text:
Penrose zigzag model of Higgs-electron interaction

1. Jul 21, 2012

johne1618

Roger Penrose in Chapter 25 of his book The Road to Reality describes a "zigzag" model of the electron that consists of a pair of massless components one with a left-handed spin (the zig) and the other with a right-handed spin (the zag).

He says that the Dirac equation can be written as a pair of equations which describe each component being continually transformed into the other. The strength of the coupling between these equations depends on the electron mass.

Thus the electron is continually jittering between a massless zig and a zag particle. The energy in this vibrating motion provides the electron with its rest mass. (This is Penrose's explanation of the well-known electron "zitterbewegung")

Penrose then goes on to say that one can think of the Higgs field as taking over the role of the electron mass. One imagines that it is the continual interaction with the Higgs field that causes the electron zig to be converted to the electron zag and vice-versa. Again it is the energy in this vibrating motion that gives rise to the electron's rest mass.

I think this picture is better than the particle-in-molasses picture that one often hears. In my view the Higgs interaction explains the origin of the rest mass/energy of a particle (and therefore its gravitational mass) but not its inertia defined as its resistance to being accelerated. I think inertia might have a "Machian" gravitational cause as outlined in Dennis Sciama's "On the origin of inertia":

Last edited: Jul 21, 2012
2. Jul 21, 2012

Bill_K

The "zig" and "zag" fields are apparently Penrose's terminology for the right- and left-handed components of the electron.
Anything would be better than the Higgs-as-molasses picture.
A pre-Einsteinian idea, intuitive but primitive, and long ago shown to be false.

3. Jul 21, 2012

Are these right- left-handed components the same as the two components of the spinor?

4. Jul 21, 2012

Bill_K

Any Dirac spinor can be decomposed into right- and left-handed parts ψR and ψL using the chirality operators PR = (1 + γ5)/2 and PL = (1 - γ5)/2. For a massless particle, PR and PL commute with H and chirality is a good quantum number. But for a particle with mass, the mass term m(ψLψR + ψRψL) couples them together. The same holds true whether m is put in by hand or generated by the Higgs field.

5. Jul 21, 2012

Ah, thanks. But what is γ5? I thought there were four gamma matrices?

6. Jul 21, 2012

tiny-tim

γ5 is short for iγ0γ1γ2γ3

7. Jul 21, 2012

johne1618

Penrose works with Pauli 2-spinors rather than the more usual Dirac 4-spinors.

As I understand it a 2-spinor directly represents a spinning particle whereas the Dirac 4-spinor decribes a state with both spin and positive and negative energy components that can't be so easily visualised as a particle.

Last edited: Jul 21, 2012
8. Jul 21, 2012

Bill_K

johne1618, Did Penrose say that, or is that your own interpretation? The massless Dirac equation decouples into two equations for two 2-component spinors. Although the equations contain the Pauli matrices, the spinors themselves are Weyl spinors. Also the solutions for each equation include both a positive energy solution and a negative energy solution.

9. Jul 22, 2012

johne1618

Sorry you're right this is largely my interpretation drawing from what Penrose writes in Chapter 24 and 25 of his book - I don't understand the mathematical details myself.

However Penrose definitely does imply that his zigzag particle picture is more directly suggested by a superposition of Weyl 2-spinors rather than the more usual 4-spinor description using the Weyl or chiral basis.

Last edited: Jul 22, 2012


text:
## Elementary Geometry for College Students (6th Edition)

$c = 5$
We can find the value of $c$: $c^2 = a^2+b^2$ $c^2 = (3)^2+(4)^2$ $c^2 = 9+16$ $c^2 = 25$ $c = \pm \sqrt{25}$ $c = -\sqrt{25}~~$ or $~~c = \sqrt{25}$ $c = -5~~$ or $~~c = 5$ Since the value of $c$ must be positive, $c = 5$

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text:
# The set related to the Cantor set

I am considering the set related to the Cantor ternary set.Let A be the set of numbers in [0,1] whose ternary expansions have only finitely many 1's.Prove that $\lambda(A)=0$ where $\lambda$ is the Lebesgue measure.

