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# zbMATH — the first resource for mathematics

Lifting modular mod $$l$$ representations. (English) Zbl 0809.11025
The article extends previous work of the authors and others. The main result is that once a modulo $$l$$ (odd prime) Galois representation has a modular lift to characteristic zero, then one can find another such lift with prescribed restriction to the decomposition groups at primes $$p\neq l$$. The proof uses a careful analysis of the possible local representations and the Jacquet-Langlands correspondence.
Reviewer: G.Faltings (Bonn)

##### MSC:
 11F11 Holomorphic modular forms of integral weight 11F80 Galois representations
Full Text:
##### References:
 [1] A. Ash and G. Stevens, Modular forms in characteristic $$l$$ and special values of their $$L$$-functions , Duke Math. J. 53 (1986), no. 3, 849-868. · Zbl 0618.10026 · doi:10.1215/S0012-7094-86-05346-9 · euclid:dmj/1077305204 [2] H. Carayol, Sur les représentations galoisiennes modulo $$l$$ attachées aux formes modulaires , Duke Math. J. 59 (1989), no. 3, 785-801. · Zbl 0703.11027 · doi:10.1215/S0012-7094-89-05937-1 [3] Fred Diamond, Congruence primes for cusp forms of weight $$k\geq 2$$ , Astérisque (1991), no. 196-197, 6, 205-213 (1992). · Zbl 0783.11022 [4] F. Diamond and R. Taylor, Non-optimal levels for $$\mod l$$ modular representations of $$\mathrm {Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$$ , to appear in Invent. Math. · Zbl 0847.11025 · doi:10.1007/BF01231768 [5] G. Faltings, Crystalline cohomology and $$p$$-adic Galois-representations , Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25-80. · Zbl 0805.14008 [6] J.-M. Fontaine and G. Laffaille, Construction de représentations $$p$$-adiques , Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 4, 547-608 (1983). · Zbl 0579.14037 · numdam:ASENS_1982_4_15_4_547_0 · eudml:82106 [7] P. Gerardin, Facteurs locaux des algèbres simples de rang $$4$$. I , Reductive groups and automorphic forms, I (Paris, 1976/1977), Publ. Math. Univ. Paris VII, vol. 1, Univ. Paris VII, Paris, 1978, pp. 37-77. [8] H. Jacquet and R. P. Langlands, Automorphic forms on $${\mathrm GL}(2)$$ , Lecture Notes in Mathematics, vol. 114, Springer-Verlag, Berlin, 1970. · Zbl 0236.12010 · doi:10.1007/BFb0058988 [9] K. A. Ribet, Congruence relations between modular forms , Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983), PWN, Warsaw, 1984, pp. 503-514. · Zbl 0575.10024 [10] K. A. Ribet, On modular representations of $${\mathrm Gal}(\overline{\mathbf Q}/{\mathbf Q})$$ arising from modular forms , Invent. Math. 100 (1990), no. 2, 431-476. · Zbl 0773.11039 · doi:10.1007/BF01231195 · eudml:143793 [11] K. A. Ribet, Report on $$\mod l$$ representations of $$\mathrm {Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$$ , to appear in proc. of the motives conference, Seattle, 1991. [12] J.-P. Serre, Sur les représentations modulaires de degré $$2$$ de $${\mathrm Gal}(\overline{\mathbf Q}/{\mathbf Q})$$ , Duke Math. J. 54 (1987), no. 1, 179-230. · Zbl 0641.10026 · doi:10.1215/S0012-7094-87-05413-5
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


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$$\require{cancel}$$

Reflect and research

1. How can a bicycle move forward? Draw the external forces on the bicycle that are required for the wheels to turn.
2. Does conservation of angular momentum play a role in being able to remain upright on a bicyle? If yes, how?
3. How does an anti-lock braking system (ABS) provide better breaking for your car? What is the physics behind this?

To try at home

1. Describe how you can qualitatively confirm conservation of angular momentum.

Reflect and research

1. Propose an experiment to measure the critical angle of an incline, above which a given object cannot roll without slipping, and compare this to a model prediction.
2. Propose an experiment to test the conservation of angular momentum of a rotating object.
3. Propose an experiment to test whether an object with constant velocity can impart angular momentum to another object.

format: plain


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# Permutations Question

Printable View

• June 30th 2008, 05:37 AM
tkexer
Permutations Question
Hi,
Could anyone please explain why the answer of 302400 is derived from the following question:

A class committee consists of 5 boys, 3 girls and 2 teachers. They have to take a group photograph for the school magazine. Find the number of ways they can be arranged in a row such that the 2 teachers are together and no 2 girls are next to each other.

Thank you.
• June 30th 2008, 06:37 AM
Isomorphism
Quote:

Originally Posted by tkexer
Hi,
Could anyone please explain why the answer of 302400 is derived from the following question:

A class committee consists of 5 boys, 3 girls and 2 teachers. They have to take a group photograph for the school magazine. Find the number of ways they can be arranged in a row such that the 2 teachers are together and no 2 girls are next to each other.

Thank you.

Soroban is brilliant at explaining such things. But let me try:

There are some tricks you should know to solve this problem:

Trick 1:

2 people always sit together, demands grouping them. Call them as single entity, that way they ill never be separated in your calculations. So group two teachers together and treat them as a boy.

Modified Problem:

A class committee consists of 6 boys, 3 girls . They have to take a group photograph for the school magazine. Find the number of ways they can be arranged in a row such that no 2 girls are next to each other.

Trick 2:

Whenever you want some set of objects always separated, then its a good idea to fix their position first. Consider the diagram below:

Code:

x1** G1**x2** G2**x3** G3***x4
G1,G2,G3 refer to the girls. This is one particular arrangement(You have 6 such arrangements).

Now lets take the 6 boys and distribute them around these girls.Since the problem does not want the girls to be together, put 1 boy between those girls. This means we can distribute the remaining four boys wherever we please. From the figure, there are x1 boys before G1, x2 boys between G1 and G2, x3 boys between G3 and G4, x4 boys after G4.

Trick 3:

We can always solve such a problem by solving an equivalent problem. Here the number of ways to distribute boys is the same as counting the number of solutions to the equation x1+x2+x3+x4 = 4.

I hope you know to solve such a problem. The number of solutions to this equation is $\binom{4+4-1}{4-1} = \binom{7}{3} = 35$

Finishing touch:

So the number of ways to the modified problem is 35. But that is not the number of arrangements.Remember that the girls can be rearranged in 6 different ways for each of the 35 boys distribution. Thus there are 6 x 35 ways to distribute boys and permute the girls. Notice again that the boys themselves can be permuted. Since there are 6 boys, they can be permuted in 6!=720 ways. So the total number of permutation is 6 x 35 x 720 ways.

Solution to the original problem:

Now lets solve the original problem. The only difference is one of the boys is actually a pair of teachers. And for each arrangement of the modified problem we can swap the teachers to get a new arrangement. Thus we have to multiply the solution by 2.

Final Answer:

So the total number of arrangements is $3! \times \binom{7}{3} \times 6! \times 2! = 302400$
• June 30th 2008, 07:03 AM
Soroban
Hello, tkexer!

Quote:

Could anyone please explain why the answer is 302,400?

A class committee consists of 5 boys, 3 girls and 2 teachers.
They have to take a group photograph for the school magazine.
Find the number of ways they can be arranged in a row so that
the two teachers are together and no two girls are next to each other.

We have ten people: . $B_1, B_2, B_3, B_4, B_5, G_1, G_2, G_3, T_1, T_2$

First, duct-tape the two teachers together . . .
. . Then we have 9 "people: to arrange: . $\boxed{T_1T_2}, B_1, B_2, B_3, B_4, B_5, G_1, G_2, G_3$

Note that the two teachers could be taped in the order $\boxed{T_2T_1}$
. . So there are ${\color{blue}2}$ choices already.

Arrange the five boys and the teacher-unit in a row.
. . There are $6!\:=\:{\color{blue}720}$ possible arrangements.

Take any one of these arrangements and insert "spaces" between the people:
. . $\_\:\boxed{TT}\:\_\:B\:\_\:B\:\_\:B\:\_\:B\:\_\:B\ :\_$

There are 7 spaces; choose 3 of them to place the girls.
. . There are: . $P(7,3) \:=\:7\cdot6\cdot5 \:=\:{\color{blue}210}$ ways.

Therefore, there are: . $2 \times 720 \times 210 \:=\:{\color{red}302,400}$ ways.

• June 30th 2008, 07:59 PM
tkexer
Thank you Isomorphism and Soroban for your help. I now understand the theory behind this question as you guys have explained it in easy to understand detail.

Thanks again,
tkexer


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### SAS FAQ  Why do I get an "Integer Divide by Zero" error when using a SAS data file?

Please Note! The problem described here only applies to Stat/Transfer Version 7 and earlier. This problem has been remedied in Stat/Transfer Version 8 and above.

Say that you used Stat/Transfer to make a SAS data file called c:\mydata\hartman.sas7bdat and you go to use it in a statistics procedure (in our example, proc reg) like below

proc reg data="c:\mydata\hartman";
model y = time cond;
run;

and you get the following error in the log file.

