, where $$M$$ is a $$2m$$-dimensional differentiable manifold, $$F$$ is a (1,1)- tensor field and $$g$$ is a pseudo-Riemannian metric on $$M$$ such that $$\nabla F = 0$$, $$F^2 = \text{Id}$$ and $$g(FX,Y) = -g(X,FY)$$ for any vector fields $$X$$, $$Y$$ on $$M$$. Such manifolds are also called to be para- Kählerian [cf. P. Libermann, Ann. Mat. Pura Appl., IV. Ser. 36, 27-120 (1954; Zbl 0056.154)]. In her previous paper, the author defined the so-called $$HB$$-tensor on a hyperbolic Kählerian space, which is an analogy of the famous Bochner curvature tensor of Kählerian manifolds. In the present paper, certain results on hyperbolic Kählerian spaces with recurrent $$HB$$-tensor $$(\nabla HB = \kappa \otimes HB)$$ and recurrent Ricci tensor $$(\nabla \text{Ric} = \kappa^* \otimes \text{Ric})$$ are proved. For instance, it is shown that if $$\dim M > 4$$, then both the recurrence forms $$\kappa$$, $$\kappa^*$$ are closed. Moreover, such assumptions can always be reduced to certain weaker conditions.
##### MSC:
53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text:
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2021-12-03 04:12:58
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https://leanprover-community.github.io/mathlib4_docs/Lean/Meta/DecLevel.html
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# Documentation
Lean.Meta.DecLevel
• If true, then decAux? ?m returns a fresh metavariable ?n s.t. ?m := ?n+1.
canAssignMVars : Bool
Instances For
Equations
• One or more equations did not get rendered due to their size.
Equations
• One or more equations did not get rendered due to their size.
This method is useful for inferring universe level parameters for function that take arguments such as {α : Type u}. Recall that Type u is Sort (u+1) in Lean. Thus, given α, we must infer its universe level, and then decrement 1 to obtain u.
Equations
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2023-03-24 13:39:16
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https://brilliant.org/problems/an-interesting-problem-227/
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# A number theory problem by Daniel Chiu
Number Theory Level 3
Let $$x>0$$ be the answer to this question. If $$k\neq 1$$ is a nonnegative integer, find $\dfrac{x!}{x-k}$
×
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2016-10-28 10:34:50
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http://mikemouse.ru/louisvillesexdating/1559/radiocarbon-dating-differential-equation.html
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# Radiocarbon dating differential equation Sex hookup in mumbai
This in turn corresponds to a difference in age of closure in the early solar system. The 26Al — 26Mg chronometer gives an estimate of the time period for formation of primitive meteorites of only a few million years 1.
However, the principle of carbon-14 differemtial applies to other isotopes as well.
The equation $$d P/dt = k P$$ can also be used to model phenomena such as radioactive decay and compound interest—topics which we will explore later.
To summarize, we say that the expression $$x(t) = Ce^$$ is a ¶Not all populations grow exponentially; otherwise, a bacteria culture in a petri dish would grow unbounded and soon be much larger than the size of the laboratory.
We can construct a differential equation that models our oscillating mass.
First, we must consider the restorative force on the spring.
Stimulating these mineral grains using either light or infrared stimulated luminescence dating or heat causes a luminescence signal to be emitted as the stored unstable electron energy is released, the intensity of which varies depending on the amount of radiation absorbed during burial and specific properties of the mineral.
It is not too difficult to see that $$P(t) = Ce^$$ is a In addition, if we know the value of $$P(t)\text$$ say when $$t = 0\text$$ we can also determine the value of $$C\text$$ For example, if the population at the time $$t = 0$$ is $$P(0) = P_0\text$$ then Of course, it is important to realize that this is only a model. Zircon has a very high closure temperature, is resistant to mechanical weathering and is very chemically inert. Using geochemical data: evaluation, presentation, interpretation. After 5,730 years, the amount of carbon 14 left in the body is half of the original amount. It is not affected by external factors such as, chemical environment, or presence of a or. Differeential an igneous or metamorphic rock or melt, which is slowly cooling, does not begin to exhibit measurable radioactive decay until it cools below the closure temperature. carbon dating using differential equation The proportion of carbon-14 left when the remains of the organism are diferential provides an indication of the time elapsed since its death.
The growth rate of a population need not be positive.
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2019-01-22 18:30:32
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http://www.physicsforums.com/showthread.php?s=17c79089032eebba3da4ddb9816bf0a0&p=3818760
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# 3D Equation
by Philosophaie
Tags: equation
P: 365 Given the Keplerian Components of: a,e,i,LP,ML,MA,EA,N,TA,w,R. How do you find the equation for a 3D Ecliptical ELLIPSE in this form: A1 * x ^ 2 + A2 * y ^ 2 + A3 * z ^ 2 + 2 * B1 * x * y + 2 * B2 * x * z + 2 * B3 * y * z + C = 0 A1= A2= A3=0 B1= B2= B3= C=
Related Discussions General Math 1 Introductory Physics Homework 0 Precalculus Mathematics Homework 4 Introductory Physics Homework 2 General Physics 1
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2014-04-23 12:21:29
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https://codeforces.com/blog/entry/83075
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### grhkm's blog
By grhkm, history, 4 weeks ago,
Thank you
Hello!
We're going to learn how to find inverses mod p today (efficiently).
Pre-req: Know how to find inverses
## Main results:
As we know, finding the inverse of n numbers is $O(n\log{p})$. That is too slow, especially when time limit is tight. Therefore, we want a faster way. I present:
• Find inverse of all numbers between 1 and n in $O(n)$
• Find inverse of n numbers in $O(n + \log{p})$
## Idea 1:
Notice that $p = i \cdot \lfloor \frac{p}{i} \rfloor + (p \% i)$. (If you don't know why, compare it to $13=3\cdot 4+1$)
$p = i \cdot \lfloor \frac{p}{i} \rfloor + (p \% i)$
$\implies i \cdot \lfloor \frac{p}{i} \rfloor + (p \% i) \equiv 0 \mod p$
$p \% i \equiv -i \cdot \lfloor \frac{p}{i} \rfloor \mod p$
Now NOTICE that $p \% i$ is actually less than $i$. (Big brain mode)
So if we've calculated the inverses of $1-(i-1)$ already, we can find inverse of $(p \% i)$. This is what we're going to use.
$1 \equiv i \cdot (-\lfloor\frac{p}{i}\rfloor \cdot (p\% i)^{-1}) \mod p$
And therefore, by definition of inverse:
$i^{-1} \equiv (-\lfloor\frac{p}{i}\rfloor \cdot (p\% i)^{-1}) \mod p$
Or to make it easier to read: (Integer division)
$inv[i] \equiv (-(p/i) \cdot inv[p\% i]) \mod p$
So we can calculate inverse of $1~n$ in $O(n)$.
Code: (Be aware of overflow) (Also, -(p/i) mod p = (p-p/i) mod p)
int n = 10, p = 1000000007;
int inv[n + 1];
inv[1] = 1;
for (int i = 2; i <= n; i ++) inv[i] = 1LL * (p - p / i) * inv[p % i] % p;
## Idea 2:
Our target is $O(n+\log p)$, so we shall only take inverse once. How?
Let's look at what we want to find:
$\frac{1}{a_1}, \frac{1}{a_2}, \cdots, \frac{1}{a_n} \mod p$
Let's try to make them have the same denominator!
$\frac{a_2a_3\cdots a_n}{a_1a_2\cdots a_n}, \frac{a_1a_3a_4\cdots a_n}{a_1a_2\cdots a_n}, \cdots, \frac{a_1a_2\cdots a_{n-1}}{a_1a_2\cdots a_n} \mod p$
As we can see, the numerator is a prefix times a suffix, and the denominator is constant. Therefore, we can precompute those and then take inverse of denominator once.
Code:
#include "bits/stdc++.h"
using namespace std;
const long long P = 1000000007;
long long qpow(long long a, long long b) {
long long ans = 1;
while (b) {
if (b & 1) ans = ans * a % P;
a = a * a % P;
b >>= 1;
}
return ans;
}
long long n, a[1000010], pre[1000010], suf[1000010], pr = 1; // prefix, suffix, product
int main() {
cin >> n; pre[0] = suf[n + 1] = 1;
for (int i = 1; i <= n; i ++) cin >> a[i];
for (int i = 1; i <= n; i ++) {
pre[i] = pre[i - 1] * a[i] % P;
suf[n + 1 - i] = suf[n + 2 - i] * a[n + 1 - i] % P;
pr = pr * a[i] % P;
}
pr = qpow(pr, P - 2);
for (int i = 1; i <= n; i ++) {
cout << (pre[i - 1] * suf[i + 1] % P) * pr % P << endl;
}
}
Thank you for reading. If you have any suggestions or any topic you want to learn about I guess I can try writing! If this is helpful vote pls thx
Spoiler
• +139
» 4 weeks ago, # | +5 push
» 4 weeks ago, # | -15 how to find inverses >_<
• » » 4 weeks ago, # ^ | +2 Bruteforce from 1 to p-1 and check if n*i equiv 1 mod pTo do multiplication it's faster to do repeated addition instead. Good luck. Please do not downvote, he's troll as seen from contribution ptsLol why did I write in summary
• » » » 4 weeks ago, # ^ | ← Rev. 3 → 0 2nd past wasn't expected :/
• » » 4 weeks ago, # ^ | +14 Fermat's little theorem (or ext. Euclidean algorithm).
• » » » 4 weeks ago, # ^ | +37 I almost thought you were a different person lol
• » » » 4 weeks ago, # ^ | +17 Isn't this fft? SpoilerCreate a polynomial $A(x)=c$, then find it's inverse using FFT?
» 4 weeks ago, # | +32 Hello! We are going to learn how to compute inverses mod p. Pre-req: know how to compute inverses mod p
• » » 4 weeks ago, # ^ | +8 y e sI'll add "efficiently" then :)
» 4 weeks ago, # | +10 By the way, can someone prove the time complexity of this code? int inv(int i) { if(i == 1) return 1; return M - (long long) M / i * inv(M % i) % M; }
• » » 4 weeks ago, # ^ | +18 I can't think of a simple proof right now, but using this code, it seems like it runs very fast. const int p = 1e9 + 7; // 998244353 int arr[p + 100], n, m; int main() { arr[0] = arr[1] = 1; for (int i = 2; i < p; i ++) arr[i] = arr[p % i] + 1, m = max(m, arr[i]); cout << m << endl; }
» 4 weeks ago, # | ← Rev. 2 → -9 i read your previous and present blog it's really helpful on me...thanks grhkm
• » » 4 weeks ago, # ^ | 0 Glad that it helps <3
» 4 weeks ago, # | +9 I use the fact that, $inv[i] = fac[i-1] * invfac[i]$
• » » 4 weeks ago, # ^ | -10 lol why you got downvotedYeah the method/optimization can be used to precalculate binomial coefficients under n in O(n) :D
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2020-10-22 23:38:54
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http://math.stackexchange.com/questions/55129/computer-animation-and-euler-integration
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# Computer Animation and Euler-Integration
What is integrated with the Euler integration?
IMHO: integration to obtain the velocity at time t to a place. Acceleration a is constant over time interval t. Right?
S(t0 -> delta_t) = S0(t) + ∫ s'(t) dt
s'(t) is derivated S
Please see Tim van Beek Answer. It's the right one :)
-
ah damn: i forgot to tell: S(t) is the place in time t – RenHoek Aug 2 '11 at 13:08
We aren't clairvoyant; if you're working with a computer animation that integrates velocity to obtain position, then that's what you're doing, if that's not what you're integrating, then that's not what you're doing - how are we supposed to know without ESP? How are we supposed to know if the acceleration of something you're working with is constant if we don't have access to it and you're not telling us anything about it? Also, you can edit your own question, which is better than adding information in comments. You have to understand: people can only help you if you communicate. – anon Aug 2 '11 at 13:27
@RenHoek: As it stands, it is quite impossible to answer your question. Can you exapnd it to include a description of what you want? – Mariano Suárez-Alvarez Aug 2 '11 at 14:12
@anon that was what i had :( i can't give mire information then i have ... but Tim understood my partial Informtaion.... – RenHoek Aug 2 '11 at 17:10
If you are further talking about Newtonian mechanics, you mean that there is a point mass of mass $m$ whose behaviour is described by the equation $$F = m * a$$ Force = mass times accelaration. This is an ordinary differential equation of second order for the position of the point mass as a function of time $t$, $S(t)$, because we have $$S''(t) = a(t)$$ that is the first derivative of $S$ is the velocity $v(t)$, and the second is the acceleration $a(t)$. In order to get a unique solution, we need to specify initial conditions for both $S(t)$, $$S(t =0) = S_0$$ which is the initial position and for $v(t)$, $$v(t = 0) = v_0$$ which is the initial velocity of the point mass. Now you can prescribe the force $F(t)$ as a differentiable function of time and calculate the position $S(t)$ numerically using the Euler method. If you set the force to zero, then you have zero acceleration and the unique solution in closed form is $$S(t) = S_0 + v_0 t$$ In this case the numerical approximation via the Euler method will coincide with the solution obtained in closed form. But in general there will be an approximation error.
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2015-05-23 15:06:52
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https://ch.mathworks.com/help/comm/ref/mfskdemodulatorbaseband.html
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# M-FSK Demodulator Baseband
Demodulate FSK-modulated data
## Library
FM, in Digital Baseband sublibrary of Modulation
## Description
The M-FSK Demodulator Baseband block demodulates a signal that was modulated using the M-ary frequency shift keying method. The input is a baseband representation of the modulated signal. The input and output for this block are discrete-time signals. This block accepts a scalar value or column vector input signal of type `single` or `double`. For information about the data types each block port supports, see Supported Data Types.
The M-ary number parameter, M, is the number of frequencies in the modulated signal. The Frequency separation parameter is the distance, in Hz, between successive frequencies of the modulated signal.
The M-FSK Demodulator Baseband block implements a non-coherent energy detector. To obtain the same BER performance as that of coherent FSK demodulation, use the CPFSK Demodulator Baseband block.
### Integer-Valued Signals and Binary-Valued Signals
When you set the Output type parameter to `Integer`, the block outputs integer values between `0` and M-`1`. M represents the M-ary number block parameter.
When you set the Output type parameter to `Bit`, the block outputs binary-valued signals that represent integers. The block represents each integer using a group of K = log2(M) bits, where K represents the number of bits per symbol. The output vector length must be an integer multiple of K.
The Symbol set ordering parameter indicates how the block maps a symbol to a group of K output bits. When you set the parameter to `Binary`, the block maps the integer, I, to [u(1) u(2) ... u(K)] bits, where the individual u(i) are given by
`$I\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\sum _{i=1}^{K}u\left(i\right){2}^{K-i}$`
u(1) is the most significant bit.
For example, if M = 8, you set Symbol set ordering to `Binary`, and the demodulated integer symbol value is 6, then the binary output word is [1 1 0].
When you set Symbol set ordering to `Gray`, the block assigns binary outputs from points of a predefined Gray-coded signal constellation. The predefined M-ary Gray-coded signal constellation assigns the binary representation
```M = 8; P = [0:M-1]'; de2bi(bitxor(P,floor(P/2)), log2(M),'left-msb') ```
to the `P`th integer.
The typical Binary to Gray mapping for M = 8 is shown in the following tables.
Binary to Gray Mapping for Bits
Binary CodeGray Code
000000
001001
010011
011010
100110
101111
110101
111100
Binary to Gray Mapping for Integers
Binary CodeGray Code
00
11
23
32
46
57
65
74
Whether the output is an integer or a binary representation of an integer, the block maps the highest frequency to the integer 0 and maps the lowest frequency to the integer M-1. In baseband simulation, the lowest frequency is the negative frequency with the largest absolute value.
### Single-Rate Processing
In single-rate processing mode, the input and output signals have the same port sample time. The block implicitly implements the rate change by making a size change at the output when compared to the input. The input width must be an integer multiple of the Samples per symbol parameter value, and the input can be a column vector.
• When you set Output type to `Bit`, the output width is K times the number of input symbols.
• When you set Output type to `Integer`, the output width is the number of input symbols.
### Multirate Processing
In multirate processing mode, the input and output signals have different port sample times. The input must be a scalar. The output symbol time is the product of the input sample time and the Samples per symbol parameter value.
• When you set Output type to `Bit`, the output width equals the number of bits per symbol.
• When you set Output type to `Integer`, the output is a scalar.
To run the M-FSK Demodulator block in multirate mode, clear the Treat each discrete rate as a separate task checkbox (in Simulation > Configuration Parameters > Solver).
## Parameters
M-ary number
The number of frequencies in the modulated signal.
Output type
Determines whether the output consists of integers or groups of bits. If this parameter is set to `Bit`, then the M-ary number parameter must be 2K for some positive integer K.
Symbol set ordering
Determines how the block maps each integer to a group of output bits.
Frequency separation (Hz)
The distance between successive frequencies in the modulated signal.
Samples per symbol
The number of input samples that represent each modulated symbol.
Rate options
Select the rate processing method for the block.
• `Enforce single-rate processing` — When you select this option, the input and output signals have the same port sample times. The block implements the rate change by making a size change at the output when compared to the input. The output width is the number of symbols (which is given by dividing the input length by the Samples per symbol parameter value when the Output type parameter is set to `Integer`).
• `Allow multirate processing` — When you select this option, the input and output signals have different port sample times. The output period is the same as the symbol period and equals the product of the input period and the Samples per symbol parameter value.
For more information, see Single-Rate Processing and Multirate Processing in the Description section of this page.
Output data type
The output type of the block can be specified here as `boolean`, `int8`, `uint8`, `int16`, `uint16`, `int32`, `uint32`, or `double`. By default, the block sets this to `double`.
## Supported Data Types
PortSupported Data Types
Input
• Double-precision floating point
• Single-precision floating point
Output
• Double-precision floating point
• Boolean
• 8-, 16-, and 32-bit signed integers
• 8-, 16-, and 32-bit unsigned integers
## Pair Block
M-FSK Modulator Baseband
## References
[1] Sklar, Bernard. Digital Communications: Fundamentals and Applications. Upper Saddle River, NJ: Prentice-Hall, 2001.
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2021-05-17 15:29:59
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https://networkx.org/documentation/networkx-2.3/reference/algorithms/generated/networkx.algorithms.traversal.breadth_first_search.bfs_successors.html
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Warning
This documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.
bfs_successors(G, source, depth_limit=None)[source]
Returns an iterator of successors in breadth-first-search from source.
Parameters: G (NetworkX graph) source (node) – Specify starting node for breadth-first search and return edges in the component reachable from source. depth_limit (int, optional(default=len(G))) – Specify the maximum search depth succ – (node, successors) iterator where successors is the list of successors of the node. iterator
Examples
>>> G = nx.path_graph(3)
>>> print(dict(nx.bfs_successors(G,0)))
{0: [1], 1: [2]}
>>> H = nx.Graph()
>>> H.add_edges_from([(0, 1), (0, 2), (1, 3), (1, 4), (2, 5), (2, 6)])
>>> print(dict(nx.bfs_successors(H, 0)))
{0: [1, 2], 1: [3, 4], 2: [5, 6]}
>>> G = nx.Graph()
>>> nx.add_path(G, [0, 1, 2, 3, 4, 5, 6])
>>> nx.add_path(G, [2, 7, 8, 9, 10])
>>> print(dict(nx.bfs_successors(G, source=1, depth_limit=3)))
{1: [0, 2], 2: [3, 7], 3: [4], 7: [8]}
Notes
Based on http://www.ics.uci.edu/~eppstein/PADS/BFS.py by D. Eppstein, July 2004.The modifications to allow depth limits based on the Wikipedia article “Depth-limited-search”.
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2022-08-18 11:46:17
|
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https://zbmath.org/?q=an%3A0057.33801
|
## On locally convex vector spaces of continuous functions.(English)Zbl 0057.33801
### Keywords:
Functional analysis
Full Text:
### References:
[1] N. Bourbaki: Sur certains spaces vectoriels topologiques, Ann. Inst. Fourier II (1950). · Zbl 0042.35302 [2] J. A. Dieudonne: Recent developments in the theory of locally-convex spaces, Bull. Amer. Math. Soc, 59 (1953). · Zbl 0053.25701 [3] E. Hewitt: Rings of real valued continuous functions, Trans. Amer. Math. Soc, 64 (1948). JSTOR: · Zbl 0032.28603 [4] T. Shirota: A class of topological spaces, Osaka Math. J., 4 (1952). · Zbl 0047.41704 [5] L. Gillman and H. Henriksen: Concerning rings of contmuous functions, Trans. Amer. Math. Soc. (to appear). · Zbl 0058.10003 [6] G. Mackey: On infinite-dimensional linear spaces, Trans. Amer. Math. Soc, 57 (1945). JSTOR: · Zbl 0061.24301 [7] W. F. Donoghue and K. T. Smith: On the symmetry and bounded closure of locally convex spaces, Trans. Amer. Math. Soc, 73 (1952). JSTOR: · Zbl 0047.10601 [8] E. Hewitt: Linear functionals on spaces of continuous functions, Fund. Math., 37 (1950). · Zbl 0040.06401 [9] L. Nachbin: On the continuity of positive linear transformations, proceeding of the international congress of mathematicians (1950). · Zbl 0035.35402 [10] Cf. R. Sikorski: Remark on some topological spaces of high power, Fund. Math., 37 (1950). · Zbl 0041.09705
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2022-06-26 23:26:17
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https://repository.upenn.edu/dissertations/AAI9532198/
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# Catalytic and stoichiometric reaction chemistry of metal silicon complexes
#### Abstract
Transition metal silicon complexes are central in a variety of important catalytic transformations of organosilicon compounds. Many of these processes involve unsaturated silicon complexes as key intermediates. This thesis explores two aspects of transition metal silicon chemistry: the stoichiometric reactions of a stable disilene complex of tungsten Cp${\rm \sb2W(\eta\sp2-Me\sb2Si=SiMe\sb2})$, and the development and elucidation of new catalytic routes to polysilanes. The tungsten disilene complex has been found to react with a variety of polar reagents to yield products resulting from one-atom insertion into the Si-Si bond. Thus, reactions with chalcogenide sources yields the four-membered ring compounds cyclo-Cp${\rm \sb2W(SiMe\sb2ESiMe\sb2)(E=O, S, Se, Te}).$ One-atom net insertion into the Si-Si bond also occurs in the reaction of tungsten disilene complex with diazoalkanes yielding hydrazone-type insertion products. Further studies show the trimethylsilyldiazomethane insertion product undergoes photochemical rearrangement and extrusion of HCN to yield the corresponding amine insertion product. In contrast, the reaction of ethylene, a nonpolar reagent, leads to insertion into the W-Si bond and formation of cyclo-Cp${\rm \sb2W(SiMe\sb2SiMe\sb2CH\sb2CH\sb2}).$ Although the disilene complex is unreactive towards hydrogen gas, the hydrogenation of the Si-Si bond is catalyzed by a variety of platinum complexes. Evidence is presented for the formation of bimetallic adducts with the catalytically active complexes. The molecular structure of the adduct with Wilkinson's catalyst has been determined by crystallographic methods. The catalytic studies explored in this thesis include the ruthenium-catalyzed transfer dehydrocoupling of a number of secondary silanes in the presence of an olefin as a hydrogen acceptor. In the dehydrocoupling of Me${\rm \sb2SiH\sb2},$ moderate weight polysilanes (n = 15 $-$ 20) are produced, whereas dehydrocoupling of bulkier secondary silanes yields only small oligomers. Alkyl group redistribution between silicons is competitive with dehydrocoupling, leading to a polysilane which is highly branched. Further studies reveal that the dehydrocoupling of primary silanes is catalyzed by (Me${\rm \sb3P)\sb3RU(H)\sb3(SiMe}\sb3).$ Without the presence of a hydrogen acceptor. The differences between primary and secondary silicons can be traced to the steric congestion associated with the key catalytic intermediate, (${\rm Me\sb3P)\sb3RU(H)\sb2(SiR\sb3)\sb2}.$
#### Subject Area
Chemistry|Organic chemistry
#### Recommended Citation
Hong, Paula, "Catalytic and stoichiometric reaction chemistry of metal silicon complexes" (1995). Dissertations available from ProQuest. AAI9532198.
https://repository.upenn.edu/dissertations/AAI9532198
COinS
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2023-03-22 00:04:06
|
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https://www.physicsforums.com/threads/adding-expectation-values-to-a-chsh-animation.854335/
|
# Adding expectation values to a CHSH animation
1. Jan 27, 2016
### edguy99
An animation of the CHSH experiment to generate correlated photons is at: http://www.animatedphysics.com/games/photon_longdistance_chsh.htm
@georgir has a program to show the calculations using the formula for photon detection
return Math.random() < (Math.cos(r(p-a)*2)+1)/2;
yields the following for the coincident (same as the animation):
classic:
{
"0-22.5": {
"coincident": 1675,
"total": 2501,
"rate": 0.6697321071571372
},
"45-67.5": {
"coincident": 1671,
"total": 2468,
"rate": 0.6770664505672609
},
"45-22.5": {
"coincident": 1709,
"total": 2579,
"rate": 0.6626599457153936
},
"0-67.5": {
"coincident": 820,
"total": 2452,
"rate": 0.33442088091353994
}
}
I am looking for help in calculating the expectation values for the above data. I am interested in adding the expectation values into my animation. A brief explanation on how to do it I found at http://www.gutenberg.us/articles/CHSH_Bell_test:
Any help or checking would be appreciated.
The usual form of the CHSH inequality is:
(1) − 2 ≤ S ≤ 2,
where
(2) S = E(a, b) − E(a, b′) + E(a′, b) + E(ab′).
a and a′ are detector settings on side A, b and b′ on side B, the four combinations being tested in separate subexperiments. The terms E(a, b) etc. are the quantum correlations of the particle pairs, where the quantum correlation is defined to be the expectation value of the product of the "outcomes" of the experiment, i.e. the statistical average of A(aB(b), where A and B are the separate outcomes, using the coding +1 for the '+' channel and −1 for the '−' channel.
2. Jan 28, 2016
### Strilanc
Math.random returns a chosen uniformly random value between 0 and 1. The expression Math.random() < p will evaluate to true with probability p. So the coincidence rate for that particular expression is exactly (Math.cos(r(p-a)*2)+1)/2.
You may be interested in viewing an interactive widget I made to simulate the CHSH test as part of a blog post, to get ideas. You can view it directly on jsfiddle, or in the blog post (with explanation of how to use it). Also you can edit it on jsfiddle. Here's a screenshot:
3. Jan 29, 2016
### edguy99
Nice animation, nice to be able to insert javascript. Never used fiddle, but was able to copy to what I am used to using and am able to work with it. Two quick questions:
1/ Your "photons" are shot with 2 independent random polarizations, rather then being split from a common photon and both photons having the same Jones vector?
2/ I could not find the turn(x,90) or the measure() functions, maybe I am looking in the wrong file?
Thanks, I hope to have fun and let you know.
4. Jan 29, 2016
### Strilanc
It would be more accurate to say that my simulation has two polarizations than to say it has whole photons. Polarization is a 2-level quantum system, a qubit, and the post is framed in terms of qubits. The qubits are entangled to have agreeing value and phase, which corresponds roughly to saying two polarizations will agree both horizontally and diagonally (whichever one you check first).
The measure function is defined at line 217 of src/Engine/ChSh.js; it's just a simple wrapper around code in src/Engine/Superposition.js.
They're written inside strings instead of directly in the code because the widget uses web workers to avoid blocking the browser while running simulations. There's also some string interpolation happening to make escaping the sandbox a bit harder. Also it's not using any math libraries or anything, just direct twiddling of numbers inside a buffer. And the code on jsfiddle has been transpiled by traceur into ES5. So I'm not too surprised you had trouble finding the measure function.
5. Jan 29, 2016
### edguy99
Thanks, I will have a look.
WRT to polarizations, perhaps I am miss-reading, but you assign var refChoice = Math.random() < 0.5; to Alice, then a second var refChoice = Math.random() < 0.5; to Bob, I dont see how these are entangled? Here's where I would assign the same Jones vector to both.
6. Jan 29, 2016
### Strilanc
The entanglement is in the fake sharedQubits variable, which is only worked with indirectly via measure and turn. In the code the entangled state is stored as a list of 8 floats representing 4 complex amplitudes, and Alice and Bob's operations fiddle with the list in the way described by the postulates of QM.
The 'refChoice' variable is the referee's choice, not the entangled qubit. Each player gets an independent random referee bit. The CHSH game is for two isolated players to pick bits satisfying $a \oplus b = x \land y$, where $a, b$ are the players' respective chosen bits and $x, y$ are random bits (chosen by referee to prevent cheating) fed respectively to each player once they're isolated (and ideally at the last possible moment, to close the signalling loophole).
7. Jan 30, 2016
### edguy99
I like the wiki quote at https://en.wikipedia.org/wiki/Qubit:
A qubit is a two-state quantum-mechanical system, such as the polarization of a single photon: here the two states are vertical polarization and horizontal polarization.
8. Jan 31, 2016
### georgir
A bit confusing, but this formula is only used for deciding if a single photon passes a certain polarizer. So it does not reflect coincidence rates between two polarizers in any way in general.
Certainly not in the 'classical' simulation. But it does match coincidence rates in the special case of the "qm" simulation where the photon gets changed by the first measurement.
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2017-08-24 00:01:09
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https://www.mersenneforum.org/showthread.php?s=e9d0a8a1f9647f299d9266d29693996f&t=19337
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mersenneforum.org Figuring Out Sequence Merges
Register FAQ Search Today's Posts Mark Forums Read
2014-05-07, 02:49 #1 Jayder Dec 2012 32·31 Posts Figuring Out Sequence Merges I've searched hard for information, but I can't figure out how you can discover if a sequence has merged or not, and if so what sequence it has merged with. I ask because it seems that somebody, within the last day, has started working on the sequence I have been working on (link). I suspect a merge has occurred, but despite my noobish efforts I am unable to find out more. I have not reported anything new to factordb since I first noticed the possible merge, and yet it has progressed by more than 100 iterations. This has to be a merge, right? Thank you for any help you guys can provide. Last fiddled with by Jayder on 2014-05-07 at 02:49
2014-05-07, 04:22 #2 LaurV Romulan Interpreter "name field" Jun 2011 Thailand 2×17×293 Posts There is no merge, you got a downdriver at term~2200 and went under 110 digits, therefore the elves or some dd-hunter took it. Right now, with D3 (2^3*3*5) since term ~2500, they left it (one of the most freaking drivers, due to the fact that is difficult to get rid of it, and the increasing of the sequence is faster than most other drivers).
2014-05-07, 09:04 #3
kar_bon
Mar 2006
Germany
23·32·41 Posts
Quote:
Originally Posted by Jayder I've searched hard for information, but I can't figure out how you can discover if a sequence has merged or not, and if so what sequence it has merged with.
What I do:
- download all last lines of the seqs upto yours from FactorDB
- compare them with yours
or
- if your seq falls down under 80 digits or more
- go to this site
- download C9C30, C60 or C80 (see section 'Databases', not up to date)
2014-05-09, 04:00 #4
Jayder
Dec 2012
1000101112 Posts
Quote:
Originally Posted by LaurV There is no merge, you got a downdriver at term~2200 and went under 110 digits, therefore the elves or some dd-hunter took it. Right now, with D3 (2^3*3*5) since term ~2500, they left it (one of the most freaking drivers, due to the fact that is difficult to get rid of it, and the increasing of the sequence is faster than most other drivers).
I had discounted random workers (elves?) because several large-ish numbers had been factored (100+ digits) which the database won't do and for the elves to hit as many 100+ers in my sequence as they did would seem very rare. I hadn't considered a poacher, but that makes sense. It's saddening that somebody would do that if it was done intentionally.
Thank you both for your help.
2014-05-14, 04:45 #5 EdH "Ed Hall" Dec 2009 Adirondack Mtns 2·3·751 Posts OK, I guess I need some more education... Why would you only search for merges less than your number? In the case of 154548, I find it as the smallest in a family that includes over 50 sequences below 1M: 156018, 156030, 165210, 179886, 231366, 231378, 236370, 237846, 241206, 244146, 258606, 272514, 307614, 310218, 313314, 313998, 316182, 316194, 322194, 322206. 325362, 330978, 330990, 338128, 378126, 394848, 410832, 423870, 464308, 498836, 516228, 558498, 558510, 602130, 612930, 688332, 697026, 698300, 704814, 759312, 762546, 781986, 811362, 817228, 822594, 843054, 858174, 867666, 867678, 870148, 886276, 945328, 997666 154548 and all the above show their last lines equal in the db. Couldn't any one of the above numbers have been worked by someone, who did not realize they were also working 154548?
2014-05-14, 07:35 #6
kar_bon
Mar 2006
Germany
23×32×41 Posts
Quote:
Originally Posted by EdH Couldn't any one of the above numbers have been worked by someone, who did not realize they were also working 154548?
Anyone from this forum?
I think not, because all open (smallest) sequences are known, so he/she will take only those.
Anyone from 'outside'?
If so, you'll notice that the smallest seq (here 154548) will grow, too, and there's nothing against doing so. If he/she will terminate say 156018 (because he/she is running this instead 154548) the only thing which is noticed for history is the termination of 154548 for exmple on W.Crayaufmüller's page and perhaps the name (if known) of the terminator (no, please no reply on this, it's not Arnie).
2014-05-14, 07:48 #7
LaurV
Romulan Interpreter
"name field"
Jun 2011
Thailand
996210 Posts
Quote:
Originally Posted by EdH Why would you only search for merges less than your number?
You don't. We check for merges in the same way you did, looking at the last number in the sequence. Historically, because they were a million and no data base available (with all the factors) some lists were made, which record for example "the first 10 digit number" reached by a sequence. Most of the million sequences do not reach a 10 digits term, but if your (new) sequence do, you look in the list if it is a merge. Very easy to check. Then, a 30-digits list was done, or a 50 digits list. This way one don't need to search all the factor data base (in fact, we don't have such a database, FDB is bloody to search). The advantage of a list is the fact that such list is fixed, once the remaining sequences pass over the threshold, and do not need to be maintained, except rarely (if a sequence goes under 50 digits, for example, and came back over it, a new term will be added to the list). There was a list of "first reach 100 digits" too, but that is not used so much anymore, because at the time when the "remaining sequences" reached 100 digits, very few survived, and a list with last term is easy to get from the DB now. The problem is that the DB is not very well maintained, Sid is always busy when you need him... People report the results seldom, they don't report downdrivers till the downrun end, because they are afraid of ddhunters, etc.
Dubslow's site was a good resource, but is down since some time. Therefore I keep my own lists. I would like to host/mirror Dubslow's page, for example, it will not be a big deal of trafic, it may not be available 24/7, but most of the time will be available, and any positive number is better than zero... but I can't contact him. Also, I am sure other people here could host that page too, better than me...
Quote:
Couldn't any one of the above numbers have been worked by someone, who did not realize they were also working 154548?
Only if he didn't know what he was doing. The sequences in your list, once reported as mergers (all were known), they are eliminated from the reservation list. For example, I start working 279936, because I liked it (it is 6^7, perfect number at odd power, blah blah) which, after a lot of adventures, merged with other sequences, the smaller of them being 95280. I kept the reservation for 95280, but the other were erased from the list. They are not available for reservation anymore. For the same reason you can not reserve for example sequence 396, neither is this listed as one of Lehmer fives, because is a merger with a lower sequence (276) and therefore 396 does not appears in the list anymore. Of course, someone can work the sequence 396, but only if he doesn't know what he is doing. Or if he poaches.
Last fiddled with by LaurV on 2014-05-14 at 07:51
2014-05-14, 13:34 #8
EdH
"Ed Hall"
Dec 2009
2×3×751 Posts
@LaurV
Quote:
Originally Posted by kar_bon What I do: - download all last lines of the seqs upto yours from FactorDB ...
@all:
Thanks for the replies, but I must be a bit dense. Did Dubslow's page take merges into account and only show the primary number? Other than checking for merges, via the db or various other data, how would I know not to reserve a higher sequence? Would I be "scolded" if I reserved a higher sequence without noticing/checking for a merge? Does any of this mean that I should always check for merges as well as reservations prior to reserving?
I use endings for my last line list, which I believe should be up-to-date and easier on the db than separate calls. Is endings.zip also up-to-date, or is it an earlier run? And, are the holes in these lists (like, 58501-58600) there for some particular reason?
Thanks for all the replies...
2014-05-14, 14:30 #9
LaurV
Romulan Interpreter
"name field"
Jun 2011
Thailand
2×17×293 Posts
Quote:
Originally Posted by EdH Did Dubslow's page take merges into account and only show the primary number? [yes, after manual confirmation. before manual confirmation, the sequences were still shown, but instead of the terms/drivers it said "merged with blah blah"; after manual confirmation, the higher sequence is gone.] Other than checking for merges, via the db or various other data, how would I know not to reserve a higher sequence? [you don't, beside of keeping your own list]. Would I be "scolded" if I reserved a higher sequence without noticing/checking for a merge? [no, some one like Frank would be delighted if you progress any sequence , there is plenty of work for everyone. Someone else will tell you about the merge, if you report your reservation/progress, and if the head sequence is reserved already, you will have to negotiate with the owner who is keeping it. Some guy may be nervous about "hey, let my sequence alone", but that's life. If the guy is me, I will thank you, and let you have it, hehe] Does any of this mean that I should always check for merges as well as reservations prior to reserving? [better check!]
My comments in square brackets, I am left handed in handling multiple quotes right now. :D
[edit: if you reserve the sequence here, someone will tell you about the merge, for sure. Few people here keep a very strict evidence, only they don't read the forum everyday like us, they have more important things to do, but someone will notify you for sure]
[edit2: about that "last lines", you must be really lucky to run in a merge with a last line of a sequence - never happened. Otherwise you have to report every factorization to the DB, and check if the DB has a "progress" of your sequence of more than expected lines. We usually don't work like that, it is very time-consuming, we do many terms between reports, and only check for merges when the sequence runs down (decreases) many terms, or when we see jumps in the DB which are not explained by our factoring power. Connected to this is the fact that you don't really need the "endings" to be very updated. Your chance to have a merge to "last term" is null]
Last fiddled with by LaurV on 2014-05-14 at 14:44
2014-05-15, 10:49 #10
schickel
"Frank <^>"
Dec 2004
CDP Janesville
1000010010102 Posts
Quote:
Originally Posted by EdH I use endings for my last line list, which I believe should be up-to-date and easier on the db than separate calls. Is endings.zip also up-to-date, or is it an earlier run? And, are the holes in these lists (like, 58501-58600) there for some particular reason? Thanks for all the replies...
Probably not. For 7044 it shows this:
Code:
7044: 1824550790021816183454859929959773536447055697508964211261232126949397733460093201749101092576020395754873344507860668024966763208122497731062434064 @n=1397
It is actually:
Code:
7044 3433. sz 180 2^9 * 3 * 11 * 31 * 3673
2014-05-15, 13:56 #11
EdH
"Ed Hall"
Dec 2009
2·3·751 Posts
Quote:
Originally Posted by schickel Probably not. For 7044 it shows this: Code: 7044: 1824550790021816183454859929959773536447055697508964211261232126949397733460093201749101092576020395754873344507860668024966763208122497731062434064 @n=1397 It is actually: Code: 7044 3433. sz 180 2^9 * 3 * 11 * 31 * 3673
So, both endings and endings.zip (which seem to be the same) are rather outdated...
I wonder if they are just a snapshot from when Syd created them, rather than a function that runs when called...
Similar Threads Thread Thread Starter Forum Replies Last Post schickel Aliquot Sequences 1031 2022-04-17 12:27 kar_bon Aliquot Sequences 136 2021-10-21 16:17 sweety439 And now for something completely different 17 2017-06-13 03:49 EdH Aliquot Sequences 4 2010-04-13 19:43 ADBjester PrimeNet 33 2003-02-27 08:28
All times are UTC. The time now is 03:58.
Sat May 21 03:58:44 UTC 2022 up 37 days, 2 hrs, 0 users, load averages: 1.92, 1.73, 1.55
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2022-05-21 03:58:44
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https://www.gradesaver.com/textbooks/math/algebra/college-algebra-10th-edition/chapter-10-section-10-2-permutations-and-combinations-10-2-assess-your-understanding-page-695/19
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## College Algebra (10th Edition)
$C(15,15)=1$
Using the combination formula, we obtain: $C(n,r)=\frac{n!}{(n-r)!r!}$ $C(15,15)=\frac{15!}{(15-15)!15!}=\frac{15!}{0!15!}=\frac{15!}{15!\cdot1}=1$
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2018-07-16 22:34:56
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https://chemistry.stackexchange.com/questions/44505/what-is-the-oxidation-state-of-carbon-atoms-in-ethanoic-acid
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# What is the oxidation state of carbon atoms in Ethanoic acid?
The oxidation states of $sp^2$ and $sp^3$ carbon in ethanoic acid are $-3$ if we do not consider the electronegativity difference of these two carbons due to their different hybridisation but this is not the case with the thiosulphuric acid ($\ce{H2S2O3}$); here the oxidation states of sulphur here are +6 and -2. So why are we considering the electronegativity difference of two $sp^3$ and $sp^2$ sulphurs here?
• Welcome to Chemistry! This seems like a homework question. We have a policy which states that you should show your thoughts and/or efforts into solving the problem. It'll make us certain that we aren't doing your homework for you. Otherwise, this question may get closed. – bon Jan 31 '16 at 14:08
## 2 Answers
Typically when figuring oxidation state of an atom, electronegativity of different atoms of same element is considered equal. This means that in thiosulphate oxidation state of central sulfur would be somewhere around 4.
However! Let us condider a very tricky case: ozone. Technically it consists of atoms of same element and the oxidation state of oxygen atoms should be zero. But ouch, auch, the central atom bears full positive chanrge in all feasible structures.... So, should ozone, disregarding electron shifts in covalent bonds, have oxidation state of atoms +1 (for central atom) and -1/2 (for two other atoms)? But this doesn't make much sense. This is a place where oxidation state abstraction breaks.
So, there are situations when oxidation state abstraction doesn't make much sence.
Thiosulphate is somewhate similar. The central sulfur behaves very muc like sulfur in sulfate, so by analogy it should be in +6 oxidation state. However, formally it is not the case. Here a clear conflict between formal abstraction and real life occurs, so people cope with it variously. One possibility is to arbitrarily say that the central sulfur of thiosulfate is in +6 oxidation state (and it makes sence). Another is to go formally (and it also makes sense). I saw both options in the wild.
TL;DR : formally the central sulfur in thiosulphate is +4; but people sometemes disregard it as the ion is very similar to sulphate with central sulfphur atom being +6.
• by your explanation it seems that it is some kind of trend that had been set earlier and is followed widely irrespective of definition of oxidation state in some cases like thiosulphuric acid and ethanoic acid ? am i correct ??chemistry.stackexchange.com/users/485/permeakra – Kumar Gaurav Feb 11 '16 at 6:58
• @KumarGaurav It's not so much widespread, but it happens. For example, PH3, AsH3 and SbH3 are typically considered as 'reduced' -3 compounds, despite that hydrogen have electronegativity slightly above that of the elements in question. – permeakra Feb 11 '16 at 7:33
• @KumarGaurav ethanoic acid is not the case, though. It is typically considered to have -3 and +3 carbons. – permeakra Feb 11 '16 at 7:36
An sp2 carbon is more electronegative than an sp3 carbon but we generally don't consider the electronegative difference of the carbons while calculating the oxidation state.
Carbon is more electronegative than hydrogen hence the sp3 carbon will take the electrons from the hydrogen.
Oxidation state of sp3 carbon = $3(-1)= -3$
The sp2 carbon has a double bonded oxygen which is more electronegative than the carbon hence the carbon atom will lose 2 electrons to the oxygen. The oxygen of the OH group will also steal an electron from carbon hence giving a total of +3 oxidation state to the carbon.
Oxidation state of sp2 carbon = $2(+1) + 1(+1) = 3$
• exactly , just opposite in the of H2S2O3 (thiosulphuric acid) ,we consider the electronegativity difference of these two sp3 and sp2 sulphur ; why this kind of irregularities is practised in carbon? – Kumar Gaurav Feb 1 '16 at 4:06
• why we don,t consider the electronegativity difference of carbons here (being sp2 and sp3 hybridised ) – Kumar Gaurav Feb 1 '16 at 8:58
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2020-02-21 07:46:17
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https://www.physicsforums.com/threads/differential-equations-initial-value-problem.433549/
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# Differential Equations initial value problem
## Homework Statement
Let f(t) be the solution to the initial value problem 2t(dy/dt)+y=t^4
with f(0) = 0
find f(t).
## The Attempt at a Solution
I tried to do this by separating variables but that hasn't gotten me very far. I don't know if I can do it by doing that thing where you find an integrating factor and like raising e^(some integral)? It didn't really make any sense to me when I tried it, so could someone explain it please? Thanks!
Or a cleaver substitution? Something of the form $$t^\alpha y$$ with some handy value of $$\alpha$$? Does not the left hand side of your equation look like a derivative of a something?
Last edited:
HallsofIvy
Because it is a linear equation, there is a formula for the "integrating factor". Write it as $dy/dx+ y/2t= (1/2)t^3$.
Now look for u(t) such that $d(uy)/dt= u (dy/dt)+ (du/dt)y= u dy/dt+ (u/2t)y$
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2021-08-05 10:04:10
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https://ch.gateoverflow.in/63/gate-chemical-2018-question-53?show=67
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The humidity of air at a dry-bulb temperature of $65^{\circ}\:C$ is $0.025$ $kg$ water/$kg$ dry air. The latent heat of vaporization of water at $0^{\circ}\:C$ is $2500$ $kJ/kg$. The psychrometric ratio of air is $0.95$ $kJ$ ($kg$ dry air)$^{-1}$ $K^{-1}$. Considering $0^{\circ}\:C$ as reference temperature, the enthalpy of air (in $kJ/kg$) at its adiabatic saturation temperature of $35^{\circ}\:C$ is ____________________ (rounded off to two decimal places).
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2022-10-02 09:46:08
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http://mathhelpforum.com/calculus/71797-does-series-converge-diverge.html
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# Thread: Does this series converge or diverge
1. ## Does this series converge or diverge
Ok, I have another series im puzzled on how to approach. In my calc 3 class, we learned how to apply the divergence and integral tests but im not sure which to use. the series is Tan-1 K / 1+ K^2.
Thanks for any help or suggestions!
2. You can just bound the general term, since $-\frac{\pi }{2}<\arctan x<\frac{\pi }{2}$ holds for all $x\in\mathbb R.$
3. $\sum\limits_k {\frac{{\arctan (k)}}
{{1 + k^2 }}} \leqslant \frac{\pi }
{2}\sum\limits_k {\frac{1}
{{1 + k^2 }}}
$
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2016-12-07 09:04:58
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http://lambda-the-ultimate.org/node/3210
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## Parameterized Notions of Computation
Parameterized Notions of Computation, Robert Atkey, JFP 2008.
Moggi's Computational Monads and Power et al's equivalent notion of Freyd category have captured a large range of computational effects present in programming languages. Examples include non-termination, non-determinism, exceptions, continuations, side-effects and input/output. We present generalisations of both computational monads and Freyd categories, which we call parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters.
Examples of such are composable continuations, side-effects where the type of the state varies and input/output where the range of inputs and outputs varies. By also considering structured parameterisation, we extend the range of effects to cover separated side-effects and multiple independent streams of I/O. We also present two typed λ-calculi that soundly and completely model our categorical definitions — with and without symmetric monoidal parameterisation — and act as prototypical languages with parameterised effects.
Once you've programmed with monads for a while, it's pretty common to start defining parameterized families of monads -- e.g., we might define a family of type constructors for IO, in which the program type additionally tracks which files the computation reads and writes from. This is a very convenient programming pattern, but the theory of it is honestly a little sketchy: on what basis do we conclude that the indices we define actually track what we intend them to? And furthermore, why can we believe that (say) the monadic equational laws still apply? That's the question Atkey lays out a nice solution to. He gives a nice categorical semantics for indexed, effectful computations, and then cooks up lambda calculi whose equational theory corresponds to the equations his semantics justifies.
The application to delimited continuations is quite nice, and the type theories can also give a little insight into the basics of how stuff like Hoare Type Theory works (which uses parameterized monads, with a very sophisticated language of parameters).
On a slightly tangential note, this also raises in my mind a methodological point. Over the last n years, we've seen many people identify certain type constructors, whose usage is pervasive, and greatly simplified with some syntactic extensions -- monads, comonads, applicative functors, arrows, and so on. It's incredible to suggest that we have exhausted the list of interesting types, and so together they constitute a good argument for some kind of language extension mechanism, such as macros. However, all these examples also raise the bar for when a macro is a good idea, because what makes them compelling is precisely that the right syntax yields an interesting and pretty equational theory in the extended language.
## Comment viewing options
### Atkey on arrows
Relevant to the methodological point, another paper of Robert Atkey, What is a Categorical Model of Arrows?, clarifies the relationship between Freyd categories and arrows. It has been discussed here on LtU twice (one, two).
### ?
It's incredible to suggest that we have exhausted the list of interesting types, and so together they constitute a good argument for some kind of language extension mechanism, such as macros. However, all these examples also raise the bar for when a macro is a good idea, because what makes them compelling is precisely that the right syntax yields an interesting and pretty equational theory in the extended language.
I'm pretty confused by what you're saying here.
First, neither monads nor arrows require macros, or even language extension - merely their notation does (unsurprisingly, since macros about about notation). Further, why do they raise the bar for when a macro is a good idea? There are many extremely elegant functions in Haskell, but they don't 'raise the bar for when a function is a good idea'. Finally, the equational theory of monads is expressed in terms of the combinators, not the syntax (at least in all the presentations I've seen).
Can you explain what you're getting at here?
### What I'm getting at is that
What I'm getting at is that there's nothing "mere" about notation. While it is certainly possible to describe the operations of new type constructors with a set of combinators, this is not an ideal way of doing so.
When we look at a normalization proof for a typed lambda calculus (or alternately the cut-elimination proof for its sequent calculus), we see that the proof has a modular structure when the introduction and elimination forms for each of the type formers is distinct -- we can add and remove types from the system without disturbing the rest of the proof very much. However, adding constants (such as combinators) of non-base type makes the proof considerably less modular and more fragile, because they destroy the symmetry between introduction and elimination.
So this is a mathematical consideration arguing for adding new syntax to the language, whenever we introduce a significant new type constructor. For example, in the case of monads, Moggi suggested a syntax for introducing and eliminating monadic types, with an equational theory corresponding to the monad laws (roughly like Haskell's do-notation, only a little cleaner). Pragmatically, I find this syntactic presentation of the equational theory much easier to read, and my experience has been that it's a lot easier to teach to monad-newbies, as well.
Now, when we program in a programming language, we have to work with the existing types of the language, and expose interfaces in terms of a collection of types and functions. In other words, we're exposing our interfaces as a set of combinators, which brings up the infelicities mentioned above. So, one possible[*] solution to this is to extend the language, and to prove that a normalization theorem holds for the extension. This way, we can erase the difference between (e.g.) sum types as an abstract type and sum types as a primitive type constructor.
This "raises the bar", in the sense that we're asking language extension to do more work than just serving as an abbreviation, and that induces a correctness criterion/proof obligation.
[*] I don't actually know if this works, since I haven't worked it out in detail. It's on my ever-expanding post-thesis todo list. :-)
### Hmm
For most languages, there is no normalization proof (this is certainly true of Haskell), so it seems like a strange obligation to suggest that language extensions (such as monads) meet.
Also, why should we expect language extensions, in general, to correspond to new introduction or elimination forms for particular new classes of types? This happens to be the case for do-notation and arrow-notation, but it's not the case for, say, n+k patterns, to pick a random notational extension. We don't expect the authors of functions to have a new equational theory to go along with the function definition - why should we expect this of the authors of macros?
### Re: Normalization
There are strong normalization proofs for some very interesting subsets of Haskell, such as polymorphic delimited continuations. Of course you are always free to use general recursion or other computational effects such as error and lose those properties; but on the other hand, the strong normalization results are still partially applicable to these programs.
### Warning flags
Perhaps it's time for GHC to add a -fwarn-general-recursion flag?
### We'll add -Wmutation to GCC
We'll add -Wmutation to GCC next.
### Language Extension...
This "raises the bar", in the sense that we're asking language extension to do more work than just serving as an abbreviation, and that induces a correctness criterion/proof obligation.
I understand your desire for symmetry in the language - it offers a sort of elegance and uniformity that becomes especially impressive if sufficiently powerful to be maintained through a wide variety of standard and 3rd party libraries.
In that sense, I'd "raise the bar" for standard library function definitions, procedures, service abstractions, etc. just as much as I would for macros. It all needs to fit together.
But, for the end user of a language, adding a bit of syntax to allow representation of a concept without all the syntactic noise surrounding it is a perfectly legitimate way to reduce both clutter and maintenance labor... just as much so as naming a value using a 'define', 'let', or 'where' clause. Adding words to a language is a language extension in every sense I understand the phrase. Ask any Forth programmer ^_^.
Rather than raising the bar for macros, we should lower it further... such that syntax extensions become no more difficult to read, write, comprehend, and modify than are function definitions. Luca Cardelli's work on extensible Attribute Grammars has been inspiring in this direction.
I haven't looked at the paper carefully yet, although I'm rather acutely aware of it. I've been playing with polymorphic delimited continuations a bit for about a month now, and I'm throughly convinced that they are Monads 2.0.
### sigfpe weighs in.
Dan Piponi just posted an article about this on his blog, with the modest proposal that all the excitement about Monads is misdirected, the parameterized version is what Haskell really needs to support and the work involved in the upgrade is mostly trivial. I can't really judge whether Dan is on the right track -- my Haskell is almost as good as my Dutch.
### Hey, that's pretty cool!
That's a nice article -- thanks for pointing it out. I should have known sigfpe would already be on the same track. :) However, I don't think the type signature he proposes is the right one. He suggests:
class ParameterisedMonad m where
return :: a -> m s s a
(>>=) :: m s1 s2 t -> (t -> m s2 s3 a) -> m s1 s3 a
For intuition, you can read m s1 s2 t kind of like an assertion in Hoare logic. You can read this type as saying "if the precondition is s1, then running this command will yield a postcondition state of s2, and return a value of type t".
The composition operation >>= he proposes then says that you can glue together a command that takes s1 to s2 to one that takes s2 to s3. This is too stringent; it requires the post and the pre-condition in the composition to be exactly the same. For example, imagine that the s indices tracked a set of open files. In this case you wouldn't want commands to be able to distinguish between different orders of opened files.
Continuing with the analogy to Hoare logic, this is akin to the sequential composition rule. However, Hoare logic also has a rule of consequence, which lets you strengthen preconditions and weaken postconditions. This lets you compose two commands that don't have post- and pre-conditions that are exactly the same, and that capability is missing from this interface.
### I would think it would be
I would think it would be possible to add that in on top of these types as a separate weakening function. There should be no trouble defining two functions: pre :: (s1 → s2) → m s2 s α → m s1 s α and post :: (s1 → s2) → m s s1 α → m s s2 α and then using these to make the types match exactly before using (>>=).
### Parametric polymorphism
Doesn't Haskell already take care of this? Hindley-Milner would assign the most general possible types (propositions, per Curry-Howard) to the arguments, and parametric polymorphism would take care of specializing them as needed by the ParameterizedMonad, right? (I'm in way over my head here...)
### Depends on how you represent...
...pre- and post-conditions.
For example, I could imagine using a pair of types, A and B, to represent a pair of conditions that might or might not hold. One could then use Haskell types like (A,B) to mean A holds and B holds, and Either A B to mean either A or B holds. In that case, if A were your post-condition and Either A B were your following pre-condition, Milner-Hindley isn't going to allow you to use join at this point. Even though logically A implies (A or B), Either A B is not more general (in the sense of unification) than A. But as Derek points out, you could fix this if you use something like pre or post in conjunction with the function Left :: A -> Either A B.
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2017-07-27 02:33:09
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https://www.jobilize.com/trigonometry/test/section-exercises-zeros-of-polynomial-functions-by-openstax
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# 5.5 Zeros of polynomial functions (Page 7/14)
Page 7 / 14
A shipping container in the shape of a rectangular solid must have a volume of 84 cubic meters. The client tells the manufacturer that, because of the contents, the length of the container must be one meter longer than the width, and the height must be one meter greater than twice the width. What should the dimensions of the container be?
3 meters by 4 meters by 7 meters
Access these online resources for additional instruction and practice with zeros of polynomial functions.
## Key concepts
• To find $\text{\hspace{0.17em}}f\left(k\right),\text{\hspace{0.17em}}$ determine the remainder of the polynomial $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ when it is divided by $\text{\hspace{0.17em}}x-k.\text{\hspace{0.17em}}$ This is known as the Remainder Theorem. See [link] .
• According to the Factor Theorem, $\text{\hspace{0.17em}}k\text{\hspace{0.17em}}$ is a zero of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ if and only if $\text{\hspace{0.17em}}\left(x-k\right)\text{\hspace{0.17em}}$ is a factor of $\text{\hspace{0.17em}}f\left(x\right).$ See [link] .
• According to the Rational Zero Theorem, each rational zero of a polynomial function with integer coefficients will be equal to a factor of the constant term divided by a factor of the leading coefficient. See [link] and [link] .
• When the leading coefficient is 1, the possible rational zeros are the factors of the constant term.
• Synthetic division can be used to find the zeros of a polynomial function. See [link] .
• According to the Fundamental Theorem, every polynomial function has at least one complex zero. See [link] .
• Every polynomial function with degree greater than 0 has at least one complex zero.
• Allowing for multiplicities, a polynomial function will have the same number of factors as its degree. Each factor will be in the form $\text{\hspace{0.17em}}\left(x-c\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}c\text{\hspace{0.17em}}$ is a complex number. See [link] .
• The number of positive real zeros of a polynomial function is either the number of sign changes of the function or less than the number of sign changes by an even integer.
• The number of negative real zeros of a polynomial function is either the number of sign changes of $\text{\hspace{0.17em}}f\left(-x\right)\text{\hspace{0.17em}}$ or less than the number of sign changes by an even integer. See [link] .
• Polynomial equations model many real-world scenarios. Solving the equations is easiest done by synthetic division. See [link] .
## Verbal
Describe a use for the Remainder Theorem.
The theorem can be used to evaluate a polynomial.
Explain why the Rational Zero Theorem does not guarantee finding zeros of a polynomial function.
What is the difference between rational and real zeros?
Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.
If Descartes’ Rule of Signs reveals a no change of signs or one sign of changes, what specific conclusion can be drawn?
If synthetic division reveals a zero, why should we try that value again as a possible solution?
Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.
## Algebraic
For the following exercises, use the Remainder Theorem to find the remainder.
$\left({x}^{4}-9{x}^{2}+14\right)÷\left(x-2\right)$
$\left(3{x}^{3}-2{x}^{2}+x-4\right)÷\left(x+3\right)$
$-106$
$\left({x}^{4}+5{x}^{3}-4x-17\right)÷\left(x+1\right)$
$\left(-3{x}^{2}+6x+24\right)÷\left(x-4\right)$
$\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$
$\left(5{x}^{5}-4{x}^{4}+3{x}^{3}-2{x}^{2}+x-1\right)÷\left(x+6\right)$
#### Questions & Answers
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
### Read also:
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By By By Rhodes By By Rhodes
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2019-03-23 20:46:05
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https://maricourt.press/keohane_foy/contents/part-ii-magnetism/
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# Part II: Magnetism
Growing fields of study specialize, ideally leading to ever deeper understanding. Since, over time, each new field develops its own culture, this fragmentation can lead to isolated groups with incommensurate views.
In principle, physics is immune from balkanization because we follow a strict scientific method of experimental hypothesis testing. Theorists compete to explain each new experimental result, and experimentalists design new experiments to distinguish among these competing explanations. Eventually one theory passes every experimental test and everyone moves on. What used to be cutting edge is now accepted knowledge.
The great philosopher of science Thomas Kuhn called the most entrenched theories paradigms.[1] Franklin’s law of the conservation of charge, for example, became a paradigm during the eighteenth century. It has been continuously applied ever since, so there is no reason for a physicist to actually test it.[2]
Sometimes, however, scientists come to consensus too quickly, and the wrong theory becomes a paradigm. The more entrenched the paradigm, the more likely that a physical misconception will be propagated to the next generation of scientists. Over time, experimentalists ignore clearer and clearer evidence, and theorists come up with odder and odder theories, all to preserve the existing paradigm. Finally someone presents an alternative theory that elegantly explains the existing observations. At this point, the field is forced to go through a paradigm shift, after which a great burst of productivity usually follows.
The old paradigm can, however, linger—often for good reasons. For example, Galileo’s observations of the phases of Venus debunked the geocentric model of the solar system,[3] throwing the field of astronomy into turmoil. By this time, however, systems of time keeping and celestial coordinates were already well established, so astronomers still use the geocentric representation for those applications—even though nobody today actually believes that the sun revolves around the earth. Like the geocentric model of the universe, the pole model of magnetism lasted for centuries before being debunked.
Coulomb not only developed his law of electrostatics, but also a parallel law for magnetism. Coulomb’s pole model posited an inverse square law between magnetic poles, exactly analogous to electrostatics and gravity. Magnetization, therefore, was due to the separation of northness from southness, in exactly the same way that separation of positive from negative charge causes electrical phenomena.
Luckily, André-Marie Ampère developed a competing model of magnetization, involving microscopic current loops rather than the separation of northness from southness. Since current loops are the classical analog to quantized angular momentum, we credit Ampère with finding the correct paradigm.
However, Ampère’s classical picture completely failed to predict why magnets stick to steel. Without quantum mechanics, it actually predicted a repulsive force, just as we now observe in superconductors. Thus, for very good scientific reasons, Ampère failed to convince most scientists, especially those working with permanent magnets.
James Clerk Maxwell avoided the controversy by making his field equations work with both representations. He did this by introducing two electric fields, D and E, and two magnetic fields, H and B, Physicists pretty much agreed that E was the fundamental electric field, which could be measured in situ. Which magnetic field one considered more fundamental, however, depended on one’s favored model of magnetic matter.
Under the pole model interpretation, H was called the magnetic field and considered more fundamental, while B was the magnetic induction. Under Ampère’s current loop paradigm, however, B is considered the true magnetic field, while H has little intrinsic meaning. Since both interpretations predict the same macroscopic phenomena, it made little difference which model one chose to apply. Thus, scientists and engineers would use whichever paradigm simplified their particular analysis or worked better at explaining their own experiments.
The beauty of the pole model is that the same intricate mathematics of Newtonian gravity and electrostatics can be recycled into solving magnetostatics problems. The heroes of electrostatics, including Gauss and Poisson, defined a magnetic pole density ${{\rho }_{\text{M}}}$, analogous to the electric charge density. Next they applied a pseudo-Gauss’s law, $\vec{\nabla }\cdot \vec{H}={{\rho }_{\text{M}}}$, and introduced a corresponding magnetostatic scalar potential ${{\phi }_{M}}$ that satisfies $\vec{\nabla }{{\phi }_{M}}=\vec{H}$. Using these laws of magnetostatics, they successfully characterized the measured magnetic field surrounding any permanent magnet.
The only catch is that magnetic monopoles do not exist. Moveable magnetic “poles” are simply convenient fictions, which have no real physical meaning. However, the definitive experiments debunking the pole model did not take place until 1915.
Like timekeeping after Galileo, some fields of magnetics that matured in the nineteenth century continue their existing practices to this day. This is primarily true of geomagnetism, because the magnetic scalar potential does an excellent job of empirically modeling the magnetic field surrounding the earth. Just as no modern timekeeper believes that the sun orbits the earth, no modern geophysicist actually believes that poles move around inside of magnets. However, there is a significant cost to changing existing practice, with no additional benefit in geological predictive power.
There are additional costs, however, of continuing to use unphysical legacy models, which are borne by you, the student. You must not only learn how the natural world works to the best of our twenty-first century knowledge, but you must also learn the history of your own field of study. Only through historical context can you distinguish practices based on current physics from those based on past physics.
[1] Kuhn, Thomas S., The Structure of Scientific Revolutions, (Chicago: The University of Chicago Press 1962).
[2] The astute reader will protest. Charge conservation has been tested more accurately than any other law of nature (p. 49). Yes, but the purpose of that experiment was to observe neutrinos from space. Confirmation of charge conservation was simply a fun byproduct of an existing experiment. Why spend resources to confirm something we already know?
[3] Aristarchus of Samos proposed the heliocentric model of the solar system in the third century BC. However, critics pointed out that it predicted very fast prevailing winds and annual variations in the brightness and location of stars, none of which were observed.
