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## Mobile OCR input: “Fully automatic” and reality

Recently I’ve been toying around with WordSnap OCR (project page, source code, app on Android Market), an app for OCR-based camera input on Android. In the process, I found out a few things about “smart” versus “fast”.

At least in data mining, “fully automatic” is an often unquestioned holy grail.  There are certainly several valid reasons for this, such as if you’re trying to scan huge collections of books such as this, or index images from your daily life like this.  In this case, you use all the available processing power to make as few errors as possible (i.e., maximize accuracy).

However, if the user is sitting right in front of your program, watching your algorithms and their output, things are a little different. No matter how smart your algorithm is, some errors will occur. This tends to annoy users. In that sense, actively involved users are a liability. However, they can also be an asset: since they’re sitting there anyway, waiting for results, you may as well get them really involved. If you have cheap but intelligent labor ready and willing, use it! The results will be better or, at the very least, no worse.  Also, users tend to remember the failures. So, even if end results were similar on average, allowing users to correct failures as early as possible will make them happier.

Instead of making algorithms as smart as possible, the goal now is to make them as fast as possible, so that they produce near-realtime results that don’t have to be perfect; they just shouldn’t be total garbage. When I started playing with the idea for WordSnap, I was thinking how to make the algorithms as smart as possible.  However, for the reasons above, I soon changed tactics.

The rest of this post describes some of the successful design decisions but,  more importantly, the failures in the balance between “automatic” and “realtime guidance”. The story begins with the following example image:

Incidentally, this image was the inspiration for WordSnap: I wanted to look up “inimical” but I was too lazy to type. Also, for the record, WordSnap uses camera preview frames, which are semi-planar YUV data at HVGA resolution (480×320). This image is a downsampled (512×384) full-resolution photograph taken with the G1 camera (2048×1536); most experiments here were performed before WordSnap existed in any usable form. Finally, I should point out that OCR isn’t really my area; what I describe below is based on common sense rather than knowledge of prior art, although just before writing this post I did try a quick review of the literature.

### Binarization

A basic operation for OCR is binarization: mapping grayscale intensities between 0 and 255 to just two values: black (0) and white (1).  Only then can we start talking about shapes (lines, words, characters, etc).  One of the most widely used binarization algorithms is Otsu’s method.  It picks a single, global threshold so that it maximizes the within-class (black/white) variance, or equivalently maximizes the across-class variance. This is very simple to implement, very fast and works well for flatbed scans, which have uniform illumination.

However, camera images are not uniformly illuminated. The example image may look fine to human eyes, but it turns out that even for this image no global threshold is suitable (click on image for animation showing various global thresholds):

If you looked at the animation carefully, you might have noticed that at some point, at least the word of interest (“inimical”) is correctly binarized in this picture.  However, if the lighting gradient were steeper, this would not be possible. Incidentally, ZXing uses Otsu’s method for binarization, because of it is fast. So, if you wondered why barcode scanning sometimes fails, now you know.

So, a slightly smarter approach is needed: instead of using one global threshold, the threshold should be determined individually for each pixel (i,j). A natural threshold t(i,j) is the mean intensity μw(i,j) of pixels within a w×w neighborhood around pixel (i,j).  The key operation here is mean filtering: convolving the original image with a w×w matrix with constant entries 1/w2.

The problem is that, using pure Java running on Dalvik, mean filtering is prohibitively slow.  First, Dalvik is fully interpreted (no JIT, yet). Firthermore, the fact that Java bytes are always signed doesn’t help: casting to int and masking off the 24 most significant bits almost doubles running time.

Method Dalvik (msec) JNI (msec) Speedup
Naïve 109,882 ± 4,813 1,712 ± 261 64×
Sliding 2,435 ± 141 71 ± 19 34×

JNI to the rescue. The table above shows speedups for two implementations. The naïve approach uses a triple nested loop and has complexity O(w2mn), where m and n is the image height and width, respectively (m = 348, n = 512 in this example). The 1-D equivalent would simply be:

for i = 0 to N-1:
s = 0
for j = max(i-r,0) to min(i+r,N-1):
s += a[j]

where w = 2r+1 is the window size. The second implementation updates the sums incrementally, based on the values of adjacent windows. The complexity now is just O(mn). An interesting aside is the relative performance of two implementations for sliding window sums (where w = 2r+1 is the window size). The first checks border conditions inside each iteration:

Initialize s = sum(a[0]..a[r])
for i = 1 to N-1:
if i > r:
s -= a[i-r-1]
if i < N-r:
s += a[i+r]

The second moves the border condition checks outside the loop which, if you think about it for a second, amounts to:

Initialize s = sum(a[0]..a[r])
for i = 1 to r:
s += a[i+r]
for i = r+1 to N-r-1:
s -= a[i-r-1]
s += a[i+r]
for i = N-r to N-1:
s -= a[i-r-1]

Among these two, the first one is faster, at least on a laptop running Sun’s JVM with JIT (I didn’t time Dalvik or JNI). I’m guessing that the second one messes loop unrolling, but I haven’t checked my guess.

It turns out that there is a very similar approach in the literature, called Sauvola’s method. Furthermore, there are efficient methods to compute it, using integral images. These are simply the 2-D generalization of partial sums. In 1-D, if partial sums are pre-computed, window sums can be estimated in O(1) time using the simple observation that sum(i…j) = sum(1..j) – sum(1..i-1).

Savuola’s method also computes local variance σw(i,j), and uses a relative threshold t(i,j) = μw(i,j)(1 + λσw(i,j)/127). WordSnap uses the global variance and an additive threshold t(i,j) = μw(i,j) + λσglobal, but after doing a contrast stretch of the original image (i.e., linearly mapping minimum intensity to 0 and maximum to 255). Doing floating point math or 64-bit integer arithmetic is much more expensive, hence the additive threshold. Furthermore, WordSnap does not use integral images because the same runtime can be achieved without the need to allocate a large buffer. Memory allocation on a mobile device is not cheap: the time needed to allocate a 480×320 buffer of 32-bit integers (about 600KB total) varies significantly depending on how much system memory is available, whether the garbage collector is triggered and so on, but on average it’s about half a second on the G1. Even though most buffers can be allocated once, startup time is important for this application: if it takes more than 2-3 seconds to start scanning, the user might as well have typed the result.

Anyway, here is the final result of locally adaptive thresholding:

Conclusion: In this case we needed the slightly smarter approach, so we invested the time to implement it efficiently. WordSnap currently uses a 21×21 neighborhood.  Altogether, binarization takes under 100ms.

### Skew

Another problem is that the orientation of the text lines may not be aligned with image edges.  This is called skew and makes recognition much harder.

Initially, I set out to find a way to correct for skew.  After a few searches on Google, I came across the Hough transform.  The idea is simple.  Sayyou want to detect a curve desribed by a set of parameters. E.g., for a line, those would be distance ρ from origin and slope θ. For each black pixel, find the parameter values for all possible curves to which this pixel may belong. For a line, that’s all angles θ from 0 to 180 degrees, and all distances ρ from 0 to sqrt(m2+n2).  Then, compute the density distribution of parameter tuples.  If a line (ρ00) is present in the image, then the parameter density distribution should have a local maximum at (ρ00).

If we apply this approach to our example image, the first maximum is detected at an angle of 20 degrees. Here is the image counter-rotated by that amount:

Success!  However, computing the Hough transform is too slow!  Typical implementations bucketize the parameter space. This would require a buffer of about 180×580 32-bit integers (for a 480×320 image), or about 410KB. In addition, it would require trigonometric operations or lookups to find the buckets for each pixel, not to mention counter-rotation. There are obvious optimizations one can try, such as computing histograms at multiple resolutions to progressively prune the parameter space.  Still, the cost implied by back-of-the envelope calculations put me off from even trying to implement this on the phone. Instead, why not just try to use the users:

Conclusion: Simple approach with help from user wins, and the computer doesn’t even have to do anything to solve the problem! Incidentally, the guideline width is determined by the size of typical newsprint text at the smallest distance that the G1’s camera can focus.

### Font size

Next, we need to detect individual words.  The approach WordSnap uses is to dilate the binary image with a rectangular structuring element (in the following image, the size 7×7), and then expand a rectangle (shown in green) until it covers the connected component which, presumably, is one word.

However, the size of the structuring element should really depend on the inter-word spacing, which in turn depends on the typeface as well as the distance of the camera from the text.  For example, if we use a 5×5 element, we would get the following:

I briefly toyed with two ideas for font size detection.  The first is to do a Fourier transform.  Presumably the first spatial frequency mode would correspond to inter-word and/or inter-line spacing and the second mode to inter-character spacing. But that assumes we apply Fourier to a “large enough” portion of the image, and things start becoming complicated.  Not to mention computationally expensive.

The second approach (which also appears to be the most common?) is to to hierarchical grouping. First expand rectangles to cover individual letters (or, sometimes, ligatures), then compute histogram of horizontal distances and re-group into word rectangles, and so on.  This is also non-trivial.

Instead, WordSnap uses a fixed dilation radius.  The implementation is optimized to allow near-realtime annotation of the detected word extent.  This video should give you an idea:

Conclusion: Simple wins again, but this time we have to do something (and let the user help with the rest). But, instead of trying to be smart and find the best parameters given the camera position, we try to be fast: fix the parameters and let the user find the camera position that works given the parameters. WordSnap uses a 5×5 rectangular structuring element, although you can change that to 3×3 or 7×7 in the preferenfces screen. Altogether, word extent detection takes about 150-200ms, although it could be significantly optimized, if necessary, by using only JNI only, instead of a mix of pure Java and JNI calls.

I’m now looking into the possibility of moving OCR into the “live” loop: as you move the camera, the phone shows not only the word extent rectangle, but also the recognized word.  Perhaps as a hyperlink to Google, or along with Google Translate results.  Then I can justifiably use the buzzword of the day, “augmented reality”!  It looks that it might just be possible, but let me get back to you in a week or two.  :)

Postscript: Some of the papers referenced were pointed out to me by Hideaki Goto, who started and maintains WeOCR. Also, skew detection and correction experiments are based on this quick-n-dirty Python script (needs OpenCV and it ain’t pretty!). Update (9/2): Fixed really stupid mistake in parametrization of line.

• Charalampos Tsourakakis

Really nice work Spiro!

