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# Pierre Curie

Pierre Curie (/ˈkjʊəri/;[1] French: [kyʁi]; 15 May 1859 – 19 April 1906) was a French physicist, a pioneer in crystallography, magnetism, piezoelectricity and radioactivity. In 1903, he received the Nobel Prize in Physics with his wife, Marie Skłodowska-Curie, and Henri Becquerel, "in recognition of the extraordinary services they have rendered by their joint researches on the radiation phenomena discovered by Professor Henri Becquerel".[2]

Pierre Curie
Pierre Curie, c. 1906
Born15 May 1859
Paris, France
Died19 April 1906 (aged 46)
Paris, France
Cause of deathFractured skull due to street accident
NationalityFrench
Alma materUniversity of Paris
Curie's law
Curie-Weiss law
Curie constant
Curie temperature
Discovery of piezoelectricity
Spouse(s)Marie Skłodowska-Curie (1867–1934; m. 1895)
ChildrenIrène Joliot-Curie
Ève Curie
AwardsDavy Medal (1903)
Nobel Prize in Physics[a] (1903)
Matteucci Medal (1904)
Elliott Cresson Medal (1909)
Scientific career
FieldsPhysics, Chemistry
Doctoral studentsPaul Langevin
André-Louis Debierne
Marguerite Catherine Perey
Signature

## Early life

Born in Paris on 15 May 1859, Pierre Curie was the son of Eugene Curie (28 August 1827 – 25 February 1910), a doctor of French Huguenot Protestant origin from Alsatia, and Sophie-Claire Depouilly Curie (15 January 1832 – 27 September 1897). He was educated by his father and in his early teens showed a strong aptitude for mathematics and geometry. When he was 16, he earned his math degree. By the age of 18, he had completed the equivalent of a higher degree, but did not proceed immediately to a doctorate due to lack of money. Instead he worked as a laboratory instructor.[3] When Pierre Curie was preparing for his bachelor of science degree, he worked in the laboratory of Jean-Gustave Bourbouze in the Faculty of Science.[4]

Pierre and Marie Skłodowska-Curie, 1895

In 1880 Pierre and his older brother Jacques (1856–1941) demonstrated that an electric potential was generated when crystals were compressed, i.e. piezoelectricity.[5] To aid this work they invented the piezoelectric quartz electrometer.[6] The following year they demonstrated the reverse effect: that crystals could be made to deform when subject to an electric field.[5] Almost all digital electronic circuits now rely on this in the form of crystal oscillators.[7] In subsequent work on magnetism Pierre Curie defined the Curie scale.[8] This work also involved delicate equipment - balances, electrometers, etc.[9]

Pierre Curie was introduced to Maria Skłodowska by their friend, physicist Józef Wierusz-Kowalski.[10] Curie took her into his laboratory as his student. His admiration for her grew when he realized that she would not inhibit his research. He began to regard Skłodowska as his muse.[11] She refused his initial proposal, but finally agreed to marry him on 26 July 1895.[3][12]

It would be a beautiful thing, a thing I dare not hope, if we could spend our life near each other, hypnotized by our dreams: your patriotic dream, our humanitarian dream, and our scientific dream. [Pierre Curie to Maria Skłodowska][3]:117

The Curies had a happy, affectionate marriage, and they were known for their devotion to each other.[13]

## Research

Propriétés magnétiques des corps à diverses temperatures
(Curie's dissertation, 1895)

Prior to his famous doctoral studies on magnetism, he designed and perfected an extremely sensitive torsion balance for measuring magnetic coefficients. Variations on this equipment were commonly used by future workers in that area. Pierre Curie studied ferromagnetism, paramagnetism, and diamagnetism for his doctoral thesis, and discovered the effect of temperature on paramagnetism which is now known as Curie's law. The material constant in Curie's law is known as the Curie constant. He also discovered that ferromagnetic substances exhibited a critical temperature transition, above which the substances lost their ferromagnetic behavior. This is now known as the Curie temperature. The Curie temperature is used to study plate tectonics, treat hypothermia, measure caffeine, and to understand extraterrestrial magnetic fields.[14]

Pierre Curie formulated what is now known as the Curie Dissymmetry Principle: a physical effect cannot have a dissymmetry absent from its efficient cause.[15][16] For example, a random mixture of sand in zero gravity has no dissymmetry (it is isotropic). Introduce a gravitational field, and there is a dissymmetry because of the direction of the field. Then the sand grains can 'self-sort' with the density increasing with depth. But this new arrangement, with the directional arrangement of sand grains, actually reflects the dissymmetry of the gravitational field that causes the separation.

Pierre and Marie Curie in their laboratory

Curie worked with his wife in isolating polonium and radium. They were the first to use the term "radioactivity", and were pioneers in its study. Their work, including Marie Curie's celebrated doctoral work, made use of a sensitive piezoelectric electrometer constructed by Pierre and his brother Jacques Curie.[17] Pierre Curie's 1898 publication with his wife Mme. Curie and also with M. G. Bémont[18] for their discovery of radium and polonium was honored by a Citation for Chemical Breakthrough Award from the Division of History of Chemistry of the American Chemical Society presented to the ESPCI ParisTech (officially the École supérieure de physique et de chimie industrielles de la Ville de Paris) in 2015.[19][20]

Curie and one of his students, Albert Laborde, made the first discovery of nuclear energy, by identifying the continuous emission of heat from radium particles.[21] Curie also investigated the radiation emissions of radioactive substances, and through the use of magnetic fields was able to show that some of the emissions were positively charged, some were negative and some were neutral. These correspond to alpha, beta and gamma radiation.[22]

The curie is a unit of radioactivity (3.7 × 1010 decays per second or 37 gigabecquerels) originally named in honor of Curie by the Radiology Congress in 1910, after his death. Subsequently, there has been some controversy over whether the naming was in honor of Pierre, Marie, or both.[23]

### Spiritualism

In the late nineteenth century, Pierre Curie was investigating the mysteries of ordinary magnetism when he became aware of the spiritualist experiments of other European scientists, such as Charles Richet and Camille Flammarion. Pierre Curie initially thought systematic investigation into the paranormal could help with some unanswered questions about magnetism.[24]:65 He wrote to his fiancée Marie: "I must admit that those spiritual phenomena intensely interest me. I think in them are questions that deal with physics."[24]:66 Pierre Curie's notebooks from this period show he read many books on spiritualism.[24]:68 He did not attend séances such as those of Eusapia Palladino in Paris in 1905–6[24]:238 as a mere spectator, and his goal certainly was not to communicate with spirits. He saw the séances as scientific experiments, tried to monitor different parameters, and took detailed notes of every observation.[24]:247 Despite studying spiritualism, Curie was an atheist.[25]

## Family

Pierre and Marie Curie's daughter, Irène, and their son-in-law, Frédéric Joliot-Curie, were also physicists involved in the study of radioactivity, and each received Nobel prizes for their work as well.[26] The Curies' other daughter, Ève, wrote a noted biography of her mother.[27] She was the only member of the Curie family to not become a physicist. Ève married Henry Richardson Labouisse, Jr., who received a Nobel Peace Prize on behalf of Unicef in 1965.[28][29] Pierre and Marie Curie's granddaughter, Hélène Langevin-Joliot, is a professor of nuclear physics at the University of Paris, and their grandson, Pierre Joliot, who was named after Pierre Curie, is a noted biochemist.[30]

## Death

Tombs of Marie (above) and Pierre Curie at Paris' Panthéon

Pierre Curie died in a street accident in Paris on 19 April 1906. Crossing the busy Rue Dauphine in the rain at the Quai de Conti, he slipped and fell under a heavy horse-drawn cart. He died instantly when one of the wheels ran over his head, fracturing his skull.[31] Statements made by his father and lab assistant imply that Curie's characteristic absent-minded preoccupation with his thoughts contributed to his death.[32]

Both the Curies experienced radium burns, both accidentally and voluntarily,[33] and were exposed to extensive doses of radiation while conducting their research. They experienced radiation sickness and Marie Curie died of aplastic anemia in 1934. Even now, all their papers from the 1890s, even her cookbooks, are too dangerous to touch. Their laboratory books are kept in special lead boxes and people who want to see them have to wear protective clothing.[34] Had Pierre Curie not been killed as he was, it is likely that he would have eventually died of the effects of radiation, as did his wife, their daughter, Irène, and her husband, Frédéric Joliot.[35][36]

In April 1995, Pierre and Marie Curie were moved from their original resting place, a family cemetery, and enshrined in the crypt of the Panthéon in Paris. Marie Curie was the first woman to be honored in this way "for her own merits".[37]

