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# Age of the universe

1. Jan 4, 2007

### heliocentricprose

I've read that the universe is 13.7 billion years old.
Also, I've read that time passes "slower" when an object is close to a large mass as opposed to an object close to a smaller mass.
Take two galaxies. One galaxy is half the mass of the other. At this moment, has 13.7 billion years passed for both galaxies? Is the heavier galaxy using up its supply of hydrogen slower than the lighter one?

2. Jan 4, 2007

### chroot

Staff Emeritus
The figure of 13.7 billion years is given for an observer co-moving with the cosmic microwave background radiation. It will be different for every other observer. For example, an observer near a black hole may have seen the entire universe to date evolve in a matter of seconds according to his own watch.

- Warren

3. Jan 4, 2007

### heliocentricprose

In that case, how old are the particles making up the earth from their own perspective?

4. Jan 4, 2007

### chroot

Staff Emeritus
We certainly don't know the history of the earth's atoms well enough to assign such a number!

The Milky Way Galaxy does not have a particularly large velocity with respect to the CMBR, so time dilation with respect to it is really pretty negligible effect.

- Warren

5. Jan 4, 2007

6. Jan 4, 2007

### chroot

Staff Emeritus
I should also point out that, in absence of forces, every particle in the unverse would be going along with the Hubble expansion, and every particle would experience the same 13.7 billion years since the big bang. In truth, gravity complicates things a bit, and pulls particles off those trajectories. The resulting movement, with respect to the Hubble flow (CMBR rest frame), causes small time dilation effects that make each particle's elapsed time a little different from 13.7 billion years -- just not substantially different. The only exception are particles near very large curvatures in spacetime, like those near black holes.

- Warren

7. Jan 4, 2007

### quantum123

Very interesting, I mean the black hole part. I have never thought of it this way.

8. Jan 5, 2007

### Chris Hillman

Debunking a misconception in "Creation Science" [sic]

Hi, heliocentricprose,

Actually, for an observer "hovering" near a massive object, both the mass of the object and the "distance" to the object are relevant.

As you may know, the so-called "Young Earth theory" [sic], a topic in the so-called "Creation Science" [sic] movement, attempts to reconcile Ussher's age based upon scripture (see http://en.wikipedia.org/wiki/Bishop_Usher), 6010 years plus a few months, with the age based upon science, about 4.5 billion years, which is a considerable discrepancy. One of the sillier arguments I have seen is the claim that gravitational time dilation explains this away!

Recall that "gravitational time dilation" is a potentially misleading shorthand for the fact that if an observer, A, located on surface of the Earth, emits time signals at the rate of one per second by his ideal clock, and if these signals are received by a second observer, B, located very far away from any massive objects, then B will find that the signals arrive at intervals longer than one second by his ideal clock.

To see what's wrong with the "Creation Science" claim, note that in the Schwarzschild vacuum (the simplest model in gtr which can be used in this situation), we have
$$\frac{dt}{ds} = \frac{1}{\sqrt{1-2 m/r}} \approx 1 + m/r$$
where $m/r \approx 6.958 \times 10^{-10}$ for the Earth. This says that in our scenario, B will measure time signals from A to be running slow at a rate of less than one part per billion (in American terminology). So this certainly does not reconcile Bishop Ussher's alleged "scriptural age" with the scientific age!

(By the way, for a neutron star, the ratio m/r can be much larger, about 0.3.)

The analogous objection about mainstream cosmology is even easier to debunk: the textbook analysis of standard cosmological models such as the FRW models does take account of all relativistic effects in computing the elapsed time measured by an ideal clock carried by an observer more comoving with an idealized galaxy (see for example D'Inverno, Introducing Einstein's Relativity for a very readable discussion of these models).

In your scenario, you need to be more specific about where in each galaxy your two observers are located. E.g. if they are both hovering outside stars, the "gravitational time dilation" (wrt distant observers) will probably be dominated by this massive nearby object, but will be tiny. The details would depend upon the m/r ratio as above.

Last edited: Jan 5, 2007
9. Jan 5, 2007

### Myst

Excuse me if it's a stupid question, but how can we compute the age of the Universe ?
Is it explainable easily ? With which experimental data ?

10. Jan 5, 2007

### chroot

Staff Emeritus
Myst,

The principle means of finding the age of the universe is to measure the rate of its expansion, and extrapolate backwards. The most precise measurements to date have been made by the WMAP spacecraft. Here's a simple, easy-to-understand page on the topic:

http://map.gsfc.nasa.gov/m_uni/uni_101age.html

- Warren

11. Jan 5, 2007

### pervect

Staff Emeritus

There should be a very small difference in the time elapsed since the big bang for the two galaxies.

To try and get an estmate of how much, first we need an estimate of the potential at the center of the galaxy

a back of the envelope calculation based on a potential energy of -G * M_galaxy / r_galaxy gives a time dilation factor of about 1.0000001 if we assume that M_galaxy is 6e11 M_sun, and that r_galaxy is about 250,000 light years (for the dark matter halo), i.e about .1 part per million

Of course we will need some constant multiplier for -GM/r. I would guess that the above result could be off by an order of magnitude, but probably (hopefully) not more than two orders of magnitude.

One might be able to get some better data out of http://arxiv.org/abs/astro-ph/0612327 for the potential.

Note that cosmology, when it assigns a constant time since the big bang, assumes that there aren't any lumps of matter (which includes galaxies). On the average this isn't a bad approximation, as one can see by the small magnitude of the time dilation factors being discussed.

12. Jan 6, 2007

### Chris Hillman

Yes, if you are Steven Weinberg. See his popular book The First Three Minutes.

13. Jan 6, 2007

### Chris Hillman

Essential qualification

Oops! I should have said, at the surface of a neutron star, the ratio can be much larger. Sorry for any confusion I might have caused.


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# Transformations: Modular tensor computation graphs in Julia

November 14, 2016

In this post I’ll try to summarize my design goals with the Transformations package for the JuliaML ecosystem. Transformations should be seen as a modular and higher-level approach to building complex tensor computation graphs, similar to those you may build in TensorFlow or Theano. The major reason for designing this package from the ground up lies in the flexibility of a pure-Julia implementation for new research paths. If you want to apply convolutional neural nets to identify cats in pictures, this is not the package for you. My focus is in complex, real-time, and incremental algorithms for learning from temporal data. I want the ability to track learning progress in real time, and to build workflows and algorithms that don’t require a gpu server farm.

### What is a tensor computation graph?

A computation graph, or data flow graph, is a representation of math equations using a directed graph of nodes and edges. Here’s a simple example using Mike Innes’ cool package DataFlow. I’ve built a recipe for converting DataFlow graphs to PlotRecipes graphplot calls. See the full notebook for complete Julia code.

We’ll compute $f(x) = w * x + b$:

g = @flow f(x) = w * x + b
plot(g)

The computation graph is a graphical representation of the flow of mathematical calculations to compute a function. Follow the arrows, and do the operations on the inputs. First we multiply w and x together, then we add the result with b. The result of the addition is our output of the function f.

When x/w/b are numbers, this computation flow is perfectly easy to follow. But when they are tensors, the graph is much more complicated. Here’s the same example for a 1D, 2-element version where w is a 2x2 weight matrix and x and b are 2x1 column vectors:

plot(@flow f(x) = out(w11*x1 + w12*x2 + b1, w21*x1 + w22*x2 + b2))

Already this computational graph is getting out of hand. A tensor computation graph simply re-imagines the vector/matrix computations as the core units, so that the first representation ($f(x) = w * x + b$) is used to represent the tensor math which does a matrix-vector multiply and a vector add.

### Transformation Graphs

Making the jump from computation graph to tensor computation graph was a big improvement in complexity and understanding of the underlying operations. This improvement is the core of frameworks like TensorFlow. But we can do better. In the same way a matrix-vector product

can be represented as simply the vector $Wx$, we can treat the tensor transformation $f(x) = wx + b$ as a black box function which takes input vector (x) and produces output vector (f(x)). Parameter nodes (w/b) are considered learnable parameters which are internal to the learnable transformation. The new transformation graph looks like:

plot(@flow f(x) = affine(x))

Quite the improvement! We have created a modular, black-box representation of our affine transformation, which takes a vector input, multiplies by weight vector and adds a bias vector, producing a vector output:

And here’s the comparison for a basic recurrent network:

g = @flow function net(x)
hidden = relu( Wxh*x + Whh*hidden + bh )
y = logistic( Why*hidden + Wxy*x + by )
end

### But… TensorFlow

The unfortunate climate of ML/AI research is: “TensorFlow is love, TensorFlow is life”. If it can’t be built into a TF graph, it’s not worth researching. I think we’re in a bit of a deep learning hype bubble at the moment. Lots of time and money is poured into hand-designed networks with (sometimes arbitrary) researcher-chosen hyperparameters and algorithms. Your average high school student can install and build a neural network to perform quite complex and impressive models. But this is not the path to human-level intelligence. You could represent the human brain by a fully connected deep recurrent neural network with a billion neurons and a quintillion connections, but I don’t think NVIDIA has built that GPU yet.

I believe that researchers need a more flexible framework to build, test, and train complex approaches. I want to make it easy to explore spiking neural nets, dynamically changing structure, evolutionary algorithms, and anything else that may get us closer to human intelligence (and beyond). See my first post on efficiency for a little more background on my perspective. JuliaML is my playground for exploring the future of AI research. TensorFlow and competitors are solutions to a very specific problem: gradient-based training of static tensor graphs. We need to break the cycle that research should only focus on solutions that fit (or can be hacked into) this paradigm. Transformations (the Julia package) is a generalization of tensor computation that should be able to support other paradigms and new algorithmic approaches to learning, though the full JuliaML design is one which empowers researchers to approach the problem from any perspective they see fit.

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# Miscellaneous#

## Per Unit System#

The bases for AC system are

• $$S_b^{ac}$$: three-phase power in MVA. By default, $$S_b^{ac}=100 MVA$$ (set by System.config.mva).

• $$V_b^{ac}$$: phase-to-phase voltage in kV.

• $$I_b^{ac}$$: current base $$I_b^{ac} = \frac{S_b^{ac}} {\sqrt{3} V_b^{ac}}$$

The bases for DC system are

• $$S_b^{dc}$$: power in MVA. It is assumed to be the same as $$S_b^{ac}$$.

• $$V_b^{dc}$$: voltage in kV.

Some device parameters are given as per unit values under the device base power and voltage (if applicable). For example, the Line model andes.models.line.Line has parameters r, x and b as per unit values in the device bases Sn, Vn1, and Vn2. It is up to the user to check data consistency. For example, line voltage bases are typically the same as bus nominal values. If the r, x and b are meant to be per unit values under the system base, each Line device should use an Sn equal to the system base mva.

Parameters in device base will have a property value in the Model References page. For example, Line.r has a property z, which means it is a per unit impedance in the device base. To find out all applicable properties, refer to the "Other Parameters" section of andes.core.param.NumParam.

After setting up the system, these parameters will be converted to per units in the bases of system base MVA and bus nominal voltages. The parameter values in the system base will be stored to the v attribute of the NumParam. The original inputs in the device base will be moved to the vin attribute. For example, after setting up the system, Line.x.v is the line reactances in per unit under system base.

