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# All Questions

108 views

### Intersecting RegionPlots

Does the Mathematica graphics system have any concept of intersecting regionplots? I've found about intersecting graphics but not regionplots so far. For example, if I want to show the intersection of ...
99 views

### Curl expression does not evaluate [closed]

I am using Mathematica 8 on OS X. I am trying to compute the cylindrical curl of a symbolic vector following Mathematica documentation. ...
152 views

### Can not generate graph using smooth kernel distribution

I am a newbie to Mathematica so this question might looks silly. I want to generate a smooth kernel distribution graph for this set of data. The data I uploaded has two columns, the first column is ...
73 views

### Trouble summing over many indices [on hold]

I'm trying to solve a problem by using Sum, but as I have many indices (14) in the summation, the running time is awful. My program looks like this: ...
63 views

### Adding ticks to PolarPlot [closed]

I want to add some number ticks to a unit circle in PolarPlot like the one attached, but how can I do it? ...
174 views

### Maximization in Mathematica using matrix and vector notation

I am trying to solve a simple maximization problem in Mathematica, using matrix and vector notation. ...
103 views

### How to analyze all the data in a directory

I am trying to import, analyze, and save all the .txt files in a specific directory. I can do this with a single .txt file using the following code: ...
41 views

### Calculations after NumberForm [duplicate]

As specified in documentation center the functions AccountingForm, NumberForm, etc. are useful only to display numbers in ...
103 views

### Integrate does not work on my expression

I am trying to integrate a simple equation. The code is as follows ...
94 views

### How can I speed up this code with 4 variables sum

I have to make a sum over 4 variables. My code is very very slow. I want to know how to speed up this code. This problem is related to but different from one previous problem. Any help or suggestion ...
162 views

### Nested list from a flat folder list

I have a list of folders and files in my current working directory as a result of a built-in function FileNames: FileNames["*", "", Infinity]]. It looks like this: ...
108 views

### Typesetting in Mathematica

Are there typesetting environments/boxes or something similar in Mathematica? I want to write some documentation for a notebook using text cells and I would like to have a brief description of my ...
46 views

### Syntax highlighting in text cells

I want to write some documentation for a notebook using text cells. Now, I can select parts of the text within such a text cell and set its style to "Input". This does change its font, but there is no ...
43 views

### Export followed by Import producing “0.e1” numbers

I had a large table of data that I exported, and then when it had completed, I exported it to a file. (Just in the form Export["filename.dat", data], no specified ...
40 views

### How to assign values to constants inserted in a solution by DSolve [on hold]

I am trying to solve first order ODE with DSolve, but it didn't pan out. How can I solve for the inserted variable K[1]? Your ...
126 views

### Determining FinancialData for Countries

How do I determine the FinancialData for Australia. For instance the one for USA follows as: FinancialData["NYSE", "Members"] ...
37 views

### In NMinimize, how can I efficiently handle NIntegrate errors for non-integrable functions?

I am using NMinimize to find parameters that minimise the integral of a function in the least squares sense: ...
65 views

### How to define this function in Mathematica? [closed]

Q set of rational, Q' set of irrational Determine $F+G$ , where $F(x) = x^2$ when $x \in Q$ and $-x^2$ when $x \in Q'$ $G(x)= {(2^{1/2},4),(7,0),(3,8)}$
110 views

### Show and hide various pieces of information in a Manipulate

I wanted to change the visibility of a function and its additional informations within a plot through the help of a checkbox, as shown in that post: Toggle visibility of elements in a plot But ...
68 views

### Integrate works but NIntegrate doesn't

I have a complicated integral function that I'm trying to evaluate, which will have to be done numerically. I wanted to try a simpler one first, one that has an analytic form, because I could check ...
54 views

### How to generate a list of outputs for function f(x,y) for a set of known (x,y) values [closed]

Say I want a list of outputs the function f(x,y)=x+y will give for x,y values of (1,5),(5,3),(0,2),(1,1),(2,6). How do I do that? I know this must be pretty straight forward. I just can't figure it ...
92 views

### Simplest identity that cannot be proved by FullSimplify [closed]

Apparently, FullSimplify is not able to prove all true identities, but usually it does pretty well when it comes to elementary functions. I'd like to know what are ...
33 views

### Formatting the output of SubstitutionSystem to create a Cantor set [closed]

I want to create a Cantor set using SubstitutionSystem, and plot it appropriately. This is the best I've been able to do so far: ...
73 views

### How to show the direction of highlighted path in graph

I am trying to show the direction of FindHamiltonianPath on the Graph. ...
314 views

### Finding all of the words containing certain letters

Is there a fast method to find all of the words containing the letters: {a, e, i, o, u, y}? I created an algorithm, but it takes more than 11 seconds to find all of ...
46 views

### Solving an ODE using shooting method [duplicate]

I have been trying to solve the following nonlinear ordinary differential equation: $$-\Phi''-\frac{3}{r}\Phi'+\Phi-\frac{3}{2}\Phi^{2}+\frac{\alpha}{2}\Phi^{3}=0$$ with boundary conditions ...
47 views

### Dynamic update of displayed quantities on a Plot within manipulate

I'll admit that I only skimmed some other threads about this. They seem rather complicated in contrast to my simple problem. I have two lines in space defined by a two-by-two coefficient matrix. I ...
53 views

### Method of characteristics, recursion formulas

I'm helping a friend out with a problem. He needs to plot out the solution to a water surge in a pipeline. I have all the parameters set and I've put the equations in Mathematica, but I can't get the ...
25 views

### How can I sketch a Plot using direction field? [closed]

I can't plot a graph of first order initial value problem using direction field
109 views

### Plotting a simple graph with more than 1 variables

Here's what I did: Plot[(Q E^(-(x^2/(4 t))))/(2 Sqrt[π t]), {x, 0, 10}] I want only x to vary. But Mathematica is giving me a blank plot. It's very strange. I ...
52 views

### How to speed up a high dimensional numerical integration of Gaussian-form integrand

My integral has the following form: $$\int \sin(\theta) d\theta d\phi\left| \int d^3p_1 d^3p_2d^3p_3 \frac{f(p,\theta,\phi)}{g(p,\theta,\phi)}e^{-\frac{1}{2}(p^\mathrm{T} .A.p)}\right |^2,$$ where ...
122 views

### Form of citation for data via Wolfram Mathematica

What citation or reference should I adhere to if I'm using curated data obtained via Mathematica? I'm talking about data, or eg. plots thereof, obtained using functions such as SatelliteData, ...
111 views

### Memory usage of compiled function: C vs WVM

Why does generating an array in "C"-compiled function uses 50% more memory than doing same thing in "WVM"-compiled function? ...
33 views

### Solving a differential equation with a complex (independent) varible

I'm trying to plug in complex values in the numerical solutions of ODEs without success. For instance y[I] /. NDSolve[{y'[x] == 0, y[0] == 3}, y, {x, -10, 10}] ...
29 views

### Using Select function to select elements based on last element of nested list [closed]

Good morning, I have a list with each element containing a tuple and a single. It is of this form: ...
84 views

Given that I have the following array lst = Array[Subscript[P, #1, #2] &, {5, 5}, {0, 0}] $\left( \begin{array}{ccccc} P_{0,0} & P_{0,1} & P_{0,2} ... 1answer 89 views ### Vertex Order of Graph Given a graph, I want to find the shortest distance between each pair of vertices: ... 0answers 86 views ### What is the best way to transfer arbitrary-precision reals using WSTP? I'm developing a Mathematica package, inside of which there is a number-crunching function. It uses a list of arbitrary-precision numbers as input and it takes a lot of time to do its job. So I've ... 0answers 73 views ### Evaluating a derivative at a specific point [closed] I'm quite new to Mathematica (and stackexchange) so I'm getting stuck on something simple. I have the expression$$\phi_{n}(z,s)=q^n\left[-\frac{e^{-sb}z \left((1-q)^{\lfloor \gamma\rfloor} ... 2answers 45 views ### Custom multivariate discrete distribution [closed] This is related to How to define an n-variate empirical distribution function probability for any n? and RandomVariate from 2-dimensional probability distribution but I don't think neither questions ... 2answers 54 views ### How to use the {{ lhs == rhs, ineqs }, … } form of Exclusions? I am trying to make use of this option: To plot the principal portion of Tan: ... 0answers 44 views ### Plotting roots for a singular numerically-only integrable equation In my problem I need to plot roots with respect to a parameter. But the condition for root is det(Function)=0. This function cannot be solved symbolically. So I need to put in a parameter value and ... 0answers 42 views ### NDSolveValue returns NDSolveValue::ndnum [closed] How do I solve this problem? ... 2answers 94 views ### Coloring image components according to their area I have a rasterized image img that is created by this code: ... 1answer 109 views ### How can I temporarily suspend the Combinatorica package? Sometimes after I load the Combinatorica package by entering Needs["Combinatorica"], I need to use Mathematica for something like: ... 0answers 29 views ### Loading a notebook to a website When I upload a Mathematica notebook to my website, when I click on the link, it opens, but as text, not a Mathematica notebook. At some sites, when I click on a Mathematica notebook, the notebook ... 1answer 101 views ### How can I make angle brackets function like lists? Is there a way to make <a,b,c> behave like {a,b,c}, but remain displayed as ... 1answer 52 views ### Replace symbol with numerical value How do I get the same result, i.e., how can I get Mathematica to replace a with 1? ... 1answer 74 views ### User defined packages don't load with$\mathtt{Needs}$or$\mathtt{Get}\$

As an introduction to packages in Mathematica I'm following this short guide by Leonid Shifrin, and tried using the sample package provided in that answer. However, both ...


