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# Proof for exponential number

## Main Question or Discussion Point

If anybody knows could they please tell me how it is known that there exists some number 'e' with the property that the derivative of ex is itself.

I have a feeling this might require some definition of ab.

Surely this isn't a complicated problem, but for some reason I can't work it out or even find a good answer online.

Thanks for the help.

## Answers and Replies

I like Serena
Homework Helper
Welcome to PF, DylanG!

Actually, the property that the derivative of ex is itself, is the definition of ex (or at least 1 of the definitions).

From wikipedia:
wikipedia said:
In mathematics, the exponential function is the function ex, where e is the number (approximately 2.718281828) such that the function ex is its own derivative.[1][2]
and also from wikipedia:
wikipedia said:
From any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity,
exp(x+y) = exp(x) exp(y),
which is why it can be written as ex.

Look at this series definition/expansion of the exponential.

$${e^p} = 1 + p + \frac{{{p^2}}}{{2!}} + \frac{{{p^3}}}{{3!}} + \frac{{{p^4}}}{{4!}} + ...$$

differentiate with respect to p

$$\frac{{d\left( {{e^p}} \right)}}{{dp}} = 0 + 1 + \frac{{2p}}{{2*1}} + \frac{{3{p^2}}}{{3*2*1}} + \frac{{4{p^3}}}{{4*3*2*1}} + ...$$

=$$1 + p + \frac{{{p^2}}}{{2!}} + \frac{{{p^3}}}{{3!}} + ... = {e^p}$$

does this help?

lurflurf
Homework Helper
Yes you would need to define ax first.
Such as f(x)=ax has two properties
f(x+y)=f(x)f(y)
and
f'(0)=1/f-1(e)
then we have
f(1)=a
f'(x)=f'(0)f(x)
so we take a=e<->f'(0)=1 as the standard case of such functions

The power series is a little like pulling a rabbit out of a hat. Doesn't show you how it's done. Just shows you it can be done. What you can do to rectify that is you can assume that the function is equal to 1 when x equals zero, and then the rest of the Taylor series is determined by that. Which gives you the thing that Studiot posted.

That just tells you there is such a function. Doesn't tell you it's a^x for some a. But you can can argue that it satisfies the property exp(x + y) = exp(x) exp (y), which implies what you want.

There's also a geometric construction of the exponential function, but I can't draw pictures on here, so it's hard to explain. Basically, you start out at the point (0,1) and you do a discrete approximation and take the limit. It leads to the definition of e as a limit. I came up with it in high school when I was studying calculus, and later I found the same argument appears in Needham's Visual Complex Analysis, somewhere in the book, as a warm-up for the complex number version.

As ILS has said, there are many ways to approach the exponential function.
It all depends upon the order you define things when you build up your theory of analysis.

Another way is to state

y = exp(x) if and only if x = log(y)

@ homeomorphic

The following properties may be proved directly from the series definition:

exp(0) = 1
exp(-x) = 1/exp(x)
exp(x) > 0 ie it never vanishes

We may also use series comparison techniques to establish the derivative from first principles by rearranging and expanding

$$\frac{{\exp (x + h) - \exp (x)}}{h}$$

But my original offering is simpler.

go well

exp(x+h)

AlephZero
Homework Helper
For a "simple" introduction to this I prefer to start by defining ln(x) as an integral:
When x > 0, $$\ln x = \int_1^x \frac{dt}{t}$$

It is then easy to see that$$\frac{d}{dx}\ln x = \frac 1 x$$ and ln(x) is a continuous and monotonic function.

From the definition ln(1) = 0, and by approximating the "area under the graph" by three rectangles, ln(4) > 1/2 + 1/3 + 1/4 > 1.

So there is a number e such that 1 < e < 4 and ln(e) = 1.

It is straightforward to show properties like ln(xy) = ln(x) + ln(y) direct from the definition, using techniqes from elementary calculus.

And finally, define $e^x$ as the inverse function of ln(x).

The advantage of this, IMO, is that it doesn't require anything more complcated than "calculus", which is much more accessible to "beginners" than a proper justification of operations on an infinite series that was "pulled out of a hat" as the definition of $e^x$.

And a few years later, if and when you have taken a course on analytic functions, you can use the power series to define $e^z$ when z is complex, which is much more useful than restricting yourself to real variables.

I definitely prefer the geometric approach (and you'll find I'm usually biased towards geometric approaches). Essentially, you're solving an ODE with an initial condition, so you can use Euler's method, which is pretty geometric. You have a slope field whose slope is equal to its height and you just follow the slopes up successively, until you get to the x-value you want, dividing it up into smaller and smaller time intervals. I think that's what my geometric construction amounts to if you unravel it. It's been a while since I've thought it through. Of course, Euler's method might be difficult to demonstrate rigorously (though it definitely can be done).

And in this context (flow along a vector field on the line or slope field), I might add that the property exp(x+y) = exp(x)exp(y) has the very elegant interpretation that time evolution by x + y is the same thing as time evolution by x followed by time evolution by y.


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# Vegetable databases usage

Good morning, I'm aligning with blast some species, against all nt database to analyse the vegetables. However, in some cases we have species of animals mixed with vegetables and want only to detect the vegetable species. Normally, we would only expect to detect the vegetables only. But this is normal since we are using the all nt. So now, could anyone point me out, the way to detect only vegetables? Thanks for the help.

• Cross posted at BioStars: biostars.org/p/421717 Feb 12 '20 at 10:51
• vegetables are plants, meat are animals, you can filter nt to the taxonomic groups you are interested in Feb 12 '20 at 11:05
• why was this question avaluated with -1? Feb 12 '20 at 15:55
• -1 means it got downvoted, probably because a user did not appreciate the question, maybe because of lack of details, language, or simply because they found it is fun. I personally do not like the principle of downvoting questions at all, as it is utterly disrespectful. Still that is how SE works. Speaking of your question, please consider being more precise in the future. Use correct terminology (plant, not vegetable, animal, not meat) and also say that you want to do this remotely, not via the web browser as the user who gave an answer assumed. Feb 12 '20 at 21:08
• Hello Atpoint, thank you for the suggestions and I will do it. Feb 13 '20 at 10:42

Using the blast command line tool, you can limit your search to a specific taxon using the –taxids option.
So, in your case -taxids 3193 should limit your search to plants.

format: plain


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Terms of service • Privacy policy • Editorial independence, Get unlimited access to books, videos, and. For policy evaluation based on solving approximate versions of a Bellman equation, we propose the use of weighted Bellman mappings. The setting of Bellman equation is the first and crucial step to solve dynamic programming problems. Metadata Show full item record. 05/25/2019 ∙ by Yihao Feng, et al. We also assume that the state changes from $${\displaystyle x}$$ to a new state $${\displaystyle T(x,a)}$$ when action $${\displaystyle a}$$ is taken, and that the current payoff from taking action $${\displaystyle a}$$ in state $${\displaystyle x}$$ is $${\displaystyle F(x,a)}$$. In value iteration, we start off with a random value function. Share Facebook Twitter LinkedIn. Solving this equation can be very challenging and is known to suffer from the “curse of dimensionality”. A Kernel Loss for Solving the Bellman Equation. Solving the Hamilton-Jacobi-Bellman Equation for a Stochastic System with State Constraints PER RUTQUIST TORSTEN WIK CLAES BREITHOLTZ Department of Signals and Systems Division of Automatic Control, Automation and Mechatronics CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden, 2014 Report No. stochastic, a powerful tool for solving in nite horizon optimization problems; 2) analyze in detail the One Sector Growth Model, an essential workhorse of modern macroeconomics and 3) introduce you in the analysis of stability of discrete dynamical systems coming from Euler Equations. O’Reilly members experience live online training, plus books, videos, and digital content from 200+ publishers. Solving the Bellman equation We can find the optimal policies by solving the Bellman optimality equation. The Bellman Equation is one central to Markov Decision Processes. To solve the diﬀerential equations that come up in economics, it is helpful to recall a few general results from the theory of diﬀerential equations. In summary, we can say that the Bellman equation decomposes the value function into two parts, the immediate reward plus the discounted future values. © 2020, O’Reilly Media, Inc. All trademarks and registered trademarks appearing on oreilly.com are the property of their respective owners. This paper recommends an alternative to solving the Bellman partial differential equation for the value function in optimal control problems involving stochastic differential or difference equations. Our agents should be able to learn many things too. Take O’Reilly online learning with you and learn anywhere, anytime on your phone and tablet. Sync all your devices and never lose your place. An introduction to the Bellman Equations for Reinforcement Learning. 1. It was something not even a Congressman could object to. Résoudre l'équation Hamilton-Jacobi-Bellman; nécessaire et suffisant pour l'optimalité? We solve a Bellman equation using two powerful algorithms: Value iteration; Policy iteration; Value iteration. Thus, I thought dynamic programming was a good name. V ˙ ( x , t ) + min u { ∇ V ( x , t ) ⋅ F ( x , u ) + C ( x , u ) } = 0. To solve the Bellman optimality equation, we use a special technique called dynamic programming. Consider a generic second order ordinary diﬀerential equation: 00()+()0()+()()=() This equation is referred to as the “complete equation.” MARTIN-DISSERTATION-2019.pdf (2.220Mb) Date 2019-06-21. This paper recommends an alternative to solving the Bellman partial differential equation for the value function in optimal control problems involving stochastic differential or difference equations. From the tee, the best sequence of actions is two drives and one putt, sinking the ball in three strokes. Guess a solution 2. 1.Choose grid of states X and a stopping threshold 2.Assume an initial V 0for each x 2X 3.For each x 2X, solve the problem: max y2(x) The goal of this thesis is to present two frameworks for the computation of the solutions of Hamilton-Jacobi-Bellman (HJB) equations. Part of the free Move 37 Reinforcement Learning course at The School of AI. View/ Open. The solution requires no global approximation of the value function and is likely to be more accurate than methods which are based on global approximations. Get Hands-On Reinforcement Learning with Python now with O’Reilly online learning. Iterate a functional operator numerically (This is the way iterative methods are used in most cases) 3 . Copyright © 2020 Elsevier B.V. or its licensors or contributors. If we start at state and take action we end up in state with probability . Our work is built on top of an alternative to the ﬁxed-point view above: given some ﬁxed distribution whose support is S, Vˇis the unique minimizer of the squared Bellman error: L Exercise your consumer rights by contacting us at donotsell@oreilly.com. Copyright © 1993 Published by Elsevier B.V. https://doi.org/10.1016/0165-1889(93)90049-X. Dynamic programming In DP, instead of solving complex problems one at a time, we break the problem into simple sub-problems, then for each sub-problem, we compute and store the solution. Many popular algorithms like Q-learning do not optimize any objective function, but are xed-point iterations of some variant of Bellman operator that is not necessarily a contraction. We use cookies to help provide and enhance our service and tailor content and ads. By continuing you agree to the use of cookies. Obviously, the random value function might not be an optimal one, so we look for a new improved... Show transcript Get quickly up to speed on the latest tech . We can find the optimal policies by solving the Bellman optimality equation. Considérons l'équation différentielle suivante où est l'état et la variable de contrôle. For a decision that begins at time 0, we take as given the initial state $${\displaystyle x_{0}}$$. ∙ Google ∙ The University of Texas at Austin ∙ 0 ∙ share Value function learning plays a central role in many state-of-the-art reinforcement-learning algorithms. Policies can be updated through policy iteration and value iteration, which represent different approaches to evaluating a policy before it is updated. Methods for Solving the Bellman Equation What are the 3 methods for solving the Bellman Equation? It’s impossible. At any time, the set of possible actions depends on the current state; we can write this as $${\displaystyle a_{t}\in \Gamma (x_{t})}$$, where the action $${\displaystyle a_{t}}$$ represents one or more control variables. However, this simple game represents a tiny fraction of human experience, and humans can learn to do many things. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Solving high dimensional HJB equation using tensor decomposition. It is represented and solved by Bellman equation method, namely, the value function method. Such mappings comprise … Abstract. Richard Bellman’s “Principle of Optimality” is central to the theory of optimal control and Markov decision processes (MDPs). V = V T. {\displaystyle V=V_ {T}} ), the Hamilton–Jacobi–Bellman partial differential equation is. 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Current and magnetic helicity of active regions