Values in the v attribute is what get utilized in computation. Writing new values directly to vin will not affect the values in v afterwards. To alter parameters after setting up, refer to example notebook 2.

## Notes#

### Modeling Blocks#

#### State Freeze#

State freeze is used by converter controllers during fault transients to fix a variable at the pre-fault values. The concept of state freeze is applicable to both state or algebraic variables. For example, in the renewable energy electric control model (REECA), the proportional-integral controllers for reactive power error and voltage error are subject to state freeze when voltage dip is observed. The internal and output states should be frozen when the freeze signal turns one and freed when the signal turns back to zero.

Freezing a state variable can be easily implemented by multiplying the freeze signal with the right-hand side (RHS) of the differential equation:

$T \dot{x} = (1 - z_f) \times f(x)$

where $$f(x)$$ is the original RHS of the differential equation, and $$z_f$$ is the freeze signal. When $$z_f$$ becomes zero the differential equation will evaluate to zero, making the increment zero.

Freezing an algebraic variable is more complicate to implement. One might consider a similar solution to freezing a differential variable by constructing a piecewise equation, for example,

$0 = (1 - z_f)\times g(y)$

where $$g(y)$$ is the original RHS of the algebraic equation. One might also need to add a small value to the diagonals of dae.gy associated with the algebraic variable to avoid singularity. The rationale behind this implementation is to zero out the algebraic equation mismatch and thus stop incremental correction: in the frozen state, since $$z_f$$ switches to zero, the algebraic increment should be forced to zero. This method, however, would not work when a dishonest Newton method is used.

If the Jacobian matrix is not updated after $$z_f$$ switches to one, in the row associated with the equation, the derivatives will remain the same. For the algebraic equation of the PI controller given by

$0 = (K_p u + x_i) - y$

where $$K_p$$ is the proportional gain, $$u$$ is the input, $$x_I$$ is the integrator output, and $$y$$ is the PI controller output, the derivatives w.r.t $$u$$, $$x_i$$ and $$y$$ are nonzero in the pre-frozen state. These derivative corrects $$y$$ following the changes of $$u$$ and $$x$$. Although $$x$$ has been frozen, if the Jacobian is not rebuilt, correction will still be made due to the change of $$u$$. Since this equation is linear, only one iteration is needed to let $$y$$ track the changes of $$u$$. For nonlinear algebraic variables, this approach will likely give wrong results, since the residual is pegged at zero.

To correctly freeze an algebraic variable, the freezing signal needs to be passed to an EventFlag, which will set an custom_event flag if any input changes. EventFlag is a VarService that will be evaluated at each iteration after discrete components and before equations.

## Profiling Import#

To speed up the command-line program, import profiling is used to breakdown the program loading time.

With tool profimp, andes can be profiled with profimp "import andes" --html > andes_import.htm. The report can be viewed in any web browser.

## What won't not work#

You might have heard that PyPy is faster than CPython due to a built-in JIT compiler. Before you spend an hour compiling the dependencies, here is the fact: PyPy won't work for speeding up ANDES.

PyPy is often much slower than CPython when using CPython extension modules (see the PyPy_FAQ). Unfortunately, NumPy is one of the highly optimized libraries that heavily use CPython extension modules.


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# Block Inference

Single-site inference in Bean Machine is a powerful abstraction that allows the inference engine to separately sample values for random variables in your model. While efficient in sampling high-dimensional models, single-site inference may not be suitable for models with highly correlated random variables. This is where Bean Machine's CompositionalInference API becomes handy: it allows us to "block" multiple nodes together and make proposals for them jointly.

## Motivation

To understand the issue better, let's walk through an example. Let's say we have two random variables $X$, $Y$ whose values are $x$ and $y$, and we'd like to move these values to $x'$ and $y'$. Using single site inference, we can move from $(x, y)$ to $(x', y')$ with either of these series of updates:

1. $(x, y) \to (x', y) \to (x', y')$
2. $(x, y) \to (x, y') \to (x', y')$

If $X$ and $Y$ are strongly correlated, e.g., if both $p(x, y)$ and $p(x', y')$ are high, but the intermediate stage $p(x', y)$ and $p(x, y')$ are low, then the single site inference methods can be stuck in $(x, y)$, because the acceptance probability for transitioning out of the initial state is going to be very low for either of these two paths. This can lead to under-exploration of $(x', y')$, which will not happen if we block $X$ and $Y$ together, which can move from $(x, y) \to (x', y')$ in a single Metropolis-Hastings step.

tip

## Configuring Block Inference

To understand how to run block inference in Bean Machine, let's consider the discrete Hidden Markov Model (discrete HMM) example below:

# alpha, beta, rho, nu, and init are externally-defined constants.@bm.random_variabledef mu(k):    return Normal(alpha, beta)@bm.random_variabledef sigma(k):    return Gamma(nu, rho)@bm.random_variabledef theta(k):    return Dirichlet(kappa)@bm.random_variabledef x(i):    return Categorical(theta(x(i - 1)) if i else init)@bm.random_variabledef y(i):    return Normal(mu(x(i)), sigma(x(i)))

This HMM describes a process with categorical latent states x, transition probabilities theta, and observed states y with emission probabilities determined by mu and sigma. Depending on the value of theta, the hidden state at each time step, x(i), can be hightly correlated with the hidden state at the previous time step, x(i-1). Therefore, we might want to block all instances of $x$ into a single block and propose new values for them jointly. Let's say we also want the parameters for emission probabilities, mu and sigma, to be updated jointly as well.

### Defining a block

You may recall that with Compositional Inference, we can mix-and-match inference methods by providing a mapping from random variable families to inference algorithms through the inference_dict argument. The syntax for "blocking" multiple nodes together is similar: instead of having a single random variable as a key, you can pass a tuple of random variable families instead. For example:

bm.CompositionalInference({    (x,): bm.SingleSiteAncestralMetropolisHastings(),    (mu, sigma): bm.GlobalNoUTurnSampler(),})

The code snippet above is going to create two blocks: one for all instances of x, which will be inferred with SingleSiteAncestralMetropolisHastings(), and another for all instances of mu and sigma, which will be inferred with GlobalNoUTurnSampler(). Random variable families that are not specified in the dictionary will fall back to the default inference methods and run without blocking.

Note that even though single site inference algorithms only update one node at a time, they can still be used to update a block of nodes. What is going to happen internally is that instead of accepting or rejecting a single site proposal immediately after it is made, we condition on it to compute the next proposal, and repeat this process for the remaining nodes in a block. After we are done with all nodes in a block, we then compute the Metropolis-Hastings acceptance probability as if the proposals are made in a single step. (Be aware that many of the single site algorithm only works well when the number of nodes in a block is low, as they might have trouble updating the samples as dimension increases.) On the other hand, multi site inference algorithms, such as GlobalNoUTurnSampler that we are using here, can make proposal for a set of nodes in one go and can take advantage of correlation between multiple nodes. To learn more about the distinctions between the two types of inference methods and how to define your own algorithms, see Custom Proposers.

### Mixing multiple inference methods in a block

Sometimes you might want to use different algorithms to update different random variable families, but stil have them group together in a single block. To do so, you can pass a tuple of inference methods as the value, one for each of the random variable families in the key, for example:

bm.CompositionalInference({    (mu, sigma): (bm.SingleSiteNewtonianMonteCarlo(), bm.SingleSiteAncestralMetropolisHastings()),})

which is equivalent to the following, more verbosed syntax with nested CompositionalInference:

bm.CompositionalInference({    (mu, sigma): CompositionalInference({        mu: bm.SingleSiteNewtonianMonteCarlo(),        sigma: bm.SingleSiteAncestralMetropolisHastings(),    }),})

In both of these snippets, we will group all instances of mu and sigma into a single block, use SingleSiteNewtonianMonteCarlo() to update all instances of mu and SingleSiteAncestralMetropolisHastings() to update all instances of sigma. Since CompositionalInference itself is just another inference method, you can use it in any places where a inference method is expected.

### Use the default inference method for a block

If you are not planning to change the default inference methods selected by CompositionalInference and only want to define a few blocks in your model, you can use Python's Ellipsis literal, or equivalently, ... (three dots), as the value. For example:

bm.CompositionalInference({    (x,): ...,    (mu, sigma): ...,})

Similar to the previous example, this is also going to create two blocks for our HMM model. However, since we didn't provide any inference method, CompositionalInference will use the default method to update each node in the block instead. Note that this is different from having Ellipsis on the left hand side of the dictionary, i.e.,

bm.CompositionalInference({    ...: bm.SingleSiteNoUTurnSampler(),})

which is used to override the default inference method and does not block the nodes automatically (unless you're using a multi-site algorithm, such as bm.GlobalNoUTurnSampler(), which always propose values jointly).

To see a live example of running block inference with CompositionalInference, check out our HMM tutorial.


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# Number of degrees of freedom of a photon

I am learning Quantum Field Theory. There they have shown that, for the four vector $A$, even though it has 4 components, it only has 2 degrees of freedom because the other 2 corresponds to gauge invariance and are therefore unphysical (i.e. there is a degeneracy).

There they mentioned that the 2 degrees of freedom corresponds to their angular momentum, with one being right handed polarised and the other left handed polarised.

My question is, why doesn't their direction of travel and their position in space contribute towards their degrees of freedom?

• Perhaps they are talking about internal degrees of freedom as opposed to external? – Mozibur Ullah Sep 10 '18 at 8:26
• Indeed there are 3 more dofs, namely $k^\mu$ under the restriction $k_\mu k^\mu$. – my2cts Sep 10 '18 at 9:24

The field operator $$A^{\mu}$$ generates two internal degrees of freedom at every point in spacetime. If we consider all the points in spacetime we see that there are actually much more than just two degrees of freedom. Remember that field operators are actually defined in terms of creation and annihilation operators $$a_s^{\dagger}(\mathbf{k})$$ and $$a_s(\mathbf{k})$$. From these operators one can directly see that they do not only carry the internal degrees of freedom, represent by the spin index $$s$$, but also spacetime degrees of freedom represented by the wavevector $$\mathbf{k}$$. The latter represents a three-dimensional vector as opposed to a four-vector, because of the dispersion relation $$\omega=c|\mathbf{k}|$$, which removes one continuous degree of freedom.

You are working with a spin-1 gauge field theory, which means you are working with the Proca equation with a fourvector $A^{\mu}$.

Having 4 entries, you start with 4 degrees of freedom, i.e. you have to make up 4 numbers in order to fully determine $A^{\mu}$.

You can reduce the number of degrees of freedom by finding equations that relate them among themselves.

The first you know is the gauge condition. $A^{\mu}$ is not the physical observable quantity, but it is related to the electric and magnetic fields via derivatives. So, $A^{\mu}$ is subject to a larger class of transformations than $E$ and $B$, i.e. you can do more stuff with it while leaving the physics unchanged. This means that you can “fix the gauge”, that is requiring $A^{\mu}$ so satisfy some condition in order to get rid of this redundancy. Common “gauges” are the Coulomb, Weyl or Lorenz gauges. For example $\nabla \cdot \mathbf{A} = 0$ is the Coulomb gauge.
This reduces the degrees of freedom from 4 to 3.