NOTE: PROCEDURE REG used:
real time           0.00 seconds
cpu time            0.00 seconds

ERROR: Integer divide by zero.
NOTE: The SAS System stopped processing this step because of errors.
20   proc reg data="c:\mydata\hartman";
21     model y = time|cond;
22   run;
23

What went wrong?  Sometimes Stat/Transfer may make a SAS file that SAS has trouble reading in a statistical procedure, but there is a workaround.  You can make a copy of the data file in SAS (because SAS can read the file in a data step) and then use the copy.  For example,

data hartman;
set "c:\mydata\hartman";
run;

proc reg data=hartman;
model y = time cond;
run;

and then the log file shows that this worked,

72   data hartman;
73     set "c:\mydata\hartman";
74   run;

NOTE: There were 9 observations read from the data set c:\mydata\hartman.
NOTE: The data set WORK.HARTMAN has 9 observations and 3 variables.
NOTE: DATA statement used:
real time           0.00 seconds
cpu time            0.00 seconds

75
76   proc reg data=hartman;
77     model y = time cond;
78   run;

NOTE: 1 observations have missing values.
NOTE: 8 observations used in computations.
79   quit;

NOTE: PROCEDURE REG used:
real time           0.02 seconds
cpu time            0.02 seconds

Now everything looks good and you can proceed to analyze the copy of your data file.

We wish to thank David Wagstaff of The Methodology Center at Pennsylvania State University for informing us of this error and providing the example data file.

The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.


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0
2013
Impact Factor

29 Peachtree Center Avenue, Science Annex, Suite 400, GA 30303-4106, Atlanta, United States of America
Department of Physics and Astronomy, Georgia State University Atlanta, GA

## Publications:

 Shilnikov A. L., Cymbalyuk G. Homoclinic bifurcations of periodic orbits en a route from tonic spiking to bursting in neuron models 2004, vol. 9, no. 3, pp.  281-297 Abstract The methods of qualitative theory of slow-fast systems applied to biophysically realistic neuron models can describe basic scenarios of how these regimes of activity can be generated and transitions between them can be made. We demonstrate how two different codimension-one bifurcations of a saddle-node periodic orbit with homoclinic orbits can explain transitions between tonic spiking and bursting activities in neuron models following Hodgkin–Huxley formalism. In the first case, we argue that the Lukyanov–Shilnikov bifurcation of a saddle-node periodic orbit with non-central homoclinics is behind the phenomena of bi-stability observed in a model of a leech heart interneuron under defined pharmacological conditions. This model can exhibit two coexisting types of oscillations: tonic spiking and bursting. Moreover, the neuron model can also generate weakly chaotic trains of bursting when a control parameter is close to the bifurcation value. In the second case, the transition is continuous and reversible due to the blue sky catastrophe bifurcation. This bifurcation provides a plausible mechanism for the regulation of the burst duration which may increases with no bound as $1/\sqrt{\alpha-\alpha_0}$, where $\alpha_0$ is the transitional value, while the inter-burst interval remains nearly constant. Citation: Shilnikov A. L., Cymbalyuk G.,  Homoclinic bifurcations of periodic orbits en a route from tonic spiking to bursting in neuron models, Regular and Chaotic Dynamics, 2004, vol. 9, no. 3, pp. 281-297 DOI:10.1070/RD2004v009n03ABEH000281


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# Number System - Natural Numbers (cardinal and ordinal numbers) - $\mathbb{N}$

## About

A natural number $\mathbb{N}$ can be used for two purposes:

• to count (ie describe the size of a set)
• to order (ie or to describe the position of an element in a sequence).
0, 1, 2, 3, 4, ... or 1, 2, 3, 4,

The natural numbers are those used for:

Operation Example Words
counting there are six coins on the table cardinal numbers
ranking this is the third largest city in the country ordinal numbers

## Number systems

N is a number system.

where:

## Representation

### Symbol

In the base 10 numeral system, the symbols for natural numbers are written using ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

### Class

In set theory, natural numbers can be represented by classes of equivalent sets. For instance, the number 3 can be represented as the class of all sets that have exactly three elements.

## Documentation / Reference

Data Science
Data Analysis
Statistics
Data Science
Linear Algebra Mathematics
Trigonometry

Powered by ComboStrap


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#### Term Iconometer

Term definition

An instrument used to measure the distance or size of an object utilizing the image of the object produced by a lens of known focal length. If the distance is being measured, then the object’s size must be known beforehand. If the size is being measured, then the distance from the lens must be known in advance.

Term etymology

from L. illuminationem (nom. illuminatio), from illuminare "to throw into light," from in- "in" (with assimilation of -n- to the following consonant) + lumen (gen. luminis) "light." Meaning "the action of lighting" is from 1563. Illuminate (M.E. enlumyen) originally meant "decorate written material with gold, silver, bright colors; sense of "shining light on" first recorded 1563. (Illumine in this sense is from 1375.)

Term image, if it exists

#### Term Illumination

Term definition

The application of light to something.

Term etymology

from L. illuminationem (nom. illuminatio), from illuminare "to throw into light," from in- "in" (with assimilation of -n- to the following consonant) + lumen (gen. luminis) "light." Meaning "the action of lighting" is from 1563. Illuminate (M.E. enlumyen) originally meant "decorate written material with gold, silver, bright colors; sense of "shining light on" first recorded 1563. (Illumine in this sense is from 1375.)

Term image, if it exists

#### Term Illusion

Term definition

Illusion is a distortion of the senses, that most people sense. Optical illusions deal with the sense of sight. Optical Illusions can use color, light, and patterns to create images that can be deceptive or misleading to our brains. The information gathered by the eye is processed by the brain, which creates a perception that does not match the true image. Perception refers to the interpretation of what we take in through our eyes. Optical illusions occur because our brain is trying to interpret what we see and make sense of the world around us. Optical illusions simply trick our brains into seeing things that may or may not be real.

For more on optical illusions, click on the link to Optical Illusions at the Optics for Kids website.

Term etymology

c.1340, "act of deception," from O.Fr. Sense of "deceptive appearance" developed in Eng. c.1374. Illusionist "conjurer, magic act performer" is from 1850.

Term image, if it exists

#### Term Image

Term definition

The reproduction of an object produced by light rays. If the beam converges to a point, a real image is formed. If the beam diverges, a virtual image is formed located where the divergent rays, if extended back, would converge to.

Term etymology

c.1225, "artificial representation that looks like a person or thing," from O.Fr. image, earlier imagene (11c.), from L. imaginem (nom. imago) "copy, statue, picture, idea, appearance," from stem of imitari "to copy, imitate" (see imitate). Meaning "reflection in a mirror" is c.1315. The mental sense was in L., and appears in Eng. c.1374.

Term image, if it exists

#### Term Impulse

Term definition

The integral of an applied force on an object with respect to time, producing a change in momentum of the object. If the force is constant and applied over a period of time then the impulse is equal to the force multiplied by the time it is applied for.

Term etymology

1432 (implied in impulsion) "an act of impelling, a thrust, push," from L. impulsus "a push against, pressure, shock," also "incitement, instigation," pp. of impellere.

Term image, if it exists

#### Term Incident

Term definition

Falling or shining upon. When light is incident on something, the light may be reflected, refracted, absorbed, or any combination of these.

Term etymology

Term image, if it exists

#### Term Incoherent

Term definition

The lack of a fixed phase relationship between two waves.

Term etymology

Term image, if it exists

#### Term Index of refraction

Term definition

A number signifying how well a material can refract light. Usually specified with the letter “n” by scientists, the index of refraction of a material depends on its composition and density. It is the ratio of the velocity of light in a vacuum to the velocity of light in a refractive material for a given wavelength.

Term etymology

Term image, if it exists

#### Term Infrared (IR)

Term definition

Electromagnetic radiation with a wavelength longer than that of visible light, measured from the nominal edge of visible red light at 0.74 μm (read as micrometer, 1 μm=10-6 m), and extending conventionally to 300 μm. These wavelengths correspond to a frequency range of approximately 1 to 400 THz. (1 THz=1012 Hz).

Term etymology

Term image, if it exists

#### Term Infrared camera

Term definition

A camera that can detect infrared light and produce images.

Term etymology

Term image, if it exists

#### Term Integrating sphere

Term definition

Also known as an Ulbricht sphere. A hollow sphere coated inside with a white diffusing material. It has openings for an incident beam, a specimen, and a detector used for measuring the diffuse reflectance or transmittance by objects.

Term etymology

Term image, if it exists

#### Term Interaction

Term definition

Any action and reaction that takes place between two objects. In physics, interactions (or forces) fall into four main categories. These categories are gravitational interactions, electromagnetic interactions (electric and magnetic forces), and the strong and weak interactions (which affect atomic nuclei and fundamental particles).

Term etymology

Term image, if it exists

#### Term Interface

Term definition

The boundary between two different materials or media.

Term etymology

1962 (n.), 1967 (v.), from inter- + face.

Term image, if it exists

#### Term Interference

Term definition

The result of waves impinging on one another, leading to the additive process where the amplitude of the two or more overlapping waves are systematically attenuated or reinforced. Constructive interference occurs when the waves are nearly in phase or when their “peaks” combine; destructive interference occurs when the waves are nearly 180˚ out of phase or when the “peaks” cancel out the troughs of the waves. (A) Constructive interference showing $\mathrm{sin}\left(x\right)+\mathrm{sin}\left(x\right)=2\text{\hspace{0.17em}}\mathrm{sin}\left(x\right)$ (B) Destructive interference showing .

Term etymology

1440, "to strike against," from M.Fr. enterferer "to strike each other," from entre- "between" + ferir "to strike," from L. ferire "to knock, strike," related to L. forare "to bore, pierce," and cognate with O.E. borian "to bore" (cf. punch (v.), which has both the sense "to hit" and "to make a hole in").

Term image, if it exists

#### Term Interferometer

Term definition

An instrument that employs the interference of lightwaves to measure the accuracy of optical surfaces. It can measure a length in terms of the length of a wave of light.

Term etymology

Term image, if it exists

#### Term Interferometry

Term definition

The study and utilization of interference phenomena.

Term etymology

Term image, if it exists

#### Term Ion

Term definition

An atom or molecule with an overall positive or negative charge.