# Current helicity of active regions as a tracer of large-scale solar magnetic helicity

## Abstract

We demonstrate that the current helicity observed in solar active regions traces the magnetic helicity of the large-scale dynamo generated field. We use an advanced 2D mean-field dynamo model with dynamo saturation based on the evolution of the magnetic helicity and algebraic quenching. For comparison, we also studied a more basic 2D mean-field dynamo model with simple algebraic alpha quenching only. Using these numerical models we obtained butterfly diagrams both for the small-scale current helicity and also for the large-scale magnetic helicity, and compared them with the butterfly diagram for the current helicity in active regions obtained from observations. This comparison shows that the current helicity of active regions, as estimated by evaluated at the depth from which the active region arises, resembles the observational data much better than the small-scale current helicity calculated directly from the helicity evolution equation. Here and are respectively the dynamo generated mean magnetic field and its vector potential. A theoretical interpretation of these results is given.

magnetohydrodynamics (MHD) – Sun: magnetic fields – dynamo – interior – surface activity – sunspots

## 1. Introduction

The solar activity cycle is believed to be a manifestation of dynamo action which somewhere in the solar interior generates waves of quasi-stationary magnetic field propagating from middle latitude towards the solar equator (”dynamo waves”). The traditional explanation of this dynamo action (Parker, 1955) is based on the joint action of differential rotation and mirror asymmetric convection which results in what has come to be known as the -effect, based on the helicity of the hydrodynamic convective flow (Krause & Rädler, 1980; Moffatt, 1978). This explanation is however not the only one discussed in the literature and, for example, meridional circulation is also suggested as an important co-factor of the -effect, see e.g., Dikpati & Gilman (2001); Choudhuri et al. (2004).

In turn, traditional dynamo scenarios based on differential rotation and the classical -effect have to include a dynamo saturation mechanism. One of the most popular saturation mechanisms is based on a contribution to the -effect from magnetic fluctuations (Pouquet et al., 1976). A relevant quantification of this effect involves considerations of magnetic helicity evolution, e.g., Kleeorin et al. (1995, 2003). A key role in the evolution of magnetic helicity is played by magnetic helicity fluxes (Kleeorin et al., 2000; Blackman & Field, 2000a; Brandenburg & Subramanian, 2005). Of course, this scenario is not the only one that has been suggested: for example Blackman & Field (2000b); Blackman & Brandenburg (2003); Brandenburg (2007) consider coronal-mass ejections as an important part of nonlinear suppression of the dynamo, and Mitra et al. (2011) discuss the effects of losses via the solar wind. Choudhuri et al. (2004) believe that the current helicity in solar active regions is substantially modified when magnetic tubes rise up to the solar surface.

A natural way to resolve such controversies is to determine relevant quantities such as the -effect through observations, this providing a check on the various scenarios. Such an option is now becoming realistic, starting from the 1990s when the first attempts to observe current helicity in solar active regions were undertaken (Seehafer, 1990; Pevtsov et al., 1994; Bao & Zhang, 1998; Hagino & Sakurai, 2004).

Twenty years of continuous efforts by several observational groups, with the main contribution coming from the Huairou Solar Station of China, has resulted (Zhang et al., 2010) in reconstruction of the current helicity time-latitude (butterfly) diagram for one full solar magnetic cycle (1988-2005). From this butterfly diagram it is apparent that the current helicity is involved in the solar activity cycle and follows a polarity law comparable with the Hale polarity law for sunspots – but rather more complicated. In other words, dynamo generated magnetic field is indeed mirror asymmetric and this mirror asymmetry is involved in the solar activity cycle and can be used to understand its nature (Kleeorin et al., 2003; Zhang et al., 2006).

What however needs clarification is which dynamo governing parameter is traced by such a surface proxy as a measure of current helicity in solar active regions. A naive idea here is to identify this part of the surface current helicity with the current and magnetic helicities of the dynamo generated small-scale magnetic fields deep inside the Sun (say, in the solar overshoot layer), which suppress dynamo action. To start with something definite, this naive interpretation was applied by Kleeorin et al. (2003) and Zhang et al. (2006).

Recent progress in observation which has resulted in butterfly diagrams for the current helicity in active regions (Zhang et al., 2010) makes it possible to go further with this topic. The aim of this paper is to argue that the current helicity in solar active regions directly reflects the magnetic helicity of the large-scale dynamo generated field. We organize our arguments as follows.

Based on the observed butterfly diagram for the current helicity, we here confront the naive interpretation with the available observations. We examine a mean-field dynamo model with dynamo suppression based on the magnetic helicity balance, obtain the corresponding current helicity butterfly diagrams, and also that of the large-scale magnetic helicity, and compare them with those observed. This comparison shows that the evolution of the large-scale magnetic helicity resembles the observational data much more closely than that of the current helicity of the small-scale fields. In order to demonstrate the robustness of this result, we also consider a more primitive dynamo model, with a simple algebraic -quenching. From this model we calculate the large-scale magnetic helicity and compare it with the observational butterfly diagram. We find that this fits observations more or less as well as that from a dynamo model based on helicity conservation. We conclude that the major part of the observed current helicity in active regions is produced in the rise of magnetic loops to the solar surface.

## 2. The role of helicities in magnetic field evolution

In the evolution of the magnetic field different helicities play different roles. Considering the small-scale velocity and magnetic fluctuations, and respectively, there are three helicities: (i) the kinetic helicity that determines the kinetic effect; (ii) the current helicity that determines the magnetic part of the effect; and (iii) the magnetic helicity , where . We use the angled brackets to denote spatial integrals over all relevant turbulent fluid. These integrated helicities are used in the sense of average helicity densities.

The total magnetic helicity, the sum of the magnetic helicities of the large- and small-scale fields, and respectively, is conserved for very large magnetic Reynolds numbers. Here is the large-scale magnetic field. Note that, on the contrary, the current helicity is not conserved. On the other hand, the kinetic helicity is conserved only for very large fluid Reynolds numbers when the large-scale magnetic field vanishes. The characteristic time for the decay of kinetic helicity is of the order of the turn-over time of turbulent eddies in the energy-containing scale, , of turbulence, while the characteristic time of the small magnetic helicity decay is of the order of (Moffatt, 1978; Zeldovich et al., 1983; Brandenburg & Subramanian, 2005), where is the magnetic Reynolds number, is the characteristic turbulent velocity and is the magnetic diffusivity due to electrical conductivity of the fluid. The small-scale current helicity, is not an integral of motion and the characteristic time of varies from a short timescale, , to much larger timescales. On the other hand, the characteristic decay times of the current helicity of large-scale field, , and of the large-scale magnetic helicity, , are of the order of the turbulent diffusion time. For weakly inhomogeneous turbulence the small-scale current helicity, , is proportional to the small-scale magnetic helicity, .

As the dynamo amplifies the large-scale magnetic field, the large-scale magnetic helicity grows in time (but not monotonically in a cyclic system). The evolution of the large-scale magnetic helicity is determined by

 ∂HM∂t+\boldmath∇⋅FM=2\boldmathE⋅B−2ηHC, (1)

(Kleeorin et al., 1995; Blackman & Field, 2000a; Brandenburg & Subramanian, 2005), where is the mean electromotive force that determines generation and dissipation of the large-scale magnetic field, is the source of the large-scale magnetic helicity due to the dynamo generated large-scale magnetic field, and is the flux of large-scale magnetic helicity that determines its transport. Since the total magnetic helicity over all scales, integrated over the turbulent fluid, is conserved for very small magnetic diffusivity, the small-scale magnetic helicity changes during the dynamo process, and its evolution is determined by the dynamic equation

 ∂Hm∂t+\boldmath∇⋅F=−2\boldmathE⋅B−2ηHc, (2)

(Kleeorin et al., 1995; Blackman & Field, 2000a; Brandenburg & Subramanian, 2005), where is the source of the small-scale magnetic helicity due to the dynamo generated large-scale magnetic field, is the flux of small-scale magnetic helicity that determines its transport and is the dissipation rate of the small-scale magnetic helicity. It follows from Eq. (1) and (2) that the source of the small-scale and the large-scale magnetic helicities is located only in turbulent regions (i.e., in our case, in the solar convective zone). The magnetic part of the effect is determined by the parameter , and for weakly inhomogeneous turbulence is proportional to the magnetic helicity: (Kleeorin & Rogachevskii, 1999; Brandenburg & Subramanian, 2005), where is the density and is the turbulent magnetic diffusion.