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What's $\lambda$? The (Lebesgue) measure? –  Gerry Myerson Oct 17 '12 at 1:30
@GerryMyerson Yes –  89085731 Oct 17 '12 at 1:33

Let $C$ be the Cantor set. Let $B$ be the set of numbers in $[0, 1]$ whose ternary expansions consist of finitely many $1$s (the rest are $0$s). Since this set consists of rational numbers only, it's countable.

Let $x \in B$. We know that $\lambda(C) = 0$. We also know that the Lebesgue measure is translation invariant. Therefore, $\lambda(C + x) = 0$. Hence: $$\lambda \left(\bigcup_{x \in B}(C + x)\right) \le \sum_{x \in B} \lambda(C + x) = 0$$

Each number in your set $A$ can be written as the sum of a number in the Cantor set and another in $B$. Hence $A \subset \bigcup_{x \in B}(C + x)$ and $\lambda(A) = 0$.

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Hint: For positive integer $m$ and subset $S$ of $\{1,2,\ldots,m\}$, let $A_{m,S}$ be the set of numbers such that of the first $m$ ternary digits $d_1 d_2, \ldots, d_m$, only $d_j$ for $j \in S$ are 1. Find $\lambda(A_{m,S})$. Then take $m$ to infinity for fixed $S$. Then take the union over all $S$.
Let $S$ be a subset of $\{1,2,\ldots,m\}$ with $k$ elements. Show that exactly $2^{m-k}$ of the $3^m$ possible digit-strings $d_1d_2\ldots d_m$ with $m$ ternary digits have $d_j = 1$ for $j \in S$ and $d_j \ne 1$ for $j \notin S$. So what is $\lambda(A_{m,S})$? Let $A_S = \bigcap_{m \ge \max(S)} A_{m,S}$. What is $\lambda(A_S)$? Now note that $A$ is the union of the sets $A_S$ for all finite subsets $S$ of the positive integers. –  Robert Israel Oct 17 '12 at 6:06


text:
# GEN_LAGUERRE_RULE Generalized Gauss-Laguerre Quadrature Rules

GEN_LAGUERRE_RULE is a MATLAB program which generates a specific generalized Gauss-Laguerre quadrature rule, based on user input.

The rule is written to three files for easy use as input to other programs.

The generalized Gauss-Laguerre quadrature rule is used as follows:

        Integral ( a <= x < +oo ) |x-a|^alpha * exp(-b*(x-a)) f(x) dx

is to be approximated by
        Sum ( 1 <= i <= order ) w(i) * f(x(i))

### Usage:

gen_laguerre_rule ( order, alpha, a, b, 'filename' )
where
• order is the number of points in the quadrature rule.
• alpha is the exponent of |x| in the weight function. The value of alpha may be any real value greater than -1.0.
• a is the left endpoint. Typically this is 0.
• b is the scale factor in the exponential, and is typically 1.
• 'filename' specifies files to be created: file_name_w.txt, file_name_x.txt, and file_name_r.txt, containing the weights, abscissas, and interval limits.

### Languages:

GEN_LAGUERRE_RULE is available in a C++ version and a FORTRAN90 version and a MATLAB version.

### Related Data and Programs:

ALPERT_RULE, a MATLAB library which can set up an Alpert quadrature rule for functions which are regular, log(x) singular, or 1/sqrt(x) singular.

CCN_RULE, a MATLAB program which defines a nested Clenshaw Curtis quadrature rule.

CHEBYSHEV1_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 1 quadrature rule.

CHEBYSHEV2_RULE, a MATLAB program which can compute and print a Gauss-Chebyshev type 2 quadrature rule.

CLENSHAW_CURTIS_RULE, a MATLAB program which defines a Clenshaw Curtis quadrature rule.

GEGENBAUER_RULE, a MATLAB program which can compute and print a Gauss-Gegenbauer quadrature rule.

GEN_HERMITE_RULE, a MATLAB program which computes a generalized Gauss-Hermite quadrature rule.

HERMITE_RULE, a MATLAB program which computes a Gauss-Hermite quadrature rule.

INT_EXACTNESS, a MATLAB program which checks the polynomial exactness of a 1-dimensional quadrature rule for a finite interval.