Term etymology

none

Term image, if it exists

#### Term Ionization

Term definition

The physical process of converting an atom or molecule into an ion by changing the difference between the number of protons and electrons. An ion is an atom or molecule that has lost or gained one or more electrons, making it negatively or positively charged. If an atom loses an electron(s), it becomes a positive ion. If an atom gains an electron(s), it becomes a negative ion.

Term etymology

Term image, if it exists

#### Term Ionize

Term definition

To change the number of electrons of an atom or molecule, causing it to become an ion.

Term etymology

Term image, if it exists

#### Term Iris

Term definition

A mechanical device designated to smoothly vary the effective diameter of a lens by controlling the amount of light allowed through.

Term etymology

From M.E., iris (the plant); from L. iris, irid-, "rainbow, iris (the plant)"; from Gk., rainbow, brightly-colored gemstone, iris of the eye. It regulates the amount of light entering the eye.

Term image, if it exists


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Multiple Sequence Alignment Python
0
0
Entering edit mode
3.0 years ago

Hi, Currently I am working on classifying coding/non coding RNAs based on sequence features. I want to try MSA score as a feature for sequence classification.

• Please help me to get MSA score (using Clustlw) for a collection of sequences, using Biopython.
• How can I a get a profile sequence after MSA of training sequences?
• Also suggest other features that can be derived from MSA
Biopython Multiple Sequence Alignment ClustlW • 1.9k views
0
Entering edit mode

What have you tried so far?

0
Entering edit mode

I have used k-mer (k-length substring) frequencies, GC content, Molecular weight ...,. I would like to include features from MSA (some variance from profile sequence) to this feature set.

1
Entering edit mode

There is a Clustal module for python I believe, though I've never personally used it. There should be a way to get the scores back out of that (though I can't help you with the specifics). In the past, I've simply parsed the STOUT output of running a commandline invocation of Clustal. It prints the pairwise alignment scores for all the sequences so you could run with that.

For building a profile sequence, you can pass your MSA through hmmer to get an HMM of the sequence.

As for other features, I can't think of anything specifically that would be valid for all sequences beyond their pairwise agreements. Others may know more...


text:
The advanced assessment editor uses a simple, text-based format to create questions. Edit questions as you would on paper. See the following multiple-choice question where the correct answer is 'c':

Type: MC
1) What is the capital of Sweden?
a. Oslo
b. Helsinki
*c. Stockholm

In addition to the question text, questions can also have properties associated with them. The properties appear above the question number. The first property must be the question type. See Reference section below for supported question types.

We describe other properties in the sections that follow.

Preview Questions

The editor includes a 'Preview Questions' button that you can click to preview the exam questions as the student would see them. For your convenience the correct answer is indicated in multiple-choice and multiple-answer questions. For fill-in-the-blank questions, you can click a 'Check Answer' button to evaluate your answer. 'Check Answer' is also helpful for questions that use math or regular expressions.

You can paste the examples from this help file into the editor and click 'Preview Questions' to see the questions. You can selectively preview questions by selecting the question text in the editor before clicking 'Preview Questions.'

Import Questions

You can import questions from other formats (including QTI) into the editor by clicking 'Import Questions' and specifying an import file. After uploading the file, the editor inserts the new questions after any existing questions. You can also import QTI .zip files that contain questions with associated resources, such as images. The import stores resources in the course's Resources folder, and the questions refer to the course resources. Note that any existing course resource with the same name is overwritten. After importing, you can validate, preview, and edit the questions before saving.

Formatting

Generally, question text is plain, unformatted text with optional special codes that indicate question structures and advanced formatting. We describe the special codes below.

Fill-in-the-blank and inline choices

You define the blank in a fill-in-the-blank question by entering a minimum of three sequential underscore (___) characters in the question text. To make the blank longer, specify more than three underscores. Each underscore widens the blank by one character width.

Type: F
1) The capital of Sweden, __________, is also known as Venice of the North.
a. Stockholm

You define the drop-down list location in "inline" multiple-choice and matching questions by entering four bracket-enclosed hyphens: [----]. The assessment player automatically determines the correct display width for the drop-down list.

Type: MC
Options: Inline
1) Michigan is located in the [----] region of the country.
a. east
b. south
*c. midwest
d. southeast
e. west
Equation

You enter equations in either ASCII Math or LaTeX format. Enclose ASCII Math equations in back quotes (), and enclose LaTeX equations in back quote and dollar ().

Type: F
1) The chemical H_2O is commonly called ______.
a. water
Media

You can include course media files such as video, images, and audio in your question with the bracket-enclosed [Media] tag. (Course media files are stored in the course media library.) You can double click on a resource in the Media Library list to insert the Media tag for that resource.

Type: MC
1) Which of these is the title song for a recent musical?
*a). [Media: mammamia.mp3]
b). [Media: dancingqueen.mp3]

The following media types are supported:

 Type Extensions Description Video .flv Adobe Flash Audio .mp3 MPEG Audio Stream, Layer III Image .gif,.jpg,.jpeg,.jfif,.png Graphic Interchange Format, JPEG/JIFF Image, and Portable Network Graphic

A media tag can have optional title and description text specified. If the text contains a comma, bracket, quote or backslash character you have to enclose the text in quotes and escape the quote and backslash with an extra backslash.

[Media: filename, title, description]
[Media: sunset.jpg]
[Media: sunset.jpg, Sunset]
[Media: sunset.jpg, Sunset, Picture displaying the sun setting]
[Media: sunset.jpg, "A \"photo\" of a sunset"]
HTML

You can specify advanced formatting by enclosing an entire question text with the tags [HTML][/HTML]. Between the tags, you can use standard HTML formatting tags. Note that you must enclose the entire question text, not just part of it.