## 3. The observed current helicity butterfly diagrams

The observed butterfly diagrams of electric current helicity for solar active regions during the last two solar cycles has been presented by Zhang et al. (2010). The general structure can be described as follows. Current helicity is involved in the solar activity cycle and follows a polarity rule comparable to (however more complicated than) the polarity rule for toroidal magnetic field which in turn comes from the Hale polarity rule for sunspot groups. Migration of the helicity pattern is clearly visible and located near the toroidal field pattern. The wings of the helicity butterflies are slightly more inclined to the equator than the magnetic field wings, but the former follow in general the latter. As a quantity quadratic in magnetic field, the current helicity in one hemisphere has the same sign for both cycles, with the opposite sign in the other hemisphere (a kind of unchanging dipolar symmetry).

Fig. 1 shows the distribution of the average helical characteristics of the magnetic field in solar active regions in the form of a butterfly diagram (latitude-time) for 1988-2005 (which covers most of the 22nd and 23rd solar cycles). These results are inferred from photospheric vector magnetograms recorded at Huairou Solar Observing Station.

This, the longest available systematic dataset covering the period of two solar cycles, comprises 6205 vector magnetograms of 984 solar active regions (most of the large solar active regions of both solar cycles). Of these, 431 active regions belong to the 22nd solar cycle and 553 to the 23rd. We have limited the latitudes of active regions to 40 and most of them are below 35. The helicity values of the active regions have been averaged over latitude by intervals of 7 in solar latitude, and over overlapping two-year periods of time (i.e., 1988-1989, 1989-1990, …, 2004-2005). By this method of averaging we were able to group sets of at least 30 data points in order to make error estimations (computed as 95 % confidence intervals) reasonably small. In this sampling we find that 66% (63%) of active regions have negative (positive) mean current helicity in the northern (southern) hemisphere over the 22nd solar cycle and 58 (57%) in the 23rd solar cycle.

There are some domains in the diagram where current helicity has the ”wrong” sign with respect to the global polarity law. These domains of ”wrong” sign are located at the very beginning and the very end of the wings.

Concerning alternative explanations of the observations (Zhang et al., 2010), Mackay & Van Ballegooijen (2005); Yeates & Mackay (2009) describe current helicity in solar active regions in terms of generation of non-potential coronal structures by surface differential rotation. Note that the surface differential rotation cannot generate large-scale and small-scale magnetic helicity. It can only redistribute the existing magnetic helicity, as can any non-uniform large-scale motions. This process is determined by the flux term in the evolutionary equation for the magnetic helicity. The local change of the magnetic helicity inside a given active region by the surface differential rotation plays an important role. In our paper we study robust global (rather than local) features of the evolution of the magnetic helicity, by averaging over an ensemble of active regions. In this case the global evolution of the magnetic helicity mainly depends on the sources of magnetic helicity inside the solar convective zone.

## 4. An estimate for the current helicity in active regions

In this section we estimate the current helicity in active regions. There is a common belief that active regions are formed due to some nonaxisymmetric instability of kG magnetic fields in the tachocline, e.g., (Gilman & Dikpati, 2000; Cally et al., 2003; Parfrey & Menou, 2007). However, the existence of such strong fields and the role of this mechanism remain questionable (Brandenburg, 2005). Another promising mechanism of formation of active regions is related to a negative effective magnetic pressure instability of the large-scale dynamo generated magnetic field. This instability was predicted theoretically (Kleeorin et al., 1996; Rogachevskii & Kleeorin, 2007) and detected recently in direct numerical simulations (Brandenburg et al., 2011). The instability is caused by the suppression of turbulent hydromagnetic pressure by the mean magnetic field. At large Reynolds numbers and for sub-equipartition magnetic fields, the resulting negative turbulent contribution can become so large that the effective mean magnetic pressure (the sum of turbulent and non-turbulent contributions) appears negative (Brandenburg et al., 2010, 2011). In a stratified turbulent convection, this results in the excitation of a large-scale instability that results in the formation of large-scale inhomogeneous magnetic structures. This mechanism is consistent with the suggestion that active regions are formed near the surface of convective zone (Kosovichev, 2010).

The spatial scale of an active region is much smaller than the solar radius, but much larger then the maximum scale of solar turbulent granulation. To estimate the current helicity in an active region, we have to relate the large-scale magnetic field and its magnetic potential inside the convective zone as well as the corresponding small-scale quantities inside the convective zone (which determine the small-scale magnetic fluctuations), with the surface magnetic field and its magnetic potential inside active regions, which are the quantities available to observations.

We base our estimate on the following idea. Consider a new-born magnetic tube at the layer in convective zone from which the active region is developing. We assume that the rise of magnetic tubes is a fast adiabatic process. Let us also assume that the mean magnetic field and the total magnetic helicity vanish at the initial instant, and take into account the magnetic helicity conservation law (as solar plasma is highly conductive, and so we consider the magnetic helicity conservation law to hold at all scales, including that of the whole Sun). If this tube rises rapidly to the surface to produce an active region, the total magnetic helicity in the tube is conserved because the process is rapid. Rising large-scale magnetic field and magnetic potential give the corresponding quantities for active regions, which may thus differ substantially from the corresponding quantities in the surrounding medium. Because the initial total magnetic helicity of the tube, which was almost nonmagnetized, was negligible, the magnetic helicity conservation law reads

 ⟨Aar⋅Bar⟩≈−\boldmathA⋅\boldmathB, (3)

where the angular brackets denote averaging over the surface occupied by the active region.

Now we relate the mean current helicity with the magnetic helicity . We rewrite it from the first principles with the use of permutation tensors as

 ⟨Bar⋅\boldmathcurlBar⟩≈1L2ar⟨Aar⋅Bar⟩+O(L2arR2⊙), (4)

(see Appendix A), where is the solar radius, and is the spatial scale of an active region. Equations (3) and (4) yield

 ⟨Bar⋅\boldmathcurlBar⟩≈−1L2ar\boldmathA⋅\boldmathB. (5)

Therefore, the observed current helicity in active regions is expected to be a proxy for . This idea will be checked using mean-field dynamo numerical modelling and comparison of the numerical results with the observed current helicity in active regions.

## 5. Dynamo models

Our approach to compare the dynamo models with observations is as follows. We consider two types of dynamo models. Both types of models are 2D mean-field models with an axisymmetric magnetic field which depends on radius and polar angle . The third (azimuthal) coordinate is , and . The dynamo action is based on differential rotation, with a rotation curve which resembles that of the solar convection zone, as known from helioseismological observations, and there is a conventional -effect.

The first type of model assumes a very naive algebraic -quenching. Then we suppose that the total magnetic helicity is locally vanishing, so the magnetic helicity of the large-scale magnetic field produced in the course of mean-field dynamo action has to be compensated by small-scale magnetic helicity. (Thus we are assuming that at an initial instant the medium is non-magnetic, so that helicity conservation means that the sum of large and small-scale helicities remains zero.) We assume also that there is a separation of scales so that characteristic turbulence scales are much smaller than the characteristic spatial scales of mean magnetic field variations. This allows a link between current and magnetic helicities (Kleeorin & Rogachevskii, 1999) to be made. This concept underlies the observational procedure for determining the current helicity of active regions, and for calculating the current helicity from the magnetic helicity of the small-scale fields. Based on the same concept we estimate the large-scale magnetic helicity as , where is the magnetic potential for the poloidal field. As a result we obtain (for a given radius ) a theoretical model for the current helicity as a function of and which we overlay on the butterfly diagram for . We compare the result with the current helicity butterfly diagram known from observations and obtained using similar underlying concepts.

A further point is that the primitive model allows a simplification to the level of the one-dimensional Parker migratory dynamo, and this opportunity has been investigated in this respect by Xu et al. (2009). We will use the results of that work for reference and comparison.

We do not consider this primitive scheme as realistic. We are sure that any more or less realistic scenario for solar dynamo suppression will have to be much more sophisticated. On the other hand, we can see whether this primitive model produces a helicity butterfly diagram that is quite similar to that observed. The only shortcoming of the model is that the maximal current helicity occurs later then the maximum , while it is observed to come up earlier. If magnetic helicity conservation determines the nonlinear dynamo suppression, we expect that a careful reproduction of this balance, including helicity fluxes and the link between magnetic helicity and -effect, will result in even a better theoretical butterfly diagram, and possibly improve the phase relations between helicity and toroidal magnetic field.

As a specific example of the second type model that takes into account the influence of magnetic helicity balance on dynamo action we use the dynamo model described by Zhang et al. (2006). Whereas simple -quenching provides a quite robust suppression of a spherical dynamo and give (more or less) steady nonlinear magnetic field oscillations for a very wide range of parameters, in contrast it is far from clear a priori that a dynamo suppression based on magnetic helicity conservation is effective enough to suppress magnetic field growth and result in steady oscillations. In fact it works more-or-less satisfactorily only in a quite narrow parameter range (Kleeorin et al., 2003; Zhang et al., 2006), which appears inadequate to fit observations.

We note two crucial points here. First of all, both types of models ignore any direct action of magnetic force on the rotation law. In the more primitive models, there is a crude parametrization of feedback onto the (purely hydrodynamic) alpha effect [see Eq. (6)] below. The latter more sophisticated model describes the back-reaction of the generated magnetic field on the dynamo process in terms of the magnetic contribution of the current helicity onto the magnetic part of -effect. On the other hand, the feedback of the generated large-scale magnetic field on turbulent convection is described in our model by the algebraic quenching of -effect, pumping velocities and turbulent magnetic diffusion.

We assume that helicity conservation is not the only mechanism of dynamo suppression. The fact that we see a manifestation of helicity on the solar surface tells us that the buoyancy indeed plays some role, and we add it to the model. We stress that the buoyancy which we include in the model transports current helicity and magnetic helicity as well as large-scale magnetic field.