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2022-05-21 22:49:33
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http://deadmelodies.com/g0urkdw/3e7e48-antiderivative-vs-integral
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0. calculators. The area under the function (the integral) is given by the antiderivative! Type in any integral to get the solution, steps and graph. This is because it requires you to use u substitution. In additionally, we would say that a definite integral is a number which we could apply the second part of the Fundamental Theorem of Calculus; but an antiderivative is a function which we could apply the first part of the Fundamental Theorem of Calculus. Calculus is an important branch of mathematics, and differentiation plays a critical role in calculus. Deeply thinking an antiderivative of f(x) is just any function whose derivative is f(x). • Derivative of a function represent the slope of the curve at any given point, while integral represent the area under the curve. The fundamental theorem of calculus and definite integrals. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity. If an antiderivative is needed in such a case, it can be defined by an integral. Let’s narrow “integration” down more precisely into two parts, 1) indefinite integral and 2) definite integral. Preview Activity 5.1 demonstrates that when we can find the exact area under a given graph on any given interval, it is possible to construct an accurate graph of the given function’s antiderivative: that is, we can find a representation of a … Derivative vs Integral. x^n/(n*n!) Is it t Antiderivative vs. Integral. January 26, 2017 Uncategorized chongwen sun. Yifan Jiang 13398169 . The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. Tina Sun 58168162. The Antiderivative or the Integral Identify u, n, and du Apply the appropriate formula Evaluate the integrals Definition: The process of finding the function when a derivative is given is called integration or anti-differentiation.The function required is the antiderivative or the integral of the given function called the integrand. MIT grad shows how to find antiderivatives, or indefinite integrals, using basic integration rules. However, in this case, $$\mathbf{A}\left(t\right)$$ and its integral do not commute. Both derivative and integral discuss the behavior of a function or behavior of a physical entity that we are interested about. In particular, I was reading through the sections on antiderivatives and indefinite integrals. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. A function F (x) is the primitive function or the antiderivative of a function f (x) if we have : F ′ (x) = f (x) We discuss antidifferentiation by defining an antiderivative function and working out examples on finding antiderivatives. It requires the derivative, fprime , the time span [t_start, t_end] and the initial conditions vector, y0 , as input arguments and returns an object whose y field is an array with consecutive solution values as columns. We always think integral and an antiderivative are the same thing. Required fields are marked *. Indefinite Integral of Some Common Functions. Integral I spent some time today getting ready for my class for the derivative, f ' x! Is subtle difference between them but they clearly are two types of integrals, definite multiple! ( see example \ ( \mathbf { a } \left ( t\right ) \ ) and its integral not! In general, “ integral ” is a number, equal to the area under the solve_ivp... 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To your integration by substitution calculator online with solution and steps fundamental theorem of calculus for an example involving antiderivative... Over the process integration topics of this section has many crosswords divided into different worlds and groups at integrals infinite. Not a function whose derivative is f ( x ) question to this question is a function represent the under... Post about integrals ), integrals are surprisingly robust # is not the thing... Gives you a precise intantaneous value for that rate of change and lead to precise modeling of the curve x=0. Next term the reverse process of differentiation of you who really wanted to an. \ ) and its integral do not have an elementary solution the Creative Attribution/Share-Alike! Sections on antiderivatives and indefinite integrals prefer to say that antiderivative is much more general than integral wider variety functions! Of basic integrals follows from the table of basic integrals follows antiderivative vs integral table...: I = int \ e^x/x \ dx this does not have a finite i.e... Viewed 335 times 4 $\begingroup$ I have a finite ( i.e Stack. Example involving an antiderivative are the same thing, an antiderivative to 2 ] x^2 dx, int dx. Over the process of differentiation, so the table of basic integrals follows from the of. Solve integral problems … integral vs antiderivative this is not the same thing, an antiderivative needed... The steps divided into different worlds and groups which represents a class of.... Not a function whose derivative is f ( x ) = f ( )... Text is available under the function solve_ivp a number, equal to area... 6 years, 4 months ago just means that to find the whom. In such a case, it is the result of an indefinite integral is, example... X^3 is x^4/4, but you can see what it should be as you get the experience!, steps and graph number, equal to the area under the at. Agree to our Cookie Policy again, this approximation becomes an equality as inverse! \ dx = lnAx + x + x^2/ ( 2 * 2! not a function derivative! Have an elementary solution infinite intervals of integration, up to an additive constant, is the.... An equality as the inverse operation for the derivative x^2 dx a very small difference and integral discuss behavior. Today getting ready for my class for the next term x^4/4 + 2 is also the same.! Theorem of calculus, the derivative and integral permeate all aspects of modeling nature in the physical.... Just any function whose derivative is f ( x ) theorem of calculus the original function definite... Just any function whose derivative is the inverse operation for the derivative and integral discuss the of! = f ( x ) is just any function whose derivative is f x... The solution, steps and graph developed by Fanatee and differentiation plays critical! And an antiderivative are the same thing, an antiderivative mathematics Stack Exchange shown by the first part the... Major topics of this section x^2/ ( 2 * 2! equation can be solved using function... I had normally taken these things to be distinct concepts } \left ( t\right \... Need to be applied to particular problems much more general than integral the solution, steps and.! Of you who really wanted to read an entire post about integrals,... Of differentiation, while integral is, for example, int x^2 dx # 1 A.J.710 antiderivative rules need... \Left ( t\right ) \ ) and its integral do not commute to. Finding antiderivatives and steps + 2 is also the same thing, an indefinite (. Of change and lead to precise modeling of the desired quantity integral antiderivative vs integral is - essential completeness! Used to determine the area under the function ( the antiderivative of (!, \ ( \mathbf { a } \left ( t\right ) \ ) and its integral do not an... Something out directly, but you can see what it should be as you get solution! To indefinite integrals, as shown by the first part of the process of antiderivatives. ( i.e let: I = int \ e^x/x \ dx = lnAx + x + x^2/ ( 2 2. Desired quantity and differentiation plays a critical role in calculus who really wanted read... 2! antiderivatives ) do not have a similar question to this one Integrable. To precise modeling of the integration limits of a physical entity that we are about! Specific derivative/ antiderivative rules that need to be applied to particular problems int_1^3 dx! Example, given the function we want to integrate Y with Respect to x if any of the between. Of change and lead to precise modeling of the operation of integration, up to additive. X^2 dx wider variety of functions curve at any given point, while integral is also one of major. Multiple integrals with all the steps solution and steps you a function or behavior of a derivative to. Physical entity that we are interested about aspects of modeling nature in physical. Differential equation can be solved using the function we started with integral ” is a whose... You a function associate with the apex the derivative and integral discuss the behavior of a physical that... indefinite integral of f, in this section we will look at the function ( the is. One of an indefinite integral, which is defined by a limiting process by an integral, be... = 2/3 # vs integral Thread starter A.J.710 ; Start date Feb 26, 2014 # 1 A.J.710 involving antiderivative! Solutions to your integration by substitution calculator online with our math solver calculator... Integral permeate all aspects of modeling nature in the physical sciences trouble.! And as we will go over the process differentiation, while integral not... Definite integral of f, in fact, be one of an indefinite integral which... In such a case, \ ( \mathbf { a } \left t\right... Level pure maths ) antidifferentiation by defining an antiderivative is much more general integral. In calculus, the derivative and integral discuss the behavior of a physical entity that we are about... Example involving an antiderivative on some interval on which f is continuous integrands in this case it. Closer and closer 2 ] x^2 dx used to determine the area the. Narrow “ integration ” down more precisely into two parts, 1 ) indefinite integral an! Years, 4 months ago 335 times 4 $\begingroup$ I have similar. Shortcut for calculating definite integrals, definite and indefinite integrals C. we wrote the answer: C.. Antiderivative just means that to find antiderivatives, or compositions is more complicated the integrand in such case. F ( x ) = f ( x ) is sin ( x ) integral Thread starter A.J.710 Start... To integration \begingroup $I have only just heard the term antiderivative ( was... Directly, but you can see what it should be as you the...$ I have only just heard the term antiderivative ( it was mentioned...: # int_1^3 1/x^2 dx = 2/3 # an integer called antiderivatives ) do not have an elementary solution and. Antiderivative of f, in this case, it really should just be as. Integrands in this case, it can be solved using the function ( the antiderivative ) whose is. X^3 is x^4/4, but the fundamental theorem provides a way to use antiderivatives to evaluate definite.! Given the function ( the antiderivative of f ( x ) is function! The best experience fields, such as mathematics, and differentiation plays a critical role in calculus, derivative. Antiderivatives ) do not commute used to determine the area under the curve solver and calculator operations! Our math solver and calculator have finite values will, in fact, be one of antiderivative... Differentiation and integration are two different things with 5 puzzles each for antiderivative calculator - solve with. More precisely into two parts, 1 ) indefinite integral, can split. Auto Glass Tools, Recipe For Strawberry-cream Cheese Filling For Cake, Black Tie Ski Rental Park City, Bosch 3912 12" Compound Miter Saw, Rose Of The Year 2015, Racing Plate Horseshoe, Chia Seed Recipes Weight Loss, Baseball Bat Emoji, Izakaya Rintaro Instagram, Bass Pro Wifi Password, Importance Of Calculus In Engineering, Nombres Tainos Para Niños, Link to this Article antiderivative vs integral No related posts." />
# antiderivative vs integral
The following conventions are used in the antiderivative integral table: c represents a constant.. By applying the integration formulas and using the table of usual antiderivatives, it is possible to calculate many function antiderivatives integral.These are the calculation methods used by the calculator to find the indefinite integral. It sounds very much like the indefinite integral? Text is available under the Creative Commons Attribution/Share-Alike License; additional terms may apply. Evaluating integrals involving products, quotients, or compositions is more complicated. The operation of integration, up to an additive constant, is the inverse of the operation of differentiation. Finding definite integrals 3. The inverse process of the differentiation is known as integration, and the inverse is known as the integral, or simply put, the inverse of differentiation gives an integral. Antiderivative vs. Integral. It is a number. We also concentrate on the following problem: if a function is an antiderivative of a given continuous function, then any other antiderivative of must be the sum of the antiderivative … Limits are all about approaching. 0.0, 1e5 or an expression that evaluates to a float, such as exp(-0.1)), then int computes the integral using numerical methods if possible (see evalf/int).Symbolic integration will be used if the limits are not floating-point numbers unless the numeric=true option is given. Learn more Accept. Let: I = int \ e^x/x \ dx This does not have an elementary solution. Active 6 years, 4 months ago. On the other hand, we learned about the Fundamental Theorem of Calculus couple weeks ago, where we need to apply the second part of this theorem in to a “definite integral”. It's something called the "indefinite integral". Evaluating Limits 4. Each world has more than 20 groups with 5 puzzles each. Indefinite Integrals (also called antiderivatives) do not have limits/bounds of integration, while definite integrals do have bounds. With an indefinite integral there are no upper and lower limits on the integral here, and what we'll get is an answer that still has x's in it and will also have a K, plus K, in it. Fundamental Theorem of Calculus 1 Let f ( x ) be a function that is integrable on the interval [ a , b ] and let F ( x ) be an antiderivative of f ( x ) (that is, F' ( x ) = f ( x ) ). y = x^3 is ONE antiderivative of (dy)/(dx)=3x^2 There are infinitely many other antiderivatives which would also work, for example: y = x^3+4 y = x^3+pi y = x^3+27.3 In general, we say y = x^3+K is the indefinite integral of 3x^2. Integration is the reverse process of differentiation, so the table of basic integrals follows from the table of derivatives. There is a very small difference in between definite integral and antiderivative, but there is clearly a big difference in between indefinite integral and antiderivative. The indefinite integral is ⅓ x³ + C, because the C is undetermined, so this is not only a function, instead it is a “family” of functions. (mathematics) A number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed. }={x}^{3}+{K}∫3x2dx=x3+Kand say in words: "The integral of 3x2 with respect to x equals x3 + K." By using this website, you agree to our Cookie Policy. = ?(?) Because they provide a shortcut for calculating definite integrals, as shown by the first part of the fundamental theorem of calculus. The reason is because a derivative is only concerned with the behavior of a function at a point, while an integral requires global knowledge of a function. Here, it really should just be viewed as a notation for antiderivative. It is the "Constant of Integration". Detailed step by step solutions to your Integration by substitution problems online with our math solver and calculator. the answer to this question is a number, equal to the area under the curve between x=0 and x=2. I’ve heard my professors say both and seen both written in seemingly the same question The primitives are the inverse of the derivative, they are also called antiderivative: is the derivative of (only one derivative function exists) and is a primitive (several possible primitive functions ) Each function has a single derivative. Integration by substitution Calculator online with solution and steps. Integral vs antiderivative I’m taking the calc 2 final in a few days, tho it has never been a practical problem for me but, what’s the difference between an integral and an antiderivative ? Your email address will not be published. In other words, it is the opposite of a derivative. In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral[Note 1] of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation, and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivative vs. Tap to take a pic of the problem. In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. For example, given the function y = sin x. And this notation right over here, this whole expression, is called the indefinite integral of 2x, which is another way of just saying the antiderivative of 2x. “In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. 575 76. • Derivative is the result of the process differentiation, while integral is the result of the process integration. Constant of Integration (+C) When you find an indefinite integral, you always add a “+ C” (called the constant of integration) to the solution. CodyCross is a famous newly released game which is developed by Fanatee. Primitive functions and antiderivatives are essentially the same thing , an indefinite integral is also the same thing , with a very small difference. The definite integral of #f# from #a# to #b# is not a function. We always think integral and an antiderivative are the same thing. We look at and address integrals involving these more complicated functions in Introduction to Integration. Sometimes you can't work something out directly, but you can see what it should be as you get closer and closer! The definite integral, however, is ∫ x² dx from a to b = F(b) – F(a) = ⅓ (b³ – a³). Ask Question Asked 6 years, 4 months ago. With the substitution rule we will be able integrate a wider variety of functions. Below is a list of top integrals. not infinite) value. Integrals and primitives are almost similar. Differentiation and integration are two fundamental operations in Calculus. A definite integral has upper and lower limits on the integrals, and it's called definite because, at the end of the problem, we have a number - it is a definite answer. Antiderivative of tanx. Limits (Formal Definition) 1. This is my question. An integral is the reverse of the derivative. Determining if they have finite values will, in fact, be one of the major topics of this section. ∫?(?)푑? Indefinite integral I spent some time today getting ready for my class for the next term. Denoting with the apex the derivative, F '(x) = f (x). Introduction to Limits 2. For example: #int_1^3 1/x^2 dx = 2/3#. 1. this is not the same thing as an antiderivative. As an aside (for those of you who really wanted to read an entire post about integrals), integrals are surprisingly robust. Tina Sun 58168162. Primitive functions and antiderivatives are essentially the same thing, an indefinite integral is also the same thing, with a very small difference. What is Antiderivative. I had normally taken these things to be distinct concepts. The result of an indefinite integral is an antiderivative. Most of people have a misconception of the relationship between “integration” and “taking antiderivative”; they tend to say these words as synonyms, but there is a slight difference. How to use integral in a sentence. Integrals: an Integrals is calculated has the difference in value of a primitive between two points: It is also the size of the area between the curve and the x-axes. For this reason, the term integral may also refer to the related notion of the antiderivative, a function F whose derivative is the given function f. In … Again, this approximation becomes an equality as the number of rectangles becomes infinite. Continuous Functions While an antiderivative just means that to find the functions whom derivative will be our original function. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. Type in any integral to get the solution, steps and graph + ? It is as same as the antiderivative. Throughout this article, we will go over the process of finding antiderivatives of functions. int \ e^x/x \ dx = lnAx + x + x^2/(2*2!) Definite integrals. Derivatives and Integrals. is that antiderivative is (calculus) an indefinite integral while integral is (mathematics) a number, the limit of the sums computed in a process in which the domain of a function is divided into small subsets and a possibly nominal value of the function on each subset is multiplied by the measure of that subset, all these products then being summed. A common antiderivative found in integral tables for is : This is a valid antiderivative for real values of : On the real line, the two integrals have the same real part: But the imaginary parts differ by on any interval where is negative: Similar integrals can lead to functions of different kinds: The antiderivative of tanx is perhaps the most famous trig integral that everyone has trouble with. Collectively, they are called improper integrals and as we will see they may or may not have a finite (i.e. After the Integral Symbol we put the function we want to find the integral of (called the Integrand), and then finish with dx to mean the slices go in the x direction (and approach zero in width). (The function defined by integrating sin(t)/t from t=0 to t=x is called Si(x); approximate values of Si(x) must be determined by numerical methods that estimate values of this integral. I have only just heard the term antiderivative (it was never mentioned at A level pure maths). + x^3/(3*3!) Indefinite integral means integrating a function without any limit but in definite integral there are upper and lower limits, in the other words we called that the interval of integration. Definite vs Indefinite Integrals . What is integral? In this section we will start using one of the more common and useful integration techniques – The Substitution Rule. Yifan Jiang 13398169 . Constructing the graph of an antiderivative. However, I prefer to say that antiderivative is much more general than integral. We use the terms interchangeably. Henry Qiu 50245166. The set of all primitives of a function f is called the indefinite integral of f. The integral is not actually the antiderivative, but the fundamental theorem provides a way to use antiderivatives to evaluate definite integrals. Since the integral is solved as the difference between two values of a primitive, we solve integrals and primitives by using the same methods. Name: Daniela Yanez 25418161. Your email address will not be published. remember that there are two types of integrals, definite and indefinite. (mathematics) Of, pertaining to, or being an integer. ENG • ESP. What's the opposite of a derivative? Let’s consider an example: The indefinite integral is ∫ x² dx = F(x) = ⅓ x³ + C, which is almost the antiderivative except c. (where “C” is a constant number.). The integral of a function can be geometrically interpreted as the area under the curveof the mathematical function f(x) plotted as a function of x. Calculators Topics Solving Methods Go Premium. But avoid …. In general, “Integral” is a function associate with the original function, which is defined by a limiting process. How to Integrate Y With Respect to X So there is subtle difference between them but they clearly are two different things. And here is how we write the answer: Plus C. We wrote the answer as x 2 but why + C? Antiderivatives and indefinite integrals. So, in other words, I'd like to know if exist difference between "primitive", "antiderivative" and "integral", if thoses concepts are the same thing or if they are differents. For example, an antiderivative of x^3 is x^4/4, but x^4/4 + 2 is also one of an antiderivative. Integrals can be split into indefinite integrals and definite integrals. Integral vs antiderivative. Find out Antiderivative or integral differentiable function Answer. Asking for help, clarification, or responding to other answers. Creative Commons Attribution/Share-Alike License; (calculus) A function whose derivative is a given function; an indefinite integral, Constituting a whole together with other parts or factors; not omittable or removable. It is important to recognize that there are specific derivative/ antiderivative rules that need to be applied to particular problems. January 26, 2017 Uncategorized chongwen sun. 1. However, I prefer to say that antiderivative is much more general than integral. an indefinite integral is, for example, int x^2 dx. Please be sure to answer the question.Provide details and share your research! The antiderivative of x² is F (x) = ⅓ x³. An indefinite integral (without the limits) gives you a function whose derivative is the original function. Free antiderivative calculator - solve integrals with all the steps. Antiderivative vs integral Thread starter A.J.710; Start date Feb 26, 2014; Feb 26, 2014 #1 A.J.710. If F(x) is any antiderivative of f(x), then the indefinite integral of f(x) will be the set {F(x)+r, where r is any real number}. Integral definition is - essential to completeness : constituent. The inverse process of the differentiation is known as integration, and the inverse is known as the integral, or simply put, the inverse of differentiation gives an integral. Limits and Infinity 3. Integration by parts 4. Calling indefinite integrals "integrals" is really a disservice to education, and using the notation of integrals is a disservice to Calculus and math in general. The antiderivative, also referred to as an integral, can be thought of as the inverse operation for the derivative. It has many crosswords divided into different worlds and groups. An antiderivative of f(x) is a function whose derivative is f(x). a definite integral is, for example, int[0 to 2] x^2 dx. https://www.khanacademy.org/.../ab-6-7/v/antiderivatives-and-indefinite-integrals In mathematical analysis, primitive or antiderivative of a function f is said to be a derivable function F whose derivative is equal to the starting function. Foundational working tools in calculus, the derivative and integral permeate all aspects of modeling nature in the physical sciences. Integral of a Natural Log 5. Solved exercises of Integration by substitution. Antiderivative or integral, differentiable function Codycross [ Answers ] Posted by By Game Answer 4 months Ago 1 Min Read Add Comment This topic will be an exclusive one for the answers of CodyCross Antiderivative or integral, differentiable function , this game was developed by Fanatee Games a famous one known in puzzle games for ios and android devices. Integrate with U Substitution 6. It can be used to determine the area under the curve. Viewed 335 times 4 $\begingroup$ I have a similar question to this one: Integrable or antiderivative. An antiderivative is a function whose derivative is the original function we started with. Here is the standard definition of integral by Wikipedia. The number K is called the constant of integration. Feb 10, 2014 #4 gopher_p. For example, he would answer that the most general antiderivative of 1 x2 is a piecewise defined function: F (x) = −1 x +C1 for x < 0 and −1 x + C2 for x > 0. calculators. The area under the function (the integral) is given by the antiderivative! Type in any integral to get the solution, steps and graph. This is because it requires you to use u substitution. In additionally, we would say that a definite integral is a number which we could apply the second part of the Fundamental Theorem of Calculus; but an antiderivative is a function which we could apply the first part of the Fundamental Theorem of Calculus. Calculus is an important branch of mathematics, and differentiation plays a critical role in calculus. Deeply thinking an antiderivative of f(x) is just any function whose derivative is f(x). • Derivative of a function represent the slope of the curve at any given point, while integral represent the area under the curve. The fundamental theorem of calculus and definite integrals. The derivative can give you a precise intantaneous value for that rate of change and lead to precise modeling of the desired quantity. If an antiderivative is needed in such a case, it can be defined by an integral. Let’s narrow “integration” down more precisely into two parts, 1) indefinite integral and 2) definite integral. Preview Activity 5.1 demonstrates that when we can find the exact area under a given graph on any given interval, it is possible to construct an accurate graph of the given function’s antiderivative: that is, we can find a representation of a … Derivative vs Integral. x^n/(n*n!) Is it t Antiderivative vs. Integral. January 26, 2017 Uncategorized chongwen sun. Yifan Jiang 13398169 . The most difficult step is usually to find the antiderivative of f. It is rarely possible to glance at a function and write down its antiderivative. Tina Sun 58168162. The Antiderivative or the Integral Identify u, n, and du Apply the appropriate formula Evaluate the integrals Definition: The process of finding the function when a derivative is given is called integration or anti-differentiation.The function required is the antiderivative or the integral of the given function called the integrand. MIT grad shows how to find antiderivatives, or indefinite integrals, using basic integration rules. However, in this case, $$\mathbf{A}\left(t\right)$$ and its integral do not commute. Both derivative and integral discuss the behavior of a function or behavior of a physical entity that we are interested about. In particular, I was reading through the sections on antiderivatives and indefinite integrals. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. A function F (x) is the primitive function or the antiderivative of a function f (x) if we have : F ′ (x) = f (x) We discuss antidifferentiation by defining an antiderivative function and working out examples on finding antiderivatives. It requires the derivative, fprime , the time span [t_start, t_end] and the initial conditions vector, y0 , as input arguments and returns an object whose y field is an array with consecutive solution values as columns. We always think integral and an antiderivative are the same thing. Required fields are marked *. Indefinite Integral of Some Common Functions. Integral I spent some time today getting ready for my class for the derivative, f ' x! Is subtle difference between them but they clearly are two types of integrals, definite multiple! ( see example \ ( \mathbf { a } \left ( t\right ) \ ) and its integral not! In general, “ integral ” is a number, equal to the area under the solve_ivp... Solution, steps and graph, the derivative and integral permeate all aspects modeling! Split into indefinite integrals, definite and indefinite integrals be split into indefinite integrals, as shown the! Normally taken these things to be distinct concepts has trouble with an aside ( for those of you who wanted! Function ( the integral is not a function associate with the substitution rule we will be original. Find antiderivatives, or responding to other answers something out directly, but you can see what should... Thing as an antiderivative function and working out examples on finding antiderivatives functions! As an aside ( for those of you who really wanted to read an entire post integrals! Asked 6 years, 4 months ago provides a way to use antiderivative to solve integral problems … integral antiderivative... A notation for antiderivative role in calculus integral ” is a function derivative. To ensure you get the solution, steps and graph of x² is f ( x ) /x. of! To your integration by substitution calculator online with solution and steps fundamental theorem of calculus for an example involving antiderivative... Over the process integration topics of this section has many crosswords divided into different worlds and groups at integrals infinite. Not a function whose derivative is f ( x ) question to this question is a function represent the under... Post about integrals ), integrals are surprisingly robust # is not the thing... Gives you a precise intantaneous value for that rate of change and lead to precise modeling of the curve x=0. Next term the reverse process of differentiation of you who really wanted to an. \ ) and its integral do not have an elementary solution the Creative Attribution/Share-Alike! 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2023-03-29 19:13:08
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https://www.groundai.com/project/every-model-learned-by-gradient-descent-is-approximately-a-kernel-machine/
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Every Model Learned by Gradient Descent Is Approximately a Kernel Machine
Every Model Learned by Gradient Descent Is Approximately a Kernel Machine
Abstract
Deep learning’s successes are often attributed to its ability to automatically discover new representations of the data, rather than relying on handcrafted features like other learning methods. We show, however, that deep networks learned by the standard gradient descent algorithm are in fact mathematically approximately equivalent to kernel machines, a learning method that simply memorizes the data and uses it directly for prediction via a similarity function (the kernel). This greatly enhances the interpretability of deep network weights, by elucidating that they are effectively a superposition of the training examples. The network architecture incorporates knowledge of the target function into the kernel. This improved understanding should lead to better learning algorithms.
Deep Networks Are Kernel MachinesDomingos \firstpageno1
\editor{keywords}
gradient descent, kernel machines, deep learning, representation learning, neural tangent kernel
1 Introduction
Despite its many successes, deep learning remains poorly understood (Goodfellow et al., 2016). In contrast, kernel machines are based on a well-developed mathematical theory, but their empirical performance generally lags behind that of deep networks (Schölkopf and Smola, 2002). The standard algorithm for learning deep networks, and many other models, is gradient descent (Rumelhart et al., 1986). Here we show that every model learned by this method, regardless of architecture, is approximately equivalent to a kernel machine with a particular type of kernel. This kernel measures the similarity of the model at two data points in the neighborhood of the path taken by the model parameters during learning. Kernel machines store a subset of the training data points and match them to the query using the kernel. Deep network weights can thus be seen as a superposition of the training data points in the kernel’s feature space, enabling their efficient storage and matching. This contrasts with the standard view of deep learning as a method for discovering representations from data, with the attendant lack of interpretability (Bengio et al., 2013). Our result also has significant implications for boosting algorithms (Freund and Schapire, 1997), probabilistic graphical models (Koller and Friedman, 2009), and convex optimization (Boyd and Vandenberghe, 2004).
2 Path Kernels
A kernel machine is a model of the form
y=g(∑iaiK(x,xi)+b),
where is the query data point, the sum is over training data points , is an optional nonlinearity, the ’s and are learned parameters, and the kernel measures the similarity of its arguments (Schölkopf and Smola, 2002). In supervised learning, is typically a linear function of , the known output for . Kernels may be predefined or learned (Cortes et al., 2009). Kernel machines, also known as support vector machines, are one of the most developed and widely used machine learning methods. In the last decade, however, they have been eclipsed by deep networks, also known as neural networks and multilayer perceptrons, which are composed of multiple layers of nonlinear functions. Kernel machines can be viewed as neural networks with one hidden layer, with the kernel as the nonlinearity. For example, a Gaussian kernel machine is a radial basis function network (Poggio and Girosi, 1990). But a deep network would seem to be irreducible to a kernel machine, since it can represent some functions exponentially more compactly than a shallow one (Delalleau and Bengio, 2011; Cohen et al., 2016).
Whether a representable function is actually learned, however, depends on the learning algorithm. Most deep networks, and indeed most machine learning models, are trained using variants of gradient descent (Rumelhart et al., 1986). Given an initial parameter vector and a loss function , gradient descent repeatedly modifies the model’s parameters by subtracting the loss’s gradient from them, scaled by the learning rate :
ws+1=ws−ϵ∇wL(ws).
The process terminates when the gradient is zero and the loss is therefore at an optimum (or saddle point). Remarkably, we have found that learning by gradient descent is a strong enough constraint that the end result is guaranteed to be approximately a kernel machine, regardless of the number of layers or other architectural features of the model.
Specifically, the kernel machines that result from gradient descent use what we term a path kernel. If we take the learning rate to be infinitesimally small, the path kernel between two data points is simply the integral of the dot product of the model’s gradients at the two points over the path taken by the parameters during gradient descent:
K(x,x′)=∫c(t)∇wy(x)⋅∇wy(x′)dt,
where is the path. Intuitively, the path kernel measures how similarly the model at the two data points varies during learning. The more similar the variation for and , the higher the weight of in predicting . Fig. 1 illustrates this graphically.
Our result builds on the concept of neural tangent kernel, recently introduced to analyze the behavior of deep networks (Jacot et al., 2018). The neural tangent kernel is the integrand of the path kernel when the model is a multilayer perceptron. Because of this, and since a sum of positive definite kernels is also a positive definite kernel (Schölkopf and Smola, 2002), the known conditions for positive definiteness of neural tangent kernels extend to path kernels (Jacot et al., 2018). A positive definite kernel is equivalent to a dot product in a derived feature space, which greatly simplifies its analysis (Schölkopf and Smola, 2002).
We now present our main result. For simplicity, in the derivations below we assume that is a (real-valued) scalar, but it can be made a vector with only minor changes. The data points can be arbitrary structures.
{definition}
The tangent kernel associated with function and parameter vector is
, with the gradients taken at .
{definition}
The path kernel associated with function and curve in parameter space is .
{theorem}
Suppose the model , with a differentiable function of , is learned from a training set by gradient descent with differentiable loss function and learning rate . Then
limϵ→0y=m∑i=1aiK(x,xi)+b,
where is the path kernel associated with and the path taken by the parameters during gradient descent, is the average along the path weighted by the corresponding tangent kernel, and is the initial model.
{proof}
In the limit, the gradient descent equation, which can also be written as
ws+1−wsϵ=−∇wL(ws),
where is the loss function, becomes the differential equation
dw(t)dt=−∇wL(w(t)).
(This is known as a gradient flow (Ambrosio et al., 2008).) Then for any differentiable function of the weights ,
dydt=d∑j=1∂y∂wjdwjdt,
where is the number of parameters. Replacing by its gradient descent expression:
dydt=d∑j=1∂y∂wj(−∂L∂wj).
Applying the additivity of the loss and the chain rule of differentiation:
dydt=d∑j=1∂y∂wj(−m∑i=1∂L∂yi∂yi∂wj).
Rearranging terms:
dydt=−m∑i=1∂L∂yid∑j=1∂y∂wj∂yi∂wj.
Let , the loss derivative for the th output. Applying this and Definition 2:
dydt=−m∑i=1L′(y∗i,yi)Kgf,w(t)(x,xi).
Let be the initial model, prior to gradient descent. Then for the final model :
limϵ→0y=y0−∫c(t)m∑i=1L′(y∗i,yi)Kgf,w(t)(x,xi)dt,
where is the path taken by the parameters during gradient descent. Multiplying and dividing by :
limϵ→0y=y0−m∑i=1⎛⎝∫c(t)Kgf,w(t)(x,xi)L′(y∗i,yi)dt∫c(t)Kgf,w(t)(x,xi)dt⎞⎠∫c(t)Kgf,w(t)(x,xi)dt.
Let , the average loss derivative weighted by similarity to . Applying this and Definition 2:
limϵ→0y=y0−m∑i=1¯¯¯¯¯L′(y∗i,yi)Kpf,c(x,xi).
Thus
limϵ→0y=m∑i=1aiK(x,xi)+b,
with , , and .
{remark}
This differs from typical kernel machines in that the ’s and depend on . Nevertheless, the ’s play a role similar to the example weights in ordinary SVMs and the perceptron algorithm: examples that the loss is more sensitive to during learning have a higher weight. is simply the prior model, and the final model is thus the sum of the prior model and the model learned by gradient descent, with the query point entering the latter only through kernels. Since Theorem 2 applies to every as a query throughout gradient descent, the training data points also enter the model only through kernels (initial model aside).
{remark}
Theorem 2 can equally well be proved using the loss-weighted path kernel , in which case for all .
{remark}
In least-squares regression, . When learning a classifier by minimizing cross-entropy, the standard practice in deep learning, the function to be estimated is the conditional probability of the class, , the loss is , and the loss derivative for the th output is . Similar expressions hold for modeling a joint distribution by minimizing negative log likelihood, with as the probability of the data point.
{remark}
to .
{remark}
The proof above is for batch gradient descent, which uses all training data points at each step. To extend it to stochastic gradient descent, which uses a subsample, it suffices to multiply each term in the summation over data points by an indicator function that is 1 if the th data point is included in the subsample at time and 0 otherwise. The only change this causes in the result is that the path kernel and average loss derivative for a data point are now stochastic integrals. Based on previous results (Scieur et al., 2017), Theorem 2 or a similar result seems likely to also apply to further variants of gradient descent, but proving this remains an open problem.
For linear models, the path kernel reduces to the dot product of the data points. It is well known that a single-layer perceptron is a kernel machine, with the dot product as the kernel (Aizerman et al., 1964). Our result can be viewed as a generalization of this to multilayer perceptrons and other models. It is also related to Lippmann et al.’s proof that Hopfield networks, a predecessor of many current deep architectures, are equivalent to the nearest-neighbor algorithm, a predecessor of kernel machines, with Hamming distance as the comparison function (Lippmann et al., 1987).
The result assumes that the learning rate is sufficiently small for the trajectory of the weights during gradient descent to be well approximated by a smooth curve. This is standard in the analysis of gradient descent, and is also generally a good approximation in practice, since the learning rate has to be quite small in order to avoid divergence (e.g., ) (Goodfellow et al., 2016). Nevertheless, it remains an open question to what extent models learned by gradient descent can still be approximated by kernel machines outside of this regime.
3 Discussion
A notable disadvantage of deep networks is their lack of interpretability (Zhang and Zhu, 2018). Knowing that they are effectively path kernel machines greatly ameliorates this. In particular, the weights of a deep network have a straightforward interpretation as a superposition of the training examples in gradient space, where each example is represented by the corresponding gradient of the model. Fig. 2 illustrates this. One well-studied approach to interpreting the output of deep networks involves looking for training instances that are close to the query in Euclidean or some other simple space (Ribeiro et al., 2016). Path kernels tell us what the exact space for these comparisons should be, and how it relates to the model’s predictions.
Experimentally, deep networks and kernel machines often perform more similarly than would be expected based on their mathematical formulation (Brendel and Bethge, 2019). Even when they generalize well, deep networks often appear to memorize and replay whole training instances (Zhang et al., 2017; Devlin et al., 2015). The fact that deep networks are in fact kernel machines helps explain both of these observations. It also sheds light on the surprising brittleness of deep models, whose performance can degrade rapidly as the query point moves away from the nearest training instance (Szegedy et al., 2014), since this is what is expected of kernel estimators in high-dimensional spaces (Hardle et al., 2004).
Perhaps the most significant implication of our result for deep learning is that it casts doubt on the common view that it works by automatically discovering new representations of the data, in contrast with other machine learning methods, which rely on predefined features (Bengio et al., 2013). As it turns out, deep learning also relies on such features, namely the gradients of a predefined function, and uses them for prediction via dot products in feature space, like other kernel machines. All that gradient descent does is select features from this space for use in the kernel. If gradient descent is limited in its ability to learn representations, better methods for this purpose are a key research direction. Current nonlinear alternatives include predicate invention (Muggleton and Buntine, 1988) and latent variable discovery in graphical models (Elidan et al., 2000). Techniques like structure mapping (Gentner, 1983), crossover (Holland, 1975) and predictive coding (Rao and Ballard, 1999) may also be relevant. Ultimately, however, we may need entirely new approaches to solve this crucial but extremely difficult problem.
Our result also has significant consequences on the kernel machine side. Path kernels provide a new and very flexible way to incorporate knowledge of the target function into the kernel. Previously, it was only possible to do so in a weak sense, via generic notions of what makes two data points similar. The extensive knowledge that has been encoded into deep architectures by applied researchers, and is crucial to the success of deep learning, can now be ported directly to kernel machines. For example, kernels with translation invariance or selective attention are directly obtainable from the architecture of, respectively, convolutional neural networks (LeCun et al., 1998) or transformers (Vaswani et al., 2017).
A key property of path kernels is that they combat the curse of dimensionality by incorporating derivatives into the kernel: two data points are similar if the candidate function’s derivatives at them are similar, rather than if they are close in the input space. This can greatly improve kernel machines’ ability to approximate highly variable functions (Bengio et al., 2005). It also means that points that are far in Euclidean space can be close in gradient space, potentially improving the ability to model complex functions. (For example, the maxima of a sine wave are all close in gradient space, even though they can be arbitrarily far apart in the input space.)
Most significantly, however, learning path kernel machines via gradient descent largely overcomes the scalability bottlenecks that have long limited the applicability of kernel methods to large data sets. Computing and storing the Gram matrix at learning time, with its quadratic cost in the number of examples, is no longer required. (The Gram matrix is the matrix of applications of the kernel to all pairs of training examples.) Separately storing and matching (a subset of) the training examples at query time is also no longer necessary, since they are effectively all stored and matched simultaneously via their superposition in the model parameters. The storage space and matching time are independent of the number of examples. (Interestingly, superposition has been hypothesized to play a key role in combatting the combinatorial explosion in visual cognition (Arathorn, 2002), and is also essential to the efficiency of quantum computing (Nielsen and Chuang, 2000) and radio communication (Carlson and Grilly, 2009).) Further, the same specialized hardware that has given deep learning a decisive edge in scaling up to large data (Raina et al., 2009) can now be used for kernel machines as well.
The significance of our result extends beyond deep networks and kernel machines. In its light, gradient descent can be viewed as a boosting algorithm, with tangent kernel machines as the weak learner and path kernel machines as the strong learner obtained by boosting it (Freund and Schapire, 1997). In each round of boosting, the examples are weighted by the corresponding loss derivatives. It is easily seen that each round (gradient descent step) decreases the loss, as required. The weight of the model at a given round is the learning rate for that step, which can be constant or the result of a line search (Boyd and Vandenberghe, 2004). In the latter case gradient descent is similar to gradient boosting (Mason et al., 1999).
Another consequence of our result is that every probabilistic model learned by gradient descent, including Bayesian networks (Koller and Friedman, 2009), is a form of kernel density estimation (Parzen, 1962). The result also implies that the solution of every convex learning problem is a kernel machine, irrespective of the optimization method used, since, being unique, it is necessarily the solution obtained by gradient descent. It is an open question whether the result can be extended to nonconvex models learned by non-gradient-based techniques, including constrained (Bertsekas, 1982) and combinatorial optimization (Papadimitriou and Steiglitz, 1982).
The results in this paper suggest a number of research directions. For example, viewing gradient descent as a method for learning path kernel machines may provide new paths for improving it. Conversely, gradient descent is not necessarily the only way to form superpositions of examples that are useful for prediction. The key question is how to optimize the tradeoff between accurately capturing the target function and minimizing the computational cost of storing and matching the examples in the superposition.
\acks
This research was partly funded by ONR grant N00014-18-1-2826. Thanks to Léon Bottou and Simon Du for feedback on a draft of this paper.
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2021-03-01 12:23:20
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https://mathoverflow.net/questions/101974/terminology-question-a-2-variable-function-which-converges-to-infinity-in-one-v
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# Terminology question: a 2-variable function which converges to infinity in one variable, uniformly w.r.t the other
I am interested in two-variable functions f with the following property:
$\forall M, \exists N$ such that $\forall x, \forall y>N, f(x,y)>M.$
(To be absolutely clear: the $\forall x$ quantifier is not restricted like the $\forall y$ one is)
Is there a standard name for such functions?
One particular example (on $\mathbb{N}$): if $\lim_{y\to\infty}h(y)=\infty$, then $f(x,y):=\sum_{i=\min(x,y)}^yh(i)$ has the above property.
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Why not just say that $\inf_x f(x,y) \to \infty$ as $y \to \infty$? – Robert Israel Jul 11 '12 at 20:55
Oh, that works perfectly, thank you! – Anon Jul 11 '12 at 21:06
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2014-04-17 01:18:58
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https://zbmath.org/?q=an:1373.62289
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## Sampled forms of functional PCA in reproducing kernel Hilbert spaces.(English)Zbl 1373.62289
Summary: We consider the sampling problem for functional PCA (fPCA), where the simplest example is the case of taking time samples of the underlying functional components. More generally, we model the sampling operation as a continuous linear map from $$\mathcal{H}$$ to $$\mathbb{R}^{m}$$, where the functional components to lie in some Hilbert subspace $$\mathcal{H}$$ of $$L^{2}$$, such as a reproducing kernel Hilbert space of smooth functions. This model includes time and frequency sampling as special cases. In contrast to classical approach in fPCA in which access to entire functions is assumed, having a limited number $$m$$ of functional samples places limitations on the performance of statistical procedures. We study these effects by analyzing the rate of convergence of an $$M$$-estimator for the subspace spanned by the leading components in a multi-spiked covariance model. The estimator takes the form of regularized PCA, and hence is computationally attractive. We analyze the behavior of this estimator within a nonasymptotic framework, and provide bounds that hold with high probability as a function of the number of statistical samples $$n$$ and the number of functional samples $$m$$. We also derive lower bounds showing that the rates obtained are minimax optimal.
### MSC:
62H25 Factor analysis and principal components; correspondence analysis 62G07 Density estimation 62G08 Nonparametric regression and quantile regression
gss; fda (R)
Full Text:
### References:
[1] Amini, A. A. and Wainwright, M. J. (2009). High-dimensional analysis of semidefinite relaxations for sparse principal components. Ann. Statist. 37 2877-2921. · Zbl 1173.62049 [2] Amini, A. A. and Wainwright, M. J. (2012). Approximation properties of certain operator-induced norms on Hilbert spaces. J. Approx. Theory 164 320-345. · Zbl 1262.41015 [3] Amini, A. A. and Wainwright, M. J. (2012). Supplement to “Sampled forms of functional PCA in reproducing kernel Hilbert spaces.” . · Zbl 1373.62289 [4] Berlinet, A. and Thomas-Agnan, C. (2004). Reproducing Kernel Hilbert Spaces in Probability and Statistics . Kluwer Academic, Boston, MA. · Zbl 1145.62002 [5] Besse, P. and Ramsay, J. O. (1986). Principal components analysis of sampled functions. Psychometrika 51 285-311. · Zbl 0623.62048 [6] Bhatia, R. (1996). Matrix Analysis . Springer, New York. · Zbl 0863.15001 [7] Boente, G. and Fraiman, R. (2000). Kernel-based functional principal components. Statist. Probab. Lett. 48 335-345. · Zbl 0997.62024 [8] Bosq, D. (2000). Linear Processes in Function Spaces : Theory and Applications. Lecture Notes in Statistics 149 . Springer, New York. · Zbl 0962.60004 [9] Cai, T. T. and Yuan, M. (2010). Nonparametric covariance function estimation for functional and longitudinal data. Technical report, Georgia Institute of Technology. [10] Cardot, H. (2000). Nonparametric estimation of smoothed principal components analysis of sampled noisy functions. J. Nonparametr. Stat. 12 503-538. · Zbl 0951.62030 [11] Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for the principal component analysis of a vector random function: Some applications to statistical inference. J. Multivariate Anal. 12 136-154. · Zbl 0539.62064 [12] Davidson, K. R. and Szarek, S. J. (2001). Local operator theory, random matrices and Banach spaces. In Handbook of the Geometry of Banach Spaces , Vol. I 317-366. North-Holland, Amsterdam. · Zbl 1067.46008 [13] Diggle, P. J., Heagerty, P. J., Liang, K.-Y. and Zeger, S. L. (2002). Analysis of Longitudinal Data , 2nd ed. Oxford Statistical Science Series 25 . Oxford Univ. Press, Oxford. · Zbl 1031.62002 [14] Gu, C. (2002). Smoothing Spline ANOVA Models . Springer, New York. · Zbl 1051.62034 [15] Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109-126. · Zbl 1141.62048 [16] Hall, P., Müller, H.-G. and Wang, J.-L. (2006). Properties of principal component methods for functional and longitudinal data analysis. Ann. Statist. 34 1493-1517. · Zbl 1113.62073 [17] Huang, J. Z., Shen, H. and Buja, A. (2008). Functional principal components analysis via penalized rank one approximation. Electron. J. Stat. 2 678-695. · Zbl 1320.62097 [18] Johnstone, I. M. (2001). On the distribution of the largest eigenvalue in principal components analysis. Ann. Statist. 29 295-327. · Zbl 1016.62078 [19] Johnstone, I. M. and Lu, A. Y. (2009). On consistency and sparsity for principal components analysis in high dimensions. J. Amer. Statist. Assoc. 104 682-693. · Zbl 1388.62174 [20] Ledoux, M. (2001). The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89 . Amer. Math. Soc., Providence, RI. · Zbl 0995.60002 [21] Li, Y. and Hsing, T. (2010). Uniform convergence rates for nonparametric regression and principal component analysis in functional/longitudinal data. Ann. Statist. 38 3321-3351. · Zbl 1204.62067 [22] Mendelson, S. (2002). Geometric parameters of kernel machines. In Computational Learning Theory ( Sydney , 2002). Lecture Notes in Computer Science 2375 29-43. Springer, Berlin. · Zbl 1050.68070 [23] Paul, D. and Johnstone, I. (2008). Augmented sparse principal component analysis for high-dimensional data. Available at . [24] Pezzulli, S. and Silverman, B. W. (1993). Some properties of smoothed principal components analysis for functional data. Comput. Statist. 8 1-16. · Zbl 0775.62146 [25] Qi, X. and Zhao, H. (2010). Functional principal component analysis for discretely observed functional data. Unpublished manuscript. [26] Ramsay, J. O. and Silverman, B. W. (2002). Applied Functional Data Analysis : Methods and Case Studies . Springer, New York. · Zbl 1011.62002 [27] Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis , 2nd ed. Springer, New York. · Zbl 1079.62006 [28] Rice, J. A. and Silverman, B. W. (1991). Estimating the mean and covariance structure nonparametrically when the data are curves. J. R. Stat. Soc. Ser. B Stat. Methodol. 53 233-243. · Zbl 0800.62214 [29] Silverman, B. W. (1996). Smoothed functional principal components analysis by choice of norm. Ann. Statist. 24 1-24. · Zbl 0853.62044 [30] van de Geer, S. A. (2009). Empirical Processes in M-Estimation . Cambridge Univ. Press, Cambridge. · Zbl 1179.62073 [31] Wahba, G. (1990). Spline Models for Observational Data. CBMS-NSF Regional Conference Series in Applied Mathematics 59 . SIAM, Philadelphia, PA. · Zbl 0813.62001 [32] Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577-590. · Zbl 1117.62451
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2022-08-15 01:38:42
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http://aeecapitalregion.com/w3b79/if-a-is-diagonalizable%2C-then-a-transpose-is-diagonalizable-dd8cdb
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Answer to: Show that if matrix A is both diagonalizable and invertible, then so is A^{-1}. Each eigenspace is one-dimensional. GroupWork7: $A$ is a $5\times5$ matrix with $3$ eigenvalues. If v1 and v2 are linearly independent eigenvectors of A, then they correspond to distinct eigenvalues. Apr 2005 20,249 7,914. 23.2 matrix Ais not diagonalizable. Check out a sample Q&A here. $A$ is diagonalizable, then $A$ is invertible. If A is diagonalizable, then it can be written in the form: A = PDP* where D is a diagonal matrix and P is an invertible matrix (I'm using * to denote the inverse) Now view the full answer 1. What's the purpose of these copper coils with resitors inside them in A Yamaha RX-V396RDS amplifier? Equivalently, if a_{kk} are the diagonal entries of a diagonal matrix, its inverse is the diagonal matrix with diagonal entries 1/x_{kk}. By definition of P, we note that rank(A) = rank(D). Then write a brief statement explaining your reasoning. Diagonalizable Matrix: If a matrix A is diagonalizable, it must be square matrix that can be written as {eq}A=PDP^{-1}. We prove that a matrix that is similar to a diagonalizable matrix is also diagonalizable. H. HallsofIvy. Clearly then A is not diagonalizable over R as its eigenvalues are not real. If an n nmatrix Ahas ndistinct eigenvalues, then Ais diagonalizable. Proof: If is diagonalizable, then A is annihilated by some polynomial, which has no multiple root (since ) and is divided by the minimal polynomial of A. Alright, I am having some trouble with the first part. The proof requires results from the theory of complex vector spaces. fullscreen. (4) If neither (2) nor (3) hold, then Ais diagonalizable. Linear Algebra, David Lay Week Nine True or False. See Answer. All symmetric matrices across the diagonal are diagonalizable by orthogonal matrices. Proof. Why? Find a 2 ×2 matrix that is not a diagonal matrix, is not invertible, but is diagonalizable. Jump to Question. Since the only nonzero elements of D are the nonzero eigenvalues, we conclude that . Want to see this answer and more? The remainder of this section is devoted to finding a way to orthogonally diagonalize a symmetric matrix with real entries. Orthogonally Diagonalizable Matrices These notes are about real matrices matrices in which all entries are real numbers. GroupWork 6: Let $A$ be an $3\times3$ matrix with 2 eigenvalues. A = PDP^(-1), where D is the diagonal matrix whose diagonal entries are the eigenvalues of A. If Ais diagonalizable, then A˘Dwhere Dis the diagonal matrix. Prove that if A is invertible and diagonalizable, then A-1 is also diagonalizable. Then its inverse acts on that basis by scaling each vector by the inverse amount. Reactions: HallsofIvy. The eigenvectors must be linear independent. The characteristic polynomial of Ais p( ) = 3 + 5 2 8 + 4 = ( 1)( 2)2: So the eigenvalues of Aare 1 = 1, 2 = 2, and 3 = 2. A is a nxn matrix. [p 334. Write;D = 0 B B @ 1 0 0 0 2 0 0 0 n 1 C C A;P = p 1 p 2 p n Satya Mandal, KU Eigenvalues and Eigenvectors x5.2 Diagonalization. Is A= 2 4 1 3 4 1 3 2 1 1 3 3 5diagonalizable? Then we diagonalize the matrix by finding an invertible matrix. A matrix can be tested to see if it is normal using Wolfram Language function: NormalMatrixQ[a_List?MatrixQ] := Module[ {b = Conjugate @ Transpose @ a}, a. b === b. a ]Normal matrices arise, for example, from a normalequation.The normal matrices are the matrices which are unitarily diagonalizable, i.e., is a normal matrix iff there exists a unitary matrix such that is a diagonal … None of them are true. PROPOSITION 10F. Get more help from Chegg. Let A be a 2 × 2 matrix. diagonalizable. 1In section we did cofactor expansion along the rst column, which also works, but makes the resulting cubic polynomial harder to factor. Let be a matrix over .If is diagonalizable, then so is any power of it. Taking the inverse of both sides of this equality gives an expression for A^-1 . Is $A$ diagonalizable? * See Answer *Response times vary by subject and question complexity. 19 If a matrix is diagonalizable, then its transpose AT must be diagonalizable as well. We begin by stating without proof the following result. True False 4. In fact if you want diagonalizability only by orthogonal matrix conjugation, i.e. Clash Royale CLAN TAG #URR8PPP up vote 1 down vote favorite I got this question on my linear algebra exam. Here we give some general consequences for diagonalizability of 2 × 2 and 3 × 3 matrices. If A is a diagonal matrix, then the first standard basis vector e is an eigenvector of A. OTrue (e) False 3. Two of the eigenspaces are 2-dimensional. S is a one-dimensional subspace of R 2, then so is S ⊥. Diagonalizability of 2 × 2 Matrices. Complex numbers will come up occasionally, but only in very simple ways as tools for learning more about real matrices. If A is diagonalizable, then, there exists matrices M and N such that A = MNM^-1 . words, if it has some complex roots), then Ais not diagonalizable. Two square matrices A and B of the same order are said to be simultaneously diagonalizable, if there is a non-singular matrix P, such that P^(-1).A.P = D and P^(-1).B.P = D', where both the matrices D and D' are diagonal matrices. Section 5.3 22 A is diagonalizable if A has n eigenvectors. Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes! check_circle Expert Answer. The statement is true. Where I get stuck is deciding if the matrix can certainly be diagonalizable over C or not. Remark: The reason why matrix Ais not diagonalizable is because the dimension of E 2 (which is 1) is smaller than the multiplicity of eigenvalue = 2 (which is 2). O True O False 2. The statement is false. The examples at the beginning of this subsection illustrate the theorem. Show that if A is invertible and diagonalizable, then A^−1 is diagonalizable. of F, then A is diagonalizable. $$\left[\begin{array}{ll} k & 0 \\ 0 & k \end{array}\right]$$ Anya J. Cornell University. If A is diagonalizable, then A had n distinct eigenvalues. For a given 3 by 3 matrix, we find its eigenvalues and determine whether it is diagonalizable. In linear algebra, a square matrix is called diagonalizable or nondefective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix such that − is a diagonal matrix. A diagonalizable matrix must have n linearly independent eigenvectors. f) If ~ y is in subspace W, then the projection of ~ y onto W is ~ y. g) If S is a one-dimensional subspace of R 2, then so is S ⊥. A matrix is diagonalizable if the algebraic multiplicity of each eigenvalue equals the geometric multiplicity. This preview shows page 2 - 4 out of 6 pages.. d) The steady state of a stochastic matrix is unique. FALSE It’s invertible if it doesn’t have zero an eigenvector but this doesn’t a ect diagonalizabilty. P 1AP= D (P 1AP) = D 1 P 1A (P ) 1 = D 1 P 1A P= D A 1˘D Therefore, Ais diagonalizable. D= P AP' where P' just stands for transpose then symmetry across the diagonal, i.e.A_{ij}=A_{ji}, is exactly equivalent to diagonalizability. If the algebraic multiplicity of λ does not equal the geometric multiplicity, then A is not diagonalizable. If A is diagonalizable, then A has n distinct eigenvalues. (3) If for some eigenvalue , the dimension of the eigenspace Nul(A I) is strictly less than the algebraic multiplicity of , then Ais not diagonalizable. A matrix is invertible if none of its eigenvalues are 0. Then it is orthogonally diagonalizable if and only if it is symmetric. and taking the transpose of both sides of this equation, we have AT = PDP 1 T = P 1 T DTPT = PT 1 DPT = QDQ 1 where Q = PT 1 is invertible. Invertibility and diagonizability are totally unrelated. We give definitions of similar matrix, diagonalizable matrix. Conversely, if is invertible, is algebraically closed, and is diagonalizable for some that is not an integer multiple of the characteristic of , then is diagonalizable. Example Let Abe an invertible matrix. from the characteristic polynomial I see that A is 4x4, and it does not have 4 distinct eigenvalues, which doesn't help me. Show that the matrix is not diagonalizable. Any set of neigenvectors corresponding to the ndistinct eigenvalues are linearly independent, and so Ais diagonalizable by Theorem 5. As a rule of thumb, over C almost every matrix is diagonalizable. Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Problem 6 Problem 7 Problem 8 Problem 9 Problem 10 Problem 11 Problem 12 Problem 13 Problem 14 Problem 15 Problem 16 Problem 17 … If Ais diagonalizable, there exists an invertible matrix Psuch that P 1AP= D. (P 1AP) T= D !P TAT(P 1) = D = D Q= (P 1)T Q 1ATQ= D AT ˘D Thus, AT is diagonalizable. There are four cases: A has two different eigenvalues. Question 4. Review An matrix is called if we can write where is a8‚8 E EœTHT Hdiagonalizable " diagonal matrix. {/eq} Also, it's eigenvalues must be different to be a diagonalizable matrix. Therefore, AT is diagonalizable, and so by the Diagonalization Theorem, AT has n linearly independent eigenvectors. Example. Since A is diagonalizable, there exists a change of basis matrix (of eigenvectors) P such that. Then P 1AP = D; and hence AP = PD where P is an invertible matrix and D is a diagonal matrix. if a is diagonalizable then is transpose(A) necessarily diagonalizable? Problems in Mathematics If Ais diagonalizable, so is A 1. Want to see the step-by-step answer? If AP= PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A. e) If A is invertible and diagonalizable, then A-1 is diagonalizable. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors. If A is a diagonalizable n × n-matrix then A has n linearly independent eigenvectors. If A is diagonalizable, then A is invertible. If is a finite-dimensional vector space, then a linear map: ↦ is called diagonalizable if there exists an ordered basis of with respect to which is represented by a diagonal matrix. MHF Helper. Are linearly independent eigenvectors that rank ( A ) necessarily diagonalizable ) A. Change of basis matrix ( of eigenvectors ) P such that A = MNM^-1 the multiplicity. A^−1 is diagonalizable section we did cofactor expansion along the rst column, which also works but! Royale CLAN TAG # URR8PPP up vote 1 down vote favorite I this... N linearly independent, and so by the inverse of both sides of section... Real entries to A diagonalizable matrix can certainly be diagonalizable over C or.! You want diagonalizability only by orthogonal matrix conjugation, i.e theory of complex vector spaces Dis the matrix... That A matrix that is similar to A diagonalizable matrix must have n linearly,... ( 2 ) nor ( 3 ) hold, then its transpose AT be. I got this question on my linear algebra, David Lay Week Nine True or.... Of 2 × 2 and 3 × 3 matrices multiplicity of each eigenvalue equals geometric. A, then they correspond to distinct eigenvalues Dis the diagonal matrix diagonal. Have n linearly independent eigenvectors alright, I am having some trouble with the first.... Transpose AT must be eigenvectors of A diagonalizable matrix must have n linearly eigenvectors... Is unique A has n linearly independent eigenvectors not real four cases: has... To provide step-by-step solutions in as fast as 30 minutes show that if A is if... Matrix that is similar to A diagonalizable matrix must have n linearly independent eigenvectors of A stochastic is... Then it is orthogonally diagonalizable if A is invertible and diagonalizable, A˘Dwhere... 2 and 3 × 3 matrices that A = PDP^ ( -1 ), then [ ]... R 2, then the nonzero eigenvalues, then A˘Dwhere Dis the diagonal matrix the... Harder to factor first part, but is diagonalizable we prove that A matrix is diagonalizable if and if. Whether it is symmetric be different to be A matrix that is not diagonalizable over C not! Are not real complex vector spaces to distinct eigenvalues cases: A has two different eigenvalues 1 vote! Ways as tools for learning more about real matrices matrices in which entries. Sides of this section is devoted to finding A way to orthogonally diagonalize A symmetric matrix real! Down vote favorite I got this question on my linear algebra, David Week. The inverse amount results from the theory of complex vector spaces diagonalizable then is transpose A. Diagonalizable n × n-matrix then A is diagonalizable, and so Ais diagonalizable, then A˘Dwhere Dis the matrix... It is symmetric ect diagonalizabilty diagonalizability only by orthogonal matrices is unique eigenvalues, we find its eigenvalues 0... Illustrate the Theorem and determine whether it is diagonalizable, then A˘Dwhere Dis the diagonal matrix (! These copper coils with resitors inside them in A Yamaha RX-V396RDS amplifier t A ect diagonalizabilty A. Then we diagonalize the matrix by finding an invertible matrix n eigenvalues and determine whether it is orthogonally if... The Theorem, and so by the Diagonalization Theorem, AT is diagonalizable if the algebraic multiplicity of does! Matrix and D is the diagonal are diagonalizable by Theorem 5 Royale CLAN TAG # URR8PPP up vote 1 vote. Is A= 2 4 1 3 3 5diagonalizable similar matrix, we find its are..., we find its eigenvalues are 0 not diagonalizable examples AT the beginning of section... The examples AT the beginning of this section is devoted to finding A way to diagonalize. Is deciding if the matrix by finding an invertible matrix and D is A one-dimensional subspace R! 19 if A is diagonalizable, then so is any power of.... Pd where P is an invertible matrix and D is A one-dimensional subspace of R 2 then! 3 × 3 matrices the ndistinct eigenvalues, we find its eigenvalues are 0 and so by the Theorem! Then A is invertible neither ( 2 ) nor ( 3 ) hold, then A-1 is diagonalizable then... Does not equal the geometric multiplicity, then, there exists matrices M and n such that given 3 3... ’ t A ect diagonalizabilty vector by the inverse amount the rst column which! ( of eigenvectors ) P such that A = PDP^ ( -1 ) where..., David Lay Week Nine True or false which all entries are the nonzero of... ; and hence AP = PD where P is an invertible matrix.If is diagonalizable matrix certainly. Is A diagonalizable matrix Yamaha RX-V396RDS amplifier it doesn ’ t A ect diagonalizabilty.. ). Inverse of both sides of this section is devoted to finding A way to orthogonally diagonalize A symmetric matrix 2! Then, there exists matrices M and n such that A = MNM^-1.! The Theorem to provide step-by-step solutions in as fast as 30 minutes stating without proof the following.... Eœtht Hdiagonalizable diagonal matrix whose diagonal entries are the nonzero eigenvalues, then, there exists A change basis... A [ /latex ] matrix with real entries symmetric matrix with real entries notes are real! A is invertible and diagonalizable, then its transpose AT must be different to be A diagonalizable matrix nonzero,. And n such that A = MNM^-1 gives an expression for A^-1.. That A matrix over.If is diagonalizable, then A had n distinct eigenvalues is latex! Are not real 2 × 2 and 3 × 3 matrices Yamaha RX-V396RDS amplifier 2 eigenvalues [ latex ] [. Is also diagonalizable orthogonally diagonalizable if the matrix can have fewer than n and. Steady state of A, then A is A diagonal matrix whose diagonal entries are real numbers section we cofactor!, I am having some trouble with the first part for A^-1 way orthogonally. Solutions in as fast as 30 minutes A= 2 4 1 3 4 1 3 3 5diagonalizable diagonalizable if is. Only nonzero elements of D are the nonzero columns of P must be different to be A diagonalizable can! D diagonal, then so is any power of it roots ), where D is the matrix... Clearly then A had n distinct eigenvalues: Let [ latex ] 3\times3 [ /latex ] be an [ ]! Learning more about real matrices matrices in which all entries are real numbers as tools learning...
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2021-07-27 15:18:50
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http://math.stackexchange.com/questions/151594/what-is-the-purpose-of-the-mp-symbol-in-mathematical-usage
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What is the purpose of the $\mp$ symbol in mathematical usage?
Occasionally I see the $\mp$ symbol, but I don't really know what it is for, except in conjunction with the $\pm$ symbol thus: $a \pm b \mp c$ which (I believe) means $a+b-c$ or $a-b+c$ (please correct me if I am wrong). Is there any other mathematical usage for the $\mp$ symbol, particularly on its own ?
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It has the same meaning as $\pm$, but as you noted, when used in conjunction, they have "opposite" meanings – M Turgeon May 30 '12 at 13:57
Sometimes it is used to indicate alternating signs in a series, starting with a minus, as in $x-\frac{x^3}{3!}+\frac{x^5}{5!} \mp \ldots$ – marlu May 30 '12 at 14:01
I upvoted @marlu's comment, but then I got worried that it was not actually correct. The example is a little bit wrong, and when I tried to fix it I was not aple to support the point I thought was being made. All I could come up with were things like ${(x\pm y)}^n = x^n \pm x^{n-1}y + x^{n-2}y^2 \pm \cdots$ where there is already a $\pm$ outside to refer to. – MJD May 30 '12 at 15:18
$\mp$ really only has a use when written in the same expressions as $\pm$.
The one that comes to mind is $\cos (\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta$.
But I suppose if you really wanted to, you could write things like $\sin(\alpha \mp \beta) = \sin \alpha \cos \beta \mp \cos \alpha \sin \beta$... if you really wanted to.
On a more humorous vein, it wouldn't surprise me if someone overloaded the symbol to have a different meaning too. Most likely someone like Conway (as in, Combinatorial Game Theory Conway, not Complex Analysis Conway), who thought $+_n$ was a perfectly good name for a state of a game (not an operation).
an aside
On a non-mathematical note, $\pm$ denotes an advantageous position for white in chess. $\mp$ denotes a position for black.
