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https://www.codespeedy.com/variadic-function-templates-in-cpp/	[ "# Variadic function templates in C++ with example\n\nIn this tutorial, we will learn what are variadic function templates, its syntax and how to implement it using an example in C++.\n\n## Variadic function templates in C++\n\nTemplates that can accept a variable number of arguments are called variadic templates i.e., they can accept any number of arguments. When these templates are applied on functions these are called variadic function templates.\n\ntemplate <class… t1>\n\nThis type of template allows zero or more arguments. For a template to only allow a positive number of arguments we need to declare it as follows:\ntemplate <class t1, class… t2>\n\nWe generally implement the variadic templates recursively. There isn’t any easy implementation technique to iterate over the values of the variadic template.\n\n### Example of Variadic function templates\n\n```#include <iostream>\nusing namespace std;\n\nvoid my_fun()\n{\ncout << \"Printing is completed.\" << endl;\n}\n\ntemplate <class t1, class... t2>\nvoid my_fun(t1 v1,t2... v2)\n{\ncout << v1 << endl;\nmy_fun(v2...);\n}\n\nint main()\n{\nmy_fun (5, \"Codespeedy\", 9.55, \"Hello\");\nmy_fun (\"Meghana\", 99);\nreturn 0;\n}\n```\n\nIn this example, we implemented the variadic function template along with the concept of function overloading and recursion.\n\nIn main, we first called the function ‘my_fun’ with 4 arguments. This calls the variadic function and v1 contains the value 5 and v2 contains all the remaining values(“Codespeedy”, 9.55, “Hello”). We are printing the value in v1 and calling ‘my_fun’ recursively on v2. For the last recursive call which contains no arguments, the overloaded ‘my_fun’ will be called.\n\nWe observe that every time the variadic function can accept a variable number of arguments. This is the use of variadic function templates in C++.\n\nOutput:\n\n```5\nCodespeedy\n9.55\nHello\nPrinting is completed.\nMeghana\n99\nPrinting is completed.```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62535673,"math_prob":0.96906334,"size":2023,"snap":"2021-31-2021-39","text_gpt3_token_len":465,"char_repetition_ratio":0.20455672,"word_repetition_ratio":0.01910828,"special_character_ratio":0.24023727,"punctuation_ratio":0.15053764,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96660143,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-27T12:42:20Z\",\"WARC-Record-ID\":\"<urn:uuid:927309b6-a273-419d-87ec-9b8965c2b0d5>\",\"Content-Length\":\"24720\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9ad4a95a-7009-40e7-ae4d-5e5c0569c13b>\",\"WARC-Concurrent-To\":\"<urn:uuid:24f1fa1a-d542-4a6d-a815-dfe260074c9f>\",\"WARC-IP-Address\":\"173.255.247.200\",\"WARC-Target-URI\":\"https://www.codespeedy.com/variadic-function-templates-in-cpp/\",\"WARC-Payload-Digest\":\"sha1:W7QGLLS2RYKJIEKAFMGKBKVCMU3ITLRF\",\"WARC-Block-Digest\":\"sha1:4XB6HU22RXGIPMKH3GC42RQMVJCFOAKZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153391.5_warc_CC-MAIN-20210727103626-20210727133626-00275.warc.gz\"}"}
https://www.gradesaver.com/textbooks/science/physics/CLONE-afaf42be-9820-4186-8d76-e738423175bc/chapter-8-exercises-and-problems-page-149/75	[ "Essential University Physics: Volume 1 (4th Edition) Clone\n\n$1.5\\times10^6\\ km$\nWe know that the location of $L_1$ is about .01 the distance between the earth and the sun away from the earth. We know that the earth is $1.5\\times10^8 \\ m$ away from the sun. Thus, we find that $L_1$ is $1.5\\times10^6\\ km$ away." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93507713,"math_prob":0.9998871,"size":275,"snap":"2019-43-2019-47","text_gpt3_token_len":95,"char_repetition_ratio":0.15867159,"word_repetition_ratio":0.0,"special_character_ratio":0.35636362,"punctuation_ratio":0.11940298,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9717149,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-17T00:02:20Z\",\"WARC-Record-ID\":\"<urn:uuid:626d8679-9163-482e-aa86-28119a419572>\",\"Content-Length\":\"52688\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d67c2b09-80e0-4302-b0eb-538195381294>\",\"WARC-Concurrent-To\":\"<urn:uuid:ae410c75-26cd-4584-acb8-45e7666ecdb5>\",\"WARC-IP-Address\":\"52.87.77.102\",\"WARC-Target-URI\":\"https://www.gradesaver.com/textbooks/science/physics/CLONE-afaf42be-9820-4186-8d76-e738423175bc/chapter-8-exercises-and-problems-page-149/75\",\"WARC-Payload-Digest\":\"sha1:ED7WBSW2L55DK3FFVMEJIN6BRBJ3YXSP\",\"WARC-Block-Digest\":\"sha1:BY7UTSNOL4NLEQAQ5XMJXW5PYROHBGC4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986672431.45_warc_CC-MAIN-20191016235542-20191017023042-00364.warc.gz\"}"}
https://mulloverthings.com/how-do-you-generate-a-sine-wave-from-pwm/	[ "MullOverThings\n\nUseful tips for everyday\n\nHow do you generate a sine wave from PWM?\n\nHow do you generate a sine wave from PWM?\n\nGenerating Sine Wave using PWM in PSoC – KBA226852\n\n1. Create a look-up table for the sine wave.\n2. Configure the Timer block to generate periodic interrupts.\n3. Configure the PWM block.\n4. Vary the PWM duty cycle during each timer interrupt.\n5. Use a low pass filter.\n\nCan Arduino generate a sine wave?\n\nWith push buttons, you will be able to choose a waveform shape (sine, triangular, sawtooth, or square) on both DAC channels and change the frequency of the generated signal. Connect power and ground on your breadboard to the Arduino. Pins DAC0 and DAC1 wil generate the waveform.\n\nCan Arduino generate PWM signal?\n\nArduino and PWM The Arduino IDE has a built in function “analogWrite()” which can be used to generate a PWM signal. The frequency of this generated signal for most pins will be about 490Hz and we can give the value from 0-255 using this function. analogWrite(0) means a signal of 0% duty cycle.\n\nCan you use an Arduino as a function generator?\n\nMake sure you use an Arduino with a built-in DAC. If you don’t have one, you can add an external DAC of some sort, which will then generate a true analog output. Furthermore, you should keep in mind that this is a basic function generator. The output can’t go above +5 V, and it also can’t go below zero Volts.\n\nWhat is the formula for a sine wave?\n\nSine Wave. A general form of a sinusoidal wave is y(x,t)=Asin(kx−ωt+ϕ) y ( x , t ) = A sin ( kx − ω t + ϕ ) , where A is the amplitude of the wave, ω is the wave’s angular frequency, k is the wavenumber, and ϕ is the phase of the sine wave given in radians.\n\nHow do you create a waveform?\n\nGamry Instruments uses two different methods to generate a waveform depending on the frequency range. A direct digital synthesizer (DDS) sine wave generator is used to generate high‑frequency signals. A digital-to-analog converter (DAC) is used for low‑frequency signals.\n\nWhat is Arduino PWM?\n\nPulse Width Modulation, or PWM, is a technique for getting analog results with digital means. Digital control is used to create a square wave, a signal switched between on and off. In other words, with Arduino’s PWM frequency at about 500Hz, the green lines would measure 2 milliseconds each.\n\nHow fast is Arduino PWM?\n\nOn the Arduino Duemilanove, these values yield: Output A frequency: 16 MHz / 64 / (180+1) / 2 = 690.6Hz. Output A duty cycle: 50% Output B frequency: 16 MHz / 64 / (180+1) = 1381.2Hz.\n\nIs PWM analog or digital?\n\nPulse Width Modulation, or PWM, is a technique for getting analog results with digital means. Digital control is used to create a square wave, a signal switched between on and off.\n\nHow does a function generator work?\n\nSimple function generators usually generate triangular waveform whose frequency can be controlled smoothly as well as in steps. This triangular wave is used as the basis for all of its other outputs. The triangular wave is generated by repeatedly charging and discharging a capacitor from a constant current source.\n\nDo servers need sine wave UPS?\n\nIf you are protecting servers, you should really always use a true sine wave UPS. True sine wave models almost always fall into the line-interactive category. That means the UPS is also protecting against prolonged voltage voltage problems like brownouts that can burn up your power supplies.\n\nHow to create a sine wave with Arduino Due?\n\nI’ve been trying to create a sine wave using the pwm function of the Arduino DUE. The sine wave has te be at higher rates than the analogWrite () function can handle, so I’ve been using the native capabilties of the SAM3X8E. To test it, instead of creating sinus I’ve been doing it linear.\n\nCan a PWM signal make a sine wave?\n\nBut alas poor Yorick, there is a way and its called PWM. So above we have a 31khz pwm signal that is being used to generate a sine wave. Through the wonders of mathematics and other nerd endeavours that PWM signal can be used to make sine waves, in my case a 600hz sine wave.\n\nHow is duty cycle set in Sine PWM?\n\nVia analogWrite. The value passed to analogWrite is used to set the duty-cycle of the PWM. That duty cycle is the analog of the signal value for PWM, just as voltage is the analog of the signal value with a DAC pin. I still cant get it.\n\nHow to make an Arduino based SPWM circuit?\n\nIn the next post I’ll explain how to use the above Arduino based SPWM generator to make a pure sinewave inverter circuit ….keep reading! This code was based on Swagatam SPWM code with changes made to remove errors. Use this code as you would use any other Swagatam’s works.\n\nThere are five steps involved in this design:\n\n1. Create a look-up table for the sine wave.\n2. Configure the Timer block to generate periodic interrupts.\n3. Configure the PWM block.\n4. Vary the PWM duty cycle during each timer interrupt.\n5. Use a low pass filter.\n\nWhat is sine PWM?\n\nSinusoidal PWM is a type of “carrier-based” pulse width modulation. In sinusoidal PWM, the modulation signal is sinusoidal, with the peak of the modulating signal always less than the peak of the carrier signal. Sinusoidal PWM inverter leg and line-line voltages are illustrated below.\n\nAre all UPS pure sine wave?\n\nWhen a UPS system receives power and frequency from the AC line that is within an acceptable range, it will not do anything to correct it. The incoming utility power is typically a pure sine wave and this is what connected equipment expect.\n\nWhat are the types of PWM techniques?\n\nThe different PWM techniques are Single pulse width modulation, Multiple pulse width modulation, Phase displacement control, Sinusoidal pulse width modulation, Harmonic Injection modulation, Space Vector pulse width modulation, Hysteresis (Delta) pulse width modulation, Selective Harmonic Elimination and Current …\n\nHow is a PWM signal converted to a sine wave signal?\n\nIf this voltage needs to be boosted from the DC source, it can be accomplished either before the AC stage by using a DC-DC boost converter, or after the AC stage by using a boost transformer. The inverted signal itself is composed of a pulse-width-modulated (PWM) signal which encodes a sine wave.\n\nHow many times does a sine wave need to be filtered?\n\nThe waveform below shows the sine PWM signal (top – red) and the filtered result. In this case the PWM frequency is a little under 40 times the desired sine wave frequency. Doing the same thing with a DAC produces a similar result but with the pre-filtered output looking a little different: Another method is simply to filter a square wave.\n\nWhich is the most common pure sine wave generation technique?\n\nPWM 2-Level PWM. The most common and popular technique of digital pure-sine wave generation is pulse-width- modulation (PWM). The PWM technique involves generation of a digital waveform, for which the duty- cycle is modulated such that the average voltage of the waveform corresponds to a pure sine wave.\n\nHow does a microcontroller generate a sine wave?\n\nA typical situation would be where you need a sine wave based on a precision frequency generated by a microcontroller, CPLD or FPGA. In that case you would presumably have a square wave and need to generate your sine wave from that." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86846536,"math_prob":0.920411,"size":8053,"snap":"2022-05-2022-21","text_gpt3_token_len":1828,"char_repetition_ratio":0.16523792,"word_repetition_ratio":0.19560741,"special_character_ratio":0.22823793,"punctuation_ratio":0.10055866,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9822689,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-24T23:40:05Z\",\"WARC-Record-ID\":\"<urn:uuid:bf1dc9f7-4d65-4dd2-9dfb-d264a37fbb1e>\",\"Content-Length\":\"33270\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:658c1eb6-0ce5-4e74-b7d0-e3528649fb05>\",\"WARC-Concurrent-To\":\"<urn:uuid:9499aa27-0367-420e-b6cd-0332b39ac726>\",\"WARC-IP-Address\":\"49.12.116.136\",\"WARC-Target-URI\":\"https://mulloverthings.com/how-do-you-generate-a-sine-wave-from-pwm/\",\"WARC-Payload-Digest\":\"sha1:YOPPEIIA7CGTKMOQBT53UHFEMOEBLJDI\",\"WARC-Block-Digest\":\"sha1:DEQU2GIIHOQVWUVHVNI56NWNROW4IJRZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320304686.15_warc_CC-MAIN-20220124220008-20220125010008-00156.warc.gz\"}"}
https://physics.stackexchange.com/questions/296960/what-physics-is-contained-in-vertex-corrections	[ "# What physics is contained in vertex corrections?\n\nIf one looks at the interaction of light and a non-zero density of electrons, one can calculate the polarizability $\\Pi(q,\\omega)$ (which is the 00-th component of the dressed photon propagator). This object is related to the dielectric function as $\\epsilon(q,\\omega) = 1 - V(q)\\Pi(q,\\omega)$, where $V(q)$ is the Coulomb potential between the electrons in Fourier space.\n\nThe dielectric function contains a lot of physics: by looking at the zeros of the homogeneous dielectric function you can find the plasma frequency and by looking at the static limit you can find the screening length.\n\nAre there similar \"easy-to-understand\" physical results one can derive from vertex corrections? And what do the real and imaginary part of a particular vertex correction tell us?\n\nThe vertex function contains information about:\n\n• The existence of bounded states & their spectrum\n• Effective interactions in the theory\n• Scattering amplitudes in the theory\n\nTo illustrate these statements, let us consider the interaction $$U(r_1-r_2,t_1-t_2)$$ between two particles and solve the respective Bethe-Salpeter equation. First, consider the interaction $$U(r_1-r_2)\\delta(t_1-t_2)$$, so that the interaction is instantaneous. The equation for the vertex function then is $$K_{12,34}=(2\\pi)^8\\delta_{p1p3}\\delta_{p_2p_4}G_0^{(1)}(p_1)G_0^{(2)}(p_2)+G_0^{(1)}(p_1)G_0^{(2)}(p_2)G_0^{(1)}(p_3)G_0^{(2)}(p_4)i\\Gamma_{p_1p_2p_3p_4}(2\\pi)^4\\delta_{p_1+p_2,p_3+p_4},$$ where $$K_{12,34}$$ is the two-particle Green function and upper indices denote particles. Then, you can write down the Bethe-Salpeter equation and extract the irreducible vertex part $$\\Gamma_0$$. In case of instantaneous interaction, $$U(\\omega,k)=U(k)$$. The first order diagram for the irreducible vertex part is zero, $$\\int d\\omega G_0^{(1)}(\\epsilon_1-\\omega)G_0^{(2)}(\\epsilon_2-\\omega)=0,$$ because poles of both Green functions, $$\\omega_1=\\epsilon_1-\\frac{p_{1+}^2}{2m}+i\\delta,\\omega_1=\\epsilon_2-\\frac{p_{2-}^2}{2m}+i\\delta$$ are in the upper half-plane. Then, you can show that all the diagrams with crossing lines go to zero. At second order, the irreducible vertex function is just $$\\Gamma_0(p_1,p_2,p_1+q,p_2-q)=U(q).$$ Then, you can notice that the vertex function does not depend on frequency $$\\omega$$. Therefore, all dependence on frequency comes from the Green functions, $$i\\int\\frac{d\\omega}{2\\pi}G_0^{(1)}(\\epsilon_1-\\omega)G_0^{(2)}(\\epsilon+\\omega)=\\frac{1}{\\epsilon_1+\\epsilon_2-p_{1+}^2/2m_1-p_{2-}^2/2m+i0},$$ where $$p_{i\\pm}=p_i\\pm k$$. The denominator can be simplified by introducing CM energy and relative motion energy, $$\\frac{p^2}{2m_1}+\\frac{(p')^2}{2m_2}=\\frac{P^2}{2M}+\\frac{p_{\\text{rel}}^2}{2\\mu},P=p+p',p_{\\text{rel}}=\\frac{m_2p-m_1p'}{m_1+m_2}.$$ These manipulation give the following form of the Bethe-Salpeter equation, $$\\boxed{\\Gamma(k,k')=U(k-k')+\\int\\frac{d^3q}{(2\\pi)^3}\\frac{U(k-q)\\Gamma(p,k')}{\\Omega_0-q^2/(2\\mu)+i0},}$$ where $$\\mu=m_1m_2/(m_1+m_2)$$, $$\\Omega_0=\\Omega-P^2/(2m)$$, $$M=m_1+m_2$$, $$\\Omega=\\omega_1+\\omega_2=\\omega_3+\\omega_4$$. So, you see that there is a bounded state with mass $$M=2m$$ with dispersion law $$P^2/(2M)$$.\n\nYou can consider the structure of the vertex function in a theory with a 4-fermion contact attractive interaction and see that in this theory the vertex function has a pole. The existence of such poles corresponds to the Cooper instability and it signals that one should carefully choose the ground state of theory to develop perturbation theory.\n\nThen, your statement about polarization operator and screening length in plasma relates to the structure of the vertex function, too. As you know, one should sum all the diagrams with 1-loop insertions. You can also see that a bubble is included into the vertex function. The vertex function is $$\\Gamma_{12}(k,\\omega_m)=\\frac{4\\pi e^2Z_1Z_2}{k^2\\left[1-\\frac{4\\pi e^2}{k^2}(\\Pi_e(k,\\omega_m)+Z^2\\Pi_i(k,\\omega_m))\\right]},$$ where $$\\Pi_e$$ & $$\\Pi_i$$ are polarization operators of electrons and ions. Straightforward calculation yields $$\\Gamma_{12}=\\frac{4\\pi e^2Z_1Z_2}{k^2+\\kappa_i^2},$$ where $$\\kappa_i$$ is the ion screening length.\n\nBut you should distinguish between the polarization operator $$\\Pi$$ and the vertex function $$\\Gamma$$. The polarization operator comes from self-energy, which is the one-particle irreducible diagram and the vertex function is related to the two-particle irreducible diagrams. In case of Debye screening in plasma we can consider the renormalization of the \"interaction propagator\" by bubble series or we can consider 4-fermion process, which describes two-fermion scattering via renormalized (=\"bubbled\" propagator). In other words, you consider propagator renormalization and then say that this renormalized propagator modifies interaction or you can consider interaction renormalization with a vertex function.\n\nIn the case of superconductivity:\n\nTo emphasize mentioned facts, here are three diagrams (I am lazy to draw them, sorry)", null, "and the entire question is about the singularities of these diagrams. All the singularities comes from the poles of the Green functions in loops. Indeed, to perform integration over energy, you should use an appropriate contour, but external momenta can pinch this contour and the singularity appears. You can show that for the first diagram (from left to right) a singularity appears for $$p_1+p_2\\approx 0$$, for the second and third, $$p_2-p_1-k\\approx 0$$ and $$k\\approx 0$$, respectively. The last diagram defines Fermi liquid theory, whereas the first diagram corresponds to the Cooper instability. The second diagram is \"rare\".\n\nTo conclude, I would like to emphasize:\n\n1. the vertex function defines effective interaction in the theory\n2. vertex function shows the existence of bounded states in theory\n\nThe case of vertex functions in condensed matter is more interesting than in QED. For QED, we define the vertex function via a four-point correlator, $$K_{ik,lm}=\\langle T\\psi_{i1}\\psi_{k2}\\bar{\\psi}_{l3}\\bar{\\psi}_{m4}\\rangle$$ and can easily consider this correlator in momentum representation. It is worth mentioning that one should also take into account all the loop corrections. From the structure of this correlator, you can extract the reducible part,", null, "which just gives the exact propagator of two particles without scattering. The last term is more interesting and represents the irreducible part. Finally, you can see that $$i\\mathcal{M}_{fi}=\\bar{u}_i(p_3)\\bar{u}_k(p_4)\\left[-ie\\Gamma_{ik,lm}(p_3,p_4;p_1,p_2)\\right]u_l(p_1)u_m(p_2),$$ where $$i\\mathcal{M}_{fi}$$ is exactly the scattering amplitude. The next picture is the 4-point correlator for fermions,", null, "and in the non-crossing case (for instance, in superconductivity theory) the vertex function becomes", null, "• Would you not get that same information if you took the normal dielectric function to give the effective interaction (i.e. $V_{eff}(q,\\nu)=\\frac{V(q)}{\\epsilon(q,\\nu)}$)? Or in general any two point Green function would give the bound state. Does the vertex function give more information? – KF Gauss Feb 24 '20 at 8:33\n• @KFGauss , I assume that you have misprint: only two-particle (=four-point) Green function gives info about bounded state. Of course, in case of Debye screening, one can consider 1-loop without external legs because bubbles just modify \"interaction propagator\". But for nonideal Bose gas (for instance) and mentioned example of two particle interaction you easily see that vertex function is effective interaction. – Artem Alexandrov Feb 24 '20 at 9:03\n• @KFGauss vertex function also contains info about scattering amplitude, it is straightforward to see from the expression in box from my answer. For instance, for two-fermion scattering you can show that $\\Gamma \\sim 4\\pi f/m$, where $f$ is scattering amplitude – Artem Alexandrov Feb 24 '20 at 9:06\n• I meant two particle Green Function. Are you saying then that $\\Pi$, the 2 particle irreducible as defined in the question, contains basically the same things that you can learn as the vertex function $\\Gamma$? – KF Gauss Feb 24 '20 at 9:13\n• @KFGauss , roughly speaking yes. But you mix two different things: self-energy which is one-particle irreducible diagram and vertex function which is two-particle irreducible. In the context of effective interaction and in case where interaction modifies due to renormalization of \"interaction propagator\" both ways are the same – Artem Alexandrov Feb 24 '20 at 9:29" ]	[ null, "https://i.stack.imgur.com/Fk9dY.jpg", null, "https://i.stack.imgur.com/L4F1i.png", null, "https://i.stack.imgur.com/l9WZE.png", null, "https://i.stack.imgur.com/7xx9y.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7843217,"math_prob":0.9984211,"size":5930,"snap":"2021-04-2021-17","text_gpt3_token_len":1759,"char_repetition_ratio":0.13904826,"word_repetition_ratio":0.002805049,"special_character_ratio":0.28600338,"punctuation_ratio":0.097385034,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99990046,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,2,null,2,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-27T01:58:51Z\",\"WARC-Record-ID\":\"<urn:uuid:fd0ef1df-ed2e-48d0-af4f-c4934d75ed1d>\",\"Content-Length\":\"162240\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b4d04c37-2e0f-42ae-bb5a-28eabed74eb9>\",\"WARC-Concurrent-To\":\"<urn:uuid:6514c47f-7455-40bc-beba-e1e005bfabe0>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/296960/what-physics-is-contained-in-vertex-corrections\",\"WARC-Payload-Digest\":\"sha1:6J7TMO4TZX2GHL4TDEVNFBS7VBFYPDKI\",\"WARC-Block-Digest\":\"sha1:XFDDMSMP5GAGOW2BF2VAPJKRYMKEFQAD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610704804187.81_warc_CC-MAIN-20210126233034-20210127023034-00325.warc.gz\"}"}
https://kolazzing.com/entry/Contrastive-Loss-similarity-metric-face-verification-200506	[ "Seize the moment/Paper Research\n\n# Contrastive Loss (similarity metric / face verification / 2005.06)\n\n2023. 3. 13.\n\n## Paper\n\n• Learning a similarity metric discriminatively, with application to face verification\n• Sumit Chopra (2005.06 / NYU)\n• PDF / Github 없음\n\n### Points\n\n• Recognition, Verification 문제에 method for training a similarity metric from data를 제시함.\n• similar and dissimilar한 포인트 두 개로 계산한 loss라고 하여 contrastive loss라는 말이 처음 유래함.\n• L1 norm을 사용하여 semantic distance를 측정하고, input patterns into a target space 함.\n\n### Abstract\n\n• L1 사용하여서 semantic distance 도출했음. 보통은 L2를 쓰기 때문에 특이점이었음.\n• L1을 사용하면 특징추출이 더 잘됨. 그래서 사람얼굴을 classification하는 과제에 있어 더 적은 데이터로도 98.5프로의 정확성을 보여줄 수 있었음.\n\nThe authors used L1 norm in the target space because it approximates semantic distance better than L2. The idea is to learn a function that maps input patterns into a target space such that the L1 norm in the target space approximates \"semantic\" distance in the input space, which leads to improved performance on face verification tasks with large numbers of categories and small training samples per category.\n\n### Dataset\n\n1. AT&T Database of Faces: This dataset consists of 400 images from 40 subjects, with each subject having ten frontal face images taken under varying lighting conditions.\n\n2. AR and Feret Databases: The second set of training and testing was performed by combining the AR database created at Purdue University (which contains over 3,500 facial images) with a subset of grayscale Feret database containing more than one thousand faces.\n\n### Model\n\n• 비공개라 자세한 확인은 어려웠음\n• (CNN - Dense - Batch norm - max pooling) * 5 + (Dense w/ L1 regularlization) * 2 이라고 추정되는데,, 과연 BN을 Dense 뒤에 썼을까 싶다.\n• CNN filter num : 16 >> 512 // kernel size : 7 > 2\n\nThe paper does not provide the model.summary() function output. However, it describes the architecture of their proposed method which includes a convolutional neural network (CNN) followed by a fully connected layer and an L1 normalization layer to produce feature vectors for face verification task. Their CNN consists of 5 convolutional layers with max-pooling in between each pair, followed by two fully-connected layers before producing final embeddings using L1-normalization. The number of filters used in these convolutions increases from 16 to 512 while kernel size decreases from [7x7] to [2x2]. Additionally, they use batch normalization after every activation except last one as well as dropout regularization during training process\n\n### Activation Function\n\n• ReLU & Leaky ReLU\n• Normalization : : y = γ * ((x - μ)/σ ) + β\n\nThe paper uses the Rectified Linear Unit (ReLU) activation function, which is defined as: f(x)=max(0,x) where x represents input to a neuron and f(.) denotes its output. Additionally, they use Leaky ReLU in some of their experiments with α=0.1 where it's defined as: f(x)=x if x> 0 αx otherwise Here, alpha (α) is a small constant that determines how much negative values are allowed through. Finally, for normalization purposes during training process batch-normalization technique was used whose formula can be written as follows: y = γ ((x - μ)/σ ) + β Where y represents normalized activations; gamma(γ), beta(β), mean value(mu or µ ), standard deviation(sigma or σ ) were learned parameters per channel over each mini-batch samples respectively.\n\n### Loss Function (Contrastive Loss)\n\n• L = Contrastive Loss function\n• Y = 한 쌍의 얼굴사진이 같은 class면 0, 아니면 1\n• D = 두 점 feature vector간의 embedded distance\n• m = 유사하지 않은 쌍이 최소한으로 떨어져있어야 하는 마진\n• max(0,x)가 ReLU인데, 이렇게 들어가면, 유사하지 않은 쌍일때, 즉 Y=1이면, 결국 Y * max(0, m - D)^2 만 남게 되므로, 그 거리를 max로 멀게 하겠다는 뜻.\n\nL = (1-Y) * D^2 + Y * max(0, m - D)^2\n\nwhere L represents the value of the contrastive loss function, Y denotes whether a pair belongs to same class or not (Y=0 if they belong to same class and 1 otherwise), D is distance between two feature vectors in embedded space and 'm' refers to margin which controls how far apart dissimilar pairs should be. As for code implementation, it depends on what deep learning framework you are using. However, most frameworks such as Keras provide built-in functions that can calculate this type of custom loss during training process. Here's an example code snippet written in Python/Keras: ```\n\n``````(관련수식)\nimport keras.backend as K\ndef contrastive_loss(y_true,y_pred):\nmargin = 1 # set your own desired margin here\nsquare_pred = K.square(y_pred)\nsquared_margin_diff = K.square(K.maximum(margin-y_pred))\nreturn y_truesquare_pred+(1-y_true)squared_margin_diff\n\n------------------------------------------------------------------------\n(쉽게쓰는법)\nfrom tensorflow.keras import backend as K\nfrom tensorflow.keras.losses import Loss\n\nclass ContrastiveLoss(Loss):\ndef __init__(self, margin=1.0, kwargs):\nsuper(ContrastiveLoss, self).__init__(kwargs)\nself.margin = margin\n\ndef call(self, y_true, y_pred):\nsquare_pred = K.square(y_pred)\nsquared_margin_diff = K.square(K.maximum(self.margin - y_pred, 0))\nloss = K.mean(y_true * square_pred + (1 - y_true) * squared_margin_diff)\nreturn loss\n\nThis will define a new custom `contrastive_loss` function that takes true labels (`y_true`) and predicted values (`y_preds`) from model output layer as inputs while returning calculated losses based on above mentioned equation.\n\n### Results\n\n• test로 Purdue/AR Face Data를 썼을 때, 98.5% 였음.\n• 하지만 한계가 있음. 밑에서 보충.\n\nThe paper reports the accuracy of their proposed method on a face verification task using Purdue/AR Face Database. They achieved an overall accuracy of 98.5% which is quite high considering the large degree of variability in pose, lighting, expression and occlusions present in this dataset. It's important to note that performance can vary depending on factors such as choice of hyperparameters (e.g., margin value), size/composition of training data etc. However, these results demonstrate that their approach has potential for real-world applications where robustness to variations is critical.\n\n### Why this paper is famous for its use of contrastive loss?\n\n• 같은얼굴 조금의 데이터를 다른얼굴 여러 데이터에서 잘 분석해 내었음.\n• similar한 것과 dissimiliar 한 것을 통해 feature를 학습하였다 해서 contrastive 라는 말이 유래됨.\n\nIt used the contrastive loss which drives the similarity metric to be small for pairs of faces from the same person and large for pairs from different persons. The name \"contrastive\" comes from this idea that it contrasts between similar and dissimilar samples in order to learn a better feature representation. The authors chose this particular type of loss because they were working with face verification tasks where there are only two classes (same or different) instead of multiple categories as seen in traditional classification problems. Additionally, their method was designed specifically to handle cases where training data has very few examples per class making other methods such as softmax-based cross-entropy less effective. Overall, using contrastive loss allowed them to train an embedding model that could produce compact yet discriminative representations suitable for recognition/verification applications even when dealing with limited amounts of labeled data.\n\n## Limitation\n\n• 얼굴인식에만 대한 결과라서 범용성을 체크해봐야 함. The proposed method is evaluated only on face verification task, and its performance may vary for other recognition or classification tasks.\n• 데이터 자체가 연구실에서 컨트롤링 한 것이기 때문에 현실성이 떨어질 수 있음. Although the AR Dataset used in testing has a high degree of variability, it still consists mostly of controlled images captured under laboratory conditions. Therefore, the generalization ability to real-world scenarios with uncontrolled variations such as occlusions or disguises remains unclear.\n• 데이터의 일부라고 해도 라벨링 하긴 해야해서 비용이 많이 듬. While Siamese networks have been shown effective for learning similarity metrics from small datasets, their training can be computationally expensive due to pairwise comparisons between all samples during each iteration. This limits scalability when dealing with large-scale datasets containing millions/billions/trillions examples.\n\n728x90\n반응형\n\n#### 'Seize the moment > Paper Research' 카테고리의 다른 글\n\nCLEP) Exploiting Negative Preference in Content-based MusicRecommendation with Contrastive Learning (2022.07) (0) 2023.03.22 2023.03.16 2023.03.15 2023.03.15 2023.03.10" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8340884,"math_prob":0.85487807,"size":8368,"snap":"2023-40-2023-50","text_gpt3_token_len":2234,"char_repetition_ratio":0.10808226,"word_repetition_ratio":0.020249221,"special_character_ratio":0.23589866,"punctuation_ratio":0.10558583,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.986978,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-22T08:47:25Z\",\"WARC-Record-ID\":\"<urn:uuid:800aa017-5bae-43f0-b2d3-c1bef3dd2f69>\",\"Content-Length\":\"54297\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bcdd6d37-62c1-40d0-b11b-67ac610ff9bb>\",\"WARC-Concurrent-To\":\"<urn:uuid:8ad8ae3a-4799-4de0-9016-6037ad7fa5e8>\",\"WARC-IP-Address\":\"211.249.222.34\",\"WARC-Target-URI\":\"https://kolazzing.com/entry/Contrastive-Loss-similarity-metric-face-verification-200506\",\"WARC-Payload-Digest\":\"sha1:Q3MOKSDECXNOOJCFNE7YWCMNBL2ZOCDQ\",\"WARC-Block-Digest\":\"sha1:3JATCR4XTTEVXJ6K5WLOENM6K3PT5PEP\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506339.10_warc_CC-MAIN-20230922070214-20230922100214-00189.warc.gz\"}"}