Below we discuss the detailed dynamo models. We use spherical coordinates and describe an axisymmetric magnetic field by the azimuthal component of magnetic field , and the component of the magnetic potential corresponding to the poloidal field.

We measure length in units of the solar radius , and time in units of a diffusion time based on the solar radius and the reference turbulent magnetic diffusivity . The magnetic field is measured in units of the equipartition field , the vector potential of the poloidal field in units of , the density is normalized with respect to its value at the bottom of the convective zone, and the basic scales of the turbulent motions and turbulent velocity at the scale are measured in units of their maximum values through the convective zone. The -effect is measured in terms of , defined below, and angular velocity in units of the maximum surface value, .

### 5.1. The primitive, alpha-quenched model

In the primitive dynamo model the -effect is given by

 α=αv=χvΦv, (6)

where is proportional to the hydrodynamic helicity, , multiplied by the turbulent correlation time , and is the model for the quenching nonlinearity. For convenience, we use for most of these computations the code of Moss & Brooke (2000) – see also Moss et al. (2011). This code has the possibility of a modest reduction in the diffusivity, to , in the innermost part of the computational shell (”tachocline”), below fractional radius 0.7. We define . We also used this primitive formulation of alpha-quenching in the (otherwise very similar) model used in Sect. 5.2 when producing Fig. 5. In the latter case, the diffusivity is everywhere uniform.

At the surface the field is matched to a vacuum external field, and ”overshoot” boundary conditions are used at the lower boundary.

### 5.2. The model based on helicity balance

Here we use the code described in Zhang et al. (2006), with two new features: we allow the possibility of meridional circulation and/or vertical motions attributed to magnetic buoyancy. The dynamo equations for and read

 ∂~A∂t+(VAθ+VMθ)r∂~A∂θ+Vr∂~A∂r=Cαα~B +ηA[∂2~A∂r2+sinθr2∂∂θ(1sinθ∂~A∂θ)], (7) ∂~B∂t+sinθr∂∂θ[(VBθ+VMθ)~Bsinθ]+∂[Vr~B]∂r =Cωsinθ[Gr∂∂θ−Gθ∂∂r]~A+sinθr2∂∂θ[ηBsinθ∂~B∂θ] +∂∂r[ηB∂~B∂r], (8)

where , and are the nonlinear drift velocities of poloidal and toroidal mean magnetic fields, is the meridional circulation velocity, is the vertical buoyancy velocity, and are the nonlinear turbulent magnetic diffusion coefficients for the mean poloidal and toroidal magnetic fields, and and the non-dimensional dynamo parameters are , . The nonlinear turbulent magnetic diffusion coefficients and the nonlinear drift velocities are given in Appendix B. The non-dimensional gradients of differential rotation are

 Gr=∂Ω∂r,Gθ=∂Ω∂θ.

In this dynamo model with magnetic helicity evolution the total -effect is given by

 α=αv+αm=χvΦv+Φmρ(z)χc, (9)

with . The magnetic part of the -effect is based on the idea of magnetic helicity conservation and the link between current and magnetic helicities. Here and are proportional to the hydrodynamic and current helicities multiplied by the turbulent correlation time, and and are quenching functions. The analytical form of the quenching functions and is given in Appendix B. The density profile is chosen in the form:

 ρ(z)=exp[−atan(0.45πz)], (10)

where and . Here corresponds to a tenfold change of the density in the solar convective zone, by a factor of about .

The equation for is

 ∂~χc∂t+~χcT=(2R⊙ℓ)2{1Cα[ηBr2∂~A∂θ∂~B∂θ+ηB∂~A∂r∂~B∂r −ηA~Bsinθr2∂∂θ(1sinθ∂~A∂θ)−ηA~B∂2~A∂r2 +(VAr−VBr)~B∂~A∂r+(VAθ−VBθ)~Br∂~A∂θ]−α~B2} −∂[~Fr+(VB+VMr)~χc]∂r−sinθr∂∂θ[~Fθ+VMθ~χcsinθ],

where , is related to the flux of the small-scale magnetic helicity and given in Appendix B, is the ratio of the solar radius to the basic scale of solar convection, is the dimensionless relaxation time of the magnetic helicity, is the magnetic Reynolds number, with the ‘basic’ magnetic diffusion due to the electrical conductivity of the fluid.

The meridional circulation (single cell in each hemisphere, poleward at surface) is determined by

 VMθ = −1sinθrρ(r)∂[rΨ(r,θ)]∂r, (12) VMr = 1sinθrρ(r)∂Ψ(r,θ)∂θ, (13)

where , , is the base of the computational shell. This is normalized so that the max of at the surface is unity. We introduce a coefficient , where the maximum surface speed.

Buoyancy is implemented by the introduction of a purely vertical velocity in Eqs. (7), (8) and (LABEL:curhel), where (Moss et al., 1999). We justify the apparent non-conservation of mass by adopting the argument of K.-H. Rädler, presented as a private communication in Moss et al. (1990), that the return velocity will be in the form of a more-or-less uniform ”rain”. In some ways the process represents pumping by a ”fountain flow”. As a result the regular velocity appears in the governing equations for the large-scale magnetic field and magnetic and current helicities but not the the equation for density. From the viewpoint of probability theory, in the first case is a mean quantity taken under the condition that an elementary volume is magnetized so it does not vanish, while in the second case this mean is taken without any condition and vanishes. In our opinion, this idea can also be constructive for other problems with magnetic helicity advective fluxes, e.g. Shukurov at al. (2006).

At the surface of the Sun, , we use vacuum boundary conditions on the field, i.e. and the poloidal field fits smoothly onto a potential external field. At the lower boundary (the bottom of the solar convective zone), , . At both and , we set , where is proportional to the current helicity [see Eq. (9)].

## 6. Simulated butterfly diagrams for current helicity

We performed an extensive numerical investigation of the models in a parametric range which is considered to be adequate for solar dynamos. We estimate the values of the governing parameters for different depths of the convective zone, using models of the solar convective zone, e.g. Baker & Temesvary (1966), Spruit (1974) – more modern treatments make little difference to these estimates. In the upper part of the convective zone, say at depth (measured from the top) cm , the parameters are cm s, cm, g cm the turbulent diffusivity cm s; the equipartition mean magnetic field is G and . At depth cm these values are , cm s, cm, g cm cm s; the equipartition mean magnetic field is G and . At the bottom of the convective zone, say at depth cm, cm s, cm, g cm cms. Here the equipartition means magnetic field G and . If we average the parameter over the depth of the convective zone, we obtain , see Kleeorin et al. (2003).

We start with the results for the primitive model. Fig. 2 presents the current helicity butterfly diagrams overlaid on those for the toroidal field. We estimate this quantity based on the idea that the observed current helicity in active regions is expected to trace , and so we plot in this section .

We see that the plots successfully represent main feature of the observed helicity patterns. The pattern presented in Fig. 2 is quite typical for the model. Of course, one can choose a set of dynamo governing parameters which is less similar to the observations. For example one can concentrate magnetic fields in the deep layer of convective zone (say, in the overshoot layer) by choosing a reduction in the nominal ”overshoot layer”, instead of (our standard case), as used in Fig. 2. This tends to make the helicity wave in the overshoot layer look more like a standing wave, but however keeps the main features of surface diagram (Fig. 3). The highly anharmonic standing patterns of the butterfly diagram that were discussed as a possible option for some stars, see Baliunas et al. (2006), look however irrelevant for the solar case.

Of course, the helicity pattern in the butterfly diagram obtained in the models for particular choices of dynamo governing parameters can be slightly different from the observed helicity patterns. Xu et al. (2009) demonstrated that meridional circulation can be used to make the simulated pattern resemble more closely that observed.

We produced the same type of plots for models based on helicity conservation Fig. 5. We also present in Fig. 4 the small-scale current helicity . Here the migration diagrams are presented for the middle radius of the dynamo region: nearer the surface displays only relatively weak vacillatory behaviour. We see that the small-scale current helicity is strongly concentrated in middle latitudes and helicity oscillations which are present in the model are almost invisible on the background of the intensive belt of constant helicity in middle latitudes. We doubt that such oscillations would be observable. We stress that, if this model produces any travelling helicity pattern, it is situated in the deep layers only. The pattern usually is much more similar to that presented in the right hand panel of Fig. 3 rather than to a travelling wave such as presented in Fig. 2.

Taking our models as a whole, determination of further details of the location of the dynamo layer or the helicity sign inversion at some latitudes and phases of the solar cycle is beyond the scope of this investigation. Such a study would require a more complex simulation, which we hope to perform in the future.

The numerical models used here are extremely simple and cannot be expected to sample the full range of solutions that are accessible to a three-dimensional, highly stratified, extremely turbulent system with complex and unknown boundary conditions. In the present paper we have only shown the theoretical possibility that the current helicity observed in solar active regions may trace the magnetic helicity of the large-scale dynamo generated field. We demonstrated that the results obtained with our simplified dynamo model are compatible with the observations. At the same time we note that such computationally complex and expensive simulations are still far from being able to reproduce the range of observed solar phenomena, and that for the immediate future mean field models will continue to play an important role. Indeed, they may provide more immediate physical insight.

The simple models which we have considered here were not intended to reproduce very fine details of the spatial-time distribution of helicity observable in the form of butterfly diagrams. We may again note that our modelling of helicity in the solar convective zone and active regions is still too simplified to be able to detect more detailed properties of the helicity dynamics.

While we feel that it is remarkable that there is a class of kinematic models that are able to reproduce the current helicity observations from vector magnetograms, we may again note that some contribution to the observed current helicity may be due to the surface effects caused by the differential rotation (Mackay & Van Ballegooijen, 2005; Yeates & Mackay, 2009) and not from the dynamo.

## 7. Conclusions

We conclude that current helicity of magnetic field in active regions is a tracer of the magnetic helicity of the large-scale magnetic field in the solar interior. We believe that this provides a unique option for tracing this quantity, which is very important for the solar dynamo. According to the observational data (Zhang et al., 2010), the current helicity in active regions is mainly negative in the Northern hemisphere. Numerical models give a negative value for in the surface layer of the convective zone in the Northern hemisphere.

