text:
python script for deleting basepair in fasta format
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Entering edit mode
8.5 years ago

I have a multiple fasta format sequence in a notepad, I want to delete rest of the base above 500 bp, in each sequence. what script I can use.

thanks

sequence python • 2.7k views
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Entering edit mode

A simple sed command might do the trick if your fasta sequences are on one line (i.e. not wrapped):

sed --regexp-extended '/^>/! s/^(.{1,500}).*/\1/' input.fasta > output.fasta
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Or using awk (I like awk!):

awk '{print /^>/ ? $0 : substr($1, 1, 500)}' input.fasta > output.fasta
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Thanks for the quick reply.

Can I use this command on CentOS Linux terminal directly or do I need to download any other package?

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These commands should work on most Linux or Unix computers without installing any additional package.

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Hi Frederic,

I have one more problem, Hope you can help me with that too

I have a single contig of around 5000 bp, which looks like

>contig 00001
ACTTTCAAAAAAATCAGCGTAAAAAACATACTAATTTGGGCAAATTCCCACCTGTTTTTAGGGACATTTT
TCTTTGAATTAGAGCCTCAGCAGCTCGTCATTGCTGAATTTTCTTGAAGTTGCCTCGTTTCTAGGCGCTT
ACGCTGGTCGTAGCCGGTTTCTGACTACGAATGATTTAAAAATTTAGTTTGTCATCTATGCTTCGTACAT
AGATAAAGATAATAATGCCTTTCCATCTATTTTAGATTGAGAGAGGCTTTTTCTGTCATGATTAAATTAG
GAAAGGAGTGAGCACTACTTTGGAAAATATTGAAGAATTATGGAGCGCTACATTAAAAAAAATTGAAGAA
AAATTGAGTAAACCTAGCTTTGATACGTGGCTTAAAAATACAAAAGCAGAAGCATTAGAAAAAGATACAC
TAATTATTTCTGCTCCAAATGAATTCGCAAGAGATTGGTTAGAAAATCAATACACGAACTTAATTTCTCA
GATGCTACTTGAAGTTACAGGATCTGAATTAAATACAAAATTTATCATTCCTGATTCATTAGAAGAAATA
GAAGAGCAAAAGCCTATGCCAAAACCAAAGCAGTCTACAGATACAGGTGACTCACCAAAATCGATGCTTA
ACTCAAAATATACTTTTGATACATTTGTCATTGGAGCTGGTAATCGTTTCGCTCATGCTGCTTCATTAGC

I want to have a multiple contigs of 100 bp, like

>contig 00001
ACTTTCAAAAAAATCAGCGTAAAAAACATACTAATTTGGGCAAATTCCCACCTGTTTTTAGGGACATTTTTCTTTGAATTAGAGCCTCAGCAGCTCGTCATTGCTGAATTTTCTTGAAGTTGCCTCGTTTCTAGGCGCTT
>contig 00002
ACGCTGGTCGTAGCCGGTTTCTGACTACGAATGATTTAAAAATTTAGTTTGTCATCTATGCTTCGTACATAGATAAAGATAATAATGCCTTTCCATCTATTTTAGATTGAGAGAGGCTTTTTCTGTCATGATTAAATTAG
>contig 00003
GAAAGGAGTGAGCACTACTTTGGAAAATATTGAAGAATTATGGAGCGCTACATTAAAAAAAATTGAAGAAAAATTGAGTAAACCTAGCTTTGATACGTGGCTTAAAAATACAAAAGCAGAAGCATTAGAAAAAGATACAC
>contig 00004
TAATTATTTCTGCTCCAAATGAATTCGCAAGAGATTGGTTAGAAAATCAATACACGAACTTAATTTCTCAGATGCTACTTGAAGTTACAGGATCTGAATTAAATACAAAATTTATCATTCCTGATTCATTAGAAGAAATA
>contig 00005
GAAGAGCAAAAGCCTATGCCAAAACCAAAGCAGTCTACAGATACAGGTGACTCACCAAAATCGATGCTTAACTCAAAATATACTTTTGATACATTTGTCATTGGAGCTGGTAATCGTTTCGCTCATGCTGCTTCATTAGC

and so on...

What command should I use?

Best!

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thanks Frederic, it works... thanks alot.

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Cross posted (to Linkedin???)

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8.5 years ago
Cytosine ▴ 460

Don't know of any scripts that do that, but it's really easy to write up one yourself. Try this one with using BioPython...

from Bio import SeqIO
import sys

fastafile, length = sys.argv[1], int(sys.argv[2])
entries = SeqIO.parse(fastafile, "fasta")

for entry in entries:
print ">"+entry.name
print entry.seq[:length]

Save it to a file (e.g. parse_fasta.py) and then run it in a terminal (or cmd if you're on windows) with:

python parse_fasta.py <input.fasta> <sequence_lengths> > <output.fasta>

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Entering edit mode

Or slightly more elegantly using SeqIO.write and a generator expression,

from Bio import SeqIO
import sys
fastafile, length = sys.argv[1], int(sys.argv[2])
trimmed = (entry[:length] for entry in SeqIO.parse(fastafile, "fasta"))
SeqIO.write(trimmed, sys.stdout, "fasta")

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Entering edit mode

I am not a bioinformatician, so I don't know where to use this command, I am using CentOS Linux system, if I open terminal window, can I directly type the above script or I need to download any package before using the script.

Thanks.

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Entering edit mode

Running this is as simple as saving the script to a file (like parse_fasta.py from my post) and running the python interpreter over that script (with the commands I mentioned at the end of my post).

If you have a fresh installation, you'll probably be missing the Biopython library. In that case you can install it by following the steps mentioned here: http://centosn00b.blogspot.be/2012/05/biopython-tip.html

Remember to prepend "sudo" to the commands if you do not have administrator privileges.

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Good information, but I would caution against advising beginning users to "make things work" but adding sudo.

0
Entering edit mode
8.5 years ago

pip install pyfaidx

python

And then paste the script, changing your file name.

format: plain


text:
# Mathematician:Mathematicians/Sorted By Nation/India

For more comprehensive information on the lives and works of mathematicians through the ages, see the MacTutor History of Mathematics archive, created by John J. O'Connor and Edmund F. Robertson.

The army of those who have made at least one definite contribution to mathematics as we know it soon becomes a mob as we look back over history; 6,000 or 8,000 names press forward for some word from us to preserve them from oblivion, and once the bolder leaders have been recognised it becomes largely a matter of arbitrary, illogical legislation to judge who of the clamouring multitude shall be permitted to survive and who be condemned to be forgotten.[1]

## India

##### Baudhayana (c. 800 BCE )

Indian mathematician, also a priest, believed to have flourished c. $800$ BCE.

Believed to have been a skilled craftsman, thus to have used his mathematical expertise in practical ways.

Did some early research into creating a circle with the same area as a given square.

Discovered $\pi$ to some degree of precision, and discovered what is now known as Pythagoras's Theorem.

Also evaluated the square root of 2 to five decimal places of accuracy.
show full page

##### Piṅgalá (c. 5th or 2nd century B.C.E. )

Indian mathematician about whom practically nothing is known, not even when he lived.

Notable for being the first in history to mention what is now known as Pascal's Triangle.
show full page

##### Aryabhata the Elder (476 – 550)

Indian mathematician and astronomer.

An early believer in the irrationality of $\pi$, and developed an approximation for it of $3.1416$.

Developed a positional system of numerals in C. 500, but it lacked a symbol for zero.
show full page

##### Varāhamihira (505 – 587)

Indian astronomer, mathematician, and astrologer.

One of several early mathematicians to discover what is now known as Pascal's triangle.

Defined the algebraic properties of zero and negative numbers.

Improved the accuracy of the sine tables of Aryabhata I.

Made some insightful observations in the field of optics.
show full page

##### Brahmagupta (598 – 668)

Indian mathematician and astronomer.

Gave definitive solutions to the general linear equation, and also the general quadratic equation.

Best known for the Brahmagupta-Fibonacci Identity.
show full page

##### Bhāskara I (c. 600 – c. 680)

Indian mathematician who was the first on record to use Hindu-Arabic numerals complete with a symbol for zero.

Gave an approximation of the sine function in his Āryabhaṭīyabhāṣya of 629 CE.
show full page

##### Mahāvīrāchārya (c. 800 – c. 870)

Mahāvīrāchārya (literally: Mahāvīrā the teacher) was an Indian mathematician best known for separating the subject of mathematics from that of astrology.

Gave the sum of a series whose terms are squares of an arithmetical progression and empirical rules for area and perimeter of an ellipse.
show full page

##### Halayudha (c. 1000 )

Halayudha was an Indian mathematician who wrote the Mṛtasañjīvanī, a commentary on Piṅgalá's Chandah-shastra, containing a clear description of Pascal's triangle (called meru-prastaara).
show full page

##### Gopāla (11th century )

Gopāla was an Indian mathematician noted for studying the Fibonacci numbers before $1135$.
show full page

##### Acharya Hemachandra (1089 – 1172)

Hemachandra Acharya Sūrī (Sanskrit: हेमचन्द्र सूरी) was an Indian all-rounder who, among other things, investigated the Fibonacci sequence, following Gopāla.
show full page

##### Bhāskara II Āchārya (1114 – 1185)

Bhāskara (Kannada: ಭಾಸ್ಕರಾಚಾರ್ಯ) was an Indian mathematician and astronomer.

He is known as Bhāskara II, Bhāskara Āchārya ("Bhāskara the teacher"), or Bhāskarāchārya, to distinguish him from Bhāskara I).
show full page

##### Narayana Pandit (c. 1340 – c. 1400)

Narayana Pandit (Sanskrit: नारायण पण्डित) was an Indian mathematician who made considerable contributions to several areas of Indian mathematics.
show full page

##### Mādhava of Saṅgamāgrama (c. 1350 – c. 1425)

Indian mathematician who made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry and algebra.

It has been suggested that his works made their way to Europe and had an influence on the later European development of calculus.
show full page

##### Srīnivāsa Aiyangār Rāmānujan (1887 – 1920)

Indian mathematician who made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions.
show full page

##### Raj Chandra Bose (1901 – 1987)

Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes.

Invented the notions of partial geometry, strongly regular graph.