Type: F
Options: Number, Multiple
4) [HTML]
Fill in the table

2	+	7	=	___

3	+	___	=	8

___	+	2	=	6

[/HTML]
a. 9
b. 5
c. 4
Feedback

You specify general question feedback by starting a line with the @ sign immediately after the question body. Feedback text can contain $-enclosed question variables. Type: F 1) The capital of Sweden is _________. @ The capital of Sweden is Stockholm a. Stockholm For multiple-choice questions, create choice-specific feedback (feedback shown when the choice is chosen by the student) by placing the feedback line below the choice. Type: MC 1) What is the capital of Sweden? *a. Stockholm @ That's correct b. Oslo @ Oslo is the capital of Norway c. Helsinki @ Helsinki is the capital of Finland Feedback can be displayed only if a condition is met. The condition is a bracket-enclosed expression that can contain question variables, literals, and the pre-defined variables named score, teacher, answer, and answer# (for multiple-answer questions). When multiple feedback lines exist, only feedback from the first satisified condition is displayed. Type: F Options: Number Var: a = 3..9 2) 2 *$a$= _____ @[teacher] This feedback is displayed only to teachers @[score>=1] This feedback is displayed only when the student's question score is greater than or equal to 1 @[answer="2+$a$"] This feedback is displayed when the student added the numbers instead of multiplying them @ This feedback is displayed only if no other feedback has been displayed. a. 2*$a$ To save some typing, you can specify the shortcut "=" instead of "answer=" in the feedback condition. The following two questions are functionally equivalent. Type: F 1) The capital of Sweden is _________. @[answer="Stockholm"] Yes. The capital of Sweden is Stockholm @[answer="Oslo"] No. Oslo is the capital of Sweden's western neighbor, Norway. @[answer="Södertälje"] No. But Södertälje is sometimes called the car capital of Sweden. @ Not even close a. Stockholm Type: F 2) The capital of Sweden is _________. @[=Stockholm] Yes. The capital of Sweden is Stockholm @[=Oslo] No. Oslo is the capital of Sweden's western neighbor, Norway. @[=Södertälje] No. But Södertälje is sometimes called the car capital of Sweden. @ Not even close a. Stockholm Your can define your own feedback condition shortcuts with the questionfeedbacktypes element in your domain customization XML. For example, we could define these Solution and Correct definitions in the domain customization: <questionfeedbacktypes> <questionfeedbacktype display="Solution" group="Solution" condition="teacher"/> <questionfeedbacktype display="Correct" condition="score>=1"/> </questionfeedbacktypes> We can then refer to those pre-defined conditions in feedback conditions: Type: F Options: Number Var: a = 3..9 2) 2 *$a$= _____ @[Solution] Feedback that only the teachers see @[Correct] Feedback for those with score >= 1 @[=2+$a$] Feedback for those who added the numbers instead of multiplying them @ Feedback that is displayed only if no other feedback has been displayed a. 2*$a$ When multiple feedback rows exist, BrainHoney displays only the first row that matches a condition. You can enable multiple-feedback-row display by grouping conditions; BrainHoney displays the first matching row from each group unless DisplayAll is specified. Start a condition with GroupName: to identify a feedback group: Type: F The capital of Sweden is _________. @[GroupA:score>=1] Corrrect @[GroupA:] Incorrect @ This line is always displayed, even though feedback from GroupA is also displayed. a. Stockholm Questions that have multiple answers, such as multiple fill-in-the-blank or matching questions use the shortcut pattern "a=" to specify feedback that applies to a specific answer. Type: MT Score: Partial, Round Points: 4 4) Capitals @[GroupA:a=a] For row 'a' (Sweden) the student answered 'a' (Stockholm). @[GroupA:a=b] For row 'a' (Sweden) the student answered 'b' (Oslo). @[GroupB:b<>b] For row 'b' (Norway) the student did not answer 'b' (Oslo). @[GroupC:c=a] For row 'c' (Finland) the student answered 'a' (Stockholm). a. Sweden = Stockholm b. Norway = Oslo c. Finland = Helsinki d. Denmark = Copenhagen You can use answercorrect variable in feedback conditions to define specific feedbacks for each blank in a multiple-fill-in-the-blank question. Use displayall to display all feedbacks with matching conditions within a group. Type: F Options: Multiple Score: Partial, Round Points: 4 @[GroupA:displayall:answercorrect1=false] Your answer to first blank is incorrect. @[GroupA:displayall:answercorrect2=false] You answer to second blank is incorrect. 1) _____ Jefferson was the ______ President of the US. a. Thomas Tom Tommy b. 3rd third Third 3 Partial score For the multiple-choice and fill-in-the-blank questions, you can specify a score for individual answer response or a correct answer. You place the score line below the multiple-choice question choices or fill-in-the-blank answers, and start the line with %. Scores are relative values comparing to the question score. See Choice Score Attribute and Answer Value Score Attribute in the Question Schema for details. Type: MC Score: Partial, Round 1) What is the capital of Sweden? *a. Stockholm % 1 b. Oslo % 0.5 c. Helsinki % 0.1 z. None of the above Type: F Options: Multiple Score: Partial, Round Points: 4 1) _____ Jefferson was the ______ President of the US. a. Thomas Tom Tommy % 0.5|0.5|0.5 b. 3rd third Third 3 % 0.5|0.5|0.5|0.4  Response labels For multiple-choice or multiple-answer questions, you can specify the type and format of labels for answer responses. See Label Attribute for the type format. Type: MC Label: A. 1) What is the capital of Sweden? *a. Stockholm b. Oslo c. Helsinki z. None of the above Question Options Keep response last To keep a given response last when randomizing responses, use the letter 'z' for the last response. Type: MC 1) What is the capital of Sweden? *a. Stockholm b. Oslo c. Helsinki z. None of the above Maintain response order Use the letter 'x' so the position of a response will remain the same while the order of the remaining responses can be randomized. Responses with the letter 'z' will be placed before any last reponses with the letter 'x' if there are any. Type: MC Label: a. 1) Which colors are primary? x. Red b. Green x. Blue d. Pink *e. both a and c Inline multiple-choice questions You can change the display of a multiple-choice question to use a drop-down list rather than bubble choices by specifying the "Inline" option and the location of the drop-down list with [----]. Response choices must be a single line of plain text. Type: MC Options: Inline 1) The capital of Sweden is [----]. *a. Stockholm b. Oslo c. Helsinki z. None of the above Matching questions Matching questions must have a left- and a right-hand side of each match. Use the equal sign (=) to separate the two sides. You can leave the left side empty to create extra match items in the right column of choices. To format a choice as HTML, enclose both sides of the match with the [HTML][/HTML] tags. Type: MT Options: Inline 1) Match the animals with their sounds. a. dog = woof b. cat = meow c. bird = chirp d. = moo You can also create drop-down boxes within the question text by specifying the "Inline" option above the question and drop-down list locations [----] in the question text. In this mode the match left-hand side is not displayed, but acts as the key to which box is the right answer. 1 = first, 2 = second, and so forth. In this mode you can also use the "MaintainOrder" option to preserve the choice order in the drop-down Type: MT Options: Inline, MaintainOrder 1) Indicate the correct animal sounds: The dog [----] chased the cat [----] that chased the bird [----]. a. 1 = woof b. 3 = chirp c. 2 = meow d. = moo Alternative fill-in-the-blank answers and case sensitivity A fill-in-the-blank question can have more than one correct answer. Specify multiple correct answers to a fill-in-the-blank-question as choices a, b, c, and so forth. To accept case-insensitive answers, specify the "IgnoreCase" option above the question. Type: F Options: IgnoreCase 1) ___________________ was the 16th president of the United States. a. Abraham Lincoln b. Abe Lincoln c. Lincoln Multiple blanks in fill-in-the-blank questions Specify multiple blanks in a fill-in-the-blank question by specifying the "Multiple" option above the question and multiple blanks in the question text. To specify multiple correct answers to a blank, put each correct answer on a different line. Type: F Options: Multiple 1) The _____ dog chased the ____ cat. a. barking b. meowing miaowing noisy Show correct answers Specify show correct answers in a fill-in-the-blank question by specifying the "ShowCorrect" option above the question and multiple blanks in the question text. Correct answers will be displayed when the question is reviewed if the display feedback setting is enabled. Type: F Options: ShowCorrect 1) The _____ dog chased the ____ cat. a. barking b. meowing miaowing noisy Show work For any question type except essay, you can set the "Workspace" option to give the student an area to show their work. If a question has the "Workspace" option and a question is answered incorrectly, the automated grading assigns a 0 score for the question and forwards the question to the teacher who can manually override the auto-assigned score. Type: MC Options: Workspace 1) What is (a+b)^2 if a = 3 and b = 2? Show your work. a. 20 *b. 25 c. 13 Composite questions Sometimes a set of question will have a common header or variables. Use a composite question to share variables and to ensure that the header is shared by keeping the question together and on one page. The questions that are part of the composite questions must have a question number with the format X.Y where is X is the composite question number and Y is number of the question part. Type: Composite Options: MaintainOrder Var: start = 2..15 1) For the following three questions start with the number$start$and apply the operations in order. Type: F Options: Number 1.1) Add 3: ________ a)$start$+3 Type: F Options: Number 1.2) Now multiply by 5 ________ a) ($start$+3)*5 Type: F Options: Number 1.3) Now subtract 11 _______ a) ($start$+3)*5-11 Custom questions A content developer can develop custom questions. To use a custom question specify the Url, resource path, or custom question name as defined in <customquestions> on the Custom property. If the question emulates a standard question specify the question type and answers like that question. For custom grading use the question type 'Custom'. Type: MC Custom: TrueFalse.xml Var: odd = 3..20,2 1)$odd$is an odd number *a. True z. False Type: Custom Custom: http://www.mywebserver.com/customquestion Var: a = 6..20 Var: b = 6..20 3)$a$*$b$= ______ Math and Science Questions You can click symbols on the Equations panel to enter the symbol at the editor's current cursor location. You can also type the symbol codes directly into the editor.  ASCII Math LaTeX Output a^2+2ab+b^2 $a^2+2ab+b^2$ sqrt((a+b)/2) $\sqrt{\frac{a+b}{2}}$ H_2O $H_2O$ H^+ $H^+$ Variables You can define variables in a question to generate data-driven questions by using the "Var" property. Variables are most useful in fill-in-the-blank questions of type Number, Match, and Equivalent. You can define answers as mathematical formulas using the variables. To define a range variable you specify the variable name, start and end of the range, and an optional step value. For example, Var: a = 3..9,2 defines the variable "a" to be 3, 5, 7, or 9. You refer to a variable by enclosing the variable name in dollar signs ($).

Type: F
Options: Number
Var: a = 4..9
Var: b = 3..9,2
1) $a$ * $b$ = _____
a) $a$*$b$

You can also specify a variable as a comma-separated list of values. Lists can contain text or numbers. Use quotes to include a comma in your variable value. You can group list variables by defining variable names as groupname.variablename. From each group variable, the question substitutes a value from the same random index into the question. In the following example, the "unit" group variables have three possible values. If the question uses km/h from unit.text, then it uses 1000/3600 from unit.conversion. Similarly, if it uses mph from unit.text, then it uses 0.44704 from unit.conversion.

Type: F
Options: Number
Var: speed = 2..50
Var: unit.text = km/h,ft/s,mph
Var: unit.conversion = 1000/3600,0.3048,0.44704
1) Convert to m/s: $speed$ $unit.text$ = ________ m/s
a) $speed$*$unit.conversion$
Fill-in-the-blank evaluation

Fill-in-the-blank questions by default evaluate as correct if the entered answer text exactly matches the keyed answer text. For math and science questions you can specify one of these types for more flexible evaluation:

 Option Description Notes Number Evaluates the answer as a number. Numerically compares the answer to the correct answer. Any number within 1% of the correct answer is considered correct. You can specify your own match range if you use two dots between the start and the end value of the correct answer. For example: 3.14..3.1416You can also specify the required number of significant figures with the "Figures" property. Students must enter exactly the number of significant figures to get the question correct. Match Compares the user-entered expression to the correct expression by matching a normalized expression parse tree. Use "Match" for simplification type questions. For example if the answer is 2a, a+a is not correct. Please see Supported operations below for available operations. Equivalent Evaluates the user-entered expression as correct if it is mathematically equivalent to the answer. For example, if the answer is 2a, a+a is correct. Correctness is computed by evaluating the expressions using sample values. The current sample values are: 0.123456789012, 0.345678901234, 0.890123456789, 5.678901234, -0.7654321. You can change the domain of these values for you question by creating a list variable named "domain".
Supported operations
 Operators Example + Addition 1+2 - Subtraction 1-2 or -4 * Multiplication 3*x or 3x / Division 3/4 ^ exponential x^2 pi 3.14159 2pi e scientific notation 1.6e26 |x| or abs(x) absolute value |-5| = 5 or abs(-3) = 3 sin, cos, tan, sec, csc, cot, asin, acos, atan (angle in radians) cos(pi) = -1 sqrt square root sqrt(y) x^(1/n) for the n root of a number x^(1/3) ln, log natual log and log base 10 ln(x) eval evaluate variable expression eval($a$ + $b$)
Expression evaluation for Match math questions

When creating expressions for Match questions that use variables it may be necessary to evaluate part of the expression. Using the built-in eval function in the expression resolves the parameters. Note that you do not need to use eval for Number of Equivalent questions, as these types are automatically evaluated.