Summarizing, we conclude that the current helicity of the magnetic field in active regions is expected to have the opposite sign to , evaluated at the depth at which the active region originates. Thus, the models presented here are consistent with the interpretation that the mechanism responsible for the sign of the observed helicity operates near the solar surface, cf. e.g., Kosovichev (2010). The mechanism of formation of the current helicity in active regions still requires further investigation.

We acknowledge the NORDITA dynamo programs of 2009 and 2011 for providing a stimulating scientific atmosphere. This collaborative work has been benefited from discussions at the Helicity Thinkshop meeting in Beijing in 2009. The work has been supported by Chinese grants NSFC 10733020, 10921303, 41174153, 11103037, NBRPC 2006CB806301, CAS KLCX2-EW-T07 and joint NNSF-RBFR grant 10911120051. K.K., D.S., N.K. and D.M. would like to acknowledge support from the National Astronomical Observatories of China during their visits, K.K. and D.S. are thankful to the joint project of NNSF of China and RFBR of Russia no. 08-02-92211, to RFBR grant 10-02-00960, to Visiting Professorship Programme 2009J2-12, as well as joint Chinese-Russian collaborative project from CAS.

## Appendix A Current helicity versus the magnetic helicity

Here we relate the mean current helicity with the magnetic helicity . First we rewrite the mean current helicity from first principles with the use of permutation tensors as

 Missing or unrecognized delimiter for \big

where is the fully antisymmetric Levi-Civita tensor, is the solar radius, is the spatial scale of an active region and . The use of the full tensor notation and limits are needed here in order to separate the large-scale and small-scale variables and to obtain a simple final answer in scalar form when there is separation of scales. Since is a small-scale variable, is a large-scale variable, the derivatives

 ∇xp≡∂∂xp=∂∂rp+12∂∂Rp=−∇yp+∂∂Rp,∇yp≡∂∂yp=−∂∂rp+12∂∂Rp=−∇xp+∂∂Rp.

This implies that

 \boldmath∇x⋅\boldmath∇y=−(∂2∂r2−14∂2∂R2),∇xp∇yq=∇yp∇xq−∇yp∂∂Rq−∇xq∂∂Rp+∂2∂Rp∂Rq.

We take into account that div [i.e. ] and div [i.e. ]. We also take into account that the characteristic scale of an active region is small compared with the thickness of the convection zone or the radius of the Sun. This implies that

 ∇xp∇yq⟨Aarq(x)Barp(y)⟩=∂2∂Rp∂Rq⟨Aarq(x)Barp(y)⟩∼O(L2arR2⊙),

and therefore this term vanishes. This yields

 ⟨Bar⋅\boldmathcurlBar⟩=−(∂2∂rp∂rp⟨Aar⋅Bar⟩)r→0+O(L2arR2⊙), (A2)

Now we take into account that the second derivative of the correlation function

 (∂2∂rp∂rp⟨Aar⋅Bar⟩)r→0

should be negative since as the correlation function has a maximum. Thus, we finally obtain

 ⟨Bar⋅\boldmathcurlBar⟩≈1L2ar⟨Aar⋅Bar⟩+O(L2arR2⊙). (A3)

Similar calculations relating the current helicity and the magnetic helicity in -space can be found in Appendix C of Kleeorin & Rogachevskii (1999).

## Appendix B Detailed description of the dynamo model

The flux of the small-scale magnetic helicity is chosen in the form

 F = ηA(B)B2{C1\boldmath∇[χvϕv(B)]+C2χvϕv(B)\boldmathΛρ}−C3κ\boldmath∇χc, (B1)

with and is the coefficient of turbulent diffusion of small-scale magnetic helicity (see below).

The quenching functions and appearing in the nonlinear effect are given by

 Φv(B) = 17[4ϕm(B)+3L(√8B)], (B2) Φm(B) = 38B2[1−arctan(√8B)√8B] (B3)

(Rogachevskii & Kleeorin, 2000), where .

The nonlinear turbulent magnetic diffusion coefficients for the mean poloidal and toroidal magnetic fields, and , are given in dimensionless form by

 ηA(B) = A1(4B)+A2(4B), (B4) ηB(B) = A1(4B)+32[2A2(4B)−A3(4B)], (B5)

(Rogachevskii & Kleeorin, 2004). For the case of weak magnetic field the turbulent diffusion coefficients are (in units of the reference value )

 ηA=1−965B2,ηB=1−32B2, (B6)

while for strong magnetic fields the scaling is

 ηA=18B2,ηB=13√2B. (B7)

The transition from one asymptotic form to the other can be thought of as occurring in the vicinity of .

The nonlinear drift velocities of poloidal and toroidal mean magnetic fields, and , are given in dimensionless form by

 \boldmathVA(B) = V1(B)\boldmathΛB2+V2(B)r(\boldmather+cotθ\boldmatheθ)+% \boldmathVρ(B), (B8) \boldmathVB(B) = V3(B)r(\boldmather+cotθ% \boldmatheθ)+\boldmathVρ(B), (B9)

where

 V1(B) = 32A3(4B)−2A2(4B), V2(B) = 12A2(4B), V3(B) = 32[A2(4B)−A3(4B)], \boldmathVρ(B) = 12\boldmathΛρ[−5A2(4B)+32A3(4B)].

The asymptotic formulae for these velocities are given by

 \boldmathVA = 325B2[\boldmathΛB+3\boldmathΛρ−\boldmather+cotθ% \boldmatheθr], \boldmathVB = 325B2[3\boldmathΛρ−%\boldmath$e$r+cotθ\boldmatheθr]

for a weak magnetic field, and

 \boldmathVA = −13√8B[\boldmathΛB+2\boldmather+cotθ\boldmatheθr]+516B2\boldmathΛρ, \boldmathVB = 43√8B\boldmather+cotθ% \boldmatheθr+516B2\boldmathΛρ

for strong fields. Here , and are unit vectors in the and directions of spherical polar coordinates, , and . See, for details, Rogachevskii & Kleeorin [2004, Eqs. (18), (19), (22)-(24)], which have been rewritten here in spherical geometry.

The functions are

 A1(y) = 65[arctanyy(1+57y2)+114L(y)−57y2], A2(y) = −65[arctanyy(1+157y2)−27L(y)−157y2], A3(y) = −2y2[arctanyy(y2+3)−3].

The nonlinear quenching of the turbulent magnetic diffusion of the magnetic helicity is given by

 κ(B)=12[1+A1(4B)+12A2(4B)]. (B10)

The coefficient of turbulent diffusion of magnetic helicity also has a dependence on , namely for weak magnetic field and

 κ(B)=12(1+3π40B) (B11)

in the strong field limit.

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How do you prove that $\int_0^\infty \frac{\sin(2x)}{1-e^{2\pi x}} dx = \frac{1}{2-2e^2}$?

3
Casimir Rönnlöf 2020-06-14 16:28.

I know the following result thanks to the technique https://isolution.pro/q/ma15818910/togo-sakunyu:

$$\int_0^\infty \frac{\sin(2x)}{1-e^{2\pi x}} dx = \frac{1}{2-2e^2}$$

So I have a proof (I might list it here later, if it turns out this question seems very hard to solve) of the result, but I wouldn't be able to solve it if I would start with the integral. I tried a few things, e.g. expanding and substitution, but I didn't come anywhere. https://www.wolframalpha.com/input/?i=integral%20from%200%20to%20infinity%20of%20sin%282t%29%2F%281-e%5E%282pit%29%29dt doesn't have the closed-form, but you can check numerically if you want.

How would you solve the integral without knowing the result?

6
Ty. 2020-06-14 23:40.

Divide the numerator and denominator by $e^{2\pi x}$: $$I=-\int_0^{\infty} \frac{e^{-2\pi x} \sin{(2x)}}{1-e^{-2\pi x}} \; dx$$ $$I=-\int_0^{\infty} \sum_{n=1}^{\infty} e^{-2\pi x n} \sin{(2x)} \; dx$$ Due to Fubini theorem we can interchange the summation and integral: $$I=-\sum_{n=1}^{\infty} \int_0^{\infty} e^{-2\pi x n} \sin{(2x)} \; dx$$ Then, use integration by parts: $$I=-\sum_{n=1}^{\infty} \frac{1}{2 \pi^2 n^2+2}$$ $$I=-\frac{1}{4} \left( \coth{1}-1\right)$$ $$I=\frac{1}{2-2e^2}$$

5
Mark Viola 2020-06-14 23:32.

HINT:

Expand the denominator as

$$\frac{1}{1-e^{2\pi x}}=-\sum_{n=0}^{\infty}e^{-2(n+1)\pi x}$$

Then note that this leaves a series

$$-\frac1{2}\,\sum_{n=1}^{\infty} \frac1{\pi^2 n^2+1}$$

The series can be found in closed form using for example contour integration or Fourier series and Parseval's theorem. See https://isolution.pro/q/ma16714851/kensaku-sum-n-1-infty-a-2-n-2-b-2-n-2-text-where-e-x-a-sum-n as an example.

3
Felix Marin 2020-07-15 11:53.

This integral was first https://archive.org/details/ACourseOfModernAnalysis/page/n129/mode/2up.

2
Claude Leibovici 2020-06-15 12:58.

For the computation of $$I=\int_0^{\infty} e^{-2\pi x n} \sin{(ax)} \; dx$$ you even do not need integration by parts. Write it as $$I=\Im\left(\int_0^{\infty} e^{-2\pi x n} e^{iax} \; dx \right)=\Im\left(\int_0^{\infty} e^{-(2\pi n-ia)x} \; dx \right)=\Im\left(\frac{1}{2 \pi n-i a}\right)=\frac{a}{4 \pi ^2 n^2+a^2}$$


text:
### Home > CCG > Chapter 10 > Lesson 10.2.1 > Problem10-70

10-70.

Find $x$ if the angles of a quadrilateral are $2x, 3x, 4x,\text{ and }5x$. Homework Help ✎

The angles in a quadrilateral have a total sum of $360º$.

$x ≈ 25.7º$


text:
# In the following question, select the one which is different from the other three responses.

1.  : Amazon kindle
3.  : Amazon
4.  : Hangouts


text:
# Logger Output¶

The logger is a particle based output (e.g. snapshot) that takes into account the large difference of timescale. If you have any question, a slack channel is available for it in SWIFT’s slack.

