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http://mathhelpforum.com/trigonometry/171880-finding-intersection-trig-non-trig-function.html | # Math Help - Finding intersection of trig and non trig function
1. ## Finding intersection of trig and non trig function
Hello
I'm having trouble figuring out the steps needed to solve the intersection between
f(x) = Cos (x)
f(x) = 1 - (2x/Pi)
I don't need the answer, only the steps used to solve such a problem. Thanks
2. Originally Posted by skyd171
Hello
I'm having trouble figuring out the steps needed to solve the intersection between
f(x) = Cos (x)
f(x) = 1 - (2x/Pi)
I don't need the answer, only the steps used to solve such a problem. Thanks
By inspection, x = 0 and x = pi are clearly solutions. There is one more solution but it cannot be found using algebra. In general equations involving such 'mixed functions' cannot be solved exactly using algebra, only approximate solutions using a CAS or numerical procedure can be found.
3. Shame, I've spent about 4 hours trying to solve it with algebra only to find that out. This professor is a real piece of work.
4. Originally Posted by skyd171
Shame, I've spent about 4 hours trying to solve it with algebra only to find that out. This professor is a real piece of work.
My previous post contains an error. The other solution is, again by inspection, x = pi/2.
If you are required to solve exactly equations like this without using technology, the solutions are generally designed to be so simple and obvious that they can be easily seen 'by inspection'. In fact, drawing accurate graphs would have put you on the right track and probably saved you 3 hours and 55 minutes of your time.
I have no problem with what your professor has asked you to do.
5. Yes, I used guess and check with graphs to find this solution. However I had been searching the web for hours for a proper algebraic way to find those roots and alternatively trying to find them numerically on my own. But it is just as informative to know that there is none.
This was only a small taste of my prof. The next problem (which I will not even bother to offend anyone here with) is
finding the area between 7 functions, the graphs of which look like a spider web. He is a swell guy.
thanks for the help
6. Originally Posted by skyd171
[snip] The next problem (which I will not even bother to offend anyone here with) is
finding the area between 7 functions, the graphs of which look like a spider web. He is a swell guy.
thanks for the help
Use of symmetry will most likely simplify the calculations. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8333371877670288, "perplexity": 500.87801220274906}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-11/segments/1424936462099.15/warc/CC-MAIN-20150226074102-00228-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.groundai.com/project/almost-isometries-between-teichmuller-spaces/ | Almost isometries between Teichmüller spaces
# Almost isometries between Teichmüller spaces
Manman Jiang and Lixin Liu and Huiping Pan Manman Jiang
Guangzhou Maritime University, 510275, Guangzhou, P. R. China
Lixin Liu
School of Mathematics and Computational Science, Sun Yat-Sen University, 510275, Guangzhou, P. R. China
Huiping Pan
School of Mathematical Science, Fudan University, 200433, Shanghai, P. R. China
July 15, 2019
###### Abstract.
We prove that the Teichmüller space of surfaces with given boundary lengths equipped with the arc metric (resp. the Teichmüller metric) is almost isometric to the Teichmüller space of punctured surfaces equipped with the Thurston metric (resp. the Teichmüller metric).
This work is partially supported by NSFC, No: 11271378.
Keywords: Teichmüller space, almost isometry, Thurston metric, Teichmüller metric, arc metric.
AMS MSC2010: 32G15, 30F60, 51F99.
## 1. introduction
Let be an oriented surface of genus with boundary components such that . The Euler characteristic of is . Throughout this paper we assume that . Recall that a marked complex structure on is a pair where is a Riemann surface and is an orientation preserving homeomorphism. Two marked complex structures and are called equivalent if there is a conformal map homotopic to . Denote by the equivalence class of . The set of equivalence classes of marked complex structures is the Teichmüller space denoted by .
Let be a Riemann surface with boundary. There exist two different hyperbolic metrics on . One is of infinite area obtained from the Uniformization theorem, the other one is of finite area obtained from the restriction to of the hyperbolic metric on its (Sckottky) double such that each boundary component is a smooth simple closed geodesic (see §LABEL:ssec:double). The second one is called the intrinsic metric on . In this paper when we mention a hyperbolic metric on a surface with nonempty boundary we mean the second one. The correspondence between complex structure and hyperbolic metric provides another approach for the Teichmüller theory. Recall that a marked hyperbolic surface is a hyperbolic surface equipped with an orientation-preserving homeomorphism , where maps each component of the boundary of to a geodesic boundary of . Two marked hyperbolic surfaces and are called equivalent if there is an isometry homotopic to relative to the boundary. The Teichmüller space is also the set of equivalence classes of marked hyperbolic surface. For simplicity, we will denote a point in by , without explicit reference to the marking or to the equivalence relation.
Let be the boundary components of . For any . Let be the set of the equivalence classes of marked hyperbolic metrics whose boundary components have hyperbolic lengths . In particular, is the Teichmüller space of surfaces with punctures. It is clear that . Let be a pants decomposition of , i.e. the complement of on consists of pairs of pants . Let be a set of disjoint simple closed curves whose restriction to any pair of pants consists of three arcs, such that any two of the arcs are not free homotopic with respect to the boundary of . The pair is called a marking of . For any , let be the corresponding Fenchel-Nielsen coordinates with respect to the marking , where represents the lengths of , represents the twists along and represents the lengths of the boundary components (for details about Fenchel-Nielsen coordinates we refer to [Bu]). The Fenchel-Nielsen coordinates induce a natural homeomorphism between Teichmüller spaces and in the following way:
ΦΓ:Tg,n(Λ) ⟶ Tg,n(0) (L,T,Λ) ⟼ (L,T,0).
The goal of this paper is to compare various metrics on the Teichmüller spaces and via the homeomorphism .
###### Definition 1.1.
Two metric spaces and are called almost isometric if there exist a map , two positive constants and , such that both of the following two conditions hold.
1. For any ,
|d2(f(x),f(y))−d1(x,y)|≤B.
2. For any , there exists such that
d2(z,f(x))≤A.
### 1.1. The Thuston metric and the arc metric
An essential simple closed curve on is a simple closed curve which is not homotopic to a single point or a boundary component. An essential arc is a simple arc whose endpoints are on the boundary and which is not homotopic to any subarc of the boundary. Let be the set of homotopy classes of essential simple closed curves on S, be the set of homotopy classes of essential arcs on S, and be the set of homotopy classes of the boundary components.
For any , define
dTh(X1,X2):=logsup[α]∈S(S)lX2([α])lX1([α])
and
dA(X1,X2):=logsup[α]∈A(S)lX2([α])lX1([α]).
From the works [Pan] and [LPST2], both and are asymmetric metric on , which are called the Thurston metric and the arc metric respectively. Moreover, the authors ([LPST2]) observed that
dA(X1,X2)=logsup[α]∈A(S)∪B(S)∪S(S)lX2([α])lX1([α]).
Our first result is the following.
###### Theorem 1.2.
and are almost isometric. More precisely, there is a constant depending on the surface and boundary lengths such that,
|dA(X1,X2)−dTh(ΦΓ(X1),ΦΓ(X2))|≤C1.
###### Remark 1.
Papadopoulos-Su ([PS]) considered the case where is close to zero, they showed that the constant in Theorem 1.2 will tend to zero if tends to zero.
###### Proof of Theorem 1.2.
To prove Theorem 1.2, it suffices to verify that they satisfy the two conditions in Definition 1.1. The first condition follows from Theorem 1.3 and Theorem LABEL:thm:thu-almost. The second condition follows from the fact that is a homeomorphism. ∎
###### Theorem 1.3.
The arc metric and the Thurston metric are almost-isometric in . More precisely, there is a constant depending on the surfaces and boundary lengths such that,
|dA(X1,X2)−dTh(X1,X2)|≤C2.
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https://math.stackexchange.com/questions/608908/prove-homotopic-attaching-maps-give-homotopy-equivalent-spaces-by-attaching-a-ce | # Prove homotopic attaching maps give homotopy equivalent spaces by attaching a cell
Prove: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ and $X\sqcup_gD^n$ are homotopy equivalent.
I think it can be proved by showing they are both deformation retracts of $X\sqcup_H(D^n\times I)$ where $H$ is the homotopy between $f$ and $g$.
However, I have difficult in proving that the deformation retracts are continuous map. In fact, I have difficulty in representing a map in quotient spaces like $X\sqcup_fD^n$. I think a map from $X\sqcup_fD^n$ to $W$ can be represented by two maps: $m_1: X\to W$, $m_2: D^n\to W$, where for $x\in S^{n-1}$, $m_1\circ f(x)=m_2\circ i(x)$.
Then I construct the deformation retract this way: $m_1: X\to X$. For $x\in H(S^{n-1},t)$, $m_1(x)=H(S^{n-1},0)$, otherwise $m_1(x)=x$.
$m_2: D^n\times I\to D^n\times {0}$: $m_2((D^n,t))=(D^n,0)$.
It is easy to verify that $m_1$ and $m_2$ define a map from $X\sqcup_H(D^n\times I)$ to $X\sqcup_fD^n$. As long as this is a continous map, obvioulsy then we find a deformation retract. But it seems such a map is not continous?