Started a systematic study of difference sets to construct symmetric block designs.

Noted with Sharadchandra Shankar Shrikhande and Ernest Tilden Parker for disproving Euler's Conjecture on Orthogonal Latin Squares.
show full page

##### Hansraj Gupta (1902 – 1988)

Indian mathematician specialising in number theory, in particular the study of the partition function.
show full page

##### Dattathreya Ramchandra Kaprekar (1905 – 1986)

Indian recreational mathematician who described several classes of natural numbers including the Kaprekar, Harshad and Self numbers and discovered the Kaprekar constant, named after him.
show full page

##### Sharadchandra Shankar Shrikhande (b. 1917 )

Indian mathematician with distinguished and well-recognized achievements in combinatorial mathematics.

Noted with Raj Chandra Bose and Ernest Tilden Parker for disproving Euler's Conjecture on Orthogonal Latin Squares.
show full page

##### Triloki Nath Bhargava (1933 – 2017)

Indian mathematician working mainly in the field of statistics.

Known for his work on permutable primes.
show full page

## References

1. Eric Temple Bell: Men of Mathematics, 1937, Victor Gollancz, London.


text:
# angle between line and plane

The angle between a line $l$ and a plane $\tau$ is defined as the least possible angle $\omega$ between $l$ and a line contained by $\tau$.

It is apparent that $\omega$ satisfies always  $0\leqq\omega\leqq 90^{\circ}$.

Let the plane $\tau$ be given by the equation (http://planetmath.org/EquationOfPlane)  $Ax\!+\!By\!+\!Cz\!+\!D=0$,  i.e. its normal vector has the components $A,\,B,\,C$. Let a direction vector of the line $l$ have the components $a,\,b,\,c$. Then the angle $\omega$ between $l$ and $\tau$ is obtained from the equation

 $\sin\omega=\frac{|Aa\!+\!Bb\!+\!Cc|}{\sqrt{A^{2}\!+\!B^{2}\!+\!C^{2}}\sqrt{a^{% 2}\!+\!b^{2}\!+\!c^{2}}}.$

In fact, the right hand side (http://planetmath.org/Equation) is the cosine of the angle $\alpha$ between $l$ and the surface normal of $\tau$ (see angle between two lines), and $\omega$ is the complementary angle of $\alpha$.

Example.  Consider the $xy$-plane and the line $l$ through the origin and the point  $(1,\,1,\,1)$. We can use the components $1,\,1,\,1$ for the direction vector of $l$ and the components $0,\,0,\,1$ for the normal vector of the plane. We have

 $\omega\;=\;\arcsin\frac{1\!\cdot\!0\!+\!1\!\cdot\!0\!+\!1\!\cdot\!1}{\sqrt{1^{% 2}\!+\!1^{2}\!+\!1^{2}}\sqrt{0^{2}\!+\!0^{2}\!+\!1^{2}}}\;=\;\arcsin\frac{1}{% \sqrt{3}}\approx 35.26^{\circ}.$
 Title angle between line and plane Canonical name AngleBetweenLineAndPlane Date of creation 2013-03-22 17:30:14 Last modified on 2013-03-22 17:30:14 Owner pahio (2872) Last modified by pahio (2872) Numerical id 13 Author pahio (2872) Entry type Definition Classification msc 51N20 Synonym slant Synonym inclination Related topic AngleBetweenTwoLines Related topic DotProduct Related topic EquationOfPlane Related topic AngleBetweenTwoPlanes Related topic NormalOfPlane Related topic ProjectionOfRightAngle Defines angle between plane and line


text:
# 1000 places of Pi

Here is the expansion of π arranged in groups of 50 digits as Aitken recited it.
3.14159265358979323846264338327950288419716939937510
58209749445923078164062862089986280348253421170679
82148086513282306647093844609550582231725359408128
48111745028410270193852110555964462294895493038196
44288109756659334461284756482337867831652712019091
45648566923460348610454326648213393607260249141273
72458700660631558817488152092096282925409171536436
78925903600113305305488204665213841469519415116094
33057270365759591953092186117381932611793105118548
07446237996274956735188575272489122793818301194912
98336733624406566430860213949463952247371907021798
60943702770539217176293176752384674818467669405132
00056812714526356082778577134275778960917363717872
14684409012249534301465495853710507922796892589235
42019956112129021960864034418159813629774771309960
51870721134999999837297804995105973173281609631859
50244594553469083026425223082533446850352619311881
71010003137838752886587533208381420617177669147303
59825349042875546873115956286388235378759375195778
18577805321712268066130019278766111959092164201989

Written by J J O'Connor and E F Robertson
Last Update January 1997


text:
A Delayed version of a function may be called to generate Delayed objects

delayed(expr, sequential = FALSE, expect_error = FALSE, timeout = NULL)

delayed_fun(fun, sequential = FALSE, expect_error = FALSE)

## Arguments

expr expression to delay if TRUE, never parallelize this task if TRUE, pass error to downstream tasks instead of specify a time limit for computation halting computation function to delay

## Examples

d <- delayed(3 + 4)
d$compute() #> [1] 7adder <- function(x, y) { x + y } delayed_adder <- delayed_fun(adder) z <- delayed_adder(3, 4) z$compute()
#> [1] 7


text:
# 9.5 Solving trigonometric equations  (Page 5/10)

 Page 5 / 10

## Solving an equation using an identity

Solve the equation exactly using an identity: $\text{\hspace{0.17em}}3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +3=2\text{\hspace{0.17em}}{\mathrm{sin}}^{2}\theta ,0\le \theta <2\pi .$

If we rewrite the right side, we can write the equation in terms of cosine:

$\begin{array}{ccc}\hfill 3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +3& =& 2{\mathrm{sin}}^{2}\theta \hfill \\ \hfill 3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +3& =& 2\left(1-{\mathrm{cos}}^{2}\theta \right)\hfill \\ \hfill 3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +3& =& 2-2{\mathrm{cos}}^{2}\theta \hfill \\ \hfill 2\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\theta +3\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +1& =& 0\hfill \\ \hfill \left(2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +1\right)\left(\mathrm{cos}\text{\hspace{0.17em}}\theta +1\right)& =& 0\hfill \\ \hfill 2\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta +1& =& 0\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\theta & =& -\frac{1}{2}\hfill \\ \hfill \theta & =& \frac{2\pi }{3},\frac{4\pi }{3}\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\theta +1& =& 0\hfill \\ \hfill \mathrm{cos}\text{\hspace{0.17em}}\theta & =& -1\hfill \\ \hfill \theta & =& \pi \hfill \end{array}$

Our solutions are $\text{\hspace{0.17em}}\theta =\frac{2\pi }{3},\frac{4\pi }{3},\pi .$

## Solving trigonometric equations with multiple angles

Sometimes it is not possible to solve a trigonometric equation with identities that have a multiple angle, such as $\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}\mathrm{cos}\left(3x\right).\text{\hspace{0.17em}}$ When confronted with these equations, recall that $\text{\hspace{0.17em}}y=\mathrm{sin}\left(2x\right)\text{\hspace{0.17em}}$ is a horizontal compression    by a factor of 2 of the function $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ On an interval of $\text{\hspace{0.17em}}2\pi ,$ we can graph two periods of $\text{\hspace{0.17em}}y=\mathrm{sin}\left(2x\right),$ as opposed to one cycle of $\text{\hspace{0.17em}}y=\mathrm{sin}\text{\hspace{0.17em}}x.\text{\hspace{0.17em}}$ This compression of the graph leads us to believe there may be twice as many x -intercepts or solutions to $\text{\hspace{0.17em}}\mathrm{sin}\left(2x\right)=0\text{\hspace{0.17em}}$ compared to $\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}x=0.\text{\hspace{0.17em}}$ This information will help us solve the equation.

## Solving a multiple angle trigonometric equation

Solve exactly: $\text{\hspace{0.17em}}\mathrm{cos}\left(2x\right)=\frac{1}{2}\text{\hspace{0.17em}}$ on $\text{\hspace{0.17em}}\left[0,2\pi \right).$

We can see that this equation is the standard equation with a multiple of an angle. If $\text{\hspace{0.17em}}\mathrm{cos}\left(\alpha \right)=\frac{1}{2},$ we know $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ is in quadrants I and IV. While $\text{\hspace{0.17em}}\theta ={\mathrm{cos}}^{-1}\frac{1}{2}\text{\hspace{0.17em}}$ will only yield solutions in quadrants I and II, we recognize that the solutions to the equation $\text{\hspace{0.17em}}\mathrm{cos}\text{\hspace{0.17em}}\theta =\frac{1}{2}\text{\hspace{0.17em}}$ will be in quadrants I and IV.

Therefore, the possible angles are $\text{\hspace{0.17em}}\theta =\frac{\pi }{3}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\theta =\frac{5\pi }{3}.\text{\hspace{0.17em}}$ So, $2x=\frac{\pi }{3}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}2x=\frac{5\pi }{3},$ which means that $\text{\hspace{0.17em}}x=\frac{\pi }{6}\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}x=\frac{5\pi }{6}.\text{\hspace{0.17em}}$ Does this make sense? Yes, because $\text{\hspace{0.17em}}\mathrm{cos}\left(2\left(\frac{\pi }{6}\right)\right)=\mathrm{cos}\left(\frac{\pi }{3}\right)=\frac{1}{2}.$

In quadrant I, $\text{\hspace{0.17em}}2x=\frac{\pi }{3},$ so $\text{\hspace{0.17em}}x=\frac{\pi }{6}\text{\hspace{0.17em}}$ as noted. Let us revolve around the circle again:

$\begin{array}{ccc}\hfill 2x& =& \frac{\pi }{3}+2\pi \hfill \\ & =& \frac{\pi }{3}+\frac{6\pi }{3}\hfill \\ & =& \frac{7\pi }{3}\hfill \end{array}$

so $\text{\hspace{0.17em}}x=\frac{7\pi }{6}.$

One more rotation yields

$\begin{array}{ccc}\hfill 2x& =& \frac{\pi }{3}+4\pi \hfill \\ & =& \frac{\pi }{3}+\frac{12\pi }{3}\hfill \\ & =& \frac{13\pi }{3}\hfill \end{array}$

$x=\frac{13\pi }{6}>2\pi ,$ so this value for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is larger than $\text{\hspace{0.17em}}2\pi ,$ so it is not a solution on $\text{\hspace{0.17em}}\left[0,2\pi \right).$

In quadrant IV, $\text{\hspace{0.17em}}2x=\frac{5\pi }{3},$ so $\text{\hspace{0.17em}}x=\frac{5\pi }{6}\text{\hspace{0.17em}}$ as noted. Let us revolve around the circle again:

$\begin{array}{ccc}\hfill 2x& =& \frac{5\pi }{3}+2\pi \hfill \\ & =& \frac{5\pi }{3}+\frac{6\pi }{3}\hfill \\ & =& \frac{11\pi }{3}\hfill \end{array}$

so $\text{\hspace{0.17em}}x=\frac{11\pi }{6}.$

One more rotation yields

$\begin{array}{ccc}\hfill 2x& =& \frac{5\pi }{3}+4\pi \hfill \\ & =& \frac{5\pi }{3}+\frac{12\pi }{3}\hfill \\ & =& \frac{17\pi }{3}\hfill \end{array}$

$x=\frac{17\pi }{6}>2\pi ,$ so this value for $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is larger than $\text{\hspace{0.17em}}2\pi ,$ so it is not a solution on $\text{\hspace{0.17em}}\left[0,2\pi \right).$

Our solutions are Note that whenever we solve a problem in the form of $\text{\hspace{0.17em}}\mathrm{sin}\left(nx\right)=c,$ we must go around the unit circle $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ times.