Type: F
Options: Match
Var: a = 2..9
Var: b = 2..9
Var: c = 2..9
1) Simplify $a$($b$x+$c$)
a) eval($a$*$b$)x+eval($a$*$c$)
Expression evaluation for text

In the question, choice, or feedback text, you can control the display format of any computed number or expression by using the $eval(expression,format) syntax, where expression is the numerical expression to be evaluated, and format is the output format of the number. Type: MC Var: a = 12..19 2) What is$a$/2? *a.$eval($a$/2,#.0)
b. $eval(($a$-1)/2,#.0) c.$eval(($a$+1)/2,#.0)

In addition to creating new questions, you can also use existing questions in your assessment. You can link to a specific question or a group of questions.

To link to a specific question use the question type Link and the question id as the id. The easiest way to link to a question is to navigate to it in the Questions Banks panel on the right side and double click the question. Referrer displays the entity id and item id to identify where the link comes from or authorizes access to the question.

For your convenience, the editor shows the question title by the number, and the full question definition enclosed in the tags [LINKEDQUESTION][/LINKEDQUESTION]. You cannot edit the title nor the linked question from the editor.

Type: Link
ID: f6ab5583a4904129a24807cca70a985a
4) What is the capital of Sweden?
Type: MC
1) What is the capital of Sweden?
a. Oslo
b. Helsinki
*c. Stockholm
[/LINKEDQUESTION]
Bank with item

You can link to an entire assessment by using the Bank type and specifying the item id as the id. Optionally you can specify the max number of questions by using the Count option. Use the Question Banks panel to choose the item. The text by the number is the title of the item you are linking to. You cannot edit the title.

Type: Bank
ID: NKRCW
Count: 5
5) Simple Algebra

Bank with query

You can link to a subset of all questions in a course by using the Bank type and a query. You should also specify a count of questions to use. The text by the number is the course title.

Type: Bank
Query: Type = F and Option = Number and Objective = 'MA.7-3'
Count: 5
6) Algebra 1

Using questions in other course

You can use each of the options above pointing to other courses. Prefix the id with COURSEID: where COURSEID is the internal id of the target course. The easiest way of creating links to other courses is to click the More Banks button in the Question Banks panel.

Reference

The complete syntax for the assessment editor follows. All properties are optional unless otherwise noted. Parentheses and vertical bars '(|)' indicate a set of mutual-exclusive items. Square brackets '[' indicate optional items. The plus sign '+' indicates the item may be repeated. All properties containing comma seperated list, such as Groups, Objectives, and Var, may contain commas in their values by surrounding the value with quotes.

Type: (MC|MA|MT|F|E|Composite|Custom|Bank|Link)
Options: [(IgnoreCase|Number|Match|Equivalent|Regex)]
[,Workspace][,MaintainOrder][,ExtraCredit][,Inline][,Multiple]
Score: [POINTS][,Partial[,Round]]
Id: ID
Query: QUERY
Referrer: REFERRER
Count: COUNT
Groups: GROUPNAME[,GROUPNAME]+
Objectives: OBJECTIVEID[,OBJECTIVEID]+
Passage: [ENTITYID:]QUESTIONID
Rubric: RUBRICPATH
Height: ESSAYHEIGHT
Var: NAME = START..END[,STEP]
Var: [GROUP.]NAME = VALUE[,VALUE]+
Figures: SIGFIGURES
Meta-NAME: VALUE
1) QUESTIONBODY
@[$[GROUPNAME:][DISPLAYALL:][CONDITION]$] QUESTIONFEEDBACK
[*]a) CHOICEBODY
@ CHOICEFEEDBACK
b) [LEFTMATCH] = RIGHTMATCH
...

PropertyValueDescription
TypeQuestion type
MCMultiple choice
MTMatching
FFill in the blank
EEssay
CompositeComposite question
CustomCustom question
BankLink to a bank of question
OptionsVarious question options
MatchUse math expression-tree matching to determine if an answer is correct.
EquivalentUse numeric substitution to determine if an answer is functionally equivalent.
RegexUse a regular expression to check if an answer is correct. See Regular Expression Syntax.
WorkspaceCreate a workspace to allow students to show their work.
MaintainOrderDo not reorder the responses, even when the exam settings allow it.
ExtraCreditQuestion counts as extra credit.
InlineUse the inline (drop-down list) view of multiple-choice and matching questions.
MultipleQuestion contains multiple blanks.
ScorePOINTSNumber of points
PartialAllow partial credit for question.
RoundIf partial credit is given, round down to the nearest whole number.
IdIDRequired property if the question type is Link, optional if Bank. It is the ID of the question or bank to use. Questions IDs have the syntax [entityid:]questionid, where entityid is optional. If not present the current course is used.
QueryStringQuery to restricts questions if question type is Bank. See XML Query for the query syntax. In addition you can use the following common properties in your query:
 Property Value Description Type MC MA MT F E Composite Custom Question type Option Normal IgnoreCase Number Match Equivalent Regex Question option Group 'GROUPNAME' Group name Objective 'Id' Learning objective Meta-NAME 'metavalue' Meta information
ReferrerID Required property if the question type is Link. It has the syntax of entityid:itemid, which indicates where the question comes from or authroizes the access to the question after the link is saved. See Item Data.
CustomUrlUrl to renderer for custom question. You can use a course resource to render by custom question by specifying the path to the resource.
CountCOUNTNumber of questions to include from a bank
GroupsGROUPNAMEComma-separated list of group names
ObjectivesOBJECTIVEIDComma-separated list of objective ids
Passage[ENTITYID:]QUESTIONIDLink to a composite question that acts as the passage for this question. Entityid is optional. If not present the current course is used.
RubricRUBRICPATHPath to the resource in this course containing the grading rubric for this essay question
HeightHEIGHTHeight of the essay area.
VarNAME = START..END[,STEP]Range variable definition. Variable is between START and END with an optional STEP.
Var[GROUP.]NAME = VALUE[,VALUE]+List variable definition. Variable is one of comma-separated list of values. Optionally the variable can have a coordination group name.
FiguresSIGFIGURESNumber of significant figures that must be in the answer.
Meta-NAMEVALUESpecify the VALUE of the meta information NAME. For example, to specify the author as John Smith type:
Meta-author: John Smith
1)BODYText of the question
@[[GROUPNAME:][DISPLAYALL:][CONDITION]]FEEDBACKAutomatic question feedback. A question may include multiple feedback rows. Feedback rows are grouped according to GROUPNAME. The first matching condition in a group is displayed unless DISPLAYALL is specified. Empty conditions are always true. So in the simplest case feedback that always displays can be created as @.
See the DLAP documentation for full syntax rules. The assessment editor supports these shortcuts:
• FEEDBACKTYPE - Use the group and condition of a pre-defined feedback type.


text:
Vol. 1, No. 2, 2006

 Recent Issues
 The Journal Subscriptions Editorial Board Research Statement Scientific Advantage Submission Guidelines Submission Form Ethics Statement Author Index To Appear ISSN: 1559-3959 Other MSP Journals
Nonlinear dynamic characteristics of a vibro-impact system under harmonic excitation

Jianlian Cheng and Hui Xu

Vol. 1 (2006), No. 2, 239–258
Abstract

Dynamical behaviors of a two-degree-of-freedom (TDOF) vibro-impact system are investigated. The theoretical solution of periodic-one double-impact motion is obtained by differential equations, periodicity and matching conditions, and the Poincaré map is established. The dynamics of the system are studied with special attention to Hopf bifurcations of the impact system in nonresonance, weak resonance, and strong resonance cases. The Hopf bifurcation theory of maps in ${ℝ}^{2}$-strong resonance is applied to reveal the existence of Hopf bifurcations of the system. The theoretical analyses are verified by numerical solutions. The evolution from periodic impacts to chaos in nonresonance, weak resonance, and strong resonance cases, is obtained by numerical simulations. The results show that dynamical behavior of the system in the strong resonance case is more complicated than that of the nonresonance and weak resonance cases.

Keywords
Hopf bifurcation, strong resonance, quasiperiodic motion, vibro-impact, chaos


text:
# The fallacious argument that ‘Ali…

The fallacious argument that ‘Ali (RAA) was coerced and thus gave Bay’ah is contradicted by the principle that if the Khilafah is indeed established for a certain person (as the Twelver Shias claim about Ghadir and ‘Ali), then this must be announced and fought for publicly and explicitly regardless of circumstances. Doing otherwise and remaining quiet is in fact religiously sinful. We would say any “narration” brought up to justify such an attitude of quiet submission to what is said to be extreme evil is a violation of these principles, and cannot be attributed to the Prophet (ﷺ), ‘Ali (RAA) or others from among the Sahabah (RAA).