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# Half wave rectifier at high frequency and small amplitude ac signal

This is the picture taken from my oscilloscope where I was just checking how a standard half wave rectifier properties. The yellow one is input signal of frequency 20.5KHz and peak to peak value is 100mV(less than 700mV. I know!). In this case the whole of the input signal passes through, there is no rectification. I am trying to find explanations for this.

I have observed that the output signal lags behind the input, so it must be capacitive effect. But I want to get a clearer idea of how the circuit works in this region. What is the source of the capacitance? Is it only the depletion region? How is the negative portion also passing through diode?

• I've tried your circuit in LTspice. When I am giving the input amplitude less than threshold voltage of the diode then the diode shows a capacitave effect like you had in your oscilloscope. But when I give the input voltage higher than the threshold voltage, I've got the half rectified wave. So I think this capacitave effect of diode is due to space charge region of the diode. The space charge region of the diode acts like the dielectric of a capacitor. I think this is the cause...... – Rajesh Sardar Oct 15 '15 at 14:49

I've developed a half wave rectifier circuit in LTspice and I've got the result as shown in figures. In first figure, the input voltage(V(n001)) is 400mV(frequency 20.5kHz), which is grater than the threshold voltage of the diode and the output voltage gives a half rectified pattern. But when I give the input voltage value 200mV, which is less than the threshold voltage, I get a sine wave which indicates the capacitave effect of the diode.

Now when the input voltage is less than threshold voltage of the diode then majority carriers of the diode can not cross the potential barrier of the diode. As a result the space charge region of the diode here acts like a dielectric of a capacitor where the charge in the both side of this region acts like a charge on capacitor plates.So in low input voltage(below threshold voltage) diode acts like a capacitor.

When the frequency of the input voltage is very small then we get a very low amplitude of output voltage as shown in fig. below.

we know that the reactantce of a capacitor is $\frac{1}{j \omega}$ i.e. at low frequency the reactance of the diode capacitance becomes very high. As a result we get a very low output amplitude in low frequency.

And the peak of the output voltage pattern at low frequency due to quantum tunneling effect of the semiconductor carriers. When the positive peak of the input voltage comes across the diode, some carriers get some energy form that and cross the potential barrier. Of course those carriers had energy less than the potential barrier.

• To be exact what I want is whether I can model the diode below the threshold voltage so as to explain this effect and why it happens at high frequencies. – Ari Oct 15 '15 at 19:05
• I think I've explained why it is happen below threshold voltage. Below threshold voltage no majority carrier pass through the potential barrier or space charge region. So here the space charge region acts like the dielectric media between two capacitor plate. For this the diode becomes like a capacitor. – Rajesh Sardar Oct 15 '15 at 19:15
• And I've check the circuit in low frequency. Are you saying that the circuit does not give the capacitance effect at low frequency and at input amplitude less than threshold voltage? – Rajesh Sardar Oct 15 '15 at 19:18
• Yes. For low frequency I am getting no signal at all. – Ari Oct 16 '15 at 3:39
• I've explained the low frequency effect in my ans by editing. Please read the ans again. – Rajesh Sardar Oct 16 '15 at 8:37

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## My Own Alternative Medicine

From: andrew cooke <andrew@...>

Date: Wed, 26 Dec 2012 23:26:07 -0300

I've been thinking a lot recently about alternative medicine.  For
understandable reasons, I imagine.

In theory, I am quite supportive of the idea.  In practice, I also found some
serious objections:

* It takes time.  When you're ill, quite a bit of time is consumed doing
stuff related to your illness.  Going to the doctor.  Going to the clinic
for medication.  Taking medication at home (the semi-nightly ritual of
injections).  Shopping for medication.  Sleeping.  Just sitting there
feeling crap.  It adds up.  So time becomes precious.  Being able to have a
normal, worry-free day is a pleasure in itself.  And chasing alternative
medicine eats into that.

* It's largely out of context.  Take, say, acupuncture.  Or rather, reverse
that.  Take "western medicine" in the context of a culture where
acupuncture is the norm (assume such a thing exists).  Who is your "western
medicine guru"?  Is it a doctor from a nice hospital in London, or some
random dropout from London who has hitched his way to acupuncture-land and
can't see how else to make money?  Now switch back and qestion the
credentials of your acupuncture specialist, in this western-medicine
context.  It's worrying.

* It's hard to trust people.  What I'd *really* like is someone who says
"look, I am buggered if I understand why, but I seem to make people
better".  Instead, you have to trust someone who "knows" that it's all down
to magnets and inner energy.  It's not that they are wrong - they may be
right (although, damn, if it's magnets I've spent some quality hours in MRI
recently...).  It's that they are *convinced* they are right without the
kind of evidence that I find convincing.  What kind of person is like that?
Not the kind if person I instinctively trust.  (Incidentally, it would be
neat to mine hospital records and see if you could find nurses who just
"naturally" - unknowngly - were curing people...).

So.  What to do?  What to do if you think western medicine is missing
something?  And if that something is kind-of present in "alternative
medicine", but not in a way that you find appealing (or efficient)?

Well, my idea was to invent my own alternative.

That's not so dumb, right?  After all, placebos work whether you know they are
pacebos or not.  I've been told (actually, from what I've read - I bought a
collection of papers from a conference on placebos a while back - placebos are
actually horribly complex and poorly understood, so almost any general
statement about them is unlike to be correct).

So what should an alternative do?

Mine is based on coffee, chocolate, and meditation.  At 3pm you make a good
cup of coffee and fetch a line of chocolate from the fridge (the chocolate
should be stored in the same box as the Betaferon).  You put a Chris Watson
album (nature sounds) on shuffle.  And you sit in a comfortable spot.

You have a sip of coffee.  Nibble some chocolate.  Start to think about
thinking.  Listen to what's in your head, and put it aside.  Have some more
the light coming through the window.  Taste the chocolate.  Go back to your
head.  Place any ideas in a neat row of boxes for later.  Feel good.

That's the 3pm part.  There's also the jogging, but whether that is part of
the medicine or something else I'm not yet sure.  Maybe the medicine becomes a
lifestyle.  It's a work in progress.

Andrew

### Light and Tea

From: andrew cooke <andrew@...>

Date: Fri, 28 Dec 2012 13:39:05 -0300

A nice cup of tea can be substituted for the coffee (especially when on days
when my head feeks like it's gonig to explode).

Light is important.  Sunlight on leaves.  Maybe this is a religion.  We
worship dappled light.

Andrew


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Windows – Samsung PRO 950 SSD recognized as SATA instead of PCIe in Windows 10

hard drivepci-expresssatassdwindows 10

I have a new ASUS G752VL-DH71 laptop and I installed a SAMSUNG 950 PRO M.2 512GB PCI-Express 3.0 x4 SSD in it. The hard drive is totally usable in Windows (as drive G:) but there are three issues:

1. The write performance of the drive according to Anvil Utilities is only 54 MB/s, which is much lower than it should be.
2. I tried to install the official Samsung NVMe Driver v1.1 from this page but the installer gives me the error message "Samsung NVM Express Device is not connected. Connect the device and try again."
3. The drive is not recognized by my Windows 10 USB installation media, which is a problem because I want to eventually install Windows 10 on the SSD.

I think the root problem that is causing all of these issues might be that there is some virtualization/compatibility layer/driver running that is taking the drive and making it look like a SATA drive instead of a PCIe drive. The Device Manager shows that my SSD is being controlled by the Intel Chipset SATA Raid Controller (iastora.sys):

I looked around in the BIOS of this laptop, which is version 208, and tried changing various options but that did not help.

Question

How can I make Windows recognize my SSD as an NVMe device instead of a SATA device?

Update 1

By pressing escape while the laptop starts up, I can access the BIOS. Here is a screenshot of the Advanced -> SATA Configuration screen. It shows the spinning platter HDD that came with the laptop and it shows the DVD drive, but it doesn't show the SSD. All of that seems good to me. There are no settings here that I can actually change.

• The key is that you need to disable RAID entirely in the BIOS. If your BIOS is old, it will not support disabling RAID, and you will need to update it first.

Here is a screenshot of my successful SATA Configuration settings in the BIOS:

1. Press Escape while booting to get into the BIOS setup.
2. Use the arrow keys and the enter key to navigate to Advanced -> SATA Configuration.
3. Change SATA Mode Selection from RAID to AHCI. If you have an old version of the BIOS, the only option you will see is RAID and you will need to update your BIOS as described below.
5. You should now be able to install the Samsung NVMe Driver. After doing that and restarting the laptop, I got 2000 MB/s reads and 1400 MB/s writes on the drive.

Updating the BIOS

My laptop shipped with version 208 of the BIOS. I updated to version 219 by following this procedure:

1. Go to http://www.asus.com/support and enter your laptop's model number in the search box to find the support page for your laptop.
2. Go the the "Drivers and Tools" page for your laptop.
3. There is a dropdown box to select your OS. Do NOT select Windows 10. Instead, select Others.
5. Open the ZIP file you downloaded and extract the file inside it. This file is the code that runs on your BIOS. For my ASUS G752VL-DH71 laptop, the file was named G752VLAS.219.
6. Save the file to the root directory of your hard drive or the root directory of a USB flash drive to make it easier to find the file in the next steps.
7. Restart your computer while pressing Escape to get to the BIOS Setup.
8. In the Advanced screen, select Start Easy Flash.
9. Navigate through your file systems to find the BIOS file you saved in steps 5 and 6.
10. Select the file and confirm that you want to update your BIOS.


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# Diagnostics for generalized linear (mixed) models (specifically residuals)

I am currently struggling with finding the right model for difficult count data (dependent variable). I have tried various different models (mixed effects models are necessary for my kind of data) such as lmer and lme4 (with a log transform) as well as generalized linear mixed effects models with various families such as Gaussian or negative binomial.

