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# April 20th Math Club – Directed Reading Program

This Wednesday, April 20th, come hear about the experiences of the first cohort of students in the Directed Reading Program. In this program undergraduate students team up with math graduate student mentors to read about some cool topic in math.

Plus, if that is not already enough, there will be pizza.  Hope to see you all there!

Here are the details:

When: Wednesday, April 20, at 5:30

Where: PHSC1105

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# Math Club meeting this week.

This is perhaps the last Math Club meeting for the calendar year 2013. Come one, come all!

Where: Physical Sciences, Room 1105

When: Wednesday, December 4th, 5:00 pm

Who: Dr. Hansmann from the Chemistry Department will talk about, “Computational Tools in Protein Folding”. Here is a link to the flyer:

mathclubDec4_2013

As always, FREE PIZZA!

# OU Citizen Science Club

We’re happy to spread the word about OU’s Citizen Science Club.They are meeting this

Monday, October 7th at 7pm in Dale Hall Room 107.

What is citizen science?

Back in the old days science was learned about and done by amateurs as well as professionals.  Heck, Fermat was a lawyer and government official during the day and came up with Fermat’s Little Theorem, Fermat’s Last Theorem, etc. at night (like Batman!).

But it’s not just in the good old days!  It’s even easier with the Internet, crowd sourcing, smartphones, etc. for people to get involved with science reearch!  One example we told you about two years ago was FolditThis page has more good examples!

Or you could just go to the meeting!

# Dr. Martin on Amazon.com!

Of course everybody thinks about the teaching when they think of professors, but profs also do  cutting edge research!

This is another installment of our very occasional series on the research being done by the faculty of the OU Math department.  This time we’re talking about our own Kimball Martin.  He recently published a book entitled “On Central Critical Values of the Degree Four \$l\$-functions for Gsp (4): The Fundamental Lemma. III”.  It was written by Dr. Martin, Masaaki Furusawa, and Joseph Shalika*.

If you’d like a copy, you can get it from amazon.com for the low, low price of \$74 (free shipping!), or you can just download the preprint version from Dr. Martin’s webpage.  So far there’s no reviews on amazon, so you’ll have to read it yourself to see if it’s as gripping as parts I and II :-).

* Fun fact about Dr. Shalika — he was the mathematical grandfather of both OU’s Luis Lomeli and OSU’s Mahdhi Asgari!

# Robots vs. Math

Earlier this semester an OU math professor, Dr. Murad Özaydin, and a former OU grad student, Dr. James Dover (now at Cameron University in Lawton), posted their most recent paper on the ArXiv.

It is entitled “Homeomorphism and Homotopy Types of Restricted Configuration Spaces of Metric Graphs”.  But really, as Dr. Özaydin explained in the topology seminar, it’s about Robots on Rails.

As part of our very occasional postings on research in the OU Math department, we wanted to tell you about their very cool theorem.  Before we do that, let’s talk robots.

In real life, robots which can move are often only allowed to move along certain paths.  They might actually move along rails; or, even if they just drive along on wheels in a factory their programming steers them along certain fixed paths so they don’t accidentally run into walls, expensive machines, people, etc.

And even non-robots often travel on real or virtual rails.  Trains, of course.  But cars on the freeway system, planes flying through the air (commercial planes can’t go any which way, air traffic control routes planes around the US on “highways in the sky”).

A typical problem, then, is to route multiple robots around a fixed set of paths on a factory floor so that they all get place to place without running into each other.  To a mathematician, this is a problem about moving around on a graph.  Each vertex is a destination, and each edge from one vertex to another is a path from one place to another.  So it might look like this (the numbers on each edge is the length of that edge):

We can then think of robots in this picture as little disks moving along these paths. Of course, the whole problem is that you don’t want the robots running in to each other. This means each robot’s disk should bigger or smaller as needed to represent the size of the robot.  A graph with distances is called a metric graph and moving robots of various sizes around on such a graph is sometimes called “Pebble Motion on Graphs Problems”.

Actually developing a routing algorithm for moving the robots around is a very challenging problem. It is already interesting and difficult to understand which routings are even possible.

How do we tackle such a problem?  Mathematicians (like Drs. Dover and Özaydın) do something bold which actually simplifies the problem (once you get your mind around it).  They consider the set of all possible configurations of robot positions on the graph.

On the one hand, this a a much bigger and more complicated gadget, but on the other hand it is topological space called a configuration space.  This means you can unleash all the tools we have from topology to study the possible configurations of robots.

For example, a path in the configuration space translates into the simultaneous motion of the robots around the graph.  So knowing that there is a path between any two configurations in the configuration space tells you that it is possible to move the robots into any positions you like.

The first question, then, is how many different configuration spaces are there for a fixed graph if you vary the size of the robots.  To go back to our example of the factory, if you you replace your robots with new and improved models which are half the size of the old ones, does your old software still work, or are their new paths and configurations possible?  For example, if you have a single track of length 1 meter:

If you have a two gigantic robots which are each 3/4 meter across, then they can’t both fit on this track.  So there is zero possible configuration spaces for two big robots on this graph.  But if you replace them with two small robots (say of size 1 cm), then they now both fit on the graph and so you have at least one space of possible configurations.

As a topological space, you count spaces the same if they are homeomorphic.  So the first problem to ask (and which Drs. Dover and Özaydın answer) is how many non-homeomorphic configuration spaces are there?

You might think that there is infinitely many.  Or at least many, many, many.  In his 2011 PhD thesis at the University of Durham, Ken Deeley proved* that if you consider the first interesting case of two robots (one robot is easy!), then the number of possible configurations is actually finite.

Dr. Deeley gave an upper bound which is an exponential function in the number of edges on the graph.  The good news is that it only depends on the number of edges, not how they are connected or how long they are or anything else.  The bad news — as we all know — is that exponential functions get very big very fast.  So if it really is an exponential number of configuration spaces, then it is surely impractical to handle them on a case by case base.

Drs. Dover and Özaydın took on the challenge.  They prove the following amazing result:

Theorem:  If you have n robots traveling around a graph, then the number of possible configuration spaces is bounded by a polynomial of degree n in the number of edges in the graph.

For example, in Deeley’s case of two robots, the bound is a quadratic polynomial!  In fact, they even compute the bound for two robots on a graph with E edges in their paper and get that the number of configuration spaces is no more than:

$9 E^{2}-5E-1$

That’s a heck of a lot less than an exponential bound!

A couple of notes before we go:

• They give an example with two robots where the number of configurations is bounded below by a quadratic polynomial.  This shows that you can’t get it down to a linear function.
• They also consider the case when the robots have various sizes.  In fact, they even allow you to set how close each pair of robots can get to each other.  Let’s say robots A, B, and C are all small, but A is carrying radioactive material.  If robot B is  sensitive and robot C is not sensitive to radioactivity, you can set the distance between A and B to be larger than the distance between A and C and B and C.  Their theorem still applies!

Very nice result!

“Imagine if all these fish were mathematical theorems…” — James Dover

* To be 100% mathematically honest, Deeley considered spaces up to homotopy, not homeomorphism.  There are fewer homotopy types, so a bound on homotopy types is not quite as strong a result as on homeomorphism types.  That is, Drs. Dover and Özaydın result implies Dr. Deeley’s but not vice versa.

# Work on the OU supercomputer!

Henry Neeman passed along this great opportunity for OU students only:

The OU Norman campus opportunity below is available to OU students only.

Henry

==================================================================

Student Employee – Supercomputing Internship (FALL)

http://jobs.ou.edu/

Requisition Number: 15162
Position #: 00022344

Job Search Category: Student or Work Study
Job Type: Part-Time
Number Needed: 1
Listing Category: Technical and Paraprofessional

Department: Information Technology (IT)
Campus: Norman

Application Deadline: Open Until Filled

Work Schedule: Normal working hours are M-F 8-5. Will depend on
class schedule.
Hours per week: 15-20

Are you interested in an Opportunity to gain real-world IT
experience?

Information Technology is offering a one-year paid Internship
that will emphasize the prevalence of Supercomputing Technology.

As an OU IT Intern, you will have an opportunity to work as a
member of a team of IT Professionals specializing in
Supercomputing technology.

You will be assigned a mentor and will learn the following
development concepts as well as apply them in real-life IT
projects:

* Account management
* Batch system/queue management
* Software management
* File system management
* Network management and troubleshooting
* User application support
* Hardware maintenance
* Backups/Restores

Note: This internship may also be eligible for college credit.

Required Education and Experience: High School diploma or GED as
well as some formal college training.

***Must have completed intro programming classes or equivalent
experience.

Must be currently enrolled in the FALL 2012 term as a student at
the University of Oklahoma. Hiring contingent upon verification of
current student status.

Required Skills and Proficiencies:

The ideal candidate demonstrates a strong work ethic, is
self-directed and is productive working independently as well as
collaboratively; Must be an analytical thinker with the ability to
identify, define, interpret, and resolve both technical and human
issues.

Must have completed intro programming classes or equivalent
experience.

* Computer Skills
* Excellent Communication Skills

Department Preferences:

* Undergraduate study in MIS, Computer Science, Engineering or
other computer related field.
* Experience in Unix system administration.
* Experience with C programming language.
* Experience with Perl programming language.

You’ll be working with OSCER not Oscar!

# It’s Alive!

As promised, Boomer is up and running!

A picture of Boomer (but where’s the blinking lights?)

Henry Neeman sent us the announcement:

OU Deploys Fastest Academic Supercomputer in Oklahoma History

May 24, 2012

NORMAN — “Boomer,” the fastest academic supercomputer in state history, was deployed today at the University of Oklahoma.

“The deployment of the state’s fastest supercomputer in state history will further enhance OU’s academic excellence,” said OU President David L. Boren.

The supercomputer clocks in at a peak speed of roughly 109
trillion calculations per second and supports OU’s research
initiatives.

“This new supercomputer represents an incredible opportunity for OU,” said Loretta Early, OU’s Chief Information Officer and University Vice President for Information Technology. “Boomer will substantially expand OU’s ability to engage in cutting-edge,
computing-intensive research — to do more, and to do it faster and better, at a lower cost.”

Researchers will employ Boomer to compute large amounts of data for a broad variety of research with emphasis on weather forecasting, molecular dynamics and high-energy physics, which explores the fundamental nature of matter and energy. Boomer also will support research in astronomy, coastal flooding, biomedical
engineering, data encoding for disk drives, petroleum engineering, nanotechnology, groundwater contamination, biofuels, and wireless networks, among many other areas.

Henry Neeman, Director of the OU Supercomputing Center for Education and Research, a division of OU Information Technology, said that OU IT focuses on the needs of researchers at a level that is almost unprecedented nationally even among top research universities.

“For the past decade, OU has been a national leader in supporting the computational research and education needs of local students, faculty and staff,” Neeman said. “We’re extremely proud to expand a great tradition with this fourth generation OU IT supercomputer,
which will enhance research capabilities by connecting scientific collaborators throughout the state and nation.”

Boomer is three times as fast as the previous fastest academic supercomputer in the state, OU’s “Sooner,” which served hundreds of undergraduates, graduate students, staff and faculty from 2008 to early 2012. It’s also 100 times as fast as OU IT’s first supercomputer, built in 2002.