To run it, you will need to use the configuration option --enable-logger. Currently the logger is implemented only for Gadget2 and the default gravity / stars, but can be easily extended to the other schemes by adding the logger structure to the particles (see src/hydro/Gadget2/hydro_part.h). The main parameters of the logger are Logger:delta_step and Logger:index_mem_frac that define the time accuracy of the logger and the number of index files. The first parameter defines the number of active steps that a particle is doing before writing and the second defines the total storage size of the index files as function of the dump file.

Unfortunately, the API is not really developed yet. Therefore if you wish to dump another field, you will need to trick the logger by replacing a field in the logger_log_part function.

For reading, the python wrapper is available through the configuration option --with-python. Once compiled, you will be able to use the file logger/examples/reader_example.py. The first argument is the basename of the index file and the second one is the time requested. During the first reading, the library is manipulating the dump file and therefore it should not be killed and may take a bit more time than usual.


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## G = C3×C32⋊2Q16order 432 = 24·33

### Direct product of C3 and C32⋊2Q16

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C12 — C3×C32⋊2Q16
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C32×C12 — C32×Dic6 — C3×C32⋊2Q16
 Lower central C32 — C3×C6 — C3×C12 — C3×C32⋊2Q16
 Upper central C1 — C6 — C12

Generators and relations for C3×C322Q16
G = < a,b,c,d,e | a3=b3=c3=d8=1, e2=d4, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, dbd-1=ebe-1=b-1, dcd-1=c-1, ce=ec, ede-1=d-1 >

Subgroups: 304 in 110 conjugacy classes, 36 normal (16 characteristic)
C1, C2, C3, C3, C3, C4, C4, C6, C6, C6, C8, Q8, C32, C32, C32, Dic3, C12, C12, C12, Q16, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C33, C3×Dic3, C3×C12, C3×C12, C3×C12, C3⋊Q16, C3×Q16, C32×C6, C3×C3⋊C8, C324C8, C3×Dic6, C3×Dic6, Q8×C32, C32×Dic3, C32×C12, C322Q16, C3×C3⋊Q16, C3×C324C8, C32×Dic6, C3×C322Q16
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, C3⋊D4, C3×D4, S32, S3×C6, C3⋊Q16, C3×Q16, D6⋊S3, C3×C3⋊D4, C3×S32, C322Q16, C3×C3⋊Q16, C3×D6⋊S3, C3×C322Q16

Smallest permutation representation of C3×C322Q16
On 48 points
Generators in S48
(1 40 46)(2 33 47)(3 34 48)(4 35 41)(5 36 42)(6 37 43)(7 38 44)(8 39 45)(9 30 19)(10 31 20)(11 32 21)(12 25 22)(13 26 23)(14 27 24)(15 28 17)(16 29 18)
(1 40 46)(2 47 33)(3 34 48)(4 41 35)(5 36 42)(6 43 37)(7 38 44)(8 45 39)(9 19 30)(10 31 20)(11 21 32)(12 25 22)(13 23 26)(14 27 24)(15 17 28)(16 29 18)
(1 40 46)(2 47 33)(3 34 48)(4 41 35)(5 36 42)(6 43 37)(7 38 44)(8 45 39)(9 30 19)(10 20 31)(11 32 21)(12 22 25)(13 26 23)(14 24 27)(15 28 17)(16 18 29)
(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 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 23 5 19)(2 22 6 18)(3 21 7 17)(4 20 8 24)(9 40 13 36)(10 39 14 35)(11 38 15 34)(12 37 16 33)(25 43 29 47)(26 42 30 46)(27 41 31 45)(28 48 32 44)

G:=sub<Sym(48)| (1,40,46)(2,33,47)(3,34,48)(4,35,41)(5,36,42)(6,37,43)(7,38,44)(8,39,45)(9,30,19)(10,31,20)(11,32,21)(12,25,22)(13,26,23)(14,27,24)(15,28,17)(16,29,18), (1,40,46)(2,47,33)(3,34,48)(4,41,35)(5,36,42)(6,43,37)(7,38,44)(8,45,39)(9,19,30)(10,31,20)(11,21,32)(12,25,22)(13,23,26)(14,27,24)(15,17,28)(16,29,18), (1,40,46)(2,47,33)(3,34,48)(4,41,35)(5,36,42)(6,43,37)(7,38,44)(8,45,39)(9,30,19)(10,20,31)(11,32,21)(12,22,25)(13,26,23)(14,24,27)(15,28,17)(16,18,29), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,40,13,36)(10,39,14,35)(11,38,15,34)(12,37,16,33)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44)>;

G:=Group( (1,40,46)(2,33,47)(3,34,48)(4,35,41)(5,36,42)(6,37,43)(7,38,44)(8,39,45)(9,30,19)(10,31,20)(11,32,21)(12,25,22)(13,26,23)(14,27,24)(15,28,17)(16,29,18), (1,40,46)(2,47,33)(3,34,48)(4,41,35)(5,36,42)(6,43,37)(7,38,44)(8,45,39)(9,19,30)(10,31,20)(11,21,32)(12,25,22)(13,23,26)(14,27,24)(15,17,28)(16,29,18), (1,40,46)(2,47,33)(3,34,48)(4,41,35)(5,36,42)(6,43,37)(7,38,44)(8,45,39)(9,30,19)(10,20,31)(11,32,21)(12,22,25)(13,26,23)(14,24,27)(15,28,17)(16,18,29), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,23,5,19)(2,22,6,18)(3,21,7,17)(4,20,8,24)(9,40,13,36)(10,39,14,35)(11,38,15,34)(12,37,16,33)(25,43,29,47)(26,42,30,46)(27,41,31,45)(28,48,32,44) );

G=PermutationGroup([[(1,40,46),(2,33,47),(3,34,48),(4,35,41),(5,36,42),(6,37,43),(7,38,44),(8,39,45),(9,30,19),(10,31,20),(11,32,21),(12,25,22),(13,26,23),(14,27,24),(15,28,17),(16,29,18)], [(1,40,46),(2,47,33),(3,34,48),(4,41,35),(5,36,42),(6,43,37),(7,38,44),(8,45,39),(9,19,30),(10,31,20),(11,21,32),(12,25,22),(13,23,26),(14,27,24),(15,17,28),(16,29,18)], [(1,40,46),(2,47,33),(3,34,48),(4,41,35),(5,36,42),(6,43,37),(7,38,44),(8,45,39),(9,30,19),(10,20,31),(11,32,21),(12,22,25),(13,26,23),(14,24,27),(15,28,17),(16,18,29)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,23,5,19),(2,22,6,18),(3,21,7,17),(4,20,8,24),(9,40,13,36),(10,39,14,35),(11,38,15,34),(12,37,16,33),(25,43,29,47),(26,42,30,46),(27,41,31,45),(28,48,32,44)]])

63 conjugacy classes

 class 1 2 3A 3B 3C ··· 3H 3I 3J 3K 4A 4B 4C 6A 6B 6C ··· 6H 6I 6J 6K 8A 8B 12A 12B 12C ··· 12N 12O ··· 12AD 24A 24B 24C 24D order 1 2 3 3 3 ··· 3 3 3 3 4 4 4 6 6 6 ··· 6 6 6 6 8 8 12 12 12 ··· 12 12 ··· 12 24 24 24 24 size 1 1 1 1 2 ··· 2 4 4 4 2 12 12 1 1 2 ··· 2 4 4 4 18 18 2 2 4 ··· 4 12 ··· 12 18 18 18 18

63 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + - + - - image C1 C2 C2 C3 C6 C6 S3 D4 D6 Q16 C3×S3 C3⋊D4 C3×D4 S3×C6 C3×Q16 C3×C3⋊D4 S32 C3⋊Q16 D6⋊S3 C3×S32 C32⋊2Q16 C3×C3⋊Q16 C3×D6⋊S3 C3×C32⋊2Q16 kernel C3×C32⋊2Q16 C3×C32⋊4C8 C32×Dic6 C32⋊2Q16 C32⋊4C8 C3×Dic6 C3×Dic6 C32×C6 C3×C12 C33 Dic6 C3×C6 C3×C6 C12 C32 C6 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 2 2 2 4 2 1 2 2 4 4 2 4 4 8 1 2 1 2 2 4 2 4

Matrix representation of C3×C322Q16 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 64 0 0 0 0 0 0 64 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 1 72
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 22 0 0 0 0 0 0 10 0 0 0 0 0 0 0 72 0 0 0 0 72 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0
,
 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 1 0 0 0 0 1 0

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,64,0,0,0,0,0,0,64,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,72,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[22,0,0,0,0,0,0,10,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C3×C322Q16 in GAP, Magma, Sage, TeX

C_3\times C_3^2\rtimes_2Q_{16}
% in TeX

G:=Group("C3xC3^2:2Q16");
// GroupNames label

G:=SmallGroup(432,423);
// by ID

G=gap.SmallGroup(432,423);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,176,1011,514,80,2028,14118]);
// Polycyclic

G:=Group<a,b,c,d,e|a^3=b^3=c^3=d^8=1,e^2=d^4,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,d*b*d^-1=e*b*e^-1=b^-1,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=d^-1>;
// generators/relations

׿
×
𝔽


text:
# MATLAB: Lsqcurvefit command window exit flag suppression

lsqcurvefitMATLABOptimization Toolboxsuppressing output

lsqcurvefit dumps its exit flags in the work space. i.e
lsqcurvefit stopped because the problem appears to be locally singular.
or
Optimization completed because the size of the gradient is less than the default value of the function tolerance.
is there any way to suppress these outputs in the command window?

opts = optimset('lsqcurvefit');optimset(opts,'Display','off');...  = lsqcurvefit(...,opts);


text:
Now showing items 1-20 of 29

• #### CÀLCUL NUMÈRIC

(Universitat Politècnica de Catalunya, 2020-11-02)
Exam
• #### CÀLCUL NUMÈRIC

(Universitat Politècnica de Catalunya, 2021-01-15)
Exam
• #### Càlcul numèric : manual de pràctiques

(Universitat Politècnica de Catalunya, 2014)
Practice
Open Access
• #### Càlcul numèric. Manual de pràctiques