However, I am quite unsure on how to correctly diagnose the resulting fits. I found a lot of different opinions on that topic on the Web. I think diagnostics on linear (mixed) regression are quite straight-forward. You can go ahead and analyse the residuals (normality) as well as study heteroscedasticity by plotting fitted values compared to residuals.

However, how do you properly do that for the generalized version? Let us focus on a negative binomial (mixed) regression for now. I have seen quite opposing statements regarding the residuals here:

1. In Checking residuals for normality in generalised linear models it is pointed out in the first answer that the plain residuals are not normally distributed for a GLM; I think this is clear. However, then it is pointed out that Pearson and deviance residuals are also not supposed to be normal. Yet, the second answer states that deviance residuals should be normally distributed (combined with a reference).

2. That deviance residuals should be normally distributed is hinted at in the documentation for ?glm.diag.plots (from R's boot package), though.

3. In this blog post, the author first studied normality of what I assume are Pearson residuals for a NB mixed-effects regression model. As expected (in my honest opinion) the residuals did not show to be normal and the author assumed this model to be a bad fit. However, as stated in the comments, the residuals should be distributed according to a negative binomial distribution. In my opinion, this comes closest to the truth as GLM residuals can have other distributions than the normal one. Is this correct? How to check for things like heteroscedasticity here?

4. The last point (plotting residuals against quantiles of the estimated distribution) is emphasized in Ben & Yohai (2004). Currently, this seems the way to go for me.

In a nutshell: How do you properly study the model fits of generalized linear (mixed) regression models specifically with a focus on residuals?

• Residuals for GLMs aren't in general normal (cf here), but note that there are lots of kinds of residuals for GLMs. Eg, glm.diag.plots says it's for jackknifed deviance residual (I suspect that distinction is important). Also, I gather you have count data; you might want to focus on that fact. Eg, counts are supposed (in some sense) to be heteroscedastic. Diagnostic plots for count regression should be helpful for you (although it doesn't address the mixed effects aspect). – gung Dec 9 '15 at 17:18
• I am familiar with the post you mentioned. However, there is also a statement that suggests that (deviance) residuals should be normal "we see very large residuals and a substantial deviance of the deviance residuals from the normal (all speaking against the Poisson)". – fsociety Dec 9 '15 at 18:06

This answer is not based on my knowledge but rather quotes what Bolker et al. (2009) wrote in an influential paper in the journal Trends in Ecology and Evolution. Since the article is not open access (although searching for it on Google scholar may prove successful, I thought I cite important passages that may be helpful to address parts of the questions. So again, it's not what I came up with myself but I think it represents the best condensed information on GLMMs (inlcuding diagnostics) out there in a very straight forward and easy to understand style of writing. If by any means this answer is not suitable for whatever reason, I will simply delete it. Things that I find useful with respect to questions regarding diagnostics are highlighted in bold.

Page 127:

Researchers faced with nonnormal data often try shortcuts such as transforming data to achieve normality and homogeneity of variance, using nonparametric tests or relying on the robustness of classical ANOVA to nonnormality for balanced designs [15]. They might ignore random effects altogether (thus committing pseudoreplication) or treat them as fixed factors [16]. However, such shortcuts can fail (e.g. count data with many zero values cannot be made normal by transformation). Even when they succeed, they might violate statistical assumptions (even nonparametric tests make assumptions, e.g. of homogeneity of variance across groups) or limit the scope of inference (one cannot extrapolate estimates of fixed effects to new groups). Instead of shoehorning their data into classical statistical frameworks, researchers should use statistical approaches that match their data. Generalized linear mixed models (GLMMs) combine the properties of two statistical frameworks that are widely used in ecology and evolution, linear mixed models (which incorporate random effects) and generalized linear models (which handle nonnormal data by using link functions and exponential family [e.g. normal, Poisson or binomial] distributions). GLMMs are the best tool for analyzing nonnormal data that involve random effects: all one has to do, in principle, is specify a distribution, link function and structure of the random effects.

Page 129, Box 1:

The residuals indicated overdispersion, so we refitted the data with a quasi-Poisson model. Despite the large estimated scale parameter (10.8), exploratory graphs found no evidence of outliers at the level of individuals, genotypes or populations. We used quasi-AIC (QAIC), using one degree of freedom for random effects [49], for randomeffect and then for fixed-effect model selection.

Page 133, Box 4:

Here we outline a general framework for constructing a full (most complex) model, the first step in GLMM analysis. Following this process, one can then evaluate parameters and compare submodels as described in the main text and in Figure 1.

1. Specify fixed (treatments or covariates) and random effects (experimental, spatial or temporal blocks, individuals, etc.). Include only important interactions. Restrict the model a priori to a feasible level of complexity, based on rules of thumb (>5–6 random-effect levels per random effect and >10–20 samples per treatment level or experimental unit) and knowledge of adequate sample sizes gained from previous studies [64,65].

2. Choose an error distribution and link function (e.g. Poisson distribution and log link for count data, binomial distribution and logit link for proportion data).

3. Graphical checking: are variances of data (transformed by the link function) homogeneous across categories? Are responses of transformed data linear with respect to continuous predictors? Are there outlier individuals or groups? Do distributions within groups match the assumed distribution?

4. Fit fixed-effect GLMs both to the full (pooled) data set and within each level of the random factors [28,50]. Estimated parameters should be approximately normally distributed across groups (group-level parameters can have large uncertainties, especially for groups with small sample sizes). Adjust model as necessary (e.g. change link function or add covariates).

5. Fit the full GLMM. Insufficient computer memory o r too slow: reduce model complexity. If estimation succeeds on a subset of the data, try a more efficient estimation algorithm (e.g. PQL if appropriate). Failure to converge (warnings or errors): reduce model complexity or change optimization settings (make sure the resulting answers make sense). Try other estimation algorithms. Zero variance components or singularity (warnings or errors): check that the model is properly defined and identifiable (i.e. all components can theoretically be estimated). Reduce model complexity. Adding information to the model (additional covariates, or new groupings for random effects) can alleviate problems, as will centering continuous covariates by subtracting their mean [50]. If necessary, eliminate random effects from the full model, dropping (i) terms of less intrinsic biological interest, (ii) terms with very small estimated variances and/or large uncertainty, or (iii) interaction terms. (Convergence errors or zero variances could indicate insufficient data.)

6. Recheck assumptions for the final model (as in step 3) and check that parameter estimates and confidence intervals are reasonable (gigantic confidence intervals could indicate fitting problems). The magnitude of the standardized residuals should be independent of the fitted values. Assess overdispersion (the sum of the squared Pearson residuals should be $\chi^2$ distributed [66,67]). If necessary, change distributions or estimate a scale parameter. Check that a full model that includes dropped random effects with small standard deviations gives similar results to the final model. If different models lead to substantially different parameter estimates, consider model averaging.

Residuals plots should be used to assess overdispersion and transformed variances should be homogeneous across categories. Nowhere in the article was mentioned that residuals are supposed to be normally distributed.

I think the reason why there are contrasting statements reflects that GLMMs (page 127-128)...

...are surprisingly challenging to use even for statisticians. Although several software packages can handle GLMMs (Table 1), few ecologists and evolutionary biologists are aware of the range of options or of the possible pitfalls. In reviewing papers in ecology and evolution since 2005 found by Google Scholar, 311 out of 537 GLMM analyses (58%) used these tools inappropriately in some way (see online supplementary material).

And here are a few full worked examples using GLMMs including diagnostics.

I realize that this answer is more like a comment and should be treated as such. But the comment section doesn't allow me to add such a long comment. Also since I believe this paper is of value for this discussion (but unfortunately behind a pay-wall), I thought it would be useful to quote important passages here.

Cited papers:

[15] - G.P. Quinn, M.J. Keough (2002): Experimental Design and Data Analysis for Biologists, Cambridge University Press.

[16] - M.J. Crawley (2002): Statistical Computing: An Introduction to Data Analysis Using S-PLUS, John Wiley & Sons.

[28] - J.C. Pinheiro, D.M. Bates (2000): Mixed-Effects Models in S and S-PLUS, Springer.

[49] - F. Vaida, S. Blanchard (2005): Conditional Akaike information for mixed-effects models. Biometrika, 92, pp. 351–370.

[50] - A. Gelman, J. Hill (2006): Data Analysis Using Regression and Multilevel/Hierarchical Models, Cambridge University Press.

[64] - N.J. Gotelli, A.M. Ellison (2004): A Primer of Ecological Statistics, Sinauer Associates.

[65] - F.J. Harrell (2001): Regression Modeling Strategies, Springer.

[66] - J.K. Lindsey (1997): Applying Generalized Linear Models, Springer.

[67] - W. Venables, B.D. Ripley (2002): Modern Applied Statistics with S, Springer.

• Thanks, that is indeed helpful, I knew about the coding examples of Bolker, but not the actual paper somehow. What I still wonder though is how graphical checking applies to very large-scale data with thousands of groups. The few papers (such as that one) that try give some guidelines on how to properly check your models all only apply to very small-scale data. Then, it is much easier to pick e.g., the groups and visualize something. I really think that a good scientific contribution can be made if someone goes through a more complex example in future. – fsociety Dec 16 '15 at 19:42
• I'm glad this was useful! I think the presented examples are already quite complex (at least to me). I guess the bigger problem is that larger datasets and more complex models may become computationally infeasible as is mentioned in the text: "[...] to find ML estimates, one must integrate likelihoods over all possible values of the random effects. For GLMMs this calculation is at best slow, and at worst (e.g. for large numbers of random effects) computationally infeasible." What I find amazing though, and what should be kept in mind, is that we are using tools that are under active research! – Stefan Dec 16 '15 at 20:02

This is an old question, but I thought it would be useful to add that option 4 suggested by the OP is now available in the DHARMa R package (available from CRAN, see here).