• On your TODO list is:
Look into porting one of the lighter-weight engines (GOCR, perhaps via Conjecture) to Android

The one you didn’t mention, Tesseract, has been ported to Android, or at least a JNI wrapper has been written:
http://www.itwizard.ro/mezzofanti-augmented-reality-through-text-recognition-146.html

• Just came across your site accidentally looking for a mobile OCR. Felt the need to comment. Really great work! Would love to see this working realtime.

• Hmm

“Binarization” seems like a bad idea to me. You’re throwing away information that you could use to read the letters more accurately. Is this normally done for OCR?

• I’ve tested WordSnap on the Samsung i7500 w/ Android 1.5… works quite well! The automated mode is not working, it switches the LED light on and off (and when it is on, the image is much too bright). The manual model works as expected, although it seems the system is biased towards dark letters on a bright background and works not at all for bright letters on a dark background.

I think that porting GoCR is a waste of time. The only thing fast enough for such small platforms would be a convolutional neural network trained to detect single letters (possibly including rotated letters). This simplifies to a large set of convolutions of the image,which can be done very efficiently in phase space, works without a large dictionary, and would be fast enough to scan paragraphs of text in real-time. However, such systems are very hard to train and the availability of sufficient training data is a real challenge…

• P.S. Given that Dalvik is only interpreting Java, I no longer wonder why Android is only as fast as J2ME on much less powerful devices… or that the same code ported via xmlvm to iPhone is about 20x faster!

• Great app! I’ve been trying to compile and deploy the app to my android phone from the source code, but I run into this problem:

03-19 00:32:56.057: ERROR/AndroidRuntime(3249): Uncaught handler: thread
OCRThread exiting due to uncaught exception
03-19 00:32:56.067: ERROR/AndroidRuntime(3249):
03-19 00:32:56.067: ERROR/AndroidRuntime(3249): at
net.bitquill.ocr.image.GrayImage.nativeMin(Native Method)

Issue further documented in google code

Any help greatly appreciated!

• luke

• jk

get forced closed all the time doesnt work for me on htc desire

• folha

free ocr is a online ocr service. You can have a try.

• vivek

hey good work!!! hey evn we are working on a similar project!!!…

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# Lagrangian of a circuit

Hello folks,

I could really do with a few hints with this. (As soon as possible!)

1. Homework Statement

For a repeated line of cells, two 'equations of motion' can be written:

$$C_{n}\dot{U_{n+1}} = I_{n}-I_{n+1}$$
$$L_{n}\dot{I_{n}} = U_{n} - U_{n+1}$$

where $$C_{n}$$ is the capacitance, $$U_{n+1}$$ the voltage after the nth cell, $$I_{n}-I_{n+1}$$ the charging current, $$L_{n}$$ the inductance.

Work out the Lagrangian that generated these equations.

You should find that

$$E = \sum_{n}\left( (1/2)\L_{n}\dot{Q_{n}}^{2} + (1/2)C_{n}U_{n+1}^{2} \right)$$

and

$$H = (1/2)\sum_{n}\left( \frac{P_{n}^{2}}{L_n}} + \frac{(Q_{n+1}-Q_{n})^{2}}{C_{n}} \right)$$

2. Homework Equations

Note that
$$Q_{n} = -\int I_{n} dt => C_{n}U_{n+1} = Q_{n+1} - Q_{n}$$

3. The Attempt at a Solution

I have suggested a Lagrangian of

$$L = (1/2)L_{n}\dot{Q_{n}}^{2} + Q_{n}(U_{n+1}-U_{n}) + (1/2)C_{n}\dot{U_{n+1}^{2}} + U_{n+1}(I_{n}-I_{n+1})$$

(which can be turned into a sum for all the repeated circuits)

but it doesn't seem convincing, despite the fact that you can recover the original 'equations of motion' using Euler-Lagrange equations (using Q and U).

I'm not sure how the energy is being derived from the Lag. in this case (in mechanics, it was always the case that L = T - U, hence E = T + U), and it seems clear that you aren't going to get either that expression for the energy or that Hamiltonian from my guess-work Lagrangian.

Any suggestions?

Many thanks!

Related Advanced Physics Homework Help News on Phys.org
I don't think Ulf would like this any more than Chris.

malawi_glenn


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Definitions

# Parameter

[puh-ram-i-ter]
In mathematics, statistics, and the mathematical sciences, a parameter (G: auxiliary measure) is a quantity that defines certain characteristics of systems or functions. In different contexts the term may have special usage.

## Examples

• In a section on frequently misused words in his book The Writer's Art, James J. Kilpatrick quoted a letter from a correspondent, giving examples to illustrate the correct use of the word parameter:

W.M. Woods...a mathematician...writes... "...a variable is one of the many things a parameter is not." ... The dependent variable, the speed of the car, depends on the independent variable, the position of the gas pedal.
[Kilpatrick quoting Woods] "Now...the engineers...change the lever arms of the linkage...the speed of the car...will still depend on the pedal position...but in a...different manner. You have changed a parameter"

• A parametric equaliser is an audio filter that allows the frequency of maximum cut or boost to be set by one control, and the size of the cut or boost by another. These settings, the frequency level of the peak or trough, are two of the parameters of a frequency response curve, and in a two-control equaliser they completely describe the curve. More elaborate parametric equalisers may allow other parameters to be varied, such as skew. These parameters each describe some aspect of the response curve seen as a whole, over all frequencies. A graphic equaliser provides individual level controls for various frequency bands, each of which acts only on that particular frequency band.
• If asked to imagine the graph of the relationship y = ax2, one typically visualizes a range of values of x, but only one value of a. Of course a different value of a can be used, generating a different graphical appearance. The a can therefore be considered to be a parameter: less variable than the variable x, but less constant than the constant 2.

## Parameters in various contexts in math and science

### Mathematical functions

Mathematical functions typically can have one or more variables and zero or more parameters. The two are often distinguished by being grouped separately in the list of arguments that the function takes:

$f\left(x_1, x_2, dots; a_1, a_2, dots\right) = cdots,$

The symbols before the semicolon in the function's definition, in this example the $x$'s, denote variables, while those after it, in this example the $a$'s, denote parameters.

Strictly speaking, parameters are denoted by the symbols that are part of the function's definition, while arguments are the values that are supplied to the function when it is used. Thus, a parameter might be something like "the ratio of the cylinder's radius to its height", while the argument would be something like "2" or "0.1".

In some informal situations people regard it as a matter of convention (and therefore a historical accident) whether some or all the arguments of a function are called parameters.

In the special case of parametric equations the independent variables are called the parameters.

### Analytic geometry

In analytic geometry, curves are often given as the image of some function. The argument of the function is invariably called "the parameter". A circle of radius 1 centered at the origin can be specified in more than one form:

• implicit form

$x^2+y^2=1$

• parametric form

$\left(x,y\right)=\left(cos ; t,sin ; t\right)$
where t is the parameter.
A somewhat more detailed description can be found at parametric equation.

### Mathematical analysis

In mathematical analysis, one often considers "integrals dependent on a parameter". These are of the form
$F\left(t\right)=int_\left\{x_0\left(t\right)\right\}^\left\{x_1\left(t\right)\right\}f\left(x;t\right),dx.$
In this formula, t is on the left-hand side the argument of the function F, and it is on the right-hand side the parameter that the integral depends on. When evaluating the integral, t is held constant, and so it considered a parameter. If we are interested in the value of F for different values of t, then, we now consider it to be a variable. The quantity x is a dummy variable or variable of integration (confusingly, also sometimes called a parameter of integration).

### Probability theory

In probability theory, one may describe the distribution of a random variable as belonging to a family of probability distributions, distinguished from each other by the values of a finite number of parameters. For example, one talks about "a Poisson distribution with mean value λ". The function defining the distribution (the probability mass function) is:

$f\left(k;lambda\right)=frac\left\{e^\left\{-lambda\right\} lambda^k\right\}\left\{k!\right\}.$
This example nicely illustrates the distinction between constants, parameters, and variables. e is Euler's Number, a fundamental mathematical constant. The parameter λ is the mean number of observations of some phenomenon in question, a property characteristic of the system. k is a variable, in this case the number of occurrences of the phenomenon actually observed from a particular sample. If we want to know the probability of observing k1 occurrences, we plug it into the function to get $f\left(k_1 ; lambda\right)$. Without altering the system, we can take multiple samples, which will have a range of values of k, but the system will always be characterized by the same λ.

For instance, suppose we have a radioactive sample that emits, on average, five particles every ten minutes. We take measurements of how many particles the sample emits over ten-minute periods. The measurements will exhibit different values of k, and if the sample behaves according to Poisson statistics, then each value of k will come up in a proportion given by the probability mass function above. From measurement to measurement, however, λ remains constant at 5. If we do not alter the system, then the parameter λ is unchanged from measurement to measurement; if, on the other hand, we modulate the system by replacing the sample with a more radioactive one, then the parameter λ would increase.

Another common distribution is the normal distribution, which has as parameters the mean μ and the variance σ².

It is possible to use the sequence of moments (mean, mean square, ...) or cumulants (mean, variance, ...) as parameters for a probability distribution.

### Statistics and econometrics

In statistics and econometrics, the probability framework above still holds, but attention shifts to estimating the parameters of a distribution based on observed data, or testing hypotheses about them. In classical estimation these parameters are considered "fixed but unknown", but in Bayesian estimation they are random variables with distributions of their own.

It is possible to make statistical inferences without assuming a particular parametric family of probability distributions. In that case, one speaks of non-parametric statistics as opposed to the parametric statistics described in the previous paragraph. For example, Spearman is a non-parametric test as it is computed from the order of the data regardless of the actual values, whereas Pearson is a parametric test as it is computed directly from the data and can be used to derive a mathematical relationship.

Statistics are mathematical characteristics of samples which can be used as estimates of parameters, mathematical characteristics of the populations from which the samples are drawn. For example, the sample mean ($overline X$) can be used as an estimate of the mean parameter (μ) of the population from which the sample was drawn.

## Other fields

Other fields use the term "parameter" as well, but with a different meaning.

### Logic

In logic, the parameters passed to (or operated on by) an open predicate are called parameters by some authors (e.g., Prawitz, "Natural Deduction"; Paulson, "Designing a theorem prover"). Parameters locally defined within the predicate are called variables. This extra distinction pays off when defining substitution (without this distinction special provision has to be made to avoid variable capture). Others (maybe most) just call parameters passed to (or operated on by) an open predicate variables, and when defining substitution have to distinguish between free variables and bound variables.