INT_EXACTNESS_LAGUERRE, a MATLAB program which checks the polynomial exactness of a Gauss-Laguerre quadrature rule.

INTLIB, a FORTRAN90 library which contains routines for numerical estimation of integrals in 1D.

JACOBI_RULE, a MATLAB program which computes a Gauss-Jacobi quadrature rule.

LAGUERRE_RULE, a MATLAB program which computes a Gauss-Laguerre quadrature rule.

LEGENDRE_RULE, a MATLAB program which computes a Gauss-Legendre quadrature rule.

LEGENDRE_RULE_FAST, a MATLAB program which uses a fast (order N) algorithm to compute a Gauss-Legendre quadrature rule of given order.

LINE_FELIPPA_RULE, a MATLAB library which returns the points and weights of a Felippa quadrature rule over the interior of a line segment in 1D.

PATTERSON_RULE, a MATLAB program which computes a Gauss-Patterson quadrature rule.

POWER_RULE, a MATLAB program which constructs a power rule, that is, a product quadrature rule from identical 1D factor rules.

QUADRATURE_RULES, a dataset directory which contains sets of files that define quadrature rules over various 1D intervals or multidimensional hypercubes.

QUADRATURE_RULES_LAGUERRE, a dataset directory which contains triples of files defining Gauss-Laguerre quadrature rules.

TEST_INT_LAGUERRE, a MATLAB library which defines test integrands for Gauss-Laguerre rules.

TRUNCATED_NORMAL_RULE, a MATLAB program which computes a quadrature rule for a normal probability density function (PDF), also called a Gaussian distribution, that has been truncated to [A,+oo), (-oo,B] or [A,B].

### Reference:

1. Milton Abramowitz, Irene Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, 1964,
ISBN: 0-486-61272-4,
LC: QA47.A34.
2. Philip Davis, Philip Rabinowitz,
Methods of Numerical Integration,
Second Edition,
Dover, 2007,
ISBN: 0486453391,
LC: QA299.3.D28.
3. Sylvan Elhay, Jaroslav Kautsky,
Algorithm 655: IQPACK, FORTRAN Subroutines for the Weights of Interpolatory Quadrature,
ACM Transactions on Mathematical Software,
Volume 13, Number 4, December 1987, pages 399-415.
4. Jaroslav Kautsky, Sylvan Elhay,
Calculation of the Weights of Interpolatory Quadratures,
Numerische Mathematik,
Volume 40, 1982, pages 407-422.
5. Roger Martin, James Wilkinson,
The Implicit QL Algorithm,
Numerische Mathematik,
Volume 12, Number 5, December 1968, pages 377-383.
6. Philip Rabinowitz, George Weiss,
Tables of Abscissas and Weights for Numerical Evaluation of Integrals of the form $\int_0^{\infty} exp(-x) x^n f(x) dx$,
Mathematical Tables and Other Aids to Computation,
Volume 13, Number 68, October 1959, pages 285-294.
7. Arthur Stroud, Don Secrest,
Prentice Hall, 1966,
LC: QA299.4G3S7.

### Examples and Tests:

• gen_lag_o4_a0.5_r.txt, the region file created by the command

gen_laguerre_rule ( 4, 0.5, 0.0, 1.0, 'gen_lag_o4_a0.5' )

• gen_lag_o4_a0.5_w.txt, the weight file created by the command

gen_laguerre_rule ( 4, 0.5, 0.0, 1.0, 'gen_lag_o4_a0.5' )

• gen_lag_o4_a0.5_x.txt, the abscissa file created by the command

gen_laguerre_rule ( 4, 0.5, 0.0, 1.0, 'gen_lag_o4_a0.5' )

You can go up one level to the MATLAB source codes.

Last revised on 24 February 2010.


