text:
# Na2So4(aq)+HCl(aq) (Any reaction?)

I am just reading what a double replacement reaction is and what are the examples.

So I came up with this question : what is a reaction actually?

For example when we add $\ce{Fe(NO3)3}$ and NaOH to water we see $\ce{Fe(OH)3}$ forms as a precipitate. So hay! A reaction has occurred!

But the reaction in the title doesn't form any precipitate depending on solubility rules. So does any reaction really takes place or the substances just dissociate into separate ions?

$\ce{Na2SO4}$ will dissolve and dissociate into separate ions. And there will be some protonation $\ce{H+ + SO4^{2-} -> HSO4-}$.

A reaction occurs when the starting chemicals (molecules, ions) are changed into some different molecules (ions). This involves making and breaking bonds. It can include transferring electrons from one species to another.

Just because a ppt is not formed does not mean that reaction has not occurred. In general chemists use a host of different techniques to determine the outcome of a reaction. Some of these are (in random order) measuring pH, quantitative analysis,titration, nuclear magnetic resonance (NMR), uv, visible, infra red & microwave spectroscopy, chromatography, xray scattering, mass spectrometry, and i've missed quite a few out. After all knowing what you have made, or if you have made it, is a major part of chemistry.

• Thanks. But which one occurs in the reaction mentioned in the title? – Rima Jul 21 '16 at 10:31

Your question seems more philosophical then chemical in that you are aware of what happens and why, but want to attach semantic labels to various states and process involved. This study is called semantics. In information science, ontologies concern the codifying of semantics. While arguably not classically chemistry per se, ontologies are valuable resources in science, particularly in the era of "big data" and involve a great deal of chemistry to develop and deploy correctly. However, analysis such as the following is definitely in the domain of recreational or pedagogical study, and the best answer is probably you know one when you see one.

Fortunately, to that end IUPAC are greatly concerned with matters of semantics and ontology, and the IUPAC Gold Book provides endless hours of gripping reading on such matters.

If you want to know the IUPAC gold books definition of chemical reaction, chemical species, and molecular entity (important, more fundamental concepts), then you can easily find it online, but ontologies can be confusing and not particularly enlightening, so please accept this user's "reader" on the relevant entries.

A chemical reaction is the result of the conversion of chemical species. So, to answer your question, the first thing we need to do is to work out the chemical species involved (whatever they are) and work out whether they are converted at the end of the reaction compared to the start.

A chemical species is a set of identical molecular entities. Jumping ahead of ourselves a bit, for example, some "ideal" water is a chemical species, but a single $\ce{H2O}$ molecule is a molecular entity. Of course, real water will at least have $\ce{H3O+}$ and $\ce{OH-}$ in it, and so each of those molecules is a different molecular entity, so together they make up other chemical species in water.

But we've just put the cart before the horse. We don't have a definition of molecular entity yet. The precision required, surprisingly, depends a lot on context, ie your theoretical and practical capabilities and purposes. For example, maybe singlet and triplet oxygen are important to you, in which case they are considered different molecular entities, or maybe you don't care and so the oxygen is all considered a single molecular entity. That may disappoint armchair philosophers but might constitute a ray of hope to people who need to just get the job done.

But the actual definition of molecular entity in the gold book is a bit of a damp squib: it merely lists some examples followed by "etc". The definition of chemical species comes to our rescue here, where it describes species as molecular entities which can explore the same set of molecular energy levels.

While IUPAC are quiet on what molecular energy levels are, that's actually quite uncontroversial: they are the energy levels of electrons in atoms, which may have been transformed by adjacent atoms, also known as a chemical bond.

So, putting it back together:

Atoms can associate with each other through chemical bonding to create a new set of energy levels, and these are known as molecular energy levels. Where there exist entities which indistinguishable (for the purposes envisaged), which have available to them (in the context of the experiment) various molecular energy levels then they are the molecular entities of of a chemical species. Should the inventory of such species change in an experiment, then you have a reaction.

Once more, with feeling:

A reaction is a process whereby, as far as the precision of the experiment is concerned, the variety of indistinguishable assemblies of atoms with distinct accessible molecular energy levels changes.

For the situation you asked about: the initial dissolution is a chemical reaction because various molecular bonds are broken, various intermolecular bonds of solvation are formed in the reactions with the solvent, and so on. This changes the inventory of the kinds of entities with distinct molecular energy levels. The precipitation is also the result of a reaction in solution, in that many of these changes in bonding on solvation are reversed. However, the settling action of the precipitation itself is largely a physical, not chemical, process concerning gravity, convection, etc. Should there be no precipitation, only the solvation reaction has taken place.

(Quotes in this answer are for clarity, not of external sources).

format: plain


text: But my book says just the opposite. But if $$v_1$$ is lower than $$v_2$$, the flow wouldn't be straight as depicted in the picture. I'd preferably go with the book's answer but I also want to know WHY that is the case.


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# Fluid

Written by Jerry Ratzlaff on . Posted in Fluid Dynamics

A fluid is a substance that deforms and changes position when put under stress.  Fluids can be both liquids and gases.  All liquids share certain characteristics, these can be used to distinguish between a liquid or a gas

• Compressibility - Liquids are only slightly compressable, but for most purposes they are non compressable.  Gasses are highly compressable.
• Molecular spacing -Liquids molequles are held quite close togeather and have a strong attraction force.  The travel distance of liquid molecules is quite short between there collisions and they also have low kinetic energy.
• Pressure - Pressure on a liquid is the same in all directions from any point in the liquid.  Any fluid pressure on a solid surface is normal to that surface.
• Resistance to motion - Since liquids are affected by viscosity, there is resistance to the change in velocity.  Gasses have a very low viscosity.
• Shape and volume - Since liquids and gasses do not support shear, they take on the shape of the container or surface of the object.  The volume of a liquid is not significantly affected by pressure or temperature and has a fixed volume no matter the size of the container.  Gases also take on the shape of the container or surface of the object but the volume of a gas can change by pressure or temperature.
• Shear resistance - Liquids and gasses deform continuously with little force which does not support shear.

## Types of Fluids

• Fluids
• Ideal Fluids - This assumes to have no viscosity, meaning no resistance or shear, and not compressable.  In an ideal fluids there is uniform velocity distribution when flowing.
• Real fluids - This fluid does not have uniform velocity distribution, it is compressable, shows finite viscosities and experience friction and turbulance in flow
• Newtonian fluids - This fluid's viscosity is constant no matter how much shear is applied for a constant temperature.
• non-Newtonian fluids - This is the opposite of Newtonian fluids and does not following Newton's law of viscosity.  These fluids having variable viscosity when shear is applied.
• Dilatant (shear thickening) fluids - Viscosity of the fluid increases with increased stress.
• Pseudoplastic (shear thinning) fluids - The opposite of dilatant fluids. Viscosity of the fluid decreases with increased stress.
• Rheopectic (time dependent) fluids - Viscosity of the fluid increases with stress over time.
• Thixotropic (time dependent) fluids - Viscosity of the fluid decreases with stress over time.
• Bingham (plastics) fluids - The fluid appears solid up to a particular stress and with further increase in stress there will be shear deformation.

## Fluid Viscosity

The viscosity, abbreviated as $$\mu$$ (Greek symbol mu), of a fluid is the measure of that's fluids resistance to flow when acted upon by an external force such as pressure differential or gravity.  When the temperature is raised most fluids will flow easier.  Fluid viscosity can be refered to as absolute viscosity or dynamic viscosity

Tags: Equations for Fluid


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A list of lists containing all WrappedModels trained in the benchmark experiment.

If models is FALSE in the call to benchmark, the function will return NULL.

getBMRModels(bmr, task.ids = NULL, learner.ids = NULL, drop = FALSE)

## Arguments

bmr (BenchmarkResult) Benchmark result. (character(1)) Restrict result to certain tasks. Default is all. (character(1)) Restrict result to certain learners. Default is all. (logical(1)) If drop is FALSE (the default), a nested list with the following structure is returned: res[task.ids][learner.ids]. If drop is set to TRUE it is checked if the list structure can be simplified. If only one learner was passed, a list with entries for each task is returned. If only one task was passed, the entries are named after the corresponding learner. For an experiment with both one task and learner, the whole list structure is removed. Note that the name of the task/learner will be dropped from the return object.

(list).

## See also

Other benchmark: BenchmarkResult, batchmark, benchmark, convertBMRToRankMatrix, friedmanPostHocTestBMR, friedmanTestBMR, generateCritDifferencesData, getBMRAggrPerformances, getBMRFeatSelResults, getBMRFilteredFeatures, getBMRLearnerIds, getBMRLearnerShortNames, getBMRLearners, getBMRMeasureIds, getBMRMeasures, getBMRPerformances, getBMRPredictions, getBMRTaskDescs, getBMRTaskIds, getBMRTuneResults, plotBMRBoxplots, plotBMRRanksAsBarChart, plotBMRSummary, plotCritDifferences, reduceBatchmarkResults


text:
DifferencePositions - Maple Help

StringTools

 DifferencePositions
 compute the character positions at which two strings differ

 Calling Sequence DifferencePositions( s, t )

Parameters

 s - string t - string

Description

 • The DifferencePositions( s, t ) function computes the positions at which the strings s and t differ.  It returns the (ordered) expression sequence of these positions.  If s and t are equal, then NULL is returned. If s and t differ in length, then the positions up to the shorter of s and t are computed and returned.
 • The length of the returned expression sequence is equal to the Hamming distance between s and t, provided that s and t are the same length. (Note that the HammingDistance command computes the generalized Hamming distance for strings of unequal length.)
 • All of the StringTools package commands treat strings as (null-terminated) sequences of $8$-bit (ASCII) characters.  Thus, there is no support for multibyte character encodings, such as unicode encodings.

Examples

 > $\mathrm{with}\left(\mathrm{StringTools}\right):$
 > $\mathrm{DifferencePositions}\left("abc","abc"\right)$
 > $\mathrm{DifferencePositions}\left("abcde","axcye"\right)$
 ${2}{,}{4}$ (1)
 > $\mathrm{HammingDistance}\left("abcde","axcye"\right)$
 ${2}$ (2)
 > $\mathrm{DifferencePositions}\left("abcde","axcyefg"\right)$
 ${2}{,}{4}$ (3)
 > $\mathrm{HammingDistance}\left("abcde","axcyefg"\right)$
 ${4}$ (4)
 > $\mathrm{DifferencePositions}\left("edit","tide"\right)$
 ${1}{,}{2}{,}{3}{,}{4}$ (5)
 > $\mathrm{IsDerangement}\left("edit","tide"\right)$
 ${\mathrm{true}}$ (6)

Compatibility

 • The StringTools[DifferencePositions] command was introduced in Maple 18.