### Engineering

In engineering (especially involving data acquisition) the term parameter sometimes loosely refers to an individual measured item. This usage isn't consistent, as sometimes the term channel refers to an individual measured item, with parameter referring to the setup information about that channel.

"Speaking generally, properties are those physical quantities which directly describe the physical attributes of the system; parameters are those combinations of the properties which suffice to determine the response of the system. Properties can have all sorts of dimensions, depending upon the system being considered; parameters are dimensionless, or have the dimension of time or its reciprocal.

The term can also be used in engineering contexts, however, as it is typically used in the physical sciences.

### Computer science

When the terms formal parameter and actual parameter are used, they generally correspond with the definitions used in computer science. In the definition of a function such as

f(x) = x + 2,

x is a formal parameter. When the function is used as in

y = f(3) + 5 or just the value of f(3),

3 is the actual parameter value that is substituted for x, the formal parameter, in the function definition. These concepts are discussed in a more precise way in functional programming and its foundational disciplines, lambda calculus and combinatory logic.

In computing, parameters are often called arguments, and the two words are used interchangeably. However, some computer languages such as C define argument to mean actual parameter (i.e., the value), and parameter to mean formal parameter.

### Linguistics

Within linguistics, the word "parameter" is almost exclusively used to denote a binary switch in a Universal Grammar within a Principles and Parameters framework.


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# Proof of time independence for normalization of wavefunction

1. Dec 29, 2015

### Jimmy87

Hi pf, I am having trouble with understanding some of the steps involved in a mathematical proof that a normalized wavefunction stays normalized as time evolves. I am new to QM and this derivation is in fact from "An introduction to QM" by Griffiths. Here is the proof:

I am fine with most of the maths but I am struggling with a few points:

Equation 21. He refers to this later and says "Ordinarily we need to justify taking a derivative inside an integral, but we can console ourselves with the knowledge that for any wave function for which this is not a valid operation, that function is also not a possible description of a physical state (for one reason or another)".

Why do you need to justify taking a partial derivative inside an integral? Is this not normally a valid operation?

Equation 25. I really don't understand how you can set a second derivative of a function equal to a first derivative? He says further on that the expression in the brackets at the start of equation 25 can be expressed as a single derivative - that doesn't make any sense to me why you can do this?

Equation 26. Why is there no integral sign on the right hand side of the equation? The expression on the right hand side is the partial derivative of psi squared with respect to time (as shown in equation 22) whereas they are implying the right hand side of equation 26 is the integral of the partial derivative with respect to time of psi squared (i.e. the integration has been done hence no integral sign)?

Thanks for any help!

2. Dec 29, 2015

### Haborix

21: By taking the derivative inside the integral you require certain assumptions on the continuity of the function and its derivatives. If the function didn't satisfy these conditions then changing the order of integration and differentiation would not hold.

25: It's basically inverting the product rule. Take the derivative on the RHS of 25 to convince yourself it is the same expression.

The reason to do the previous step is that you would like to appeal to the first fundamental theorem of calculus. By integrating a derivative of a function you are able to just evaluate the function at the end points. This leads to 26. The integral has been done using the first fundamental theorem (note the evaluation at the boundary).

3. Dec 29, 2015

### Jimmy87

25. I think I get what you are saying. On the RHS there is a d/dx and if you carry this out then you get back to the LHS. However, this d/dx seems to disappear in equation 26 and without the d/dx surely it is not equal to the LHS of equation 25?

26. I still don't follow. The RHS of this equation hasn't been integrated. The limits have just been stuck on the end. From how I see it, its like saying the integral of (x^2) evaluated at two limits is (x^2) with the limits added whereas the integral of x^2 is (x^3/3) evaluated at those limits.

Unless integrating the RHS of equation 25 makes the d/dx go away? Is that what you are saying? I am just making guesses based on the outcomes.

4. Dec 29, 2015

### Haborix

The d/dx does disappear and that is because you have integrated the expression. Notice that the LHS of 26 is the integration of the LHS of 25. Similarly the integration of the RHS of 25 has already been done when it is written in 26. This was done using the following: $\int_a^bF'(x) dx=F(x)|_a^b$

5. Dec 30, 2015

### Jimmy87

Thanks again. So on the RHS of 26 do you insert the limits where the curly d's are? So if you were to write it out fully when you evaluate the limits would it be:

I have called them a and b for plugging in the limits + and - infinity. And both of these equations would be equal to zero, is that right?

Could you also confirm the reason for having an integration and a partial derivative at the same time. I believe it is because the normalization needs an integration since it is the integral over all of space of psi squared. The reason for the partial derivative is because you want to see how the normalized wave function evolves with time. Is that the right way to think about it? Do you recommend any sources that discuss integrating derivatives because we haven't covered it and the book seems to assume the reader knows how to do this. All the other maths I have done but I have never come across integrals involving partial derivatives.

6. Dec 30, 2015

### vanhees71

I don't understand what the infinity symbols should mean in the just quoted equations. I guess, it's just the usual proof of the conservation of the norm of the wave function. It follows from unitarity of the time evolution, i.e., the self-adjointness of the Hamiltonian. In the position representation ("wave mechanics"), you just use Schrödinger's equation
$$\hbar \mathrm{i} \partial_t \psi(t,\vec{x})=\hat{H}(t,\vec{x}).$$
The norm is
$$\langle \psi|\psi \rangle=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} |\psi(t,\vec{x})|^2.$$
The time derivative is
$$\frac{\mathrm{d}}{\mathrm{d} t} = \langle \psi|\psi \rangle=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} [\psi^{*} \partial_t \psi+(\partial_t \psi)^* \psi] = (\partial_t \langle \psi|)|\psi \rangle + \langle \psi|\partial_t \psi \langle=\langle -\mathrm{i} \hat{H}/\hbar|psi \rangle + \langle \psi|-\mathrm{i} \hat{H}/\hbar \rangle = \langle \psi|+\mathrm{i} \hat{H}/\hbar \rangle+ \langle \psi|-\mathrm{i} \hat{H}/\hbar \rangle=0.$$


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# How can I position long tables at multicolumn

I am trying to print two long table which are can't be fit into one page side to side at multicolumns. The problem is whenever realize the table which is continuing along pages with longtable and multicol, they start not to be positioned side to side.

Thanks.

• you can add space between the two halves by @{\hspace{3cm}} you would have to use \cline rather than \hline so lines didn't cross the whole thing. Not ideal but TeX doesn't really support two parallel flows both taking part in page breaking. It is possible to have a custom output routine for one table that saves everything in boxes and then a custom output routine for the second that adds one page box from the first table each time it outputs a page but it would mean extensive rewrites to longtable internals. – David Carlisle Jun 27 '13 at 10:46


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In Bayesian nonparametrics, many models address the problem of density regression, including covariate dependent processes. These were settled by the pioneering works by [current ISBA president] MacEachern (1999) who introduced the general class of dependent Dirichlet processes. The literature on dependent processes was developed in numerous models, such as nonparametric regression, time series data, meta-analysis, to cite but a few, and applied to a wealth of fields such as, e.g., epidemiology, bioassay problems, genomics, finance. For references, see for instance the chapter by David Dunson in the Bayesian nonparametrics textbook (edited in 2010 by Nils Lid Hjort, Chris Holmes, Peter Müller and Stephen G. Walker). With Kerrie Mengersen and Judith Rousseau, we have proposed a dependent model in the same vein for modeling the influence of fuel spills on species diversity (arxiv).

Several densities can be plotted on the same 3D plot thanks to the Plotly R library, “an interactive, browser-based charting library built on the open source JavaScript graphing library, plotly.js.”

In our ecological example, the model provides a series of densities on the Y axis (in our case, posterior density of species diversity), indexed by some covariate X (a pollutant). See file density_plot.txt. The following Plotly R code

library(plotly)
df = as.data.frame(mydata)
plot_ly(df, x = Y, y = X, z = Z, group = X, type = "scatter3d", mode = "lines")

provides a graph as below. For the interactive version, see the RPubs page here.


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# Fundamental Theorem of Arithmetic

Posted on Sun 21 March 2021 in math

Proof of the Fundamental Theorem of Algebra's uniqueness part without utilizing Bézout's lemma.

Lemma (Euclid)

Let $p$ be prime. From $p \mid ab \implies p \mid a$ or $p \mid b$.

Proof:

Let $np = ab$ and $ab$ a minimal counterexample. We assume, that $p \nmid a$ and show, that $p \mid b$.

\begin{aligned} np &= ab &|-bp\\ (n-b)p &= (a-p)b \end{aligned}

$a=p$:

trivial

$a>p$:

$\textrm{From } p \nmid a \implies p \nmid (a-p) \textrm{ and with } (a-p)b < ab \implies p \mid b$

$a: \begin{aligned} (n-b)p &= (a-p)b & &|\cdot(-1)\\ (b-n)p &= (p-a)b,\quad p:=xa+y \quad (y

Theorem (Fundamental Theorem of Arithmetic)

Every $n \in \mathbb{N}$ has an unique prime factorization.

Proof:

Let \begin{aligned} a = \prod_{i=1}^n p_i^{k_i} = \prod_{i=1}^m q_i^{l_i} \end{aligned} and a the smallest number with a non-unique decomposition by primes. From Euclid's lemma follows, that for some (s,t) $p_s = q_t$. With $a := p_s\hat{a}$: \begin{aligned} \hat{a} = p_s^{k_s-1}\prod_{i=1, i\neq s}^n p_i^{k_i} = q_t^{l_t-1}\prod_{i=1, i\neq t}^m q_i^{l_i} \end{aligned} Because $\hat{a}


text:
Math Help - need urgent help

1. need urgent help

I need help with these, step by step solution PLEASE HELP

3-2/y
1/y+4

2x- x 2 - x
2-x

x-1 - x+1
x+1 x-1
x-1
+ x+1
x+1 x-1

2. Re: need urgent help

Originally Posted by flor
I need help with these, step by step solution PLEASE HELP

3-2/y
1/y+4

2x- x 2 - x
2-x

x-1 - x+1
x+1 x-1
x-1
+ x+1
x+1 x-1
None of these make any sense. Please rewrite it using parenthesis.

-Dan

3. Re: need urgent help

Hello, flor!

I can guess what the last one says.
The others need parentheses and careful typing.