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text:
Convergence of probability measures. 2nd ed.(English)Zbl 0944.60003

Wiley Series in Probability and Statistics. Chichester: Wiley. x, 277 p. (1999).
It seems safe to assume that most probabilists are familiar with the first edition of the author’s classical book on “Weak convergence of probability measures” (1968; Zbl 0172.21201). This second edition is more than just a reprinted version. It takes up the classical theory, but presents it in an often simplified and streamlined manner and adds a number of topics that have been developed since the appearance of the first edition more than 30 years ago.
The book consists of five chapters. Chapters 1 (Weak convergence in metric spaces), Chapter 2 (The space $$C$$) and Chapter 3 (The space $$D$$) are fairly similar to the first edition although the order of topics has been changed in some places and the overall presentation has been tightened. The material on separable stochastic processes and the entire old Section 7 (Characteristic functions, local limit theorems, some examples) have been removed. Prokhorov’s theorem is proved by a different method, empirical processes are now only treated within the setting of the space $$D$$ and the maximal inequalities for dependent random variables in Chapter 3 are proved much more efficiently. In addition, there are some new sections. In Chapter 1, Section 4 treats the asymptotics of long cycles of a random permutation and of large prime divisors of a random integer, and Section 6 contains miscellaneous useful results, including Skorokhod’s representation theorem and the Prokhorov metric. Chapter 2 presents in Section 11 functional limit theorems for lacunary series, and Chapter 3 has new Sections 15 on uniform topologies with applications to empirical processes and 16 on the space $$D[0,\infty)$$. The Poisson limit case is now also briefly treated in Chapter 3. Chapter 4 (Dependent variables) has changed quite drastically. By using Gordin’s method of approximating stationary sequences by martingales, the number of pages here has been cut by more than half even after inclusion of a new second section on prime divisors of a random integer. The final Chapter 5 (Other modes of convergence) is entirely new. It treats Strassen’s theorem and a version of Donsker’s theorem with convergence in probability instead of weak convergence.
At \$ 80 or £65, the book is not low-priced. But it seems destined to become another classic and is of interest even to those who already own the first edition.

MSC:

 60-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory 60B10 Convergence of probability measures 28A33 Spaces of measures, convergence of measures 60B05 Probability measures on topological spaces

Zbl 0172.21201


text:
Tophat Protocol Error Occured At Step 4
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10.7 years ago
ashis.csedu ▴ 100

Hi all, I was doing differential gene and transcript expression analysis by following the Tophat protocol (published in Nature protocol in March 2012) on the same example D. melanogaster as it is mentioned in the paper. But I am confused of an error/warning message at step 4: Running Cuffmerge on all the assemblies to create a single merged transcriptome annotation using the command:

cuffmerge -g genes.gtf -s genome.fa -p 2 assemblies.txt

Here, "genes.gtf" was collected from iGenome package of Ensembl_BDGP5.25 of the species. The reference genome "genome.fa" was also from the same package. And I was using a 2-core processor, so I limitted my thread spawning to 2 at most. And the "assemblies.txt" are the list of GTF files produced at step 2 by cufflinks on the mentioned six experiments.

While executing the above command, the following error/warning message is shown and after 2/3 minutes, it completes where it supposed to be completed in almost 4 hours in my machine. That's why I was thinking that the error must be solved before going to the next step.

[Tue May 29 11:13:43 2012] Beginning transcriptome assembly merge
-------------------------------------------

[Tue May 29 11:13:43 2012] Preparing output location ./merged_asm/
[Tue May 29 11:13:46 2012] Converting GTF files to SAM
[Tue May 29 11:13:51 2012] Quantitating transcripts
Warning: Your version of Cufflinks is not up-to-date. It is recommended that you upgrade to Cufflinks v2.0.0 to benefit from the most recent features and bug fixes (http://cufflinks.cbcb.umd.edu).
Command line:
cufflinks -o ./merged_asm/ -F 0.05 -g genes.gtf -q --overhang-tolerance 200 --library-type=transfrags -A 0.0 --min-frags-per-transfrag 0 --no-5-extend -p 2 ./merged_asm/tmp/mergeSam_fileCA810u
File ./merged_asm/tmp/mergeSam_fileCA810u doesn't appear to be a valid BAM file, trying SAM...
[11:13:53] Inspecting reads and determining fragment length distribution.
Processed 11320 loci.
> Map Properties:
>    Total Map Mass: 67742.00
>    Fragment Length Distribution: Truncated Gaussian (default)
>                  Default Mean: 200
>               Default Std Dev: 80
[11:13:54] Assembling transcripts and estimating abundances.
Processed 11320 loci.
[Tue May 29 11:15:43 2012] Comparing against reference file genes.gtf
Warning: Your version of Cufflinks is not up-to-date. It is recommended that you upgrade to Cufflinks v2.0.0 to benefit from the most recent features and bug fixes (http://cufflinks.cbcb.umd.edu).
Warning: couldn't find fasta record for '2LHet'!
Warning: couldn't find fasta record for '2RHet'!
Warning: couldn't find fasta record for '3LHet'!
Warning: couldn't find fasta record for '3RHet'!
Warning: couldn't find fasta record for 'U'!
Warning: couldn't find fasta record for 'XHet'!
Warning: couldn't find fasta record for 'YHet'!
Warning: couldn't find fasta record for 'dmel_mitochondrion_genome'!
[Tue May 29 11:15:52 2012] Comparing against reference file genes.gtf  ..........