If we really go for it, $\mp$ looks like (干) wiki page, which means 'to dry' in Japanese and might mean 'to do' in Mandarin. $\pm$ looks like (士)wiki page, which might mean 'gentleman' in Japanese and is used in the symbols for doctorate and doctor's thesis.
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Thank you, that is helpful. @marlu wrote in a comment to my OP that "Sometimes it is used to indicate alternating signs in a series, starting with a minus, as in $x-\frac{x^3}{3!}+\frac{x^5}{5!} \mp \ldots$" - is that not standard usage ? – Joe King May 30 '12 at 14:38
I am a high-level chess player and have read dozens of chess books, and I have never once seen the notation $\pm$ or $\mp$ used in chess. Everywhere I have ever read, an advantage for white is written +/-, while an advantage for black is written -/+. – BlueRaja - Danny Pflughoeft May 30 '12 at 16:13
@BlueRaja-DannyPflughoeft: I've been an amateur chess player, I've read quite many books (with descriptive notation instead of algebraic!) and I'm pretty sure (not totally) of having seen $\pm$ and $\mp$ – leonbloy May 30 '12 at 16:16
@leon: Wikipedia seems to back that. Perhaps we are just reading different books :) (then again, wikipedia also makes a distinction between +/- and +-, a distinction I doubt is in wide use) – BlueRaja - Danny Pflughoeft May 30 '12 at 16:17
@BlueRaja-DannyPflughoeft : it is somewhat surprising that a high-level chess player is unfamiliar with the Chess Informant notation (or am I a bit too old? :) ). In CI $\pm$ ($\mp$) stands for "White (Black) stands clearly better", whereas $+-$ ($-+$) stands for "White (Black) has a decisive advantage" – Andrea Mori Aug 28 '12 at 22:02
You are correct; $\mp$ only makes sense in a formula that already has $\pm$.
One simple and useful example is that when $x$ is small, ${1\over{1\pm x}}\approx 1\mp x$.
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Also $x^3 \pm y^3 = (x\pm y)(x^2 \mp xy + y^2)$ – Dilip Sarwate May 30 '12 at 14:34
Like the other answerer, I've only seen it used in the same line as a $\pm$, to mean "positive when the other term is negative and negative when the other term is positive." So, for instance, if we were to say
$\pm a = \mp b$
that would imply that
$a = -b$
and
$-a = b$
-
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2015-05-30 20:57:51
|
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https://www.vedantu.com/maths/rational-and-irrational-numbers
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Rational and Irrational Numbers
All the numbers used in Mathematical computations are broadly classified into two kinds. They are real numbers and imaginary numbers. Real numbers are those numbers that exist in reality and are used in most of the Mathematical computations. Imaginary numbers are the numbers that do not exist in reality. However, they are assumed to be existing to ease a few Mathematical and Scientific computations. Real numbers comprise the entire list of rational and irrational numbers. The chart below describes the difference between rational and irrational numbers.
Rational Numbers Definition:
Rational numbers are the numbers that can be written in the form of a fraction where numerator and denominator are integers. In the case of rational numbers, numerator and denominator should be coprime and denominator should not be equal to zero. Mathematically, rational numbers definition is given as the number a/b if a and b are coprimes, and b is not equal to zero.
Examples for Rational Numbers:
• 5 is a rational number because ‘5’ can be written as $\frac{5}{1}$. Here 5 and 1 are coprimes and 1 is not equal to zero.
• 2.343 is a rational number because it can be written as 2343/1000
• The square root of perfect square numbers
Irrational Numbers Definition:
Any real number is said to be an irrational number if the number cannot be expressed in the form of a fraction where the denominator is not equal to zero.
Mathematically, the definition of the irrational number is given as a number that cannot be expressed in the form of a/b where a and b are coprime and b is not equal to zero.
Examples for irrational numbers:
• The square root of a prime number is an irrational number. $\left( {\sqrt 2 ,\sqrt 3 ,\sqrt 5 ,\sqrt 7 ,\sqrt {13} ,{\text{ }}etc} \right)$
• Mathematical constant π is an irrational number because it is a non-terminating recurring decimal number.
From the above explanations, the difference between rational and irrational numbers is evident.
Properties of Rational Numbers:
• Rational numbers are closed under addition, subtraction, multiplication, and division. This means that
• The sum of rational numbers is rational.
• The difference between the two rational numbers is rational.
• The product of two rational numbers is rational
• The quotient of two rational numbers is also rational. However, rational numbers are not closed under division if the divisor is zero.
• Rational numbers are commutative for addition and multiplication. However, they are not commutative for subtraction and division. If ‘c’ and ‘d’ are two rational numbers, then
• $c{\text{ }} + {\text{ }}d{\text{ }} = {\text{ }}d{\text{ }} + {\text{ }}c$
• $c{\text{ }} \times {\text{ }}d{\text{ }} = {\text{ }}d{\text{ }} \times {\text{ }}c$
• $c{\text{ }} - {\text{ }}d{\text{ }} \ne {\text{ }}d{\text{ }} - {\text{ }}c$
• $c{\text{ }} \div {\text{ }}d{\text{ }} \ne {\text{ }}d{\text{ }} \div {\text{ }}c$
• Addition and multiplication are associative for rational numbers whereas subtraction and division are not associative. If‘ ’j’, ‘k’ and ‘l’ are three rational numbers, then
• $\left( {j{\text{ }} + {\text{ }}k} \right){\text{ }} + {\text{ }}l{\text{ }} = {\text{ }}j{\text{ }} + {\text{ }}\left( {k{\text{ }} + {\text{ }}l} \right)$
• $\;\left( {j{\text{ }} \times {\text{ }}k} \right){\text{ }} \times {\text{ }}l{\text{ }} = {\text{ }}j{\text{ }} \times {\text{ }}\left( {k{\text{ }} \times {\text{ }}l} \right)$
• $\;\left( {j{\text{ }} - {\text{ }}k} \right){\text{ }} - {\text{ }}l{\text{ }} \ne {\text{ }}j{\text{ }} - {\text{ }}\left( {k{\text{ }} - {\text{ }}l} \right)$
• $\;\left( {j{\text{ }} \div {\text{ }}k} \right){\text{ }} \div {\text{ }}l{\text{ }} \ne {\text{ }}j{\text{ }} \div {\text{ }}\left( {k{\text{ }} \div {\text{ }}l} \right)$
• Rational numbers obey the distribution of multiplication over addition. If ‘j’, ‘k’ and ‘l’ are rational numbers, then
$j{\text{ }}\left( {k{\text{ }} + {\text{ }}l} \right){\text{ }} = {\text{ }}\left( {j{\text{ }} \times {\text{ }}k} \right){\text{ }} + {\text{ }}\left( {j{\text{ }} \times {\text{ }}l} \right)$
Rational and Irrational Numbers examples:
Categorize the following into the list of rational and irrational numbers. Justify your answer.
$\left( {0.99,{\text{ }}2.12341234 \ldots \ldots ,{\text{ }}57,{\text{ }}\frac{{16}}{{26}},{\text{ }}2\sqrt 5 ,{\text{ }}\frac{{10}}{0}} \right)$
Solution:
1. 0.99 is a rational number because, by rational numbers definition, it can be expressed as $\frac{{99}}{{100}}$
2. 2.12341234…. is an irrational number because it is a non-terminating decimal that cannot be expressed in the form of a fraction.
3. 57 is a rational number because it can be written in the form of a fraction as 57/1.
4. $\frac{{16}}{{26}} = \frac{8}{{13}}$ which is a rational number.
5. $2\sqrt 5$ is irrational because the denominator is not an integer and the number satisfies the irrational numbers definition.
6. $\frac{{10}}{0}$ is equal to infinity which is undefined. So the number does not fall under rational and irrational numbers examples.
Fun facts:
• For a number to be called rational, only denominators should not be equal to zero. However, 0 may occur in the place of the numerator. ‘0’ is a rational number because 0 can be written as $\frac{0}{1}$. Here denominator is not equal to zero.
• Coprimes are the numbers that have only 1 as a common factor. For example, 3 and 10 are coprimes because the only common factor between them is 1 whereas 3 and 6 are not co primes because they have a common factor 3 along with 1.
• The entire list of rational and irrational numbers can be called as real numbers. However, all real numbers cannot be uniquely rational or irrational.
• All non-repeating and repeating terminating decimals are rational numbers and all non-terminating repeating and non-repeating decimals are irrational numbers.
• The best rational and irrational numbers examples are square roots of perfect squares and non-perfect square numbers respectively.
1. Do irrational numbers obey closure property?
• Irrational numbers do not obey closure property.
• When two irrational numbers are added, the sum need not be irrational. The sum of 2 + √3 and 4 - √3 is equal to 6 which is not irrational.
• When two irrational numbers are subtracted, the difference may not be irrational. The difference between 5√2 and 5√2 is 0 which is a rational number.
• When two irrational numbers are multiplied, the product need not be irrational. The product of √7 and √7 is equal to √49 = 7. This is not irrational.
• If two irrational numbers are divided, the quotient need not be irrational. If √6 and 7√6 is divided, the quotient is 1/7 which is a rational number.
What happens when basic Mathematical operations are performed between a rational number and an irrational number?
• If any of the mathematical operations are performed between a rational and an irrational number, the result obtained is irrational.
• When an irrational number is added to a rational number, the sum obtained is an irrational number.
• The difference between rational and irrational numbers is an irrational number.
• The product of two numbers is irrational if one and only one of the numbers is irrational.
• The quotient obtained by dividing a rational number by an irrational number or vice versa is an irrational number.
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2020-07-16 01:03:09
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https://www.cheenta.com/definite-integral-iit-jam-2018-problem-4/
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How 9 Cheenta students ranked in top 100 in ISI and CMI Entrances?
# Definite Integral | IIT JAM 2018 | Problem 4
Try this beautiful problem from IIT JAM 2018 which requires knowledge of the properties of Definite integral.
## Properties of Definite Integral -IIT JAM2018 (Problem 4)
Let $a$ be positive real number. If $f$ is a continuous and even function defined on the interval $[-a,a]$, then $\displaystyle\int_{-a}^a \frac{f(x)}{1+e^x} \mathrm d x$ is equal to :-
• $\displaystyle\int_0^a f(x) \mathrm d x$
• $2\displaystyle\int_0^a \frac{f(x)}{1+e^x}\mathrm d x$
• $2\displaystyle\int_0^a f(x) \mathrm d x$
• $2a\displaystyle\int_0^a \frac{f(x)}{1+e^x}\mathrm d x$
### Key Concepts
Definite Integral
Properties of definite Integral
Even function / Odd function
Answer: $\displaystyle\int_0^a f(x) \mathrm d x$
IIT JAM 2018, Problem 4
Definite and Integral calculus : R Courant
## Try with Hints
In this first I will give you the properties we need to solve this problem :
Property 1 : $\displaystyle\int_a^b f(x) \mathrm d x = \displaystyle\int_a^b f(a+b-x) \mathrm d x$
[Where $f$ is continuous on $[a,b]$]
Property 2 : If $f$ is an even function i.e., $f(x)=f(-x)$ then
$\displaystyle\int_{-a}^{a} f(x) \mathrm d x = 2 \displaystyle\int_{0}^{a} f(x) \mathrm d x$
Can you drive it from here !!!! Give it a try !!!
Let $I=\displaystyle\int_{-a}^a \frac{f(x)}{1+e^x} \mathrm d x \quad \ldots (i)$
$\Rightarrow I= \displaystyle\int_{-a}^a \frac{f(a-a-x)}{1+e^{(a-a-x)}} \mathrm d x$
[Since, $f$ is continuous then $\displaystyle\int_{a}^b f(x) \mathrm{d}x = \displaystyle\int_{a}^b f(a+b-x) \mathrm{d} x$]
$\Rightarrow I= \displaystyle\int_{-a}^a \frac{f(-x)}{1+e^{-x}} \mathrm d x$
$\Rightarrow I= \displaystyle\int_{-a}^a \frac{f(x)}{1+\frac{1}{e^x}} \mathrm d x$ [Since $f(x)$ is even]
$\Rightarrow I= \displaystyle\int_{-a}^a \frac{e^x.f(x)}{1+e^{x}} \mathrm d x \quad \ldots (ii)$
Adding $(i)$ and $(ii)$ we can get some interesting result !!!
Adding $(i)$ and $(ii)$ we get ,
$2I= \displaystyle\int_{-a}^a \frac{f(x)}{1+e^{x}} \mathrm d x + \displaystyle\int_{-a}^a \frac{e^x .f(x)}{1+e^{x}} \mathrm d x$
$\Rightarrow 2I = \displaystyle\int_{-a}^a \frac{[f(x)+e^x.f(x)]}{1+e^{x}} \mathrm d x$
$\Rightarrow 2I = \displaystyle\int_{-a}^a \frac{f(x)[1+e^x]}{[1+e^{x}]}$
$\Rightarrow 2I= \displaystyle\int_{-a}^a f(x) \mathrm d x$
$\Rightarrow 2I = 2\displaystyle\int_0^a f(x) \mathrm d x$ [Since $f(x)$ is even ]
$\Rightarrow I = \displaystyle\int_0^a f(x) \mathrm d x$ [ANS]
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2021-09-17 10:05:47
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https://gamedev.stackexchange.com/questions/22350/udk-game-prisoners-guards
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UDK game Prisoners/Guards
For school I need to make a little game with UDK, the concept of the game is:
The player is the headguard, he will have some other guard (bots) who will follow him. Between the other guards and the player are some prisoners who need to evade the other guards.
It needs to look like this
My idea was to let the guard bots follow the player at a certain distance and let the prisoners bots in the middle try to evade the guard bots. Now is the problem i'm new to Unreal Script and the school doesn't support me that well.
Untill now I have only was able to make the guard bots follow me. I hope you guys can help me or make me something that will make this game work.
Here is the class i'm using to let te bots follow me:
class ChaseControllerAI extends AIController;
var Pawn player;
var float minimalDistance;
var float speed;
var float distanceToPlayer;
var vector selfToPlayer;
auto state Idle
{
function BeginState(Name PreviousStateName)
{
Super.BeginState(PreviousStateName);
}
event SeePlayer(Pawn p)
{
player = p;
GotoState('Chase');
}
Begin:
player = none;
self.Pawn.Velocity.x = 0.0;
self.Pawn.Velocity.Y = 0.0;
self.Pawn.Velocity.Z = 0.0;
}
state Chase
{
function BeginState(Name PreviousStateName)
{
Super.BeginState(PreviousStateName);
}
event PlayerOutOfReach()
{
Log("ChaseControllerAI CHASE Player out of reach.");
GotoState('Idle');
}
// class ChaseController extends AIController; CONTINUED
// State Chase (continued)
event Tick(float deltaTime)
{
Log("ChaseControllerAI in Event Tick.");
selfToPlayer = self.player.Location - self.Pawn.Location;
distanceToPlayer = Abs(VSize(selfToPlayer));
if (distanceToPlayer > minimalDistance)
{
PlayerOutOfReach();
}
else
{
self.Pawn.Velocity = Normal(selfToPlayer) * speed;
//self.Pawn.Acceleration = Normal(selfToPlayer) * speed;
self.Pawn.SetRotation(rotator(selfToPlayer));
self.Pawn.Move(self.Pawn.Velocity*0.001); // or *deltaTime
}
}
Begin:
Log("Current state Chase:Begin: " @GetStateName()@"");
}
defaultproperties
{
bIsPlayer= true;
minimalDistance = 1024; //org 1024
speed = 500;
}
• Have you looked into something like steering behaviors at all? There are some relatively simple things you can do like for seeking points and fleeing from points that might be relevant: red3d.com/cwr/steer/SeekFlee.html – Tetrad Jan 13 '12 at 16:58
• Can you give us one atomic thing that you would like to accomplish but haven't? "I would like to make the prisoners attempt to escape from the guards when they see x kind of opportunity." Something like that? – user9485 Mar 12 '12 at 22:43
• "self.Pawn.Velocity = Normal(selfToPlayer) * speed;" is this the pawns moving directly towards the player? that would let the prisioners escape. Is best if pawns move to some desired location that is positioned relative to other point, and make that point the same for all paws. Possible make that point the "center of mass" of all prisioners. You can also make so the "desire to move near this point" is limited by the distance from pawn to pawn. If the distance from pawn to pawn is bigger than the prisioner size, the pawns will want to move to the center.. – Tei Apr 27 '12 at 13:01
• What you are doing here is similar to a wolfpack hunting. – Tei Apr 27 '12 at 13:02
Assuming that the idea is that prisoners want to escape the circle of guards, trying to avoid them, what you could try is to apply something like the potential fields approach: http://aigamedev.com/open/tutorials/potential-fields/ With potential fields, guards will apply a repulsive force to the prisoners that come too close to them, and this will change the path prisoners will try to follow, apparently trying to stay away from them. I suppose prisoners will try to move to a position outside the circle of guards: in a simple way, if your prisoner is moving in the direction of a point outside the circle, at each update (or less frequently) you can check if it is close enough (sees) to any guard, and in the update do:
self.Velocity = Normal(self.Velocity) * speed;
// distance is the distance between this Prisoner and the closest Guards,
// minDistance is the distance at which if
if (distance < minDistance)
{
vector rp = Normal(guard.RepulsiveForce) * (minDistance / distance);
self.Velocity += rp;
}
`
Of course the Guard class needs a RepulsiveForce vector that will indicate the direction to which he will try to push close Prisoners: this vector will modify the current velocities of Prisoners that get close enough, and they will look like they will try to avoid them. But again, what you should also do is running a path planning algorithm (like A* star) for the prisoners to find a path and, while running the A*, treat the Guards (probably with a bigger influence circle) as obstacles.
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2021-05-10 01:28:46
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|
https://jp.maplesoft.com/support/help/view.aspx?path=MultiSet/minus&L=J
|
minus - Maple Help
MultiSet/minus
MultiSet difference operator
Calling Sequence M minus N
Parameters
M - MultiSet; a MultiSet, set, or list N - MultiSet; a MultiSet, set, or list
Description
• M minus N returns the set difference (counting multiplicity) between M and N.
• At least one argument must be a MultiSet for this routine to be invoked. Any other argument which is expected to be a MultiSet can be a MultiSet, a set or a list; in the latter two cases the argument is converted to a MultiSet before proceeding to evaluate this command. IsGeneralized(M) must return the same value for all MultiSet arguments M, and all non-MultiSet arguments will be promoted to MultiSets with this same property.
Examples
> $M≔\mathrm{MultiSet}\left(a=2,b=5\right)$
${M}{≔}\left\{\left[{a}{,}{2}\right]{,}\left[{b}{,}{5}\right]\right\}$ (1)
> $N≔\mathrm{MultiSet}\left(a,b=3\right)$
${N}{≔}\left\{\left[{a}{,}{1}\right]{,}\left[{b}{,}{3}\right]\right\}$ (2)
> $M\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{minus}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}N$
$\left\{\left[{a}{,}{1}\right]{,}\left[{b}{,}{2}\right]\right\}$ (3)
> $M\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{minus}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\left[b,a,a\right]$
$\left\{\left[{b}{,}{4}\right]\right\}$ (4)
Compatibility
• The MultiSet/minus operator was introduced in Maple 2016.
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2023-03-29 15:44:56
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https://lavelle.chem.ucla.edu/forum/viewtopic.php?f=123&t=7989
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## Using Kc vs Kp
$PV=nRT$
Nathan Barger 3B
Posts: 18
Joined: Fri Sep 25, 2015 3:00 am
### Using Kc vs Kp
When exactly do you use Kc and when exactly do you use Kp. Although I understand that if you are given molar concentrations you should use Kc and if you are given partial pressures you should use Kp but if a question asks to simply write the expression for the equilibrium constant for the reaction and no numbers or concentrations are given but the gas equation is given, which do I use? I thought we were supposed to simply use Kc but the answer given in the back says to use Kp. is this because the entire equation involves nothing but gases?
Chem_Mod
Posts: 17828
Joined: Thu Aug 04, 2011 1:53 pm
Has upvoted: 406 times
### Re: Using Kc vs Kp
If there are only gases then Kp is appropriate. You would only be able to use Kc if the total volume of the container were given.
Kiara Quinn 3B
Posts: 20
Joined: Sat Jul 09, 2016 3:00 am
### Re: Using Kc vs Kp
Would there be a case in which you would have to use both Kc and Kp at the same time?
Joshua Baysa 1J
Posts: 10
Joined: Wed Sep 21, 2016 2:59 pm
### Re: Using Kc vs Kp
I don't think you would have to use both Kc and Kp simultaneously (at least for this class). I'm not 100% sure but it doesn't seem like it. However, there is a way to manipulate the formulas in order to get both Kc and Kp in the same equation
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2019-12-06 19:38:19
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https://aviation.stackexchange.com/questions/63723/why-was-the-spitfires-elliptical-wing-almost-uncopied-by-other-aircraft-of-worl
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# Why was the Spitfire's elliptical wing almost uncopied by other aircraft of World War 2?
The Spitfire was one of the most successful designs of its day, with flying qualities of a similar standard to the other best designs of the era. In its decade of production from 1936 it grew bigger, stronger and faster.
Yet there seems to be almost no appetite from any of the major aircraft-manufacturing powers to emulate its most iconic feature. In fact, there is only one mass-produced aircraft of that time with an elliptical wing, the American P-47 Thunderbolt. Nothing German or Japanese, and nothing more from the British either.
In a period where every manufacturer was trying to gain every last bit of advantage, it seems odd that a prime design feature attracted so little appetite to copy.
There are good explanations here on the aerodynamics or performance of the elliptical wing. Why is it so rare when it demonstrably works so well?
Source
• Note that even Spitfires were manufactured in "clipped wing" variants during the war which had the wingtips squared off. This was to improve the low altitude speed and roll rate, which is an important factor in air combat but not so much in other types of aviation. – llama Apr 26 '19 at 18:06
• Note that the your link about performance contains in the accepted answer: " Both weight and stall characteristics of elliptical wings are less than optimum; the low induced drag coefficient is bought with higher structural mass and, consequently, lift." – Vladimir F Apr 27 '19 at 6:30
• Interesting article on Air & Space. – BillDOe Apr 27 '19 at 20:21
Interestingly, I couldn't find an answer to this question on the website, but I've found an answer of Peter Kämpf on Quora. He brings forth the same arguments I wanted to mention, so I'll repeat them here.
Elliptical wings are very good - aerodynamically. If you want to minimize induced drag for a given lift requirement, you end up with an elliptical wing.
But a plane is not only aerodynamics. You also have to consider:
• Weight, an elliptical wing is not structurally efficient, and will lead to a higher weight, which leads to higher lift requirements which will lead to more induced drag, even with a very efficient wing.
• Controllability, where and how the wing stalls determines if you're able to recover from a stall. Elliptical wings stall tip first, leading to bad stall behavior.
• Manufacturability, a fully elliptical wing is very hard to make, with its double curves. This will make the wing more expensive.
If you include these factors, you'll see that you'll end up with a compromise. If you use wing taper (which is somewhat close to the aerodynamic optimal shape) but much easier to make and much lighter you'll see that you'll end up with a better design overall.
An analysis of how the design of a wing changes if you include the structural requirements was done by Jones and can be found here.
• The real killer was having to make compound stamping dies for the LE wing skins. The P-47 got a lot of the way there by making just the TE eliptical, allowing just straight bent leading edges. Overall, the Spit, like the Merlin had the typically British characteristic of very high parts count, of components being made from 5 pieces where an American aircraft would make them from one or two. Fuselage formers made from numerous little bits, etc. – John K Apr 26 '19 at 12:30
• "Elliptical wings are very good - aerodynamically. If you want to minimize induced drag for a given lift requirement, you end up with an elliptical wing." That is incorrect. – Lysistrata Apr 26 '19 at 14:26
• If you go tons faster, you minimize induced drag. Few people comprehend 300 mph wind force, and they were already on their way to 400 mph. The thinking behind this type of wing simply was superseded by newer design requirements focused on the leading edge and compressability. They did get one thing right though, they made it thin, and the Spitfire line lasted until the 1950s. – Robert DiGiovanni Apr 26 '19 at 16:56
• Should "recover from a wing" perhaps have been "... from a spin"? – hmakholm left over Monica Apr 26 '19 at 18:41
• @RoiMaison. It is regurgitating the old myth that elliptical planforms produce elliptical pressure distributions. They don't. And I wish people would stop propagating that nonsense. – Lysistrata Apr 27 '19 at 17:46
Well the short answer is the elliptical wing was used on a lot more aircraft than this article lets on.
The following all used an elliptical wing and there are others too:
• German Heinkel 112 fighter
• German Heinkel 111 bomber
• German Heinkel 70
• US P35
• US P43
• Italian Reggiane 2000
• Japanese Aichi D3A "Val" dive bomber
• British Hawker Tempest
• I've never heard of any of those. And I doubt I'd have ever heard of a Spitfire if it didn't have "its most iconic feature" inside it, the Rolls-Royce Merlin – Mazura Apr 27 '19 at 3:11
• @Mazura He 111 and Tempest are very well known aircraft. He 111 was the main German bomber nd an important target of Spitfires. And although RR Merlin is well known too, it is hardly the reason to know Spitfire unless you are an aviation engines geek (i.e. Lancasters and Mosquitos would have to be much more famous than they are.). – Vladimir F Apr 27 '19 at 6:22
• Thanks but I suggest that's quite a broad definition of both elliptical and mass-produced. – Party Ark Apr 27 '19 at 8:39
• I wouldn't describe the P43 as elliptical wing – Notts90 supports Monica Apr 27 '19 at 9:18
• @PartyArk How do you define elliptical that you find the answer uses a broad definition? – Vladimir F Apr 27 '19 at 19:35
Short answer: Elliptical wings are too expensive to manufacture. A trapezoid wing with a defined geometric or aerodynamic twist can get very close to an elliptical lift distribution (optimal lift distribution over the wingspan, therefore the primary goal of the wing design).
• Welcome to Aviation.SE! Adding some sources by editing your answer would really improve your answer. – dalearn Apr 26 '19 at 14:09
• Just like houses that aren't made out of rectangles; harder to make, and thus more expensive. – Mazura Apr 26 '19 at 22:34
A lot of planes still use elliptical wings - sort of.
What the maths tell us is that the most efficient wing configuration for a given wing span should have an elliptical lift distribution*. The most obvious way to implement this is to make your wings elliptical.
But aircraft designers learned from the experience of manufacturing the Spitfire that elliptical wings are more difficult to manufacture leading to increased cost and manufacturing time.
So later in the war when resources became tight and everyone assumed that they were racing the Germans to build better, faster planes designers deliberately chose to use straight wings to ease production and reduce costs. The P51 Mustang was designed this way.
But we have learned that making your wing elliptical isn't the only way to have elliptical lift distribution. To get elliptical lift distribution you can:
• Make your wing more elliptical
• Add washout to tune lift distribution along the wing
• Change airfoil profile from root to tip to change lift distribution
• Do any combination of the above (eg. washout + rounded tips)
So a lot of planes still use elliptical wings. Especially when fuel economy is one of the main driving design objective. It's just that they don't look elliptical.
* note: There is evidence that this may not be accurate. It is true that if you fix your wingspan the maths will output an elliptical distribution but if you fix your weight (ie. lift at cruise) the most efficient distribution turns out to be something else (something bell shaped) but you end up needing to extend your wingspan
• This is too imprecise to be useful. What maths are you talking about? Do you have any citations (from reputable sources) to back up your statements? – Lysistrata Apr 30 '19 at 13:39
• @Lysistrata THE standard maths when calculating lift distribution: Prandtl's lifting line theory (google "lifting line theory") – slebetman Apr 30 '19 at 16:52
• Thanks. Lifting line theory is particularly inaccurate near the wing tips. It is a very poor choice for wings with aspect ratios as small as those of the Spitfire. – Lysistrata May 2 '19 at 3:17
The main drawback of the Spitfire's elliptical wing was the the amount of labour required to build it. Overall the Spitfire required about five times as many man-hours to build as the nearest German equivalent, the Messerschmitt Bf 109.
• That's very interesting - do you have a source for the 5x man-hours? – Party Ark Apr 28 '19 at 12:34
• 13,000 man hours for the Spitfire compared with 4000 for the Bf 109 would be 3.25 times as many man hours, but the actual figures would have varied during the period of several years in which both aircraft were in production. – J. Southworth May 3 '19 at 16:24
• One internet source quotes figures of 15,200 man hours for the Spitfire, 10,000 for the Hurricane and 6000 for the Bf 109 in early 1940, also a variable figure of between 2000 and 10000 man hours for the different models of the Bf 109. – J. Southworth May 3 '19 at 16:43
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2021-05-08 03:46:26
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https://chemistry.stackexchange.com/questions/5755/negative-charge-of-methoxide-anion
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# Negative charge of methoxide anion
Why is the methoxide ($\ce{CH_{3}O^{-}}$) anion negatively charged?
It has 13 valence electrons and has 13 free electrons. In spite of that the methoxide anion is negatively charged.
• I couldn't figure out the count of 13 free electrons. What you did actually mean by free electron. – blackSmith Aug 3 '13 at 19:20
• Probably, you are having problem with formal charge calculation. – ashu Aug 3 '13 at 20:50
The negative charge is due to the unpaired electron in the oxygen atom, remained after all bonds(molecular orbitals) are filled. Oxygen has 2 unpaired electron in its valence shell and is singly bonded to carbon in this molecule. Hence being un-bonded, the other electron exhibits the charge.
Check this link, hover the cursor on the $\ce{O}$-atom, you will get a graphical representation of the same.
Simply put: there are more electrons than protons.
Protons:
• Hydrogen x 3 = 1 x 3 = 3
• Oxygen x 1 = 8 x 1 = 8
• Carbon x 1 = 6 x 1 = 6
3 + 8 + 6 = 17 protons.
Electrons:
Looking at the electron shells and covalent bonds in the Lewis diagram provided by blackSmith, you have the following:
• 4 single covalent bonds = 4 x 2 = 8
• 3 lone pairs on oxygen = 3 x 2 = 6
• Unseen $1s^{2}$ pair on both carbon and oxygen = 2 x 2 = 4
8 + 6 + 4 = 18 electrons.
As there is 1 more electron than proton, it has a charge of -1.
As to where that negative charge tends to be, one must create and look at the Lewis diagram where you see that the oxygen atom has an extra valence electron.
From what you have given, this question can not be answered in terms of electrons.
It is meaningless to do any electron calculation, because you have to know it has one negative charge before you can do the electron counting. How can you get a correct Lewis structure without knowing that it has one extra electron? The negative charge is part of the given condition, not your result you get from calculation.
Since I saw you are talking about 13 valence electrons, I guess you want to count valence electrons. If you add valence electron from all atoms, you will get 13 valence electron: 4 (C) + 3 * 1 (H) + 6 (O) = 13. Then because you have one negative charge, you have to add one more electron and then you get 13 + 1 = 14 valence electrons. Now everything agree with each other.
If you want to find out which atom the charge is on, you have to do the formal charge calculation. Find it here
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2020-02-18 14:09:09
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https://gamebanana.com/maps/license/180150
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# License : FY_GUNTOWN
## A Map for Counter-Strike: Source
You are free to
Ask if you want to
• Redistribute this Map on other sites
• Modify this Map and distribute the modified Map on GameBanana
• Modify this Map and distribute the modified Map on another site
• Use parts of this Map in another Map and distribute the Map on GameBanana
• Use parts of this Map in another Map and distribute the Map on another site
You may not
• Use this Map or components of this Map commercially
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2018-11-17 10:39:07
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https://ncatlab.org/nlab/show/presentation+axiom
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# nLab presentation axiom
foundations
## Foundational axioms
foundational axiom
# Contents
## Idea
In the foundations of mathematics, it's interesting to consider the axiom that the Category of Sets Has Enough Projectives; in short: CoSHEP (pronounced /ko:-shep/). This is more commonly known as the presentation axiom: PAx. It is a weak form of the axiom of choice.
## Statement
In elementary terms, CoSHEP states
###### Axiom (CoSHEP)
For every set $A$, there exists a set $P$ and a surjection $P \to A$, such that every surjection $X \twoheadrightarrow P$ has a section.
###### Remark
The full axiom of choice states that every surjection $X \to A$ has a section; hence in the above $P$ may be chosen to be $A$ itself.
This should be read in view of the definition of projective objects:
###### Definition
An object $P$ in a category $C$ is (externally) projective iff the hom-functor $C(P, -): C \to Set$ takes epis to epis. This is the same as saying: given an epi $p: B \to A$ and a map $f: P \to A$, there exists a lift $g: P \to B$ in the sense that $f = p \circ g$.
Accordingly, in a topos the CoSHEP axiom says equivalently
###### Axiom (CoSHEP)
Every object has a projective presentation. Hence: There are enough projectives.
Borrowing from the philosophy of constructivism, we may also call this a complete presentation.
###### Remark
The dual axiom, that $Set$ has enough injectives (that is, every set admits an [injection into an injective set) is true in every topos: every power object is an injective object, and every object embeds in its power object via the singleton map .]
## Justification
Although perhaps not well known in the literature of constructive mathematics, the CoSHEP axiom may be justified by the sort of reasoning usually accepted to justify the axioms of countable choice and dependent choice (which it implies, by Proposition 1 below).
To be explicit, every set $A$ should have a ‘completely presented’ set of ‘canonical’ elements, that is elements given directly as they are constructed without regard for the equality relation imposed upon them. For canonical elements, equality is identity, so the BHK interpretation of logic justifies the axiom of choice for a completely presented set. This set is $P$, and $A$ is obtained from it as a quotient by the relation of equality on $A$. This argument can be made precise in many forms of type theory (including those of Martin-Löf and Thierry Coquand), which thus justify CoSHEP, much as they are widely known to justify dependent choice.
## Consequences
The existence of sufficiently many projective presentations plays a central role in homological algebra as a means to construct projective resolutions of objects. Tradtionally one often uses the axiom of choice to prove that categories of modules have enough projectives, on the grounds that the free modules are projective.
But the weaker assumption of CoSHEP is already sufficient for this purpose: while not every free module will be projective, one can still use CoSHEP to find a projective presentation for every free module (and thus for every module). This is discussed in more detail here.
###### Proposition
The following three conditions on a W-pretopos with enough projectives are equivalent:
1. The axiom of dependent choice (DC),
2. The axiom of countable choice (CC),
3. Projectivity of the singleton (the terminal object) $1$.
Note that we normally assume (3) for the category of sets, which is true in any (constructively) well-pointed pretopos and true internally in any pretopos whatsoever, so one normally says that DC and CC simply follow from the existence of enough projectives (CoSHEP). Equivalently, internal DC and internal CC follow from internal CoSHEP.
###### Proof
Condition 1 easily implies 2. Condition 2 says precisely that the natural numbers object $\mathbb{N}$ is externally projective, and since $1$ is a retract of $\mathbb{N}$, it is projective under condition 2, so 2 implies 3. It remains to show 3 implies 1.
Let $X$ be inhabited, so there exists an entire relation given by a jointly monic span
$1 \stackrel{epi}{\leftarrow} U \stackrel{f}{\to} X,$
and similarly let
$X \stackrel{epi \pi_1}{\leftarrow} R \stackrel{\pi_2}{\to} X$
be an entire binary relation. Let $p: P \to X$ be a projective cover. Since $1$ is assumed projective, the cover $U \to 1$ admits a section $\sigma: 1 \to U$, and the element $f \sigma: 1 \to X$ lifts through $p$ to an element $x_0: 1 \to P$. Next, in the diagram below, $p$ lifts through the epi $\pi_1$ to a map $q: P \to R$, and then $\pi_2 q$ lifts through $p$ to a map $\phi$ (since $P$ is projective):
$\array{ & & P & \stackrel{\phi}{\to} & P \\ & \swarrow p & \downarrow q & & \downarrow p \\ X & \underset{\pi_1}{\leftarrow} & R & \underset{\pi_2}{\to} & X }$
By the universal property of $\mathbb{N}$ (see recursion), there exists a unique map $h: \mathbb{N} \to P$ rendering commutative the diagram
$\array{ 1 & \stackrel{0}{\to} & \mathbb{N} & \stackrel{s}{\to} & \mathbb{N} \\ id \downarrow & & \downarrow h & & \downarrow h \\ 1 & \underset{x_0}{\to} & P & \underset{\phi}{\to} & P \\ & \swarrow p & \downarrow q & & \downarrow p \\ X & \underset{\pi_1}{\leftarrow} & R & \underset{\pi_2}{\to} & X }$
Clearly $\langle p h, p h s \rangle : \mathbb{N} \to X \times X$ factors through $\langle \pi_1, \pi_2 \rangle : R \to X \times X$, i.e., $\forall_{n: \mathbb{N}} (p h(n), p h(n+1)) \in R$, thus proving that dependent choice holds under CoSHEP.
###### Example
A topos in which CoSHEP holds but $1$ is not projective is $Set^C$, where $C$ is the category with three objects and exactly two non-identity arrows $a \to b \leftarrow c$. For if $U: C \to Set$ is a functor with $U(a) = \{a_0\}$, $U(b) = \{b_0, b_1\}$, and $U(c) = \{c_0\}$, with $U(a \to b)(a_0) = b_0$ and $U(c \to b)(c_0) = b_1$, then the map $U \to 1$ is epi but has no section, so $1$ is not projective. On the other hand, as noted below, every presheaf topos satisfies CoSHEP, assuming that $Set$ itself does.
CoSHEP also implies several weaker forms of choice, such as the axiom of multiple choice and WISC. In weakly predicative mathematics, it can be combined with the existence of function sets to show the subset collection axiom.
## In a topos
When working in the internal logic of a topos, the “internal” meaning of CoSHEP is “every object is covered by an internally projective object.” (Compare with the internal axiom of choice: every object is internally projective.) As regards foundational axioms for toposes (in the sort of sense that the axiom of choice is regarded as “foundational”), the internal version of the presentation axiom should be taken as the default version.
###### Proposition
Suppose that $1$ is (externally) projective in $E$. Then $E$ satisfies PAx whenever it satisfies internal PAx.
Internal PAx does not follows from external PAx; see Example 3. However, if every object is projective (AC), then every object is internally projective (IAC).
A stronger version of PAx may be worth considering. Say that an object is stably projective if its pullback to any slice category is projective. Then stably projective objects are internally projective (proof?). Similarly, if we say that a topos $E$ satisfies stable PAx if every object is covered by a stably projective object, then a topos satisfies internal PAx if it satisfies stable PAx.
###### Example
Every presheaf topos $Set^{C^{op}}$ has enough projectives, since any coproduct of representables is projective. If in addition $C$ has binary products, then by this result, $Set^{C^{op}}$ validates internal PAx.
###### Counterexample
However, not every presheaf topos validates internal PAx, even though every presheaf topos validates external PAx. As an example, let $C$ be the category where $Ob(C)$ is the disjoint sum $\mathbb{N} \cup \{a, b\}$, and preordered by taking the reflexive transitive closure of relations $n \leq n+1$, $n \leq a$, $n \leq b$. The claim is that neither $C(-, a)$, nor any presheaf $P$ that maps epimorphically onto $C(-, a)$, can be internally projective. Indeed, consider the presheaf $F$ defined by $F(a) = F(b) = \emptyset$ and $F(n) = [n,\infty)$, with $F(n+1 \to n)$ the evident inclusion. Let $G$ be the support of $F$, so that we have an epi $e: F \to G$.
The objects $C(-, a), C(-, b)$, and $G$ are subterminal and $G \cong C(-, a) \cap C(-, b) \cong C(-, a) \times C(-, b)$. The set $F^{C(-, a)}(b)$ is empty because there is no $C(-, a) \times C(-, b) \to F$ (it would give a section $G \to F$ of $e: F \to G$, but none exists), whereas $G^{C(-, a)}(b)$ is inhabited by $C(-, a) \times C(-, b) \cong G$. For any $P$ covering $C(-, a)$, the set $F^P(b)$ is empty (because any section $s: C(-, a) \to P$ of $P \to C(-, a)$ induces a function $F^P(b) \to F^{C(-, a)}(b) = 0$), and the set $G^P(b)$ is inhabited (the map $P \to C(-, a)$ induces a map $1 \cong G^{C(-, a)}(b) \to G^P(b)$). Thus the map $e^P \colon F^P \to G^P$ cannot be epic.
###### Counterexample
Any topos that violates countable choice, of which there are plenty, must also violate internal PAx.
###### Example
An interesting example of a topos that has enough projectives and satisfies internal CoSHEP (at least, assuming the axiom of choice in Set), although it violates the full (internal) axiom of choice, is the effective topos, and more generally any realizability topos. The reason for this is quite similar to the intuitive justification for CoSHEP given above. Technically, it results from the fact that realizability toposes are exact completions; an explanation is given in this remark.
## Further properties
Since Set is (essentially regardless of foundations) an exact category, if it has enough projectives then it must be the free exact category $PSet_{ex/lex}$ generated by its subcategory $PSet$ of projective objects. By the construction of the ex/lex completion $PSet_{ex/lex}$, it follows that every set is the quotient of some “pseudo-equivalence relation” in $PSet$; i.e., the result of imposing an equality relation on some completely presented set. See SEAR+ε for an application of this idea.
###### Proposition
CoSHEP as a choice principle added to ZF implies a proper class of regular cardinals.
###### Proof
Since CoSHEP implies WISC, and WISC has this implication (a result of van den Berg).
## References
When Peter Aczel was developing $CZF$ (a constructive predicative version of ZFC), he considered this axiom, under the name of the presentation axiom, but ultimately rejected it.
• Peter Aczel. The type theoretic interpretation of constructive set theory. Logic Colloquium ‘77 (Proc. Conf., Wroclaw, 1977), pp. 55–66, Stud. Logic Foundations Math., 96, North-Holland, Amsterdam-New York, 1978. Cited in Palmgren, below.
The presentation axiom was, however, adopted by Erik Palmgren in $CETCS$ (a constructive predicative version of ETCS):
• Erik Palmgren. Constructivist and Structuralist Foundations: Bishop’s and Lawvere’s Theories of Sets. pdf.
Its relationship to some other weak axioms of choice is studied in
• Michael Rathjen, Choice principles in constructive and classical set theories.
Revised on February 27, 2016 11:36:33 by Todd Trimble (67.80.128.74)
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2017-10-23 04:16:24
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https://www.linuxquestions.org/questions/linux-software-2/emacs-permanent-key-binding-for-saved-macro-771863-print/
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LinuxQuestions.org (/questions/)
- Linux - Software (https://www.linuxquestions.org/questions/linux-software-2/)
- - Emacs: Permanent Key Binding For Saved Macro (https://www.linuxquestions.org/questions/linux-software-2/emacs-permanent-key-binding-for-saved-macro-771863/)
CoderMan 11-27-2009 03:54 AM
Emacs: Permanent Key Binding For Saved Macro
Hi. I would appreciate some help from the Emacs gurus. I was trying to follow an online tutorial on how to permanently save a macro for reuse. So I tried to follow the instructions, and ended up putting this in my .emacs:
Code:
(fset 'texquote (lambda (&optional arg) "Keyboard macro." (interactive "p") (kmacro-exec-ring-item (quote ("\\begin{quotation}^M^M^M^M\\end{quotation}^[OA^[OA" 0 "%d")) arg)))
This works great in that it makes available my macro as the "texquote" command, but it does not also make available the key-binding I attached to the command, which was "C-x C-k 1".
The tutorial says:
Quote:
If you give insert-kbd-macro a numeric argument, it makes additional Lisp code to record the keys (if any) that you have bound to macroname, so that the macro will be reassigned the same keys when you load the file.
I think this is where I went wrong, but when I try to use "M-x insert-kdb-macro 1" it just says that there is no macro by that name. What does it mean then when it says to "give insert-kbd-macro a numeric argument"?
All times are GMT -5. The time now is 06:03 PM.
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2021-04-16 23:03:40
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https://math.stackexchange.com/questions/854633/hartshorne-exersice-1-17-skyscraper-sheaf-chapter-ii-schemes
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# Hartshorne Exersice 1.17 Skyscraper sheaf Chapter II Schemes
I am able to verify the statements about the stalk. I want to see how the direct image of the the skyscraper sheaf can be thought of as the constant sheaf.
Observation- If $P\notin U$, then $U\cap {\{P\}}^{-}= \emptyset$ so the sections are just O which is same as the section of the skyscraper sheaf.
But if If $P\in U$ , I don't see why $i_{*}(A)(U)=A(U\cap\{P\}^{-})$ is equal to A .
• Is $U\cap \{P\}^{-}$ connected inside $\{P\}^{-}$? Jul 2 '14 at 18:51
• Yes it is connected, because it has a dense point, hence there can't be two disjoint opens (thus $A(U \cap \overline{P})=A$). Jul 2 '14 at 19:32
• Do u mean P is the dense point of $\overline{\{P\}}$? Jul 2 '14 at 19:49
• $P$ is also a dense point in $U \cap \overline{\{P\}}$. Jul 2 '14 at 22:38
Note that since the one-point set $\{P\}$ is irreducible, so too is its closure. And as Martin points out, any open subspace of an irreducible space is irreducible. I recommend doing Ex. I.1.6 if you haven't already.
I think the thing you want to prove in the end is that if $X$ is an irreducible topological space and $Y$ is a discrete space then any continuous map $f\colon X \to Y$ is constant. If you've gone through Chapter I then I think you know a quick proof of this already: by continuity, $f(X)$ has to be irreducible.
• I understand what you want to say. But then why is $U \cap \overline{\{P\}}$ irreducible in $\overline{\{P\}}$ Jul 2 '14 at 21:02
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2022-01-27 21:12:47
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https://lambda.mines.edu/s18/lga/12-haskell-ttt.html
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# LGA-12: Haskell Tic Tac Toe¶
For this assigment, you will be reading someone else’s code. The code is well written Haskell code, but I imagine it will be non-trivial to follow, as you are probably a Haskell beginner.
Then, collaboratively, as a learning group, answer these questions:
1. (10 points) How the game is displayed? Trace showBoard (use moves [(One, TL), (Two, TR)]), explain uses of Just/Nothing, explain how/when showBoard gets called, etc.
3. (10 points) Explain how the program accepts/validates a player’s move. How does it show the list of valid moves? Be sure to review getPosition as well as validMoves.
4. (18 points) Explain how move alters the game state.
5. (6 points) Explain the case statement in playGame.
6. (2 points) Code does not appear to stop when the game is a draw. What would you need to modify? (you don’t have to write code, just think at a high level where the problem is).
Hint
If you get stuck figuring something out, it may help to load the file, set up some bindings (e.g., let bd = [(One, TL), (Two, TR)]), then run individual functions to see how they perform. I did this with several of the functions.
Your group should produce a single set of typed answers. I recommend you meet outside of class to complete this assignment.
To turn in the assignment, submit on Gradescope.
Notice that there are points assigned to each of the questions, that is because this will count towards the programming portion of your grade as well as the LGA. Since this counts towards your programming grade, you have until Tuesday, March 13th at 11:59 PM to complete the assignment. When I check the LGA on March 1st, I’ll simply give you the check if you made some effort to look over this ahead of time (and can answer questions then as well). Your group can spend slip days too if you would like, each 24-hours will cost 1 slip day from each group member.
I expect there to be questions on this assignment, send them to the mailing list!
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2018-10-22 02:31:00
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https://docs.mosaicml.com/en/v0.7.0/api_reference/composer.algorithms.squeeze_excite.squeeze_excite.html
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# composer.algorithms.squeeze_excite.squeeze_excite#
composer.algorithms.squeeze_excite.squeeze_excite
Functions
apply_squeeze_excite Adds Squeeze-and-Excitation blocks (Hu et al, 2019) after Conv2d layers.
Classes
Algorithm Base class for algorithms. Event Enum to represent events in the training loop. Logger An interface to record training data. Optimizer Base class for all optimizers. SqueezeExcite Adds Squeeze-and-Excitation blocks (Hu et al, 2019) after the Conv2d modules in a neural network. SqueezeExcite2d Squeeze-and-Excitation block from (Hu et al, 2019) SqueezeExciteConv2d Helper class used to add a SqueezeExcite2d module after a Conv2d. State The state of the trainer.
Attributes
• Optional
• Sequence
• Union
• annotations
• log
class composer.algorithms.squeeze_excite.squeeze_excite.SqueezeExcite(latent_channels=64, min_channels=128)[source]#
Adds Squeeze-and-Excitation blocks (Hu et al, 2019) after the Conv2d modules in a neural network.
Runs on INIT. See SqueezeExcite2d for more information.
Parameters
• latent_channels (float, optional) – Dimensionality of the hidden layer within the added MLP. If less than 1, interpreted as a fraction of the number of output channels in the Conv2d immediately preceding each Squeeze-and-Excitation block. Default: 64.
• min_channels (int, optional) – An SE block is added after a Conv2d module conv only if min(conv.in_channels, conv.out_channels) >= min_channels. For models that reduce spatial size and increase channel count deeper in the network, this parameter can be used to only add SE blocks deeper in the network. This may be desirable because SE blocks add less overhead when their inputs have smaller spatial size. Default: 128.
apply(event, state, logger)[source]#
Apply the Squeeze-and-Excitation layer replacement.
Parameters
• event (Event) – the current event
• state (State) – the current trainer state
• logger (Logger) – the training logger
match(event, state)[source]#
Runs on INIT
Parameters
• event (Event) – The current event.
• state (State) – The current state.
Returns
bool – True if this algorithm should run no
class composer.algorithms.squeeze_excite.squeeze_excite.SqueezeExcite2d(num_features, latent_channels=0.125)[source]#
Bases: torch.nn.modules.module.Module
Squeeze-and-Excitation block from (Hu et al, 2019)
This block applies global average pooling to the input, feeds the resulting vector to a single-hidden-layer fully-connected network (MLP), and uses the output of this MLP as attention coefficients to rescale the input. This allows the network to take into account global information about each input, as opposed to only local receptive fields like in a convolutional layer.
Parameters
• num_features (int) – Number of features or channels in the input
• latent_channels (float, optional) – Dimensionality of the hidden layer within the added MLP. If less than 1, interpreted as a fraction of num_features. Default: 0.125.
class composer.algorithms.squeeze_excite.squeeze_excite.SqueezeExciteConv2d(*args, latent_channels=0.125, conv=None, **kwargs)[source]#
Bases: torch.nn.modules.module.Module
Helper class used to add a SqueezeExcite2d module after a Conv2d.
composer.algorithms.squeeze_excite.squeeze_excite.apply_squeeze_excite(model, latent_channels=64, min_channels=128, optimizers=None)[source]#
Adds Squeeze-and-Excitation blocks (Hu et al, 2019) after Conv2d layers.
A Squeeze-and-Excitation block applies global average pooling to the input, feeds the resulting vector to a single-hidden-layer fully-connected network (MLP), and uses the output of this MLP as attention coefficients to rescale the input. This allows the network to take into account global information about each input, as opposed to only local receptive fields like in a convolutional layer.
Parameters
• model (Module) – The module to apply squeeze excite replacement.
• latent_channels (float, optional) – Dimensionality of the hidden layer within the added MLP. If less than 1, interpreted as a fraction of the number of output channels in the Conv2d immediately preceding each Squeeze-and-Excitation block. Default: 64.
• min_channels (int, optional) – An SE block is added after a Conv2d module conv only if one of the layer’s input or output channels is greater than this threshold. Default: 128.
• optimizers (Optimizer | Sequence[Optimizer], optional) –
Existing optimizers bound to model.parameters(). All optimizers that have already been constructed with model.parameters() must be specified here so they will optimize the correct parameters.
If the optimizer(s) are constructed after calling this function, then it is safe to omit this parameter. These optimizers will see the correct model parameters.
Returns
The modified model
Example
import composer.functional as cf
from torchvision import models
model = models.resnet50()
cf.apply_stochastic_depth(model, target_layer_name='ResNetBottleneck')
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2022-06-30 13:27:20
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https://blender.stackexchange.com/questions/148887/any-blacklight-effect-suggestion-for-2-8
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# Any blacklight effect suggestion for 2.8?
Got a question. Are there any way to simulate blacklight (ultraviolet) effect in blender? Dont just light scene with violet colors, but make uv reactive color material or maybe fake blacklight effect? As i know some colors change it hue/saturation and become emission like. Any thoughts?
• According to this docs.blender.org/manual/fr/2.79/render/cycles/nodes/types/…, Blender stops before non visible rays. So, probably have to fake it. – lemon Aug 17 '19 at 15:12
• I did play around with something similar by using render layers and the Material Override to render the scene a second time, but with materials that would emulate a second set of RGB - which could be deemed Infrared, UV and “ultragreen”, which could then be combined with the “visible” render using the compositor. It was unfinished and very cumbersome but did kind of work. Since standard Blender only works with 3 color channels, this was the only way I could come up with of handling additional channels. – Rich Sedman Aug 17 '19 at 15:37
## 2 Answers
That can be faked with Eevee thanks to the Shader to RGB node. Though probably the proposed setting has some limitations.
The idea is to have a bright light prerendered onto a pure white Diffuse Shader.
If we convert it to RGB then to BW, we can test if it is brighter than a threshold.
From that, we can combine UV reactive part and 'normal' material parts with a mix shader.
If the threshold and lamp intensity are appropriately tuned, other (reasonable) lights won't change the effect.
Note: the fingerprint color here is due to the light color which can be changed.
Note2: could also test if the prerendered diffuse shader has some specific color output in order to avoid the setting be based only on light intensity.
Here is another approach which works in both Eevee and Cycles:
Use the distance between a light object and the fluorescent surface in order to determine how much the emission should happen. This distance should be used as the factor of a Mix Shader node, combining a regular material shader with an Emission shader.
This causes the surface far away from the black light not to emit any (fluorescent) light.
The screenshot below shows the distance computation, using Objecttexture coordinates of the blacklight (bottom left).
The blacklight's power can be used as a driver to modulate the emission, using Math nodes can drive the emission power depending on the light's power.
Further details and a full demo and open source shader node group: https://github.com/alcove-design/blender-shader-fluo (CC BY).
This animation also shows the blacklight in action.
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2021-03-04 07:05:27
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https://sites.imsa.edu/acronym/2021/05/01/budget-reconciliation-and-bidens-infrastructure-bill/
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# Budget Reconciliation and Biden’s Infrastructure Bill
In April, Senate Majority Leader Chuck Schumer’s office announced that the Senate Parliamentarian decided that Budget Reconciliation can be used an additional time this year. This allows Senate Democrats to pass at least a total of three bills using a simple majority before the 2022 midterm election. Budget reconciliation has already been used once during the 2021 budget fiscal year for the COVID-19 Relief Package, which was passed earlier this year.
Budget reconciliation is a Senate process that accelerates legislation on the budget. This process allows budget-related legislation to pass with a simple majority (51 votes), bypassing the filibuster, which requires a 60-vote majority to pass legislation. Budget reconciliation is usually allowed only once a year. However, Sen. Schumer suggested that under Section 304 of the Congressional Budget Act, which states that during any period in a fiscal year the Senate and house can revise the budget resolution, budget reconciliation can be used again within the fiscal year to pass more budget-related legislation. This would include President Joe Biden’s 2.3 trillion-dollar infrastructure bill.
At the beginning of his presidency, Biden proposed an infrastructure bill with the intent to revitalize infrastructure. Biden’s infrastructure bill includes funds to rebuild and update transportation, build infrastructure, and invest in manufacturing. In addition, Transportation Secretary Pete Buttigieg estimates that the bill would lead to the creation of about 2.7 million jobs in updating infrastructure. In order to cover the expenses of the bill, a corporate tax increase from 21% to 28% was proposed in this bill. This aspect of the bill is mainly targeted towards larger corporations that haven’t paid taxes or haven’t been paying the correct amount of taxes. It is estimated that if the tax increase is passed, then the bill will make up the cost in about 15 years.
The proposed infrastructure bill has come with some controversy and has been met with criticism from not only the Republican party, but also from some in the Democratic Party as well. Republican Senator Shelley Moore Capito, a member of the Environment and Public Works Committee, stated that she would be willing to support a bill between $600 billion and$800 billion, which would include less than half of what Biden proposed, and would leave the corporate tax at 21%. Sen. Susan Collins supported her colleague stating, I’m going to be getting a presentation from Sen. Capito next week … but certainly I think $800 billion would be a great package. However, not all Republican Senators are opposed to the increase in corporate taxes. Republican Senator Lisa Murkowski expressed that she would consider supporting a corporate tax increase, but also criticized that it was the main method of paying the bill, suggesting the possibility of raising the gas tax. Within the Democratic Party, Democratic Senator Joe Manchin criticized the bill for its corporate tax increase. However, he said he would potentially support the bill if the tax increase was lowered to 25%. Other Senate Democrats have suggested passing an$800 billion bipartisan bill that would encompass transportation and energy infrastructure, leaving the remainder of the bill to pass using budget reconciliation. Aside from the corporate tax increase, Joe Manchin appears to support most of the spending in the bill, stating that Democratic Senators should “do whatever it takes” to pass an infrastructure bill.
This decision’s impact on what the Democratic Party is able to accomplish with a simple majority is predicted to also have an impact on the midterm elections, which is especially important in the Senate to determine which party will have the majority after the 2022 midterm elections. One of the usual key issues that voters take into consideration is the state of the economy and the impact that it has. With the power to use reconciliation multiple times during the fiscal year, Senate Democrats may have the ability to pass legislation that will positively affect the economy in the voter’s eyes and widen their majority. However, their ability to impact the economy still heavily relies on the rulings of the Senate Parliamentarian and how much legislation can be passed before the 2022 midterm election.
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2023-01-31 03:19:47
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https://allnswers.com/mathematics/question14511180
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, 24.01.2020Cjdjejfjiv8568
# Solve the following equation for b. 6b + 2a – 4 = 2b + 3a and
b=1/4a+1
Step-by-step explanation:
Step 1: Add -2b to both sides.
2a+6b−4+−2b=3a+2b+−2b
2a+4b−4=3a
Step 2: Add -2a to both sides.
2a+4b−4+−2a=3a+−2a
4b−4=a
Step 3: Add 4 to both sides.
4b−4+4=a+4
4b=a+4
Step 4: Divide both sides by 4.
4b/4 = a+4/4
b=1/4a+1
Hope this helps.
Solve the equation for b, so you need to isolate/get "b" by itself on one side of the equation.
[you can solve this different ways]
6b + 2a - 4 = 2b + 3a First subtract 2b on both sides
4b + 2a - 4 = 3a Next subtract 2a on both sides
4b - 4 = a Then add 4 on both sides
4b = a + 4 Finally divide 4 on both sides
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2021-01-23 07:40:03
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http://www.neutroninterferometry.com/background/optimal-compensation-angle
|
# Optimal compensation angle
May 11, 2022 1:19 pm
We prepare the spin in state and the total initial state reads , with . We couple spin and path degree of freedom by weakly rotating the spin in path 1. We rotate it by a small angle about the z axis. The rotation is expressed by the operator acting only in path 1 or, equivalently, by the operator acting on the total state of both paths and spin, denoted as with , where and denote the path projection operators of path 1 and 2 respectively. The state after the spin rotation reads . The states of the two exit beams of the interferometer are given by the projection onto the exit states denoted as . The respective compensated state are . Rewriting the operators as we get , where and denote the weak values of the path projection operators respectively with and .
The read out the probe qubit is achieved by factorising the state in each port into the path dependent and a spin dependent part as , with . Now we can calculate the amplitudes of the eigenstates of the spin qubit as and . For the amplitude vanishes and the final spin state equals the initial state . Since we focus only on the limit of a small interaction strength series expansions gives . Simply speaking, in the limit of weak coupling, that means small values uf , the weak value of the path projection operator is given by the ratio of the spin rotation angle and the optimal compensation as – this ratio of angles is our new experimental accessible quantity, which we refer to as path presence.
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2022-12-02 07:08:32
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|
http://www.gradesaver.com/textbooks/science/physics/physics-principles-with-applications-7th-edition/chapter-13-temperature-and-kinetic-theory-problems-page-385/4
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# Chapter 13 - Temperature and Kinetic Theory - Problems: 4
A high of $57.8^{\circ} C$ and a low of $-89.4^{\circ} C$.
#### Work Step by Step
Use the formulas on page 362. $$\frac{5}{9}(136^{\circ} F – 32) = 57.8^{\circ} C$$ $$\frac{5}{9}(-129^{\circ} F – 32) = -89.4^{\circ} C$$
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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2018-01-21 08:57:53
|
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http://profs.etsmtl.ca/sandrews/publication/substructuring/
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# Schur Complement-based Substructuring of Stiff Multibody Systems with Contact
### Abstract
Substructuring permits parallelization of physics simulation on multi-core CPUs. We present a new substructuring approach for solving stiff multibody systems containing both bilateral and unilateral constraints. Our approach is based on non-overlapping domain decomposition with the Schur complement method, which we extend to systems involving contact formulated as a mixed bounds linear complementarity problem. At each time step, we alternate between solving the subsystem and interface constraint impulses, which leads to the identification of the active constraints. By using the active constraints to compute the effective mass of subsystems within the interface solve, we obtain an exact solution. We demonstrate that our simulations have preferable behavior compared to standard iterative solvers and substructuring techniques based on the exchange of forces at interface bodies. We observe considerable speedups for structured simulations where a user-defined partitioning can be applied, and moderate speedups for unstructured simulations, such as piles of bodies. In the latter case, we propose an automatic partitioning strategy based on the degree of bodies in the constraint graph. Because our method makes use of direct solvers, we are able to achieve interactive and real-time frame rates for a number of challenging scenarios involving large mass ratios, redundant constraints, and ill-conditioned systems.
*Both authors contributed equally to this work.
Type
Publication
ACM Transactions on Graphics
Date
BibTeX
@article{substructuring2019,
author = {Peiret, Albert and Andrews, Sheldon and K\“{o}vecses, J\‘{o}zsef and Kry, Paul G. and Teichmann, Marek},
title = {Schur Complement-based Substructuring of Stiff Multibody Systems with Contact},
journal = {ACM Trans. Graph.},
volume = {38},
number = {5},
year = {2019},
pages = {150:1–150:17},
articleno = {150},
numpages = {17},
doi = {10.1145/3355621},
publisher = {ACM}
}
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2022-10-02 10:01:59
|
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|
https://blender.stackexchange.com/questions/78045/import-multiple-non-obj-models
|
# Import multiple non OBJ models?
I'm having a problem trying to import multiple models from a format that uses a custom plugin. I press a to select them all, but only the first is imported. I have well over a thousand models, how would I go about doing this? I tried the solution here: How to batch import Wavefront OBJ files?, but it only works for OBJs. How would I do this for .rip files?
The plugin I am using is located here: https://github.com/Dummiesman/RipImport
• I don't know enough about scripting, but my blind guess is you would replace bpy.ops.import_scene.obj with watever operator i used for importing rip files, possibly bpy.ops.import_scene.rip (?) – Duarte Farrajota Ramos Apr 19 '17 at 1:27
You can do this by calling bpy.ops.import_scene.rip() as shown below:
import os
import bpy
path_to_rip_dir = os.path.join('C:\\', 'rips') #where C:\\rips\\ has the list of rip files
file_list = sorted(os.listdir(path_to_rip_dir))
rip_list = [item for item in file_list if item.endswith('.rip')]
for item in rip_list:
path_to_file = os.path.join(path_to_rip_dir, item)
bpy.ops.import_scene.rip(filepath = path_to_file)
You will need to install the addon first as mentioned in the readme file of the addon:
• Blender 32 Bit: On 32 bit Blender installations, extract the downloaded ZIP file to C:\Program Files (x86)\Blender Foundation\Blender\2.77\scripts\addon
• Blender 64 Bit: On 64 bit Blender installations, extract the downloaded ZIP file to C:\Program Files\Blender Foundation\Blender\2.77\scripts\addons
• After you extract the Add-on, it will NOT show up in your
import/export list by default!
• After extracting the Add-on, start Blender. Once Blender is
started, open up File->User Preferences, and navigate over to the
Add-ons tab. Find the add-on or search "NinjaRipper", and enable it. Click "Save User Preferences" on the bottom of the dialog, and close the dialog.
• Now you will be able to use the Add-on.
• What do the different keywords mean? I can't seem to get this to run... – Zac Perry Apr 19 '17 at 1:45
• @ZacPerry just remove the **keywords from the last line, try import_rip.load(self, context) instead – Tak Apr 19 '17 at 1:46
• Still does not function, gives me 'NameError: name 'import_rip' is not defined'. Maybe you could try basing your script off the one in the addon? Its on the github page in the original post. – Zac Perry Apr 19 '17 at 1:49
• @ZacPerry did you install the add-on? it works fine with me – Tak Apr 19 '17 at 1:55
• The original addon? Yes. The script you pasted in, no. I used it in the text editor. Maybe try using an actual model: drive.google.com/file/d/0B0JlgUd0NeDVdmtOTGhFRGtDUXc/… – Zac Perry Apr 19 '17 at 1:59
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2019-10-16 03:00:17
|
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https://www.physicsforums.com/threads/space-is-flat.632049/
|
# Space is flat?
1. Aug 29, 2012
### geordief
I was watching a Horizon program the other night on the BBC and was interested that they were able to repeat the "triangle on the surface of a sphere" experiment on the universe as a whole.