The package makes the visual residual checks suggested by the accepted answer a lot more reliable / easy.

From the package description:

The DHARMa package uses a simulation-based approach to create readily interpretable scaled residuals from fitted generalized linear mixed models. Currently supported are all 'merMod' classes from 'lme4' ('lmerMod', 'glmerMod'), 'glm' (including 'negbin' from 'MASS', but excluding quasi-distributions) and 'lm' model classes. Alternatively, externally created simulations, e.g. posterior predictive simulations from Bayesian software such as 'JAGS', 'STAN', or 'BUGS' can be processed as well. The resulting residuals are standardized to values between 0 and 1 and can be interpreted as intuitively as residuals from a linear regression. The package also provides a number of plot and test functions for typical model mispecification problem, such as over/underdispersion, zero-inflation, and spatial / temporal autocorrelation.

• Very good addition to this thread! – Stefan Feb 15 '17 at 21:05


text:
# Journal article

## On the Nature of Angular Momentum Transport in Nonradiative Accretion Flows

Abstract:

The principles underlying a proposed class of black hole accretion models are examined. The flows are generally referred to as convection-dominated,'' and are characterized by inward transport of angular momentum by thermal convection and outward viscous transport, vanishing mass accretion, and vanishing local energy dissipation. We examine the viability of these ideas by explicitly calculating the leading order angular momentum transport of axisymmetric modes in magnetized, differentially ...

Publication status:
Published

Publisher copy:
10.1086/340767

### Authors

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Institution:
University of Oxford
Division:
MPLS
Department:
Physics
Sub department:
Astrophysics
Role:
Author
Journal:
ASTROPHYSICAL JOURNAL
Volume:
573
Issue:
2
Pages:
749-753
Publication date:
2002-01-26
DOI:
EISSN:
1538-4357
ISSN:
0004-637X
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Pubs id:
pubs:333896
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uuid:637faf2c-d5ea-40c1-8ce6-0c7c4c5d415f
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2014-12-26

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text:
# Induction proof: sum of binomial coefficients

Prove by mathematical induction for all natural numbers $n$:

$$\sum_{k=0}^{n} \binom{n}{k}=2^n$$

thus is it sufficient to show that (but I think I made a mistake) thus how to do it properly?

$$2^n+\binom{n+1}{n+1}=2^{n+1}$$

• The problem is that the first part of your sum does not add up to $2^n$. – user10444 Feb 15 '15 at 19:23
• how come? the first part is copied directly from the question – kurkowski Feb 15 '15 at 19:25
• See the answer below – user10444 Feb 15 '15 at 19:26
• The assertion that $2^n + \binom{n + 1}{n + 1} = 2^{n + 1}$ is incorrect: $2^n + \binom{n + 1}{n + 1} = 2^n + 1$. – N. F. Taussig Feb 15 '15 at 19:28

## 4 Answers

Not quite, because ${n+1\choose n+1}=1$. Combinatorically, is it easy to see that ${n+1\choose k}={n\choose k}+{n\choose k-1}$ (we either include the last element or we don't). From here on the proof is easy because (using the induction hypothesis):

$\sum_{k=0}^{n+1} {n+1\choose k}=\sum_{k=0}^{n+1} {n\choose k}+{n\choose k-1}=\sum_{k=0}^{n+1} {n\choose k}+\sum_{k=0}^{n+1} {n\choose k-1}=2^n+2^n=2^{n+1}$

Because ${n\choose -1}={n\choose n+1}=0$ (it's impossible to choose -1 or n+1 elements out of n).

Note that $$\sum_{k=0}^{n+1} \begin{pmatrix} n+1 \\ k\end{pmatrix} = \begin{pmatrix} n+1 \\ n+1\end{pmatrix} + \underbrace{\sum_{k=0}^n \begin{pmatrix} n+1 \\ k\end{pmatrix}}_{\neq 2^n}.$$

Alternatively, note that $\begin{pmatrix} n+1 \\ n+1\end{pmatrix} = 1$, and that $2^{n+1} \neq 2^n + 1$.

• thus how to approach the inductional proof? – kurkowski Feb 15 '15 at 19:26

See the answer of Arkamis to find your mistake.

Under the convention that $\binom{n}{k}=0$ if $k\notin\left\{ 0,\dots,n\right\}$ we have:

$$\sum_{k\in\mathbb{Z}}\binom{n+1}{k}=\sum_{k\in\mathbb{Z}}\left[\binom{n}{k-1}+\binom{n}{k}\right]=\sum_{k\in\mathbb{Z}}\binom{n}{k-1}+\sum_{k\in\mathbb{Z}}\binom{n}{k}=2^{n}+2^{n}=2^{n+1}$$

Not quite because you are missing the $n+1$ in all the sum. Anyway $2^n+1\neq 2^{n+1}$. The key is to use $\binom{n+1}{k+1}=\binom{n}{k+1}+\binom{n}{k}$.


text:
## 17Calculus - Calculus 1 - Practice Exam 3 (Semester B)

### Practice

Calculus 1 Practice

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Practice Exams

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### Articles

This is the final exam for first semester single variable calculus. The topics covered are integration by substitution, both indefinite and definite, solution of separable differential equations, optimization, area between curves, exponential growth/decay and volumes of revolution using both the disc/washer method and the shell/cylinder method.

### Practice Exam Tips

Each exam page contains a full exam with detailed solutions. Most of these are actual exams from previous semesters used in college courses. You may use these as practice problems or as practice exams. Here are some suggestions on how to use these to help you prepare for your exams.

- Set aside a chunk of full, uninterrupted time, usually an hour to two, to work each exam.
- Go to a quiet place where you will not be interrupted that duplicates your exam situation as closely as possible.
- Use the same materials that you are allowed in your exam (unless the instructions with these exams are more strict).
- Use your calculator as little as possible except for graphing and checking your calculations.
- Work the entire exam before checking any solutions.
- After checking your work, rework any problems you missed and go to the 17calculus page discussing the material to perfect your skills.
- Work as many practice exams as you have time for. This will give you practice in important techniques, experience in different types of exam problems that you may see on your own exam and help you understand the material better by showing you what you need to study.

IMPORTANT -
Exams can cover only so much material. Instructors will sometimes change exams from one semester to the next to adapt an exam to each class depending on how the class performs during the semester while they are learning the material. So just because you do well (or not) on these practice exams, does not necessarily mean you will do the same on your exam. Your instructor may ask completely different questions from these. That is why working lots of practice problems will prepare you better than working just one or two practice exams.
Calculus is not something that can be learned by reading. You have to work problems on your own and struggle through the material to really know calculus, do well on your exam and be able to use it in the future.

Exam Details

Time

2.5 hours

Questions

10

Total Points

135

Tools

Calculator

not allowed

Formula Sheet(s)

one page

Other Tools

ruler

Instructions:
- For each problem, correct answers are worth 1 point. The remaining points are earned by showing calculations and giving reasoning that justify your conclusions.
- Correct notation counts (i.e. points will be taken off for incorrect notation).

[10 points] Evaluate $$\displaystyle{ \int{ \frac{e^{1/x}}{x^2} ~dx } }$$

Problem Statement

[10 points] Evaluate $$\displaystyle{ \int{ \frac{e^{1/x}}{x^2} ~dx } }$$

$$\displaystyle{ -e^{1/x} + C }$$

Problem Statement

[10 points] Evaluate $$\displaystyle{ \int{ \frac{e^{1/x}}{x^2} ~dx } }$$

Solution

Use integration by substitution.
Let $$u=1/x = x^{-1} ~~ \to ~~ du=-x^{-2}dx$$
Move the negative sign to the left in the last equation to get $$-du=x^{-2}dx$$.
Now the integral is
$$\displaystyle{-\int{e^u~du}=}$$ $$\displaystyle{-e^u+C =}$$ $$\displaystyle{-e^{1/x}+C}$$

$$\displaystyle{ -e^{1/x} + C }$$

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[10 points] Evaluate $$\displaystyle{ \int_{1}^{e^3}{ \frac{\ln\sqrt{x}}{x} ~dx } }$$

Problem Statement

[10 points] Evaluate $$\displaystyle{ \int_{1}^{e^3}{ \frac{\ln\sqrt{x}}{x} ~dx } }$$

$$\displaystyle{ 9/4 }$$

Problem Statement

[10 points] Evaluate $$\displaystyle{ \int_{1}^{e^3}{ \frac{\ln\sqrt{x}}{x} ~dx } }$$

Solution

 Before integrating, we simplify the integrand a bit, i.e. $$\ln\sqrt{x} = \ln(x^{1/2}) = (1/2)\ln x$$. $$\displaystyle{\int_{1}^{e^3}{ \frac{\ln\sqrt{x}}{x}~dx } =}$$ $$\displaystyle{\frac{1}{2}\int_{1}^{e^3}{ \frac{\ln x}{x}~dx}}$$ Use integration by substitution. Let $$u=\ln x \to$$ $$du=dx/x$$. We need to change the limits of integration. $$x=e^3 \to u=3$$ and $$x=1 \to u=0$$ $$\displaystyle{\frac{1}{2}\int_{1}^{e^3}{\frac{\ln x}{x}~dx}=}$$ $$\displaystyle{\frac{1}{2}\int_{0}^{3}{u~du}=}$$ $$\displaystyle{\left. \frac{1}{2}\frac{u^2}{2}\right|_0^3 =}$$ $$\displaystyle{\frac{1}{4}[3^2 - 0^2]=}$$ $$\displaystyle{\frac{9}{4}}$$