OneNet, Oklahoma’s statewide research, education and government network, will deliver Boomer’s capabilities from OU IT’s high-speed campus network to OU research teams, and 24 other Oklahoma institutions and more than 150 out-of-state and international collaborators will also be connected through OneNet.

“We’re very proud of our role in facilitating research, one of
OU’s key missions and a crucial engine for statewide economic development,” Early said. “With this new resource, we improve our potential to attract a growing number of research projects and increase external funding, and therefore attract and retain the best and brightest researchers, both faculty and students. Boomer
is both a logical next step and a major breakthrough for
researchers on campus.”

In addition, Boomer will connect to the Oklahoma PetaStore, which has the capacity to store multiple Petabytes (millions of Gigabytes) of research data, allowing OU researchers to create and maintain very large research data collections. Keith Brewster, acting Director of the Center for Analysis and Prediction of Storms, is looking forward to improving forecasting models with Boomer’s capabilities. “Severe weather, including tornadoes and hurricanes, kills hundreds of people and destroys billions of dollars of property every year. OU’s new supercomputer will help us to improve forecasts of these events, allowing us to resolve features half the size we could resolve previously.” Making large-scale, accessible and professionally managed advanced computing capability available to OU’s researchers also ensures that investigators will meet the requirements of federal research funding programs. Through deployment of Boomer, the University’s goal is to strengthen OU’s grant applications, leading to improved outcomes for researchers, students and Oklahoma’s economy.

To paraphrase Dr. Boren when approving the construction of Boomer: “But the deciding factor was when we learned that Texas was working along similar lines, and we were afraid of a doomsday gap.”


text:
Links lead to another part of an app, other pages, or help articles. They can also be used to initiate commands.

For more details and examples visit the official docs. The R package can not handle each and every case, so for advanced use cases you need to work using the original docs to achieve the desired result.

Link(...)

## Arguments

as

string | React.ComponentClass | React.FunctionComponent
A component that should be used as the root element of the link returned from the Link component.

componentRef

IRefObject<ILink>
Optional callback to access the ILink interface. Use this instead of ref for accessing the public methods and properties of the component.

disabled

boolean

keytipProps

IKeytipProps

styles

IStyleFunctionOrObject<ILinkStyleProps, ILinkStyles>
Call to provide customized styling that will layer on top of the variant rules.

theme

ITheme
Theme (provided through customization.)

## Best practices

### Layout

• Links visually indicate that they can be clicked, typically by being displayed using the visited or unvisited link system colors. Traditionally, links are underlined as well, but that approach is often reserved for body copy links within an article.

### Content

• People should be able to accurately predict the result of selecting a link based on its link text and optional tooltip.

• Use descriptive, actionable link text when possible. Avoid using URLs as link text.

• Don't use if the action is destructive or irreversible. Links aren't appropriate for commands with significant consequences.

• Keep discrete links far enough apart that people can differentiate between them and easily select each one.

• Use sentence-style capitalization—only capitalize the first word. For more info, see Capitalization in the Microsoft Writing Style Guide.

## Examples

library(shiny.fluent)

if (interactive()) {
shinyApp(
ui = Link(href = "https://appsilon.com", "Appsilon"),
server = function(input, output) {}
)
}

format: plain


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# Math Help - Question on Percent

1. ## Question on Percent

This is really silly, but I am stuck on a simple percent!

Could you please tell me what 0.4 percent of 650 is? Is it just multiplied as it is, or does the decimal move?

Thanks so much!
RainieB

2. ## 0.4 percent of 650

Could you please tell me what 0.4 percent of 650 is? Is it just multiplied as it is, or does the decimal move?

Thanks so much!
RainieB

RainieB,
Just Multiply. But be carefull about "0.4 percent"
"0.4 percent" means = 0.4% and also equal to (0.4/100)
Therefor, 0.4% of 650 will be (0.4/100) x 650 = 2.6

3. ## Thank you

Thank you very much for your quick response!!!

RainieB

4. very simple.. :
0.4 % of 650.

$\frac {0.4}{100}$ x $650$

= $\frac {\frac {4}{10}}{100}$ x $650$

= $\frac {4}{10}$ x $\frac {1}{100}$ x $650$

= $2.6$

hope this helps!!


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#jsDisabledContent { display:none; } My Account | Register | Help

# Maupertuis' principle

Article Id: WHEBN0004040947
Reproduction Date:

 Title: Maupertuis' principle Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Maupertuis' principle

In classical mechanics, Maupertuis' principle (named after Pierre Louis Maupertuis), is that the path followed by a physical system is the one of least length (with a suitable interpretation of path and length). It is a special case of the more generally stated principle of least action. Using the calculus of variations, it results in an integral equation formulation of the equations of motion for the system.

## Contents

• Mathematical formulation 1
• Jacobi's formulation 2
• Comparison with Hamilton's principle 3
• History 4
• References 6

## Mathematical formulation

Maupertuis' principle states that the true path of a system described by N generalized coordinates \mathbf{q} = \left( q_{1}, q_{2}, \ldots, q_{N} \right) between two specified states \mathbf{q}_{1} and \mathbf{q}_{2} is an extremum (i.e., a stationary point, a minimum, maximum or saddle point) of the abbreviated action functional

\mathcal{S}_{0}[\mathbf{q}(t)] \ \stackrel{\mathrm{def}}{=}\ \int \mathbf{p} \cdot d\mathbf{q}

where \mathbf{p} = \left( p_{1}, p_{2}, \ldots, p_{N} \right) are the conjugate momenta of the generalized coordinates, defined by the equation

p_{k} \ \stackrel{\mathrm{def}}{=}\ \frac{\partial L}{\partial\dot{q}_{k}}

where L(\mathbf{q},\dot{\mathbf{q}},t) is the Lagrangian function for the system. In other words, any first-order perturbation of the path results in (at most) second-order changes in \mathcal{S}_{0}. Note that the abbreviated action \mathcal{S}_{0} is not a function, but a functional, i.e., something that takes as its input a function (in this case, the path between the two specified states) and returns a single number, a scalar.

## Jacobi's formulation

For many systems, the kinetic energy T is quadratic in the generalized velocities \dot{\mathbf{q}}

T = \frac{1}{2} \frac{d\mathbf{q}}{dt} \cdot \mathbf{M} \cdot \frac{d\mathbf{q}}{dt}

although the mass tensor \mathbf{M} may be a complicated function of the generalized coordinates \mathbf{q}. For such systems, a simple relation relates the kinetic energy, the generalized momenta and the generalized velocities

2 T = \mathbf{p} \cdot \dot{\mathbf{q}}

provided that the potential energy V(\mathbf{q}) does not involve the generalized velocities. By defining a normalized distance or metric ds in the space of generalized coordinates

ds^{2} = d\mathbf{q} \cdot \mathbf{M} \cdot d\mathbf{q}

one may immediately recognize the mass tensor as a metric tensor. The kinetic energy may be written in a massless form

T = \frac{1}{2} \left( \frac{ds}{dt} \right)^{2}

or, equivalently,

2 T dt = \mathbf{p} \cdot d\mathbf{q} = \sqrt{2m T} \ ds.

Hence, the abbreviated action can be written

\mathcal{S}_{0} \ \stackrel{\mathrm{def}}{=}\ \int \mathbf{p} \cdot d\mathbf{q} = \int ds \sqrt{2m}\sqrt{E_{tot} - V(\mathbf{q})}

since the kinetic energy T = E_{tot} - V(\mathbf{q}) equals the (constant) total energy E_{tot} minus the potential energy V(\mathbf{q}). In particular, if the potential energy is a constant, then Jacobi's principle reduces to minimizing the path length s = \int ds in the space of the generalized coordinates, which is equivalent to Hertz's principle of least curvature.

## Comparison with Hamilton's principle

Hamilton's principle and Maupertuis' principle are occasionally confused and both have been called the principle of least action. They differ from each other in three important ways:

• their definition of the action...
Hamilton's principle uses \mathcal{S} \ \stackrel{\mathrm{def}}{=}\ \int L \, dt, the integral of the Lagrangian over time, varied between two fixed end times t_{1}, t_{2} and endpoints q_1, q_2. By contrast, Maupertuis' principle uses the abbreviated action integral over the generalized coordinates, varied along all constant energy paths ending at q_1 and q_2.
• the solution that they determine...
Hamilton's principle determines the trajectory \mathbf{q}(t) as a function of time, whereas Maupertuis' principle determines only the shape of the trajectory in the generalized coordinates. For example, Maupertuis' principle determines the shape of the ellipse on which a particle moves under the influence of an inverse-square central force such as gravity, but does not describe per se how the particle moves along that trajectory. (However, this time parameterization may be determined from the trajectory itself in subsequent calculations using the conservation of energy.) By contrast, Hamilton's principle directly specifies the motion along the ellipse as a function of time.
• ...and the constraints on the variation.
Maupertuis' principle requires that the two endpoint states q_{1} and q_{2} be given and that energy be conserved along every trajectory. By contrast, Hamilton's principle does not require the conservation of energy, but does require that the endpoint times t_{1} and t_{2} be specified as well as the endpoint states q_{1} and q_{2}.

## History

Maupertuis was the first to publish a principle of least action, where he defined action as \int v \, ds, which was to be minimized over all paths connecting two specified points. However, Maupertuis applied the principle only to light, not matter (see the 1744 Maupertuis reference below). He arrived at the principle by considering Snell's law for the refraction of light, which Fermat had explained by Fermat's principle, that light follows the path of shortest time, not distance. This troubled Maupertuis, since he felt that time and distance should be on an equal footing: "why should light prefer the path of shortest time over that of distance?" Accordingly, Maupertuis asserts with no further justification the principle of least action as equivalent but more fundamental than Fermat's principle, and uses it to derive Snell's law. Maupertuis specifically states that light does not follow the same laws as material objects.

A few months later, well before Maupertuis' work appeared in print, Euler independently defined action in its modern abbreviated form \mathcal{S}_{0} \ \stackrel{\mathrm{def}}{=}\ \int m v \, ds \ \stackrel{\mathrm{def}}{=}\ \int p \, dq and applied it to the motion of a particle, but not to light (see the 1744 Euler reference below). Euler also recognized that the principle only held when the speed was a function only of position, i.e., when the total energy was conserved. (The mass factor in the action and the requirement for energy conservation were not relevant to Maupertuis, who was concerned only with light.) Euler used this principle to derive the equations of motion of a particle in uniform motion, in a uniform and non-uniform force field, and in a central force field. Euler's approach is entirely consistent with the modern understanding of Maupertuis' principle described above, except that he insisted that the action should always be a minimum, rather than a stationary point.

Two years later, Maupertuis cites Euler's 1744 work as a "beautiful application of my principle to the motion of the planets" and goes on to apply the principle of least action to the lever problem in mechanical equilibrium and to perfectly elastic and perfectly inelastic collisions (see the 1746 publication below). Thus, Maupertuis takes credit for conceiving the principle of least action as a general principle applicable to all physical systems (not merely to light), whereas the historical evidence suggests that Euler was the one to make this intuitive leap. Notably, Maupertuis' definitions of the action and protocols for minimizing it in this paper are inconsistent with the modern approach described above. Thus, Maupertuis' published work does not contain a single example in which he used Maupertuis' principle (as presently understood).