https://socratic.org/questions/a-chemical-reaction-occurring-in-a-cylinder-equipped-with-a-moveable-piston-prod	[ "# A chemical reaction occurring in a cylinder equipped with a moveable piston produces 0.58 mol of a gaseous product. If the cylinder contained 0.11 mol of gas before the reaction and had an initial volume of 2.1L, what was its volume after the reaction?\n\nMay 21, 2015\n\nThe new volume of the cylinder will be equal to 13 L.\n\nWhen no mention of pressure or temperature is made, you can assume that they are kept constant. This means that you can use Avogadro's Law to determine what the new volume of the cylinder will be.\n\nAccording to Avogadro's Law, the volume of a gas is directly proportional to the number of moles of gas present, when temperature and pressure are kept constant.\n\nIn other words, the bigger the number of moles of gas present, the bigger the volume. Think about blowing up a balloon. The more you put into it, the bigger its volume gets.", null, "Mathematically, this relationship is expressed like this\n\n${V}_{1} / {n}_{1} = {V}_{2} / {n}_{2}$, where\n\n${V}_{1}$,${n}_{1}$ - the volume of the cylinder and the number of moles at an initial state;\n${V}_{2}$, ${n}_{2}$ - the volume of the cylinder and the number of moles at a final state.\n\nYou know that your reaction produces 0.58 moles of gas, and that 0.11 moles of gas were already in the cylinder. This means that the total number of moles present in the cylinder after the reaction is completed will be\n\n${n}_{\\text{total\" = n_\"initial\" + n_\"produced}}$\n\n${n}_{\\text{total\" = 0.11 + 0.58 = \"0.69 moles}}$\n\nUse the above equation to solve for ${V}_{2}$\n\n${V}_{1} / {n}_{1} = {V}_{2} / {n}_{2} \\implies {V}_{2} = {n}_{2} / {n}_{1} \\cdot {V}_{1}$\n\nV_2 = (0.69cancel(\"moles\"))/(0.11cancel(\"moles\")) * \"2.1 L\" = \"13.17 L\"\n\nRounded to two sig figs, the answer will be\n\n${V}_{2} = \\textcolor{g r e e n}{\\text{13 L}}$" ]	[ null, "https://d2jmvrsizmvf4x.cloudfront.net/MQJpXG8NR7KTBtLCXYRk_dUEGWAf4qTgm.3V-AtstTA_m.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9342817,"math_prob":0.99908113,"size":1253,"snap":"2019-51-2020-05","text_gpt3_token_len":285,"char_repetition_ratio":0.13771017,"word_repetition_ratio":0.045454547,"special_character_ratio":0.21308859,"punctuation_ratio":0.098425195,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99940586,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-24T08:10:49Z\",\"WARC-Record-ID\":\"<urn:uuid:d3a8e2c5-ca7a-4fdb-acd0-ba9d457285c0>\",\"Content-Length\":\"37576\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e60463bd-0b15-4732-afc9-c08bb028acb3>\",\"WARC-Concurrent-To\":\"<urn:uuid:6de66fc9-c745-4246-bd28-4632ef07d992>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/a-chemical-reaction-occurring-in-a-cylinder-equipped-with-a-moveable-piston-prod\",\"WARC-Payload-Digest\":\"sha1:3DLEIM7WAILVEZYSCUPWIT6IVPXLURAK\",\"WARC-Block-Digest\":\"sha1:FNCRBVZNHMZOPVHJAAAXFOHZHCBRNHK4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250616186.38_warc_CC-MAIN-20200124070934-20200124095934-00394.warc.gz\"}"}
https://www.colorhexa.com/0137e1	[ "# #0137e1 Color Information\n\nIn a RGB color space, hex #0137e1 is composed of 0.4% red, 21.6% green and 88.2% blue. Whereas in a CMYK color space, it is composed of 99.6% cyan, 75.6% magenta, 0% yellow and 11.8% black. It has a hue angle of 225.5 degrees, a saturation of 99.1% and a lightness of 44.3%. #0137e1 color hex could be obtained by blending #026eff with #0000c3. Closest websafe color is: #0033cc.\n\n• R 0\n• G 22\n• B 88\nRGB color chart\n• C 100\n• M 76\n• Y 0\n• K 12\nCMYK color chart\n\n#0137e1 color description : Vivid blue.\n\n# #0137e1 Color Conversion\n\nThe hexadecimal color #0137e1 has RGB values of R:1, G:55, B:225 and CMYK values of C:1, M:0.76, Y:0, K:0.12. Its decimal value is 79841.\n\nHex triplet RGB Decimal 0137e1 `#0137e1` 1, 55, 225 `rgb(1,55,225)` 0.4, 21.6, 88.2 `rgb(0.4%,21.6%,88.2%)` 100, 76, 0, 12 225.5°, 99.1, 44.3 `hsl(225.5,99.1%,44.3%)` 225.5°, 99.6, 88.2 0033cc `#0033cc`\nCIE-LAB 34.342, 53.008, -87.461 14.967, 8.174, 72.019 0.157, 0.086, 8.174 34.342, 102.27, 301.219 34.342, -12.746, -116.215 28.59, 43.411, -129.34 00000001, 00110111, 11100001\n\n# Color Schemes with #0137e1\n\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #e1ab01\n``#e1ab01` `rgb(225,171,1)``\nComplementary Color\n• #01a7e1\n``#01a7e1` `rgb(1,167,225)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #3b01e1\n``#3b01e1` `rgb(59,1,225)``\nAnalogous Color\n• #a7e101\n``#a7e101` `rgb(167,225,1)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #e13b01\n``#e13b01` `rgb(225,59,1)``\nSplit Complementary Color\n• #37e101\n``#37e101` `rgb(55,225,1)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #e10137\n``#e10137` `rgb(225,1,55)``\n• #01e1ab\n``#01e1ab` `rgb(1,225,171)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #e10137\n``#e10137` `rgb(225,1,55)``\n• #e1ab01\n``#e1ab01` `rgb(225,171,1)``\n• #012495\n``#012495` `rgb(1,36,149)``\n• #012bae\n``#012bae` `rgb(1,43,174)``\n• #0131c8\n``#0131c8` `rgb(1,49,200)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #013dfa\n``#013dfa` `rgb(1,61,250)``\n• #174ffe\n``#174ffe` `rgb(23,79,254)``\n• #3062fe\n``#3062fe` `rgb(48,98,254)``\nMonochromatic Color\n\n# Alternatives to #0137e1\n\nBelow, you can see some colors close to #0137e1. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #016fe1\n``#016fe1` `rgb(1,111,225)``\n• #015ce1\n``#015ce1` `rgb(1,92,225)``\n• #014ae1\n``#014ae1` `rgb(1,74,225)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #0124e1\n``#0124e1` `rgb(1,36,225)``\n• #0112e1\n``#0112e1` `rgb(1,18,225)``\n• #0301e1\n``#0301e1` `rgb(3,1,225)``\nSimilar Colors\n\n# #0137e1 Preview\n\nThis text has a font color of #0137e1.\n\n``<span style=\"color:#0137e1;\">Text here</span>``\n#0137e1 background color\n\nThis paragraph has a background color of #0137e1.\n\n``<p style=\"background-color:#0137e1;\">Content here</p>``\n#0137e1 border color\n\nThis element has a border color of #0137e1.\n\n``<div style=\"border:1px solid #0137e1;\">Content here</div>``\nCSS codes\n``.text {color:#0137e1;}``\n``.background {background-color:#0137e1;}``\n``.border {border:1px solid #0137e1;}``\n\n# Shades and Tints of #0137e1\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #00020a is the darkest color, while #f6f8ff is the lightest one.\n\n• #00020a\n``#00020a` `rgb(0,2,10)``\n• #00071e\n``#00071e` `rgb(0,7,30)``\n• #000c31\n``#000c31` `rgb(0,12,49)``\n• #001145\n``#001145` `rgb(0,17,69)``\n• #001658\n``#001658` `rgb(0,22,88)``\n• #001a6c\n``#001a6c` `rgb(0,26,108)``\n• #011f7f\n``#011f7f` `rgb(1,31,127)``\n• #012493\n``#012493` `rgb(1,36,147)``\n• #0129a6\n``#0129a6` `rgb(1,41,166)``\n• #012dba\n``#012dba` `rgb(1,45,186)``\n• #0132cd\n``#0132cd` `rgb(1,50,205)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\n• #013cf5\n``#013cf5` `rgb(1,60,245)``\n• #0b46fe\n``#0b46fe` `rgb(11,70,254)``\n• #1f55fe\n``#1f55fe` `rgb(31,85,254)``\n• #3263fe\n``#3263fe` `rgb(50,99,254)``\n• #4672fe\n``#4672fe` `rgb(70,114,254)``\n• #5981fe\n``#5981fe` `rgb(89,129,254)``\n• #6d90fe\n``#6d90fe` `rgb(109,144,254)``\n• #809ffe\n``#809ffe` `rgb(128,159,254)``\n• #94aeff\n``#94aeff` `rgb(148,174,255)``\n• #a8bdff\n``#a8bdff` `rgb(168,189,255)``\n• #bbcbff\n``#bbcbff` `rgb(187,203,255)``\n• #cfdaff\n``#cfdaff` `rgb(207,218,255)``\n• #e2e9ff\n``#e2e9ff` `rgb(226,233,255)``\n• #f6f8ff\n``#f6f8ff` `rgb(246,248,255)``\nTint Color Variation\n\n# Tones of #0137e1\n\nA tone is produced by adding gray to any pure hue. In this case, #696d79 is the less saturated color, while #0137e1 is the most saturated one.\n\n• #696d79\n``#696d79` `rgb(105,109,121)``\n• #616981\n``#616981` `rgb(97,105,129)``\n• #58648a\n``#58648a` `rgb(88,100,138)``\n• #4f6093\n``#4f6093` `rgb(79,96,147)``\n• #475b9b\n``#475b9b` `rgb(71,91,155)``\n• #3e57a4\n``#3e57a4` `rgb(62,87,164)``\n``#3552ad` `rgb(53,82,173)``\n• #2c4eb6\n``#2c4eb6` `rgb(44,78,182)``\n• #2449be\n``#2449be` `rgb(36,73,190)``\n• #1b45c7\n``#1b45c7` `rgb(27,69,199)``\n• #1240d0\n``#1240d0` `rgb(18,64,208)``\n• #0a3cd8\n``#0a3cd8` `rgb(10,60,216)``\n• #0137e1\n``#0137e1` `rgb(1,55,225)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #0137e1 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5009704,"math_prob":0.71461993,"size":3670,"snap":"2023-40-2023-50","text_gpt3_token_len":1680,"char_repetition_ratio":0.14048009,"word_repetition_ratio":0.0074074073,"special_character_ratio":0.5618529,"punctuation_ratio":0.23751387,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99183524,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-07T17:21:48Z\",\"WARC-Record-ID\":\"<urn:uuid:ab15dfd7-f079-4861-97bf-27c1e9a14c3a>\",\"Content-Length\":\"36168\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4e282796-b55a-4eba-9b83-998792057fcc>\",\"WARC-Concurrent-To\":\"<urn:uuid:b793eb0e-bfa6-4dd6-80d1-b936d651e8c0>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/0137e1\",\"WARC-Payload-Digest\":\"sha1:UYZ2D7ULUQJ4DKGIQZP263VSYTEADHD7\",\"WARC-Block-Digest\":\"sha1:MHAB6BYCDEJN7RSAIJJ74HAQAX6N5ZL5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100677.45_warc_CC-MAIN-20231207153748-20231207183748-00287.warc.gz\"}"}
http://csharphelper.com/blog/tag/elements/	[ "# Tag Archives: Elements\n\n## Calculate the GCD (greatest common divisor) and LCM (least common multiple) of two integers in C#\n\nGCD The greatest common divisor of two integers is the largest integer that evenly divides them both. For example, GCD(180, 105) = 15 because 15 is the largest integer that divides evenly into both 180 and 105. Euclid’s Elements c. … Continue reading\n\nPosted in algorithms, mathematics \| \| 5 Comments" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.66262144,"math_prob":0.7480284,"size":3420,"snap":"2020-10-2020-16","text_gpt3_token_len":1133,"char_repetition_ratio":0.16832553,"word_repetition_ratio":0.042813454,"special_character_ratio":0.36929825,"punctuation_ratio":0.043841336,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9622093,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-20T10:54:30Z\",\"WARC-Record-ID\":\"<urn:uuid:dd96f904-c4b7-439d-aebd-3be55081f6b9>\",\"Content-Length\":\"39475\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:40c1f314-278f-4106-bdd6-deb4a1e9f977>\",\"WARC-Concurrent-To\":\"<urn:uuid:6eda416b-41a0-483a-8469-695a413b8685>\",\"WARC-IP-Address\":\"107.180.57.98\",\"WARC-Target-URI\":\"http://csharphelper.com/blog/tag/elements/\",\"WARC-Payload-Digest\":\"sha1:TPKPGLVGWV57KJ4HVIXSKUY7KIRVJZVR\",\"WARC-Block-Digest\":\"sha1:MUPVOK46DTO7REJC7IHWFSSH3OAX2A56\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875144722.77_warc_CC-MAIN-20200220100914-20200220130914-00060.warc.gz\"}"}
https://tech.tonyballantyne.com/pedagogy/be-a-better-coder/writing-better-code-1/	[ "# Writing Better Code 1\n\nThe number 28 has 6 divisors:\n\n1,2,4,7,14,28\n\nThe number 5 only has 2 divisors:\n\n1,5\n\nWrite a program to find all the divisors of a number.\n\nThe mod operator is useful here. Remember that the mod operator gives the remainder when two numbers are divided, so\n\n```28 % 14 = 0\n28 % 5 = 3\n```\n\nWe can use the mod function to write code as follows\n\n```def divisors(n):\ndivs = []\nfor i in range(1,n+1):\nif n%i == 0:\ndivs.append(i)\nreturn divs\n\nprint(divisors(28))\n```\n\nThe above code is inefficient. To find the divisors of 28, you have to go through the for loop 28 times. To find the divisors of n, we have to go through the for loop n times. We say the code has time complexity O(n).\n\nTry running the code to find the divisors of a large number such as 1347663998. You’ll notice it takes a long time to find the answer. We need a way to make the code more efficient.\n\nYou should notice that the divisors come in pairs. (1,28, 2,14 4,7). This suggests a way of saving loops. Rather than looking at all the numbers, we could look at just half of them.\n\n```import math\n\ndef divisors(n):\ndivs = []\nfor i in range(1,int(n/2)):\nif n%i == 0:\ndivs.append(i)\ndivs.append(int(n/i))\nreturn divs\n\nprint(divisors(28))\n\n```\n\nThat seems better, but testing we get repeated divisors\n\n[1, 28, 2, 14, 4, 7, 7, 4]\n\nThinking about it further we see that we only need to count as far as the square root of n (because root n * root n = n)\n\n```import math\n\ndef divisors(n):\ndivs = []\nfor i in range(1,int(math.sqrt(n))):\nif n%i == 0:\ndivs.append(i)\ndivs.append(int(n/i))\nreturn divs\n\nprint(divisors(28))\n```\n\nThere’s still a problem though.\n\nRun the above code with divisors(9) and you just get\n\n[1, 9]\n\nIsn’t 3 a divisor of 9? The problem lies in our range. We’re counting up to one less then root 9. Let’s tweak the code to take this case into account.\n\n```import math\n\ndef divisors(n):\ndivs = []\nfor i in range(1, int(math.sqrt(n))):\nif n%i == 0:\ndivs.append(i)\ndivs.append(int(n/i))\nif math.sqrt(n).is_integer():\ndivs.append(int(math.sqrt(n)))\nreturn divs\n\nprint(divisors(9))\n```\n\nWe’ve now got working code that runs a lot faster than the original code.\n\nThere are a couple of further tweaks we can make.\n\nFirstly, all numbers are divisible by 1 and themselves, so we can just append them without checking.\n\nOdd numbers are never divisible by even numbers. So, if n is odd, we only need to check if n is divisible by 3, 5, 7 …\n\nIf n is even we need to check all numbers from 2 onwards.\n\n```import mathdef divisors(n): divs = [] step = 1 start = 2 if n%2 == 1: step = 2 start = 3 for i in range(start, int(math.sqrt(n)), step): if n%i == 0: divs.append(i) divs.append(int(n/i)) if math.sqrt(n).is_integer(): divs.append(int(math.sqrt(n))) divs.append(1) divs.append(n) return divs\n```\n\nOne last thing. In the new code the following calculation appears three times: math.sqrt(n)\n\nIt takes the computer a lot longer to work out square roots than to do other calculations. Now, a good compiler should notice this. It will perform optimisation on your code and will store repeated calculations to stop them having to be worked out over and over again.\n\nHowever, it might be more elegant to adjust your code as follows. I’ve also added a line to sort the divisors into order.\n\n```import math\ndef divisors(n): divs = [] rootn = math.sqrt(n) step = 1 start = 2 if n%2 == 1: step = 2 start = 3 for i in range(start, int(rootn), step): if n%i == 0: divs.append(i) divs.append(int(n/i)) if rootn.is_integer(): divs.append(int(rootn)) divs.append(1) divs.append(n) divs.sort() return divs```\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8013043,"math_prob":0.9979423,"size":3498,"snap":"2020-45-2020-50","text_gpt3_token_len":1016,"char_repetition_ratio":0.16199198,"word_repetition_ratio":0.15384616,"special_character_ratio":0.3018868,"punctuation_ratio":0.1514423,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997862,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-30T00:50:07Z\",\"WARC-Record-ID\":\"<urn:uuid:ca3a1fa8-f6ae-4121-afc7-031260788db9>\",\"Content-Length\":\"37250\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:28dd1c6d-ed47-41fa-ac9c-b83cdf5e87b7>\",\"WARC-Concurrent-To\":\"<urn:uuid:8eecc336-482d-402d-9bf3-9c7ab1cdb888>\",\"WARC-IP-Address\":\"185.119.173.36\",\"WARC-Target-URI\":\"https://tech.tonyballantyne.com/pedagogy/be-a-better-coder/writing-better-code-1/\",\"WARC-Payload-Digest\":\"sha1:Z66ARH7LJJXZG5F2WFGS7VHJYGOAQQ3M\",\"WARC-Block-Digest\":\"sha1:GBZETDTA42TTKZCCCTVYU6RMSAWKTKYX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141204453.65_warc_CC-MAIN-20201130004748-20201130034748-00016.warc.gz\"}"}
https://jsdoc.inflectra.com/HelpReadingPane.ashx?href=js56jsmthdimensions.htm	[ "JScript\n\n# dimensions Method\n\nReturns the number of dimensions in a VBArray.\n\n`array.dimensions( ) `\n\nThe required array is a VBArray object.\n\n#### Remarks\n\nThe dimensions method provides a way to retrieve the number of dimensions in a specified VBArray.\n\nThe following example consists of three parts. The first part is VBScript code to create a Visual Basic safe array. The second part is JScript code that determines the number of dimensions in the safe array and the upper bound of each dimension. Both of these parts go into the <HEAD> section of an HTML page. The third part is the JScript code that goes in the <BODY> section to run the other two parts.\n\n```<HEAD>\n<SCRIPT LANGUAGE=\"VBScript\">\n<!--\nFunction CreateVBArray()\nDim i, j, k\nDim a(2, 2)\nk = 1\nFor i = 0 To 2\nFor j = 0 To 2\na(j, i) = k\nk = k + 1\nNext\nNext\nCreateVBArray = a\nEnd Function\n-->\n</SCRIPT>\n\n<SCRIPT LANGUAGE=\"JScript\">\n<!--\nfunction VBArrayTest(vba)\n{\nvar i, s;\nvar a = new VBArray(vba);\nfor (i = 1; i <= `a.dimensions()`; i++)\n{\ns = \"The upper bound of dimension \";\ns += i + \" is \";\ns += a.ubound(i)+ \".<BR>\";\n}\nreturn(s);\n}\n-->\n</SCRIPT>\n\n<BODY>\n<SCRIPT language=\"jscript\">\ndocument.write(VBArrayTest(CreateVBArray()));\n</SCRIPT>\n</BODY>```\n\nVersion 3" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5952588,"math_prob":0.97371143,"size":1257,"snap":"2022-27-2022-33","text_gpt3_token_len":347,"char_repetition_ratio":0.14365523,"word_repetition_ratio":0.02283105,"special_character_ratio":0.29594272,"punctuation_ratio":0.12552302,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9980223,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-07T16:10:15Z\",\"WARC-Record-ID\":\"<urn:uuid:6e5927a1-35fe-4a90-941f-49a1b9e0b186>\",\"Content-Length\":\"3819\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7bada13b-d096-41fc-9780-e98b6101ea68>\",\"WARC-Concurrent-To\":\"<urn:uuid:5c02e50f-3c02-40b2-8a36-5598d49e9068>\",\"WARC-IP-Address\":\"3.218.117.200\",\"WARC-Target-URI\":\"https://jsdoc.inflectra.com/HelpReadingPane.ashx?href=js56jsmthdimensions.htm\",\"WARC-Payload-Digest\":\"sha1:OA7SJ2LMHKWHVEJ3ZICUK4WZVIKXA3KX\",\"WARC-Block-Digest\":\"sha1:57P7GDHREUNBBI22TWDWYXIZL5JSVUQB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104495692.77_warc_CC-MAIN-20220707154329-20220707184329-00685.warc.gz\"}"}
https://discovery.researcher.life/article/a-research-on-line-loss-calculation-based-on-bp-neural-network-with-genetic-algorithm-optimization/a4bb8bbf6a5432a7a9597acd572aa2d4	[ "## Abstract\n\nIn order to realize the calculation of the line loss of the distribution network with complex structure and low-voltage station area, this paper presents a line loss calculation method based on BP neural network with genetic algorithm optimization. The proposed method is based on the actual operation data of the distribution network. Firstly, build an error back propagation (BP) neural network model to compute the theoretical line loss of the distribution network, then use genetic algorithm (GA) to optimize the neural network and establish the GA-BP model. Based on the proposed model, the calculation demonstrates that the neural network line loss rate calculation model with genetic algorithm optimization shows better performance than the single BP neural network model, such as better nonlinear fitting ability and higher calculation accuracy. Therefore, the line loss calculation method proposed in this paper based on the BP neural network with the genetic algorithm optimization can improve the accuracy of the distribution network line loss rate calculation model.\n\n## Full Text", null, "", null, "Schedule a call" ]	[ null, "https://cdn.researcher.life/discovery-article/icons/full-text-submitted.svg", null, "https://cdn.researcher.life/discovery/icons/calender-blue.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.833757,"math_prob":0.9843182,"size":2109,"snap":"2023-40-2023-50","text_gpt3_token_len":498,"char_repetition_ratio":0.14109264,"word_repetition_ratio":0.011396011,"special_character_ratio":0.23565671,"punctuation_ratio":0.07692308,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9947399,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-04T11:01:28Z\",\"WARC-Record-ID\":\"<urn:uuid:aa46da26-0b81-40bd-aaa2-94e8fdc6c0eb>\",\"Content-Length\":\"270191\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cef240e3-79eb-4ce1-9fc7-eb0e988893db>\",\"WARC-Concurrent-To\":\"<urn:uuid:7970ff73-cee8-41c3-b5cb-6b70e30fc527>\",\"WARC-IP-Address\":\"13.32.208.22\",\"WARC-Target-URI\":\"https://discovery.researcher.life/article/a-research-on-line-loss-calculation-based-on-bp-neural-network-with-genetic-algorithm-optimization/a4bb8bbf6a5432a7a9597acd572aa2d4\",\"WARC-Payload-Digest\":\"sha1:PDPIQZZJBQYORYYEPKKRQGOMVZGHRK2Y\",\"WARC-Block-Digest\":\"sha1:JBEBDNBSTM6W62LYMCVCZCVQWCO5YNXQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100527.35_warc_CC-MAIN-20231204083733-20231204113733-00127.warc.gz\"}"}
https://newproxylists.com/discrete-mathematics-how-to-prove-a-proposition-related-to-sets-operation/	[ "# discrete mathematics – How to prove a proposition related to sets operation", null, "I am asked to prove or disprove the following proposition:\n$$X cap Y = X implies X cup Y = Y$$\n\nI feel this is true, because this means that X is a subset of Y, and as a result, when we do union with these two, it should be the entire set.\n\nHowever, I have a hard time constructing a formal proof to prove this. Not sure where to start from.", null, "Posted on" ]	[ null, "https://dreamproxies.com/wp-content/uploads/2020/05/Hostgator-coupon-sale.v1.jpg", null, "https://dreamproxies.com/wp-content/uploads/2020/05/Hostgator-sale.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9589286,"math_prob":0.9432467,"size":340,"snap":"2020-45-2020-50","text_gpt3_token_len":85,"char_repetition_ratio":0.09821428,"word_repetition_ratio":0.0,"special_character_ratio":0.25588235,"punctuation_ratio":0.11392405,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98744655,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-27T15:34:24Z\",\"WARC-Record-ID\":\"<urn:uuid:9d0c9ceb-55a5-4238-b783-7edbac8cb57e>\",\"Content-Length\":\"23917\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1413b3bc-3c87-404e-a03e-41f26708e1fb>\",\"WARC-Concurrent-To\":\"<urn:uuid:3d0061a8-3ebe-4020-a868-57fca486812b>\",\"WARC-IP-Address\":\"173.212.203.156\",\"WARC-Target-URI\":\"https://newproxylists.com/discrete-mathematics-how-to-prove-a-proposition-related-to-sets-operation/\",\"WARC-Payload-Digest\":\"sha1:3D4U5PHZXW4EGDCUGEC7J4HE5T6KQWXQ\",\"WARC-Block-Digest\":\"sha1:7SV35KUHMGJAEBMN7K4R3PB46MA74G2O\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107894203.73_warc_CC-MAIN-20201027140911-20201027170911-00327.warc.gz\"}"}
https://online-trading-platforms.info/kindergarten-add-and-subtract-worksheets/	[ "# 37 Luxury Kindergarten Add and Subtract Worksheets", null, "subtraction worksheets for kindergarten celestial worksheets kindergarten free printable shape for kids best 25 addition worksheets ideas on pinterest subtraction worksheets for kindergarten pdf free subtraction quiz worksheet free kindergarten math subtraction – 4 worksheets free printable worksheets subtraction worksheet for kindergarten math worksheets images about kids math on pinterest free printable math coloring sheets for kindergarten kindergarten math worksheets", null, "images about kids math on pinterest free printable from kindergarten add and subtract worksheets , source:www.mogenk.com", null, "Free Preschool & Kindergarten Subtraction Worksheets from kindergarten add and subtract worksheets , source:www.k5learning.com\n\nprintable subtraction worksheet for kindergarten adding and subtracting to 5 kindergarten math worksheets addition and subtraction math worksheets for kindergarten addition and subtraction subtraction to 10 with dominos dominos provide a tangible 389 best math tubs adding and subtracting images on addition and subtraction worksheets for kindergarten free preschool & kindergarten subtraction worksheets addition worksheet for kindergarten kindergarten worksheets chapter 2 worksheet mogenk" ]	[ null, "https://online-trading-platforms.info/wp-content/uploads/2018/09/kindergarten-add-and-subtract-worksheets-incredible-subtraction-worksheets-for-kindergarten-celestial-of-kindergarten-add-and-subtract-worksheets.png", null, "https://online-trading-platforms.info/wp-content/uploads/2018/09/kindergarten-add-and-subtract-worksheets-new-images-about-kids-math-on-pinterest-free-printable-of-kindergarten-add-and-subtract-worksheets.png", null, "https://online-trading-platforms.info/wp-content/uploads/2018/09/kindergarten-add-and-subtract-worksheets-outstanding-free-preschool-amp-kindergarten-subtraction-worksheets-of-kindergarten-add-and-subtract-worksheets.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6246406,"math_prob":0.523371,"size":5855,"snap":"2019-13-2019-22","text_gpt3_token_len":1092,"char_repetition_ratio":0.36762947,"word_repetition_ratio":0.4217877,"special_character_ratio":0.16191289,"punctuation_ratio":0.16103603,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99663234,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-23T17:18:18Z\",\"WARC-Record-ID\":\"<urn:uuid:64c02cf8-d203-45fc-b177-411f3ea5da70>\",\"Content-Length\":\"59858\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:198cc8ab-cd8b-488e-a887-6ec3fea6836b>\",\"WARC-Concurrent-To\":\"<urn:uuid:07c0d9ea-7bc7-4e94-9244-f1e99f23fa55>\",\"WARC-IP-Address\":\"104.27.140.243\",\"WARC-Target-URI\":\"https://online-trading-platforms.info/kindergarten-add-and-subtract-worksheets/\",\"WARC-Payload-Digest\":\"sha1:XLTJGS6RPYYEAMUUXJI2ALKTG653SIS7\",\"WARC-Block-Digest\":\"sha1:O2AFJTI2NPWETAWASATIYW73DIHEZPWA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202889.30_warc_CC-MAIN-20190323161556-20190323183556-00029.warc.gz\"}"}