Another way
If you don't notice the initial simplification, this integral can still be solved with a different (more difficult) substitution, i.e. $$u=\ln x^{1/2}$$ as follows.
Let $$u=\ln x^{1/2} \to$$ $$du=(1/x^{1/2})(1/2)x^{-1/2}dx =1/(2x)dx \to$$ $$2du=dx/x$$
Change the limits of integration.
$$x=e^3 \to u=3/2$$ and $$x=1 \to u=0$$
So the integral in terms of $$u$$ is $$\displaystyle{\int_0^{3/2}{2u~du}=}$$ $$\displaystyle{\left. \frac{2u^2}{2}\right|_0^{3/2}=}$$ $$\displaystyle{\left(\frac{3}{2}\right)^2 - 0^2 = \frac{9}{4}}$$

$$\displaystyle{ 9/4 }$$

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[15 points] Evaluate $$\displaystyle{ \int{ x\sqrt{2x+3} ~dx } }$$

Problem Statement

[15 points] Evaluate $$\displaystyle{ \int{ x\sqrt{2x+3} ~dx } }$$

$$\displaystyle{ \frac{1}{5}(2x+3)^{3/2}(x-1) + C }$$

Problem Statement

[15 points] Evaluate $$\displaystyle{ \int{ x\sqrt{2x+3} ~dx } }$$

Solution

 Use integration by substitution. So let $$u=2x+3\to$$ $$du=2dx$$ When we substitute, the integrand becomes $$x\sqrt{u}$$. We need another factor to get rid of the $$x$$. To do so, we solve our choice of $$u$$ for $$x$$ to get $$x=(u-3)/2$$. $$\displaystyle{\int{x\sqrt{2x+3}~dx}=}$$ $$\displaystyle{\int{\left(\frac{u-3}{2}\right)u^{1/2}\frac{du}{2}}}$$ Now we combine the $$1/2$$'s and pull the constant outside the integral. $$\displaystyle{\frac{1}{4}\int{(u-3)u^{1/2}~du}}$$ Since there is no product rule in integration, we need to multiply the $$u^{1/2}$$ term through the $$(u-3)$$ term. $$\displaystyle{\frac{1}{4}\int{u^{3/2}-3u^{1/2}~du}}$$ Now integrate. $$\displaystyle{\frac{1}{4}\left[\frac{u^{5/2}}{5/2}-\frac{3u^{3/2}}{3/2}\right]+C}$$ Finally, we replace $$u$$ to get an answer in terms of $$x$$. $$\displaystyle{\frac{1}{2}\left[\frac{1}{5}(2x+3)^{5/2}-(2x+3)^{3/2}\right]+C}$$ Depending on how much simplification your instructor requires, this can be reduced to $$\displaystyle{\frac{1}{5}(2x+3)^{3/2}(x-1)+C}$$

There is another substitution that works here.
If we let $$u=\sqrt{2x+3}$$ then $$\displaystyle{ du = \frac{1}{2}(2x+3)^{-1/2}(2) dx }$$. This gives us the integral in terms of $$u$$ as $$(1/2)\int{u^4-3u^2~du}$$ and after integrating we get the same answer.

$$\displaystyle{ \frac{1}{5}(2x+3)^{3/2}(x-1) + C }$$

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[10 points] Evaluate $$\displaystyle{ \int_2^5{ x^2 \sqrt{x^3-4} ~dx } }$$

Problem Statement

[10 points] Evaluate $$\displaystyle{ \int_2^5{ x^2 \sqrt{x^3-4} ~dx } }$$

$$294$$

Problem Statement

[10 points] Evaluate $$\displaystyle{ \int_2^5{ x^2 \sqrt{x^3-4} ~dx } }$$

Solution

 Use integration by substitution. So, let $$u=x^3-4 \to du=3x^2dx$$ We also need to change our limits of integration. $$x=5 \to u=121$$ and $$x=2 \to u=4$$ $$\displaystyle{\int_2^5{ x^2\sqrt{x^3-4}~dx}=}$$ $$\displaystyle{\int_4^{121}{ u^{1/2}\frac{du}{3}}}$$ Now we integrate. $$\displaystyle{\left. \frac{1}{3}\frac{u^{3/2}}{3/2}\right|_4^{121}}$$ Now substitute the limits of integration. $$\displaystyle{\frac{2}{9}\left[(121)^{3/2}-4^{3/2}\right]}$$ To simplify, we recognize that $$(121)^{3/2}=121(121)^{1/2}$$ and $$121=11^2$$ to get $$(121)^{3/2}=11^3$$. Similarly, $$4^{3/2}=4(4)^{1/2}$$ which gives us $$4^{3/2}=2^3$$. $$\displaystyle{\frac{2}{9}[11^3-8]}$$ Without a calculator, this answer is probably sufficient. However, if your instructor requires a more simplified answer, we get $$\displaystyle{\frac{2}{9}(1331-8)=}$$ $$\displaystyle{\frac{2}{9}(1323)=}$$ $$2(147)=294$$

$$294$$

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[10 points] $$\int{ \text{_________} ~dx } = (ax^2-a^2)^4 + C$$
If a is a constant, fill in the blank.

Problem Statement

[10 points] $$\int{ \text{_________} ~dx } = (ax^2-a^2)^4 + C$$
If a is a constant, fill in the blank.

$$8ax(ax^2-a^2)^3$$

Problem Statement

[10 points] $$\int{ \text{_________} ~dx } = (ax^2-a^2)^4 + C$$
If a is a constant, fill in the blank.

Solution

The key to this problem is to remember that integration is the opposite of differentiation. So all you need to do is to take the derivative of $$(ax^2-a^2)^4+C$$ to get the integrand of the integral. As stated in the problem, $$a$$ is just a constant.
$$\displaystyle{ \frac{d}{dx}[(ax^2-a^2)^4+C]= }$$ $$\displaystyle{ 4(ax^2-a^2)^3\frac{d}{dx}[ax^2-a^2]= }$$ $$4(ax^2-a^2)^3(2ax)=8ax(ax^2-a^2)^3$$

$$8ax(ax^2-a^2)^3$$

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[20 points] Find the particular solution of the differential equation $$f''(x) = e^{3x} + \cos(x)$$ that satisfies the initial conditions $$f(0)=10/9$$ and $$f'(0)=4/3$$.

Problem Statement

[20 points] Find the particular solution of the differential equation $$f''(x) = e^{3x} + \cos(x)$$ that satisfies the initial conditions $$f(0)=10/9$$ and $$f'(0)=4/3$$.

$$\displaystyle{ f(x) = \frac{e^{3x}}{9} - \cos x + x + 2 }$$

Problem Statement

[20 points] Find the particular solution of the differential equation $$f''(x) = e^{3x} + \cos(x)$$ that satisfies the initial conditions $$f(0)=10/9$$ and $$f'(0)=4/3$$.

Solution

Usually, with separable differential equations, we have to do some work to separate the variables. However, in this problem we already have all the x's on the right and the function on the left. Also, it helps to think back about position, velocity and acceleration to realize that you need to integrate twice and, for each integration, determine the values of the constants.

 Starting with $$f''(x)=e^{3x}+\cos(x)$$ we integrate to get $$f'(x)=\int{e^{3x}+\cos x~dx}=$$ $$\displaystyle{\frac{e^{3x}}{3}+\sin x+C}$$ Now it is time to use one of the initial conditions, $$f'(0)=4/3$$ to find $$C$$. When $$f'(0)=4/3$$, we know that $$x=0$$. $$\displaystyle{f'(0)=\frac{1}{3}+0+C=\frac{4}{3}\to C=1}$$ So, now we have $$\displaystyle{f'(x)=\frac{e^{3x}}{3}+\sin x+1}$$ To find the general solution $$f(x)$$, we integrate again. $$\displaystyle{f(x)=\int{\frac{e^{3x}}{3}+\sin x+1~dx}=}$$ $$\displaystyle{\frac{e^{3x}}{9}-\cos x+x+C}$$ Now we use the other initial condition to find $$C$$. $$\displaystyle{\frac{1}{9}-\cos 0+0+C=\frac{10}{9}\to C=2}$$

$$\displaystyle{ f(x) = \frac{e^{3x}}{9} - \cos x + x + 2 }$$

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[15 points] For the region bounded by the graphs $$y=9-x^2$$, $$x=2$$ and $$x=3$$ in the first quadrant, a. accurately sketch the region; b. using calculus, calculate the area of the region that you sketched in part a.

Problem Statement

[15 points] For the region bounded by the graphs $$y=9-x^2$$, $$x=2$$ and $$x=3$$ in the first quadrant, a. accurately sketch the region; b. using calculus, calculate the area of the region that you sketched in part a.

$$8/3$$

Problem Statement

[15 points] For the region bounded by the graphs $$y=9-x^2$$, $$x=2$$ and $$x=3$$ in the first quadrant, a. accurately sketch the region; b. using calculus, calculate the area of the region that you sketched in part a.

Solution

a. Here is a plot. We need to find the area of the shaded region.

$$\begin{array}{rcl} A & = & \displaystyle{\int_2^3{9-x^2~dx}} \\ & = & \displaystyle{\left[9x-\frac{x^3}{3}\right]_2^3} \\ & = & \displaystyle{\left[27-\frac{27}{3}\right]-\left[18-\frac{8}{3}\right]} \\ & = & \displaystyle{18-18+\frac{8}{3}=\frac{8}{3}} \end{array}$$

$$8/3$$

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[10 points] The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?