In 1751, Maupertuis' priority for the principle of least action was challenged in print (Nova Acta Eruditorum of Leipzig) by an old acquaintance, Johann Samuel Koenig, who quoted a 1707 letter purportedly from Voltaire and the King of Prussia, Frederick II were engaged in the quarrel. However, no progress was made until the turn of the twentieth century, when other independent copies of Leibniz's letter were discovered. The present scholarly consensus seems to be that the quotations from Leibniz are indeed genuine, i.e., that he had invented Maupertuis' principle and applied it to several mechanical problems by 1707 (37 years before Maupertuis and Euler) but did not publish his findings.

## References

• Pierre Louis Maupertuis, Accord de différentes loix de la nature qui avoient jusqu'ici paru incompatibles (original 1744 French text); Accord between different laws of Nature that seemed incompatible (English translation)
• Leonhard Euler, Methodus inveniendi/Additamentum II (original 1744 Latin text); Methodus inveniendi/Appendix 2 (English translation)
• Pierre Louis Maupertuis, Les loix du mouvement et du repos déduites d'un principe metaphysique (original 1746 French text); Derivation of the laws of motion and equilibrium from a metaphysical principle (English translation)
• Leonhard Euler, Exposé concernant l'examen de la lettre de M. de Leibnitz (original 1752 French text); Investigation of the letter of Leibniz (English translation)
• König JS. "De universali principio aequilibrii et motus", Nova Acta Eruditorum, 1751, 125-135, 162-176.
• J.J. O'Connor and E.F. Robertson, "The Berlin Academy and forgery", (2003), at The MacTutor History of Mathematics archive.
• C.I. Gerhardt, (1898) "Über die vier Briefe von Leibniz, die Samuel König in dem Appel au public, Leide MDCCLIII, veröffentlicht hat", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, I, 419-427.
• W. Kabitz, (1913) "Über eine in Gotha aufgefundene Abschrift des von S. König in seinem Streite mit Maupertuis und der Akademie veröffentlichten, seinerzeit für unecht erklärten Leibnizbriefes", Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, II, 632-638.
• H. Goldstein, (1980) Classical Mechanics, 2nd ed., Addison Wesley, pp. 362–371. ISBN 0-201-02918-9
• L.D. Landau and E.M. Lifshitz, (1976) Mechanics, 3rd. ed., Pergamon Press, pp. 140–143. ISBN 0-08-021022-8 (hardcover) and ISBN 0-08-029141-4 (softcover)
• G.C.J. Jacobi, Vorlesungen über Dynamik, gehalten an der Universität Königsberg im Wintersemester 1842-1843. A. Clebsch (ed.) (1866); Reimer; Berlin. 290 pages, available online 8Œuvres complètes volume at Gallica-Math from the Gallica Bibliothèque nationale de France.
• H. Hertz, (1896) Principles of Mechanics, in Miscellaneous Papers, vol. III, Macmillan.
• V.V. Rumyantsev (2001), "Hertz's principle of least curvature", in Hazewinkel, Michiel,
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.

Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.


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# Tag Info

First, $l(\tau)(y)$ is a function of both $\tau$ and $y$, and the dependence on $\tau$ is essential; this is how communication happens. Since $T$ and $Y$ are finite, there is no point in introducing integrals. $\mathbb{E}_{l(\tau)}g(F(y))$ is the expectation over the function when the random value $y$ is distributed according to $l(\tau)$. That is, \mathbb{...
We have that ${\cal I} = ((X^i)_i, \mu)$ and ${\cal J} = ((Y^i)_i, \nu)$ are two information structures. An Interpretation mapping for player $i$ is a mapping $\phi^i: X^i \to \Delta(Y^i)$ so it associates with every $x^i$ a distribution over $Y^i$. Let $x^i \in X^i$. Then $\phi^i(x^i)$ is a distribution over $Y^i$ so $\phi(x^i)(y^i)$ is the ...


text:
# Show that $V = W \oplus U$.

The Problem

Let $V=k^3$ for some field $k$. Let $W$ be the subspace spanned by $(1,0,0)$ and let $U$ be the subspace spanned by $(1,1,0)$ and $(0,1,1)$. Show that $V= W \oplus U$. Explain your argument in detail.

What I Know

1. I know that a field $k^n=n$-tuples of elements of $k$.

2. I know that a subset $W$ of a vector space $V$ over a field $k$ is a subspace if the operations of $V$ make $W$ into a vector space over $k$.

3. I know that if span$(S)=V$ for a set $S$ in a vector space $V$, where $S$ is linearly independent, then $S$ is a basis for $V$.

4. I know that the external direct sum $V \oplus W$ for vector spaces $V$ and $W$ over a field $k$ is defined as the set of all ordered pairs $(v,w)$ such that $v\in V$ and $w \in W$.

What I Don't Know

1. How to apply what I listed above to help me solve the problem. I am absolutely atrocious at this material and struggle so much in simply starting these problems.

2. If everything I listed above is even relevant to the problem at hand.

3. If what I listed above is insufficient to complete the problem.

Text: Abstract Linear Algebra by Curtis

• It sounds like you need to know the internal direct sum. – Slade Feb 28 '17 at 11:45
• Alright, but why? – Thy Art is Math Mar 1 '17 at 8:09

Let $V$ be a vector space and $U, W$ subspaces. Define
$$\phi:U\oplus W\to V$$ $$\phi(u, w)=u+w$$
Obviously $\phi$ is linear. When is it an isomorphism? How can you apply it to your case?
Take the span of $W$ as $a(1,0,0) = (a,0,0)$ for $a$ in your field, and the span of $U$ as $b(1,1,0)+c(0,1,1)=(b,b+c,c)$ with $b,c$ in the field. Now take an element from $V$ say it is $(x,y,z)$ and decompose it in sum of an element in $W$ and an element in $U$ as follows: $(x,y,z) = (x+z-x,0,0) + (y-z,y,z)$ and you can see the first element is in $W$ and the second in $U$ so now, if you proof that it is the unique way of decomposing an element of $V$ as sum of an element in each subspace you would prove $V$ is direct sum of $W$ and $V$
An alternative is to proof the intersection of $W$ and $V$ is ${(0,0,0)}$ and that dim ($W$) =1 and dim ($U$)=2 so you have dim ($V$)=dim ($W$) + dim ($U$) and their intersection is ${(0,0,0)}$ this implies the result.


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# Gilgamath

## Twelve Dimensional Chess is Stupid

Tue 16 October 2018 by Steven

Chess and the Curse of Dimensionality

## Another Confidence Limit for the Markowitz Signal Noise ratio

Wed 28 March 2018 by Steven

Another confidence limit on the Signal Noise ratio of the Markowitz portfolio.

## Markowitz Portfolio Covariance, Elliptical Returns

Mon 12 March 2018 by Steven E. Pav

In a previous blog post, I looked at asymptotic confidence intervals for the Signal to Noise ratio of the (sample) Markowitz portfolio, finding them to be deficient. (Perhaps they are useful if one has hundreds of thousands of days of data, but are otherwise awful.) Those confidence intervals came from revision four of my paper on the Asymptotic distribution of the Markowitz Portfolio. In that same update, I also describe, albeit in an obfuscated form, the asymptotic distribution of the sample Markowitz portfolio for elliptical returns. Here I check that finding empirically.

Suppose you observe a $$p$$ vector of returns drawn from an elliptical distribution with mean $$\mu$$, covariance $$\Sigma$$ and 'kurtosis factor', $$\kappa$$. Three times the kurtosis factor is the kurtosis of marginals under this assumed model. It takes value $$1$$ for a multivariate normal. This model of returns is slightly more realistic than multivariate normal, but does not allow for skewness of asset returns, which seems unrealistic.

Nonetheless, let $$\hat{\nu}$$ be the Markowitz portfolio built on a sample of $$n$$ days of independent returns:

$$\hat{\nu} = \hat{\Sigma}^{-1} \hat{\mu},$$

where $$\hat{\mu}, \hat{\Sigma}$$ are the regular 'vanilla' estimates of mean and covariance. The vector $$\hat{\nu}$$ is, in a sense, over-corrected, and we need to cancel out a square root of $$\Sigma$$ (the population value). So we will consider the distribution of $$Q \Sigma^{\top/2} \hat{\nu}$$, where $$\Sigma^{\top/2}$$ is the upper triangular Cholesky factor of $$\Sigma$$, and where $$Q$$ is an orthogonal matrix ($$Q Q^{\top} = I$$), and where $$Q$$ rotates $$\Sigma^{-1/2}\mu$$ onto $$e_1$$, the first basis vector:

$$Q \Sigma^{-1/2}\mu = \zeta e_1,$$

where $$\zeta$$ is the Signal to Noise ratio of the population Markowitz portfolio: $$\zeta = \sqrt{\mu^{\top}\Sigma^{-1}\mu} = \left\Vert … ## A Lack of Confidence Interval Thu 15 February 2018 by Steven E. Pav For some years now I have been playing around with a certain problem in portfolio statistics: suppose you observe \(n$$ independent observations of a $$p$$ vector of returns, then form the Markowitz portfolio based on those returns. What then is the distribution of what I call the 'signal to noise ratio' of that Markowitz portfolio, defined as the true expected return divided by the true volatility. That is, if $$\nu$$ is the Markowitz portfolio, built on a sample, its 'SNR' is $$\nu^{\top}\mu / \sqrt{\nu^{\top}\Sigma \nu}$$, where $$\mu$$ is the population mean vector, and $$\Sigma$$ is the population covariance matrix.

This is an odd problem, somewhat unlike classical statistical inference because the unknown quantity, the SNR, depends on population parameters, but also the sample. It is random and unknown. What you learn in your basic statistics class is inference on fixed unknowns. (Actually, I never really took a basic statistics class, but I think that's right.)

Paulsen and Sohl made some progress on this problem in their 2016 paper on what they call the Sharpe Ratio Information Criterion. They find a sample statistic which is unbiased for the portfolio SNR when returns are (multivariate) Gaussian. In my mad scribblings on the backs of envelopes and scrap paper, I have been trying to find the distribution of the SNR. I have been looking for this love, as they say, in all the wrong places, usually hoping for some clever transformation that will lead to a slick proof. (I was taught from a young age to look for slick proofs.)

Having failed that mission, I pivoted to looking for confidence intervals for the SNR (and maybe even prediction intervals on the out-of-sample Sharpe ratio of the in-sample Markowitz portfolio). I realized that some of the work I had done …


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# ordered topology on integers and local compactness

Hi I am trying to make sure my logic is sound,

Let's suppose that we declared the discrete topology on $\mathbb{Z}$. Let us consider the set $\{1,2,3\}$. This set is open. However, this set is also closed because it has no limit points (the open set $\{3\}$ contains 3 but does not intersect the set $\{1,2,3\}$, the case is similar for $\{1\}$ and $\{2\}$) , also, under the discrete topology the $\mathbb{Z}$ is hausdorff so any finite point set has to be closed.

Is what I said correct?

Also, consider $\mathbb{Z}$ in the order topology. I would guess that order topology on $\mathbb{Z}$ would equal the discrete topology on $\mathbb{Z}$. However, I considered the order topology for the sake that I was reading the Munkres Topology book (p.183 Example 3) where it stated that "Every simply ordered set $X$ having the least upper bound property is locally compact. Given a basis element for $X$ it is contained in a closed interval in $X$, which is compact.