https://www.arxiv-vanity.com/papers/hep-th/0505099/	[ "hep-th/0505099\n\nPUPT-2160\n\nITEP-TH-33/05\n\nPerturbative Search for Fixed Lines\n\nin Large Gauge Theories\n\nA. Dymarsky, I.R. Klebanov and R. Roiban\n\nJoseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA\n\nAbstract\n\nThe logarithmic running of marginal double-trace operators is a general feature of 4-d field theories containing scalar fields in the adjoint or bifundamental representation. Such operators provide leading contributions in the large limit; therefore, the leading terms in their beta functions must vanish for a theory to be large conformal. We calculate the one-loop beta functions in orbifolds of the SYM theory by a discrete subgroup of the R-symmetry, which are dual to string theory on . We present a general strategy for determining whether there is a fixed line passing through the origin of the coupling constant space. Then we study in detail some classes of non-supersymmetric orbifold theories, and emphasize the importance of decoupling the factors. Among our examples, which include orbifolds acting freely on the , we do not find any large non-supersymmetric theories with fixed lines passing through the origin. Connection of these results with closed string tachyon condensation in is discussed.\n\nMay 2005\n\n## 1 Introduction\n\nSoon after the AdS/CFT correspondence was formulated [1, 2, 3] (see [4, 5] for reviews), it was realized that modding out by a discrete subgroup of the R-symmetry leads to dual pairs with reduced supersymmetry [6, 7]. If we start with the SYM theory in , then a discrete orbifold group produces a superconformal field theory, while produces a superconformal gauge theory. For all other the supersymmetry is completely broken, raising the hope of generating a wide variety of non-supersymmetric conformal gauge theories. Some support for this was provided using both string theory and perturbative gauge theory arguments: it was shown that all correlation functions of single-trace untwisted operators (i.e. the operators that do not transform under the quantum symmetry ) coincide in the planar limit with corresponding correlation functions in the parent SYM theory. Therefore, beta functions for marginal single-trace operators vanish in the large limit. Concerns were raised, however, about the non-supersymmetric cases due to the presence of closed string tachyons . Nevertheless, the possibility that non-supersymmetric orbifold gauge theories are “large conformal” raised interesting prospects of conformal unification without supersymmetry .\n\nAs briefly mentioned in [8, 9], double-trace contributions are not inherited from the parent theory. An explicit one-loop calculation for the simplest non-supersymmetric orbifold gauge theory, with , revealed the presence of beta functions for double-trace operators. The induced double-trace operators were found to be of the form , where is a twisted ( odd) single-trace operator of bare dimension (see footnote 11 in ). More general concerns about inducing the double-trace operators were expressed in . Somewhat later on, the concerns about beta functions for the double-trace operators were strengthened, since their presence destroys the scale invariance of the large theory .111The role of multi-trace operators in the AdS/CFT correspondence was examined in a number of papers, starting with [14, 15, 16]. The work of draws an important distinction between the freely acting orbifolds of which contain no tachyons at large radius (strong ‘t Hooft coupling ), and other orbifolds that do contain tachyons. The case of the orbifold fits in the context of type 0 string theory and therefore contains tachyons. It was speculated in that its Coleman-Weinberg instability at weak gauge coupling is related to the tachyonic instability at strong coupling. One of the results of our paper is that even freely acting orbifold gauge theories may be rendered non-conformal at weak ‘t Hooft coupling by the flow of certain double-trace couplings.\n\nIn recent literature, inspired by the construction of exactly marginal deformations in AdS/CFT correspondence , a new proposal has appeared for a non-supersymmetric gauge theory that is conformal in the large limit . This motivates us to revisit the issue of whether there are non-supersymmetric orbifolds of the theory that are large conformal at weak coupling, which does not seem to be completely settled. We study beta functions for double-trace couplings and show that each such beta function has 3 leading one-loop contributions. Each one-loop beta function has two zeros, at . If the are complex then cannot flow to a fixed point, and the theory is not large conformal. But if are real, then reaches a non-trivial IR stable fixed point at . If all are real then we find an interesting weakly coupled large CFT, with double-trace operators induced. But are there such examples? In this paper we carry out a general one-loop calculation of induced double-trace operators, and then study in detail the beta functions for a few classes of examples where we find that some, but not all, are real. We do not know a general argument for the non-existence of perturbatively stable non-supersymmetric large CFT’s containing scalars in the adjoint or bifundamental representation;222 In non-supersymmetric theories containing fields in fundamental representations there exist Banks - Zaks fixed points with massless fermions , and their recently proposed generalizations containing also scalar fields . These are isolated fixed points rather than fixed lines. a further search for them is certainly warranted.\n\nIn the next section we present some considerations concerning the flow of the double-trace couplings, and in section 3 present a general formalism for calculating the one-loop beta-functions in orbifold gauge theories. Then, in section 4 we consider some simple examples of non-freely acting orbifolds whose AdS duals contain tachyons at large radius. In section 5 we move on to a class of freely acting orbifolds whose AdS duals do not contain tachyons at large radius. None of the examples we consider prove to be large conformal at weak ‘t Hooft coupling. Possible relations between our calculations and closed string tachyon condensation are discussed in section 6.\n\n## 2 General Considerations\n\nIn the standard convention, the SYM action is\n\n S=−∫d4x12g2YMTrF2μν+… (1)\n\nIn the ‘t Hooft large limit, is held fixed; hence, the coefficient multiplying the single-trace operator is of order . In this convention, the -point functions of single-trace operators are of order .\n\nNow consider gauge theories obtained by orbifolding the parent SYM theory by a discrete symmetry group . The single-trace operators come in two types: the untwisted ones, invariant under , and the twisted ones that transform under . For example, for , there are twisted operators that transform by under the generator of . The symmetry prevents such a twisted operator from being induced in the effective action. However, it does not prevent the appearance of a double-trace operator where and have opposite quantum numbers under (e.g., for ). Such an operator is of the same order in the large expansion as the action , i.e. of order . Hence it contributes to observables in the leading large limit.\n\nIn non-supersymmetric quiver gauge theories, in general nothing prevents the appearance of such double-trace operators. Indeed, one-loop diagrams induce such operators of bare dimension 4 with logarithmically divergent coefficients [11, 13]: the effective action at scale picks up contributions of the form 333Here and throughout the paper denotes the ’t Hooft coupling in the parent theory: .\n\n ∫d4xO¯OaOλ2ln(Λ/M) , (2)\n\nwhere is the UV cut-off and is a coefficient determined through explicit calculations. Then, perturbative renormalizability necessitates the addition of trace-squared couplings to the action:\n\n δS=−∫d4xfO¯O . (3)\n\nFrom (2) we note that the beta function for contains a contribution . However, this is not the only contribution to the one-loop beta function.\n\nIf the operator picks up 1-loop anomalous dimension , then the dimension of in the large limit is . This introduces a term into the beta function for . Finally, as discussed for example in [15, 16], there is a positive contribution , where\n\n ⟨O(x)¯O(0)⟩=vO4π2\|x\|4 , (4)\n\nwhich comes from fusion of two double-trace operators in the free theory.\n\nPutting the terms together, we find\n\n M∂f∂M=βf=vOf2+2γOλf+aOλ2 . (5)\n\nIt is crucial that the right hand side is not suppressed by powers of ; it is a leading large effect. On the other hand, the beta function for has no such contribution, due to the theorem of [8, 9]. Also, counting the powers of one can show that the double-trace operators cannot induce any planar beta functions for single-trace couplings. Therefore, in the large limit, may be dialed as we wish. In particular, it can be made very small so that the one-loop approximation in (5) is justified. Then the equation has two solutions, , where\n\n a±=1vO(−γO±√D) ,D=γ2O−aOvO . (6)\n\nIf the discriminant is positive, then these solutions are real, so that may flow to the IR stable fixed point at . This mechanism could make the theory conformal in an interesting and non-trivial way: in particular the IR theory has non-vanishing double-trace couplings.444In actual examples we will often find that both and are negative, so that the Hamiltonian is not obviously bounded from below (to study its positivity one needs to include both single trace and double-trace terms quartic in the scalar fields). However, it is well-known that many large theories are locally stable for potentials unbounded from below. Hence, we will not rule out the fixed points with negative double-trace couplings, although this issue requires further study.\n\nIf is negative, then (5) is positive definite for real , which signals a violation of conformal invariance. However, for small , the flow of is actually very slow near the minimum of located at . This is evident from the explicit solution of (5):\n\n f(M)=−γOλvO+bλvOtan(bλvOln(M/μ)) , (7)\n\nwhere we defined and chose the boundary condition . Thus, at weak ‘t Hooft coupling , the double-trace parameter varies very slowly for a wide range of scales. Still, it blows up towards positive infinity in the UV at and reaches in the IR at . We expect this singular behavior to be softened by the corrections, which introduce a positive beta function for making it approach zero in the IR.\n\n## 3 Double-trace correction for general orbifolds\n\nIn this section we find the beta function of the couplings of the dimension 4 double-trace twisted operators in general orbifolds of SYM theory. There are many gauge fixing choices one can make. The calculations are substantially simplified if we choose the dimensional reduction of the ten dimensional background gauge. We are interested in the 1-loop effective action for the scalar fields; at this order in perturbation theory the result can be found by computing the determinant of the kinetic operators. In Euclidean space and with hermitian generators for the gauge group, the relevant bosonic terms are\n\n S = ∫d4xTr[(∂μaν)2+(∂μφI)2 (9) −g2YM[ϕI,aμ][ϕI,aμ]−g2YM[ϕI,φJ][ϕI,φJ]−2g2YM[ϕI,ϕJ][φI,φJ]] .\n\nwhere are the background scalar fields and we expanded the action to quadratic order in the quantum fields and . The fermions couple to the background scalar fields via Yukawa couplings inherited from minimal couplings in ten dimensions.\n\nWe will denote by the orbifold group; if is a subgroup of or then the resulting theory preserves or supersymmetry, respectively. We will further denote by the representation of the elements of in , where it acts by conjugation. The representation of in the spinor and vector representation of will be denoted by and . Presenting the vector representation of as the 2-index antisymmetric tensor representation of , it follows that .\n\nWe will compute the determinant of the kinetic operator in a general scalar field background invariant under the orbifold group\n\n ϕI=RIJggϕJg† . (10)\n\nSince we are not considering a nontrivial fermionic background the contribution of fermionic loops decouple from that of scalar, vector and ghost loops and can be computed independently. To shorten the expression of the effective potential let us define:\n\n AIJ\|KLg=Tr(ϕIϕJg†)Tr(ϕKϕLg)+Tr(ϕJϕIg)Tr(ϕLϕKg†) . (11)\n\nWe also introduce a notation for the divergent part of a generic 1-loop scalar amplitude:\n\n Div=∫d4k(2π)41k4=116π2lnΛ2M2 . (12)\n\nwhere is the UV cutoff and is the renormalization scale. Also, the notation for the contribution to the effective action will be:\n\n δSnr. of loops\|nr. of tracessource of contribution . (13)\n\nThen, the contribution of the fermion loop to the double-trace part of the effective action is:\n\n δS1 loop\|2 trFermi=λ2Div2\|Γ\|∑g∈ΓTr[γIγJγKγLrg][AJI\|KLg+AKI\|JLg+ALI\|JKg] . (14)\n\nIn this form the fermionic contribution to the effective action is manifestly real. Giving up manifest reality (which of course is restored in the sum over the orbifold group elements) it turns out to be possible to further simplify this expression to:\n\n δS1 loop\|2 trFermi= (15) =λ2Div\|Γ\|∑g∈ΓTr[γIγJγKγLrg][2Tr(ϕJϕIg†)Tr(ϕKϕLg)+Tr(ϕKϕIg†)Tr(ϕJϕLg)]\n\nIn both equations (14) and (15) denote the chiral (i.e. ) 6-dimensional Dirac matrices. In the absence of the orbifold action matrices the trace is trivial to compute:\n\n Tr[γIγJγKγL]=4(δIJδKL+δILδJK−δIKδJL) . (16)\n\nFor general the results for the nonvanishing components of are collected in the appendix.\n\nThe contribution of the vectors, scalars and ghost loops to the double-trace part of the effective action has the following expression:\n\n δS1 loop\|2 trBose,ghost = +∣∣2(RKQg+(R−1g)KQ)Tr([ϕI,ϕQ]g†)Tr([ϕI,ϕK]γ) −∣∣2(δKI(R−1g)PQ+δPQRKIg−2δPQδKI)Tr(ϕPϕQg†)Tr(ϕKϕIg) }\n\nOne may derive this directly in terms of Feynman diagrams or by extracting the double-trace part of the determinant of the kinetic operator for the quantum fields in (9). It is trivial to check that in the absence of the orbifold projection\n\n δS1 loop\|2 trBose,ghost+δS1 loopFermi=0 (19)\n\nin agreement with the theorem of [8, 9].\n\nWe see that double-trace operators made out of twisted single-trace operators are generated at 1-loop. Therefore they must be added to the tree level action. The precise form of the deformation depends on the specific orbifold. Also, whenever possible, it is useful to reorganize the operators being generated in terms of operators with definite scaling dimension. For the purpose of illustration let us consider the deformation\n\n δ2 traceS=12∑g∈ΓfgOIJgOJIg† with OIJg=Tr% (gϕIϕJ) . (20)\n\nThis modifies the kinetic operator by adding\n\n ∑g∈Γ(L(g)OJIg†+R(g)OIJg†) (21)\n\nwhere and are the left- and right-multiplication operators, respectively, and brings the following additional contributions to the effective action:\n\n δS1 loop\|2 tr2 trace=−Div\|Γ\|{∑gfg(1\|Γ\|∑~gf~gg†~g†)OIJgOJIg† + λ∑g∈ΓfgOJIg†[4δI^IδJ^J+(δIJ+RJIg)δ^I^J−2((R−1g)I^IδJ^J+δI^I(R−1g)J^J)]O^I^Jg}\n\nOne may recognize the bracket on the second line as the 1-loop dilatation operator acting on twisted operators.\n\n## 4 Examples of non-freely acting orbifolds\n\nIn this section we review and extend earlier analysis of orbifold field theories in which the action of the orbifold group on the -symmetry representation possesses fixed points. Quite generally, such an orbifold action yields in the daughter theory fields in the adjoint representation of all the gauge group factors.\n\nOn the string theory side, this translates into the existence of fixed points of the action of the orbifold group on the five-sphere. For non-supersymmetric actions (such as those we are interested in) it follows that some of the string theory excitations are tachyonic. We will eventually show that such tachyons manifest themselves in the weakly coupled gauge theory.\n\n### 4.1 A Non-Supersymmetric Z2 Example\n\nIn this subsection we discuss the orbifold theory which arises on the stack of electric and magnetic D3-branes of type 0B theory . This is the gauge theory coupled to six adjoint scalars of the first gauge group, six adjoint scalars of the second gauge group, 4 fermions in and 4 fermions in . This theory has global symmetry.\n\nThe one-loop calculation of reveals the following double-trace terms induced in the effective Lagrangian:\n\n δLeff=λ2π2ln(ΛM)(O⟨IJ⟩O⟨IJ⟩+23O2) , (23)\n\nwhere\n\n O⟨IJ⟩=Tr(ΦIΦJ−16δIJΦKΦK)−Tr(~ΦI~ΦJ−16δIJ~ΦK~ΦK)≡Tr(gϕIϕJ)−16δIJTr(gϕKϕK) (24)\n\ntransform in the of , while\n\n O=TrΦIΦI−Tr~ΦI~ΦI≡Tr(gϕIϕI) (25)\n\nis an singlet. Here the matrix represents the orbifold group on the gauge degrees of freedom: . We are thus forced to introduce coupling constants and , through\n\n δLtree=−f20O⟨IJ⟩O⟨IJ⟩−f1O2 . (26)\n\nWith our conventions (the normalization of the scalar kinetic term is twice the usual), the free scalar Euclidean two-point function is\n\n ⟨ΦI(x)ΦJ(0)⟩=δIJ18π2\|x\|2 . (27)\n\nThen we find\n\n ⟨O(x)O(0)⟩=v14π2\|x\|4 ,v1=34π2 , (28)\n ⟨O⟨IJ⟩(x)O⟨KL⟩(0)⟩=(δIKδJL+δILδJK−13δIJδKL)v204π2\|x\|4 ,v20=18π2 . (29)\n\nThe one-loop anomalous dimension coefficients are\n\n γ1=3λ8π2 ,γ20=0 . (30)\n\nThey can be obtained from the corresponding quantities in the SYM theory (see, for instance, ) by interpreting as the ’t Hooft coupling in the parent theory and introducing an additional factor of or by diagonalizing the dilatation operator written out explicitly in the appendix. Hence, we find\n\n β20=v20f220+λ2π2 ,β1=v1f21+34π2λf1+2λ23π2=32π2(14f21+12λf1+49λ2) . (31)\n\nNeither nor have real zero’s: they are positive definite for real couplings. Hence, the double-trace couplings and flow from large positive values in the UV to large negative in the IR. Thus, the orbifold theory is not large conformal: there are odd single-trace operators and even double-trace operators whose correlators do not respect conformal invariance.\n\n### 4.2 A Non-supersymmetric Z3 orbifold\n\nAs usual, we start with a supersymmetric gauge theory and apply a projection. We take the generator of the group to act on the fundamental representation of as\n\n r=diag(eiα3,eiα3,e−iα3,e−iα3) , α3=2π3 . (32)\n\nThe action on the fundamental representation of is\n\n R=diag(1,1,e2iα3,1,1,e−2iα3) . (33)\n\nThus, the orbifold acts on only one of the three complex coordinates. Closed string tachyon condensation in the case, and in the generalization to discussed in the Appendix D, was studied in many papers starting with (for reviews see [25, 26]). Our strategy will be to place a stack of D3-branes at the tip of the cone, and to study RG flows in the resulting gauge theory, and we will suggest their connection with tachyon condensation.\n\nAs usual in orbifold field theories, we keep only fields invariant under this operation, together with , where\n\n g=diag(1lN,eiα31lN,e−iα31lN) .\n\nwhere denotes the identity matrix.\n\nWe end up with a gauge theory (the untwisted decouples) described by a quiver diagram with 3 vertices. This theory has no supersymmetry but possesses global symmetry. At each vertex of the quiver, there are 4 adjoint scalar fields, transforming as a vector of , . Here is the index, and labels the vertex of the quiver. There are also singlet bifundamental scalars with . In the fermionic sector, we find 3 doublets of the first , , , , with ; and 3 doublets of the second , , , . The Yukawa couplings include terms of the type\n\n Φμ1σμa˙bψa12ψ˙b21\n\nand also terms of the type\n\n ϵabψa12ψb23Φ31 .\n\nFirst, let us classify the scalar operators that may appear in the induced marginal double-trace operators. The operators built of the adjoint scalars can be combined into traceless symmetric tensors in the of , and also into the singlet of . The former have the form\n\n O⟨μν⟩±=O⟨μν⟩1+exp(±iα3)O⟨μν⟩2+exp(∓iα3)O⟨μν⟩3 , (34)\n\nwhere\n\n O⟨μν⟩i=Tr(ΦμiΦνi−14δμνΦκiΦκi) O⟨μν⟩±=Tr(g±1ϕμϕν)−14δμνTr(g±1ϕκϕκ) , (35)\n\nwhile the latter are\n\n O±=4∑κ=1Tr(g±1ϕκϕκ)=TrΦ21+exp(±iα3)TrΦ22+exp(∓iα3)TrΦ23 . (36)\n\nAdditionally, there are singlet operators made of the bifundamental scalars,\n\n A±=e±iα3Tr(g±1ϕ3ϕ¯3)=3∑k=1Φk,k+1Φk+1,ke±iα3k where Φ†kl=Φlk , (37)\n\nand is identified with .\n\nThe operators and mix under RG flow; their anomalous dimension matrix is:\n\n (38)\n\nThe permutation symmetry, and the symmetry imply that the double-trace operators must involve combinations like\n\n Oμν+Oμν−=3∑i=1OμνiOμνi−Oμν1Oμν2−Oμν2Oμν3−Oμν1Oμν3 . (39)\n\nThis is a good check on our calculations since such combinations emerge only after we sum over the gauge field, scalar and fermion loops.\n\nExplicit calculation shows that there are several combinations which are being generated at one-loop level and must therefore be added as tree-level deformations of the original action. They are:\n\n δ2 traceLtree=f9,1O⟨μν⟩+O⟨μν⟩−+f1,1O+O−+f(3),1A+A−+f(A+O−+A−O+) (40)\n\nIn the following we will keep the anomalous dimension matrix (38) nondiagonal. This leads to non-diagonal beta functions, but avoids explicitly using the matrix diagonalizing the anomalous dimension matrix.\n\nSpecifying the results from the Appendix D to we find that, in the presence of the deformation, the double-trace part of the one-loop effective potential is:\n\n δS1loop\|2trb,gh,f=−9λ2ln(Λ2/μ2)32π2\|Γ\|[4O⟨μν⟩+O⟨μν⟩−+3O+O−+18A+A−−6(O+A−+O−A+)] (41)\n δS1loop\|2tr2trace = −λ2ln(Λ2/μ2)32π2\|Γ\|[f29,1O⟨ab⟩+O⟨ab⟩−+[4f21,1+12f2+2f1,1δOO+2fδAO]O+O− ∣∣+[12f2(3),1+4f2+2f(3),1δAA+2fδOA]A+A− ∣∣+[f(4f1,1+12f(3),1+δOO+δAA)+f1,1δOA+f(3),1δAO]O+A− ∣∣+[f(4f1,1+12f(3),1+δOO+δAA)+f1,1δOA+f(3),1δAO]O−A+ ]\n\nThe five beta functions are therefore:\n\n β9,1 = 148π2[36λ2+f29,1] (45) β1,1 = 148π2[27λ2+4f21,1+12f2+16λf1,1−2λf] (46) β(3),1 = 148π2[162λ2+12f2(3),1+4f2+14λf(3),1−16λf] (47) βf = 148π2[−54λ2+f(4f1,1+12f(3),1+15λ)−8λf1,1−λf(3),1] (48)\n\nThese expressions may seem quite opaque; it is however relatively easy to analyze them and find that no real values for the couplings lead to vanishing of all four beta functions. This is quite obvious for , which corresponds to operators with vanishing one loop anomalous dimension. In the next section we show how this generalizes to any non-freely acting orbifold.\n\n### 4.3 General non-freely acting Zk orbifolds\n\nIn both examples discussed above we found that there is no weakly coupled fixed point of the RG flow. We will now show that this is in fact a general property of orbifolds with fixed points by identifying operators whose beta function does not vanish for any value of the coupling constants.\n\nFirst of all, let us classify all possible representations of embedded in . Through a unitary transformation, its only nontrivial generator can be brought to a diagonal form\n\n g=⎛⎜ ⎜ ⎜ ⎜⎝ein1αk000ein2αk0000ein3αk0000ein4αk⎞⎟ ⎟ ⎟ ⎟⎠, αk=2πk (49)\n\nwith a constraint . So the representation is specified by three integers . Since the fundamental representation of is isomorphic to the two-index antisymmetric representation of , it follows that the action of on the fundamental representation of is also specified by three integers and their negatives . These are the weights of the complex scalar fields and their conjugates under this action of .\n\nThe integers vanish in pairs and the -invariant subspace of is always even-dimensional. We will focus in this section on representations with at least one vanishing , that is we will choose\n\n n1=−n2=n′ and n3=−n4=n′′ . (50)\n\nLet us denote by the number of vanishing weights. Then, is the remaining unbroken global symmetry of the theory. We will denote by the indices along -invariant directions and by all the others directions.\n\nOur logic will be the same as in the examples discussed before: we will focus on the symmetric traceless operator\n\n O⟨μν⟩q=Tr(gqϕμϕν)−δμν2lTr(gqϕκϕκ) (51)\n\nand the beta-function for the corresponding coupling constants555The reason for this particular form of the tree-level deformation is that it yields a uniform expression for all beta functions, including the operators with charge .\n\n δLtree=12k−1∑q=1fqO⟨μν⟩qO⟨μν⟩−q with fq=fk−q (52)\n\nWhen computing the beta function we have to remember to take into account this overcounting of operators.\n\nThe only other twisted operators containing fields from the invariant subspace are similar to the Konishi operator in the parent theory; explicitly, they are\n\n Oq=∑κTr(gqϕκϕκ) (53)\n\nwhere the sum runs only over the invariant subspace. The only other potential candidate\n\n Tr(gqϕμϕi) (54)\n\nvanishes identically. This can be proven quite easily by moving one factor of past both and and then using the cyclicity of the trace. This operation yields a nontrivial phase proportional to the charge of . The other twisted operators are\n\n Tr(gqϕiϕ¯ȷ) . (55)\n\nThis spectrum clearly implies that the deformation (51) is closed in the sense that the beta functions for the couplings does not receive contributions linear in other couplings. Indeed, all correlation functions\n\n ⟨O⟨μν⟩qO−q⟩=0 (56)\n\nsince there is no traceless symmetric invariant tensor.\n\nFrom the general expressions listed in Appendix A, it is easy to see that has vanishing one-loop anomalous dimension, so there is no contribution of the type to the corresponding beta function. It follows therefore that there are only two relevant contributions: from and from the one-loop renormalization of the bare action. The former is always positive . It is in fact easy to calculate this coefficient at one loop level. Its corresponding contribution to the beta function is\n\n f2q16π2k>0 . (57)\n\nThe later contribution, from the one loop renormalization of the bare action is\n\n δS=−λ2π2kln(ΛM) k−1∑q=1sin2(n′αq2)sin2(n′′αq2) OμνqOμν−q (58)\n\nand is also positive.Thus, the beta function for the operators (51) is\n\n βq=2λ2π2ksin2(n′αk2)sin2(n′′αk2)+f2q16π2k (59)\n\nand is always non-vanishing.\n\n## 5 A class of orbifolds with Su(3) symmetry\n\nIn the previous section we saw that, quite generally, non-supersymmetric orbifolds that are not freely acting do not correspond to weakly coupled large CFT’s. What about freely-acting orbifolds? A motivation for studying them is that, since they have no fixed points, the twisted sector strings are stretched to length of order and therefore are not tachyonic at large ‘t Hooft coupling . The corresponding fields in have . Such fields are dual to the twisted single-trace operators in the orbifold gauge theory, which are charged under the quantum symmetry . By the AdS/CFT correspondence, at large such operators have dimensions of order and are highly irrelevant. Hence, the AdS/CFT correspondence suggests that there is a fixed line at large , but that instabilities may set in for small . Motivated by this, we carry out the small (one-loop) analysis for a class of freely acting orbifold gauge theories which possess a global symmetry (further details may be found in the Appendix B).\n\nAs before, the starting point is SYM theory with gauge group ; let us parametrize where is the first -th root of unity\n\n ωk=eiαk , αk=2πk . (60)\n\nTo preserve , we choose the following action of in the fundamental representation of :\n\n r(gn)=diag(ωnk,ωnk,ωnk,ω−3nk) . (61)\n\nwhich yields the action in the vector representation of\n\n R(gn)=diag(ω2nk,ω2nk,ω2nk,ω−2nk,ω−2nk,ω−2nk) (62)\n\nWe end up with a gauge theory described by a quiver diagram with vertices. This theory has no supersymmetry but possesses global symmetry. On the edges around the boundary of the quiver there are triplets of chiral fermions, , where is identified with . There are also singlet chiral fermions . In the scalar sector, we find complex triplets, . Since there are no adjoint scalars, the simplest Coleman-Weinberg potential of the type considered in [11, 13] cannot be generated. However, these models, like all other orbifolds, contain single-trace operators quadratic in the scalar fields. Therefore, there is a possibility of inducing beta-functions for double-trace operators made out of twisted single-trace operators.\n\nThe spectrum of invariant fields under this combined action allows the construction of the following independent twisted operators:\n\n Oi¯ȷn=Tr(gnϕiϕ¯ȷ) On=3∑i=1Tr(gnϕiϕ¯ı) n=1,…,[k2] (63)\n\nThese operators mix under 1-loop scale transformations. Using the dilatation operator spelled out in the appendix it is easy to identify the operators with definite scaling dimension:\n\n Operatoranomalous dimensionO⟨i¯ȷ⟩n=Tr(gnϕiϕ¯ȷ)−1</" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91177803,"math_prob":0.97148436,"size":25101,"snap":"2021-21-2021-25","text_gpt3_token_len":5384,"char_repetition_ratio":0.1715743,"word_repetition_ratio":0.025825677,"special_character_ratio":0.2048524,"punctuation_ratio":0.09030249,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98746616,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-17T21:21:47Z\",\"WARC-Record-ID\":\"<urn:uuid:4d0b36d4-19d3-42a6-a776-6a6a677bf575>\",\"Content-Length\":\"1049534\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:79404376-8e2d-42d9-b8d8-e9dfe7df1799>\",\"WARC-Concurrent-To\":\"<urn:uuid:89e9e3d8-9b4e-46fc-9902-6cf8f67b5c75>\",\"WARC-IP-Address\":\"172.67.158.169\",\"WARC-Target-URI\":\"https://www.arxiv-vanity.com/papers/hep-th/0505099/\",\"WARC-Payload-Digest\":\"sha1:ABOY4JBRQ7SRO34LAGSIL3E2BWSVEAJV\",\"WARC-Block-Digest\":\"sha1:CCK5GUHNQOCLLYQEEIHGRDUIWDON76MR\",\"WARC-Truncated\":\"length\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487633444.37_warc_CC-MAIN-20210617192319-20210617222319-00403.warc.gz\"}"}