$\displaystyle \frac{\;\dfrac{x-1}{x+1} - \dfrac{x+1}{x-1}\;}{\dfrac{x-1}{x+1} + \dfrac{x+1}{x-1}}$

Multiply top and bottom by $(x-1)(x+1)$

. . $\dfrac{(x-1)(x+1)\left(\dfrac{x-1}{x+1} - \dfrac{x+1}{x-1}\right)}{(x-1)(x+1)\left(\dfrac{x-1}{x+1} + \dfrac{x+1}{x-1}\right)} \;=\;\dfrac{(x-1)^2 - (x+1)^2}{(x-1)^2 + (x+1)^2}$

. . $=\;\frac{(x^2-2x+1) - (x^2 + 2x + 1)}{(x^2-2x+1) + (x^2 + 2x + 1)} \;=\;\frac{\text{-}4x}{2x^2+2} \;=\;\frac{\text{-}2x}{x^2+1}$


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# American Institute of Mathematical Sciences

June  2008, 21(2): 445-475. doi: 10.3934/dcds.2008.21.445

## Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity

 1 Departamento de Fisica Aplicada I, Escuela Universitaria Politénica, C/ Virgen de Africa, 7, University of Sevilla, 41011 Sevilla, Spain 2 Maxwell Institute and Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, United Kingdom 3 Department of Mathematics, University of the Aegean, Karlovassi 83200, Samos, Greece

Received  January 2007 Revised  December 2007 Published  March 2008

We consider the question of existence of periodic solutions (called breather solutions or discrete solitons) for the Discrete Nonlinear Schrödinger Equation with saturable and power nonlinearity. Theoretical and numerical results are proved concerning the existence and nonexistence of periodic solutions by a variational approach and a fixed point argument. In the variational approach we are restricted to DNLS lattices with Dirichlet boundary conditions. It is proved that there exists parameters (frequency or nonlinearity parameters) for which the corresponding minimizers satisfy explicit upper and lower bounds on the power. The numerical studies performed indicate that these bounds behave as thresholds for the existence of periodic solutions. The fixed point method considers the case of infinite lattices. Through this method, the existence of a threshold is proved in the case of saturable nonlinearity and an explicit theoretical estimate which is independent on the dimension is given. The numerical studies, testing the efficiency of the bounds derived by both methods, demonstrate that these thresholds are quite sharp estimates of a threshold value on the power needed for the the existence of a breather solution. This it justified by the consideration of limiting cases with respect to the size of the nonlinearity parameters and nonlinearity exponents.
Citation: J. Cuevas, J. C. Eilbeck, N. I. Karachalios. Thresholds for breather solutions of the discrete nonlinear Schrödinger equation with saturable and power nonlinearity. Discrete & Continuous Dynamical Systems, 2008, 21 (2) : 445-475. doi: 10.3934/dcds.2008.21.445
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# Tag Archives: Avoided Crossing

## Screening, Plasmons and LO-TO Splitting: One Last Time

I hope you are not by now fed up with my posts on this topic, but there is a great paper by Mooradian and Wright, which I’ve actually linked before in a different context, that is worth tackling. In this paper, they discuss the concept of plasmon-longitudinal optical (LO) phonon coupling.

To my mind, there is a significant aspect of the data which they do not explicitly address in their paper. Of course, I’m referring to degeneracy, LO-TO splitting, and screening. An image of their data is shown below (click to enlarge):

A quick run-down of the experiment: they are using Raman scattering on several different samples of GaAs with different doping levels. The carrier density can be read off in the image above.

It can be seen that for lower doped samples of GaAs, that there is strong LO-TO splitting. This is because of the long-ranged nature of the Coulomb interaction, as detailed here. As the carrier concentration is increased beyond the plasmon-phonon mixing region, the LO-TO splitting starts to disappear. This observation is noteworthy because there exists a “critical carrier density”, beyond which the LO and TO phonons are degenerate.

One can think of this in the following way: the plasmon energy is a measure of how quickly the free carriers can respond to an electric field. Therefore, for the highly doped GaAs samples, where the plasmon is at a significantly higher energy than the phonons, the free carriers can quickly screen the Coulomb field set up by the polar lattice. The electric field that is set up by the phonons can hence be approximated by a screened electric field (of the Thomas-Fermi kind) in this limit, and the Coulomb interaction is hence no longer long-ranged.

While the points I have made above will be quite obvious to many of you, I still find the data and its implications from a historical perspective quite profound.

Aside: I was heartened by Sarang’s post on the concept of emergence and upward heritability. One tends to think harder about one’s stance when there is an opposing view. He made some extremely important points regarding this topic, though I have to admit that I still lean towards Wilczek-ian concepts at present.

## A Little More on LO-TO Splitting

In my previous post, I addressed the concept of LO-TO splitting and how it results from the long-ranged nature of the Coulomb interaction. I made it a point to emphasize that while the longitudinal and transverse optical phonons are non-degenerate near $\textbf{q}=0$, they are degenerate right at $\textbf{q}=0.$ This scenario occurs because of the retarded nature of the Coulomb interaction (i.e. the finite speed of light).

What exactly goes on? Well, it so happens that in a very narrow momentum window near $\textbf{q}=0$, the transverse optical phonon is strongly coupled to light and forms a polariton. This is a manifestation of the avoided crossing or level repulsion principle that I have blogged about previously. Since light is a transverse wave, it interacts with the transverse optical phonon (but not the longitudinal one).

In a tour-de-force experiment at Bell Labs by Henry and Hopfield (the same Hopfield of Hopfield neural networks), Raman scattering was conducted at grazing incident angles to measure the dispersion of the lower polariton branch as shown below:

The dispersing solid lines represent the transverse optical (TO) phonon interacting with light. The straight solid line is the unaffected longitudinal optical (LO) phonon branch. The dotted line labelled with the angles are the incident beam angles in the Raman experiment. The remaining dotted lines represent the non-interacting TO phonon and the non-interacting light dispersion.

Usual Raman measurements are taken in a backscattering as opposed to a grazing incidence geometry, hence the momentum transfers are ordinarily too high to observe the low-$\textbf{q}$ dispersion. Because of this, the authors mentioned that some Raman exposures in this experiment took up to seven hours!

The takeaway from the plot above is that the transverse optical phonon at $\textbf{q}=0$ is degenerate with the longitudinal one and “turns into a photon” at higher momenta, while the photon branch at $\textbf{q}=0$ “turns into the transverse optical phonon” at higher momenta.

Unfortunately, the paper does not contain their raw data, only the dispersion. Publishing standards seem to have been different back then. Nonetheless, this is a very clever and illuminating experiment.

## Avoided Crossings, Level Repulsion or Anti-Crossing

The question of avoided crossing arises frequently in the study of condensed matter systems and always seems surprising to younger graduate students. In condensed matter physics, avoided crossings are ubiquitous, arising in phonon spectra, electronic band spectra and often when different types of quasi-particles interact.

Let me first say that avoided crossings are the norm and degenerate energies only occur because of some strict constraint (usually due to symmetry). As an example, let’s take the simple s-d Hamiltonian of a solid:

$-t\sum_i c^+_i c_{i+1} + h.c. + E_0 \sum_i d^+_i d_i + \sum_i \Delta c^+_id_i + h.c.$

where the $c^+_i$ creates an electron in an s-orbital, $d^+_i$ creates an electron in a d-orbital, $t$ is the hopping parameter, $E_0$ is the energy of an electron in a d-orbital and $\Delta$ is the hopping amplitude from an s- to a d-orbital. One can obtain the energy dispersion of this Hamiltonian by diagonalizing the following matrix:

$h(k) = \left( \begin{array}{cc} 2t\cos(ka) & \Delta \\ \Delta^* & E_0 \end{array} \right)$

Here are the resulting bands for a couple different values of $\Delta$ (click to enlarge):

While in the literature this is usually called a gap, this is nothing more than an avoided crossing and will occur for any finite $\Delta$. Again, this avoided crossing is not limited to electron energy bands, but can occur in many situations. Here is an old experiment by Mooradian and Wright from the 60’s which shows coupling between a plasmon and an optical phonon (click to enlarge):

One can very clearly see the plasmon (the broad peak) “becomes a logitudinal optical phonon” (the narrow peak) and the optical phonon “becomes the plasmon”. Notably, one can see that there is no coupling to the other (transverse) optical phonon because of symmetry reasons (plasmons cannot couple to transverse phonons), and it therefore stays put.


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# Sample of asymptotic mean distribution

## The asymptotic distribution of the bootstrap sample mean

The asymptotic distribution of the bootstrap sample mean. Asymptotic Normality of the GMM Estimator To develop the asymptotic distribution of the estimator, we require an asymptotically valid closed form representation for в€љ T(bОё T в€’Оё0) This representation comes from an application of the Mean Value Theorem. Let GT = 1 T XT t=1 в€‚g(v,Оё) в€‚Оё0 The Mean Value Theorem implies that g(bОё T) = g, ) be the ordinary sample mean vector; then we have S n = X n &X n,(2) provided that X n {0, so that indeed S n is the sample mean direction. We study the asymptotic behavior of the empirical mean direction under much weaker conditions than, e.g., those in Watson (1983). In particular, the assumption of rotational symmetry (around some axis) for.

### Asymptotic distribution of GMM Estimator

Point estimation of the mean Statlect. 4.3 The Large Sample Region with Plug-in Values for the Asymptotic Variances and an F Distribution This third approximation to the exact region is merely a modi cation of the second. If plug-ins are used for the asymptotic variances, it might be prudent to change the asymptotic distribution from Лњ2 2to 2F;nв€’ (Douglas 1993). 1 Лњ 2 and Лњ, Asymptotic Normality of the GMM Estimator To develop the asymptotic distribution of the estimator, we require an asymptotically valid closed form representation for в€љ T(bОё T в€’Оё0) This representation comes from an application of the Mean Value Theorem. Let GT = 1 T XT t=1 в€‚g(v,Оё) в€‚Оё0 The Mean Value Theorem implies that g(bОё T) = g.