text:
# hall effect derivation engineering physics

The Hall effect is the production of a voltage difference (the Hall voltage) across an electrical conductor, transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. The photo either does not exist, or is private For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. ... Electromagnetic field theory has allowed a more refined interpretation of the physics responsible for the Hall effect. Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. Share sensitive information only on official, secure websites. Classical Hall effect, Hall constants. – Energy levelsEnergy levels. It provides a direct determination of both the sign of the charge carriers, e.g.electron or holes (appendix A), and their density in a given sample. Engineering Physics (Common to all Branches) Time : 3 hrs Max. This demonstration shows Hall effect in semiconductor materials and shows how n-type and p-type semiconductors can be identified. Hall Effect, deflection of conduction carriers by an external magnetic field, was discovered in 1879 by Edwin Hall. Hall Effect definition. Crystal Structures: Lattice points ... Hall Effect and Applications. Hall effect, development of a transverse electric field in a solid material when it carries an electric current and is placed in a magnetic field that is perpendicular to the current. Animation of interference of waves coming from two point sources. Material shown in Figure $$\PageIndex{1}$$ is a p-type semiconductor (doped with acceptor atoms). HALL effect - SemiConductors - and it's Applications - Engineering Physics 1. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. The derivation of physics formulas will help students to retain the concept for a … you are here->home->Physical Sciences->Solid State Physics Virtual Lab->Hall effect experiment:- Determination of charge carrier density.. Hall effect experiment:- Determination of charge carrier density.. The current carriers experience a force (Lorents force) in the downward direction. Here you can download the free lecture Notes of Engineering Physics Pdf Notes materials with multiple file links to download. Importance of Hall effect Hall effect proved that band theory of solids is more accurate than free electron theory. Legal. When a magnetic field is applied to a current carrying conductor in a direction perpendicular to that of the flow of current, a potential difference or transverse electric field is created across a conductor. This was first experimentally observed by Edwik H. Hall in 1879. The Michelson interferometer produces interference fringes by splitting a beam of monochromatic light so that one beam strikes a fixed mirror and the other a movable mirror. When an electron moves along the electric field direction perpendicular to an applied magnetic field, it experiences a magnetic force - q v X B acting normal to both directions. Assignment . We develop a hydrodynamic field theory of the three-dimensional fractional quantum Hall effect, which was recently proposed to exist in magnetic Weyl semimetals, when the Weyl nodes are gapped by strong repulsive interactions. The Hall Effect Principle has been named after an American physicist Edwin H. Hall (1855–1938). Similarly the deficiency of electrons make the upper face positive. It was discovered by Edwin Hall in 1879. – Sommerfeld theory. The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. Band theory of solids, (qualitative approach). Procedure . This looks like a completely classical derivation. Hall effect, development of a transverse electric field in a solid material when it carries an electric current and is placed in a magnetic field that is perpendicular to the current. A locked padlock) or https:// means you've safely connected to the .gov website. Here you can download the free lecture Notes of Engineering Physics Pdf Notes materials with multiple file links to download. This will provide a useful background for our discussion of the quantum Hall eﬀect. In the presence of a magnetic field B 0 along the z-direction, the holes will experience a force (the Lorentz force) driving them towards the bottom of the sample as shown in Figure 1. If $$\mathbf{B}$$ is perpendicular to current flow direction as seen in Figure $$\PageIndex{1}$$, then the direction of the force is found by right hand rule and magnitude is expressed as scalar multiplication of $$q$$, $$v$$ and $$B$$. It has been known that moving carriers in a magnetic field are accelerated by the Lorentz Force, and the magnitude and the direction of the applied force on the carriers are given as in Equation (1): $\mathbf{F} = q\boldsymbol{v}\times \mathbf{B} \label{1}$. This phenomenon was discovered in 1879 by the U.S. physicist Edwin Herbert Hall. Fig. More advanced than Hook and Hall •Solid State Physics, by N. W. Ashcroft and D. N. Mermin, Holt-Sanders This is the standard complete introduction to solid state physics. EH is Hall field. The Hall effect refers to the situation in which the Lorentz force acts on the electrons moving through a conductor, suc… inspired engineering; What is the Hall-effect? This effect consists in the appearance of an electric field called Hall field EH r, due to the deviation of the charge carrier trajectories by an external magnetic field. Physics 545 Electrons in metalsElectrons in metals. The theory of the Hall effect is discussed in Omar, Sections 4.10, 5.19, 6.8, Hutchison and Baird, the Physics of Engineering Solids, Section 11.6, and Kittel, Elementary Solid State Physics, pp. Since $$R_H$$ is found to be positive for p-type material and negative for the n-type, Hall coefficient can be measured to analyze the type of majority carrier of an unknown material. Hall Effect was named after Edwin Hall, its discoverer. Without application of any magnetic field, holes flow through by experiencing no force along the y-direction. It directly determines both the sign and density of charge carriers in a given sample. ,4th ed.Prentice Hall, 1995. As a result, an electric field is developed along Y-axis. However, this derivation stipulates that the force is downward facing because of the magnetic field (equal to the upward electric force), in the case of equilibrium. All of we know that when we place a current carrying conductor inside a magnetic field , the conductor experiences a mechanical force to a direction depending upon the direction of magnetic field and the direction of current in the conductor. Explain the variation of Fermi factor with temperature. The electric field developed is called Hall field EH. In this case, Lorentz force on electrons and holes can be written as: $\varepsilon_{e}=\dfrac{J_{e}B}{qn} \label{9}$, $\varepsilon_{h}=\dfrac{J_{h}B}{qp} \label{10}$. In atomic physics, we observe the interaction of the magnetic field with a magnetic dipole moment. Sign in to view the content . Problem with quantum Hall effect and Berry curvature. When an electron moves along the electric field direction perpendicular to an applied magnetic field, it experiences a magnetic force - q v X B acting normal to both directions. Learn about the derivation of Compton effect equation here. Where one end is connected from the positive end of a battery to one end of the plate and another end is connected from the negative end of a battery to … The Hall voltage divided by the current is the Hall resistance. This effect is called Hall effect and the emf thus developed is called Hall voltage VH. They will make you ♥ Physics. This phenomenon was discovered in 1879 by the U.S. physicist Edwin Herbert Hall. The Hall effect can be used to measure magnetic fields. •Solid-State Physics, 4ed, by H. Ibach and H. Luth, Springer-Verlag Another very popular book on the subject, with quite a bit of information in it. The Hall effect is basic to solid-state physics and an important diagnostic tool for the characterization of materials – particularly semi-conductors. MA: Addison Wesley Publishing Company, 1993, pp. Therefore, one has to consider the following components of Hall effect expression components to have a better understanding of the derivation – Theory . An internal electrical field ($$\varepsilon_y$$) starts building. Hall effect physics involves a metal body which contains a single form of charge carriers, like electrons. The production of transverse voltage across a current carrying conductor when placed in a perpendicular magnetic field,is called Hall effect.The voltage developed across the conductor is called Hall voltage. Answer FIVE full questions, choosing one full question from each module. Mainly Lorentz force is responsible for Hall effect . Hall Effect Principle suggested that the magnetic force in a current results in a crowding at the end of Pipe or (wire). Recommended for you Where is the Quantum Mechanics? Hall Effect was discovered by Edwin Herbert Hall in 1879. If w is the width of … d is the width of the metal slab. Table $$\PageIndex{1}$$ shows Hall coefficient of a number of metals. Being very simple and straight forward phenomena in physics, Hall effect is a fundamental principle in magnetic field sensing and have many practical applications in our daily life. Engineering Physics I B.Tech CSE/EEE/IT & ECE GRIET 1 I.B.Tech (CSE/EEE/IT & ECE) Engineering Physics Syllabus UNIT-I 1. Measurement of Hall voltage and Hall coefficient. We will study the Hall effect in a parallelepipedic semiconductor sample of sizes a, b, c Electromagnetic principle has now further explained the science behind Hall Effect with a much better appreciation. The physics derivation formulas will help learners to get in-depth knowledge about the related topic. ... Derivation: Consider a crystal in which the three axes are orthogonal and the intercepts are same. In steady state, current density (net charge flow) along the y-axis has to be zero, which results in Hall coefficient of: $R_{H}=\dfrac{p\mu_{h}^{2}-n\mu_{e}^{2} }{q(p\mu_{h}+n\mu_{e})^{2}} \label{12}$. Where $$q$$ is electron charge, $$\mathbf{v}$$ is drift velocity and $$\mathbf{B}$$ is magnetic flux density. 3: Hall effect geometry again; the strip has a thickness δ, length l, and height h. Only an authenticated user can view this page. For the Love of Physics - Walter Lewin - May 16, 2011 - Duration: 1:01:26. Drawbacks of QFET, Hall Effect, derivation of Hall coefficient in metals. If both holes and electrons are conduction carriers, then a different derivation has to be done to solve for Hall coefficient. The Hall Eﬀect 1 Background In this experiment, the Hall Eﬀect will be used to study some of the physics of charge transport in metal and semiconductor samples. Figure $$\PageIndex{2}$$ Hall effect in presence of both holes (h) and electrons (e) $$^{[3]}$$. The Hall effect is an extension of the Lorentz force, which describes the force exerted on a charged particle—such as an electron—moving through a magnetic field. 4. Feedback . Explanation of Hall effect with Hall voltage and Hall field, derivation of the expression for Hall coefficient. Simulator . M. A. Omar, Elementary Solid State Physics: Principles and Applications. Hall Effect, deflection of conduction carriers by an external magnetic field, was discovered in 1879 by Edwin Hall. Where $$\mu_p$$ is hole mobility. Obtain the expression for Hall voltage in terms of Hall co-efficient (6 marks). An excellent, detailed but highly readable explanation can be found in "The Hall Effect Sensors; … Interpretation of the Kubo formula for Hall conductance. 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How to know if $\mathbb{P}(f(P_a) > \tau)$ is increasing or decreasing function of $P_a$ without actually computing it?