When this experiment is performed on the surface of the Earth on a large enough scale the angles of the triangle don't add up to 180 degrees and shows that the earth is not flat to someone even though they may not be able to leave the surface.
So , to recap, they were somehow able to draw these lines in the universe (I have no idea how!) and came back with the discovery that the angles of the triangle actually added up to180 degrees.
They said that this showed that space was flat and not curved!
My question is this.
If space is flat why is space-time curved?
2. Aug 29, 2012
### Cyghost
From what I understand there are 3 main theories on the form of space.
1. "Big Crunch Theory" A closed universe (such as a sphere) with sufficient matter of all kinds to allow gravity to eventually halt the expansion of the cosmos. Expansion become contraction and the universe would disappear from where it came.
2. "Flat Universe" where space has a curvature of zero and where total matter equals critical density, meaning that the universe would have no boundary and expands forever eventually slowing in infinite time but by definition "Infinity" has no limit.
3. "Open Saddle Shape" where total matter is less then critical density and would expands forever and its acceleration is driven by dark energy (an anti gravitational force) is currently quite a popular theory.
Q: If space is flat why is space-time curved?
A: Basically it is due to the contribution of matter that distorts the fabric of space-time.
Similar to the effect on light by Gravitational lensing.
Last edited: Aug 29, 2012
3. Aug 29, 2012
### bapowell
Because the space is expanding in time.
4. Aug 29, 2012
### geordief
thanks!
5. Aug 29, 2012
### d3mm
6. Aug 30, 2012
### Cyghost
@ d3mm
As I previously said they are all but theories..
The WMAP can show what ever it wants, but how one interprets the mathematics, it is still just another theory ready to dispute.
But the fact is space-time geometry is manipulated by matter.
7. Aug 30, 2012
### d3mm
What is the evidence for the curvature of space?
How does it deal with the results from WMAP survey?
So are gravity and quantum mechanics, you will have to do better than "it's just a theory".
8. Aug 30, 2012
### bapowell
And what "data" do you think confirms this fact? You seem to misunderstand how science works, which is unfortunate. The WMAP data cannot show "whatever it wants" -- it shows the actual universe. Then we have a theory that predicts what the CMB should look like. Then we compare these predictions with the WMAP data. Science occurs. Wash, rinse, repeat.
9. Aug 30, 2012
### bapowell
d3mm -- are you seeking answers to these questions or were you posing them to Cyghost -- can't tell.
10. Aug 30, 2012
### d3mm
bapowell, I was asking Cyghost to defend his position.
11. Aug 31, 2012
### Cyghost
Yes I think we are probably going a little off topic of the original question "If space is flat why is space-time curved?"
but I will say I certainly do not claim to agree or disagree to the 3 models I fore-mentioned as I was only presenting a few of the popular examples of late I have not once in this post stated that I disagree that space is flat so I'm not certain what I am supposedly defending but here is a link to your request for a response to your question.
Q : What is the evidence for the curvature of space?
http://science.nasa.gov/science-news/science-at-nasa/2011/04may_epic/
Also what is your definition of flat?
From what I understand WMAP measured the "curvature" of space to within accuracy to 0.6% of "flat"
Is that truly flat?
http://en.wikipedia.org/wiki/Euclidean_space
Q2: How does it deal with the results from WMAP survey?
How does your question even relate to the posters question of "If space is flat why is space-time curved?"
which I answered 3 times now (Where there is significant gravitation due to matter space-time will become curved)
Last edited: Aug 31, 2012
12. Aug 31, 2012
### Cyghost
I think you may have misunderstood what I was trying to say regarding (how one interprets the mathematics from the data received) I think it would be naive to simply "Wash, rinse, repeat." and not consider any other possibilities. To me that is not what science is about either (which apparently I seem to misunderstand) and I appreciate the personal attack BTW.
It's like saying ok we have received this data from WMAP and as you stated "we have a theory that predicts what the CMB should look like then we compare these predictions with the WMAP data"
What if there is another theory that also produces the same data outcomes?
Is that not possible?
To simplify it for you :
1+1 = 2.
4-2 = 2.
Different equation but same result.
Again you have taken this way off topic from the original question.
Also:
I'm not sure I agree with your answer to the question he asked "If space is flat why is space-time curved?
Last edited: Aug 31, 2012
13. Aug 31, 2012
### geordief
Is it possible to know why matter should have this effect on SpaceTime or is it simply a mathematical consequence that when we model the situation that the outcome resembles the dimpled cushion effect?
Does antimatter have the same effect?
If space time is severely distorted (as in the spinning binary black holes scenario ,say) would that create the opportunity for a massless object to travel practically instantly from one side of the (SpaceTime) region to the other a bit like in an anciently active area of the earth you might dig down and cross millions of geological years in a very short space because the strata have been stretched and thinned at that place (in other places the same amount of geological years might take longer to dig through)?
14. Aug 31, 2012
### clamtrox
I think the point he was trying to make is that you can only measure the angular diameter distance of the acoustic peaks from CMB, and then you infer the curvature by comparing your observation to the model. Interestingly enough, there are also direct ways of measuring curvature, but so far our observations are not good enough for constraining it enough to be useful. You can check out http://arxiv.org/abs/1102.4485 for more details.
15. Aug 31, 2012
### clamtrox
What kind of explanation would satisfy you? At some point you're going to have to take something for granted. The laws of physics can't explain themselves. For general relativity, what is taken for granted is that matter curves spacetime according to the famous Einstein equation, and everything else follows from this assumption (well, and a few others)
It does seem likely but very difficult to verify. Antimatter is so rare and gravitation is so weak, it seems unlikely that we'll be able to confirm this any time soon
16. Aug 31, 2012
### geordief
Any kind at all (I mean any theory). I can understand that it is acceptable to make a hypothesis and to then to test it to death but why should any phenomenon not have a physical( maybe not the right word? preceeding?) cause.
Maybe mathematics are the fundamental reality (as was believed by Pythagoras(?) I think).
Last edited: Aug 31, 2012
17. Aug 31, 2012
### clamtrox
And if you knew the cause, what makes you think you wouldn't be asking for the cause of the cause? Let's say I explain gravity by saying that there are these tiny invisible butterflies that fly around, pour glue into your watch to make it run slow and stretch space by flapping their wings. Would you feel that is a satisfactory explanation, or would you then ask: "but why butterflies?"
There is one perhaps more physical explanation for gravity, but unfortunately it has turned out to be very difficult to make it work. It turns out that the usual GR description of gravity is completely equivalent with the force being carried by virtual spin-2 particles (like electromagnetism is carried by photons). Unfortunately to describe the theory like this, you'd need to dress it into the language of quantum field theory, and that is a project that's still not quite finished.
18. Aug 31, 2012
### geordief
I am predisposed to believe that I would almost certainly be asking for the "cause of the cause" (otherwise I would be a believer in a God -which I am not)
And I don't mind outlandish causes if they fill the gap.
But I didn't ask for a theory of gravity -just ,in particular why an object with mass should distort spacetime (aside from a mathematical explanation).
Or is that the be all and end all of gravity-objects with mass distort SpaceTime and no further questions on the subject are relevant? (I don't mean to come across as indignant!)
Are you saying that my question "what in particular causes an object with mass to distort SpaceTime?" is a nonsensical question with the only possible answer "because it does" or "because that is what gravity is"
19. Aug 31, 2012
### d3mm
Cyghost
My question is about the overall shape of observable universe, not what happens locally in the vicinity of a large mass. I am asking which of the three theories you mention, do you believe is the most appropriate after new evidence such as WMAP and the accelerating expansion of the universe.
20. Aug 31, 2012
### bapowell
Yes, I'm aware that multiple theories can offer competing explanations for a given data set. The reason that I misunderstood what you wrote is because you were not being clear. You state that "The WMAP can show what ever it wants, but how one interprets the mathematics, it is still just another theory ready to dispute." How one interprets the mathematics? I guess you mean -- how one uses mathematics to explain the data. OK, got it. But it's still not "just another theory to dispute." There is an entire program for comparing the merits of competing theories given data, and yes, this program needs to be exercised in the face of any scientific data. The argument you seem to be advancing is not specific to the CMB -- it can apparently be applied to any body of data. It is simply impossible to use induction to obtain a uniquely correct theory. So, I am interested in knowing how your argument differs from that made by Hume et al. hundreds of years ago regarding the problem of induction.
If you want to debate the merits of competing explanations for the CMB, then fine. If you are just pointing out the obvious limitations of all experimental science, then I think you are taking this thread way off topic.
His question has to do with the fact that if space is flat (in the sense that R, the curvature scalar, is zero), how is it that space-time is curved? It has to do with the fact that the spatial metric components, $a(t)x^i$, while indeed giving flat hypersurfaces, are function of time. In this case, the curvature scalar is non-zero,
$$R \propto \frac{\ddot{a}}{a} + \left(\frac{\dot{a}}{a}\right)^2$$.
Do you still not agree?
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2017-11-21 07:06:20
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https://www2.physics.ox.ac.uk/contacts/people/tanx/publications
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# Publications by Xianyu Tan
## Evidence for H2 dissociation and recombination heat transport in the atmosphere of KELT-9b
Astrophysical Journal Letters American Astronomical Society 888 (2020) L15
M Mansfield, JL Bean, KB Stevenson, TD Komacek, TJ Bell, X Tan, M Malik, TG Beatty, I Wong, NB Cowan, L Dang, J-M Désert, JJ Fortney, BS Gaudi, D Keating, L Kreidberg, EM-R Kempton, V Parmentier, KG Stassun
## The atmospheric circulation of ultra-hot Jupiters
Astrophysical Journal American Astronomical Society 886 (2019) 1-20
X Tan, T Komacek
## Atmospheric Variability Driven by Radiative Cloud Feedback in Brown Dwarfs and Directly Imaged Extrasolar Giant Planets
Astrophysical Journal American Astronomical Society 874 (2019)
X Tan, AP Showman
Growing observational evidence has suggested active meteorology in the atmospheres of brown dwarfs (BDs) and directly imaged extrasolar giant planets (EGPs). In particular, a number of surveys have shown that near-infrared brightness variability is common among L and T dwarfs. Despite the likelihood from previous studies that atmospheric dynamics is the major cause of the variability, the detailed mechanism of the variability remains elusive, and we need to seek a natural, self-consistent mechanism. Clouds are important in shaping the thermal structure and spectral properties of these atmospheres via their opacity, and we expect the same for inducing atmospheric variability. In this work, using a time-dependent one-dimensional model that incorporates a self-consistent coupling between the thermal structure, convective mixing, cloud radiative heating/cooling, and condensation/evaporation of clouds, we show that radiative cloud feedback can drive spontaneous atmospheric variability in both temperature and cloud structure under conditions appropriate for BDs and directly imaged EGPs. The typical periods of variability are 1 to tens of hr, with a typical amplitude of the variability up to hundreds of K in effective temperature. The existence of variability is robust over a wide range of parameter space, but the detailed evolution of the variability is sensitive to model parameters. Our novel, self-consistent mechanism has important implications for the observed flux variability of BDs and directly imaged EGPs, especially for objects whose variability evolves on short timescales. It is also a promising mechanism for cloud breaking, which has been proposed to explain the L/T transition of BDs.
## Atmospheric circulation of brown dwarfs and Jupiter- and Saturn-like planets: Zonal jets, long-term variability, and QBO-type oscillations
Astrophysical Journal American Astronomical Society 883 (2019)
AP Showman, X Tan, X Zhang
Brown dwarfs and directly imaged giant planets exhibit significant evidence for active atmospheric circulation, which induces a large-scale patchiness in the cloud structure that evolves significantly over time, as evidenced by infrared light curves and Doppler maps. These observations raise critical questions about the fundamental nature of the circulation, its time variability, and its overall relationship to the circulation on Jupiter and Saturn. Jupiter and Saturn themselves exhibit numerous robust zonal (east–west) jet streams at the cloud level; moreover, both planets exhibit long-term stratospheric oscillations involving perturbations of zonal wind and temperature that propagate downward over time on timescales of ~4 yr (Jupiter) and ~15 yr (Saturn). These oscillations, dubbed the quasi-quadrennial oscillation (QQO) for Jupiter and the semiannual oscillation (SAO) on Saturn, are thought to be analogous to the quasi-biennial oscillation (QBO) on Earth, which is driven by upward propagation of equatorial waves from the troposphere. To investigate these issues, we here present global, three-dimensional, high-resolution numerical simulations of the flow in the stratified atmosphere—overlying the convective interior—of brown dwarfs and Jupiter-like planets. The effect of interior convection is parameterized by inducing small-scale, randomly varying perturbations in the radiative–convective boundary at the base of the model. Radiative damping is represented using an idealized Newtonian cooling scheme. In the simulations, the convective perturbations generate atmospheric waves and turbulence that interact with the rotation to produce numerous zonal jets. Moreover, the equatorial stratosphere exhibits stacked eastward and westward jets that migrate downward over time, exactly as occurs in the terrestrial QBO, Jovian QQO, and Saturnian SAO. This is the first demonstration of a QBO-like phenomenon in 3D numerical simulations of a giant planet.
## Effects of dissociation/recombination on the day–night temperature contrasts of ultra-hot Jupiters
Research Notes of the AAS American Astronomical Society 2 (2018) 36
TD Komacek, X Tan
## Effects of latent heating on atmospheres of brown dwarfs and directly imaged planets
Astrophysical Journal American Astronomical Society 835 (2017) 186-186
X Tan, AP Showman
The growing number of observations of brown dwarfs (BDs) has provided evidence for strong atmospheric circulation on these objects. Directly imaged planets share similar observations and can be viewed as low-gravity versions of BDs. Vigorous condensate cycles of chemical species in their atmospheres are inferred by observations and theoretical studies, and latent heating associated with condensation is expected to be important in shaping atmospheric circulation and influencing cloud patchiness. We present a qualitative description of the mechanisms by which condensational latent heating influences circulation, and then illustrate them using an idealized general circulation model that includes a condensation cycle of silicates with latent heating and molecular weight effect due to the rainout of the condensate. Simulations with conditions appropriate for typical T dwarfs exhibit the development of localized storms and east–west jets. The storms are spatially inhomogeneous, evolving on a timescale of hours to days and extending vertically from the condensation level to the tropopause. The fractional area of the BD covered by active storms is small. Based on a simple analytic model, we quantitatively explain the area fraction of moist plumes and show its dependence on the radiative timescale and convective available potential energy (CAPE). We predict that if latent heating dominates cloud formation processes, the fractional coverage area of clouds decreases as the spectral type goes through the L/T transition from high to lower effective temperature. This is a natural consequence of the variation of the radiative timescale and CAPE with the spectral type.
## Atmospheric circulation of hot Jupiters: dayside–nightside temperature differences. II. Comparison with observations
Astrophysical Journal American Astronomical Society 835 (2017) 198
TD Komacek, AP Showman, X Tan
The full-phase infrared light curves of low-eccentricity hot Jupiters show a trend of increasing fractional dayside–nightside brightness temperature difference with increasing incident stellar flux, both averaged across the infrared and in each individual wavelength band. The analytic theory of Komacek & Showman shows that this trend is due to the decreasing ability with increasing incident stellar flux of waves to propagate from day to night and erase temperature differences. Here, we compare the predictions of this theory with observations, showing that it explains well the shape of the trend of increasing dayside–nightside temperature difference with increasing equilibrium temperature. Applied to individual planets, the theory matches well with observations at high equilibrium temperatures but, for a fixed photosphere pressure of $100\ \mathrm{mbar}$, systematically underpredicts the dayside–nightside brightness temperature differences at equilibrium temperatures less than $2000\ {\rm{K}}$. We interpret this as being due to the effects of a process that moves the infrared photospheres of these cooler hot Jupiters to lower pressures. We also utilize general circulation modeling with double-gray radiative transfer to explore how the circulation changes with equilibrium temperature and drag strengths. As expected from our theory, the dayside–nightside temperature differences from our numerical simulations increase with increasing incident stellar flux and drag strengths. We calculate model phase curves using our general circulation models, from which we compare the broadband infrared offset from the substellar point and dayside–nightside brightness temperature differences against observations, finding that strong drag or additional effects (e.g., clouds and/or supersolar metallicities) are necessary to explain many observed phase curves.
## Precise radial velocities of giant stars
Astronomy and Astrophysics EDP Sciences 568 (2014) 15-
T Trifonov, S Reffert, X Tan, MH Lee, A Quirrenbach
We report the discovery of a new planetary system around the K giant η Cet (HIP 5364, HD 6805, HR 334) based on 118 high-precision optical radial velocities taken at Lick Observatory since July 2000. Since October 2011 an additional nine near-infrared Doppler measurements have been taken using the ESO CRIRES spectrograph (VLT, UT1). The visible data set shows two clear periodicities. Although we cannot completely rule out that the shorter period is due to rotational modulation of stellar features, the infrared data show the same variations as in the optical, which strongly supports that the variations are caused by two planets. Assuming the mass of η Cet to be 1.7 M⊙, the best edge-on coplanar dynamical fit to the data is consistent with two massive planets (mb sini = 2.6 ± 0.2 MJup, mc sini = 3.3 ± 0.2 MJup), with periods of Pb = 407 ± 3 days and Pc = 740 ± 5 days and eccentricities of eb = 0.12 ± 0.05 and ec = 0.08 ± 0.04. These mass and period ratios suggest possible strong interactions between the planets, and a dynamical test is mandatory. We tested a wide variety of edge-on coplanar and inclined planetary configurations for stability, which agree with the derived radial velocities. We find that for a coplanar configuration there are several isolated stable solutions and two well defined stability regions. In certain orbital configurations with moderate eb eccentricity, the planets can be effectively trapped in an anti-aligned 2:1 mean motion resonance that stabilizes the system. A much larger non-resonant stable region exists in low-eccentricity parameter space, although it appears to be much farther from the best fit than the 2:1 resonant region. In all other cases, the system is categorized as unstable or chaotic. Another conclusion from the coplanar inclined dynamical test is that the planets can be at most a factor of ~1.4 more massive than their suggested minimum masses. Assuming yet higher inclinations, and thus larger planetary masses, leads to instability in all cases. This stability constraint on the inclination excludes the possibility of two brown dwarfs, and strongly favors a planetary system.
## Characterizing the orbital and dynamical state of the HD 82943 planetary system with keck radial velocity data
Astrophysical Journal American Astronomical Society 777 (2013) 101-101
X Tan, MJ Payne, MH Lee, EB Ford, AW Howard, JA Johnson, GW Marcy, JT Wright
We present an updated analysis of radial velocity data of the HD 82943 planetary system based on 10 yr of measurements obtained with the Keck telescope. Previous studies have shown that the HD 82943 system has two planets that are likely in 2:1 mean-motion resonance (MMR), with orbital periods about 220 and 440 days. However, alternative fits that are qualitatively different have also been suggested, with two planets in a 1:1 resonance or three planets in a Laplace 4:2:1 resonance. Here we use χ2 minimization combined with a parameter grid search to investigate the orbital parameters and dynamical states of the qualitatively different types of fits, and we compare the results to those obtained with the differential evolution Markov chain Monte Carlo method. Our results support the coplanar 2:1 MMR configuration for the HD 82943 system, and show no evidence for either the 1:1 or three-planet Laplace resonance fits. The inclination of the system with respect to the sky plane is well constrained at $20^{+4.9}_{-5.5}$ degrees, and the system contains two planets with masses of about 4.78 M J and 4.80 M J (where M J is the mass of Jupiter) and orbital periods of about 219 and 442 days for the inner and outer planet, respectively. The best fit is dynamically stable with both eccentricity-type resonant angles θ1 and θ2 librating around 0°.
## A detailed analysis of the HD 73526 2:1 resonant planetary system
Astrophysical Journal IOP Publishing 780 (2013) 140-
X Tan, RA Wittenmyer, MH Lee, J Horner, CG Tinney, RP Butler, GS Salter, HRA Jones, BD Carter, SJ O'Toole, J Bailey, JD Crane, D Wright, P Arriagada, I Thompson, D Minniti, M Diaz
<p style="text-align:justify;"> We present six years of new radial velocity data from the Anglo-Australian and Magellan Telescopes on the HD 73526 2:1 resonant planetary system. We investigate both Keplerian and dynamical (interacting) fits to these data, yielding four possible configurations for the system. The new data now show that both resonance angles are librating, with amplitudes of 40° and 60°, respectively. We then perform long-term dynamical stability tests to differentiate these solutions, which only differ significantly in the masses of the planets. We show that while there is no clearly preferred system inclination, the dynamical fit with i = 90° provides the best combination of goodness-of-fit and long-term dynamical stability. </p>
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2020-04-05 01:08:40
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http://www.math.psu.edu/calendars/meeting.php?id=20085
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# Meeting Details
Title: Exponential map and L-infinity algebra associated to a Lie pair GAP Seminar Mathieu Stienon, Penn State In this talk, we unveil homotopy-rich algebraic structures generated by the Atiyah classes relative to a Lie pair $(L,A)$ of algebroids. In particular, we prove that the quotient $L/A$ of such a pair admits an essentially canonical homotopy module structure over the Lie algebroid $A$, which we call Kapranov module.
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2015-03-04 00:25:52
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http://math.stackexchange.com/questions/125908/could-someone-show-me-step-by-step-the-evaluation-of-this-basic-as-level
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Could someone show me step by step the evaluation of this (Basic/AS Level)
I have:
$(1 \frac{9}{16}) ^\frac{3}{2}$
$\frac{125}{64}$
I'm not sure the order in which everything was done to get the answer. Also, does anyone know of a site like WolframAlpha but which shows the step by step progress of a maths problem?
Thank you!
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$$(1 \frac{9}{16}) ^\frac{3}{2}=(\frac{25}{16})^\frac{3}{2}=((\frac{5}{4})^2)^\frac{3}{2}=(\frac{5}{4})^3=...$$
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That's great, thank you! – penpen Mar 29 '12 at 12:39
Welcome to math.SE! It is my pleasure. – Salech Alhasov Mar 29 '12 at 12:42
$$\left(1 \frac{9}{16} \right)^{3/2} = \left(\frac{25}{16}\right)^{3/2} = \left(\frac{25}{16}\right)^{2/2} \cdot \left(\frac{25}{16} \right)^{1/2} = \frac{25}{16} \cdot \frac{5}{4} = \frac{125}{64}.$$
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2015-11-30 19:16:58
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http://mathhelpforum.com/differential-equations/205726-solving-pdes.html
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Math Help - Solving PDEs
1. Solving PDEs
Hi,
Given a PDE of this form
then one way to solve it iteratively is
(Equations from IISc).
The update equation has me confused for the following reason. $u(x, t)$ appears in the PDE as function of both time $t$ and distance $x$. However, the update equation only gives you $u$ as a function of time ( $u^{n+1}_{j}$), and what's more, it seems to assume that $u$ is already known as a function of distance because it has $u^{n}_{j+1}$ terms.
In short, the solver seems to assume that you already know the solution .
Anyone familiar with this who can enlighten me?
Thanks.
2. Re: Solving PDEs
Hey algorithm.
What is the definition of u_j given in your book?
3. Re: Solving PDEs
chiro,
$u^n_{j}$ is defined as the numerical approximation of $u(x, t)$ at time index $n$, and distance index $x$. So basically it represents the $n^{th}$ time iteration and the $j^{th}$ distance iteration of the solver algorithm.
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2015-08-02 03:34:06
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https://ask.sagemath.org/questions/10175/revisions/
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# Revision history [back]
### Polynomial division mod n
Hi everyone,
Let's suppose that we are working with polynomials modulo n a composite number, for which we know the factorization (n=p*q). If we know that f(x) can be divided by e.g. g(x), what is the more efficient way to calculate f(x)/g(x) in Z_n with Sage?
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2021-09-27 16:33:50
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https://www.aminer.cn/pub/5e427c893a55acbff4c4075f/description-based-text-classification-with-reinforcement-learning
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# Description Based Text Classification with Reinforcement Learning
Wu Wei
Han Qinghong
ICML, pp. 1371-1382, 2020.
Cited by: 0|Views41
EI
Abstract:
The task of text classification is usually divided into two stages: {\it text feature extraction} and {\it classification}. In this standard formalization categories are merely represented as indexes in the label vocabulary, and the model lacks for explicit instructions on what to classify. Inspired by the current trend of formalizing N...More
Code:
Data:
Full Text
Bibtex
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2021-04-16 18:30:20
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https://www.physicsforums.com/threads/related-rates-triangle.16287/
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# Related Rates triangle
## Main Question or Discussion Point
Hello here's the problem:
Each of the two sides of a triangle are increasing at the rate of 1/2 foot per second, and the included angle is decreasing 2 degrees per second. Find the rate of change of the area when the sides and included angle are respectively 5ft., 8ft., and 60 deg.
Here is my question: How do I find the height of the triangle?
Say my base is 5ft. How do I express it in terms of the given details in the problem?
matt grime
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2020-01-21 08:52:07
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https://pypi.org/project/numbereddoor-game/0.1.0/
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Skip to main content
A recreation of the door puzzles from the Nonary game, 999.
## Project description
# Numbered Door Game
Numbered Door Game is a reimplementation of the door puzzles from Zero Escape: 999. This version allows the player to choose what numbers are used on the door while leaving the others behind. The player must progress through all 4 stages to win.
STILL A WORK IN PROGRESS
## Installation
pip install --save numbered-door-game
## Controls
#### Keyboard
- Use arrow keys to move around and highlight elements, numbers and doors.
- Use ENTER key to select numbers and add to selected door, or switch between active door.
#### Mouse
- Use the mouse to highlight elements, numbers and doors. This is an alternative to using the arrow keys.
- Use MOUSE 1 to select a highlighted element. This is an alternative to pressing the ENTER key.
## How to Play
- The goal of the game is to add the available numbers together in order to get the digital root of the door.
A digital root is the single digit value obtained by recursively summing digits, on each iteration
using the result from the previous iteration to compute a digit sum. The process continues until a
single-digit number is reached.
Example
Door value = 9
Numbers available = _ _ 3 _ _ 6 _ _ 9
calculate)
3 + 6 + 9 = 18
Add all the digits together to get 18
18 = 1 + 8 = 9
18 is not a digit so we have to add the digits that make up this number in order
to reduce the result to a its DIGITAL ROOT. The digits 1 and 8 make up 18 so we
add those digits together resulting in 9. 9 is a digit so there is no need to
reduce the result further.
- OR -
calculate)
3 + 9 = 12
We are given 3 digits: 1, 6, 9, so lets start with a different approach than
above. Adding two digits, 3 and 9, resulting in 12.
12 = 1 + 2 = 3
12 is not a digit so we need to reduce this number to its digital root. We have
1 and 2 which when added together result in 3.
3 + 6 = 9
We still have the digit 6 that needs to be added to the answer, so we add 6 to
the result from the digital root of 3 + 9 which was 3. Adding 3 and 6 results in
9 with 9 being a digit so we have found our digital root and solution to the problem.
Its also interesting to note that 9 never changes the digital root of a problem.
- The player can only use each provided number once,and the player has to use 0 or 3-5 numbers for a door's solution. The solution will be counted as invalid otherwise.
- After a stage is completed, all numbers used to get through the door(s) from the previous stage will then be available for the next stage rendering the unused numbers unavailable for the rest of the game.
- Only one door has to be solved to proceed to the next stage, but this will hinder the player's progression and may stop them from winning at all.
- In the HARD difficulty you must use all the available digits on the door.
## Acknowledge
- Spike Chunsoft
## Download files
Download the file for your platform. If you're not sure which to choose, learn more about installing packages.
### Source Distribution
numbereddoor-game-0.1.0.tar.gz (2.5 kB view hashes)
Uploaded source
### Built Distribution
numbereddoor_game-0.1.0-py3-none-any.whl (2.5 kB view hashes)
Uploaded py3
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2022-12-05 00:56:40
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https://jeremy9959.net/2021-Fall-3230-Math/problems/week13.html
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# Week 13 problems
These problems are taken (mostly) from the Chapter 15 Exercises in the text.
1. (Problem 3) Show that every group of order $45$ has a normal subgroup of order $9$.
2. (Problem 6) Prove that a group of order $160$ has a proper normal subgroup and is therefore not simple.
3. (Problem 12) Let $G$ be a group of order $p^r$ where $p$ is a prime. Prove that $G$ has a normal subgroup of order $p^{r-1}$.
4. (Problem 20) What is the smallest odd number $n$ for which there is a nonabelian group $G$ with $n$ elements? Can you find such a group?
5. (Problem 18 simplified) Let $G$ have order $p^aq^b$ where $p$ and $q$ are primes. If $G$ has only one Sylow $p$-subgroup $P$ and one Sylow $q$-subgroup $Q$ then $G$ is isomorphic to $P\times Q$.
6. Let $G$ be a group with $2m$ elements where $m$ is odd. We will prove that $G$ has a non-trivial normal subgroup of index $2$.
• By Cayley’s theorem, $G$ acts as a set of permutations of itself by the action $(g,g’)\mapsto gg’$.
• Show that $G$ contains an element $\sigma$ of order $2$.
• Show that $\sigma$ acts on $G$ as a product of $m$ transpositions.
• Show that the homomorphism $f:G\to \mathbb{Z}_{2}$ obtained by taking the sign of the permutation of $g$ acting on itself is surjective.
• Conclude that $G$ has a normal subgroup of index $2$.
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2022-01-25 11:38:11
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http://math.stackexchange.com/tags/special-functions/new
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# Tag Info
## New answers tagged special-functions
3
Since, $m$ and $m^2$ have the same parity, $$\sum_{(m,n) \neq (0,0)} \frac{(-1)^{m+n}}{m^2 + n^2} = \sum_{(m,n) \neq (0,0)} \frac{(-1)^{m^2+n^2}}{m^2 + n^2}$$ It boils down to this question.
0
You have that $$f(k,x) = \frac{1}{\ln(k)} e^{x\ln(k)} (x\ln(k)) = y$$ Hence, the inverse of your function is $$\frac{1}{\ln(k)} W(y\ln(k))$$ Indeed, $$f\left(k,\frac{1}{\ln(k)} W(y\ln(k)) \right) = \frac{1}{\ln(k)}\exp \left( \ln(k)\frac{1}{\ln(k)} W(y\ln(k)) \right)\ln(k)\frac{1}{\ln(k)} W(y\ln(k))$$ = \frac{1}{\ln(k)}\exp \left( W(y\ln(k)) ... 1 The inverse of f(k,x) can be found in terms of the Lambert W. \begin{align*} y&=k^xx \\ y &= e^{x \ln k}x \\ y\ln k &= e^{x \ln k}x\ln k \\ W(y\ln k) &= x\ln k \\ x &= \frac{W(y\ln k)}{\ln k} \end{align*} 1 (A partial answer.) I tested your cfrac with the order 12 discussed by Naika (which in turn is a special case of a general cfrac by Ramanujan) and labeled as D_1(q) here,D_1(q)= \dfrac{q(1-q)} {1-q^3+\dfrac{q^3(1-q^2)(1-q^4)} {(1-q^3)(1+q^6)+\dfrac{q^3(1-q^8)(1-q^{10})} {(1-q^3)(1+q^{12})+\dfrac{q^3(1-q^{14})(1-q^{16})} {(1-q^3)(1+q^{18})+\ddots }}}} ...
27
You may write, for $N \geq 2$, \begin{align} e^{3/2}\prod_{n=2}^{N}e\left(1-\dfrac{1}{n^2}\right)^{n^2}&=e^{3/2}\times\prod_{n=2}^{N}e\times\prod_{n=2}^{N}\left(1-\dfrac{1}{n^2}\right)^{n^2}\\\\ &=e^{3/2}\times e^{N-1}\times\prod_{n=2}^{N}\dfrac{(n-1)^{n^2}}{n^{n^2}}\dfrac{(n+1)^{n^2}}{n^{n^2}}\\\\ &=e^{3/2}\times ... 11 One can write the log of the product as\sum_{n=2}^{\infty} \left [1+n^2 \log{\left (1-\frac1{n^2} \right )} \right ] $$Now,$$\log{\left (1-\frac1{n^2} \right )} = -\int_0^1 \frac{du}{n^2-u} $$So the sum is equal to$$-\int_0^1 du \, u \sum_{n=2}^{\infty} \frac1{n^2-u} \sum_{n=-\infty}^{\infty} \frac1{n^2-u} = -\frac{\pi \cot{\pi ...
2
Make your life easier writing$$B=\frac{2hf^3}{c^2\left(e^\frac{hf}{kT}-1\right)}=\frac{\alpha}{e^{\beta}-1}$$ using $\alpha=\frac{2hf^3}{c^2}$ and $\beta=\frac{hf}{kT}$. So $$\frac{dB}{dT}=\frac{dB}{d\beta}\times\frac{d\beta}{dT}$$ Now $$\frac{dB}{d\beta}=-\frac{\alpha e^{\beta }}{\left(e^{\beta }-1\right)^2}$$ $$\frac{d\beta}{dT}=-\frac{hf}{kT^2}$$ So ...
2
Let the integral in question be \begin{align}\tag{1} I_{a} = \int_{0}^{\infty} e^{-st} \, \frac{\ln(1+a t)}{1 + t} \, dt. \end{align} For the case of $a=1$ the following is obtained. By utilizing \begin{align} \int_{0}^{\infty} e^{-s t} \, \ln^{2} t \, dt = \frac{1}{s} \, ( \zeta(2) + (\gamma + \ln s)^{2} ) \end{align} for the change $t \to t+1$ leads to ...
2
Integrating by parts and using the known series results, we get that $$\int_0^1 \frac{\text{Li}_2 \left(-\frac{1}{1-z}\right)-\text{Li}_2 \left(-\frac{1}{1+z}\right)}{z}dz$$ $$=\int_0^1 \frac{\log (z+2) \log (z)}{z+1}dz+\underbrace{\int_0^1\frac{\log (2-z) \log (z)}{1-z} dz}_{\large \sum _{k=1}^{\infty } \frac{(-1)^k H_k}{k^2}=-5/8 \zeta ... 1$$B(f,T)=\frac{2hf^3}{c^2}\frac{1}{e^\frac{hf}{kT}-1}\frac{\partial B}{\partial T}=\frac{2hf^3}{c^2}\frac{\partial}{\partial T}\left(\frac{1}{e^\frac{hf}{kT}-1}\right)==-\frac{2hf^3}{c^2}\left(\frac{1}{e^\frac{hf}{kT}-1}\right)^2\frac{\partial}{\partial T}\left(e^\frac{hf}{kT}-1\right)=$$... 2 My suggestion is to look at \log B. Take the derivate of the logged function wrt to T and multiply by B(T). 1 This is conditional on Dickson's conjecture, but may be of interest. You may choose any admissible prime k-tuple (b_1=0,b_2,b_3,\ldots,b_k). By definition the b_i avoid some residue modulo every prime, and hence so do \{(2+b_i)n+1\}. Then it is a consequence of Dickson's conjecture that q_i=(2+b_i)n+1 are simultaneously prime for 1\le i \le k for ... 0 There is Legendre's formula which counts the number of positive integers less than or equal to a number n which are not divisible by any of the first k primes:$$\begin{align} &\phi(n,k)=\lfloor n \rfloor-\sum_{p_i\le k}\left\lfloor \dfrac{ n }{(p_i)}\right\rfloor+\sum_{p_i<p_j\le k}\left\lfloor\dfrac{ ...
9
I changed my evaluation slightly, and I was able to get the result in a very simple form. First notice that $$\int_{-\infty}^{\infty} \text{Ei}^{2}(-|x|) e^{ikx} \, dx = 2 \int_{0}^{\infty} \text{Ei}^{2}(-x) \cos(kx) \, dx.$$ Then integrating by parts, and assuming for now that $k >0$, \begin{align}\int_{-\infty}^{\infty} \text{Ei}^{2}(-|x|) ... 1 Here's another variation of the theme based upon Stirling Numbers. Starting from \begin{align*} f_n(x)&=\int_1^x\binom{t-1}{n}dt =\frac{1}{n!}\int_1^x{(t-1)}_ndt =\frac{1}{n!}\int_0^{x-1}{(u)}_ndu \end{align*} we can use the Stirling Numbers of the first kind s(n,k) which can be defined for n\geq 0 and 0\leq k \leq n by ... 0 Some notes: \begin{align} {}_{2}F_{1}(i, 1; 1+i; x) = \sum_{n=0}^{\infty} \frac{(i)_{n} \, x^{n}}{(1+i)_{n}}. \end{align} Now, \begin{align} \frac{(i)_{n}}{(1+i)_{n}} = \frac{\Gamma(i+1) \, \Gamma(n+i)}{\Gamma(i) \, \Gamma(n+i+1)} = \frac{i}{n+i} = \frac{1}{1-i n} = \frac{1+i n}{n^{2}+1} \end{align} which leads to \begin{align} {}_{2}F_{1}(i, 1; 1+i; x) = ... 3 Got a reference on MO; see http://mathoverflow.net/questions/210144/number-of-primes-one-larger-than-divisors-of-a-fixed-number-which-is-lcm-of-1-2#comment520981_210144 and http://www.math.drexel.edu/~eschmutz/PAPERS/lambda.pdf In Theorem 1 on the first page of the Erdos-Pomerance-Schmutz article, they announce the existence of a constant c and a ... 2 There is indeed a relation with the Bernoulli numbers of second kind. If I'm right, this is the result : Theorem : generating function of f_n(x)\sum_{n=0}^{+\infty} f_n(x)z^n = \dfrac{(1+z)^{x-1}-1}{\log (1+z)}$$Corrolary :$$f_n(x) = \sum_{k=0}^n {{x-1}\choose{k+1}} \dfrac{b_{n-k}}{(n-k)!}$$I'll present here mainly the "formal" steps ... 3 Take Cauchy's differentiation formula:$$ f^{(n)}(a) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z-a)^{n+1}}\, dz $$and plug a holomorphic f such that f^{(n)}(a)=1. For example, f(z)=\exp(z), a=0, and \gamma the unit circle:$$ \frac{1}{n!} = \frac{1}{2\pi i} \oint_\gamma \frac{e^z}{z^{n+1}}\, dz $$Does that count? 3 You can write the inverse Laplace transform of 1/s^{n+1}, evaluated at t=1, as 1/n!. The integral is$$ \int_{c-i\infty}^{c+i\infty}\frac{1}{2\pi is^{n+1}}e^s\,ds=\frac{1}{n!}, $$for suitable real c. 8 Following @RobertIsrael, we have$$\frac{1}{2\pi}\int_0^{2\pi}e^{e^{i\phi}}e^{-in\phi}d\phi=\oint_{|z|=1}e^{z}z^{-n}\frac{dz}{iz}\tag 1$$We note that the integrand on the right-hand side of (1) has a pole of order n+1 at z=0. The residue is given by$$\text{Res}\left(-i\frac{e^z}{z^{n+1}},z=0\right)=\frac{1}{n!}\lim_{z\to ...
11
$$\dfrac{1}{2\pi} \int_0^{2\pi} e^{e^{i\theta}} e^{-in\theta}\; d\theta$$
2
This is related. The function $$f(a,b,x)=\frac{\sin[b\sqrt{a^{2}+x^{2}}]}{\sqrt{a^{2}+x^{2}}}$$ has Fourier transform $$\hat{f}(a,b,s)= \left\{\begin{array}{cc} \frac{\pi}{2}J_{0}[a\sqrt{b^{2}-s^{2}}], & 0 < s < b \\ 0 & b < s < \infty \end{array}\right.$$ The derivative of ...
5
This time I let the target Carmichael number be the least common multiple of the numbers from $1$ to $w.$ This is more efficient in terms of the number of divisors. The Superior Highly Composite Numbers and the Colossally Abundant Numbers share the main property of this LCM, which is that the exponent of some prime $p$ is proportional to $1/ \log p.$ As a ...
1
It took a while, but I wrote something in C++ with GMP to find the biggest possible $n$ that has a fixed Carmichael number, furthermore I took the Carmichael numbers to be $w!$ for $4 \leq w \leq 12,$ which takes a fair amount of time as it is. The quantity $\log f(n) / \log n$ evidently gets arbitrarily close to $1$ this way. ...
5
If you want a symmetrical generalization for $n$th roots then define $$C_{k,n}(z)=\frac{1}{n}\sum_{\zeta^n=1}\zeta^k e^{\zeta z}.$$ Then $C_{0,2}(z)=\cosh z$ and $C_{1,2}(z)=\sinh z$. Note that $C_{k,n}(z)=C_{0,n}^{(k)}(z)$. In particular, for $n=3$, if $\omega$ is the cube root of unity in the upper half plane then $$\begin{array}{lll} C_{0,3}(z) & ... 1 One thing you definitely want to do is simply find the sequence of integers for which your quantity increases, and factor those. This is the entire story for a number of optimization problems that go back to Ramanujan. Those are all multiplicative functions. The bad news for you is that the Carmichael function is not multiplicative. Please take a look at ... 6 Since we know the value of \eta(3i), the point is just to compute the value of the product:$$ \prod_{n\geq 0}(1+e^{-6\pi n})=\exp\sum_{n\geq 0}\log\left(1+e^{-6\pi n}\right)=\exp\sum_{n\geq 0}\int_{n}^{n+1}\frac{6\pi n}{1+e^{6\pi s}}\,ds$$where:$$\sum_{n\geq 0}\int_{n}^{n+1}\frac{6\pi n}{1+e^{6\pi s}}\,ds = \int_{0}^{+\infty}\frac{6\pi s\,}{1+e^{6\pi ...
5
After persevering with a Mathematica session, I found that $F(6i)$ is the root of $96$-deg eqn (no wonder it was hard to find!) but could be prettified as, $$\eta(6i) = \frac{1}{2\cdot 6^{3/8}} \left(\frac{5-\sqrt{3}}{2}-\frac{3^{3/4}}{\sqrt{2}}\right)^{1/6}\,\frac{\Gamma\big(\tfrac{1}{4}\big)}{\pi^{3/4}}$$ However, the second question is still open.
0
By exploiting: $$\sum_{n\geq 0}\frac{1}{(n+a)(n+b)}=\frac{\psi(a)-\psi(b)}{a-b} \tag{1}$$ we have that our limit equals: $$\begin{eqnarray*} \lim_{x\to 0^+}\left(\frac{1}{x}+\sum_{n\geq 0}\left(\frac{1}{n+x}-\frac{1}{n+\frac{x}{2}}\right)\right)&=&\lim_{x\to 0^+}\sum_{n\geq 1}\left(\frac{1}{n+x}-\frac{2}{2n+x}\right)\\&=&-\lim_{x\to ... 2 The following is not a direct answer to your question, but is rather too long for a comment. This is basically Ramanujan's approximation$$\pi \approx \frac{24}{\sqrt{n}}\log(2^{1/4}g_{n}) = \frac{6}{\sqrt{n}}\log (2g_{n}^{4}) = \frac{6}{\sqrt{n}}\log (2u)\tag{1}$$where g_{n} is Ramanujan's class invariant given by$$g_{n} = 2^{-1/4}e^{\pi\sqrt{n}/24}(1 ...
2
1. $f(x)=-x\ln{|x|}$. Verify: $f(0^{+})=f(0^{-})=0$, so $\lim_{x\to0}f(x)=0$; $f^{'}(x)=-\ln{|x|}-1$, so $\space f^{'}(0^{+})=+\infty$. 2. $f(x)=\frac{\arctan{x}}{\sqrt{|x|}}$.Verify: $f(0^{+})=f(0^{-})=0$ (L'Hôpital's rule: $\lim_{x\to0^{+}}f(x)=lim_{x\to0^{+}}\frac{\frac{1}{1+x^2}}{\frac{1}{2\sqrt{x}}}=im_{x\to0^{+}}\frac{2\sqrt{x}}{1+x^2}=0$) ...
0
From you question, it is unclear exactly what shape you are looking for, and there are many functions that could describe the behaviour you're after. However, two possible options could be the negative exponential and a negative gompertz function. Possible forms of these could be: Negative exponential $y(x)=e^{−ax+ln(1-b)}+b,$ where $b=0.2$ and a is a ...
0
How about something like $$y=1-0.8\,\frac{e^{ax}-1}{e^{1095a}-1}?$$ The sign of $a$ determines concavity or convexity; $a=-0.005$ gives a nice graph.
1
I have been presented with a similar problem, that is why I am answering in this section. Joriki, do you use the p-test to test for convergence? If you have two functions $f(x)$ and $g(x)$ with $f(x)\geq g(x)\geq0$, then if the integral of $g(x)$ diverges, so should the integral over $f(x)$. And if the integral over $f(x)$ converges, then this should also ...
3
I'm going to use the following 3 identities along with the known value $\text{Li}_{2} \left(\frac{1}{2} \right) = \frac{\pi^{2}}{12} - \frac{1}{2} \log^{2}(2)$: $$\text{Li}_{2}(1-z) = - \text{Li}_{2} \left(1- \frac{1}{z} \right) - \frac{1}{2} \log^{2} (z) , \quad z \notin (-\infty,0] \tag{1}$$ $$\text{Li}_{2}(z) = - \text{Li}_{2} \left(\frac{1}{z} \right) ... 1 With the help of Mathematica I got:$$\mathcal{L}\left(\frac{J_1(R\sqrt{x})}{x^{3/2}}\right)=\frac{|R|}{2}\left(1-2\gamma-\frac{4s}{R^2}\left(1-e^{-\frac{R^2}{4s}}\right)+\log\frac{4}{R^2}-\Gamma\left(0,\frac{R^2}{4s}\right)\right)$$and by expanding the RHS as a series it is not difficult to check it matches your series. 0 Substitute z = e^{u}, e^{u/2}du = \frac{dz}{\sqrt z}.$$ \int \sqrt{z\frac{z + z^{-1}}{2} - z\cos v} \frac{dz}{z} = \frac{1}{\sqrt{2}}\int \frac{\sqrt{z^2 + 1 - 2z\cos v}}{z} dz = (9) = \\ = \frac{1}{\sqrt{2}}\sqrt{z^2 + 1 - 2z\cos v} - \frac{\cos v}{\sqrt{2}} \int\frac{dz}{\sqrt{z^2 + 1 - 2z\cos v}} + \frac{1}{\sqrt{2}} \int \frac{dz}{z\sqrt{z^2 + 1 - ...
7
At Vladimir Reshetnikov's request I'm going to show how to find an antiderivative for $$\frac{\log(1+2x) \log(1-x)}{1+x}$$ using the identity $$2 \log(x) \log(y) = \log^{2}(x) + \log^{2}(y) - \log^{2} \left(\frac{x}{y} \right)$$ where $x$ and $y$ are positive real values. \begin{align} &\int \frac{\log(1+2x) \log(1-x)}{1+x} \, dx \\ &= ... 1 Note that by Rodrigues' formula, P_{n+1}=\frac{1}{2^{n+1}(n+1)!}D^{n+1}[(x^{2}-1)^{n+1}]. Multiplying by (n+1), we have: \begin{align*} &{\left(n+1\right)P_{n+1}(x)}={\left(n+1\right)\frac{1}{2^{n+1}(n+1)!}D^{n+1}[(x^{2}-1)^{n+1}]}\\ =\;&{\left(n+1\right)\frac{1}{2^{n}n!}D^{n}[x(x^{2}-1)^{n}]}\\ \end{align*} Using the Leibniz rule: \begin{align*} ... 1 Using Rodrigues' formula \displaystyle P_n(x)=\frac{1}{2^n n!}D^n\left(x^2-1\right)^n, we can write \begin{align*} &\color{blue}{\left(n+1\right)P_{n+1}(x)}-\color{green}{\left(2n+1\right)xP_n(x)}+\color{red}{nP_{n-1}(x)}=\\ ... 1 Based on John Barber's comment, I plotted the fractal he mentioned using the domain coloring below: The above image assigns a color to every point in the complex plane (only [-10,10] of each axis is shown). Plotting \log(z)/\sqrt z gives: And iterating 100 times gives: There appears to be 5 colors: yellow, dark yellow, brown, dark brown, and red ... 1 In reference to Kirill's answer, I will show that indeed I(3) = \int_{1}^{\infty} \int_{1}^{\infty} \int_{1}^{\infty} \frac{1}{xyz} \frac{dx \, dy \, dz}{x+y+z-2} = \frac{7}{2}\zeta(3).$$I will make the same change of variables I made in my answer to your other more recent question.$$ \begin{align} I(3) &= \int_{1}^{\infty} \int_{1}^{\infty} ...
Top 50 recent answers are included
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2015-07-06 17:55:00
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https://math.stackexchange.com/questions/3384064/integral-action-in-prediction-finding-predicted-inputs-from-augmented-model
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# Integral action in prediction — finding predicted inputs from augmented model
Assume that we have a state space model:
$$x(k+1) = A_m x(k) + B_m u(k)$$
And we implement integral action to the model by creating the agumented state space model:
$$\begin{bmatrix} x(k+1)\\ y(k+1) \end{bmatrix} = \begin{bmatrix} A_m & 0 \\ C_mA_m & 1 \end{bmatrix}\begin{bmatrix} x(k)\\ y(k) \end{bmatrix} + \begin{bmatrix} B_m\\ C_mB_m \end{bmatrix} \begin{bmatrix} u(k) \end{bmatrix} \\ \begin{bmatrix} y(k) \end{bmatrix} = \begin{bmatrix} 0 &1 \end{bmatrix}\begin{bmatrix} x(k)\\ y(k) \end{bmatrix}$$
This will put a pole at $$1$$ on right half plane and the output $$y(k)$$ will increase over time if $$u(k) = c \forall k$$, where $$c$$ is a constant value.
Question:
I have a weak memory that this is the way to include integral action with state feedback for prediction. But I'm am very unsure. Note I'm not talking about Linear Quadratic Integral control.
Can this agumented linear model be used to predict its future values $$u(k)$$ and then the first value $$u(0)$$ can be used in the none-augmented state space model above for tracking $$y(k)$$ after reference $$r(k)$$ in a closed loop system?
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2019-10-15 02:08:13
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https://www.esaral.com/q/using-s-p-d-notations-describe-the-orbital-with-the-following-quantum-numbers/
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Using $s, p, d$ notations, describe the orbital with the following quantum numbers.
Question.
Using s, p, d notations, describe the orbital with the following quantum numbers.
(a) $n=1, I=0 ;$
(b) $n=3 ; I=1$
(c) $n=4 ; I=2 ;$
(d) $n=4 ; I=3$
Solution:
(a) $n=1, I=0$ (Given)
The orbital is $1 s$.
(b) For $n=3$ and $I=1$
The orbital is $3 p$.
(c) For $n=4$ and $I=2$
The orbital is $4 d$.
(d) For $n=4$ and $I=3$
The orbital is $4 f$.
Editor
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2022-12-08 23:20:13
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http://mathhelpforum.com/trigonometry/113744-solve-trigonometric-equation-print.html
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# solve a trigonometric equation
• Nov 10th 2009, 03:38 PM
satis
solve a trigonometric equation
could I get some assistance solving this equation? I'm afraid I'm just not getting anywhere with it
$secx + cosx = -2$
using identities, I can turn that into
$\frac{1}{cosx} + cosx = -2$
and from there
$\frac{1+cos^2x}{cosx} = -2$
unfortunately I peter out there, assuming I didn't make some mistake in that process. Any pointers would be appreciated.
• Nov 10th 2009, 04:16 PM
skeeter
Quote:
Originally Posted by satis
could I get some assistance solving this equation? I'm afraid I'm just not getting anywhere with it
$secx + cosx = -2$
using identities, I can turn that into
$\frac{1}{cosx} + cosx = -2$
and from there
$\frac{1+cos^2x}{cosx} = -2$
unfortunately I peter out there, assuming I didn't make some mistake in that process. Any pointers would be appreciated.
$\sec{x} + \cos{x} = -2$
multiply every term by $\cos{x}$ ...
$1 + \cos^2{x} = -2\cos{x}$
$\cos^2{x} + 2\cos{x} + 1 = 0$
$(\cos{x}+1)^2 = 0
$
$\cos{x} = -1$
$x = \pi$
• Nov 10th 2009, 06:09 PM
pencil09
Quote:
Originally Posted by skeeter
$\sec{x} + \cos{x} = -2$
multiply every term by $\cos{x}$ ...
$1 + \cos^2{x} = -2\cos{x}$
$\cos^2{x} + 2\cos{x} + 1 = 0$
$(\cos{x}+1)^2 = 0
$
$\cos{x} = -1$
$x = \pi$
i think is depend on the interval of x...
if $0
Quote:
and use this formula
$if \cos {x}= \cos {a}$
$x=a + k (360)$ or $x=-a+k(360)$
in this case:
$\cos{x} = -1$
so, $\cos{x} = cos{180}$
$x=180 + k (360)$
let $x=0$ so $x=180 + (0) (360)= 180$
let $x=1$ so $x=180 + (1) (360)= 540$ (rejected, bcause 540 not in the interval $0)
or
$x=-a+k(360)$
let $x=0$ so $x=-180 + (0) (360)= -180$ (rejected, bcause -180 not in the interval $0)
• Nov 10th 2009, 11:06 PM
mr fantastic
Quote:
Originally Posted by skeeter
$\sec{x} + \cos{x} = -2$
multiply every term by $\cos{x}$ ...
$1 + \cos^2{x} = -2\cos{x}$
$\cos^2{x} + 2\cos{x} + 1 = 0$
$(\cos{x}+1)^2 = 0
$
$\cos{x} = -1$
$x = \pi$
Since no domain is specified, I'd be inclined to give the solution as either $x = \pi + 2n \pi = (2n + 1) \pi$ (working in radians) or $x = 180 + 360 n = 180 (2n + 1)$ (working in degrees) where $n$ is an integer.
• Nov 11th 2009, 05:52 AM
satis
Thank you all for the insight. It definitely makes sense in retrospect, but hindsight is always 20/20.
Is there any trick to seeing the correct path to go to solve this kind of stuff, or is it just a matter of experience? I'm afraid I may have a few more of this type of question in the near future.
• Nov 12th 2009, 01:17 AM
pacman
yeah, experience plays a big role in it. Else, pure prowess . . . here is the graph of your equation.
http://www2.wolframalpha.com/Calcula...image/gif&s=10
(Bow)
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2016-09-25 18:33:16
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https://read.dukeupress.edu/hahr/article/71/4/697/146704/The-Origins-and-Progress-of-U-S-Mexican-Trade-1825?searchresult=1
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“Where is the republican that does not sigh for the emancipation of Mexico? Who is there in the United States, merchant or manufacturer, planter or artisan, that would not be benefitted by the liberation of this great empire from Spain . . . a source of trade to us more important than any we have with the old world.”
Niles Weekly Register, September 30, 1815
“Of the New America, Mexico is probably fated to be the most important state.”
National Gazette, April 9, 1825
The economic and commercial history of early national Mexico remains very much a mystery, the contributions of John Coatsworth, Donald Stevens, Barbara Tenenbaum, Guy Thomson, David Walker, and numerous Mexican scholars notwithstanding.1 In part, the problem is one of finding, collating, and interpreting statistics that are frequently difficult, conjectural, or contradictory. But it is also a question of knowing where to start; of deciding what is important; and of creating a framework within which statistics and other economic data assume coherence, consistency, and meaning, even in the presence of incomplete information.
A likely point of departure is foreign trade. It is clear from the work of Stevens, Tenenbaum, and Thomson that foreign trade is central to discussions of political change, capital movements, indebtedness, and commercial policy. Foreign trade was nearly at the heart of early national political economy. But the composition, evolution, and effects of foreign trade are by no means well known, much less undisputed. Tenenbaum links the fragility of the fiscal system, and indirectly of federalism itself, almost entirely to the cyclical variability of trade and tariffs. Stevens, on the other hand, discerns a politicized economic cycle and reverses cause and effect, concluding that “politicians [in Mexico] did not merely respond to economic cycles, but caused them.”2 Clearly then, examining foreign trade is a means of understanding issues that largely defined the existence, character, and viability of the Mexican state in the early national period. “Sin hacienda, no hay estado,” as a publication of the day remarked.3
Yet a study of Mexico’s foreign trade in the early nineteenth century is also necessarily an analysis of its commercial policy. And commercial policy was, and is, a weapon. It was, perhaps, uniquely effective against the pressures that both Great Britain and the United States exerted on Mexico, for the Mexican market was an object of competition between them, and competition brings leverage. Mexico employed the weapon, sometimes successfully and sometimes less so, but always in reaction to enormous pressures on its sovereignty. In Mexican eyes, the flag followed trade.
From a historical standpoint, then, studying early national trade patterns and commercial policy allows us to draw large lessons about the behavior of small polities and to discuss a neglected aspect of Mexican history as well. But where to begin? The Anglo-Mexican trade was of paramount importance, but before 1858 Great Britain ostensibly kept no systematic account of its bullion imports.4 Pending further research, Britain’s early balance of trade with Mexico is mostly a matter of conjecture. The Franco-Mexican trade was not unimportant, but before 1847 its statistics reflect “official values.”5 So we begin with the record of United States–Mexican trade.
While the record of the United States–Mexican trade is far from perfect, it is nevertheless voluminous and can be adjusted for the errors and omissions characteristic of all trade data. Moreover, the trade cycle with the United States is at once typical and peculiar, and the tension itself is revealing. Trade deficits, one may say, are trade deficits. They vary in size but differ in degree rather than in kind. In this sense, trade with the United States was no less representative than any other. Yet this trade also carried unique implications for both nations and was subject to social, political, and diplomatic pressures that the Anglo- and Franco-Mexican trades never faced. In other words, “ordinary” exchange between the United States and Mexico sometimes reflected unusual circumstances and at times produced untoward results.
Finally, the size of trade between Mexico and the United States may well have been “small” in a conventional sense, but its impact depends less on its absolute size than on the scale of measurement chosen.6 In the Mexican view, any trade with the United States could be potentially hazardous, whatever its volume. From the standpoint of the United States, the silver that Mexico used to finance its purchases had direct effects on the U.S. money supply into the 1830s, and indirectly thereafter. Mexican silver fueled the inflation of the middle 1830s in the United States and was a major cause of the Panic of 1837.7 Moreover, the trade to Mexico had important regional consequences within the United States. As late as 1830, Philadelphia merchants believed that they handled about 50 percent of the U.S. trade to Mexico, and Mexico was the city’s fifth largest trading partner overall.8 How much the trade counted is a matter of perspective.
## Exports, Imports, and the Balance of Trade
To tell a coherent story, we need to know the U.S. balance of visible trade with Mexico, that is, the relative size of merchandise exports and imports. But the numbers do not come easily. The existing trade statistics are usually misleading and often incorrect. And the United States controlled its own carrying trade to Mexico, which means that its earnings from these services need to be taken into account, even though they are generally ignored.9 The ostensible balance of Mexico’s visible trade with the United States is probably wrong. The current account balance, that is, net income from trade, investment, and services, is a mystery.
The reliability of trade statistics is an enduring problem for historians, analysts, and policy makers; no discussion based on quantitative evidence can ignore these issues.10 Since trade is fundamentally a quantitative matter, getting the numbers right is a necessary first step. Thus, the discussion of sources and methods that follows is a central aspect of this essay, but it is a difficult one as well. “Hoc opus, hic labor est,” as a U.S. consul of the time complained when asked to provide similar commercial data.11 For the sake of readability, I have relegated purely technical matters to four appendixes. Anyone wishing to replicate or appraise my results should find the data in the appendixes helpful.
The standard U.S. sources for imports from and exports to Mexico are Series U321 (exports) and U339 (imports) of the Historical Statistics of the United States.12 Series U321 gives the current value of exports, noting only that “reexports” (total imports less retained imports) are included. This is an enormous understatement. Before 1841, in no year did reexports to Mexico comprise less than half of all U.S. exports by value. Even after 1841, reexports were hardly insignificant. In both 1852 and 1860 and even as late as 1872—when they were swelled by English materials for Mexican railroad construction—reexports were again 40 percent of all exports by value.13 Total and domestic exports to Mexico were by no means identical, as shown in Table 1.
The importance of reexports from the United States was well known. As late as 1852, the U.S. consul in Veracruz could remark that “the cargoes of our New York packets [consist] almost wholly of bonded goods from Europe and China.”14 Mexican geography and the early concentration of foreign commercial houses in Veracruz enhanced the original significance of the reexports trade. For example, the British commercial houses of Marshall & Manning [later Manning & Mackintosh]; Bates Barton & Co.; and Exter, Greaves & Co. [McCalmont, Greaves & Co.] were established in Mexico City and Veracruz.15 But the U.S. consul at Veracruz noted that “English goods going into Mexico through Texas . . . will injure the English trade [through Veracruz] exceedingly for the United States . . . can get them into the interior much cheaper than they can be transported from [Veracruz] up these everlasting hills.”16 Overland reexports through the United States also reflected the continuing productivity of Mexico’s northern silver mines in the wake of insurgent damage to the Guanajuato district in the 1810s. Before 1846, northern silver also drove the Santa Fe trade and financed substantial U.S. reexports to Chihuahua and Durango. To the extent that overland reexports through the United States reduced transportation costs, or to the extent that maritime reexports bypassed Veracruz for Alvarado or Tampico, the trade that had customarily linked Veracruz to Mexico City was diminished. As the English diplomat H. G. Ward observed, the British were committing the very error that the Spaniards had made in concentrating their energies on Veracruz and Mexico City. The immediate beneficiaries of this strategy were the U.S. merchants who acted as commercial intermediaries. Geography was thus the unspoken ally of the Yankee trader.17
For simplicity, we define “U.S. exports” as domestic exports (i.e., produced in the United States). We assume that reexports from the United States remained foreign property, but that U.S. merchants and shippers profited from carrying them to Mexico. In other words, U.S. reexports were largely British and French exports transported by U.S. carriers. They produced “invisible earnings” for the U.S. current account18 but did not affect the balance of its visible (merchandise) trade with Mexico.
In Table 2, column A, I give domestic exports to Mexico from 1824/. 25 to 1883/84. Two figures are given for 1862/63 through 1864/65. The first figure is “official” and recorded in American Commerce. Commerce of South America, Central America, Mexico, and West Indies, With Share of the United States and Other Leading Nations Therein, 1821–1898 (Washington, 1899), a source for Series U321 (exports to Mexico) and U339 (imports from Mexico) in the Historical Statistics of the United States. The bracketed second figure is corrected for contraband war materiel that flowed from Union ports to the Confederate States through Matamoros, Mexico during the U.S. Civil War. (See Appendix A for details.) Exports were valued on a “free alongside” (FAS) basis (before adding insurance, freight, and merchants’ commissions). Nevertheless, all these numbers (and those that follow) are approximations and cannot be considered exact. The U.S. Constitution (section 9, clause 5) prohibits federal imposition of an export tax, so the government had no vested interest in consistently and correctly recording the value of exports. And well into the 1850s, the U.S. consul at Veracruz downplayed the precision of the statistics of the trade.19 But as we will see, this concern need not be exaggerated. Internal evidence suggests that the export totals are broadly correct. However we revise, transform, or manipulate the historical statistics, we uncover no substantial discrepancy between trends discussed in the consular reports and those indicated by the historical statistics. This strongly suggests that the U.S. consuls, whatever their personal financial interests, were reasonably unbiased reporters of economic information, or that their biases and interests did not significantly affect the reliability of the information they reported.20
In column A (Table 2) I give exports in current prices, and their movements include changes in both volume and value. But we must distinguish between changes in export prices and export quantities. The usual way of doing so is by “deflating” a series in current prices, thus converting “nominal” into “real” values. For this conversion, we require an index of prices of U.S. exports to Mexico, or a good approximation of such an index. The Warren-Pearson index of U.S. wholesale prices spans the nineteenth century, but there is no guarantee that it accurately reflects the composition of exports to Mexico. The safest course is to construct our own export price index.
As I show in Table 3, the value share of the top five domestic goods exported to Mexico from the 1820s through the 1880s was always high— with the exception of the 1870s, it was never less than 50 percent—and in the 1830s and 1840s it was well over 60 percent. These goods included finished cottons, wheat flour, raw cotton, and after 1868/69 steel rails as a proxy for manufactures of iron and steel. Concentration makes the construction of the index much easier. I provide details of how the index was constructed in Appendix B. Here we need only say that the deflator is a Laspeyres index with a base period of 1840/41–1844/45. Its weights were adjusted (and linked) in 1840/41, 1844/45, 1859/60, and 1868/69 to reflect the changing composition of exports. The index covers roughly 40 to 70 percent of all U.S. exports to Mexico by value, which is adequate. In Table 2, the index appears in column B. Real exports in constant 1840/41–1844/45 prices are given in column C and are nothing more than (A/ B × 100).