## Awards

1903 Nobel Prize diploma

## Notes

1. ^ Awarded jointly to Pierre Curie and wife Marie Skłodowska-Curie

## References

1. ^ Jones, Daniel (2011). Roach, Peter; Setter, Jane; Esling, John (eds.). Cambridge English Pronouncing Dictionary (18th ed.). Cambridge University Press. ISBN 978-0-521-15253-2.
2. ^ a b "The Nobel Prize in Physics 1903". NobelPrize.org. Retrieved 8 July 2016.
3. ^ a b c d Quinn, Susan (1996). Marie Curie : a life. Reading, Mass.: Addison-Wesley. ISBN 978-0201887945.
4. ^ Marie Curie et les conquérants de l'atome : 1896-2006, par Jean-Pierre Poirier
5. ^ a b "This Month in Physics History: March 1880: The Curie Brothers Discover Piezoelectricity". ACS News. March. 2014. Retrieved 8 July 2016.
6. ^ Molinié, Philippe; Boudia, Soraya (May 2009). "Mastering picocoulombs in the 1890s: The Curies' quartz–electrometer instrumentation, and how it shaped early radioactivity history". Journal of Electrostatics. 67 (2–3): 524–530. doi:10.1016/j.elstat.2009.01.031. Retrieved 8 July 2016.
7. ^ Manbachi, A. and Cobbold R.S.C. (November 2011). "Development and Application of Piezoelectric Materials for Ultrasound Generation and Detection". Ultrasound. 19 (4): 187–196. doi:10.1258/ult.2011.011027. Archived from the original on 22 July 2012.
8. ^ Kürti, N.; Simon, F. (1938). "LXXIII. Remarks on the "Curie" scale of temperature". The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science. 26 (178): 849–854. doi:10.1080/14786443808562176.
9. ^ Nobel Lectures, Physics 1901-1921, Elsevier Publishing Company, Amsterdam, 1967.[1]
10. ^ Redniss, Lauren (2011). Radioactive. New York, New York: HarperCollins. p. 26.
11. ^ Redniss, Lauren (2011). Radioactive. New York, New York: HarperCollins. p. 33.
12. ^ Estreicher, Tadeusz (1938). Curie, Maria ze Skłodowskich (in Polish) (vol. 4 ed.). In Polski słownik biograficzny. p. 111.
13. ^ Goldsmith, Barbara (16 May 2011). Obsessive Genius: The Inner World of Marie Curie (Great Discoveries). W. W. Norton & Company. ISBN 9780393079760.
14. ^ Redniss, Lauren (2011). Radioactive. New York, New York: HarperCollins. p. 30.
15. ^ Castellani, Elena; Ismael, Jenann (16 June 2016). "Which Curie's Principle?" (PDF). Philosophy of Science. doi:10.1086/687933. Retrieved 8 July 2016.
16. ^ Berova, Nina (2000). Circular dichroism : principles and applications. New York, NY: Wiley-VCH. pp. 43–44. ISBN 0471330035. Retrieved 8 July 2016.
17. ^ "Marie and Pierre Curie and the Discovery of Polonium and Radium". Nobelprize.org. 2014.
18. ^ P. Curie, Mme. P. Curie, and M. G. Bémont, Comptes Rendus de l'Académie des Sciences, Paris, 1898 (26 December), vol. 127, pp. 1215-1217.
19. ^ a b "2015 Awardees". American Chemical Society, Division of the History of Chemistry. University of Illinois at Urbana-Champaign School of Chemical Sciences. 2015. Retrieved 1 July 2016.
20. ^ a b "Citation for Chemical Breakthrough Award" (PDF). American Chemical Society, Division of the History of Chemistry. University of Illinois at Urbana-Champaign School of Chemical Sciences. 2015. Retrieved 1 July 2016.
21. ^ Abbott, Steve; Jensen, Carsten; Aaserud, Finn; Kragh, Helge; Rudinger, Erik; Stuewer, Roger H. (July 2000). "Controversy and Consensus: Nuclear Beta Decay 1911-1934". The Mathematical Gazette. 84 (500): 382. doi:10.2307/3621743. Retrieved 9 July 2016.
22. ^ Lagowski, Joseph J. (1997). Macmillan encyclopedia of chemistry. 2. New York: Macmillan Reference USA. p. 1293. ISBN 0028972252.
23. ^ Paul W. Frame. "How the Curie Came to Be". Retrieved 30 April 2008.
24. Hurwic, Anna (1995). Pierre Curie, Translated by Lilananda Dasa and Joseph Cudnik. Paris: Flammarion. ISBN 9782082115629.
25. ^ Warren Allen Smith (2000). Who's who in hell: a handbook and international directory for humanists, freethinkers, naturalists, rationalists, and non-theists. Barricade Books. p. 259. ISBN 9781569801581. Retrieved 4 February 2017. Curie, Pierre (1859—1906) A codiscoverer of radium, Pierre Curie was an atheist.
26. ^ "The Nobel Prize in Chemistry 1935: Frédéric Joliot, Irène Joliot-Curie". Nobel Foundation. 2008. Retrieved 4 September 2008.
27. ^ Curie, Eve (1937). Madame Curie. London: William Heinemann.
28. ^ Fox, Margalit (25 October 2007). "Eve Curie Labouisse, Mother's Biographer, Dies at 102". The New York Times. Retrieved 9 July 2016.
29. ^ Smith, Warren Allen (2000). "Curie, Pierre (1859—1906)". Who's who in hell : handbook and international directory for humanists, freethinkers, rationalists, and non-theists. New York: Barricade Books. p. 259. ISBN 9781569801581.
30. ^ "Hélène Langevin-Joliot: A Granny, More Than a Physicist". Peking University News. 19 May 2014. Retrieved 9 July 2016.
31. ^ "Prof. Curie killed in a Paris street" (PDF), The New York Times, 20 April 1906
32. ^ "Marie Curie - Tragedy and Adjustment (1906-1910)", Marie Curie and the Science of Radioactivity, 2000
33. ^ Mould, R.F. (2007). "Pierre Curie, 1859–1906". Current Oncology. 14 (2): 74–82. doi:10.3747/co.2007.110. PMC 1891197. PMID 17576470.
34. ^ Tasch, Barbara (31 August 2015). "These personal effects of 'the mother of modern physics' will be radioactive for another 1500 years". Business Insider. Retrieved 9 July 2016.
35. ^ Redniss, Lauren (2010). Radioactive : Marie And Pierre Curie : a tale of love and fallout (1st ed.). New York: HarperEntertainment. ISBN 978-0061351327.
36. ^ Bartusiak, Marcia (11 November 2011). ""Radioactive: Marie & Pierre Curie — A Tale of Love and Fallout" by Lauren Redniss". The Washington Post. Retrieved 9 July 2016.
37. ^ "Marie Curie Enshrined in Pantheon". The New York Times. 21 April 1995. Retrieved 9 July 2016.
38. ^ ""Matteucci" Medal". Accademia Nazionale delle Scienza. Archived from the original on 7 March 2016. Retrieved 9 July 2016.


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# Thread: Why are the two paralellograms equal in this construction?

1. ## Why are the two paralellograms equal in this construction?

Triangle ABC is an arbitrary triangle in $E^2$, not necessarily right. Draw parallelograms ABDE and ACFG on the outside of triangle ABC s.t. ABDE meets triangle ABC along the edge AB and ACFG meets triangle ABC along the edge AC. Let P be the point where the lines DE and FG meet. Draw a third parallelogram BCHI on the outside of ABC s.t. the vectors PA = BI.

I have the picture, but I am sorry I dont know how to Latex it to post it here.

My book says the the area of ABDE = area of ABPQ = area JKIB (since they have the same base and height).

Looking at the picture, this makes no sense to me.

what is the height of ABDE?
what is the base of ABDE?
what is the height of ABPQ?
what is the base of ABPQ?
what is the height of JKIB?
what is the base of JKIB?

and why they are equal?

I honestly don't understand why this is true by the picture. Thank you for your help.

2. Originally Posted by Sheld
Triangle ABC is an arbitrary triangle in $E^2$, not necessarily right. Draw parallelograms ABDE and ACFG on the outside of triangle ABC s.t. ABDE meets triangle ABC along the edge AB and ACFG meets triangle ABC along the edge AC. Let P be the point where the lines DE and FG meet. Draw a third parallelogram BCHI on the outside of ABC s.t. the vectors PA = BI.

I have the picture, but I am sorry I dont know how to Latex it to post it here.

My book says the the area of ABDE = area of ABPQ = area JKIB <--- what is J, K, Q? and where?(since they have the same base and height).

Looking at the picture, this makes no sense to me.

what is the height of ABDE?
what is the base of ABDE?
what is the height of ABPQ?
what is the base of ABPQ?
what is the height of JKIB?
what is the base of JKIB?

and why they are equal?

I honestly don't understand why this is true by the picture. Thank you for your help.
Below your editor window you'll find "Additional Options" where you can attach images.

3. I apologize, first the topic title should say equal in area.

Q,R,J,K are defined as follows

Q = DE $\cap$ BI

R = FG $\cap$ CH

J = PA $\cap$ BC

K = PA $\cap$ IH

I will try to draw the picture in paint.

4. Originally Posted by Sheld
Triangle ABC is an arbitrary triangle in $E^2$, not necessarily right. Draw parallelograms ABDE and ACFG on the outside of triangle ABC s.t. ABDE meets triangle ABC along the edge AB and ACFG meets triangle ABC along the edge AC. Let P be the point where the lines DE and FG meet. Draw a third parallelogram BCHI on the outside of ABC s.t. the vectors PA = BI.

I have the picture, but I am sorry I dont know how to Latex it to post it here.

My book says the the area of ABDE = area of ABPQ = area JKIB (since they have the same base and height).

Looking at the picture, this makes no sense to me.

what is the height of ABDE? <--- h
what is the base of ABDE? <--- AB
what is the height of ABPQ? <--- h, because DE is the same straight line as PQ
what is the base of ABPQ? <--- AB
what is the height of JKIB? <--- d
what is the base of JKIB? <--- IB
...
Take PA as base of ABPQ. According to the text of the question $PA \parallel BI$ and PA and BI have the same length. Since IBQ are located on a parallel to PA the bas BI has the same height to JK as the base PA to BQ.
Thus the areas of ABPQ and BIJK must be equal.

I've attached a sketch and I hope now that I understood your description correctly.