Empirical Distributions, Exact Sampling Distributions, Asymptotic Sampling Distributions The Mean of the Empirical Distribution In the special case where g(x) = x, we get the mean of the a nite population from which we sample, or as the formula of the sample mean, if x1, :::, xn is considered a sample from a speci ed population. 3. Remember that the support of the Poisson distribution is the set of non-negative integer numbers: To keep things simple, we do not show, but we rather assume that the regularity conditions needed for the consistency and asymptotic normality of the maximum likelihood estimator of are satisfied. The likelihood function

ASYMPTOTIC DISTRIBUTION OF MAXIMUM LIKELIHOOD ESTIMATORS 1. INTRODUCTION For a random sample, X = (X1... Xn), the likelihood function is product of the individual density func-tionsand the log likelihood function is the sum of theindividual likelihood functions, i.e., ASYMPTOTIC DISTRIBUTION OF MAXIMUM LIKELIHOOD ESTIMATORS 1. INTRODUCTION For a random sample, X = (X1... Xn), the likelihood function is product of the individual density func-tionsand the log likelihood function is the sum of theindividual likelihood functions, i.e.,

The asymptotic distribution of the bootstrap sample mean of an infinitesimal array J. A. Cuesta-Albertos ; C. MatrГЎn Annales de l'I.H.P. ProbabilitГ©s et statistiques (1998) Asymptotic and Finite-Sample Properties in Statistical Estimation The sample mean XN n has n 1I thus XN n is heavy-tailed with the tail index m for any n <1: 2.2 Estimation of Shift Parameter, Non-identically Distributed Observations Asymptotic and Finite-Sample Properties 383

On the Asymptotic Joint Normality of Quantiles From a M ultivariateDistribution 1 Lionel Weiss 2 (Decembe~ 18, 1963) A simple proof is given of the asymptotic joint normality of sample quantiles from a multivariate population, under very mild conditions. вЂ¦ The asymptotic distributions of the empirical quantle process (Cdrgo and RCvCsz [6]) and of a fixed set of specified quantiles (Mosteller [ 111) in one dimension are well known. Of particular interest is the joint asymptotic distribution of the marginal sample medians tm, . ..Y fpct,, (1.10)

Lecture 3 Properties of MLE: consistency, asymptotic normality. converges in distribution to normal distribution with zero mean and variance n в†’, where П•0 is the вЂ™trueвЂ™ unknown parameter of the distribution of the sample. 2. Asymptotic Normality. We say that П•Л†is asymptotically normal if 20.05.2015В В· An Introduction to the Asymptotic Behaviour of Estimators Mean and Variance of Normal Distribution - Duration: Deriving the Mean and Variance of the Sample Mean - Duration:

Important note the Xn and Yn in the equation have an upper bar and are the sample average of xi and yi. So I looked at the following first: it is known a the central limit theorem that the sample mean of any distribution has an asymptotic gaussian distribution. 4.3 The Large Sample Region with Plug-in Values for the Asymptotic Variances and an F Distribution This third approximation to the exact region is merely a modi cation of the second. If plug-ins are used for the asymptotic variances, it might be prudent to change the asymptotic distribution from Лњ2 2to 2F;nв€’ (Douglas 1993). 1 Лњ 2 and Лњ

Remember that the support of the Poisson distribution is the set of non-negative integer numbers: To keep things simple, we do not show, but we rather assume that the regularity conditions needed for the consistency and asymptotic normality of the maximum likelihood estimator of are satisfied. The likelihood function that the asymptotic distribution of Efron's bootstrap sample mean is random, if the parent sample is obtained from a distribution in the domain of attraction of a non-normal stable law and the resampling and the parent sample sizes coincide. Thus, the bootstrap does not work in this case.

sample of size, let us say, mn=n1=2, the bootstrap mean will mimic the asymptotic distribution of the sample mean. However, even assuming that the bootstrap works in degenerate distribution since it converges on a single value. However, it is often possible to study the distribution of ON as it approaches a degenerate distribution. Studies of these asymptotic or limiting distributions are useful in situations where the finite sample вЂ¦

Lecture 3 Properties of MLE: consistency, asymptotic normality. converges in distribution to normal distribution with zero mean and variance n в†’, where П•0 is the вЂ™trueвЂ™ unknown parameter of the distribution of the sample. 2. Asymptotic Normality. We say that П•Л†is asymptotically normal if , so the asymptotic distribution of the left-hand side is identical to the asymptotic distribution of the term 2 2 1 1 n i i nX n PV В§В· ВЁВё В©В№, which leads to eq.[1]. But for a uniform dichotomous random variable this result is not valid, because we have aU a b PV22 ( ) 2 2, a constant, whether U takes the value or b.

A Primer on Asymptotics Eric Zivot Department of Economics University of Washington September 30, 2003 Revised: January 7, 2013 1 Introduction The two main concepts in вЂ¦ , so the asymptotic distribution of the left-hand side is identical to the asymptotic distribution of the term 2 2 1 1 n i i nX n PV В§В· ВЁВё В©В№, which leads to eq.[1]. But for a uniform dichotomous random variable this result is not valid, because we have aU a b PV22 ( ) 2 2, a constant, whether U takes the value or b.

Asymptotic distributions of sample mean and ACVF. The asymptotic distribution of the bootstrap sample mean of an infinitesimal array J. A. Cuesta-Albertos ; C. MatrГЎn Annales de l'I.H.P. ProbabilitГ©s et statistiques (1998), But when in practice when there is only one sample, asymptotic properties must be established. The aim is then to study the behavior of estimators as n, or the sample population size, increases. The asymptotic properties an estimator may possess include asymptotic вЂ¦.

### A.R.E. of Sample mean to Sample Median

Asymptotic distributions of the sample mean. The asymptotic distributions of the empirical quantle process (Cdrgo and RCvCsz [6]) and of a fixed set of specified quantiles (Mosteller [ 111) in one dimension are well known. Of particular interest is the joint asymptotic distribution of the marginal sample medians tm, . ..Y fpct,, (1.10), On the Asymptotic Joint Normality of Quantiles From a M ultivariateDistribution 1 Lionel Weiss 2 (Decembe~ 18, 1963) A simple proof is given of the asymptotic joint normality of sample quantiles from a multivariate population, under very mild conditions. вЂ¦.

Asymptotic distributions of the sample mean DeepDyve. degenerate distribution since it converges on a single value. However, it is often possible to study the distribution of ON as it approaches a degenerate distribution. Studies of these asymptotic or limiting distributions are useful in situations where the finite sample вЂ¦, Important note the Xn and Yn in the equation have an upper bar and are the sample average of xi and yi. So I looked at the following first: it is known a the central limit theorem that the sample mean of any distribution has an asymptotic gaussian distribution..

### An estimate of the asymptotic standard error of the sample

Asymptotic and Finite-Sample Properties in Statistical. But when in practice when there is only one sample, asymptotic properties must be established. The aim is then to study the behavior of estimators as n, or the sample population size, increases. The asymptotic properties an estimator may possess include asymptotic вЂ¦ Asymptotic distribution of the sample average value-at-risk Stoyan V. Stoyanov Svetlozar T. Rachev September 30, 2007 Abstract In this paper, we prove a result for the asymptotic distribution of the sample average value-at-risk (AVaR) under certain regularity assumptions. The asymptotic distribution can be used to derive as-.

degenerate distribution since it converges on a single value. However, it is often possible to study the distribution of ON as it approaches a degenerate distribution. Studies of these asymptotic or limiting distributions are useful in situations where the finite sample вЂ¦ Important note the Xn and Yn in the equation have an upper bar and are the sample average of xi and yi. So I looked at the following first: it is known a the central limit theorem that the sample mean of any distribution has an asymptotic gaussian distribution.

The asymptotic distribution of the bootstrap sample mean of an infinitesimal array J. A. Cuesta-Albertos ; C. MatrГЎn Annales de l'I.H.P. ProbabilitГ©s et statistiques (1998) Asymptotic joint distribution of sample mean and sample variance 1 Is a t distribution for a certain degree of freedom equivalent to the sample mean distribution for the corresponding sample size?

Important note the Xn and Yn in the equation have an upper bar and are the sample average of xi and yi. So I looked at the following first: it is known a the central limit theorem that the sample mean of any distribution has an asymptotic gaussian distribution. Important note the Xn and Yn in the equation have an upper bar and are the sample average of xi and yi. So I looked at the following first: it is known a the central limit theorem that the sample mean of any distribution has an asymptotic gaussian distribution.

The asymptotic distribution of the bootstrap sample mean of an infinitesimal array J. A. Cuesta-Albertos ; C. MatrГЎn Annales de l'I.H.P. ProbabilitГ©s et statistiques (1998) , so the asymptotic distribution of the left-hand side is identical to the asymptotic distribution of the term 2 2 1 1 n i i nX n PV В§В· ВЁВё В©В№, which leads to eq.[1]. But for a uniform dichotomous random variable this result is not valid, because we have aU a b PV22 ( ) 2 2, a constant, whether U takes the value or b.

Asymptotic Normality of the GMM Estimator To develop the asymptotic distribution of the estimator, we require an asymptotically valid closed form representation for в€љ T(bОё T в€’Оё0) This representation comes from an application of the Mean Value Theorem. Let GT = 1 T XT t=1 в€‚g(v,Оё) в€‚Оё0 The Mean Value Theorem implies that g(bОё T) = g But when in practice when there is only one sample, asymptotic properties must be established. The aim is then to study the behavior of estimators as n, or the sample population size, increases. The asymptotic properties an estimator may possess include asymptotic вЂ¦

Asymptotic and Finite-Sample Properties in Statistical Estimation The sample mean XN n has n 1I thus XN n is heavy-tailed with the tail index m for any n <1: 2.2 Estimation of Shift Parameter, Non-identically Distributed Observations Asymptotic and Finite-Sample Properties 383 Asymptotic and Finite-Sample Properties in Statistical Estimation The sample mean XN n has n 1I thus XN n is heavy-tailed with the tail index m for any n <1: 2.2 Estimation of Shift Parameter, Non-identically Distributed Observations Asymptotic and Finite-Sample Properties 383

arXiv:1207.4242v1 [math.PR] 18 Jul 2012 Asymptotic Joint Distribution of Extreme Eigenvalues of the Sample Covariance Matrix in the Spiked Population Model Dai Shiв€— Abstract In Asymptotic and Finite-Sample Properties in Statistical Estimation The sample mean XN n has n 1I thus XN n is heavy-tailed with the tail index m for any n <1: 2.2 Estimation of Shift Parameter, Non-identically Distributed Observations Asymptotic and Finite-Sample Properties 383

Empirical Distributions, Exact Sampling Distributions, Asymptotic Sampling Distributions The Mean of the Empirical Distribution In the special case where g(x) = x, we get the mean of the a nite population from which we sample, or as the formula of the sample mean, if x1, :::, xn is considered a sample from a speci ed population. 3. sample of size, let us say, mn=n1=2, the bootstrap mean will mimic the asymptotic distribution of the sample mean. However, even assuming that the bootstrap works in

Consequently, confidence intervals based on interpolated adjacent order statistics or various approximations of the asymptotic distribution of the sample median can be used to achieve the desired Consequently, confidence intervals based on interpolated adjacent order statistics or various approximations of the asymptotic distribution of the sample median can be used to achieve the desired

RAC, v. 5, n. 3, Set./Dez. 2001 171 Convergent and Discriminant Validity of the Perceived Risk Scale suggested that the consequence dimension may be taken in consideration more seriously by buyersвЂ™ than the uncertainty dimension. However, despite some effort to determine the weighting relationships between these two factors, the Convergent validity pdf Blenheim Convergent Validity Convergent validity is problematic as various measures of impulsivity (e.g., continuous performance tests, matching familiar figure tests, and parent or teacher ratings) tend to correlate poorly with one another (Milich & Kramer, 1984).