• Your idea is good. However, since you do not know anything about $X$, you will have to have $m_1={\rm id}_X$, since you do not really know any other continuous maps from $X$ to $X$ (constant maps aside). So $m_2$ will require more attention (we do know $D^n$!). Draw a picture for $n=1$ and go from there. – Carsten S Dec 16 '13 at 10:58
There is a retraction of $D^n\times I\twoheadrightarrow D^n×\{0\}\cup S^{n-1}×I$ defined via $$r(x,t)=\begin{cases} \left(\frac{2x}{2-t},\ 0\right) &\text{, if }t\le2(1-||x||) \\ \left(\frac x{||x||},2-\frac{2-t}{||x||}\right)&\text{, if }t\ge2(1-||x||) \end{cases}$$ It is easy to prove that this map is well-defined and continuous and a retraction. Then $$d:D^n×I×I\to D^n×I\\ d(x,t,s)=sr(x,t)+(1-s)(x,t)$$ is a homotopy between the identity and $r$, so $r$ is a deformation retraction. But then $(D^n×I)\cup_F X$ deformation retracts onto $(D^n×\{0\}\cup S^{n-1}×I)\cup_H X=(D^n×\{0\})\cup_f X$
Note that a pushout square ($A,X$, and $B$ are arbitrary spaces)
$\$ gives rise to a pushout square$\$
because the quotient map $q:X\sqcup B\to X\cup_f B$ induces a quotient map $q\times 1:X\times I\sqcup B\times I\to(X\cup_f B)\times I$.
This means that a pair of homotopies $F_t:X→Y$, $G_t:B→Y$, such that $F_ti=G_t f$ for all $t\in I$, induces a homotopy $H_t:X∪_f B→Y$
That's the reason why a deformation retraction on $D^n×I$ induces a deformation retraction on the pushout $(D^n×I)\cup_F X$
There is more general result: If $(X,A)$ is cofibered, then $X×I$ deformation retracts to $X×\{0\}\cup A×I$, so if $X$ is glued via two homotopic maps $f$ and $g$ to a space $B$, then $X\cup_f B$ and $X\cup_g B$ are homotopy equivalent.
• Is $r$ or $d$ a deformation retraction? Also can you elaborate more on why $(D^n×\{0\}\cup S^{n-1}×I)\cup_F X=(D^n×\{0\})\cup_f X$? I guess the result follows by using the same argument with deformation retracting the solid cylinder onto $(D^n×\{1\}\cup S^{n-1}×I)$ which in turn gives the identification space with $g$ as the attaching map correct? I suppose $F$ is the homotopy between $f$ and $g$? – Alp Uzman Jan 2 '16 at 16:54
• @A.AlpUzman: The space $(D^n×\{0\}∪S^{n−1}×I)∪_F X$ is homeomorphic to $(D^n×\{0\})∪_f X$ by the map sending $D^n×\{0\}$ and $X$ to itself and $S^{n-1}×I$ to $F(S^{n-1}×I)$, where $F$ is the homotopy from $f$ to $g$ (The asker used $H$ instead of $F$, I've edited my answer to fit the notation in the question). And yes, the result follows exactly by the argument you give. – Stefan Hamcke Jan 3 '16 at 16:36
This is also proved in Topology and Groupoids (as it was in the 1968 edition, "Elements of Modern Topology"); this has some pictures of the crucial mapping cylinder construction $$M(f) \cup X$$ which, if $$i: A \to X$$ is a cofibration, is a useful model of the adjunction space $$B \cup _f X$$ for $$f: A \to B$$. Here is a coloured picture of the homotopy as Fig 7.10:
I just learned that the fact that both adjunction spaces are homotopy equivalent to each other can be seen as an immediate consequence of a general property:
Let $h\mathbf{Top}^B$ denote the homotopy category under $B$, the quotient category of $(B\downarrow\mathbf{Top})$ where we identify $f\sim g:i\to j$ if there is a homotopy $H:f\simeq g$ under $B$, that is $H(i\times 1)=j$. Let $\pi B^A$ denote the track groupoid whose objects are maps $A→B$ and whose arrows are homotopies $H:f\simeq g$ where $H$ and $K$ are identified if there's a continuous deformation between them which leaves $f$ and $g$ fixed.
The statement is that if $j:A\to X$ is a cofibration, there exists a contravariant functor $\beta$ from the track groupoid $\pi B^A$ to the category $h\mathbf{Cof}^B$, the full subcategory of $h\mathbf{Top}^B$ whose objects are cofibrations. This $\beta$ assigns to an $f:A→B$ the cofibration $j_f:B\to X\cup_f B$, and to a morphism $[\phi]:f→g$ the homotopy class $[k]$ of maps $j_g\to j_f$, where $k$ is induced by extending the homotopy $\phi:A→B$ to a homotopy $\Phi:X→X\cup_f B$ and setting $k=\Phi_1\cup j_f$.
You can find the proof in tom Dieck's Algebraic Topology where it is theorem $5.1.9$
Note that $[k]=\beta[\phi]$ is an isomorphisms by functoriality, hence a homotopy equivalence.
• People may like to compare the proof in tom Dieck's book with that in "Topology and Groupoids". They both use in this area the operation of a track groupoid. – Ronnie Brown Nov 15 '18 at 12:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.983460009098053, "perplexity": 102.08173119463001}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560627999273.79/warc/CC-MAIN-20190620210153-20190620232153-00360.warc.gz"} |
https://cs.stackexchange.com/questions/29833/how-do-i-solve-interdependent-recurrence-relations/29834 | # How do I solve interdependent recurrence relations?
I have three functions with values given as
\begin{align*} P(0) &= 0 \quad & P(i+1) &= 5M(i)\\ M(0) &= 1 \quad & M(i+1) &= R(i) + 2P(i)\\ R(0) &= 3 \quad & R(i+1) &= R(i) + 3P(i)\,. \end{align*}
If it was a linear recurrence with one function I could solve it using matrices. But here I cannot bring it to a single relationship. Each one is interrelated with the others.
How do I approach this problem?
Eliminate one recurrence at a time, by plugging in.
For instance, you can use the equation $P(i)=5 M(i-1)$ to eliminate every instance of $P(\cdot)$ in each of the other recurrences. Plugging in, we get
$$M(i) = R(i-1) + 10 M(i-2)$$ $$R(i) = R(i-1) + 15 M(i-2).$$
Now you have two inter-related recurrence relations, instead of three; you've eliminated one.
Do it one more time, say to remove each instance of $M(\cdot)$. You get something like
$$R(i) = R(i-1) + 15 R(i-3) + 150 M(i-4).$$
Do it again, to remove the $M(\cdot)$:
$$R(i) = R(i-1) + 15 R(i-3) + 150 R(i-5) + 1500 M(i-7).$$
Keep doing this, and you'll get a summation, say something like:
$$R(i) = R(i-1) + 15 R(i-3) + 150 R(i-5) + \cdots + 15 \times 10^{(i-3)/2} \times M(0)$$
(if $i$ is odd.)
Now you're down to a single recurrence relation. Solve it. Then, use the equations above to derive the solutions for $P(\cdot)$ and $M(\cdot)$.
• Thanks for the quick reply. I have tried this, I am writing a program for this and calculating the value of R,M,P for a particular value i. Is there any fast way algorithmically. – katori Sep 11 '14 at 9:07
• @katori Yes, there is a fast way algorithmically. Solve the recurrences, as D.W. has shown you how, and then just write a program that calculates $P(i)=\frac{4}{5}i(\log i)^2$ or whatever the real answer is (that function's just an example that I made up). – David Richerby Sep 11 '14 at 9:21
• @D.W. I solved the problem. Thanks to your idea. But instead of R(i) I calculated M(i) which was an independent linear recurrence. Then with the help of given relation ships I calculate R(i) and P(i). – katori Sep 16 '14 at 6:39 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9921322464942932, "perplexity": 733.4613750081973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141717601.66/warc/CC-MAIN-20201203000447-20201203030447-00207.warc.gz"} |
https://planetmath.org/StructuralStability | # structural stability
Given a metric space $(X,d)$ and an homeomorphism $f\colon X\to X$, we say that $f$ is structurally stable if there is a neighborhood $\mathscr{V}$ of $f$ in $\operatorname{Homeo}(X)$ (the space of all homeomorphisms mapping $X$ to itself endowed with the compact-open topology) such that every element of $\mathscr{V}$ is topologically conjugate to $f$.
If $M$ is a compact smooth manifold, a $\mathcal{C}^{k}$ diffeomorphism $f$ is said to be $\mathcal{C}^{k}$ structurally stable if there is a neighborhood of $f$ in $\operatorname{Diff}^{k}(M)$ (the space of all $\mathcal{C}^{k}$ diffeomorphisms from $M$ to itself endowed with the strong $\mathcal{C}^{k}$ topology) in which every element is topologically conjugate to $f$.
If $X$ is a vector field in the smooth manifold $M$, we say that $X$ is $\mathcal{C}^{k}$-structurally stable if there is a neighborhood of $X$ in $\mathscr{X}^{k}(M)$ (the space of all $\mathcal{C}^{k}$ vector fields on $M$ endowed with the strong $\mathcal{C}^{k}$ topology) in which every element is topologically equivalent to $X$, i.e. such that every other field $Y$ in that neighborhood generates a flow on $M$ that is topologically equivalent to the flow generated by $X$.
Remark. The concept of structural stability may be generalized to other spaces of functions with other topologies; the general idea is that a function or flow is structurally stable if any other function or flow close enough to it has similar dynamics (from the topological viewpoint), which essentially means that the dynamics will not change under small perturbations.
Title structural stability StructuralStability 2013-03-22 13:48:45 2013-03-22 13:48:45 Koro (127) Koro (127) 8 Koro (127) Definition msc 37C20 msc 34D30 structurally stable | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 32, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9385907053947449, "perplexity": 135.07924836954612}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039749562.99/warc/CC-MAIN-20181121173523-20181121195523-00355.warc.gz"} |
https://inquiryintoinquiry.com/2014/06/08/peirces-1870-logic-of-relatives-%E2%80%A2-comment-11-24/ | ## Peirce’s 1870 “Logic Of Relatives” • Comment 11.24
We come to the end of the “number of” examples that we noted at this point in the text.