Problem Setup Consider the following optimization problem \begin{equation*} \begin{aligned} & \underset{P_{a},\lambda_{a}}{\text{maximize}} & & P_{cov}(P_{a},\lambda_{a}) \\ & \text{subject to} & & 0 \leq P_a \leq \min\{P_{a,max},\frac{\rho}{\lambda_a}\} \\ & & & 0 \leq \lambda_a \leq \lambda_{a,max} \\ \end{aligned} \tag{1} \end{equation*} with $$P_{cov}(P_{a},\lambda_{a})= P_{cov, a}(P_{a},\lambda_{a}) \, \mathcal{A}_a(P_{a},\lambda_{a}) + P_{cov, {\rm g}}(P_{a},\lambda_{a}) \, \mathcal{A}_ {\rm g}(P_{a},\lambda_{a})$$ where $$$$P_{cov, {\rm g}}(P_{a},\lambda_{a})= \mathbb{P} \left( \frac{P_{{\rm g}} {R_{{\rm g},0}}^{-\eta_{\rm g}} h_{{\rm g},0}}{\sum_{i=1}^{\infty} P_{{\rm g}} {R_{{\rm g},i}}^{-\eta_{\rm g}} h_{{\rm g},i}+\sum_{i=0}^{\infty} P_{a} {R_{a,i}}^{-\eta_a} h_{a,i}} > \tau \right),$$$$ $$$$P_{cov, a}(P_{a},\lambda_{a})= \mathbb{P} \left( \frac{P_{a} {R_{a,0}}^{-\eta_{a}} h_{{a},0}}{\sum_{i=1}^{\infty} P_{{a}} {R_{{a},i}}^{-\eta_{a}} h_{{a},i}+\sum_{i=0}^{\infty} P_{{\rm g}} {R_{{\rm g},i}}^{-\eta_{\rm g}} h_{{\rm g},i}} > \tau \right),$$$$ $$$$\mathcal{A}_a =\mathbb{P} \left( P_{a} R_a^{-{\eta_a}} > P_{{\rm g}} {R_{\rm g}}^{-\eta_{\rm g}} \right), \quad \text{and} \quad \mathcal{A}_{\rm g} =\mathbb{P} \left( P_{{\rm g}} {R_{\rm g}}^{-\eta_{\rm g}} > P_{a} R_a^{-{\eta_a}} \right).$$$$ where $$R_{a,i}, R_{\rm g,i}, h_{a,i}, h_{\rm g,i}$$ are all random variables and $$P_a, P_{\rm g}, \eta_a, \eta_{\rm g}, \rho$$ are all constants. $$\lambda_a$$ is a parameter that characterizes the pdf of $$R_a$$.

Questions

• Is $$P_{cov}$$ is an increasing function of $$P_a$$?

• If yes, how to prove it without computing the probabilities?

• If no, how can approach such an optimization problem? My initial approach was to utilize the note below, then perform a numerical search over a compact subset of candidate $$\lambda_a$$ values on the $$P_a \lambda_a = \rho$$ curve to find the optimum $$\lambda_a$$.

Note

If $$P_{cov}$$ is an increasing function of $$P_a$$ we can, without loss of optimality, reduce the problem to the following one-dimensional constrained optimization problem: \begin{equation*} \begin{aligned} & \underset{\lambda_{a}}{\text{maximize}} & & P_{cov}(\min\{P_{a,max},\frac{\rho}{\lambda_a}\},\lambda_{a}) \\ & \text{subject to} & & 0 \leq \lambda_a \leq \lambda_{a,max} \\ \end{aligned} \tag{2} \end{equation*} since any $$P^{\star}$$ solving $$(1)$$ must achieve the power constraint with equality.

• There is quite a number of variables here. Would you mind clarify the definitions of those variables, and the relationship between them, if any? – BGM Oct 20 '18 at 3:04
• @BGM I did. Kindly see the updated post. – Lod Oct 20 '18 at 22:39
• have you tried coupling? it's a general method allowing you to show certain probabilities are increasing (as a function of a certain parameter) – mathworker21 Oct 23 '18 at 9:36
• @mathworker21 No I haven't, I'll look into that. – Lod Oct 23 '18 at 12:38
• @AlexFrancisco Yes non-negative. $R_{a,i},R_{g,i},h_{a,i},h_{g,i}$ are random variables with pdfs having a support $[0,\infty)$. – Lod Oct 23 '18 at 22:52


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Can anyone show me some maths to explain the train problem with relative simultaneity

posts 164 and 165

1 agrees with me and the other doesnt
 Recognitions: Gold Member Then the video goes back and repeats the sequence of the platform observer seeing the lightning flashes by panning up above him (unfortunately, I can only put three videos on a single post so this will be scattered among several posts): Attached Thumbnails
 Recognitions: Gold Member Notice how in the first image here, they depict the progress of the two flashes of light hitting the window in which the observer is (although they don't show her, it's the second window from the front of the train car-the second window from the right). They hit simultaneously. Then in the next image, the two flashes hit the platform observer: Attached Thumbnails

Mentor
 Quote by solarflare posts 164 and 165 1 agrees with me and the other doesnt
They are both pointing out that you made a mistake in calling the distance between the track and the platform "r1". Do you understand that r1 is not the same as the distance between the track and the platform, which I suggest we call "d"?
 Recognitions: Gold Member Here's where they show that the platform observer is equidistant from the two lightning strikes that hit the moving train car (at an earlier time but they have not shown this): Attached Thumbnails

Mentor
Blog Entries: 1
 Quote by ghwellsjr Here is the transcript for the first minute of the video devoted exclusively to the platform frame:Imagine two observers, one seated in the center of a speeding train car, and another standing on the platform as the train races by. As the center of the car passes the observer on the platform, he sees two bolts of lightning strike the car - one on the front, and one on the rear. The flashes of light from each strike reach him at the same time, so he concludes that the bolts were simultaneous, since he knows that the light from both strikes traveled the same distance at the same speed, the speed of light. He also predicts that his friend on the train will notice the front strike before the rear strike, because from her perspective on the platform the train is moving to meet the flash from the front, and moving away from the flash from the rear.
I agree, sloppy use of the term 'see' in the second sentence. And sloppy illustration of the light flash reaching the platform observer while the train just sits there.
 Recognitions: Gold Member Now here's where they repeat the sequence again but this time it is an animation and it shows something different than what they showed earlier. And note the platform observer is not present so we can't tell exactly when he is supposed to see the flashes of light. This is further complicated by the fact that they are panning the image from left to right so the perspective is changing making it impossible to know where the platform observer is. The last image is where they "light up" the passenger to show that she sees the front flash first: Attached Thumbnails
 Recognitions: Gold Member And now with further panning, they show the rear flash arriving at the train passenger in the last of these images but note the lightning strikes have disappeared: Attached Thumbnails
 Mentor Blog Entries: 1 Why don't we scrap the video and just discuss the scenario that the video was trying (badly) to illustrate, which is what we've actually been discussing for the most part. It's the standard Einstein train thought experiment, of course. A train passes by a person on the platform. At the instant the center of the train passes the person, lightning strikes both ends of the train according to the platform frame. (These are the two events that I label #1 and #2 in post 155.) Solarflare, given this setup, can you comment on my statements in post #155?
 how can you say scrap the video when my whole point is that the video is wrong ?
 Mentor Your statements are also wrong, independently of any errors in the video. Your original mistakes were not even involving the same scenario as the one in the video.

 Quote by solarflare how can you say scrap the video when my whole point is that the video is wrong ?
The situation the video describes is essentially correct; we want to do away with it anyway because the small flaws do not seriously jeopardize the larger argument, which is what you seem to have an issue with.

Nevertheless, it may help to start from a clean slate so we can pick out exactly what the issue is. So let's draw up a new scenario.

A man is rafting on a narrow stream. He moves at a constant velocity. There are two other boats, one downstream of the man and one upstream, both with mirrors. These boats (and the mirrors attached to them) move in some arbitrary, unspecified manner. They may decelerate and accelerate at will.

The man uses two lasers to shine beams off both mirrors. If these beams both return to him at the same moment, then he concludes that he must have been equidistant from both mirrors at the time the beams hit the mirrors. If he shines both beams at some time $t=0$ and the beams return to him at some time $t = 2 \delta$, then he concludes that the mirrors were each a distance $\delta$ from him at time $t = \delta$ according to his watch.

Now, let us presume that, at the man's time $t = \delta$, there is a child in another raft just beside him, except the child has some constant velocity downstream relative to the man.

Now, solarflare, some questions for you:
a) Would the child believe the man emitted both laser pulses at the same moment?
b) Would the child say that the pulses reflected off both mirrors at the same time according to his (the child's) watch?
c) Would the child receive both reflected pulses at his boat at the same moment?
 take the position of the train when it is in the centre of the platform - and run the scenario for both observers - the result comes out the same. if the strikes occur when r1 = r2 then a simple triangle shows that they must occur in both frames simultaneously but at different times. take a spaceship with two lasers - one on each wing observer 1 is in the centre moving directly away from the ship in another smaller ship observer 2 is stationary also in the centre but at a greater distance. the space ship fires its lasers - observer 1 sees the two lasers pass simultaneously before observer 2 observer 2 sees the two lasers pass him simultaneously also
 they both say the lasers pass them simultaneously but they disagree on the time that they pass

Mentor
Blog Entries: 1
 Quote by solarflare take the position of the train when it is in the centre of the platform - and run the scenario for both observers - the result comes out the same.
Don't know what you mean by 'run the scenario for both observers'. There is just one scenario, described from two different frames of reference.

 if the strikes occur when r1 = r2 then a simple triangle shows that they must occur in both frames simultaneously but at different times.

 take a spaceship with two lasers - one on each wing observer 1 is in the centre moving directly away from the ship in another smaller ship observer 2 is stationary also in the centre but at a greater distance. the space ship fires its lasers - observer 1 sees the two lasers pass simultaneously before observer 2 observer 2 sees the two lasers pass him simultaneously also

Do you mean:

(A) At the moment that the ship fires its two lasers towards the middle, there is an observer sitting in the middle of the ship (observer 2) and a second observer (observer 1) in a small ship moving parallel to the big ship just passing the middle of the ship at that moment (according to the big ship frame).

In this case the light flashes from each laser reach the middle of the big ship at the same time, but they reach the small ship at different times. In any case, the observers in the small ship do not agree that the lasers were fired at the same time.

Or do you mean:

(B) The ship fires its two lasers towards the middle. There is an observer sitting in the middle of the ship (observer 2) and a second observer (observer 1) in a small ship moving parallel to the big ship who happens to pass by the middle point just as the light reaches the middle point.

In this case both observers see the light simultaneously (since they are at the middle when the light arrives). But the observers in the small ship do not agree that the lasers were fired at the same time.