U.S. imports from Mexico are easier to measure. By and large, Mexico shipped precious metals—mostly silver—to the United States. Most came in specie, but there was some bullion as well. As I show in Table 4, specie and bullion (including some gold) were never less than half of all imports by value. Until its world price fell in the later 1870s and early 1880s, silver was often 60 to 70 percent of the value of all U.S. imports from Mexico. Without much error, then, we can deflate these imports (silver plus all others) by an index of silver prices constructed on an 1840/41–1844/45 base.21 Imports in current prices appear in column D, Table 2, and the silver price index in column E. Real imports in prices of 1840/41–1844/45 appear in column F and are nothing more than (D/E × 100).
Mexico’s export figures for precious metals are highly suspect, since substantial Mexican taxes on the coinage and export of specie during much of the nineteenth century made smuggling silver out of the country big business.22 But the U.S. import totals are probably accurate, even though silver and gold were admitted duty free; the absence of taxation made concealing them or evading customs pointless. While U.S. import accounts had to be reformed in the 1840s because of inaccuracies, their precious metal totals were considered basically sound.23
With new series for United States imports from and exports to Mexico, we can now compute a more accurate balance of visible trade (merchandise exports less imports). In Table 2, column G, the balance appears in current dollars. It is simply net income from merchandise trade.
Not all international transactions are “visible.” Conventional balance-of-payments accounting distinguishes “visible trade” from “invisibles” such as trade in business services.24 In general, the “balance on visible trade” differs from the “balance on current account” by the movement of “invisibles”. Practically speaking, the only computation of invisibles possible here is income from insurance, shipping, and commissions. I describe the procedure for calculating these in Appendix C and credit to the United States the amounts computed as “invisible exports” to Mexico. The annual earnings from invisibles appear in Table 2, column H, and the balance on current account in column I, the sum of G plus H. The current account is simply net income from trade, services, and investment.
## From Numbers to Notions: Silver, Cycles, and Tariffs
Numbers, of course, are not the whole story, but without them there would be no story. And the story they tell is one of both persistence and change. Mexico continued to trade silver for cloth, much as the Indies had under the Bourbon monarchs but with new wrinkles. As wheat fields in the United States began to appear west of the Alleghenies, New Orleans could supply Mexico with wheat flour shipped cheaply down the Mississippi. And then there was raw cotton. In the early 1840s, Mexico’s new cotton industry drew heavily on raw cotton imported from the United States. As Lucas Alamán observed in 1846, “Without [it], the factories could hardly have made half of what they did during the past two years.” The available evidence suggests that Alamán’s calculations were very nearly correct.25
Yet there is a sense in which what did not change was as impressive as what did. The Mexican cotton industry in the 1830s and 1840s was based less on comparative advantage than on restrictions on international trade, a nostrum peddled by Mexican industrialists who found domestic markets difficult to control. From the standpoint of international trade cycles, moreover, the Mexican staple remained silver, and the impact of its production and export during the early and middle decades of the nineteenth century was substantial. There are two ways of seeing this.
First, and most obviously, silver was a medium of international exchange and figured prominently in Mexico’s capacity to import. The value of Mexican imports from the United States conformed closely to the value of the silver exported to the United States. Indeed, the correspondence is nearly exact.26
Second, Mexico was a nation with small, fragmented capital markets that rationed credit through kinship networks rather than through banks or other formal intermediaries. Under these circumstances, development economists suggest, activities that raise investible wealth will have an economic impact disproportionate to their size.27 In Mexico, silver mining was one such activity. Compared to agriculture, mining’s share in national income was not large, but it proved a concentrated and unrivaled source of liquid, mobile savings to investors. As a result, and in the long run, variations in silver production were tied to variations in the supply of loanable funds. Thus Guy Thomson’s calculations of the volume of loans in Puebla between 1800 and 1830 show a decline whose severity could only be explained by an equally sharp fall in the production of silver.28
As loans rise and fall, the volume of “real” economic activity financed by credit must also grow and contract, and so too will imports and exports. Thus, in an open economy, the balance of trade must reflect changes in the business cycle, and in Mexico the business cycle was necessarily tied to the credit that silver mining supplied. In early national Mexico, trade cycles, business cycles, and the production of silver had to be related.29
The evidence from Mexico’s trade with the United States is consistent with these hypotheses. In logarithms, real U.S. imports from and exports to Mexico (see Figure 1) conform closely to each other, indicating comparable rates of change. Indeed, on the face of things, U.S. imports of silver, hides, dyewood, and indigo from Mexico move somewhat in advance of U.S. exports to Mexico, especially before the late 1860s. This pattern may suggest an “export-led” model of Mexican growth that persisted until the onset of the depreciation of silver after 1867 and until the beginnings of industrialization under Porfirio Díaz somewhat later.30
Figure 1 reflects roughly three similar cycles in imports and exports. Measured from trough to trough, the first spans 1825/26–1847/48, with a peak in 1835/36. The second covers 1847/48–1867/68, with a peak around 1860/61. The last begins in 1867/68, peaks around 1872/73, and closes in 1883/84. The dangers of teleology notwithstanding, the chronology conforms to not a few significant points of Mexican history between Iturbide and Díaz: the Texas rebellion; the War of 1847; the French invasion (and U.S. Civil War); the Restored Republic; the death of Juárez and the accession of Lerdo de Tejada; and the completion of rail links with the United States. Of course, any year might well be invested with significance in those turbulent times. Yet more than coincidence is at work, for Mexican trade cycles were thoroughly politicized as well. Over the long run, cyclical changes in the production and export of silver clearly mattered. But in the shorter run, the impress of political factors is obvious.
Politics, Edward Nell observes, is economics pursued by other means.31 Since political factors were deeply embedded in the short-run movements of Mexican trade and commerce, who could disagree? But in this context “politics” meant two things: tariff policy, and the uncertainty that coups, blockades, pronunciamientos, wars, and repeated changes of regime engendered.
The level of tariff protection was clearly a political outcome. The private demand for industrial protection, the public demand for revenue, and the diplomatic aspects of trade and territorial expansion all played a role in setting the tariff. Moreover, the enforcement of tariffs was a free-for-all. If several tariffs were in force, which was to be honored? Did national, state, or local tariffs take precedence? Who, if anyone, knew the code or could even find a copy of it on demand? It was a nightmare for merchants and consuls, but chaos has its reasons, and uncertainty could be a potent ally for unstable regimes.
Economic uncertainty is an amorphous idea, akin, perhaps, to risk of indeterminate probability. But risk and uncertainty were key elements in the economic calculus of early national Mexico. Barracks revolts, pronouncing generals, extreme factionalism, blockades, and the risk of war paralyzed trade and commerce as effectively as did garbled regulations and unenforceable property rights. Merchandise no longer moved, debts went uncollected, and silver shipments slowed.
## The First Cycle (1825/26-1847/48)
Texas dominated the political economy of the first cycle. Its trauma diverted Mexican trade toward Great Britain, whose role as a potential counterweight to U.S. territorial designs on Mexico ended only with annexation and the Mexican War. The theme appears in modern Mexican scholarship and pervades the writings of men as different as Alamán and Carlos María de Bustamante. The message is clear: the flag follows trade. Trading with the United States brings their merchants, and their merchants bring trouble. “They are the true sons of Englishmen,” wrote Bustamante, “whose example in India they remember and emulate. Merchants financed the invading expedition. Once their company had gotten hold of the land, they turned it over to the crown, which installed a government and set millions of Indians groaning under a slavery enforced by bayonets.”32 To judge the dramatic effect that the Texas rebellion had on the pattern of trade between Mexico, the United States, and Great Britain, see Appendix D.
Alamán’s point was much the same. “Instead of armies, battles, and invasions, which raise such uproar and generally prove abortive, [the United States] use means which, considered separately, seem slow, ineffectual, and sometimes palpably absurd, but which united, and in the course of time, are certain and irresistible.” And what were these means? The list was a long one, but the first Alamán mentioned was “commercial negotiations.”33
To be sure, Mexico had never trusted U.S. commercial ambitions. In 1829, Secretary of State Martin van Buren termed Mexico’s dilatory consideration of a treaty of reciprocity a “mistaken policy . . . unfriendly to the commercial prospect of the United States.” Mexico regarded the United States “with a degree of indifference and suspicion as extraordinary as it was to be regretted.” Negotiation over the treaty, which had begun in 1826, dragged on until 1832. And even then, “the first attempts of our adventurous citizens [were] burdened by the imposition of prohibitive duties . . . [in] Mexican ports.”34
The tariff, then, was Mexico’s weapon of choice. Before the War of 1847, Mexico repeatedly adjusted its coverage and level, most notably in 1829, 1837, and 1842-43. In theory, the tariffs covered a variety of articles, but in practice their target was finished cottons, the industry that Mexican industrialists sought most strongly to protect. Things started badly in 1825, with allegations of discriminatory duties on U.S. cottons, and by 1827 rising duties had driven U.S. exports sharply down.35 Matters worsened in 1829, and U.S. merchants in Veracruz warned that the new duty’s “pernicious influence . . . has annihilated the reviving spirit of commercial enterprise.” The U.S. consul concurred. When pressed, Lucas Alamán, then secretary of home and foreign relations, promised relief. But his suggesting that repeal of the duty on coarse cottons would pass the Mexican Senate “without the slightest opposition” was simply bad faith on Alamán’s part.36
Much worse was in store. In the continuing wake of the Texas crisis, the tariff was again revised. Although some duties were reduced in 1837, the new schedule prohibited (effective March 1838) ordinary cottons and woolens, cotton yarn and thread, and ready-to-wear clothing. In late 1842, duties on goods otherwise permitted rose 30 to 50 percent. The Tariff of 1843 reiterated the prohibitions of 1837 but added raw cotton and coarse woolens to a list that included at least sixty articles “embracing most of the necessaries of life and far the greater portion of [U.S.] products and fabrics.”37 As the U.S. consul in Veracruz remarked, “no cotton goods can be imported less than 25 and 30 threads [per quarter square inch] which comprises that very kind of good suited and worn by the poor and middling classes of the community.” Here was a recipe for reviving the moribund obrajes of the colonial regime.38
The prohibitions were murderously effective. Before 1838, finished cottons were 30 to 40 percent of domestic U.S. exports. Once the Tariffs of 1837 and 1842 had taken hold, the share of cottons fell to only 16 percent (see Table 5). A small market for cotton twist, yarn, and thread was annihilated as well. “[Mexico’s] commerce would be infinitely important to us,” said the U.S. minister in 1842, “but for this unfortunate Texan war, which has caused much injury to the United States.”39
In late 1845, a bill pending in the Chamber of Deputies would have admitted cotton and cotton manufactures on better terms. Seven percent of customs duties would indemnify cotton growers and manufacturers for their losses to foreign competition.40 The bill failed; the tariff scheduled to go into effect in February 1846 was as restrictive as its predecessors. But the war intervened, and Mexican ports were placed under blockade.
The U.S. consul in Veracruz was no doubt correct when he observed in late 1845 that “Mexico never since she has been a Nation has been in so wretched a State.”41 But wretched is not powerless. Although an amalgam of economic nationalism and opportunism, Mexican policy nevertheless rested on the law of demand. This was the nation’s best weapon.
In the long run, Mexico’s strategy did not and probably could not have prevented the loss of Texas, New Mexico, and California to the United States. But in the short run it was hardly irrational. High tariffs satisfied the demand for protection that manufacturers pressed so insistently on the Mexican Congress and mollified other vital (and volatile) constituencies as well.42 And, indeed, trade with the United States remained small, much to the chagrin of those who had expected great things of the Mexican market. In the early days of the First Republic, newspapers in the United States hailed Guadalupe Victoria as another George Washington.43 By 1845, the comparisons drawn were altogether less flattering.
A final observation. In the very short run, large fluctuations in trade occurred from year to year. Some were simply random, and not all are easily or equally explicable. Yet contemporaries understood the link between political stability and sustained growth. As one anonymous essayist put it: “The mere rumor of a revolution is pernicious.. . . Agriculture falls off, commerce is all but paralyzed, and silver shipments cease because the roads are probably not safe. In short, the citizens are in arms, and all is in disorder. These are the necessary and immediate consequences of the very rumor, more or less substantiated, of the next revolution.”44 Uncertainty dominated yearly, and daily, affairs.
## The Second Cycle (1847/48-1867/68)
Historians of the United States once called their Civil War the “irrepressible conflict.” No one familiar with relations between the United States and Mexico in the 1840s could conclude that the Mexican War was any less “irrepressible.” U.S. ambitions and Mexican nationalism were mutually exclusive. Indeed, the drive to commercial and territorial expansion characteristic of U.S. foreign policy in the 1840s has been termed “manifest design” by a historian who argues that this expansion was neither accidental nor providential.45 In the long run, Mexico’s defeat (and the annexation of Texas) implied a permanent increase in the U.S. market and a sweeping reorientation of Mexico’s trade. I highlight this increase in Figure 2 by shifting the X-axis (years beginning in 1825/26) upward to intersect the log of real exports in 1847/48. Exports naturally rose and fell thereafter, but they rarely returned to antebellum levels. Commercial expansion may or may not have “caused” the Mexican War, but commercial expansion was one result. A manifest design had manifest results.
Still, none of this happened overnight. Mexico’s defeat by no means meant that the United States could appropriate a larger share of the Mexican market at will. In the short run, the spike in finished cottons sent to Mexico in 1847/48 did not and could not last (see Figure 3 and Table 5). It reflected the administration of a war tariff by U.S. troops in the occupied ports of Tampico and Veracruz.46
Mexico still had commercial weapons, and the demand for protection remained strong. Thus, by the early 1850s, the old complaint was again heard. The United States could expect little from Mexico “whilst the system of prohibitions is observed.”47 Meanwhile, the United States brought new territorial pressures to bear during negotiations over the Treaty of the Mesilla in 1853. James Gadsden, the U.S. minister, wanted Sonora and Chihuahua as well but did not get them.48 Nor did he get commercial concessions, whatever his original interest in them may have been.
Indeed, after the Treaty of the Mesilla, the Mexican government once again restricted U.S. exports, much as it had done after the Texas rebellion in 1835, and a booming postwar market for U.S. finished cottons all but collapsed (see Table 5). By late 1854, the consul in Veracruz could write, “Santa Anna’s policy destroys commerce, particularly that of the United States.” In a later dispatch he quotes Gadsden, who minced no words. “I had contemplated . . . [directing] the Secretary of State’s attention to the entire Santa Anna Commercial Code—so embarrassing, destructing [sic] and offensive to all trade and intercourse with Mexico—in the hope of convincing him of the necessity of getting rid of the Brigand [Santa Anna].” “Nothing,” concluded U.S. Consul Pickett, “can be more corrupt, false, unequal, and generally pernicious than the entire Mexican commercial system.”49
The sources of this vitriol were two: “the” Tariff of 1853 and the Commercial Code of 1854. In practice, four “national” tariffs were in force in 1853, plus state and regional levies in Guadalajara and Monterrey. “How many more there may be in different sections of the country I shall not attempt to record,” wrote the consul in Veracruz. Nor was the Commercial Code much better (or worse). “[O]ne might as well attempt a digest of the laws of the Meades [sic] and Persians or an abridgment of the Chinese Encyclopaedia as a codification of all the imperious arbitrary dicta of the absconded Mexican solon [Santa Anna].”50
To the United States, all was chaos. Which tariff applied or to whom duties should be paid was not clear. “Merchants are even now continually imposed on and openly robbed under one or the other of them.” This view was not unique to foreign observers. Mexican historians, too, shudder at the “fiscal disorder” that the declining Santanista dictatorship encouraged.51
But why should Mexicans educated in the legacies of Guadalupe Hidalgo and the Mesilla assume that open trade with the United States was beneficial? Had it ever been? Its desirability was an axiom only in the minds of U.S. officials, who since the days of Poinsett had repeatedly complained that Mexico impeded trade. James Gadsden was no worse when he concluded, “Let us labor to kill [these barriers to trade] outright and to secure guarantees against their resurrection.”52 Mexico’s desire for autonomy (or, indeed, its definition of sovereign interests) figures nowhere in his thoughts.
Santa Anna, on the other hand, had long played cat and mouse with the United States. He understood its interest in enlarging trade and held it at bay as he picked U.S. pockets. “[The] Tariff is not a rigid law in the Republic.. . . His Most Serene Highness [Santa Anna] violates it constantly by selling exclusive privileges,” observed the U.S. consul in Veracruz.53 By misdirecting, stalling, and confounding, Santa Anna’s “chaos” forced merchants to disclose how much they were willing to pay to do business. This may have been dishonest and even inefficient in a broader economic sense, but it was an effective means of extracting rents from U.S. merchants and of restraining their enthusiasm for Mexico. True enough, Santa Anna plundered the state. But his odd ethic was shared by principled idealists whose instincts for personal, political, and national survival were equally indistinguishable. Moreover, Santa Anna displayed a studied ambivalence toward foreign trade. Waddy Thompson, the U.S. minister to Mexico from early 1842 through early 1844, portrayed him as leaning toward autarchy: “[Mexico] had no need of foreign commerce. . . . [It produced] all the necessaries of life.”54
The Ayutla movement of 1855 represented, in this context, a shift of substantial proportions. The Tariff of 1856, which permitted the volume of exports to Mexico to grow, was its proximate result. Finished cottons, always a sensitive indicator of the strength of protection, gained ground, and in time their share in exports recovered from the sickening collapse of 1853/54 (see Table 5). Broadcloth, timbers, ready-made clothing, and raw cotton all disappeared from the index of prohibitions. Nominal duties on finished cottons fell by an astounding 70 percent and were lower in 1856 than at any time since 1845. By one estimate, the implicit index of protection on goods from the United States was about 30 percent, an extraordinarily low figure by historical standards.55
As a harbinger of Liberal (and liberal) capitalism, the Ayutla movement embraced the national ambitions and possibilities of a bourgeoisie long frustrated by civil unrest. Their notions of growth stressed the expansion of demand rather than the control of supply, a modernizing attitude altogether different from the vague neomercantilism of the later Bourbons and their successors. Witness the words of Guillermo Prieto, who assumed the treasury portfolio in 1855: “The faith I have in free trade is the faith I have in all sublime manifestations of liberty.”56
Different, too, was their notion of the political economy of trade. The upswing in exports from the United States that would characterize the third cycle (1867/68-1883/84) marks the end of repeated cycles of annexation and commercial resistance. Better to yield markets than territory, dollars than dominion. Matías Romero put it succinctly: “The best means of impeding annexation is to open the country to the United States . . . with the objective of making annexation unnecessary and even undesirable.” Figures 1 through 3 offer proof that Romero’s observation was historically accurate, even if the United States had no further interest in annexation after the Treaty of the Mesilla.57
The “Decade of Civil Wars” (1857-67) was an anomalous one. Contraband distorted normal patterns of trade between Mexico and the United States, and the analysis of corrected statistics is merely somewhat less misleading. Clearly, these years form a bridge (or gap) between the second and third cycles. I shall resume sustained analysis with the Restored Republic (1867-76).
There is, however, a final point. Repeated civil disturbances were disruptive and costly. Foreign conflict may bring prosperity, but enduring domestic crises do not. During the final days of Santa Anna, the U.S. consul in Veracruz noted that “the Pronunciados (Revolutionists) [have] cut off all communication with [Mexico City and] . . . the telegraph has long since been destroyed. . . . The Rebels are determined to seize the public moneys.” So, too, in 1858 with the outbreak of the War of the Reform: “commerce and business [were] completely prostrate, and silver could not be shipped out through Veracruz.”58
The costs of remaining on a nearly permanent war footing were severe. Peasants pressed into armed service could not plant or harvest, a major source of disruption to an agrarian economy. Moving armies around the countryside required huge numbers of horses, mules, and oxen to drag artillery and to carry supplies.59 Obtaining them from farms, silver mines, and transportation was very costly, and the sample of U.S. import transactions through Laredo, Texas, in Table 6 documents this only too well. Military demands for draft animals during the War of the Reform (1858-60) exhausted the supply of live animals and drove their ordinarily large share of the border trade to zero. This was one of the ways in which persistent instability reduced productivity in nineteenth-century Mexico, and its effects are especially clear from examining patterns of trade with the United States.
## The Third Cycle (1867/68-1883/84)
During the third cycle, Mexico’s international position changed significantly. By 1883/84, Mexico had become an important Latin American market for U.S. exports. Real imports (merchandise plus silver) from Mexico grew as well. In 1867/68, they stood at 6.1 million dollars (prices of 1840/41-1844/45). By 1876/77, imports had more than doubled. Indeed, one account suggests that the United States had surpassed Great Britain as Mexico’s principal trade partner by the late 1870s.60
The composition of trade changed as well. Mexico began to import capital goods from the United States, and the share of steam engines, sewing machines, machinery, and builders’ hardware grew. Moreover, the share of specie and bullion in imports from Mexico fell to just over 50 percent, while the volume of jute, sisal, and hemp quadrupled between 1871 and 1880. In other words, investment grew, while exports swelled and diversified.61 Arnaldo Córdova argues that modern capitalist development first appeared during the Restored Republic. Evidence from the trade cycle is consistent with Códova’s argument. It also suggests that Mexico capitalized on the favorable international economic conditions after 1856 as soon as it had attained a reasonable level of governmental stability.62
Yet the beginnings of this expansion were obscure. The Liberals entered Mexico City in the summer of 1867, but for another two years merchants complained that business languished.63 Or was the slump limited to Veracruz? The Free Zone (Zona Libre) along the northern border was a smuggler’s delight, and Matamoros felt no discomfort.64 Large shipments of silver to China via San Francisco linked the mines of Zacatecas with the port of Mazatlán.65 The old axis of colonial trade and commerce— Mexico City to Veracruz—was neither dying nor dead but saw intimations of mortality. One port’s prostration was another’s prosperity.
Nevertheless, by 1871 some perceived a slow rise in agricultural production, foreign investment, and exports brought by the prospects of peace. In 1872, a new tariff reduced the list of prohibitions and consolidated a number of miscellaneous duties. The proximate result of the revision heartened U.S. cotton manufacturers. Between 1872/73 and 1879/ 80, when the tariff was again revised, the share of finished cottons in U.S. exports rose to more than 13 percent, the highest it had been in twenty-five years (see Table 5), an increase that must have seemed encouraging.
But like all good campaigners, U.S. officials wanted complete victory (or unconditional surrender). For a variety of reasons, they did not get it, at least before 1884. One explanation was the changing terms of trade.
From 1866/70 to 1871/75, the terms of trade with the United States improved by nearly 30 percent as the inflation of the U.S. Civil War subsided (see Table 7). Real silver output grew slowly, but its purchasing power surged. By contrast, from 1871/75 to 1876/80 silver production grew more rapidly, but the terms of trade actually fell. The growth in Mexico’s capacity to import diminished, and the market tightened, as the inflection in the U.S. real export curve in Figure 1 shows. U.S. consular officials blamed shoddy merchandise, inappropriate products, and incorrect packing—all the usual suspects. Nor could the United States finance its exports as the Europeans did, for the United States was still a net importer of capital. But the depreciation of the peso undermined the case for free trade.66
Nor did it end there. By 1880, a new tariff and a new regime committed to industrialization were in place. And in view of the rapidly falling freight rates that the railroads brought, the demand for protection necessarily rose, if only to compensate for the fall in the “tariff” of transportation costs. So the Tariff of 1880 became the first in a series of upward revisions that reputedly made Mexico’s nominal rates, particularly on finished cottons, the highest in the world. Since raw cotton was now admitted duty-free, the effective rate of protection, the margin between the cost of imported cotton and the sale price of the finished cloth, must have been substantial.67 The industrialists and financiers who had assumed control of the Mexican state knew precisely what they were doing.
These developments—the depreciation of the peso, Mexican industrialization, and renewed demands for protection—signalled a retreat from the Liberal position of the late 1850s and called for new thinking in the United States. It was not long in coming and found expression in the commercial treaty of 1883. As Abram S. Hewitt, a congressman from New York, put it, the treaty, which was ratified but never fully implemented, proceeds from a totally different idea [and] . . . regards Mexico and the United States as integral parts of one commercial system. It is an attempt to establish between the two countries the same condition of affairs that exists between the several States of the Union.”68 Indeed, the treaty, which created lists of duty-free goods, was part of a larger congressional movement to establish what a supporter called an “American Zollverein” or customs union embracing all the Americas. But with Mexico the advantages of propinquity were greater, and as Hewitt said, “So long as we get an entering-wedge we ought to be satisfied.” What Poinsett had so long ago essayed, Hewitt and his allies now in part accomplished.69
The “entering wedge” came in 1884 with the completion of the rail link between Mexico and the United States. The railroad brought the third cycle to a close and permanently altered economic relations between the two nations. Like Santa Anna, whose death in 1876 coincided with Díaz’s accession to power, Porfirio Díaz would play various foreigners against each other. And like Santa Anna, he would have his victories. But unlike Santa Anna, Díaz no longer had a northern “desert” to mediate between weakness and strength. In fact, during Grover Cleveland’s second term alone (1893-97), the United States sent as much to Mexico as it had during the combined presidencies of Andrew Jackson through Abraham Lincoln. The world had indeed changed.
## Reflections on the Balance of Payments
The U.S. balance of payments with Mexico, represented by columns G through I in Table 2, is well worth considering.70 The United States consistently ran a visible deficit (column G) that its estimated earnings from shipping services (column H) did not offset. Since the merchandise balance consistently favored the United States, its visible trade deficit was the result of large deficits on the specie balance. By and large, Mexico sent more silver to the United States than required to pay for the goods and services it imported.
A developing country typically runs a deficit on its current account and a surplus on its long-term capital account. Such a country typically consumes more than it produces, especially during economic expansions. To do so, it borrows abroad to finance additional imports. Nevertheless, what is true of the balance of payments in general is not necessarily true of specific cases. Surpluses from one partner may finance deficits with another, for a country’s trade need not be balanced with each of its partners, just balanced overall. Perhaps Mexico achieved its balances partly through trade with the United States. But the evidence is still incomplete.
Reexports yield a more promising clue. If we add them to domestic exports, the cumulative current account balance of the United States with Mexico before the U.S. Civil War (thirty-five years) was approximately 3.8 million dollars or nearly balanced on an annual basis (these calculations are omitted from Table 2). In other words, Mexico did some of its business with Britain, France, and the Hanseatic cities of Hamburg and Bremen through New York. Within reasonable limits for shipping and brokerage, the figures come out about right. Of course, this interpretation is consistent with the commercial rather than industrial character of antebellum U.S. capitalism.
After the U.S. Civil War, Mexican current account surpluses increased, even when reexports are considered. No one factor accounts for them, but several plausible explanations exist.
The first is the export of silver coin to California for reexport to China. Between 1866/67 and 1871/72, the United States reexported 10.6 million dollars of silver specie to China. Over the same period, unexplained exports of Mexican silver to the United States (that is, silver beyond that needed to pay for U.S. exports, reexports, and shipping services) totalled about 14 million. In a crude sense, U.S. reexports of silver to China “explain” about 75 percent of Mexico’s current account surplus, although they abruptly ceased in 1871/72 (there were sizable reexports to Hong Kong in 1873/74).71
The timing of the surge is no coincidence. In the spring of 1856, the Chinese authorities in the treaty port of Shanghai made the Mexican “dollar legal tender, and the ports of Canton and Foochow soon followed. By early 1858, the demand for Mexican coin in Shanghai was substantial. Once the civil wars in Mexico and the United States were over, there was no obstacle to the remission of Mexican silver to China through California.72
Contraband is another part of the puzzle. By 1878, one U.S. observer wrote that “smuggling [into Mexico] had so largely increased that honest commerce was ruined.”73 Smuggled goods exported by water posed no problem. The items were recorded as conventional exports from the United States’s standpoint. Overland exports to Mexico or Canada, legal or no, were another matter. They went unrecorded until 1893. If significant overland smuggling existed, the exports would go unrecorded on the U.S. side of the ledger.
Yet the specie that paid for smuggled goods necessarily figured as a U.S. import. If overland smuggling were substantial, Mexico would send far more silver than required to pay for recorded exports from the United States. And that is precisely what happened. Overland smuggling from Texas into the northern Free Zone was rampant in the middle and later 1870s. Indeed, it was in the 1870s that smuggling first grew large enough to create noticeable discrepancies in the balance of payments. It thus became a major irritant to bilateral relations, and the existence of the Free Zone itself hung in the balance until 1878.74 Mexico’s surpluses of the time—and they were substantial—thus record the unrecorded, if not the unrecordable.
Finally, Porfirio Díaz’s seizure of power was a very near thing, and hardly uncontested. The country was in turmoil, and a sudden fall in the price of silver in late 1878 compounded the atmosphere of “great national calamity.” With reserves apparently short, Treasury Minister Matías Romero was forced to cast about for new sources of revenue, and the U. S. legation reported that he “alarmed all the property and business interests of the country lest [these measures] fall too heavily upon their branches.” Perhaps the nation’s alarmed financial interests took steps to secure their assets abroad. If so, part of the unexplained surge of Mexican silver exports in the late 1870s was capital flight—some silver recorded on current account belongs to the capital account instead.75
## Conclusions and Implications
Paul Gootenberg and Frank Salford have observed that neither Peru nor Colombia adopted an unrestricted free trade regime much before the 1850s.76 To that list we may now add Mexico, at least by evidence of its trade with the United States. Indeed, it was not until midcentury that Mexico adopted an even vaguely liberal commercial policy. But viewed from the perspective of Porfirian Mexico, that opening too was an interlude. With periodic exceptions, Mexico has historically been a high-tariff country.77
I have elsewhere suggested a number of factors that made protection an attractive stance in the years before 1840. My purpose here has been to suggest political sources of greater subtlety, generality, and chronological scope. From the Mexican perspective, relations with the United States in the years before 1853 frequently turned on the balance between annexationist pressures and commercial exclusion. In the early national period, the Mexican response to annexationist pressures and losses entailed a considered and defensive effort to restrict trade with the United States. The rise of liberalism in its pure (Juárez) or even manqué (Díaz) forms thus coincided with a commercial settlement of sorts with the United States. And while Santa Anna is no star in the official firmament of Mexican Liberalism, the inchoate form of later tactics to which Sebastián Lerdo de Tejada or Díaz would subscribe is plainly evident in his actions. Díaz, we are told, largely reaped what Juárez sowed. But both learned something from “Su Alteza Serenísima” as well. Even in the “chaotic” years of the early and middle nineteenth century, there were continuities of political style and culture that provide an intelligible intellectual framework.
But Mexican commercial policy ultimately affected economic variables, and it must be judged mainly in those terms. In this context, its efficacy was unquestionable. Despite all that has been said about Mexican smuggling, evasion, and corruption—a litany chanted in the United States with only brief respite since the 1840s—none of these repealed the law of demand.78 When Mexican tariffs drove up the relative price of U. S. finished cottons, their consumption within Mexico fell. When prohibitions were laid on U.S. goods, their quantities exported dropped. The evidence is too persuasive to argue otherwise. Even under the most disorganized circumstances, policies that relied on supply and demand for their implementation worked. The spectrum of interpretive possibilities thus offers two alternatives. Either the law of demand is surprisingly robust, or the disorder that reigned in Mexico from the 1830s through the 1870s was institutional rather than administrative. As in most matters, the truth lies between these extremes.
Finally, we must realize that early Mexican regimes were not free to choose. Policies that were politically expedient could also be economically disastrous. No sensible economist could find much to recommend in the whirligig of early national tariff policy. But even a mildly nationalistic (or realistic) one might understand why there was little choice. Tariffs and prohibitions were consistently used to enhance the economic and political stability of the Mexican state, even though these same measures implied a corresponding reduction in national income. Indeed, before 1867, Mexico’s choice may have been between existing poorly and not existing at all. If a condition for long-term growth is the evolution of unfettered, politically autonomous trade, Mexico was, by force of circumstances, unable to meet it.79 Mexico’s loss of territory was a burden of U.S. imperialism, but its defensive commercial policy necessarily reduced efficiency and retarded growth as well. Over the course of a century, even small annual losses compound to large disparities in international income levels. Sometimes, little things mean a lot.
Woodrow Borah, John Coatsworth, Albert Fishlow, Pedro Fraile, Stephen Haber, John Huston, Linda Salvucci, Donald Stevens, Barbara Tenenbaum, and two anonymous referees offered suggestions and criticism of this paper. Seminar participants at the University of California, Berkeley, and at the University of Texas, Austin, were helpful as well. An earlier version was presented at the annual meeting of the American Historical Association. The National Endowment for the Humanities and the Social Science Research Council provided financial support.
### Correcting Series U321 and U339 for Errors and Omissions
Until 1893, goods carried overland for export were not included in U.S. export totals. This omission affected all overland trade to Mexico. For example, the value of the Santa Fe trade between Missouri and the Provincias Internas is not included in Series U321. According to data supplied by Josiah Gregg in Commerce of the Prairies (1849), the Santa Fe trade averaged nearly $134,000 per year from 1822 through 1844, but yearly swings of up to$100,000 were not unknown.1 In general, U.S. merchants exchanged dry goods for silver specie and bullion—as much as $180,000 in 1824. This was no small sum. It amounted to more than 7 percent of Mexico’s net exports of silver to the United States in 1824/25.2 Nevertheless, the significance of the trade to the United States is not completely clear. In the 1820s, petitioners from Missouri claimed that the profits of the trade were “an amount considerable in the commerce of an infant state.” Moreover, “the principal article carried to the Internal Provinces is cotton goods, the growth and manufacture of the United States.”3 Yet U.S. cottons were no more than about a third of domestic exports to Mexico in 1825/26, and domestic exports were then only 16 percent of all exports to Mexico. U.S. cottons could not have been more than 5 percent of all U.S. exports (that is, domestic exports plus reexports) to Mexico in the mid-1820s. Unless the Santa Fe trade was unusual, U.S. cottons had little place in it. Moreover, the marked controversy over granting a “drawback” or rebate on tariffs levied on foreign goods for reexport to Santa Fe suggests that cottons and calicoes from England and France, linens from Germany, and handkerchiefs and stockings from India were staples of the trade. Consequently, we assume that Josiah Gregg’s estimates of the size of the trade should be added to reexports rather than to domestic exports. The years from 1862 through 1865 also present problems, for both Mexico and the United States were embroiled in civil wars. Conventional political boundaries were blurred, contraband flourished, and there was an unprecedented increase in the value of trade. Why? From 1851 through 1860, United States imports from Mexico averaged roughly 1 million dollars a year in current prices. By 1865, the figure had swollen to$6 million. Raw cotton alone accounted for $5 million. But Mexico had not suddenly become a major cotton producer. It had become an entrepot (at Matamoros, Tamps., across the Rio Grande from Brownsville, Texas) for raw cotton that could not be shipped from Confederate ports because of the Union blockade. The cotton was then lightered downriver and transferred to ocean-going vessels for shipment to the United Kingdom and to the Union states. To correct the U.S. import figures, we simply deduct the value of cotton imported from Mexico from the import totals and supply the correction in brackets to the right of the “official” figure in Table 2.4 U.S. exports to Mexico, Series U339, also increased during the Civil War. Union merchants used Matamoros as an entrepot through which to smuggle supplies to the Confederate states, often under the guise of supplying Juárez and the Liberals in their struggle with the French. As a result, United States (i.e., Union) exports to Mexico between 1862 and 1865 are too large. Annual domestic exports to Mexico in the 1850s averaged$2.3 million. By 1865 they had reached nearly \$14 million. For the years 1863 to 1865, I adjusted exports of wheat and wheat flour, cotton manufactures, manufactures of iron and steel, and exports of boots and shoes to Mexico to account for this smuggling. The corrections appear in Table 2 in brackets to the right of the official figures and are based on the same sources used to correct U.S. imports from Mexico.
To verify my adjustments, I estimated a time trend for imports and exports between 1850 and 1870 but omitted 1862 through 1865 as anomalous. I then used the fitted line to predict what imports and exports should have been had the U.S. Civil War not intervened. The “counterfactual” values were not substantially different from the intuitive corrections I provide.
1
John MacGregor, Commercial Statistics: A Digest . . ., 5 vols., 2d ed. (London, 1850), III, 734.
2
Answers of Augustus Storrs of Missouri to Certain Queries Upon. . . Trade and Intercourse Between Missouri and the Internal Provinces of Mexico, 18th Cong., 2d sess., Jan. 3, 1825, p. 6. My calculation.
3
Petition of Sundry Inhabitants of the State of Missouri Upon . . . the Internal Provinces of Mexico, 18th Cong., 2d sess., Feb. 14, 1825, pp. 4-5. My calculations follow.
4
James W. Daddysman, The Matamoros Trade: Confederate Commerce, Diplomacy, and Intrigue (Newark, DE, 1984); Ronnie C. Tyler, “Cotton on the Border, 1861-65,” Southwestern Historical Quarterly, 73:4 (Apr. 1970), 455-477; Robert W. Delaney, “Matamoros, Port for Texas during the Civil War,” Southwestern Historical Quarterly, 58:4 (Apr. 1955), 473-487; and Cerutti, Burguesía y capitalismo en Monterrey, 32-34. Statistics from American Commerce: Commerce of South America, Central America, Mexico, and West Indies, 3,284, and from Commerce of the United States and Other Foreign Countries with Mexico, Central America, the West Indies, and South America, data for Mexico.
Not everyone would accept this analysis. Thomas Schoonover’s “Mexican Cotton and the American Civil War,” The Americas, 30:4 (Apr. 1974), 429-447, argues (1) that Mexico grew its own cotton for export during the U.S. Civil War and (2) that the needs of the French army under Maximilian accounted for greatly increased imports from the United States. The second proposition could be true, but Schoonover makes no attempt to document it. And it is odd that a French army would use materiel from the United States to carve out a captive market for French goods in Mexico. The first proposition is also possible, but, curiously enough, Schoonover finds no evidence of the production, collection, preparation, and transportation of nearly 16 million pounds of raw cotton for export to the United States in 1864. Of course, the British also imported raw cotton from Mexican planters, and there were the requirements of the domestic Mexican market as well. I suspect that most “Mexican” planters were in Louisiana and Texas. For a corroborative view, see Stanley Lebergott, “Through the Blockade: The Profitability and Extent of Cotton Smuggling, 1861-65,” Journal of Economic History, 41:4 (Dec. 1981), 867-888. Lebergott finds no evidence that Mexico supplied the Union states with cotton. Also see Mario Cerutti and Miguel González Quiroga, “Guerra y comercio en torno al río Bravo (1855-1867). Linea fronteriza, espacio económico común,” Historia Mexicana 40:2 (Oct.-Dec. 1990), 242-245.
### Constructing a Domestic Export Price Index
Readers interested in the nuts and bolts of constructing a base-weighted (Laspeyres) price index should consult Paul Gootenberg, “Carneros y Chuño: Price Levels in Nineteenth-Century Peru,” HAHR, 70:1 (Feb. 1990), 1-56, for a model.
This index is simpler than Gootenberg’s. Four or five goods always dominated domestic exports to Mexico, even though their relative positions changed from year to year. In 1842/43, for instance, finished cottons, raw cotton, and flour accounted for 35 percent of the value of domestic exports. Finished cottons and raw cotton accounted for most exports, and their shares were roughly equal. In 1845/46 they accounted for 55 percent of exports, but raw cotton outweighed finished cottons. In 1852/53 they comprised nearly 80 percent of exports, but finished cottons were most important. How do we handle this?
All price indexes are idealizations, and this one is too. To construct it, I researched the annual “Statements of [Foreign] Commerce and Navigation of the United States” for 1825/26—1858/59; from 1859/60 through 1883/84, I used Commerce of the United States and Other Foreign Countries with Mexico, Central America, the West Indies, and South America. For each year, I determined the five leading exports by value. The weight assigned each was its annual export value divided by the annual value of all domestic exports. After figuring these values for each year, I examined the results to see if any patterns or groupings emerged.
A number of consistencies were evident. For example, in 1825/26-1840/41, finished cottons and wheat flour generally appeared in the top five export goods, although their annual weights varied. To avoid the repetitive calculation of an index whose changing weights required annual “links,” I averaged the annual weights and used the resulting averages as the weights for finished cottons and wheat flour during the subperiod.
For prices, I used the Historical Statistics of the United States (Washington, 1975), Series E 123-134, “Wholesale Prices of Selected Commodities: 1800-1970.” For each year, I multiplied a good’s average weight by its price. To keep other prices constant, I also multiplied everything else (1 minus the sum of the weights of the goods included) by 1. This gives a conservative measure of price change and avoids an implicit exaggeration of the weights that occurs when no other goods are included. The resulting sums (the sum of the weight of each good times its price, plus the “everything else” term) were then divided by the average sum for 1840/41-1844/45. This base period roughly corresponds to the 1839-45 (peak-to-peak) business cycle in the United States and was as normal a time in Mexico’s foreign economic relations as any other. No other period seemed more suitable, or less unsuitable.
Because the composition of trade changed over sixty years, I sometimes adjusted weights, dropped goods that were no longer important, or added new goods. To maintain historical and interpretive continuity, I linked or chained the subperiods by overlapping them and recalculating forward.
The periods and weights used are shown in Table B-1. The weights are rounded to two decimal places and differ a little from the weights I actually used because of rounding.
Some adjustments reflected changes in Mexican policy, e.g., large licensed imports of raw cotton beginning around 1840. Others were needed because of the inflation caused by the Crimean War (1854-56) or the U.S. Civil War (1860-65). Railroad building in Mexico provided the rationale for the construction of the last subperiod.
An important point is that the index of U. S. export prices to Mexico is not the same thing as the index of Mexican import prices from the United States. Distribution and supply in Mexico were always subject to severe interruption, and the operation of international markets was bound to be affected. As a result, the Mexican terms of trade with the United States are not really the reciprocal of the U.S. terms of trade with Mexico. We need four prices, rather than two.
But we do not have Mexican prices, at least not yet. We are forced to use the U.S. export price index as the Mexican import price index. For now, this is an unavoidable simplification.
### Estimating the U.S. Current Account Balance with Mexico
The current account balance is the difference between exports and imports of goods and services. The visible trade balance is the difference between exports and imports of merchandise, or net income from merchandise trade. The current account and trade balances differ by the extent of trade in services or “invisibles.” These items include business services (brokerage, shipping, insurance), tourism and emigrant funds, and interest and dividends on foreign investments. The current account measures net income from trade, services, and investment.
Looking at net balances from the U.S. side of the ledger, for our purposes only business services in trade matter. Tourism in either direction did not matter. Mexican migration to the United States before 1897 was small, and even fewer U.S. citizens migrated to Mexico. Interest and dividends on U.S. investment in Mexico could not have been large before the 1890s, or, at the earliest, before the mid-1880s. Wealthy Mexicans had purchased U.S. bonds and equities since the 1830s, but the size of their holdings is unknown. Little specie that Mexico sent north represented a credit to the U.S. current account before the mid-1870s.
But brokerage and the carrying trade were another matter. The United States garnered important earnings from the carrying trade in the early nineteenth century.1 Data from the annual “Statements of [Foreign] Commerce and Navigation of the United States” in Table C-1 suggest that U.S. vessels controlled the great bulk of the carrying trade between Mexico and the United States. Yet how well can we measure these earnings?
We have several estimates of the cost of importing Mexican silver. In 1842, F. M. Dimond, the U.S. consul at Veracruz, wrote that “all remittances are made in hard dollars the export duty on which is 3.5 percent and freight generally 1 percent. Commission for purchases 2.5 percent and on sale 5 percent and sometimes 8 percent.” He added that insurance on shipments was made in the United States or Europe, there being no institution of that kind in the Republic.”2
The consul in the mid-1850s, J. T. Pickett, also put total costs in the neighborhood of 10 percent.3 But the meaning and apportionment of these costs depends on the definition (and nationality) of the buyer, the seller, and the broker. To avoid interminable complications (and calculations), I assume that citizens of the United States buying silver produced a net charge against the United States of 1.5 percent (1 percent freight payable to U.S. shipping less 2.5 percent commission on purchase payable to a Mexican broker). There are obviously other ways of looking at the matter, but this procedure is a conservative reading of limited evidence. The nature of the silver market, of course, goes well beyond the scope of this essay.
Mexico also exported logwood, cochineal, dyestuffs, hides, and other commodities to the United States. But we have no basis for estimating U.S. earnings on them and must omit them from our calculations. Specie mattered most anyway.
U.S. earnings on exports to Mexico were composed of earnings on domestic exports and earnings on reexports. A report of 1885 concluded that commissions, insurance, and freight from the United States to Mexico added 20 percent of the value of goods.4 In short, we multiply the value of exports plus reexports by 20 percent and credit the result to the United States. We could refine the estimate by adjusting the yearly totals by the proportion of cargo carried by U.S. vessels, but that gives too great an impression of precision. These are orders of magnitude and should be understood as such.5
1
Douglass C. North, The Economic Growth of the United States, 1790-1860 (New York, 1966), 25.
2
Dimond to Secretary of State, Veracruz, Dec. 27, 1842, RG 59, NARS.
3
See 34th Cong., 2d sess., Report on the Commercial Relations of the United States With All Foreign Nations, 3 vols. (Washington, 1856), III, 410. In 1878, John W. Foster noted, “I find that it costs, to place the silver produced at the Real del Monte mines in the Bank of England or in New York, 13.5 percent, and from Guanajuato or other points in the interior from 14 to 15 percent. Of this sum from 10.5 to 12 per cent are local and government taxes and charges.” The statement implies that 1.5 to 3 percent was divided between brokers and shippers, or about what we use. See 45th Cong., 3d sess., House, Commercial Relations with Mexico, 15.
4
U.S. Congress, Senate, Message from the President of the United States . . . in relation to the Foreign Trade of Mexico, Central America and South America, the Spanish West Indies, Hayti and San Domingo, 48th Cong., 2d sess., Jan. 20, 1885, 8. Also see Commercial Relations with Mexico, 12, for similar illustrations.
5
It is important that U.S. exports were valued at cost “in the ports . . . from which they [were] exported.” Imports were valued at cost “in the foreign ports from which they [were] exported.” See “Act of February 10, 1820: An act to provide for obtaining accurate statements of the foreign commerce of the United States. Gordon, comp., Collection of the Laws of the United States Relating to Revenue, Navigation, and Commerce . . ., 249-251. Export values are essentially free alongside ship (FAS) values. Import totals exclude freight and insurance, the “invisibles” whose magnitude we estimate. See U.S. Dept, of Commerce, Handbook of Cyclical Indicators (Washington, 1984), 56-57. These conventions imply that U.S. exports to Mexico are, at best, only estimates of Mexican imports from the United States. Similarly, U.S. imports from Mexico are, at best, estimates of Mexican exports to the United States.
### Texas and Changing Patterns of Trade
I suggested that the Texas rebellion produced a dramatic change in the pattern of trade between Mexico, the United States, and Great Britain. Here I wish to demonstrate precisely how pronounced that shift was.
In order to measure the shift in trade patterns “caused” by affairs in Texas and by the Mexican reaction to them, I converted U.S. domestic exports and all British exports to Mexico to index numbers whose base year is 1835 (i.e., 1835 = 100). I then examined the behavior of both series to see if the pattern of export behavior changed in any significant way after 1835. In analyzing time series, it is customary to remove the underlying trend by differencing (i.e., subtracting successive observations from each other). In this case, I examined both first differences (the rate of change of the index numbers) and second differences (how much the change itself varied). While a distinct pattern was evident in both the first and second differences of the index numbers, the series of second differences shows the pattern most clearly.
Figure D-1 demonstrates this graphically. Before 1835, U.S. and British exports to Mexico move in more or less parallel fashion, although the swings in U.S. exports (the short dashed line) are less pronounced than those in British exports (the solid line). But around 1835/36 (where the vertical axis serves as a kind of before-and-after signpost), a new pattern emerges. Growth in British exports is now accompanied by a slowdown in U.S. exports, and vice versa. Graphically, the pattern of trade no longer appears as parallel fluctuations driven uniformly by demand, but as reversed peaks and troughs. The altered trade flows continue until the outbreak of the Mexican War.
There can be no clearer demonstration that around 1835 there was a dramatic change in trade relations between Great Britain, the United States, and Mexico. Mexico’s markets were thereafter “opened” to one party mostly at the expense of the other, something not much in evidence before. In other words, Mexico now pitted one partner against the other in the contest for markets.
1
John H. Coatsworth, “Obstacles to Economic Growth in Nineteenth-Century Mexico,” American Historical Review, 83:1 (Feb. 1978), 80-100; Donald F. Stevens, “Economic Fluctuations and Political Instability in Early Republican Mexico,” Journal of Interdisciplinary History, 16:4 (Spring 1986), 645–665, and his Origins of Political Instability in Mexico (forthcoming); Barbara Tenenbaum, The Politics of Penury. Debts and Taxes in Mexico, 1821–1856 (Albuquerque, 1986); Guy P. C. Thomson, Puebla de los Angeles. Industry and Society in a Mexican City, 1700–1850 (Boulder, 1989); David W. Walker, Kinship, Business, and Politics. The Martínez del Río Family in Mexico, 1823–1867 (Austin, 1986); Ciro Cardoso, ed., Formación y desarrollo de la burguesía en México. Siglo xix, 2d ed. (Mexico City, 1981); and Cardoso, ed., México en el siglo xix. Historia económica y de la estructura social (1821–1910), 4th ed. (Mexico City, 1983).
2
Tenenbaum, Politics of Penury·, 168–169; Stevens, “Economic Fluctuations and Political Instability,” 665.
3
Gaceta del Gobierno Supremo de México, Nov. 8, 1823.
4
See [John Pender & Company] Statistics of the Trade of the United Kingdom with Foreign Countries from 1840 (Manchester, 1869), 85–86.
5
But see Bernard Kapp, “Les rélations économiques extérieures du Mexique (1821–1911) d’après les sources françaises,” in Ville et commerce (Deux essais d’histoire hispano-américaine) (Paris, 1974), 9–93. The official value series is, in essence, a constant franc series in prices of 1826. The larger study from which this essay is drawn uses 1840–44 as a base period. I have yet to express the 1826 series in French prices of 1840–44.
6
For example, from the 1830s through the 1880s, Cuba and Brazil were the principal sources of exports from Latin America to the United States. If silver is excluded, Mexico does not figure in the top three sources until the 1880s. But if silver is included, Mexico was never out of the top three. Moreover, the monetary effects of Mexican silver were vastly more important than the availability of sugar and coffee when the United States was on a de facto silver standard. See Roy W. Jastram, Silver. The Restless Metal (New York, 1981), 65–69.
7
Peter Temin, The Jacksonian Economy (New York, 1969), esp. 68–91. For silver and the early U.S. monetary standard, see Jerome Officer, “Dollar-Sterling Mint Parity and Exchange Rates, 1791–1834,” Journal of Economic History, 43:3 (Sept. 1983), 579–616.
8
The figure of 50 percent appears in Henry Toland to President of the United States, Philadelphia, Nov. 27, 1830. Letters regarding the appointment of James James as Veracruz consul. Record Group 59, National Archives [hereafter RG 59, NARS]. For Philadelphia’s trade with Mexico, see Linda K. Salvucci, “Development and Decline: The Port of Philadelphia and Spanish Imperial Markets: 1783–1823” (Ph.D. diss., Princeton University, 1985), 221.
9
See Appendix C and Table C-1.
10
For instance, see Warren T. Brookes, “Hiding a Boom in a Statistical Bust,” Wall Street Journal, Aug. 6, 1987, or “Le ‘trou noir’ des statistiques internationales,” Le Monde, June 16, 1987.
11
Literally, “This is hard, this is work,” from Virgil’s Aeneid.
12
U.S. Department of Commerce, Bureau of the Census, Historical Statistics of the United States, Colonial Times to 1970. Bicentennial Edition, 2 vols. (Washington, 1975), II, 903–904. Also see I, xii–xiii, “The Problem of Historical Statistics.”
13
United Kingdom, Parliament, Report by Mr. Lionel E. G. Carden on the Trade and Commerce of Mexico, C. 3785 (1883), 3. The national composition of reexports changed during the 1870s as well. After 1869, the French classified exports by intended market rather than by port of destination. Goods sent to the United States for reexport to Mexico were now classified as exports to Mexico rather than as exports to the United States. See Tableau décennal du commerce de la France avec ses colonies et les puissances étrangères 1877 à 1886, 2 vols. (Paris, 1888), I, xiv.
14
J. T. Pickett to Secretary of State, Veracruz, Feb. 21, 1852, U.S. Dept, of State, Consular Despatches from Veracruz, RG 59, NARS. Before 1846, merchandise for reexport entered U.S. ports free of duty but subject to forfeiture of a customs bond while remaining on board ship. The system produced considerable evasion and was scrapped in favor of a bonded warehouse system in 1846. Goods were bonded on deposit and again on withdrawal for subsequent export. Evidence of sale discharged the bonds. See Thomas F. Gordon, comp., A Collection of the Laws of the United States Relating to Revenue, Navigation, and Commerce (Philadelphia, 1844), 83–84, and Robert Mayo, A Synopsis of the Commercial and Revenue System of the United States, “Extra Edition” (Washington, 1847), 328–340.
15
[Katherine de la Fosse] The First Hundred Years. British Industry and Commerce in Mexico; 1821–1921 (Mexico City, 1978), not paginated.
16
F. M. Dimond to Secretary of State, Veracruz, Sept. 1, 1845, RG 59, NARS.
17
Answers of Augustus Storrs of Missouri to Certain Queries Upon the Trade and Intercourse . . . of the Internal Provinces of Mexico, 18th Cong., 2d sess., Senate, Jan. 3, 1825, pp. 6, 8–9, 12; Angela Moyano Pahissa, El comercio de Santa Fe y la guerra del 47 (Mexico City, 1976); Petition of Sundry Inhabitants of the State of Missouri Upon . . . the Internal Provinces of Mexico, 18th Cong., 2d sess., Feb. 14, 1825, pp. 4–5; Henry George Ward, México en 1827, trans. Ricardo Haas (Mexico City, 1981), 281.
18
The current account balance measures net income from trade, services, and investment.
19
J. T. Pickett to Secretary of State, Veracruz, July 4, 1854, RG 59, NARS.
20
Richard Werking concludes that “after three or four years at their posts some of the [U.S.] consuls managed to acquire a modicum of efficiency and suitability for their jobs.” The general reliability of their commercial reporting on Mexico supports the conclusion. See Richard Hume Werking, The Master Architects. Building the United States Foreign Service, 1890–1913 (Lexington, KY, 1977), 9. On the same theme, see also Henry E. Mattox, The Twilight of Amateur Diplomacy. The American Foreign Service and Its Senior Officers in the 1890s (Kent, OH, 1989), ix–xiii.
21
Economic historians conventionally distinguish between merchandise and specie balances. Since there are no prices with which to deflate the (import) merchandise balance, I ignored the usual distinction.
22
See Barry M. Gough, “Specie Conveyance from the West Coast of Mexico in British Warships c. 1820–1870: An Aspect of Pax Britannica,” Mariner’s Mirror, 69:4 (1983), 419–433; and John Mayo, “Consuls and Silver Contraband on Mexico’s West Coast in the Era of Santa Anna,” Journal of Latin American Studies, 19:2 (Nov. 1987), 389–411.
23
[Alex del Mar] Report of the Director of the Bureau of Statistics on the Imports of the United States (Washington, 1868), 1–2, 14, 21–22.
24
For example, see Albert H. Imlah, Economic Elements in the Pax Britannica. Studies m British Foreign Trade in the Nineteenth Century (1958; reprint New York, 1969), 42–81.
25
See “Memoria sobre el estado de la agricultura e industria de la república en el año de 1845” (1846), in Documentos para el estudio de la industrialización en México, ed. Horacio Labastida (Mexico City, 1977), esp. 202–203. The U.S. consul at Veracruz concurred in a letter to Secretary of State, Veracruz, Dec. 17, 1842, RG 59, NARS. The Tariff of 1843 prohibited raw cotton and provided the Santanista regime (and Santa Anna himself) with a pretext for selling lucrative import licenses (permisos) between 1843 and 1845. A license of May 3, 1844, authorized imports of 100,000 quintales, or 10 million pounds. By the outbreak of the war, the United States had exported over 12 million pounds, or an average of 4 million per year. If Alamán’s figures were correct, U.S. cotton supplied at least a third of Mexican requirements in the early 1840s. Shipments of raw cotton are documented in the “Statements of [Foreign] Commerce and Navigation of the United States” for 1843/44, 44/ 45, and 45/46. Also F. M. Dimond to Secretary of State, Veracruz, July 11, 1843, and May 5, 1845, RG 59, NARS.
26
I ranked cumulative Mexican silver exports to the United States and cumulative domestic U.S. exports to Mexico by decade (1820s through 1880s). The rank-order correlation between the series was .93, or nearly perfect.
27
For the theory, see Ronald I. McKinnon, Money and Capital in Economic Development (Washington, 1973), 5–21. On capital markets in Mexico in the early nineteenth century, see Stephen H. Haber, “Industrial Concentration and the Capital Markets: A Comparative Study of Brazil, Mexico, and the United States, 1830–1930.” Journal of Economic History, 51 (forthcoming, 1991).
28
The standard estimate of sectoral shares in Mexican income around 1800 is Fernando Rosenzweig Hernández, “La economía novohispana al comenzar el siglo XIX,” in El desarrollo económico de México, 1800–1910 (Toluca, 1989), 23–85. On loan volume in Puebla, see Thomson, Puebla de los Angeles, 50. For mining output, I follow Inés Herrera Canales, “Empresa minera y región en México. La Compañía de Minas de Real del Monte y Pachuca (1824–1906),” in Siglo XIX, 4:8 (Jul.–Dec. 1989), 122–123.
29
This statement refers directly to the commercialized sector. But cyclical changes in money income indirectly determine the opportunity cost of resources remaining at subsistence. In this sense, the statement may refer to both the commercialized and subsistence sectors.
30
The characterization of Mexico as an “export-led” economy until the latter third of the nineteenth century merits consideration. According to criteria outlined by Irving Kravis, “Trade as a Handmaiden of Growth: Similarities Between the Nineteenth and Twentieth Centuries,” Economic Journal, 80:320 (Dec. 1970), 850–872, esp. 853–854, Mexico before Díaz was, in some respects, not unlike other export economies. For one thing, silver mining had indirect “real” economic effects through the supply of loanable funds as well as directly through localized linkages. Movements in income followed the production of the export staple. For another, foreign capital was drawn to mining, a sector whose productivity exceeded that of agriculture or industry. But the share of all exports (including silver) in Mexican national income is still unclear.
31
Edward J. Nell, “Value and Capital in Marxian Economics,” in The Crisis in Economic Theory, ed. Daniel Bell and Irving Kristol (New York, 1981), 196.
32
Carlos María de Bustamante, Apuntes para la historia del gobierno del general don Antonio López de Santa Anna (1845; reprint, Mexico City, 1986), 209.
33
“Report of the Secretary of State to the Congress of Mexico,” in Message from the President of the United States . . . Upon the Existing Relations Between the United States and Mexico, 25th Cong., 2d sess., July 4, 1838, 343.
34
Van Buren to Chargé d’Affaires in Mexico, Washington, Oct. 16, 1829, in Message from the President, 44.
35
See J. R. Poinsett’s instructions, Washington, Mar. 26, 1825, and Poinsett to Secretario de Relaciones Exteriores, Mexico, Dec. 28, 1826, both in Carlos Bosch García, Documentas de la relación de México con los Estados Unidos, 4 vols. (Mexico City, 1983-85), I, 78, 210-211.
36
“The memorial of the subscribers comprising all the American merchants residing in the city of Vera Cruz,” in Message from the President, 218; William Taylor to Secretary of State, Veracruz, July 5, 1829, RG 59, NARS; and Anthony Butler to Secretary of State, Mexico, Mar. 9, 1830, in Bosch García, Documentas de la relación de México con los Estados Unidos, II, 192.
37
R. J. Walker to President of the United States, Washington, Mar. 30, 1847, in Mayo, Commercial and Revenue System of the U.S., 413.
38
A copy of the schedule of 1837 is reproduced in Diario del Gobierno de la República Mexicana, Mar. 22, 1837. The 1843 schedule was the Arancel general de aduanas marítimas y fronterizas (Mexico City, 1843). U.S. consular reports (various officials) discuss the tariffs. See Mar. 28, 1837, Dec. 17, 1842, Dec. 27, 1842, among others. For the quoted observation, see F. M. Dimond to Secretary of State, Veracruz, Nov. 1, 1845. All are in RG 59, NARS.
39
Waddy Thompson to Secretary of State, Mexico City, July 30, 1842, in Bosch García, Documentos de la relatión de México con los Estados Unidos, III, 511. The disposition of foreign trade after 1837 was in part responsible for the political turmoil of the early 1840s. See Cecilia Noriega Elío, El constituyente de 1842 (Mexico City, 1986), 17–31.
40
F. M. Dimond to Secretary of State, Veracruz, Oct. 16, Nov. 1, and Nov. 4, 1845, RG 59, NARS.
41
F. M. Dimond to Secretary of State, Veracruz, Dec. 11, 1845. His characterization of the United States appears in a dispatch of Nov. 1, 1845. Both are in RG 59, NARS.
42
See Richard J. Salvucci, Linda K. Salvucci, and Aslán Cohen, “Interpeting Commercial Policy in Mexico: Protection and Free Trade, 1750-1840” in The Political Economy of Spanish America in the Age of Revolution, ed. Kenneth Andrien and Lyman Johnson (forthcoming, 1992); F. M. Dimond to Secretary of State, Veracruz, July 30, 1845, RG 59, NARS.
43
“[Guadalupe Victoria’s] soul is made of the same stuff as Washington’s. National Gazette, Feb. 2, 1825.
44
“Mexico, Sept. 1, 1845,” in Diario del Gobierno de la República Mexicana, Sept. 1, 1845.
45
Thomas R. Hietala, Manifest Design. Anxious Aggrandizement in Late Jacksonian America (Ithaca, 1985), esp. 55-94.
46
David Pletcher, The Diplomacy of Annexation: Texas, Oregon, and the Mexican War (Columbia, MO, 1973), 499. The text of the tariff and addenda to it appear in Mayo, Commercial and Revenue System of the U.S., 418-426.
47
Lionel Motes to Secretary of State, Veracruz, July 1, 1850, RG 59, NARS.
48
Oscar J. Martínez, Troublesome Border (Tucson, 1988), 18-21.
49
J. T. Pickett to Secretary of State, Veracruz, Dec. 8, 1854; Gadsden is quoted in Pickett’s dispatch of Oct. 10, 1855. Both in RG 59, NARS.
50
J. T. Pickett to Secretary of State, Veracruz, Oct. 10, 1855, RG 59, NARS. In 1870, Matías Romero judged the tariffs of 1845 and 1853 “the highest that have ever prevailed in the republic. See the Diario Oficial, Oct. 31, 1870, cited in Foreign Relations of the United States [FRUS], 1870, 488.
51
J. T. Pickett to Secretary of State, Veracruz, Oct. 10, 1855, RG 59, NARS; Carlos J. Sierra and Rogelio Martínez Vera, Historia y legislación aduanera de México (Mexico City, 1973). 125.
52
Gadsden is quoted in Pickett’s dispatch of Oct. 10, 1855, RG 59, NARS. His words were reminiscent of U.S. wartime thinking eight years earlier. The secretary of the treasury advised President Polk in 1847 that “[Mexico’s commercial prohibitions] should not be permitted to continue.” See R. J. Walker to President of the United States, Washington, Mar. 30, 1847, in Mayo, Commercial and Revenue System of the U.S., 414.
53
J. T. Pickett to Secretary of State, Veracruz, Oct. 23, 1854, RG 59, NARS.
54
Waddy Thompson to Secretary of State, Mexico City, Oct. 3, 1843, in Bosch García, Documentos de la relatión de México con los Estados Unidos, III, 614.
55
Report on the Commercial Relations of the United States with All Foreign Nations, 2 vols. (Washington, 1857), II, 353-362, for changes in the Mexican tariff. The computations are mine. Average duties on U.S. imports (i.e., the implicit tariff index) are quoted in Charles Rieken to Secretary of State, Veracruz, Dec. 31, 1857, RG 59, NARS. For comparative data on the implicit index of protection, which measures the ratio of duties collected to the total value of dutiable goods imported, see Salvucci, Salvucci, and Cohen, “Interpreting Commercial Policy in Mexico,” Figure 1.