#### NOF 4.5
It is to be observed that
$[\mathit{1}] ~=~ 1.$
Boole was the first to show this connection between logic and probabilities. He was restricted, however, to absolute terms. I do not remember having seen any extension of probability to relatives, except the ordinary theory of expectation.
Our logical multiplication, then, satisfies the essential conditions of multiplication, has a unity, has a conception similar to that of admitted multiplications, and contains numerical multiplication as a case under it.
(Peirce, CP 3.76 and CE 2, 376)
There are problems with the printing of the text at this point. Let us first recall the conventions we are using in this transcription, in particular, $\mathit{1}$ for the italic 1 that signifies the dyadic identity relation and $\mathfrak{1}$ for the “antique figure one” that Peirce defines as $\mathit{1}_\infty = \text{something}.$
CP 3 gives $[\mathit{1}] = \mathfrak{1},$ which I cannot make sense of. CE 2 gives the 1’s in different styles of italics, but reading the equation as $[\mathit{1}] = 1,$ makes the best sense if the “1” on the right hand side is read as the numeral “1” that denotes the natural number 1, and not as the absolute term “1” that denotes the universe of discourse. In this reading, $[\mathit{1}]$ is the average number of things related by the identity relation $\mathit{1}$ to one individual, and so it makes sense that $[\mathit{1}] = 1 \in \mathbb{N},$ where $\mathbb{N}$ is the set of non-negative integers $\{ 0, 1, 2, \ldots \}.$
With respect to the relative term $\mathit{1}"$ in the syntactic domain $S$ and the number $1$ in the non-negative integers $\mathbb{N} \subset \mathbb{R},$ we have:
$v(\mathit{1}) ~=~ [\mathit{1}] ~=~ 1.$
And so the “number of” mapping $v : S \to \mathbb{R}$ has another one of the properties that would be required of an arrow $S \to \mathbb{R}.$
This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 18, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.979194164276123, "perplexity": 638.8158813950078}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187823114.39/warc/CC-MAIN-20171018195607-20171018215607-00638.warc.gz"} |
http://clay6.com/qa/15896/at-10-c-the-value-of-the-density-of-a-fixed-mass-of-an-ideal-gas-divided-by | At $10^{\circ}C$, the value of the density of a fixed mass of an ideal gas divided by its pressure is 'x' . At $110^{\circ}C$ this ratio is :
$\begin {array} {1 1} (1)\;\large\frac{10}{110}x & \quad (2)\;\large\frac{383}{283}x \\ (3)\;\large\frac{110}{10}x & \quad (4)\;\large\frac{283}{383}x \end {array}$
1 Answer
(4) $\large\frac{283}{383}x$
answered Nov 7, 2013 by
0 answers
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1 answer | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9833409190177917, "perplexity": 3312.814010694272}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267861456.51/warc/CC-MAIN-20180618222556-20180619002556-00137.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/182633-kolmogorov-equations.html | # Math Help - Kolmogorov equations
1. ## Kolmogorov equations
Where can i find some information on solving continuous markov chains with the kolmogorov equations.
We have been specifically told that there will be a question of this type on the exam.
The only example we have been given is a simple two state system where you can basically find the eigenvalues and solve it quite easily. I have spent heaps of time on the internet and even in the library trying to find information on solving it for a 3 state, 4 state system etc, but all i can find is the same example in every book over and over again, the simple two state system
my lecturer told me that we will get a three state system in the exam and it will have symetry, where can i find some worked examples, or can someone give me an example and show me how to solve it, This is a pretty neat concept but it is frustrating that all the books give a simple example and assume that you are advanced enough to figure the rest out,
so if theres a mathematical maestro out there i challenge you to explain this concept and gain my admiration and praise! by the way none of my fellow students understand this either!
And the lecturer refuses to explain it in detail for some reason. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8894062042236328, "perplexity": 261.6511877315349}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447266.73/warc/CC-MAIN-20151124205407-00195-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/is-this-a-good-substitution-that-will-work.184439/ | # Is this a good substitution that will work
1. Sep 13, 2007
### rock.freak667
1. The problem statement, all variables and given/known data
Prove $$\int_0^{1} \frac{1}{\sqrt{x^2+6x+25}} = ln(\frac{1+\sqrt{2}}{2})$$
2. Relevant equations
3. The attempt at a solution
$$\int_0^{1} \frac{1}{\sqrt{x^2+6x+25}} = \int_0^{1} \frac{1}{\sqrt{(x+3)^2+16}}$$
Let $$x+3=4tan\theta$$ so that $$dx=4sec^2\theta d\theta$$
and so the problem becomes
$$\int \frac{4sec^2\theta}{\sqrt{16tan^2\theta+16}} d\theta$$
giving $$\int sec\theta d\theta = ln|sec\theta + tan\theta|+ K$$
Last edited: Sep 13, 2007
2. Sep 14, 2007
### danago
Isolate theta in the substitution you made i.e. $$\theta=arctan\frac{x+3}{4}$$. From there, you should be able to evaluate the definite integral, and come to the required solution.
3. Sep 14, 2007
### Gib Z
You are correct so far, now just change the bounds accordingly to your substitutions. The first bound, x=1, so put that into your subsitution, tan theta = 1, ie theta = pi/4. Do the same for the other bound, and evaluate from your last line.
4. Sep 14, 2007
### HallsofIvy
Staff Emeritus
It is not absolutely necessary to let
$$\theta=arctan\frac{x+3}{4}$$
(and then use trig identities). Imagine a right triangle with one angle $\theta$ since you know
$$tan(\theta)= \frac{x+3}{4}$$,
the triangle has "opposite side" of length x+3 and "near side" of 4.
By the Pythagorean theorem, the square of the hypotenuse is $(x+3)^2+ 16= x^2+ 6x+ 25$.
Then $sec(\theta)$, hypotenuse over near side is $$\frac{x^2+ 6x+ 25}{4}$$
and
$$tan(\theta)$$
is, of course,
[tex]\frac{x+3}{4}[/itex].
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https://brilliant.org/problems/interesting-integral-2/ | # Interesting Integral
Calculus Level 5
If the value of the integral
$\displaystyle\int_{0}^{\infty}\dfrac{(x^{2}+4)\ln x}{x^{4}+16}\text{ }\text{d}x$
can be expressed as $$\dfrac{\pi\ln a}{b\sqrt{c}}$$ where $$a$$ is a prime number and $$c$$ is square free find $$a+b+c$$.
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9631812572479248, "perplexity": 377.7494331001797}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719033.33/warc/CC-MAIN-20161020183839-00553-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://mathoverflow.net/questions/127845/eigenfunctions-of-fourth-order-differential-operator | # Eigenfunctions of fourth-order differential operator
This a question where I have thought quite long about:
The eigenfunctions (or also normal modes) of an dry Euler beam subject to free-free boundary conditions are given by
$$\frac{\partial^4\psi}{\partial x^4}=\lambda_k^4\psi\qquad(0\le x\le1)\,,$$ $$\frac{\partial^2\psi}{\partial x^2}=\frac{\partial^3\psi}{\partial x^3}=0\qquad(x=0\text{ and }x=1)\,,$$
The solution of this problem is given by $$\psi_0(x)=1\qquad\text{and}\qquad\psi_1(x)=\sqrt3(2x-1),$$ $$\psi_k(x)=\frac{\cosh((\frac12-x)\lambda_k)}{\cosh(\frac12\lambda_k)}+\frac{\cos((\frac12-x)\lambda_k)}{\cos(\frac12\lambda_k)}\qquad (k\ge 2 \text{ even}),$$ $$\psi_k(x)=\frac{\sinh((\frac12-x)\lambda_k)}{\sinh(\frac12\lambda_k)}+\frac{\sin((\frac12-x)\lambda_k)}{\sin(\frac12\lambda_k)}\qquad (k\ge 2\text{ odd}),$$ where $$\cosh(\lambda_k)\cos(\lambda_k)=1\qquad (k\ge2)\,,$$ This is a complete set of eigenfunctions. These eigenfunctions are orthogonal (it can be shown that they are even orthonormal) in terms of the standard scalar product, since the fourth order derivative is a self-adjoint operator subject to the given boundary conditions and a standard smoothness condition for $\psi$. I would like to know if this set of eigenfunctions is also an orthonormal basis in $L_2([0,1])$. There seems to be a Sturm-Liouville theory for fourth order differential operators. Is there any standard book which discusses such a problem?
You need to specify $\lambda_k$ more clearly. More precisely, what is the spectrum of this operator? The equation
$$\cos x\cosh x =1$$
seems to have a unique solution $mu_k$ on any interval of the form $(k\pi/2, k\pi/2+\pi)$, $k\in\mathbb{Z}$, so I assume the spectrum might be $\mu_k^4$, $k\in\mathbb{Z}$?!? (Please edit you question to remove this ambiguity.) In any case if the boundary value problem is elliptic (please check that) then the spectrum is discrete. In particular it can be determined by finding the eigenfunctions which means solving some ode's. My guess is that you found all the eigenfunctions, i.e., the system you found is complete.