Note that in either scenario the lasers are only fired simultaneously in the frame of the big ship. The small ship will think that they were fired at different times.

Mentor
Blog Entries: 1
 Quote by solarflare they both say the lasers pass them simultaneously but they disagree on the time that they pass
If you say they reach the observers simultaneously, then you are talking about the second version (B) of the scenario (per my last post).

Again, you are hung up on the times at which the light flashes reach the observer. But the real interesting deal is what they conclude about whether the lasers fired at the same or different times. The different frames disagree about that!
 Solar, the lasers have to point in opposite directions to to see the difference we're talking about. You understand that the point if the train example is that the light from the strikes approach the observers from two different directions, right? You also keep talking about triangles. There are no triangles necessary. Put the platform and train observers right next to each other.


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# A question on convergence of series

Suppose $(z_i)$ is a sequence of complex numbers such that $|z_i|\to 0$ strictly decreasing. If $(a_i)$ is a sequence of complex numbers that has the property that for any $n\in\mathbb{N}$

$$\sum_{i}a_iz_i^{n}=0$$ does this imply that $a_i=0$ for any $i$?

Edit: For the "finite dimensional" case, when we have $n$ distinct $(z_i)$, then $(a_i)$ must be $0$. This amounts to solving a homogeneous system of $n$ equations with $n$ unknowns, which only has the trivial solution in the case of distinct $(z_i)$. I am really curious what happens in the infinite dimensional case. My intuition tells me the same must be true, but I don't have a proof for it.

Edit 2: Very interesting, looking were this question originated, the fact that all $a_i=0$ when $(a_i)\in l_1$ is "expected". I was hoping to get a counterexample otherwise. However, if a non-trivial sequence $a_i$ exists (at least for some sequences $z_i$), I would "expect" to be able to choose it in $l_2$. Looking at Davide and Julien answers below, it seems $(a_i)\in l_1$ is an essential assumption in their argument.

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For the sake of clarity: am I right that the series is not assumed absolutely convergent? –  user53153 Dec 21 '12 at 6:44
No, it is not. All it is known is that $(z_i)$ converge to $0$, decreasing. So I cannot assume a priori anything about the convergence of $\sum a_i z_i$. –  Theo Dec 21 '12 at 20:06
@picakhu Your example shows that $\sum a_i z_i=0$, but not $\sum a_i z_i^n$ when $n\in\mathbb{N}$, $n\geq 2$. –  Theo Dec 24 '12 at 21:29
The following special cases already looks interesting: the case $z_i = \frac{i}$ is equivalent to showing that if a Dirichlet series vanishes on the integers then it is identically zero. –  Ofir Dec 25 '12 at 22:15
If we know that $(a_i)\in \ell^1$, can we conclude? Just for curiosity, if we define $T:\ell^1\rightarrow\ell^{\infty}$ by $Ta=(\sum_i a_iz_i,\sum_i a_iz_i^2,...,\sum_i a_iz^n,...)$,where $a=(a_1,a_2,...,a_n,...)$, can we conclude that the only solution of the equation $Ta=0$ is $a=0$ –  Tomás Dec 26 '12 at 19:25

Under the assumption that $\sum_{i=0}^\infty |a_i| < \infty$, we can use power series. This assumption can be weakened: it is enough to assume that there exists $n \in \mathbb{N}$ such that $(a_i z_i^n) \in \ell^1$ (since we can work with $a'_i = a_i z_i^n$).

For $|\lambda| < |z_0|^{-1}$, define $$f(\lambda) = \sum_{i=0}^\infty \frac{a_i}{1 - \lambda z_i} = \sum_{n=0}^\infty \sum_{i=0}^\infty (\lambda z_i)^na_i =0.$$ If $k = \min\{i \in \mathbb{N} \mid a_i \neq 0\}$ exists, then $f(\lambda) \sim \dfrac{a_k}{1-\lambda z_k}$ when $\lambda \to z_k^{-1}$, wich is absurd.

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While this does not completely answer my question, it is enough to settle the problem from which the question originated. My sequences $(z_i)$ converge very fast to $0$, so in particular are in $l_1$. In this situation it is impossible to find a non-trivial, bounded (let alone in $l_2$ as I require) sequence $(a_i)$. By your observation, $(a_i z_i)$ will be in $l_1$, thus $(a_i)$ is necessarily trivial. Thanks guys, this was very helpful! –  Theo Dec 28 '12 at 4:36
Just one more comment. I would be still very much interested what happens in the general case. For example, if $z_i=1/\log(i+1)$ as Pavel M suggested in the comments, can one find a sequence $(a_i)$ which is in $l_2$? Perhaps I should formulate this as a different question. –  Theo Dec 28 '12 at 4:47

It's a partial answer to @Tomas' comment which ask what happens if we assume the sequence $\{a_j\}$ in $\ell^1$, and it would be too long for a comment.

I interpret the hypothesis $\sum_{j=0}^{+\infty}a_jz_j^n=0$ for all $n$ as "the sequence of partial sums converges and its limit is $0$". In particular, it implies that $\{a_j\}$ is convergent to $0$, hence $\{|a_j|\}$ is in particular a bounded sequence, say by $M$. We show by induction that $a_j=0$ for all $j$. First, we have for all integer $p$ that $$|a_0z_0^p|\leqslant\sum_{j\geqslant 1}|a_j||z_j|^p$$ so $$|a_0|\leqslant \sum_{j\geqslant 1}|a_j|\left(\frac{|z_j|}{|z_0|}\right)^p.$$ Let $b_{j,p}:=|a_j|\left(\frac{|z_j|}{|z_0|}\right)^p$. We have $|b_{j,p}|\leqslant |a_j|\left(\frac{|z_j|}{|z_0|}\right)^p$, and as $\{a_j\}\in\ell^1$, we can conclude by monotone convergence theorem that $a_0=0$.

Now assume that $a_0=\dots=a_n=0$. Then $$|a_{n+1}|\leqslant \sum_{j\geqslant n+2}|a_j|\left(\frac{|z_j|}{|z_{n+1}|}\right)^p,$$ and an other application of monotone convergence theorem yields $a_{n+1}=0$.

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I think you dont need monotone convergence theorem. In the second inequality you can use the fact that $$\Big(\frac{|z_j|}{|z_0|}\Big)^p\leq \Big(\frac{|z_1|}{|z_0|}\Big)^p,\ \forall\ i\geq 1$$ and by hypothesis $\frac{|z_1|}{|z_0|}<1$, hence, by making $p\rightarrow\infty$ you can conclude. Am I right? –  Tomás Dec 27 '12 at 12:37
@Tomás I think yes. So the case $\{a_j\}\in\ell^1$ seems to be too simple with respect to the complexity of the problem. –  Davide Giraudo Dec 27 '12 at 13:21
Maybe it is possible to show that $(a_i)\in\ell^1$. –  Tomás Dec 27 '12 at 13:25
@DavideGiraudo Just an observation: when I say $\sum a_i z_i^n=0$, for all $n$, I actually mean $n>0$. So the conclusion in your second sentence follows from the added assumption that $(a_i)\in l_1$, but not from the original hypothesis alone. –  Theo Dec 27 '12 at 14:30
@Theo Ok. (in France, $\Bbb N$ usually mean the set of integers greater or equal than $0$) –  Davide Giraudo Dec 27 '12 at 16:52

Suppose that the statement is not true, and let $N$ be the smallest index such that $a_N \neq 0$. Then $-a_N z_N^n= \sum_{i > N} a_i z_i^n \neq 0$ for all $n$. Then $0 < |a_N| \leq |\sum_{i \geq N} a_i \frac{z_i^n}{z_N^n}| \leq \sum_{i > N}|a_i|(|z_i|/|z_N|)^n$. Taking $n \to \infty$ in this strict inequality would yield $0 < 0$, a contradiction.

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I don't think you can interchange the limits to conclude the RHS converges to $0$. Or am I missing something? –  Theo Dec 19 '12 at 17:33
Indeed the RHS could even be infinite for all n. –  user53153 Dec 20 '12 at 7:51


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# How to work with LaTeX such that files are interchangeable linux/unix/windows?

I am mosly using `pdflatex`. One recurring problem is that the encoding used on Linux (my system), UTF8, is different from on Windows, I use

``````\inputencoding{utf8}
``````

on Windows one must use (well, to write spanish ...)

``````\inputencoding{latin1}
``````

and files must be converted when moved between systems. Is there a way to organize the workflow such that the same input file can be used on both systems? without conversion and without any editing? (possibly invoking change to XelaTeX/LuaTeX, if necessary)

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As long as you're using a Windows editor that supports Unicode (pretty much anything except WinEdt) you should be able to use `utf8` on all machines. – Alan Munn Jun 26 '12 at 0:26
@AlanMunn: there's good news regarding WinEdt and Unicode support --- version 7, released ca 3 months ago, is fully Unicode-capable. :-) – Mico Jun 26 '12 at 0:33
@Mico Good to know. I've made this a little clearer in the big LaTeX editors/IDE question. – Alan Munn Jun 26 '12 at 0:40
@AlanMunn: I guess it's pretty much anything except TeXnicCenter (except for the alpha version). – Caramdir Jun 26 '12 at 1:07

Its always better to write `\usepackage[utf8]{inputenc}` or `\usepackage[latin1]{inputenc}` to reveal your input encoding for third person so that he opens the document in the same encoding when using pdflatex.
On Windows, with the help of cross-platform Unicode editors one can use `latin1`, `utf8` or even `utf8x` encoding and share it in Linux without any trouble.
LyX isn't a TeX editor: it's a separate system that really shouldn't be recommended as an equivalent to an editor. Also, if you're using XeLaTeX or LuaLaTeX you shouldn't load `inputenc`. – Alan Munn Jun 26 '12 at 1:13