You can get rid of one more degree of freedom because the field is massless. Some maths can should you that the product $k_{\mu} A^{\mu}$ should be proportional to the mass $m$. Having $m=0$ entails you have yet another equation that relates the entries of $A$ among themselves.

So for a spin-1 massless gauge field theory, you only have 2 degrees of freedom. This makes sense in the context of light in free space, since we know it can be described as L and R polarisation, or H and V.

This doesn’t mean that the polarisation is always perpendicular to the direction of propagation. In a medium, there will be a component of the electric field along $z$. But you can still re-cast the problem as a 2 degree of freedom problem.

In the special case of free space, you can interpret these 2 as two orthogonal polarisations. But in a waveguide, for instance, there is also $E_{\perp}$ so this intuitive correspondence does not apply.


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# Generate a unique 64 bit numeric database key without collisions at $2^{32}$ birthday threshold?

What is the best way to generate a unique 64 bit numeric database key from a string?

For example: if the string is a URL, you could generate a 64 bit key using SipHash or by truncating a SHA256 hash to 64 bits, but collisions are likely at $2^{32}$ (~ 4 billion) entries.

However, if instead you truncated the hash to (64 bits - X bits), and concatenated the resulting hash with the first X bits of the input string, you would reduce the chance of collisions since two similar strings are unlikely to produce the same hash code:

key_bits = 64
x_bits = 8
hash_code = sha256(url_string)
key = truncate(hash_code, key_bits - x_bits) + truncate(url_string, x_bits)

Is this correct? Is there a better way?

• Why do you think that "two similar strings are unlikely to produce the same hash code"? If you look at MD5 collisions that are out there, it is the opposite. Many of the (MD5) colliding strings are very similar. – mikeazo Jun 10 '14 at 12:56
• What do you REALLY want to solve? If by database key you mean data useful as a primary key in a RDBMS, never, ever create primary keys that are somehow related to the data (here URL) since the data may change and you do not want to update the primary key that by then may be used elsewhere as a foreign key. Best regards from a DBA. – AHalvar Jun 10 '14 at 13:51
• @AHalvar Using Hadoop to bulk load a massive dataset into a graph DB built on HBase that requires 64bit keys, the data is immutable so no chance of it changing. Need key generation to be deterministic to avoid reads to find out if key exists. – espeed Jun 10 '14 at 14:12
• @mikeazo Confusion and Diffusion / Avalanche effect. – espeed Jun 10 '14 at 14:12
• Related: Best way to reduce chance of hash collisions: Multiple hashes, or larger hash? – Your truncation setup can be considered as two hash functions (one of which is quite poor on its own), so it is in general worse than a single longer hash. – otus Jun 10 '14 at 14:45

2. Ignore duplicate keys, e.g. set things up so that only one of the two entries is kept. As long as you don't go much above $2^{32}$ entries, the expected number of collisions will be very low, and depending on your task, it may be acceptable to ignore a small fraction of the entries.


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# Prony’s method (Polyreference)

## Introduction

I am re-reviewing the bible-like book “Introduction to Modal Analysis,” which every Japanese vibration engineer has read at least once.

I would like to explain what I have learned in the re-review, showing the MATLAB code. This time, we will review from section 6.3.2, “Prony Method,” p. 367. The “poly-reference method” is an extension of the Prony method to the multipoint reference method. (also called “Vold’s method” since it was drafted by Vold).

The mode identification method with a function called modalfit, introduced in the Signal Processing Toolbox of MATLAB2017a, can also be solved by the “Prony method”. Please refer to MathWorks’ website for details. Incidentally, “FitMethod”=”lsce”, the default mode identification method of modalfit, is explained as the Prony method.

The following link is a summary to a description of vibration theory in English, if you are interested.

TopPage 振動騒音 研究所
MATLABとNVH(Noise, Vibration and Harshness)をこよなく愛する人生の先輩、それが私（MATLABパイセン）です。NVHを極めるにあたり、周辺の知識も網羅的に勉強することになり、その知識を共有すべく、本HPを運営しています。日本の製造業を応援すべく、機械エンジニアの「車輪の再開発」の防止と業務効率化の手助けをモットーに活動。専門である振動騒音工学や音響工学について、プログラムを示しながら解説。大学の授業よりもわかりやすく説明することを目指す。質問頂ければ、回答したいと思います。

## 6.3.2 Prony’s method

In previous articles, I have explained that FRFs and vibration shapes can be expressed as a superposition of modal characteristics (resonant frequencies & vibration modes, etc.). It is the concept of modal analysis. And extracting mode characteristics from FRF is called mode identification or curve fitteing.

The mode circle fitting (curve fitting) and half-width method were applicable only when the adjacent resonance frequencies are far enough apart, as in the red frequency band in Figure 1, and the system can be regarded as a one-degree-of-freedom system.

Figure 1 FRF

On the other hand, Prony’s method is a mode identification method for multi-degree-of-freedom systems. In other words, it is a method that can extract the mode characteristics of each resonance frequency even if it contains multiple resonance frequencies, such as the 0 to 100 Hz data in Figure 1.

From here, the explanation becomes a bit more technical.
Prony’s method is a method to estimate modal characteristics from impulse response data. Although it is possible to use time history data of the response of impact excitation with an impact hammer, etc., it is known that the signal-to-noise ratio is generally bad, and in many cases, data obtained by IFFT of FRF averaged over the number of measurements and then the data, converted back to impulse response, is used as input for the Prony’s method.

Since the impulse response can be expressed as a superposition of mode characteristics, it can be expressed as in Equation (6.52).

$$h(t)=\sum_{r=1}^{N} a_{r}(je^{λ_r t} – je^{λ_r^* t} ) \tag{6.52}$$

where λr is the eigenvalue, ωdr is the damped natural angular frequency, and σr is the modal damping factor, detailed in the equation below.

$$a_r=\frac{ Ω_r^2 }{2ω_{dr}K_r}, λ_r=-σ_r + jω_{dr}, λ_r^*=-σ_r – jω_{dr} \tag{6.51}$$

However, rather than the above equation, it is better to just remember that ar is the residu, λr is the eigenvalue, ωdr is the damped natural angular frequency, and σr is the modal damping rate.

Again, mode identification means finding the distn ar and eigenvalue λr.
There are various methods for mode identification, but they differ only in their approach to finding the abscissa ar and eigenvalue λr.

Since the data are discretized, equation (6.53) can be expressed in a discretized form for time t as in equation (6.54). Note that i is the data id. It is assumed that only n (n<N) of N are to be identified, rather than all N, the number of all resonance frequencies.

$$h(iΔt)=\sum_{r=1}^{n} a_{r}(e^{λ_r iΔt} – e^{λ_r^* iΔt} ) \tag{6.52}$$

Here, since the character expression is complicated, it is replaced by the following for simplicity.

$$x_r=e^{λ_r Δt}, x_r^*=e^{λ_r^* Δt} \tag{6.55}$$

Then it can be expressed as in equation (6.56).

$$h(iΔt)=\sum_{r=1}^{n} a_{r}(x_r^i + x_r^{i*} )=\sum_{r=1}^{n} real( 2a_{r}(x_r^i ) )=\sum_{r=1}^{2n} a_{r}(x_r^i ) \tag{6.56}$$

I think some of you may not understand the equation expansion of equation (6.56). Originally, the expansion was based on the assumption that n resonance frequencies are to be identified, but since it is more coherent to identify 2n resonance frequencies, the number of resonance frequencies is 2n.

You can manage to get to this point.
Equation (6.56) is just a repeated transformation of the equation, and as it is, we can’t extract the distnumber ar and eigenvalue λr, can we?

For now, let’s consider equation (6.56) in a simplified form, with n=2 and i=0,1,2,3,4.

$$h(0)=a_1 x_1^0 + a_2 x_2^0 + a_3 x_3^0 + a_4 x_4^0 \\ h(1)=a_1 x_1^1 + a_2 x_2^1+ a_3 x_3^1 + a_4 x_4^1 \\ h(2)=a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2 + a_4 x_4^2 \\ h(3)=a_1 x_1^3 + a_2 x_2^3 + a_3 x_3^3 + a_4 x_4^3 \\ h(4)=a_1 x_1^4 + a_2 x_2^4 + a_3 x_3^4 + a_4 x_4^4$$
$$\tag{6.56.0}$$

In this situation, we can’t extract the abstention number ar and eigenvalue λr, can we?

So, consider the following.

$$(x-x_1)(x-x_1^*)(x-x_2)(x-x_2^*) =0 \tag{6.57.3}$$

Since it is difficult to calculate by hand, we replace it with the following so that it can be solved by Maxima.

$$(x-a1)(x-c1)(x-a2)(x-c2) =0$$

Expanding the above equation, we get

$$x^4 \\ +(-c2-c1-a2-a1)x^3 \\ +(c1*c2+a2*c2+a1*c2+a2*c1+a1*c1+a1*a2)x^2 \\ +(-a2*c1*c2-a1*c1*c2-a1*a2*c2-a1*a2*c1)x \\ +a1*a2*c1*c2 =0$$
$$\tag{6.57.0}$$

Replace the above with the following. Note that b4=1 here.

$$b_4 x^4 \\ +b_3 x^3 \\ +b_2 x^2 \\ +b_1 x^1 \\ +b_0 x^0 =0$$
$$\tag{6.57.1}$$

Equation (6.57.1) is a generalization of equation (6.57).

$$\sum_{i=0}^{2n}b_i x^i =0 \tag{6.57}$$

Now we will consider how to extract the abstention number ar and the eigenvalue λr, comparing equations (6.56.0) and (6.57.1). For now, let us multiply both sides of equation (6.56.0) by bi.

$$b_0 h(0)=b_0 (a_1 x_1^0 + a_2 x_2^0 + a_3 x_3^0 + a_4 x_4^0) \\ b_1 h(1)=b_1 (a_1 x_1^1 + a_2 x_2^1+ a_3 x_3^1 + a_4 x_4^1) \\ b_2 h(2)=b_2 (a_1 x_1^2 + a_2 x_2^2 + a_3 x_3^2 + a_4 x_4^2) \\ b_3 h(3)=b_3 (a_1 x_1^3 + a_2 x_2^3 + a_3 x_3^3 + a_4 x_4^3) \\ b_4 h(4)=b_4 (a_1 x_1^4 + a_2 x_2^4 + a_3 x_3^4 + a_4 x_4^4)$$
$$\tag{6.60.0}$$

All of equation (6.60.0) can be added together and summarized as follows

$$\sum_{i=0}^{2*2}b_i h(i) = a_1 \sum_{i=0}^{2*2}b_i x_1^i +a_2 \sum_{i=0}^{2*2}b_i x_1^i +a_3 \sum_{i=0}^{2*2}b_i x_3^i +a_4 \sum_{i=0}^{2*2}b_i x_4^i \tag{6.60.1}$$

rom equation (6.57), equation (6.60) becomes equation (6.61.0).

$$\sum_{i=0}^{2*2}b_i h(i) = 0 \tag{6.61.0}$$

A slight variant of equation (6.61.0)

$$b_4 h(4) + \sum_{i=0}^{2*2-1}b_i h(i) = 0 \tag{6.61.1}$$

Here, b4=1, so the equation below holds.

$$-h(4) = \sum_{i=0}^{2*2-1}b_i h(i) \tag{6.62.0}$$

In the Introduction to Modal Analysis, there is a generalized explanation of the above. However, if you are a gory mode identification enthusiast like me, I would like to add a supplemental explanation that is a little different.