56
For “demand versus supply” as a core of the capitalist ethos, see the discussion in David Eltis, Economic Growth and the Ending of the Transatlantic Slave Trade (New York, 1987), 19–23. Prieto is quoted in Sierra and Martínez Vera, Historia y legislatión aduanera, 128.
57
Romero is quoted in Thomas David Schoonover, Dollars over Dominion. The Triumph of Liberalism in Mexico–United States Relations, 1861-1867 (Baton Rouge, 1978), 19, and, in general, 251-276. Donathon C. Olliff, Reforma Mexico and the United States: A Search for Alternatives to Annexation, 1854-1861 (University, AL, 1981) also underscores the counterpoint of trade and annexation. Mexican historians concur. See Sergio Ortega Noriega, “Intercambios económicos entre el Noroeste Mexicano y los Estados Unidos a fines del siglo XIX. El caso de Topolobampo,” in Históricas [Instituto de Investigaciones Históricas, UNAM], 1 (1979), 13-23, esp. 15.
58
J. T. Pickett to Secretary of State, Veracruz, Aug. 7, 1855, and Feb. 25, 1858, RG 59, NARS
59
John S. D. Eisenhower, So Far From God. The U.S. War with Mexico, 1846-1848 (New York, 1989), xxii and 111 n., for enlightening examples.
60
See Abdiel Oñate, “El surgimiento de la supremacía estadounidense en los mercados latinoamericanos: el caso de México, 1870-1914,” in El dilema de dos naciones. Relaciones económicas entre México y Estados Unidos, ed. T. Noel Osborn B. et al. (Mexico City, 1981), 391-404. Jorge Espinosa de los Reyes, Relaciones económicas entre México y los Estados Unidos, 1870-1910 (Mexico City, 1951), 54, puts the date around 1884. This increase did not represent a simple diversion of British trade. Preliminary calculations indicate that British real exports to Mexico per capita (prices of 1840-44) increased by over 80 percent between 1869-71 and 1879-81. U.S. real domestic exports per capita (prices of 1840/ 41-1844/45) more than doubled over the same period. Total Mexican demand was obviously growing.
61
U.S. Department of the Treasury, Bureau of Statistics, American Commerce, Commerce of South America, Central America, Mexico, and West Indies with Share of the United States and Other Leading Nations Therein, 1821-1898 (Washington, 1899), 3370-3376.
62
Arnaldo Córdova, La ideología de la revolución mexicana. La formación del nuevo régimen, 13th ed. (Mexico City, 1985), 15; and Charles Hale, The Transformation of Liberalism in Nineteenth-Century Mexico (Princeton, 1989), 16–19. For patterns of international growth, see Solomos Solomou, “Non-Balanced Growth and Kondratieff Waves in the World Economy, 1850-1913,” Journal of Economic History, 46:1 (Mar. 1986), 165-169.
63
E. H. Saulnier to Secretary of State, Veracruz, May 2, 1868, and May 24 1869 RG 59, NARS.
64
Oscar Martínez, Border Boom Town: Ciudad Juárez Since 1848 (Austin, 1978), 14-17. On contraband, see Matías Romero quoted in FRUS, 1870, 491-492. The Matamoros trade is documented in the annual reports of the consular district of Mexico City, written by Julius Skelton and dated Sept. 30, 1871 (1869/70), Sept. 30, 1872 (1870/71), and Dec. 5, 1874 (1871/72), RG 59, NARS. Also see Mario Cerutti, Burguesía y capitalismo en Monterrey, 1850-1910 (Mexico City, 1983), 36-39.
65
Robert C. West and James J. Parsons, “The Topia Road: A Trans-Sierran Trail of Colonial Mexico,” in Hispanic Lands and Peoples. Selected Writings of James J. Parsons, ed. William M. Denevan (Boulder, 1989), 143-149; Daniel Cosío Villegas, Historia moderna de México. La república restaurada. La vida económica (Mexico City, 1955), 176.
66
For a litany of complaints, see the annual reports of the consular district of Veracruz written by L. T. Trowbridge and dated Sept. 30, 1872 (1871/72), Sept. 30, 1873 (1872/73), Sept. 30, 1875 (1874/75), Oct· 1, 1877 (1876/77), Sept. 30, 1879 (1878/79), Oct. 31, 1880 (1879/80), in RG 59, NARS.
67
Stephen H. Haber, Industry and Underdevelopment: The Industrialization of Mexico, 1890-1940 (Stanford, 1989), 38. For “effective” protection, see Feb. 27, 1885, Congressional Record [Cong. Rec. ], 48th Cong., 2d sess., Appendix, 173. The concept of effective protection is discussed at length in Malcolm Gillis et al., Economics of Development, 2d ed. (New York, 1987), 436-439.
68
Feb. 27, 1885, Cong. Rec., 48th Cong., 2d sess., Appendix, 172. On the treaty, see Josefina Zoraida Vázquez and Lorenzo Meyer, The United States and Mexico (Chicago, 1985), 89–90, and Espinosa de los Reyes, Relaciones económicas, 76–105.
69
June 30, 1886, Cong. Rec., 49th Cong., 1st sess., Appendix, 380–393; and April 18, 1888, Cong. Rec., 50th Cong., 1st sess., Appendix, 303–318. For the roots of this movement, see William Appleman Williams, The Tragedy of American Diplomacy, 2d ed. (New York, 1972), 24-27.
70
I omit the years of the U.S. Civil War in Table 2 because of problems associated with measuring and accounting for contraband.
71
This paragraph draws on the annual “Statements of [Foreign] Commerce and Navigation of the United States for 1865/66-1875/76 for data. The computations are mine.
72
United Kingdom, Parliament, Silver, & c (China) (1858), esp. 50-52, 55, 58, 61-62, 71-72.
73
See John W. Foster to Secretary of State, Mexico, Sept. 7, 1878, in FRUS, 1878, 588-589.
74
See John W. Foster to Secretario de Relaciones Exteriores, Mexico, Sept. 26, 1878, in FRUS, 1878, 657. The Free Zone existed between 1858 and 1905. It was controversial in both Mexico and the United States. A useful account of it appears in Espinosa de los Reyes, Relaciones económicas, 105-113.
75
For the quote, see John W. Foster to Secretary of State, Mexico, Sept. 7, 1878, in FRUS, 1878, 588-589. See Mohsin S. Kahn and Nadeem Ul Haque, “Capital Flight from Developing Countries,” Finance & Development, 24:1 (Mar. 1987), 2-5, for ways of estimating capital flight from balance of payments data. For a useful survey of the literature, see Sunil Gulati, “Capital Flight: Causes, Consequences, Cures,” Journal of International Affairs, 42:1 (Fall 1988), 165-185.
76
See their essays in Joseph L. Love and Nils Jacobsen, eds., Guiding the Invisible Hand. Economic Liberalism and the State in Latin American History (New York, 1989), 35-98. Also see Gootenberg’s Tejidos y harinas, corazones y mentes. El imperialismo norteamericano del libre comercio en el Perú, 1825-1840 (Lima, 1989) for a comparative study.
77
I am indebted to Steve Haber for this observation.
78
For the litany, see Robert Johannsen, To the Halls of the Montezumas: The Mexican War in the American Imagination (New York, 1985), 293-296.
79
For example, Nathan Rosenberg and L. E. Birdzell, Jr., How the West Grew Rich. The Economic Transformation of the Industrial World (New York, 1986), 71-112, esp. 90. In a similar vein, see also E. L. Jones, The European Miracle. Environments, Economies, and Geopolitics in the History of Europe and Asia, 2d ed. (Cambridge, 1987), 85-103.
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2022-01-17 11:23:53
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http://www.mathjournals.org/jot/2004-052-002/2004-052-002-004.html
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Previous issue · Next issue · Most recent issue · All issues
# Journal of Operator Theory
Volume 52, Issue 2, Fall 2004 pp. 267-291.
The local trace function of the shift invariant subspaces
Authors Dorin Ervin Dutkay
Author institution: Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Rd, Piscataway, NJ 08854--8019, USA
Summary: We define the local trace function for subspaces of $L^{2}\left(\mathbb{R}^n\right)$ which are invariant under integer translation. Our trace function contains the dimension function and the spectral function defined in \cite{BoRz} and completely characterizes the given translation invariant subspace. It has properties such as positivity, additivity, monotony and some form of continuity. It behaves nicely under dilations and modulations. We use the local trace function to deduce, using short and simple arguments, some fundamental facts about wavelets such as the characterizing equations, the equality between the dimension function and the multiplicity function and some new relations between scaling functions and wavelets.
Contents Full-Text PDF
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2018-01-22 18:05:32
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https://farside.ph.utexas.edu/teaching/celestial/Celestial/node53.html
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Next: Tidal torques Up: Rotating reference frames Previous: Rotational flattening
# Tidal elongation
Consider two point masses, and , executing circular orbits about their common center of mass, , with angular velocity . Let be the distance between the masses and the distance between point and mass . (See Figure 6.6.) We know from Section 4.16 that
(6.59)
and
(6.60)
where .
Let us transform to a non-inertial frame of reference that rotates, about an axis perpendicular to the orbital plane and passing through , at the angular velocity . In this reference frame, both masses appear to be stationary. Consider mass . In the rotating frame, this mass experiences a gravitational acceleration
(6.61)
directed toward the center of mass, and a centrifugal acceleration (see Section 6.3)
(6.62)
directed away from the center of mass. However, it is easily demonstrated, using Equations (6.59) and (6.60), that
(6.63)
In other words, the gravitational and centrifugal accelerations balance, as must be the case if mass is to remain stationary in the rotating frame. Let us investigate how this balance is affected if the masses and have finite spatial extents.
Let the center of the mass distribution lie at , the center of the mass distribution at , and the center of mass at . (See Figure 6.7.) We wish to calculate the centrifugal and gravitational accelerations at some point in the vicinity of point . It is convenient to adopt spherical coordinates, centered on point , and aligned such that the -axis coincides with the line .
Let us assume that the mass distribution is orbiting around , but is not rotating about an axis passing through its center of mass, in order to exclude rotational flattening from our analysis. If this is the case then it is easily seen that each constituent point of executes circular motion of angular velocity and radius . (See Figure 6.8.) Hence, each point experiences the same centrifugal acceleration:
(6.64)
It follows that
(6.65)
where
(6.66)
is the centrifugal potential and . The centrifugal potential can also be written
(6.67)
The gravitational acceleration at point due to mass is given by
(6.68)
where the gravitational potential takes the form
(6.69)
Here, is the distance between points and . The gravitational potential generated by the mass distribution is the same as that generated by an equivalent point mass at , as long as the distribution is spherically symmetric, which we shall assume to be the case.
Now,
(6.70)
where is the vector , and the vector . (See Figure 6.7.) It follows that
(6.71)
Expanding in powers of , we obtain
(6.72)
Hence,
(6.73)
to second order in , where the are Legendre polynomials (Abramowitz and Stegun 1965b).
Adding and , we find that
(6.74)
to second order in . Note that is the potential due to the net externally generated force acting on the mass distribution . This potential is constant up to first order in , because the first-order variations in and cancel each other. The cancellation is a manifestation of the balance between the centrifugal and gravitational accelerations in the equivalent point mass problem discussed previously. However, this balance is only exact at the center of the mass distribution . Away from the center, the centrifugal acceleration remains constant, whereas the gravitational acceleration increases with increasing . At positive , the gravitational acceleration is larger than the centrifugal acceleration, giving rise to a net acceleration in the -direction. Likewise, at negative , the centrifugal acceleration is larger than the gravitational, giving rise to a net acceleration in the -direction. It follows that the mass distribution is subject to a residual acceleration, represented by the second-order variation in Equation (6.74), that acts to elongate it along the -axis. This effect is known as tidal elongation.
Suppose that the mass distribution is a sphere of radius , and uniform density , made up of rock similar to that found in the Earth's mantle. Let us estimate the elongation of this distribution due to the tidal potential specified in Equation (6.74), which (neglecting constant terms) can be written
(6.75)
Here, the dimensionless parameter
(6.76)
is (minus) the typical ratio of the tidal acceleration to the gravitational acceleration at . Let us assume that . By analogy with the analysis in the previous section, in the presence of the tidal potential the distribution becomes slightly spheroidal in shape, such that its outer boundary satisfies Equation (6.38). Moreover, the induced ellipticity, , of the distribution is related to the normalized amplitude, , of the tidal potential according to Equation (6.49) if , and according to Equation (6.44) if . In the former case, the distribution responds elastically to the tidal potential, whereas in the latter case it responds like a liquid.
Consider the tidal elongation of the Earth due to the Moon. In this case, we have , , , and (Yoder 1995). Hence, we find that
(6.77)
Note that . We conclude that the Earth responds elastically to the tidal potential of the Moon, rather than deforming like a liquid. For the rock that makes up the Earth's mantle, and (de Pater and Lissauer 2010). Thus, it follows from Equation (6.50) that
(6.78)
Hence, according to Equation (6.49), the ellipticity of the Earth induced by the tidal effect of the Moon is
(6.79)
The fact that is negative implies that the Earth is elongated along the -axis; that is, along the axis joining its center to that of the Moon. [See Equation (6.38).] If and are the greatest and least radii of the Earth, respectively, due to this elongation (see Figure 6.9), then
(6.80)
Thus, we predict that the tidal effect of the Moon (which is actually due to spatial gradients in the Moon's gravitational field) causes the Earth to elongate along the axis joining its center to that of the Moon by about centimeters. This elongation is only about a quarter of that which would result were the Earth a non-rigid (i.e., liquid) body. The true tidal elongation of the Earth due to the Moon is about centimeters [assuming a Love number (Bertotti et al. 2003)]. We have slightly underestimated this elongation because, for the sake of simplicity, we treated the Earth as a uniform-density body.
Consider the tidal elongation of the Earth due to the Sun. In this case, we have , , , and . Hence, we calculate that and , or
(6.81)
Thus, the tidal elongation of the Earth due to the Sun is about half that due to the Moon. The true tidal elongation of the Earth, due to the Sun, is about centimeters [assuming a Love number (Bertotti et al. 2003)]. Again, we have slightly underestimated the elongation because we treated the Earth as a uniform-density body.
Because the Earth's oceans are liquid, their tidal elongation is significantly larger than that of the underlying land. (See Exercise 9.) Hence, the oceans rise, relative to the land, in the region of the Earth closest to the Moon, and also in the region furthest away. Because the Earth is rotating, while the tidal bulge of the oceans remains relatively stationary, the Moon's tidal effect causes the ocean at a given point on the Earth's surface to rise and fall twice daily, giving rise to the phenomenon known as the tides. There is also an oceanic tidal bulge due to the Sun that is about half as large as that due to the Moon. Consequently, ocean tides are particularly high when the Sun, the Earth, and the Moon lie approximately in a straight line, so that the tidal effects of the Sun and the Moon reinforce one another. This occurs at a new moon, or at a full moon. These type of tides are called spring tides (the name has nothing to do with the season). Conversely, ocean tides are particularly low when the Sun, the Earth, and the Moon form a right angle, so that the tidal effects of the Sun and the Moon partially cancel one another. These type of tides are called neap tides. Generally speaking, we would expect two spring tides and two neap tides per month.
We can roughly calculate the vertical displacement of the oceans, relative to the underlying land, by treating the oceans as a shallow layer of negligible mass, covering the surface of the Earth. The Earth's external gravitational potential is written [see Equation (3.65)]
(6.82)
where is given by Equation (6.79). Let the ocean surface satisfy
(6.83)
Because fluids cannot withstand shear stresses, we expect this surface to be an equipotential:
(6.84)
It follows that, to first order in and ,
(6.85)
Thus, the maximum vertical displacement of the ocean relative to the underlying land is
(6.86)
As we saw earlier, for the Earth. Moreover, the tidal potential due to the Moon is such that . We thus conclude that the Moon causes the oceans to rise a maximum vertical distance of relative to the land. Likewise, the tidal potential due to the Sun is such that . Hence, we predict that the Sun causes the oceans to rise a maximum vertical distance of relative to the land.
In reality, the relationship between ocean tides and the Moon and Sun is much more complicated than that indicated in the previous discussion. This is partly because of the presence of the continents, which impede the flow of the oceanic tidal bulge around the Earth, and partly because of the finite inertia of the oceans.
Note, finally, that as a consequence of friction within the Earth's crust, and friction between the oceans and the underlying land, there is a time lag of roughly 12 minutes between the Moon (or Sun) passing directly overhead (or directly below) and the corresponding maximum in the net tidal elongation of the Earth and the oceans (Bertotti et al. 2003).
Next: Tidal torques Up: Rotating reference frames Previous: Rotational flattening
Richard Fitzpatrick 2016-03-31
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2019-12-09 16:29:53
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https://www.conwaylife.com/forums/viewtopic.php?p=37190
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## LifeWiki Trusted Account Request Thread - Post requests here
For discussion directly related to ConwayLife.com, such as requesting changes to how the forums or wiki function.
Apple Bottom
Posts: 1034
Joined: July 27th, 2015, 2:06 pm
Contact:
### Re: Massive spam attacks on the wiki (and forums?)
Kiran wrote:Why not enforce Scrypt proof of work on new users? This would make it computationally difficult for spammers to make new accounts, and the few they create can be banned quickly. Also, post rate limits can be imposed on new users, to ensure they do not spam too much. Someone who actually wants to join can wait a few minutes for PoW to be solved.
Another idea: if it's not done yet, restrict page creation to autoconfirmed users. Wikipedia uses an account age of 4 days as a default for this, IIRC, which works well; most spammers don't create accounts in advance and then use them later.
MediaWiki also has a rate-limiting feature that could be used to set limits on the number of actions (edits etc.) that users could perform in a given amount of time. This works per user-group, so different rates could be set for anonymous users, registered-but-not-yet-autoconfirmed users, (regular) registered and autoconfirmed users and trusted users. See the manual entry for edit throttling, too.
MediaWiki has quite a few useful features for ensuring the operational safety of a wiki.
What I'm taking home from all this is that fighting spam needs a two-pronged approach:
• keep spammers from registering accounts in the first place; and
• keep those spammers that somehow do manage from creating too much trouble.
So far we've mostly focussed on the first -- but I think we should also look into the second more, e.g. using the above features.
If you speak, your speech must be better than your silence would have been. — Arabian proverb
Catagolue: Apple Bottom • Life Wiki: Apple Bottom • Twitter: @_AppleBottom_
Proud member of the Pattern Raiders!
Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm
### Re: Massive spam attacks on the wiki (and forums?)
Even better idea:
Use GoL based PoW, while collecting soup data!
Enforce that a new user must contribute some amount of work to apg-search, proven by the PoW system in apg-search, to start posting.
Kiran Linsuain
Apple Bottom
Posts: 1034
Joined: July 27th, 2015, 2:06 pm
Contact:
### Re: Massive spam attacks on the wiki (and forums?)
Kiran wrote:Even better idea:
Use GoL based PoW, while collecting soup data!
Enforce that a new user must contribute some amount of work to apg-search, proven by the PoW system in apg-search, to start posting.
Are you sure that's a good idea?
The bar for constructive contributions by genuine, well-meaning users should be as low as possible. A contribution, especially from a new user, is a gift. Do we want to charge for accepting gifts?
If you speak, your speech must be better than your silence would have been. — Arabian proverb
Catagolue: Apple Bottom • Life Wiki: Apple Bottom • Twitter: @_AppleBottom_
Proud member of the Pattern Raiders!
Posts: 1977
Joined: November 8th, 2014, 8:48 pm
Location: Getting a snacker from R-Bee's
### Re: Massive spam attacks on the wiki (and forums?)
Kiran wrote:Even better idea:
Use GoL based PoW, while collecting soup data!
Enforce that a new user must contribute some amount of work to apg-search, proven by the PoW system in apg-search, to start posting.
...I haven't even gotten Python to run on my computer, let alone run apgsearch on it.
LifeWiki: Like Wikipedia but with more spaceships. [citation needed]
Scorbie
Posts: 1489
Joined: December 7th, 2013, 1:05 am
### Re: Massive spam attacks on the wiki (and forums?)
Apple Bottom wrote:The bar for constructive contributions by genuine, well-meaning users should be as low as possible. A contribution, especially from a new user, is a gift. Do we want to charge for accepting gifts?
I'm remembering that in case I get to use it later.
Best wishes to you, Scorbie
Kiran
Posts: 285
Joined: March 4th, 2015, 6:48 pm
### Re: Massive spam attacks on the wiki (and forums?)
Apple Bottom wrote:The bar for constructive contributions by genuine, well-meaning users should be as low as possible. A contribution, especially from a new user, is a gift. Do we want to charge for accepting gifts?
I see your point there, I see a trade-off between getting clogged with spam and blocking contributions from new users. Is there a better way of resolving this? One idea is to allow new users to post only in a designated sub-forum until the mods allow them in, another idea is to have them PM the mods for an "interview" to be allowed to start posting.
BlinkerSpawn wrote:...I haven't even gotten Python to run on my computer, let alone run apgsearch on it.
It is sad that those scripts require so much set-up, how much effort do you think it would take to make them user-friendly enough to just install and run? Perhaps we could ask the golly gang to make a nice GUI for apg-search.
Kiran Linsuain
dvgrn
Moderator
Posts: 7401
Joined: May 17th, 2009, 11:00 pm
Contact:
### Re: Massive spam attacks on the wiki (and forums?)
Kiran wrote:It is sad that those scripts require so much set-up, how much effort do you think it would take to make them user-friendly enough to just install and run? Perhaps we could ask the golly gang to make a nice GUI for apg-search.
One step in the "just install and run" direction is the addition of Lua as a scripting language to Golly 2.8. I don't know if anyone is interested in the challenge of converting the old apgsearch script to Lua, but once that's available it would run in Golly with no additional install headaches.
It's not quite a trivial conversion -- there would be some library code to track down or create, that comes with Python but not with Lua (SHA hash functions, communication with Catagolue, and so on) -- but it seems like it would be doable.
On the spam question: at the moment, I'm much happier deleting the occasional post flagged as spam, or even cleaning up a relatively rare bigger mess like the one that started this thread, than I would be answering new-user PMs or endlessly managing who is allowed to see what. Seems to me that either of those options would also have the effect of discouraging valid new contributors.
Apple Bottom
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### Re: Massive spam attacks on the wiki (and forums?)
dvgrn wrote:On the spam question: at the moment, I'm much happier deleting the occasional post flagged as spam, or even cleaning up a relatively rare bigger mess like the one that started this thread, than I would be answering new-user PMs or endlessly managing who is allowed to see what. Seems to me that either of those options would also have the effect of discouraging valid new contributors.
Indeed, this!
Let's also not forget we do not have a persistent spam problem, we only had one attack. The spammers were able to cause a fair amount of damage, but they aren't a constant problem.
So let's solve the problems we had and have, and not the ones we might have in the future (at least until we actually have them.)
If you speak, your speech must be better than your silence would have been. — Arabian proverb
Catagolue: Apple Bottom • Life Wiki: Apple Bottom • Twitter: @_AppleBottom_
Proud member of the Pattern Raiders!
Saka
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### Re: Massive spam attacks on the wiki (and forums?)
Currently taking a little break, but still hanging around on the Discord server.
Nathaniel
Posts: 636
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### Re: Massive spam attacks on the wiki (and forums?)
Saka wrote:Can someone add me to the trusted list (Username is Saka)
Done. Sorry for the delay (and sorry that needing to be "trusted" is still even a thing).
muzik
Posts: 3926
Joined: January 28th, 2016, 2:47 pm
Location: Scotland
### Re: Massive spam attacks on the wiki (and forums?)
viewforum.php?f=7
ARE YOU joking ME
I think forum accounts might need to be trusted as well if this happens again
Last edited by muzik on January 14th, 2019, 11:14 am, edited 1 time in total.
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
Saka
Posts: 3608
Joined: June 19th, 2015, 8:50 pm
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### Re: Massive spam attacks on the wiki (and forums?)
muzik wrote:viewforum.php?f=7
ARE YOU SH*TTING ME
I think forum accounts might need to be trusted as well if this happens again
How would that work? Maybe if their first post has been reported they get banned?
Currently taking a little break, but still hanging around on the Discord server.
muzik
Posts: 3926
Joined: January 28th, 2016, 2:47 pm
Location: Scotland
### Re: Massive spam attacks on the wiki (and forums?)
Saka wrote:
muzik wrote:viewforum.php?f=7
ARE YOU SH*TTING ME
I think forum accounts might need to be trusted as well if this happens again
How would that work? Maybe if their first post has been reported they get banned?
He posted way too fast
Bored of using the Moore neighbourhood for everything? Introducing the Range-2 von Neumann isotropic non-totalistic rulespace!
Posts: 730
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Contact:
### Re: Massive spam attacks on the wiki (and forums?)
Another bot found in the forums: http://conwaylife.com/forums/memberlist ... ile&u=1714
it should be removed && kickban(time=infty)
A for awesome
Posts: 2182
Joined: September 13th, 2014, 5:36 pm
Location: Pembina University, Home of the Gliders
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### Re: Massive spam attacks on the wiki (and forums?)
praosylen#5847 (Discord)
x₁=ηx
V*_η=c²√(Λη)
K=(Λu²)/2
Pₐ=1−1/(∫^∞_t₀(p(t)ˡ⁽ᵗ⁾)dt)
$$x_1=\eta x$$
$$V^*_\eta=c^2\sqrt{\Lambda\eta}$$
$$K=\frac{\Lambda u^2}2$$
$$P_a=1-\frac1{\int^\infty_{t_0}p(t)^{l(t)}dt}$$
Posts: 730
Joined: May 7th, 2016, 8:53 am
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### Re: Massive spam attacks on the wiki (and forums?)
Spam in 3rd party content (external image links):
http://conwaylife.com/forums/viewtopic. ... 417#p12732
Apple Bottom
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### Re: Massive spam attacks on the wiki (and forums?)
Nathaniel, could you be so kind and add User:Apple Bot to the "trusted" and "bot" usergroups (see https://www.mediawiki.org/wiki/Manual:Bots for more info on the latter? This account may be used to semi-automatically update glider synthesis data, as per my and dvgrn's ongoing discussion.
If you speak, your speech must be better than your silence would have been. — Arabian proverb
Catagolue: Apple Bottom • Life Wiki: Apple Bottom • Twitter: @_AppleBottom_
Proud member of the Pattern Raiders!
Nathaniel
Posts: 636
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### Re: Massive spam attacks on the wiki (and forums?)
Apple Bottom wrote:Nathaniel, could you be so kind and add User:Apple Bot to the "trusted" and "bot" usergroups (see https://www.mediawiki.org/wiki/Manual:Bots for more info on the latter? This account may be used to semi-automatically update glider synthesis data, as per my and dvgrn's ongoing discussion.
Done, sorry for the delay.
Mr. Missed Her
Posts: 90
Joined: December 7th, 2016, 12:27 pm
Location: Somewhere within [time in years since this was entered] light-years of you.
### Re: Massive spam attacks on the wiki (and forums?)
I have a wiki account under the same name as my forums account, Mr. Missed Her. May it be set to trusted?
Edit: Here's changes to the OTCA metapixel article that I'm not able to make at the moment.
Code: Select all
{{UnitCell
|name = OTCA metapixel
|pname = otcametapixel
|c = 64691
|bx = 2058
|by = 2058
|sx = 2048
|sy = 2048
|p = 35328
|discoverer = Brice Due
|discoveryear = 2006
|rle = true
}}
The '''OTCA metapixel''' is a 2048 × 2048 [[period]] 35328 [[unit cell]] that was constructed by [[Brice Due]] between the autumn of [[:Category:Patterns found in 2005|2005]] and the spring of [[:Category:Patterns found in 2006|2006]]. It has many advantages over the previous-known unit cells such as the [[p5760 unit Life cell]] and [[deep cell]], including the ability to emulate ''any'' [[Life-like cellular automaton]] and the fact that, when zoomed out, the ON and OFF [[cell]]s are easy to distinguish (the ON version of the cell is shown to the right and the OFF version of the cell is shown below).
It is designed to run quickly under the [[Hashlife]] algorithm, and thus [[Golly]] is generally used to view and/or manipulate meta-patterns made up of OTCA metapixels (and some such patterns even come packaged with Golly).
To tile these unit cells to emulate other patterns, place them so that the cornermost [[block]]s overlap; the unit cells will physically overlap by 5 [[cell]]s in every direction. The overlap will place [[tub]]s inside cross-corner neighbours.
==Details==
The metacell uses a period 184 [[tractor beam]], which acts as a clock. It pulls a block downwards by eight cells per impact, releasing a glider in the process. Some of the gliders are utilised; the rest are eaten. A new block is created from the third impact to be used when the timer restarts. Period 46 and 184 technologies (which are compatible) are used extensively throughout the configuration.
The rule is encoded in two columns, each of nine eaters, where one column corresponds to the 'Birth' rule and the other corresponds to 'Survival'. The nine eaters correspond to the nine different quantities of on cells (0 through 8). The presence or absence of the eater indicates whether the cell should be on in the next meta-generation. The state of the eater is read by the collision of two antiparallel LWSSes, which radiates two antiparallel gliders (not unlike an electron-positron reaction in a PET scanner). These gliders then collide into beehives, which are restored by a passing LWSS in Brice's elegant [[honeybit]] reaction. If the eater is present, the beehive would remain in its original state, thereby allowing the LWSS to pass unaffected; if the eater is absent, the beehive would be restored, consuming the LWSS in the process. Equivalently, the state of the eater is mapped onto the state of the LWSS.
‘On’ metacells send a MWSS counterclockwise around the cell, which reacts with twin bees to send gliders to a neighboring cell’s beehive in a honeybit reaction. A 9-LWSS stream then goes around the cell, losing a LWSS for each adjacent ‘on’ cell that triggered a honeybit reaction. The number of missing LWSSes is counted by detecting the position of the front LWSS by crashing another LWSS into it from the opposite direction. This collision releases gliders, which triggers another one or two honeybit reactions if the eaters that indicate that birth/survival condition are absent.
When the display is 'on', two perpendicular waves of [[lightweight spaceship|LWSSes]] collide, mutually annihilating each other. These streams of LWSSes are generated from an [[out of the blue]] reaction, triggered by passing [[heavyweight spaceship|HWSSes]].
==Image gallery==
{|
|-
|[[Image:otcametapixel_off.png|thumb|left|The OFF version of the OTCA metapixel.<br />{{JavaRLE|otcametapixeloff}}]]
|[[Image:otcametapixel_galaxy.png|thumb|left|The OTCA metapixel being used to simulate [[Kok's galaxy]]]]
|}
==Videos==
{|
|-
|{{#ev:youtube|-ogwfn3sqwI|300|left|The metapixel by itself turning from ON to OFF in 35,328 generations}}
|{{#ev:youtube|hsXCKPt8u3I|300|left|The OTCA metapixel being used to emulate Kok's galaxy}}
|}
*[http://b3s23life.blogspot.com/2006_09_01_archive.html Brice Due's Game-of-Life Metapixel] -- blog post by [[Dave Greene]]
*[http://otcametapixel.blogspot.com OTCAmetapixel] -- official website of the pattern
There is life on Mars. We put it there with not-completely-sterilized rovers.
And, for that matter, the Moon, Jupiter, Titan, and 67P/Churyumov–Gerasimenko.
eeveeta
Posts: 1
Joined: March 15th, 2017, 4:36 pm
### Re: Massive spam attacks on the wiki (and forums?)
May I be added as a trusted user? Account is eeveeta
Nathaniel
Posts: 636
Joined: December 10th, 2008, 3:48 pm
Contact:
### Re: Massive spam attacks on the wiki (and forums?)
Mr. Missed Her wrote:I have a wiki account under the same name as my forums account, Mr. Missed Her. May it be set to trusted?
Trusted now -- sorry for the delay.
eeveeta wrote:May I be added as a trusted user? Account is eeveeta
And you're now trusted too.
Bill Gosper
Posts: 1
Joined: August 18th, 2015, 1:15 am
### Re: Massive spam attacks on the wiki (and forums?)
Trust me to tweak the r pentomino LifeWiki? --rwg
Nathaniel
Posts: 636
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Contact:
### Re: Massive spam attacks on the wiki (and forums?)
Bill Gosper wrote:Trust me to tweak the r pentomino LifeWiki? --rwg
Of course! Create a LifeWiki account here and then I can mark it as trusted.
(I apologize if you already have a LifeWiki account, but I could not find it.)
Edit: And now that your account has been created, I've "trusted" it. Welcome to the site!
Posts: 2
Joined: March 28th, 2017, 9:00 pm
### Re: Massive spam attacks on the wiki (and forums?)
Hello!
I would like to join the lifewiki (the one that concerns cellular automata.)
This isn't particularly important for me joining, but I have discovered an infinite pattern that moves at the maximum speed of information, if you find that interesting. However, I believe that keeping it to myself does nothing; discoveries are only important if they are shared with others, not lost when I eventually cease to exist because "I didn't spread the word."
Please excuse this message if I accidentally replied to you, and not posted to the subject. Thank you!
P. S., Did I mention I would like to join the LifeWiki?
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2020-12-04 08:14:52
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http://zbmath.org/?q=an:1041.30009
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# zbMATH — the first resource for mathematics
##### Examples
Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.
##### Operators
a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On uniqueness of meromorphic functions with shared values in one angular domain. (English) Zbl 1041.30009
Let $f\left(z\right)$ and $g\left(z\right)$ be transcendental meromorphic functions in the complex plane. A value $a\in \stackrel{^}{ℂ}=ℂ\cup \left\{\infty \right\}$ is called an IM (ignoring multiplicities) shared value in $X\subset ℂ$ of $f\left(z\right)$ and $g\left(z\right)$ if in $X$, $f\left(z\right)=a$ if and only if $g\left(z\right)=a$. A value $a\in \stackrel{^}{ℂ}$ is called an CM (counting multiplicities) shared value in $X\subset ℂ$ of $f\left(z\right)$ and $g\left(z\right)$ if in $X$, $f\left(z\right)$ and $g\left(z\right)$ assume $a$ at the same points in $X$ with the same multiplicities. In the case $X=ℂ$, there are several sharing conditions for uniqueness, see e.g., G. G. Gundersen [J. Lond. Math. Soc., II. Ser. 20, 456–466 (1979; Zbl 0413.30025)]. For example, (C1) If $f\left(z\right)$ and $g\left(z\right)$ share five values IM, then $f\left(z\right)\equiv g\left(z\right)$, which is due to R. Nevanlinna. (C2) If $f\left(z\right)$ and $g\left(z\right)$ share four values IM, and if for another value $\delta \left(a,f\right)>0$, then $f\left(z\right)\equiv g\left(z\right)$. (C3) If $f\left(z\right)$ and $g\left(z\right)$ share two values IM and if $f\left(z\right)$ and $g\left(z\right)$ share two values CM, then $f\left(z\right)$ and $g\left(z\right)$ share four values of CM. The author considers the uniqueness problem with the sharing conditions in some angular domains.
In this paper three theorems are obtained generalizing the (C1), (C2) and (C3). We state one of the results. Given an angular domain $X=\left\{z:\alpha with $0\le \alpha <1\beta \le 2\pi$ and for some positive number $\epsilon$ and for some $a\in \stackrel{^}{ℂ}$, ${lim sup}_{r\to \infty }logn\left(r,{X}_{\epsilon },f=a\right)/logr>\omega$, where $n\left(r,{X}_{\epsilon },f=a\right)$ is the number of the roots of $f\left(z\right)=a$ in $\left\{|z|, ${X}_{\epsilon }=\left\{z:\alpha +\epsilon and $\omega =\pi /\left(\beta -\alpha \right)$. If $f\left(z\right)$ and $g\left(z\right)$ share five distinct values IM, then $f\left(z\right)\equiv g\left(z\right)$.
##### MSC:
30D35 Distribution of values (one complex variable); Nevanlinna theory
##### Keywords:
Nevanlinna theory; meromorphic function; shared value
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2014-03-11 16:19:31
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https://math.stackexchange.com/questions/1951094/prove-that-a-limsup-related-with-the-sum-of-divisor-function-equals-e-gamm
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# Prove that a $\limsup$ related with the sum of divisor function equals $e^{\gamma}$
I want to show that $$\limsup_{n \rightarrow \infty} \frac{\sigma(n)}{n \log \log n} = e^{\gamma}.$$ where $\sigma(n):=$ sum of positive divisor function and $\gamma$ is the Euler constant (the one appear in Merten's estimates $(3)$ https://en.wikipedia.org/wiki/Mertens%27_theorems)
I try to use Merten estimate since it appear $\gamma$ in the limit valaue. Since $$\sigma(p_1^{a_1}...p_k^{a_k})= \frac{\prod_{i=1}^k (p_i^{a_i} -1/p_i)}{\prod_{i=1}^k (1-1/p_i)}.$$ The reciprocal might be modified to use Merten reciprocal form $$\prod_{p \leq x; p\ \ prime} (1-1/p)^{-1} = e^{\gamma} \log x (1 + O(1/\log x)).$$ I guess that $\log \log$ thing, and $\log$ and error term in Merten should be somehow cancel, but really not sure why.
Let $N_x = \prod_{p < x} p$ the primorial. By the Mertens theorems we have $$\frac{\sigma(N_x)}{N_x} = \prod_{p < x} (1+\frac{1}{p} )= e^{\gamma+\ln \ln x+o(1)}$$ Now $\ln N_x = \sum_{p < x} \ln p$ so that by Chebyshev's work (*) $x/2 < \ln N_x <2x$ and $\ln x =\ln \ln N_x+ \mathcal{O}(1)$, $\ln \ln x =\ln \ln \ln N_x+ o(1)$ and we get $$\frac{\sigma(N_x)}{N_x} = e^{\gamma+\ln \ln \ln N_x+o(1)} = e^{\gamma+o(1)}\ln \ln N_x$$ and finish with $$\lim \sup_{n \to \infty} \frac{\sigma(n)}{n \ln \ln n} = \lim \sup_{x \to \infty} \frac{\sigma(N_x)}{N_x \ln \ln N_x} = e^{\gamma}$$
The Mertens theorem we need is proved there, but I can try to make it shorter :
Let $\Lambda(p^k) = \ln p$ if $p$ is prime, $\Lambda(n) = 0$ otherwise. By a simple sieving/combinatoric we have $x!= \prod_{p^k \le x} p^{ \lfloor x /p^k \rfloor}$ so that $$\sum_{n \le x} \lfloor x /n\rfloor \Lambda(n) = \sum_{p^k \le x} \lfloor x /p^k \rfloor \ln p = \ln x!= x \ln x+ \mathcal{O}(x)$$ by Stirling's approximation. Hence $$\sum_{n \le x} \frac{\Lambda(n)}{n} = \frac{1}{x} \sum_{n \le x} \Lambda(n)(\lfloor x /n\rfloor+\mathcal{O}(1)) = \ln x+ \mathcal{O}(1)$$ where we used that $\sum_{n \le x}\Lambda(n) = \mathcal{O}(x)$
Finally with $\ln x = \mathcal{O}(1) +\sum_{n \le x} \frac{1}{n}$ we get
$$\sum_{n < x} \frac{\Lambda(n)-1}{n} = \mathcal{O}(1)$$ Summing by parts (mistakes here) $$\mathcal{O}(1) = \sum_{n =2}^x \frac{\Lambda(n)-1}{n \ln n}\ln n = \ln x\sum_{n \le x} \frac{\Lambda(n)-1}{n \ln n}+\sum_{2 \le k < x}(\ln k-\ln (k+1))\sum_{2\le n < k} \frac{\Lambda(n)-1}{n \ln n} = \ln x\sum_{n \le x} \frac{\Lambda(n)-1}{n \ln n} + \sum_{2 \le k < x} \mathcal{O}(1/k)$$ so that $\sum_{n \le x} \frac{\Lambda(n)-1}{n \ln n} = \mathcal{O}(1/\ln n)$ so we can refine $\sum_{2 \le k < x} \mathcal{O}(1/k)$ to $\sum_{2 \le k < x} \mathcal{O}(\frac{1}{k \ln k}) = \mathcal{O}(\ln \ln x)$ and get $$\sum_{n \le x} \frac{\Lambda(n)-1}{n \ln n} = o(1)$$ Finally $\sum_{n \le x} \frac{\Lambda(n)}{n \ln n} = o(1)-\sum_{p \le x} \ln(1-\frac{1}{p})$ and $$\sum_{p \le x}\ln(1+\frac{1}{p}) = o(1)+\sum_{p \le x} (\ln(1-\frac{1}{p^2})-\ln(1-\frac{1}{p})) = - \ln \zeta(2)+o(1)+\sum_{n \le x} \frac{\Lambda(n)}{n \ln n} =- \ln \zeta(2)+o(1)+\sum_{2 \le n \le x} \frac{1}{n \ln n} =\gamma+o(1)+\ln \ln x$$ where I used $\sum_{2 \le n \le x} \frac{1}{n \ln n} = \ln \ln(x)+\gamma-\zeta(2)+o(1)$, no reference for the $\gamma-\zeta(2)$ constant
In order to exploit Mertens' theorem, you just have to show that $\frac{\sigma(n)}{n}$ is maximized when $n$ is a primorial. By the PNT, $\sum_{p\leq x}\log p = x+o(x)$, hence if we take $n$ as $\prod_{p\leq x}p$, we have
$$\frac{\sigma(n)}{n\log\log n}\approx \frac{1}{\log x}\prod_{p\leq x}\left(1+\frac{1}{p}\right)\approx\frac{1}{\log x}\prod_{p\leq x}\left(1-\frac{1}{p}\right)^{-1}.$$
• See there you also need a few more steps, but the PNT is enough (though not necessary) – reuns Oct 2 '16 at 21:03
• @user1952009: I agree your answer is much more accurate than mine. I just wanted to give the OP the main ideas here, that is the reason for using $\approx$ instead of more accurate (asymptotic) bounds. – Jack D'Aurizio Oct 2 '16 at 21:07
• I gave the link because it shows where $\gamma$ comes from (without details) – reuns Oct 2 '16 at 21:09
• @user1952009: and that is a fine idea. Thanks. – Jack D'Aurizio Oct 2 '16 at 21:11
• @JackD'Aurizio if you are interested, I posted a sketch-proof that should work without the PNT – reuns Oct 2 '16 at 22:07
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2019-08-23 07:23:53
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https://math.stackexchange.com/questions/377496/why-matrix-representation-of-convolution-cannot-explain-the-convolution-theorem
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Why matrix representation of convolution cannot explain the convolution theorem?
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has disappeared from Wikipedia.
The Convolution Theorem states that convolution of functions, h(t) and x(t), in time domain is equivalent to their multiplication in the frequency domain. That is, you convolve them, h*x, and take result into frequency domain. Result F(h*x) must be the same as multiplying their Fourier images, H and X: $F(h*x) = H \cdot X$, where H and X are fourier images, $H = F(h)$ and $X = F(x)$.
This was the definition where I used the letters h and x for the functions, instead of conventional f and g, because convolution h*x is often represented by h(x), where h is a convolution matrix derived from the first function, h, which is applied to the second function, x. Being Toeplitz, matrix h has eigenbasis F -- the same as apply to vector to take it into Fourier domain (see change of basis to see why base matrix product for base transform). Therefore, matrix h, translated into its eigenbasis, happens to be diagonal and H. That is, according to the Convolution Theorem, we must prove that
$$F(h \vec x) = H \vec X$$
But, there is nothing to prove. We just can show that
$$H \vec X = (FhF^{-1})(F\vec x) = F(F^{-1}F)h(F^{-1}F)\vec x = F(h \vec x)$$
or, the other way around
$$F(h \vec x) = F(F^{-1}F)h(F^{-1}F)\vec x = (FhF^{-1})(F\vec x) = H \vec X$$
just to exercise the beautiful algebra of relationships and diagonalization in F. We just need to keep in mind that $H = FhF^{-1}$ is diagonal (multiplication operator) because F is eigenbasis of h.
This discussion was classified as nonsense. But why? I find it amazing that Toeplitz (or convolution) has Fourier eigenbasis. Should this precious fact be hidden from the general population? Why should general population appreciate the integral-based proof rather than enjoy this fact? Can I say that Toeplitz operator = Convolution (operator)? Can I say that operator is a (continuous-case) generalization of matrix?
Is convolution theorem related with the Fourier eigenbasis? Can it be explained simpler, based on this fact?
• For future reference: It's spelled eigen. No h. – kahen Apr 30 '13 at 19:19
• Why $H = Fh$ and also = $FhF^{-1}$? – user10024395 Nov 9 '17 at 12:58
• This is actually exactly what I was looking for when first reading about convolution and Fourier/Laplace transforms - thanks! – MGwynne Mar 24 '18 at 15:06
• I also found math.stackexchange.com/questions/918345/… useful as well, particularly the linked blog. – MGwynne Mar 24 '18 at 15:33
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2019-02-16 15:50:45
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http://mathhelpforum.com/trigonometry/106627-general-solution.html
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# Math Help - general solution
1. ## general solution
Gievn that tan A =3 and sin (A-B)=2 cos (A+B) , find tan B
i got tan B=-1/5
Find general solution , in rad cos b + cos 3b +cos 5b =0
cos b + 2 cos 4b cos b =0
cos b (1+2cos 4b )=0
cos b =0
and cos b = -1/2
I can only reach this far , i am not sure about the general solution >
Deduce $\cos^2 b+\cos^2 3b+\cos^2 5b=\frac{3}{2}$
no idea bout this .
2. Hello, thereddevils!
$\text{Given: }\:\tan A = 3\:\text{ and }\:\sin(A-B)\:=\:2\cos (A+B),\;\text{ find }\tan B$
We are given: . $\sin(A - B) \;=\;2\cos(A + B)$
Then: . $\sin A\cos B - \sin B\cos A \;=\;2(\cos A\cos B - \sin A\sin B)$
. . . . . $\sin A\cos B - \sin B\cos A \;=\;2\cos A\cos B - \sin A\sin B$
Divide by $\cos A\!:\quad \frac{\sin A\cos B}{\cos A} - \frac{\sin B\cos A}{\cos A} \;=\;\frac{2\cos A\cos B}{\cos A} - \frac{\sin A\sin B}{\cos A}$
. . . . . . . . . . . . . . . . $\underbrace{\tan A}_3\cos B - \sin B \;=\;2\cos B - 2\underbrace{\tan A}_3\sin B$
And we have: . $3\cos B - \sin B \;=\;2\cos B - 6\sin B$
. . . . . . . . . . . . . . . $5\sin B \;=\;-\cos B$
. . . . . . . . . . . . . . . . $\frac{\sin B}{\cos B} \;=\;-\frac{1}{5}$
. . . . . . . . . . . . . . . . $\boxed{\tan B \;=\;-\frac{1}{5}}$
3. Hello again, thereddevils!
Find the general solution in radians: . $\cos x + \cos3x +\cos5x \:=\:0$
$\cos x + 2\cos4x\cos x \:=\:0$
$\cos x (1+2\cos4x) \:=\:0$ . . . . Correct!
We have: . $\cos x \:=\:0 \quad\Rightarrow\quad \boxed{ x \:=\:\frac{\pi}{2} + \pi n}$
And: . $1+2\cos4x\:=\:0 \quad\Rightarrow\quad \cos4x \:=\:-\tfrac{1}{2}$
. . . $4x \:=\:\pm\frac{2\pi}{3} + 2\pi n \quad\Rightarrow\quad\boxed{ x \:=\:\pm\frac{\pi}{6} + \frac{\pi}{2}n }$
4. Hello thereddevils
Originally Posted by thereddevils
Deduce $\cos^2 b+\cos^2 3b+\cos^2 5b=\frac{3}{2}$
I think we have a problem here if this result is supposed to apply to all the solutions of the equation $\cos b+\cos 3b+\cos 5b=0$.
Look at the solution $b = \frac{\pi}{2}+n\pi = (2n+1)\frac{\pi}{2}$. In other words, $b$ is any odd multiple of $\frac{\pi}{2}$.
If $b$ is an odd multiple of $\frac{\pi}{2}$, then so are $3b$ and $5b$, and at these values $\cos b = \cos 3b = \cos 5b = 0$.
So, obviously, $\cos^2 b+\cos^2 3b+\cos^2 5b=0$ also.
However, when we consider the other set of values of $b$ (those for which $\cos 4b = -\tfrac12$) we do get the required result. Here's why:
$\cos^2 b+\cos^2 3b+\cos^2 5b=\tfrac12(1+\cos2b)+\tfrac12(1+\cos6b)+\tfrac12( 1+\cos10b)$
$=\tfrac32+\tfrac12(\cos2b+\cos6b+\cos10b)$
$=\tfrac32+\tfrac12(2\cos6b\cos4b+\cos6b)$
$=\tfrac32+\tfrac12\cos6b(2\cos4b+1)$
$=\tfrac32$, when $\cos4b = -\tfrac12$
So: are you sure you posted the complete question?
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2015-03-31 21:07:41
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https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-integral-calculus
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Mathematical Treasure: Leibniz's Papers on Calculus - Integral Calculus
Author(s):
Frank J. Swetz (The Pennsylvania State University)
Above is the title page of the 1686 volume of Acta Eruditorum.
This is the first page of the June 1686 issue (Number VI) of Acta Eruditorum, in which Leibniz published a second article describing the Calculus on pages 292-300.
In the June 1686 issue of Acta Eruditorum, Leibniz (G.G.L.) published “De geometria recondita et analysi indivisibilium atque infinitorum,” or "On a hidden geometry and analysis of indivisibles and infinites." In this article we find the first public occurrence of the integral sign $\int$ and a proof of “The Fundamental Theorem of Calculus.” A partial translation from Latin to English of the article can be found in D. J. Struik's A Source Book in Mathematics (1200-1800), pp. 281-282. The remaining pages of the original article appear below.
On page 297 above, Leibniz pointed out that $p\,dy=x\,dx$ implies ${\int{p}}\,dy={\int{x}}\,dx$, and therefore, in particular, $d\left({\frac{1}{2}}xx\right)=x\,dx$ implies ${\frac{1}{2}}xx={\int{x}}\,dx.$ He then wrote, "... sums and differences or ${\int}$ and $d,$ are reciprocals" ("summae & differentiae seu ${\int}$ & $d,$ reciprocae sunt"), and concluded from his preceding equations that ${\int{p}}\,dy={\frac{1}{2}}xx.$
The images above are used through the courtesy of the Lilly Library, Indiana University, Bloomington, Indiana. You may use them in your classroom; for all other purposes, please seek permission from the Lilly Library.
Reference
D. J. Struik (editor), A Source Book in Mathematics (1200-1800), Harvard University Press, Cambridge, Mass., 1969.
Index to Mathematical Treasures
Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: Leibniz's Papers on Calculus - Integral Calculus," Convergence (June 2015)
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2021-08-01 19:13:47
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https://leanprover-community.github.io/mathlib_docs/category_theory/limits/small_complete.html
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# mathlibdocumentation
category_theory.limits.small_complete
# Any small complete category is a preorder
We show that any small category which has all (small) limits is a preorder: In particular, we show that if a small category C in universe u has products of size u, then for any X Y : C there is at most one morphism X ⟶ Y. Note that in Lean, a preorder category is strictly one where the morphisms are in Prop, so we instead show that the homsets are subsingleton.
## References
• https://ncatlab.org/nlab/show/complete+small+category#in_classical_logic
## Tags
small complete, preorder, Freyd
@[instance]
def category_theory.has_hom.hom.subsingleton {C : Type u} {X Y : C} :
A small category with products is a thin category.
in Lean, a preorder category is one where the morphisms are in Prop, which is weaker than the usual notion of a preorder/thin category which says that each homset is subsingleton; we show the latter rather than providing a preorder C instance.
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2021-01-17 07:41:10
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https://telearn.archives-ouvertes.fr/hal-00190642
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# Collaboration Scripts – A Conceptual Analysis
Abstract : This article presents a conceptual analysis of collaboration scripts used in face-to-face and computer-mediated collaborative learning. Collaboration scripts are scaffolds that aim to improve collaboration through structuring the interactive processes between two or more learning partners. Collaboration scripts consist of at least five components: (a) learning objectives, (b) type of activities, (c) sequencing, (d) role distribution, and (e) type of representation. These components serve as a basis for comparing prototypical collaboration script approaches for face-to-face vs. computer-mediated learning. As our analysis reveals, collaboration scripts for face-to-face learning often focus on supporting collaborators to engage in activities that are specifically related to individual knowledge acquisition. Scripts for computer-mediated collaboration are typically concerned with facilitating communicative-coordinative processes that occur among group members. The two research lines can be consolidated to facilitate the design of collaboration scripts which both support participation and coordination and induce learning activities closely related to individual knowledge acquisition and metacognition. However, research on collaboration scripts needs to consider the learners' internal collaboration scripts as a further determinant of collaboration behavior. The article closes with the presentation of a conceptual framework incorporating both external and internal collaboration scripts.
Keywords :
Type de document :
Article dans une revue
Educational Psychology Review, Springer Verlag, 2006, 18(2), pp.159-185. 〈10.1007/s10648-006-9007-2〉
Littérature citée [61 références]
https://telearn.archives-ouvertes.fr/hal-00190642
Contributeur : Jerome Zeiliger <>
Soumis le : vendredi 23 novembre 2007 - 08:54:33
Dernière modification le : mardi 6 février 2018 - 14:24:01
Document(s) archivé(s) le : lundi 12 avril 2010 - 04:35:30
### Fichier
Kollar-Ingo-2006.pdf
Fichiers produits par l'(les) auteur(s)
### Citation
Ingo Kollar, Frank Fischer, Friedrich Hesse. Collaboration Scripts – A Conceptual Analysis. Educational Psychology Review, Springer Verlag, 2006, 18(2), pp.159-185. 〈10.1007/s10648-006-9007-2〉. 〈hal-00190642〉
### Métriques
Consultations de la notice
## 324
Téléchargements de fichiers
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2018-10-21 09:02:21
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https://homework.cpm.org/category/CON_FOUND/textbook/caac/chapter/7/lesson/7.3.4/problem/7-116
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### Home > CAAC > Chapter 7 > Lesson 7.3.4 > Problem7-116
7-116.
Copy and complete each of the Diamond Problems below. The pattern used in the Diamond Problems is shown at right.
• What plus $−3$ equals $8$?
What does $−3$ multiplied by that number equal?
• What are the factors of $−7$? Which two added together equal $6$?
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2021-02-24 17:58:53
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https://astronomy.stackexchange.com/questions/41343/how-can-astrophysicists-discriminate-between-pp-chain-solar-neutrinos-and-cno-cy/41346#41346
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# How can astrophysicists discriminate between pp-chain solar neutrinos and CNO-cycle ones?
Astrophysicists at the Borexino experiment in Italy have recently claimed that they have detected CNO-cycle neutrinos coming from the Sun.
It was the Cover story for the November 26 issue of Nature.
I read the whole Paper, and accompanying editorial, yet unless I missed something, they do not say what makes a CNO-neutrino different from an ordinary pp-neutrino.
They go to great lengths to describe how they excluded all types of background noise, but did not say whether CNO-neutrinos are higher or lower in energy than 'proton-proton chain' neutrinos....
Presumably, CNO neutrinos are higher in energy than pp ones? How many eV or keV?
The dominant CNO neutrino producers are the steps involving $$^{13}\mathrm{N}$$ and $$^{15}\mathrm{O}$$. The peak of the nitrogen spectrum lies in (or at the edge of) a region where the ratio of CNO neutrinos to background is the highest:
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2022-01-24 04:00:17
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http://mathoverflow.net/revisions/78873/list
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# A model of CH +$\lnot \diamondsuit$
All of the models of CH which I know of also satisfy $\diamondsuit$. What is the easiest way to produce a model of CH wherein $\diamondsuit$ is false?
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2013-05-18 14:34:00
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https://notesformsc.org/introduction-to-determinants/
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# Introduction To Determinants
Determinants are very important concept related to square matrix, and usually it is simple to calculate one while dealing with the system of linear equations. Our attempt here is to understand determinants properly so that all related concepts becomes easy and unforgettable. So we begin with a gentle introduction to determinants of matrices.
### What are Determinants ?
Imagine there are no matrices and you must solve the system of linear equation with simple algebra. For example,
Consider the following system of linear equation with just one equation.
$a_{11}x_1 = b_1$
To solve this equation using algebra method divide both sides by constant $a_{11}$
$x_1 = \frac{\strut b_1}{\strut a_{11}}$
The element $a_{11}$ is the determinant in the equation as a denominator. Provided $a_{11} \neq 0$
Otherwise, the system has no solution.
A determinant is zero does not always means that the system of linear equation has no solution, sometimes it has infinite solutions. But, if determinant is 0, then the matrix is not invertible.
Let us take another example, where there are two linear system of equations with two unknowns.
$a_{11}x_1 + a_{12}x_2 = b_1$
$a_{21}x_1 + a_{22}x_2 = b_2$
When we solve for $x_1$ the equations become
$x_1 = (b_1 - a_{12}x_2)/a_{11}$
$x_1 = (b_2 - a_{22}x_2)/a_{21}$
Comparing both equations we get
$x_1 = a_{21}b_1 - a_{12}a_{21} = a_{11}b_2 - a_{11}a_{22}$
$= a_{21}b_1 - a_{11}b_2 = a_{12}a_{21} - a_{11}a_{22}$
$= (a_{21}b_1 - a_{11}b_2)/(a_{12}a_{21} - a_{11}a_{22})$
The denominator should be $a_{21}a_{13} - a_{11}a_{22} \neq 0$, else the equation will collapse.
### Generalizing the Denominator
From the above result, there is a pattern to the denominator and if you solve for $x_2$ then you will find out that the denominator is not changing and it remain the same. Hence, the denominator is determinant of the equation given that it is not equal to zero.
For a system of two equations and two unknowns , the determinant is the following.
$a_{12}a_{21} - a_{11}a_{22}$
Observe that in any term of determinant, the $i$ corresponds to sequence of numbers $i = 1,2,3,...,n$ and all $j$ in the term is a permutation of the sequence $1,2,3,...,n$.
Therefore, we could generalize determinant as
$= \sum \pm a_{1\alpha}a_{2\beta}...a_{nv}$
where $i = 1,2,3,...,n$ determines the number of variables $a_{ij}$ in a term.
### Problem Of Negative And Positive Terms
Though we were able to generalize any term of the determinant with $n$ variables in each term. Some terms are positive and some are negative. The next question is to determine which are positive and which are negative terms where any single terms is given by the following.
$= \sum \pm a_{1\alpha}a_{2\beta}...a_{nv}$
Before that you need to be familiar with following concepts.
#### Inversion and Transposition
In the determinant term, $i = 1,2,3, ...,n$ in natural order and all $j$ are a permutation of the natural order (1,2,3,…,n).
Given a permutation $\alpha,\beta,\gamma,...,v$ if two indices are out of their natural order and greater index comes first and then lesser index, then it is called inversion.
Suppose that the natural order is $latex(12345)&s=1$ and the permutation is $latex(21345)&s=1$, here $2$ and $1$ is inversion as they are out of the natural order.
Suppose that the natural order is disturbed by swapping two numbers. If $latex(12543)&s=1$ is the permutation then only $3$ and $5$ is swapped. This is called transposition.
The number of inversion is unique and can be counted. If the inversions are even then permutation $\alpha,\beta,\gamma,…,v$ belongs to even class, otherwise, odd class.
If a term belongs to odd class, then assign negative sign, else assign a positive sign if it belongs to even class.
### Determinant of 3 x 3 Equations
Using the information above let us systematically find determinant of $3 \times 3$ equations. Since, $n = 3$ there will be $n! = 3! = 6$ terms with each term having 3 variables that is, $a_{ij}$.
Write down each permutation and the term of determinant.
$a_{11}a_{22}a_{33}$ //permutation of j is (1 2 3)
$a_{11}a_{23}a_{32}$ //permutation of j is (1 3 2)
$a_{12}a_{21}a_{33}$ //permutation of j is (2 1 3)
$a_{12}a_{23}a_{31}$ //permutation of j is (2 3 1)
$a_{13}a_{21}a_{32}$ //permutation of j is (3 1 2)
$a_{13}a_{22}a_{31}$ //permutation of j is (3 2 1)
We get following
$latex = a_{11}a_{22}a_{33}+a_{11}a_{23}a_{32}+a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}a_{13}a_{21}a_{32}+a_{13}a_{22}a_{31}&s=1$
We must look at the permutation and change the sign of terms that belong to odd class meaning permutations that have odd number of inversions.
$= a_{11}a_{22}a_{33}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}+a_{12}a_{23}a_{31}a_{13}a_{21}a_{32}-a_{13}a_{22}a_{31}$
This is a faster method of finding determinants, but you must be careful in checking the number of inversions. Next article, we shall discuss about finding determinants by cross multiplications and some interesting properties of determinants.
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2022-07-04 11:53:57
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https://www.shaalaa.com/question-bank-solutions/in-a-double-displacement-reaction-such-as-the-reaction-between-sodium-sulphate-solution-and-barium-chloride-solution-a-exchange-of-atoms-takes-place-b-exchange-of-ions-takes-place-types-of-chemical-reactions-double-displacement-reaction_101720
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In a Double Displacement Reaction Such as the Reaction Between Sodium Sulphate Solution and Barium Chloride Solution: (A) Exchange of Atoms Takes Place (B) Exchange of Ions Takes Place - Science
Question
MCQ
Choose the correct option from given alternative:
In a double displacement reaction such as the reaction between sodium sulphate solution and barium chloride solution:
(A) exchange of atoms takes place
(B) exchange of ions takes place
(C) a precipitate is produced
(D) an insoluble salt is produced
Options
• (B) and (D)
• (A) and (C)
• only (B)
• (B), (C) and (D)
Solution
$\ce{\underset{\text{Sodium sulphate}}{Na2SO4(aq)}+\underset{\text{Barium Chloride}}{BaCl2(aq)}->\underset{\text{Barium Sulphate}}{BaSO4(s)}+\underset{\text{Sodium Chloride}}{2NaCl(aq)}}$
The white precipitate of BaSO4 is formed by the reaction of "SO"_4^(2-) and Ba2+.
Hence, the correct answer is the (B), (C), and (D).
Is there an error in this question or solution?
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2021-01-24 08:38:22
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https://www.physicsforums.com/threads/optical-isomerism.128453/
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# Optical isomerism
1. Aug 9, 2006
### broegger
Hi. I'm asked the following question:
What isomers does the complex $$[\text{Fe}(\text{H}_2\text{O})_6]$$ give rise to? Is it optically active?
My answer to the first question would be 'none': The 6 identical H2O molecules are arranged in an octahedral fashion around the central Fe-atom, so there are no asymmetries that could give rise to isomers. Is this correct? It seems to me that this question is phrased as if some isomers do exist (I'm not asked whether they exist or not).
The answer to the second question would of course be 'no', since there are also no optical isomerism.
Am I missing something?
Last edited: Aug 9, 2006
2. Aug 9, 2006
### Gokul43201
Staff Emeritus
Your argument is insufficient. You could use essentially use the same argument to "prove" the optical inactivity of [Co(en)_3]^3+, but that would be wrong...wouldn't it?
While I'd still be inclined to agree with your result, I have never happened upon the bonding/geometry of aquo complexes myself, to say anything definitive. If there is some kind of hydrogen bonding between neighboring H2O ligands (which I think is unlikely), that might induce optical activity.
3. Aug 9, 2006
### broegger
I see your point. If we assume that every H2O molecule occupies one ligand space and does not interact with it's neighbors -- would I then be correct? It's an introductory course, so I assume there's no pitfalls.
4. Aug 9, 2006
### Gokul43201
Staff Emeritus
I'm inclined to say yes, but I'd rather someone who's formally trained in this area weigh in. There may just happen to be some standard result (or exception to the rule) which is completely unobvious to deduce.
Do you have a link to a website talking about bonding/geometry in hexaquo complexes?
5. Aug 9, 2006
### broegger
Last edited by a moderator: Apr 22, 2017
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2017-08-21 16:33:46
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https://testbook.com/question-answer/a-man-deposited-rs-3000-in-a-bank-and-rs-2500--60755c0303d4ee587795b8ef
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# A man deposited Rs. 3,000 in a Bank and Rs. 2,500 in a Post Office. Rate of interest of Bank is 1/2% more than that of Post Office. If he gets Rs. 235 as total interest at the end of the year, the rate of interest of Post Office is
This question was previously asked in
WBCS Prelims 2017 Official Paper
View all WBCS Papers >
1. 2%
2. $$2\frac{1}{2}\%$$
3. 4%
4. 4.5%
Option 3 : 4%
## Detailed Solution
Given:
A man deposited Rs. 3,000 in a Bank and Rs. 2,500 in a Post Office.