## Solving right triangle problems

We can now use all of the methods we have learned to solve problems that involve applying the properties of right triangles and the Pythagorean Theorem    . We begin with the familiar Pythagorean Theorem, $\text{\hspace{0.17em}}{a}^{2}+{b}^{2}={c}^{2},$ and model an equation to fit a situation.

## Using the pythagorean theorem to model an equation

Use the Pythagorean Theorem, and the properties of right triangles to model an equation that fits the problem.

One of the cables that anchors the center of the London Eye Ferris wheel to the ground must be replaced. The center of the Ferris wheel is 69.5 meters above the ground, and the second anchor on the ground is 23 meters from the base of the Ferris wheel. Approximately how long is the cable, and what is the angle of elevation (from ground up to the center of the Ferris wheel)? See [link] .

Using the information given, we can draw a right triangle. We can find the length of the cable with the Pythagorean Theorem.

The angle of elevation is $\text{\hspace{0.17em}}\theta ,$ formed by the second anchor on the ground and the cable reaching to the center of the wheel. We can use the tangent function to find its measure. Round to two decimal places.

$\begin{array}{ccc}\hfill \mathrm{tan}\text{\hspace{0.17em}}\theta & =& \frac{69.5}{23}\hfill \\ & & \\ & & \\ \hfill {\mathrm{tan}}^{-1}\left(\frac{69.5}{23}\right)& \approx & 1.2522\hfill \\ & \approx & 71.69°\hfill \end{array}$

The angle of elevation is approximately $\text{\hspace{0.17em}}71.7°,$ and the length of the cable is 73.2 meters.

what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
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Miranda
I am living in india
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Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
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Jeffrey
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Miranda
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Miranda
Jeffrey
Jeffrey
Miranda
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Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
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C'est comment
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Hi
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ANSHU
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h r u friends
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yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
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SORIE
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lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function


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# Math Help - log and exponential functions

1. ## log and exponential functions

for the function y=0.5^x

a) identity intervals for which the function is positive and intervals for which it is negative for both the the function and its inverse

b) intervals for which the function is increasing and intervals for which it is decreasing for both the function and its inverse

I don't get how to do these questions. How do I figure out if the function and its inverse is positive, negative, increasing or decreasing? If you could provide me the info that I need to know to figure out these questions, it would be really helpful! thank you so much!

2. Originally Posted by skeske1234
for the function y=0.5^x

a) identity intervals for which the function is positive and intervals for which it is negative for both the the function and its inverse

b) intervals for which the function is increasing and intervals for which it is decreasing for both the function and its inverse

I don't get how to do these questions. How do I figure out if the function and its inverse is positive, negative, increasing or decreasing? If you could provide me the info that I need to know to figure out these questions, it would be really helpful! thank you so much!
$y = \left(\frac{1}{2}\right)^x = 2^{-x}$

you should know what the graph looks like.

using that graph, you should be able to sketch its inverse, then answer the questions.


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# Refinement (category theory)

In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called envelope.

## Definition

Suppose ${\displaystyle K}$ is a category, ${\displaystyle X}$ an object in ${\displaystyle K}$, and ${\displaystyle \Gamma }$ and ${\displaystyle \Phi }$ two classes of morphisms in ${\displaystyle K}$. The definition[1] of a refinement of ${\displaystyle X}$ in the class ${\displaystyle \Gamma }$ by means of the class ${\displaystyle \Phi }$ consists of two steps.

Enrichment
• A morphism ${\displaystyle \sigma :X'\to X}$ in ${\displaystyle K}$ is called an enrichment of the object ${\displaystyle X}$ in the class of morphisms ${\displaystyle \Gamma }$ by means of the class of morphisms ${\displaystyle \Phi }$, if ${\displaystyle \sigma \in \Gamma }$, and for any morphism ${\displaystyle \varphi :B\to X}$ from the class ${\displaystyle \Phi }$ there exists a unique morphism ${\displaystyle \varphi ':B\to X'}$ in ${\displaystyle K}$ such that ${\displaystyle \varphi =\sigma \circ \varphi '}$.
Refinement
• An enrichment ${\displaystyle \rho :E\to X}$ of the object ${\displaystyle X}$ in the class of morphisms ${\displaystyle \Gamma }$ by means of the class of morphisms ${\displaystyle \Phi }$ is called a refinement of ${\displaystyle X}$ in ${\displaystyle \Gamma }$ by means of ${\displaystyle \Phi }$, if for any other enrichment ${\displaystyle \sigma :X'\to X}$ (of ${\displaystyle X}$ in ${\displaystyle \Gamma }$ by means of ${\displaystyle \Phi }$) there is a unique morphism ${\displaystyle \upsilon :E\to X'}$ in ${\displaystyle K}$ such that ${\displaystyle \rho =\sigma \circ \upsilon }$. The object ${\displaystyle E}$ is also called a refinement of ${\displaystyle X}$ in ${\displaystyle \Gamma }$ by means of ${\displaystyle \Phi }$.

Notations:

${\displaystyle \rho =\operatorname {ref} _{\Phi }^{\Gamma }X,\qquad E=\operatorname {Ref} _{\Phi }^{\Gamma }X.}$

In a special case when ${\displaystyle \Gamma }$ is a class of all morphisms whose ranges belong to a given class of objects ${\displaystyle L}$ in ${\displaystyle K}$ it is convenient to replace ${\displaystyle \Gamma }$ with ${\displaystyle L}$ in the notations (and in the terms):

${\displaystyle \rho =\operatorname {ref} _{\Phi }^{L}X,\qquad E=\operatorname {Ref} _{\Phi }^{L}X.}$

Similarly, if ${\displaystyle \Phi }$ is a class of all morphisms whose ranges belong to a given class of objects ${\displaystyle M}$ in ${\displaystyle K}$ it is convenient to replace ${\displaystyle \Phi }$ with ${\displaystyle M}$ in the notations (and in the terms):

${\displaystyle \rho =\operatorname {ref} _{M}^{\Gamma }X,\qquad E=\operatorname {Ref} _{M}^{\Gamma }X.}$

For example, one can speak about a refinement of ${\displaystyle X}$ in the class of objects ${\displaystyle L}$ by means of the class of objects ${\displaystyle M}$:

${\displaystyle \rho =\operatorname {ref} _{M}^{L}X,\qquad E=\operatorname {Ref} _{M}^{L}X.}$

## Examples

1. The bornologification[2][3] ${\displaystyle X_{\operatorname {born} }}$ of a locally convex space ${\displaystyle X}$ is a refinement of ${\displaystyle X}$ in the category ${\displaystyle \operatorname {LCS} }$ of locally convex spaces by means of the subcategory ${\displaystyle \operatorname {Norm} }$ of normed spaces: ${\displaystyle X_{\operatorname {born} }=\operatorname {Ref} _{\operatorname {Norm} }^{\operatorname {LCS} }X}$
2. The saturation[4][3] ${\displaystyle X^{\blacktriangle }}$ of a pseudocomplete[5] locally convex space ${\displaystyle X}$ is a refinement in the category ${\displaystyle \operatorname {LCS} }$ of locally convex spaces by means of the subcategory ${\displaystyle \operatorname {Smi} }$ of the Smith spaces: ${\displaystyle X^{\blacktriangle }=\operatorname {Ref} _{\operatorname {Smi} }^{\operatorname {LCS} }X}$

5. ^ A topological vector space ${\displaystyle X}$ is said to be pseudocomplete if each totally bounded Cauchy net in ${\displaystyle X}$ converges.


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# Why is the SVI parameterization in terms of variance?

The Gatheral SVI parameterization reads $$\sigma^2 = a + b \left[\rho(k-m) + \sqrt{(k-m)^2+s^2}\right]\,.$$ Why is it expressed in terms of variance $$\sigma^2$$ and not directly in terms of volatility $$\sigma$$ or in terms of total variance $$\sigma^2 T$$ ?