5. Thank you so much for your help. I understand now. Sorry I couldn't get the picture to look nice at all in paint. Your posts were very helpful!

format: plain


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# Writing Rationals with the Fundamental Theorem of Arithmetic

The Fundamental Theorem of Arithmetic says that all integers greater than one can be written uniquely as the product of prime numbers. Another way of stating this is that, if $$P = (p_1, p_2, p_3, …)$$ is the (infinite) set of all primes, in order from least to greatest, and $$K = (k_1, k_2, k_3, …)$$ is a set of non-negative integers with the same number of elements, then all positive integers can be expressed in the form $$p_1^{k_1} p_2^{k_2} p_3^{k_3} …$$. The key here is “non-negative”: Since $$6 = 2 \cdot 3$$, for instance, $$k_1 = k_2 = 1$$ while all other elements of $$K$$ are 0. This restatement allows us to include the value of 1 in the Theorem: It is expressed as the set $$K$$ containing all elements of 0.

Because most elements of $$K$$ will be zero, particularly for small numbers, we can use a short-hand. We currently do this for standard place-value: $$517 = (5 \cdot 100) + (1 \cdot 10) + (7 \cdot 1)$$. We don’t bother with the initial zeroes because they’re irrelevant, but also because there would an infinite number of them.

In the new system, each place represents the corresponding element of $$K$$. Since $$P = (2, 3, 5, 7, 11, 13, …)$$, the rightmost digit of a short-hand number is the power of 2, the next digit to the left is the power of 3, and so on as high as needed.

Here are the first ten positive integers: $$0, 1, 10, 2, 100, 11, 1000, 3, 20, 101$$. This is short hand for, respectively, $$2^0, 2^1, 3^12^0, 2^2, 5^13^02^0, 3^12^1, 7^15^03^02^0, 2^3, 3^2, 5^13^02^1$$.

We can modify this system for all positive rational numbers by underlining negative exponents. For instance, $$\frac{1}{2} = \underline{1}$$, while $$\frac{2}{3} = \underline{1}1$$. Note that this eliminates the idea of equivalent fractions: Each rational number would have a unique representation.

We could also modify this system for all non-zero rational numbers by placing a negative sign in front for negative values. We could even modify it for all non-zero complex numbers with rational coefficients.

What we can’t do is represent a value of zero without creating a special symbol. Because of the empty product, $$\emptyset$$ is a representation of one.

Adding values written in this form as if they were in standard place-value notation has the effect of multiplying them. For instance, $$2 + 100 = 102$$ in prime place notation corresponds to $$4 \cdot 5 = 20$$ in decimal notation.

The first number whose prime place notation matches standard decimal is 12. Are there any others?

A challenge to this system comes at $$2^{10} = 1024$$. We could work with this by extending the set of digits (as done in standard hexadecimal representations), but eventually we’ll run out of symbols. For instance, using all upper and lower case letters as well as the ten digits, we run out of characters at $$2^{62} \approx 4.6 \times 10^{18}$$.


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$$\require{cancel}$$

# Special Relativity

The Theory of Special Relativity is a branch of physics dealing with the behavior of inertial frames of reference moving at very different speeds. It does not cover the behavior of non-inertial frames of reference which are instead covered by General Relativity.

It is an axiomatic theory, derived from two main axioms:

1. The laws of physics are the same in all inertial frames (the principle of relativity)
2. The speed of light is determined by the laws of physics

From these it can be concluded that in all inertial frames the speed of light must be measured to be the same constant value. In order to accommodate these several assumptions which seem commonsense must be thrown away. All the derivations can be performed purely through thought experiments combined with calculation

### The major changes of special relativity

Note: it is important for this section to know that

$\gamma=\dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}$

Where $$v$$ is the speed of the object or frame under consideration.

1. Different inertial frames measure the distance between two points to be different; the proper distance and the measured distance will (see length contraction)
2. Different inertial frames measure the time between two events to be different (see Lorentz transformations)
3. Mass and energy are interchangeable through the formula $$E=mc^2$$
4. The overall energy of an object including it's Kinetic is $$E^2=m^2c^4+p^2c^2 =(\gamma mc^2)^2$$
5. The momentum of an object is defined as $$p=\gamma mv^2$$
6. The formula $$F=ma$$ no longer holds, however the alternative $$F=dp/dt$$ still does.


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Please use this identifier to cite or link to this item: http://arks.princeton.edu/ark:/88435/dsp01k930bx22r
 Title: The Effects of Poly(ethylene glycol) Chain Length on the Presentation of Species on the Surface of Poly(styrene-b-ethylene oxide) Nanoparticles Authors: Triana, Brian Prescott Advisors: Prud'homme, Robert K. Contributors: Chen, Christopher Vicente S. Department: Chemical and Biological Engineering Class Year: 2014 Abstract: Nanoparticles (NPs) are increasingly attractive in therapeutic and diagnostic medical applications, such as drug delivery or imaging contrast agents. In addition to enabling multiple methods of delivery (oral and inhalation), NPs can increase bioavailability, allow for controlled release, and reduce potential side effects. Applications in medicine may be enhanced through NP surface modifications, including functional chain ends such as peptides, antibodies, or drugs. Functionalized NPs allow drug aggregation towards specific sites in the body, potentially reducing side effects and drug toxicity. This approach may be particularly effective in targeting tumors, which can cause significant damage to healthy cells in the body. However, functionalized NP effectiveness is dependent on the presentation of molecules attached to chain ends at the surface of NPs, allowing for binding to target surfaces or proteins. Hence, the effectiveness of surface modifications may be reduced by migration or incorporation of functionalized ends to the NP corona or NP core. In order to verify that functionalized nanoparticle surface chains are available, PEG chains were modified with Nile Red and the fluorescence of NP constructed with these chains was measured. Because the emitted wavelengths depend on the local environment of the dye, this technique allows for the determination of chain end migration to the either the NP core or corona. These experiments were able to differentiate between dye aggregation in various NP regions during formation and were used to measure migration of Nile Red after synthesis. Chain length effects on dye accumulation within NPs were determined to inform future work in designing and implementing NPs for more effective medical applications. Extent: 48 pages URI: http://arks.princeton.edu/ark:/88435/dsp01k930bx22r Type of Material: Princeton University Senior Theses Language: en_US Appears in Collections: Chemical and Biological Engineering, 1931-2017

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How can I calculate the visible area of my camera in world space and take zoom into account?

I am trying to get the bounds from an orthographic camera in Unity which take into account the current orthographic size (zoom).

I thought that I could use the cameraToWorldMatrix to transform the camera pixel rectangle into world coordinates.

var m = camera.cameraToWorldMatrix;

var bL = m.MultiplyPoint(new Vector3(camera.pixelRect.xMin, camera.pixelRect.yMin));
var uL = m.MultiplyPoint(new Vector3(camera.pixelRect.xMin, camera.pixelRect.yMax));
var uR = m.MultiplyPoint(new Vector3(camera.pixelRect.xMax, camera.pixelRect.yMax));

var rect = new Rect(uL.x, uL.y, Math.Abs(uR.x - uL.x), Math.Abs(bL.y - uL.y));

However, this does not take into account the orthographic size (zoom).

Multiplying the cameraToWorldMatrix with the camera's projectionMatrix property does not give me the correct results either.

Is there a sensible way that this can be solved?

1 Answer

Turns out that the answer is more simple than I originally thought.

        // Calculate orthographic camera bounds
float height = 2f * camera.orthographicSize;
float width = height * camera.aspect;


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Browse Questions

# What is the area of the plates of 1F parallel capacitor, if the separation between the plates is 1mm?

Can you answer this question?

$(A) 4 mm$


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# From year to year

How many positive integers cannot be expressed in the form $$2015a + 2016b$$ for non-negative integers $$a$$ and $$b$$?

×


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# Can you l2 normalize word2vec vectors for density clustering?

I have a situation where I have to cluster word2vec vectors (200 length dimension vectors on a very large corpus). I decided to use Density based clustering (DBSCAN, HDBSCAN) because my dataset is very high in noise, and I do not want it to be part of my clusters. My knowledge on Cosine distance is limited but I find that Density based clustering algorithms do not have a direct implementation using cosine distance (pairwise_distance calculation is too memory intensive).

My question here is can I normalize the Word2vec vectors using L2 normalization using: norm_data = normalize(vector_array, norm='l2') from the sklearn library to normalize these vectors, and then use euclidean distance over the normalized vectors?

Can someone suggest any other better technique to cluster word vectors when there is noise in the dataset?

• Clustering is problematic in such high dimensions. For visualization purposes I'd use t-SNE on a 3D projection. If the goal is not visualization specify it and maybe we can suggest a workaround. – Emre Jul 26 '17 at 16:39
• What amount of data you are talking about? Have executed a pipeline of word2Vec, clustering and random forest on text corpus data consisting of 40,000 blogs. Took me less than 20 sec was doing it with Spark ML-Lib, one node cluster. You can explore Spark if performance is the only bottleneck here. – vaibhavbarmy Jul 27 '17 at 9:43

1. You can do DBSCAN, OPTICS, HDBSCAN with cosine similarity - they do not expect or require Euclidean distances.

2. Yes, one version of cosine distance corresponds nicely to Euclidean distance of L2 normalized vectors. So there is some theoretical support to L2 normalize the vectors and try Euclidean then.

3. Given how word2vec is trained, I don't think Euclidean nor cosine distance is the right thing to do. Instead, use the raw dot product as a similarity measure (beware that it is not limited to [0;1] but allows negative similarity - values may be too spread out to use them intuitively).

4. t-SNE, which was discussed in the comments and another answer, also relies on distances to the nearest neighbors. So it does suffer from the same problem - cosine, Euclidean, dot product? The fast ELKI implementation does support cosine distance as well as Euclidean, but you appear to be set on Python already. Also, if you have a large vocabulary, t-SNE may give you some scalability trouble. The standard algorithm will need O(n²) memory and time.