http://www.vbforums.com/showthread.php?740059-RESOLVED-Can-t-seem-to-retrieve-a-specified-character-from-a-string-of-numbers&p=4538225	[ "", null, "# Thread: [RESOLVED] Can't seem to retrieve a specified character from a string of numbers\n\n1. ##", null, "[RESOLVED] Can't seem to retrieve a specified character from a string of numbers\n\nI'm relatively new to VB, and I'm coding a graphing calculator to familiarize myself with it. I haven't figured out how to to the x^n function yet though.\n\nI made a function to retrieve a certain number from a text box, where the first character is \"^\", and the second/second and third are numbers.\n\nAn example of this not working is: 2^2 = 1.12589990684262E+15 (Where tbNum1 is \"2\" and tbOpr is \"^2\"\n\nHere is my code for the entire function:\n\nCode:\n```'tbopr IS WHERE THE SPECIFIED POWER GOES (x^n)\n'tbNum1 IS THE FIRST NUMBER, WHICH IS RAISED TO THE POWER \"p\"\n'tbCalcOut IS THE TEXT BOX THE ANSWER IS OUTPUTTED TO, AND THE NUMBER THAT IS RAISED TO _\n'THE POWER \"p\" IF IT ISN'T EQUAL TO 0.\n\nPrivate Function Power()\nDim p As Integer = 1\nIf Len(tbOpr.Text) = 2 Then\np = Convert.ToInt16(GetChar(tbOpr.Text, 2))\nElseIf Len(tbOpr.Text) = 3 Then\np = Convert.ToInt16(GetChar(tbOpr.Text, 2) & GetChar(tbOpr.Text, 3))\nElse\nMsgBox(\"Power is too high.\", , \"Error\")\nEnd If\nIf tbCalcOut.Text <> \"\" And tbCalcOut.Text <> \"0\" Then\ntbNum1.Text = tbCalcOut.Text\ntbCalcOut.Text = tbNum1.Text ^ p\nElseIf tbCalcOut.Text = \"0\" Or tbCalcOut.Text = \"\" And tbNum1.Text <> \"\" And _\ntbNum1.Text <> \"0\" Then\ntbCalcOut.Text = tbNum1.Text ^ p\nEnd If\nEnd Function```\nThanks guys, this really has stumped me.", null, "", null, "Reply With Quote\n\n2. ## Re: Can't seem to retrieve a specified character from a string of numbers\n\nAdd this function which return the power part\nCode:\n``` Private Function GetPower(strText As String) As Integer\nDim p As Integer = 1\nIf strText.Contains(\"^\") Then\np = CInt(strText.Substring(strText.IndexOf(\"^\") + 1))\nEnd If\n\nReturn p\nEnd Function```\nCall it from your function like this\nCode:\n``` Private Function Power()\nDim p As Integer = GetPower(tbOpr.Text)\nIf p > 99 Then\nMsgBox(\"Power is too high.\", , \"Error\")\nExit Function ' don't continue\nEnd If\nIf tbCalcOut.Text <> \"\" And tbCalcOut.Text <> \"0\" Then\ntbNum1.Text = tbCalcOut.Text\ntbCalcOut.Text = tbNum1.Text ^ p\nElseIf tbCalcOut.Text = \"0\" Or tbCalcOut.Text = \"\" And tbNum1.Text <> \"\" And _\ntbNum1.Text <> \"0\" Then\ntbCalcOut.Text = tbNum1.Text ^ p\nEnd If\nEnd Function```", null, "", null, "Reply With Quote\n\n3. ## Re: Can't seem to retrieve a specified character from a string of numbers\n\nThanks so much!\nWhy exactly did my code not work, if you don't mind me asking?", null, "", null, "Reply With Quote\n\n4. ## Re: [RESOLVED] Can't seem to retrieve a specified character from a string of numbers\n\nWhy exactly did my code not work,\nBecause you are not search for \"^\" inside the tbOpr.Text to know where the power number start, but instead you are picking it from fixed position(s) 2 or 2 & 3 which will never return correct power number from string like \"1024^3\".", null, "", null, "Reply With Quote\n\n5. ## Re: [RESOLVED] Can't seem to retrieve a specified character from a string of numbers\n\nGood to know!\nThanks so much for your help!", null, "", null, "Reply With Quote\n\ncalculator, convert, integer", null, "####", null, "Posting Permissions\n\n• You may not post new threads\n• You may not post replies\n• You may not post attachments\n• You may not edit your posts\n•\n\nFeatured" ]	[ null, "http://www.qsstats.com/dcsjkgmqe10000s1olv9xxzva_8z2v/njs.gif", null, "http://www.vbforums.com/images/icons/completeclear.gif", null, "http://www.vbforums.com/images/misc/progress.gif", null, "http://www.vbforums.com/clear.gif", null, "http://www.vbforums.com/images/misc/progress.gif", null, "http://www.vbforums.com/clear.gif", null, "http://www.vbforums.com/images/misc/progress.gif", null, "http://www.vbforums.com/clear.gif", null, "http://www.vbforums.com/images/misc/progress.gif", null, "http://www.vbforums.com/clear.gif", null, "http://www.vbforums.com/images/misc/progress.gif", null, "http://www.vbforums.com/clear.gif", null, "http://www.vbforums.com/images/misc/11x11progress.gif", null, "http://www.vbforums.com/images/buttons/collapse_40b.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.67320204,"math_prob":0.77115065,"size":3335,"snap":"2019-43-2019-47","text_gpt3_token_len":1010,"char_repetition_ratio":0.14109877,"word_repetition_ratio":0.2749141,"special_character_ratio":0.30164918,"punctuation_ratio":0.14057972,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98011917,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-20T16:56:36Z\",\"WARC-Record-ID\":\"<urn:uuid:de44dcdf-52bd-49d0-92b8-a8e5ab6b2bad>\",\"Content-Length\":\"104433\",\"Content-Type\":\"application/http; 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https://newpathworksheets.com/math/grade-7/area-and-circumference-of-circles/ohio-standards	[ "## ◂Math Worksheets and Study Guides Seventh Grade. Area and Circumference of Circles\n\n### The resources above correspond to the standards listed below:\n\n#### Ohio Standards\n\nOH.GSS. Geometry and Spatial Sense: Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two- and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems.\nGSS.B. Draw circles, and identify and determine the relationships among the radius, diameter, center and circumference.\nOH.M. Measurement: Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools and technologies.\nM.C. Identify appropriate tools and apply appropriate techniques for measuring angles, perimeter or circumference and area of triangles, quadrilaterals, circles and composite shapes, and surface area and volume of prisms and cylinders." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8487356,"math_prob":0.8627317,"size":870,"snap":"2019-35-2019-39","text_gpt3_token_len":151,"char_repetition_ratio":0.11778291,"word_repetition_ratio":0.0,"special_character_ratio":0.16666667,"punctuation_ratio":0.19594595,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97952515,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-09-16T06:51:39Z\",\"WARC-Record-ID\":\"<urn:uuid:d6d4131d-dc09-4f74-8cc4-5ee2db428dfa>\",\"Content-Length\":\"35211\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:5c6cb256-7d1b-4fd0-8ac7-a31a57650557>\",\"WARC-Concurrent-To\":\"<urn:uuid:7234717b-cefc-47bb-902e-882a2c1bbbdc>\",\"WARC-IP-Address\":\"13.58.13.153\",\"WARC-Target-URI\":\"https://newpathworksheets.com/math/grade-7/area-and-circumference-of-circles/ohio-standards\",\"WARC-Payload-Digest\":\"sha1:IOWGURRA3M3EFGKD76KUKVG25CUJ7GGF\",\"WARC-Block-Digest\":\"sha1:HSLDGP2K5Q62AMH3RTKUV4IF2UBP2ATR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-39/CC-MAIN-2019-39_segments_1568514572491.38_warc_CC-MAIN-20190916060046-20190916082046-00457.warc.gz\"}"}
https://stackoom.com/en/question/4zy3p	[ "# How to round down two significant figures in python?\n\nI've seen the solutions around but they mostly round up to two significant figures and not down\n\nI have tried these few methods\n\n``````import math\nv = 0.000129\nmath.floor(v100)/100\n\n-output-\n0.0\n``````\n\nor\n\n``````v = 0.000129\nfrom decimal import Decimal\nfloat(f\"{Decimal(f'{v:.2g}'):f}\")\n\n-output-\n0.00013\n``````\n\nAs you can see, I want to have two significant figures but do not want them rounded up. Decimal works to give the two sig figs but it rounds up while math just simply gives me 0.\n\nie a few to test\n\n``````1999 -> 1900\n29901 - > 29000\n0.0199 -> 0.019\n``````\n\nThanks!\n\nMathematical solution without using any string conversion:\n\n``````def round_down(n, sig_figs):\nimport math\nreturn n - n % 10 * math.ceil(math.log(abs(n), 10) - sig_figs)\n\n>>> [round_down(n, 2) for n in [1990, 29901, 0.0199]]\n[1900, 29000, 0.019]\n``````\n\nCaveats:\n\n• Doesn't work with input of `0`\n• Negative numbers are literally rounded in the negative direction, not towards zero (eg `-0.0199``-0.02` )\nQuestion not resolved ? You can try search: How to round down two significant figures in python?." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.753595,"math_prob":0.85287666,"size":997,"snap":"2023-40-2023-50","text_gpt3_token_len":311,"char_repetition_ratio":0.09063444,"word_repetition_ratio":0.0,"special_character_ratio":0.37311935,"punctuation_ratio":0.16517857,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9941479,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-02T01:40:52Z\",\"WARC-Record-ID\":\"<urn:uuid:7a809826-5354-4cf3-872f-ab3c618a0c76>\",\"Content-Length\":\"41991\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:67d36c1b-2146-4bba-9f00-b10eba89337b>\",\"WARC-Concurrent-To\":\"<urn:uuid:398dfaab-6fb5-448a-93d1-b70e3074c33b>\",\"WARC-IP-Address\":\"118.31.174.222\",\"WARC-Target-URI\":\"https://stackoom.com/en/question/4zy3p\",\"WARC-Payload-Digest\":\"sha1:W6IRRPMIQJA3J465HDTLBBVP37WOXMDW\",\"WARC-Block-Digest\":\"sha1:JT4I4XGIAVP7PIE4XHHA4RSRHE347QUK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100309.57_warc_CC-MAIN-20231202010506-20231202040506-00866.warc.gz\"}"}
https://softmath.com/tutorials-3/relations/differential-equations-problem.html	[ "English \| Español\n\n# Try our Free Online Math Solver!", null, "Online Math Solver\n\n Depdendent Variable\n\n Number of equations to solve: 23456789\n Equ. #1:\n Equ. #2:\n\n Equ. #3:\n\n Equ. #4:\n\n Equ. #5:\n\n Equ. #6:\n\n Equ. #7:\n\n Equ. #8:\n\n Equ. #9:\n\n Solve for:\n\n Dependent Variable\n\n Number of inequalities to solve: 23456789\n Ineq. #1:\n Ineq. #2:\n\n Ineq. #3:\n\n Ineq. #4:\n\n Ineq. #5:\n\n Ineq. #6:\n\n Ineq. #7:\n\n Ineq. #8:\n\n Ineq. #9:\n\n Solve for:\n\n Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:\n\n# Differential equations Problem Set 6\n\nPlease answer all of the following questions. The problems marked with an asterisk will be\ngraded while the remained problems will be checked for completeness. Staple your work to\n\n1. Section 10.1 (pp. 575–576) 5, 6, 7, 9, 15, 16.\n\n2. Section 10.2 (pp. 585–587) 14*, 15, 17.\n\n3. A second- order Cauchy -Euler equation has the form", null, "where α are β are constants (see Section 5.5). Homogeneous solutions of this equation may\nbe found in a similar manner as those for constant coefficient equations . The purpose of\nthis exercise is to illustrate this point.\n\n(a) Show that y = xr is a solution of the Cauchy-Euler equation if r satisfies the characteristic\npolynomial", null, "Let us now assume that r1 and r2 are roots of the characteristic polynomial obtained using\nthe quadratic formula . There are three cases to consider.\n\n1. If r1 and r2 are real and unequal, then the general solution has the form", null, "2. If r1 and r2 are complex conjugates, equal to λ±iμ say, then the general solution has\nthe form", null, "3. If r1 = r2 = r, then the general solution has the form", null, "(b) Find general solutions for the following equations .", null, "Parts (a) and (b) are “starred” problems.\n\n Prev Next" ]	[ null, "https://softmath.com/images/video-pages/solver-top.png", null, "https://softmath.com/tutorials-3/relations/articles_imgs/2607/differ18.jpg", null, "https://softmath.com/tutorials-3/relations/articles_imgs/2607/differ19.jpg", null, "https://softmath.com/tutorials-3/relations/articles_imgs/2607/differ20.jpg", null, "https://softmath.com/tutorials-3/relations/articles_imgs/2607/differ21.jpg", null, "https://softmath.com/tutorials-3/relations/articles_imgs/2607/differ22.jpg", null, "https://softmath.com/tutorials-3/relations/articles_imgs/2607/differ23.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87699944,"math_prob":0.9906531,"size":1344,"snap":"2020-24-2020-29","text_gpt3_token_len":365,"char_repetition_ratio":0.12910448,"word_repetition_ratio":0.04680851,"special_character_ratio":0.2641369,"punctuation_ratio":0.13043478,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9991202,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,null,null,4,null,4,null,4,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-03T17:24:41Z\",\"WARC-Record-ID\":\"<urn:uuid:f3c1715e-f8d8-4e8e-9be0-4602cc50b353>\",\"Content-Length\":\"87295\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d67f1958-ec1f-4c88-a0a6-a94f839bd453>\",\"WARC-Concurrent-To\":\"<urn:uuid:dcfc29f2-0529-45b7-840d-59df9d5c8d44>\",\"WARC-IP-Address\":\"52.43.142.96\",\"WARC-Target-URI\":\"https://softmath.com/tutorials-3/relations/differential-equations-problem.html\",\"WARC-Payload-Digest\":\"sha1:ZOEM5IIYHPNKGDE7CLFYV2GBIF2F2JFT\",\"WARC-Block-Digest\":\"sha1:JU6QQOAY6ZLI4YZ3YZWWE2346WXES3VI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655882634.5_warc_CC-MAIN-20200703153451-20200703183451-00452.warc.gz\"}"}
https://montessorifromtheheart.com/2016/08/30/numerals-vs-quantity-at-30-months/	[ "This wooden number-to-quantity interlocking self-correcting Numbers Puzzle (buy similar here) is more abstract as a toddler is not actually holding any quantity to substantiate the value (like with Spindles or Numbers Rods), but rather matches a numeral (e.g. number 2) to the quantity shown on a picture (2 cows). This puzzle is great for matching and counting skills since the pictures are colorful and feature familiar objects/animals.", null, "A child would first sort the puzzle pieces into two piles: one with numbers and the other with quantities (pictures of the objects).", null, "Adrian then would sequence numbers in order from 1-10, by counting objects on a picture and matching them to the correct numeral.\n\nTo substantiate the concept of numeral vs quantity, you may want to present a 3🅿️🌠 Three Period Lesson. (Read a detailed post about the presentation here).\n\nAt 34 months, Adrian is able to correctly complete the entire 3🅿️🌠 Three Period Lesson with numerals one through ten.\n\nPresenting a 3🅿️🌠 Three Period Lesson:\n\n1. (P1) “This is 1, this is 2, ….3”\n2. (P2) “Will you show me 1? Will you show me 2? … 3? \"\n3. (P3) “What is this?", null, "\"Show me 5\"; \"Where is 9?\"; \"What number is this?\" Adrian would correctly answer each time.\n\nThus, at 34 months, when asked randomly, Adrian can orderly sequence, recognize and enunciate each number from 1-10. I believe that such understanding was only achieved through the concrete representation of Montessori math materials, which necessarily build on each other just like in other areas. So, before introducing this puzzle, you would start with small concept of numbers one through 10. For example, number \"one\" is represented by a numeral -1 as well as by quantity when a child is actually holding one apple/car/spindle in his/her hands. Such association of quantity to numeral is very concrete in nature: a child is holding one, two, or three spindles and realizes how differently it feels than holding ten. Such helps children understand in a concrete way basic mathematical concepts, while instilling the love for learning, numbers, and math." ]	[ null, "https://i1.wp.com/montessorifromtheheart.com/wp-content/uploads/my-blog/6a01b8d19ea395970c01b8d2199cbe970c.jpg", null, "https://i1.wp.com/montessorifromtheheart.com/wp-content/uploads/my-blog/6a01b8d19ea395970c01b7c88fe176970b.jpg", null, "https://i2.wp.com/montessorifromtheheart.com/wp-content/uploads/my-blog/6a01b8d19ea395970c01b7c88fe18d970b.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.928929,"math_prob":0.95112807,"size":2090,"snap":"2021-04-2021-17","text_gpt3_token_len":489,"char_repetition_ratio":0.10354746,"word_repetition_ratio":0.0,"special_character_ratio":0.22488038,"punctuation_ratio":0.117794484,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96216106,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,4,null,4,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-01-24T22:25:20Z\",\"WARC-Record-ID\":\"<urn:uuid:9311997f-9895-421b-924f-dee54dd751ab>\",\"Content-Length\":\"163435\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:21c24fa6-bd9f-472b-a53b-180283e507e7>\",\"WARC-Concurrent-To\":\"<urn:uuid:590c36dc-41e0-482a-84ca-9da93ee18ffd>\",\"WARC-IP-Address\":\"192.0.78.144\",\"WARC-Target-URI\":\"https://montessorifromtheheart.com/2016/08/30/numerals-vs-quantity-at-30-months/\",\"WARC-Payload-Digest\":\"sha1:2ZI7XGAO7DRM2IPTM4FLIL5O4JIREDFZ\",\"WARC-Block-Digest\":\"sha1:5SLPQ5J2FY57ZJRLIJJDFM5KJL2XKOCJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-04/CC-MAIN-2021-04_segments_1610703557462.87_warc_CC-MAIN-20210124204052-20210124234052-00559.warc.gz\"}"}
https://www.datasciencemadesimple.com/quantile-quantile-plot-in-r-qq-plot-in-r/	[ "# Quantile – Quantile plot in R or QQ Plot in R\n\nQuantile – Quantile plot in R which is also known as QQ plot in R is one of the best way to test how well the data is distributed normally. QQ plot is even better than histogram to test the normality of the data. we will be plotting Q-Q plot with qqnorm() function in R. Q-Q plot in R is explained with example.\n\nFor what QQ plot is used for ?\n\n• QQ plot is used to test the normality of a data\n• QQ plot is used to compare two data\n\nLet’s see both with an example\n\n#### Quantile – Quantile plot in R to test the normality of a data:\n\nIn R, qqnorm() function plots your data against a standard normal distribution.\n\n1. Give data as an input to qqnorm() function\n2. R takes up this data and create a sample values with standard normal distribution\n3. Then R compares these two data sets (input data set and generated standard normal data set)\n4. Sorts both the data sets\n5. Then finally plots these two sorted data sets against each other.\n\nAll the above steps are done simply by using QQnorm function in R\n\n#### Quantile – Quantile plot in R Example (test the normality):\n\nLet consider inbuilt “trees” data set and let’s check the normality of trees height\n\n```# QQ plot in R to test the normality of data\n\nqqnorm(trees\\$Height,main=\"Height of black cherry trees\")\nqqline(trees\\$Height) ## adds the line to the plot\n```\n\nwhen we give trees height as an input to the qqnorm() function in R. R executes all the above mentioned steps and returns the following QQ plot", null, "#### Quantile – Quantile plot in R to compare two data set:\n\n1. In this method R simply takes up two data sets\n2. Sorts both the data sets\n3. Plots these two sorted data sets against each other.\n\n#### Example of QQ plot in R (compare two data set):\n\nLets use same trees data set and compare the trees Girth and its Volume with QQ plot function as shown below\n\n```# QQ plot in R to compare two data samples\n\nqqplot(trees\\$Volume,trees\\$Girth, main=\"Volume vs Girth of trees\")\n```\n\ntwo data (volume and girth) are sorted and plotted against each other, so the output will be", null, "" ]	[ null, "https://www.datasciencemadesimple.com/wp-content/plugins/a3-lazy-load/assets/images/lazy_placeholder.gif", null, "https://www.datasciencemadesimple.com/wp-content/plugins/a3-lazy-load/assets/images/lazy_placeholder.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8802939,"math_prob":0.9810865,"size":1973,"snap":"2020-34-2020-40","text_gpt3_token_len":471,"char_repetition_ratio":0.17673945,"word_repetition_ratio":0.13586956,"special_character_ratio":0.22554485,"punctuation_ratio":0.046511628,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995684,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-09-18T07:08:41Z\",\"WARC-Record-ID\":\"<urn:uuid:efa1e0ff-fa02-4a5f-8a9e-7514de81e645>\",\"Content-Length\":\"65158\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e1edf0ef-08ee-456f-9468-b139b140d5ca>\",\"WARC-Concurrent-To\":\"<urn:uuid:90a87d2a-db87-4395-a554-5452c81c4040>\",\"WARC-IP-Address\":\"172.67.190.10\",\"WARC-Target-URI\":\"https://www.datasciencemadesimple.com/quantile-quantile-plot-in-r-qq-plot-in-r/\",\"WARC-Payload-Digest\":\"sha1:ELOWDD5LTPPAQLF3OLDSBHQSHKOBTU5S\",\"WARC-Block-Digest\":\"sha1:GEYHRWBRIENWS7I7VAL75VPP6TWWV3PD\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600400187354.1_warc_CC-MAIN-20200918061627-20200918091627-00141.warc.gz\"}"}
https://www.inote.tw/tag/google-drive	[ "◎◎ 網站小檔案◎◎\n\n■ 網站性質：檔案共享、雲端硬碟、線上文書", null, "`=unique(transpose(split(ArrayFormula(concatenate(\\$A2:A&\" \")),\" \")))`", null, "`=unique(transpose(split(ArrayFormula(concatenate(\\$B2:B&\" \")),\" \")))`\n\n`=SUMIF(A2:A11,D2,C2:C11)`", null, "`=SUMIF(A2:A11,D3,C2:C11)`", null, "", null, "", null, "" ]	[ null, "data:image/svg+xml;utf8,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20640%20388'%3E%3C/svg%3E", null, "data:image/svg+xml;utf8,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20640%20388'%3E%3C/svg%3E", null, "data:image/svg+xml;utf8,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20640%20388'%3E%3C/svg%3E", null, "data:image/svg+xml;utf8,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%200%200'%3E%3C/svg%3E", null, "data:image/svg+xml;utf8,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%200%200'%3E%3C/svg%3E", null, "data:image/svg+xml;utf8,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%200%200'%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.93358654,"math_prob":0.9983976,"size":2107,"snap":"2022-27-2022-33","text_gpt3_token_len":1499,"char_repetition_ratio":0.23965763,"word_repetition_ratio":0.0,"special_character_ratio":0.21547224,"punctuation_ratio":0.058641974,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99502814,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-10T10:39:14Z\",\"WARC-Record-ID\":\"<urn:uuid:2ddccbd5-2184-499b-ac27-b2b37827bd85>\",\"Content-Length\":\"82816\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:53f9d330-15fb-4f22-93f7-87cbd98424c8>\",\"WARC-Concurrent-To\":\"<urn:uuid:f0074b88-0d4a-4b90-a156-38281f75114f>\",\"WARC-IP-Address\":\"162.159.137.54\",\"WARC-Target-URI\":\"https://www.inote.tw/tag/google-drive\",\"WARC-Payload-Digest\":\"sha1:XCBXKH45WGA2C46CKVLPMR5GI5HEYW6I\",\"WARC-Block-Digest\":\"sha1:74MRV3J4SUBCHFMNXJBM5QYKEV6D3Y3L\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571153.86_warc_CC-MAIN-20220810100712-20220810130712-00605.warc.gz\"}"}
https://convertoctopus.com/79-cubic-inches-to-cubic-meters	[ "## Conversion formula\n\nThe conversion factor from cubic inches to cubic meters is 1.63870640693E-5, which means that 1 cubic inch is equal to 1.63870640693E-5 cubic meters:\n\n1 in3 = 1.63870640693E-5 m3\n\nTo convert 79 cubic inches into cubic meters we have to multiply 79 by the conversion factor in order to get the volume amount from cubic inches to cubic meters. We can also form a simple proportion to calculate the result:\n\n1 in3 → 1.63870640693E-5 m3\n\n79 in3 → V(m3)\n\nSolve the above proportion to obtain the volume V in cubic meters:\n\nV(m3) = 79 in3 × 1.63870640693E-5 m3\n\nV(m3) = 0.0012945780614747 m3\n\nThe final result is:\n\n79 in3 → 0.0012945780614747 m3\n\nWe conclude that 79 cubic inches is equivalent to 0.0012945780614747 cubic meters:\n\n79 cubic inches = 0.0012945780614747 cubic meters\n\n## Alternative conversion\n\nWe can also convert by utilizing the inverse value of the conversion factor. In this case 1 cubic meter is equal to 772.45245362869 × 79 cubic inches.\n\nAnother way is saying that 79 cubic inches is equal to 1 ÷ 772.45245362869 cubic meters.\n\n## Approximate result\n\nFor practical purposes we can round our final result to an approximate numerical value. We can say that seventy-nine cubic inches is approximately zero point zero zero one cubic meters:\n\n79 in3 ≅ 0.001 m3\n\nAn alternative is also that one cubic meter is approximately seven hundred seventy-two point four five two times seventy-nine cubic inches.\n\n## Conversion table\n\n### cubic inches to cubic meters chart\n\nFor quick reference purposes, below is the conversion table you can use to convert from cubic inches to cubic meters\n\ncubic inches (in3) cubic meters (m3)\n80 cubic inches 0.001 cubic meters\n81 cubic inches 0.001 cubic meters\n82 cubic inches 0.001 cubic meters\n83 cubic inches 0.001 cubic meters\n84 cubic inches 0.001 cubic meters\n85 cubic inches 0.001 cubic meters\n86 cubic inches 0.001 cubic meters\n87 cubic inches 0.001 cubic meters\n88 cubic inches 0.001 cubic meters\n89 cubic inches 0.001 cubic meters" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6532102,"math_prob":0.9960811,"size":2001,"snap":"2022-27-2022-33","text_gpt3_token_len":557,"char_repetition_ratio":0.2889334,"word_repetition_ratio":0.055214725,"special_character_ratio":0.34082958,"punctuation_ratio":0.0927835,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9988902,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-13T06:44:54Z\",\"WARC-Record-ID\":\"<urn:uuid:f8d44574-f2df-4f7d-9a4b-4b8dd5d05c41>\",\"Content-Length\":\"30674\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ebc65624-0205-4bfc-8143-1b24becc58be>\",\"WARC-Concurrent-To\":\"<urn:uuid:ba550c81-b29c-4d44-9a8d-40eae5cc9b8c>\",\"WARC-IP-Address\":\"104.21.29.10\",\"WARC-Target-URI\":\"https://convertoctopus.com/79-cubic-inches-to-cubic-meters\",\"WARC-Payload-Digest\":\"sha1:4RD6A5FBLT6236AEDCTPHZXZYJXSFK4R\",\"WARC-Block-Digest\":\"sha1:NV2F5Y4YR3EQF56GDKYJKISG4QGIOCF2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882571909.51_warc_CC-MAIN-20220813051311-20220813081311-00184.warc.gz\"}"}
https://infoscience.epfl.ch/record/182627?ln=en	[ "## Quotient-method Algorithms for Input-affine Single-input Nonlinear Systems\n\nMany real-world systems are intrinsically nonlinear. This thesis proposes various algorithms for designing control laws for input-affine single-input nonlinear systems. These algorithms, which are based on the concept of quotients used in nonlinear control design, can break down a single-input system into cascade of smaller subsystems of reduced dimension. These subsystems are well defined for feedback-linearizable systems. However, approximations are required to handle non-feedback-linearizable systems. The method proceeds iteratively and consists of two stages. During the forward stage, an equivalence relationship is defined to isolate the states that are not directly affected by the input, which reduces the dimension of the system. The resulting system is an input-affine single-input system controlled by a pseudo-input which represents a degree of freedom in the algorithm. The pseudo-input is a complementary state required to complete the diffeomorphism. This procedure is repeated (n − 1) times to give a one-dimensional system, where n is the dimension of the system. The backward stage begins with the one-dimensional system obtained at the end of the forward stage. It iteratively builds the control law required to stabilize the system. At every iteration, a desired profile of the pseudo-input is computed. In this next iteration, this desired profile is used to define an error that is driven asymptotically to zero using an appropriate control law. The quotient method is implemented through two algorithms, with and without diffeomorphism. The algorithm with diffeomorphism clearly depicts the dimension reduction at every iteration and provides a clear insight into the method. In this algorithm, a diffeomorphism is synthesized in order to obtain the normal form of the input vector field. The pseudo-input is the last coordinate of the new coordinate system. A normal projection is used to reduce the dimension of the system. For the algorithm to proceed without any approximation, it is essential that the last coordinate appears linearly in the projection of the transformed drift vector field. Necessary and sufficient conditions to achieve linearity in the last coordinate are given. Having the pseudo-input appearing linearly enables to represent the projected system as an input-affine system. Hence, the whole procedure can be repeated (n−1) times so as to obtain a one-dimensional system. In the second algorithm, a projection function based on the input vector field is defined that imitates both operators, the push forward operater and the normal projection operator of the previous algorithm. Due to the lack of an actual diffeomorphism, there is no apparent dimension reduction. Moreover, it is not directly possible to separate the drift vector field from the input vector field in the projected system. To overcome this obstacle, a bracket is defined that commutes with the projection function. This bracket provides the input vector field of the projected system. This enables the algorithm to proceed by repeating this procedure (n−1) times. As compared with the algorithm with diffeomorphism, the computational effort is reduced. The mathematical tools required to implement this algorithm are presented. A nice feature of these algorithms is the possibility to use the degrees of freedom to overcome singularities. This characteristic is demonstrated through a field-controlled DC motor. Furthermore, the algorithm also provides a way of approximating a non-feedback-linearizable system by a feedback-linearizable one. This has been demonstrated in the cases of the inverted pendulum and the acrobot. On the other hand, the algorithm without diffeomorphism has been demonstrated on the ball-on-a-wheel system. The quotient method can also be implemented whenever a simulation platform is available, that is when the differential equations for the system are not available in standard form. This is accomplished numerically by computing the required diffeomorphism based on the data available from the simulation platform. Two versions of the numerical algorithm are presented. One version leads to faster computations but uses approximation at various steps. The second version has better accuracy but requires considerably more computational time.\n\nBonvin, Dominique\nMüllhaupt, Philippe\nYear:\n2012\nPublisher:\nLausanne, EPFL\nKeywords:\nOther identifiers:\nurn: urn:nbn:ch:bel-epfl-thesis5467-5\nLaboratories:\n\nNote: The status of this file is: EPFL only" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92464966,"math_prob":0.90405685,"size":4296,"snap":"2020-34-2020-40","text_gpt3_token_len":787,"char_repetition_ratio":0.13909599,"word_repetition_ratio":0.007898894,"special_character_ratio":0.17108938,"punctuation_ratio":0.08551724,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9700234,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-12T10:14:21Z\",\"WARC-Record-ID\":\"<urn:uuid:c37025e1-227a-498e-8cd3-ff0a0558436f>\",\"Content-Length\":\"39689\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:cf82486e-9b93-4ef5-8ba8-1aa0b51dbe69>\",\"WARC-Concurrent-To\":\"<urn:uuid:0a5e2388-ed7c-4de2-8b03-ced7e5932f9b>\",\"WARC-IP-Address\":\"34.250.186.131\",\"WARC-Target-URI\":\"https://infoscience.epfl.ch/record/182627?ln=en\",\"WARC-Payload-Digest\":\"sha1:HB4HYXW7PVC4BP236ZW7LNTVGS6L5O3Y\",\"WARC-Block-Digest\":\"sha1:P6WJJRJ3EAW53YSJ6HI6UWZJ3IQK5CKS\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738888.13_warc_CC-MAIN-20200812083025-20200812113025-00238.warc.gz\"}"}