• If you look at Gatheral's paper then it is expressed in terms of total variance, were you looking at Zeliade's paper? – raptor22 Oct 2 '19 at 15:09
• Which paper are you referring to? It is in terms of variance in his initial presentation in Madrid "A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives". – jherek Oct 3 '19 at 8:25

One main characteristic of the SVI parameterization is to be linear in variance in the wings. It is a desirable property, since the criteria to obeys Lee's Moment Formula for Implied Volatility at Extreme Strikes translates then a simple condition on the asymptotic slopes, that is on $$a$$ and $$b$$.
Let $$\tilde{a} = at$$ and $$\tilde{b} = bt$$ and you can jump from a parametrization to another. In Gatheral and Jacquier's paper (Arbitrage-free SVI volatility surfaces) https://arxiv.org/pdf/1204.0646.pdf they parametrize total variance directly whereas in Zeliade's 2+3 optimization (Quasi-Explicit Calibration of Gatheral’s SVI model) http://www.zeliade.com/whitepapers/zwp-0005.pdf they parametrize for variance.
• Well the only difference should be in the interpretation of these two parameters. Increasing $a$ translates either the variance (= total variance by $at$) or total variance (=variance by $a/t$) and same for $b$ with the slope of put/call wings (see p. 5 of the linked Gatheral paper). It seems clear that working with total variance is more natural for deriving arbitrage conditions. However I guess that it is easier to interpret the former parametrization. If you look at p. 6/7 they introduce another paramtrization (SVI-JW) of the variance that is more interpretable. – raptor22 Oct 3 '19 at 8:42


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## Du, Zengji

Compute Distance To:
 Author ID: du.zengji Published as: Du, Zengji; Du, Zeng-ji; Du, Zeng-Ji more...less
 Documents Indexed: 107 Publications since 2001 Co-Authors: 62 Co-Authors with 101 Joint Publications 2,642 Co-Co-Authors
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### Co-Authors

 5 single-authored 24 Lin, Xiaojie 23 Ge, Weigao 8 Bai, Zhanbing 6 Xue, Chunyan 5 Liu, Jiang 4 Feng, Zhaosheng 4 Lv, Yansen 4 Meng, Fanchao 3 Cai, Guolan 3 Kong, Lingju 3 Liu, Wenbin 3 Mo, Jiaqi 3 Shang, Xiaohui 3 Xu, Dongcheng 3 Xu, Ying 3 Yan, Shuling 3 Yin, Jian 3 Zhao, Bensheng 3 Zhuang, Kaige 2 Cai, Xiaojing 2 Chen, Shuting 2 Chen, Xiao 2 Fu, Zhengmin 2 Ge, Jianjiang 2 Li, Ji 2 Li, Wenbin 2 Li, Xiaowan 2 Liang, Jing 2 Lin, Wantao 2 Ma, Dexiang 2 Ren, Yulin 2 Wei, Lei 2 Xu, Min 2 Xu, Shengmei 2 Zhang, Qin 2 Zhang, Xiaoni 2 Zhou, Mingru 1 Babenko, Mikhail 1 Chen, Lihua 1 Chen, Yangquan 1 Chervyakov, Nikolay 1 Dong, Yanyan 1 Han, Bang-Sheng 1 Han, Xianglin 1 Ji, Shuguan 1 Jiang, Tingting 1 Kuchukov, Viktor 1 Li, Shuang 1 Li, Xiaofeng 1 Lian, Wen 1 Liang, Xiangqian 1 Miranda-López, Vanessa 1 Peng, Rui 1 Qiao, Qi 1 Rivera-Rodriguez, Raul 1 Romanovskiĭ, Valeriĭ Georgievich 1 Sun, Sujing 1 Talbi, El-Ghazali 1 Tchernykh, Andrei 1 Tian, Yu 1 Tisdell, Christopher C. 1 Wang, Zhi Cheng 1 Wei, Dandan 1 Wu, Ranchao 1 Yang, Pinghua 1 Zhang, Fubao 1 Zhang, Shuo 1 Zhang, Xiang 1 Zhu, Huaiping
all top 5

### Serials

 8 Applied Mathematics and Computation 6 Journal of Differential Equations 6 Qualitative Theory of Dynamical Systems 4 Computers & Mathematics with Applications 4 Journal of Mathematical Analysis and Applications 3 Mathematics in Practice and Theory 3 Applied Mathematics Letters 3 Abstract and Applied Analysis 3 Nonlinear Analysis. Real World Applications 3 Communications on Pure and Applied Analysis 3 Journal of Applied Mathematics and Computing 3 Discrete and Continuous Dynamical Systems. Series S 2 Mathematical Methods in the Applied Sciences 2 Rocky Mountain Journal of Mathematics 2 Journal of Computational and Applied Mathematics 2 Acta Mathematicae Applicatae Sinica. English Series 2 Applications of Mathematics 2 Journal of Difference Equations and Applications 2 Nonlinear Dynamics 2 Nonlinear Analysis. Modelling and Control 2 Discrete and Continuous Dynamical Systems. Series B 2 Journal of Xuzhou Normal University. Natural Science Edition 2 Carpathian Journal of Mathematics 1 Journal of Mathematical Biology 1 Archiv der Mathematik 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Functional Analysis 1 Mathematische Nachrichten 1 Programming and Computer Software 1 Journal of Mathematical Research & Exposition 1 Applied Mathematics and Mechanics. (English Edition) 1 Journal of Systems Science and Mathematical Sciences 1 Annals of Differential Equations 1 Applied Mathematical Modelling 1 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 1 Journal of Nonlinear Science 1 Applied Mathematics. Series B (English Edition) 1 Journal of Mathematical Sciences (New York) 1 Electronic Journal of Differential Equations (EJDE) 1 Filomat 1 Communications in Applied Analysis 1 Differential Equations and Dynamical Systems 1 Taiwanese Journal of Mathematics 1 Electronic Journal of Qualitative Theory of Differential Equations 1 Wuhan University Journal of Natural Sciences (WUJNS) 1 Fractional Calculus & Applied Analysis 1 Communications in Contemporary Mathematics 1 JIPAM. Journal of Inequalities in Pure & Applied Mathematics 1 Journal of Liaoning Normal University. Natural Science Edition 1 Differential Equations 1 Mathematical Modelling and Analysis 1 Acta Mathematica Scientia. Series A. (Chinese Edition) 1 Acta Mathematica Scientia. Series B. (English Edition) 1 Journal of Beijing Institute of Technology 1 Boundary Value Problems 1 Acta Mathematica Sinica. Chinese Series 1 International Journal of Mathematical Analysis (Ruse) 1 Communication on Applied Mathematics and Computation 1 Journal of Applied Analysis and Computation
all top 5

### Fields

 83 Ordinary differential equations (34-XX) 31 Partial differential equations (35-XX) 23 Biology and other natural sciences (92-XX) 22 Operator theory (47-XX) 9 Dynamical systems and ergodic theory (37-XX) 7 Difference and functional equations (39-XX) 3 Global analysis, analysis on manifolds (58-XX) 3 Fluid mechanics (76-XX) 1 Integral equations (45-XX) 1 Computer science (68-XX) 1 Mechanics of deformable solids (74-XX) 1 Systems theory; control (93-XX) 1 Information and communication theory, circuits (94-XX)