An estimate of the asymptotic standard error of the sample. , so the asymptotic distribution of the left-hand side is identical to the asymptotic distribution of the term 2 2 1 1 n i i nx n pv в§в· вёвё в©в№, which leads to eq.[1]. but for a uniform dichotomous random variable this result is not valid, because we have au a b pv22 ( ) 2 2, a constant, whether u takes the value or b., asymptotic distribution of maximum likelihood estimators 1. introduction for a random sample, x = (x1... xn), the likelihood function is product of the individual density func-tionsand the log likelihood function is the sum of theindividual likelihood functions, i.e.,).

On the Asymptotic Joint Normality of Quantiles From a M ultivariateDistribution 1 Lionel Weiss 2 (Decembe~ 18, 1963) A simple proof is given of the asymptotic joint normality of sample quantiles from a multivariate population, under very mild conditions. вЂ¦ The asymptotic distributions of the empirical quantle process (Cdrgo and RCvCsz [6]) and of a fixed set of specified quantiles (Mosteller [ 111) in one dimension are well known. Of particular interest is the joint asymptotic distribution of the marginal sample medians tm, . ..Y fpct,, (1.10)

Remember that the support of the Poisson distribution is the set of non-negative integer numbers: To keep things simple, we do not show, but we rather assume that the regularity conditions needed for the consistency and asymptotic normality of the maximum likelihood estimator of are satisfied. The likelihood function The distribution of these means, or averages, is called the "sampling distribution of the sample mean". This distribution is normal (, /) (n is the sample size) since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not (see central limit theorem).

Asymptotic joint distribution of sample mean and sample variance 1 Is a t distribution for a certain degree of freedom equivalent to the sample mean distribution for the corresponding sample size? Asymptotic joint distribution of sample mean and sample variance. Ask Question _T-\mu_r \\ \bar{r^2}_T-E[r_t^2]\end{pmatrix}$$is sufficient to get the asymptotic joint distribution of the sample mean and sample variance? To get from here to the joint вЂ¦ But when in practice when there is only one sample, asymptotic properties must be established. The aim is then to study the behavior of estimators as n, or the sample population size, increases. The asymptotic properties an estimator may possess include asymptotic вЂ¦ Asymptotic distribution of the sample average value-at-risk Stoyan V. Stoyanov Svetlozar T. Rachev September 30, 2007 Abstract In this paper, we prove a result for the asymptotic distribution of the sample average value-at-risk (AVaR) under certain regularity assumptions. The asymptotic distribution can be used to derive as- Important note the Xn and Yn in the equation have an upper bar and are the sample average of xi and yi. So I looked at the following first: it is known a the central limit theorem that the sample mean of any distribution has an asymptotic gaussian distribution. Asymptotic and Finite-Sample Properties in Statistical Estimation The sample mean XN n has n 1I thus XN n is heavy-tailed with the tail index m for any n <1: 2.2 Estimation of Shift Parameter, Non-identically Distributed Observations Asymptotic and Finite-Sample Properties 383 The sample mean has smaller variance. In fact, since the sample mean is a suп¬ѓcient statistic for the mean of the distri-bution, no further reduction of the variance can be obtained by considering also the sample median. (2) The logistic: ПЂ2/34log2 4log2 4. Again the mean has smaller asymptotic variance. Point estimation of the mean. by Marco Taboga, PhD. This lecture presents some examples of point estimation problems, focusing on mean estimation, that is, on using a sample to produce a point estimate of the mean of an unknown distribution. Asymptotic distributions of sample mean and ACVF 11 Asymptotic normality of the sample mean and Edgeworth. point estimation of the mean. by marco taboga, phd. this lecture presents some examples of point estimation problems, focusing on mean estimation, that is, on using a sample to produce a point estimate of the mean of an unknown distribution., p.k. bhattacharya, prabir burman, in theory and methods of statistics, 2016. 12.9.3 asymptotic results in principal components analysis. asymptotic distributions of sample eigenvalues and sample eigenvectors are somewhat complicated. it is important to point out that the asymptotic distributions depend on the distribution of the multivariate population from which the observations are taken.); 4.3 the large sample region with plug-in values for the asymptotic variances and an f distribution this third approximation to the exact region is merely a modi cation of the second. if plug-ins are used for the asymptotic variances, it might be prudent to change the asymptotic distribution from лњ2 2to 2f;nв€’ (douglas 1993). 1 лњ 2 and лњ, asymptotic distribution of maximum likelihood estimators 1. introduction for a random sample, x = (x1... xn), the likelihood function is product of the individual density func-tionsand the log likelihood function is the sum of theindividual likelihood functions, i.e.,. Asymptotic distributions of sample mean and ACVF An estimate of the asymptotic standard error of the sample. 11 asymptotic normality of the sample mean and edgeworth expansions so far we have concentrated on showing asymptotic normality for anything under the sun. but this result, as the name suggests, is asymptotic. for п¬ѓnite samples, indeed small samples, this approximation can be quite poor. we recall if the original data appears to have the, that the asymptotic distribution of efron's bootstrap sample mean is random, if the parent sample is obtained from a distribution in the domain of attraction of a non-normal stable law and the resampling and the parent sample sizes coincide. thus, the bootstrap does not work in this case.). Poisson distribution Maximum likelihood estimation On the Asymptotic Behavior of the Contaminated Sample Mean. the distribution of these means, or averages, is called the "sampling distribution of the sample mean". this distribution is normal (, /) (n is the sample size) since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not (see central limit theorem)., the asymptotic distribution of the bootstrap sample mean of an infinitesimal array j. a. cuesta-albertos ; c. matrгўn annales de l'i.h.p. probabilitг©s et statistiques (1998)). Asymptotic Distribution YouTube Joint Con dence Sets for the Mean and Variance of a Normal. consequently, confidence intervals based on interpolated adjacent order statistics or various approximations of the asymptotic distribution of the sample median can be used to achieve the desired, ) be the ordinary sample mean vector; then we have s n = x n &x n,(2) provided that x n {0, so that indeed s n is the sample mean direction. we study the asymptotic behavior of the empirical mean direction under much weaker conditions than, e.g., those in watson (1983). in particular, the assumption of rotational symmetry (around some axis) for). [1406.4151] On the asymptotic distribution of the mean Asymptotic and Finite-Sample Properties in Statistical. the asymptotic distribution of the bootstrap sample mean of an infinitesimal array j. a. cuesta-albertos ; c. matrгўn annales de l'i.h.p. probabilitг©s et statistiques (1998), asymptotic distributions of sample mean and acvf felix dietrich and gundelinde wiegel university of trento 29th of october 2013 felix dietrich and gundelinde wiegel (utn) asymptotic distributions of вђ¦). Consequently, confidence intervals based on interpolated adjacent order statistics or various approximations of the asymptotic distribution of the sample median can be used to achieve the desired The sample mean has smaller variance. In fact, since the sample mean is a suп¬ѓcient statistic for the mean of the distri-bution, no further reduction of the variance can be obtained by considering also the sample median. (2) The logistic: ПЂ2/34log2 4log2 4. Again the mean has smaller asymptotic variance. , so the asymptotic distribution of the left-hand side is identical to the asymptotic distribution of the term 2 2 1 1 n i i nX n PV В§В· ВЁВё В©В№, which leads to eq.[1]. But for a uniform dichotomous random variable this result is not valid, because we have aU a b PV22 ( ) 2 2, a constant, whether U takes the value or b. But when in practice when there is only one sample, asymptotic properties must be established. The aim is then to study the behavior of estimators as n, or the sample population size, increases. The asymptotic properties an estimator may possess include asymptotic вЂ¦ 4.3 The Large Sample Region with Plug-in Values for the Asymptotic Variances and an F Distribution This third approximation to the exact region is merely a modi cation of the second. If plug-ins are used for the asymptotic variances, it might be prudent to change the asymptotic distribution from Лњ2 2to 2F;nв€’ (Douglas 1993). 1 Лњ 2 and Лњ The asymptotic distributions of the empirical quantle process (Cdrgo and RCvCsz [6]) and of a fixed set of specified quantiles (Mosteller [ 111) in one dimension are well known. Of particular interest is the joint asymptotic distribution of the marginal sample medians tm, . ..Y fpct,, (1.10) The correct asymptotic variance for the sample mean of a homogeneous Poisson Marked Point Process William Garner and Dimitris N. Politis Department of Mathematics University of California at San Diego La Jolla, CA 92093, USA Emails: wgarner@ucsd.edu; dpolitis@ucsd.edu Abstract The asymptotic variance of the sample mean of a homogeneous Poisson Asymptotic joint distribution of sample mean and sample variance. Ask Question _T-\mu_r \\ \bar{r^2}_T-E[r_t^2]\end{pmatrix}$$ is sufficient to get the asymptotic joint distribution of the sample mean and sample variance? To get from here to the joint вЂ¦

05.02.2019В В· Abstract. An observation of a cumulative distribution function F with finite variance is said to be contaminated according to the inflated variance model if it has a large probability of coming from the original target distribution F, but a small probability of coming from a contaminating distribution that has the same mean and shape as F, though a larger variance. On the Asymptotic Joint Normality of Quantiles From a M ultivariateDistribution 1 Lionel Weiss 2 (Decembe~ 18, 1963) A simple proof is given of the asymptotic joint normality of sample quantiles from a multivariate population, under very mild conditions. вЂ¦