Update. You need to check two things: 1) the boundary value problem is elliptic 2) it is symmetric. I'll deal with the 2nd issue first because it is faster. Denote by $A$ the operator
$$A=\frac{d^4}{dx^4}.$$
A simple integration by parts shows that for any $u,v\in C^4([0,1])$ we have
$$\int_0^1 \bigl(\; (Au) -u(Av)\;\bigr) dx=\sum_{j=0}^3(-1)^j\bigl( u^{(3-j)}(1)v^{(j)}(1)- u^{(3-j)}(0) v^{j}(0)\;\bigr).$$
If the function $u$ satisfies your boundary conditions $u^{(k)}(x)=0$ for $k=2,3$, $x=0,1$ the above equality simplifies a bit
$$\int_0^1 \bigl(\; (Au) -u(Av)\;\bigr) dx= \sum_{j=2}^3(-1)^j\bigl( u^{(3-j)}(1)v^{(j)}(1)- u^{(3-j)}(0) v^{j}(0)\;\bigr).$$
If the function $v$ satisfies the same boundary conditions as $u$, then the last equality takes the very simple form
$$\int_0^1 \bigl(\; (Au) -u(Av)\;\bigr) dx= 0.$$
This says that the boundary value problem is symmetric, or formally selfadjoint.
The ellipticity of this problem is another issue. The most readable account I could find is in Chap. 20 vol.3 of the book The Analysis of Linear Partial Differential Operators by the late great Lars Hormander.
The ellipticity of the boundary value problem requires that the symbol of your operator $A$ be elliptic (which it is) and that the boundary value conditions should satisfy the so called Lopatinskii-Schapiro conditions.
In your case they are trivially satisfied because you work on a one-dimensional space $[0,1]$. The upshot is that in your case the boundary conditions are elliptic. We can form the unbounded operator $\newcommand{\bD}{\boldsymbol{D}}$
$$\hat{A}: \bD(\hat{A})\subset L^2(0,1)\to L^2(0,1), u\mapsto \frac{d^4 u}{dx^4}$$
Where the domain $\bD(\hat{A})$ of $\hat{A}$ consists of functions in the Sobolev space $L^{4,2}(0,1)$ (four weak derivatives in $L^2$) such that $u^{(j)}(x)=0$ for $x=0,1$, $j=2,3$.
The results in the above monograph show that $\hat{A}$ viewed as an unbounded operator on the Hilbert space $L^2(0,1)$, is closed, densely defined, selfadjoint and has compact resolvent. This is all you need. Arguably, the above argument is a bit heavy, and it feels like hunting a mosquito using a bazooka.
There is a direct, more elementary approach to proving that $\hat{A}$ has compact resolvent. Observe first that the above integration by parts formulae show that the operator $\hat{A}+1$ is positive, i.e.,
$$(\hat{A}u,u)_2+(u,u)_2>0,\;\;\forall u\in \bD(\hat{A})\setminus 0,$$
where $(-,-)_2$ denotes the $L^2$-inner product. Hence $\hat{A}+1$ is injective. Then follow the strategy in the proof of Theorem 8.22 in Brezis' book Functional Analysis, Sobolev Spaces and Partial Differential Equations to prove that $\hat{A}+1$ is invertible and it's inverse is compact as an operator $L^2(0,1)\to L^2(0,1)$.
• Thanks, I have edited the question. What do you mean with the problem is elliptic? I know only that the operator is self-adjoint and non-negative. However, the spectrum is discrete. And the solution displayed is also complete. – Moritz Reinhard Apr 17 '13 at 18:02
• Because of lack of comment space I'll give my reply as an update to my original answer. – Liviu Nicolaescu Apr 17 '13 at 18:54
• I know the term self-adjoint. – Moritz Reinhard Apr 17 '13 at 19:24
• Thank you! I do not mind to shoot with a bazooka on a mosquito. I will study the chapter you have cited. – Moritz Reinhard Apr 19 '13 at 21:35
Choose some $\lambda \ne \lambda_k$ for all $k$. Consider the resolvent $R_\lambda = (\lambda-\partial_x^4)^{-1}$. This is a compact operator on the Hilbert space $H = \{ \psi \in L^2([0,1]) \; | \; \psi''(0) = \psi''(1) = \psi'''(0) = \psi'''(1) = 0\}$. Thus $H$ has a basis of eigenvectors of $R_\lambda$. But $R_\lambda$ and $\partial_x^4$ share eigenvectors, thus $H$ has a basis of eigenvectors of $\partial_x^4$ as desired. The key bit is using compactness to show that a maximizing sequence for the Rayleigh quotient converges to an eigenvector. The argument can be found in Lax's Functional Analysis Ch 28 Thm 3.
• How do you know that the operator $R_\lambda=(\lambda^4-\partial_x^4)^{-1}$ is compact? If I know compactness, then Hilbert-Schmidt theorem directly implies that my set of eigenfunctions is an orthonormal basis. How can I show compactness of $R_\lambda$ or is there a reference? I do not understand your last two sentences – Moritz Reinhard Apr 17 '13 at 18:38
• Furthermore, your space $H$ does not make sense. – Moritz Reinhard Apr 17 '13 at 19:06
• (1) You are absolutely right about the space. $H$ makes no sense. Instead of $L^2$ I should have $H^4$. (2) I was confused about where you were stuck. The last two sentences are probably not relevant. (3) Perhaps I'm being quite sloppy here, but: given $R_\lambda f = g$ we have $\|g\|_{L^2} \le C\|f\|_{L^2}$ and $\|g''''\|_{L^2} \le \lambda\|g\|_{L^2} + \|f\|_{L^2}$ which implies \|g\|_{H^4} \le C'\|f\|_2$. Since bounded sets in$H^4([0,1])$are compact in$L^2([0,1])\$, the resolvent is compact. – Aaron Hoffman Apr 18 '13 at 14:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9515092968940735, "perplexity": 176.8182390909582}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178361808.18/warc/CC-MAIN-20210228235852-20210301025852-00416.warc.gz"} |
https://www.thinkib.net/mathhlsl/page/14110/calculus | # Calculus
#### Syllabus Content - SL & HL
The Calculus Topic is the largest syllabus topic in terms of the recommended teaching hours for both SL and HL. In SL, 40 teaching hours are recommended while it is 48 hours for HL. Although there are more hours recommended for teaching calculus in HL than in SL, it is worth noting that the Calculus Topic takes up a larger percentage (≈ 27%) of the total teaching hours for the SL course (40 out of a total of 150 hours) than the HL course where the Calculus Topic is 48 out of 240 hours (20%). Regardless of the details of recommended teaching hours, it is clear that the Calculus Topic is a very important one for Maths SL and Maths HL students.
The syllabus content that is in HL but not in SL are primarily the following items:
♦ informal idea of continuity
♦ derivatives of reciprocal trig functions
♦ derivatives of general exponential and logarithmic functions
♦ derivatives of inverse trig functions
♦ implicit differentiation
♦ related rates of change
♦ area of a region enclosed by a curve and the y-axis
♦ volume of revolution about the y-axis
♦ integration by substitution involving sophisiticated ('non-standard') substitutions
♦ integration by parts
Of these items, it could be argued that the most important in terms of teaching effort will be implicit differentiation, related rates, sophisticated integration by substitution, and integration by parts.
It is also worth noting how the content of the Calculus Topic changed from the previous syllabus (last exams 2013) to the current syllabus (first exams 2014).
► main changes for the HL core syllabus:
• differential equations (solution by separation of variables) removed and put in the Calculus Option Topic
• informal idea of continuity added
• oblique asymptotes no longer specifically mentioned
• although previously implied the following are now specifically stated:
- relationship between graphs of f, f´ and f´´
- calculating total distance travelled (linear motion)
► main changes for the SL syllabus:
• relationship between graphs of f, f´ and f´´ added
• calculating total distance travelled (linear motion) added
• integration by substitution of the form $$\int {f\left( {g\left( x \right)} \right)\,} g'\left( x \right)dx$$ added
#### Teaching Approaches
Although the mathematical background of students entering Maths SL or HL can vary greatly, it is fair to say that calculus is one area of mathematics that few students have any significant experience with before entering either course. Many, if not most, students will have had exposure to some of the content in each of the other five syllabus topics (algebra, functions & equations, trigonometry, vectors, statistics & probability). It is also fair to say that gaining a sufficient fluency with many of the concepts and skills in calculus is often more difficult for students compared to concepts and skills in other syllabus topics. Because the content in the Calculus Topic is mostly, if not completely, new to students and because it often takes students a bit more time to adequately absorb the ideas and techniques in calculus it is strongly recommended that a teacher plan substantial review of calculus material during the 2nd year of the course.
Although there are several topics in HL that are not in SL (e.g. related rates, integration by parts), I would still recommend that the teaching of the calculus syllabus topic in both SL and HL is approached by dividing the content into three different teaching units: Differential Calculus I - Fundamentals, Differential Calculus II - Further Techniques & Applications, and Integral Calculus. It's possible to simply teach the Calculus Topic in two parts - differential and integral calculus, but I feel it's worth breaking up differential calculus into two parts for a few reasons. Perhaps the best reason is that since the topic is very new to students it's helpful to not rush too quickly through initial concepts such as limits of functions (and notation), the limit definition of the derivative (and finding slope from 'first principles'), and the rate of change of a function.
The support and teaching materials on this site are organized into these three teaching units and smaller teaching topics as folllows:
Differential Calculus I - Fundamentals
• differentiation basics
• maxima & minima
• tangents & normals
Differential Calculus II - Further Techniques & Applications
• further differentiation methods
• optimization
• implicit differentiation & related rates (HL only)
Integral Calculus
• integration basics
• integration by substitution
• further integration methods (HL only)
• areas & volumes
• modelling linear motion
go here for a set of unit tests on calculus - differential or integral calculus - for both HL and SL
## Selected Pages
free
### tangents & normals8 March 2018
Quick links:► downloadable teaching materials for tangents & normals► syllabus content for the Calculus Topic: SL syllabus...
more
### Calculus Tests & Review10 November 2018
This page contains several unit tests on calculus - differential calculus & integral calculus - for both HL and SL. Most...
more
### areas & volumes17 September 2018
Quick links:► downloadable teaching materials for areas & volumes► syllabus content for the Calculus Topic: SL syllabus...