### Citations contained in zbMATH Open

71 Publications have been cited 780 times in 557 Documents Cited by Year
Existence and iteration of monotone positive solutions for multipoint boundary value problem with $$p$$-Laplacian operator. Zbl 1095.34009
Ma, De-Xiang; Du, Zeng-Ji; Ge, Wei-Gao
2005
Existence of solutions for a class of third-order nonlinear boundary value problems. Zbl 1053.34017
Du, Zengji; Ge, Weigao; Lin, Xiaojie
2004
A priori $$L^\infty$$ estimates for solutions of a class of reaction-diffusion systems. Zbl 1338.35220
Du, Zengji; Peng, Rui
2016
Some higher-order multi-point boundary value problem at resonance. Zbl 1059.34010
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2005
The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach. Zbl 1392.35223
Du, Zengji; Li, Ji; Li, Xiaowan
2018
On a third-order multi-point boundary value problem at resonance. Zbl 1072.34012
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2005
Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative. Zbl 1074.34011
Du, Zengji; Xue, Chunyan; Ge, Weigao
2005
Singular perturbations for third-order nonlinear multi-point boundary value problem. Zbl 1084.34014
Du, Zengji; Ge, Weigao; Zhou, Mingru
2005
Positive solutions for some second-order four-point boundary value problems. Zbl 1115.34016
Bai, Zhanbing; Du, Zengji
2007
Permanence and almost periodic solution of a Lotka-Volterra model with mutual interference and time delays. Zbl 1351.34080
Du, Zengji; Lv, Yansen
2013
Existence results for discrete Sturm-Liouville problem via variational methods. Zbl 1129.39007
Tian, Yu; Du, Zengji; Ge, Weigao
2007
Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse. Zbl 1392.34089
Chen, Xiao; Du, Zengji
2018
Existence and global attractivity of positive periodic solution to a Lotka-Volterra model. Zbl 1205.34058
Wang, Xiaoli; Du, Zengji; Liang, Jing
2010
Existence and asymptotic behaviors of traveling waves of a modified vector-disease model. Zbl 1397.35314
Du, Zengji; Feng, Zhaosheng
2018
Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays. Zbl 1330.37074
Du, Zengji; Feng, Zhaosheng
2014
Existence and global attractivity of a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response. Zbl 1231.37052
Lv, Yansen; Du, Zengji
2011
Traveling wave phenomena of $$n$$-dimensional diffusive predator-prey systems. Zbl 1387.35091
Du, Zengji; Feng, Zhaosheng; Zhang, Xiaoni
2018
Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations. Zbl 1133.34011
Du, Zengji; Liu, Wenbin; Lin, Xiaojie
2007
A multiplicity result for $$p$$-Lapacian boundary value problems via critical points theorem. Zbl 1173.34007
Du, Zengji; Lin, Xiaojie; Tisdell, Christopher C.
2008
Traveling waves for nonlocal Lotka-Volterra competition systems. Zbl 1437.35141
Han, Bang-Sheng; Wang, Zhi-Cheng; Du, Zengji
2020
Solvability of multipoint boundary value problems at resonance for higher-order ordinary differential equations. Zbl 1081.34017
Lin, Xiaojie; Du, Zengji; Ge, Weigao
2005
Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation. Zbl 1349.37071
Xu, Ying; Du, Zengji; Wei, Lei
2016
Solvability of a second-order multi-point boundary value problem at resonance. Zbl 1168.34310
Meng, Fanchao; Du, Zengji
2009
Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem. Zbl 1166.34008
Lin, Xiaojie; Du, Zengji; Liu, Wenbin
2008
Solutions to $$m$$-point boundary value problems of third order ordinary differential equations at resonance. Zbl 1070.34031
Xue, Chunyan; Du, Zengji; Ge, Weigao
2005
Solvability of functional differential equations with multi-point boundary value problems at resonance. Zbl 1142.34357
Du, Zengji
2008
The dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system. Zbl 1440.35291
Du, Zengji; Qiao, Qi
2020
Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems. Zbl 1200.34061
Du, Zengji; Kong, Lingju
2010
A second order differential equation with generalized Sturm-Liouville integral boundary conditions at resonance. Zbl 1474.34192
Du, Zengji; Yin, Jian
2014
A note on a third-order multi-point boundary value problem at resonance. Zbl 1229.34028
Lin, Xiaojie; Du, Zengji; Meng, Fanchao
2011
Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Zbl 1308.34107
Du, Zengji; Chen, Xiao; Feng, Zhaosheng
2014
Nonlocal boundary value problem of higher order ordinary differential equations at resonance. Zbl 1145.34006
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2006
The solitary wave solutions of the nonlinear perturbed shallow water wave model. Zbl 1440.35006
Ge, Jianjiang; Du, Zengji
2020
Triple positive solutions for some second-order boundary value problem on a measure chain. Zbl 1133.34015
Bai, Zhanbing; Liang, Xiangqian; Du, Zengji
2007
Singularly perturbed third-order boundary value problem for nonlinear systems. Zbl 1131.34043
Du, Zengji
2007
Varieties and analytic normalizations of partially integrable systems. Zbl 1357.34072
Du, Zengji; Romanovski, Valery G.; Zhang, Xiang
2016
Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method. Zbl 1345.35007
Zhuang, Kaige; Du, Zengji; Lin, Xiaojie
2015
A class of third order multi-point boundary value problem. Zbl 1080.34006
Du, Zengji; Cai, Guolan; Ge, Weigao
2005
Global asymptotic stability of almost periodic solution for a multispecies competition-predator system with time delays. Zbl 1418.37132
Lin, Xiaojie; Du, Zengji; Lv, Yansen
2013
Positive solutions of M-point boundary value problem for second-order dynamic equations on time scales. Zbl 1149.34014
Lin, Xiaojie; Du, Zengji
2008
Traveling waves in a generalized nonlinear dispersive-dissipative equation. Zbl 1342.35312
Shang, Xiaohui; Du, Zengji
2016
Existence of three solutions for sytems of multi-point boundary value problems. Zbl 1201.34030
Du, Zengji; Kong, Lingju
2009
Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach. Zbl 1452.35219
Du, Zengji; Liu, Jiang; Ren, Yulin
2021
Iterative method for a class of fourth-order $$p$$-Laplacian beam equation. Zbl 1458.34046
Bai, Zhanbing; Du, Zengji; Zhang, Shuo
2019
Traveling wave solutions of $$n$$-dimensional delayed reaction-diffusion systems and application to four-dimensional predator-prey systems. Zbl 1336.92072
Shang, Xiaohui; Du, Zengji; Lin, Xiaojie
2016
Existence of solution of a three-point boundary value problem via variational approach. Zbl 1441.34035
Lian, Wen; Bai, Zhanbing; Du, Zengji
2020
A third-order multi-point boundary value problem at resonance with one three dimensional kernel space. Zbl 1324.34035
Lin, Xiaojie; Zhao, Bensheng; Du, Zengji
2014
Positive solutions for a second-order three-point discrete boundary value problem. Zbl 1155.39005
Du, Zengji
2008
Dynamics of nonconstant steady states of the Sel’kov model with saturation effect. Zbl 1443.35076
Du, Zengji; Zhang, Xiaoni; Zhu, Huaiping
2020
Solvability of a third-order multipoint boundary value problem at resonance. Zbl 1474.34125
Du, Zengji; Zhao, Bensheng; Bai, Zhanbing
2014
Solutions to third-order multi-point boundary-value problems at resonance with three dimensional kernels. Zbl 1290.34022
Li, Shuang; Yin, Jian; Du, Zengji
2014
Positive solutions for higher-order boundary value problems with sign changing nonlinear terms. Zbl 1129.34018
Du, Zengji; Zhang, Fubao; Ge, Weigao
2006
On eigenvalue intervals for discrete second order boundary value problems. Zbl 1087.39009
Du, Zengji; Xue, Chunyan; Ge, Weigao
2005
Traveling wave solution for a reaction-diffusion competitive-cooperative system with delays. Zbl 1359.92094
Du, Zengji; Xu, Dongcheng
2016
Positive periodic solutions of $$n$$-species neutral delayed Lotka-Volterra competition system with impulsive perturbations. Zbl 1335.92075
Du, Zengji; Xu, Min
2014
Periodic boundary value problem for first order impulsive differential equation at resonance. Zbl 1146.34025
Cai, Guolan; Du, Zengji; Ge, Weigao
2007
Existence and uniqueness results for third-order nonlinear differential systems. Zbl 1251.34032
Du, Zengji
2011
Asymptotic solutions for a second-order differential equation with three-point boundary conditions. Zbl 1120.34311
Du, Zengji; Bai, Zhanbing
2007
Existence of solitary wave solutions for a nonlinear fifth-order KdV equation. Zbl 1434.35164
Li, Xiaofeng; Du, Zengji; Liu, Jiang
2020
The existence of solutions for third-order vector Robin problems. Zbl 1003.34501
Zhou, Mingru; Du, Zengji
2001
Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation. Zbl 1477.35160
Du, Zengji; Li, Ji
2022
The Green function for a class of Caputo fractional differential equations with a convection term. Zbl 1474.34200
Bai, Zhanbing; Sun, Sujing; Du, Zengji; Chen, YangQuan
2020
Existence of solution for a class of $$n$$th-order multi-point boundary value problem. Zbl 1197.34025
Fu, Zhengmin; Du, Zengji
2010
Existence and uniqueness for second-order vector boundary value problem of nonlinear systems. Zbl 1086.34508
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2005
Existence of solutions for higher order multi-point boundary value problem at resonance. Zbl 1156.34304
Lin, Xiaojie; Du, Zengji; Ge, Weigao
2006
Solutions to a second-order multi-point boundary value problem at resonance. Zbl 1240.34055
Du, Zengji; Meng, Fanchao
2010
Traveling wave solution of a reaction-diffusion predator-prey system. Zbl 1415.35165
Liu, Jiang; Xu, Dongcheng; Du, Zengji
2019
Solvability of a second-order differential equation with integral boundary condition at resonance on an unbounded domain. Zbl 1299.34067
Lin, Xiaojie; Du, Zengji; Zhang, Qin
2013
Solvability of a third order nonlocal boundary value problem at resonance. Zbl 1126.34317
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2006
The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Zbl 1492.35254
Chen, Shuting; Du, Zengji; Liu, Jiang; Wang, Ke
2022
Solitary wave solutions for a generalized KdV-mKdV equation with distributed delays. Zbl 1451.35159
Du, Zengji; Wei, Dandan; Xu, Ying
2014
Geometric singular perturbation analysis to Camassa-Holm Kuramoto-Sivashinsky equation. Zbl 1477.35160
Du, Zengji; Li, Ji
2022
The dynamic properties of a generalized Kawahara equation with Kuramoto-Sivashinsky perturbation. Zbl 1492.35254
Chen, Shuting; Du, Zengji; Liu, Jiang; Wang, Ke
2022
Traveling pulse solutions of a generalized Keller-Segel system with small cell diffusion via a geometric approach. Zbl 1452.35219
Du, Zengji; Liu, Jiang; Ren, Yulin
2021
Traveling waves for nonlocal Lotka-Volterra competition systems. Zbl 1437.35141
Han, Bang-Sheng; Wang, Zhi-Cheng; Du, Zengji
2020
The dynamics of traveling waves for a nonlinear Belousov-Zhabotinskii system. Zbl 1440.35291
Du, Zengji; Qiao, Qi
2020
The solitary wave solutions of the nonlinear perturbed shallow water wave model. Zbl 1440.35006
Ge, Jianjiang; Du, Zengji
2020
Existence of solution of a three-point boundary value problem via variational approach. Zbl 1441.34035
Lian, Wen; Bai, Zhanbing; Du, Zengji
2020
Dynamics of nonconstant steady states of the Sel’kov model with saturation effect. Zbl 1443.35076
Du, Zengji; Zhang, Xiaoni; Zhu, Huaiping
2020
Existence of solitary wave solutions for a nonlinear fifth-order KdV equation. Zbl 1434.35164
Li, Xiaofeng; Du, Zengji; Liu, Jiang
2020
The Green function for a class of Caputo fractional differential equations with a convection term. Zbl 1474.34200
Bai, Zhanbing; Sun, Sujing; Du, Zengji; Chen, YangQuan
2020
Iterative method for a class of fourth-order $$p$$-Laplacian beam equation. Zbl 1458.34046
Bai, Zhanbing; Du, Zengji; Zhang, Shuo
2019
Traveling wave solution of a reaction-diffusion predator-prey system. Zbl 1415.35165
Liu, Jiang; Xu, Dongcheng; Du, Zengji
2019
The existence of solitary wave solutions of delayed Camassa-Holm equation via a geometric approach. Zbl 1392.35223
Du, Zengji; Li, Ji; Li, Xiaowan
2018
Existence of positive periodic solutions for a neutral delay predator-prey model with Hassell-Varley type functional response and impulse. Zbl 1392.34089
Chen, Xiao; Du, Zengji
2018
Existence and asymptotic behaviors of traveling waves of a modified vector-disease model. Zbl 1397.35314
Du, Zengji; Feng, Zhaosheng
2018
Traveling wave phenomena of $$n$$-dimensional diffusive predator-prey systems. Zbl 1387.35091
Du, Zengji; Feng, Zhaosheng; Zhang, Xiaoni
2018
A priori $$L^\infty$$ estimates for solutions of a class of reaction-diffusion systems. Zbl 1338.35220
Du, Zengji; Peng, Rui
2016
Geometric singular perturbation method to the existence and asymptotic behavior of traveling waves for a generalized Burgers-KdV equation. Zbl 1349.37071
Xu, Ying; Du, Zengji; Wei, Lei
2016
Varieties and analytic normalizations of partially integrable systems. Zbl 1357.34072
Du, Zengji; Romanovski, Valery G.; Zhang, Xiang
2016
Traveling waves in a generalized nonlinear dispersive-dissipative equation. Zbl 1342.35312
Shang, Xiaohui; Du, Zengji
2016
Traveling wave solutions of $$n$$-dimensional delayed reaction-diffusion systems and application to four-dimensional predator-prey systems. Zbl 1336.92072
Shang, Xiaohui; Du, Zengji; Lin, Xiaojie
2016
Traveling wave solution for a reaction-diffusion competitive-cooperative system with delays. Zbl 1359.92094
Du, Zengji; Xu, Dongcheng
2016
Solitary waves solutions of singularly perturbed higher-order KdV equation via geometric singular perturbation method. Zbl 1345.35007
Zhuang, Kaige; Du, Zengji; Lin, Xiaojie
2015
Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays. Zbl 1330.37074
Du, Zengji; Feng, Zhaosheng
2014
A second order differential equation with generalized Sturm-Liouville integral boundary conditions at resonance. Zbl 1474.34192
Du, Zengji; Yin, Jian
2014
Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms. Zbl 1308.34107
Du, Zengji; Chen, Xiao; Feng, Zhaosheng
2014
A third-order multi-point boundary value problem at resonance with one three dimensional kernel space. Zbl 1324.34035
Lin, Xiaojie; Zhao, Bensheng; Du, Zengji
2014
Solvability of a third-order multipoint boundary value problem at resonance. Zbl 1474.34125
Du, Zengji; Zhao, Bensheng; Bai, Zhanbing
2014
Solutions to third-order multi-point boundary-value problems at resonance with three dimensional kernels. Zbl 1290.34022
Li, Shuang; Yin, Jian; Du, Zengji
2014
Positive periodic solutions of $$n$$-species neutral delayed Lotka-Volterra competition system with impulsive perturbations. Zbl 1335.92075
Du, Zengji; Xu, Min
2014
Solitary wave solutions for a generalized KdV-mKdV equation with distributed delays. Zbl 1451.35159
Du, Zengji; Wei, Dandan; Xu, Ying
2014
Permanence and almost periodic solution of a Lotka-Volterra model with mutual interference and time delays. Zbl 1351.34080
Du, Zengji; Lv, Yansen
2013
Global asymptotic stability of almost periodic solution for a multispecies competition-predator system with time delays. Zbl 1418.37132
Lin, Xiaojie; Du, Zengji; Lv, Yansen
2013
Solvability of a second-order differential equation with integral boundary condition at resonance on an unbounded domain. Zbl 1299.34067
Lin, Xiaojie; Du, Zengji; Zhang, Qin
2013
Existence and global attractivity of a positive periodic solution to a Lotka-Volterra model with mutual interference and Holling III type functional response. Zbl 1231.37052
Lv, Yansen; Du, Zengji
2011
A note on a third-order multi-point boundary value problem at resonance. Zbl 1229.34028
Lin, Xiaojie; Du, Zengji; Meng, Fanchao
2011
Existence and uniqueness results for third-order nonlinear differential systems. Zbl 1251.34032
Du, Zengji
2011
Existence and global attractivity of positive periodic solution to a Lotka-Volterra model. Zbl 1205.34058
Wang, Xiaoli; Du, Zengji; Liang, Jing
2010
Asymptotic solutions of singularly perturbed second-order differential equations and application to multi-point boundary value problems. Zbl 1200.34061
Du, Zengji; Kong, Lingju
2010
Existence of solution for a class of $$n$$th-order multi-point boundary value problem. Zbl 1197.34025
Fu, Zhengmin; Du, Zengji
2010
Solutions to a second-order multi-point boundary value problem at resonance. Zbl 1240.34055
Du, Zengji; Meng, Fanchao
2010
Solvability of a second-order multi-point boundary value problem at resonance. Zbl 1168.34310
Meng, Fanchao; Du, Zengji
2009
Existence of three solutions for sytems of multi-point boundary value problems. Zbl 1201.34030
Du, Zengji; Kong, Lingju
2009
A multiplicity result for $$p$$-Lapacian boundary value problems via critical points theorem. Zbl 1173.34007
Du, Zengji; Lin, Xiaojie; Tisdell, Christopher C.
2008
Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem. Zbl 1166.34008
Lin, Xiaojie; Du, Zengji; Liu, Wenbin
2008
Solvability of functional differential equations with multi-point boundary value problems at resonance. Zbl 1142.34357
Du, Zengji
2008
Positive solutions of M-point boundary value problem for second-order dynamic equations on time scales. Zbl 1149.34014
Lin, Xiaojie; Du, Zengji
2008
Positive solutions for a second-order three-point discrete boundary value problem. Zbl 1155.39005
Du, Zengji
2008
Positive solutions for some second-order four-point boundary value problems. Zbl 1115.34016
Bai, Zhanbing; Du, Zengji
2007
Existence results for discrete Sturm-Liouville problem via variational methods. Zbl 1129.39007
Tian, Yu; Du, Zengji; Ge, Weigao
2007
Multiple solutions to a three-point boundary value problem for higher-order ordinary differential equations. Zbl 1133.34011
Du, Zengji; Liu, Wenbin; Lin, Xiaojie
2007
Triple positive solutions for some second-order boundary value problem on a measure chain. Zbl 1133.34015
Bai, Zhanbing; Liang, Xiangqian; Du, Zengji
2007
Singularly perturbed third-order boundary value problem for nonlinear systems. Zbl 1131.34043
Du, Zengji
2007
Periodic boundary value problem for first order impulsive differential equation at resonance. Zbl 1146.34025
Cai, Guolan; Du, Zengji; Ge, Weigao
2007
Asymptotic solutions for a second-order differential equation with three-point boundary conditions. Zbl 1120.34311
Du, Zengji; Bai, Zhanbing
2007
Nonlocal boundary value problem of higher order ordinary differential equations at resonance. Zbl 1145.34006
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2006
Positive solutions for higher-order boundary value problems with sign changing nonlinear terms. Zbl 1129.34018
Du, Zengji; Zhang, Fubao; Ge, Weigao
2006
Existence of solutions for higher order multi-point boundary value problem at resonance. Zbl 1156.34304
Lin, Xiaojie; Du, Zengji; Ge, Weigao
2006
Solvability of a third order nonlocal boundary value problem at resonance. Zbl 1126.34317
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2006
Existence and iteration of monotone positive solutions for multipoint boundary value problem with $$p$$-Laplacian operator. Zbl 1095.34009
Ma, De-Xiang; Du, Zeng-Ji; Ge, Wei-Gao
2005
Some higher-order multi-point boundary value problem at resonance. Zbl 1059.34010
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2005
On a third-order multi-point boundary value problem at resonance. Zbl 1072.34012
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2005
Multiple solutions for three-point boundary value problem with nonlinear terms depending on the first order derivative. Zbl 1074.34011
Du, Zengji; Xue, Chunyan; Ge, Weigao
2005
Singular perturbations for third-order nonlinear multi-point boundary value problem. Zbl 1084.34014
Du, Zengji; Ge, Weigao; Zhou, Mingru
2005
Solvability of multipoint boundary value problems at resonance for higher-order ordinary differential equations. Zbl 1081.34017
Lin, Xiaojie; Du, Zengji; Ge, Weigao
2005
Solutions to $$m$$-point boundary value problems of third order ordinary differential equations at resonance. Zbl 1070.34031
Xue, Chunyan; Du, Zengji; Ge, Weigao
2005
A class of third order multi-point boundary value problem. Zbl 1080.34006
Du, Zengji; Cai, Guolan; Ge, Weigao
2005
On eigenvalue intervals for discrete second order boundary value problems. Zbl 1087.39009
Du, Zengji; Xue, Chunyan; Ge, Weigao
2005
Existence and uniqueness for second-order vector boundary value problem of nonlinear systems. Zbl 1086.34508
Du, Zengji; Lin, Xiaojie; Ge, Weigao
2005
Existence of solutions for a class of third-order nonlinear boundary value problems. Zbl 1053.34017
Du, Zengji; Ge, Weigao; Lin, Xiaojie
2004
The existence of solutions for third-order vector Robin problems. Zbl 1003.34501
Zhou, Mingru; Du, Zengji
2001
all top 5