I am not sure why it could not find the fasta record for "2LHet", "2RHet", etc. What do you think would be the problem in my case? Do I have to do anything with the "genome.fa" to solve this?
Thanks.

tophat rna-seq • 6.5k views
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Most obvious thing to first check is to see if the name of the references sequences are different between the fasta file and gtf file. For example does the fasta file have a 2LHet header?

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Thanks for your answer. I looked for the headers in the "genome.fa" and got 7 sequences having headers - 2L,2R,3L,3R,4,M,X. And also I looked in to the "gene.gtf" and found that 46265 entries have header "2L", but only 89 "2LHet". Similarly, there are 60521 "2R" entries, and 470 "2RHet" entries. Cuffmerge did not fail entirely due to the absence of those headers, it actually skipped those entries and showed me warnings, that's all! I did not check at the first time whether it actually did produce the "merged.gtf", but now checked, it certainly did. Thanks Dk for the help. Now I'm doing step 5.

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Glad to see this, because I was trying to run the files from that paper too. I had a question, why in procedure 1,

tophat -p 8 -G genes.gtf -o C1_R1_thout genome C1_R1_1.fq C1_R1_2.fq
tophat -p 8 -G genes.gtf -o C1_R2_thout genome C1_R2_1.fq C1_R2_2.fq
tophat -p 8 -G genes.gtf -o C1_R3_thout genome C1_R3_1.fq C1_R3_2.fq
tophat -p 8 -G genes.gtf -o C2_R1_thout genome C2_R1_1.fq C1_R1_2.fq ＃why this is C2_R1_1.fq C1_R1_2.fq not C2_R1_1.fq C2_R1_2.fq, I thought this is a paired end dataset, so the first three commands above seems right, but I do not understand this command, and the two below. help me, please!
tophat -p 8 -G genes.gtf -o C2_R2_thout genome C2_R2_1.fq C1_R2_2.fq
tophat -p 8 -G genes.gtf -o C2_R3_thout genome C2_R3_1.fq C1_R3_2.fq

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I think you are right. I changed the C1 into C2 for the last three lines.

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I got the same error in tophat protocol error occurred at step 4, I knew the name of the references sequences are different between the fasta file and gtf file. Should I delete the 2LHet header and so on in the gtf file?


text:
Prove by contradiction that a real number that is less than every positive real number cannot be positive

having troubles with this

could someone give me a small hint to get started?

pasmith
Homework Helper
Prove by contradiction that a real number that is less than every positive real number cannot be positive

having troubles with this

could someone give me a small hint to get started?

Suppose $x > 0$ is such that every positive real number is greater than or equal to $x$. What then is $x/2$?

Suppose $x > 0$ is such that every positive real number is greater than or equal to $x$. What then is $x/2$?

I'm sorry but I can't seem to proceed even with this hint :(

pasmith
Homework Helper
I'm sorry but I can't seem to proceed even with this hint :(

Given that $x$ is a positive real number, is $x/2$ a positive real number?

HallsofIvy
Homework Helper
I wouldn't even bother with x/2. Suppose x is positive. Can it then be "less than every positive number"

berkeman
Mentor
Prove by contradiction that a real number that is less than every positive real number cannot be positive

having troubles with this

could someone give me a small hint to get started?

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