Euclidean distances between l2-normalized word vectors is equivalent to the angular distance between the word vectors. This is not the same as cosine distance, but it is very similar. It is worth noting that cosine distance is not actually a metric, and that forces many algorithms, including the ones discussed here, to resort to more expensive O(n^2) asymptotic performance.

With that in mind I have done quite well clustering similar dimension word vectors trained with word2vec using HDBSCAN using exactly such an l2-normalizing approach as you suggest here. In general I did find it beneficial to use the leaf cluster extraction method rather than the traditional excess of mass approach to get tight clusters. I found the results to be quite good, with clusters providing very clean "topics" for the corpus on which word2vec was trained.

Normalizing the vectors maps them all down to a unit sphere. That definitely changes their Euclidean distances (not cosine distances though), so, no I don't think that's valid, nor does it help the dimensionality.

Clustering in high dimensions is indeed problematic and I agree with @Emre that t-SNE is an option to get more meaningful clusters out for visualization purposes. Simple PCA followed by clustering might be effective too.

• You assume that the Euclidean distances are worth preserving, but it seems to be more common to use cosine with word2vec. Also, t-SNE will require a good distance metric on the input data, so it suffers from the same problem. Usually it's applied to Euclidean distances, so the discussed approach may improve t-SNE results unless you use e.g. ELKIs t-SNE which supports cosine. – Has QUIT--Anony-Mousse Aug 3 '17 at 7:02
• The OP says he uses Euclidean distance. – Sean Owen Aug 3 '17 at 11:10
• Not quite, he suggest to use L2 normalization and Euclidean to approximate cosine. – Has QUIT--Anony-Mousse Aug 6 '17 at 13:02


text:
Resistors

Two resistors connected in series give the resulting resistance 216Ω and 48Ω in parallel. Determine the resistance of these resistors.

Result

R1 =  72 Ω
R2 =  144 Ω

Solution:

$\ \\ R_1+R_2 = 216 \ \\ \dfrac{1}{R_1}+\dfrac{1}{R_2} = \dfrac{1}{ 48} \ \\ R_2 = 216 - R_1 \ \\ \ \\ \dfrac{1}{R_1}+\dfrac{1}{ 216 - R_1} = \dfrac{1}{ 48} \ \\ \ \\ R_1^2-216 R_1 + 216\cdot 48 = 0 \ \\ R_1^2-216 R_1 + 10368 = 0 \ \\ (R_1-72)\cdot (R_1-144) =0 \ \\ \ \\ R_1 = 72 \Omega \ \\ R_2 = 144 \Omega \ \\$

Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!

Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):

Be the first to comment!

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text:
# Managing migrations with sqitch

Managing database migrations with sqitch

From Pexels Wendy Wei

I do have one little problem with sqitch. I am so used to typing a letter u after a letter q that I regularly mistype the command as squitch, and then end up staring at the terminal for so long I wonder if I should have typed sqint. Very minor naming issues aside, the project has a website at www.sqitch.org. I’ve found it a simple and specific database migration tool which does as much as I want and no more. It’s written in Perl and available directly from Ubuntu repositories with

sudo apt-get install sqitch

sqitch is designed to work very closely with git. Indeed, as with using git init to create a new repository, executing sqitch init in a git repository will initialize a data managed migration project. This blog post documents a couple of departures I made from the official tutorial. Optional arguments to the init call give the name of the project, a remote git repository address and the database engine of choice. A full call to init would therefore contain the following information:

sqitch init <MY PROJECT> --uri <MY REMOTE GIT REPO> --engine pg

This will generate two local files, sqitch.conf and sqitch.plan. These files need to be kept under version control, along with the sql files that will be generated under three folders, deploy, revert and verify. The manual recommends adding information on the location of the local psql as well as a user name and email address.

sqitch config --user engine.pg.client /usr/bin/psql
sqitch config --user.name 'Paul Hewson'
sqitch config --user.email '<MY EMAIL ADDRESS>

However, this has the effect of adding a “global” config file in ~/.sqitch/sqitch.conf. I find this annoying partly because I like the idea of keeping dotfiles under ~/.config. More importantly, I prefer keeping all config files for a specific project under the folders for that project and hence under the version control for that project. I would rather my project git history knew if I had swapped psql locations at some stage. One change I found I had to make manually was to specify the target databases. This means adding a section to the local sqitch.conf file. The name “official_surveys” becomes a proxy for the full database server specification. The preamble db:pg indicates that this is a PostgreSQL database. This makes is simple to specify test and production databases as necessary. Currently, my project is a research project working with nationally commissioned surveys and I only have one working database. I may in due course decide to run a “production” database in a container.

[target "official_surveys"]

Having set up the project, the first step is to add files locally specifying the DDL required for the first migration. sqitch add with an optional (short) comment sets up the required sql files locally and makes an adjustment to the sqitch.plan.

sqitch add first_migration -n 'My first migration'

This generates files in three folders, namely

deploy/first_migration.sql
revert/first_migration.sql
verify/first_migration.sql

Here’s where the fun starts. The next task is to write into these files all the DDL necessary to set up schema, tables, views and functions in the deploy script, all the corresponding DDL to remove these objects in the revert script. The verify script is intended to provide tests that the database structure is correct. In a complex migration scheme on a live database there are lots of ways this could go wrong. In my case, I am somewhat misusing this by writing anonymous scripts to ensure that data has been loaded correctly into the tables. This is an illustration from the National Travel Survey. Data is supplied in a tab separated value, with the column name HomeIUse_B01ID and the responses are coded as integers, including -8, -9 for missing data values. This is manipulated into an enum in the target table. There are therefore several steps where something could go wrong, and so a number of variables are chosen at random and checks are made that the data has been transformed correctly.

DO $$DECLARE home_internet int; BEGIN home_internet := ( select count(*) from dft.nts_individual_2002_2019 where internet_frequency_home::text = 'Several times a day' ); ASSERT home_internet = 25388; END$$;

Running the command

sqitch verify official_surveys

will execute these functions on the target database. As noted, this is an abuse of the design intentions of sqitch as the command could be executed along with the deploy script as

sqitch deploy --verify official_surveys

would both deploy a migration and verify the subsequent database structure. However, this slight misuse of sqitch serves a purpose at present until I find a way to more closely link data import steps to data verification steps.

Developing a new migration consists of repeatedly cyling through

sqitch deploy official_surveys

and

sqitch revert official_surveys --to @HEAD^ -y

until the migration looks acceptable, fixing any errors which arise. At that stage, I can import the data and verify the migration and data transformation. One point of note is that the --to @HEAD^ reverts the last migration only (i.e., the migration currently being worked on). It is possible to use alternatives such as @HEAD^^ and @HEAD^3 which remove all migrations up to the immediately previous one, and two migrations previous. @ROOT may be a great England cricketer, but reverting --to @ROOT will remove all migrations. It is possible to be more precise; by issuing:

sqitch log official_surveys

we obtain a log of all migrations applied to the target database, with the SHA for each migration and can use the SHA codes to limit the reversion.

Finally, two features I haven’t used yet. At a certain point, it is possible to tag a migration using

sqitch tag v1.0.0 -n 'A useful comment'

and there is also a sqitch bundle command which bundles all the migrations in a bundle folder ready for zipping and distributing. I can’t see my self every using this, as I can only imagine using sqitch from a git repository.

Anyway, I’m finding this an extremely useful utility when developing databases to handle official surveys which are large scale and messy. It is only by working through the surveys you discover features such as a survey boost and then have to make a decision whether to break the table into a main and boost table (because only a smaller sub-sample answered the boost questions and to include these in the main table would include a lot of empty rows). One example of this is the Health Survey for England where there is a specific battery of questions which require input from a nurse and are applied to a smaller subset of the whole. For this reason, the database migration is closely linked to the data import and verification steps. I would really like to split them out, partly to increase my skill level is using Great Expectations a suite of “unit” style tests intended for data rather than code.

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text:
# The Bézout-Theorem and Pascal's Hexagon

This posting contains large animations.

Given a polynomial $p(x,y): \mathbb{R}^2 \to \mathbb{R}$ of degree $3$, a cubic is defined to be the plane curve

$$\{(x,y) \in \mathbb{R}^2 : p(x,y) = 0\}.$$

The same way, we define a quadratic or conic to be a second-order plane curve. Now suppose that we have two cubics, defined by $p_1$ and $p_2$, respectively, and a conic, given by $q$. Then by Bézout’s theorem, the cubics intersect in at most $9$ points. By the same theorem, both cubics intersect with the conic in at most $6$ points. We have $18$ degrees of freedom, of which we spend $12$ to request the two cubics to intersect at exactly $6$ points lying on the conic, and we use the remaining $6$ degrees to force the cubics to intersect in exactly $3$ more points in the plane.

This construction makes sure that $p_1$, $p_2$, and $q$ do not share a common component. We introduce one more degree of freedom by taking a linear combination of $p_1$ and $p_2$, say $\beta p_1 + (1-\beta) p_2$, and we fix this new parameter by claiming the additional condition

$$\beta p_1(x_0,y_0) + (1-\beta) p_2(x_0,y_0) = 0$$

for some $(x_0,y_0)$ satisfying $q(x_0,y_0) = 0$. As an effect, the curve defined by $\beta p_1(x,y) + (1-\beta) p_2(x,y) = 0$ and the conic now intersect in $7$ points, but the linear combination of $p_1$ and $p_2$ still defines a cubic, which, again due to Bézout’s theorem, is only possible if $\beta p_1 + (1-\beta) p_2$ and $q$ share a common component. This component can only be $q$ itself, so that there exists an affine function $l$ such that

$$\beta p_1 + (1-\beta)p_2 = q \cdot l$$

This proves Pascal’s and Pappus’ (here, $q = s \cdot t$ with affine functions $s$ and $t$) theorem at the same time, as it says that the $3$ points outside the conic are collinear, that is, they are located on the same line, the Pascal-line.