I guess what I am asking, if we define the order topology on $\mathbb{Z}$ (because it is simply ordered) , it surely is locally compact because any closed subset of it, is also open, correct?

thank you.

• $\{1,2,3\}$ is also closed in $\mathbb{Z}$ because it is the complement of the set $U=\mathbb{Z}\smallsetminus \{1,2,3\}$ which is open as all subsets are open. Yes, $\mathbb{Z}$ is Hausdorff under the discrete topology. Every set is Hausdorff under the discrete topology. – Jeremy Upsal Nov 12 '13 at 23:10

The order topology on $\Bbb Z$ is indeed the discrete topology, and it is locally compact. It’s locally compact because for each $n\in\Bbb Z$, $\{n\}$ is a compact open nbhd of $n$ that is contained in every open nbhd of $n$.
• There are different definitions of local compactness. Munkres states that $X$ is locally compact if every point has a compact set which contains a neighborhood of that point. (Which is of course true of $\{n\}$. ) – Cheerful Parsnip Nov 13 '13 at 0:07


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# 16.6: Nucleic Acids- Parts, Structure, and Function

Learning Objectives

• Describe the two types of nucleic acids and the function of each type.
• Describe the secondary structure of DNA and the importance of complementary base pairing.
• Describe how a new copy of DNA is synthesized.

The repeating, or monomer, units that are linked together to form nucleic acids are known as nucleotides. The deoxyribonucleic acid (DNA) of a typical mammalian cell contains about 3 × 109 nucleotides. Nucleotides can be further broken down to phosphoric acid (H3PO4), a pentose sugar (a sugar with five carbon atoms), and a nitrogenous base (a base containing nitrogen atoms).

$\mathrm{nucleic\: acids \underset{down\: into}{\xrightarrow{can\: be\: broken}} nucleotides \underset{down\: into}{\xrightarrow{can\: be\: broken}} H_3PO_4 + nitrogen\: base + pentose\: sugar}$

If the pentose sugar is ribose, the nucleotide is more specifically referred to as a ribonucleotide, and the resulting nucleic acid is ribonucleic acid (RNA). If the sugar is 2-deoxyribose, the nucleotide is a deoxyribonucleotide, and the nucleic acid is DNA.

The nitrogenous bases found in nucleotides are classified as pyrimidines or purines. Pyrimidines are heterocyclic amines with two nitrogen atoms in a six-member ring and include uracil, thymine, and cytosine. Purines are heterocyclic amines consisting of a pyrimidine ring fused to a five-member ring with two nitrogen atoms. Adenine and guanine are the major purines found in nucleic acids (Figure $$\PageIndex{1}$$).

The formation of a bond between C1′ of the pentose sugar and N1 of the pyrimidine base or N9 of the purine base joins the pentose sugar to the nitrogenous base. In the formation of this bond, a molecule of water is removed. Table $$\PageIndex{1}$$ summarizes the similarities and differences in the composition of nucleotides in DNA and RNA.

The numbering convention is that primed numbers designate the atoms of the pentose ring, and unprimed numbers designate the atoms of the purine or pyrimidine ring.

Table $$\PageIndex{1}$$ Composition of Nucleotides in DNA and RNA.
Composition DNA RNA
pyrimidine bases cytosine and thymine cytosine and uracil
pentose sugar 2-deoxyribose ribose
inorganic acid phosphoric acid (H3PO4) H3PO4

The names and structures of the major ribonucleotides and one of the deoxyribonucleotides are given in Figure $$\PageIndex{2}$$.

Apart from being the monomer units of DNA and RNA, the nucleotides and some of their derivatives have other functions as well. Adenosine diphosphate (ADP) and adenosine triphosphate (ATP), shown in Figure $$\PageIndex{3}$$, have a role in cell metabolism. Moreover, a number of coenzymes, including flavin adenine dinucleotide (FAD), nicotinamide adenine dinucleotide (NAD+), and coenzyme A, contain adenine nucleotides as structural components.

Nucleic acids are large polymers formed by linking nucleotides together and are found in every cell. Deoxyribonucleic acid (DNA) is the nucleic acid that stores genetic information. If all the DNA in a typical mammalian cell were stretched out end to end, it would extend more than 2 m. Ribonucleic acid (RNA) is the nucleic acid responsible for using the genetic information encoded in DNA to produce the thousands of proteins found in living organisms.

## Primary Structure of Nucleic Acids

Nucleotides are joined together through the phosphate group of one nucleotide connecting in an ester linkage to the OH group on the third carbon atom of the sugar unit of a second nucleotide. This unit joins to a third nucleotide, and the process is repeated to produce a long nucleic acid chain (Figure $$\PageIndex{4}$$). The backbone of the chain consists of alternating phosphate and sugar units (2-deoxyribose in DNA and ribose in RNA). The purine and pyrimidine bases branch off this backbone.

Each phosphate group has one acidic hydrogen atom that is ionized at physiological pH. This is why these compounds are known as nucleic acids.

Like proteins, nucleic acids have a primary structure that is defined as the sequence of their nucleotides. Unlike proteins, which have 20 different kinds of amino acids, there are only 4 different kinds of nucleotides in nucleic acids. For amino acid sequences in proteins, the convention is to write the amino acids in order starting with the N-terminal amino acid. In writing nucleotide sequences for nucleic acids, the convention is to write the nucleotides (usually using the one-letter abbreviations for the bases, shown in Figure $$\PageIndex{4}$$) starting with the nucleotide having a free phosphate group, which is known as the 5′ end, and indicate the nucleotides in order. For DNA, a lowercase d is often written in front of the sequence to indicate that the monomers are deoxyribonucleotides. The final nucleotide has a free OH group on the 3′ carbon atom and is called the 3′ end. The sequence of nucleotides in the DNA segment shown in Figure $$\PageIndex{4}$$ would be written 5′-dG-dT-dA-dC-3′, which is often further abbreviated to dGTAC or just GTAC.

## Secondary Structure of DNA

The three-dimensional structure of DNA was the subject of an intensive research effort in the late 1940s to early 1950s. Initial work revealed that the polymer had a regular repeating structure. In 1950, Erwin Chargaff of Columbia University showed that the molar amount of adenine (A) in DNA was always equal to that of thymine (T). Similarly, he showed that the molar amount of guanine (G) was the same as that of cytosine (C). Chargaff drew no conclusions from his work, but others soon did.

At Cambridge University in 1953, James D. Watson and Francis Crick announced that they had a model for the secondary structure of DNA. Using the information from Chargaff’s experiments (as well as other experiments) and data from the X ray studies of Rosalind Franklin (which involved sophisticated chemistry, physics, and mathematics), Watson and Crick worked with models that were not unlike a child’s construction set and finally concluded that DNA is composed of two nucleic acid chains running antiparallel to one another—that is, side-by-side with the 5′ end of one chain next to the 3′ end of the other. Moreover, as their model showed, the two chains are twisted to form a double helix—a structure that can be compared to a spiral staircase, with the phosphate and sugar groups (the backbone of the nucleic acid polymer) representing the outside edges of the staircase. The purine and pyrimidine bases face the inside of the helix, with guanine always opposite cytosine and adenine always opposite thymine. These specific base pairs, referred to as complementary bases, are the steps, or treads, in our staircase analogy (Figure $$\PageIndex{5}$$).

The structure proposed by Watson and Crick provided clues to the mechanisms by which cells are able to divide into two identical, functioning daughter cells; how genetic data are passed to new generations; and even how proteins are built to required specifications. All these abilities depend on the pairing of complementary bases. Figure $$\PageIndex{6}$$ shows the two sets of base pairs and illustrates two things. First, a pyrimidine is paired with a purine in each case, so that the long dimensions of both pairs are identical (1.08 nm).

If two pyrimidines were paired or two purines were paired, the two pyrimidines would take up less space than a purine and a pyrimidine, and the two purines would take up more space, as illustrated in Figure $$\PageIndex{7}$$. If these pairings were ever to occur, the structure of DNA would be like a staircase made with stairs of different widths. For the two strands of the double helix to fit neatly, a pyrimidine must always be paired with a purine. The second thing you should notice in Figure $$\PageIndex{6}$$ is that the correct pairing enables formation of three instances of hydrogen bonding between guanine and cytosine and two between adenine and thymine. The additive contribution of this hydrogen bonding imparts great stability to the DNA double helix.

We previously stated that deoxyribonucleic acid (DNA) stores genetic information, while ribonucleic acid (RNA) is responsible for transmitting or expressing genetic information by directing the synthesis of thousands of proteins found in living organisms. But how do the nucleic acids perform these functions? Three processes are required: (1) replication, in which new copies of DNA are made; (2) transcription, in which a segment of DNA is used to produce RNA; and (3) translation, in which the information in RNA is translated into a protein sequence.

## DNA: Self-Replication

New cells are continuously forming in the body through the process of cell division. For this to happen, the DNA in a dividing cell must be copied in a process known as replication. The complementary base pairing of the double helix provides a ready model for how genetic replication occurs. If the two chains of the double helix are pulled apart, disrupting the hydrogen bonding between base pairs, each chain can act as a template, or pattern, for the synthesis of a new complementary DNA chain.

The nucleus contains all the necessary enzymes, proteins, and nucleotides required for this synthesis. A short segment of DNA is “unzipped,” so that the two strands in the segment are separated to serve as templates for new DNA. DNA polymerase, an enzyme, recognizes each base in a template strand and matches it to the complementary base in a free nucleotide. The enzyme then catalyzes the formation of an ester bond between the 5′ phosphate group of the nucleotide and the 3′ OH end of the new, growing DNA chain. In this way, each strand of the original DNA molecule is used to produce a duplicate of its former partner (Figure $$\PageIndex{8}$$). Whatever information was encoded in the original DNA double helix is now contained in each replicate helix. When the cell divides, each daughter cell gets one of these replicates and thus all of the information that was originally possessed by the parent cell.

Example $$\PageIndex{1}$$

A segment of one strand from a DNA molecule has the sequence 5′‑TCCATGAGTTGA‑3′. What is the sequence of nucleotides in the opposite, or complementary, DNA chain?

Solution

Knowing that the two strands are antiparallel and that T base pairs with A, while C base pairs with G, the sequence of the complementary strand will be 3′‑AGGTACTCAACT‑5′.

Exercise $$\PageIndex{1}$$

A segment of one strand from a DNA molecule has the sequence 5′‑CCAGTGAATTGCCTAT‑3′. What is the sequence of nucleotides in the opposite, or complementary, DNA chain?

3′‑GGTCACTTAACGGATA‑5′.

## Summary

• Nucleotides are composed of phosphoric acid, a pentose sugar (ribose or deoxyribose), and a nitrogen-containing base (adenine, cytosine, guanine, thymine, or uracil). Ribonucleotides contain ribose, while deoxyribonucleotides contain deoxyribose.
• DNA is the nucleic acid that stores genetic information. RNA is the nucleic acid responsible for using the genetic information in DNA to produce proteins.
• Nucleic acid sequences are written starting with the nucleotide having a free phosphate group (the 5′ end).
• Two DNA strands link together in an antiparallel direction and are twisted to form a double helix. The nitrogenous bases face the inside of the helix. Guanine is always opposite cytosine, and adenine is always opposite thymine.
• In DNA replication, each strand of the original DNA serves as a template for the synthesis of a complementary strand.
• DNA polymerase is the primary enzyme needed for replication.