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# It's a New Car

One of our vehicles was nearing the end of its useful, economic life and so we began the process of searching out a suitable replacement.

The vehicle to be replaced was a Mazda Tribute - a 3.0 litre gas-guzzling SUV that has transported us safely but expensively for the past seven years. It was bought second hand in 2011 when it had around 60K on the clock and we’d taken it up to around 255K in those seven years of ownership.

We thought about buying a small or medium SUV as a replacement and had thought we’d buy another second hand one - something around 3 or 4 years old…

But some visits to local car yards pushed us towards something new instead as the market for second-hand, late-model SUVs is quite limited (and therefore relatively expensive). We looked at a couple of Mazdas (CX3, CX5) and also Hondas (CR-V and HR-V) and immediately dismissed the more expensive incarnations (the CX5 and CR-V).

We quite liked the CX3 but were told it was built on the same “platform” as the Mazda 2 - so you’re effectively paying three thousand dollars for a larger cabin than the sedan or hatch offers. We also liked the Honda HR-V but it was similarly three or four thousand dollars more expensive than its sedan or hatch cousin/brother/sister.

The smaller SUVs also had poorer fuel consumption than the sedan or hatch. I’d worked out that our Mazda Tribute, at around 11 litres/100 km had cost us something like $8,000 extra in fuel than a vehicle using 8 litres/100 km over the time we had it. So we opted for a sedan/hatch rather than SUV. We finally chose the Mazda 3 (same platform as the CX5) on the basis that you get more car for your money with a sedan or hatch than the equivalent SUV. These days we rarely need the extra space offered by an SUV so can save a few thousand dollars on the purchase price, and the same again over the life of the vehicle in fuel costs. The final decision was a Mazda 3 hatch “Maxx Sport” in “eternal blue mica”. The Maxx Sport is the first model up from the base Neo Sport and offers a range of safety features over the base. The next model up from that - the “Touring” was offering more aesthetic choices but we needed a vehicle, not a statement. We also opted for an automatic - which added$2K to the price, but the salesman suggested that is reflected in the resale as the market for manuals is small and diminishing.

Being close to the end of year the deal was sweetened (if that’s possible when paying someone over $20K) by going from a five to seven year warranty, and our$500 worth of accessories were also “free”.

So there you have it. It’s in the driveway and has its first thousand km on the clock.

And the Tribute was traded in.


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# What causes polarised materials to change colour under stress?

Our physics teacher showed the class a really interesting demonstration. He used two polarised filters in opposite orientations, then he took some antistatic tape and stretched it under the two plates. The resulting image was projected on the wall using an overhead projector unit.

Under stress, the originally clear looking tape (as it looked between the polarised filters) turned all sorts of weird colours, and apparently different colors correlate to different levels of stress. What causes this odd effect?

• In case it's not clear from the answers of sigoldberg1 and Flaviu Cipcigan below, the title of this question isn't quite right. It's not "polarised materials" which change color under stress, it's birefringent materials viewed under crossed polarizers. – j.c. Nov 5 '10 at 14:52

Stress implies changing the thickness, therefore the Bragg reflection change. Or shorter: Interference

edit Also, as sigoldberg1 states, Birefringence might occur, too

• Nice short answer. I think you mean 'reflection' though, 'reflexes' doesn't make sense here. – Noldorin Nov 5 '10 at 16:20
• @downvoter: no comment? – Tobias Kienzler Dec 6 '10 at 9:49

I agree with the answer given by sigoldberg1 -- it is most probably stress induced birefringence. Bragg scattering would not (at least to first order) change the polarisation of the incoming light beam, thus making the crossed polarisers pretty useless. Also, changes in colour due to Bragg scattering would be observed on reflection and would be angle dependent.

What actually happens in experiments of the type shows in this video is the following:

• The first polariser is used to produce linearly polarised light.
• The second polariser is rotated at 90 degrees with respect to the first, and thus in the absence of any material, no light will pass through (from Malus' law, $cos^2(90^\circ) = 0$
• However, if the material is birefringent (the refractive index depends on polarisation) the polarisation angle of light passing through will be rotated, and this rotation will depend on the wavelength of the light.
• Thus, the final polarisation angle will be different than 90 degrees, and that difference will depend on the wavelength of the light, giving the patterns you see in such an experiment.

Stress induced birefringence is of technical importance due to the fact that "similar effects occur in bent optical fibers, and also due to thermal effects in laser crystals, which can lead to depolarization loss" (ref).

More likely that the stress induced strain, i.e. stretching of the tape. When the tape stretches, the polymer molecules in it tend to orient or also stretch. This leads to a third layer of polarizing material in the middle, or even birefringence, which makes the colors between the crossed polarizing filters you saw on the screen. At least that's the way I remember it.

• You should have left a comment at my answer, I didn't see your post until now. I should definitely also have included Birefringence in my answer – Tobias Kienzler Nov 8 '10 at 8:11

This is a little off-topic, in that the answers to the question by sigoldberg1 and Flaviu Cipcigan address the relevant physics in plenty of detail already, but I can't resist linking to this video which shows a related phenomenon: beautiful swirls of colors in liquid crystals viewed under crossed-polarizers.

Here the principle is close to what is described by Flaviu Cipcigan, but in the systems of liquid crystals showed in the movie, the stresses arise from the interaction between flow and temperature gradients in the liquid, and instead of birefringence, it's the fact that the molecular orientations themselves are rotating and changing in response to the stresses, and thus letting different amounts of light of different colors through the two polarizers.

Well they don't change color under stress. It is strain, not stress that distorts the polymer molecules and renders them bi-refringent, typically resulting in a rotation of the plane of polarization. The first (plane) polarizer renders the random light plane polarized. Without the intervening material, the second crossed polarizer then eliminates the only light which was left, leaving darkness (with perfect polarizers)

After insertion of the strained material, the plane polarized light has its plane of polarization rotated by the strain bi-refringence, so some of it now passes through the second crossed plane polarizer. Since the rotation is wavelength dependent, different colors rotate different amounts giving the color fringes.

It is extremely difficult to determine stress in a material. Only strain gives observable changes, that can be measured.

Stress of course has units of Newtons per metre squared, while strain is dimensionless (meters per meter).

Birefringence. Many substances such as cellophane or antistatic tape have two indices of refraction, a 'fast' axis, and a 'slow' axis, not necessarily at right angles to each other. Minute changes in the thickness of a birefringent material will appear to produce different colors between crossed linear polarizers because different thicknesses of birefringent materials result in different polarization phase angles for different colors of light.

Try it yourself by criss-crossing layers of cellophane tape at different angles, and then view this between crossed linear polarizers. You will see a stained glass effect that changes the color of each different thickness of tape, depending on the angle of the polarizers (rotate them with respect to each other).

This is the principle used instead of stains in polarization microscopes, and also is the trick for making color flat screen television and computer displays (try using a polarizer in front of those, too!)


text:
# Run a macro when Rhino starts

Hi Everybody,

I would like to run a macro at startup to make Pipe behave the way I prefer for the rest of the session but the following does not work for me.

Can anybody assist?

! _-NoEcho
Setredrawoff
Line
0,0,0
S
0,0,1
Sellast
Pipe
C
R
1
1
Enter
delete
selprev
sellast
Delete
Setredrawon
Echo

Hello - make an alias for your macro, then but the macro in your start up commands (Options > General)
Not sure what the ‘S’ is for in your macro though on line 5.
-Pascal

Sweet !
I can finally make my commands sticky !
Thank you Pascal


text:
1. ## Musings concerning maximal expressions leading to a logarithmic equation

Upon awakening this morning I had this idea about the largest expression that could be formed from a fixed collection of digits, all the same number, using standard operations such as multiplication, exponentiation, etc.

For example suppose, the collection is 2,2,2,2. My first thought was that 2^2^2^2 would be the answer for the maximum value, however $22^2^2$ is larger, and $2^2^2^2$ is larger still.

Then I noticed that $22^2^2$ is closer to $2^2^2^2$ (by ratio) than $33^3^3$ is to $3^3^3^3$. The situation is further divergent as the value of the digit climbs to 9.

Define d*d as 10·d+d and d*d*d as d·100+d·10+d etc., where d is not only allowed to be a digit, but any real number. (Numbers less than or equal to zero, although allowed, may or may not be reasonable).

So this thought formed: what would be the value of d such that $(d*d)^(^d^*^d^)$ = $d^(^d^*^d^*^d^)$?

It would be a solution to the equation: $(10x+x) log_x (10x+x)= 100x+10x+x$.

Not wishing to look for an analytic solution to the equation prior to determining a numerical result; I proceeded to do that obtaining

x = 1.3018267624863938102032556203448

using my Windows calculator after 39 iterations (I know, I could have written a C program to do this in less than half the time, but I'm on vacation for three weeks, I like doing it, and I probably would not have discovered the following result had I written the program):

I noticed this as a convergent during my calculations: $log_x$ (10x+x) = 111/11.

It is not obvious to me why this should be so; perhaps it would fall out of an analytic solution quite easily, but I have yet to look for one, and I'm posting this so that all and sundry may have a shot at it as I work on it myself.

Any feedback would be greatly appreciated.

2. ## Re: Musings concerning maximal expressions leading to a logarithmic equation

$x=11^\frac{11}{100}$ solves the given equation.


text:
# Solution of linear Navier-Stokes equations in a cylindrical coordinates

+ 0 like - 0 dislike
1240 views

Can you advise me please how to solve analytically the linear Navier-Stokes equations in a cylindrical coordinate system. Or share the link please if this solution exists somewhere.

Also there is сontinuity equation

asked Dec 20, 2016

This set of differential equation lacks the initial and boundary conditions for its variables $p$ and $\bf{v}$. As there is no time derivative, this set describes a steady flow within some "geometry" - in a pipe, cone, etc., not necessarily with cylindrical symmetry.

In this task a solid ball rotates around Oy with a constant angular velocity and it is streamlined with velocity U on the Ox by flow of liquid. In cylindric coordinates z is Oy.

Maybe there is some symmetry which reduces equations.