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http://repub.eur.nl/pub/14064/ | We develop empirical tests for stochastic dominance efficiency of a given investment portfolio relative to all possible portfolios formed from a given set of assets. Our tests use multivariate statistics, which result in superior statistical power properties compared to existing stochastic dominance efficiency tests and increase the comparability with existing mean-variance efficiency tests. Using our tests, we demonstrate that the mean-variance inefficiency of the CRSP all-share index relative to beta-sorted portfolios can be explained by tail risk not captured by variance. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8005345463752747, "perplexity": 1343.8229432073372}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440644062327.4/warc/CC-MAIN-20150827025422-00193-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://brilliant.org/problems/co-prime-or-not-co-prime/ | # Co-prime or Not Co-prime?
$\begin{cases} a &=& 100 + 85i \\ b &=& 208 + 39i \\ c &=& 188 + 22i \\ \end{cases}$
Given that $a, b, c$ are Gaussian integers shown above, if $d$ is the greatest common divisor (GCD) of $a, b, c,$ what is the value of $|d|^{2}?$
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https://www.encyclopediaofmath.org/index.php?title=Radon-Nikod%C3%BDm_theorem&oldid=27238 | ##### Actions
generalized measure, real valued measure
2010 Mathematics Subject Classification: Primary: 28A33 [MSN][ZBL] $\newcommand{\abs}[1]{\left|#1\right|}$
A classical theorem in measure theory first established by J. Radon and O.M. Nikodym, which has the following statement.
Let $\mathcal{B}$ be a $\sigma$-algebra of subsets of a set $X$ and let $\mu$ and $\nu$ be two measures on $\mathcal{B}$. If $\nu$ is absolutely continuous with respect to $\mu$, i.e. $\nu (A)=0$ whenever $\mu (A) = 0$, then there is a $\mathcal{B}$-measurable nonnegative function $f$ such that $$\label{e:R-N} \nu (B) = \int_B f\, d\mu \qquad \forall B\in \mathcal{B}\, .$$ The function $f$ is uniquely determined up to sets of $\mu$-measure zero. The theorem can be generalized to signed measures, $\mathbb C$-valued measures and, more in general, vector-valued measures (see Signed measure). More precisely, let $\mu$ be a (nonnegative real-valued) measure on $\mathcal{B}$, $V$ be a finite-dimensional vector-space and $\nu:\mathcal{B}\to V$ a $\sigma$-additive function such that $\nu (A) = 0$ whenever $\mu (A) =0$. Then there is a function $f\in L^1 (\mu, V)$ such that \ref{e:R-N} hold. See also Vector measure for more general statements.
#### References
[AmFuPa] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001 [Bo] N. Bourbaki, "Elements of mathematics. Integration" , Addison-Wesley (1975) pp. Chapt.6;7;8 (Translated from French) MR0583191 Zbl 1116.28002 Zbl 1106.46005 Zbl 1106.46006 Zbl 1182.28002 Zbl 1182.28001 Zbl 1095.28002 Zbl 1095.28001 Zbl 0156.06001 [DS] N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) MR0117523 [Bi] P. Billingsley, "Convergence of probability measures" , Wiley (1968) MR0233396 Zbl 0172.21201 [He] E. Hewitt, K.R. Stromberg, "Real and abstract analysis" , Springer (1965) [Ma] P. Mattila, "Geometry of sets and measures in euclidean spaces". Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. MR1333890 Zbl 0911.28005 [Ni] O. M. Nikodym, "Sur une généralisation des intégrales de M. J. Radon". Fund. Math. , 15 (1930) pp. 131–179 [Ra] J. Radon, "Ueber lineare Funktionaltransformationen und Funktionalgleichungen", Sitzungsber. Acad. Wiss. Wien , 128 (1919) pp. 1083–1121
How to Cite This Entry: | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9482601881027222, "perplexity": 1038.2367246045887}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-30/segments/1563195526254.26/warc/CC-MAIN-20190719140355-20190719162355-00328.warc.gz"} |
https://pub.uni-bielefeld.de/publication/2394939 | # Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath
Wetzorke I (2001)
Bielefeld: Fakultät für Physik.
Bielefeld Dissertation | English
Author
Supervisor
Karsch, Frithjof
Department
Year
PUB-ID
### Cite this
Wetzorke I. Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath. Bielefeld: Fakultät für Physik; 2001.
Wetzorke, I. (2001). Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath. Bielefeld: Fakultät für Physik.
Wetzorke, I. (2001). Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath. Bielefeld: Fakultät für Physik.
Wetzorke, I., 2001. Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath, Bielefeld: Fakultät für Physik.
I. Wetzorke, Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath, Bielefeld: Fakultät für Physik, 2001.
Wetzorke, I.: Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath. Fakultät für Physik, Bielefeld (2001).
Wetzorke, Ines. Lattice QCD Calculations of Hadron Spectra and Spectral Functions in the Vacuum and in a Thermal Heat Bath. Bielefeld: Fakultät für Physik, 2001.
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https://sassafras13.github.io/Ex11_1/ | # Example 11.1 (Nise)
In this post I am going to design a proportional controller to meet a set of specifications using the Bode plot of the plant. This is presented as Example 11.1 in Nise [1]. The question asks for the gain required to yield a 9.5% overshoot in the transient response, and requires the solution to be presented in the frequency domain [1]. Here’s the flow of the answer [1]:
(1) Draw out the block diagram of the system and pick some nominal gain to start designing. (A good trick is to pick the gain so that the numerator is equal to the factored out coefficients of the denominator, or offset by an order of magnitude. In this case, the gain is set to 3.6 so that the numerator and denominator are a factor of 10 away from each other.) Draw the Bode plot for the uncompensated system.
(2) Calculate the desired damping coefficient for the given percent overshoot. We find that the coefficient is about 0.6 in this example.
(3) Calculate the phase margin required to meet the damping coefficient specification.
(4) Check the current phase margin on the uncompensated system. Now find the frequency that corresponds to the desired phase margin. In this case, we want a phase margin of about 60 degrees and the Bode plot tells us that this phase margin occurs at a frequency of 10 rad/s.
(5) Look at the magnitude plot. How much additional gain do you need to add to the system to move the crossover frequency to 10 rad/s? I find that I need to add about 40dB, or a gain of 100, to my current system. This means that my new compensator gain needs to be 3.6 * 100 = 360.
Next time I will go through more complex examples of designing lead and lag compensators. But before then we are going to take a quick detour into the fundamentals of Nyquist plots because I am still finding them quite challenging.
Figure 1
#### References:
[1] Nise, Norman S. Control Systems Engineering, 4th Ed. John Wiley & Sons, Inc. 2004.
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http://link.springer.com/article/10.1023%2FA%3A1025780028846 | International Journal of Theoretical Physics
, Volume 42, Issue 7, pp 1461–1478
# Quantum Algorithm for Hilbert's Tenth Problem
• Tien D Kieu
Article
DOI: 10.1023/A:1025780028846
Kieu, T.D. International Journal of Theoretical Physics (2003) 42: 1461. doi:10.1023/A:1025780028846
## Abstract
We explore in the framework of Quantum Computation the notion of Computability, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle—that is, if certain Hamiltonian and its ground state can be physically constructed according to the proposal—quantum computability would surpass classical computability as delimited by the Church—Turing thesis. It is thus argued that computability, and with it the limits of Mathematics, ought to be determined not solely by Mathematics itself but also by Physical Principles. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9326413869857788, "perplexity": 579.4685665655634}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721027.15/warc/CC-MAIN-20161020183841-00216-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.atptree.com/en/glossary/available-water/ | water remaining in the soil after gravitational water has drained and before the permanent wilting point has been reached. (compare to field capacity. gravitational water. permanent wilting point. and saturation point).
available water (Wikipedia)
Water activity or aw is the partial vapor pressure of water in a substance divided by the standard state partial vapor pressure of water. In the field of food science, the standard state is most often defined as the partial vapor pressure of pure water at the same temperature. Using this particular definition, pure distilled water has a water activity of exactly one. As temperature increases, aw typically increases, except in some products with crystalline salt or sugar.
Higher aw substances tend to support more microorganisms. Bacteria usually require at least 0.91, and fungi at least 0.7. See also fermentation.
Water migrates from areas of high aw to areas of low aw. For example, if honey (aw ≈ 0.6) is exposed to humid air (aw ≈ 0.7), the honey absorbs water from the air. If salami (aw ≈ 0.87) is exposed to dry air (aw ≈ 0.5), the salami dries out, which could preserve it or spoil it.
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http://en.wikipedia.org/wiki/Fast_Syndrome_Based_Hash | # Fast Syndrome Based Hash
Fast Syndrom-based hash Function (FSB)
General
Designers Daniel Augot, Matthieu Finiasz, Nicolas Sendrier
First published 2003
Derived from McEliece cryptosystem and Niederreiter cryptosystem
Successors Improved Fast Syndrome Based Hash Function
Related to Syndrom-based Hash Function
Detail
Digest sizes Scalable
In cryptography, the Fast Syndrome-based hash Functions (FSB) are a family of cryptographic hash functions introduced in 2003 by Daniel Augot, Matthieu Finiasz, and Nicolas Sendrier. [1] Unlike most other cryptographic hash functions in use today, FSB can to a certain extent be proven to be secure. More exactly, it can be proven that breaking FSB is at least as difficult as solving a certain NP-complete problem known as Regular Syndrome Decoding so FSB is provably secure. Though it is not known whether NP-complete problems are solvable in polynomial time, it is often assumed that they are not.