For example, suppose that the impulse response is as shown in Figure 2 with i = 0 to 2n-1 and the time history data is for 0.1 second.
Assuming a time resolution of 0.0001, the way to take 0.1 second of the impulse response is
0 to 0.1 second, or
0.0001 to 1.0001 seconds is also fine.

Therefore, a known method is to calculate b such that the error is minimized using a large number of 0.1 second data. Specifically, we solve as in Equations (6.62.1)-(6.62.3).

$$\left( \begin{array}{ccc} h(0) & \ldots & h(2n-1) \\ \vdots & \ddots & \vdots \\ h(2n-1) & \ldots & h(4n-2) \end{array} \right) \left( \begin{array}{c} b_0 \\ \vdots \\ b_{2n-1} \end{array} \right)= \left( \begin{array}{c} h(2n) \\ \vdots \\ h(4n-1) \end{array} \right) \tag{6.62.1}$$

Equation (6.62.2) is a simplified version of Eq(6.62.1).

$$Hb=y \tag{6.62.2}$$

Considering that y contains an error, equation (6.62.3) is the solution of b by the least-squares method.

$$b=(H^tH)^{-1} H^t y \tag{6.62.2}$$

Now you have obtained b.
Once b is obtained, solving the algebraic equation in Eq. (6.57) yields xr in Eq. (6.55).
From xr, we obtain the damped angular resonance frequency ωdr and the modal damping factor σr as in the equation below.

$$σ_r=-\frac{In|x_r|}{Δt} \tag{6.100.1}$$

$$ω_{dr}=\frac{1}{Δt}atan(\frac{imag(x_r)}{real(x_r)}) \tag{6.100.2}$$

Theoretical formulas up to the above may be found in modal analysis handbooks and other sources of modal identification.
However, the above theoretical equations alone cannot identify the mode characteristics. Rather, the Prony method requires the identification of a larger number of resonance frequencies. For example, a waveform with only two resonance frequencies excited is assumed to have 50 resonance frequencies by the Prony method to estimate the mode characteristics.
Therefore, it is necessary to extract the two that seem to be correct from the estimated 50. In some cases, the 50 may not have been estimated correctly in the first place. We need to make these judgments, but the indicators for making these judgments are not described in any commercially available reference book.

I would like to explain the above issues using examples.

## Example of MATLAB

#### Solution from eq(6.52) to eq(6.62.2)

Let us now solve the example problem in MATLAB.
Let us find the modal characteristics from the impulse response waveforms with resonance frequencies of 10Hz and 50Hz as shown in Figure 3. Note that the modal damping ratios are 5%@10Hz and 1%@50Hz.

Figure 3 Impulse response

Now, let’s make the waveform shown in Figure 4, and the MATLAB program will look like the following.

clear all;clc;close all;

j=sqrt(-1);
time=0:0.0001:1;

% % Example of time response vector X
freq1=10; omega1=freq1*2*pi;
freq2=50; omega2=freq2*2*pi;
zeta1=0.05; sgm1=zeta1*omega1;
zeta2=0.01; sgm2=zeta2*omega2;
phi1=0; phi2=0;
X=exp(-zeta1*omega1*time).*sin(omega1*time+phi1)......
+exp(-zeta2*omega2*time).*sin(omega2*time+phi2);

figure
plot(time,X)
xlabel('Time [s]')
ylabel('Ampli [mm]')

Now that we have created the impulse response waveform, let’s program it up to the Prony method equation (6.62.2). By the way, the program is as follows.

% Number of resonance frequencies N to be sought (the trick is to set a larger number for now)
% N-1 to be sought
N=10; n=N/2;

% % eq(6.62.1) H
H=zeros(2*n);
for ii1=1:2*n
H(ii1,:)=X(ii1:(ii1+2*n-1)).';
end

% % Right side of Eq. (6.62.1) (y-vector in Eq. (6.62.2))
y=-X(2*n+1:n*4).';

% % eq(6.62.2)
b=(H.'*H)(H'*y);

You have found b in equation (6.62.2).
From b, you can find the mode characteristic by finding x that satisfies equation (6.57).

#### Estimation of mode properties using “roots” MATLAB function (doesn’t work)

Now, here comes the hard part. How do we solve equation (6.57)?
MATLAB has a function “roots” to find the roots of polynomials.
Let’s try it here (as I mentioned earlier, this method does not work)

r = roots(b.'); %apply roots function to eq(6.57.3)......
% figure
% plot(real(r),imag(r),'.')
% axis equal
% grid on

omega_prony = angle(r)/time(2); %eq(6.100.2)
fn_prony = omega_prony/(2*pi);
sgm_prony = -log(abs(r))/time(2); %eq(6.100.1)
zeta_prony = sgm_prony./omega_prony;

The resonant frequency and mode damping ratio obtained by the above program are as follows

Figure 4 Obtained mode characteristics (resonant frequency and mode attenuation ratio)

Theoretical values are fn=10, 50; zeta=0.05, 0.01.
Since the Prony method requires a larger order to be identified, this example estimates 9 resonant frequencies and mode damping ratios, but I do not see any combination that matches the theoretical values.

If we increase the number of resonant frequencies to be estimated, we may find a combination that matches the theoretical value. (It is very likely that they will.) However, it is difficult to select the most plausible combination from a huge number of combinations of resonant frequencies and mode damping ratios. To begin with, we are estimating without prior information on how many resonance frequencies are included in the impulse response waveform, so we would like to have a criterion for selecting a combination.

This is the difficult part of mode identification. (This is where mode identification maniacs get hooked.)
Most of the theory of mode identification does not work well even if you program the theoretical equation as it is written in a paper or a reference book. In order to create a program, it is necessary to unravel the information hidden behind the theoretical equation.

#### Proposed Method

Again, once b is obtained, it is considered possible to extract the modal characteristics by finding xi that satisfies the equality in equation (6.57).

$$\sum_{i=0}^{2n}b_i x^i =0 \tag{6.57}$$

So what value is xi likely to be?
To begin with, xi was the following, right? (xr and xi are different orders of resonance, r-order or i-order.)

$$x_r=e^{λ_r Δt} \tag{6.55}$$

Also, λr is the following.
$$λ_r=-σ_r + jω_{dr} \tag{6.51}$$

Thus,

$$x_r=e^{(-σ_r + jω_{dr})Δt}=e^{(-ω_{dr}ζ + jω_{dr})Δt}$$

What is the realistic range of modal attenuation ratio ζ?
Empirically, it should be as follows
0<ζ<0.1

The next thing to consider is the range of resonant frequencies we want to find.
Since this is an example problem, we know the answer to the resonant frequency of the impulse response waveform.
However, (unless you have checked the spectrum in advance and have an estimate of the resonance frequency), you would not normally perform mode identification with the resonance frequency known.

When performing mode identification, the key point is to specify the frequency range you want to find and to limit the frequency range to be sought using the Prony method.
For example, in this example, let’s say you want to find resonance frequencies below 100 Hz. (Assume that you are solving without knowing the number of resonance frequencies in the impulse response waveform or the resonance frequencies themselves.)

In other words, the range where resonant frequencies are likely to exist is as follows.

0<fn<100
0<ωn<2*pi*100

Therefore, the range of ζ and ωn determines the range that x_r can also take.
In MATLAB, x (x_r is defined as x for search) is defined as follows.
Here, x is made to correspond to a change in ζ in the row direction and a change in ωn in the column direction.

We then find the left-hand side of Eq. (6.57).

fn_min=0;
fn_max=100;
zeta_min=0.00001; %モード減衰比が0になることはないので、限りなく小さくしておく
zeta_max=0.1;
dt=time(2);

fn_range=fn_min:0.5:fn_max;
wn_range=2*pi*fn_range;
zeta_range=linspace(zeta_min,zeta_max,1000);

x=exp((-zeta_range.'+j)*wn_range*dt);

Eq_6_57=1*x.^(2*n); %b_2n*x.^(2*n)
for ii1=1:2*n
Eq_6_57=Eq_6_57+b(ii1)*x.^(ii1-1); %式(6.57)
end

If we can find x such that Eq_6_57, the left side of Eq. (6.57), becomes 0, i.e., a minimum value, we can extract the modal characteristics that are consistent with the theoretical values.

The minimum value is considered to be obtained as follows.

% Determination and search for poles
surf_Eq_6_57 = abs(Eq_6_57);
surf_pole_logical = islocalmin(surf_Eq_6_57,1) & islocalmin(surf_Eq_6_57,2); % Finding the smallest point on a plane (matrix)

xr = x(surf_pole_logical); %pole

omega_prony = angle(xr)/dt; %eq(6.100.2)
fn_prony=omega_ans/(2*pi);
sgm_prony = -log(abs(xr))/dt; %eq(6.100.1)
zeta_prony = sgm_prony./omega_prony;

The results are as follows. The theoretical value is firmly obtained.

## MATLAB code (full ver.)

Check that the results of the resonance frequency and mode damping ratio estimation are correct when the amplitudes A1 and A2 are changed, etc.

clear all;clc;close all;

j=sqrt(-1);
time=0:0.0001:1;
dt=time(2);

% Number of resonance frequencies N to be sought (the trick is to set a larger number for now)
% N-1 to be sought
N=100; n=N/2;

% % Example of time response vector X
A1=1; freq1=10; omega1=freq1*2*pi;
A2=2; freq2=50; omega2=freq2*2*pi;
zeta1=0.05; sgm1=zeta1*omega1;
zeta2=0.01; sgm2=zeta2*omega2;
phi1=0; phi2=0;
X=A1*exp(-zeta1*omega1*time).*sin(omega1*time+phi1)......
+A2*exp(-zeta2*omega2*time).*sin(omega2*time+phi2);

figure
plot(time,X)
xlabel('Time [s]')
ylabel('Ampli [mm]')

% % eq(6.62.1) H
H=zeros(2*n);
for ii1=1:2*n
H(ii1,:)=X(ii1:(ii1+2*n-1)).';
end

% % Right side of eq.(6.62.1) (equation (6.62.2) y-vector)
y=-X(2*n+1:n*4).';

% % eq(6.62.2)
b=(H.'*H)(H'*y);

fn_min=0;
fn_max=100;
zeta_min=0.00001; % Keep the modal damping ratio as small as possible since it will never be zero.
zeta_max=0.1;

fn_range=fn_min:0.5:fn_max;
wn_range=2*pi*fn_range;
zeta_range=linspace(zeta_min,zeta_max,1000);

x=exp((-zeta_range.'+j)*wn_range*dt);

Eq_6_57=1*x.^(2*n); %b_2n*x.^(2*n)
for ii1=1:2*n
Eq_6_57=Eq_6_57+b(ii1)*x.^(ii1-1); %eq(6.57)
end

% Determination and search for poles
surf_Eq_6_57 = abs(Eq_6_57);
surf_pole_logical = islocalmin(surf_Eq_6_57,1) & islocalmin(surf_Eq_6_57,2); % 平面上（行列上）の極小点を探す

xr = x(surf_pole_logical); %pole

omega_prony = angle(xr)/dt; %eq(6.100.2)
fn_prony=omega_prony/(2*pi)
sgm_prony = -log(abs(xr))/dt; %eq(6.100.1)
zeta_prony = sgm_prony./omega_prony

MATLABパイセンが教える振動・騒音・音響・機械工学


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# zbMATH — the first resource for mathematics

The centre of the spaces of Banach lattice-valued continuous functions on the generalized Alexandroff duplicate. (English) Zbl 1221.46025
Summary: We characterize the centre of the Banach lattice of Banach lattice $$E$$-valued continuous functions on the Alexandroff duplicate of a compact Hausdorff space $$K$$ in terms of the centre of $$C(K,E)$$, the space of $$E$$-valued continuous functions on $$K$$. We also identify the centre of $$CD_0(Q,E)= C(Q, E)+ c_0(Q,E)$$ whose elements are the sums of $$E$$-valued continuous and discrete functions defined on a compact Hausdorff space $$Q$$ without isolated points, which was given by S. Alpay and Z. Ercan [Positivity 4, No. 3, 213–225 (2000; Zbl 0973.46026)].