The boundary conditions on this photo:

## 1 Answer

+ 0 like - 0 dislike

I do not know what is the analytical solution is in this case, but it may exist in textbooks. What I know is that there is the pressure difference between co-current and counter-current sides. This pressure difference creates a force acting perpendicularly the body velocity and it is known as a Magnus effect. Football players and tennis players use it. Also, cylindrical tubes (rotors) were installed on some ships. I personally can throw a round stone in water and this stone will first enter the water and then reappear above the surface again (jump out of water).

Maybe you can pick up (guess) a solution from qualitative behaviour and dimensional reasoning.

answered Dec 21, 2016 by (132 points)
edited Dec 21, 2016

## Your answer

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text:
# Intrinsic superstatistical components of financial time series

and

## Abstract

Time series generated by a complex hierarchical system exhibit various types of dynamics at different time scales. A financial time series is an example of such a multiscale structure with time scales ranging from minutes to several years. In this paper we decompose the volatility of financial indices into five intrinsic components and we show that it has a heterogeneous scale structure. The small-scale components have a stochastic nature and they are independent 99% of the time, becoming synchronized during financial crashes and enhancing the heavy tails of the volatility distribution. The deterministic behavior of the large-scale components is related to the nonstationarity of the financial markets evolution. Our decomposition of the financial volatility is a superstatistical model more complex than those usually limited to a superposition of two independent statistics at well-separated time scales.

## Authors

Călin Vamoş
“Tiberiu Popoviciu” Institute of Numerical Analysis

Maria Crăciun
“Tiberiu Popoviciu” Institute of Numerical Analysis

## Keywords

Computational Methods; superstatistics; econophysics; financial time series analysis; volatility; multiscale structure; nonstationarity of financial markets evolution

## Paper coordinates

C. Vamoş, M. Crăciun, Intrinsic superstatistical components of financial time series, The European Physical Journal B 87 (2014): 301, pp. 9,
doi: 10.1140/epjb/e2014-50596-y

### References

see the expanding block below.

## PDF

soon

##### Journal

The European Physical Journal B

##### Publisher Name

Springer Berlin Heidelberg

1434-6028

1434-6036

soon


text:
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## G = C3×D20⋊6C4order 480 = 25·3·5

### Direct product of C3 and D20⋊6C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — C3×D20⋊6C4
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C60 — C6×D20 — C3×D20⋊6C4
 Lower central C5 — C10 — C20 — C3×D20⋊6C4
 Upper central C1 — C2×C6 — C2×C12 — C3×C4⋊C4

Generators and relations for C3×D206C4
G = < a,b,c,d | a3=b20=c2=d4=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd-1=b11, dcd-1=b15c >

Subgroups: 432 in 100 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C5, C6, C6, C8, C2×C4, C2×C4, D4, C23, D5, C10, C12, C12, C2×C6, C2×C6, C15, C4⋊C4, C2×C8, C2×D4, C20, C20, D10, C2×C10, C24, C2×C12, C2×C12, C3×D4, C22×C6, C3×D5, C30, D4⋊C4, C52C8, D20, D20, C2×C20, C2×C20, C22×D5, C3×C4⋊C4, C2×C24, C6×D4, C60, C60, C6×D5, C2×C30, C2×C52C8, C5×C4⋊C4, C2×D20, C3×D4⋊C4, C3×C52C8, C3×D20, C3×D20, C2×C60, C2×C60, D5×C2×C6, D206C4, C6×C52C8, C15×C4⋊C4, C6×D20, C3×D206C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, D5, C12, C2×C6, C22⋊C4, D8, SD16, D10, C2×C12, C3×D4, C3×D5, D4⋊C4, C4×D5, D20, C5⋊D4, C3×C22⋊C4, C3×D8, C3×SD16, C6×D5, D10⋊C4, D4⋊D5, Q8⋊D5, C3×D4⋊C4, D5×C12, C3×D20, C3×C5⋊D4, D206C4, C3×D10⋊C4, C3×D4⋊D5, C3×Q8⋊D5, C3×D206C4

Smallest permutation representation of C3×D206C4
On 240 points
Generators in S240
(1 91 46)(2 92 47)(3 93 48)(4 94 49)(5 95 50)(6 96 51)(7 97 52)(8 98 53)(9 99 54)(10 100 55)(11 81 56)(12 82 57)(13 83 58)(14 84 59)(15 85 60)(16 86 41)(17 87 42)(18 88 43)(19 89 44)(20 90 45)(21 106 61)(22 107 62)(23 108 63)(24 109 64)(25 110 65)(26 111 66)(27 112 67)(28 113 68)(29 114 69)(30 115 70)(31 116 71)(32 117 72)(33 118 73)(34 119 74)(35 120 75)(36 101 76)(37 102 77)(38 103 78)(39 104 79)(40 105 80)(121 206 176)(122 207 177)(123 208 178)(124 209 179)(125 210 180)(126 211 161)(127 212 162)(128 213 163)(129 214 164)(130 215 165)(131 216 166)(132 217 167)(133 218 168)(134 219 169)(135 220 170)(136 201 171)(137 202 172)(138 203 173)(139 204 174)(140 205 175)(141 231 196)(142 232 197)(143 233 198)(144 234 199)(145 235 200)(146 236 181)(147 237 182)(148 238 183)(149 239 184)(150 240 185)(151 221 186)(152 222 187)(153 223 188)(154 224 189)(155 225 190)(156 226 191)(157 227 192)(158 228 193)(159 229 194)(160 230 195)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20)(21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100)(101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140)(141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160)(161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180)(181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200)(201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220)(221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240)
(1 20)(2 19)(3 18)(4 17)(5 16)(6 15)(7 14)(8 13)(9 12)(10 11)(21 40)(22 39)(23 38)(24 37)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(41 50)(42 49)(43 48)(44 47)(45 46)(51 60)(52 59)(53 58)(54 57)(55 56)(61 80)(62 79)(63 78)(64 77)(65 76)(66 75)(67 74)(68 73)(69 72)(70 71)(81 100)(82 99)(83 98)(84 97)(85 96)(86 95)(87 94)(88 93)(89 92)(90 91)(101 110)(102 109)(103 108)(104 107)(105 106)(111 120)(112 119)(113 118)(114 117)(115 116)(121 135)(122 134)(123 133)(124 132)(125 131)(126 130)(127 129)(136 140)(137 139)(141 145)(142 144)(146 160)(147 159)(148 158)(149 157)(150 156)(151 155)(152 154)(161 165)(162 164)(166 180)(167 179)(168 178)(169 177)(170 176)(171 175)(172 174)(181 195)(182 194)(183 193)(184 192)(185 191)(186 190)(187 189)(196 200)(197 199)(201 205)(202 204)(206 220)(207 219)(208 218)(209 217)(210 216)(211 215)(212 214)(221 225)(222 224)(226 240)(227 239)(228 238)(229 237)(230 236)(231 235)(232 234)
(1 151 21 136)(2 142 22 127)(3 153 23 138)(4 144 24 129)(5 155 25 140)(6 146 26 131)(7 157 27 122)(8 148 28 133)(9 159 29 124)(10 150 30 135)(11 141 31 126)(12 152 32 137)(13 143 33 128)(14 154 34 139)(15 145 35 130)(16 156 36 121)(17 147 37 132)(18 158 38 123)(19 149 39 134)(20 160 40 125)(41 191 76 176)(42 182 77 167)(43 193 78 178)(44 184 79 169)(45 195 80 180)(46 186 61 171)(47 197 62 162)(48 188 63 173)(49 199 64 164)(50 190 65 175)(51 181 66 166)(52 192 67 177)(53 183 68 168)(54 194 69 179)(55 185 70 170)(56 196 71 161)(57 187 72 172)(58 198 73 163)(59 189 74 174)(60 200 75 165)(81 231 116 211)(82 222 117 202)(83 233 118 213)(84 224 119 204)(85 235 120 215)(86 226 101 206)(87 237 102 217)(88 228 103 208)(89 239 104 219)(90 230 105 210)(91 221 106 201)(92 232 107 212)(93 223 108 203)(94 234 109 214)(95 225 110 205)(96 236 111 216)(97 227 112 207)(98 238 113 218)(99 229 114 209)(100 240 115 220)

G:=sub<Sym(240)| (1,91,46)(2,92,47)(3,93,48)(4,94,49)(5,95,50)(6,96,51)(7,97,52)(8,98,53)(9,99,54)(10,100,55)(11,81,56)(12,82,57)(13,83,58)(14,84,59)(15,85,60)(16,86,41)(17,87,42)(18,88,43)(19,89,44)(20,90,45)(21,106,61)(22,107,62)(23,108,63)(24,109,64)(25,110,65)(26,111,66)(27,112,67)(28,113,68)(29,114,69)(30,115,70)(31,116,71)(32,117,72)(33,118,73)(34,119,74)(35,120,75)(36,101,76)(37,102,77)(38,103,78)(39,104,79)(40,105,80)(121,206,176)(122,207,177)(123,208,178)(124,209,179)(125,210,180)(126,211,161)(127,212,162)(128,213,163)(129,214,164)(130,215,165)(131,216,166)(132,217,167)(133,218,168)(134,219,169)(135,220,170)(136,201,171)(137,202,172)(138,203,173)(139,204,174)(140,205,175)(141,231,196)(142,232,197)(143,233,198)(144,234,199)(145,235,200)(146,236,181)(147,237,182)(148,238,183)(149,239,184)(150,240,185)(151,221,186)(152,222,187)(153,223,188)(154,224,189)(155,225,190)(156,226,191)(157,227,192)(158,228,193)(159,229,194)(160,230,195), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)(141,145)(142,144)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(161,165)(162,164)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)(196,200)(197,199)(201,205)(202,204)(206,220)(207,219)(208,218)(209,217)(210,216)(211,215)(212,214)(221,225)(222,224)(226,240)(227,239)(228,238)(229,237)(230,236)(231,235)(232,234), (1,151,21,136)(2,142,22,127)(3,153,23,138)(4,144,24,129)(5,155,25,140)(6,146,26,131)(7,157,27,122)(8,148,28,133)(9,159,29,124)(10,150,30,135)(11,141,31,126)(12,152,32,137)(13,143,33,128)(14,154,34,139)(15,145,35,130)(16,156,36,121)(17,147,37,132)(18,158,38,123)(19,149,39,134)(20,160,40,125)(41,191,76,176)(42,182,77,167)(43,193,78,178)(44,184,79,169)(45,195,80,180)(46,186,61,171)(47,197,62,162)(48,188,63,173)(49,199,64,164)(50,190,65,175)(51,181,66,166)(52,192,67,177)(53,183,68,168)(54,194,69,179)(55,185,70,170)(56,196,71,161)(57,187,72,172)(58,198,73,163)(59,189,74,174)(60,200,75,165)(81,231,116,211)(82,222,117,202)(83,233,118,213)(84,224,119,204)(85,235,120,215)(86,226,101,206)(87,237,102,217)(88,228,103,208)(89,239,104,219)(90,230,105,210)(91,221,106,201)(92,232,107,212)(93,223,108,203)(94,234,109,214)(95,225,110,205)(96,236,111,216)(97,227,112,207)(98,238,113,218)(99,229,114,209)(100,240,115,220)>;