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Prev Next Index-> contents reference index search external Up-> CppAD ADFun Reverse reverse_two CppAD-> Install Introduction AD ADFun preprocessor multi_thread utility ipopt_solve Example speed Appendix ADFun-> record_adfun drivers Forward Reverse sparsity_pattern sparse_derivative optimize abs_normal FunCheck check_for_nan Reverse-> reverse_one reverse_two reverse_any subgraph_reverse reverse_two-> reverse_two.cpp hes_times_dir.cpp Headings-> Syntax Purpose x^(k) W f w dw ---..First Order Partials ---..Second Order Partials Vector Hessian Times Direction Example

Second Order Reverse Mode

Syntax
dw = f.Reverse(2, w)

Purpose
We use $F : B^n \rightarrow B^m$ to denote the AD function corresponding to f . Reverse mode computes the derivative of the Forward mode Taylor coefficients with respect to the domain variable $x$.

x^(k)
For $k = 0, 1$, the vector $x^{(k)} \in B^n$ is defined as the value of x_k in the previous call (counting this call) of the form       f.Forward(k, x_k)  If there is no previous call with $k = 0$, $x^{(0)}$ is the value of the independent variables when the corresponding AD of Base operation sequence was recorded.

W
The functions $W_0 : B^n \rightarrow B$ and $W_1 : B^n \rightarrow B$ are defined by $$\begin{array}{rcl} W_0 ( u ) & = & w_0 * F_0 ( u ) + \cdots + w_{m-1} * F_{m-1} (u) \\ W_1 ( u ) & = & w_0 * F_0^{(1)} ( u ) * x^{(1)} + \cdots + w_{m-1} * F_{m-1}^{(1)} (u) * x^{(1)} \end{array}$$ This operation computes the derivatives $$\begin{array}{rcl} W_0^{(1)} (u) & = & w_0 * F_0^{(1)} ( u ) + \cdots + w_{m-1} * F_{m-1}^{(1)} (u) \\ W_1^{(1)} (u) & = & w_0 * \left( x^{(1)} \right)^\R{T} * F_0^{(2)} ( u ) + \cdots + w_{m-1} * \left( x^{(1)} \right)^\R{T} F_{m-1}^{(2)} (u) \end{array}$$ at $u = x^{(0)}$.

f
The object f has prototype       const ADFun<Base> f  Before this call to Reverse, the value returned by       f.size_order()  must be greater than or equal two (see size_order ).

w
The argument w has prototype       const Vector &w  (see Vector below) and its size must be equal to m , the dimension of the range space for f .

dw
The result dw has prototype       Vector dw  (see Vector below). It contains both the derivative $W^{(1)} (x)$ and the derivative $U^{(1)} (x)$. The size of dw is equal to $n \times 2$, where $n$ is the dimension of the domain space for f .

First Order Partials
For $j = 0 , \ldots , n - 1$, $$dw [ j * 2 + 0 ] = \D{ W_0 }{ u_j } \left( x^{(0)} \right) = w_0 * \D{ F_0 }{ u_j } \left( x^{(0)} \right) + \cdots + w_{m-1} * \D{ F_{m-1} }{ u_j } \left( x^{(0)} \right)$$ This part of dw contains the same values as are returned by reverse_one .

Second Order Partials
For $j = 0 , \ldots , n - 1$, $$dw [ j * 2 + 1 ] = \D{ W_1 }{ u_j } \left( x^{(0)} \right) = \sum_{\ell=0}^{n-1} x_\ell^{(1)} \left[ w_0 * \DD{ F_0 }{ u_\ell }{ u_j } \left( x^{(0)} \right) + \cdots + w_{m-1} * \DD{ F_{m-1} }{ u_\ell }{ u_j } \left( x^{(0)} \right) \right]$$

Vector
The type Vector must be a SimpleVector class with elements of type Base . The routine CheckSimpleVector will generate an error message if this is not the case.

Hessian Times Direction
Suppose that $w$ is the i-th elementary vector. It follows that for $j = 0, \ldots, n-1$ $$\begin{array}{rcl} dw[ j * 2 + 1 ] & = & w_i \sum_{\ell=0}^{n-1} \DD{F_i}{ u_j }{ u_\ell } \left( x^{(0)} \right) x_\ell^{(1)} \\ & = & \left[ F_i^{(2)} \left( x^{(0)} \right) * x^{(1)} \right]_j \end{array}$$ Thus the vector $( dw[1], dw[3], \ldots , dw[ n * q - 1 ] )$ is equal to the Hessian of $F_i (x)$ times the direction $x^{(1)}$. In the special case where $x^{(1)}$ is the l-th elementary vector , $$dw[ j * 2 + 1 ] = \DD{ F_i }{ x_j }{ x_\ell } \left( x^{(0)} \right)$$

Example
The files reverse_two.cpp and hes_times_dir.cpp contain a examples and tests of reverse mode calculations. They return true if they succeed and false otherwise.
Input File: omh/reverse/reverse_two.omh


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# What is the correct time complexity of the following code

I was wondering what is the correct time complexity expressed in terms of big O on this type of loop:

for(i=1;i<=n;i++)
for(int j=1;j*i<=n;j++)
// O(1) code here

The inner loops will make $$n + \frac{n}{2} + \frac{n}{3} + \dots + \frac{n}{n}$$. I know that this type of formula is $$O(n \log n)$$, so is this the correct time complexity on this piece of code?

$$n+\frac{n}{2}+\cdots +\frac{n}{n}$$

$$= n(1+\frac{1}{2}+\cdots +\frac{1}{n})$$

Now it is a well-known fact that $$1+\frac{1}{2}+\cdots +\frac{1}{n} \le c \log n$$, where $$c$$ is some constant.

$$\le cn(\log n)$$

So the overall Runtime is $$\mathcal{O}{(n \log n)}$$.

The number of times that the body of the inner loop runs is exactly $$T(n) = \sum_{i=1}^n \left\lfloor \frac{n}{i} \right\rfloor.$$ This is sequence A006218 in OEIS, where it is stated that $$T(n) = n(\log n + 2\gamma - 1) + \tilde O(n^{131/416}).$$ The exact magnitude of the error term isn't known, but it cannot be reduced below $$\tilde O(n^{1/4})$$.

For more references, see the Wikipedia article on Dirichlet's divisor problem.


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computer science, math, programming and other stuff
a blog by Christopher Swenson

# Learning Rails

Since I have been learning Ruby lately, I thought it might be fun to pick up Rails. Normally, while doing this, I would be inclined to search around the Internet for some tutorials and documentation, and start digging in.

However, I was not able to find any acceptable tutorials on the Internet. I mean, sure, I found a ton of "Build your own blog" kind of tutorials, but Rails typically just needs more of an explanation, from what I could get. Sure, the tutorial might tell me to insert a line into 004addstuff.rb and run rake migrate (more properly, as I have learned, rake db:migrate), but what the hell does that mean? And so on.

Another problem is that Rails 2.0 appears to be just broken, everywhere. It came out less a month ago, and I cannot get critical parts of it working properly on any box I am running (OS X and Linux). But that doesn't matter, because there's no decent (see previous paragraph) tutorials on it.

So, my current solution is to use my Gentoo Linux box (which currently only supports Rails 1.2, since it also sees 2.0 as totally broken), and I bought Agile Web Development with Rails. Looking through all of the book options, this one best suited what I wanted. It's a bit chatty, but that can be good, since Rails does an awful lot of complicated stuff without you knowing, but you have to know in order to change any of it.

The biggest issue is that the book does not cover Rails 2.0, and came out slightly before Rails 1.2, even though it is only about a year old! I don't hold that against the book, especially since I can only seem to get the older Rails to work on my boxes.

In a nutshell: having the book is about a billion times better than not having one. Though I may still give up on this whole Ruby thing and switch to Python and Pylons at some point.


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J. Korean Ceram. Soc. > Volume 31(10); 1994 > Article
 Journal of the Korean Ceramic Society 1994;31(10): 1176.
 플라즈마 용사된 부분안정화 지르코니아 코팅의 상온 마모거동에 미치는 열처리온도 및 분위기의 영향 김장엽, 임대순 고려대학교 재료공학과 The Effects of the Annealing Temperature and Environments on Room Temperature Wear Behavior of Plasma-Sprayed Partially Stabilized Zirconia Coatings ABSTRACT The sliding wear behavior of the plasma-sprayed zirconia containing 3 mol% yttria was investigated after the annealing at room temperature to 80$0^{circ}C$ in various concentrations of humid atmosphere as laboratory, humid, dry argon atmosphere. Both of the friction coefficient and the wear loss increased with increasing temperature up to 80$0^{circ}C$. Surface morphology of the worn samples changed with annealing temperature. The change of monoclinic/tetragonal (m/t) x-ray peak intensity ratio effected the wear behavior. The m/t ratio had maximum value at 20$0^{circ}C$ and decreased with increased temperature in laboratory and humid atmosphere. In argon atmosphere the m/t ratio had no maximum value and decreased with increasing temperature. At all the annealing temperature humid atmosphere had more the m/t ratio value than any other atmosphere. The change of toughness was showed the inversed result of m/t ratio change. The results indicated that the resudial stress which was induced by the different amount of phase transformation takes a detrimental role in wear behavior.
TOOLS


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Back-End Reductions¶

The reductions listed here are specific to the choice of back end, i.e., solver. Currently, we support two types of back ends: conic solvers and quadratic program solvers. When a problem is solved through the solve() method, CVXPY attempts to find the best back end for your problem. The Dcp2Cone reduction converts DCP-compliant problems into conic form, while the Qp2SymbolicQp converts problems with quadratic, piecewise affine objectives, affine equality constraints, and piecewise-linear inequality constraints into a form that is closer to what is accepted by solvers. The problems output by both reductions must be passed through another sequence of reductions, not documented here, before they are ready for to be solved.

Dcp2Cone¶

class cvxpy.reductions.dcp2cone.dcp2cone.Dcp2Cone(problem=None)[source]

Bases: cvxpy.reductions.canonicalization.Canonicalization

Reduce DCP problems to a conic form.

This reduction takes as input (minimization) DCP problems and converts them into problems with affine objectives and conic constraints whose arguments are affine.

accepts(problem)[source]

A problem is accepted if it is a minimization and is DCP.

apply(problem)[source]

Converts a DCP problem to a conic form.

Qp2SymbolicQp¶

class cvxpy.reductions.qp2quad_form.qp2symbolic_qp.Qp2SymbolicQp(problem=None)[source]

Bases: cvxpy.reductions.canonicalization.Canonicalization

Reduces a quadratic problem to a problem that consists of affine expressions and symbolic quadratic forms.

accepts(problem)[source]

Problems with quadratic, piecewise affine objectives, piecewise-linear constraints inequality constraints, and affine equality constraints are accepted.

apply(problem)[source]

Converts a QP to an even more symbolic form.


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# Category:Definitions/Prime Decompositions

This category contains definitions related to Prime Decompositions.
Related results can be found in Category:Prime Decompositions.

Let $n > 1 \in \Z$.