##### MSC:
 4.6e+06 Lattices of continuous, differentiable or analytic functions 4.6e+41 Spaces of vector- and operator-valued functions
##### Keywords:
centre; Banach lattice; Alexandroff duplicate
Full Text:
##### References:
 [1] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Dordrecht, The Netherlands, 2006. · Zbl 1156.46001 [2] P. Meyer-Nieberg, Banach Lattices, Universitext, Springer, Berlin, Germany, 1991. · Zbl 0743.46015 [3] Z. Ercan and A. W. Wickstead, “Banach lattices of continuous Banach lattice-valued functions,” Journal of Mathematical Analysis and Applications, vol. 198, no. 1, pp. 121-136, 1996. · Zbl 0869.46012 · doi:10.1006/jmaa.1996.0072 [4] M. \cCaglar, Z. Ercan, and F. Polat, “Generalized Alexandroff duplicates and CD0(K) spaces,” Central European Journal of Mathematics, vol. 4, no. 3, pp. 371-375, 2006. · Zbl 1139.46032 · doi:10.2478/s11533-006-0018-5 [5] R. Engelking, “On the double circumference of Alexandroff,” Bulletin de l’Académie Polonaise des Sciences, vol. 16, pp. 629-634, 1968. · Zbl 0167.21001 [6] P. S. Alexandroff and P. S. Urysohn, “Memoire sur les espaces topologiques compacts,” Koninklijke Nederlandse Akademie van Wetenschappen te Amsterdam, vol. 14, pp. 1-96, 1929. · JFM 55.0960.02 [7] Y. A. Abramovich and A. W. Wickstead, “Remarkable classes of unital AM-spaces,” Journal of Mathematical Analysis and Applications, vol. 180, no. 2, pp. 398-411, 1993. · Zbl 0792.46004 · doi:10.1006/jmaa.1993.1408 [8] \cS. Alpay and Z. Ercan, “CD0(Q,E) and CD\omega (Q,E)-spaces as Banach lattices,” Positivity, vol. 4, no. 3, pp. 213-225, 2000. · Zbl 0973.46026 · doi:10.1023/A:1009878527795 [9] Z. Ercan, “A concrete description of CD0(K)-spaces as C(X)-spaces and its applications,” Proceedings of the American Mathematical Society, vol. 132, no. 6, pp. 1761-1763, 2004. · Zbl 1050.46022 · doi:10.1090/S0002-9939-03-07235-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
- Current Issue - Ahead of Print Articles - All Issues - Search - Open Access - Information for Authors - Downloads - Guideline - Regulations ㆍPaper Submission ㆍPaper Reviewing ㆍPublication and Distribution - Code of Ethics - For Authors ㆍOnlilne Submission ㆍMy Manuscript - For Reviewers - For Editors
 Remarks on the infinity wave equation Bull. Korean Math. Soc.Published online December 29, 2020 Hyungjin Huh Chung-Ang University Abstract : We propose the infinity wave equation which can be derived from the exponential wave equation through the limit $p \to \infty$. The solution of infinity Laplacian equation can be considered as a static solution of the infinity wave equation. We present basic observations and find some special solutions. Keywords : Infinity wave equation, Exponential wave equation, Infinity Laplacian. MSC numbers : 35L80, 35L72, 34A05 Full-Text :


text:
## Easy Question, What does this symbol mean?

What does the upside down hat mean in vector notation. I stumbled across this symbol in my fluid mechanics book compressible flow, Munson seems to like use lots of non standard notations to begin with but I figure this symbol is still commonly used somewhere. Hat usually implies a unit vector but upside down hat?

$$c_{v}=\frac{\partial\check{u}}{\partial T}$$

I would guess that Munson uses the check as latex calls it to make sure you notice that he means internal energy rather than velocity as u often represents. I was curious what the history of this symbol is as this is the first time I've seen it used.

 PhysOrg.com science news on PhysOrg.com >> Heat-related deaths in Manhattan projected to rise>> Dire outlook despite global warming 'pause': study>> Sea level influenced tropical climate during the last ice age


text:
# Can rewriterules work without carets or dollar signs?

Currently when I remap my friendly to non-friendly URLs on my website, I normally use lines like these in my .htaccess:

RewriteEngine On
RewriteCond %{ENV:REDIRECT_STATUS} !^$RewriteRule .* - [L] RewriteRule ^afolder/subfolder/(.*)$ /internal.php?Q1=$1 [L] RewriteRule ^afolder2/subfolder2/(.*)$ /internal.php?Q2=$1 [L] .... RewriteRule ^something/else/(.*)$ /internal.php?Qn=$1 [L]  As you can see, I'm remapping http://example.com/whatever/whatever/value to http://example.com/internal.php?Q(something)=value. The point I'm making is that with every RewriteRule statement, I'm used to starting the search with a caret and ending it with a dollar sign, even for the simplest rules. For example to map http://example.com/anything to http://example.com/something.php, I use: RewriteRule ^anything$  /something.php [L]

My question is, when could I get away with NOT using carets or dollar signs to remap my URLs?

I'm asking because I want to boost the overall speed of my website, and every millisecond I shave off of processing time makes clients happier, (P.S. I need to shave off about 12ms loading time from the other end of North America then I'll meet the needs of google) and if I can remove those carets and/or dollar signs, then I feel I'll get a boost since the mod_rewrite engine will have a simpler string to process.

• The carets and dollar signs are anchors. Without them, the match can happen anywhere along the value to be matched against. If you want a very specific match, one or more anchor should be used. If you want to match anywhere, then neither anchor should be used. There is little or no gain in speed in using anchors or not. It is better to use an anchor for safety. Anchors do not require recursion during matches so they are technically faster, though you are shaving extremely small amounts of time. Go for safety first, convenience second, speed last. – closetnoc Jan 17 '16 at 18:43
• So technically removing an anchor is asking for trouble. Thanks. – Mike -- No longer here Jan 17 '16 at 18:47
• Not always. It just means that a recursive match anywhere within the string can happen. If there is only one case, then it will match only once. But if you go temporarily senile or insane and forget a case, you can get more than one match. That is why I stress safety first. It can be more risky to not include at least one anchor (often the caret) unless you intend to match anywhere. – closetnoc Jan 17 '16 at 18:52

You are using the ^ and $ (anchors in regex speak) because you are matching the whole URL, which is what most people want to do, so this is the most common example you see. If you omit the ^ and/or $ anchors then you are only going to be matching part of the URL. eg. anything$ is going to match "anything" at the end of the URL - this could match too many URLs (depending on your URL structure) and possibly allow invalid URLs to be matched. However, if you need to match "anything" at the end of the URL then this is the correct regex to use. You basically need to use the appropriate regex for what you are trying to match. Speed/efficiency, for the most part, doesn't really come into it. RewriteRule ^afolder/subfolder/(.*)$ /internal.php?Q1=$1 [L] If you omitted the ^ anchor from this pattern then a user could potentially request /foo/afolder/subfolder/... and it will be caught by this rule. Incidentally, the $ (end of string anchor) is superfluous here (because of the .* pattern) and can be omitted.
Removing the anchors is not going to make any difference with speed. In fact, removing the ^ (start) anchor could even make it less efficient, since the regex parser now needs to search the entire URL. With a ^ anchor it can fail on the first character mismatch, which is more efficient.


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# March 30 Exercise (b) and (c) from lecture

Exercise: Complete parts (b) and (c) below for your team.

Team parentheses:
Let $C_n$ be the number of ways of parenthesizing a product of $n+1$ numbers.

Team binary rooted trees:
Let $C_n$ be the number of full, binary, rooted trees with $n+1$ leaves.

Team triangulated polygons:
Let $C_n$ be the number of ways of triangulating a polygon with $n+2$ sides.

(b) Find a recursive formula for $C_n$. (That is, find a formula for $C_n$ in terms of $C_0, C_1, C_2, \dots, C_{n-1}$.)

(c) Use your formula from (b) to determine $C_5$ and $C_6$.

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Last edited by Tajora
Monday, July 27, 2020 | History

2 edition of An inequality for self-adjoint operators on a Hilbert space found in the catalog.

# An inequality for self-adjoint operators on a Hilbert space

## by Herbert J. Bernstein

Published by Courant Institute of Mathematical Sciences, New York University in New York .
Written in English

The Physical Object
Pagination5 p.
ID Numbers
Open LibraryOL25400585M

On the other hand, it is shown in that if f: J → R is an operator convex function on the interval J, then for any self-adjoint operators S and T with spectra in J we have the inequality f (S + T 2) ≤ ∫ 0 1 f ((1 − t) S + t T) d t ≤ f (S) + f (T) 2. according algebra apply assume asymptotic basis belongs Boolean Borel bounded called channel Chapter Chapter IV closed commute complete concept Consequently consider containing continuous converges corresponding defined Definition denote dense derive differential domain easily element energy equal equation equivalence establish Euclidean space.

Some reverses of the Jensen inequality for functions of self-adjoint operators in Hilbert spaces under suitable assumptions for the involved operators are given. Applications for particular cases of interest are also by:   He also presented some Chebyshev inequalities for self-adjoint operators acting on a Hilbert space in. In this paper we provide several operator extensions of the Chebyshev inequality. In the second section, we present our main results dealing with the Hadamard product of Hilbert space operators and weighted operator geometric by: 5.

However, since the spectrum of a a self-adjoint operator is real the theorem should be true in real Hilbert spaces. I can imagine an argument by complexification but there a . Definition. Let be a Hilbert space and () be the set of bounded operators , an operator ∈ is said to be a compact operator if the image of each bounded set under is relatively compact.. Some general properties. We list in this section some general properties of compact operators. If X and Y are Hilbert spaces (in fact X Banach and Y normed will suffice), then T: X → Y is compact.

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### An inequality for self-adjoint operators on a Hilbert space by Herbert J. Bernstein Download PDF EPUB FB2

Mond-Pecaric Method in Operator Inequalities (Inequalities for bounded selfadjoint operators on a Hilbert space) (Monographs in Inequalities) Hardcover – January 1, by Takayuki Furuta (Author), Jadranka Micic Hot (Author), Josip Pecaric (Author), Yuki Seo (Author) & 1 moreCited by: An inequality is proved in abstract separable Hilbert space H where A and B are bounded self-adjoint positive operators defined in H such that R(A)=R(B) and R(A) is closed.