G:=Group( (1,91,46)(2,92,47)(3,93,48)(4,94,49)(5,95,50)(6,96,51)(7,97,52)(8,98,53)(9,99,54)(10,100,55)(11,81,56)(12,82,57)(13,83,58)(14,84,59)(15,85,60)(16,86,41)(17,87,42)(18,88,43)(19,89,44)(20,90,45)(21,106,61)(22,107,62)(23,108,63)(24,109,64)(25,110,65)(26,111,66)(27,112,67)(28,113,68)(29,114,69)(30,115,70)(31,116,71)(32,117,72)(33,118,73)(34,119,74)(35,120,75)(36,101,76)(37,102,77)(38,103,78)(39,104,79)(40,105,80)(121,206,176)(122,207,177)(123,208,178)(124,209,179)(125,210,180)(126,211,161)(127,212,162)(128,213,163)(129,214,164)(130,215,165)(131,216,166)(132,217,167)(133,218,168)(134,219,169)(135,220,170)(136,201,171)(137,202,172)(138,203,173)(139,204,174)(140,205,175)(141,231,196)(142,232,197)(143,233,198)(144,234,199)(145,235,200)(146,236,181)(147,237,182)(148,238,183)(149,239,184)(150,240,185)(151,221,186)(152,222,187)(153,223,188)(154,224,189)(155,225,190)(156,226,191)(157,227,192)(158,228,193)(159,229,194)(160,230,195), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140)(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160)(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180)(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200)(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220)(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(41,50)(42,49)(43,48)(44,47)(45,46)(51,60)(52,59)(53,58)(54,57)(55,56)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)(81,100)(82,99)(83,98)(84,97)(85,96)(86,95)(87,94)(88,93)(89,92)(90,91)(101,110)(102,109)(103,108)(104,107)(105,106)(111,120)(112,119)(113,118)(114,117)(115,116)(121,135)(122,134)(123,133)(124,132)(125,131)(126,130)(127,129)(136,140)(137,139)(141,145)(142,144)(146,160)(147,159)(148,158)(149,157)(150,156)(151,155)(152,154)(161,165)(162,164)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(172,174)(181,195)(182,194)(183,193)(184,192)(185,191)(186,190)(187,189)(196,200)(197,199)(201,205)(202,204)(206,220)(207,219)(208,218)(209,217)(210,216)(211,215)(212,214)(221,225)(222,224)(226,240)(227,239)(228,238)(229,237)(230,236)(231,235)(232,234), (1,151,21,136)(2,142,22,127)(3,153,23,138)(4,144,24,129)(5,155,25,140)(6,146,26,131)(7,157,27,122)(8,148,28,133)(9,159,29,124)(10,150,30,135)(11,141,31,126)(12,152,32,137)(13,143,33,128)(14,154,34,139)(15,145,35,130)(16,156,36,121)(17,147,37,132)(18,158,38,123)(19,149,39,134)(20,160,40,125)(41,191,76,176)(42,182,77,167)(43,193,78,178)(44,184,79,169)(45,195,80,180)(46,186,61,171)(47,197,62,162)(48,188,63,173)(49,199,64,164)(50,190,65,175)(51,181,66,166)(52,192,67,177)(53,183,68,168)(54,194,69,179)(55,185,70,170)(56,196,71,161)(57,187,72,172)(58,198,73,163)(59,189,74,174)(60,200,75,165)(81,231,116,211)(82,222,117,202)(83,233,118,213)(84,224,119,204)(85,235,120,215)(86,226,101,206)(87,237,102,217)(88,228,103,208)(89,239,104,219)(90,230,105,210)(91,221,106,201)(92,232,107,212)(93,223,108,203)(94,234,109,214)(95,225,110,205)(96,236,111,216)(97,227,112,207)(98,238,113,218)(99,229,114,209)(100,240,115,220) );

G=PermutationGroup([[(1,91,46),(2,92,47),(3,93,48),(4,94,49),(5,95,50),(6,96,51),(7,97,52),(8,98,53),(9,99,54),(10,100,55),(11,81,56),(12,82,57),(13,83,58),(14,84,59),(15,85,60),(16,86,41),(17,87,42),(18,88,43),(19,89,44),(20,90,45),(21,106,61),(22,107,62),(23,108,63),(24,109,64),(25,110,65),(26,111,66),(27,112,67),(28,113,68),(29,114,69),(30,115,70),(31,116,71),(32,117,72),(33,118,73),(34,119,74),(35,120,75),(36,101,76),(37,102,77),(38,103,78),(39,104,79),(40,105,80),(121,206,176),(122,207,177),(123,208,178),(124,209,179),(125,210,180),(126,211,161),(127,212,162),(128,213,163),(129,214,164),(130,215,165),(131,216,166),(132,217,167),(133,218,168),(134,219,169),(135,220,170),(136,201,171),(137,202,172),(138,203,173),(139,204,174),(140,205,175),(141,231,196),(142,232,197),(143,233,198),(144,234,199),(145,235,200),(146,236,181),(147,237,182),(148,238,183),(149,239,184),(150,240,185),(151,221,186),(152,222,187),(153,223,188),(154,224,189),(155,225,190),(156,226,191),(157,227,192),(158,228,193),(159,229,194),(160,230,195)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20),(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100),(101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140),(141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160),(161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180),(181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200),(201,202,203,204,205,206,207,208,209,210,211,212,213,214,215,216,217,218,219,220),(221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240)], [(1,20),(2,19),(3,18),(4,17),(5,16),(6,15),(7,14),(8,13),(9,12),(10,11),(21,40),(22,39),(23,38),(24,37),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(41,50),(42,49),(43,48),(44,47),(45,46),(51,60),(52,59),(53,58),(54,57),(55,56),(61,80),(62,79),(63,78),(64,77),(65,76),(66,75),(67,74),(68,73),(69,72),(70,71),(81,100),(82,99),(83,98),(84,97),(85,96),(86,95),(87,94),(88,93),(89,92),(90,91),(101,110),(102,109),(103,108),(104,107),(105,106),(111,120),(112,119),(113,118),(114,117),(115,116),(121,135),(122,134),(123,133),(124,132),(125,131),(126,130),(127,129),(136,140),(137,139),(141,145),(142,144),(146,160),(147,159),(148,158),(149,157),(150,156),(151,155),(152,154),(161,165),(162,164),(166,180),(167,179),(168,178),(169,177),(170,176),(171,175),(172,174),(181,195),(182,194),(183,193),(184,192),(185,191),(186,190),(187,189),(196,200),(197,199),(201,205),(202,204),(206,220),(207,219),(208,218),(209,217),(210,216),(211,215),(212,214),(221,225),(222,224),(226,240),(227,239),(228,238),(229,237),(230,236),(231,235),(232,234)], [(1,151,21,136),(2,142,22,127),(3,153,23,138),(4,144,24,129),(5,155,25,140),(6,146,26,131),(7,157,27,122),(8,148,28,133),(9,159,29,124),(10,150,30,135),(11,141,31,126),(12,152,32,137),(13,143,33,128),(14,154,34,139),(15,145,35,130),(16,156,36,121),(17,147,37,132),(18,158,38,123),(19,149,39,134),(20,160,40,125),(41,191,76,176),(42,182,77,167),(43,193,78,178),(44,184,79,169),(45,195,80,180),(46,186,61,171),(47,197,62,162),(48,188,63,173),(49,199,64,164),(50,190,65,175),(51,181,66,166),(52,192,67,177),(53,183,68,168),(54,194,69,179),(55,185,70,170),(56,196,71,161),(57,187,72,172),(58,198,73,163),(59,189,74,174),(60,200,75,165),(81,231,116,211),(82,222,117,202),(83,233,118,213),(84,224,119,204),(85,235,120,215),(86,226,101,206),(87,237,102,217),(88,228,103,208),(89,239,104,219),(90,230,105,210),(91,221,106,201),(92,232,107,212),(93,223,108,203),(94,234,109,214),(95,225,110,205),(96,236,111,216),(97,227,112,207),(98,238,113,218),(99,229,114,209),(100,240,115,220)]])

102 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3A 3B 4A 4B 4C 4D 5A 5B 6A ··· 6F 6G 6H 6I 6J 8A 8B 8C 8D 10A ··· 10F 12A 12B 12C 12D 12E 12F 12G 12H 15A 15B 15C 15D 20A ··· 20L 24A ··· 24H 30A ··· 30L 60A ··· 60X order 1 2 2 2 2 2 3 3 4 4 4 4 5 5 6 ··· 6 6 6 6 6 8 8 8 8 10 ··· 10 12 12 12 12 12 12 12 12 15 15 15 15 20 ··· 20 24 ··· 24 30 ··· 30 60 ··· 60 size 1 1 1 1 20 20 1 1 2 2 4 4 2 2 1 ··· 1 20 20 20 20 10 10 10 10 2 ··· 2 2 2 2 2 4 4 4 4 2 2 2 2 4 ··· 4 10 ··· 10 2 ··· 2 4 ··· 4

102 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C3 C4 C6 C6 C6 C12 D4 D4 D5 D8 SD16 D10 C3×D4 C3×D4 C3×D5 C4×D5 D20 C5⋊D4 C3×D8 C3×SD16 C6×D5 D5×C12 C3×D20 C3×C5⋊D4 D4⋊D5 Q8⋊D5 C3×D4⋊D5 C3×Q8⋊D5 kernel C3×D20⋊6C4 C6×C5⋊2C8 C15×C4⋊C4 C6×D20 D20⋊6C4 C3×D20 C2×C5⋊2C8 C5×C4⋊C4 C2×D20 D20 C60 C2×C30 C3×C4⋊C4 C30 C30 C2×C12 C20 C2×C10 C4⋊C4 C12 C12 C2×C6 C10 C10 C2×C4 C4 C4 C22 C6 C6 C2 C2 # reps 1 1 1 1 2 4 2 2 2 8 1 1 2 2 2 2 2 2 4 4 4 4 4 4 4 8 8 8 2 2 4 4