### Cited by 708 Authors

 47 Du, Zengji 41 Ge, Weigao 22 Lin, Xiaojie 17 Jiang, Weihua 13 Bai, Zhanbing 11 Liu, Jiang 10 Huo, Hai-Feng 10 Xia, Yonghui 9 Feng, Meiqiang 9 Galewski, Marek 9 Lei, Chengxia 9 Pei, Minghe 9 Xiang, Hong 8 Feng, Zhaosheng 8 Ji, Dehong 8 Liu, Wenbin 7 Agarwal, Ravi P. 7 Kong, Lingju 7 Ma, Dexiang 7 Sun, Bo 6 Chen, Fengde 6 Cui, Renhao 6 Feng, Hanying 6 Graef, John R. 6 Han, Bang-Sheng 6 Han, Wei 6 Heidarkhani, Shapour 6 Imaga, Ogbu Famous 6 Iyase, Samuel Azubuike 6 Liu, Xiping 6 Tian, Yu 6 Yang, Liu 6 Zhang, Xuemei 6 Zhou, Mingru 5 Bai, Yuzhen 5 Bao, Xiongxiong 5 Jia, Mei 5 Li, Bo 5 Li, Ji 5 Liu, Lishan 5 Ni, Mingkang 5 O’Regan, Donal 5 Sang, Yanbin 5 Su, Hua 5 Wang, Guangwa 5 Wang, Youyu 5 Xu, Fuyi 5 Yang, Zhilin 5 Zhu, Wenjing 4 Chang, Meng-Xue 4 Chang, Sung Kag 4 Chen, Haibo 4 Dogan, Abdulkadir 4 Geng, Fazhan 4 Li, Wan-Tong 4 Li, Xiaowan 4 Nieto Roig, Juan Jose 4 Shen, Chunfang 4 Sun, Jianping 4 Webb, Jeffrey Ronald Leslie 4 Wei, Yongfang 4 Wu, Yonghong 4 Zhang, Zizhen 4 Zhou, Hui 3 Afrouzi, Ghasem Alizadeh 3 Ahmad, Bashir 3 Caristi, Giuseppe 3 Du, Bo 3 Fu, Zhengmin 3 Guo, Zhongkai 3 Jin, Zhen 3 Kang, Shugui 3 Khan, Rahmat Ali 3 Lian, Hairong 3 Lian, Wen 3 Lin, Guo 3 Moradi, Shahin 3 Pang, Huihui 3 Peng, Rui 3 Shen, Jianhe 3 Smejda, Joanna 3 Sun, Bingzhi 3 Sun, Li 3 Sun, Yongping 3 Verma, Amit Kumar 3 Wang, Libo 3 Wang, Qiru 3 Wang, Yuanming 3 Wieteska, Renata 3 Wu, Xiao 3 Xie, Dapeng 3 Yamazaki, Kazuo 3 Yang, Caixia 3 Yang, Huizhong 3 Yang, Qian 3 Yang, Youyuan 3 Zhang, Jihui 3 Zhang, Youwei 3 Zhao, Junfang 3 Zhao, Yulin ...and 608 more Authors
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### Cited in 124 Serials