From the Fundamental Theorem of Arithmetic, $n$ has a unique factorization of the form:

 $\displaystyle n$ $=$ $\displaystyle \prod_{p_i \mathop \divides n} {p_i}^{k_i}$ $\quad$ $\quad$ $\displaystyle$ $=$ $\displaystyle {p_1}^{k_1} {p_2}^{k_2} \cdots {p_r}^{k_r}$ $\quad$ $\quad$

where:

$p_1 < p_2 < \cdots < p_r$ are distinct primes
$k_1, k_2, \ldots, k_r$ are (strictly) positive integers.

This unique expression is known as the prime decomposition of $n$.

## Pages in category "Definitions/Prime Decompositions"

The following 3 pages are in this category, out of 3 total.


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Wednesday, January 21, 2015

5 New High-Level Math Jokes

We here at the Analysisters enjoy puns, and are happy to contribute to the already-gigantic list of math jokes every once in a while. Here are some we've come up with over the years!

1. A topologist walks into $\bar A$. It's closed.

2. (3 variants) Q: If chocolate is a Hilbert space and peanut butter is its dual, why can every element in peanut butter be written as an inner product $\langle y,x \rangle$, where $y$, $x$ are chocolates and $x$ is uniquely fixed?
A: Reese's Representation Theorem.

Q: Why is peanut butter the adjoint of chocolate? Why is chocolate the adjoint of peanut butter?
A: Reese's Representation Theorem.
(Thanks, Paul!)

Q: What's a mathematician's favorite candy?
A: Riesz's Pieces.

3. What did the analyst have for dinner?
Limsup.

4. Your mama isn't Lebesgue integrable because she doesn't vanish at infinity!

5. Eight mathematicians walk into a diner. The first one says they're not hungry, and orders nothing. The next three order a beer, a hamburger, and french fries, respectively. The next three order a burger and fries, fries and a beer, and a burger and a beer, respectively. The last one orders a burger, fries, and a beer.

"I'm sorry, I can't fill your order," she says.

"Why is that?"

"This is only a partial order."

BONUS: Not necessarily in joke format, but I refuse to refer to
$$F_n = \frac{1}{\sqrt{5}}(\phi^n - \bar{\phi}^{n})$$
as anything except "that formula that shoots water up your butt".


text:
Public Group

# Random Numbers

This topic is 5027 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.

## Recommended Posts

Im reading the book beginning C++ proggraming and it does not explain very well how the formula works for creating random numbers, it just tells me that it works. Here is 2 examples: int die = (randomNumber % 6) + 1; // creats number between 1 and 6 theNumber = rand() % 100 + 1; //random number betwwen 1 and 100 I would like to understand why this happens so I can make code that for example creates random numbers between 1 and 33 or 45 and 60. Thaks :)

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Modulo [the % operator] is used for finding the "remainder" of division. 5/3==1 5%3==2.

In more complex terms, the modulo reduces a number to its position on a finite number line. Thus to make a finite number line, the modulo operator is used.

x % 5 means that 0,1,2,3,4 are the only possible results. A finite number line. To move that finite number line, simply add or subtract the offset. (x % 5) + 4 means that 0+4,1+4,2+4,3+4,4+4 are the only results possible [4-8].

Note that (x+4) % 5 will still only result in [0-4] as possible results.

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Thanks for the awsome post, I thought I understood it untill I got these crazy reuslts.

Im trying to make a game were you pick a number 1-100 and then the computer randomly pics a number. Then i want it to say that if the number picked is to high or low. For example if I chose the number 33 and the computer picks 56, I wanted it to make it say, now pick a number between 1-55. This is my code that I have looked at over and over and cant find out why it does not work.

#include <iostream>
#include <cstdlib>
#include <ctime>

using namespace std;
int main()
{
srand(time(0));

int myNumber;
int computerGuess;
int tries = 0;
int base = 100;
char again = 'y';

while (again == 'y')
{

tries = 0;
cout << "\n\nChose A Number For The Computer To Guess from 1-100: ";
cin >> myNumber;

do
{
computerGuess = rand() % base + add;
++tries;

cout << "\n\n" << computerGuess;

if (computerGuess > myNumber)
cout << "\n\nThe Computer Guessed To High And Tried Again";
base = 100;
base = computerGuess - 1;

if (computerGuess < myNumber)
cout << "\n\nThe Computer Guessed To Low And Tried Again";
base = 100;

} while (computerGuess != myNumber);

cout << "\n\nIt Took Your Computer " << tries << " guesses to get the your number";
cout << "\n\nWould you like to try again? (y/n): ";
cin >> again;
}

cout << "\nOkay, CYA!!!";
return 0;
}

// The problem is for some reason when it says... the computerGuess is bigger then myNumber I should pick a higher number. I am getting resluts like this, I pick 44 and it picks 56 then 67 then 72 then 92 then 99 and 100 then it dies... lol I appreiciate your help :)

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I'm a little confused here, sorry. Are you saying you want the player to choose a number and then let the computer try to guess at it?

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The app. crashes because the variable base is getting set to 0. Here is the fatal line:
computerGuess = rand() % base + add;

You never want to mod by 0. There are many ways you can correct this error so I will leave that up to your discretion.

CAB

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Here is what I think you want. If not you can only learn from it:

#include <iostream>#include <cstdlib>#include <ctime>using namespace std;unsigned int gPlayerNum = 0;unsigned int gCompGuesses = 0;unsigned int gatherNum( void );void compGuessNum( void );void results( void );short int again( void );int main( void ){	int answer = 0;	do	{		gPlayerNum = gatherNum( );		compGuessNum( );		results( );		gCompGuesses = 0;		answer = again( );	} while ( answer != 2 );		return 0;}unsigned int gatherNum( void ){	unsigned int temp = 0;	puts( "Welcome. Choose a number between 1 and 10." );	cin >> temp;	return temp;}void compGuessNum( void ){	unsigned int temp = 0;	srand( time( NULL ) );	puts( "Now the computer is going to guess the number you just picked. Please wait..." );	do	{		temp = ( rand( ) % 10 ) + 1;		gCompGuesses += 1;	} while ( temp != gPlayerNum );	return;}void results( void ){	cout << "The computer took " << gCompGuesses << " guess(es) to find out your number (" << gPlayerNum << ")." << endl;	puts( "Have a nice day!" ); // programs should always be nice	return;}short int again( void ){	short int temp = 0;	puts( "Do you want to play again? [1] for yes and [2] for no." );	cin >> temp;	while ( temp != 1 && temp != 2 )	{		puts( "Huh?" );		cin >> temp;	}	return temp;}

I lowered it down to 10 for 100 just took way to long.

[Edited by - V E C T O R on August 21, 2004 9:43:58 PM]

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Scrime show me this code here:
(works great for me, thank you again!)

credits to Scrime!
namespace random {	template<class T>	class TRandom	{	public:		static T MinMax(T minimum, T maximum) {			// do a bit of sanity on the min/max values			if( minimum == maximum ) 				return minimum;			else if( minimum < maximum )				return static_cast<T>( ((static_cast<float>(rand()) / static_cast<float>(RAND_MAX+1)) * (maximum-minimum)) + minimum);			else				return static_cast<T>( ((static_cast<float>(rand()) / static_cast<float>(RAND_MAX+1)) * (minimum-maximum)) + maximum);		}	};}#ifndef RANDOM_FLOAT#define RANDOM_FLOAT(min, max) (random::TRandom<float>::MinMax((min),(max)))#endif#ifndef RANDOM_INT#define RANDOM_INT(min, max) (random::TRandom<int>::MinMax((min),(max)))#endif#ifndef RANDOM_DIE#define RANDOM_DIE(min, max) (random::TRandom<int>::MinMax((min),(max)+1))#endif#ifndef RANDOM_DOUBLE#define RANDOM_DOUBLE(min, max) (random::TRandom<double>::MinMax((min),(max)))#endif

use:
const double rnd_double = RANDOM_DOUBLE( 1.11, 11.11 );const int rnd_int = RANDOM_INT( 1, 100 );

Credits to Scrime! :)

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## ordinal logistic regression likelihood function

December 6, 2020 in Uncategorized

Another critical fact is that the difference of two type-1 extreme-value-distributed variables is a logistic distribution, i.e. Even though income is a continuous variable, its effect on utility is too complex for it to be treated as a single variable. diabetes) in a set of patients, and the explanatory variables might be characteristics of the patients thought to be pertinent (sex, race, age. These factors mayinclude what type of sandwich is ordered (burger or chicken), whether or notfries are also ordered, and age of the consumer. Minitab provides three link functions: logit (the default), normit, and gompit. ... Link function. The gompit function, also known as complementary log-log, is the inverse of the Gompertz distribution function. ) The table of concordant, discordant, and tied pairs is calculated by forming all possible pairs of observations with different response values. 2ologit— Ordered logistic regression ... an underlying score is estimated as a linear function of the independent variables ... log likelihood = -85.908161 Ordered logistic regression Number of obs = 66 LR chi2(1) = 7.97 ) One can also take semi-parametric or non-parametric approaches, e.g., via local-likelihood or nonparametric quasi-likelihood methods, which avoid assumptions of a parametric form for the index function and is robust to the choice of the link function (e.g., probit or logit). Till here, we have learnt to use multinomial regression in R. As mentioned above, if you have prior knowledge of logistic regression, interpreting the results wouldn’t be too difficult. [32] Linear regression assumes homoscedasticity, that the error variance is the same for all values of the criterion. Nevertheless, the Cox and Snell and likelihood ratio R²s show greater agreement with each other than either does with the Nagelkerke R². To this data, one fits a length-p coefficient vector w and a set of thresholds θ1, ..., θK−1 with the property that θ1 < θ2 < ... < θK−1. ) 1. [46] Pearl and Reed first applied the model to the population of the United States, and also initially fitted the curve by making it pass through three points; as with Verhulst, this again yielded poor results. Z 1 , Note that this general formulation is exactly the softmax function as in. We can also interpret the regression coefficients as indicating the strength that the associated factor (i.e. In this article, we discuss the basics of ordinal logistic regression and its implementation in R. Ordinal logistic regression is a widely used classification method, with applications in variety of domains. Select the options that you want. It may be too expensive to do thousands of physicals of healthy people in order to obtain data for only a few diseased individuals. Statistics >Ordinal outcomes >Ordered logistic regression 1. This formulation is common in the theory of discrete choice models and makes it easier to extend to certain more complicated models with multiple, correlated choices, as well as to compare logistic regression to the closely related probit model. π = ... F. S. Pancotto, T. T. Long III, F. E. Harrell, K. L. Lee, M. P. Tyor, and D. B. Pryor. Types of Logistic Regression. machine learning and natural language processing. For example, if a data set includes the factors gender and race and the covariate age, the combination of these predictors may contain as many different covariate patterns as subjects. {\displaystyle \chi _{s-p}^{2},} Many other medical scales used to assess severity of a patient have been developed using logistic regression. The probit model influenced the subsequent development of the logit model and these models competed with each other. In such a model, it is natural to model each possible outcome using a different set of regression coefficients. As a result, the model is nonidentifiable, in that multiple combinations of β0 and β1 will produce the same probabilities for all possible explanatory variables. − Pr {\displaystyle 1-L_{0}^{2/n}} explanatory variable) has in contributing to the utility — or more correctly, the amount by which a unit change in an explanatory variable changes the utility of a given choice. These observations are denoted by y 1, ..., y n, where yi = (y i1, ..., yik ) and Î£ j yij = mi is fixed for each i. Since the logarithm is a monotonic function, any maximum of the likelihood function will also be a maximum of the log likelihood function and vice versa. The link functions allow you to fit a broad class of ordinal response models. This is the approach taken by economists when formulating discrete choice models, because it both provides a theoretically strong foundation and facilitates intuitions about the model, which in turn makes it easy to consider various sorts of extensions. [citation needed] To assess the contribution of individual predictors one can enter the predictors hierarchically, comparing each new model with the previous to determine the contribution of each predictor. Thus, it is necessary to encode only three of the four possibilities as dummy variables. Pearson isn't useful when the number of distinct values of the covariate is approximately equal to the number of observations, but is useful when you have repeated observations at the same covariate level. Two measures of deviance are particularly important in logistic regression: null deviance and model deviance. The likelihood ratio R² is often preferred to the alternatives as it is most analogous to R² in linear regression, is independent of the base rate (both Cox and Snell and Nagelkerke R²s increase as the proportion of cases increase from 0 to 0.5) and varies between 0 and 1. and is preferred over R²CS by Allison. p ∞ [50] The logit model was initially dismissed as inferior to the probit model, but "gradually achieved an equal footing with the logit",[51] particularly between 1960 and 1970. As shown above in the above examples, the explanatory variables may be of any type: real-valued, binary, categorical, etc. The p-value is the probability of obtaining a test statistic that is at least as extreme as the actual calculated value, if the null hypothesis is true. an unobserved random variable) that is distributed as follows: i.e. is the true prevalence and For a predictor with 2 levels x 1 and x 2, the cumulative odds ratio is: The large sample confidence interval for Î²i is: To obtain the confidence interval of the odds ratio, exponentiate the lower and upper limits of the confidence interval. In general, the presentation with latent variables is more common in econometrics and political science, where discrete choice models and utility theory reign, while the "log-linear" formulation here is more common in computer science, e.g. ~ ( = The model is usually put into a more compact form as follows: This makes it possible to write the linear predictor function as follows: using the notation for a dot product between two vectors.