Matrix representation of C3×D206C4 in GL4(𝔽241) generated by

 225 0 0 0 0 225 0 0 0 0 1 0 0 0 0 1
,
 1 2 0 0 240 240 0 0 0 0 52 1 0 0 240 0
,
 1 0 0 0 240 240 0 0 0 0 52 1 0 0 189 189
,
 0 22 0 0 11 0 0 0 0 0 200 156 0 0 85 41
G:=sub<GL(4,GF(241))| [225,0,0,0,0,225,0,0,0,0,1,0,0,0,0,1],[1,240,0,0,2,240,0,0,0,0,52,240,0,0,1,0],[1,240,0,0,0,240,0,0,0,0,52,189,0,0,1,189],[0,11,0,0,22,0,0,0,0,0,200,85,0,0,156,41] >;

C3×D206C4 in GAP, Magma, Sage, TeX

C_3\times D_{20}\rtimes_6C_4
% in TeX

G:=Group("C3xD20:6C4");
// GroupNames label

G:=SmallGroup(480,87);
// by ID

G=gap.SmallGroup(480,87);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-5,365,92,2524,1271,102,18822]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^20=c^2=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d^-1=b^11,d*c*d^-1=b^15*c>;
// generators/relations

׿
×
𝔽


text:
# Conditional dependence

A Bayesian network enunciating conditional dependence

In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.[1][2] For example, if A and B are two events that individually increase the probability of a third event C, and do not directly affect each other, then initially (when it has not been observed whether or not the event C occurs)

${\displaystyle {\text{P}}(A\mid B)={\text{P}}(A){\text{ and }}{\text{P}}(B\mid A)={\text{P}}(B)}$[3][4] (A and B are independent).

But suppose that now C is observed to occur. If event B occurs the probability of occurrence of the event A will decrease because its positive relation to C is less needed as an explanation of the occurrence of C. (Similarly, event A occurring will decrease the probability of occurrence of B). Hence, now the two events A and B are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have

${\displaystyle {\text{P}}(A\mid C,B)<{\text{P}}(A\mid C).}$[5]

Conditional dependence is different from conditional independence. In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.[6]

## Example

In essence probability is influenced by a person's information about the possible occurrence of an event. For example, let the event A be 'I have a new phone'; event B be 'I have a new watch'; and event C be 'I am happy'; and suppose that having either a new phone or a new watch increases the probability of my being happy. Let us assume that the event C has occurred – meaning 'I am happy'. Now if another person sees my new watch, he/she will reason that my likelihood of being happy was increased by my new watch, so there is less need to attribute my happiness to a new phone.

To make the example more numerically specific, suppose that there are four possible states, given in the four columns of the following table, in which the occurrence of event A is signified by a 1 in row A and its non-occurrence is signified by a 0 (and likewise for B and C):

probability 1/4 1/4 1/4 1/4
A 0 1 0 1
B 0 0 1 1
C 0 1 1 1

In this example, C occurs if and only if at least one of A, B occurs. Unconditionally (i.e., without reference to C), A and B are independent of each other because P(A)—the sum of the probabilities associated with a 1 in row A—is ${\displaystyle {\tfrac {1}{2}},}$ while P(A|B) = P(A and B) / P(B) = ${\displaystyle {\tfrac {1/4}{1/2}}={\tfrac {1}{2}}}$ = P(A). But conditional on C having occurred (the last three columns in the table), we have P(A|C) = P(A and C) / P(C) = ${\displaystyle {\tfrac {1/2}{3/4}}={\tfrac {2}{3}}}$ while P(A|C and B) = P(A and C and B) / P(C and B) = ${\displaystyle {\tfrac {1/4}{1/2}}={\tfrac {1}{2}}}$ < P(A|C). Since in the presence of C the probability of A is affected by the presence or absence of B, A and B are mutually dependent conditional on C.


text:
# Testing Hypotheses on Spectrum Parameters of a Gaussian Time Series

• K. Dzhaparidze
Chapter
Part of the Springer Series in Statistics book series (SSS)

## Abstract

Following the general ideas of LeCam [80–82] (cf. also [110]) we shall consider a sequence of experiments
$${E_n} = \{ {X_n},{\mathfrak{A}_n},{P_{n,}}_\theta ,\theta \in \theta \} ,\,n = 1,2,...,$$
where the family of distributions Pnθ, θ ∈ θ for some choice of random vector Δn,θ = Δn, θ(x), xXn, and the nonrandom matrices Γθ satisfy the conditions (D1)–(D4) for τn = $$\mathop n\limits^\_$$ of asymptotic differentiability as well as the condition (D5) which assures the asymptotic normality of the vector Δn,θ (cf. the Introduction, page 21 and Section 1 of Chapter III). Assume for definiteness that the set θ ∈ Rp of possible values of the vector-valued parameter θ contains the origin and consider the problem of testing the hypothesis H0 that the parameter θ takes on the value 0. A test for this hypothesis is given by a sequence of test functions Φn = Φn(x) defined on the sample space xXn. Any measurable function taking on values 0 ⩽ Φn ⩽ 1 may serve as a test function which determines the probability Φn(x) that hypothesis H0 will be rejected when x is observed.

## Keywords

Random Vector Critical Region Consistent Estimator Spectrum Parameter Noncentrality Parameter
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


text:
## Monday, October 13, 2008

### Crisis watch, 13 October

Please note: LaTeX mathematics works. This means that if you want to say $10 you have to say \$10.


text:
## Jonas Azzam

##### University of Edinburgh Lecturer

JCMB 4613, Kings Buildings
j.azzam$@$ed.ac.uk

### Brief CV

2011: University of California-Los Angeles, PhD

##### PhD Students
Jean Burnazyan
Matthew Hyde
Michele Villa (PhD 2020, now at U. Helsinki)

## Research

Rectifable Sets, Harmonic Measure, General Geometric Measure Theory, Miscellaneous

### Rectifiable Sets

Given a set (or metric space) how can I tell if and how well the set resembles Rd? There are various degrees of resemblance you could demand, like being isometric. One variant that I consider is the property of being rectifiable: a set is d-rectifiable if it can be covered up to d-dimensional measure zero by Lipschitz images of Rd. One can think of these sets as measure theoretic versions of smooth manifords: a rectifiable set is one that is "locally" parametrized by Lipschitz chart maps. There are even different degrees of rectifiability (like uniform rectifiability). Much of my work concerns with classifying when a set is rectifiable and studying the properties of rectifiable sets. There is a particular emphasis on quantitative analysis, and this only became more important around the 90s in connection with boundedness of singular integrals and Painleve's probelm.

We show that Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ where $\mathscr{H}^{d}_{\infty}(E\cap B(x,r))<(2r)^{d}(1-\epsilon)$ is a Carleson set for every $\epsilon>0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\epsilon,\rho\in (0,1)$, $A>1$, and $X_{n}$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathscr{B}=\{B(x,2^{-n}):x\in X_{n},n\in \mathbb{N}\}$, then $\sum \left\{r_{B}^{s}: B\in \mathscr{B}, \frac{\mathscr{H}^{s}_{\rho r_{B}}(X\cap AB)}{(2r_{B})^{s}}>1+\epsilon\right\} \lesssim_{C,A,\epsilon,\rho} \mathscr{H}^{s}(X).$ This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most $1$ $\mathscr{H}^{s}$-almost everywhere.
We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincaré inequality with respect to surface measure is uniformly rectifiable.
Arxiv
We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, $\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \lesssim ({\rm diam} R)^{d}.$ We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if $\beta_{E}(R)$ denotes the square sum of $\beta$-numbers over subcubes of $R$ as in the Traveling Salesman Theorem for higher dimensional sets [AS18], then $\mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}\sim \beta_{E}(R).$ We prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.
Arxiv.
We characterize Radon measures $\mu$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by $d$-dimensional Lipschitz graphs and $\mu \ll \mathscr{H}^{d}$. The characterization is in terms of the a.e. finiteness Jones function involving the so-called $\alpha$-numbers. We also give an example of an unrectifiable measure that is not pointwise doubling for which the Jones function is finite almost everywhere. This answers a question left open in a former work by Azzam, David, and Toro. Slides

Epilogue: The counterexample mentioned above means that we cannot characterize rectifiable measures using $\alpha$-numbers we define. However, Damian Dąbrowski has developed a full characterization using $\alpha$-numbers based on the $W_{2}$-Wasserstein distance.

We develop a quantity $\beta_{E}^{d}(Q)$ that measures the $d$-flatness of a set $E\subseteq \mathbb{R}^{n}$ inside a cube $Q$ so that, if $E$ is a Reifenberg flat surface, $(\mbox{diam } E)^{d}+\sum_{Q\subseteq \mathbb{R}^{n} \atop Q\cap E\neq\emptyset} \beta_{E}(3Q)^{2}\ell(Q)^{d}\sim_{n} \mathscr{H}^{d}(E).$ The result also holds for surfaces $E\subseteq \mathbb{R}^{n}$ satisfying Condition B, that is, every ball centered on $E$ has two large balls in different components of $E^{c}$. This gives a generalization of the (Euclidean) Analyst's Traveling Salesman Theorem of Peter Jones and Kate Okikiolu, which originally held only for curves.

Note: The published version has many annoying typos and small mistakes that make for hard reading. For this reason, I am keeping an up-to-date version here where I correct any mistakes that I happen to find or that people point out. If you have any questions about the paper or notice an error, let me know so I can sort it. Slides, Video.