Several versions of FSB have been proposed, the latest of which was submitted to the SHA-3 cryptography competition but was rejected in the first round. Though all versions of FSB claim provable security, some preliminary versions were eventually broken. [2] The design of the latest version of FSB has however taken this attack into account and remains secure to all currently known attacks.
As usual, provably security comes at a cost. FSB is slower than traditional hash functions and uses quite a lot of memory, which makes it impractical on memory constrained environments. Furthermore, the compression function used in FSB needs a large output size to guarantee security. This last problem has been solved in recent versions by simply compressing the output by another compression function called Whirlpool. However, though the authors argue that adding this last compression does not reduce security, it makes a formal security proof impossible. [3]
## Description of the hash function
We start with a compression function $\phi$ with parameters ${n,r,w}$ such that $n > w$ and $w \log(n/w) > r$. This function will only work on messages with length $s = w\log(n/w)$; $r$ will be the size of the output. Furthermore, we want $n,r,w,s$ and $\log(n/w)$ to be natural numbers, where $\log$ denote the binary logarithm. The reason for $w \cdot \log(n/w) > r$ is that we want $\phi$ to be a compression function, so the input must be larger than the output. We will later use the Merkle-Damgård construction to extend the domain to inputs of arbitrary lengths.
The basis of this function consists of a (randomly chosen) binary $r \times n$ matrix $H$ which acts on a message of $n$ bits by matrix multiplication. Here we encode the $w\log(n/w)$-bit message as a vector in $(\mathbf{F}_2)^n$, the $n$-dimensional vector space over the field of two elements, so the output will be a message of $r$ bits.
For security purposes as well as to get a faster hash speed we want to use only “regular words of weight $w$” as input for our matrix.
### Definitions
• A message is called a word of weight $w$ and length $n$ if it consists of $n$ bits and exactly $w$ of those bits are ones.
• A word of weight $w$ and length $n$ is called regular if in every interval $[(i-1)w, i w)$ it contains exactly one nonzero entry for all $0 < i . More intuitively, this means that if we chop up the message in w equal parts, then each part contains exactly one nonzero entry.
### The Compression Function
There are exactly $(n/w)^w$ different regular words of weight $w$ and length $n$, so we need exactly $\log((n/w)^w)= w \log(n/w) = s$ bits of data to encode these regular words. We fix a bijection from the set of bit strings of length $s$ to the set of regular words of weight $w$ and length $n$ and then the FSB compression function is defined as follows :
1. input: a message of size $s$
2. convert to regular word of length $n$ and weight $w$
3. multiply by the matrix $H$
4. output: hash of size $r$
This version is usually called Syndrome Based Compression. It is very slow and in practice done in a different and faster way resulting in Fast Syndrome Based Compression. We split $H$ into sub-matrices $H_i$ of size $r \times n/w$ and we fix a bijection from the bit strings of length $w\log(n/w)$ to the set of sequences of $w$ numbers between 1 and $n/w$. This is equivalent to a bijection to the set of regular words of length $n$ and weight $w$, since we can see such a word as a sequence of numbers between 1 and $n/w$. The compression function looks as follows:
1. Input: message of size $s$
2. Convert $s$ to a sequence of $w$ numbers $s_1,\dots,s_w$ between 1 and $n/w$
3. Add the corresponding columns of the matrices $H_i$ to obtain a binary string a length $r$
4. Output: hash of size $r$
We can now use the Merkle-Damgård construction to generalize the compression function to accept inputs of arbitrary lengths.
### Example of the compression
Situation and initialization: Hash a message $s = 010011$ using $4 \times 12$ matrix H
$H = \left(\begin{array}{llllcllllcllll} 1&0&1&1 &~& 0&1&0&0 &~& 1&0&1&1 \\ 0&1&0&0 &~& 0&1&1&1 &~& 0&1&0&0 \\ 0&1&1&1 &~& 0&1&0&0 &~& 1&0&1&0 \\ 1&1&0&0 &~& 1&0&1&1 &~& 0&0&0&1 \end{array}\right)$
that is separated into $w = 3$ sub-blocks $H_1$, $H_2$, $H_3$.
Algorithm:
1. We split the input $s$ into $w = 3$ parts of length $\log_2(12/3) = 2$ and we get $s_1 = 01$, $s_2 = 00$, $s_3 = 11$.
2. We convert each $s_i$ into an integer and get $s_1 = 1$, $s_2 = 0$, $s_3 = 3$.
3. From the first sub-matrix $H_1$, we pick the column 2, from the second sub-matrix $H_2$ the column 1 and from the third sub-matrix the column 4.
4. We add the chosen columns and obtain the result $r = 0111 \oplus 0001 \oplus 1001 = 1111$.
## Security Proof of FSB
The Merkle-Damgård construction is proven to base its security only on the security of the used compression function. So we only need to show that the compression function $\phi$ is secure.
A cryptographic hash function needs to be secure in three different aspects:
1. Pre-image resistance: Given a Hash h it should be hard to find a message m such that Hash(m)=h
2. Second pre-image resistance: Given a message m1 it should be hard to find a message m2 such that Hash(m1) = Hash(m2)
3. Collision resistance: It should be hard to find two different messages m1 and m2 such that Hash(m1)=Hash(m2)
Note that if an adversary can find a second pre-image, than he can certainly find a collision. This means that if we can prove our system to be collision resistant, it will certainly be second-pre-image resistant.
Usually in cryptography hard means something like “almost certainly beyond the reach of any adversary who must be prevented from breaking the system”. We will however need a more exact meaning of the word hard. We will take hard to mean “The runtime of any algorithm that finds a collision or pre-image will depend exponentially on size of the hash value”. This means that by relatively small additions to the hash size, we can quickly reach high security.
### Pre-image resistance and Regular Syndrome Decoding (RSD)
As said before, the security of FSB depends on a problem called Regular Syndrome Decoding (RSD). Syndrome Decoding is originally a problem from coding theory but its NP-Completeness makes it a nice application for cryptography. Regular Syndrome Decoding is a special case of Syndrome Decoding and is defined as follows:
Definition of RSD: Given $w$ matrices $H_i$ of dimension $r \times (n/w)$ and a bit string $S$ of length $r$ such that there exists a set of $w$ columns, one in each $H_i$, summing to $S$. Find such a set of columns.
This problem has been proven to be NP-Complete by a reduction from 3-dimensional matching. Again, though it is not known whether there exist polynomial time algorithms for solving NP-Complete problems, none are known and finding one would be a huge discovery.
It is easy to see that finding a pre-image of a given hash $S$ is exactly equivalent to this problem, so the problem of finding pre-images in FSB must also be NP-Complete.
We still need to prove collision resistance. For this we need another NP-Complete variation of RSD: 2-Regular Null Syndrome Decoding.
### Collision resistance and 2-Regular Null Syndrome Decoding (2-NRSD)
Definition of 2-NRSD: Given $w$ matrices $H_i$ of dimension $r \times (n/w)$ and a bit string $S$ of length $r$ such that there exists a set of $w'$ columns, two or zero in each $H_i$, summing to zero. $(0 < w' < 2w)$. Find such a set of columns.
2-NRSD has also been proven to be NP-Complete by a reduction from 3-dimensional matching.
Just like RSD is in essence equivalent to finding a regular word $w$ such that $Hw = S$, 2-NRSD is equivalent to finding a 2-regular word $w'$ such that $Hw'=0$. A 2-regular word of length $n$ and weight $w$ is a bit string of length $n$ such that in every interval $[(i-1)w , iw)$ exactly two or zero entries are equal to 1. Note that a 2-regular word is just a sum of two regular words.
Suppose that we have found a collision, so we have Hash(m1) = Hash(m2) with $m_1\neq m_2$. Then we can find two regular words $w_1$ and $w_2$ such that $Hw_1=Hw_2$ . We then have $H(w_1+w_2)= Hw_1 + Hw_2 =2Hw_1=0$; $(w_1 + w_2)$ is a sum of two different regular words and so must be a 2-regular word of which the hash is zero, so we have solved an instance of 2-NRSD. We conclude that finding collisions in FSB is at least as difficult as solving 2-NRSD and so must be NP-Complete.
The latest versions of FSB use the compression function Whirlpool to further compress the hash output. Though this cannot be proven, the authors argue that this last compression does not reduce security. Note that even if one were able to find collisions in Whirlpool, one would still need to find the collisions pre-images in the original FSB compression function to find a collision in FSB.
### Examples
Solving RSD, we are in the opposite situation as when hashing. Using the same values as in the previous example, we are given $H$ separated into $w=3$ sub-blocks and a string $r = 1111$. We are asked to find in each sub-block exactly one column such that they would all sum to $r$. The expected answer is thus $s_1 = 1$, $s_2 = 0$, $s_3 = 3$. This is known to be hard to compute for large matrices.
In 2-NRSD we want to find in each sub-block not one column, but two or zero such that they would sum up to 0000 (and not to $r$). In the example, we might use column (counting from 0) 2 and 3 from $H_1$, no column from $H_2$ column 0 and 2 from $H_3$. More solutions are possible, for example might use no columns from $H_3$.
### Linear cryptanalysis
The provable security of FSB means that finding collisions is NP-complete. But the proof is a reduction to a problem with asymptotically hard worst-case complexity. This offers only limited security assurance as there still can be an algorithm that easily solves the problem for a subset of the problem space. For example, there exists a linearization method that can be used to produce collisions of in a matter of seconds on a desktop PC for early variants of FSB with claimed 2^128 security. It is shown that the hash function offers minimal pre-image or collision resistance when the message space is chosen in a specific way.