The animations below show how $p_1 = 0$ deforms to $p_2 = 0$ for $\beta$ running from $0$ to $1$ while passing through $q\cdot l = 0$.

The so-called 6-point-Pascal consists of 6 distinct points on the conic.

The Pascal-line does not need to pass the conic.

We get a 5-point-Pascal, when two intersection-points, connected via one of the $6$ lines defining $p_1$ and $p_2$, coincide. Then, we have an intersection point with multiplicity $2$, or, in other words, one of the 6 lines is tangential to the conic.

Two of the 6 lines are tangential to the conic:

Three of the 6 lines are tangential to the conic:

Degenerated case: The cubics and the conic all share one linear component. This component is the Pascal-line and it is now tangential to the conic.


text:
Path Homotopy in a Topological Annulus

Let $C_1$ and $C_2$ be simple, closed curves in $\mathbb{R}^2$ such that $C_1$ lies in the region bounded by $C_2$, and the origin $O$ lies in the region bounded by $C_1$. Define an annulus $A$ as the region bounded by and including $C_1$ and $C_2$.

Let $\gamma_0,\gamma_1:[0,1]\rightarrow A$ be paths such that $\gamma_0(0)=\gamma_1(0)$ is a point on $C_1$ and $\gamma_0(1)=\gamma_1(1)$ is a point on $C_2$.

Define winding number $W(\gamma,O)$ about the origin as $$W(\gamma,O)=\frac{\theta(1)-\theta(0)}{2\pi}$$ where $\theta:[0,1] \rightarrow \mathbb{R}$ is the continuous angular coordinate of $\gamma$.

Proposition: $\gamma_0$ and $\gamma_1$ are homotopic $\iff$ $W(\gamma,O)=W(\gamma,O)$.

I have managed to show for the part $\implies$ using the lifting lemma. However, for the converse part, it gets difficult to find the homotopy, especially $C_1$ and $C_2$ can be wobbly curves. I asked people around and they said that the proposition is true, but none gives me the rigorous proof. I even looked in some books but the settings are different, ie for closed paths, circular annulus, or a punctured plane.

I would like to get some ideas, references or proper proof for that from anyone here. Thank you so much in advance.

Edited:

Possible idea: Since $A$ is a doubly connected region, by theorem in conformal theory, there exists a bijective analytic conformal map $\phi: A \rightarrow A^r$ where $A^r$ is a circular annulus (provided $C_1$ and $C_2$ are analytic curves). Then, obtain a homotopy between $A^r$ and $S^1$. Then, find the path homotopy between the images of the transformed curves. We can return to original $A$ and obtain the path homotopy.

Do you think the idea is possibly correct?

• Do you know how to do it for $C_1=C_2$, i.e. loops in the circle? If not, read about the fundamental group of the circle in relation to the group of covering transformations of the universal cover. Once you are done reading, see if the same proof works for the annulus (or use the fact that one deformation retracts to the other). – Moishe Kohan Sep 19 '14 at 10:49


text:
# Use the $\varepsilon$-$\delta$ definition of a limit to prove this.

I know to how prove normal limits using the epsilon-delta definition, say:

$$\lim_{x\to a}f(x) = L$$

But, there was a question on my textbook which I couldn't quite figure out to do, even though I've thought about it for a while I don't even know how to go about starting it.

Use $\varepsilon$-$\delta$ definition of a limit to prove that

$$\lim \limits _{x\to c}f(x) = 0$$ iff $$\lim \limits_{x\to c}|f(x)| = 0$$

Could anyone help me with this, even a hint on where to start? Thank you in advance.

• Do you know that you're required to prove that $$\forall \varepsilon >0\exists\delta >0 \forall x\in D_f(|x-c|<\delta\implies |f(x)-L|<\varepsilon) \\ \Updownarrow \\ \forall \varepsilon >0\exists\delta >0 \forall x\in D_{|f|}(|x-c|<\delta\implies ||f(x)|-L|<\varepsilon)?$$ – Git Gud Sep 25 '13 at 18:23
• yes, i realize by that by the definition, but I've never encountered a situation where two limits are involved.. – Daniel Cook Sep 25 '13 at 18:29
• You assume one of the statements to be true and prove the other. The direction $\Uparrow$ is easier. – Git Gud Sep 25 '13 at 18:32
• Will I later have to prove that my assumption was indeed true? – Daniel Cook Sep 25 '13 at 18:33
• No. What you want to prove is something that looks like $P\iff Q$. To prove something like this, it suffices to prove that $P\implies Q$ and $Q\implies P$. To prove that $P\implies Q$, you assume that $P$ is true and try to concldue that so is $Q$. Similarly for $Q\implies P$. – Git Gud Sep 25 '13 at 18:35

With $L=0$ you wish to prove that $$\forall \varepsilon >0\exists\delta >0 \forall x\in D_f(|x-c|<\delta\implies |f(x)-L|<\varepsilon) \\ \Updownarrow \\ \forall \varepsilon >0\exists\delta >0 \forall x\in D_{|f|}(|x-c|<\delta\implies ||f(x)|-L|<\varepsilon)$$
which is equivalent to proving that $$\forall \varepsilon >0\exists\delta >0 \forall x\in D_f(|x-c|<\delta\implies |f(x)|<\varepsilon) \\ \Updownarrow \\ \forall \varepsilon >0\exists\delta >0 \forall x\in D_{|f|}(|x-c|<\delta\implies |\color{green}|f(x)\color{green}||<\varepsilon).$$
Now you need to prove that $D_f=D_{|f|}$, once this is done your initial problem becomes equivalent to proving $$\forall \varepsilon >0\exists\delta >0 \forall x\in D_f(|x-c|<\delta\implies |f(x)|<\varepsilon) \\ \Updownarrow \\ \forall \varepsilon >0\exists\delta >0 \forall x\in D_f(|x-c|<\delta\implies |\color{green}|f(x)\color{green}||<\varepsilon).$$
Now just prove that for all $x\in D_f$ the equality $|\color{green}|f(x)\color{green}||=|f(x)|$ holds and you get
$$\forall \varepsilon >0\exists\delta >0 \forall x\in D_f(|x-c|<\delta\implies |f(x)|<\varepsilon) \\ \Updownarrow \\ \forall \varepsilon >0\exists\delta >0 \forall x\in D_f(|x-c|<\delta\implies |f(x)|<\varepsilon)$$


text:
# Welcome to Event-B Sequent Prover

The Event-B sequent prover is a plug-in for the Eclipse platform which provides different proof rules to be used within the Rodin Tool. The following sections discuss the issues when extending the Event-B sequent prover.

# Sequent Prover Core

TODO

TODO

TODO

## Reasoners

### Concepts

Reasoners are used in the Event-B sequent prover to generate proof rules. A reasoner is provided with the sequent to prove as well as some extra optional input. The reasoner is implemented by a computer program and it is the job of the developers to ensure that the reasoners either fail or generate logically valid proof rules.

An example of a reasoner is the "conjI" reasoner that generates the conjI proof rules, which splits a conjunctive goal into two sub-goals. Here, assuming that the input sequent is $A, B, C \vdash D \land E$, the conjI reasoner produces the following proof rules.

 $\frac{\quad A, B, C \vdash D ~~~ A, B, C \vdash E\quad}{A, B, C \vdash D \land E}$

A reasoner is added into the Event-B sequent prover using the extension point org.eventb.core.seqprover.reasoners. Below is an example of how to contribute to the extension point.

<extension point="org.eventb.core.seqprover.reasoners">
<reasoner
class="org.eventb.contributors.seqprover.reasoners.Hyp"
id="hyp"
name="%hypName"/>
</extension>

The above declaration defines a reasoner with a specific id (which will be automatically prefixed by the project name, e.g. org.eventb.contributors.seqprover). The name is an externalised string which is used for display purpose.

The class attribute must be a valid Java class name which will be used to create an instance of the reasoner. This class must implement the org.eventb.core.seqprover.IReasoner interface.

However, most of the time, developers only need to extend one of the abstract implementations (depending on the type of the reasoner). For this implementation purpose, the type of the reasoners are specified below:

1. Automatic rewriting reasoners: Reasoners which apply some rewriting rules automatically to simplify the sequent.
2. Automatic inference reasoners: Reasoners which apply an inference rule automatically.
3. Manual rewriters: Reasoners which apply a rewriting rule at a given position.
4. Manual inference: Reasoner which apply an inference rule manually.
5. General reasoners: If the reasoner does not fall into the above categories, clients need to implement the interface IReasoner directly.

Implementing a reasoner is only a first step of contributing to the sequent prover. In order to use a reasoner, the reasoner needs to be wrapped around by a tactic.