In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of $\sigma$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if the boundary of a corkscrew domain in $\mathbb{R}^{d+1}$ has $\sigma$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $d$-measure. We also give a simpler proof of the well-known fact that, if $\Omega\subseteq \mathbb{R}^{d+1}$ is an exterior corkscrew domain whose boundary has locally finite $\mathscr{H}^{d}$-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary.
Arxiv.
Let $\mu$ be a doubling measure in $\mathbb{R}^n$. We introduce two parts of the support of $\mu$ where $\mu$ has some quantitative self-similarity properties, measured with a variant of the $L^1$ Wasserstein distance, and prove that at each point of these two parts, all the tangent measures to $\mu$ are multiples of some flat measure (the Lebesgue measure on a vector subspace). We use this to decompose the two parts into rectifiable pieces of various dimensions.
Arxiv.
A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation is $\omega_{f}(x,r)=\inf_{A}\left(\frac{1}{|B(x,r)|}\int_{B(x,r)}\left(\frac{|f-A|}{|A'|r}\right)^{2}\right)^{\frac{1}{2}},$ where the infimum ranges over all non-zero affine transformations $A$. This is natural insofar as it is invariant under rescaling $f$ in either its domain or image. We show that if $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is quasisymmetric and its image has a sufficient amount of rectifiable structure (although not necessarily $\mathcal{H}^{d}$-finite), then $\omega_{f}(x,r)^{2}\frac{dxdr}{r}$ is a Carleson measure on $\mathbb{R}^{d}\times(0,\infty)$. Moreover, this is an equivalence: if this is a Carleson measure, then, in every ball $B(x,r)\subseteq \mathbb{R}^{d}$, there is a set $E$ occupying 90$\%$ of $B(x,r)$, say, upon which $f$ is bi-Lipschitz (and hence guaranteeing rectifiable pieces in the image). En route, we make a minor adjustment to a theorem of Semmes to show that quasisymmetric maps of subsets of $\mathbb{R}^{d}$ into $\mathbb{R}^{d}$ are bi-Lipschitz on a large subset quantitatively.
Arxiv., Slides.
Garnett, Killip, and Schul have exhibited a doubling measure $\mu$ with support equal to $\mathbb{R}^{d}$, $d\geq 2$, which is $1$-rectifiable, meaning there are countably many curves $\Gamma_{i}$ of finite length for which $\mu(\mathbb{R}^{d}\backslash \bigcup \Gamma_{i})=0$. In this note, we give a characterisation of when a doubling measure with connected support in a metric space has a $1$-rectifiable subset of positive measure and show this set coincides up to a set of $\mu$-measure zero with the set of $x\in \mbox{supp} \mu$ for which $\liminf_{r\rightarrow 0} \mu(B(x,r))/r>0$.
Arxiv., Slides.
Let $\mu$ be a doubling measure in $\mathbb{R}^n$. We investigate quantitative relations between the rectifiability of $\mu$ and its distance to flat measures. More precisely, for $x$ in the support $\Sigma$ of $\mu$ and $r > 0$, we introduce a number $\alpha(x,r)\in (0,1]$ that measures, in terms of a variant of the $L^1$-Wasserstein distance, the minimal distance between the restriction of $\mu$ to $B(x,r)$ and a multiple of the Lebesgue measure on an affine subspace that meets $B(x,r/2)$. We show that the set of points of $\Sigma$ where $\int_0^1 \alpha(x,r) {dr \over r} < \infty$ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of $\mu$ when we assume that some Carleson measure estimates hold.
Arxiv..
We generalize the results of David and Semmes by providing sufficient $\beta$-number-type conditions for rectifiability of Radon measures, requiring only that their upper densities are positive and finite almost everywhere. More precisely, let $n>0$ and let $\mu$ be a Radon measure in $\mathbb{R}^{d}$. Suppose that for some $1\leq n\neq d$ we have $\theta^{n,*}(\mu,x)\in(0,\infty)$ $\mu$-a.e. and that $\int_{0}^{1}\beta_{\mu,2}(x,r)^{2}\frac{dr}{r}<\infty\;\; \mu-a.e.$ where $\beta_{\mu,2}(x,r)^{2}=\inf_{L\in G(n,d)}\int_{B(x,r)}\left(\frac{dist(x,L)}{r}\right)^{2}\frac{dr}{r^{n}}$ and $G(n,d)$ denotes the set of $n$-dimensional planes in $\mathbb{R}^{d}$. Then $\mu$ may be covered up to a set of $\mu$-measure zero by Lipschitz images. Combined with a forthcoming paper by Tolsa, this $\beta$-number condition completely characterizes $n$-rectifiable Radon measures with positive and finite upper $n$-densities.
Arxiv..

Epilogue: This result was generalized by Edelen, Naber and Valtorta. They have been used in a vareity of contexts, like line defects in liquid quasicrystals, obstacle problems, calculus of variations, and singular integrals.
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of $\mathbb{R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz $n$-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of $\mathbb{R}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension $N-n$. To establish the big pieces criterion, we prove new extension theorems for almost affine'' maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.
Arxiv..
The purpose of this note is to point out a simple consequence of some earlier work of the authors, Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson packing condition.
Article..
All Lipschitz maps from $R^7$ to $R^3$ are orthogonal projections". This is of course quite false as stated. It turns out however, that there is a surprising grain of truth in this statement. We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:\mathbb{R}^n\times \mathbb{R}^m\to\mathbb{R}^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $\bar{g}:\mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}^n\times \mathbb{R}^m$ such that $f\circ \bar{g}$ will satisfy, when we restrict to a large portion of the domain $E\subset \mathbb{R}^n\times \mathbb{R}^m$, that $f\circ \bar{g}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $\bar{g}$ distorts $\mathbb{R}^{n+m}$ in a controlled manner so that the fibers of $f$ are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set $E$ on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a $C^1$ map from $[0,1]^3$ onto $[0,1]^2$ with rank $\leq 1$ everywhere.

On route we prove an extension theorem which is of independent interest. We show that for any $D\geq n$, any Lipschitz function $f:[0,1]^n\to \mathbb{R}^D$ gives rise to a large (in an appropriate sense) subset $E\subset [0,1]^n$ such that $f|_E$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of $\mathbb{R}^n$. This extends results of P. Jones and G. David, from 1988. As a simple corollary, we show that $n$-dimensional Ahlfors-David regular spaces lying in $\mathbb{R}^{D}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $\mathbb{R}^{D}$. This was previously known only for $D\geq 2n+1$ by a result of G. David and S. Semmes.
Arxiv, Slides.
Epilogue:This result was the main part of my thesis. It was improved upon by Raanan Schul and Guy C. David in 2020, see here for video seminar about their result, and the video also starts by giving a synopsis of our result.
For a given connected set $\Gamma$ in $d-$dimensional Euclidean space, we construct a connected set $\tilde\Gamma\supset \Gamma$ such that the two sets have comparable Hausdorff length, and the set $\tilde\Gamma$ has the property that it is quasiconvex, i.e. any two points $x$ and $y$ in $\tilde\Gamma$ can be connected via a path, all of which is in $\tilde\Gamma$, which has length bounded by a fixed constant multiple of the Euclidean distance between $x$ and $y$. Thus, for any set $K$ in $d-$dimensional Euclidean space we have a set $\tilde\Gamma$ as above such that $\tilde\Gamma$ has comparable Hausdorff length to a shortest connected set containing $K$. Constants appearing here depend only on the ambient dimension $d$. In the case where $\Gamma$ is Reifenberg flat, our constants are also independent the dimension $d$, and in this case, our theorem holds for $\Gamma$ in an infinite dimensional Hilbert space. This work closely related to $k-$spanners, which appear in computer science.
Arxiv, Slides.

### Harmonic Measure

For a domain $\Omega\subseteq \mathbb{R}^{n}$, its harmonic measure with pole at $x$, denoted $\omega_{\Omega}^{x}$ can be defined two ways: one is that, for $A\subseteq \partial\Omega$, $\omega_{\Omega}^{x}(A)$ is the probability that a Brownian motion starting at $x\in \Omega$ first hits $\partial \Omega$ in the set $A$. Alternatively (and assuming our domain is "nice" enough), for each continuous function $f$ one can solve the Dirichlet problem to find a harmonic function $u_{f}$ on $\Omega$ that agrees with $f$ on the boundary; for $x\in \Omega$, the map $f\mapsto u_{f}(x)$ is a bounded linear functional and the Riesz Representation Theorem supplies a measure $\omega_{\Omega}^{x}$ so that $u_{f}(x) = \int_{\Omega}fd\omega_{\Omega}^{x}$. This is quite a natural measure to give the boundary of a domain, and so it's natural to ask what the relationship between the geometry of the domain and the behavior of its harmonic measure is. My work in this area has mostly focused on the relationship between the reguarity of harmonic measure and the rectifiable structure of the boundary.