 43 Boundary Value Problems 42 Applied Mathematics and Computation 28 Nonlinear Analysis. Theory, Methods & Applications. Series A: Theory and Methods 27 Journal of Mathematical Analysis and Applications 24 Advances in Difference Equations 23 Journal of Applied Mathematics and Computing 21 Applied Mathematics Letters 20 Abstract and Applied Analysis 19 Qualitative Theory of Dynamical Systems 16 Discrete and Continuous Dynamical Systems. Series B 15 Computers & Mathematics with Applications 12 Discrete Dynamics in Nature and Society 12 Nonlinear Analysis. Real World Applications 12 Mathematical Biosciences and Engineering 10 Journal of Computational and Applied Mathematics 10 Journal of Differential Equations 8 Journal of Difference Equations and Applications 8 Communications in Nonlinear Science and Numerical Simulation 7 International Journal of Biomathematics 6 Applicable Analysis 6 ZAMP. Zeitschrift für angewandte Mathematik und Physik 6 Mathematical and Computer Modelling 6 Communications on Pure and Applied Analysis 6 Journal of Biological Dynamics 5 Mathematical Methods in the Applied Sciences 5 Discrete and Continuous Dynamical Systems. Series S 5 Differential Equations and Applications 4 International Journal of Computer Mathematics 4 International Journal of Bifurcation and Chaos in Applied Sciences and Engineering 4 Journal of Applied Mathematics 4 Journal of Applied Analysis and Computation 4 Journal of Function Spaces 4 AIMS Mathematics 3 Journal of Mathematical Biology 3 Mathematische Nachrichten 3 Electronic Journal of Differential Equations (EJDE) 3 Nonlinear Dynamics 3 Nonlinear Analysis. Modelling and Control 3 Mathematical Modelling and Analysis 3 Bulletin of the Malaysian Mathematical Sciences Society. Second Series 3 International Journal of Differential Equations 2 Journal of the Franklin Institute 2 Physica A 2 Demonstratio Mathematica 2 Proceedings of the American Mathematical Society 2 Quaestiones Mathematicae 2 Results in Mathematics 2 Acta Applicandae Mathematicae 2 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 2 Applied Mathematics. Series B (English Edition) 2 Filomat 2 Complexity 2 Discrete and Continuous Dynamical Systems 2 Mathematical Problems in Engineering 2 Journal of Inequalities and Applications 2 Journal of Shanghai University 2 Acta Mathematica Sinica. English Series 2 Kragujevac Journal of Mathematics 2 Portugaliae Mathematica. Nova Série 2 Mediterranean Journal of Mathematics 2 Journal of Fixed Point Theory and Applications 2 Afrika Matematika 2 Journal of Mathematics 2 International Journal of Applied and Computational Mathematics 1 Bulletin of the Australian Mathematical Society 1 International Journal of Control 1 International Journal of Mathematical Education in Science and Technology 1 International Journal of Theoretical Physics 1 Mathematical Biosciences 1 Mathematical Notes 1 Rocky Mountain Journal of Mathematics 1 Chaos, Solitons and Fractals 1 Annales Polonici Mathematici 1 Applied Mathematics and Optimization 1 Calcolo 1 Journal of Functional Analysis 1 Journal of Optimization Theory and Applications 1 Mathematics and Computers in Simulation 1 Mathematische Zeitschrift 1 Ricerche di Matematica 1 Bulletin of the Korean Mathematical Society 1 Applied Mathematics and Mechanics. (English Edition) 1 Chinese Annals of Mathematics. Series B 1 Acta Mathematica Hungarica 1 Journal of the Nigerian Mathematical Society 1 Physica D 1 Acta Mathematicae Applicatae Sinica. English Series 1 Applications of Mathematics 1 Applied Mathematical Modelling 1 SIAM Journal on Mathematical Analysis 1 Journal of Dynamics and Differential Equations 1 Calculus of Variations and Partial Differential Equations 1 Journal of Mathematical Sciences (New York) 1 Computational and Applied Mathematics 1 Fractals 1 Opuscula Mathematica 1 Advances in Differential Equations 1 Arab Journal of Mathematical Sciences 1 Differential Equations and Dynamical Systems 1 Positivity ...and 24 more Serials
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### Cited in 25 Fields

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text:
# 70 6-sided dice Question re: Gaming

I play a game where a d6 (a regular, fair 6 sided die) is used to determine whether or not a weapon hits an opponent's vehicle. There is one possible scenario where the attacker can roll 70d6 in one attack step. A "hit" is scored on a 4, 5 or 6 on each die. In other words, you roll all dice, look at each and count up the 4's, 5's and 6's. A 4 counts as 1 hit and a 5 counts as 1 hit while a 6 counts as 2 hits.

Question 1 - What is the probability of scoring at least 20 hits when rolling this 70d6 attack?

Question 2 - How does the probability change if a 3 is also 1 hit?

Thanks!

• Note that not everyone is familiar with die game notation. 30d6 means you throw 30 standard (assumed fair) six-sided dice. – Arthur Sep 26 '18 at 11:29
• Generating functions work as well as anything here. You are interested in the coefficients of $\frac {(4+2x+x^2)^{30}}{6^{30}}$. You'll want to use a computer – lulu Sep 26 '18 at 11:52
• Would anyone be able to explain why this post got voted down? I would like to make sure I meet the criteria expected of this forum next time I post. Thank you. – MattN Sep 26 '18 at 11:59
• @Lulu - THank you. By coefficients, do you mean solving for X when the equation = 0? – MattN Sep 26 '18 at 12:02
• Note: I mistyped in my first comment. It should have been $\frac {(3+2x+x^2)^{30}}{6^{30}}$. If you really want $70$ dice then replace $30$ with $70$, but with that many dice the probability of getting a score of at least $20$ is essentially $1$. – lulu Sep 26 '18 at 12:09

The expected number of hits is just $$70$$ times the expected number for one die. For $$4,5,6$$ hitting, you expect $$\frac 46$$ hit per die or $$\frac {4\cdot 70}6=41\frac23$$ hits on average per roll. If a $$3$$ hits as well, you now expect $$\frac 56$$ from one die, for a total of $$58\frac 13$$ per roll. The chance of scoring at least $$20$$ is very high in either case. We can approximate the distribution with a normal one. For the first case the variance of one die is $$\frac 16\cdot (4+1+1)-(\frac 23)^2=\frac 59$$ so the variance of the whole roll is $$70\cdot \frac 59=39\frac 19$$ and the standard deviation is about $$6$$. The first is over $$3$$ standard deviations low, which happens less than once in $$600$$ tries. The other is even more likely to yield at least $$20$$ hits.

Here is an approximation by simulation: In a million sessions of 70 rolls of the die according to Question 1, I found that it is almost sure to score at least 20 hits. [Notice I have labeled die faces with the appropriate numbers of hits; R treats the three 0's as different faces.]

set.seed(926); m = 10^6
die = c(0,0,0,1,1,2); rolls=70
hits = replicate(m, sum(sample(die,rolls,rep=T)))
mean(hits >= 20)
[1] 0.999998

If you intended 30 rolls of the die (as mentioned in a comment), then its a little over 50%.

set.seed(926); m = 10^6
die = c(0,0,0,1,1,2); rolls=30
hits = replicate(m, sum(sample(die,rolls,rep=T)))
mean(hits >= 20)
[1] 0.540681

Either way, I suggest you try the CLT method proposed by @awkward. Seventy rolls, or even 30, should be enough to get a reasonably good approximation. Then you can compare results from the CLT with results from simulation.

Note: Here is a histogram (blue) for the number of hits in 70 rolls, along with the best-fitting normal curve (orange). The fit is not perfect, but is surely good enough to show that there is almost no probability below 20.

# for rolls=70
mu = mean(hits);  sg = sd(hits);  mu;  sg
[1] 46.66131
[1] 6.235426
hist(hits, prob=T, col="skyblue2", ylim=c(0,.07))
curve(dnorm(x, mu, sg), add=T, col="orange", lwd=2)


text:
How can I add arrows onto gnuplot contour plot?

I have a contour plot for a doublet flow, but I'm trying to add arrows onto the contour plots.

 \begin{axis}[ xshift=8cm,
ticks = none,
view={0}{90},
xmin = -1, ymin = -3,
xmax = 5, ymax= 3,
axis background/.style={fill=white} ]

\addplot3 [ samples=101,smooth,  contour gnuplot={levels={-2.5,-1.75,-1,-0.75,-0.5,-0.4,0,0.4,0.5,0.75,1,1.75,2.5}, draw color = black, labels=false, }] {0-y/((x-2)^2 + y^2)};
\addplot3 [ samples=101,smooth, contour gnuplot={levels={-2.5,-1.75,-1,-0.75,-0.5,-0.4,0,0.4,0.5,0.75,1,1.75,2.5}, draw color = red, labels=false}] {(x-2)/((x-2)^2 + y^2)};

\end{axis}

I have tried using decorate but it doesn't work. Here is an example of what it should look like with the red arrows.