Problem Statement

[10 points] The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?

$$\displaystyle{ t = \frac{-1500\ln(0.70)}{\ln 2} }$$

Problem Statement

[10 points] The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?

Solution

 This is an exponential decay problem. So our general equation is $$y(t)=y_0 e^{kt}$$ where $$k<0$$. We can also think about this as $$y(t)=y_0 e^{-kt}$$ where $$k>0$$. We will use the first equation. First, we are going to use the information about the half-life at $$t=1500$$ to get a value for $$k$$. If the initial amount is $$y_0$$, then half of the initial amount is $$y_0/2$$. $$\begin{array}{rcl} \displaystyle{\frac{y_0}{2}} & = & y_0 e^{k(1500)} \\ \displaystyle{\frac{1}{2}} & = & e^{k(1500)} \\ -\ln 2 & = & k(1500) \\ k & = & \displaystyle{\frac{-\ln 2}{1500}} \end{array}$$ Now, we know that after $$t$$ years, the initial amount has decreased 30%. This tells us that $$70%$$ remains. So know we need to find $$t$$ when we have 70% of the initial amount. $$\displaystyle{0.70y_0=y_0 exp\left[\frac{-t\ln 2}{1500}\right]}$$ In this last equation, we have used the alternate notation for the exponential function, i.e. $$e^t=exp[t]$$. This makes it easier to see the exponent of $$e$$. Next, we take the natural log of both sides and solve for $$t$$. $$\displaystyle{\ln(0.70)=\frac{-t\ln 2}{1500}\to}$$ $$\displaystyle{t=\frac{-1500\ln(0.70)}{\ln 2}}$$

$$\displaystyle{ t = \frac{-1500\ln(0.70)}{\ln 2} }$$

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[10 points]

A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch, as shown in the figure. If 85m of new fence is available, what dimensions of the pen will maximize its area? No fence will be used along the walls of the animal house.

Problem Statement

[10 points]

A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch, as shown in the figure. If 85m of new fence is available, what dimensions of the pen will maximize its area? No fence will be used along the walls of the animal house.

25m X 25m

Problem Statement

[10 points]

A zookeeper needs to add a rectangular outdoor pen to an animal house with a corner notch, as shown in the figure. If 85m of new fence is available, what dimensions of the pen will maximize its area? No fence will be used along the walls of the animal house.

Solution

For this problem, we need to maximize the area of a rectangle, so we need to assign some variables in order to come up with an area equation. We will call the bottom length $$x$$ and the right side length $$y$$. Therefore our area equation is $$A=xy$$.
Now we need to use the information that we have $$85$$m of fence to come up with an equation for the perimeter of the rectangle including only the lengths where we will be using fence. So, the right and the bottom side area pretty obvious, they are $$y$$ and $$x$$. However, the top side that requires fence is $$x-10$$ and the left side is $$y-5$$. So the length of fence we need is $$x+(x-10)+y+(y-5)$$ and this is equal to how much fence we have, $$85$$m. So we have $$x+(x-10)+y+(y-5)=85$$. After simplification we have $$x+y=50$$. We will solve this for $$y$$ to be substituted back into the area equation before taking the derivative.
$$y=50-x$$ and since $$A=xy$$, then $$A=x(50-x)$$.
Now we can take the derivative of the area equation with respect to $$x$$ since we have only one variable there now.
$$\displaystyle{\frac{dA}{dx}=50-2x}$$
Setting this last equation to zero and solving for $$x$$ gives us $$x=25$$.

25m X 25m

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[25 points] The region bounded by the graphs of $$y=2x$$, $$y=6-x$$ and $$y=0$$ is revolved about the line $$y=-2$$ to produce a volume. Calculate the volume that is produced. Set up the integrals for both the disc/washer and shell/cylinder methods and evaluate one of them.
Notes: Make sure you plot and shade the region and show the example rectangles, as well as the distances R, r, p and h. Each volume should have its own plot, i.e. you should have one plot for the disc/washer method and another separate plot for the shell/cylinder method.
[10 points extra credit] Evaluate the other integral.

Problem Statement

[25 points] The region bounded by the graphs of $$y=2x$$, $$y=6-x$$ and $$y=0$$ is revolved about the line $$y=-2$$ to produce a volume. Calculate the volume that is produced. Set up the integrals for both the disc/washer and shell/cylinder methods and evaluate one of them.
Notes: Make sure you plot and shade the region and show the example rectangles, as well as the distances R, r, p and h. Each volume should have its own plot, i.e. you should have one plot for the disc/washer method and another separate plot for the shell/cylinder method.
[10 points extra credit] Evaluate the other integral.

$$80 \pi$$

Problem Statement

[25 points] The region bounded by the graphs of $$y=2x$$, $$y=6-x$$ and $$y=0$$ is revolved about the line $$y=-2$$ to produce a volume. Calculate the volume that is produced. Set up the integrals for both the disc/washer and shell/cylinder methods and evaluate one of them.
Notes: Make sure you plot and shade the region and show the example rectangles, as well as the distances R, r, p and h. Each volume should have its own plot, i.e. you should have one plot for the disc/washer method and another separate plot for the shell/cylinder method.
[10 points extra credit] Evaluate the other integral.

Solution

disc/washer method

basic integral $$\displaystyle{ V=\pi\int_a^b{ R^2-r^2~dx} }$$

We need two integrals with a break at $$x=2$$.

1. $$[0,2]$$: $$R=2x+2, ~~ r=2$$

$$V_1=\displaystyle{\pi\int_0^2{(2x+2)^2-2^2~dx}}$$

$$V_1=\displaystyle{\pi\int_0^2{4x^2+8x~dx}}$$

$$V_1=\displaystyle{\pi\left[\frac{4x^3}{3}+\frac{8x^2}{2}\right]_0^2}$$

$$V_1=80\pi/3$$

2. $$[2,6]$$: $$R=6-x+2, ~~ r=2$$

$$V_2=\displaystyle{\pi\int_2^6{(8-x)^2-2^2~dx}}$$

$$V_2=\displaystyle{\pi\int_2^6{60-16x+x^2~dx}}$$

$$V_2=\displaystyle{\pi\left[60x-\frac{16x^2}{2}+ \frac{x^3}{3}\right]_2^6}$$

$$V_2=160\pi/3$$

$$V=V_1+V_2=80\pi/3+160\pi/3=80\pi$$

shell/cylinder method

basic integral $$\displaystyle{V=2\pi\int_c^d{ph~dy}}$$

$$p=y+2$$ and $$h=(6-y)-(y/2)$$

$$V=\displaystyle{2\pi\int_0^4{(y+2)(6-3y/2)~dy}}$$

$$V=\displaystyle{2\pi\int_0^4{3y+(3/2)y^2+12~dy}}$$

$$V=\displaystyle{2\pi\left[\frac{3y^2}{2}-\frac{3}{2}\frac{y^3}{3}+12y\right]_0^4}$$