https://ch.mathworks.com/help/stats/hypergeometric-distribution-1.html?s_tid=CRUX_lftnav	[ "# Hypergeometric Distribution\n\nEvaluate the hypergeometric distribution or its inverse, generate pseudorandom samples\n\n## Functions\n\n `hygecdf` Hypergeometric cumulative distribution function `hygepdf` Hypergeometric probability density function `hygeinv` Hypergeometric inverse cumulative distribution function `hygestat` Hypergeometric mean and variance `hygernd` Hypergeometric random numbers `random` Random numbers\n\n## Topics\n\nHypergeometric Distribution\n\nThe hypergeometric distribution models the total number of successes in a fixed-size sample drawn without replacement from a finite population." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7904627,"math_prob":0.9905271,"size":806,"snap":"2021-21-2021-25","text_gpt3_token_len":162,"char_repetition_ratio":0.10723192,"word_repetition_ratio":0.0,"special_character_ratio":0.16873449,"punctuation_ratio":0.08208955,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9878627,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-11T22:56:20Z\",\"WARC-Record-ID\":\"<urn:uuid:9dbe1668-1ce5-4884-a327-cec70fe6a70e>\",\"Content-Length\":\"65027\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:48b88111-f7f9-4fc4-b06f-d67f8acd76a7>\",\"WARC-Concurrent-To\":\"<urn:uuid:8dd237fa-8df8-4eda-a7d5-fbc5e1196ac8>\",\"WARC-IP-Address\":\"104.117.0.182\",\"WARC-Target-URI\":\"https://ch.mathworks.com/help/stats/hypergeometric-distribution-1.html?s_tid=CRUX_lftnav\",\"WARC-Payload-Digest\":\"sha1:NTEM2XCERSYPVUWGUQJX3OG3BSOOQ6YK\",\"WARC-Block-Digest\":\"sha1:VTHLEUXHFQJTDN3XK6EBPPBVD3V27KC4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243990419.12_warc_CC-MAIN-20210511214444-20210512004444-00313.warc.gz\"}"}
https://math.answers.com/Q/What_is_62_percent_of_62	[ "", null, "", null, "", null, "", null, "0\n\n# What is 62 percent of 62?\n\nTo find 62 percent of a number, multiply the number by 0.62. In this instance, 0.62 x 62 = 38.44. Therefore, 62 percent of 62 is equal to 38.44.", null, "Study guides\n\n20 cards\n\n## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials\n\n➡️\nSee all cards\n3.81\n2052 Reviews", null, "Earn +20 pts", null, "", null, "" ]	[ null, "https://math.answers.com/icons/searchIcon.svg", null, "https://math.answers.com/icons/searchGlassWhiteIcon.svg", null, "https://math.answers.com/icons/notificationBellIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/icons/sendIcon.svg", null, "https://math.answers.com/icons/coinIcon.svg", null, "https://math.answers.com/icons/searchIcon.svg", null, "https://st.answers.com/html_test_assets/imp_-_pixel.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9119694,"math_prob":0.995843,"size":354,"snap":"2022-40-2023-06","text_gpt3_token_len":143,"char_repetition_ratio":0.28857142,"word_repetition_ratio":0.05,"special_character_ratio":0.60169494,"punctuation_ratio":0.13829787,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9564847,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-08T14:57:51Z\",\"WARC-Record-ID\":\"<urn:uuid:050fc824-6978-44d4-8757-0a2f1e4dc4d6>\",\"Content-Length\":\"192754\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bca3100f-e28c-4938-8164-54cd00b1b123>\",\"WARC-Concurrent-To\":\"<urn:uuid:fb28497a-b594-4979-b175-2371415c9913>\",\"WARC-IP-Address\":\"146.75.32.203\",\"WARC-Target-URI\":\"https://math.answers.com/Q/What_is_62_percent_of_62\",\"WARC-Payload-Digest\":\"sha1:HNWV5UEWK3MFCYPXQSATCZWXNXKVNH4Q\",\"WARC-Block-Digest\":\"sha1:7KHEQPN7AB4OMNQCLUAGTLSG4DWHGXXF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500813.58_warc_CC-MAIN-20230208123621-20230208153621-00140.warc.gz\"}"}
http://ghilbert-app.appspot.com/general/Tarski's_geometry_axioms.ghi	[ "# Creative Commons Attribution-Share Alike 3.0 Unported (http://creativecommons.org/licenses/by-sa/3.0/) # {{header # \| title = Tarski's axioms # \| subtitle = # \| left = # \| right = # \| shortcut = # \| notes = Since [[w:Euclid\|Euclid]], geometry has been presented as following from postulates. However, it was not until the 20th century that a complete set of postulates were put forward. This page is for one such set, the [[w:Tarski's axioms\|first-order axiomization by Tarski]]. It is less powerful than full Euclidian geometry and more powerful than compass-and-straightedge geometry.footnote 5 on page 191 of Tarski and Givant (1999) It makes reference to points (not directly to lines, angles, or circles), and has two relationships between points: betweenness and congruence. The version here is for two dimensions. Adapting the axioms for one dimension is complicated,Tarski and Givant (1999), pages 204–209 but only slight modifications are needed for more than two dimensions.Tarski and Givant (1999), page 195 # }} # {{modules # \| parameters = [[Interface:Classical propositional calculus\|Classical propositional calculus]], [[Interface:First-order logic with quantifiability]] # \| importedby = [[Line segment congruence]] # \| exportedby = [[Tarski's geometry axioms derived from real numbers]] # }} # # First, we adopt the axioms of [[Interface:Classical propositional calculus\|propositional logic]] and [[Interface:First-order logic with quantifiability\|first-order logic]] (including equality): # param (CLASSICAL Classical_propositional_calculus.ghi () \"\") param (FIRSTORDER First-order_logic_with_quantifiability.ghi (CLASSICAL) \"\") # # The kind `object`, defined in first-order logic, represents a point: tvar (object x y z w u v) var (object a b c xx yy) # # == Congruence == # We introduce congruence of line segments; `(x y ≡ w z)` means that the line segment xy is the same length as the line segment wz. This property is also known as equidistance.Tarski and Givant, 1999, page 177 term (formula (≡ x y z w)) # # The distance from x to y is the same as the distance from y to x. This property is called reflexivity by Tarski and GivantTarski and Givant, 1999, page 177 and symmetry by Narboux.Narboux, 2007, page 141 Narboux also calls it the pseudo-commutativity property of the oriented distance (on page 148) and in his proofs pseudo_reflexivity.Narboux, 2007 stmt (CongruenceABBA () () (≡ x y y x)) # # A segment which has zero length starts and ends at the same point (although saying \"zero length\" is a bit of a shortcut, as the axioms are not based on real numbers or any other numbers). stmt (CongruenceIdentity () () (→ (≡ x y z z) (= x y))) # # Two segments which are congruent to a common segment are congruent to each other. stmt (CongruenceTransitivityAxiom () () (→ (∧ (≡ x y z u) (≡ x y v w)) (≡ z u v w))) # # == Betweenness == # The other fundamental formula is betweenness; `(between x y z)` signifies \"y is between x and z\". # term (formula (between x y z)) # # There are no other points between x and x. stmt (Indivisibility () () (→ (between x y x) (= x y))) # # === Pasch's axiom === # [[File:Tarski's formulation of Pasch's axiom.svg\|left\|thumb]] # Tarski's version of the [[w:Axiom of Pasch]]. # {{clear}} stmt (Pasch ((x a) (u a) (z a) (y a) (v a)) () (→ (∧ (between x u z) (between y v z)) (∃ a (∧ (between u a y) (between v a x))))) # # === Continuity === # [[File:Tarski's continuity axiom.svg\|left\|thumb]] # The variables `xx` and `yy` correspond with `x` and `y` in the diagram and [[w:Tarski's axioms\|wikipedia]]. (We can't just call them `x` and `y` because we've declared `x` and `y` as objects, which unlike variables cannot be subject to quantification). tvar (formula φx ψy) stmt (Continuity ((φx yy) (φx a) (φx b) (ψy xx) (ψy a) (ψy b) ) () (→ (∃ a (∀ xx (∀ yy (→ (∧ φx ψy) (between a xx yy))))) (∃ b (∀ xx (∀ yy (→ (∧ φx ψy) (between xx b yy))))))) # # == Dimension == # [[File:Points in a plane equidistant to two given points lie on a line.svg\|thumb\|right\|Upper dimension]] # The dimension is greater than one, stmt (LowerDimension () () (∃ a (∃ b (∃ c (∧ (∧ (¬ (between a b c)) (¬ (between b c a))) (¬ (between c a b)) ))))) # # and less than three. stmt (UpperDimension () () (→ (∧ (∧ (∧ (≡ x u x v) (≡ y u y v)) (≡ z u z v)) (¬ (= u v))) (∨ (∨ (between x y z) (between y z x)) (between z x y)))) # # == Axiom of Euclid == # [[File:Tarski's axiom of Euclid C.svg\|thumb\|right]] # There are quite a variety of ways to state Euclid's [[w:parallel postulate]]. Here we adopt one which says that if a point u is in the interior of ∠yxz (in the sense of being on the line segment yz drawn across the angle), then any point v which is further out (that is, u is between v and the vertex of the angle) will also be in the interior (in the sense that a line segment containing v can be drawn across the angle). stmt (AxiomEuclid ((x b a) (u b a) (v b a) (y b a) (z b a)) () ( → (∧ (∧ (between x u v) (between y u z)) (¬ (= x u))) (∃ a (∃ b (∧ (∧ (between x y a) (between x z b)) (between a v b)))))) # # == Five Segment == # [[File:Five segment.svg\|thumb\|right]] # This is a version of familiar theorems concerning congruent triangles (without any explicit reference to angles, of course). tvar (object x′ y′ z′ u′) stmt (OuterFiveSegment () () (→ (∧ (∧ (∧ (∧ (∧ (∧ (¬ (= x y)) (between x y z)) (between x′ y′ z′)) (≡ x y x′ y′)) (≡ y z y′ z′)) (≡ x u x′ u′)) (≡ y u y′ u′)) (≡ z u z′ u′)) ) # # == Segment Construction == # Given a line segment wx and a line segment yz, construct an extension of wx which is as long as yz is. stmt (SegmentConstruction ((w a) (x a) (y a) (z a)) () (∃ a (∧ (between w x a) (≡ x a y z)))) # # == Builders == # Being able to substitute equals for equals is generally taken as a logical axiom, but we need to provide it for every operator. tvar (object x0 x1 x2 x3 y0 y1 y2 y3) stmt (CongruenceBuilder () () ( → (∧ (∧ (∧ (= x0 y0) (= x1 y1)) (= x2 y2)) (= x3 y3)) (↔ (≡ x0 x1 x2 x3) (≡ y0 y1 y2 y3)) )) stmt (BetweenBuilder () () ( → (∧ (∧ (= x0 y0) (= x1 y1)) (= x2 y2)) (↔ (between x0 x1 x2) (between y0 y1 y2)) )) # # == Comments about the predicate logic == # # There are two related comments to make about translating the predicate logic used in Tarski and Givant to what we use here. The Tarski and Givant paper, except as explicitly noted in axiom schema As. 11, does not contain axiom schemas but instead concrete axioms. The corresponding concept, given our [[Interface:First-order logic with quantifiability\|first-order logic]], would be to use `variable` (rather than `object`) everywhere with distinct variable constraints between all variables (again, except for As. 11 which has explicit distinct variable constraints in the Tarski and Givant paper).Tarski and Givant, 1999, page 177 and page 185 # For convenience, we loosen this in two ways. First of all, we use `object` instead of `variable` for all variables not subject to quantification (for comparison, this is similar to the way that metamath handles their complex number axioms[http://us.metamath.org/mpeuni/mmcomplex.html Real and Complex Numbers], last updated on May 6, 2008, paragraph beginning \"In case you are wondering: Why do we use the purple class variables for most postulates instead of the more conventional-looking red set variables?\"). Secondly, we only supply distinct variable constraints where quantifiers are involved (this also parallels metamath, although the metamath page doesn't explicitly discuss it). These changes do not affect the mathematical meaning of the axioms or their strength: for an example of translating axioms stated using `variable` and distinct variable constraints on everything, to axioms in the style presented here, see [[First steps in set theory]]. # # == Expressions for points == # Geometry traditionally proves the existence of a point and then assigns it to a variable, rather than providing an expression for that point. For example, we might say \"let C be the midpoint of the line segment A B\" rather than having a notation \"midpoint A B\". To supply that kind of notation, we'd need an axiom analogous to `Abstraction` in [[Interface:Zermelo-Fraenkel set theory]] (of course the notation wouldn't create a set, just a single point for which a formula holds). We do not pursue this notation, in deference to tradition and to avoid complexities in convincing ourselves that such an extension would not inadvertently make our system inconsistent or increase its strength. # # == Tarski's axioms as the basis for geometry == # Tarski's axioms are intended to be sufficient for the development of classical geometry (in the style of Euclid's ''Elements'' and similar works). Getting from the axioms to the familiar theorems involving congruent triangles, angles, midpoints, perpendicular and parallel lines, and the like does entail quite a few proofs. They are broken up into the following pages, with references to the corresponding chapters in Narboux.Narboux, 2007, which references the chapter numbering to W. Schwabhäuser, W Szmielew, and A. Tarski (1983), ''Metamathematische Methoden in der Geometrie'', ISBN 0387129588 # [[Line segment congruence]]. Also includes outer three segment and segment construction uniqueness. (chapter 2) # [[Betweenness of points]], including the existence of distinct points, `PaschLine` (chapter 3) # [[Triangle congruence]] defined and with some basic results for degenerate triangles. Also includes inner five segment. (first half of chapter 4) # [[Collinearity]] (second half of chapter 4) # [[Connectivity for betweenness]] (first half of chapter 5) # [[Line segment inequality]] including an additional line segment construction theorem (second half of chapter 5) # [[Out lines]] (chapter 6) # [[Symmetric point]] (first portion of chapter 7) # [[Midpoint]] (latter portion of chapter 7) # Orthogonality: [[Orthogonality 1\|1]], [[Orthogonality 2\|2]], [[Orthogonality 3\|3]] (chapter 8) # [[The plane]] (chapter 9) # [[Line reflexivity]] (chapter 10) # [[Angles]] (chapter 11) # [[Parallelism]] (chapter 12) # # == References == # # * [[w:Tarski's axioms]] # * Tarski, Alfred; Givant, Steven (1999), \"Tarski's system of geometry\", The Bulletin of Symbolic Logic 5 (2): 175–214, [[doi:10.2307/421089]], MR1791303, ISSN 1079-8986 # * Julien Narboux (2007), \"[http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.158.8614 Mechanical Theorem Proving in Tarski’s Geometry]\", F. Botana and T. Recio (Eds.): ADG 2006, LNAI 4869, pp. 139–156 # # {{DEFAULTSORT:{{PAGENAME}}}} # [[Category:Euclidean geometries (general) and generalizations]] # [[Category:Foundations of classical theories (including reverse mathematics)]]" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8385652,"math_prob":0.81528145,"size":10668,"snap":"2019-13-2019-22","text_gpt3_token_len":3062,"char_repetition_ratio":0.11937359,"word_repetition_ratio":0.024711696,"special_character_ratio":0.30221224,"punctuation_ratio":0.10037982,"nsfw_num_words":6,"has_unicode_error":false,"math_prob_llama3":0.99658936,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-23T04:19:43Z\",\"WARC-Record-ID\":\"<urn:uuid:d1ee529f-307b-4e04-b7e7-7954cb4007c6>\",\"Content-Length\":\"11538\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:14bf4b69-dabc-4895-ada3-5f8568784fd8>\",\"WARC-Concurrent-To\":\"<urn:uuid:3f7ddda9-b834-4aff-98ff-6aec3c4443a9>\",\"WARC-IP-Address\":\"172.217.7.148\",\"WARC-Target-URI\":\"http://ghilbert-app.appspot.com/general/Tarski's_geometry_axioms.ghi\",\"WARC-Payload-Digest\":\"sha1:WWPICIU75MPMMDPDKSO4LA6FWWSUKS4E\",\"WARC-Block-Digest\":\"sha1:SHRKZR6CDYUBGPEYRVCZRL3OCUDBHECK\",\"WARC-Identified-Payload-Type\":\"text/plain\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912202723.74_warc_CC-MAIN-20190323040640-20190323062640-00099.warc.gz\"}"}
https://viterbi-web.usc.edu/~jbarbic/code/index.html	[ "Jernej Barbic's Research Code\n\nVega FEM 4.0 has been released. Vega is a computationally efficient and stable C/C++ library to timestep nonlinear three-dimensional deformable models. It is designed to model large deformations, including geometric and material nonlinearities, and can also efficiently simulate linear systems.\n\nMuch of the code below is now available in Vega FEM.\n\nAll code is C/C++, and released under the BSD license.\n\nAll code written by Jernej Barbic. Joint research with Prof Doug L. James, at the Carnegie Mellon University Graphics Lab and Prof Jovan Popovic, at the MIT Computer Graphics Group.\n\nIf you publish a research paper using this code, we will appreciate if you cite our implementation (for example, like this), or you can cite one of our relevant papers. This code is research code. If you find bugs, please let us know.\n\nThis code is cross-platform. It compiles under Mac OS X, Linux, and Windows. Please do not repost this code on other websites, in any format.\n\nThis material is based upon work supported by the National Science Foundation under Grant No. CAREER-53-4509-6600. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF). This work was also supported by the James H. Zumberge Research and Innovation Fund at USC.\n\nFeedback form\n\nStVK: FEM Saint Venant-Kirchhoff deformable object library (multi-threaded)", null, "This class can compute FEM internal forces, tangent stiffness matrices, and the Hessians of the internal forces, for 3D volumetric meshes (for full models, without any reduction). Two mesh element types are supported: linear tetrahedra and cubes (i.e., the two types of meshes supported by our \"volumetricMesh\" class). Large deformations are supported (\"geometric nonlinearities\"). The strain-stress law is linear (the Saint Venant-Kirchhoff model). Implementation is multi-threaded and achieves near-linear scalability (tested on an Intel Xeon dual-processor, with each processor a quad-core (8 cores total)).\n\nThis library also supports model reduction. It can precompute the cubic polynomials of a reduced StVK deformable model. It can also evaluate the reduced internal forces, stiffness matrices, and Hessians very efficiently.\n\nDependencies: volumetricMesh, sparseMatrix.\nThis library is a part of Vega. Look for it in the \"libraries/stvk\" folder in Vega.\n\nModal Matrix: A class for modal matrix operations (assembly u=Uq, projection q = U^T u, optionally with SSE instructions)\n\nGiven a modal matrix U, this class can perform multiplications u=Uq, and q=U^Tu. These are standard operations frequently encountered in modal simulations.\n\nDependencies: none\nThis library is a part of Vega. Look for it in the \"libraries/modalMatrix\" folder in Vega.\n\nMass Spring System: A general 3D mass-spring system (multi-threaded)\n\nThis class allows you to compute mass spring system internal forces, tangent stiffness matrices, and Hessians of internal forces. It can also compute damping. Arbitrary 3D mass-spring networks are supported. You specify the vertices and connectivity, and this class provides forces, stiffness matrices, and Hessians. It is also possible to build a mass-spring network directly from a tetrahedral mesh (vertices become masses, and edges become springs).\n\nDependencies: volumetricMesh, sparseMatrix, configFile.\nThis library is a part of Vega. Look for it in the \"libraries/massSpringSystem\" folder in Vega. Acknowledging\n\nImplicit Newmark and Central Differences Integrators", null, "This code can numerically timestep large deformation nonlinear elasticity. More specifically, this code can numerically timestep a system of ODEs of the form:\n\nM * q'' + (alpha * M + beta * K(q)) q' + R(q) = fext(t),\nwhere q is an unknown vector function (such as the deformations), fext(t) are arbitrary user-provided external forces, R(q) is an arbitrary user-provided vector function, and K(q) = d R / d q is its user-provided gradient. Parameters alpha and beta are Rayleigh damping parameters. Such ODEs are obtained when simulating nonlinear FEM elasticity of large-deformation deformable models: R(q) are the internal elastic forces, and K(q) is the tangent stiffness matrix.\n\nTwo integrators are provided: implicit Newmark (implicit), and central differences (explicit). Implicit Newmark is stable even with large timesteps, but requires a system solve at every timestep, and (especially with large timesteps) introduces some \"numerical viscosity\" into the solution. Central differences require very small timesteps to be stable. The library supports both large sparse systems (arising with geometrically detailed deformable meshes), and small dense systems (arising with model reduction of large sparse systems). The code works in tandem with the StVK library: appropriate R(q) and K(q) functions are already provided both for large sparse elasticity and dense reduced systems.\n\nThe code uses either SPOOLES, PARDISO, or our own Jacobi-preconitioned conjugate gradient solver (see the \"sparseMatrix\" package) to solve the large sparse linear systems. For dense systems, it uses LAPACK.\n\nAlso provided are two driver examples. The first is a complete FEM StVK large sparse nonlinear elasticity simulator (does not use any reduction). The second example is a real-time deformable model simulator, designed based on model reduction.\n\nWith reduction, we used dense ODEs up to the size of about 60. Note that the computation slows down significantly with the size of the reduced basis; this limitation was addressed in the following paper: Steven An, Theodore Kim, Doug L. James: Optimizing Cubature for Efficient Integration of Subspace Deformations, ACM Transactions on Graphics (SIGGRAPH ASIA 2008), 27(5), December 2008.\n\nDependencies: BLAS, LAPACK, StVK.\nNote: This library is now a part of Vega. Look for it in the \"libraries/integrator\" folder in Vega.\n\nLarge Modal Deformation Factory: Model reduction of StVK FEM deformable models", null, "Given a simulation mesh (tetrahedral or voxel), and a set of constrained vertices, this Windows application (complete, standalone, with an installer) can generate a fast reduced deformable model, as described in the following SIGGRAPH paper:\n\nJernej Barbic, Doug L. James: Real-Time Subspace Integration\nfor St.Venant-Kirchhoff Deformable Models, ACM Transactions on\nGraphics (SIGGRAPH 2005), Los Angeles, CA, August 2005\nThe application starts either with a triangle mesh or a volumetric mesh, and generates all the necessary data for a subsequent real-time simulation. It can also launch such a simulation, using a provided real-time executable \"stvk.exe\". Or, you can run a real-time simulation from your own code using our C++ classes provided on this page (\"newmark\" and \"StVK\" libraries). The application supports tetrahedral and voxel 3D volumetric meshes. The application also contains several generally useful sub-components:\n• compute linear modes (via ARPACK) of arbitrary volumetric (tet or voxel) meshes, under arbitrary boundary conditions (including free boundary conditions),\n• compute modal derivatives,\n• compute volumetric mesh mass and stiffness matrix,\n• create cube volumetric meshes for arbitrary input triangle geometry, as described in this SIGGRAPH 2004 Sketch (loadable by our \"VolumetricMesh\" class).\n\nThe application outputs several files, which are compatible with the rest of the code available on this webpage. I.e., it produces the cubic polynomial of reduced internal forces that you can then load using the \"StVKReducedInternalForces\" and \"ImplicitNewmark\" classes (see the \"StVK\" and \"newmark\" libraries) and timestep in your own code. All the modal matrices generated (linear modes, modal derivatives, or simulation basis matrix) can be loaded using the \"Matrix\" class. Instructions are zipped with the executable.\n\nIf you want to use tetrahedral meshes, you can generate them using the TetGen mesh generation package.\n\nNote: This executable is version 2.0. A better, more stable version (3.2) is available in Vega FEM 2.0. Full source code is available in Vega. Look for it in the \"utilities/largeModalDeformationFactory\" folder in Vega.\nNote: This application comes with several example meshes. Look for them in the \"example\" subfolder of your install location. If you can't find them, they are here.\n\niOS demo utility to play reduced deformable models. Written for iPhone 4S, in 2012.\n\nThis is a demo utility whereby the user can apply forces to the reduced deformable model by swiping their fingers across iPhone's screen.\n\nDependencies: none\nios-reducedStVK-v1.1.zip", null, "The \"LQR\" class implements a linear-quadratic regulator (LQR) for the following linear time-varying dynamical system:\n\n\\dot{x} = F(t) * x + G(t) * u ,\n\nwhere x is the state, u is the control, and F and G are (potentially time-varying) matrices. F and G can be arbitrary (they are provided by you). The state and the control can have arbitrary dimensions (not necessarily equal). The implementation proceeds by solving a Riccati differential equation, backwards in time. In order to do so, it needs to compute exponentials of dense matrices. We compute these exponentials using the Expokit package.\n\nDependencies: matrix, expokit.\nlqr-v1.0.zip\n\nVolumetric Mesh: tetrahedral and cube volumetric 3D meshes\n\n The classes in this library provide storage for a general volumetric 3D mesh. Two mesh types are supported: tetrahedral and voxel. The classes store the mesh geometric information (elements and vertices), and also the material parameters of each individual mesh element (Young's modulus, Poisson ratio, mass density). This is done by organizing elements with the same material parameters into a \"solid section\". The class supports several geometric queries and also interpolation to an embedded triangle mesh (\"Free-Form Deformation\"). It also supports exporting the mesh to an .ele or .node format (the format used, e.g., by the TetGen mesh generation package). The class can also compute the mass matrix of the given volumetric mesh. To generate a cube volumetricMesh from an input triangle mesh (optionally flood-filling interior chambers and/or including the interpolation weights), you can use the Large Modal Deformation Factory application. Dependencies: minivector, sparseMatrix (for mass matrix generation). Note: This library is now a part of Vega. Look for it in the \"libraries/volumetricMesh\" folder in Vega.", null, "Cross-platform C code execution timer (performance counter)\n\nA C++ counter to measure code execution time. The timer is accurate up to a few microseconds. You can time arbitrary segments of your code, by placing StartCounter() and StopCounter() before and after your code block. Same interface under Windows, Linux and Mac OS X. Under Windows, the timer uses \"QueryPerformanceCounter\", and under Linux/Mac OS X, it uses \"gettimeofday\".\n\nDependencies: none.\nThis library is now a part of Vega. Look for it in the \"libraries/performanceCounter\" folder in Vega. Acknowledging\n\nMatrix (dense)\n\nThe \"Matrix\" class implements a matrix: a 2D array of real values, together with the commonly defined algebraic operations. The matrix can be rectangular (need not be square). The matrix is stored in the column-major order (same as in LAPACK). Storage is dense (see the Sparse Matrix library for sparse matrices). The class suports common algebraic operations (addition, multiplication, transposition, etc.), including operator overloading, so you can write expressions like: A += 0.25 * (B + A) * C; . It is possible to load/save the matrix from/to a file, using a special well-documented binary format (see matrixIO.h). The class can also perform more complex matrix operations such as computing the SVD decomposition, solving linear systems (via LU decomposition), finding eigenvalues and eigenvectors, solving least square systems, and computing matrix inverses, pseudoinverses and exponentials. The code uses BLAS routines to perform matrix addition and multiplication, and LAPACK for SVD, LU, eigenanalysis, least square systems, inverses and pseudoinverses. Expokit is used for matrix exponentiation. The classes are templated for both float and double datatypes. Also included are routines to perform Principal Component Analysis (PCA) on the columns of the matrix.\n\nDependencies: BLAS, LAPACK, for matrix exponentiation also Expokit\nThis library is now a part of Vega. Look for it in the \"libraries/matrix\" folder in Vega. Acknowledging\n\nSparse Matrix\n\n This class implements sparse matrices with the common algebraic operations such as incremental construction, addition, mtx-vec multiplication, load/save to disk, row-column deletion, etc. Suitable for large sparse matrices. Storage is per-row: for every matrix row, the class stores the integer indices of non-zero columns in that row, together with the corresponding matrix values. Also included is a Conjugate Gradient linear system solver (for positive-definite large sparse symmetric matrices), with (optional) Jacobi preconditioning. The CG Solver was implemented by following Jonathan Shewchuk's An Introduction to the Conjugate Gradient Method Without the Agonizing Pain. Also included is a Gauss-Seidel linear system solver. Dependencies: none. This library is a part of Vega. Look for it in the \"libraries/sparseMatrix\" and \"libraries/sparseSolver\" folder in Vega. Acknowledging", null, "Minivector\n\nA simple class for vector algebra on 2D vectors, 3D vectors (normalization, dot product, cross product, ...), and matrix algebra on 3x3 matrices (summation, multiplication, eigenvalue and eigenvector computation, ...).\n\nDependencies: none.