The asymptotic distributions of the empirical quantle process (Cdrgo and RCvCsz [6]) and of a fixed set of specified quantiles (Mosteller [ 111) in one dimension are well known. Of particular interest is the joint asymptotic distribution of the marginal sample medians tm, . ..Y fpct,, (1.10) degenerate distribution since it converges on a single value. However, it is often possible to study the distribution of ON as it approaches a degenerate distribution. Studies of these asymptotic or limiting distributions are useful in situations where the finite sample вЂ¦

Asymptotic joint distribution of sample mean and sample variance. Ask Question _T-\mu_r \\ \bar{r^2}_T-E[r_t^2]\end{pmatrix} is sufficient to get the asymptotic joint distribution of the sample mean and sample variance? To get from here to the joint вЂ¦ Remember that the support of the Poisson distribution is the set of non-negative integer numbers: To keep things simple, we do not show, but we rather assume that the regularity conditions needed for the consistency and asymptotic normality of the maximum likelihood estimator of are satisfied. The likelihood function

Asymptotic distributions of the sample mean


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Main / Cards & Casino / Latex tabularx package

# Latex tabularx package

 Name: Latex tabularx package File size: 526mb Language: English Rating: 5/10 Download

tabularx – Tabulars with adjustable-width columns. The package and so on.) The package is part of the latex-tools bundle in the LaTeX required distribution. This package provides a table environment called tabularx, which is similar to the tabular* environment except that it has a. \documentclass{article} \usepackage{tabularx} X | X | X |} \begin{tabularx}{\ textwidth}{bss} \hline Alpha & Beta & Gamma \\ \hline 0 & 2 & 4.

7 Jan This package implements a version of the tabular environment in which which is useful in narrow columns is ragged right, however LATEX's. The tabularx package provides support for tables. The tabularx packagetabular. 20 Feb regular LATEX commands, a large number of packages can be used to . Table 4: Example of the standard LATEX multicolumn and cline.

1 Introduction; 2 Creating a simple table in LaTeX; 3 Tables with fixed length . entire table and then distribute the space within evenly, use the package tabu. The original source files were: %% %% pagineparole.com (with options: package') This file was generated from file(s) of the Standard LaTeX Tools Bundle'. LaTeX unfortunately does not include a good separation of text and style in tables. This package tries to solve this problem by defining reusable table commands. 29 Jul The tabularx package defines a tabular environment that generates a to the standard LaTeX \tabular* command; please refer to the package. tabularx package. tabularxp:: PackageName Source #. tabularx package. Use it to import it like this: usepackage [] tabularxp.

My default packages for a LaTeX document are the standard amsmath, of LaTeX falls to pieces when confronted with any unusual table requirements. The powerful new ctable package is supported and especially recommended when footnotes are needed. LaTeX::Table ships with some predefined, good. How to create a table in LaTeX with fixed width? The alignment of the cells could be changed by using the package tabularx and define three new row-types . Quickly create even complex LaTeX tables online with our generator -- cells merging professionally looking tables you probably want to try booktabs package.

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Math Home

# 4. Product Rule

### Product Rule

Let $$f$$ and $$g$$ be differentiable functions at the point $$a.$$ The product rule says that $(fg)'(a) = f'(a)g(a) + f(a)g'(a)$

Proof:
By the definition of derivative, $(fg)'(a) = \lim_{h \rightarrow 0} \frac{f(a+h)g(a+h) - f(a)g(a)}{h}$ Add and subtract $$f(a)g(a+h),$$ then factor. \begin{align} \lim_{h \rightarrow 0} \frac{f(a+h)g(a+h) - f(a)g(a)}{h} & = \lim_{h \rightarrow 0} \frac{f(a+h)g(a+h) - f(a)g(a+h) + f(a)g(a+h) - f(a)g(a)}{h} \\ & = \lim_{h \rightarrow 0} \frac{(f(a+h) - f(a))g(a+h) + f(a)(g(a+h) - g(a))}{h} \\ & = \lim_{h \rightarrow 0} \frac{(f(a+h) - f(a))g(a+h)}{h} + \lim_{h \rightarrow 0} \frac{f(a)(g(a+h) - g(a))}{h} \end{align} Now use $$\lim_{h \rightarrow 0} g(a+h) = g(a)$$ and the definition of the derivatives of $$f$$ and $$g$$ to get $\lim_{h \rightarrow 0} \frac{(f(a+h) - f(a))g(a+h)}{h} + \lim_{h \rightarrow 0} \frac{f(a)(g(a+h) - g(a))}{h} = f'(a)g(a)+f(a)g'(a)$

### Examples

Find the derivative of $$x^2\text{sin}(x).$$

Solution:
By the power rule, the derivative of $$x^2$$ is $$2x.$$ The derivative of $$\text{sin}(x)$$ is $$\text{cos}(x).$$ So, by the product rule, $\frac{d}{dx}x^2\text{sin}(x) = 2x\text{sin}(x) + x^2\text{cos}(x)$

Find the derivative of $$x^3e^x.$$

Solution:
By the power rule, the derivative of $$x^3$$ is $$3x^2.$$ The derivative of $$e^x$$ is $$e^x.$$ So, by the product rule, $\frac{d}{dx}x^3e^x = 3x^2e^x + x^3e^x$

Find the derivative of $$e^x\text{sin}(x).$$

Solution:
The derivative of $$e^x$$ is $$e^x$$ and the derivative of $$\text{sin}(x)$$ is $$\text{cos}(x).$$ So, by the product rule, $\frac{d}{dx}e^x\text{sin}(x) = e^x\text{sin}(x) + e^x\text{cos}(x)$


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# Multiple Integrals

• ### Introduction

The function $y=f(x)$ is a function of single variable $x$. $y$ is the dependent variable and $x$ is the independent variable. The indefinite integral

${\int ydx = g(x)}$

is a function, such that

${\frac {d}{dx} g(x) = y = f(x)}$

The definite integral

${\int \limits_a^b ydx}$

gives the area under the curve $y=f(x)$, bounded by the lines $x=a$, $x=b$ and the $X$ axis.

These are examples of single integrals, where only 1 independent variable is present. If the number of independent variables in a function is greater than 1, multiple integrals are used.

For functions with 2 independent variables, $f_1 (x,y)$, double integrals are evaluated and for functions with 3 independent variables, $f_2 (x,y,z)$, triple integrals are evaluated. It will soon become clear that

I) Single integrals are evaluated over a line.

II) Double integrals are evaluated over an area and

III) Triple integrals are evaluated over a 3D space.

• ### Double Integrals

Let $f(x,y)$ be a function of 2 independent variables $x$ and $y$, defined over a region $R$, bounded by a closed curve $C$. Let this region $R$ be divided into $n$ smaller regions, such that $r$th region has area $\delta A_r$. Let $P(x_r,y_r)$ be a point inside the $r$th region. Then the double integral over the region $R$ is defined as

${\iint_R f(x,y)dxdy = \lim \limits_{n \to \infty, \delta A \to 0} \sum \limits_{r=1}^{n} f(x_r,y_r) \delta A_r}$

A critical part in evaluating multiple integrals is choosing the upper and lower limits. For double integrals, the limits of the inner integral are to be expressed as functions of the other variable, which corresponds to the outer integral. For example, consider the figure below:

${\iint_R f(x,y)dxdy = \int \limits_{x=a}^{x=b} \int \limits_{y=f_1 (x)}^{y=f_2(x)} f(x,y)dydx}$

Thus, the inner integral is to be evaluated w.r.t. $y$, considering $x$ as a constant. Having obtained the inner integral, now, we should evaluate the outer integral as a single integral (the usual one). The limits of outer integral will be constants ($a$ and $b$ in this case).

If the limits of the inner integrals are functions of $y$, the outer integral will be w.r.t $y$ and the inner integral must be evaluated w.r.t. $x$.

Note that order of integration DOES NOT matter, as long as the limits are appropriately put.

I) Sometimes, it is required to change the order of integration, when the inner integral is impossible to integrate. In such cases, the limits are obtained by plotting the graph.

II) If the integrand involves terms of the form $x^2+y^2$ or $x^2y^2$, it becomes easier, if we transform the integrand to polar coordinates,

${x= r \ cos (\theta), y = r \ sin (\theta), r^2=x^2 + y^2, \theta = tan^{-1} \frac y x}$

Using Jacobians,

${dxdy = |J| d\theta dr = \begin {vmatrix} \frac {\partial (x,y)}{\partial (r, \theta)} \end {vmatrix} d \theta dr = r d \theta dr}$

Generally, the inner integral is obtained w.r.t. $r$.

III) If $f(x,y) =1$, we get the area of the region.

• ### Illustrated Examples

Question :

Evaluate the following over the region ${y \ge x^2, x \ge 1}$.

${\iint \frac {1}{x^4+y^2} dxdy}$

Solution :

The region of integration is plotted below :

Clearly, to the left of line ${x=1}$, the region is absent, hence the lower limit on ${x}$ will be ${1}$. The upper limit will be ${\infty}$. No restriction on the highest value of ${x}$.

Let’s now consider a strip parallel to ${Y}$ axis. It will first intersect the region at ${y=x^2}$. So, the lower limit on ${y}$ is ${x^2}$. The upper limit will be ${\infty}$ because no curve bounds the region in that direction. Hence, given integral will be

${\int \limits_{x=1}^{\infty} \int \limits_{y=x^2}^{\infty} \frac {1}{x^4+y^2} dydx = I \ say}$

Since inner limits are of ${y}$, let’s first integrate w.r.t. ${y}$.