• As it seems this is not doable in an easy way. At least all my attempts to add the arrows sooner or later ended in a "dimension too large" error ... – Stefan Pinnow Jan 27 at 21:04


text:
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Min Hamming distance of a given string from substrings of another string

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text:
My Math Forum Probability paradoxes and other difficulties

 Probability and Statistics Basic Probability and Statistics Math Forum

 April 14th, 2017, 11:21 AM #11 Senior Member   Joined: Mar 2017 From: . Posts: 213 Thanks: 3 Math Focus: Number theory Good example.. the card problem. Lets say there are 52 cards. You are told to choose one at random. The probability of choosing a card is 1. The probability of choosing an ace is 1/52. What the solution to the question was to clarify is that it is not a paradox. Perhaps it was wrongly phrased by using the word random by which you have deduced it is impossible to pick any number randomly in that case. Either way we arrive to the same conclusion. The problem is not a paradox as the writer claimed it is.
April 15th, 2017, 12:41 AM   #12
Senior Member

Joined: Jun 2015
From: England

Posts: 569
Thanks: 150

Quote:
 Most particularly how can something be random and have a probability of 1? You have completely failed to answer, or even offer any answer to that question I asked. Why not?

 April 15th, 2017, 04:52 AM #13 Math Team   Joined: Dec 2013 From: Colombia Posts: 6,783 Thanks: 2197 Math Focus: Mainly analysis and algebra It is perfectly possible to randomly pick a natural number. Two such probability distributions for this are given in the other thread. What is not possible is a uniform probability distribution over an infinite domain. Over an infinite domain, the probability of picking a number $n$ must tend to zero as $|n| \to \infty$. This is required for the cumulative probability function to converge to unity without which we don't have a probability distribution at all. Thanks from Joppy Last edited by v8archie; April 15th, 2017 at 04:55 AM.
April 15th, 2017, 05:38 AM   #14
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Joined: Mar 2017
From: .

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Math Focus: Number theory
Quote:
I don't really know whether your question is really relevant here but I'll still answer it. You can choose something randomly and it's probability would be 1 if we only have one choice.
Read #11. There are two different probabilities we are considering.

April 15th, 2017, 12:02 PM   #15
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From: England

Posts: 569
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Quote:
 Originally Posted by Mariga I don't really know whether your question is really relevant here but I'll still answer it. You can choose something randomly and it's probability would be 1 if we only have one choice. Read #11. There are two different probabilities we are considering.
No this is not an answer or an explanation or a proof or a derivation.

I asked for an explanation since I assert the opposite.

Yes I agree it is possible to construct a process wherein the outcome is randomly dependent on a list of possiblities with preassigned probabilites, one of which is 1.

To me this would also indicate that the process was rigged.

But I do not agree when there is only one choice as in the last sweet in my finite bag of sweets.
In that case the process is not random, it is deterministic.

Do you think a process can simultaneously be both random and deterministic?

In response to your second point,

I do not understand the point of saying you are told to chose an outcome and then saying therefore the probability of there being an outcome is 1.

Yes this is true at the end of the process, but so what.

What if you never reach the end of the process?

April 15th, 2017, 01:05 PM   #16
Senior Member

Joined: Jun 2014
From: USA

Posts: 286
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Quote:
 Originally Posted by v8archie It is perfectly possible to randomly pick a natural number. Two such probability distributions for this are given in the other thread. What is not possible is a uniform probability distribution over an infinite domain. Over an infinite domain, the probability of picking a number $n$ must tend to zero as $|n| \to \infty$. This is required for the cumulative probability function to converge to unity without which we don't have a probability distribution at all.
$\frac{1}{\infty} = 0$ is not technically true. It is undefined.

For example, consider:

$0 = 0 + 0 + 0 + ... = \frac{1}{\infty} + \frac{1}{\infty} + \frac{1}{\infty} + ... = \frac{1}{\infty} ( \, 1 + 1 + 1 + ... ) \, = \frac{\infty}{\infty} = 1$

Also, given:

1) the axiom of choice,
2) a randomly selected infinite binary sequence (created via the theoretical flipping of a coin infinitely many times), and
3) a definition of "select an element of an infinite set uniformly at random" that simply means, in layman's terms, that one and only one element of an infinite set will be selected using a process where all elements of the set have an equal chance of being selected, then

it is possible to select a natural number uniformly at random as well as a real number (and not just a real number on a closed interval, I mean from all of $\mathbb{R}$).

I am happy to post such a proof sometime in the next two weeks. It involves partitioning [0, 1] into an uncountable number of countable subsets, using choice to select a single element from each subset, and then defining a bijection between $\mathbb{N}$ and each subset based on the element selected from it using choice. I struggled with this a while back with Maschke, but have a proof now that works with the above criteria.

 April 15th, 2017, 01:24 PM #17 Senior Member     Joined: Sep 2015 From: CA Posts: 1,238 Thanks: 638 [QUOTE=AplanisTophet;567341]$\frac{1}{\infty} = 0$ is not technically true. It is undefined. For example, consider: $\frac{\infty}{\infty} = 1$ no, you cannot just aribtarily assert this.
April 15th, 2017, 01:40 PM   #18
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From: USA

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[QUOTE=romsek;567343]
Quote:
 Originally Posted by AplanisTophet $\frac{1}{\infty} = 0$ is not technically true. It is undefined. For example, consider: $\frac{\infty}{\infty} = 1$ no, you cannot just aribtarily assert this.
I didn't. $\frac{\infty}{\infty} = 1$ is also undefined.

You don't have to take my word for it though:

https://www.mathsisfun.com/calculus/...-infinity.html

April 15th, 2017, 01:52 PM   #19
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Joined: Sep 2015
From: CA

Posts: 1,238
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Quote:
 Originally Posted by AplanisTophet $\frac{1}{\infty} = 0$ is not technically true. It is undefined. For example, consider: $0 = 0 + 0 + 0 + ... = \frac{1}{\infty} + \frac{1}{\infty} + \frac{1}{\infty} + ... = \frac{1}{\infty} ( \, 1 + 1 + 1 + ... ) \, = \frac{\infty}{\infty} = 1$
despite being couched with the term "consider" this sure looks like an assertion to me.

Quote:
 I am happy to post such a proof sometime in the next two weeks.
of course you are. I will gladly pay you on Tuesday.

Quote:
 It involves partitioning [0, 1] into an uncountable number of countable subsets, using choice to select a single element from each subset, and then defining a bijection between $\mathbb{N}$ and each subset based on the element selected from it using choice.
of course it does. It's clearly much more valid now than it was a few weeks back.

Quote:
 I struggled with this a while back with Maschke, but have a proof now that works with the above criteria.
of course you do. Maybe Kellyanne and Sean can announce it for you.

April 15th, 2017, 02:15 PM   #20
Senior Member

Joined: Aug 2012

Posts: 1,377
Thanks: 327

Quote:
 Originally Posted by AplanisTophet it is possible to select a natural number uniformly at random...
If your probability distribution is countably additive then you can't possibly do that. If the probability of a singleton is zero, then the total probability of all the natural numbers must be zero; and if it's positive, the total probability of all the natural numbers must be infinite. So either you are mistaken or you've broken math. I'll let you form your own conclusion.

However the Vitali set is always a fascinating topic of discussion and actually bears on the Banach-Tarski paradox, which has been a topic of conversation lately. So feel free to have another go at it and I'm sure I'll be unable to resist chiming in.

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text:
# Break a simple compression function of a cryptographic hash function

I have a very simple compression function, which looks like this in C++:

unsigned long long compress(unsigned long long current_block, unsigned long long last){
for(int i = 0; i<16; i++){
last += current_block;
last = last ^ (last << 13);
last = last ^ (last >> 11);
}
return last;
}

It's my own first try of a compression function :)
Now, this function is the core of a cryptographic hash function. All values are 64-bit. The function splits the message into 64-bit blocks, which get passed to the compression function (current_block) together with the sum (addition modulo $2^{64}$) of the last compressions (last). The initial value (the value that is used for last when the compression function is called the first time) is 0.
About the hash function itself: It is a Merkle–Damgård construction and the messages are padded with the length of the original message (64-bit value).

So, now I want to prove that this compression function is not preimage resistant (first and second) and not collision resistant. In order to do that I want to ignore the padding and just try to find the weaknesses of the compression function itself and then think about improves I could make.
In these days it is of course computationally feasible to brute force a 64-bit hash (particularly with a birthday attack), so I understand the term "collision resistant" as "an attacker needs to try all $2^{64}$ possibilities" (in case of a preimage attack).

What I've done so far
It's obvious, that compress(0, 0) == 0, so I could add as many 0-blocks as I want to the beginning of the message. This problem could be solved by using a different initial value.
It seems to be quite difficult to reverse the compression function, because it does 16 additions and 16 mixings, which creates quite a complex relationship and every bit of the input (current_block) can affect every bit of the output.
A general problem I thought of: Couldn't it happen, that we run into a loop? That would happen, if the result after the mixing in one round would be equals the starting value last. In this case, we would have quite a simple relationship as we need only to break one (or a few) mixing rounds. If this problem exists, I think it could be solved by adding a round constant (changing every round) in each round.

Questions I have
Does it make sense to have a compression function that uses 64-bit of the message to create an output of 64-bit? I see no problem with it, but I'm not sure about it as many common hash function use a much bigger message part to compress (MD5 does 512-bit => 128-bit).
Can we expect collisions with the same last-value, like compress(x, 0) == compress(x', 0) ?

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This sounds like a homework assignment. A general hint: Consider how current_block and last are combined - it's simple addition. The next two steps are mixing the state using a reversible operation (assuming << and >> are not cyclical, which they are not in the provided language). –  B-Con Sep 6 '13 at 17:36
As a trivial break, consider the 0 case. What obvious input hashes to 0? If you wanted to generate collisions for 0, how could you force the internal state to reach a value of 0? –  B-Con Sep 6 '13 at 17:49
So we know that compress(0, 0) = 0. Obviously, this can be iterated for an arbitrary number of blocks. Which of the hash properties can we use this fact to break? –  B-Con Sep 10 '13 at 21:12
Let's craft a less trivial collision. We can attack this general scheme, ignoring most of the details of compress. First, this is a Merkle–Damgård construction, so a collision for the compression function breaks the scheme. Note that compress is a permutation (16 rounds of a permutation): If you know the current_block that yielded a new value for last, you can run entire thing backward and get the original last that was updated by current_block. So let's model the for loop as a block cipher E(key, msg), using current_block as the key and last being encrypted/permuted.[...] –  B-Con Sep 11 '13 at 7:56
This question appears to be off-topic because it does not comply with our policy about homework. It does not show prior research or any research effort. It is just a "Please solve my homework for me", with a plain problem copied out of a textbook/homework problem. –  D.W. Sep 12 '13 at 21:13