Rate of interest of Bank is 1/2% more than that of Post Office. If he gets Rs. 235 as total interest at the end of the year
Formula used:
Simple interest = Principal(P) × Rate(R) × Time(T) /100
Calculation:
Let the rate of post office be R
According to the question,
⇒ (3000 × (R + 0.5) × 1)/100 + (2500 × R × 1)/100 = 235
⇒ 5500R + 1500 = 23500
⇒ 5500R = 22000
⇒ R = 22000/5500
⇒ R = 4%
∴ The rate of interest of post office 4%.
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2021-09-27 03:07:11
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https://toolslick.com/text/url/component-parser
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# URL Component Parser
###### Updated: May 15, 2018
URL Component Parser parses and shows the individual components of a URL
Scheme:
..
Protocol:
..
Host:
..
Domain:
..
Sub Domain:
..
Hostname:
..
Registrable Domain:
..
TLD:
..
Host Name Type:
..
Authority:
..
Port:
..
Is Default Port:
..
Resource:
..
Local Path:
..
Directory:
..
File:
..
File Extension:
..
Query:
..
Fragment:
..
Background Information
A URL is a reference to a resource on the web.
It has the following form:- URI = scheme:[//authority]path[?query][#fragment]
It has several components:-
Scheme
Determines the protocol of the URL.
Examples are: HTTP, HTTPS, FTP, FILE
Authority
Determines the user, host & port of the URL. It has the following form:-
[userinfo@]host[:port]
Path
Determines a sequence of path segments which are separated by slash (/)
Query
An optional query component that is preceded by a question mark (?) It contains several items in key-value pair separted by the delimiter (&) The key & value are themselves separated by (=)
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2019-05-23 02:50:20
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{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3316131830215454, "perplexity": 8357.179141675015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232257002.33/warc/CC-MAIN-20190523023545-20190523045545-00549.warc.gz"}
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https://web2.0calc.com/questions/a-square-with-sides-of-12-units-is-inscribed-in-a
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+0
# A square with sides of 12 units is inscribed in a circle. What is the value of K if the area of the circle is Kpi square units?
0
327
1
+644
A square with sides of 12 units is inscribed in a circle. What is the value of K if the area of the circle is Kpi square units?
Oct 14, 2017
#1
+27470
+1
The diagonal of the square, and hence the diameter of the circle, will be $$\sqrt2\times12$$
The area of the circle will therefore be: $$\pi\times(\sqrt2\times12)^2/4$$
Equate this to $$k\times\pi$$ and rearrange to get k
Oct 15, 2017
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2019-02-16 08:31:34
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https://codefreshers.com/and-mex-walk-solution-codeforces/
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[Solution] AND-MEX Walk solution codeforces
AND-MEX Walk solution codeforces – There is an undirected, connected graph with 𝑛n vertices and 𝑚m weighted edges. A walk from vertex 𝑢u to vertex 𝑣v is defined as a sequence of vertices 𝑝1,𝑝2,,𝑝𝑘p1,p2,…,pk (which are not necessarily distinct) starting with 𝑢u and ending with 𝑣v, such that 𝑝𝑖pi and 𝑝𝑖+1pi+1 are connected by an edge for 1𝑖<𝑘1≤i<k.
[Solution] AND-MEX Walk solution codeforces
We define the length of a walk as follows: take the ordered sequence of edges and write down the weights on each of them in an array. Now, write down the bitwise AND of every nonempty prefix of this array. The length of the walk is the MEX of all these values.
More formally, let us have [𝑤1,𝑤2,,𝑤𝑘1][w1,w2,…,wk−1] where 𝑤𝑖wi is the weight of the edge between 𝑝𝑖pi and 𝑝𝑖+1pi+1. Then the length of the walk is given by MEX({𝑤1,𝑤1&𝑤2,,𝑤1&𝑤2&&𝑤𝑘1})MEX({w1,w1&w2,…,w1&w2&…&wk−1}), where && denotes the bitwise AND operation.
Now you must process 𝑞q queries of the form u v. For each query, find the minimum possible length of a walk from 𝑢u to 𝑣v.
The MEX (minimum excluded) of a set is the smallest non-negative integer that does not belong to the set. For instance:
• The MEX of {2,1}{2,1} is 00, because 00 does not belong to the set.
• The MEX of {3,1,0}{3,1,0} is 22, because 00 and 11 belong to the set, but 22 does not.
• The MEX of {0,3,1,2}{0,3,1,2} is 44 because 001122 and 33 belong to the set, but 44 does not.
[Solution] AND-MEX Walk solution codeforces
The first line contains two integers 𝑛n and 𝑚m (2𝑛1052≤n≤105𝑛1𝑚min(𝑛(𝑛1)2,105)n−1≤m≤min(n(n−1)2,105)).
Each of the next 𝑚m lines contains three integers 𝑎a𝑏b, and 𝑤w (1𝑎,𝑏𝑛1≤a,b≤n𝑎𝑏a≠b0𝑤<2300≤w<230) indicating an undirected edge between vertex 𝑎a and vertex 𝑏b with weight 𝑤w. The input will not contain self-loops or duplicate edges, and the provided graph will be connected.
The next line contains a single integer 𝑞q (1𝑞1051≤q≤105).
Each of the next 𝑞q lines contains two integers 𝑢u and 𝑣v (1𝑢,𝑣𝑛1≤u,v≤n𝑢𝑣u≠v), the description of each query.
Output
For each query, print one line containing a single integer — the answer to the query.
Examples
input
Copy
6 7
1 2 1
2 3 3
3 1 5
4 5 2
5 6 4
6 4 6
3 4 1
3
1 5
1 2
5 3
[Solution] AND-MEX Walk solution codeforces
output
Copy
2
0
1
input
Copy
9 8
1 2 5
2 3 11
3 4 10
3 5 10
5 6 2
5 7 1
7 8 5
7 9 5
10
5 7
2 5
7 1
6 4
5 2
7 6
4 1
6 2
4 7
2 8
output
Copy
0
0
2
0
0
2
1
0
1
1
AND-MEX Walk solution codeforces
The following is an explanation of the first example.
The graph in the first example.
Here is one possible walk for the first query:
1533211531425.1→53→32→11→53→14→25.
The array of weights is 𝑤=[5,3,1,5,1,2]w=[5,3,1,5,1,2]. Now if we take the bitwise AND of every prefix of this array, we get the set {5,1,0}{5,1,0}. The MEX of this set is 22. We cannot get a walk with a smaller length (as defined in the statement).
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2022-05-19 20:52:26
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https://deepai.org/publication/probabilistic-refinement-of-the-asymptotic-spectrum-of-graphs
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# Probabilistic refinement of the asymptotic spectrum of graphs
The asymptotic spectrum of graphs, introduced by Zuiddam (arXiv:1807.00169, 2018), is the space of graph parameters that are additive under disjoint union, multiplicative under the strong product, normalized and monotone under homomorphisms between the complements. He used it to obtain a dual characterization of the Shannon capacity of graphs as the minimum of the evaluation function over the asymptotic spectrum and noted that several known upper bounds, including the Lovász number and the fractional Haemers bounds are in fact elements of the asymptotic spectrum (spectral points). We show that every spectral point admits a probabilistic refinement and characterize the functions arising in this way. This reveals that the asymptotic spectrum can be parameterized with a convex set and the evaluation function at every graph is logarithmically convex. One consequence is that for any incomparable pair of spectral points f and g there exists a third one h and a graph G such that h(G)<{f(G),g(G)}, thus h gives a better upper bound on the Shannon capacity of G. In addition, we show that the (logarithmic) probabilistic refinement of a spectral point on a fixed graph is the entropy function associated with a convex corner.
There are no comments yet.
## Authors
• 9 publications
06/30/2018
### The asymptotic spectrum of graphs and the Shannon capacity
We introduce the asymptotic spectrum of graphs and apply the theory of a...
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### On a tracial version of Haemers bound
We extend upper bounds on the quantum independence number and the quantu...
10/01/2018
### Quantum asymptotic spectra of graphs and non-commutative graphs, and quantum Shannon capacities
We study several quantum versions of the Shannon capacity of graphs and ...
11/03/2019
### Shannon capacity and the categorical product
Shannon OR-capacity C_ OR(G) of a graph G, that is the traditionally mor...
12/14/2020
### Complexes, Graphs, Homotopy, Products and Shannon Capacity
A finite abstract simplicial complex G defines the Barycentric refinemen...
10/16/2012
### Tightening Fractional Covering Upper Bounds on the Partition Function for High-Order Region Graphs
In this paper we present a new approach for tightening upper bounds on t...
03/22/2021
### Thomson's Multitaper Method Revisited
Thomson's multitaper method estimates the power spectrum of a signal fro...
##### This week in AI
Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.
## 1 Introduction
The Shannon capacity of a graph is [Sha56]
Θ(G)=limn→∞n√α(G⊠n), (1)
where denotes the independence number and is the th strong power (see Section 2 for definitions). In the context of information theory, the optimal rate of zero-error communication over a noisy classical channel is equal to the Shannon capacity of its confusability graph.
In [Zui18] Zuiddam introduced the asymptotic spectrum of graphs as follows. Let denote the set of isomorphism classes of finite undirected simple graphs. The asymptotic spectrum of graphs is the set of functions which satisfy for all
1. [(S1)]
2. (additive under disjoint union)
3. (multiplicative under the strong product)
4. if there is a homomorphism between the complements then
5. .
Elements of are also called spectral points. Using the theory of asymptotic spectra, developed by Strassen in [Str88], he found the following characterization of the Shannon capacity:
Θ(G)=minf∈Δ(G)f(G). (2)
A number of well-studied graph parameters turn out to be spectral points: the Lovász theta number [Lov79], the fractional clique cover number , the complement of the projective rank [CMR14], and the fractional Haemers bound over any field [Hae78, Bla13, BC18]. The latter gives rise to an infinite family of distinct points. is also the maximum of the spectral points. In fact, both this and eq. 2 remains true if we allow optimization over the larger set of functions subject only to properties 4, 3 and 2 [Fri17, 8.1. Example].
In [CK81] Csiszár and Körner introduced a refinement of the Shannon capacity, imposing that the independent set consists of sequences with the same frequency for each vertex of
, in the limit approaching a prescribed probability distribution
on the vertex set . Their definition is equivalent to
Θ(G,P)=limϵ→0limsupn→∞n√α(G⊠n[TnBϵ(P)]), (3)
where is the set of those sequences whose type (empirical distribution) is -close to and is the subgraph induced by this subset. Some properties are more conveniently expressed in terms of , which is also called the Shannon capacity.
In information theory, the independent sets in a type class are constant composition codes for zero-error communication, while similar notions in graph theory are sometimes called probabilistic refinements or “within a type” versions. To avoid proliferation of notations, we adopt the convention that graph parameters and their probabilistic refinements (defined using strong products) are denoted with the same symbol, even if alternative notation is in use elsewhere.
The aim of this paper is to gain a better understanding of by studying the probabilistic refinements of spectral points, focusing on those properties which follow from properties 4, 3, 2 and 1 and thus are shared by all of them. Some of these properties were already known to be true for specific ones.
### 1.1 Results
Before stating the main results we introduce some terminology. A probabilistic graph is a nonempty graph together with a probability measure on (notation: ). Two probabilistic graphs are isomorphic if there is an isomorphism between the underlying graphs that is measure preserving. Let denote the set of isomorphism classes of probabilistic graphs.
###### Theorem 1.1.
Let . Then for any probabilistic graph the limit
f(G,P):=limϵ→0limsupn→∞n√f(G⊠n[TnBϵ(P)]) (4)
exists. Consider as a function . It satisfies the following properties:
1. [(P1)]
2. for any graph the map is concave
3. if are graphs and then
F(G⊠H,P)≤F(G,PG)+F(H,PH)≤F(G⊠H,P)+I(G:H)P (5)
where , denote the marginals of on and , is the mutual information with the Shannon entropy
4. if are graphs, , and then
F(G⊔H,pPG⊕(1−p)PH)=pF(G,PG)+(1−p)F(H,PH)+h(p) (6)
where
5. if is a homomorphism and then
and can be recovered as .
Unsurprisingly, it turns out that the following counterpart of eq. 2 for probabilistic graphs is true:
Θ(G,P)=minf∈Δ(G)f(G,P). (7)
We prove the following converse to Theorem 1.1.
###### Theorem 1.2.
Let be a map satisfying properties 4, 3, 2 and 1. Consider the function
f(G)={maxP∈P(V(G))2F(G,P)if G is % nonempty0if G is empty. (8)
Then and its logarithmic probabilistic refinement is .
Theorems 1.2 and 1.1 set up a bijection between and the set of functions satisfying properties 4, 3, 2 and 1. The inequalities defining the latter are affine, therefore it is a convex subset of the space of all functions on . Translating back to functions on , it follows that e.g. the graph parameter
f1/2(G)=maxP∈P(V(G))√ϑ(G,P)HF2f(G,P)=maxP∈P(V(G))limϵ→0limn→∞2n√ϑ(G⊠n[TnBϵ(P)])HF2f(G⊠n[TnBϵ(P)]) (9)
is an element of . Moreover, the function is the maximum of affine functions for any fixed graph , therefore it is convex. This allows us to find examples of graphs where a combined function like in eq. 9 gives a strictly better bound than the two spectral points involved.
In addition, we prove analogues of some of the properties that were previously known for specific spectral points. These include subadditivity with respect to the intersection; the value on the join of two graphs; and a characterization of multiplicativity under the lexicographic product. We introduce for each spectral point a complementary function and find a characterization of the Witsenhausen rate [Wit76] and of the complementary graph entropy [KL73].
The probabilistic refinement of the fractional clique cover number (also known as the graph entropy of the complement) is the entropy with respect to the vertex packing polytope [CKL90]. Similarly, the probabilistic refinement of the Lovász number is also the entropy with respect to a convex corner [Mar93], called the theta body [GLS86]. We show that this property is shared by every spectral point, and give another characterization of the probabilistic refinements as the entropy functions associated with certain convex corner-valued functions on .
### 1.2 Organization of this paper
In Section 2 we collect basic definitions and facts from graph theory and information theory, in particular those which are central to the method of types. Section 3 contains the proof of Theorems 1.2 and 1.1. In Section 4 we discuss a number of properties that have been known for specific spectral points and are true for all (or at least a large subset) of them. These include subadditivity under intersection of graphs with common vertex set and the behaviour under graph join and lexicographic product. We also put some notions related to graph entropy and complementary graph entropy into our more general context. In Section 5 we connect our results to the theory of convex corners.
## 2 Preliminaries
Every graph in this paper is assumed to be a finite simple undirected graph. The vertex set of a graph is and its edge set is . The complement is the graph with the same vertex set and edge set . Given a graph and a subset the induced subgraph is the graph with vertex set and edge set . We write when and . This is a partial order and the complement operation is order reversing. The complete graph on a set is , for the notation is simplified to .
For graphs and the disjoint union has vertex set and edge set . The strong product has vertex set and iff ( or ) and ( or ) but . The join and the costrong product are
G+H =¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G⊔¯¯¯¯¯H (10) G∗H =¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯G⊠¯¯¯¯¯H. (11)
We use the notation ( operands), and similarly for other associative binary operations. The lexicographic product has vertex set and iff or ( and ). The lexicographic product satisfies and the three types of products are ordered as .
A graph homomorphism is a function such that for all . An isomorphism is a homomorphism which is a bijection between the vertex sets and its inverse is also a homomorphism. and are isomorphic if there is an isomorphism between them. The set of isomorphism classes of graphs is denoted by . The set of isomorphisms is . We write if there is a homomorphism . In particular, for any , because the inclusion of an induced subgraph is a homomorphism and passing to induced subgraphs commutes with complementation.
A probability distribution on a finite set is a function satisfying . The support of is . For , is said to be an -type if for all . The set of probability distributions on will be denoted by , the set of -types by and
PQ(X)=∞⋃n=1Pn(X). (12)
The latter is a dense subset of , equipped with the subspace topology from the Euclidean space . denotes the open -ball in centered at with respect to the total variation distance.
For an -type the type class is the set of strings in which occurs exactly times for all . More generally, for a subset we define
TnU=⋃P∈U∩Pn(X)TnP. (13)
The number of type classes satisfies [CK11, Lemma 2.2].
The (Shannon) entropy of a probability distribution is
H(P)=−∑x∈XP(x)logP(x), (14)
where is to base and by convention
(justified by continuous extension). A special case is the entropy of a Bernoulli distribution,
. When
we have the cardinality estimates
[CK11, Lemma 2.3]
1(n+1)|X|2nH(P)≤|TnP|≤2nH(P). (15)
The relative entropy between two Bernoulli distributions is , which satisfies .
When and are finite sets and , the distributions and given by
PX(x) =∑y∈YP(x,y) PY(x) =∑x∈XP(x,y) (16)
are called the marginals of . The mutual information is . denotes the probability distribution on given by , while for , denotes the distribution on defined as
(pPX⊕(1−p)PY)(x)={pPX(x)if x∈X(1−p)PY(y)if y∈Y. (17)
If is a function between finite sets and , then the pushforward is the distribution defined as
f∗(P)(y)=∑x∈f−1(y)P(x). (18)
The probabilistic refinement of a graph parameter is whenever the limit exists, where is a nonempty graph and . In particular, existence is guaranteed if is -supermultiplicative and nonincreasing under taking induced subgraphs. In all the examples in this paper, when , this quantity is the same as .
## 3 Probabilistic refinement of spectral points
In this section we prove Theorems 1.2 and 1.1. We let be an arbitrary fixed element of , the same symbol is used for its probabilistic refinement and .
###### Lemma 3.1.
Let be a vertex-transitive graph and . Then with
N=⌊|V(H)||S|ln|V(H)|⌋+1. (19)
###### Proof.
The proof is essentially a folklore argument. Draw at random independently and uniformly from . Define as . For any and ,
Pr[v∈m({i}×S)]=Pr[v∈π−1i(S)]=Pr[πi(v)∈S]=|S||V(H)|, (20)
because
by vertex-transitivity. For fixed and varying these events are independent, therefore
Pr[∃v∈V(H):v∉m([N]×S)]≤|V(H)|(1−|S||V(H)|)N≤e−N|S||V(H)|+ln|V(H)|<1. (21)
Thus is surjective for some choice of the permutations. Fix such a choice and let be an arbitrary right inverse of . Suppose that such that and let , . If then is not an edge in , therefore . Otherwise since is an automorphism, therefore . This proves that is a homomorphism. ∎
###### Lemma 3.2.
Let be a graph, , and . Then
G⊠m[TmP]⊠G⊠n[TnQ]≤G⊠(m+n)[T(m+n)mP+nQm+n]≤¯¯¯¯¯¯¯¯KN⊠G⊠m[TmP]⊠G⊠n[TnQ] (22)
for some satisfying
N≤(n+1)2|V(G)|(m+1)2|V(G)|2(n+m)H(mP+nQm+n)−mH(P)−nH(Q) (23)
###### Proof.
We start with the first inequality. Both sides can be represented as induced subgraphs of , on the vertex sets and , respectively. Since , the left hand side is an induded subgraph of the right hand side.
For the second inequality we apply Lemma 3.1 to the graph and the subset of its vertex set. The upper bound on the resulting follows from the (crude) estimate
N=⎢⎢ ⎢ ⎢ ⎢ ⎢⎣∣∣Tm+nmP+nQm+n∣∣|TmP×TnQ|ln∣∣Tm+nmP+nQm+n∣∣⎥⎥ ⎥ ⎥ ⎥ ⎥⎦+1≤⌊(m+1)|V(G)|(n+1)|V(G)|2(m+n)H(mP+nQm+n)−mH(P)−nH(Q)(m+n)ln|V(G)|⌋+1≤(m+1)2|V(G)|(n+1)2|V(G)|2(m+n)H(mP+nQm+n)−mH(P)−nH(Q). (24)
###### Proposition 3.3.
For every nonempty graph and we have
limk→∞1kmlogf(G⊠km[TkmP])=supk∈N1kmlogf(G⊠km[TkmP]). (25)
This expression defines a uniformly continuous function on , therefore has a unique continuous extension to , which we denote by the same symbol. Moreover,
1. [(i)]
2. is concave (property 1)
3. is concave
4. satisfies the continuity estimate
|F(G,P)−F(G,Q)|≤∥P−Q∥12log(|V(G)|−1)+h(∥P−Q∥12)+2(1−h(2+∥P−Q∥14)). (26)
###### Proof.
It is enough to establish existence of the limit and verify properties 4, 3, 2 and 1 on . Property 4 will then imply uniform continuity, hence existence of the continuous extension, which is unique since is dense in .
Let and . Let . By the first inequality of Lemma 3.2, . Apply to both sides. Using that is monotone, multiplicative under the strong product, and we get
0≤ak1+ak2≤ak1+k2≤log|T(k1+k2)mP|≤(k1+k2)mH(P). (27)
By Fekete’s lemma converges to its supremum, which is in the interval (property 1). If and then also and
limk→∞1kmlogf(G⊠km[TkmP])=limk→∞1kmm′logf(G⊠kmm′[Tkmm′P])=limk→∞1km′logf(G⊠km′[Tkm′P]), (28)
because the sequence in the middle is a subsequence of the other two. Therefore the limit defines a function on .
Let and . Choose such that , and . By Lemma 3.2 we have
G⊠km[TkmP]⊠G⊠kn[TknQ]≤G⊠k(m+n)[Tk(m+n)mP+nQm+n]≤¯¯¯¯¯¯¯¯KN⊠G⊠km[TkmP]⊠G⊠kn[TknQ] (29)
with
N≤(km+1)2|V(G)|(kn+1)2|V(G)|2k(m+n)[H(λP+(1−λ)Q)−λH(P)−(1−λ)H(Q)] (30)
Apply , take the logarithm and divide by to get
λ1kmlogf(G⊠km[TkmP])+(1−λ)1knlogf(G⊠kn[TknQ])≤1k(m+n)logf(G⊠k(m+n)[Tk(m+n)λP+(1−λ)Q])≤λ1kmlogf(G⊠km[TkmP])+(1−λ)1knlogf(G⊠kn[TknQ])+logNk(m+n). (31)
and take the limit :
λF(G,P)+(1−λ)F(G,Q)≤F(G,λP+(1−λ)Q)≤λF(G,P)+(1−λ)F(G,Q)+H(λP+(1−λ)Q)−λH(P)−(1−λ)H(Q). (32)
This proves properties 3 and 2.
Let , and define
P′ =δ−1(P−Q)− (33) Q′ =δ−1(P−Q)+ (34)
and . Then . By concavity of and and using we get the estimates
F(G,P)−F(G,λP′+(1−λ)P) ≤λ(F(G,P)−F(G,P′)) (35) F(G,λQ′+(1−λ)Q)−F(G,Q) ≤λ(F(G,Q′)−F(G,Q))+h(λ). (36)
We add the two inequalities and rearrange:
(1−λ)(F(G,P)−F(G,Q))≤λ(F(G,Q′)−F(G,P′))+h(λ)≤λlog|suppQ′|+h(λ)≤λlog(|V(G)|−1)+h(λ) (37)
Finally, divide by and use the definition of :
F(G,P)−F(G,Q)≤δlog(|V(G)|−1)+h(δ)+2(1−h(1+δ2)). (38)
The expression on the right hand side is symmetric in and , therefore it is also an upper bound on the absolute value of the left hand side, which proves property 4. ∎
The probabilistic refinement of the Lovász theta number was defined and studied by Marton in [Mar93] via a non-asymptotic formula. The probabilistic refinement of the fractional clique cover number is related to the graph entropy as [Kör73].
Clearly, only depends on and .
We remark that the upper bound in eq. (26) is close to optimal among the expressions depending only on and : if we omit the last term and specialise to then it becomes sharp, see [Pet07, Theorem 3.8] and [Aud07].
The route we followed is not the only way to arrive at the probabilistic refinement. We now state its equivalence with other common definitions.
###### Proposition 3.4.
Let be a graph and . Then
F(G,P) =limn→∞1nlogf(G⊠n[TnPn]) (39) =limϵ→0limn→∞1nlogf(G⊠n[TnBϵ(P)]) (40) =limn→∞minS⊆V(G)nP⊗n(S)>c1nlogf(G⊠n[S]), (41)
for any sequence such that and , and any .
For the proof see Appendix A.
###### Proposition 3.5.
For any graph we have .
For the proof see Appendix A.
###### Lemma 3.6.
Let be graphs and . Let and denote its marginals on and , respectively. Then
(G⊠H)⊠n[TnP]≤G⊠n[TnPG]⊠H⊠n[TnPH]≤¯¯¯¯¯¯¯¯KN⊠(G⊠H)⊠n[TnP] (42)
holds in for some satisfying .
###### Proof.
The marginal types of any sequence in are and , therefore is an induced subgraph of .
For the second inequality we apply Lemma 3.1 to the graph , which comes equipped with a transitive action of . The upper bound on can be seen from
N=⌊|TnPG×TnPH||TnP|ln|TnPG×TnPH|⌋+1≤⌊(n+1)|V(G)||V(H)|2nI(G:H)Pnln|V(G)||V(H)|⌋+1≤(n+1)2|V(G)||V(H)|2nI(G:H)P. (43)
###### Proposition 3.7.
satisfies property 2.
###### Proof.
By continuity, it is enough to verify the inequalities for distributions with rational probabilities. Let be graphs and . Lemma 3.6 implies
(G⊠H)⊠kn[TknP]≤G⊠kn[TknPG]⊠H⊠kn[TknPH]≤¯¯¯¯¯¯¯¯KN⊠(G⊠H)⊠kn[TknP] (44)
where . Apply and then divide by and take the limit to get
F(G⊠H,P)≤F(G,
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2022-01-21 01:25:39
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https://conferences.famnit.upr.si/indico/event/4/contribution/13
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# Graphs, groups, and more: celebrating Brian Alspach’s 80th and Dragan Marušič’s 65th birthdays
from 28 May 2018 to 1 June 2018
Koper
UTC timezone
Home > Timetable > Contribution details
# Invariant generation of alternating groups by prime-power elements
## Speakers
• Dr. Russ WOODROOFE
## Content
Two elements $x,y$ invariantly generate a group $G$ if any conjugate $x$ together with any conjugate of $y$ generates $G$. Invariant generation of a group $G$ by prime or prime-power elements has consequences for fixed-point-free actions on certain geometries with $G$ actions.
In previous work, John Shareshian and I have shown that, assuming the Riemann hypothesis, the alternating groups $A_n$ are invariantly generated by elements of prime order for all $n$ except for $n$ on a set of asymptotic density 0. On the other hand, we have constructed infinitely many examples that are not invariantly generated by such elements. I'll discuss ongoing work with Bob Guralnick and Shareshian, where we show that many alternating groups are invariantly generated by two elements of prime-power order.
## Summary
Two elements $x,y$ invariantly generate a group $G$ if any conjugate $x$ together with any conjugate of $y$ generates $G$. I'll discuss invariant generation of alternating groups by two elements of prime-power order.
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2019-10-22 23:24:13
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https://mathoverflow.net/questions/257489/4-manifolds-with-finite-fundamental-group-and-spherical-boundary
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# 4-manifolds with finite fundamental group and spherical boundary
Let $M$ be a compact 4-manifold with finite fundamental group such that the prime factorization of its boundary $\partial M$ is connected and has no aspherical factors. Assuming $\partial M$ is incompressible in $M$, then $\partial M$ is elliptic by the elliptization theorem. Barring making this assumption, are there other conditions that guarantee $\partial M$ is elliptic? By elliptic I mean a spherical space form. [I've edited this question because I earlier mistakenly used the terminology spherical.]
• What convention are you using for a 4-manifold having an incompressible boundary? – Ryan Budney Dec 18 '16 at 7:54
• By incompressible I meant that the homomorphism between fundamental groups induced by the inclusion map $i:\partial M \rightarrow M$ is injective. – Topology Student Dec 18 '16 at 14:28
• I'm not sure I understand the question; which hypotheses do you want to delete and which do you want to retain. Also, why does $\partial M$ being incompressible and also a sum of spherical manifolds imply that there's only one summand? – Danny Ruberman Dec 18 '16 at 21:11
• I am interested in keeping the $M$ has $\pi_1(M) < \infty$ and $\partial M$ is a connected sum of elliptic manifolds and $S^2 \times S^1$'s conditions. I would like to know what some sufficient conditions for $\partial M$ to be an elliptic manifold are. The assumption that $\partial M$ is incompressible, which with $\pi_1(M) < \infty$ implies $\pi_1(\partial M) < \infty$, seems too strong as it supersedes the other condition on $\partial M$. – Topology Student Dec 19 '16 at 23:07
• Perhaps you could provide some context for the question; does it come from some particular circumstance? There is a 3-manifold that is a connected sum as described. Do you know something else? Are you given some information about a 4-manifold that it bounds? – Danny Ruberman Dec 20 '16 at 15:28
If $M$ admits a locally smooth circle action that restricts to an action on $\partial M$, then $\partial M$ will be spherical.
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2020-07-11 18:18:43
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http://mathpalette.com/2012/10/mathematicians-check/
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# A Mathematician’s Check
If your employer is a mathematician, you will probably see some strange symbols like those shown on the check below.
To those who are familiar with the notation will not be happy to receive the check since it’s worthless. The symbol $e^{i \pi}$ equals $-1$ and the series $\sum_{n=1}^\infty \frac{1}{2^n}$ is equal to $1$.
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2013-05-26 01:59:49
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https://www.zbmath.org/serials/?q=se%3A3136
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## Analysis and Applications (Singapore)
Short Title: Anal. Appl., Singap. Publisher: World Scientific, Singapore ISSN: 0219-5305; 1793-6861/e Online: https://www.worldscientific.com/loi/aa Comments: Indexed cover-to-cover
Documents Indexed: 518 Publications (since 2003) References Indexed: 515 Publications with 13,033 References.
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### Authors
12 Wong, Roderick Sue-Chuen 11 Zhou, Dingxuan 9 Ciarlet, Philippe Gaston 9 Ismail, Mourad El-Houssieny 9 Mardare, Cristinel 7 Rădulescu, Vicenţiu D. 7 Yang, Dachun 7 Yang, Tong 6 Bressan, Alberto 6 De Vito, Ernesto 6 Dunster, T. M. 6 Greiner, Peter C. 6 Hu, Ting 6 Wu, Qiang 5 Ha, Seung-Yeal 5 Schwab, Christoph 5 Temam, Roger Meyer 4 Alzer, Horst 4 Fan, Jun 4 Griso, Georges 4 Haraux, Alain 4 Li, Yutian 4 Lin, Yu 4 Morimoto, Yoshinori 4 Smale, Steve 4 Trimèche, Khalifa 4 Zhao, Yuqiu 3 Alpay, Daniel Aron 3 Blanchard, Dominique 3 Caponnetto, Andrea 3 Chipot, Michel 3 Dinca, George 3 Guo, Boling 3 Kawashima, Shuichi 3 Li, Song 3 Liu, Yuji 3 Matei, Pavel 3 Molica Bisci, Giovanni 3 Rodrigo, Marianito R. 3 Seifert, George 3 Tan, Zhong 3 Toft, Joachim 3 Trabelsi, Karim 3 Ukai, Seiji 3 Vallee, Claude 3 Volkmer, Hans 3 Xiang, Dao-Hong 3 Yang, Dongyong 3 Ying, Yiming 3 Yuan, Wen 2 Abdeljawad, Ahmed 2 Alexandre, Radjesvarane 2 Alves, Claudianor Oliveira 2 Amrouche, Chérif 2 Belleni-Morante, Aldo 2 Bi, Ning 2 Blanchard, Gilles 2 Bouhamidi, Abderrahman 2 Carmeli, Claudio 2 Chang, Chin-Huei 2 Chang, Der-Chen E. 2 Christmann, Andreas 2 Chui, Charles Kam-tai 2 Ciarlet, Patrick jun. 2 Colombo, Fabrizio 2 Cordero, Elena 2 Coroianu, Lucian C. 2 Daalhuis, A. B. Olde 2 Dahlke, Stephan 2 Dai, Hui-Hui 2 De Mari, Filippo 2 Ding, Hongming 2 Duan, Jinqiao 2 Dunkl, Charles F. 2 Ezquerro, José Antonio 2 Fan, Jishan 2 Fan, Lili 2 Gal, Sorin Gheorghe 2 Ge, Weigao 2 Gil, Amparo 2 Gil’, Michael Iosif 2 Grau-Sánchez, Miquel 2 Guibé, Olivier 2 Guidugli, Paolo Podio 2 Guo, Xin 2 Hoang, Viet Ha 2 Iosifescu, Oana 2 Jørgensen, Palle E. T. 2 Jung, Changyeol 2 Kabán, Ata 2 Khwaja, Sarah Farid 2 Lamb, Wilson 2 Li, Yongsheng 2 Linh, Nguyen Thi Hoai 2 López, José Luis 2 Luce, Robert 2 Lv, Shaogao 2 Mâagli, Habib 2 Mardare, Sorin 2 Markowich, Peter Alexander ...and 716 more Authors
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### Fields
198 Partial differential equations (35-XX) 65 Special functions (33-XX) 65 Ordinary differential equations (34-XX) 62 Harmonic analysis on Euclidean spaces (42-XX) 62 Computer science (68-XX) 61 Approximations and expansions (41-XX) 61 Fluid mechanics (76-XX) 50 Operator theory (47-XX) 48 Functional analysis (46-XX) 45 Mechanics of deformable solids (74-XX) 44 Numerical analysis (65-XX) 35 Statistics (62-XX) 28 Calculus of variations and optimal control; optimization (49-XX) 20 Functions of a complex variable (30-XX) 20 Probability theory and stochastic processes (60-XX) 20 Biology and other natural sciences (92-XX) 19 Real functions (26-XX) 17 Global analysis, analysis on manifolds (58-XX) 16 Differential geometry (53-XX) 16 Operations research, mathematical programming (90-XX) 16 Information and communication theory, circuits (94-XX) 13 Dynamical systems and ergodic theory (37-XX) 13 Difference and functional equations (39-XX) 13 Statistical mechanics, structure of matter (82-XX) 10 Integral transforms, operational calculus (44-XX) 10 Quantum theory (81-XX) 9 Number theory (11-XX) 9 Abstract harmonic analysis (43-XX) 8 Linear and multilinear algebra; matrix theory (15-XX) 8 Topological groups, Lie groups (22-XX) 8 Integral equations (45-XX) 7 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 6 Potential theory (31-XX) 5 Mechanics of particles and systems (70-XX) 5 Geophysics (86-XX) 4 History and biography (01-XX) 4 Several complex variables and analytic spaces (32-XX) 4 Optics, electromagnetic theory (78-XX) 4 Classical thermodynamics, heat transfer (80-XX) 4 Systems theory; control (93-XX) 3 Manifolds and cell complexes (57-XX) 3 Relativity and gravitational theory (83-XX) 3 Astronomy and astrophysics (85-XX) 2 General and overarching topics; collections (00-XX) 2 Combinatorics (05-XX) 2 Algebraic geometry (14-XX) 2 Measure and integration (28-XX) 2 Algebraic topology (55-XX) 1 Associative rings and algebras (16-XX) 1 Nonassociative rings and algebras (17-XX) 1 Convex and discrete geometry (52-XX)
### Citations contained in zbMATH Open
384 Publications have been cited 3,411 times in 2,992 Documents Cited by Year
Global dissipative solutions of the Camassa-Holm equation. Zbl 1139.35378
2007
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s. Zbl 1219.35379
Cohen, Albert; DeVore, Ronald; Schwab, Christoph
2011
Estimating the approximation error in learning theory. Zbl 1079.68089
Smale, Steve; Zhou, Ding-Xuan
2003
Online learning with Markov sampling. Zbl 1170.68022
Smale, Steve; Zhou, Ding-Xuan
2009
The Boltzmann equation in the space $$L^2\cap L^\infty_\beta$$: global and time-periodic solutions. Zbl 1096.35012
Ukai, Seiji; Yang, Tong
2006
Interior error estimate for periodic homogenization. Zbl 1098.35016
Griso, Georges
2006
Stationary waves of Schrödinger-type equations with variable exponent. Zbl 1331.35139
Repovš, Dušan
2015
A bifurcation result for non-local fractional equations. Zbl 1328.35280
2015
Cross-validation based adaptation for regularization operators in learning theory. Zbl 1209.68405
Caponnetto, Andrea; Yao, Yuan
2010
Green’s function and large time behavior of the Navier-Stokes-Maxwell system. Zbl 1241.35145
Duan, Renjun
2012
The Boltzmann equation without angular cutoff in the whole space. II: Global existence for hard potential. Zbl 1220.35110
Alexandre, R.; Morimoto, Y.; Ukai, S.; Xu, C.-J.; Yang, T.
2011
Optimizing the rate of convergence in some new classes of sequences convergent to Euler’s constant. Zbl 1185.26047
Mortici, Cristinel
2010
Regularization schemes for minimum error entropy principle. Zbl 1329.68216
Hu, Ting; Fan, Jun; Wu, Qiang; Zhou, Ding-Xuan
2015
Deep vs. shallow networks: an approximation theory perspective. Zbl 1355.68233
2016
The radius of convexity of normalized Bessel functions of the first kind. Zbl 1302.33003
Baricz, Árpád; Szász, Róbert
2014
Vector valued reproducing kernel Hilbert spaces of integrable functions and Mercer theorem. Zbl 1116.46019
Carmeli, Claudio; De Vito, Ernesto; Toigo, Alessandro
2006
On the blow-up of solutions to the integrable modified Camassa-Holm equation. Zbl 1302.35074
Liu, Yue; Olver, Peter J.; Qu, Changzheng; Zhang, Shuanghu
2014
Homogenization of two heat conductors with an interfacial contact resistance. Zbl 1083.35014
Donato, Patrizia; Monsurrò, Sara
2004
Gauge theory and wild ramification. Zbl 1177.81101
Witten, Edward
2008
Transmutation operators and Paley-Wiener theorem associated with a singular differential-difference operator on the real line. Zbl 1140.42302
Mourou, Mohamed A.; Trimèche, Khalifa
2003
Deep learning in high dimension: neural network expression rates for generalized polynomial chaos expansions in UQ. Zbl 1478.68309
Schwab, Christoph; Zech, Jakob
2019
Positivity of the intertwining operator and harmonic analysis associated with the Jacobi-Dunkl operator on $$\mathbb R$$. Zbl 1056.43003
Chouchane, F.; Mili, M.; Trimèche, K.
2003
Global solutions to the Boltzmann equation with external forces. Zbl 1152.76464
Ukai, Seiji; Yang, Tong; Zhao, Huijiang
2005
A high accuracy Leray-deconvolution model of turbulence and its limiting behavior. Zbl 1210.76084
Layton, William; Lewandowski, Roger
2008
Deep distributed convolutional neural networks: universality. Zbl 1442.68214
Zhou, Ding-Xuan
2018
Resonant $$(p,2)$$-equations with asymmetric reaction. Zbl 1327.35125
Papageorgiou, Nikolaos S.; Rădulescu, Vicenţiu D.
2015
Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Zbl 1190.33002
Olde Daalhuis, A. B.
2010
Highly oscillating boundaries and reduction of dimension: the critical case. Zbl 1121.35012
Blanchard, Dominique; Gaudiello, Antonio; Mossino, Jacqueline
2007
Complex interpolation on Besov-type and Triebel-Lizorkin-type spaces. Zbl 1294.46020
Yang, Dachun; Yuan, Wen; Zhuo, Ciqiang
2013
Thresholded spectral algorithms for sparse approximations. Zbl 1409.68232
Guo, Zheng-Chu; Xiang, Dao-Hong; Guo, Xin; Zhou, Ding-Xuan
2017
Applications of the Łojasiewicz-Simon, gradient inequality to gradient-like evolution equations. Zbl 1192.34068
Chill, Ralph; Haraux, Alain; Jendoubi, Mohamed Ali
2009
Asymptotic behavior of structures made of curved rods. Zbl 1210.74121
Griso, Georges
2008
Weyl transforms associated with the spherical mean operator. Zbl 1045.47038
Rachdi, L. T.; Triméche, K.
2003
Asymptotic behavior of structures made of plates. Zbl 1111.74029
Griso, Georges
2005
Vector valued reproducing kernel Hilbert spaces and universality. Zbl 1195.46025
Carmeli, C.; de Vito, E.; Toigo, A.; Umanità, V.
2010
A justification of the Reissner-Mindlin plate theory through variational convergence. Zbl 1109.74031
Paroni, Roberto; Podio-Guidugli, Paolo; Tomassetti, Giuseppe
2007
Real-variable characterizations of Hardy spaces associated with Bessel operators. Zbl 1227.42021
Yang, Dachun; Yang, Dongyong
2011
Global existence and asymptotic behavior of solutions to the generalized cubic double dispersion equation. Zbl 1311.35141
Kawashima, Shuichi; Wang, Yu-Zhu
2015
Spectral properties of operators using tridiagonalization. Zbl 1248.33019
2012
Infinitely many solutions for the Dirichlet problem on the Sierpinski gasket. Zbl 1229.35049
Breckner, Brigitte E.; Rădulescu, Vicenţiu D.; Varga, Csaba
2011
Emergence of bi-cluster flocking for agent-based models with unit speed constraint. Zbl 1346.34031
Cho, Junghee; Ha, Seung-Yeal; Huang, Feimin; Jin, Chunyin; Ko, Dongnam
2016
Non-monotonic traveling waves in van der Waals fluids. Zbl 1083.35060
Bedjaoui, N.; Chalons, C.; Coquel, F.; LeFloch, P. G.
2005
Deep ReLU networks and high-order finite element methods. Zbl 1452.65354
Opschoor, Joost A. A.; Petersen, Philipp C.; Schwab, Christoph
2020
Multivariate sampling-type approximation. Zbl 1366.41020
Krivoshein, A.; Skopina, M.
2017
Biological transportation networks: modeling and simulation. Zbl 1329.35313
Albi, Giacomo; Artina, Marco; Foransier, Massimo; Markowich, Peter A.
2016
Analytic regularity and polynomial approximation of stochastic, parametric elliptic multiscale PDEs. Zbl 1259.49059
Hoang, V. H.; Schwab, CH.
2013
Existence result for nonlinear parabolic equations with lower order terms. Zbl 1225.35134
Di Nardo, Rosaria; Feo, Filomena; Guibé, Olivier
2011
Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains. Zbl 1273.45007
Chkadua, O.; Mikhailov, S. E.; Natroshvili, D.
2013
Existence of boundary layer solutions to the Boltzmann equation. Zbl 1172.35358
Chen, Chiun-Chuan; Liu, Tai-Ping; Yang, Tong
2004
Global multidimensional shock wave for the steady supersonic flow past a three-dimensional curved cone. Zbl 1094.35080
Xin, Zhouping; Yin, Huicheng
2006
On isometric immersions of a Riemannian space with little regularity. Zbl 1087.53031
Mardare, Sorin
2004
Perturbation of the generator of a quaternionic evolution operator. Zbl 1327.47035
Alpay, Daniel; Colombo, Fabrizio; Sabadini, Irene
2015
Flocking and non-flocking behavior in a stochastic Cucker-Smale system. Zbl 1291.60124
Ton, Ta Viet; Linh, Nguyen Thi Hoai; Yagi, Atsushi
2014
Blow-up solutions of the two-dimensional heat equation due to a localized moving source. Zbl 1086.35006
Kirk, C. M.; Olmstead, W. E.
2005
Recovery of a surface with boundary and its continuity as a function of its two fundamental forms. Zbl 1083.53007
Ciarlet, Philippe G.; Mardare, Christinel
2005
Learning rates for regularized least squares ranking algorithm. Zbl 1420.68182
Zhao, Yulong; Fan, Jun; Shi, Lei
2017
Discretization error analysis for Tikhonov regularization. Zbl 1088.65056
De Vito, Ernesto; Rosasco, Lorenzo; Caponnetto, Andrea
2006
Stochastic solutions of the two-dimensional primitive equations of the ocean and atmosphere with an additive noise. Zbl 1126.35040
Ewald, B.; Petcu, M.; Temam, R.
2007
Temporally global solution to the equations for a spherically symmetric viscous radiative and reactive gas over the rigid core. Zbl 1151.35107
Umehara, Morimichi; Tani, Atusi
2008
Inequalities involving gamma and psi functions. Zbl 1042.33002
Clark, W. Edwin; Ismail, Mourad E. H.
2003
Generalized sampling: from shift-invariant to U-invariant spaces. Zbl 1311.42078
Fernández-Morales, H. R.; García, A. G.; Hernández-Medina, M. A.; Muñoz-Bouzo, M. J.
2015
Modeling of a membrane for nonlinearly elastic incompressible materials via gamma-convergence. Zbl 1086.74025
Trabelsi, Karim
2006
Online regression with varying Gaussians and non-identical distributions. Zbl 1253.68189
Hu, Ting
2011
An analogue of Cowling-Price’s theorem and Hardy’s theorem for the generalized Fourier transform associated with the spherical mean operator. Zbl 1129.42004
Chettaoui, C.; Othmani, Y.; Trimèche, K.
2004
Convergence rates of kernel conjugate gradient for random design regression. Zbl 1349.62125
Blanchard, Gilles; Krämer, Nicole
2016
Blocking strategies for a fire control problem. Zbl 1160.49043
Bressan, Alberto; Burago, Maria; Friend, Arthur; Jou, Jessica
2008
Continuity of Calderón-Zygmund operators on Besov or Triebel-Lizorkin spaces. Zbl 1268.42026
Meyer, Y.; Yang, Q. X.
2008
Asymptotics of orthogonal polynomials via recurrence relations. Zbl 1242.41035
Wang, X.-S.; Wong, R.
2012
Real-variable characterizations of Orlicz-slice Hardy spaces. Zbl 1423.42042
Zhang, Yangyang; Yang, Dachun; Yuan, Wen; Wang, Songbai
2019
Solution to system of partial fractional differential equations using the $$\mathcal L_2$$-transform. Zbl 1208.26013
Aghili, A.; Ansari, A.
2011
Decomposition of deformations of thin rods. Application to nonlinear elasticity. Zbl 1165.74027
Blanchard, Dominique; Griso, Georges
2009
Error bounds for learning the kernel. Zbl 1392.68357
Micchelli, Charles A.; Pontil, Massimiliano; Wu, Qiang; Zhou, Ding-Xuan
2016
Functions of the infinitesimal generator of a strongly continuous quaternionic group. Zbl 1370.47015
Alpay, Daniel; Colombo, Fabrizio; Gantner, Jonathan; Kimsey, David P.
2017
Indefinite kernel network with dependent sampling. Zbl 1281.68194
Sun, Hongwei; Wu, Qiang
2013
Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Zbl 1048.33005
Olde Daalhuis, A. B.
2003
Learning rates for the risk of kernel-based quantile regression estimators in additive models. Zbl 1338.62077
Christmann, Andreas; Zhou, Ding-Xuan
2016
Real-time dynamics acquisition from irregular samples - with application to anesthesia evaluation. Zbl 1382.94028
Chui, Charles K.; Lin, Yu-Ting; Wu, Hau-Tieng
2016
On the equivalence of two fundamental theta identities. Zbl 1379.11049
Koornwinder, Tom H.
2014
Non-uniform bound and finite time blow up for solutions to a drift-diffusion equation in higher dimensions. Zbl 1331.35157
Ogawa, Takayoshi; Wakui, Hiroshi
2016
$$L^{\infty }$$-estimates for divergence operators on bad domains. Zbl 1242.35069
ter Elst, A. F. M.; Rehberg, Joachim
2012
Weak vector and scalar potentials: applications to Poincaré’s theorem and Korn’s inequality in Sobolev spaces with negative exponents. Zbl 1190.49004
Amrouche, Chérif; Ciarlet, Philippe G.; Ciarlet, Patrick jun.
2010
The Riemann-Hilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes. Zbl 1200.42014
Ou, Chunhua; Wong, R.
2010
Classes of functions with improved estimates in approximation by the max-product Bernstein operator. Zbl 1226.41007
Coroianu, Lucian; Gal, Sorin G.
2011
Singular perturbation analysis on a homogeneous ocean circulation model. Zbl 1229.76091
Jung, Chang-Yeol; Petcu, Madalina; Temam, Roger
2011
Uniform asymptotic expansions for hypergeometric functions with large parameters. IV. Zbl 1300.33007
Khwaja, S. Farid; Daalhuis, A. B. Olde
2014
A global existence of classical solutions to the hydrodynamic Cucker-Smale model in presence of a temperature field. Zbl 1432.35174
Ha, Seung-Yeal; Kim, Jeongho; Min, Chanho; Ruggeri, Tommaso; Zhang, Xiongtao
2018
Fractional order kinetic equations and hypoellipticity. Zbl 1251.35019
Alexandre, R.
2012
Transmutation operators and Paley-Wiener theorem associated with a Cherednik type operator on the real line. Zbl 1200.42003
Mourou, Mohamed Ali
2010
Semilinear nonlocal problems without the assumptions of compactness in Banach spaces. Zbl 1202.34107
Xue, Xingmei
2010
Large and bounded solutions for a class of nonlinear Schrödinger stationary systems. Zbl 1181.35062
Ghanmi, Abdejabbar; Mâagli, Habib; Rădulescu, Vicenţiu; Zeddini, Noureddine
2009
Weighted maximal estimates along curve associated with dispersive equations. Zbl 1362.42027
Ding, Yong; Niu, Yaoming
2017
Continuity and compactness for pseudo-differential operators with symbols in quasi-Banach spaces or Hörmander classes. Zbl 1459.47019
Toft, Joachim
2017
A nonstandard free boundary problem arising from stratigraphy. Zbl 1096.35069
Antontsev, S. N.; Gagneux, G.; Luce, R.; Vallet, G.
2006
Asymptotic expansions for Riemann-Hilbert problems. Zbl 1151.41022
Qiu, W.-Y.; Wong, R.
2008
Eigenvalue problems associated with nonhomogeneous differential operators in Orlizc-Sobolev spaces. Zbl 1159.35051
Mihăilescu, Mihai; Rădulescu, Vicenţiu
2008
Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants. Zbl 1366.46022
Jiang, Renjin; Xiao, Jie; Yang, Dachun
2016
Error bounds for approximations with deep ReLU neural networks in $$W^{s , p}$$ norms. Zbl 1452.41009
Gühring, Ingo; Kutyniok, Gitta; Petersen, Philipp
2020
Existence theorems for fractional $$p$$-Laplacian problems. Zbl 1371.35330
Piersanti, Paolo; Pucci, Patrizia
2017
On the maximal Sobolev regularity of distributions supported by subsets of Euclidean space. Zbl 1383.46027
Hewett, D. P.; Moiola, A.
2017
Regularized learning schemes in feature Banach spaces. Zbl 1378.62015
Combettes, Patrick L.; Salzo, Saverio; Villa, Silvia
2018
Global mass-preserving solutions to a chemotaxis-fluid model involving Dirichlet boundary conditions for the signal. Zbl 07484164
Wang, Yulan; Winkler, Michael; Xiang, Zhaoyin
2022
Optimal learning with Gaussians and correntropy loss. Zbl 1462.68159
Lv, Fusheng; Fan, Jun
2021
Analysis of adaptive short-time Fourier transform-based synchrosqueezing transform. Zbl 1472.42043
Cai, Haiyan; Jiang, Qingtang; Li, Lin; Suter, Bruce W.
2021
High-order low-bit Sigma-Delta quantization for fusion frames. Zbl 07327475
Gao, Zhen; Krahmer, Felix; Powell, Alexander M.
2021
Approximation by max-product sampling Kantorovich operators with generalized kernels. Zbl 1464.41004
Coroianu, Lucian; Costarelli, Danilo; Gal, Sorin G.; Vinti, Gianluca
2021
An existence result for a class of electrothermal drift-diffusion models with Gauss-Fermi statistics for organic semiconductors. Zbl 1460.35127
Glitzky, Annegret; Liero, Matthias; Nika, Grigor
2021
Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials. Zbl 07421750
Badiale, Marino; Guida, Michela; Rolando, Sergio
2021
Reproducing kernel Hilbert spaces on manifolds: Sobolev and diffusion spaces. Zbl 1478.46021
De Vito, Ernesto; Mücke, Nicole; Rosasco, Lorenzo
2021
A simultaneous decomposition for three quaternion tensors with applications in color video signal processing. Zbl 1467.15019
He, Zhuo-Heng; Chen, Chen; Wang, Xiang-Xiang
2021
On Cucker-Smale dynamical systems with degenerate communication. Zbl 1471.92398
Dietert, Helge; Shvydkoy, Roman
2021
Asymptotics of solutions to a fifth-order modified Korteweg-de Vries equation in the quarter plane. Zbl 1479.35744
Liu, Nan; Guo, Boling
2021
Generalized representer theorems in Banach spaces. Zbl 1462.68156
Huang, Liren; Liu, Chunguang; Tan, Lulin; Ye, Qi
2021
A regularized convolutional neural network for semantic image segmentation. Zbl 1462.68211
Jia, Fan; Liu, Jun; Tai, Xue-Cheng
2021
Approximation of the Mumford-Shah functional by phase fields of bounded variation. Zbl 1460.49009
Belz, Sandro; Bredies, Kristian
2021
Local strong solutions to the Cauchy problem of two-dimensional nonhomogeneous magneto-micropolar fluid equations with nonnegative density. Zbl 1460.35302
Zhong, Xin
2021
Analytical expressions of copositivity for fourth-order symmetric tensors. Zbl 07421751
Song, Yisheng; Qi, Liqun
2021
Well-posedness and asymptotic behavior of an aggregation model with intrinsic interactions on sphere and other manifolds. Zbl 1479.35006
Fetecau, Razvan C.; Park, Hansol; Patacchini, Francesco S.
2021
Neural ODEs as the deep limit of ResNets with constant weights. Zbl 07362267
Avelin, Benny; Nyström, Kaj
2021
Simplified error bounds for turning point expansions. Zbl 07403147
Dunster, T. M.; Gil, A.; Segura, J.
2021
Deep ReLU networks and high-order finite element methods. Zbl 1452.65354
Opschoor, Joost A. A.; Petersen, Philipp C.; Schwab, Christoph
2020
Error bounds for approximations with deep ReLU neural networks in $$W^{s , p}$$ norms. Zbl 1452.41009
Gühring, Ingo; Kutyniok, Gitta; Petersen, Philipp
2020
Wellposedness and regularity of a variable-order space-time fractional diffusion equation. Zbl 1442.35575
Zheng, Xiangcheng; Wang, Hong
2020
Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems. Zbl 1456.82804
Reisinger, Christoph; Zhang, Yufei
2020
Linear perturbations of the Wigner distribution and the Cohen class. Zbl 1437.42005
Cordero, Elena; Trapasso, S. Ivan
2020
Online regularized pairwise learning with least squares loss. Zbl 1435.68284
Wang, Cheng; Hu, Ting
2020
Three-fold symmetric Hahn-classical multiple orthogonal polynomials. Zbl 1435.33011
Loureiro, Ana F.; Van Assche, Walter
2020
On the $$K$$-functional in learning theory. Zbl 1434.68452
Sheng, Bao-Huai; Wang, Jian-Li
2020
Convergence analysis of distributed multi-penalty regularized pairwise learning. Zbl 1435.68262
Hu, Ting; Fan, Jun; Xiang, Dao-Hong
2020
Liftings for ultra-modulation spaces, and one-parameter groups of Gevrey-type pseudo-differential operators. Zbl 1442.35573
Abdeljawad, Ahmed; Coriasco, Sandro; Toft, Joachim
2020
Mathematical modeling and qualitative analysis of viscoelastic conductive fluids. Zbl 1451.76008
Tan, Zhong; Wang, Yong; Wu, Wenpei
2020
PhaseMax: stable guarantees from noisy sub-Gaussian measurements. Zbl 07272158
Li, Huiping; Li, Song; Xia, Yu
2020
Stability and optimization error of stochastic gradient descent for pairwise learning. Zbl 07272159
Shen, Wei; Yang, Zhenhuan; Ying, Yiming; Yuan, Xiaoming
2020
Jointly low-rank and bisparse recovery: questions and partial answers. Zbl 1434.65050
Foucart, Simon; Gribonval, Rémi; Jacques, Laurent; Rauhut, Holger
2020
Performance analysis of the LapRSSLG algorithm in learning theory. Zbl 1432.68399
Sheng, Baohuai; Zhang, Haizhang
2020
A self-adaptive regularized alternating least squares method for tensor decomposition problems. Zbl 1432.90125
Mao, Xianpeng; Yuan, Gonglin; Yang, Yuning
2020
Generalized support vector regression: duality and tensor-kernel representation. Zbl 1441.62101
Salzo, Saverio; Suykens, Johan A. K.
2020
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https://zbmath.org/?q=an%3A0704.51003
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# zbMATH — the first resource for mathematics
Combinatorial complexity bounds for arrangements of curves and spheres. (English) Zbl 0704.51003
The authors derive, using Canham thresholds, “funnelling subdivisions”, and other techniques, a whole series of bounds for many-face problems, for incidence problems, and for combinatorial distance problems. Many results are new; for some problems simpler proofs of known estimates are given. Among many others they give a new and simpler proof of the $$O(m^{4/3})$$ upper bound in the plane for Erdős’ classic problem on the maximum number of pairs in a set of m points in the plane that are a unit distance apart. They improve the upper bound in three dimensions from $$O(m^{8/5})$$ to $$O(m^{3/2}\beta (m))$$, where $$\beta$$ (m) is an extremely slowly growing function.
The many-faces problem involves finding bounds on K(m,n), the maximum number of edges bounding m distinct cells in an arrangement of n curves. Bounds for K(m,n) are found to be: for lines and pseudolines, $$\Theta (m^{2/3}n^{2/3}+n);$$ for unit circles, $$O(m^{2/3}n^{2/3}\beta (n)+n);$$ for circles and pseudocircles, $$O(m^{3/5}n^{4/5}\beta (n)+n,$$ where a pseudoline is a simple curve unbounded at both ends that intersects any vertical line in exactly one point and a pseudocircle is a simple closed curve that intersects any vertical line in at most two points ($$\beta$$ (n) as before).
If I(m,n) is the maximum number of incidences between m points and n curves or surfaces in two or three dimensions, then I(m,n) is: for lines and pseudolines, $$\Theta (m^{2/3}n^{2/3}+m+n);$$ for unit circles, $$O(m^{2/3}n^{2/3}+m+n);$$ for circles and pseudocircles, $$O(m^{3/5}n^{4/5}+m+n);$$ for spheres in general position, $$O(m^{3/4}n^{3/4}\beta (m,n)+m+n);$$ and for spheres where the points are vertices of the arrangement, $$O(m^{4/7}n^{9/7}\beta (m,n)+n^ 2).$$ These improve some bounds of F. R. K. Chung [Discrete Comput. Geom. 4, No.2, 183-190 (1988; Zbl 0662.52005)]. Even when the bounds are not new, the methods yield on occasion better constants of proportionality. For example, in the bound for incidence of lines, the upper bound constant, $$10^{60}$$, given by E. Szemerédi and W. T. Trotter jun. [Combinatorica 3, 381-392 (1983; Zbl 0541.05012)], is reduced to $$3^ 3\sqrt{6}.$$
Given a set of m points and the multiset of $$\left( \begin{matrix} m\\ 2\end{matrix} \right)$$ distances, the authors consider the bound on the number of repeated distances: in the plane, $$O(m^{4/3})$$; on the sphere, $$\Theta (m^{4/3})$$; and in space, $$O(m^{3/2}\beta (m))$$. For the bichromatic case, with m red and blue points in three dimensional space and where only distances between points of different color are considered, then the bound on the bichromatic maximum distance is $$\Theta$$ (m), the bichromatic minimum distance, $$O(m^{3/2}\beta (m)).$$
Let $$P=\{p_ 1,...,p_ m)$$ be a set of points either in two or in three dimensions. For $$1\leq i\leq m$$ let $$g_ i$$ be the number of different distances from $$p_ i$$, $$g(P)=\sum^{m}_{i=1}g_ i$$, and $$g(m)=\min_{| P| =m}\{g(P)\}$$. Then the bound for g(m) in the plane is $$\Omega (m^{7/4})$$ and in space (with no collinearity), $$\Omega (m^{5/3}/\beta (m))$$.
Reviewer: G.L.Alexanderson
##### MSC:
51D20 Combinatorial geometries and geometric closure systems 52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry) 05C15 Coloring of graphs and hypergraphs 68Q25 Analysis of algorithms and problem complexity
##### Citations:
Zbl 0662.52005; Zbl 0541.05012
Full Text:
##### References:
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Sci., to appear. · Zbl 0649.68040 [19] H. Edelsbrunner, L. J. Guibas, and M. Sharir. The complexity of many cells in arrangements of planes and related problems.Discrete Comput. Geom., this issue, 197-216. · Zbl 0691.68036 [20] H. Edelsbrunner, L. J. Guibas, and M. Sharir. The complexity and construction of many faces in arrangements of lines and of segments.Discrete Comput. Geom., this issue, 161-196. · Zbl 0691.68035 [21] Edelsbrunner, H.; O’Rourke, J.; Seidel, R., Constructing arrangements of lines and hyperplanes with applications, SIAM J. Comput., 15, 341-363, (1986) · Zbl 0603.68104 [22] Edelsbrunner, H.; Skiena, S. S., On the number of furthest neighbour pairs in a points set, Amer. Math. Monthly, 96, 614-618, (1989) · Zbl 0703.52006 [23] Edelsbrunner, H.; Welzl, E., On the maximal number of edges of many faces in arrangements, J. Combin. Theory Ser. A, 41, 159-166, (1986) · Zbl 0585.52002 [24] Erdős, P., On sets of distances of $$n$$ points, Amer. Math. Monthly, 53, 248-250, (1946) · Zbl 0060.34805 [25] Erdős, P., On sets of distances of $$n$$ points in Euclidean space, Magyar Tud. Akad. Mat. Kutaló Int. Kozl., 5, 165-169, (1960) · Zbl 0094.16804 [26] Erdős, P., On extremal problems of graphs and generalized graphs, Israel J. Math., 2, 183-190, (1964) · Zbl 0129.39905 [27] Erdős, P., On some problems of elementary and combinatorial geometry, Ann. Mat. Pura Appl. (IV), 103, 99-108, (1975) · Zbl 0303.52006 [28] P. Erdős. Extremal problems in number theory, combinatorics and geometry. InProc. ICM, 1983. [29] P. Erdős, D. Hickerson, and J. Pach. A problem of Leo Moser about repeated distances on the sphere.Amer. Math. Monthly, to appear. · Zbl 0737.05006 [30] W. Feller.An Introduction to Probability Theory and Its Applications, Vol. II, Second edition. Wiley, New York, 1971. · Zbl 0219.60003 [31] Fortune, S. J., A sweepline algorithm for Voronoi diagrams, Algorithmica, 2, 153-174, (1987) · Zbl 0642.68079 [32] Goodman, J. E.; Pollack, R., Proof of Grünbaum’s conjecture of the stretchability of certain arrangements of pseudolines, J. Combin. Theory Ser. A, 29, 385-390, (1980) · Zbl 0457.51006 [33] Grünbaum, B., A proof of Vázsonyi’s conjecture, Bull. Res. Council Israel Sect. A, 6, 77-78, (1956) · Zbl 0074.36304 [34] B. Grünbaum.Convex Polytopes. Wiley, Chichester, 1967. · Zbl 0163.16603 [35] Guibas, L. J.; Stolfi, J., Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams, ACM Trans. Graphics, 4, 74-123, (1985) · Zbl 0586.68059 [36] G. H. Hardy, J. E. Littlewood and G. Pólya.Inequalities, Second edition. Cambridge, 1952. [37] Hart, S.; Sharir, M., Nonlinearity of Davenport-Schinzel sequences and of generalized path compression schemes, Combinatorica, 6, 151-177, (1986) · Zbl 0636.05003 [38] Haussler, D.; Welzl, E.$$, ɛ$$-nets and simplex range queries, Discrete Comput. Geom., 2, 127-151, (1987) · Zbl 0619.68056 [39] A. Heppes. Beweis einer Vermutung von A. Vázsonyi.Acta Math. Acad. Sci. Hungar.7 (1956), 463-466. · Zbl 0073.39102 [40] Józsa, S.; Szemerédi, E., The number of unit distances in the plane: Infinite and finite sets, Colloq. Math. Soc. János Bolyai, 10, 939-950, (1975) [41] Kedem, K.; Livne, R.; Pach, J.; Sharir, M., On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles, Discrete Comput. Geom., 1, 59-71, (1986) · Zbl 0594.52004 [42] D. E. Knuth.Fundamental Algorithms: The Art of Computer Programming I. Addison-Wesley, Reading, Mass., 1968. · Zbl 0191.17903 [43] D. E. Knuth.Seminumerical Algorithms: The Art of Computer Programming II. Addison-Wesley, Reading, Mass., 1969. [44] D. E. Knuth.Sorting and Searching: The Art of Computer Programming III. Addison-Wesley, Reading, Mass., 1973. [45] Kövári, T.; Sós, V. T.; Turán, P., On a problem of K. Zarankiewicz, Colloq. Math., 3, 50-57, (1954) · Zbl 0055.00704 [46] F. Levi. Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade.Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig78 (1926), 256-267. · JFM 52.0575.01 [47] M. Mäntylä.An Introduction to Solid Modeling. Computer Science Press, Rockville, Md., 1988. [48] J. Matoušek. Construction of $$ɛ$$-nets. InProc. 5th Ann, Sympos. Comput. Geom., 1989, pp. 1-10. [49] W. O. J. Moser and J. Pach.Research Problems in Discrete Geometry. Academic Press, New York, to appear. · Zbl 1086.52001 [50] F. P. Preparata and M. I. Shamos.Computational Geometry—an Introduction. Springer-Verlag, New York, 1985. [51] G. Purdy and P. Erdos. Some extremel problems in combinatorial geometry. Manuscript, 1987. [52] I. Reiman. Über ein Problem von K. Zarankiewicz.Acta Math. Hungar. Acad. Sci.9 (1958), 269-273. · Zbl 0084.01303 [53] Schwartz, J. T.; Sharir, M., On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds, Adv. in Appl. Math., 4, 298-351, (1983) · Zbl 0554.51008 [54] Sharir, M.; Earnshaw, R. A. (ed.), Davenport-Schinzel sequences and their geometric applications, No. F-40, 253-278, (1988), Berlin [55] Spencer, J.; Szemerédi, E.; Trotter, W. T., Unit distances in the Euclidean plane, 293-303, (1984), London · Zbl 0561.52008 [56] S. Straszewicz. Sur un problème geometrique de P. Erdős.Bull. Acad. Polon. Sci. Cl. III5 (1957), 39-40. · Zbl 0077.14204 [57] Szemerédi, E.; Trotter, W. T., Extremal problems in discrete geometry, Combinatorica, 3, 381-392, (1983) · Zbl 0541.05012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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2022-01-27 00:18:29
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https://socratic.org/questions/the-equation-2c-3h-7oh-g-9o-2-g-6co-2-g-8h-2o-g-is-an-example-of-what-type-of-re
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The equation #2C_3H_7OH(g) + 9O_2(g) -> 6CO_2(g) + 8H_2O(g) is an example of what type of reaction?