${I = \int \limits_{x=1}^{\infty} \left [\frac {1}{x^2} tan^{-1} \left (\frac {y}{x^2} \right) \right]_{x^2}^{\infty}dx}$

${= \int \limits_{x=1}^{\infty} \frac {1}{x^2} \left[ \frac {\pi}{2} - \frac {\pi}{4} \right] dx}$

${= \frac {\pi}{4} \int \limits_1^{\infty} x^{-2}dx}$

${= \frac {\pi}{4} \left [\frac {x^{-2+1}}{-2+1} \right ]_{1}^{\infty}}$

${= \frac {\pi}{4} \times \frac {1}{-1} \times \left[\frac 1 x \right]_1^{\infty}}$

${= \frac {\pi}{4} \times -1 \times [0 - 1]}$

${= \frac {\pi}{4}}$

• ### Triple Integrals

As stated earlier, a triple integral is evaluated over a $3D \ space$. Let there be a function $f(x,y,z)$ defined over a closed region having volume $V$. Let the region be divided into $n$ subregions, such that the $r$th subregion has volume $\delta V_r$. Let $f(x_r,y_r,z_r$ be a point in $r$th subregion. Then the triple integral is given by

${\iiint \limits_{V} f (x,y,z) dxdydz = \lim \limits_{n \to \infty, \delta V_r \to 0} f(x_r,y_r,z_r) \delta V_r}$

The limits of integrals are equally important. The innermost integral (say w.r.t. $z$) has limits expressed in terms of functions of the other 2 variables, $x,y$. Once it is evaluated treating $x$ and $y$ as constants, it simplifies to a double integral problem and can be solved as explained earlier.

I) Transforming the integral into spherical polar coordinates requires the following replacement:

${dxdydz = |J|d \theta d \phi dr = r^2 sin (\theta) d \theta d \phi dr}$

II) Transforming the integral into cylindrical polar coordinates requires the following replacement:

${dxdydz = |J|d \rho d \phi dz = \rho d \rho d \phi dz}$

III) For a sphere $x^2 + y^2 + z^2 = a^2$, the limits are $r=0$ to $r = a$, $\theta = 0$ to $\theta = \pi$, $\phi = 0$ to $\phi = 2 \pi$

IV) For a hemisphere $x^2 + y^2 + z^2 = a^2, z \ge 0$, the limits are $r=0$ to $r = a$, $\theta = 0$ to $\theta = \frac \pi 2$, $\phi = 0$ to $\phi = 2 \pi$

IV) Dirichlet’s Theorem for triple integrals :

If $x+y+z \le 1$, then

${\iiint x^{a-1}y^{b-1} z^{c-1} dxdydz = \frac {\Gamma (a) \Gamma (b) \Gamma (c)}{\Gamma (a+b+c+1)}}$

• ### Areas and Volumes

I) When in a double integral, the function $f(x,y)=1$ for all values of $(x,y)$ in the region, the double integral gives the area of the region.

II) When in a triple integral, the function $f(x,y,z)=1$ for all $(x,y,z)$ in the closed region, the triple integral gives the volume of the region.

III) Other applications include obtaining center of mass, center of gravity, mean values, RMS values, moment of inertia etc.


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Researchers increase carbon efficiency of BtL process from 38 to >90% by adding hydrogen from renewable sources; 2.4x more fuel produced

12 August 2018

Researchers in Norway report that the carbon efficiency of a conventional Biomass-to-Liquid (BtL) process can be increased from 38 to more than 90% by adding hydrogen from renewable energy sources. The increased carbon efficiency is possible because the water gas shift reaction is avoided and instead a reversed water gas shift is introduced to convert CO2 to CO.

This means that the amount of fuel can be increased by a factor of 2.4 with the same amount of biomass. In addition, the emitted amount of CO2 per produced unit can be reduced by a factor of 16.

They also determined that there is a near linear relationship between added hydrogen and production of surplus FT-products—meaning that maximum hydrogen addition is favorable.

Based on their evaluation of process technology and economics, the researchers concluded that converting excess renewable electric power to advanced biofuels in a power and biomass to Fischer-Tropsch liquid fuels (PBtL) plant is a sensible way of storing energy as a fuel with a relatively high energy density. Their open-access paper is published in the journal Fuel.

Energy flow of the staged PBtL concept. The steam produced from the FT reactors (FT cooling) is energy that partly can be utilized.

In their process, hydrogen is produced through high temperature steam electrolysis in a solid oxide electrolysis cell (SOEC), with high temperature steam generated from the hot syngas from the Fischer-Tropsch biomass-to-liquids process. The oxygen produced from the SOEC is sufficient as oxidant in the gasifier, thereby eliminating the need for a cryogenic air separation unit.

The use of high-temperature electrolysis allows some of the required energy to be supplied by available heat.

The required electrical power for the extra production is estimated to be 11.6 kWh per liter syncrude (C5+). By operating the SOEC iso-thermally close to 850 °C the electric energy may be reduced to 9.5 kWh per liter—close to the energy density of jet fuel.

With an electrical power price of 0.05 $/kWh and with SOEC investment cost of the 1000$/kW(el), the levelized cost of producing advanced biofuel with the PBtL concept is $1.70/liter ($6.40/gallons US)—approximately 30% lower than for conventional BtL.

Resources

• M. Hillestad, M. Ostadi, G.d. Alamo Serrano, E. Rytter, B. Austbø, J.G. Pharoah, O.S. Burheim (2018) “Improving carbon efficiency and profitability of the biomass to liquid process with hydrogen from renewable power,” Fuel, Volume 234, Pages 1431-1451 doi: 10.1016/j.fuel.2018.08.004

About time someone got a clue about this.  Fixed carbon has always been the limiting factor in biofuels production; any process which wastes it as CO2 is throwing away the most precious part of the feedstock.

But at $6.40/gallon production cost, I don't think this scheme is going anywhere. I posted years ago, add more hydrogen you get more fuel. I have no idea where the authors are going to find a solid oxide electrolysis cell (SOEC) investment cost of 1000$/kWe.

However, I agree that utilizing green hydrogen is a great way to improve the technoeconomic viability of any X-to-fuels process.

Best,

--Keith
@KeithDPatch

Looking more closely at the numbers, this scheme requires about 44 kWh per gallon of product.  That is roughly what it takes to produce one GGe of hydrogen (1 kg), and the hydrogen can be used much more efficiently.

As electricity it can be used far more efficiently still.  43.9 kWh will send a Tesla Model S 115 miles, or a Chevy Bolt 188 miles.  One gallon of synthetic gasoline will drive a Prius some 50 miles or so.

Note that this is WITH the assistance of the embodied energy of the biomass!  Any practical scheme has to do MUCH better than this.

This concept was a well-known in Sweden already at least 15 years ago from work by Chemrec and others. If you synthetize "hydrogen-rich" fuels, such as e.g. methanol or DME, instead of hydrocarbon fuels (gasoline and diesel fuels), you need – or can utilize – even more hydrogen. (For various reasons, Chemrec did not succeed commercially but this is a completely different story.)

In a situation where the refinery needs to shift the balance from gasoline to diesel fuel, you also need more hydrogen. This is the current situation in Europe. Perhaps this will change if (?) market penetration of diesel cars decreases. However, the most energy-efficient solution would be increasing the share of diesel cars, particularly now when we already know that the NOx problem has been solved. Well, at least this problem will be solved no later than September 1 this year, when the new Euro 6d-TEM regulation is enforced. In a scenario of increased use of diesel fuel, more hydrogen would be needed in the refineries.

From an energy efficiency perspective, it makes more sense to use any non-fossil hydrogen in (fossil) refineries and biorefineries rather than opting for direct use of hydrogen in vehicles. If such hydrogen could be produced cost-effectively, there would, for sure, be a big market waiting without any need for change in the fuel supply chain. However, I think there will be some problems with production cost…

There are several "flaws" in this study that everyone has pointed out.
The use of Hydrogen (or Hydrotreating) is a standard practice in the petrochemical industry for fuel upgrading or removing impurities, e.g. Sulfur or heavy metals. Though this Hydrogen comes from Steam Methane Reforming not renewable sources.
Also, why use SOEC for Electrolysis? Maybe this has something to do with SINTEF research where one of the authors works. Why not include Zero-Gap Alkaline Water Electrolysis? Nel Hydrogen (a Norwegian company) and Thyssen Krupp already have production scale equipment used in the Ammonia Industry and probably is cheaper.
BTW studies in the Ammonia Industry say if Renewable Electricity is to be competitive it needs to cost \$25/MWh.

SOEC has the advantage that the waste heat from the electrolysis step is at a high enough temperature to help drive gasification and the reverse water gas shift.  Presumably the energetics would be much worse without this.

E-P you are correct about the use of the waste heat.
After reading the article in more depth and one of the references (Bernical et al. [4] ) there are references to alkaline water electrolysis, e.g. "Compared to alkaline water electrolysis, high temperature steam electrolysis requires less electrical power. "
Also, in Section 3.4. Modeling the Solid Oxide Electrolysis Cell
"excess heat from the gasifier is available and can be used to heat steam to temperatures well above 1000 °C. When feeding this hot steam to an SOEC . . . allows for electric input that is lower than what is needed at room temperature.".
The article probably more importantly points out that the cost of producing advanced biofuel with the PBtL concept even with Hydrogen upgrading would still be very high compared to current fuel prices.

In a refinery where high yield of gasoline is the objective, there is a surplus of hydrogen, even though some of it is used for hydrogenation and desulfurization. However, in European refineries, where high yield of diesel fuel is the objective, there is lack of hydrogen. As correctly mentioned by gryf, this hydrogen is mostly made from methane but this is from fossil CNG/LNG, so there would be some scope of replacing this hydrogen. Efficiency in a Btl refinery could be greatly increased if hydrogen from another source would be available.

Interesting to find that work on DME/MeOH is conducted:
http://www.greencarcongress.com/2018/08/20180814-meohdme.html

Net producers of hydrogen, Peter?  Do you have a source for this?  I am very skeptical, given the hydrogen required to turn today's heavy crudes into light products like gasoline.

If gasoline production was a net producer of hydrogen there wouldn't be any great reason to move refining to places where natural gas (hydrogen) is cheap, but that is exactly what we see.

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text:
# ThermoOpticalCorrection¶

Type: section multiple

The thermo-optical correction describes the temperature () dependency of the relative permittivity in a linear sense near a specified temperature

is the thermo-optical coefficient. It is a - matrix to cover anisotropic effects. The total relative permittivity is then given by

For too large temperature differences the linear correction term may loose its validity. To prevent obscure relative permittivity definitions one may restrict the absolute value of the correction term by a threshold value ,

The JCM - syntax looks like this:

# define the relative permittivity with thermo-optical correction
RelPermittivity {
# define temperature independent terms
Constant = ...
...
ThermoOpticalCorrection {
T0 = ... # set correction free temperature
C = ... # set thermo-optical coefficients
CutOff .. # set threshold here

}
}

Warning

Often, the thermo-optical coefficient is defined as a refractive index - temperature relation,

In a linear sense both definitions are equivalent:

The approximation in the last line is due to linearization.