### Automatic Rewriting Reasoners

#### Concepts

The reasoners of this type do not have any extra input (except the sequent) and automatically rewrite all occurrences of certain formulas into other ones. An example of this is the rewriter which rewrites according to the following rules:

• $\btrue \limp P \defi P$
• $P \limp \btrue \defi \btrue$
• $\bfalse \limp P \defi \btrue$
• $P \limp \bfalse \defi \lnot P$

#### Implementation

Step by step:

• Create a new plug-in project, e.g. org.eventb.contributor.seqprover.arith
• Add an extension to the point org.eventb.core.seqprover.reasoners, e.g.,
class="org.eventb.core.seqprover.arith.ArithRewrites"
id="org.eventb.core.seqprover.arithmetic"
name="%arithRewritesName"

• Implement the class: Click on the class attribute of the extension, a dialog will appear for new class wizard.
• This class should extend the import org.eventb.internal.core.seqprover.eventbExtensions.rewriters.AbstractAutoRewrites. For the moment, the class is internal within the plug-in and subject to change. Clients can make a copy of this class to develop their own implementation.
An example is as follows:
import org.eventb.core.ast.IFormulaRewriter;
import org.eventb.core.seqprover.SequentProver;

public class ArithRewrites extends AbstractAutoRewrites {
private static final IFormulaRewriter rewriter = new ArithRewriterImpl();

public AutoRewrites() {
super(rewriter, true);
}

public static String REASONER_ID = SequentProver.PLUGIN_ID + ".autoRewrites";

public String getReasonerID() {
return REASONER_ID;
}

@Override
protected String getDisplayName() {
return "simplification rewrites";
}
}

The constructor shall call the super constructor super(IFormulaRewriter, boolean) to specify the formula rewriter to be used, and the reasoner may hide the source of the rewritten hypothesis or not. Clients need to implement two methods:
1. The first one, namely getReasonerID(), returns the string ID of this reasoner. It must be the same ID as the one declared in the XML file (do not forget the project name prefix).
2. The second method is getDisplayName(). It returns the display name of the reasoner, which will be used for display purpose in the UI.
In order to implement the formula rewriter, clients are suggested to use Tom. The instructions are as follows:
1. Create a file named build-tom.xml within your project. The best way to do this is to copy from org.eventb.core.seqprover, and modify this accordingly.
2. Right click on the name of your project and choose "Properties".
3. Choose "Builders", then click on "New" and choose "Ant Build" and click "OK". A dialog will open for the new builder wizard.
4. At the "Main" tab:
1. Enter the name of the builder (e.g., Tom).
2. Buildfile: "Browse Workspace" to your build-tom.xml.
3. Base Directory: "Variables" and choose "build_project".
5. At the "Targets" tab:
1. Auto Build: Click "Set Targets" and choose "tom" from the list.
2. During a "clean": Click "Set Targets" and choose "clean" (unchecked "tom").
You will come back to this builder later on.
• Implementing the rules:
• Create a file called ArithRewriterImpl.t (the name should be consistent with the step above).
• The content of the file should look like this:
package org.eventb.core.seqprover.arith;

import java.util.ArrayList;
import java.util.Collection;
import java.math.BigInteger;

import org.eventb.core.ast.AssociativeExpression;
import org.eventb.core.ast.AssociativePredicate;
import org.eventb.core.ast.AtomicExpression;
import org.eventb.core.ast.BinaryExpression;
import org.eventb.core.ast.BinaryPredicate;
import org.eventb.core.ast.BoolExpression;
import org.eventb.core.ast.BoundIdentDecl;
import org.eventb.core.ast.BoundIdentifier;
import org.eventb.core.ast.DefaultRewriter;
import org.eventb.core.ast.Expression;
import org.eventb.core.ast.Formula;
import org.eventb.core.ast.FormulaFactory;
import org.eventb.core.ast.FreeIdentifier;
import org.eventb.core.ast.Identifier;
import org.eventb.core.ast.IntegerLiteral;
import org.eventb.core.ast.LiteralPredicate;
import org.eventb.core.ast.Predicate;
import org.eventb.core.ast.QuantifiedExpression;
import org.eventb.core.ast.QuantifiedPredicate;
import org.eventb.core.ast.RelationalPredicate;
import org.eventb.core.ast.SetExtension;
import org.eventb.core.ast.SimplePredicate;
import org.eventb.core.ast.UnaryExpression;
import org.eventb.core.ast.UnaryPredicate;
import org.eventb.core.seqprover.eventbExtensions.Lib;

@SuppressWarnings("unused")
public class ArithRewriterImpl extends DefaultRewriter {

public ArithRewriterImpl() {
super(true, FormulaFactory.getDefault());
}
...
}

See for example org.eventb.internal.core.seqprover.eventbExtensions.rewriters.AutoRewriterImpl.t, which implements most of the automatic rewriting rules supplied by the Rodin platform.

#### Testing

The abstract automatic rewriting reasoners have been tested, so clients only need to test if the reasoner has been declared correctly (testing the reasoner ID), and if the formula rewriter used for this reasoner has been implemented correctly.

##### Testing the reasoner ID

This test is based on the abstract class org.eventb.core.seqprover.rewriterTests.AbstractAutomaticReasonerTests. An example is as follows.

import org.eventb.core.seqprover.SequentProver;

public class ArithRewriterReasonerTests extends AbstractAutomaticReasonerTests {

@Override
protected SuccessfulTest[] getSuccessfulTests() {
// No need to test this. This should be guaranteed by testing the
// abstract automatic rewrite reasoner and the formula rewriter itself.
return new SuccessfulTest [] {
};
}

@Override
protected String[] getUnsuccessfulTests() {
// No need to test this. This should be guaranteed by testing the
// abstract automatic rewrite reasoner and the formula rewriter itself.
return new String [] {
};
}

@Override
public String getReasonerID() {
return "org.eventb.core.seqprover.arithmetic";
}
}

In this test, clients only need to provide the reasoner ID under test.

##### Testing the formula rewriter

This test is based on the abstract class org.eventb.core.seqprover.rewriterTests.AbstractFormulaRewriterTests. An example is as follows.

import org.eventb.internal.core.seqprover.eventbExtensions.rewriters.AutoRewriterImpl;
import org.junit.Test;

public class ArithFormulaRewriterTests extends AbstractFormulaRewriterTests {
// The automatic rewriter for testing.
private static final IFormulaRewriter rewriter = new ArithRewriterImpl();

/**
* Constructor.
*
* Create an abstract formula rewriter test with the input is the automatic
* rewriter.
*/
public ArithFormulaRewriterTests() {
super(rewriter);
}

/**
* Tests for rewriting conjunctions.
*/
@Test
public void testRewrite() {
// Test rewriting predicate
predicateTest("⊥", "x = 1 ∧ ⊥");
// Test rewriting expression
expressionTest("1", "card({x + 1})");
}
}

More instructions about how to use the method predicateTest and expressionTest (along with some other utility methods) can be found in the org.eventb.core.seqprover.rewriterTests.AbstractFormulaRewriterTests class.

### Automatic Inference Reasoners

#### Concepts

The reasoners of this type do not have any extra input (except the sequent) and implements an inference rule or a collection of inference rules. For example, a reasoner can implement the following inference rules which deal with conjunction in the hypothesis and in goal.

 $\frac{\quad H, P \vdash Q \quad}{H \vdash P \limp Q}$

#### Implementation

In order to implement a reasoner of this type, clients extend the abstract class org.eventb.core.seqprover.reasonerInputs.EmptyInputReasoner. Examples include org.eventb.internal.core.seqprover.eventbExtensions.ImpI which implements the above proof rule.

TODO

### Manual Rewriting Reasoners

#### Concepts

The reasoners of this type have an input that refers to a predicate of the sequent (either a hypothesis or null if the goal is rewritten) and the position in that predicate where the rewriting occurs. An example of this is the rewriter which rewrites according to the following rules:

TODO

TODO

TODO

### Manual Inference Reasoners

#### Concepts

The reasoners of this type have an input that refers to a predicate of the sequent (either a hypothesis or null if the goal is rewritten) and the position in that predicate where the rule is applied. For example, a reasoner can implement the following inference rules which deal with conjunction in the hypothesis and in goal.

 $\frac{\quad H, P \vdash Q \quad}{H \vdash P \limp Q}$

TODO

TODO

### General Reasoners

#### Concepts

If the reasoner does not fall into the above categories, clients need to implement the interface IReasoner directly. It includes how to serialize/deserialize the reasoner input. More information to follow ...

TODO

TODO

TODO

## Tactics

### Concepts

Tactics provide a way to structure strategic or heuristic knowledge about proof search. They provide control structures to call reasoners or other tactics to make modifications to the proof trees. In the Rodin platform, the users are dealing with tactics, not reasoners.

A tactic can just be the wrapper around a reasoner, but it can be more complicated by combining different tactics.

Tactics are divided into two classes: automatic and manual tactics.

##### Automatic Tactics

A tactic is added into the Event-B sequent prover using the extension point org.eventb.core.seqprover.autoTactics.

 <extension point="org.eventb.core.seqprover.autoTactics">
<autoTactic
class="org.eventb.contributors.seqprover.TrueGoalTac"
description="%trueGoalTacDesc"
id="trueGoalTac"
name="%trueGoalTacName">
</autoTactic>
</extension>

The above declaration defines an automatic tactic with a specific id (which will automatically be prefixed by the project name, e.g. org.eventb.contributors.seqprover). The name attribute is an externalised string used for display purposes. The description attribute is an externalised string that gives the detailed description of the tactic (what it does).

The class attribute must be a valid Java class name which will be used to create an instance of the tactic. This class must implement the org.eventb.core.seqprover.ITactic interface.

For automatic tactics, they must have no input, and most of the time they are just a wrapper around an automatic reasoner. Below is an example of such a tactic.

public static class AutoTac extends AbsractLazilyConstrTactic {
@Override
protected ITactic getSingInstance() {
return BasicTactics.reasonerTac(new AutoReasoner(), EMPTY_INPUT);
}
}

Here, the implementation extends the abstract class org.eventb.core.seqprover.eventbExtensions.AutoTactics#AbstractLazilyConstrTactic. AutoReasoner is the name of the class which implements an automatic reasoner expecting an empty input (See reasoners).

TODO

# Event-B Tactics

POM tactics are tactics that are run by the auto POM. Only automatic tactics (See tactics) can be declared as POM tactics.