We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit Corona decompositions for harmonic measure if and only if the square sum $\beta_{\partial\Omega}$ of the generalized Jones $\beta$-numbers is finite. Using this, we give estimates on the fluctuation of Green's function in a uniform domain in terms of the $\beta$-numbers. As a corollary, for bounded NTA domains , if $B_{\Omega}=B(x_{\Omega},c\mathrm{diam} \Omega)$ is so that $2B_{\Omega}\subseteq \Omega$, we obtain that $(\mathrm{diam} \partial\Omega)^{d} + \int_{\Omega\backslash B_{\Omega}} \left|\frac{\nabla^2 G_{\Omega}(x_{\Omega},x)}{G_{\Omega}(x_{\Omega},x)}\right|^{2} \mathrm{dist}(x,\Omega^c)^{3} dx \sim \mathscr{H}^{d}(\partial\Omega).$ Secondly, we also use $\beta$-numbers to estimate how much harmonic measure fails to be $A_{\infty}$-weight for semi-uniform domains with Ahlfors regular boundaries.
Slides
We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary with $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.
Article.
It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_\infty$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $Ω\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $Ω$, with data in $L^p(\partialΩ)$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_\infty$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are in the nature of best possible: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds).
Arxiv.
For domains with Ahlfors-David regular boundaries, it is known that the $A_{\infty}$ property of harmonic measure implies uniform rectifiability of the boundary \cite{MT15,HLMN17}. Since $A_{\infty}$-weights are doubling, this also implies the domain is semi-uniform if the domain is John by the work of Aikawa and Hirata \cite{AH08}. In this paper, we firstly remove the John condition from their result, and secondly, we show that these two properties, semi-uniformity and uniformly rectifiable boundary, also imply the $A_{\infty}$-property for harmonic measure, thus classifying geometrically all domains for which this holds. This requires finding suitable substitutes for classical estimates for nontangentially accessible domains that work in semi-uniform domains with the capacity density condition, which may be of independent interest.
Arxiv.
We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy a uniform thickness condition and the intersection of their boundaries $F$ has positive harmonic measure. Then we show that in a fixed ball $B$ centered on $F$, if the harmonic measure of $\Omega_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the harmonic measure of $\Omega_2$ in $B$, then there exists a uniformly $n$-rectifiable set $\Sigma$ so that the harmonic measure of $\Sigma \cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable feature of our result is that the harmonic measures do not need to satisfy any doubling condition.
Arxiv.
Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains $\Omega\subseteq \mathbb{R}^{n+1}$:
1. We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for any domains implies the tangent measures are a.e. flat and the elliptic measures have dimension $n$.
2. We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials.
3. In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower $n$-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to $n$-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17]. Finally, we generalize one of the main results of [Bad11] by showing that if $\omega$ is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials.
Arxiv.
Let $\Omega\subseteq \mathbb{R}^{n+1}$, $n\geq 2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator L in divergence form associated with a matrix $A$ with real, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^∗$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partial\Omega$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^∗v=0$ in $\Omega$ is $\epsilon$-approximable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^∗v=0$ in $\Omega$ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called $S$<$N$ estimates, and another in terms of a suitable corona decomposition involving $L$-harmonic and $L^∗$-harmonic measures. We also prove that if $L$-harmonic measure and $L^∗$-harmonic measure satisfy a weak $A_\infty$-type condition, then $\partial\Omega$ is $n$-uniformly rectifiable. In the process we obtain a version of Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.
Arxiv.
We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $\mathbb{R}^{d+1}$ into two NTA domains then $\omega_{\Omega}\ll \mathscr{H}^{d}$ on $\Gamma\cap \partial\Omega$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $\Gamma\cap\partial\Omega$ is a cone point if $\Gamma$ is a Lipschitz graph. Combining these results and a result from [AHMMMTV], we characterize sets of absolute continuity with finite $\mathscr{H}^{d}$-measure both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. This generalizes the results of McMillan in [McM69] and Pommerenke in [Pom86]. Finally, we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix.
Arxiv, Slides.
We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition.
Arxiv.
For uniform domains $\Omega\subseteq \mathbb{R}^{d+1}$ satisfying the capacity density condition (or the CDC), we show that harmonic measure has a tangent measure whose support is the boundary of a uniform domain with the CDC. More importantly, the harmonic measure on these domains can be expressed as the weak limit of blown-up copies of the original domain $\Omega$. Our main application is to obtain bounds on $s$ when harmonic measure has positive and finite upper and lower $s$-densities on a set of positive harmonic measure. In the special case that $\Omega$ is nontangentially accessible, our bounds imply that $s=d$.
Arxiv.

Epilogue: Tolsa recently generalized the results of this paper by showing that harmonic measure for any domain whose boundary has positive and finite $s$-dimensional measure is singular with respect to the $s$-dimensional surface measure on the boundary.
We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries.
Arxiv, Slides.
For $d>2$, we show there exists locally Reifenberg flat domains $\Omega\subseteq \mathbb{R}^{d+1}$ with $\mathscr{H}^{d}(\partial\Omega)$<$\infty$ and a subset $E\subseteq \partial\Omega$ with positive harmonic measure yet zero $\mathscr{H}^{d}$-measure. This generalizes a counter-example of Wu and answers a conjecture of Badger. The main mart of the proof consists in showing how, for a prescribed compact subset $E$ of $\partial \Omega$, where $\Omega$ is a Reifenberg flat domain, to construct a larger domain $\Omega^{+}$ whose boundary is also Reifenberg flat and intersects $\partial\Omega$ on $E$.
Arxiv.
We show that if $\Omega\subset\mathbb{R}^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(\sigma)$, where $\sigma$ is the surface measure of $\Omega$, then the outer unit normal to $\partial\Omega$ belongs to $\textrm{VMO}(\sigma)$ too. The analogous result fails for dimensions larger than $3$. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.
Arxiv.
We show that if $\Omega \subseteq \mathbb{R}^{n+1}$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $\Omega$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $\Omega$ is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition. We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems.
Arxiv.
If $\Omega\subseteq\mathbb{R}^{d+1}$ is an NTA domain with harmonic measure $w$ and $E\subseteq \partial\Omega$ is contained in an Ahlfors regular set, then for all $\tau>0$ there is $E'\subseteq E$ $d$-rectifiable with $w(E\backslash E')$<$\tau w(E)$ and $w|_{E'}\ll \mathscr{H}^{d}|_{E'}\ll w|_{E'}$, so in particular, $w|_{E}\ll \mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that $w$ obeys an $A_{\infty}$-type condition with respect to $\mathscr{H}^{d}$ on $E'$, even though $\partial\Omega$ may not even be locally $\mathscr{H}^{d}$-finite. We also show that if $(\Omega^{c})^{\circ}$ is also NTA, and if $E\subseteq\partial \Omega$ is in a Lipschitz image of $\mathbb{R}^{d}$, then for all $\tau>0$ there is $E'\subseteq E$ with $\mathscr{H}^{d}(E\backslash E')$<$\tau\mathscr{H}^{d}(E)$, $w|_{E'}\ll\mathscr{H}^{d}|_{E'}\ll w|_{E'}$, and such that a similar $A_{\infty}$-condition holds.
Arxiv, Slides.
We show that if $n\geq 1$, $\Omega\subset \mathbb{R}^{n+1}$ is a connected domain, and $E\subset \partial\Omega$ is a set of finite and positive $\mathscr{H}^{n}$-measure upon which the harmonic measure $\omega$ is absolutely continuous with respect to $\mathscr{H}^{n}$, then $\omega|_E$ is $n$-rectifiable set.
Arxiv, Slides

### General Geometric Measure Theory

Most of my work falls into the broad category of geometric measure theory. Here I list a few one-off results in this area that aren't connected to my more specific subareas areas.

We show that if $0$<$t$<$s\leq n-1$, $\Omega\subseteq \mathbb{R}^{n}$ with lower $s$-content regular complement, and $z\in \Omega$, there is a chord-arc domain $\Omega_{z}\subseteq \Omega$ with center $z$ so that $\mathscr{H}^{t}_{\infty}(\partial\Omega_{z}\cap \partial\Omega)\gtrsim_{t} \textrm{dist}(z,\Omega^{c})^{t}$. This was originally shown by Koskela, Nandi, and Nicolau with John domains in place of chord-arc domains when $n=2$, $s=1$, and $\Omega$ is a simply connected planar domain. As an application, our result combined with works of Koskela and Lehrbäck give a characterization of which domains support pointwise $(p,\beta)$-Hardy inequalities for $\beta$<$p-1$.
Arxiv.
A set in the Euclidean plane is constructed whose image under the classical Radon transform is Lipschitz in every direction. It is also shown that, under mild hypotheses, for any such set the function which maps a direction to the corresponding Lipschitz constant cannot be bounded.
Arxiv.
For a compact connected set $X\subseteq \ell^{\infty}$, we define a quantity $\beta'(x,r)$ that measures how close $X$ may be approximated in a ball $B(x,r)$ by a geodesic curve. We then show there is $c>0$ so that if $\beta'(x,r)>\beta>0$ for all $x\in X$ and $r\leq r_{0}$, then ${\rm dim} X\geq 1+c\beta^{2}$. This generalizes a theorem of Bishop and Jones (which was originally for Euclidean spaces) and another of by Bishop and Tyson about the dimension of antenna sets.
Arxiv.

### Miscellaneous

We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we provide a significantly simplified proof to known uniqueness results for the 2D Euler equations in $L^1 \cap BMO$ and provide a mild improvement to the recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge, the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for weak solutions to the Vlasov-Poisson equations, extending the results of Robert and Loeper to include densities in $L^1 \cap BMO$, and the strongest known uniqueness results for the Patlak-Keller-Segel models. We obtain these results via technical refinements of energy methods which are well-known in the $L^2$ setting but are less well-known in the $\dot{H}^{-1}$ setting. The $\dot{H}^{-1}$ method can be considered a generalization of Yudovich's classical method and is naturally applied to equations such as the Patlak-Keller-Segel models with nonlinear diffusion, and other variants and the Vlasov-Poisson equation and other kinetic models. The $\dot{H}^{-1}$ method is a variant of the classical method due to Yudovich for the 2D Euler equations. An important tool in our analysis is a Sobolev embedding lemma which shows that velocity fields $v$ with $\nabla v \in BMO$ are locally log-Lipschitz. Hence, they satisfy the Osgood uniqueness criterion for ODE, which is known to play an important role in the uniqueness of active scalars.
Arxiv.


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How do you multiply -\frac { 2} { 5} x ^ { 2} y ^ { 3} \cdot \frac { 5} { 7} x ^ { 4} y ^ { 4}?

May 15, 2017

$- \frac{2}{5} {x}^{2} {y}^{3} \cdot \frac{5}{7} {x}^{4} {y}^{4} = - \frac{7}{25} {x}^{6} {y}^{7}$

Written in decimal form: $- \frac{7}{25} {x}^{6} {y}^{7} = - 0.29 {x}^{6} {y}^{7}$

Explanation:

Given: $- \frac{2}{5} {x}^{2} {y}^{3} \cdot \frac{5}{7} {x}^{4} {y}^{4}$

Performing multiplication within an expression, we can move the fractions together:

$\left(- \frac{2}{5}\right) \left(\frac{5}{7}\right) {x}^{2} {y}^{3} \cdot {x}^{4} {y}^{4}$

Now we can add exponents (by the exponent rule of multipication):

$\left(- \frac{2}{\cancel{5}}\right) \left(\frac{\cancel{5}}{7}\right) {x}^{6} {y}^{7} = - \frac{2}{7} {x}^{6} {y}^{7} = - 0.29 {x}^{6} {y}^{7}$