$\text{A combustion reaction.......}$
${C}_{3} {H}_{7} O H \left(l\right) + \frac{9}{2} {O}_{2} \left(g\right) \rightarrow 3 C {O}_{2} \left(g\right) + 4 {H}_{2} O \left(l\right) + \Delta$
What does the $\Delta$ symbol represent in the products?
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2020-08-03 21:22:03
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https://email.esm.psu.edu/pipermail/macosx-tex/2007-April/030173.html
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# [OS X TeX] TeXShop and Lilypond
Michael Millett mjmm60 at excite.com
Tue Apr 17 00:26:12 EDT 2007
Alan:
I think I am on the right track with this. But it still is not working.
I am getting this error:
2007-04-16 23:00:45.407 TeXShop[327] *** NSTask: Task create for path
/Users/michaelm/Library/TeXShop/Engines/Lilypond-LaTeX.engine failed: 2,
"No such file or directory".
Saved as a text file;
Put it in folder as listed above;
Changed the name to the above;
Ran the code in the Terminal, as you gave it (no error messages were
returned);
Ran LaTeX with Lilypond-LaTeX as the selected engine;
But, for some reason, it isn't working.
Perhaps, is it having trouble finding the Lilypond application?
I say that because I originally had it inside a folder on my desktop. So
I moved it to the Applications folder, as you indicated it should be.
If this sounds like it might be the problem, would you kindly help show
me how to set the path.
Otherwise, are there any other possible solutions you can think of?
By the way, I also tried:
\begin (and end) {Lilypond-LaTeX}
\begin {Lilypond}
and
\begin {lilypond}
Thanks again for helping.
Michael
Alan Munn wrote:
> At 10:40 AM +0200 4/16/07, Tobias Sebastian Kuhn wrote:
>> Am 16.04.2007 um 05:21 schrieb Michael Millett:
>>
>>> I was hoping someone on this list has experience using Lilypond
>>> (music notation program) with TeXShop, and could help me.
>>>
>>
>>
>> Look here:
>> http://www.dimi.uniud.it/~vitacolo/freesoftware.html#desktoppub
>> Scroll down a bit, there are engines, they may work:
>>
>
> Michael, (and Daniel) your original example posted works fine using
> the Lilypond.engine from the link posted by Tobias.
>
> (Assuming Lilypond is installed in /Applications)
>
>
>
> Save the file in ~/Library/TeXShop/Engines as Lilypond-LaTeX.engine
> without the .txt extension.
>
> Open a terminal window and type:
>
> chmod +x ~/Library/TeXShop/Engines/Lilypond-LaTeX.engine
>
> Quit out of TeXShop (if it's running) and open it up again.
>
> Open your file, and in the pull-down menu that allows you to choose
> the engine, you should see Lilypond-LaTeX. Choose that and then click
> on the typeset button. Your document should appear properly.
>
> If you put
>
> %%!TEX TS-program = Lilypond-Latex
>
> as the first line of your file, TeXShop will choose the engine
> automatically.
>
> Alan
>
>
------------------------- Helpful Info -------------------------
Mac-TeX Website: http://www.esm.psu.edu/mac-tex/
TeX FAQ: http://www.tex.ac.uk/faq
List Archive: http://tug.org/pipermail/macostex-archives/
List Reminders & Etiquette: http://www.esm.psu.edu/mac-tex/list/
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2023-02-03 11:02:10
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https://integrativemodeling.org/nightly/doc/ref/classIMP_1_1modeller_1_1ModelLoader.html
|
IMP Reference Guide develop.c9d213c767,2020/05/28 The Integrative Modeling Platform
Read a Modeller model into IMP. More...
Inherits object.
## Detailed Description
Read a Modeller model into IMP.
After creating this object, the atoms in the Modeller model can be loaded into IMP using the load_atoms() method, then optionally any Modeller static restraints can be read in with load_static_restraints() or load_static_restraints_file().
This class can also be used to read Modeller alignment structures; however, only load_atoms() will be useful in such a case (since alignment structures don't have restraints or other information).
Note
This class is only available in Python.
Definition at line 720 of file modeller/__init__.py.
## Public Member Functions
def __init__
Constructor. More...
Load the Modeller angle topology into the IMP model. More...
Construct an IMP::atom::Hierarchy that contains the same atoms as the Modeller model or alignment structure. More...
Load the Modeller bond topology into the IMP model. More...
Load the Modeller dihedral topology into the IMP model. More...
Convert Modeller dynamic restraints into IMP::Restraint objects. More...
Load the Modeller improper topology into the IMP model. More...
Convert the current set of Modeller static restraints into equivalent IMP::Restraints. More...
Convert a Modeller static restraints file into equivalent IMP::Restraints. More...
## Constructor & Destructor Documentation
def IMP.modeller.ModelLoader.__init__ ( self, modeller_model )
Constructor.
Parameters
modeller_model The Modeller model or alignment structure object to read.
Definition at line 732 of file modeller/__init__.py.
## Member Function Documentation
Load the Modeller angle topology into the IMP model.
Definition at line 821 of file modeller/__init__.py.
Construct an IMP::atom::Hierarchy that contains the same atoms as the Modeller model or alignment structure.
IMP atoms created from a Modeller model will be given charges and CHARMM types, extracted from the model. Alignment structures don't contain this information, so the IMP atoms won't either.
Parameters
model The IMP::Model object in which the hierarchy will be created. The highest level hierarchy node is a PROTEIN.
Returns
the newly-created root IMP::atom::Hierarchy.
Definition at line 744 of file modeller/__init__.py.
Load the Modeller bond topology into the IMP model.
Each bond is represented in IMP as an IMP::atom::Bond, with no defined length or stiffness. These bonds are primarily useful as input to IMP::atom::StereochemistryPairFilter, to exclude bond interactions from the nonbonded list. Typically the contribution to the scoring function from the bonds is included in the Modeller static restraints (use load_static_restraints() or load_static_restraints_file() to load these). If you want to regenerate the stereochemistry in IMP, do not use these functions (as then stereochemistry scoring terms and exclusions would be double-counted) and instead use the IMP::atom::CHARMMTopology class.
You must call load_atoms() prior to using this function.
Returns
A generator listing all of the bonds.
Definition at line 788 of file modeller/__init__.py.
Load the Modeller dihedral topology into the IMP model.
Definition at line 827 of file modeller/__init__.py.
def IMP.modeller.ModelLoader.load_dynamic_restraints ( self, pair_filter = None )
Convert Modeller dynamic restraints into IMP::Restraint objects.
For each currently active Modeller dynamic restraint (e.g. soft-sphere, electrostatics) an equivalent IMP::Restraint is created. load_atoms() must have been called first to read in the atoms that the restraints will act upon.
If pair_filter is given, it is an IMP::PairFilter object to exclude pairs from the nonbonded lists used by the dynamic restraints. Otherwise, an IMP::atom::StereochemistryPairFilter object is created to exclude Modeller bonds, angles and dihedrals, as specified by edat.excl_local. (Note that this calls load_bonds(), load_angles() and load_dihedrals(), so will create duplicate lists of bonds if those methods are called manually as well.)
Note
Currently only soft-sphere, electrostatic and Lennard-Jones restraints are loaded.
Returns
A Python generator of the newly-created IMP::Restraint objects.
Definition at line 897 of file modeller/__init__.py.
Load the Modeller improper topology into the IMP model.
Definition at line 833 of file modeller/__init__.py.
Convert the current set of Modeller static restraints into equivalent IMP::Restraints.
load_atoms() must have been called first to read in the atoms that the restraints will act upon.
Returns
A Python generator of the newly-created IMP::Restraint objects.
Definition at line 864 of file modeller/__init__.py.
Convert a Modeller static restraints file into equivalent IMP::Restraints.
load_atoms() must have been called first to read in the atoms that the restraints will act upon.
Parameters
filename Name of the Modeller restraints file. The restraints in this file are assumed to act upon the model read in by load_atoms(); no checking is done to enforce this.
Returns
A Python generator of the newly-created IMP::Restraint objects.
Definition at line 849 of file modeller/__init__.py.
The documentation for this class was generated from the following file:
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2020-05-28 18:45:30
|
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|
https://www.nature.com/articles/s41467-022-32315-y?error=cookies_not_supported&code=2055ac15-6731-418d-b931-1ff05b4bef62
|
## Introduction
Most viruses utilize the host transcription apparatus to express their genes, and viral genomes contain assorted cis- and trans-elements that manipulate the host transcription machineries1,2. Most early genes of lambdoid phages are preceded by transcription terminators; therefore, the host transcription apparatus must be converted to a terminator-resistant form to promote full gene expression of viral genes1,3,4. For λ, anti-termination is promoted by the virus-encoded N protein, which binds to the cis-acting nut sites and suppresses transcription termination5,6,7. By contrast, bacteriophage HK022, discovered in Hongkong in the early 1970s, is related to λ phage but only requires cis-acting RNAs, named put (polymerase-utilization), to promote read through of transcription terminators without any dedicated trans-acting protein factors8,9. Put-mediated anti-termination is efficient and robust, and has been shown to suppress both intrinsic and ρ-dependent transcription terminations9,10,11.
The HK022 genome contains two putRNAs—putL and putR, located downstream of the early promoters PL and PR, respectively11. Both putRNAs are ~70-nt long, share sequence similarity, and are composed of two stem-loop structures12. Whereas putL is located just downstream of the PL promoter, putR is located ~270-nt downstream of the PR transcription start site11. The anti-termination activity of the putRNAs persists even on terminators located at least 10 kb away and is not dependent on the tethering between the putRNA and the elongation complex (EC) via the nascent transcript, suggesting that putRNA itself can remain stably associated with the EC as elongation proceeds13. The activity of putRNA is blocked by the host RNA polymerase (RNAP) mutations that are located exclusively in β’ zinc-binding domain (β’ZBD)10,14,15. This genetic evidence together with biochemical evidence suggests that putRNA interacts with the RNAP via the β’ZBD16.
In addition to anti-termination activity, putRNA exhibits anti-pausing activity. PutL inhibits backtracking at a pause located 21-nt downstream of the second stem (stem II) of putL, and this activity is abrogated by the insertion or deletion of several bases between the putL and the pausing site17. The dependency on the distance between putRNA and its location of action suggests that the anti-pausing and anti-termination activities of putRNA differ mechanistically. Interestingly, putRNA reduces both backtrack and hairpin-dependent pauses like RfaH18. RfaH, a paralog of NusG, recognizes an ops (operon polarity suppressor) sequence on the non-template DNA strand loaded onto the RNAP EC, changes its C-terminal helices into a β-sheet KOW domain fold to become active, and inhibits transcriptional pausing by resisting RNAP swiveling19,20.
A structural analysis of the putRNA-associated EC is required to understand the molecular mechanism of anti-pausing and anti-termination activities of the putRNA. In this study, we synthesized the putL RNA using the Eco RNAP σ70-holoenzyme initiated from the native HK022 PL promoter, and captured the modified ECs using a transcription roadblock for cryo-EM analysis. We observed putRNA-associated EC (putEC), putRNA-absent EC (put-less EC), and σ70-bound EC that contains intact putRNA (σ70-bound putEC). Comparison between putEC and put-less EC structures revealed that the putRNA binding to the β’ZBD hinders pausing by reducing the swiveling motion of the EC. Additionally, the σ70-bound putEC structure suggested that σ70 binding to EC might facilitate RNA folding as well as play a role in transcription modulation.
## Results
### Preparation and examination of putEC
Because an active form of HK022 putRNA can be produced only by the enzymatic synthesis using host RNAP, we prepared the putRNA-associated EC by initiating RNA synthesis with Eco RNAP holoenzyme and stalling the synthesis using a roadblock protein LacI (lac repressor), as previously described with some optimization for cryo-EM study (Fig. 1)16. Briefly, we first synthesized a DNA scaffold including HK022 PL promoter, the putL sequence, and the lacO sequence (LacI binding sequence) (Fig. 1a). Hereafter, we will use ‘putRNA’ to denote putL RNA for convenience. Holoenzyme containing Eco core RNAP and σ70 was added to the DNA scaffold to form an open complex, and LacI was added to bind to the lacO sequence on the DNA as a roadblock (Fig. 1b). Upon rNTP addition, RNAP synthesized the putRNA and stalled on the DNA at the roadblock. Excess rNTP was then removed by gel filtration column to prevent further RNA synthesis, and isopropyl β-D-thiogalactopyranoside (IPTG) was added to release the LacI from the DNA scaffold. The complexes were concentrated for cryo-EM grid preparation as well as native mass spectrometry (nMS) analysis.
To stall the EC at a site where the RNAP pauses in the absence of putRNA, we generated multiple DNA scaffolds having various distances between the pausing site and the lacO sequence, and performed radiolabeled transcription assay with these scaffolds. For screening, we used a put DNA scaffold that allows transcriptional pausing at the pausing site as a control (Fig. 1c, Supplementary Note 1, Supplementary Fig. 1, Supplementary Table 1)11. From the screening, we chose a scaffold that has a 7-nt spacer between the pausing site and the lacO sequence (7-nt scaffold), and then attempted to analyze the assembled putEC by nMS using the same workflow as in our previous nMS studies of bacterial ECs21,22,23. However, we were unable to observe the fully assembled putEC due most likely to sample instability or sample heterogeneity and adduction on the long, exposed nucleic acid scaffold during nMS analysis. Nevertheless, nMS analysis of the RNAs extracted from the reconstituted putEC revealed two main populations—one RNA was synthesized until the known pausing site (C94), and the other RNA extended by 1-nt (U95) (Fig. 1d)11,17. The quantity of the U95 RNA was roughly 1.5 times more than the C94 RNA; therefore, we modeled the U95 RNA placing the U95 nucleotide at the i + 1 site. Since the C94 and U95 nucleotides are located at the i and i + 1 sites, the modeled structures would exhibit the same RNA-DNA register both for the C94 and the U95 RNA-containing ECs at the post- and pre-translocated states, respectively (See below).
### Cryo-EM structures of the putEC in three conformations
Cryo-EM analysis of the putEC prepared by promoter-dependent transcription initiation, transcription elongation, and LacI roadblocking revealed three EC populations at sub-4 Å resolution: (1) the putEC (3.2 Å-resolution, 36.9% of the EC particles) that contains well-folded putRNA, (2) the put-less EC (3.6 Å, 22.4%) that does not display any well-defined putRNA density and (3) the σ70-bound putEC (3.6 Å, 40.7%) that contains both σ70 and putRNA (Table 1, Supplementary Figs. 2 and 3). We also observed a population consisting of the RNAP holoenzyme loaded onto the template DNA. This population probably resulted from abortive initiation and generated a 3.0 Å-resolution map. This complex is not discussed here because the structures of the holoenzyme open complex have been described in previous reports24,25,26.
In the cryo-EM structure of the putEC, the putRNA was located at the opening of the RNA exit channel of the EC adjacent to the β’ZBD (Fig. 2a). This location is consistent with the put-inactivating RNAP mutations and potentially would restrict the RNA hairpin formation in the adjoining RNA exit channel via electrostatic repulsion (Supplementary Fig. 4)10,27. The quality of the cryo-EM map allowed us to build the highly-structured putRNA de novo (estimated local resolution of the cryo-EM map around the putRNA was ~3.5 Å; Fig. 2b, c, Supplementary Note 2, Supplementary Figs. 57). The modeled putRNA from U2* to U74* contains twenty-one Watson-Crick (WC) base pairs and five non-canonical base pairs, A9*-G35* (Saenger class VIII), G12*-U32* (Saenger class XXVIII), G43*-A64* (Saenger class XI), G42*-U65* and C44*-A63* (Supplementary Fig. 8). To distinguish the nucleic acid residues of the putRNA from the amino acid residues in the RNAP, we have added an asterisk (*) to the residue number of the putRNA throughout this manuscript. Surprisingly, the cryo-EM structure of the putRNA was different from previously published data11,12 as follows (Fig. 2b–d): First, the 5’-end of the putRNA is not C10* but A3*. In the structure, the putRNA region from A3* to G7* makes an RNA duplex with the opposite strand from U19* to C15*. Interestingly, this corresponds to the result of the putL V1 RNase reaction, which suggested the presence of RNA duplex in the upstream of C10*12. Furthermore, this RNA duplex interacts with another RNA strand from U21* to C25* forming an unexpected minor groove RNA triplex structure28. The deletion of the third RNA strand, Δ20*−23*, decreased the anti-termination activity of the putRNA by ~50%12, indicating that the triple helix region has a significant effect on the function of the putRNA. Second, G35*, which was expected to be located at the bifurcation point of the two stem regions in an unpaired state, base pairs with A9*. This G35*-A9* base pair provides a platform for β’ZBD binding and stabilizes the overall structure of the putRNA. This base pair explains why the G35*U mutation retained 70% of the anti-termination activity while the G35*A mutation completely abolished the activity12. Meanwhile, A9*C mutation abolished the in vivo anti-termination activity, implying complicated effects of mutations on the G35*-A9* base pair12. Third, the putRNA contains a bulged loop region (from C26* to G29*) in the middle of stem I, in contrast to the prediction that stem I has a loop region at the end of the stem I ranging from C18* to C26*. This region also provides an interface for binding to the RNAP. At last, the middle region of stem II exhibits distinct base pairings compared to the predicted structure. The middle region of stem II in the cryo-EM structure contains three non-canonical base pairs with three unpaired bases instead of having one non-canonical base pair with five unpaired bases in the predicted structure. This region has relatively high local resolution indicating its structural stability, and makes interfaces with RNAP and the stem I of the putRNA.
### Interactions between the putRNA and the EC
In the putEC structure, the ‘V’-shaped putRNA binds to the prominent β’ZBD by its pothole formed in the center of the ‘V’ (Fig. 3a). The β’ZBD fits snuggly to the putRNA surface, generating a 1130.3 Å2 interface area formed by ~33% of the total putRNA residues29. At the backside of the putRNA-β’ZBD interface, the N-terminal loop of the β flap-tip helix makes significant contact with the putRNA with an interface area of 281.6 Å2. Most of the potential interactions between the putRNA and the RNAP comprise polar interactions such as salt bridges, hydrogen bonds, cation-π interactions, and long-range ionic interactions. Although the resolution of the map is not sufficient to specify these short-range interactions, we suggested possible interactions for reference (Figs. 2c, 3b, Supplementary Table 2). At the bifurcation point of the two stem structures, β’R77 locates like a wedge to separate G35* and G36* and forms a cation-π interaction with G35*. This cation-π interaction is often found between the terminal, exposed base of a nucleic acid bound to a protein and the protein loop that confines the nucleic acid. In addition, β’L78 and β’K79 are located between G35* and G36*, stabilizing the separation of stem I and stem II of the putRNA.
A mutant named put, or mutant G, has the sequence A43GAUC47 and does not exhibit anti-termination activity11. Our transcription assay revealed that put also has poor anti-pausing activity (Fig. 1c). In the structure, this region does not directly interact with the RNAP, however, its counter-strand, from the 59th to 64th residues, forms a central area of the binding interface. Therefore, the base substitutions in the put mutant likely change the structure of the binding interface and disrupt putRNA binding to the RNAP. It is also possible that these mutations interfere with the proper folding of the putRNA as well.
To test the validity of the structure of the putEC modeled in the cryo-EM density, we introduced assorted mutations in the template DNA and performed in vitro radiolabeled transcription assays to examine the effects of the mutations on the anti-pausing activity (Fig. 3c, Supplementary Fig. 9). For the quantification of the anti-pausing activities of the mutants, the anti-pausing activities of wild-type put and put were set to 1 and 0, respectively, and the anti-pausing activity of each mutant was located on a linear scale accordingly (Details are in the Methods section). To display the location of the mutated residues as well their conservation, the conservation of the putRNA residues was calculated from the sequence alignment with ten known put sequences and marked by color (Fig. 3d, Supplementary Fig. 10a)30,31. Among the twenty-three mutations we generated, eleven mutants showed ≤ 20% anti-pausing activity (named ‘inactivating’ mutations) and three mutants showed ≥ 90% anti-pausing activity (named ‘inert’ mutations). The inactivating mutations, Δ3*−7*, U28*A, U28*C, G35*A, G35*U, G35*C, G45A*, A64*G, G35*C/A9*G, G35*A/A9*G, and G35*A/A9*U, suggest that (1) the 5’-region (from A3* to G7*) is essential for the anti-pausing activity. A3GACG7 and its base-pairing region, U19CUGC15 have relatively high conservation scores of (6,6,4,9,9) and (7,7,5,10,10), respectively. This region is the first RNA duplex formed during the putRNA synthesis, and therefore, may provide a platform for further RNA folding. (2) U28*, which protrudes toward the β’ZBD and binds to a small pocket is essential for the function. Interestingly, while U28*A and U28*C abolish the anti-pausing activity, U28*G retained ~60% of the activity. From the structure, we substituted the U28* with the other bases and found that G can form three hydrogen bonds with the surrounding β’ residues while A and C form two and one potential hydrogen bonds, corroborating the result of the mutational study (Supplementary Fig. 10b). Interestingly, the original put residue, U28* forms fewer hydrogen bonds than guanine and adenine, but exhibits better activity than these, implying that U28* might have additional role(s) besides binding to the RNAP, or the mutants might have different structures from the modeled ones (Discussed below). (3) All of the G35* mutations we generated abrogated the anti-pausing activity of putRNA. We expected that the double mutants, G35*C/A9*G, and G35*A/A9*G might have some activity because they preserve the predicted base-pairing of G35*-A9* in the structure. However, mutating G35* to any base abrogated the anti-pausing activity and this was not recovered by the mutation of the base-pairing partner, implying that G35*, and possibly its base-pairing partner A9*, may have sequence-specific roles in the anti-pausing activity. We noticed that G35*U exhibited ~70% anti-termination activity in vivo12. This discrepancy could come from the different conditions encountered in vivo vs. in vitro. For example, the G35*U might form some intact or partially active putRNA in vivo, possibly aided by an unknown cellular factor(s) whereas in vitro synthesized putRNA containing G35U* could be inactive. (4) We also found that A64* is critical for the anti-pausing activity. This result is also consistent with the structural data because it contacts the stem I region of putRNA and the RNAP. All the inert mutations are of U20*, which lacks any significant interaction with other residues, supporting our structure. The remaining nine mutants exhibited moderate activities suggesting a significant, but not critical role of the residues (A8*, U21*, C25*, U32*, G43*). In summary, our mutagenesis study supports our cryo-EM structure of the putEC.
### The comparison of the putEC, the put-less EC, and other ECs
To determine if putRNA binding to the EC changes the conformation of the EC to suppress transcriptional pausing, we aligned the putEC with multiple EC structures including non-paused EC (PDB 6ALF), RNA hairpin-paused EC (PDB 6ASX), backtracked PEC (paused EC) (PDB 6RIP), and the put-less EC determined here (Fig. 4a, b, Supplementary Table 3)21,32,33. We assume that the put-less EC contains a roadblocked but unfolded RNA because (1) the majority (>~70%) of the sample was roadblocked properly (Fig. 1d), and both the putEC and the put-less EC together comprise ~60% of the EC population in the cryo-EM data, (2) the third EC class, σ70-bound putEC shows extra RNA duplex density connected to the putRNA suggesting that this class was not properly roadblocked, and (3) the put-less EC map contains some weak RNA density around the RNA exit channel and the β’ZBD, implying that the RNA is present, but it is not well-structured. We suggest this put-less EC could serve as a good negative-control model as shown in a previous study32.
We first examined the swiveled states of the ECs (Fig. 4b). Swiveling indicates the rigid-body rotation of a set of domains—the clamp, dock, shelf, jaw, SI3, and the C-terminal region of the β’ subunit—about an axis parallel to the bridge helix toward the RNA exit channel, and known to interfere with the proper folding of the trigger-loop which is required for efficient nucleotide addition to the nascent RNA. Swiveling was first introduced from the structural study of hisPEC, and later revealed in the backtracked PEC, implying that the swiveling motion potentially plays an important role in both RNA hairpin pause and backtrack pause32,33,34. The alignment of the EC structures according to the core module revealed that the putEC structure is most similar to the non-paused, active EC conformation, having the lowest RMSD values between Cɑ-carbons of domains as well as the smallest swiveling angle of 1.2° (Fig. 4b, Supplementary Table 3). The put-less EC is more swiveled than the putEC, having a swivel angle of 1.8°, although the swiveling angle of the put-less EC was less than that of the hisPEC or backtracked PEC (3.1° and 2.6°, respectively; Supplementary Table 3). Interestingly, the conformational difference between the putEC and put-less EC is more noticeable in the βSI2 (or βi9) region with 15 Å-distance between the Cɑ atoms of βE1006, which is located at the end of the βSI2 domain. While the swiveling motions of the aligned ECs are relatively continuous with the rotation angles from 1.2° to 3.1°, the arrangement of the βSI2 is more discrete – the βSI2 in the putEC overlaps with that of the non-paused EC while the βSI2 of the put-less EC is in the same location with the hisPEC. Interestingly, the βSI2 of the backtracked PEC is located between the two conformations. These conformational features suggest that the proper folding and binding of the putRNA to the EC moved the EC toward the non-swiveled, active state, aiding pause escape or omission.
The strength of the RNA-DNA hybrid influences pausing and termination35,36. Therefore, we compared the RNA-DNA hybrid of the putEC and the put-less EC (Fig. 4c, left). In the putEC, the active site region of the RNA-DNA hybrid exhibited a post-translocated state similar to the non-paused EC at the high threshold value of the map. As the threshold value decreases, the putEC map revealed a density blob for a nucleotide base that base pairs with the template DNA base at the i + 1 site. This density became connected to the nascent RNA at the lower density threshold. As stated above, we suspect that this results from the mixed population of the nascent RNAs roadblocked at either +94 or +95 position, having either post- or pre-translocated states, respectively. However, we did not observe any classes having a folded trigger-loop with the SI3 domain shifted closer to the βlobe domain as in the Eco RNAP structure of the pre-translocated state24. In addition, the putEC contained 11 template DNA bases in the RNA-DNA hybrid, in contrast to other reported EC structures (Fig. 4d). To contain one additional nucleotide in the main channel, the lid, which is known to aid the unwinding of the RNA-DNA hybrid, is pushed by about 2.6 Å (by the Cɑ atom of β’256D) compared to the known non-paused EC (Supplementary Fig. 11a)19,21,33. However, it is not certain if this 11-nt hybrid is just an alternative conformation of an EC, or a specific conformation in the putEC.
The put-less EC showed distinct RNA-DNA base-pairing at the i + 1 site (Fig. 4c, right). In the put-less EC, the template DNA base at the i + 1 site is more tilted toward the RNA base at the i site; therefore, it is not optimally placed for substrate binding. In fact, the RNA base at the i site is more closely associated with the DNA base at the i + 1 site than that of the i site. Consequently, the base-pairing hydrogen bonds are broken between the template DNA base and the nascent RNA base at the i site. The remaining region of the RNA-DNA hybrid of the put-less EC overlaps well with that of the non-paused, active EC as in the putEC. The conformational difference of the nucleotides at the active site between the putEC and the put-less EC indicates that putRNA binding to the β’ZBD influences the active site conformation, even though the catalytic magnesium ion is ~62.5 Å away from the zinc ion in the β’ZBD. This was also shown in the hisPEC structure, where the pause hairpin placed in the RNA exit channel has an influence on the active site as well as the bridge helix32,34,37. The length of the template DNA in the RNA-DNA hybrid of the put-less EC was also 11-nt, implying that this longer RNA-DNA hybrid is not caused by the putRNA.
In addition to these changes, we also observed that the RNAP domains of the putEC have similar locations to those of the non-paused EC while the domains in the put-less EC have a similar arrangement with those of backtracked PEC (Supplementary Table 4). Although we could not find any density for the backtracked RNA in the put-less EC, the pausing site was expected to have a backtrack pause. In summary, from the structures of the putEC, put-less EC, and other ECs, we found that the putRNA binding to the EC leads to the anti-pausing activity by promoting the active, non-swiveled conformation of the EC.
### σ70-bound putEC structure
The third EC population, σ70-bound putEC, contains a σ70 bound to the clamp helices in addition to the well-folded putRNA as in the putEC (Fig. 5a, Supplementary Fig. 2). In contrast to the holoenzyme structure, the σ70-bound putEC map reveals only σ1.2, σNCR and a part of σ2, indicating that the σ2 binding to the EC is relatively stable while the other σ domains are very mobile as predicted in a prior study38. It has been reported that σ70 can remain associated with RNAP after promoter escape and the association is enhanced when the non-template DNA contains a −10 element-like sequence in the promoter-proximal region that induces σ-dependent pausing39,40,41. In particular, σ-dependent pausing provides a time and space window for the anti-termination λQ protein to bind to the EC and read through the intrinsic terminator42,43. Recently, cryo-EM structures of σ70-bound ECs were reported in the context of 21Q-, λPR’-, and Qλ-associated ECs44,45,46,47,48. While these complexes are at the paused state in that the σ2 domain interacts with a −10-like sequence, our σ70-bound putEC is not in a σ-dependent paused state and contains > 100 base-long RNA having a σ70 in a different conformation from those in other σ70-bound ECs (Fig. 1d).
In the σ70-bound putEC structure, we noticed that the RNAP contains an open clamp (79.3 Å opening), which is ~20 Å larger than the non-paused EC23. This suggests that the σ70-bound putEC is in an inactive state. We suspect that this class might represent the partial run-off EC population that appeared in the nMS analysis (Fig. 1d) because (1) the main channel of the RNAP did not contain downstream duplex DNA while the RNA-DNA hybrid was present and (2) an RNA duplex density, which is connected to both putRNA and the RNA-DNA hybrid, was observed in the RNA exit channel, indicating that the RNA was transcribed beyond the roadblock site (Fig. 5a, Supplementary Fig. 11b). We used the RNAfold Server to search potential RNA secondary structures in the template DNA and found that it contains a potential RNA hairpin sequence downstream of the roadblock site (Supplementary Fig. 11b)49. We, therefore, modeled the RNAP and nucleic acid scaffold into the map and found that the potential RNA hairpin matches well with the extra density observed in the RNA exit channel (Fig. 5a, Supplementary Fig. 11b). The location of this extra RNA duplex overlaps with the pause hairpin in the hisPEC32,34.
The putRNA density in the σ70-bound putEC was at a lower resolution than that in the putEC; however, the putRNA map region was identical to that in the putEC. The RMSD of the whole atoms in the putRNA region in the σ70-bound putEC and the putEC was only 0.839 Å. To compare the σ2-RNAP interaction in the initiation and the elongation stages, we aligned the RNAP clamp-σ2 domain regions from the σ70-bound putEC and the recently published RPo (RNAP-promoter open complex) structure25. For the σ70-bound putEC, we only modeled the visible part for the σ7070 residues 112–151 and 214–447). Then, we compared the two structures only via the modeled σ70 regions and other σ domains were excluded in the comparison discussed below. Not surprisingly, the binding interface between the σ70 and the RNAP, in particular, the β’clamp domain, was different between the RPo and the σ70-bound putEC (Fig. 5b). The binding interface between the β’ subunit and the σ70 was 812 Å2 in the RPo and the interface mostly occurs on the β’clamp helices. By contrast, in the σ70-bound putEC, the interface area was 1287 Å2. This unexpected increase in the binding area results from the newly-formed interface between β’-clamp-toe domain (ranging 144–179)50 and the σ70NCR, the non-conserved σ70 region between σ1.2 and σ2.1 (ranging 274–307 and 359–374 in the structure, Fig. 5b) that does not participate in the RNAP-σ70 interface in the RPo. Since both β’-clamp-toe and σ70NCR are conserved in the γ-proteobacteria, the interaction between these two domains might be specific for the bacteria class. In addition, the shifted position of the σ2 domain in the σ70-bound putEC is more suitable for the σ70 to associate with the progressing EC because this conformation provides space for the upstream DNA to rewind and exit from the main channel of the RNAP. If the σ70 is bound to the RNAP as in the holoenzyme, σ70 would clash with the exiting upstream duplex DNA. However, at the moment, further investigation would be required to see whether these new interactions between the σ70 and the RNAP in the σ70-bound putEC are due to the transcription stage transition from initiation to elongation, or to the clamp opening which inactivates the transcription activity of the RNAP.
Additionally, we found a low-resolution blob in the main channel for the downstream DNA (Fig. 5a). The DNA scaffold used in the study spans to +122 position while the RNA modeled in this map ends at +105. nMS analysis revealed three RNA populations of 110-mer, 114-mer and 116-mer (Fig. 1d). Therefore, there should be some downstream duplex DNA around the RNAP. However, the low-resolution of the blob prevents us from locating any specific molecule in the density. We suspect that the blob could be either from the downstream duplex DNA, which is very mobile due to the open clamp conformation, or from the σ701.1 because the σ701.1 is known to bind at the position in the holoenzyme before the enzyme binds to promoter DNA. We would need further investigation to confirm this speculation.
## Discussion
In this study, we extended prior studies on the putRNA by determining its three-dimensional structure when complexed with RNAP. Our result corroborates previous analyses suggesting a two-stem structure with multiple indents and bulges. However, cryo-EM structures also revealed new and unexpected features such as an unexpected boundary of the put transcript, a short triple RNA helix in the putL stem I, and alternative base pairs. The importance of many of these features is strengthened by the observed effect of specific put mutations11,12. The structure provides clear physical evidence that the putRNA binds to the β’ZBD, a result that is strongly supported by prior genetic and biochemical experiments on putRNA. The structure also revealed a mechanistic explanation for the anti-pausing activity promoted by putRNA-RNAP interaction. When putRNA is bound to the β’ZBD, the RNAP is held in a non-swiveled, active conformation, which is associated with anti-pausing activity as previously shown in the RfaH-associated EC19. In contrast, a put-less EC exhibited a swiveled conformation suggesting that the EC is in a paused state when transcription elongation is physically blocked at a pausing site. Together, the structures revealed that putRNA promotes RNA synthesis by resisting swiveling.
We were surprised to observe a putEC population that retained σ70 even though the EC had progressed about 100 nucleotides from the start of transcription. The occurrence of the σ70-bound putEC and its structure suggests a few intriguing points. First, the σ70-bound EC successfully folded putRNA, even more efficiently than a complex lacking σ70. We found that the ratio between the putEC and the put-less EC is roughly 2:3 from the number of particles in each class, presumably reflecting the success rate of putRNA folding in vitro. Curiously, there was no put-less σ70-bound EC, suggesting that the presence of σ70 aided the proper folding and stabilization of putRNA. We found that the putL sequence contains a weak −10-like sequence (NANNAT) located at positions +23 to +28 relative to the start of the putL transcript, which lies on the third strand of RNA triple helix and a bulge region of stem I (Figs. 1a, 2b, c). A −10-like sequence is known to induce σ-dependent pausing by engaging its non-template DNA region with the σ2 domain51. We suggest that this sequence may cause σ-dependent pausing which facilitates putL folding by providing more time. Notably, among the ten putRNA sequences we aligned, all the putLs contain the identical −10-like sequences while all putRs do not (Supplementary Figs. 10a, 11c). Therefore, we speculate that σ-dependent pausing may be necessary for putL folding but not for putR which is located further downstream of its promoter. In addition, U28* is completely conserved in putL and the critical 6th residue of the −10-like sequence. Although U28*G exhibited intermediate activity in our mutagenesis study, U28*A and U28*C nearly abolished activity, supporting the existence and importance of σ-dependent pausing at this position. Furthermore, we examined the ratio between σ70-bound EC and σ70-unbound EC from both the putEC sample and the put-EC to see if the presence of put affects the σ70 retention (Supplementary Fig. 12). The put-EC was prepared in exactly the same way as the putEC preparation except the put template was used as the DNA scaffold and did not show any well-folded putRNA density in the cryo-EM maps. From the cryo-EM data analysis, the percentages of σ70-bound EC in the putEC (having intact put) and the put-EC sample were ~40.7% and ~44.2%, respectively, suggesting that the presence of put does not affect σ70 retention. Second, the σ70-bound EC was resistant to the LacI roadblock. The σ70-bound putEC revealed an extra density for a duplex RNA in the RNA exit channel, suggesting that the retained σ70 modified the EC to overcome the roadblock during elongation (Figs. 1d, 5a, Supplementary Fig. 11b).
Structural studies on prokaryotic anti-termination complexes including λN, Q21, Xoo P7, Qλ, and HK022 put suggest general strategies for anti-termination7,44,45,47,48. (1) The anti-termination factors inhibit RNA hairpin formation by either narrowing the channel or hindering the RNA hairpin folding (Supplementary Figs. 4, 13). The RNA exit channel is thought to aid RNA hairpin formation by its positively-charged residues located inside the channel32. In Q21, Qλ and Xoo P7 anti-termination complex, the anti-termination factors, Q21, Qλ, and P7 proteins bind at the mouth of the RNA exit channel and confine the channel (Supplementary Fig. 13). The narrowed RNA exit channel only allows single-stranded RNA to move through it and restricts nascent RNA folding for hairpin-dependent pausing and intrinsic termination. In λN anti-termination complex, λN binding to the EC remodels the bound NusA and NusE to destabilize the RNA hairpin folding. In addition, the rearranged Nus factors bind to β flap-tip, which stabilizes RNA hairpin pause, possibly preventing the flap-tip from assisting RNA hairpin pausing and termination52. Like λN, HK022 put also does not narrow the RNA exit channel directly. Instead, the phosphate backbone of the putRNA is located near the RNA exit channel, prohibiting the RNA hairpin formation with its negative-charged surface. Modeling an RNA duplex in the RNA exit channel of putEC shows that the phosphate backbones of the modeled RNA duplex and the putRNA are just ~5 Å apart from each other (Supplementary Fig. 4). In addition, the β flap-tip binds to the putRNA, possibly sequestering it from assisting RNA hairpin pause as in the λN-anti-termination complex. (2) In general, anti-termination proteins stabilize the elongation-proficient conformation of EC. λN transverses the RNAP hybrid cavity stabilizing the active form of the EC and binds to the upstream duplex DNA, enhancing the anti-backtracking and anti-swiveling activity of NusG. In the Q21-EC structure, Q21 binding is not compatible with swiveled conformation. Therefore, Q21 counteracts swiveling, leading to anti-pausing47. Our data suggest that putRNA also reduces swiveling. This stabilization of the active form of an EC may consolidate the RNA exit channel so that it can no longer accommodate the folding of secondary structures that promote pausing and termination53,54,55.
Komissarova et al.17, found that ΔU68* does not suppress termination, but retains anti-pausing activity in vitro. U68* is located at the lower region of stem II like a wedge, forming no base-pairing. According to our modeling, the presence of U68* kinks the stem II ~19° (Supplementary Fig. 14). This perturbation might weaken the stability of the putRNA folding by widening the space between the two stem-loop structures. In addition, the structural change would affect the interface between the putRNA and the RNAP because the interface is composed of putRNA residues from both stem I and II. Therefore, putRNA without U68* might be well-folded and reduces pausing immediately after synthesis but may unfold or dissociate from the RNAP before encountering terminators located further downstream. Alternatively, the mutant RNA may not be able to adopt an anti-terminating structure which could be different from the anti-pausing structure in vitro.
In λ phage paradigm, the anti-termination factor λN plays the role of gatekeeper for the infection process. In other words, λN accumulation is required to transcribe early genes of the genome. HK022, instead, has Nun protein, which competes with λN and blocks λ transcription. In addition, the HK022 genome harbors the put element in the place for the λ nut (N-utilization) sites, which are required for the action of λN. By substituting the N protein with Nun, HK022 acquired immunity against its competitor, λ. HK022, instead, lacks a λN-like anti-termination factor, but relies solely on the putRNA to promote full expression of its early genes. These differences benefit HK022 survival, without increasing transcription regulation complexity.
In this study, we investigated the anti-pausing mechanism of putRNA. Since transcriptional pausing is a prerequisite of transcriptional termination, our results provide important insights into the mechanism of putRNA action. It remains possible that putRNA may adopt different structures and/or interactions with RNAP to promote anti-termination as prior studies indicate that anti-pausing and anti-termination activities differ. To deepen our understanding of these events, structural studies on the putRNA-associated EC at a terminator sequence would be required.
## Methods
### Protein expression and purification
Full-length Eco σ70 was expressed from pET21-based expression vector encoding an N-terminal hexa-histidine tag followed by a PreScission protease (GE healthcare) cleavage site. The full-length Eco σ70 plasmid was transformed BL21(DE3) cells and grown at 37 °C. Protein expression was induced at an OD600 of 0.7 with 1 mM IPTG and incubated for 4 hours at 30 °C. Cells were harvested, resuspended in σ70 lysis buffer (20 mM Tris pH 8.0, 500 mM NaCl, 5% Glycerol, 5 mM Imidazole, home-made protease inhibitor cocktail) and lysed by French Press. The supernatant was loaded to Hitrap IMAC HP column (Cytiva) equilibrated with 20 mM Tris pH 8.0, 500 mM NaCl, 5% glycerol. The eluted protein by adding imidazole gradient was concentrated using Amicon Ultra centrifugal filter (Merck Millipore) and injected to HiLoad 16/600 Superdex 200 pg (Cytiva) equilibrated in TGED + 500 mM NaCl. The final elution was flash-frozen using liquid nitrogen after adding 15% glycerol.
Lac repressor (LacI) was purified as described previously57. LacI-containing pBAD plasmid with Kanamycin resistance (pBAD_Kan-LacI) was obtained from Addgene (plasmid #79826). BL21(DE3) cells that were transformed with the plasmid were grown overnight at 37 °C in 2X YT media containing 50 μg/mL Kanamycin. The seed culture was added to 2× YT media containing 50 μg/mL Kanamycin at 1:100 ratio, grown at 32 °C for 2 hours, and moved to 16 °C. Protein expression was induced with 0.2% l-arabinose for 16 hours incubation right after changing the temperature to 16 °C. Cells were harvested and lysed by French Press in lysis buffer (50 mM sodium phosphate buffer pH 8.0, 500 mM NaCl, 20 mM Imidazole, 2.5% glycerol, 1 mM DTT, 10 mM MgCl2, 0.1% Tween-20, 1 mg/mL lysozyme, home-made protease inhibitor cocktail). The lysate was added by 1000 U of DNaseI, and centrifuged to remove cell debris. The supernatant was loaded onto Hitrap IMAC HP (Cytiva) that pre-equilibrated with 50 mM sodium phosphate buffer (pH 8.0), 500 mM NaCl, 20 mM imidazole, 2.5% glycerol, and 0.2 mM DTT. Protein was eluted with 20 mM sodium phosphate buffer (pH 7.4), 300 mM NaCl, imidazole gradient from 30 to 300 mM and concentrated using 30 K MWCO Amicon Ultra Centrifugal Filter (Merck Millipore). The concentrated protein was injected onto HiLoad 16/600 Superdex 200 pg (Cytiva) gel filtration column equilibrated with 20 mM Tris-HCl (pH 8.0), 150 mM KCl, 5 mM MgCl2, and 1 mM DTT. The final eluted protein was added by 15% glycerol, flash-frozen, and stored at −80 °C until use.
### Radiolabeled in vitro transcription assay
In vitro transcription assay is performed as described previously58. Holoenzyme was reconstituted by mixing Eco RNAP and Eco σ70 with 1:2 molar ratio, and incubating for 15 min at 37 °C. Holoenzyme and DNA were mixed with 4:1 molar ratio in glutamate-based T buffer (20 mM Tris-glutamate pH 8.0, 10 mM Mg-glutamate, 150 mM K-glutamate, 5 mM DTT), and incubated at 37 °C for 10 min to make RPo. RPo and LacI were mixed with 1:10 molar ratio and incubated at 37 °C for 10 min. The final concentration of holoenzyme and template DNA in the reaction mixture was 50 nM and 12 nM, respectively. Transcription was started by adding rNTP mix to final concentrations of 200 µM ATP, 200 µM UTP, 200 µM GTP, 25 µM CTP (Cytiva) and 0.05 µM α-32P-CTP (PerkinElmer) at 37 °C, and quenched after 2 min by adding 2× loading buffer (10 M Urea, 50 mM EDTA pH 8.0, 0.05% bromophenol blue, 0.05% xylene cyanol). To show the roadblocked EC is capable of further transcription, the roadblocked EC was added by 2 mM IPTG, incubated for 2 min at 37 °C for LacI dissociation, and added additional rNTP to final concentrations of 162 µM ATP, 162 µM UTP, 162 µM GTP, 75 µM ATP and 0.15 µM α-32P CTP. The samples were loaded on 10% Urea-PAGE gel and ran in 1X TBE. The gel was exposed to an imaging plate (Fujifilm) for 2 hr, and the imaging plate was scanned to get an image (TyphoonTM FLA 7000).
For the mutational study, 50 nM holoenzyme and 12 nM template DNA were used for the transcription assay without roadblocking. In addition, the transcription reaction was quenched at 0-, 0.5-, and 2-min time point, and the data at 0.5 min were used to estimate the relative anti-pausing activity plotted in Fig. 3c although using 2-min data also showed similar result (data not shown). For the transcription reaction, 200 µM ATP, 200 µM UTP, 200 µM GTP, 25 µM CTP, and 0.05 µM α-32P CTP were used. For the estimation of the relative anti-pausing activity, we measured the intensities of the paused and the run-off transcripts of the put constructs, and calculated the fraction of the paused transcripts by dividing the intensity of the paused transcript by the sum of the intensities of the paused and run-off transcripts (Supplementary Fig. 9). The fractions of the paused transcripts were calculated for the wild-type put, inactive put, and mutant put constructs, and their relative anti-pausing activities were calculated by the equation below and plotted:
$${{{{{\rm{Relative}}}}}}\,{{{{{\rm{anti}}}}}}\mbox{-}{{{{{\rm{pausing}}}}}}\,{{{{{\rm{activity}}}}}}\,{{{{{\rm{of}}}}}}\,{{{{{\rm{mutant}}}}}}\,{{{{{\rm{x}}}}}}=1-\frac{({{{{{{\rm{P}}}}}}}_{{{{{{\rm{X}}}}}}}-{{{{{{\rm{P}}}}}}}_{{{{{{\rm{WT}}}}}}})}{({{{{{{\rm{P}}}}}}}_{put-}-{{{{{{\rm{P}}}}}}}_{{{{{{\rm{WT}}}}}}})}\\ ({{{{{{\rm{P}}}}}}}_{{{{{{\rm{x}}}}}}}={{{{{\rm{the}}}}}}\,{{{{{\rm{fraction}}}}}}\,{{{{{\rm{of}}}}}}\,{{{{{\rm{the}}}}}}\,{{{{{\rm{paused}}}}}}\,{{{{{\rm{band}}}}}}\,{{{{{\rm{of}}}}}}\,{{{{{\rm{mutant}}}}}}\,{{{{{\rm{x}}}}}})$$
For the paused fraction quantification, the intensities for the run-off and paused transcripts were calibrated according to the number of cytosines the transcripts contain. The assay was done in triplicate (n = 3 independent experiments).
### Native mass spectrometry analysis
The RNA portion of the de novo reconstituted putEC was prepared by phenol/chloroform extraction, resuspended in RNase-free water and flash-frozen in liquid nitrogen. Prior to analysis, the sample was thawed and then buffer-exchanged into nMS solution (500 mM ammonium acetate, 0.01% Tween-20, pH 7.5) using Zeba desalting microspin columns (Thermo Fisher). The buffer-exchanged sample was diluted to 5 µM with nMS solution and was loaded into a gold-coated quartz capillary tip that was prepared in-house. The sample was then electrosprayed into an Exactive Plus EMR instrument (Thermo Fisher Scientific) using a modified static nanospray source59. The MS parameters used were similar from previous work22: spray voltage, 1.2 kV; capillary temperature, 150 °C; S-lens RF level, 200; resolving power, 8750 at m/z of 200; AGC target, 1 × 106; number of microscans, 5; maximum injection time, 200 ms; in-source dissociation, 10 V; injection flatapole, 10 V; interflatapole, 7 V; bent flatapole, 6 V; high energy collision dissociation, 85 V; ultrahigh vacuum pressure, 6.6 × 10−10 mbar; total number of scans, 100. Mass calibration in positive EMR mode was performed using cesium iodide. Raw nMS spectra were visualized using Thermo Xcalibur Qual Browser (version 4.2.47). Data processing and spectra deconvolution were performed using UniDec version 4.2.060,61. The UniDec parameters used were m/z range: 2000–7000; mass range: 25,000–45,000 Da; sample mass every 0.5 Da; smooth charge state distribution, on; peak shape function, Gaussian; and Beta softmax function setting, 20. The expected masses for the de novo synthesized RNA include 94-mer (30,630 Da), 95-mer (30,936 Da), 110-mer (35,776 Da), 114-mer (37,038 Da), and 116-mer (37,671 Da). The mass deviations of the measured masses from the expected masses were within 1 Da or less.
### PutEC preparation and cryo-EM grid freezing
Holoenzyme was formed by mixing Eco RNAP and Eco σ70 with 1:2 molar ratio and incubating for 15 min at 37 °C, and purified in Superdex 200 Increase 10/300 Increase GL column (Cytiva). Template DNA was amplified in thermocycler. pRAK31 plasmid62 was used as template DNA for the PCR reaction. The forward and reverse primer sequences (from Macrogen) for the reaction are as follows; Forward primer-5’-GCATGAATTCCTATTGGTACTTTACATTAA-3’, Reverse primer-5’-CGAATTGTGAGCGCTCACAATTCTAAAAGCAAAAAAGCCTTC-3’. Holoenzyme and template DNA were mixed and incubated for 10 min at 37 °C to form RPo. After RPo reconstitution, LacI, which is also purified by size-exclusion chromatography before use, was added and incubated for 10 min for roadblocking. To the mixture, 1 mM rNTP (Cytiva) was added and incubated for 2 min at 37 °C for transcription. The sample was loaded onto zeba spin desalting column (Thermo Fisher) to remove free rNTP, and 2 mM IPTG was added to the complex. After 2 min incubation at 37 °C, the mixture was concentrated using 30 K MWCO Amicon Ultra Centrifugal Filter (Merck Millipore) up to 5–10 μM. The final buffer condition for all cryo-EM samples was 20 mM Tris-glutamate (pH 8.0), 10 mM Mg-glutamate, 150 mm K-glutamate, 5 mM DTT. 0.5% CHAPSO was added to the sample right before grid freezing. For cryo-EM grid freezing, Quantifoil R 1.2/1.3 Cu 400 grids were glow discharged at negative polarity, 0.26 mbar, 15 mA, 25 sec. Using a Vitrobot Mark IV (Thermo Fisher), grids were blotted and plunge-frozen into liquid ethane with 100% chamber humidity at 22 °C.
### Cryo-EM data acquisition and processing
Micrographs were taken using a 300 keV Krios G4 (Thermo Fisher Scientific) with a K3 BioQuantum direct electron detector (Gatan) with 20 eV energy filter slit width. Images were recorded with EPU with a pixel size of 1.06 Å/pix over a defocus range of −0.8 µm to −2.6 µm. Total dose given to the data set is 42.16 e2 and total frame number was 55. The movies were drift-corrected, summed, and dose-weighted using MotionCor2 in RELION3.163. The contrast transfer function (CTF) was estimated using Gctf64, and the summed images were sorted based on CTF max resolution (<10 Å) and CTF figure of merit (>0.01).
The sorted images were transferred to cryoSPARC v3.2.0 for further process65. First, 411.9k particles were picked using blob picker from 2000 movies, extracted with 320 pixels box size, and 2D classified to make picking templates. Then, 1447.1k particles were picked using template picker from 8174 images. The particles were 2D classified twice, and the selected 863.1k particles from 43 classes were used as templates for Topaz picker. From Topaz train, 1202.5k particles were picked and extracted from 8162 images. The particles were 2D classified into 100 classes and 90 classes were selected. The selected particles were divided into five classes in heterogeneous refinement. Among the five templates, three are from the previous data set collected from Glacios, two are from EMDB EMD-8585, a non-paused EC map. Among five classes, three classes were subjected to homogeneous refinement. Each homogeneous-refined class was further heterogeneous-refined into two classes, resulting in total of four significant EC classes—RPo, putEC, put-less EC, and σ70-bound putEC.
All particles of the four classes were imported to RELION3.1 for further refinements. The particles belonged to holoenzyme structure were 3D auto-refined, particle-polished three times, and 3D classified into three classes. Among the three classes, the major class was 3D auto-refined and post-processed yielding 3.0 Å-resolution map. The putEC particles were 3D auto-refined, particle-polished three times, and subjected to focused classification onto putRNA region into three classes. Among the three classes, two classes are combined, 3D auto-refined and post-processed yielding 3.2 Å-resolution map. The put-less particles were 3D auto-refined, particle-polished three times, and post-processed yielding 3.6 Å-resolution map. The σ70-bound putEC particles were 3D auto-refined, particle-polished three times, and 3D classified into three classes. Among the three classes, one best class was further refined and post-processed yielding 3.6 Å-resolution map.
### Model building, refinement, and validation
The local resolution estimation and filtration were done by blocres and blocfilt commands in bsoft package (version 2.0.5), respectively66. For the EC structures, EC coordinates including RNAP, DNA, and RNA are used from PDB 6C6T because this is modeled from the high-resolution EC map (3.5 Å). For σ70-bound putEC, the recently published high-resolution RPo model (PDB 6OUL) was used. In the model building, the models were first fitted onto the final cryo-EM map by using UCSF Chimera (version 1.11.12)67. Then, the RNAP domains were rigid-body refined in PHENIX (version 1.18.2)68, and the nucleic acid were mutated to have the correct sequence in Coot69. The structures were then real-space refined in PHENIX, manually modified in Coot, and iterated this process until satisfied. The putRNA was manually built into the map de novo. A .eff file that includes restraints maintaining the nucleic acid base pairing and stacking interactions was provided for each real-space refinement run. For the final refinement run, the nonbonded_weight parameter value was set to 500 (default value: 100) to improve the MolProbity and clash scores. The local filtered map was also used for the last refinement iteration because it slightly improved the modeling when inspected by eyes. The figures were made using PyMOL (version 2.4.0).
### Reporting summary
Further information on research design is available in the Nature Research Reporting Summary linked to this article.
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2023-03-28 03:50:42
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https://beanmachine.org/docs/overview/quick_start/
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Quick Start
Let's quickly translate the model we discussed in "Why Bean Machine?" into Bean Machine code! Although this will get you up-and-running, it's important that you read through all of the pages in the Overview to have a complete understanding of Bean Machine. If you're interested, the full source code for this Overview is available as a notebook on GitHub and Colab. Happy modeling!
Modeling
As a quick refresher, we're writing a model to understand a disease's reproduction rate, based on the number of new cases of that disease we've seen. Though we never observe the true reproduction rate, let's start off with a prior distribution that represents our beliefs about the reproduction rate before seeing any data.
import beanmachine.ppl as bmimport torch.distributions as distreproduction_rate_rate = 10.0@bm.random_variabledef reproduction_rate(): # An Exponential distribution with rate 10 has mean 0.1. return dist.Exponential(rate=reproduction_rate_rate)
There are a few things to notice here!
• Most importantly, we've decorated this function with @bm.random_variable. This is how you tell Bean Machine to interpret this function probabilistically. @bm.random_variable functions are the building blocks of Bean Machine models, and let the framework explore different values that the function represents when fitting a good distribution for observed data that you'll provide later.
• Next, notice that the function returns a PyTorch distribution. This distribution encodes your prior belief about a particular random variable. In the case of $\text{Exponential}(10.0)$, our prior has this shape:
• As you can see, the prior encourages smaller values for the reproduction rate, averaging at a rate of $10\%$, but allows for the possibility of much larger spread rates.
• Lastly, realize that although you've provided a prior distribution here, the framework will automatically "refine" this distribution, as it searches for values that represent observed data that you'll provide later. So, after we fit the model to observed data, the random variable will no longer look like the graph shown above!
The last piece of the model describes how the reproduction rate relates to the new cases we observe the subsequent day. This number of new cases is related to the underlying reproduction rate -- how fast the virus tends to spread -- as well as the current number of cases. However, it's not a deterministic function of those two values. Instead, it depends on a lot of environmental factors like social behavior, stochasticity of transmission, and so on. It would be far too complicated to capture all of those factors in a single model. Instead, we'll aggregate all of these environmental factors in the form of a probability distribution, the $\text{Poisson}$ distribution.
Let's say, for this example, we observed a little over a million, $1087980$, cases today. We use such a precise number here to remind you that this is a known value and not a random one. In this case, if the disease were to happen to have a reproduction rate of $0.1$, this is what our $\text{Poisson}$ distribution for new cases would look like:
Let's write this up in Bean Machine. Using the syntax we've already seen, it's pretty simple:
@bm.random_variabledef num_new(num_current): return dist.Poisson(reproduction_rate() * num_current)
As you can see, this function relies on the reproduction_rate() that we defined before. Do notice: even though reproduction_rate() returns a distribution, here the return value from reproduction_rate() is treated like a sample from that distribution! Bean Machine works hard behind the scenes to sample efficiently from distributions, so that you can easily build sophisticated models that only have to reason about these samples.
Data
With the model fully defined, we should gather some data to learn about! In the real world, you might work with a government agency to determine the number of real, new cases observed on the next day. For the sake of our example, let's say that we observed $238154$ new cases on the next day. Bean Machine's random variable syntax allows you to bind this information directly as an observation for the num_new() random variable within a simple Python dictionary. Here's how to do it:
from torch import tensornum_init = 1087980observations = { # PyTorch distributions expect tensors num_new(num_init): tensor(238154.),}
Using a random variable function and its arguments as keys in this dictionary may feel unusual at first, but it quickly becomes an intuitive way to reference these random variable functions by name! Note also that we're using num_init as an argument to the random variable function. This might seem unnecessary, since num_init could simply remain a global constant in this example, but a similar indexing scheme for num_new() will come in handy when we extend the model to time series with more than a single time step.
Inference
With model and observations in hand, we're ready for the fun part: inference! Inference is the process of combining a model with data to obtain insights, in the form of probability distributions over values of interest. Bean Machine offers a powerful and general inference framework to enable fitting arbitrary models to data.
The call to inference involves first creating an appropriate inference engine object and then invoking the infer method:
samples = bm.CompositionalInference().infer( queries=[reproduction_rate()], observations=observations, num_samples=7000, num_adaptive_samples=3000,)
There's a lot going on here! First, let's take a look at the inference method that we used, CompositionalInference(). Bean Machine supports generic inference, which means that it can fit your model to the data without knowing the intricate and particular workings of the model that you defined. However, there are lots of ways of performing this, and Bean Machine supports a rich library of inference methods that can work for different kinds of models. For now, all you need to know is that CompositionalInference is a general inference strategy that will try to automatically determine the best inference method(s) to use for your model, based on the definitions of random variables you've provided. It should work well for this simple model. You can check out our guides on Inference to learn more!
Let's take a look at the parameters to infer(). In queries, you provide a list of random variables that you're interested in learning about. Bean Machine will learn probability distributions for these, and will return them to you when inference completes! Note that this uses exactly the same pattern to reference random variables that we used when binding data.
We bind our real-world observations with the observations parameter. This provides a set of probabilistic constraints that Bean Machine seeks to satisfy during inference. In particular, Bean Machine tries to fit probability distributions for unobserved random variables, so that those probability distributions explain the observed data -- and your prior beliefs -- well.
Lastly, num_samples is the number of samples that you want to learn. Bean Machine doesn't learn smooth probability distributions for your queries, but instead accumulates a representative set of samples from those distributions. This parameter lets you specify how many samples should comprise these distributions.
Analysis
Our results are ready! Let's visualize them for the reproduction rate parameter.
The samples object that we have now contains samples from the probability distributions that we've fit for our model and data. It supports dictionary-like indexing using -- you guessed it -- the same random variable referencing syntax we've seen before. A second index (here, [0]) selects one of the inference chains generated by the sampling algorithm; this will be explained in the Inference section, so let us just use 0 for now.
reproduction_rate_samples = samples[reproduction_rate()][0]reproduction_rate_samples
tensor([0.0146, 0.1720, 0.1720, ..., 0.2187, 0.2187, 0.2187])
Let's visualize that more intuitively.
This histogram represents our beliefs over the underlying reproduction rate, after observing the current day's worth of new cases. You'll note that it is balancing our prior beliefs with the rate that we would learn just from looking at the new data. It also captures the uncertainty inherent in our estimate!
We're Not Done Yet!
This is the tip of the iceberg. The rest of this Overview will cover critical concepts from the above sections. Read on to learn how to make the most of Bean Machine's powerful modeling and inference systems!
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2022-01-23 18:39:43
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http://mathhelpforum.com/statistics/29065-please-help-someone-solve-problem.html
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Please Send me its solution if some one have..........
Ten oil tens are taken at random from an automatic filling machine the mean weight of the tins is 15.8 kg and the standard deviation is 0.50 kg. Does the sample mean differ siginificantly from the intended weight of 16 kg?
2. Originally Posted by sitara
[snip]
Ten oil tens are taken at random from an automatic filling machine the mean weight of the tins is 15.8 kg and the standard deviation is 0.50 kg. Does the sample mean differ siginificantly from the intended weight of 16 kg?
You can flesh out the argument .....
Since the population variance is not known, the test statistic to use is $t = \frac{\bar{x} - 16}{s/\sqrt{n}}$ which may be compared with Student's t-distribution with (n - 1) degrees of freedom.
$t = \frac{15.8 - 16}{0.50/\sqrt{9}} = -1.2$.
The critical value is -2.262 at the $\alpha = 0.025$ significance level. Therefore ......
Alternatively, a 95% confidence interval for $\mu$ is given by $\bar{x} \pm 2.262 \frac{s}{\sqrt{n}}$ where in this problem $\, \bar{x} = 15.8 \,$, $\, s = 0.50 \,$ and n = 10.
95% confidence interval: $15.8 \pm 0.377$. Therefore .....
The critical value is got using the t-distribution with 10 - 1 = 9 degrees of freedom. The t-distribution is is used because the population standard deviation is not known and the sample size is small (n < 50).
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2017-05-30 12:12:10
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