(2014-07)
External research report
Open Access
• #### Confinement around the Vertical Family of Periodic Orbits close L4 and L5

(2001)
Article
Open Access
We consider the spatial restricted three body problem (RTBP) for values of the mass parameter close to µRouth. We analyse the dynamics around the vertical periodic orbits which are born at the equilateral equilibrium ...
• #### Controllability of continuous bimodal linear systems

(2013-05-01)
Article
Open Access
We consider bimodal linear systems consisting of two linear dynamics acting on each side of a given hyperplane, assuming continuity along the separating hyperplane. We prove that the study of controllability can be reduced ...
• #### Description of characteristic non-hyperinvariant subspaces in GF(2)

(Elsevier, 2013-12-16)
Article
Restricted access - publisher's policy
Given a square matrix A , an A -invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T ) that commute ...
• #### Description of characteristic non-hyperinvariant subspaces over the field GF(2)

(Elsevier, 2013-10-26)
Article
Restricted access - publisher's policy
Given a square matrix A, an A-invariant subspace is called hyperinvariant (respectively, characteristic) if and only if it is also invariant for all matrices T (respectively, nonsingular matrices T) that commute with A. ...
• #### Dynamics and bifurcation near the transition from stability to complex instability

(2001)
Article
Open Access
• #### Dynamics and bifurcation near the transition from stability to complex instability: an analytical approach using normal forms

(1998)
Article
Open Access
We consider a Hamiltonian of three degrees of freedom and a family of periodic orbits with a transition from stability to complex instability, such that there is an irrational collision of the Floquet eigenvalues of opposite ...
• #### Dynamics close to a non semi-simple 1: -1 resonant periodic orbit.

(2003)
Article
Open Access
In this work, our target is to analyze the dynamics around the $1:-1$ resonance which appears when a family of periodic orbits of a real analytic three-degree of freedom Hamiltonian system changes its stability from ...
• #### Hopf bifurcation for the hydrogen atom in a circularly polarized microwave field

(2018-09-01)
Article
Open Access
We consider the Rydberg electron in a circularly polarized microwave field, whose dynamics is described by a Hamiltonian depending on one parameter, K¿>¿0. The corresponding Hamiltonian system of ODE has two equilibrium ...
• #### How does the hopf bifurcation appear in the hydrogen atom in a Circularly Polarized (CP) microwave field?

(Birkhäuser, 2019)
Part of book or chapter of book
Restricted access - publisher's policy
The dynamics of the Rydberg atom in a rotating electric field can be described by a two degree of freedom Hamiltoniandepending on a parameterK. This Hamiltonian has two equilibrium pointsL1andL2. WhileL1is a saddle-center ...
• #### KAM aspects of the quasi-periodic Hamiltonian Hopf bifurcation: summary of results

(s.n., 2007-08-20)
Conference report
Open Access
In this work we consider a 1:-1 non semi-simple resonant periodic orbit of a three-degrees of freedom real analytic Hamiltonian system. From the formal analysis of the normal form, it is proved the branching off a ...
• #### KAM aspects of the quasiperiodic Hamiltonian Hopf bifurcation

(2004)
Article
Open Access
In this work we consider a normal form around a 1:-1 non semi-simple resonant periodic orbit of a three-degree of freedom Hamiltonian system. From the formal analysis it can be proved the branching off a two-parameter ...
• #### Miniversal Deformations of Bimodal Picewise Linear Systems

(Servicio de publicaciones de la UPV, 2010)
Conference lecture
Open Access
Bimodal linear systems are those consisting of two linear systems on each side of a given hyperplane, having continuous dynamics along that hyperplane. In this work, we focus on the derivation of (orthogonal) miniversal ...
• #### Motion close to the hopf bifurcation of the vertical family of periodic orbits of L^4

(2003)
Article
Open Access
The paper deals with diﬀerent kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that takes place ...
• #### Motion near the transition to complex instability: some examples

(1997)
Article
Open Access
Complex instability is a generic kind of instability in Hamiltonian systems with three degrees of freedom. In this work, some examples of such instability are shown, together with a numerical analysis of the dynamics ...
• #### On the Hamiltonian Andronov-Hopf bifurcation

(2001)
Article
Open Access
• #### On the quantitative estimates of the remainder in normal forms

(2002)
Article
Open Access
We consider an analytic Hamiltonian system with three degrees of freedom and having a family of periodic orbits with a transition stability complex instability. We reduce the Hamiltonian to a normal form around a transition ...


text:
# How many positive integers between 100 and 999 inclusive are odd?

I found the answer to this in a pdf online but don't understand their method: Every 2nd number is odd. 1000 div 2 − 100 div 2 = 500 − 50 = 450

The method I thought I could use didn't work either. If someone can explain to me why my logic doesn't make sense I would appreciate it. I simply created an inequality as such:

100 ≤ 2k+1 ≥ 999

and then solved for k

99 ≤ 2k ≥ 998

49.5 ≤ k ≥ 449.5

since its integers only:

50 ≤ k ≥ 449

and then the numbers in this range would be (449-50)+1=400

• When you divide $998/2$ you should get $499$, not $449$ – Ross Millikan Mar 29 '16 at 3:12
• @RossMillikan d'oh! – ohbrobig Mar 29 '16 at 3:24

$$100 \leq 2k+1 \leq 999$$ $$99 \leq 2k \leq 998$$ $$\frac{99}{2} \leq k \leq 499$$
$\frac{900}{2} = 450$


text:
# Approximation of topological dynamical systems?

I'm trying to find references to approximations of topological dynamical systems in the following sense:

A topological dynamical system $(X, f)$ consists of a topological space (typically compact Hausdorff, or even metrisable) and a continuous $f : X \to X$. We can topologise $C(X, X)$ in various ways, for instance with pointwise / compact-open / uniform topologies. Given such a topology $\tau$, we denote the topological space by $C_\tau(X, X)$.

Are there theorems along the lines of "Given a sequence $(f_n) \subseteq C_\tau(X, X)$, if $(f_n) \to f$ then the dynamical systems $(X, f_n)$ 'approximate' $(X, f)$"? Perhaps $f_n \in C_\tau(X_n, X_n)$ for some subspace $X_n$ and we can consider 'approximations' from subsystems? Or $\{f \in C_\tau(X, X) : (X, f) \text{ has property } P\} \subseteq \overline{\{f \in C_\tau(X, X) : (X, f) \text{ has property } Q\}}$ for dynamical properties $P$ and $Q$?

• You may be looking for the notion of Structural Stability: en.wikipedia.org/wiki/Structural_stability – Lasse Rempe-Gillen Aug 21 '15 at 13:39
• Of course, there are situations where you may not have structural stability, but still have "dynamical convergence" along certain sequence. For example, as pointed out below, the Julia set of a quadratic polynomial is not continuous at c=1/4. However, if you approach through the interval $c\in (0,1/4)$, then the Julia sets will converge (and the maps are even topologically conjugate on the Julia set - where they are topologically just the doubling map on the circle). – Lasse Rempe-Gillen Aug 21 '15 at 13:42
• This seems like the right concept, although I'm probably asking for something weaker than being 'locally topologically conjugate'. But this is a nice direction to look in. Thank you! – Giraffro Aug 21 '15 at 13:55
• Another situation that comes to mind is the convergence of families of unicritical polynomials, parameterised as $z\mapsto (1+z/d)^d+c$, to the exponential family $z\mapsto e^z+c$. The convergence seems "dynamical" in a certain sense. There is an old preprint by Devaney-Goldberg-Hubbard, now published in two parts with additional co-authors. Also, additional results in a similar spirit by Kriete, Krauskopf and others; see arxiv.org/abs/0910.0743 by my former student Helena Mihaljević-Brandt. However, I am not sure this "dynamical convergence" has ever been fully formalised as a concept. – Lasse Rempe-Gillen Aug 24 '15 at 11:11

In general, dynamical properties of dynamical systems are often quite unstable. The Mandelbrot set makes one nice parameter: the base equation $z \mapsto z^2 + c$ is varies continuously with respect to any reasonable topology on continuous functions. The Mandelbrot set contains large open regions (for instance, the central cardoid, or all the disks touching it). In the interior of these, the dynamics is stable: nearby maps are actually topologically conjugate. But outside of these almost all properties of the dynamics will depend quite sensitively on where you are. For instance, look up plots of Julia sets of the function $z \mapsto z^2 + c$ for $c$ close to $1/4$.


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A compromise approach could be to give a problem (with parts), as you might on a guided worksheet. Such as: A function $f$ has domain $(2,4)$. We define $g$ by $g(x) = f(x-2)$. a) Is $g(3)$ defined? b) Is $g(5)$ defined? c) What is the domain of $g$? You might even give a follow up problem (perhaps as extra credit) along the lines of: A function $f$ ...

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I think the given example is highly appropriate. You cannot cover every possible combination of ideas in class. Students display understanding of a concept (rather than "recipe following") by showing the ability to adapt at least a little bit to novel conditions. I think the problem you gave is a great homework problem. I personally like homework to be a ...

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I would frame this issue a little differently than you have. I think it's unreasonable, at least in the context of courses which aren't well into a math major, to ask students to do something they have not been taught to do. That is, the problems on an assessment should be the same as problems they've seen already. The catch is that "the same" is actually ...

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My advice is to minimize the amount of such synthesis required. Don't make it a large fraction of your tests, if at all. Teach the students the methods you expect them to display on the exam. Not something requiring some spark of creativity. Program for success. Creativity is tough in general and even tougher under test conditions. If you push too ...

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What follows is a critique of the paper cited in another answer here: Setren, et. al., "Effects of the Flipped Classroom: Evidence from a Randomized Trial", Annenberg Institute at Brown University (2019). That paper doesn't, in fact, refute the value of the flipped classroom in any way and is an example of poor research design. In fact, it says nothing about ...

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I would like to echo Peter Saveliev's point in the comments: You are asking about "higher results" but you do not specify what qualities you are interested in measuring. Do you care about grades? Performance on standardized tests of content knowledge? Student success in courses which depend on the given course as a prerequisite? Student perception of ...