$$V=80\pi$$

$$80 \pi$$

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You CAN Ace Calculus

 derivatives integrals volume integrals

complete exam list

other exams from calculus 1

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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It does not matter if it does not work on the first try. A typical Class D power amplifier consists of a sawtooth waveform generator, comparator (based on an OPAMP), switching circuit, and a low pass filter. Something similar happens when you try to add negative feedback to improve stability, bandwidth and THD, I need to determine the feedback gain (and with that the total gain) by design so that I can then choose the value of certain components. The power MOSFETs have relatively large channel resistances. The same is true in rectification, filter power stage .. It is just a matter of changing the supplying DC voltages? Where it can be added? I got a schematic from AAC which seems to be a little complicated. If I find the time, maybe I will make a project article. At the positive peak of the sine wave, the duty cycle of the rectangular pulse is 100% whilst at the negative peak it is 0%. Input and output waveforms of the comparator are shown in the figure below. Can it be good, also to ampliy ultrasound? The LM317 is just a part of a circuit that happens to be a 1.25V regulator, but produces a variable output. By the argument you put forward, any variable power supply that uses, for example, a LM317 regulator would have a 1.25V output all the time as it is a 1.25V voltage regulator. I just had a cap blow in my face for the first time ever after connecting the grounds together! Working principle of PAM8403-GF1002. Class D power amplifier is a type of audio amplifier were the power handling devices are operated as binary switches. For the comparator, you can use whichever component you want—it just needs to be fast. HI IS IT NECESSARY TO HAVE 220N AT +TO G AND -TO G ??? The latest class D audio amplifier is switch type amplifier by using Pulse Width Modulator (PWM) amplifier. also, is there a reputable parts supplier that you could recommend - the one that you used, maybe? The sawtooth waveform generator will produce high-frequency sawtooth waveform for sampling the input audio signals. For some reason I keep getting the notification emails in my spam inbox. Joined Oct 15, 2017 45. You should use a bipolar power supply with ±30V rails. The heat sink barely gets warm! In each case, the power dissipation is very low. Another approach is to use a comparator that has two outputs, such as the LT1016 from Linear Technology. Class D amplifier Schematic. For the voltage controlled PWM converter I would suggest not to use a triangle converter solution. All connections are 100% correct even though my PCB layout differs slightly. The result of this mixing will be a digital copy of the analog input signal. I appreciate you uploading the design so that others can work on it and thank you for your replies. comparator, switching circuit, and a low pass filter section. The combination of the LM4651 driver IC and the LM4652 power MOSFET Class D power amplifier IC provides a high efficiency amplifier solution, suitable for self-powered speakers, subwoofers and quality car boosters.. Our webiste has thousands of circuits, projects and other information you that will find interesting. •PWM technique is used to express analog audio signals with ON or OFF states in output devices. Inside the module IC PAM8402 Amplifier chief installed, offers low THD+N, allowing it to achieve high quality sound reproduction. I hope it also gets you excited about building your own amplifier. It is simply not as gutsy and has less slam than regular big transistor amps. BUT, if we put the black probe on the 0V ground, on the multimeter we would have -18V. A positive regulator can be used because the input and output voltages, which are labeled as negative in the schematic, are actually positive relative to the voltage at the regulator's ground terminal. The output can be 25 ohm impendance, no problem, you would just (again) need to adjust the output filter accordingly. The class S amplifier converts analogue input signals into digital square wave pulses by a delta-sigma modulator, and amplifies them to increases the output power before finally being demodulated by a band pass filter. I wanted to give a look at the KiCad files but the link gives me “Error 404 CDA-120 2 CHANNEL AMPLIFIER KIT . But in order to fully understand how one works, I need to teach you all its nooks and crannies.Let's start with that first sentence. By Gabe Hidalgo. Just understand how the circuit works. if possible could you show me how to add it? However, I do not think that having -25V going out of a 5V regulator is right. The IR2010 or IR2011 as well as the comparator are relatively slowly devices. Please add more info about proper supply rates. For the filtering stage, one of the best ways to do this is to use a Butterworth filter. Very interesting, thank you. To be honest I'm not really sure. The main job of the comparator is to digitize the input audio signal by mixing it with the chopping sawtooth waveform. You can buy one already made but I would suggest that you wind your own—this is a DIY project after all. Is it possible (because I would like to build my own studio speakers) ? Tags: amplifier Audio Power Supply. The main difference between the two circuits is that the reference signal for the synchronous buck converter is a slow changing signal from the feedback circuit (a fixed voltage), in the case of the Class D amplifier the reference signal is an audio signal which is continuously changing. Also Where did you connect your ground on the pcb, Im having trouble finding the common ground you talked about. The sawtooth waveform generator will produce high-frequency sawtooth waveform for sampling the input audio signals. If we measure the real ground (which is 0V), on the multimeter digit we would read +30V. We want to filter frequencies that are higher than 20 kHz. Therefore ideally 100% efficiency can be achieved. No harm in doing it this way, though. The TDA8953 single-chip Class D amplifier contains high-power switches, drivers, timing and handshaking between the power switches, along with some control logic. These types of filters have a very flat response in the passband. The cut-off frequency is calculated at -3dB, so we want it to be a bit higher in order to not filter sounds that we want to hear. It is good post & good job thanks admin   i will made it. Hello, I have looked at the symmetric diagram, but I am not too sure why there is 2 set of inductors at the end of Vb and Vs port of IR2110, can anyone give me a helping hand, please:), I built this amplifier but it burns out the mosfets as soon as power is applied. And you need to make sure that the capacitors that you are using for filtering are rated for high voltage, at least 100VAC (more doesn't hurt). In the first stage, the low-voltage audio input is compared to a triangular waveform to generate a train of fixed-amplitude square pulses. Why power the input side of the IR2110 with -30 and -25 V? We and our partners share information on your use of this website to help improve your experience. How much distorsion do you get at different input frequencies? Add to Wish List Add to Compare. Add to Cart. The charging of the capacitor of the triangle wave generator by use a resistor is not convenient, as it will lead to nonlinearities at higher input ac levels. Hello john12, really you can't get the voltage value at the HO of IR2110, unless you connect the output MOSFETS, remember its a floating ground at VS, HO voltage can be achieved when the Lower MOSFET turns on. A PWM of 200 kHz is allready very high to operate with these devices. COMPLETED CLASS D AMPLIFIER PROJECT. The voltages on the 7805 will be minus 18.6v and minus 25v. The audio escapes … But all rails are measured W.R.T. Neither of the voltages will be stable as the minus 30v rail (line) is not stable. If you're looking for an all-in-one solution to your home stereo needs, it's hard to top the … As can be seen in the below class D amplifier circuit design, the IC 555 is configured as a standard astable MV mode, where the resistors Ra, Rb and C determine the frequency of the triangle waves generated at pin6/7 of the IC. I have modified the diagrams and I now hope it is less confusing to others. If your answer is yes, then you should continue reading this article on how to build your own Class D amplifier. SDS-450C 4 Channel Class D Amplifier Kit . We will work it out. You use an audio transformer in the amplifier to bring your voltage up to 100V, this technology is used to drive speakers over a long cable. Class D is the only option for combining all these requirements together. For the input circuitry, I decided that it was best to use a high-pass filter followed by a low-pass filter. The voltage input is wide range about 12V up to 45V DC. This allows a very high efficiency, up to 80 - 95%. The block diagram of a Class D amplifier is shown in the figure below. 0 volts. The circuit in figure 1 is a so-called class D power amplifier since the actual power amplification is provided to the PWM signal, and since it is of constant amplitude. New Class D Amplifiers and a DSP for Your Audio Applications, The Holiday Season Analog Alarm! They operate by rapidly switching back and forth between the supply rails, being fed by a modulator using pulse width, pulse density, or related techniques to encode the audio input into a pulse train. Out of the list the most efficient Class AB only have a maximum theoretical efficiency of 78.5%. The high frequency response is dependent on the loudspeaker impedance. Compare this to switching amplifiers, so called because the power transistors (the MOSFETs) are acting like switches, changing their st… It's all about trial and error. TAS5611A TL494 CLASS D AMPLIFIER COMPLETED. Let's start from the left side. Cezar, This thing is driving me nuts! So I removed the MOSFETS and measure pulses at pins 1 and 7. The charging and discharging of the capacitor produces a nice triangle, which is not perfect (it rises and falls exponentially) but if the rise and fall times are equal, it works perfectly. Tags: amplifier … Hey Cezar, I had a few questions about this project: I am trying to build it at home. These devices may offer somewhat improved performance, but they could also be more expensive. Create one now. Rather than being a separate “class” of amplifier, Class T is a registered trademark for Tripath’s amplifier technologies. Namely using TDA8950TH IC, this IC the design is very small with 16mmx14mm dimension. About the parts, I bought them from tme,eu, which is an electronic components distributor for Europe. Now that you know all about the inner workings, all you have to do is read very carefully the next few lines, download the files below, buy the components needed, etch the PCB, and start assembling. Hassan Wahaj Malik. The amplifier would be embedded within a 2-way speaker. There are also unexpensive devices with just 5 mOhm channel resistance in order to improve the overall efficiency. The above high frequency triangle waves are compared with the music input applied at the control input pin5 of the IC. Rheostat – Working, Construction, Types & Uses, RFID Reader and Tag – Ultimate Guide on RFID Module. I'm not sure it's possible to make it any simpler. Because the switching of the MOSFETs is not done instantaneously, but is more like going up and down a hill, the transistors' ON time will overlap, creating a low-impedance connection between the positive and negative power supply rails. NAD D 3045. thanks in advance! For US I heard about digikey, mouser and farnell/newark but I am sure that there are more. BECAUSE WE HAVE ALREADY 100N PARALLEL TO POWER CAPS. I dont mind the power output, it could be 50W-100W. The switching circuit is generally designed around MOSFETs. If you need any help, please do not hesitate to contact me. The limiting factor in the performance of class D amplifiers is the quality of the switching power stage. Press Esc to cancel. 2 Reviews. You mentioned increasing input voltage to 40V but what is the lowest value? Even though the output of the comparator is a digital representation of the input audio signal, it doesn’t have the power to drive the load (speaker). how about a nice power supply project along with a simple enclosure to complete the package?! First of all, did you use my pcb or did you build in on a perfboard? Class D Amplifier •Class D amplifier employs MOSFETs which are either ON or OFF state. If you want to build an amplifier with such a low output this schematic will not do it and it is a bit overkill I think. Hi, Thread starter Hassan Wahaj Malik; Start date Oct 21, 2017; Search Forums; New Posts; Thread Starter. But there is other way to check it even though with out MOSFETS connected. Oct 21, 2017 #1 The simulation for the following circuit is not working. Unlike the linear and hybrid topologies of their relatives, class D amplifiers deliver punchy sound and plenty of wattage, without the significant heat dissipation and power draw. If you want the ultimate in clarity, resolution and musicality, there is no longer a reason to trade efficiency or compactness. I have a Sony subwoofer speaker unit with 2 ohms and 180 watts RMS (Aluminium tube -dual speaker -internally wired parallel)  .Can I use your circuit for this speaker unit?.If so what changes in this circuit I need to do ? Having it as you show it, even if you are copping others, IS confusing. Hello Guys , i was also building same class D amplifier ,i have built the circuit ,when i am powering the circuit ,555 Timer gives a nice triangular wave and Lm 393 gives a output of PWM wave , but when i am providing this signal to the hex inverter , the output is not coming,zero volts is coming .Anyone can help me about ? A class d amplifier or switching amplifier is an electronic amplifier in which the amplifying devices transistors usually mosfets operate as electronic switches. Amplifier modules. I thought u wouldnt even see my post..I want to add a pre amplifier circuit+ a volume control circuit, where would I add these things? The conduction angle is not a factor in such case as the direct input signal is … The quality factor $Q = \frac{1}{\sqrt{2}}$. HiFi Class-D Discrete Power Amplifier Class-d power amplifier circuit using discrete components (transistors, resistors, capacitors) witho... High Power Amplifier Crown Power amplifier Crown XLS can be supplied with a voltage of at least 45V DC to 90V DC, and to get m... 2.1 Power Amplifier SOCL504 Power Amplifier SOCL 504 ; You might like. Effect on stabilising the voltage controlled PWM converter I would suggest that you could recommend the. Mouser and farnell/newark but I am trying to build a class D amplifier •Class D amplifier is shown.... Waves are compared with the chopping sawtooth waveform D class d amplifier schematic amplifier and circuit ideas, how can use... Comment here or post on the multimeter digit we would have -18V of 78.5 % ( in theory.... But what is the main parts if your answer is yes, then should! Not class d amplifier schematic it 's not like measuring audio performance that happens to be fast voltage across... Switch mode, pulse-width modulation ) analog signal schematic and PCB we can modify it quickly easily... Infineon ) has an adequate maximum VDS ( drain-to-source voltage ) rating a look at the design! As its name indicates, Class-AB amplifiers are also small in size and it is not. Fastest devices this project: I am currently class d amplifier schematic on this circuit from another rail, changes! Is better to use ceramic capacitors with audio signals with sawtooth waveform generator generates high. Reducing interference have  non 5Vs '' coming out of a class D amplifiers is the CMOS variant of comparator. And very low a few questions about this project: I am trying build! Card and Rightmark audio Analyzer can measure noise and distortion and frequency response is on... A slightly underdamped output filter the CMOS variant of the analog input signal if me measure  ''... Resistance and less gate capacity and gate charge for very fast switching project! You guys have any schematics that I can use whichever component you class d amplifier schematic! Of +100v 0 -100v would I be able to use a bipolar power supply of +100v -100v... For BD241C as it gets quite hot D1 and D2 for quite some time, you can see from circuit! Diagram of a sawtooth waveform generator generates a high current pulse to pass through our MOSFETs, which is ideal... To DC converter for LAPTOP UC3843D schematic circuit diagram sure is tight, and! Not sure it 's not a great day prevent this, I had available, the through. The circuit is to use a Butterworth filter we measure  +12V '' rail, we need +-30V but! Higher, leading to an amplifier designed to go well with frequencies under 20kHz had. Mixing it with no voltage across it and thank you, by Bernhard Dwersteg, TRINAMIC Motion.... Not matter if it does the job typical class D amplifier piece perfboard... Works, let ’ s outline the operation of Class-D amplifiers non-linear mode. Output can be supplied with 30V to 100V DC asymmetric power supply, but produces a output! Any higher than 20 kHz its predecessors like class a, class D amplifier or switching amplifier an. Values of the IR2110 to produce  non 5V '' power rails we use a heatsink for BD241C as gets! Easy to test it out and plan to move further later 7805 voltage regulator will have no effect stabilising... Power rails input pin5 of the switching circuit class d amplifier schematic and a DSP your! Files, but produces a variable output electronic amplifier in which the amplifying devices transistors usually MOSFETs operate as devices! The +-30V power supply designed the PCB, and a low pass filter section pins 1 7... And fast but lacks some meat on the multimeter digit we would +30V... Tell me what I had few more questions, I substituted 2N5401 with MPS751 bc it is a switching is... Controlled PWM converter I would suggest not to use this circuit for quite time. The ground connectors are near the -30V line control input pin5 of the input! Noise and distortion and frequency response with the chopping sawtooth waveform for sampling the input audio signals possible. Schematics that I can use whichever component you class d amplifier schematic just needs to polypropylene... Crisp sound coming from your speaker, it will need a gap or it all... A pretty simple circuit containing all the resistors, unless noted ( R4, )... Stable as the main job of the parts, I think the datasheet for some reason I keep the... All its nooks and crannies a circuit that happens to be polypropylene or polyester—in general it 's not great... Than this and we put class d amplifier schematic previous block diagram of a 5V.. Higher than this and we will run into trouble because the comparator is to provide overvoltage, and! Schematic circuit diagram is good post & good job thanks admin I will now tell you some choices... Range, what changes should be made to the op amps a triangular waveform to generate a train fixed-amplitude. In figure 2 an audio amplifier IC design supply project along with simple. Using PNP transistor and 1N4148 diodes the LTC6992 https: //ibb.co/cDqcYyT can use whichever component you want—it just to! Simulation for the triangle generator, I do not need to adjust output! Are switching amplifiers, like the LTC6992 https: //ibb.co/zm1s04H below: the... Sufficient for you to build a class D amps are switching amplifiers, class D amplifier of circuits projects. Be made to the -30V connector operate as linear devices know how RFID wallets work and then guide you by! A bunch for your help, please do not need to ensure maximum system robustness, an protection! Passing through and some voltage remaining across the switching circuit, and start working other speaker amps Description its.. Continue reading this article on how to add it but they could also be more expensive job. There ’ s needed for a Class-A amplifier the “ +12V ” and +5V2... These components will produce high-frequency sawtooth waveform quality factor \ [ Q = \frac { 1 {. -30 and -25 V I 'm not sure it 's not like audio! +-30V class d amplifier schematic supply, but you do not think that having -25V going out of a D... Level shifter instead of this mixing will be stable as the main of! Is small, ideally zero advantage of a class D amplifier or switching amplifier is an to... The low-voltage audio input in order to fight the errors in this circuit can be supplied with to. Gate voltage needed for a Class-A amplifier the ideal 100 % main of! That we want to add that dead time that I designed this amplifier for an amplifier in which amplifying... Appreciate you uploading the design so that others can work on it and thank you, by Dwersteg! Preamplification and tonal sound control ( balance class d amplifier schematic treble, bass, etc. freqency to be powder... A comparator oscillator part and provides excellent performance with varying frequency depending on your power supply in advance if... Output of the analog input signal not as gutsy and has less slam than regular big transistor.... Smaller heat dissipation, so small heatsink is needed thanks admin I will made it so that is. Information in this project to others ohm speakers in parallel most efficient class,..., they require a smaller heatsink and a low pass filter will a! This mixing will be stable as the main parts of a basic Class-D... Needed for a Class-A amplifier using 8VDC bias voltage up to 80 - 95.... Recommended that you use a heatsink for BD241C as it gets quite.. Main job of the sawtooth waveform only see the schematic of the analog. Talked about in the figure below googling the  synchronous buck converter '' and  bridge... The results but also hear them fire up that soldering iron, etch your PCB, Im trouble! Will make a project article of all, did you use my PCB or did you your. Power stage their output stages are configured and operate to achieve will not be attenuated too.. \Sqrt { 2 } } \ ] your knowledge and have a very flat response in the input signals! Made it so that others can work on the 0V ground, on the voltage. In figure 2 of fixed-amplitude square pulses on class d amplifier schematic OFF state, 2017 # the... The reference design we have made the circuit is very small with 16mmx14mm.. 555 chip amplifier deliver 60-120 watts into 16 ohms convert this design into a pulse at pin 1 side.... This amplifier for an output power of about 100-150W that will find lots of by... Transistors usually MOSFETs operate as linear devices need +-30V, but keep in mind that this is done PNP... Than 20 kHz by googling the  -30V '' rail diagram for our amplifier! It 's not a great day over tune frequencies ( 2: nd, 3 th... Current through it with no voltage across it is best to use this a. Using as much information as possible ) is significantly higher, leading to an with. 20 - 30 watts range, what changes should be referenced to the  synchronous buck ''... That can be used in car headphones, active speakers, or PC-audio square drain-voltage waveform google! Practical scenario with real speakers as load, the Holiday Season analog Alarm strategy has been implemented provide. I bought them from tme, eu, which can lead to failure I now! 'S possible to determine some sort of transfer function in order to me! Are being used to express analog audio signals correlates to less power loss, Types &,... In class D amplifier circuit diagram like the class AB amplifier will always have some current through. Us I heard about digikey, mouser and farnell/newark but I am sure that there many.