text:
eLibrary of Mathematical Instituteof the Serbian Academy of Sciences and Arts
 > Home / All Journals / Journal /

 Matematički VesnikPublisher: Društvo matematičara Srbije, BeogradISSN: 0025-5165 (Print), 2406-0682 (Online)Issue: 64_4Date: 2012Journal Homepage

 Fixed point theorems for occasionally weakly compatible mappings in Menger spaces 267 - 274 B. D. Pant and Sunny Chauhan
AbstractKeywords: Menger space, occasionally weakly compatible, common fixed point.MSC: 54H25 47H10
 Asymptotic distribution of robust k-nearest neighbour estimator for functional nonparametric models 275 - 285 Mohammed Kadi Attouch and Tawfik Benchikh
AbstractKeywords: Asymptotic distribution, functional data, k-nearest neighbour, robust estimation, small balls probability.MSC: 62G05 62G08 62G20 62G35
 Score lists in bipartite multi hypertournaments 286 - 296 S. Pirzada
AbstractKeywords: Hypertournaments, bipartite hypertournaments, score, losing score.MSC: 05C65
 On $\pi$-images of separable metric spaces and a problem of Shou Lin 297 - 302 Tran Van An and Luong Quoc Tuyen
AbstractKeywords: $cs^*$-network, $cs^*$-cover, $cs$-cover, $sn$-cover, separable metric space, Cauchy $sn$-symmetric, $\sigma$-strong network, compact map, $\pi$-map.MSC: 54C10 54D55 54E40 54E99
 Global smoothness preservation by some nonlinear max-product operators 303 - 315 Lucian Coroianu and Sorin G. Gal
AbstractKeywords: Max-product Bernstein approximation operator, max-product Hermite-Féjer interpolation operator, max-product Lagrange interpolation operator, Chebyshev nodes of the first and second kind, global smoothness preservation.MSC: 41A05 41A20 41A17
 Harmonic starlike functions of complex order involving hypergeometric functions 316 - 325 G. Murugusundaramoorthy
AbstractKeywords: Harmonic functions, harmonic starlike functions, hypergeometric functions, Dziok-Srivastava operator.MSC: 30C45 30C50
 A note on sequence-covering $\pi$-images of metric spaces 326 - 329 Zhaowen Li and Tusheng Xie
AbstractKeywords: Sequence-covering mappings, $\pi$-mappings, $cs$-covers, $sn$-covers, $\sigma$-strong networks, Cauchy $sn$-symmetric spaces.MSC: 54D55 54E40 54E99
 Factorization of weakly compact operators between Banach spaces and Fréchet or (LB)-spaces 330 - 335 José Bonet and J. D. Maitland Wright
AbstractKeywords: Weakly compact operators, reflexive operators, factorization, reflexive Banach spaces, (LB)-spaces, Fréchet spaces.MSC: 46A04 46A03 46A25 46B10 47B07
 Ostrowski inequalities for cosine and sine operator functions 336 - 346 George A. Anastassiou
AbstractKeywords: Ostrowski inequality, Cosine operator function, Sine operator function.MSC: 26D99 47D09 47D99
 Finite dimensions defined by means of $m$-coverings 347 - 360 Vitaly V. Fedorchuk
AbstractKeywords: Dimension, dimension $(m,n)\text{-}x{\rm dim}$, metrizable space, hereditarily normal space.MSC: 54F45

 Remote Address: 54.227.186.112 • Server: elib.mi.sanu.ac.rsHTTP User Agent: CCBot/2.0 (https://commoncrawl.org/faq/)


text:
Algebra and Trigonometry 10th Edition

Quadratic Equation : $-5x^2-4x+1=0$ Use discriminant to determine the number of real solutions. Discriminant = $(-4)^2-4(-5)(1)=36$ which is greater than zero, hence there are 2 unique real solutions.


text:
## A general factorization approach to the extension theory of nonnegative operators and relations.(English)Zbl 1164.47003

Summary: The Kreĭn-von Neumann and the Friedrichs extensions of a nonnegative linear operator or relation (i.e., a multivalued operator) are characterized in terms of factorizations. These factorizations lead to a novel approach to the transversality and equality of the Kreĭn-von Neumann and the Friedrichs extensions and to the notion of positive closability (the Kreĭn-von Neumann operator being an operator). Furthermore, all extremal extensions of a nonnegative operator or relation are characterized in terms of analogous factorizations. This approach for the general case of nonnegative linear relations in a Hilbert space extends the applicability of such factorizations. In fact, the extension theory of densely and nondensely defined nonnegative relations or operators fits in the same framework. In particular, all extremal extensions of a bounded nonnegative operator are characterized.

### MSC:

 47A06 Linear relations (multivalued linear operators) 47A57 Linear operator methods in interpolation, moment and extension problems 47A63 Linear operator inequalities 47B25 Linear symmetric and selfadjoint operators (unbounded) 47A07 Forms (bilinear, sesquilinear, multilinear) 47B65 Positive linear operators and order-bounded operators


text:
# Proving a function is onto

1. Nov 10, 2012

### gottfried

1. The problem statement, all variables and given/known data

Let G be a group and define λg:G→G to be λg(x)=g.x , x$\in$G.

Show that λg is onto and one-to-one.

3. The attempt at a solution
Suppose g.x=g.x' g-1.g.x=g-1.g.x' which means x=x'.

How should I show that the function is onto?

2. Nov 10, 2012

### HallsofIvy

Staff Emeritus
If y is any member of G, then $\lambda(g^{-1}y)= y$


text:
# Vacuum Crystallizer

## Principle

In a vacuum crystallizer, supersaturation is obtained by adiabatic evaporative cooling. When the warm saturated solution is introduced into the crystallizer, due to high vacuum the solution undergoes flashing. A part of the solvent gets evaporated, thereby causing the cooling of the solution. From the resulting supersaturation, crystals are produced.

## Construction

The construction of a vacuum crystallizer is shown in Figure. A vacuum crystallizer is a cylindrical body with a conical bottom. A condenser is attached to the crystallizer with a vacuum pump in between. The bottom of the crystallizer is attached to a discharge pipe. Internally, the body of the crystallizer can be lined with acid-resistant material such as lead or rubber. Two propellers are placed above the discharge pipe to prevent a short circuit of the feed (to the discharge pipe).

## Working

A high vacuum is created using a vacuum pump. The vacuum so created must correspond to the boiling point of the solution. but lower than the feed temperature. A hot saturated solution is fed into the crystallizer at a convenient point. Solution undergoes flashing, which results in evaporation of the solvent. This process is allowed adiabatically so that the crystallizer body is cooled. The resultant cooling causes supersaturation and crystallization. The evaporation of the solvent enhances the yield. Flashing of the solution in the crystallizer leads to ebullition, which keeps the crystals in suspension until they become large enough to fall into the discharge pipe. The propellers mix the contents thoroughly and prevent the contents from reaching the discharge pipe without flashing. With the help of a pump, the product is collected and subjected to filtration or centrifugation to obtain crystals. The filtrate returns to the feed.

Uses: It is suitable for thermolabile substances, due to low-temperature conditions.

Make sure you also check our other amazing Article on : Falling Film Molecular Still


text:
# How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum?

How does the Gordon Decomposition of Dirac Current give rise to spin angular momentum?

I used the Gordon Decomposition to split the Probability Current of the Dirac Field into its orbital current and its spin current. I multiplied the currents by $mc$ to covert them into momentum and then crossed the momentum with position to obtain the orbital angular moment and the spin angular momentum. However the spin angular momentum was twice as large as the accepted value of spin angular momentum. I am stuck on how to get rid of the extra factor of two.

The Gordon Decomposition splits the probability current into two terms \begin{align} mc\langle \gamma^{0n} \rangle = \langle \gamma^{0} i\hbar \partial^{n} \rangle + \tfrac{\hbar}{2}\partial_{j}\langle i\gamma^{0nj} \rangle_{n\neq j} , \end{align} where $\langle \gamma^{0} i\hbar \partial^{n} \rangle$ looks like a orbital momentum current and $\tfrac{\hbar}{2}\partial_{j}\langle i\gamma^{0nj} \rangle_{n\neq j}$ looks like a spin momentum current?

The angular moments of $mc\langle \gamma^{0n} \rangle$ can be computed by crossing then with position to obtain the following: \begin{align} mc \left( x^m\langle \gamma^{0n} \rangle - x^n\langle \gamma^{0m} \rangle \right) = \langle \gamma^{0} i\hbar (x^m\partial^{n}-x^n\partial^{m}) \rangle + \tfrac{\hbar}{2}\partial_{j} \left( x^m\langle i\gamma^{0nj} \rangle_{n\neq j} - x^m\langle i\gamma^{0nj} \rangle_{n\neq j} \right) , \end{align}

Be careful: The spin contribution to the number current is proportional to $\nabla \times S$ where $S$ is the spin density. The spin contribution to the momentum density is $(\nabla\times S)/2$ because the $g=2$ gyromagnetic ratio makes spin twice as effective at contributing to the number (electric) current as to the momentum density. See my contribution at "Gordon Decomposition" for more details.


text:
# In the following question, a matrix of certain characters is given. These characters follow a certain trend, row-wise or column-wise. Find out this trend and choose the missing character accordingly. Option 1) 6Option 2) 7Option 3) 10Option 4) 9Option 5) 8

In this matrix operation occurred column-wise like in first column = $\dpi{80} \frac{(3+5)}{2}=4$
In second column = $\dpi{80} \frac{(6+8)}{2}=7$
In third column = $\dpi{80} \frac{(8+4)}{2}=6$


text:
# End-to-End Example: Using Logistic Regression for predicting Diabetes

March 23, 2018

In this tutorial, we will see how to predict whether a person has diabetes or not, based on information like blood pressure, body mass index (BMI), age, etc.