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Consider a paper from this year: Setren, et. al., "Effects of the Flipped Classroom: Evidence from a Randomized Trial", Annenberg Institute at Brown University (2019). In their introduction, the authors write: Despite the proliferation of the flipped classroom, little well-identified evidence exists on its impact on student learning. In this study a ...

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The Education Endowment Foundation has a 'Teaching and Learning Toolkit' which aims to assess the effectiveness of various teaching methods or types of intervention, including collaborative learning, altering teaching for individual learning styles, and small group learning. It is focused on teaching 5–16 year olds, and not specific to mathematics education, ...

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# Finding the current and voltage for a resistor

#### Davidllerenav

Homework Statement
For the circuit in Figure 1, find the current and voltage at resistor R1 considering that the values of the voltage sources
they are V1 [V] and V2 [V], and the resistance values are R1 [Ω], R2 [Ω] and R3 [Ω] respectively.
Homework Equations
Kirchhoff's Laws:
$\sum_i I_i=0$
$\sum_i V_i=0$

I tried to solve it by loop currents. So on the left mesh the loop current $I_1$ goes clockwise and on the right mesh the loop current $I_2$ goes counterclockwise.

I ended up with the following equations:
1) $V_1-R_1(I_1+I_2)-R_2I_1=0$;
2) $V_2-R_3I_2-R_1(I_1+I_2)=0$.
To find the current on $R_1$, I tried to find $I_1$ and $I_2$. I first solved 1) for $I_1$: $I_1=\frac{V_1-R_1I_2}{R_1+R_2}$. Then I solved 2) for $I_2$ replacing the expression that I found for $I_1$ :

$V_2-R_3I_2-R_1(I_1+I_2)=V_2-I_2(R_3-R_1)-R_1I_1$
$=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)$
$=V_2(R_1+R_2)-I_2(R_3-R_1)(R_1+R_2)-R_1(V_1-R_1I_2)$
$=V_2(R_1+R_2)-I_2[(R_3-R_1)(R_1+R_2)-R_1^2]-R_1V_1$
$I_2=\frac{V_2(R_1+R_2)-R_1V_1}{(R_1+R_2)(R_3-R_1)-R_1^2}$
Finally, I replaced $I_2$ on $I_1$: $I_1=\frac{V_1-R_1\left(\frac{V_2(R_1+R_2)-R_1V_1}{(R_1+R_2)(R_3-R_1)-R_1^2}\right)}{R_1+R_2}$. Am I correct?

I'm not sure how to find the voltage on $R_1$, but I think that I need to find $R_1(I_1+I_2)$, right?

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#### berkeman

Mentor
TBH, I never liked working problems with KVL. Can you try solving it with KCL instead? My KCL equations for this problem are a lot simpler than what you posted...

#### haruspex

Homework Helper
Gold Member
2018 Award
Homework Statement: For the circuit in Figure 1, find the current and voltage at resistor R1 considering that the values of the voltage sources
they are V1 [V] and V2 [V], and the resistance values are R1 [Ω], R2 [Ω] and R3 [Ω] respectively.
Homework Equations: Kirchhoff's Laws:
$\sum_i I_i=0$
$\sum_i V_i=0$

View attachment 252658
I tried to solve it by loop currents. So on the left mesh the loop current $I_1$ goes clockwise and on the right mesh the loop current $I_2$ goes counterclockwise.

View attachment 252659
I ended up with the following equations:
1) $V_1-R_1(I_1+I_2)-R_2I_1=0$;
2) $V_2-R_3I_2-R_1(I_1+I_2)=0$.
To find the current on $R_1$, I tried to find $I_1$ and $I_2$. I first solved 1) for $I_1$: $I_1=\frac{V_1-R_1I_2}{R_1+R_2}$. Then I solved 2) for $I_2$ replacing the expression that I found for $I_1$ :

$V_2-R_3I_2-R_1(I_1+I_2)=V_2-I_2(R_3-R_1)-R_1I_1$
$=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)=V_2-I_2(R_3-R_1)-R_1\left(\frac{V_1-R_1I_2}{R_1+R_2}\right)$
$=V_2(R_1+R_2)-I_2(R_3-R_1)(R_1+R_2)-R_1(V_1-R_1I_2)$
$=V_2(R_1+R_2)-I_2[(R_3-R_1)(R_1+R_2)-R_1^2]-R_1V_1$
$I_2=\frac{V_2(R_1+R_2)-R_1V_1}{(R_1+R_2)(R_3-R_1)-R_1^2}$
Finally, I replaced $I_2$ on $I_1$: $I_1=\frac{V_1-R_1\left(\frac{V_2(R_1+R_2)-R_1V_1}{(R_1+R_2)(R_3-R_1)-R_1^2}\right)}{R_1+R_2}$. Am I correct?

I'm not sure how to find the voltage on $R_1$, but I think that I need to find $R_1(I_1+I_2)$, right?
You have a sign error in $V_2-R_3I_2-R_1(I_1+I_2)=V_2-I_2(R_3-R_1)-R_1I_1$.
An easier way to proceed from (1) and (2) is to find a linear combination of them such that all references to currents are in the form $I_1+I_2$.

#### Davidllerenav

TBH, I never liked working problems with KVL. Can you try solving it with KCL instead? My KCL equations for this problem are a lot simpler than what you posted...
That would just change the $I_1+I_2$, right? It would give an equation of $I_1+I_2=I_3$ I think.

#### Davidllerenav

You have a sign error in $V_2-R_3I_2-R_1(I_1+I_2)=V_2-I_2(R_3-R_1)-R_1I_1$.
An easier way to proceed from (1) and (2) is to find a linear combination of them such that all references to currents are in the form $I_1+I_2$.
You're right, it would be $V_2-R_3I_2-R_1(I_1+I_2)=V_2-I_2(R_3+R_1)-R_1I_1$ , right?

How do I find a linear combination?

Last edited:

#### haruspex

Homework Helper
Gold Member
2018 Award
You're right, it would be $V_2-R_3I_2-R_1(I_1+I_2)=V_2-I_2(R_3+R_1)-R_1I_1$ , right?

How do I find a linear combination?
It just means multiplying each equation by some variable (different ones) and adding them.
In the equations, some terms are already in the form I1+I2, but in one you have an R2I1 and the other an R3I2. What two values do you need to multiply by so that both currents have the same coefficient?

#### Davidllerenav

It just means multiplying each equation by some variable (different ones) and adding them.
In the equations, some terms are already in the form I1+I2, but in one you have an R2I1 and the other an R3I2. What two values do you need to multiply by so that both currents have the same coefficient?
$R_2I_2$ would need to be multiplied by $\left(\frac{1}{R_2}+\frac{I_2}{R_2I_1}\right)$ and $R_3I_2$ by $\left(\frac{1}{R_3}+\frac{I_1}{R_3I_2}\right)$, am I correct?

#### haruspex

Homework Helper
Gold Member
2018 Award
$R_2I_2$ would need to be multiplied by $\left(\frac{1}{R_2}+\frac{I_2}{R_2I_1}\right)$ and $R_3I_2$ by $\left(\frac{1}{R_3}+\frac{I_1}{R_3I_2}\right)$, am I correct?
You mean $R_2I_1$, not $R_2I_2$. But no, it's much simpler than that. A single variable in each case.

#### Davidllerenav

You mean $R_2I_1$, not $R_2I_2$. But no, it's much simpler than that. A single variable in each case.
Yes, sorry for the typo. How would it be? Maybe $R_3$ and $R_2$? then both would have $R_2R_3$.

#### haruspex

Homework Helper
Gold Member
2018 Award
Yes, sorry for the typo. How would it be? Maybe $R_3$ and $R_2$? then both would have $R_2R_3$.
Right. That should give you an equation from which you can find $I_1+I_2$ immediately.

#### Davidllerenav

Right. That should give you an equation from which you can find $I_1+I_2$ immediately.
Ok, then I would have:
$V_1R_3-R_1R_3(I_1+I_2)-R_2R_3I_1=0$
$V_2R_2-R_1R_2(I_1+I_2)-R_2R_3I_2=0$
Then I can subtract, right?

#### haruspex

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Gold Member
2018 Award
Ok, then I would have:
$V_1R_3-R_1R_3(I_1+I_2)-R_2R_3I_1=0$
$V_2R_2-R_1R_2(I_1+I_2)-R_2R_3I_2=0$
Then I can subtract, right?
No, if you subtract you will have a term $R_2R_3(I_1-I_2)$.

#### Davidllerenav

No, if you subtract you will have a term $R_2R_3(I_1-I_2)$.
Then I'll add them, obtaining $V_1R_3+V_2R_2-R_1R_3(I_1+I_2)-R_1R_2(I_1+I_2)-R_2R_3I_1-R_2R_3I_2=0$ thus $V_1R_3+V_2R_2-(I_1+I_2)[R_1R_3+R_1R_2]-R_2R_3(I1+I_2)=0$
$\Rightarrow V_1R_3+V_2R_2-(I_1+I_2)[R_1R_3+R_1R_2+R_2R_3]=0$. Am I correct?

#### haruspex

Homework Helper
Gold Member
2018 Award
Then I'll add them, obtaining $V_1R_3+V_2R_2-R_1R_3(I_1+I_2)-R_1R_2(I_1+I_2)-R_2R_3I_1-R_2R_3I_2=0$ thus $V_1R_3+V_2R_2-(I_1+I_2)[R_1R_3+R_1R_2]-R_2R_3(I1+I_2)=0$
$\Rightarrow V_1R_3+V_2R_2-(I_1+I_2)[R_1R_3+R_1R_2+R_2R_3]=0$. Am I correct?
That looks better.

#### Davidllerenav

+I
That looks better.
Then I only have to solve for $I_1+I_2$ and to find the voltage on $R_1$ I just multiply $I_1+I_2$ times $R_1$, right?

#### haruspex

Homework Helper
Gold Member
2018 Award
+I
Then I only have to solve for $I_1+I_2$ and to find the voltage on $R_1$ I just multiply $I_1+I_2$ times $R_1$, right?
Yes.

#### Davidllerenav

Thank you. I guess that if I continued simplifying my original expressions for $I_1$ and $I_2$ nd adding them I would arrive to the same answer, but this helped me a lot. Thanks!