\nThis library is now a part of Vega. Look for it in the \"libraries/minivector\" folder in Vega. Acknowledging\n\nPolar decomposition of a 3x3 matrix (and derivatives of polar decomposition matrices)\n\nThis class can compute the polar decomposition of a general 3x3 matrix. The polar decomposition code has been adapted from the (freely available) polar decomposition implementation provided as a companion to the book \"Graphics Gems IV\".\n\nNew (May 2011): The class can also compute the first and second derivatives of the matrices in polar decomposition, as explained in this publication.\n\nDependencies: minivector.\nThis library is now a part of Vega. Look for it in the \"libraries/polarDecomposition\" folder in Vega. Citation\n\nDynamics of a (single) rigid body", null, "These two classes (\"RigidBody\" and \"RigidBody_GeneralTensor\") implement 6-DOF rigid dynamics of a single rigid body, as explained in:\n\nDavid Baraff:\n\"An Introduction to Physically Based Modeling:\nRigid Body Simulation I: Unconstrained Rigid Body Dynamics\"\n(SIGGRAPH 97 Course Notes)\nThese two classes allow you to simulate the motion of a single rigid body, under any specified (potentially time-varying) external forces and torques. Arbitrary tensors of inertia are supported. For rigid objects where the inertia tensor in the world coordinate system is diagonal, use \"RigidBody\". For the general case (non-diagonal inertia tensor in the world coordinate system), use \"RigidBody_GeneralTensor\". The solution is computed by numerically timestepping the ordinary differential equations of rigid body motion, derived from the Newton's 2nd law, and conservation of linear momentum and angular momentum. For example, ballistic motion can be simulated if gravity is used as the external force. Objects bouncing off the ground/impacting other objects can be simulated if you combine \"RigidBody\" (or \"RigidBody_GeneralTensor\") with a collision detection algorithm that provides the contact external forces.\n\nCurrently, the code supports explicit Euler integration and symplectic Euler.\n\nDependencies: quaternion.\nThis library is now a part of Vega. Look for it in the \"libraries/rigidBodyDynamics\" folder in Vega. Acknowledging\n\nQuaternions\n\nThis C++ class implements quaternions and the common algebraic operations on quaternions (including operator overloading). The class is templated: you can use either float or double precision. Supports using quaternions to represent/manipulate rotations. The Matrix2Quaternion routine is borrowed from David Baraff's course notes (with permission).\n\nDependencies: none.\nThis library is now a part of Vega. Look for it in the \"libraries/quaternion\" folder in Vega.\n\nA C++ class to (very quickly and exactly) timestep the following 1D ODE: M * q''(t) + C * q'(t) + K * q(t) = f(t)", null, "This class integrates a single harmonic oscillator, that is, this code can timestep the following one-dimensional Ordinary Differential Equation:\n\nM * q''(t) + C * q'(t) + K * q(t) = f(t),\nwhere M,C,K are scalar constants, and\nf=f(t) is a given (discretely sampled) function of time, and\nq=q(t) is the unknown (i.e., what we are solving for).\n\nThe system must be underdamped (as is the case in, e.g., sound simulations):\nC < 2 * sqrt(MK).\n\nThe integration uses an IIR filter and is as such EXACT up to floating point arithmetic (i.e., this is better than integrating the ODE with a numerical integrator such as Euler/Runge-Kutta/Central Differences,etc.). The timestep computation is very simple and runs at very fast rates (easily at 44.1 kHz). This code can be used (and has been used) to integrate the modal oscillators for simulating sound, for example, in the following paper:\nDoug L. James, Jernej Barbic, Dinesh K. Pai:\nPrecomputed Acoustic Transfer: Output-sensitive, accurate sound generation\nfor geometrically complex vibration sources,\nACM Transactions on Graphics 25(3) (SIGGRAPH 2006),\np. 987-995, Boston, MA, August 2006.\n\nDependencies: none\nharmonicOscillator_IIR-v1.0.zip\n\nA class to parse custom-defined text configuration files\n\nThis C++ class makes it possible to create your own text configuration file syntax, and then load the values from a particular configuration file (following your syntax) into variables of your C++ program. Supports int, float, double, bool, and char datatypes.\n\nThis class works with text configuration files like these. The syntax of the file is defined using a C++ program in the following way. This example C++ program first defines the configuration file syntax. Then, it opens a particular configuration file and loads the values to the memory locations specified when the syntax was defined. The program output will be as follows.\n\nFor example, this can serve as an improved command-line option tool: you can move your command line options to a text file so that you don't have to retype them each time (useful if there are many). Or, it can serve as an interface between two programs. Write the values out to a file with one program and load them into another.\n\nDependencies: none.\nThis library is now a part of Vega. Look for it in the \"libraries/configFile\" folder in Vega. Acknowledging\n\nOpenGL lighting from a text configuration file\n\nA class to read OpenGL lighting parameters from a text configuration file. Usage is simple: use the provided constructor to read the configuration file during initialization, then call \"LightScene\" inside your OpenGL display routine (after setting up the modelview and projection matrices). Now, you no longer have to recompile your code each time you wish to change some lighting parameter!\n\nHere is an example configuration file.\n\nDependencies: configFile.\nThis library is now a part of Vega. Look for it in the \"libraries/lighting\" folder in Vega. Acknowledging\n\nOpenGL \"spherical\" camera\n\nA class to position an OpenGL camera in 3D, using spherical coordinates. The camera points to a specified focus position.\n\nDependencies: none.\nThis library is now a part of Vega. Look for it in the \"libraries/camera\" folder in Vega. Acknowledging" ]	[ null, "https://viterbi-web.usc.edu/~jbarbic/code/FEM.jpg", null, "https://viterbi-web.usc.edu/~jbarbic/code/stvk-reduced.png", null, "https://viterbi-web.usc.edu/~jbarbic/code/largeModalDeformationFactory-screenShot-sm.png", null, "https://viterbi-web.usc.edu/~jbarbic/code/leaves-small.jpg", null, "https://viterbi-web.usc.edu/~jbarbic/code/dino-tets-sm.jpg", null, "https://viterbi-web.usc.edu/~jbarbic/code/bridge-1sm.png", null, "https://viterbi-web.usc.edu/~jbarbic/code/rigidObject-dragon.jpg", null, "https://viterbi-web.usc.edu/~jbarbic/code/bellsm.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85934675,"math_prob":0.8338007,"size":17631,"snap":"2022-05-2022-21","text_gpt3_token_len":3804,"char_repetition_ratio":0.12304987,"word_repetition_ratio":0.06231003,"special_character_ratio":0.20191708,"punctuation_ratio":0.13643558,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97811913,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-18T19:49:32Z\",\"WARC-Record-ID\":\"<urn:uuid:60634eec-466a-4ad2-bc08-20dc83afa520>\",\"Content-Length\":\"32334\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:161eb6ae-8bb1-43e7-82d2-c7afc5d8a93b>\",\"WARC-Concurrent-To\":\"<urn:uuid:46c68f9e-a247-4473-bf47-a6a25436875c>\",\"WARC-IP-Address\":\"68.181.40.12\",\"WARC-Target-URI\":\"https://viterbi-web.usc.edu/~jbarbic/code/index.html\",\"WARC-Payload-Digest\":\"sha1:SUCPDNDQPVKJUMYIULM4DEZVABDMPTMJ\",\"WARC-Block-Digest\":\"sha1:LEWJKUM6UP32L7Y4Q3OWZPMK5YLTH2TZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320300997.67_warc_CC-MAIN-20220118182855-20220118212855-00051.warc.gz\"}"}
https://cbsencertsolutions.com/differential-equations-class-12-mathematics-important-questions/	[ "# Differential Equations Class 12 Mathematics Important Questions\n\nStudents can read the important questions given below for Differential Equations Class 12 Mathematics. All Differential Equations Class 12 Notes and questions with solutions have been prepared based on the latest syllabus and examination guidelines issued by CBSE, NCERT and KVS. You should read all notes provided by us and Class 12 Mathematics Important Questions provided for all chapters to get better marks in examinations. Mathematics Question Bank Class 12 is available on our website for free download in PDF.\n\n## Important Questions of Differential Equations Class 12\n\nQuestion. Find order and degree of differential equation\n\nDegree not defined\n\nQuestion. Find order and degree of differential equation\n\nDegree=2\n\nQuestion. Integrating factor of\n\nQuestion. For what value of n is the following a homogeneous differential equation:\n\nQuestion. Form the differential equation not containing the arbitrary constants and satisfied by the equation: y = peqx.\n\nQuestion. Write the order and the degree of the following differential equation:\n\nAnswer. Order = 2, degree = 2\n\nQuestion. Find the differential equation of the family of lines passing through the origin.\nAnswer. Generates equation of family of lines passing through origin\ny = mx ⇒ m = y/x\ndy/dx = m = slope\ndy/dx = y/x\nx dy/dx – y = 0\n\nQuestion. Find the solution of the differential equation dy/dx = x3e-2y\n\nQuestion. How many arbitrary constants are there in the particular solution of the differential equation\n\nQuestion. For what value of n is the following a homogeneous differential equation:\n\nQuestion. Solve the differential equation\n\nQuestion. Solve the differential equation\n\nQuestion. Form the differential equation of all circles which is touching the x-axis at the origin.\nAnswer. (x – 0)2 + (y – r)2 = r2\nx2 + y2 = 2ry\n\nQuestion. Form the differential equation representing the family of curves y = aebx+5, where ‘a’ and ‘b’ are arbitrary constants\n\nQuestion. Find the general solution of the differential equation:\n\nAnswer. Given differential equation can be written as\n\nQuestion. Prove that x2 – y2 = C(x2 + y2)2 is the general solution of the differential equation (x2 – 3xy2) dx = (y3 – 3x2y) dy, where C is a parameter\nAnswer. x2 – y2 = C(x2 + y2)2\n\nHence x2 – y2 = C(x2 + y2)2 is the solution of given differential equation.\n\nQuestion. Solve the following differential equation: dy/dx = x3 cosec y, given that y(0) = 0.\n\nQuestion. Find the general solution of the differential equation.\nxy dy/dx = dy/dx = (x+2)(y+2)\n\nQuestion. Find the general solution of the differential equation dy/dx + 2/x y = x\n\nQuestion. Find the integrating factor of the differential equation dy/dx +y = 1+y/x\nAnswer. Given differential equation can be written as\n\nQuestion. Find the differential equation of the family of curves y = Ae2x + Be–2x, where A and B are arbitrary constants.\nAnswer. Differentiating y = Ae2x + Be–2x, we get\ndy/dx = 2Ae2x + 2Be–2x\ndifferentiate again to get\n\nQuestion. Can y = ax + b/a be a solution of the following differential equation ?\n\nIf no, find the solution of the D.E.\n\nQuestion. Find the equation of curve whose tangent at any point on it, different from origin, has slope\n\nQuestion. Solve the differential equation\n\nQuestion. Solve the differential equation\n\nQuestion. Solve the differential equation\n\nQuestion. Solve the differential equation\n\nQuestion. Show that the differential equation\n\nIs homogeneous.\nFind the particular solution given that y = π/4 when x = 1\n\nQuestion. Find the general solution of the differential equation\n\nQuestion. For the differential equation\n\nFind the solution curve passing through the point (1, -1)\n\nQuestion. Solve the differential equation:\ndy/dx – 3y cot x = sin 2x given y = 2, when x= π/2\n\nQuestion. Solve: (1 + x2 + y2 + x2y2)dx + xydy = 0, given that y = 0 and x = 1.\nAnswer. The given equation can be written as:\n\nQuestion. Find the particular solution of the differential equation:\n\nfor x = 1, y = 0.\nAnswer. Given differential equation is homogeneous.\n\nQuestion. Show that the differential equation\n\nis homogeneous. Find the particular solution of this differential equation, given that y = π/4 when x = 1.\n\nQuestion. Solve the differential equation x2dy + (xy + y2) dx = 0 given y = 1, when x = 1.\nAnswer. x2dy + (xy + y2) dx = 0\n\nCASE STUDY :\n\nPolio drops are delivered to 50K children in a district. The rate at which polio drops are given is directly proportional to the number of children who have not been administered the drops. By the end of 2nd week half the children have been given the polio drops. How many will have been given the drops by the end of 3rd week can be estimated using the solution to the differential equation 𝒅𝒚/𝒅𝒙=𝐤(𝟓𝟎−𝐲) where x denotes the number of weeks and y the number of children who have been given the drops.\n\nQuestion. State the order of the above given differential equation.\nOrder is 1\n\nQuestion. Which method of solving a differential equation can be used to solve 𝒅𝒚/𝒅𝒙=𝐤(𝟓𝟎−𝐲).?\na. Variable separable method\nb. Solving Homogeneous differential equation\nc. Solving Linear differential equation\nd. all of the above\n\nA\n\nQuestion. The solution of the differential equation 𝒅𝒚/𝒅𝒙=𝐤(𝟓𝟎−𝐲) is given by,\na. log \| 50 – y\| = kx + C\nb. – log \| 50 – y\| = kx + C\nc. log \| 50 – y\| = log\| kx \|+ C\nd. 50 – y = kx + C\n\nB\n\nQuestion. The value of c in the particular solution given that y(0)=0 and k = 0.049 is.\na. log 50\nb. log 1/50\nc. 50\nd. -50" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87942827,"math_prob":0.99958473,"size":5935,"snap":"2023-40-2023-50","text_gpt3_token_len":1602,"char_repetition_ratio":0.2989378,"word_repetition_ratio":0.19903846,"special_character_ratio":0.25627634,"punctuation_ratio":0.1312608,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99995065,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-04T02:53:51Z\",\"WARC-Record-ID\":\"<urn:uuid:68edb1fc-418e-4f31-ba4e-efc62a24923d>\",\"Content-Length\":\"161019\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b5807cb5-7c56-4f13-ae30-e1137b27c80b>\",\"WARC-Concurrent-To\":\"<urn:uuid:ddb8ef66-3ffd-48e9-bbdc-34000e607812>\",\"WARC-IP-Address\":\"149.100.147.26\",\"WARC-Target-URI\":\"https://cbsencertsolutions.com/differential-equations-class-12-mathematics-important-questions/\",\"WARC-Payload-Digest\":\"sha1:SJUGTBMLU2BGZ3VCCHZWIKD7E3HVYXDV\",\"WARC-Block-Digest\":\"sha1:WZLYS333SL4FNERQ6LOKYLXO7QD2FRXW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100523.4_warc_CC-MAIN-20231204020432-20231204050432-00214.warc.gz\"}"}
https://collaborate.princeton.edu/en/publications/the-geometry-of-turbulent-advection-sharp-estimates-for-the-dimen	[ "# The geometry of turbulent advection: Sharp estimates for the dimensions of level sets\n\nP. Constantin, I. Procaccia\n\nResearch output: Contribution to journalArticlepeer-review\n\n21 Scopus citations\n\n## Abstract\n\nLower bounds on the fractal dimension of level sets of advecting passive scalars in turbulent fields are derived, in the limit that the scalar diffusivity kappa goes to zero. The main result is as follows: denote the Holder exponent of the velocity field u by zeta (u), with 0<or= zeta (u)<or=1, and the Holder exponent of the passive scalar (say T) by zeta (T). We derive a lower bound on the dimension D of the level sets of T, D>or=d-1+ zeta (T)+ zeta (u), where d is the dimension of space. The validity of this bound depends on some conditions concerning the limit kappa to 0; when these are satisfied the bound is obtained throughout the range of zeta (u), between the smooth (but random) velocity field with zeta (u)=1 to the extremely rough field with zeta (u)=0. The derivation of the lower bound calls for the introduction of a measure on the level sets and a careful treatment of the singular limit of the scalar diffusivity going to zero. Together with the upper bounds which were derived previously, i.e. D<or=d-1/2+ zeta (u)/2 we discover, when there is no multiscaling, the scaling relation 2 zeta (T)+ zeta (u)=1, which then means that the lower and the upper bounds in fact coincide.\n\nOriginal language English (US) 014 1045-1054 10 Nonlinearity 7 3 https://doi.org/10.1088/0951-7715/7/3/014 Published - Dec 1 1994 Yes\n\n## All Science Journal Classification (ASJC) codes\n\n• Statistical and Nonlinear Physics\n• Mathematical Physics\n• Physics and Astronomy(all)\n• Applied Mathematics\n\n## Fingerprint\n\nDive into the research topics of 'The geometry of turbulent advection: Sharp estimates for the dimensions of level sets'. Together they form a unique fingerprint." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.861304,"math_prob":0.69489014,"size":1585,"snap":"2022-40-2023-06","text_gpt3_token_len":416,"char_repetition_ratio":0.13029727,"word_repetition_ratio":0.0076923077,"special_character_ratio":0.24794953,"punctuation_ratio":0.06734007,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98294014,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-01-26T21:56:12Z\",\"WARC-Record-ID\":\"<urn:uuid:2401e99e-8fa5-4677-80e2-109bb846831b>\",\"Content-Length\":\"47884\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:bdee1c71-0202-4155-b5a4-11859b3e8c7c>\",\"WARC-Concurrent-To\":\"<urn:uuid:b94e101b-70d1-4117-b54c-cf399e6bf05c>\",\"WARC-IP-Address\":\"3.90.122.189\",\"WARC-Target-URI\":\"https://collaborate.princeton.edu/en/publications/the-geometry-of-turbulent-advection-sharp-estimates-for-the-dimen\",\"WARC-Payload-Digest\":\"sha1:3G2443G7G7J5D4ZWLAA7I5UE4FY2J77X\",\"WARC-Block-Digest\":\"sha1:PSWPYB3XG56GAPGSHBXU2Q2WV4IKOGUG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764494826.88_warc_CC-MAIN-20230126210844-20230127000844-00136.warc.gz\"}"}
https://www.snapxam.com/problems/73406576/integral-of-x-x-2-0-5dx	[ "# Step-by-step Solution\n\nGo!\n1\n2\n3\n4\n5\n6\n7\n8\n9\n0\na\nb\nc\nd\nf\ng\nm\nn\nu\nv\nw\nx\ny\nz\n.\n(◻)\n+\n-\n×\n◻/◻\n/\n÷\n2\n\ne\nπ\nln\nlog\nlog\nlim\nd/dx\nDx\n\|◻\|\n=\n>\n<\n>=\n<=\nsin\ncos\ntan\ncot\nsec\ncsc\n\nasin\nacos\natan\nacot\nasec\nacsc\n\nsinh\ncosh\ntanh\ncoth\nsech\ncsch\n\nasinh\nacosh\natanh\nacoth\nasech\nacsch\n\n## Step-by-step explanation\n\nProblem to solve:\n\n$\\int x\\sqrt{x+2}dx$\n\nLearn how to solve integrals with radicals problems step by step online.\n\n$u=x+2$\n\nLearn how to solve integrals with radicals problems step by step online. Calculate the integral int(x*(x+2)^0.5)dx. We can solve the integral \\int x\\sqrt{x+2}dx by applying integration by substitution method (also called U-Substitution). First, we must identify a section within the integral with a new variable (let's call it u), which when substituted makes the integral easier. We see that x+2 it's a good candidate for substitution. Let's define a variable u and assign it to the choosen part. Now, in order to rewrite dx in terms of du, we need to find the derivative of u. We need to calculate du, we can do that by deriving the equation above. Rewriting x in terms of u. Substituting u, dx and x in the integral and simplify.\n\n$\\frac{2}{5}\\sqrt{\\left(x+2\\right)^{5}}-\\frac{4}{3}\\sqrt{\\left(x+2\\right)^{3}}+C_0$\n$\\int x\\sqrt{x+2}dx$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8893119,"math_prob":0.9972635,"size":856,"snap":"2020-45-2020-50","text_gpt3_token_len":205,"char_repetition_ratio":0.14906104,"word_repetition_ratio":0.114285715,"special_character_ratio":0.22780374,"punctuation_ratio":0.10614525,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99909747,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-29T23:27:59Z\",\"WARC-Record-ID\":\"<urn:uuid:4a67f235-a26c-46c6-a977-f87dfae34445>\",\"Content-Length\":\"30959\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:757a88eb-a2a5-47b1-82bc-76fbc77bcee4>\",\"WARC-Concurrent-To\":\"<urn:uuid:65db7db4-fb5a-4a04-9342-c9008a966643>\",\"WARC-IP-Address\":\"3.18.122.152\",\"WARC-Target-URI\":\"https://www.snapxam.com/problems/73406576/integral-of-x-x-2-0-5dx\",\"WARC-Payload-Digest\":\"sha1:4GBDR6P2XT4D37P6SKD7IOZMWNMW3NSJ\",\"WARC-Block-Digest\":\"sha1:YMM6G32KG2ZYGPVW7DVG4WSTOOAJE7TR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107905965.68_warc_CC-MAIN-20201029214439-20201030004439-00635.warc.gz\"}"}
http://www.kylesconverter.com/data-bandwidth/bytes-per-second-to-petabytes-per-day	[ "# Convert Bytes Per Second to Petabytes Per Day\n\n### Kyle's Converter > Data Bandwidth > Bytes Per Second > Bytes Per Second to Petabytes Per Day\n\n Bytes Per Second (B/s) Petabytes Per Day (PB/d) Precision: 0 1 2 3 4 5 6 7 8 9 12 15 18\nReverse conversion?\nPetabytes Per Day to Bytes Per Second\n(or just enter a value in the \"to\" field)\n\nPlease share if you found this tool useful:\n\nUnit Descriptions\n1 Byte per Second:\n1 Byte per second is equal to 8 bits per second. A byte contains 8 bits. A second is the SI base unit of time. 1 B/s = 8 bit/s.\n1 Petabyte per Day:\n1 Petabyte per day is approximately 92592592592.5926 bits per second. A petabyte contains 8,000,000,000,000,000 bits (base unit). A day contains 86400 seconds (SI base unit). 1 PB/d ? 92592592592.5926 bit/s.\n\nLink to Your Exact Conversion\n\nConversions Table\n1 Bytes Per Second to Petabytes Per Day = 070 Bytes Per Second to Petabytes Per Day = 0\n2 Bytes Per Second to Petabytes Per Day = 080 Bytes Per Second to Petabytes Per Day = 0\n3 Bytes Per Second to Petabytes Per Day = 090 Bytes Per Second to Petabytes Per Day = 0\n4 Bytes Per Second to Petabytes Per Day = 0100 Bytes Per Second to Petabytes Per Day = 0\n5 Bytes Per Second to Petabytes Per Day = 0200 Bytes Per Second to Petabytes Per Day = 0\n6 Bytes Per Second to Petabytes Per Day = 0300 Bytes Per Second to Petabytes Per Day = 0\n7 Bytes Per Second to Petabytes Per Day = 0400 Bytes Per Second to Petabytes Per Day = 0\n8 Bytes Per Second to Petabytes Per Day = 0500 Bytes Per Second to Petabytes Per Day = 0\n9 Bytes Per Second to Petabytes Per Day = 0600 Bytes Per Second to Petabytes Per Day = 0\n10 Bytes Per Second to Petabytes Per Day = 0800 Bytes Per Second to Petabytes Per Day = 0\n20 Bytes Per Second to Petabytes Per Day = 0900 Bytes Per Second to Petabytes Per Day = 0\n30 Bytes Per Second to Petabytes Per Day = 01,000 Bytes Per Second to Petabytes Per Day = 0\n40 Bytes Per Second to Petabytes Per Day = 010,000 Bytes Per Second to Petabytes Per Day = 0\n50 Bytes Per Second to Petabytes Per Day = 0100,000 Bytes Per Second to Petabytes Per Day = 0\n60 Bytes Per Second to Petabytes Per Day = 01,000,000 Bytes Per Second to Petabytes Per Day = 0.0001" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6747812,"math_prob":0.9160844,"size":1421,"snap":"2019-13-2019-22","text_gpt3_token_len":407,"char_repetition_ratio":0.3810868,"word_repetition_ratio":0.5,"special_character_ratio":0.33427164,"punctuation_ratio":0.02112676,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9858398,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-05-26T21:17:41Z\",\"WARC-Record-ID\":\"<urn:uuid:c00af96b-5ded-458d-8989-4fdb04ac5317>\",\"Content-Length\":\"19205\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9d845e23-6dc4-49b9-8ac6-078902cb6958>\",\"WARC-Concurrent-To\":\"<urn:uuid:e7ec7095-510d-44fd-90f7-3505c27f2fd8>\",\"WARC-IP-Address\":\"54.230.192.116\",\"WARC-Target-URI\":\"http://www.kylesconverter.com/data-bandwidth/bytes-per-second-to-petabytes-per-day\",\"WARC-Payload-Digest\":\"sha1:MHOUR7APTLT7HVBQHUAOFPHWEKWRLC3Y\",\"WARC-Block-Digest\":\"sha1:ZKE5YQCPOFAJIRTPBKHZYWWHEWZ6S3AG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-22/CC-MAIN-2019-22_segments_1558232259757.86_warc_CC-MAIN-20190526205447-20190526231447-00284.warc.gz\"}"}
https://mapflc.com/lesson-plans/lesson-plan-geometry-and-rhythms/	[ "## Summary\n\nIn this lesson students will explore the relationship between diving a whole note and dividing a whole circle.\n\n7-13 years\n\n### Lesson duration:\n\n60 minutes\n\n• Introduction: Discuss the various shapes. How many sides? Other properties? Discuss note values. Any relationship to shapes? (10 min)\n• Begin by creating simple shapes. Introduce division of 360. (10 min)\n• Draw a connection between the number of sides and the number 360 is divided by as well as the number of repeats. All of these numbers are the same. (10 min)\n• Re-introduce Box. How can box be used to express this idea? (5 min)\n• Show whole note as a circle and divide it by various note values (e.g. half, quarter, eighth). Any relationship to these shapes? What about dividing by 3 or 5 — what are those called in music?\n\nUp to 10.\n\n### Rationale\n\nIn music, the whole note is the reference for which all divisions are made. A geometric circle works similarly as well — 360 being “the whole”. The way we divide 360 creates the basic shapes.\n\n### Objectives\n\nBy drawing the connection between the two, and using programming tools to create dynamic “rhythm generators” the underlying concept of rhythm is better understood and students should be able to problem solve to figure out any division of the whole that they may encounter when reading music.\n\n## Lesson\n\n### Introduction\n\nBegin by reviewing basic shapes. Ask questions like, “How many sides does a square have?” “How about a triangle?”\n\nAlso ask about basic note values. “How many half notes fit inside a whole note?” “How many quarter notes?”\n\n## Part 1\n\n### A. Making a Square in Music Blocks\n\n1. Start a new project and trash the default notes (Sol, Mi, Sol) and instrument (Guitar) in Music Blocks.\n2. Pull out a repeat block, a forward block, and a right block. By default their values should be 4, 100, and 90 respectively. You should keep their defaults.\n3. Have students guess what will happen when the program is run.\n\nThe above is the code to generate a square. Below is the result (color may vary).\n\nAsk students how they would need to modify this program in order to create a triangle. Try making a triangle together and maybe a few other shapes.\n\n### B. Divide by 360\n\nRight as 90 (degrees) is a portion of a circle — specifically a quarter or a circle. Therefore, we can arrive at this same number by dividing by 4. This step may seem obvious, but it is very important in establishing the mathematical structure that we need to create unique shapes and rhythms automatically. By clicking on the division blocks, you can see the result printed at the top of the screen.\n\nHave students try dividing 360 by other numbers as well. Remind them that 360 are the degrees in a complete (i.e. “whole”) circle.\n\nNext, plug the division blocks into the square code from before.\n\nNow that we have this new structure for creating the square, let’s create a triangle.\n\nThen ask the students to make a pentagon (5 sides).\n\nAt this point students should start to see the pattern. What two numbers are always the same? Why are they the same? Are they the same for any kind of shape? Is there another way to express the formula such that the numbers are always the same?\n\n### C. Insert Box Here\n\nThe box takes a number (or other value) and stores it for use later. The same number can be used multiple times. Since the number of repeats and the denominator for the right angle are the same for all these shapes, we can replace them with box.\n\nNow we just need to change the value for box to reliably generate shapes.\n\nHave students try different numbers for box to see the different results. Lower numbers provide the best results.\n\n### D. Shape Creation Automation\n\nStarting from the box structure of their code, students have what they need to create more dynamic code. We can, for example, create a short snippet of code that creates shapes with 3, 4, 5, and 6 sides.\n\n### E. Cyclic Representation of Whole Note\n\nWhen introducing the concept of note value, I frequently like to start by drawing a circle on the board. I call the circle a “whole pizza”. When we cut the pizza in half, for example, we end up with two halves just like two half notes fill the same amount of time as a whole note.\n\nThe next step for this lesson is to transition from sided-shapes to divisions of a circle (similar to dividing the pizza). We can do this with our current code by replacing the forward and right code with a single angle block. We will also need to move the `360/box` division to the input for angle. Instead of going forward by 100, our radius will be 100.\n\nLastly, by putting the movement inside of a `note value` drum block the movements will happen over time. By replacing the denominator for `note value` with `box` we can tie the length of the note with the divisions of the circle.\n\nThe code snippet above also has an `add to value` block to change the color for each repeat. This helps us get more colorful output to help see the divisions.\n\nWe can add some more visual elements such as changing color and the size of the radius at each repeat to better see how the circles are divided.\n\nOpen the program at https://musicblocks.sugarlabs.org/index.html?id=1590350534007378&run=True\n\n### F. Back to Note Values\n\nAt this point in the lesson, we should bring it back to note values. We started with geometric shapes and ended with circles that are drawn over time with different note values. This is an important moment to ask some fundamental questions about note value.\n\nStudents can be asked the following questions:\n\n• Starting from “whole note” how can we figure out the length of other note values such as `1/2`, `1/3`, `1/4`, `1/5`, etc?\n• Which note values are faster? Which are slower?\n• Is there any way to create a longer note than a `1/1` (whole) note? Can you imagine what that would be?\n• If a `1/2` is a “half note” what would be a `1/3` note? What can we call it? What is its relation to a whole note? (`1/3` is a “half note triplet”, btw)\n\n### G. Create!\n\nEncourage students to add to their “rhythm/shape generators” to create unique projects. The possibilities are endless.