View full-text Article. Even more, the concept of convex function of operators, its properties including inequalities and applications, defined for continuous functions on bounded self-adjoint operators in Hilbert space.

Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share. The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators.

Furuta, J. Mićić Hot, J. Pečarić, Y. Seo, Mond-Pečarić Method in Operator Inequalities. Inequalities for bounded self adjoint operators on a Hilbert space, element, Zagreb () Google Scholar.

inequality. We will show that under some assumptions this relation is anti-symmetric. Introduction Let f(t) be a continuous, increasing concave function on the real line R and let A be a bounded self-adjoint operator on some Hilbert space H with an in-ner product h,i.

Then for each unit vector ξ ∈ H, we have so-called Jensen inequality. for any and. Since in the operator order, then which gives that, that is, for any, which implies that for ore, which together with () prove the last part of ().

The case of Lipschitzian functions is as follows. Theorem Let be a self-adjoint operator in the Hilbert space with the spectrum for some real numbers, and let be its spectral by: 2.

Statement of the inequality. The Cauchy–Schwarz inequality states that for all vectors and of an inner product space it is true that |, | ≤, ⋅, where ⋅, ⋅ is the inner es of inner products include the real and complex dot product; see the examples in inner lently, by taking the square root of both sides, and referring to the norms of the vectors, the.

Some norm inequalities for weighted power means of Hilbert space operators are proved for the general class of unitarily invariant norms.

These inequalities generalize a recent inequality of : Danko Jocic. The book is a graduate text on unbounded self-adjoint operators on Hilbert space and their spectral theory with the emphasis on applications in mathematical physics (especially, Schrödinger operators) and analysis (Dirichlet and Neumann Laplacians, Sturm-Liouville operators Reviews: 1.

In a similar way given a Hilbert space H, L(H) will be the algebra of all linear bounded operators on H, L sa(H) the real vector space of self-adjoint operators, and L(H)+ the cone of all positive operators.

For every C∈ L(H) its range will be denoted by R(C), its null space will. sa denotes the convex set of bounded self-adjoint operators on the Hilbert space H with spectra in a real interval I. A partial order is naturally equipped on B(H)sa by deﬁning A ≤ B if and only if B − A ∈ B(H)+.

We write A > 0 to mean that A is a strictly positive operator, or equivalently, A ≥ 0 and A is invertible. operator L deﬁned in equation() is self-adjoint.

The following result gives a useful condition for telling when an operator on a complex Hilbert space is self-adjoint. Proposition Let H be a complex Hilbert space (i.e., F = C), and let A ∈ B(H) be given. Then: A is self-adjoint ⇐⇒ hAf,fi ∈ R ∀f ∈ H.

\$\begingroup\$ I think you forgot to write self-adjoint in your first sentence, even though you write self-adjoint in the title.

\$\endgroup\$ – DisintegratingByParts Sep 9. Spectral Theory of Self-Adjoint Operators in Hilbert Space (Mathematics and its Applications) Hardcover – by Michael Sh. Birman (Author), M.Z. Solomjak (Author) See all 3 formats and editions Hide other formats and editions.

Price New from Used from Cited by:   This paper conducts a further study to the development of the existing theory of Jensen’s inequality for self-adjoint operators in a Hilbert space.

The main contribution is the obtained complementary to Jensen’s inequality for general real-valued twice differentiable functions. In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N: H → H that commutes with its hermitian adjoint N*, that is: NN* = N*N.

Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood. Examples of normal operators are. Hardy-Hilbert's inequality and power inequalities for Berezin numbers of operators Author: Mubariz T.

Garayev, Mehmet Gürdal and Arzu Okudan Subject: Math. Inequal. Appl., 19, 3 () Keywords: 47A63, Hardy inequality, Hardy-Hilbert inequality, Berezin symbol, Berezin number, positive operator, self-adjoint operator Created Date.

The operator A∗ is called the adjoint operator of A. If A = A∗, we say that A is self-adjoint. By the deﬁnition of A∗ we have that the self-adjoint operators on a real ﬁnite dimensional Hilbert space are precisely those operators that are represented by symmetric matrices w.r.t.

an. Operators on Hilbert Space Topics to be covered • Operators on Hilbert spaces Special families of operators: adjoints, projections, Hermitian, unitaries, partial isometries, polar decomposition Density matrices and trace class operators B(H) as dual of trace class • Spectral Theory Spectrum and resolvent.The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions.A Hilbert space is an abstract vector space possessing the structure of an inner product that allows.Theorem A self-adjoint operator in a rigged Hilbert space has a complete system of generalized eigenvectors, corresponding to real generalized eigenvalues.

This is Theorem 5' in Subsection I of Volume 4 of I. M. Gelfand's Generalized Functions (on pg. ). They define generalized eigenvectors and eigenvalues as follows.


text:
# nLab Alexander Beilinson

Alexander (or Sasha) Beĭlinson is currently a professor at University of Chicago. He was student of Yuri Manin at Moscow State University, with main works in algebraic geometry. He has made visionary contributions to the study of algebraic cycles, automorphic forms and L-functions, algebraic K-theory, Hodge theory, motives and motivic cohomology. He conjectured the category of motivic sheaves with remarkable cohomological properties, what is often called the Beĭlinson dream. Some of his works, especially those in collaboration with Vladimir Drinfel'd are of large importance to mathematical physics, especially on their concept of chiral algebras which are an approach to a chiral part of the conformal field theory on a curve, which is a geometric counterpart of the theory of vertex operator algebras. In late 1980s Beĭlinson proposed a geometric analogue of the Langlands program, now called geometric Langlands program, which has been continued in his collaboration with Drinfel’d and also by Ed Frenkel, Dennis Gaitsgory, Ivan Mirković, Kari Vilonen, and more recently taken up by mathematical physics community lead by Witten.

Beĭlinson has substantial contributions to geometric representation theory, which has been revolutionized after two discoveries: his proof (with input from Bernstein) of Kazhdan-Lusztig’s connjectures in 1980, and his paper with Bernstein on what is now called Beĭlinson-Bernstein localization theorem, which lead to influx of the algebraic methods involving algebraic D-modules to representation theory. With Kazhdan, Beĭlinson has used D-modules in the proof of Jantzen’s conjecture, where he introduced the noton of D-affinity and the geometric viewpoint via D-schemes. In the work on Hitchin fibration and Hitchin integrable system (with Drinfel’d) much of technique of algebraic geometry on ind-schemes, including study of D-modules is developed and used. In similar spirit to D-modules, he was also using perverse modules; with Ofer Gabber, Bernstein and Deligne he developed their basic theory including deep and extremely powerful theorem for usage in representation theory, the decomposition theorem (see a survey: pdf). On technical side, he also described appropriate gluing procedure for the derived categories of perverse sheaves, involving t-structures.

Beĭlinson has shown a remarkable structure of the bounded derived category of coherent sheaves on projective spaces, and its connections to quivers. This work, together with subsequent work with Joseph Bernstein and also later works of Mikhail Kapranov and Alexei Bondal marked the birth of the derived noncommutative algebraic geometry. With Victor Ginzburg, Manin, Wolfgang Soergel and others, Beĭlinson introduced a wide picture of “Koszul duality patterns” in representation theory.

His other works concentrated on motives, higher regulators, epsilon-factors?, and so on.

• geometric Langlands homepage

• wikipedia: Beilinson

• Alexander BeilinsonHigher regulators and values of L-functions, Journal of Soviet Mathematics 30 (1985), 2036-2070, (mathnet (Russian), DOI)

• Alexander Beilinson, Higher regulators of curves, Funct. Anal. Appl. 14 (1980), 116-118, mathnet (Russian).

• Alexander Beilinson, Height pairing between algebraic cycles, in K-Theory, Arithmetic and Geometry, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, DOI.

• A. Beilinson, J. Bernstein, Localisations de $\mathfrak{g}$–modules, C. R. Acad. Sci. Paris 292 (1981), 15–18.

• A. A. Beilinson, V. Drinfeld, Chiral Algebras, AMS 2004 (a preprint in various forms since around 1995, cf. here).

• A. A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, J. Amer. Math. Soc. 9 (2): 473–527 (1996).

• A. A. Beĭlinson, V. A. Ginsburg, V. V. Schechtman, Koszul duality, J. Geom. Phys. 5 (1988), no. 3, 317–350.

• A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1980).

• A. Beilinson, V. Ginzburg, Wall-crossing functors and $D$-modules, Representation Theory 3 (electronic), 1–31 (1999)

• A. Beĭlinson, J. Bernstein, A proof of Jantzen conjectures, I. M. Gelʹfand Seminar, 1–50, Adv. Soviet Math., 16, Part 1, Amer. Math. Soc. 1993, pdf

Last revised on March 11, 2015 at 07:27:43. See the history of this page for a list of all contributions to it.


text:
💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

## Educators

+ 6 more educators

### Problem 1

Use the given graph to estimate the value of each derivative. Then sketch the graph of $f'$:

(a) $f'(-3)$
(b) $f'(-2)$
(c) $f'(-1)$
(d) $f'(0)$
(e) $f'(1)$
(f) $f'(2)$
(g) $f'(3)$

Eduard S.

### Problem 2

Use the given graph to estimate the value of each derivative. Then sketch the graph of $f'$:

(a) $f'(0)$
(b) $f'(1)$
(c) $f'(2)$
(d) $f'(3)$
(e) $f'(4)$
(f) $f'(5)$
(g) $f'(6)$
(h) $f'(7)$

Daniel J.

### Problem 3

Match the graph of each function in (a)-(d) with the graph of its derivative in I-IV. Give reasons for your choices.

Daniel J.

### Problem 4

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

DM
David M.

### Problem 5

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

DM
David M.

### Problem 6

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

Aparna S.

### Problem 7

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

DM
David M.

### Problem 8

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

Aparna S.

### Problem 9

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

Aparna S.

### Problem 10

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

Daniel J.

### Problem 11

Trace or copy the graph of the given function $f$. (Assume that the axes have equal scales.) Then use the method of Example 1 to sketch the graph of $f'$ below it.

Aparna S.

### Problem 12

Shown is the graph of the population function $P(t)$ for yeast cells in a laboratory culture. Use the method of Example 1 to graph the derivative $P'(t)$. What does the graph $P'$ tell us about the yeast population?

DM
David M.

### Problem 13

A rechargeable battery is plugged into a charger. The graph shows $C(t),$ the percentage of full capacity that the battery reaches as a function of time $t$ elapsed (in hours).
(a) What is the meaning of the derivative $C^{\prime}(t) ?$
(b) Sketch the graph of $C^{\prime}(t) .$ What does the graph tell you?

DM
David M.

### Problem 14

The graph (from the US Department of Energy) shows how driving speed affects gas mileage. Fuel economy $F$ is measured in miles per gallon and speed $v$ is measured in miles per hour.

(a) What is the meaning of the derivative $F'(v)$?

(b) Sketch the graph of $F'(v)$.