### Practical security results
The following table shows the complexity of the best known attacks against FSB.
Output size (bits) Complexity of collision search Complexity of inversion
160 2100.3 2163.6
224 2135.3 2229.0
256 2190.0 2261.0
384 2215.5 2391.5
512 2285.6 2527.4
## Genesis
FSB is a speed-up version of Syndrom-based hash function (SB). In the case of SB the compression function is very similar to the encoding function of Niederreiter's version of McEliece cryptosystem. Instead of using the parity check matrix of a permuted Goppa code, SB uses a random matrix $H$. From the security point of view this can only strengthen the system.
## Other properties
• Both the block size of the hash function and the output size are completely scalable.
• The speed can be adjusted by adjusting the number of bitwise operations used by FSB per input bit.
• Bad instances exist and one must take care when choosing the matrix $H$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 122, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9222990274429321, "perplexity": 1284.5903299470297}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678682243/warc/CC-MAIN-20140313024442-00070-ip-10-183-142-35.ec2.internal.warc.gz"} |
https://www.idea.int/node/309166 | # Electoral system for national legislature - Kiribati
Country:
Kiribati
Question:
Electoral system for national legislature
TRS
Source:
Kiribati, Laws of Kiribati, Revised Edition 1979; “Chapter 29b Elections (Elections Regulations)”, accessed 15 April 2020
Declaration of result
25. When the result of the poll has been ascertained, the Electoral Officer of result shall forthwith declare to be elected the candidate, or in the case of an election for which 2 or 3 members are to be elected, the 2 or 3 candidates as the case may be, for whom the greatest number of votes have been cast, and shall also declare the number of votes for each and every candidate, whether elected or not:
Provided that the Electoral Officer shall not declare any candidate to be elected unless the number of votes cast in favour of that candidate is in excess of one-half of the total number of ballot papers counted in accordance with regulation 19.
Further election in certain circumstances
26. (2) Where another election is to be held by virtue of the operation of the proviso to regulation 25, of the candidates who failed to secure a number of votes in excess of one-half of the total number of ballot papers counted, the following shall be entitled to be candidates at such election -
(a) where 1 vacancy remains to be filled, the 3 candidates for whom the greatest number of votes were cast in the 1st election;
(b) where 2 vacancies remain to be filled, the 4 candidates for whom the greatest number of votes were cast in the 1st election;
(c) where 3 vacancies remain to be filled, the 5 candidates for whom the greatest number of votes were cast in the 1st election, and
in ascertaining the candidates for whom the greatest number of votes were cast for the purpose of this paragraph, a candidate who received a number of votes in excess of one-half of the total number of ballot papers counted and thereby was declared to be elected in the 1st election shall be ignored, and where there is an equality of votes between candidates such that a determination of the correct number of candidates for whom the greatest number of votes were cast is not possible, all of the candidates between whom there was the equality of votes shall be candidates. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8309365510940552, "perplexity": 850.7617070299804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323583423.96/warc/CC-MAIN-20211016043926-20211016073926-00402.warc.gz"} |
http://mathoverflow.net/questions/121173/perturbation-of-morse-functions-at-critical-points-leaving-stable-manifolds-inva | # Perturbation of Morse functions at critical points leaving stable manifolds invariant
Let $f$ be a given Morse-Smale function $f$ on $\mathbb R^n$ with finite many critical points and sufficient growth at $\infty$ like $\langle x, f(x)\rangle \geq |x|$ (cp. MO120858).
Is there a way to find a perturbation $\tilde f$ of $f$ such that:
1. the eigenvalues of the Hessian are controlled in a quantitative way without changing the index
2. and the stable manifolds of $\tilde f$ and $f$ are the same?
One naive way is to use the local stable manifold theorem to linearize the coordinates of the Morse function $f$ around a critical point $p$ and then do a quadratic perturbation. But this is only sufficient, if there are no other stable manifolds in the neighborhood of the critical point $p$, i.e. only if $W^u_{\mathrm{loc}}(p)\cap W^s(q) =\emptyset$ for all $q\in\mathrm{Crit(f)}$.
On the other hand the Morse-Smale property ensures that $W^u(p)\cap W^s(q)$ is again a manifold and one can use the fact in the construction. From this point of view such a construction looks possible but technical involved. Is there a more simple approach?
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https://aferro.dynu.net/research_engineering/ilp_predicates/ | # Summary
The goal of this post is to showcase how to encode predicates into Linear Program (LP) constraints by solving two real-worldish problems, a simpler and a harder one. Examples of what I mean with predicates are:
\begin{aligned} \neg (a \wedge b), \quad a, b \in \{0, 1\} \quad &\text{i.e. solution is invalid ifa$and$b$are$1$}\\ a = b, \quad a, b \in \mathbb{Z} \quad &\text{i.e. solution is valid if and only if$a = b} \end{aligned}
Encoding such rules into LPs can be extremely useful, allowing us to marry the expressiveness of predicates with the theoretical guarantees and existing high-quality software related to LP optimization (e.g. CVXPY, Gurobi and CPLEX).
Despite their intuitively simple appearance, encoding predicates as LP constraints can be tricky sometimes. While the examples and explanations here could be useful by themselves, I hope that the thought process can also help others connecting the overall goals with the actual constraints when doing LP. It can also be useful to understand related literature.
I provided explanations and working Python code for two examples:
• A regular LP where specific combinations of parameters are forbidden Python script
• A multi-group matching problem with constraints on user preferences and group properties Python script
If you find the explanations too verbose, you can skip straight to the code samples. At the end I included some considerations regarding LP performance and some interesting connections with Graph Theory and Machine Learning, but please keep in mind that this is a practical post, I could never hope or dare to cover LP or convex optimization sufficiently.
# General Background
The general idea in a Linear Program is to solve an optimization objective in the form
\begin{aligned} \min_{x} \quad c^Tx \quad \textrm{s.t.} \quad \begin{cases} Gx \leq h\\ Ax = b \end{cases} \end{aligned}
usually all numbers are real- or complex-valued, but in the case of a (Mixed-) Integer Linear Programs (ILPs), we have the further constraint that all or some entries of the $x$ vector must be boolean or in $\mathbb{Z}$. This constraint is not a minor detail, and in general makes the problem much more difficult to solve than the real-valued counterpart (ILPs are NP-complete in the general case). Still, they are convex and one way of leveraging that is by expressing them as ILPs and feeding them to a high-quality solver. In a practical way, we can think of LPs as follows: if we have an objective in the form of a linear combination, and we are able to express all the required constraints in the form of linear (in)equalities, the problem is convex and can be cast in the form of an LP.
The problem is therefore defined by the vectors and matrices that form the objective and constraints. See e.g. the docstring for cvxopt.glpk.ilp as an example of an ILP Python interface: we only provide matrices and vectors for the objective and constraints, hit optimize, and the solver does the rest. All we need to know for today is how to encode our higher-level goals into such objectives and constraints!
ilp(...)
Solves a mixed integer linear program using GLPK.
(status, x) = ilp(c, G, h, A, b, I, B)
PURPOSE
Solves the mixed integer linear programming problem
minimize c'*x
subject to G*x <= h
A*x = b
x[k] is integer for k in I
x[k] is binary for k in B
ARGUMENTS
c nx1 dense 'd' matrix with n>=1
G mxn dense or sparse 'd' matrix with m>=1
h mx1 dense 'd' matrix
A pxn dense or sparse 'd' matrix with p>=0
b px1 dense 'd' matrix
I set of indices of integer variables
B set of indices of binary variables
status if status is 'optimal', 'feasible', or 'undefined',
a value of x is returned and the status string
gives the status of x. Other possible values of status are: 'invalid formulation',
'infeasible problem', 'LP relaxation is primal
infeasible', 'LP relaxation is dual infeasible',
'unknown'.
x a (sub-)optimal solution if status is 'optimal',
'feasible', or 'undefined'. None otherwise
For more background on LPs and convex optimization, I absolutely recommend the lecture series by Prof. Stephen Boyd at Stanford university, which can be found in YouTube (full video list). Lecture 5 introduces LP:
Details on how exactly do (I)LP solvers tackle the optimization are even further out of scope for this practical post. Interested readers may want to take a look at algorithm families like branch-and-cut and techniques like interior-point methods.
The elegance, computational efficiency and theoretical guarantees of LPs, together with the fact that a vast amount of problems can be naturally expressed via linear objectives and constraints, makes LPs one of the most popular tools in optimization. Typical examples are:
• The budget for water and iron must be equal to the budget for nitrogen, and can’t surpass the budget for ethanol.