POM tactics are added using the extension point org.eventb.core.pomTactic. An example is given below:

<extension point="org.eventb.core.pomTactics">
<tactic id="org.eventb.core.seqprover.trueGoalTac"/>
</extension>

Each tactic element only contains an attribute id, which must be a valid tactic ID (including the project prefix) as declared using the org.eventb.core.seqprover.autoTactics extension point.

Once declared as a POM tactic, the information about the tactic will appear in the preferences for the POM tactic and the users of the platform. However, this tactic will not be a default POM tactic and the users must change the preferences to enable this tactic.

Post tactics are tactics that are run after every manual proof steps in interactive mode. Only automatic tactics (See tactics) can be declared as post tactics.

Post tactics are added using the extension point org.eventb.core.postTactic. An example is as follows:

<extension point="org.eventb.core.postTactics">
<tactic id="org.eventb.core.seqprover.trueGoalTac"/>
</extension>

Each tactic element only contains an attribute id which must be a valid tactic ID (including the project prefix) as declared using the org.eventb.core.seqprover.autoTactics extension point.

Once declared as a post tactic, the information about the tactic will appear in the preferences for the post tactic and the users of the platform. However, this tactic will not be a default post tactic and the users must change the preferences to enable this tactic.

# UI Tactic Providers

Tactic providers allow to define how manual tactics contribute to the User Interface.

TODO


text:
# zbMATH — the first resource for mathematics

Bayesian modeling of the dependence in longitudinal data via partial autocorrelations and marginal variances. (English) Zbl 1277.62088
Summary: Many parameters and positive-definiteness are two major obstacles in estimating and modeling a correlation matrix for longitudinal data. In addition, when longitudinal data are incomplete, incorrectly modeling the correlation matrix often results in bias in estimating the mean regression parameters. We introduce a flexible and parsimonious class of regression models for a covariance matrix parameterized using marginal variances and partial autocorrelations. The partial autocorrelations can freely vary in the interval $$(-1, 1)$$ while maintaining positive definiteness of the correlation matrix, so the regression parameters in these models will have no constraints. We propose a class of priors for the regression coefficients and examine the importance of correctly modeling the correlation structure on estimation of longitudinal (mean) trajectories and the performance of the deviance information criterion (DIC) in choosing the correct correlation model via simulations. The regression approach is illustrated on data from a longitudinal clinical trial.

##### MSC:
 62F15 Bayesian inference 62J12 Generalized linear models (logistic models) 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62H20 Measures of association (correlation, canonical correlation, etc.) 62P10 Applications of statistics to biology and medical sciences; meta analysis 65C60 Computational problems in statistics (MSC2010)
covreg
Full Text:
##### References:
 [1] Agresti, A.; Hitchcock, D. B., Bayesian inference for categorical data analysis, Statistical Methods and Applications, 14, 297-330, (2005) · Zbl 1124.62307 [2] Albert, J. H.; Chib, S., Bayesian analysis of binary and polychotomous response data, Journal of the American Statistical Association, 88, 669-679, (1993) · Zbl 0774.62031 [3] Barnard, J.; McCulloch, R.; Meng, X.l., Modeling covariance matrices in terms of standard deviations and correlations, Statistica Sinica, 10, 1281-1312, (2000) · Zbl 0980.62045 [4] Chiu, T. Y.M.; Leonard, T.; Tsui, K. W., The matrix-logarithmic covariance model, Journal of the American Statistical Association, 91, 198-210, (1996) · Zbl 0870.62043 [5] Clyde, M.; George, E. I., Flexible empirical Bayes estimation for wavelets, Journal of the Royal Statistical Society. Series B (Statistical Methodology), 62, 681-698, (2000) · Zbl 0957.62006 [6] Czado, C., Multivariate regression analysis of panel data with binary outcomes applied to unemployment data, Statistical Papers, 41, 281-304, (2000) · Zbl 1047.62509 [7] Daniels, M. J., Bayesian modeling of several covariance matrices and some results on propriety of the posterior for linear regression with correlated and/or heterogeneous errors, Journal of Multivariate Analysis, 97, 1185-1207, (2006) · Zbl 1089.62025 [8] Daniels, M. J.; Hogan, J. W., Missing data in longitudinal studies: strategies for Bayesian modeling and sensitivity analysis, (2008), Chapman and Hall/CRC · Zbl 1165.62023 [9] Daniels, M. J.; Kass, R. E., Nonconjugate Bayesian estimation of covariance matrices and its use in hierarchical models, Journal of the American Statistical Association, 94, 1254-1263, (1999) · Zbl 1069.62508 [10] Daniels, M. J.; Kass, R. E., Shrinkage estimators for covariance matrices, Biometrics, 57, 1173-1184, (2001) · Zbl 1209.62132 [11] Daniels, M. J.; Normand, S. L., Longitudinal profiling and health care units based on mixed multivariate patient outcomes, Biometrics, 7, 1-15, (2006) · Zbl 1169.62370 [12] Daniels, M. J.; Pourahmadi, M., Bayesian analysis of covariance matrices and dynamic models for longitudinal data, Biometrika, 89, 553-566, (2002) · Zbl 1036.62019 [13] Daniels, M. J.; Pourahmadi, M., Modeling covariance matrices via partial autocorrelations, Journal of Multivariate Analysis, 100, 2352-2363, (2009) · Zbl 1175.62090 [14] Daniels, M. J.; Zhao, Y. D., Modeling the random effects covariance matrix in longitudinal data, Statistics in Medicine, 22, 1631-1647, (2003) [15] Efron, B.; Morris, C., Multivariate empirical Bayes and estimation of covariance matrices, The Annals of Statistics, 4, 22-32, (1976) · Zbl 0322.62041 [16] George, E. I.; Foster, D. P., Calibration and empirical Bayes variable selection, Biometrika, 87, 731-747, (2000) · Zbl 1029.62008 [17] Hoff, P. D.; Niu, X., A covariance regression model, Statistica Sinica, 22, 729-753, (2012) · Zbl 1238.62065 [18] Joe, H., Generating random correlation matrices based on partial correlations, Journal of Multivariate Analysis, 97, 2177-2189, (2006) · Zbl 1112.62055 [19] Lapierre, Y. D.; Nair, N. P.V.; Chouinard, G.; Awad, A. G.; Saxena, B.; Jones, B.; McClure, D. J.; Bakish, D.; Max, P.; Manchanda, R.; Beaudry, P.; BIoom, D.; Rotstein, E.; Ancill, R.; Sandor, P.; Sladen-Dew, N.; Durand, C.; Chandrasena, R.; Horn, E.; Elliot, D.; Das, M.; Ravindran, A.; Matsos, G., A controlled dose-ranging study of remoxipride and haloperidol in schizophrenia — a Canadian multicentre trial, Acta Psychiatrica Scandinavica, 82, 72-77, (1990) [20] Leonard, T.; Hsu, J. S.J., Bayesian inference for a covariance matrix, The Annals of Statistics, 20, 1669-1696, (1992) · Zbl 0765.62031 [21] Liechty, J. C.; Liechty, M. W.; Muller, P., Bayesian correlation estimation, Biometrika, 91, 1-14, (2004) · Zbl 1132.62314 [22] Little, R. J.A.; Rubin, D. B., Statistical analysis with missing data, (2002), John Wiley New York · Zbl 1011.62004 [23] McCullagh, P.; Nelder, J. A., Generalized linear models, (1989), Chapman and Hall · Zbl 0744.62098 [24] Munoz, A.; Carey, V.; Schouten, J. P.; Segal, M.; Rosner, B., A parametric family of correlation structures for the analysis of longitudinal data, Biometrics, 48, 733-742, (1992) [25] Nelsen, R. B., An introduction to copulas, (1999), Springer · Zbl 0909.62052 [26] Pourahmadi, M., Joint mean-covariance models with applications to longitudinal data: unconstrained parameterisation, Biometrika, 86, 677-690, (1999) · Zbl 0949.62066 [27] Pourahmadi, M., Maximum likelihood estimation of generalised linear models for multivariate normal covariance matrix, Biometrika, 87, 425-435, (2000) · Zbl 0954.62091 [28] Pourahmadi, M.; Daniels, M. J., Dynamic conditionally linear mixed models for longitudinal data, Biometrics, 58, 225-231, (2002) · Zbl 1209.62152 [29] Smith, M.; Kohn, R., Parsimonious covariance matrix estimation for longitudinal data, Journal of the American Statistical Association, 97, 1141-1153, (2002) · Zbl 1041.62044 [30] Spiegelhalter, D. J.; Best, N. G.; Carlin, B. P.; Linde, A., Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society. Series B (Statistical Methodology), 64, 583-639, (2002) · Zbl 1067.62010 [31] Verbyla, A. P., Modelling variance heterogeneity: residual maximum likelihood and diagnostics, Journal of the Royal Statistical Society. Series B, 55, 493-508, (1993) · Zbl 0783.62051 [32] Wang, C.; Daniels, M. J., A note on MAR, identifying restrictions, model comparison, and sensitivity analysis in pattern mixture models with and without covariates for incomplete data (correction, vol. 68, p. 994), Biometrics, 67, 810-818, (2011) · Zbl 1272.62096 [33] Wong, F.; Carter, C. K.; Kohn, R., Efficient estimation of covariance selection models, Biometrika, 90, 809-830, (2003) · Zbl 1436.62346 [34] Yang, R.; Berger, J. O., Estimation of a covariance matrix using the reference prior, The Annals of Statistics, 22, 1195-1211, (1994) · Zbl 0819.62013 [35] A. Zellner, On assessing prior distributions and Bayesian regression analysis with $$g$$-prior distributions, in: Bayesian Inference and Decision Techniques: Essays in Honor of Bruno de Finetti, 1986, pp. 233-243.
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text:
# Math Book 2: Rates: Work Time

Pipes Cisterns and Work Time

Pipes cisterns and work time will account for about 1 or 2 questions in the GMAT math section. Questions from this topic appear in both variants - problem solving and data sufficient.