(c) At what speed should you drive if you want to save on gas?

DM
David M.

### Problem 15

The graph shows how the average age of first marriage of Japanese men varied in the last half of the 20th century. Sketch the graph of the derivative function $M'(t)$. During which years was the derivative negative?

DM
David M.

### Problem 16

Make a careful sketch of the graph of $f$ and below it sketch the graph of $f'$ in the same manner as in Exercises 4-11. Can you guess a formula for $f'(x)$ from its graph?

$f(x) = \sin x$

Aparna S.

### Problem 17

Make a careful sketch of the graph of $f$ and below it sketch the graph of $f'$ in the same manner as in Exercises 4-11. Can you guess a formula for $f'(x)$ from its graph?

$f(x) = e^x$

Daniel J.

### Problem 18

Make a careful sketch of the graph of $f$ and below it sketch the graph of $f'$ in the same manner as in Exercises 4-11. Can you guess a formula for $f'(x)$ from its graph?

$f(x) = \ln x$

Daniel J.

### Problem 19

Let $f(x) = x^2$.

(a) Estimate the values of $f'(0)$, $f'(\frac{1}{2})$, $f'(1)$, and $f'(2)$ by using a graphing device to zoom in on the graph of $f$.

(b) Use symmetry to deduce the values of $f'(-\frac{1}{2})$, $f'(-1)$, and $f'(-2)$.

(c) Use the results from parts (a) and (b) to guess a formula for $f'(x)$.

(d) Use the definition of derivative to prove that your guess in part (c) is correct.

DM
David M.

### Problem 20

Let $f(x) = x^3$.

(a) Estimate the values of $f'(0)$, $f'(\frac{1}{2})$, $f'(1)$, $f'(2)$, and $f'(3)$ by using a graphing device to zoom in on the graph of $f$.

(b) Use symmetry to deduce the values of $f'(-\frac{1}{2})$, $f'(-1)$, $f'(-2)$, and $f'(-3)$.

(c) Use the values from parts (a) and (b) to graph $f'$.

(d) Guess a formula for $f'(x)$.

(e) Use the definition of derivative to prove that your guess in part (d) is correct.

DM
David M.

### Problem 21

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x) = 3x - 8$

Leon D.

### Problem 22

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x) = mx + b$

Leon D.

### Problem 23

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(t) = 2.5t^2 + 6t$

Leon D.

### Problem 24

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x) = 4 + 8x - 5x^2$

Leon D.

### Problem 25

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x) = x^2 - 2x^3$

Evelyn C.

### Problem 26

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$g(t) = \dfrac{1}{\sqrt{t}}$

Ma. Theresa A.

### Problem 27

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$g(x) = \sqrt{9 - x}$

Ma. Theresa A.

### Problem 28

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x) = \dfrac{x^2 - 1}{2x - 3}$

Evelyn C.

### Problem 29

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$G(t)=\frac{1-2 t}{3+t}$

Ma. Theresa A.

### Problem 30

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x) = x^{3/2}$

DM
David M.

### Problem 31

Find the derivative of the function using the definition of derivative. State the domain of the function and the domain of its derivative.

$f(x) = x^4$

Ma. Theresa A.

### Problem 32

(a) Sketch the graph of $f(x) = \sqrt{6 - x}$ by starting with the graph of $y = \sqrt{x}$ and using the transformations of Section 1.3.

(b) Use the graph from part (a) to sketch the graph of $f'$.

(c) Use the definition of a derivative to find $f'(x)$. What are the domains of $f$ and $f'$?

(d) Use a graphing device to graph $f'$ and compare with your sketch in part (b).

Linda H.

### Problem 33

(a) If $f(x) = x^4 + 2x$, find $f'(x)$.
(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of $f$ and $f'$.

Daniel J.

### Problem 34

(a) If $f(x) = x + 1/x$, find $f'(x)$.
(b) Check to see that your answer to part (a) is reasonable by comparing the graphs of $f$ and $f'$.

Daniel J.

### Problem 35

The unemployment rate $U(t)$ varies with time. The table gives the percentage of unemployed in the US labor force from 2003 to 2012.

(a) What is the meaning of $U'(t)$? What are its units?

(b) Construct a table of estimated values for $U'(t)$.

Daniel J.

### Problem 36

The table gives the number $N(t)$, measured in thousands, of minimally invasive cosmetic surgery procedures performed in the United States for various years $t$.

(a) What is the meaning of $N'(t)$? What are its units?

(b) Construct a table of estimated values for $N'(t)$.

(c) Graph $N$ and $N'$.

(d) How would it be possible to get more accurate values for $N'(t)$?

DM
David M.

### Problem 37

The table gives the height as time passes of a typical pine tree grown for lumber at a managed site.

If $H(t)$ is the height of the tree after $t$ years, construct a table of estimated values for $H'$ and sketch its graph.

DM
David M.

### Problem 38

Water temperature affects the growth rate of brook trout. The table shows the amount of weight gained by brook trout after 24 days in various water temperatures.

If $W(x)$ is the weight gain at temperature $x$, construct a table of estimated values for $W'$ and sketch its graph. What are the units for $W'(x)$?

DM
David M.

### Problem 39

Let $P$ represent the percentage of a city's electrical power that is produced by solar panels $t$ years after January 1, 2000.

(a) What does $dP/dt$ represent in this context?
(b) Interpret the statement $$\frac{dP}{dt} \bigg|_{t = 2} = 3.5$$

Leon D.

### Problem 40

Suppose $N$ is the number of people in the United States who travel by car to another state for a vacation this year when the average price of gasoline is $p$ dollars per gallon. Do you expect $dN/dp$ to be positive or negative? Explain.

Leon D.

### Problem 41

The graph of $f$ is given. State, with reasons, the numbers at which $f$ is $not$ differentiable.

JT
Jessie T.

### Problem 42

The graph of $f$ is given. State, with reasons, the numbers at which $f$ is $not$ differentiable.

Aparna S.

### Problem 43

The graph of $f$ is given. State, with reasons, the numbers at which $f$ is $not$ differentiable.

DM
David M.

### Problem 44

The graph of $f$ is given. State, with reasons, the numbers at which $f$ is $not$ differentiable.

Aparna S.

### Problem 45

Graph the function $f(x) = x + \sqrt{| x |}$. Zoom in repeatedly, first toward the point $(-1, 0)$ and then toward the origin. What is different about the behavior of $f$ in the vicinity of these two points? What do you conclude about the differentiability of $f$?

Daniel J.

### Problem 46

Zoom in toward the points $(1, 0)$, $(0, 1)$, and $(-1, 0)$ on the graph of the function
$g(x) = (x^2 -1)^{2/3}$. What do you notice? Account for what you see in terms of the differentiability of $g$.

Daniel J.

### Problem 47

The graphs of a function $f$ and its derivative $f'$ are shown. Which is bigger, $f'(-1)$ or $f''(1)$?

Aparna S.

### Problem 48

The graphs of a function $f$ and its derivative $f'$ are shown. Which is bigger, $f'(-1)$ or $f''(1)$?

Aparna S.

### Problem 49

The figure shows the graphs of $f$, $f'$, and $f''$. Identify each curve, and explain your choices.

DM
David M.

### Problem 50

The figure shows graphs of $f$, $f'$, $f''$, and $f'''$. Identify each curve, and explain your choices.

Aparna S.

### Problem 51

The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Identify each curve, and explain your choices.

DM
David M.

### Problem 52

The figure shows the graphs of four functions. One is the position function of a car, one is the velocity of the car, one is its acceleration, and one is its jerk. Identify each curve, and explain your choices.

Daniel J.

### Problem 53

Use the definition of a derivative to find $f'(x)$ and $f''(x)$. Then graph $f$, $f'$, and $f''$ on a common screen and check to see if your answers are reasonable.

$f(x) = 3x^2 + 2x + 1$

Daniel J.

### Problem 54

Use the definition of a derivative to find $f'(x)$ and $f''(x)$. Then graph $f$, $f'$, and $f''$ on a common screen and check to see if your answers are reasonable.

$f(x) = x^3 - 3x$

Daniel J.

### Problem 55

If $f(x) = 2x^2 - x^3$, find $f'(x)$, $f''(x)$, $f'''(x)$, and $f^{(4)}(x)$. Graph $f$, $f'$, $f''$, and
$f'''$ on a common screen. Are the graphs consistent with the geometric interpretations of these derivatives?

Leon D.

### Problem 56

(a) The graph of a position function of a car is shown, where $s$ is measured in feet and $t$ in seconds. Use it to graph the velocity and acceleration of the car. What is the acceleration at $t = 10$ seconds?

(b) Use the acceleration curve from part (a) to estimate the jerk at $t = 10$ seconds. What are the units for jerk?

Daniel J.

### Problem 57

Let $f(x) = \sqrt[3]{x}$.
(a) If $a \neq 0$, use Equation 2.7.5 to find $f'(a)$.
(b) Show that $f'(0)$ does not exist.
(c) Show that $y = \sqrt[3]{x}$ has a vertical tangent line at $(0, 0)$. (Recall the shape of the graph of
$f$. See Figure 1.2.13)

Anupa D.

### Problem 58

(a) If $g(x) = x^{2/3}$, show that $g'(0)$ does not exist.
(b) If $a \neq 0$, find $g'(a)$.
(c) Show that $y = x^{2/3}$ has a vertical tangent line at $(0, 0)$.
(d) Illustrate part (c) by graphing $y = x^{2/3}$.

Daniel J.

### Problem 59

Show that the function $f(x) = | x - 6 |$ is not differentiable at 6. Find a formula for $f'$ and sketch its graph.

Leon D.

### Problem 60

Where is the greatest integer function $f(x) = [ x ]$ not differentiable? Find a formula for $f'$ and sketch its graph.

Darshan M.

### Problem 61

(a) Sketch the graph of the function $f(x) = x | x |$.
(b) For what values of $x$ is $f$ differentiable?
(c) Find a formula for $f'$.

Daniel J.

### Problem 62

(a) Sketch the graph of the function $g(x) = x + | x |$.
(b) For what values of $x$ is $g$ differentiable?
(c) Find a formula for $g'$.

Daniel J.

### Problem 63

Recall that a function $f$ is called \textit{even} if $f(-x) = f(x)$ for all $x$ in its domain and \textit{odd} if $f(-x) = -f(x)$ for all such $x$. Prove each of the following.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.

Daniel J.

### Problem 64

The left-hand and right-hand derivatives of $f$ at $a$ are defined by
$$\begin{array}{c}{f_{-}^{\prime}(a)=\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h}} \\ {\text { and } \quad f_{+}^{\prime}(a)=\lim _{h \rightarrow 0^{-}} \frac{f(a+h)-f(a)}{h}}\end{array}$$
if these limits exist. Then $f^{\prime}(a)$ exists if and only if these one-sided derivatives exist and are equal. (a) Find $f^{\prime}-(4)$ and $f^{\prime}+(4)$ for the function
$$f(x)=\left\{\begin{array}{l}{0} \\ {5-x} \\ {\frac{1}{5-x}}\end{array}\right.$$
if $x \leq 0$
if $0< x<4$
if $x \geqslant 4$
(b) Sketch the graph of $f$ .
(c) Where is $f$ discontinuous?
(d) Where is $f$ not differentiable?