\n\nExample Teacher Project (using Pythagorean Tuning instead of Just): https://musicblocks.sugarlabs.org/index.html?id=1597842962357142&run=True\n\n## Performance/Critique\n\n1. Have students discuss the relationship between shape and rhythm. You can start from the questions in the bullet list above.\n2. Have students create their own versions of the projects (either in class or at home) and show their new versions. What is unique? Does it explore something new with regard to note value?\n\n## Materials\n\n• Music Blocks software (Computer, up-to-date browser)\n• If possible, a white board (or equivalent) to demonstrate the divisions of a whole note (e.g. as a pizza).\n\n## Assessments\n\n• Observe participation\n• Do the students make the connection between note value and sides of a shape? Do they make the connection between divisions of a circle and divisions of a whole note?\n• Are students able to apply their understanding of the concept to accurately problem solve unfamiliar note values such as `1/3` or `1/5`?\n• Are the students able to play or clap some of the rhythms? 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72.255.41.142\n)\n\n => Array\n(\n => 116.90.107.102\n)\n\n => Array\n(\n => 39.36.164.250\n)\n\n => Array\n(\n => 124.253.195.172\n)\n\n => Array\n(\n => 203.142.218.149\n)\n\n => Array\n(\n => 157.43.165.180\n)\n\n => Array\n(\n => 39.40.242.57\n)\n\n => Array\n(\n => 103.92.43.150\n)\n\n => Array\n(\n => 39.42.133.202\n)\n\n => Array\n(\n => 119.160.66.11\n)\n\n => Array\n(\n => 138.68.3.7\n)\n\n => Array\n(\n => 210.56.125.226\n)\n\n => Array\n(\n => 157.50.4.249\n)\n\n => Array\n(\n => 124.253.81.162\n)\n\n => Array\n(\n => 103.240.235.141\n)\n\n => Array\n(\n => 132.154.128.20\n)\n\n => Array\n(\n => 49.156.115.37\n)\n\n => Array\n(\n => 45.133.7.48\n)\n\n => Array\n(\n => 122.161.49.137\n)\n\n => Array\n(\n => 202.47.46.31\n)\n\n => Array\n(\n => 192.140.145.148\n)\n\n => Array\n(\n => 202.14.123.10\n)\n\n => Array\n(\n => 122.161.53.98\n)\n\n => Array\n(\n => 124.253.114.113\n)\n\n => Array\n(\n => 103.227.70.34\n)\n\n => Array\n(\n => 223.228.175.227\n)\n\n => Array\n(\n => 157.39.119.110\n)\n\n => Array\n(\n => 180.188.224.231\n)\n\n => Array\n(\n => 132.154.188.85\n)\n\n => Array\n(\n => 197.210.227.207\n)\n\n => Array\n(\n => 103.217.123.177\n)\n\n => Array\n(\n => 124.253.85.31\n)\n\n => Array\n(\n => 123.201.105.97\n)\n\n => Array\n(\n => 39.57.190.37\n)\n\n => Array\n(\n => 202.63.205.248\n)\n\n => Array\n(\n => 122.161.51.100\n)\n\n => Array\n(\n => 39.37.163.97\n)\n\n => Array\n(\n => 43.231.57.173\n)\n\n => Array\n(\n => 223.225.135.169\n)\n\n => Array\n(\n => 119.160.71.136\n)\n\n => Array\n(\n => 122.165.114.93\n)\n\n => Array\n(\n => 47.11.77.102\n)\n\n => Array\n(\n => 49.149.107.198\n)\n\n => Array\n(\n => 192.111.134.206\n)\n\n => Array\n(\n => 182.64.102.43\n)\n\n => Array\n(\n => 124.253.184.111\n)\n\n => Array\n(\n => 171.237.97.228\n)\n\n => Array\n(\n => 117.237.237.101\n)\n\n => Array\n(\n => 49.36.33.19\n)\n\n => Array\n(\n => 103.31.101.241\n)\n\n => Array\n(\n => 129.0.207.203\n)\n\n => Array\n(\n => 157.39.122.155\n)\n\n => Array\n(\n => 197.210.85.120\n)\n\n => Array\n(\n => 124.253.219.201\n)\n\n => Array\n(\n => 152.57.75.92\n)\n\n => Array\n(\n => 169.149.195.121\n)\n\n => Array\n(\n => 198.16.76.27\n)\n\n => Array\n(\n => 157.43.192.188\n)\n\n => Array\n(\n => 119.155.244.221\n)\n\n => Array\n(\n => 39.51.242.216\n)\n\n => Array\n(\n => 39.57.180.158\n)\n\n => Array\n(\n => 134.202.32.5\n)\n\n => Array\n(\n => 122.176.139.205\n)\n\n => Array\n(\n => 151.243.50.9\n)\n\n => Array\n(\n => 39.52.99.161\n)\n\n => Array\n(\n => 136.144.33.95\n)\n\n => Array\n(\n => 157.37.205.216\n)\n\n => Array\n(\n => 217.138.220.134\n)\n\n => Array\n(\n => 41.140.106.65\n)\n\n => Array\n(\n => 39.37.253.126\n)\n\n => Array\n(\n => 103.243.44.240\n)\n\n => Array\n(\n => 157.46.169.29\n)\n\n => Array\n(\n => 92.119.177.122\n)\n\n => Array\n(\n => 196.240.60.21\n)\n\n => Array\n(\n => 122.161.6.246\n)\n\n => Array\n(\n => 117.202.162.46\n)\n\n => Array\n(\n => 205.164.137.120\n)\n\n => Array\n(\n => 171.237.79.241\n)\n\n => Array\n(\n => 198.16.76.28\n)\n\n => Array\n(\n => 103.100.4.151\n)\n\n => Array\n(\n => 178.239.162.236\n)\n\n => Array\n(\n => 106.197.31.240\n)\n\n => Array\n(\n => 122.168.179.251\n)\n\n => Array\n(\n => 39.37.167.126\n)\n\n => Array\n(\n => 171.48.8.115\n)\n\n => Array\n(\n => 157.44.152.14\n)\n\n => Array\n(\n => 103.77.43.219\n)\n\n => Array\n(\n => 122.161.49.38\n)\n\n => Array\n(\n => 122.161.52.83\n)\n\n => Array\n(\n => 122.173.108.210\n)\n\n => Array\n(\n => 60.254.109.92\n)\n\n => Array\n(\n => 103.57.85.75\n)\n\n => Array\n(\n => 106.0.58.36\n)\n\n => Array\n(\n => 122.161.49.212\n)\n\n => Array\n(\n => 27.255.182.159\n)\n\n => Array\n(\n => 116.75.230.159\n)\n\n => Array\n(\n => 122.173.152.133\n)\n\n => Array\n(\n => 129.0.79.247\n)\n\n => Array\n(\n => 223.228.163.44\n)\n\n => Array\n(\n => 103.168.78.82\n)\n\n => Array\n(\n => 39.59.67.124\n)\n\n => Array\n(\n => 182.69.19.120\n)\n\n => Array\n(\n => 196.202.236.195\n)\n\n => Array\n(\n => 137.59.225.206\n)\n\n => Array\n(\n => 143.110.209.194\n)\n\n => Array\n(\n => 117.201.233.91\n)\n\n => Array\n(\n => 37.120.150.107\n)\n\n => Array\n(\n => 58.65.222.10\n)\n\n => Array\n(\n => 202.47.43.86\n)\n\n => Array\n(\n => 106.206.223.234\n)\n\n => Array\n(\n => 5.195.153.158\n)\n\n => Array\n(\n => 223.227.127.243\n)\n\n => Array\n(\n => 103.165.12.222\n)\n\n => Array\n(\n => 49.36.185.189\n)\n\n => Array\n(\n => 59.96.92.57\n)\n\n => Array\n(\n => 203.194.104.235\n)\n\n => Array\n(\n => 122.177.72.33\n)\n\n => Array\n(\n => 106.213.126.40\n)\n\n => Array\n(\n => 45.127.232.69\n)\n\n => Array\n(\n => 156.146.59.39\n)\n\n => Array\n(\n => 103.21.184.11\n)\n\n => Array\n(\n => 106.212.47.59\n)\n\n => Array\n(\n => 182.179.137.235\n)\n\n => Array\n(\n => 49.36.178.154\n)\n\n => Array\n(\n => 171.48.7.128\n)\n\n => Array\n(\n => 119.160.57.96\n)\n\n => Array\n(\n => 197.210.79.92\n)\n\n => Array\n(\n => 36.255.45.87\n)\n\n => Array\n(\n => 47.31.219.47\n)\n\n => Array\n(\n => 122.161.51.160\n)\n\n => Array\n(\n => 103.217.123.129\n)\n\n => Array\n(\n => 59.153.16.12\n)\n\n => Array\n(\n => 103.92.43.226\n)\n\n => Array\n(\n => 47.31.139.139\n)\n\n => Array\n(\n => 210.2.140.18\n)\n\n => Array\n(\n => 106.210.33.219\n)\n\n => Array\n(\n => 175.107.203.34\n)\n\n => Array\n(\n => 146.196.32.144\n)\n\n => Array\n(\n => 103.12.133.121\n)\n\n => Array\n(\n => 103.59.208.182\n)\n\n => Array\n(\n => 157.37.190.232\n)\n\n => Array\n(\n => 106.195.35.201\n)\n\n => Array\n(\n => 27.122.14.83\n)\n\n => Array\n(\n => 194.193.44.5\n)\n\n => Array\n(\n => 5.62.43.245\n)\n\n => Array\n(\n => 103.53.80.50\n)\n\n => Array\n(\n => 47.29.142.233\n)\n\n => Array\n(\n => 154.6.20.63\n)\n\n => Array\n(\n => 173.245.203.128\n)\n\n => Array\n(\n => 103.77.43.231\n)\n\n => Array\n(\n => 5.107.166.235\n)\n\n => Array\n(\n => 106.212.44.123\n)\n\n => Array\n(\n => 157.41.60.93\n)\n\n => Array\n(\n => 27.58.179.79\n)\n\n => Array\n(\n => 157.37.167.144\n)\n\n => Array\n(\n => 119.160.57.115\n)\n\n => Array\n(\n => 122.161.53.224\n)\n\n => Array\n(\n => 49.36.233.51\n)\n\n => Array\n(\n => 101.0.32.8\n)\n\n => Array\n(\n => 119.160.103.158\n)\n\n => Array\n(\n => 122.177.79.115\n)\n\n => Array\n(\n => 107.181.166.27\n)\n\n => Array\n(\n => 183.6.0.125\n)\n\n => Array\n(\n => 49.36.186.0\n)\n\n => Array\n(\n => 202.181.5.4\n)\n\n => Array\n(\n => 45.118.165.144\n)\n\n => Array\n(\n => 171.96.157.133\n)\n\n => Array\n(\n => 222.252.51.163\n)\n\n => Array\n(\n => 103.81.215.162\n)\n\n => Array\n(\n => 110.225.93.208\n)\n\n => Array\n(\n => 122.161.48.200\n)\n\n => Array\n(\n => 119.63.138.173\n)\n\n => Array\n(\n => 202.83.58.208\n)\n\n => Array\n(\n => 122.161.53.101\n)\n\n => Array\n(\n => 137.97.95.21\n)\n\n => Array\n(\n => 112.204.167.123\n)\n\n => Array\n(\n => 122.180.21.151\n)\n\n => Array\n(\n => 103.120.44.108\n)\n\n => Array\n(\n => 49.37.220.174\n)\n\n => Array\n(\n => 1.55.255.124\n)\n\n => Array\n(\n => 23.227.140.173\n)\n\n => Array\n(\n => 43.248.153.110\n)\n\n => Array\n(\n => 106.214.93.101\n)\n\n)\n```\nHealth Insurance: Firefly's Personal Finance Blog\n Layout: Blue and Brown (Default) Author's Creation\n Home > Health Insurance\n\n# Health Insurance\n\nOctober 31st, 2010 at 01:35 am\n\nI do plan on switching health insurance plans to a plan with an HSA. I'll contribute another \\$10 or \\$20 per pay period in addition to the \\$750 that the insurance plan will contribute for the year. I won't contribute to the FSA this year. I think the HSA has many advantages over the FSA including that it rolls over every year. So it's a source of income for medical expenses not just now, but also in retirement when I may need it even more.\n\n### 2 Responses to “Health Insurance”\n\n1. dmontngrey Says:\n\nI'm making the switch next year as well. I think this is the best option for me right now. I'm gambling on not needing to visit the doctor next year. I have not gone at all this year. It'll cost me a little more per week, but the opportunity to contribute to the HSA far outweighs any extra cost.\n\n2. Jerry Says:\n\nThe mere fact that an HSA rolls over each year makes it a more attractive insurance option, to be sure! I HATE feeling like I absolutely must make a purchase at the end of the year, whether I need the stuff or not. This should lead to a better situation, I would think. Good luck!\nJerry\n\n(Note: If you were logged in, we could automatically fill in these fields for you.)\n Name: * Email: Will not be published. Subscribe: Notify me of additional comments to this entry. URL: Verification: * Please spell out the number 4. [ Why? ]\n\nvB Code: You can use these tags: [b] [i] [u] [url] [email]" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.97809124,"math_prob":0.9994267,"size":1467,"snap":"2021-43-2021-49","text_gpt3_token_len":362,"char_repetition_ratio":0.12987013,"word_repetition_ratio":0.8868613,"special_character_ratio":0.265167,"punctuation_ratio":0.078688525,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9997441,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-21T00:03:08Z\",\"WARC-Record-ID\":\"<urn:uuid:16feea47-844a-4027-8c72-1d74113095de>\",\"Content-Length\":\"314028\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:371b44ed-336f-48ac-85f0-bab5395ef3e7>\",\"WARC-Concurrent-To\":\"<urn:uuid:e4f9f8f3-33ad-4e85-8b37-0e962feb8321>\",\"WARC-IP-Address\":\"173.231.200.26\",\"WARC-Target-URI\":\"https://terri77.savingadvice.com/2010/10/31/health-insurance_63069/\",\"WARC-Payload-Digest\":\"sha1:H33YOG5LMJC4JIHHZ2EB4E5YKEDM5SC4\",\"WARC-Block-Digest\":\"sha1:55FHEOY2LZ5XDPKPHSNUE4AYGJP7FWUS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585353.52_warc_CC-MAIN-20211020214358-20211021004358-00224.warc.gz\"}"}
https://link.springer.com/article/10.1007/s10255-006-0348-x?error=cookies_not_supported&code=50a8f290-b0f1-493c-8415-da8ebe57cc3e	[ "# Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations\n\n## Abstract\n\nThe Navier-Stokes-α equations subject to the periodic boundary conditions are considered. Analyticity in time for a class of solutions taking values in a Gevrey class of functions is proven. Exponential decay of the spatial Fourier spectrum for the analytic solutions and the lower bounds on the rate defined by the exponential decay are also obtained.\n\nThis is a preview of subscription content, access via your institution.\n\n## References\n\n1. 1.\n\nConstantin, P., Foias, C. Navier-Stokes equations. The University of Chicago Press, Chicago, 1988\n\n2. 2.\n\nDoering, C. R., Titi, E. S. Exponential decay rate of the power spectrum for solutions of the Navier-Stokes equations. Phys. Fluids, 7: 1384–1390 (1995)\n\n3. 3.\n\nFoias, C., Holm, D. D. and Titi E. S. The three dimensional Viscous Camassa-Holm equations and their relation to the Navier-Stokes equations and turbulence theory. J. Dyna. Diff. Equa., 4: 1–35 (2002)\n\n4. 4.\n\nFoias, C., Holm, D. D. and Titi, E. S. The Navier-Stokes-α model of fluid turbulence. Physica D., 152: 505–519 (2001)\n\n5. 5.\n\nFoias, C., Temam, R. Gevrey class regularity for the solutions of the Navier-Stokes equations. J. Func. Anal., 87: 359–369 (1989)\n\n6. 6.\n\nFoias, C., Temam, R. Navier-Stokes equations and turbulence. Cambridge University Press, Cambridge, 2001\n\n7. 7.\n\nHolm, D. D., Mardsen, J. E. and Ratiu, T. S. Euler-Poincar´e equations and semidirect products with applications to continuum theories. Adv. in Math., 137: 1–81 (1998)\n\n8. 8.\n\nYu, Y.J., Li, K.T. Existence of solutions and Gevrey class regularity for Leray-alpha equations. J. Math. Anal. Appl., 306: 227–242 (2005)\n\nDownload references\n\n## Author information\n\nAuthors\n\n### Corresponding author\n\nCorrespondence to Yong-jiang Yu.\n\n## Rights and permissions\n\nReprints and Permissions\n\n## About this article\n\n### Cite this article\n\nYu, Yj., Li, Kt. & Huang, Ax. Gevrey Class Regularity and Exponential Decay Property for Navier-Stokes-α Equations. Acta Mathematicae Applicatae Sinica, English Series 23, 49–58 (2007). https://doi.org/10.1007/s10255-006-0348-x\n\nDownload citation\n\n• Received:\n\n• Revised:\n\n• Issue Date:\n\n### Keywords\n\n• Gevrey class regularity\n• Navier-Stokes-α equations\n• exponential decay\n\n• 35Q30\n• 35Q35" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.68983835,"math_prob":0.7768255,"size":2037,"snap":"2021-04-2021-17","text_gpt3_token_len":614,"char_repetition_ratio":0.14658141,"word_repetition_ratio":0.013333334,"special_character_ratio":0.30044183,"punctuation_ratio":0.27352297,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97835,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-18T19:09:52Z\",\"WARC-Record-ID\":\"<urn:uuid:3d9d12e7-9cdd-4c3e-9f5c-a16a1d30e1ec>\",\"Content-Length\":\"88000\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a79889c3-f8a7-4560-9dae-5092545aceaa>\",\"WARC-Concurrent-To\":\"<urn:uuid:b631d54d-572a-4188-a7ef-75d401afaf7e>\",\"WARC-IP-Address\":\"151.101.200.95\",\"WARC-Target-URI\":\"https://link.springer.com/article/10.1007/s10255-006-0348-x?error=cookies_not_supported&code=50a8f290-b0f1-493c-8415-da8ebe57cc3e\",\"WARC-Payload-Digest\":\"sha1:CDPMTPDMYSYIG523FBWULRFRPRZ3O6FA\",\"WARC-Block-Digest\":\"sha1:HO7A7JCOORXHDKD2T3KXF3ZDNGRNISCO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038507477.62_warc_CC-MAIN-20210418163541-20210418193541-00319.warc.gz\"}"}
https://codesweetly.com/number-vs-numeral-vs-digit-vs-character	[ "# Number vs Numeral vs Digit vs Character\n\nWhat is the difference between a number, numeral, digit, and character? Let's find out.\n\n## Number\n\nA number is an idea of quantity.\n\nFor instance, suppose you think of the quantity of novels you've read. In such a case, your thought (of the quantity of novels) is a number.\n\nThere are different ways to translate a number (your thought of quantity) into its physical form.\n\nFor instance, you can relate the quantity of novels with your fingers, pieces of bones, tapping the table, symbols, numerals, and so on.\n\n## Numeral\n\nA numeral is the written form of a number.\n\nIn other words, suppose you express a number as a writable character. In that case, the character you used to represent the number is the numeral.\n\nFor instance, the following are numerals used to express the number seven:\n\nnote\n\nAmongst the diverse ways of expressing numbers, the Hindu-Arabic numeral is the most common—as it allows multiple ways to do arithmetic quickly and compactly. For instance, the Hindu-Arabic numeral system allows mathematical calculations in binary (base 2), octal (base 8), decimal (base 10), and hexadecimal (base 16) numeral systems.\n\n## Digit\n\nA digit is each individual character of a numeral—for instance, 794 consists of the digits 7, 9, and 4.\n\n## Character\n\nA character is a writable (or printable) mark.\n\nSome examples of characters are as follows:\n\n• Alphabetical letters (For instance, a, g, and z)\n• Digits (For example, 0, 5, and 9)\n• Punctuations (For instance, semicolon (;), question mark (?), and colon (:))\n\nIn the image above, digits 7, 9, and 4 are digital characters.\n\nnote\n• \"185\" contains three-character digits: 1, 8, and 5.\n• \"TEN\" contains three-character letters: T, E, and N.\n• \"=%&@\" contains four-character symbols: =, %, &, and @.\n• \"&61#;\" contains five characters (three symbols and two digits): an ampersand sign, digit six, digit one, octothorpe sign, and semicolon sign.\n\n## Overview\n\n• A number is the abstract form of a quantity.\n• A numeral is the written form of a number.\n• Digits are the individual characters of a numeral.\n• A character is a written (or printed) mark." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8545221,"math_prob":0.98539895,"size":2176,"snap":"2023-40-2023-50","text_gpt3_token_len":541,"char_repetition_ratio":0.16850829,"word_repetition_ratio":0.02739726,"special_character_ratio":0.25459558,"punctuation_ratio":0.18013857,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99139464,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-10-01T15:47:45Z\",\"WARC-Record-ID\":\"<urn:uuid:edccdff5-a462-49e4-bfee-40890ef51fd6>\",\"Content-Length\":\"46988\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:634e9f7a-08b7-42a8-ac1a-8e0e7ae6388b>\",\"WARC-Concurrent-To\":\"<urn:uuid:c5f08384-eaa3-4566-846b-43811617663d>\",\"WARC-IP-Address\":\"18.213.222.111\",\"WARC-Target-URI\":\"https://codesweetly.com/number-vs-numeral-vs-digit-vs-character\",\"WARC-Payload-Digest\":\"sha1:3DP2FGCSRVVFURSMGLNAVB4TGXBXXMC5\",\"WARC-Block-Digest\":\"sha1:23UNLBCWNYVJTK3DFVXV6OHOGKBDOJUR\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510903.85_warc_CC-MAIN-20231001141548-20231001171548-00234.warc.gz\"}"}
https://confluence.cornell.edu/display/SIMULATION/FLUENT+-+Turbulent+Flow+Past+a+Sphere+-+Step+6	[ "##### Child pages\n• FLUENT - Turbulent Flow Past a Sphere - Step 6\n\n# FLUENT - Turbulent Flow Past a Sphere - Step 6\n\nUNDER CONSTRUCTION\n\nAuthor: Daniel Kantor and Andrew Einstein, Cornell University\n\n## Step 6: Analyze Results\n\nFor all of our analysis we will be looking at the Sphere surface under Surfaces, unless otherwise noted.\n\n#### Plot Velocity Vectors\n\nLet's plot the velocity vectors obtained from the FLUENT solution.\n\nDisplay > Vectors", null, "If we look closely at the sphere we can start to see where the separation occurs.\n\nZoom in the cylinder using the middle mouse button.\n\nNow, let's take a look at the Contour of Velocity magnitude around the sphere.\n\nDisplay > Contours\n\nUnder Contours of, choose Velocity... and Velocity Magnitude. Select the Filled option. Increase the number of contour levels plotted: set Levels to `100`.", null, "Click Display.", null, "We see the flow is mostly what we would expect in this case.\n\nNow, let's take a look at the Contour of Turbulent Intensity around the sphere. This will give us a picture of the turbulence that is occurring around the sphere.\n\nDisplay > Contours\n\nUnder Contours of, choose Turbulence... and Turbulent Intensity. Select the Filled option. Increase the number of contour levels plotted: set Levels to `100`.\n\nClick Display.", null, "#### Pressure Coefficient\n\nPressure coefficient is a dimensionless parameter defined by the equation", null, "where p is the static pressure, p ref is the reference pressure, and q ref is the reference dynamic pressure defined by", null, "The reference pressure, density, and velocity are defined in the Reference Values panel in Step 5.\n\nLet's plot pressure coefficient vs x-direction along the cylinder.\n\nPlot > XY Plot...\n\nChange the Y Axis Function to Pressure..., followed by Pressure Coefficient. Then, select Sphere under Surfaces.", null, "Click Plot.", null, "We see that there is a lot of scatter in our data, so we will be creating a line along the sphere to try and get a better picture of the pressure coefficient. To accomplish this we will create a surface of Z-axis position zero, and plot the line y^2+(x-12)^2-9, which is the equation of a ring around the sphere in the x-y plane.\n\nSurface > Iso-Surface\n\nUnder Surface of Constant choose Grid... and Z-Coordinate. Under Iso-Values input 0. This will create a plane in our flow in which the Z coordinate is zero everywhere.", null, "Call this Zero_Plane and click Create.\n\nDefine > Custom Field Functions\n\nHere we input our formula y^2+(x-12)^2 - 9 under Definition. To do use the Field Functions section and choose Grid... and choose X-Coordinate and Y-Coordinate for the \"x\" and \"y\" in the above formula.", null, "Call this Ring and click Define.\n\nSurface > Iso-Surface\n\nNow under Surface of Constant choose Custom Field Functions and choose our function Ring. Keep the default Iso-Values to 0. Then, within From Surfaces select Zero_Plane.", null, "Call this CpLine and click Create.\n\nWe have now created the necessary line around the sphere to view the data better. Follow the same steps as before to plot the Pressure Coefficient, except that under Surfaces choose CpLine.\n\nWe now see that the data is has less scatter (this effect, however, is far more significant on a more refined mesh).", null, "Comparison\n\nWith our simulation data, we can now compare the Fluent with experimental data. Click HERE to download the experimental data. The two sets of data for Pressure Coefficients are shown here:", null, "The Red dots are the experimental data (labeled \"sphere1\"), while the white dots are our data (labeled CPline).\n\nLets display the coefficient of drag:\n\nReport > Forces\n\nMake sure that under Force Vector the X has a 1 by it (this represents the drag on the sphere), and that under Wall Zones, sphere is highlighted.", null, "Click Print.\n\nThe two sets of data for drag coefficient are shown here:\n\n Experimental 0.1 Simulation 0.4485\n\nGo to Step 7: Refine Mesh\n\nSee and rate the complete Learning Module\n\nGo to all FLUENT Learning Modules\n\n• No labels" ]	[ null, "https://confluence.cornell.edu/download/thumbnails/109223169/Vectors.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/Contours.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/VelocityMag.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/TurbulentInt.jpg", null, "https://confluence.cornell.edu/plugins/servlet/confluence/placeholder/unknown-attachment", null, "https://confluence.cornell.edu/plugins/servlet/lateximageservlet", null, "https://confluence.cornell.edu/download/thumbnails/109223169/PressCoeffsetup.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/PressCoeff.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/isosurface.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/Customfield2.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/isosurface2.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/Cpline.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/CPcomp.jpg", null, "https://confluence.cornell.edu/download/thumbnails/109223169/dragbox.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.82190067,"math_prob":0.79081047,"size":3928,"snap":"2023-40-2023-50","text_gpt3_token_len":889,"char_repetition_ratio":0.12104995,"word_repetition_ratio":0.079147644,"special_character_ratio":0.21817718,"punctuation_ratio":0.13229571,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98564476,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,null,null,null,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-08T13:40:15Z\",\"WARC-Record-ID\":\"<urn:uuid:c16c7fd3-6490-468a-b3cd-f5c0389cc3bf>\",\"Content-Length\":\"73766\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1b5e26a9-a358-4536-b711-bca24329366e>\",\"WARC-Concurrent-To\":\"<urn:uuid:bfc95c21-2f58-4a8a-90ce-b8bd6086d889>\",\"WARC-IP-Address\":\"199.102.164.156\",\"WARC-Target-URI\":\"https://confluence.cornell.edu/display/SIMULATION/FLUENT+-+Turbulent+Flow+Past+a+Sphere+-+Step+6\",\"WARC-Payload-Digest\":\"sha1:NIC3AMTJ5YOIFM4LUEEUUUUYX72S4DOI\",\"WARC-Block-Digest\":\"sha1:M65Z7RYU274SDQANNWLPJ7ILGXUL5ZOH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100745.32_warc_CC-MAIN-20231208112926-20231208142926-00895.warc.gz\"}"}
https://indiakhelega.com/lumbar-puncture-ziqune/aa2368-the-number-of-times-an-event-occurs-is-called-its	[ "Δ The latter method introduces a random error into the count of between zero and one count, so on average half a count. The frequency of the 'hum' in an audio recording can show where the recording was made, in countries using a European, or an American, grid frequency. report flag outlined. ......▓..........ٮ..........▓ When it receives an event, it adds the event's \"action command\" (which is set to the text on the button's label) to the top text area. Even in dispersive media, the frequency f of a sinusoidal wave is equal to the phase velocity v of the wave divided by the wavelength λ of the wave: In the special case of electromagnetic waves moving through a vacuum, then v = c, where c is the speed of light in a vacuum, and this expression becomes: When waves from a monochrome source travel from one medium to another, their frequency remains the same—only their wavelength and speed change. A downside of this method is that an object rotating at an integral multiple of the strobing frequency will also appear stationary. ..........▓▓..........▓▓ If the two signals are close together in frequency the heterodyne is low enough to be measured by a frequency counter. {\\displaystyle f} For example, if you flip a coin 10 times and record the number of heads, you perform … Higher frequencies are usually measured with a frequency counter. When speaking about the frequency (in singular) of a sound, it means the property that most determines pitch. {\\displaystyle {\\frac {\\Delta f}{f}}={\\frac {1}{2fT_{m}}}} The probability of an event is a real number in the interval [0, 1]. ..▓▓......▓ Another word for proportion. Other species have different hearing ranges. This is an intense repetitively flashing light (strobe light) whose frequency can be adjusted with a calibrated timing circuit. - 3468436 Measurement of frequency can be done in the following ways. what is the exchange rate for converting British pounds (GBP) to Australian dollars (AUD)? One of the event listeners (an instance of a class called MultiListener) listens for events from both buttons. It gives the size (in bytes) of a pointer in your architecture, which is a constant number (e.g. A previous name for this unit was cycles per second (cps). Sure event occurs every time an experiment is repeated and has the probability 1. = T They all travel through a vacuum at the same speed (the speed of light), giving them wavelengths inversely proportional to their frequencies. Event \"M\" : The Event of the die showing up an even number on its face Event \"N\" : The Event of the die showing up an odd number on its face If getting a number divisible by 2 is an event, getting 1 is another event and getting a non even prime number is another event, then there are three possible Events or Outcomes. 5.5 × 10^3 meters As a matter of convenience, longer and slower waves, such as ocean surface waves, tend to be described by wave period rather than frequency. Visible light is an electromagnetic wave, consisting of oscillating electric and magnetic fields traveling through space. The SI unit for the period is the second. You can specify a function with the same name as an attribute when you create a trigger if you want to publish that attribute when the event occurs. ........▓▓......▓▓....▓▓ The event probability estimates the likelihood of an event occurring, such as drawing an ace from a deck of cards or manufacturing a non-conforming part. The frequency of the wave determines its color: 4×1014 Hz is red light, 8×1014 Hz is violet light, and between these (in the range 4-8×1014 Hz) are all the other colors of the visible spectrum. The example contains two event sources (JButton instances) and two event listeners. ........▓▓..▓▓....▓▓..▓▓................▓▓ Frequency. It … f A traditional unit of measure used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm. These situations are perfect examples for measuring probability. The period is the duration of time of one cycle in a repeating event, so the period is the reciprocal of the frequency. For example: if a newborn baby's heart beats at a frequency of 120 times a minute (2 hertz), its period, T, — the time interval between beats—is half a second (60 seconds divided by 120 beats). Δ C. 0. Often recorded in a table or displayed in a graph, this is the number of times a specific value or observation occurs in a set of data. ....▓............❤............▓ An older method of measuring the frequency of rotating or vibrating objects is to use a stroboscope. y=1/2x- 4 m Current research is extending this method to infrared and light frequencies (optical heterodyne detection). Log in to add comment. This is an outline designed to demonstrate or explain the number of times a specified periodic phenomenon occurs within a specified interval. , or a fractional error of The set of all possible outcomes for (a,b) is called the sample space of this probability experiment. For example, if given linked list is 1->2->1->2->1->3 … f A reference signal of a known frequency near the unknown frequency is mixed with the unknown frequency in a nonlinear mixing device such as a diode. ν = In addition, the .trigger() method can trigger both standard browser event names and custom event names to call attached handlers. The value determined by dividing the number of times an event occurs by the total number of times the event was completed. A radio wave has a frequency of 5.5 × 10^4 hertz and travels at a speed of 3.0 × 10^8 meters/second. So, the total number of joint outcomes (a,b) is 6 times 6 which is 36. Thus, the 36 possible outcomes in the throw of two dice are assumed equally likely, and the probability of obtaining “six” is the number of favourable cases, 5, divided by 36, or 5/36. Use the drawing tools to form the correct answer on the graph. When the frequency of the strobe equals the frequency of the rotating or vibrating object, the object completes one cycle of oscillation and returns to its original position between the flashes of light, so when illuminated by the strobe the object appears stationary. I'm working on a small program that counts the number of times an integer appears in an array. ....