### Concepts Covered

A typical question that appear from this topic in the GMAT quant section is a word problems. The core idea covered in this chapter is similar to the one covered in Speed Time Distance. So, if you have a good grasp of Speed Time Distance, you will be able to understand and solve questions in this topic with a lot of ease. Wizako's GMAT Math Lesson Book in this chapter covers the following concepts:

1. The chapter starts by explaining the relation between the time taken to complete a task (fill a tank) and rate at which the task is completed with illustrative examples. The reciprocal relation between time and fraction of work completed is explained in depth.
2. There are 17 solved examples in questions related to pipes and cisterns. Some of the tough math questions in this topic are featured here.
3. How to frame equations for work time questions when time taken to complete a task is provided?
4. There are 14 solved examples for questions related to work and time. Shorcuts to few hard math questions in work time are provided.
5. 24 exercise problems, 12 each in pipes & cisterns and work & time have been provided with the answer key and explanatory answers.
6. A multiple choice test with 35 GMAT level questions with explanatory answers and answer key is provided in the Math work book.

Here is a typical solved example from this chapter.

### Sample Question

Pipe A fills a tank in 30 minutes. Pipe B can fill the same tank 5 times as fast as pipe A. If both the pipes were kept open when the tank is empty, in how many minutes will the tank be filled?

Pipe B fills the tank 5 times as fast as pipe A.
Therefore, pipe B will fill the tank in one-fifth of the time that pipe A takes.
Pipe B will fill the tank in $\frac{30}{5}$ = 6 minutes.

In 1 minute, pipe A will fill $\frac{1}{30}^{th}$ of the tank and pipe B will fill $\frac{1}{6}^{th}$ of the tank.
Therefore, together, the two pipes will fill $\frac{1}{30}$ + $\frac{1}{6}$ = $\frac{1 + 5}{30}$ = $\frac{6}{30}$ = $\frac{1}{5}^{th}$ of the tank in a minute

Hence, the two pipes working together will take 5 minutes to fill the tank.

### Chapterwise details of Wizako's Math Lesson Books

Math Lesson Book 1

 1 Linear Equations Details 2 Quadratic Equations Details 3 Set Theory Details 4 Sequences & Series Details 5 Number Properties & Theory Details 6 Inequalities Details 7 Functions

Math Lesson Book 2

 1 Descriptive Statistics Details 2 Ratio Proportion Details 3 Mixtures 4 Interest 5 Rates: Speed Distance Details 6 Rates: Races Details 7 Rates: Work Time Details

Math Lesson Book 3

 1 Percents Details 2 Profits Details 3 Permutation Combination Details 4 Probability Details 5 Geometry Details 6 Solid Geometry 7 Coordinate Geometry Details


text:
[XeTeX] dealing with \$ character

Hans Hagen pragma at wxs.nl
Thu Jul 8 10:12:03 CEST 2004

```Jonathan Kew wrote:

> On 7 Jul 2004, at 11:40 pm, Jon Breitenbucher wrote:
>
>> I actually found that \\$ used the cm fonts to create the \$ glyph and
>> not Garamond Pro. That was my reason for starting the post about
>> \textdollar a while ago. I guess it is the same issue as \" and friends.
>>
>
> Well, in plain TeX \\$ is simply defined as \char"24, so it should
> print a dollar sign in any normal font. But LaTeX may do something
> different with it.

hm, the problem is that computer modern si non standard and that in some
cmr fonts there is a pound symbol instead; this is one of the reasons
why font systems in tex are often so complex ... they start with cmr and
then have to handle all kind of exceptions

Hans

-----------------------------------------------------------------


text:
# Exploring a SPARQL endpoint

In this case, semanticweb.org.

In the second edition of my book Learning SPARQL, a new chapter titled “A SPARQL Cookbook” includes a section called “Exploring the Data,” which features useful queries for looking around a dataset that you know little or nothing about. I was recently wondering about the data available at the SPARQL endpoint http://data.semanticweb.org/sparql, so to explore it I put several of the queries from this section of the book to work.

An important lesson here is how easy SPARQL and RDF make it to explore a dataset that you know nothing about. If you don’t know about the properties used, or whether any schema or schemas were used and how much they was used, you can just query for this information. Most hypertext links below will execute the queries they describe using semanticweb.org’s SNORQL interface.

I started with what is generally my favorite query, listing which predicates are used in the data, because that’s the quickest way to get a flavor for what kind of data is available. Several of the predicates that got listed immediately told me some interesting things:

• rdfs:subClassOf shows me that there’s probably some structure worth exploring.

• dcterms:subject (and dc:subject) shows that things have probably been tagged with keywords.

• ical properties such as dtstart shows that events are recorded.

• FOAF properties show that there is probably information about people.

• dcterms:title, swrc:booktitle, dc:title, src:title, and swrc:subtitle show me that works are covered.

An RDF dataset may or may not have explicit structure, and the use of rdfs:subClassOf in this data showed me that there was, so my next query asked what classes were subclasses of what classes so that I could get an overview of how much structure the dataset included. The result showed me that the ontology seemed to be mostly in the swc namespace, which turns out to be the semanticweb.com conference ontology. The site does include nice documentation for this ontology.

The use of the FOAF vocabulary showed me that there are probably people described, but if the properties foaf:name, foaf:lastName, foaf:familyName, foaf:family_name, and foaf:surname are all in there, which should I try first? A quick ego search showed foaf:family_name being used. It also showed that the URI used to represent me is http://data.semanticweb.org/person/bob-ducharme, and because they’ve published this data as linked data, sending a browser to that URL showed that it described me as a member of the 2010 ISWC program committee.

It also showed me to be a proud instance of the foaf:Person class, so I did a query to find out how many persons there were in all: 10,982.

Given the domain of the ontology and the reason that I was listed, I guessed that it was all about ISWC conferences, so I listed the dc:title values to see what would show up. The query took long enough that I added a LIMIT keyword to create a politer version of that query. Looking at the complete data for one work showed all kinds of interesting information, including an swrc:year value to indicate the year of this paper’s conference. A list of all year values showed a range from 2001 right up to 2014, so it’s nice to see that they’re keeping the data up to date.

Next, I listed all papers that mention “SPARQL” in their title, with their years. After listing the number of papers with SPARQL in their title each year, I used sgvizler (which I described here last September) to create the chart of these figures shown above.

The use of dcterms:subject and dc:subject was interesting because these add some pretty classic metadata for navigating content. Listing triples that used either, I included LIMIT 100 to be polite to the server in case these properties were used a lot. They are. Doing this with dc:subject shows subjects such as “ontology alignment” and “controlled natural language” assigned to articles. Doing it with dcterms:subject showed it used more the way I might use rdf:type, indicating that something is an instance of a particular class: for example, swc:Chair and swc:Delegate each have dcterms:subject values of http://dbpedia.org/resource/Role.

My interest in taxonomies (spurred by my work with TopQuadrant’s TopBraid EVN) led me to look harder at the dc:subject values. They’re string values, and not instances of something like skos:Concept, so they have no hierarchical relationship or other metadata themselves. I’m guessing that this is because key phrases assigned to conference papers are more of a folksonomy, in which people can make up their own key phrases as they wish. Either some people must have been aware of other key phrases in use or some were added automatically, because, while counting how many different ones there were came up with 3,594, a query to see which were the most popular showed that “Corpus (creation, annotation, etc.)” was far and away the most used, with 506 papers having that subject.

I could go on. Call me a SPARQL geek, but I really enjoy looking around a data set like this, especially when (as the presence of the papers for ISWC 2014 shows) the data is kept up to date. For people interested in any aspect of semantic web technology, the ability to look around this particular dataset and count up which data falls into which patterns is a great resource.


text:
# Beer–Lambert law from Maxwell equations?

1. Oct 20, 2011

### Gavroy

hi

is it possible to derive the Beer-Lambert law directly from Maxwell's equations? cause i have to derive it and i have only seen some geometrically motivated derivations but i need a proper one.

Last edited: Oct 20, 2011
2. Oct 20, 2011

### chrisbaird

Exponential decay laws (of which the Beer-Lambert law is one example) are not unique to electromagnetics, so Maxwell's equations will not help you. Exponential decay laws are simply the solution to the differential equation where the change in a variable is proportional to the variable (for instance, if the number of donuts I eat every hour is proportional to the number of donuts left, then the total number of donuts as a function of time will decay exponentially).

You could show that plane waves with complex-valued k (which includes absorption) are a solution to Maxwell's equations, and that plane waves with non-zero imaginary part decay exponentially, and therefore obey the Beer-Lambert law. But I would not consider that deriving the law from Maxwell's equation. It's like asking someone to prove 2+2=4 using Maxwell's equations. While number addition is surely obeyed and used in Maxwell's equations, it is a mathematical entity that holds true beyond electromagnetics.


text:
# Compute the amount of cash on the balance sheet at February 2, 2008. 1 answer below »

Cash flow relations. The statement of cash flows for Target Corporation, a U.S. retailer, for the year ended February 2, 2008 (fiscal 2007), showed a net cash inflow from operations of $4,125, a net cash outflow for investing of$6,195, and a net cash inflow for financing of $3,707. The balance sheet at February 3, 2007, showed a balance in cash of$813. Target reports all amounts in millions of U.S. dollars (\$). Compute the amount of cash on the balance sheet at February 2, 2008.