The data was collected and made available by “National Institute of Diabetes and Digestive and Kidney Diseases” as part of the Pima Indians Diabetes Database. Several constraints were placed on the selection of these instances from a larger database. In particular, all patients here belong to the Pima Indian heritage (subgroup of Native Americans), and are females of ages 21 and above.

# Overview

We will be using Python as our programming language, and making use of some popular python machine learning and data science related packages. First of all, we will import pandas to read our data from a CSV file and manipulate it for further use. We will also use numpy to convert out data into a format suitable to feed our classification model. We’ll use seaborn and matplotlib for visualizations. We will then import Logistic Regression algorithm from sklearn. This algorithm will help us build our classification model. Lastly, we will use joblib available in sklearn to save our model for future use. If you’re missing any packages, you can install them using the pip3 install command, so pip3 install pandas numpy seaborn matplotlib sklearn

import pandas as pd
import numpy as np
import seaborn as sns
import matplotlib.pyplot as plt
from sklearn.linear_model import LogisticRegression
from sklearn.externals import joblib

# Data Description

We have our data saved in a CSV file called diabetes.csv which you can download here. We first read our dataset in a pandas dataframe called diabetesDF, and then use the head() function to show the first five records from our dataset.

diabetesDF = pd.read_csv('diabetes.csv')
print(diabetesDF.head())

First 5 records in the Pima Indians Diabetes Database

The following features have been provided to help us predict whether a person is diabetic or not:

• Pregnancies: Number of times pregnant
• Glucose: Plasma glucose concentration over 2 hours in an oral glucose tolerance test
• BloodPressure: Diastolic blood pressure (mm Hg)
• SkinThickness: Triceps skin fold thickness (mm)
• Insulin: 2-Hour serum insulin (mu U/ml)
• BMI: Body mass index (weight in kg/(height in m)2)
• DiabetesPedigreeFunction: Diabetes pedigree function (a function which scores likelihood of diabetes based on family history)
• Age: Age (years)
• Outcome: Class variable (0 if non-diabetic, 1 if diabetic)

Let’s also make sure that our data is clean (has no null values, etc).

diabetesDF.info() # output shown below
<class 'pandas.core.frame.DataFrame'>
RangeIndex: 768 entries, 0 to 767
Data columns (total 9 columns):
Pregnancies                 768 non-null int64
Glucose                     768 non-null int64
BloodPressure               768 non-null int64
SkinThickness               768 non-null int64
Insulin                     768 non-null int64
BMI                         768 non-null float64
DiabetesPedigreeFunction    768 non-null float64
Age                         768 non-null int64
Outcome                     768 non-null int64
dtypes: float64(2), int64(7)
memory usage: 54.1 KB

Note that the data does have some missing values (see Insulin = 0) in the samples in the previous figure. For the model we will be using, (logistic regression), values of 0 automatically imply that the model will simply be ignoring these values. Ideally we could replace these 0 values with the mean value for that feature, but we’ll skip that for now.

# Data Exploration

Let us now explore our data set to get a feel of what it looks like and get some insights about it.

Let’s start by finding correlation of every pair of features (and the outcome variable), and visualize the correlations using a heatmap.

corr = diabetesDF.corr()
print(corr)
sns.heatmap(corr,
xticklabels=corr.columns,
yticklabels=corr.columns)

Output of feature (and outcome) correlations

Heatmap of feature (and outcome) correlations

In the above heatmap, brighter colors indicate more correlation. As we can see from the table and the heatmap, glucose levels, age, BMI and number of pregnancies all have significant correlation with the outcome variable. Also notice the correlation between pairs of features, like age and pregnancies, or insulin and skin thickness.

Let’s also look at how many people in the dataset are diabetic and how many are not. Below is the barplot of the same:

Barplot visualization of number of non-diabetic (0) and diabetic (1) people in the dataset

It is also helpful to visualize relations between a single variable and the outcome. Below, we’ll see the relation between age and outcome. You can similarly visualize other features. The figure is a plot of the mean age for each of the output classes. We can see that the mean age of people having diabetes is higher.

Average age of non-diabetic and diabetic people in the dataset

# Dataset preparation (splitting and normalization)

When using machine learning algorithms we should always split our data into a training set and test set. (If the number of experiments we are running is large, then we can divide our data into 3 parts, namely - training set, development set and test set). In our case, we will also separate out some data for manual cross checking.

The data set of record consists of 767 patients in total. To train our model we will be using 650 records. We will be using 100 records for testing, and the last 17 records to cross check our model.

dfTrain = diabetesDF[:650]
dfTest = diabetesDF[650:750]
dfCheck = diabetesDF[750:]

Next, we separate the label and features (for both training and test dataset). In addition to that, we will also convert them into NumPy arrays as our machine learning algorithm process data in NumPy array format.

trainLabel = np.asarray(dfTrain['Outcome'])
trainData = np.asarray(dfTrain.drop('Outcome',1))
testLabel = np.asarray(dfTest['Outcome'])
testData = np.asarray(dfTest.drop('Outcome',1))

As the final step before using machine learning, we will normalize our inputs. Machine Learning models often benefit substantially from input normalization. It also makes it easier for us to understand the importance of each feature later, when we’ll be looking at the model weights. We’ll normalize the data such that each variable has a 0 mean and a standard deviation of 1.

means = np.mean(trainData, axis=0)
stds = np.std(trainData, axis=0)
trainData = (trainData - means)/stds
testData = (testData - means)/stds
# np.mean(trainData, axis=0) => check that new means equal 0
# np.std(trainData, axis=0) => check that new stds equal 1

# Training and Evaluating Machine Learning Model

We can now train our classification model. We’ll be using a simple machine learning model called logistic regression. Since the model is readily available in sklearn, the training process is quite easy and we can do it in few lines of code. First, we create an instance called diabetesCheck and then use the fit function to train the model.

diabetesCheck = LogisticRegression()
diabetesCheck.fit(trainData, trainLabel)

Next, we will use our test data to find the accuracy of the model.

accuracy = diabetesCheck.score(testData, testLabel)
print("accuracy = ", accuracy * 100, "%")

The print statement will print accuracy = 78.0 %.

# Interpreting the ML model

To get a better sense of what is going on inside the logistic regression model, we can visualize how our model uses the different features and which features have greater effect.

coeff = list(diabetesCheck.coef_[0])
labels = list(dfTrain.drop('Outcome',1).columns)
features = pd.DataFrame()
features['Features'] = labels
features['importance'] = coeff
features.sort_values(by=['importance'], ascending=True, inplace=True)
features['positive'] = features['importance'] > 0
features.set_index('Features', inplace=True)
features.importance.plot(kind='barh', figsize=(11, 6),color = features.positive.map({True: 'blue', False: 'red'}))
plt.xlabel('Importance')

Visualization of the weights in the Logistic Regression model corresponding to each of the feature variables

From the figure above, we can draw the following conclusions:

1. Glucose level, BMI, pregnancies and diabetes pedigree function have significant influence on the model, specially glucose level and BMI. It is good to see our machine learning model match what we have been hearing from doctors our entire lives!
2. Blood pressure has a negative influence on the prediction, i.e. higher blood pressure is correlated with a person not being diabetic. (also, note that blood pressure is more important as a feature than age, because the magnitude is higher for blood pressure).
3. Although age was more correlated than BMI to the output variables (as we saw during data exploration), the model relies more on BMI. This can happen for several reasons, including the fact that the correlation captured by age is also captured by some other variable, whereas the information captured by BMI is not captured by other variables.

Note that the interpretations above require that our input data be normalized. Without that, we can’t claim that importance is proportional to weights.

# Save Model

Now we will save our trained model for future use using joblib.

joblib.dump([diabetesCheck, means, stds], 'diabeteseModel.pkl')

To check whether we have saved the model properly or not, we will use our test data to check the accuracy of our saved model (we should observe no change in accuracy if we have saved it properly).

diabetesLoadedModel, means, stds = joblib.load('diabeteseModel.pkl')
print("accuracy = ",accuracyModel * 100,"%")

# Predicting using the model

We will now use our unused data to see how predictions can be made. We have our unused data in dfCheck.

print(dfCheck.head())

We will now use the first record to make our prediction.

sampleData = dfCheck[:1]
# prepare sample
sampleDataFeatures = np.asarray(sampleData.drop('Outcome',1))
sampleDataFeatures = (sampleDataFeatures - means)/stds
# predict
print('Probability:', predictionProbability)
print('prediction:', prediction)

From above code we get:

Probability: [[ 0.4385153,  0.5614847]]
prediction: [1]

The first element of array predictionProbability 0.438 is the probability of the class being 0 and second element 0.561 is the probability of the class being 1. The probability sum up to 1. As we can see that the 1 is more probable class, we get [1] as our prediction, which means that the model predicts that the person has diabetes.

# Next steps

There are lots of ways to improve the above model. Here are some ideas:

1. Input feature bucketing should help, i.e. create new variables for blood pressure in a particular range, glucose levels in a particular range, and so on.
2. You could also improve the data cleaning, by replacing 0 values with the mean value.
3. Read a bit about what metrics doctors rely on the most to diagnose a diabetic patient, and create new features accordingly.

See if you can get to 85-90% accuracy. You can get started with the jupyter notebook for this tutorial: pima_indians.ipynb. You can also view the entire code in one place here: Code for End-to-End Example: Using Logistic Regression for predicting Diabetes

The complete notebook with all the cells executed is available via Google Colaboratory using the link above. Google Colab is a free tool that lets you run small Machine Learning experiments through your browser. You should read this 1 min tutorial if you’re unfamiliar with Google Colaboratory. Note that, for this project, you’ll have to upload the dataset diabetes.csv (which you can download here) to Google Colab after saving the notebook to your own system.


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# Some endpoint estimates for bilinear paraproducts and applications

In this talk we will present some endpoint estimates for bilinear paraproducts of Bony’s type. That is, operators of the form $\Pi(f,g)(x)= \int_0^\infty Q_t f(x)\, P_tg(x)\, m(t)\frac{\mathrm{d}t}{t},$ where $P_t$ and $Q_t$ represent frequency localisation operators near the ball $|{\xi}|\lesssim 1/t$ and the annulus $|{\xi}|\thickapprox 1/t$, respectively. More precisely, we will present some new boundedness estimates for bilinear paraproducts operators on local {\rm bmo} spaces.

We will motivate this study by giving some applications to the investigations on the boundedness of bilinear Fourier integral operators and bilinear Coifman-Meyer multipliers.

This is a joint work with W. Staubach (Uppsala University).

This talk is part of the Partial Differential Equations seminar series.