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## Description

 Title: ULTRAVIORET AND INFRARED PHOTODISSOCIATION SPECTROSCOPY OF HYDRATED ANILINIUM ION Author(s): Kurusu, Itaru Contributor(s): Ishikawa, Haruki; Kasahara, Yasutoshi; Yagi, Reona Subject(s): Clusters/Complexes Abstract: To understand the temperature effect on the microscopic hydration, we have been carrying out the laser spectroscopy of temperature-controlled hydrated phenol cation clusters using our temperature-variable ion trap apparatusfootnote{R.~Yagi, Y.~Kasahara, H.~Ishikawa, the 70th International Symposium on Molecular Spectroscopy (2015).}$^,$footnote{H.~Ishikawa, T.~Nakano, T.~Eguchi, T.~Shibukawa, K.~Fuke textit{Chem. Phys. Lett.} textbf{514}, 234 (2011).}. In the present study, we have chosen an anilinium ion (AnH$^+$) as a solute. Since the phenol cation has $(pi)^{-1}$ configuration, the phenyl ring does not play as a proton-acceptor. On the contrary, the $pi$-orbitals in the AnH$^+$ are fulfilled and both the NH$_3^+$ and phenyl groups can behave as hydrogen-bonding sites. Thus, hydration structures around the AnH$^+$ are expected to be different from those of the phenol cation. Since there is no spectroscopic report on the hydrated AnH$^+$ clusters, we have carried out the UV and IR photodissociation spectroscopy of AnH$^+$(H$_2$O) clusters. par In the present study, the AnH$^+$(H$_2$O) is produced by an electrospray ionization method. As the first step, spectroscopic measurements are carried out without temperature control. In the UV photodissociation spectrum, the 0-0 band appears at 36351,cm$^{-1}$ which is red-shifted by 1863,cm$^{-1}$ from that of the AnH$^+$ monomerfootnote{G.~F'{e}raud, textit{et al.} textit{Phys. Chem. Chem. Phys.} textbf{16}, 5250 (2014).}. The band pattern is similar to that of the AnH$^+$ monomer. This indicates that the structure of the AnH$^+$ is not so affected by the single hydration. In the IR photodissociation spectrum, OH stretching band of the H$_2$O moiety and free NH stretching band of AnH$^+$ moiety are observed. Comparison with the results of the DFT calculation at M05-2X/6-31++G(d,p) level, we determined the structure of the AnH$^+$(H$_2$O) cluster. Issue Date: 24-Jun-15 Publisher: International Symposium on Molecular Spectroscopy Citation Info: ACS Genre: Conference Paper / Presentation Type: Text Language: English URI: http://hdl.handle.net/2142/79254 Date Available in IDEALS: 2016-01-05


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# The inability to continue a fibration sequence even when a delooping exists

$\newcommand{\i}{\iota}$ The general notion that I am trying to disprove is that if we are given a fibration $X \to Y$ with fiber $F$ such that the delooping $BF$ exists, that there is a map $Y \to BF$ such that $F \to X \to Y \to BF$ is a fibration sequence.

This question came up when I was trying to see what went wrong when trying to define postnikov towers for nonsimply connected spaces.

Here is my proposed counterexample:

Let $X$ be a nonsimply connected space with the action of $\pi_1(X) \curvearrowright \pi_2(X)$ nontrivial. Since the meaning of a postnikov tower here is ambiguous let me explicitly define $X_1=K(\pi_1(X),1)$ and $X_2$ the space obtained by killing all homotopy groups above dimension $2$, and let $X_2 \to X_1$ be the map obtained by just adding cells to $X_2$ to kill homotopy at dimension 2.

I want to show that there is no map $X_1 \to K(\pi_2(X),3)=BK(\pi_2(X),2)$ such that $K( \pi_2(X),2) \to X_2 \to X_1 \to K(\pi_2(X),3)$ is a fibration sequence.

Intuitively, the only candidate for such a map would be an element of $[X_1, K(\pi_2(X),3)]=H^3(X_1, \pi_2(X))$ arising as the transgression of the fundamental class of $K(\pi_2(X),2)$ in the Serre Spectral Sequence of fibration $K( \pi_2(X),2) \hookrightarrow X_2 \to X_1$.

In order for this to make sense the fundamental class $\i$ of $K(\pi_2(X),2)$ needs to be in the domain of the transgression, and it is not:

We can view $\i =Id \in Hom(\pi_2(X), \pi_2(X))$, $E_2^{0,2}=Hom(\pi_2(X), \pi_2(X))^{\pi_1(X)}$ where $\pi_1(X)$ acts on the first $\pi_2(X)$, and the transgression is $d_3$, which in view of the vanishing of $E_2^{1,1}$, acts on all of $E_2^{0,2}$.

Then we have $\iota \in Hom(\pi_2(X), \pi_2(X))$ is not $\pi_1(X)$ invariant, and so it is not in the domain of the transgression.

This leaves the following question on how I can make this argument rigorous:

How can I show that any candidate map $X_1 \to K(\pi_2(X),3)$ would have to arise as the transgression of the fundamental class of $K(\pi_2(X),2)$?

• You might like to check out T. Ganea's paper "Induced Fibrations and Cofibrations". Dec 18, 2016 at 15:26
• $\mathbb Z/3$ is a group. Consider a fibration over a wedge of circles with fiber $\mathbb Z/3$ such that the full monodromy group is $S_3$. Obviously it can't be represented by a map to $B\mathbb Z/3$. In general, any fibration $F\to Y \to X$ can be represented by $X \to B\mathrm{Aut}\ F$, where $\mathrm{Aut}\ F$ is the group of homotopy autoequivalences. If $F$ is a group then we have $BF \stackrel{f}{\to} B\mathrm{Aut}\ F$ (left self-action), and if the classifying map doesn't factor through $f$, then representation by $X \to BF$ is impossible. Dec 20, 2016 at 5:54

I haven't thought about the example that you describe, but here is a different kind of example.

Consider a topological group $F$, and let $M$ denote the monoid of weak equivalences $F\to F$, so there is an evident inclusion $F\to M$. Let $X$ be any based space, and let $i\colon X\to CX$ be the inclusion of the base of the cone. Now suppose we have a based map $f\colon X\to M$. Take two copies of $CX\times F$ and identify $(i(x),u)$ in the first copy with $(i(x),f(x)(u))$ in the second. This gives a space $E(f)$ with evident maps $F\xrightarrow{j}E(f)\xrightarrow{q}\Sigma X$. I think that this is always a fibration up to homotopy. If the map $f$ does not factor (up to homotopy) through $F$, then we have a fibration that is not classified by a map $\Sigma X\to BF$.

Note here that $f$ is supposed to be a based map when we take the identity as the basepoint in $f$. One way to define such an $f$ is to take a based map $m\colon X\wedge F\to F$, and put $f(x)(u)=m(x\wedge u).u$.

For a minimal example of this, take $F=S^3=SU(2)$ and $X=S^1$. Let $\eta\colon S^3\to S^2$ be the Hopf map, and take $m=\Sigma\eta\colon X\wedge F\to F$. The resulting map $f\colon X\to M$ is nontrivial in homotopy, but $[X,F]=\pi_1(S^3)=0$.

• Dear Professor Strickland, Thank you for giving me a better idea of when a fibration is classified by a map from the base space. I think I have been failing to generalize this to my example because this criterion of lifting to something that has a multiplication, seems specific to spaces that are suspensions. I hope to come back later when I have something more intelligent to say about this wonderful answer. Best Regards, Hari (B.A. 2017 Courant Institute) Dec 22, 2016 at 4:59

By hypothesis in my question, $\pi_1(K(\pi_1(X),1))=\pi_1(X)$ acts nontrivially on $H^2(K(\pi_2(X),2),\pi_2(X))$.

The path fibration $K(\pi_2(X),2) \hookrightarrow P \to K(\pi_2(X),3)$ has trivial monodromy and so does any pullback of this fibration. Therefore my fibration is not a pullback of a path loop fibration. In other words there is no map that classifies my fibration.


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## On Divergence of Puiseux Series Asymptotic Expansions of Solutions to the Third Painlevé Equation

math. arxive. Cornell University, 2017. No. 1702.05758.
Parusnikova A., Vasilyev A. V.
In this paper we present a family of values of the parameters of the third Painlevé equation such that Puiseux series formally satisfying this equation -- considered as series of z^{2/3} -- are series of exact Gevrey order one. We prove the divergence of these series and provide analytic functions which are approximated by them in sectors with the vertices at infinity.


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## Untitled

... an admissible decision rule is a rule for making a decision that is better in some sense than any other rule that may compete with it.

This sentence seems highly inaccurate to me. First, "in some sense" can be read to mean "in a vague sense," while the intended meaning here is "in a specific sense discussed below."

Second, what do we mean by "better"? Sure, they are better for at least one value of $\theta$, but this doesn't make them universally better.

I have changed the sentence to read:

... an admissible decision rule is a rule for making a decision that is "better" than any other rule that may compete with it, in a specific sense defined below.

--Zvika 14:28, 18 February 2006 (UTC)

This helps a little - but not much. As you point out, just because something is better for one value doesn't mean it is universally better. What is stated below in the article is that admissible rules are better than inadmissible ones. Even if this is true, (I think this is not always true), the natural way to read this sentence is that an admissible rule is better than ANY other rule - as if there couldn't be more than one admissible rule. Obviously if there is more than one admissible rule, then it is false that any admissible rule is preferable to any other rule.
As for "An admissible rule should be preferred over an inadmissible rule..." this makes it sound like ANY admissible rule should be preferable to ANY inadmissible rule. Personally, I think this is wrong. For example, here is an admissible rule for estimating $\theta$: say "seven". This rule cannot be dominated because when theta really is seven, it has zero error. Here is an inadmissible rule: use maximum likelihood estimation and then subtract epsilon from your estimate. This will be better than the constant rule in almost every case. The comparison to Pareto optimality is apt - just because you can't make a Pareto improvement doesn't mean you can't make an improvement. I am changing these but I would be happy to see other uses make further edits (or corrections with explanations if they think I am wrong). --Jdvelasc (talk) 23:30, 5 September 2009 (UTC)

Is it possible to provide a simple example? Or some words about the history of the concept? Bo Jacoby (talk) 03:59, 4 July 2010 (UTC).

I wrote the following in the "Bayesian probability" article:

Wald's result also established the Bayesian formalism as a fundamental technique in such areas of frequentist statistics as point estimation, hypothesis testing, and confidence intervals.

Wald characterized admissible procedures as Bayesian procedures (and limits of Bayesian procedures), making the Bayesian formalism a central technique in such areas of frequentist statistics as parameter estimation, hypothesis testing, and computing confidence intervals, as established by the following articles and then textbooks:

• Kiefer, J. and Schwartz, R. (1965). "Admissible Bayes character of T2-, R2-, and other fully invariant tests for multivariate normal problems". Annals of Mathematical Statistics 36: 747–770.
• Schwartz, R. (1969). "Invariant proper Bayes tests for exponential families". Annals of Mathematical Statistics 40: 270–283.
• Hwang, J. T. and Casella, George (1982). "Minimax confidence sets for the mean of a multivariate normal distribution". Annals of Statistics 10: 868–881.

These examples could be expanded here, but I lack the time (for some months), alas. Best regards, Kiefer.Wolfowitz (talk) 18:01, 4 July 2010 (UTC)