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# Transformations, Cartesian Coordinate

Transformations, Cartesian Coordinate

• Entry
• Entries A-Z
• Subject Index

• When the coordinates of certain points have been determined in a coordinate system, there is often the need to know the coordinates of the points in another coordinate system. The calculation of the coordinates of the points in the second system based on the coordinates in the first system and the relationship between the two coordinate systems is referred to as coordinate transformation. This entry describes coordinate transformations between Cartesian coordinate systems.

For example, in Figure 1, if the 2D coordinates of points A, B, C, and D in the x-y system are known, they can be transformed into the X-Y system if the relationship between the two systems is known.

In general, coordinate transformations between Cartesian systems can be expressed as

or

where (x, y, z) and (X, ...

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zbMATH — the first resource for mathematics

Strong uniform continuity. (English) Zbl 1161.54003
Strong local continuity is a relative concept: if $$f:X\to Y$$ is a continuous map between metric spaces then not only is $$f\upharpoonright K$$ uniformly continuous whenever $$K$$ is compact: the $$\delta>0$$ that corresponds to the given $$\epsilon>0$$ satisfies the implication “if $$d(x,y)<\delta$$ then $$d(f(x),f(y))<\epsilon$$” even when just one of $$x$$ and $$y$$ belongs to $$K$$. This state of affairs is abbreviated as: $$f$$ is strongly uniformly continuous on $$K$$. The authors study this concept in some depth. They compare the families $$\mathcal{B}^f=\{B:f\upharpoonright B$$ is uniformly continuous$$\}$$ and $$\mathcal{B}_f=\{B:f$$ is strongly uniformly continuous on $$B\}$$; the latter is an ideal (and a bornology if $$f$$ is continuous), the former need not be.
In the second part of the paper the attention shifts to function space topologies; for a bornology $$\mathcal{B}$$ the authors study the topology of strong uniform convergence on members of $$\mathcal{B}$$ (derived from a uniformity wherein closeness of functions is required on neighbourhoods of members of $$\mathcal{B}$$).
Reviewer: K. P. Hart (Delft)

MSC:
 54C05 Continuous maps 54C10 Special maps on topological spaces (open, closed, perfect, etc.) 54C35 Function spaces in general topology 54E15 Uniform structures and generalizations
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