Will E.

### Problem 65

Nick starts jogging and runs faster and faster for 3 minutes, then he walks for 5 minutes. He stops at an intersection for 2 minutes, runs fairly quickly for 5 minutes, then walks for 4 minutes.
$$\begin{array}{l}{\text { (a) Sketch a possible graph of the distance } s \text { Nick has covered }} \\ {\text t \text { minutes. }} \\ {\text { (b) Sketch a graph of } d s / d t}\end{array}$$

Pawan Y.

### Problem 66

When you turn on a hot-water faucet, the temperature $T$ of the water depends on how long the water has been running.
$$\begin{array}{l}{\text { (a) Sketch a possible graph of } T \text { as a function of the time } t} \\ {\text { that has elapsed since the faucet was turned on. }} \\ {\text { (b) Describe how the rate of change of } T \text { with respect to } t} \\ {\text { varies as } t \text { increases. }} \\ {\text { (c) Sketch a graph of the derivative of } T .}\end{array}$$

Carson M.

### Problem 67

Let $\ell$ be the tangent line to the parabola $y=x^{2}$ at the point
$(1,1)$ . The angle of inclination of $\ell$ is the angle $\phi$ that $\ell$
makes with the positive direction of the $x$ -axis. Calculate $\phi$
correct to the nearest degree.


text:
### Archives For Coxeter groups

The Joint Mathematics Meetings took place last week in Baltimore, MD. The JMM is a joint venture between the American Mathematical Society and the Mathematical Association of America. Held each January, the JMM is the largest annual mathematics meeting in the world. Attendance in 2013 was an incredible 6600. I don’t know what the numbers were this year, but my impression was that attendance was down from last year.

As usual, the JMM was a whirlwind of talks, meetings, and socializing with my math family. My original plan was to keep my commitments to a minimum as I’ve been feeling stretched a bit thin lately. After turning down a few invitations to give talks, I agreed to speak as part of panel during a Project NExT session. The title of the panel was “Tried & True Practices for IBL & Active Learning”. Here, IBL is referring to inquiry-based learning. Last year I gave several talks and co-organized workshops about IBL, so I knew preparing for the panel wouldn’t be a huge burden.

The other speakers on the panel were Angie Hodge (University of Nebraska, Omaha) and Anna Davis (Ohio Dominican University). Each of us were given 15 minutes to discuss our approach to IBL and active learning and then we had about 20 minutes for questions. For my portion, I summarized my implementation of IBL in my proof-based courses followed by an overview of my IBL-lite version of calculus. Angie discussed in detail the way in which UNO is successfully adopting IBL in some of their calculus sections. Anna spent some time telling us about an exciting project called the One-Room Schoolhouse, which attempts to solve the following problem: Small colleges and universities often struggle to offer a variety of upper-level courses due to low enrollment. According to the ORS webpage, the ORS setting allows schools to offer low-enrollment courses every semester at no additional cost to the institution. ORS is made possible by flipping the classroom.

If you are interested in my slides, you can find them below.

I’m really glad that I took the time to speak on the panel. The audience asked some great questions and I had several fantastic follow-up conversations with people the rest of the week.

As a side note, one thing I was reminded of is that speaking early during the conference (which is when the panel occurred) is much better than speaking later. The past couple JMMs, I spoke on the last day or two, and in this case, my talk is always taking up bandwidth in the days leading up to it. It’s nice to just get it out of the way and then be able to concentrate on enjoying the conference.

One highlight of the conference was having a pair of my current undergraduate research students (Michael Hastings and Sarah Salmon) present a poster during the Undergraduate Student Poster Session. Michael and Sarah have been working on an original research project involving diagrammatic representations of Temperley-Lieb algebras. I’ve been extremely impressed with the progress that they have made and it was nice to be able to see them show off their current results. The title of their poster was “A factorization of Temperley–Lieb diagrams” (although this was printed incorrectly in the program). Here is their abstract:

The Temperley-Lieb Algebra, invented by Temperley and Lieb in 1971, is a finite dimensional associative algebra that arose in the context of statistical mechanics. Later in 1971, R. Penrose showed that this algebra can be realized in terms of certain diagrams. Then in 1987, V. Jones showed that the Temperley–Lieb Algebra occurs naturally as a quotient of the Hecke algebra arising from a Coxeter group of type $A$ (whose underlying group is the symmetric group). This realization of the Temperley-Lieb Algebra as a Hecke algebra quotient was generalized to the case of an arbitrary Coxeter group. In the cases when diagrammatic representations are known to exist, it turns out that every diagram can be written as a product of “simple diagrams.” These factorizations correspond precisely to factorizations in the underlying group. Given a diagrammatic representation and a reduced factorization of a group element, it is easy to construct the corresponding diagram. However, given a diagram, it is generally difficult to reconstruct the factorization of the corresponding group element. In cases that include Temperley–Lieb algebras of types $A$ and $B$, we have devised an efficient algorithm for obtaining a reduced factorization for a given diagram.

Perhaps I’m biased, but I think their poster looks pretty darn good, too.

All in all, I would say that I had a successful JMM. I had a lot more meetings than I’ve had in the past, so I missed out on several talks I really would have liked to attend. There’s always next year.

On March 1-3, Jeff Rushall (he’s the Patrick Stewart-looking guy in the photo above) and I, together with a group of 8 Northern Arizona University mathematics majors, attended the 2013 Southwestern Undergraduate Mathematics Research Conference (SUnMaRC) at the University of New Mexico in Albuquerque. SUnMaRC is an annual conference that provides undergraduates in the Southwest an opportunity to give 20 minute talks in order to showcase the work they have been doing for their undergraduate research projects. Three of the 8 NAU students are currently conducting research projects under my guidance. That’s them in the picture above (taken during a break at the conference).

Two of my students, Dane Jacobson and Michael Woodward, have spent the spring semester studying the mathematics behind Spinpossible, which is a free game that is available for iOS and Android devices. Alternatively, you can just play the game in any modern web browser. The game is played on a 3 by 3 board of scrambled tiles numbered 1 to 9, each of which may be right-side-up or up-side-down. The objective of the game is to return the board to the standard configuration where tiles are arranged in numerical order and right-side-up. This is accomplished by a sequence of “spins”, each of which rotates a rectangular region of the board by 180 degrees. The goal is to minimize the number of spins used. It turns out that the group generated by the set of allowable spins is identical to the symmetry group of the 9 dimensional hyper-cube (equivalently, a Coxeter group of type $B_9$).

In a 2011 paper, Alex Sutherland and Andrew Sutherland (a father and son team) present a number of interesting results about Spinpossible and list a few open problems. You can find the paper here. As a side note, Alex is one of the developers of the game and his father, Andrew, is a mathematics professor at MIT. Using brute-force, the Sutherlands verified that every scrambled board can be solved in at most 9 moves. One of the goals of my students’ research project is to find a short proof of this fact. During their talk at SUnMaRC, Dane and Michael presented their current progress on this unexpectedly difficult problem. Here are their slides.

A week after SUnMaRC, Dane and Michael gave a similar and improved talk (which includes some random images of Samuel L. Jackson) during NAU’s Friday Afternoon Undergraduate Mathematics Seminar (FAMUS). As the name of the seminar suggests, the target audience for the FAMUS talks are undergraduates. You can find the slides for their FAMUS talk here.

My third student, Selina Gilbertson, has spent the spring semester working on extending the results of some of my previous students. During the 2011-2012 academic year, I mentored Ryan Cross, Katie Hills-Kimball, and Christie Quaranta at Plymouth State University on an original research project aimed at exploring the “T-avoiding” elements in Coxeter groups of type $F$. Coxeter groups can be thought of as generalized reflections groups. In particular, a Coxeter group is generated by a set of elements of order two. Every element of a Coxeter group can be written as an expression in the generators, and if the number of generators in an expression is minimal, we say that the expression is reduced. We say that an element $w$ of a Coxeter group is T-avoiding if $w$ does not have a reduced expression beginning or ending with a pair of non-commuting generators. My PSU students successfully classified the T-avoiding elements in the infinite Coxeter group of type $F_5$, as well as the finite Coxeter group of type $F_4$. At the time, we had conjectured that our classification holds more generally for arbitrary $F_n$. However, Selina has recently discovered that this is far from true. In Selina’s SUnMaRC talk, she relayed her current progress on classifying the T-avoiding elements in type $F_n$. Here are her slides.

Both sets of students gave wonderful talks. In fact, all 8 NAU students gave fantastic talks. I was impressed. Below are some photos from our trip.

On Tuesday, February 5 (my birthday!), I gave a talk titled “A diagrammatic representation of the Temperley-Lieb algebra” in the NAU Department of Mathematics and Statistics Colloquium. Here is the abstract.

One aspect of my research involves trying to prove that certain associative algebras can be faithfully represented using “diagrams.” These diagrammatic representations are not only nice to look at, but they also help us recognize things about the original algebra that we may not otherwise have noticed. In this talk, we will introduce the diagram calculus for the Temperley-Lieb algebra of type $A$. This algebra, invented by Temperley and Lieb in 1971, is a certain finite dimensional associative algebra that arose in the context of statistical mechanics in physics. We will show that this algebra has dimension equal to the nth Catalan number and discuss its relationship to the symmetric group. If time permits, we will also briefly discuss the diagrammatic representation of the Temperley-Lieb algebra of type affine $C$.

And here are the slides.

Despite the fact that this was a 50-minute talk, it was intended to be an overview of one aspect of a long and complex story. The subject matter is intimately related to my PhD thesis, as well as a series of papers that I have written.

• Ernst, D. C. (2010). Non-cancellable elements in type affine $C$ Coxeter groups. Int. Electron. J. Algebra, 8, 191–218. [arXiv]
• Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley-Lieb algebra, I. J. Pure Appl. Alg. (to appear). [arXiv]
• Ernst, D. C. (2012). Diagram calculus for a type affine $C$ Temperley–Lieb algebra, II. [arXiv]

In addition, there is (at least) a part III that goes with the last two papers that is in progress.

On Friday, September 12, 2012, I gave a 25 minute talk titled “An open problem of the symmetric group” during NAU’s Friday Afternoon Mathematics Undergraduate Seminar (FAMUS). Here is the open problem that I discussed.

How many commutation classes does the longest element in the symmetric group have?

The main goal of the talk was to understand what this question is asking. The secondary goal was to illustrate that mathematics is a lively field with open questions and to provide an example of what research in mathematics looks like. Here’s the abstract.

Many people are often surprised to hear that mathematicians do research. What is mathematical research? Research in mathematics takes many forms, but one common theme is that the research seeks to answer an open question concerning some collection of mathematical objects. The goal of this talk will be to introduce you to one of the many open questions in mathematics: how many commutation classes does the longest element in the symmetric group have? This problem has been nicknamed “Heroin Hero” by my advisor in honor of a game from the TV show “South Park” in which the character Stan obsesses over chasing a dragon. We will review the basics of the symmetric group and introduce all of the necessary terminology, so that we can understand this problem.

Here are the slides.

I really enjoy this short video introduction to group theory by James Grime (aka Singing Banana), especially since it mentions Coxeter groups (one of research interestes).