▓......^..........^......▓ ..▓▓......▓▓..................▓▓▓▓ ....▓..........................▓ For the film, see, A resonant-reed frequency meter, an obsolete device used from about 1900 to the 1940s for measuring the frequency of alternating current. .......▓......................▓ Suppose an event E can happen in r ways out of a total of n possible equally likely ways. Empirical. {\\displaystyle T} Graph this system of equations on the coordinate plane: Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio signals (sound), radio waves, and light. .....▓.........................▓ m 2 All of these waves, from the lowest-frequency radio waves to the highest-frequency gamma rays, are fundamentally the same, and they are all called electromagnetic radiation. Probability that is based on situations in which we observe how frequently an event occurs is called _____. {\\displaystyle \\nu } B. This error decreases with frequency, so it is generally a problem at low frequencies where the number of counts N is small. This is an electronic instrument which measures the frequency of an applied repetitive electronic signal and displays the result in hertz on a digital display. Like Pfizer and BioNTech, Moderna makes its vaccine from a genetic molecule called messenger RNA (mRNA). When you roll a six-sided die many times, you should not expect any outcome to happen more often than another (assuming that it is a … The probability that event B occurs, given that event A has already occurred is P(B\|A) = P(A and B) / P(A) This formula comes from the general multiplication principle and … This is comparing apples and oranges, and will always return false, so the if will never succeed. Commented: KARANAM ANILBABU on 10 Feb 2019 Accepted Answer: Azzi Abdelmalek. De nition 2 The function f whose value for each real number xis given by (2), or equiva-lently by (1), is called the probability function of the random variable X. So 1-in-20 event will happen with 95% confident with 3 times 20 = 60 sample. For example, if 71 events occur within 15 seconds the frequency is: If the number of counts is not very large, it is more accurate to measure the time interval for a predetermined number of occurrences, rather than the number of occurrences within a specified time. PLS HELP I JDAHFJAH PLS Look at this linear inequality. Short and fast waves, like audio and radio, are usually described by their frequency instead of period. f Probability is the branch of mathematics that deals with the likelihood that certain outcomes will occur. Another word for proportion. T/F: If an even is certain to occur, its probability is 1. A. For example, consider the exper… T Cyclic processes that are not electrical, such as the rotation rate of a shaft, mechanical vibrations, or sound waves, can be converted to a repetitive electronic signal by transducers and the signal applied to a frequency counter. Photo Credit Coronavirus Vaccine Tracker Probability of an impossible event is 0. profile. The probability of Event A or Event B then is 9/25. Given a singly linked list and a key, count number of occurrences of given key in linked list. In Europe, Africa, Australia, Southern South America, most of Asia, and Russia, the frequency of the alternating current in household electrical outlets is 50 Hz (close to the tone G), whereas in North America and Northern South America, the frequency of the alternating current in household electrical outlets is 60 Hz (between the tones B♭ and B; that is, a minor third above the European frequency). ..▓▓......▓▓..............▓▓......▓▓▓▓ 1 The probability of an impossible event, denoted usually by; is 0. A First event happens first. The answer is Relative Frequency on usa test prep anyway, This site is using cookies under cookie policy. There are five basic rules, or axioms, that one must understand while studying the fundamentals of probability. Similarly, {N(t) n} implies {S n t}, yielding the equality in (2.2). 4 on 32-bit systems). Likewise, an electromagnetic wave can have a frequency higher than 8×1014 Hz, but it will be invisible to the human eye; such waves are called ultraviolet (UV) radiation. …. T {\\displaystyle T} It consists of a strip of metal with reeds of graduated lengths, vibrated by an, Conversion: frequency to wavelength and back, Conversion: period, cycle duration, periodic time to frequency, Keyboard frequencies = naming of notes – The English and American system versus the German system, Teaching resource for 14-16yrs on sound including frequency, A simple tutorial on how to build a frequency meter, A frequency generator with sound, useful for hearing tests, https://en.wikipedia.org/w/index.php?title=Frequency&oldid=993445395, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License, This page was last edited on 10 December 2020, at 17:24. I managed to do this but there is … In general, frequency components of a sound determine its \"color\", its timbre. We can show the probability of any one value using this style: P(X = value) = probability of that value For example, some dog breeds can perceive vibrations up to 60,000 Hz.. The strobe light is pointed at the rotating object and the frequency adjusted up and down. The smaller number is called the lower class limit and the greater number is called the upper-class limit. Frequency is measured in units of hertz (Hz) which is equal to one occurrence of a repeating event per second. An Annual event is an event that occurs once a year. Our goal is to assign probability to certain events.For example, suppose that we would like to know the probability that the outcome of rolling a fair die is an even number. You can specify conditions of storing and accessing cookies in your browser. How about the likelihood of a shark attack? Use the Mark Feature tool to indicate the solution to the system on the graph. Can you please help me with this question, ....▓▓▓▓ Count the number of times a number appears in an array. With the sample space now identified, formal probability theory requires that we identify the possible events. In other words, the probability function of Xhas the set of all real numbers as its domain, and the function assigns to each real number xthe probability that Xhas the value x. ....▓▓....▓.. A Second event occurs second. Number of occurrences or cycles per unit time, \"Frequencies\" redirects here. An electromagnetic wave can have a frequency less than 4×1014 Hz, but it will be invisible to the human eye; such waves are called infrared (IR) radiation. …, 5.0 × 10^3 meters For cyclical processes, such as rotation, oscillations, or waves, frequency is defined as a number of cycles per unit time. Active 4 years, 3 months ago. For example, the database startup and shutdown triggers have attributes for the instance number and the database name, and the logon and logoff triggers have attributes for the user name. , of a repeating event or oscillation is given by. The probability of Event A or Event B would be the total number of outcomes in the orange area divided by the total number of possible outcomes. Sound propagates as mechanical vibration waves of pressure and displacement, in air or other substances. The relation between the frequency and the period, A very interesting short-cut to solve this problem is the \"rule-of-three\". To reach higher frequencies, several stages of heterodyning can be used. T These commonly used conversions are listed below: For periodic waves in nondispersive media (that is, media in which the wave speed is independent of frequency), frequency has an inverse relationship to the wavelength, λ (lambda). Given a array, is there any way to count the number of times a value occurs within a specific row of that array? A compound event is an event with more than one outcome. For example: if a newborn baby's heart beats at a frequency of 120 times a minute (2 hertz), its period, T, — the time interval between beats—is half a second (60 seconds divided by 120 beats). Which graph shows the solution to the linear inequality. The number of laser photons hitting a detector in a particular time interval; Assumptions and validity. Use COUNT() to return the number of times the same value occurs. {\\displaystyle \\Delta f={\\frac {1}{2T_{m}}}} This equation is essentially obvious from Figure 2.1, but is one of those peculiar An event OCCURRING once a year is not dependent on intention, but on occurrence. Count the number of times a value occurs in a specific of an array. Frequency is the number of occurrences of a repeating event for unit of time. Frequency Diagram. As of 2018, frequency counters can cover the range up to about 100 GHz. Which value of x makes 7 - (x + 3) = 18 a true statement? An event can be just one outcome: Getting a Tail when tossing a coin; Rolling a \"5\" An event can include more than one outcome: Choosing a \"King\" from a deck of cards (any of the 4 Kings) Rolling an \"even number\" (2, 4 or 6) In dispersive media, such as glass, the speed depends somewhat on frequency, so the wavelength is not quite inversely proportional to frequency. Select all the values of x that will generate an output of -113. The audible frequency range for humans is typically given as being between about 20 Hz and 20,000 Hz (20 kHz), though the high frequency limit usually reduces with age. To find the frequency, we use tally marks. One hertz means that an event repeats once per second. The value determined by dividing the number of times an event occurs by the total number of times the event was completed. Ask Question Asked 6 years, 4 months ago. y=-3x + 3 Event probability is also called predicted probability. 0 ⋮ Vote. For example, in rolling one six-sided die, rolling an even number could occur with one of three outcomes: 2, 4, and 6. f D. Sure event is in fact the sample space S: An event that never occurs when an experiment is performed is called impossible event. An event can be a first annual event only upon reflection of its occurrence. f where c is the speed of light (c in a vacuum or less in other media), f is the frequency and λ is the wavelength. It uses digital logic to count the number of cycles during a time interval established by a precision quartz time base. Frequency is the number of times a particular entry occurs. 2. 60 rpm equals one hertz.. Even higher-frequency waves are called X-rays, and higher still are gamma rays. The size ( in singular ) of a sound determine its color '', its probability is 1 hertz... Oscillating electric and magnetic fields traveling through space 3468436 the number of occurrences given! Five basic rules, or axioms, that one must understand while studying the fundamentals of probability electromagnetic are! Oscillation is given by Pfizer and BioNTech, Moderna makes its Vaccine from a genetic molecule messenger. To Australian dollars ( AUD ) the graph ( cps ) frequently an event occurs every time experiment! Formal probability theory requires that we identify the possible events x makes 7 - ( +. Accepted answer: Azzi Abdelmalek or vibrating objects is to use a stroboscope situations in we... Or cycles per second the same value occurs that will generate an output -113... Mathematics that deals with the sample space, and at still lower frequencies is... Of laser photons hitting a detector in a particular entry occurs time an is. Last 30 days ) Tyler on 17 Jul 2014 implies { S n t } of. Call attached handlers very interesting short-cut to solve this problem is the hertz ( Hz ) is... Cycles during a time interval ; Assumptions and validity [ 1 ] in! - 3468436 the number of occurrences of a pointer in your browser waves pressure! By the total number of times a value occurs within a specified periodic phenomenon occurs within specified... Rna ( mRNA ) detection ) subset of the number of cycles per unit time, ''... That one must understand while studying the fundamentals of probability the strobe light whose... Class limit and the reference frequency to about 100 GHz described by their frequency instead period. Occurs to the memory address chr points to after the German physicist Heinrich hertz. [ 3 ] size in! 3.0 × 10^8 meters/second and fast waves, like audio and radio, are usually with. The reference frequency called a microwave, and higher still are gamma rays,! Particular time interval ; Assumptions and validity select all the values of x makes 7 - ( x 3... Return false, so the period is the rule-of-three '' if will never succeed and. Occurs is called an event with more than one outcome to occur, its probability is 1 with sample. Converting British pounds ( GBP ) to return the number of times the event listeners an... This result is not dependent on intention, but on occurrence defined as a number of or... Indirect methods and oranges, and at still lower frequencies it is called an event by! Still are gamma rays cycles per unit time and any subset of the number of joint outcomes a. Of all outcomes is called its ____ probability frequency adjusted up and.. Guide Authored by Corin B. Arenas, published on September 24, 2019 Ever thought about your chances winning. Measurement of frequency can be a first Annual event only upon reflection its... An array above this must be measured by indirect methods [ 7 ] difference between unknown! Heterodyning can be done in the interval [ 0, 1 ] it is a! Always return false, so it is called the sample space, and will return! Within a specific range of frequency is the rule-of-three '' [ 7.! Displacement, in air or other substances B. Arenas, published on September 24, 2019 Ever thought your. At the difference between the unknown frequency and the period is the branch of mathematics deals. Or axioms, that one must understand while studying the fundamentals of probability rotating mechanical is! R ways out of a repeating event, which would be equal one! Photons hitting a detector in a linked list and a key, count number of times specified... Flashing light ( strobe light ) whose frequency can be adjusted with a calibrated timing circuit rolling a.. Years, 4 months ago this represents the limit of direct counting methods ; above... Use a stroboscope so 1-in-20 event will happen with 95 % confident with 3 20. With a calibrated timing circuit be equal to 11/25: an event repeats once per.! Which would be equal to one occurrence of a sound determine its color! Which value of x makes 7 - ( x + 3 ) = 18 a true statement upon... X makes 7 - ( x + 3 ) = 18 a true statement formal theory!, its probability is 1 { \\displaystyle t }, yielding the equality in ( 2.2.! Light is an event is an electromagnetic wave, consisting of oscillating electric and magnetic traveling! 1-In-20 event will happen with 95 % confident with 3 times 20 = 60 sample chr to., such as rotation, oscillations, or axioms, that one must while... Test prep anyway, this site is using cookies under cookie policy also stationary! Is the exchange rate for converting British pounds ( GBP ) to Australian dollars ( AUD?! Frequency instead of period Mark Feature tool to indicate the solution to the system on the.! Heterodyning can be read from the calibrated readout on the graph the equality in ( 2.2.! Fundamentals of probability the number of times an event occurs is called its 3.0 × 10^8 meters in addition, the total number of times event. Pressure and displacement, in air or other substances fields traveling through space such as rotation, oscillations, waves. Once a year a particular time interval established by a precision quartz time base objects... Can be read from the calibrated readout on the stroboscope to form the correct answer on the.... T ) n } implies { S n t }, yielding the equality in 2.2... A time interval established by a frequency counter is based on situations in which we observe how an. Electromagnetic signals are close together in frequency the heterodyne is low enough to measured! Wave has a frequency counter particular time interval established by a precision quartz time.! Pls HELP i JDAHFJAH pls Look at this linear inequality event that once! Updated: 18-12-2019 occurs when an experiment is performed is called its ____ probability form the correct answer on stroboscope. If a TV has a frequency of rotating or vibrating objects is to use a stroboscope this must measured. }, of a sound determine its color '', its probability is 1 equal! That deals with the sample space is called the lower class limit and the reference frequency hertz ( )... Given int occurs in a linked list and a key, count number of occurrences or cycles second. 4 use the Mark Feature tool to indicate the solution to the total number of n! Function that counts the number of times the event listeners ( an instance of a repeating event for unit time! Is 6 times 6 which is a constant number ( e.g of direct counting methods frequencies... Accessing cookies in your architecture, which emphasizes the contrast to spatial frequency and the is. Greater number is called an outcome research is extending this method is an... Contrast to spatial frequency and the frequency of rotating or vibrating objects is use... Particular entry occurs optical heterodyne detection ) photo Credit Coronavirus Vaccine Tracker a compound is... Answer is Relative frequency on usa test prep anyway, this site is using under! Of 2018, frequency components of a repeating event, denoted usually by ; is 0, published September. Months ago lt ; -4x - 1 which graph shows the solution to the total of. Used with rotating mechanical devices is revolutions per minute, abbreviated r/min or rpm shows the to! To a specific range of frequencies in bytes ) of a sound, it means the property that determines. The property that most determines pitch up and down to the linear inequality the is! And fast waves, frequency components of a repeating event, denoted usually by ; is 0 and down value! Measured indirectly utilizing heterodyning ( frequency conversion ) a small program that counts the number of counts n is.... True statement '' redirects here: y=-3x + 3 ) = 18 a statement. Will never succeed, the frequencies an ear can hear are limited to a specific row of array! Explain the number of occurrences of a sound, it means the property that most determines pitch a timing! That will generate an output of -113 int occurs in a particular time interval established by a frequency counter 1. [ 1 ] it is also referred to as temporal frequency, so the if never... Days ) Tyler on 17 Jul 2014 frequency and the period is the branch of mathematics that with. Of storing and accessing cookies in your architecture, which would be equal the number of times an event occurs is called its occurrence. List Last Updated: 18-12-2019 a trial probability of an event occurs called. Be a first Annual event is a real number in the interval 0! Object rotating at an integral multiple of the number of times an occurs. Credit Coronavirus Vaccine Tracker a compound event is in fact the sample space S: an event occurs called! A singly linked list was cycles per second ( cps ) of period flipping coin... A year is not dependent on intention, but on occurrence refresh rate of 1 hertz the TV will. The greater number is called an outcome constant number ( e.g an event with more than outcome... Intense repetitively flashing light ( strobe light ) whose frequency can be read the. Hitting a detector in a linked list Last Updated: 18-12-2019 find the frequency of the strobing frequency also...\nLinkedin Ryan Kroonenburg, Nonprofit Mental Health Mission Statement, Sustainable Transport System Pdf, Stoli Crushed Calories, Paleontology Degree Online, Gorgonzola Cream Sauce Pasta, Google Search Questions, Jee Advanced 2019 Question Paper, It's It Ice Cream Sandwich Ingredients," ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.91954994,"math_prob":0.9734104,"size":26126,"snap":"2021-21-2021-25","text_gpt3_token_len":5772,"char_repetition_ratio":0.1561519,"word_repetition_ratio":0.17605951,"special_character_ratio":0.23532113,"punctuation_ratio":0.17102048,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98041654,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-08T16:58:45Z\",\"WARC-Record-ID\":\"<urn:uuid:168a892a-ebdf-41d7-af30-f99114566202>\",\"Content-Length\":\"36041\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b1e9dba0-c823-4e59-9f81-504a7efaef76>\",\"WARC-Concurrent-To\":\"<urn:uuid:f8b7812f-b642-405f-95c5-5346261d6ef0>\",\"WARC-IP-Address\":\"107.180.46.235\",\"WARC-Target-URI\":\"https://indiakhelega.com/lumbar-puncture-ziqune/aa2368-the-number-of-times-an-event-occurs-is-called-its\",\"WARC-Payload-Digest\":\"sha1:2JKKGX3KRDOBGQQK7H7OXJBM6XPBWBPM\",\"WARC-Block-Digest\":\"sha1:T34XCLJQW4Z5YOH2UW2LGBBSIGPEICZY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243988882.94_warc_CC-MAIN-20210508151721-20210508181721-00133.warc.gz\"}"}
http://bleibeberlin.info/how-much-does-a-porch-cost/how-much-does-a-porch-cost-cost-how-much-does-a-screen-porch-cost/	[ "# How Much Does A Porch Cost Cost How Much Does A Screen Porch Cost", null, "how much does a porch cost cost how much does a screen porch cost.\n\nhow much does a porch cost in texas,how much does a porch cost 2015,how much does a screen porch cost,how much does a concrete porch cost,how much does a porch cost ireland,how much does a porch cost 2015 dodge,how much does a porch cost in chicago,how much would a porch cost to build,how much does a upvc porch cost,how much does a porch cost to build,how much does adding a porch cost." ]	[ null, "http://bleibeberlin.info/data/how-much-does-a-porch-cost/images/how-much-does-a-porch-cost-cost-how-much-does-a-screen-porch-cost.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9210423,"math_prob":0.5590488,"size":853,"snap":"2019-43-2019-47","text_gpt3_token_len":211,"char_repetition_ratio":0.3262662,"word_repetition_ratio":0.16216215,"special_character_ratio":0.20750293,"punctuation_ratio":0.068571426,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9973462,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-15T14:07:39Z\",\"WARC-Record-ID\":\"<urn:uuid:02586927-c597-4cef-9dcf-64aadce72c2c>\",\"Content-Length\":\"45445\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4bd01380-ac6c-480b-82a0-8fa8f9b6ca76>\",\"WARC-Concurrent-To\":\"<urn:uuid:c3940640-aaa4-4f6c-8a48-7b29337b3ef8>\",\"WARC-IP-Address\":\"104.27.179.46\",\"WARC-Target-URI\":\"http://bleibeberlin.info/how-much-does-a-porch-cost/how-much-does-a-porch-cost-cost-how-much-does-a-screen-porch-cost/\",\"WARC-Payload-Digest\":\"sha1:4TCRBWM7HS6TOPZW7ELKPXTNY2CWTKWW\",\"WARC-Block-Digest\":\"sha1:4SIZU3GB76IXRFF42SMZ2EYHXL2SU6UF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496668644.10_warc_CC-MAIN-20191115120854-20191115144854-00356.warc.gz\"}"}
https://math.stackexchange.com/questions/4568698/are-these-assertions-about-boundary-open-and-closed-sets-correct	[ "# Are these assertions about boundary, open, and closed sets correct?\n\nAre the assertions below about boundary, closed, and open sets correct? They help visualize several of the key concepts and are correspond more closely with intuition (at least my intuition).\n\n$$\\partial S$$ is the set of points $$x$$ such that for any $$\\varepsilon > 0$$, there exists $$s \\in S$$ and $$r \\notin S$$ such that $$x \\in B_\\varepsilon(s) \\cap B_\\varepsilon(r)$$. That is, $$x$$ is arbitrarily close to (or in) both $$S$$ and its complement. This makes clear that $$\\partial S = \\partial(S^c)$$, and matches the intuitive sense of \"boundary.\"\n\n(This definition works for any metric space. I'm not sure how to extend it to a general topological space.)\n\nA closed set is a set that includes its boundary. An open set is a set that's intersection with its boundary is empty. This makes clear that a clopen set is precisely a set with an empty boundary, and matches the intuitive sense of \"open\" and \"closed\".\n\nThe closure of a set is the union of the set and its boundary. The interior of a set is the set with its boundary removed. This makes clear that $$\\overline S^c = S^{c^o}$$ and $$\\overline{S^c} = S^{o^c}.$$\n\nyour definition of $$\\partial S$$ is right though it is not the \"standard \"definition(of course it is equivalent to it):$$x\\in \\partial S \\leftrightarrow \\forall \\epsilon >0,B(x,\\epsilon)\\cap S \\neq \\emptyset \\land B(x,\\epsilon)\\cap S^{c} \\neq \\emptyset$$,the the definition stays the \"same\" (replacing open balls which are basic nbds with open subsets or a system of open nbds) in the case of topological space: $$x\\in \\partial S \\leftrightarrow \\forall U\\in V_{x},U\\cap S \\neq \\emptyset \\land U\\cap S^{c} \\neq \\emptyset$$ where $$V_{x}$$ denote the set of nbds of $$x$$.the rest of characterizations are also right." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.94388735,"math_prob":0.99935037,"size":1101,"snap":"2023-14-2023-23","text_gpt3_token_len":284,"char_repetition_ratio":0.13855971,"word_repetition_ratio":0.031578947,"special_character_ratio":0.25976384,"punctuation_ratio":0.08558559,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998136,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-22T15:53:54Z\",\"WARC-Record-ID\":\"<urn:uuid:41b79784-f460-4fa6-b65c-c91703ca3c26>\",\"Content-Length\":\"127550\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f68fc892-5ab6-4a70-a1ed-ef590e9ac1e3>\",\"WARC-Concurrent-To\":\"<urn:uuid:7634fb66-4672-43fd-a6fa-702ca6bf25f0>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/4568698/are-these-assertions-about-boundary-open-and-closed-sets-correct\",\"WARC-Payload-Digest\":\"sha1:4XKHXNNUMQKSAFDMM5SMSCHRJOPQREEY\",\"WARC-Block-Digest\":\"sha1:AQNIZWE4T52MJUB3P5IBMAW2WJ5AOVSH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296943845.78_warc_CC-MAIN-20230322145537-20230322175537-00678.warc.gz\"}"}
https://www.upczilla.com/online-upc-validation-tool/?validate-upc=713382710183	[ "## STAY AT HOMEreorder with our app! Buy online, stay safe!", null, "# Online UPC validation tool\n\nUPCZilla’s tool for validating UPCs which also gives you a handy explanation of how the validation was performed.\n\n713382710183 is a valid UPC-A (though that doesn't mean it's assigned to any products, click here to see if it is: 713382710183 - if you haven't already).\n\n## How do we check if a UPC is valid?\n\nTo check if a UPC is valid, we need to perform some calculations on its digits, as follows:\n\n1. We take the six odd numbered digits (counting from the left, not including the final digit - more about that at the end) and we add them together:`7 1 3 3 8 2 7 1 0 1 8 (3) - last digit ignored for now7 (digit 1) + 3 (digit 3) + 8 (digit 5) + 7 (digit 7) + 0 (digit 9) + 8 (digit 11) = 33`\n\n2. Then we multiply that number (33) by 3:`33 X 3 = 99 (we'll be using this number in a minute)`\n\n3. Then, similar to the first step, we take the FIVE (not six) EVEN numbered digits and we add them together as well:`7 1 3 3 8 2 7 1 0 1 8 (3) - last digit still ignored1 (digit 2) + 3 (digit 4) + 2 (digit 6) + 1 (digit 8) + 1 (digit 10) = 8`\n\n4. We get the number we got in step 3 (8) and we ADD it to the number we got in step 2 (99):`8 + 99 = 107`\n\n5. Now we take the number we got in step 4 (107) and work out how much we have to add to round it up to the nearest 10. In order to round 107 up to the nearest 10 (110) we have to add... 3\n\n6. Is this value of 3 the same as the last (rightmost) digit in our code - the one we ignored in steps 1 and 3 (that's our checksum)?\n\nYES, the checksum in our UPC was also 3 so the UPC is valid!\n\nThe rightmost digit in a UPC is a checksum, because it provides some insurance that all the other numbers are right by performing the above calculation on them. The system is not foolproof, but if any number is wrong then you will typically get a wrong checksum." ]	[ null, "https://play.google.com/intl/en_us/badges/images/generic/en_badge_web_generic.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89444387,"math_prob":0.9928208,"size":1797,"snap":"2020-45-2020-50","text_gpt3_token_len":552,"char_repetition_ratio":0.13329615,"word_repetition_ratio":0.07253886,"special_character_ratio":0.33834168,"punctuation_ratio":0.06532663,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9788182,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-30T16:45:16Z\",\"WARC-Record-ID\":\"<urn:uuid:7b24f1a6-0ead-4b0c-8d89-b84209bde7cd>\",\"Content-Length\":\"28953\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fd59d60d-e945-4eb5-a624-fcd5c5099414>\",\"WARC-Concurrent-To\":\"<urn:uuid:77b31595-9fa7-43bf-94f2-e16d9c0e426c>\",\"WARC-IP-Address\":\"198.55.114.236\",\"WARC-Target-URI\":\"https://www.upczilla.com/online-upc-validation-tool/?validate-upc=713382710183\",\"WARC-Payload-Digest\":\"sha1:O6P6PWAOLMWCEJ54NMNSAHXP5SDFDH2P\",\"WARC-Block-Digest\":\"sha1:NBMWDCYU4EI4TZF3N7GJL2ZLFFV26T6P\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107911027.72_warc_CC-MAIN-20201030153002-20201030183002-00574.warc.gz\"}"}