• Overall costs can’t surpass $1000 • Every liter of mercury weights ~13.53 times a liter of water • For every 2 atoms of hydrogen, we need exactly 1 atom of oxygen • Typical objectives: minimize weight, costs, durations… where each element contributes separately to a weighted sum Examples like the above can be abundantly found online and in the literature. Less documented are constraints like the following ones: • If$x_a$is active,$x_b$must be active as well • If$\{x_a, x_b\}$and$\{x_b, x_c\}$are connected,$\{x_a, x_c\}$must be connected as well (transitivity) • We want the$K$-best solution (i.e. forbid the$K-1$best ones) Constraints like these can increase the expressivity of our LP repertoire, e.g. through the encoding of non-linear behaviour. Conversely, solving problems like these via LP can be very advantageous since we can leverage theoretical and practical advantages associated with LPs. But in many cases, they involve operators like absolute values and some boolean logic that, despite their conceptual simplicity, can be surprisingly unintuitive to implement as LPs. To help the intuition and provide a bit of systematic, I’ve gathered a series of tips in the next section, followed by examples. # Practical Tips for ILP programming As we’ve seen, an encoded ILP has basically three sets of matrices and vectors: • The$c$and$x$vectors compose our minimization objective$c^Tx$. While we put specific weights in$c$, for$x$we only define the domain (e.g. boolean, real-valued, integer…), leaving the specific values to be found by the solver. In a typical ILP all objects that aren’t$x$are real-valued. • The$G$matrix and$h$vector define our inequality constraints, so that$Gx \leq h$must hold for any valid solution. • The$A$matrix and$b$vector define our equality constraints, so that$Ax \leq b$must hold for any valid solution. By this convention, the$i^{th}$column of$G, A, c^T$corresponds to the$x_i$variable. Then, each row of$G, h$corresponds to an inequality constraint, and each row of$A, b$to an equality constraint. When we are coding, we typically create these objects based on the known size of the problem, populate the contents of$c, G, h, A, b$, and feed everything to the solver. The only step that is really error-prone and challenging is encoding and populating the contents. For that, there are a few tricks&tips that may be helpful (the next sections will exemplify most, if not all of them): • Encoded constraints can look very differently to the original goals. Furthermore, the tiniest mistake in the encoding can easily lead to infeasible or inexact programs. Such bugs can be very difficult to catch! Even if we think all is well, I strongly recommend investing some time in testing the results against the original goals. • Linear constraints can be split by columns, inequality rows, and equality rows. Conventionally, columns correspond to individual variables, and rows to (in)equality constraints. • Since they are common to all types of constraints, it is a good idea to start partitioning the columns based on how many variables do we have, and group them by variable type (i.e. all booleans together, all integers together…). Then, partition the column indexes based on semantics (e.g. columns 1-10 are the variables for the 10 clients we have). • Once we have the column partitions, we can start developing our constraints in the rows. Start with the simpler, more basic ones, and add them incrementally. Usually it makes sense to expand both equalities and inequalities in parallel, whichever makes most sense at a given point. Here it also helps immensely to keep track of the number of constraints needed for each goal, and extracting the indexes beforehand. • When designing the columns, make sure that the defined variables are sufficient to encode everything. Existing variables can be composed into new ones to create more complex behaviour. Usually, these auxiliary or control variables are only there to model behaviour and do not contribute to the objective, i.e. the correspoding$c$entries are zero. • Running the solver at any point is a good way of ensuring we aren’t encoding anything unfeasible (that would mean our program doesn’t have any possible solution, so we likely have a bug in our encoding). At the beginning, we may not have enough constraints and the solver may return unbonded results. That’s fine, keep adding constraints until it provides bounded solutions. • Make sure that columns have consistent semantic meaning across all$A, G, c, x$encodings. E.g. the$10^{th}$column always represents the$10^{th}$client and nothing else. This is kind of obvious but overlooking this can result in hard-to-find bugs. • Remove redundancies in the sets of constraints: sometimes, removing a constraint doesn’t alter the problem space. E.g. if we have a set of nonnegative variables, we don’t need to require that their sum is nonnegative. While some solvers do this automatically, some optimizations are beyond their current capabilities. Coming up with the minimal set of constraints can be a creative and fun task that leads to (sometimes substantial) size reduction of the LPs and subsequent speedups, so it’s always worth looking for! • Plotting the encoded constraints can help tremendously with debugging. Particularly, I’d recommend to choose a diverging colormap like the one in the image below, where white corresponds to zero, red to positive values and blue to negatives. It is particularly helpful to clip the colormap to reasonably small values, otherwise it can saturate and smaller entries will look too similar to zero. Many times, a simple look at the map reveals bugs and design issues. More specifically to the thought process of converting predicates into specific (in)equality constraints, I found that the following systematic was of great help: 1. Identify relevant changepoints: Note that linear constraints have linear boundaries, so we need to “draw a line” at some point. Observing how the pertinent variables change, and where would we like to have a changepoint is a good first step. Sometimes changepoints are exposed by particular linear combinations of variables. 2. Define control variables and set their fixed/don’t care behaviour via further inequality constraints: we define new variables that don’t contribute to the objective (i.e. they have zero associated cost,$c_c=0$for control variable$x_c$). They are only there to enforce constraints among other variables. We then adapt the changepoints from step 1 so that the variable has a fixed value on one side of the boundary (e.g.$0$or$1$), and an unconstrained value on the other (i.e. “don’t care” or$DC$). 3. Enforce the condition via equality constraints: In this step we add further constraints to associate the fixed$x_c$values from step 2 with feasible or infeasible conditions. E.g. if we add an equality constraint like$x_c = 1$, we know that all solutions with$x_c = 0$will be infeasible. Note that we still need to provide goals that present specific changepoints in a meaningful way, which by itself can be already unintuitive. This systematic helps converting them to specific constraints, allowing us to focus on the higher-level modeling part. # Example 1: Avoiding Collisions Consider the following scenario: 1. We’re hiring software developers for our killer app. We found 6 candidates:$a, b, c, d, e, f$. 2. Each person has agreed to a different salary, respectively:$1000, 1500, 1200, 1350, 900, 1300$. 3. Furthermore, our Agile specialists at the interview team were able to quantify the value they would bring to the company in terms of scrum story points:$3, 3.5, 3.1, 3.3, 2.5, 3$. 4. We are seeking to maximize the number of scrum points acquired, while keeping the budget below$5000$. 5. But there is a catch: We got a note from HR, we can’t hire both$a$and$b(better not ask). Which subset of the candidates satisfies our objective and the constraints? We can solve this automatically, efficiently and with guarantees via LP. Points 2 to 4 can be directly translated into an LP as follows: \begin{aligned} c \quad := \quad &(-3, -3.5, -3.1, -3.3, -2.5, -3)\\ x \quad := \quad &(x_1, x_2, x_3, x_4, x_5, x_6)^T \quad \in \quad \{ 0, 1 \}^6\\ G \quad = \quad &\begin{pmatrix} 1000 & 1500 & 1200 & 1350 & 900 & 1300 \end{pmatrix}\\ h \quad = \quad &5000\\ &\min_{x} \quad c^Tx \quad \textrm{s.t.} \quad Gx \leq h \end{aligned} But adding the constraint in point 5 requires a bit more thought, as it doesn’t translate directly into a linear (in)equality like the others. Time to identify, define and enforce! ### 1. Identify changepoints A relevant changepoint occurs whenever bothx_1$and$x_2$are$1$: we want that to be infeasible. If we add them, that case would reach a value of$2$, while all other cases have less value. We are then able to draw a line between our relevant scenario and all others via the$(x_1 + x_2) \leq 1$constraint:$x_1x_2x_1 + x_2(x_1 + x_2) \leq 1000$feasible$011$feasible$101$feasible$112$infeasible This way we’ve achieved our goal of identifying the relevant changepoint. In fact, since this is a simpler problem involving just one changepoint, we can solve it without the addition of any control variables. The following threshold would ensure the infeasibility of$a \wedge b$: $$x_1 + x_2 \leq 1$$ In this case,$x_1=x_2=1$can’t happen, while all other possibilities are allowed, so we’re done. Control variables are only needed when we’re composing multiple constraints and some sort of memory is needed. Still, let’s see how would that play out! ### 2. Define control variable$x_e$The next step would be to add an extra boolean variable to our LP,$x_e$, and decide which region is fixed and which one is don’t care. In our model, we’d like$x_1 = x_2 = 1$to result in$x_e = 1$being fixed. The rest is$DC$. This can be enforced with: $$x_1 + x_2 - x_e \leq 1$$ It can help to make a table with the$0, 1, DC$values as follows:$x_1x_2x_e$with$x_1 + x_2 - x_e \leq 100DC01DC10DC111$### 3. Enforce the constraint: Now$x_e$behaves as we want, but we still need to add a further constraint to ensure the infeasibility of$x_e = 1$. In this case it is simple because$x_e$is boolean. We enforce: $$x_e = 0$$ And we’re done. All the$DC$combinations will collapse then to$0$, but this won’t harm the solution because all of them are still feasible. We can now encode the complete LP with the control variable$x_eas follows: \begin{aligned} c \quad := \quad &(-3, -3.5, -3.1, -3.3, -2.5, -3, 0)\\ x \quad := \quad &(x_1, x_2, x_3, x_4, x_5, x_6, x_e)^T \quad \in \quad \{ 0, 1 \}^6\\ G \quad = \quad &\begin{pmatrix} 1000 & 1500 & 1200 & 1350 & 900 & 1300 & 0\\ 1 & 1 & 0 & 0 & 0 & 0 & -1 \end{pmatrix}\\ h \quad = \quad &\begin{pmatrix} 5000\\ 1 \end{pmatrix}\\ A \quad = \quad &\begin{pmatrix} 0 & 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}\\ b \quad = \quad &0\\ &\min_{x} \quad c^Tx \quad \textrm{s.t.} \begin{cases} Gx \leq h\\ Ax = b \end{cases} \end{aligned} Note that in this case we added 2 extra constraints and 1 extra variable, while in the shorter version we would just add 1 extra constraint and 0 extra variables. Although not always precise, a popular rule of thumb for measuring the size of an LP is number of variables times number of constraints. Let’s see that in action! import cvxpy as cp import numpy as np FLOAT_DTYPE = np.float64 # objective x = cp.Variable(7, boolean=True) c = FLOAT_DTYPE([-3, -3.5, -3.1, -3.3, -2.5, -3, 0]) # inequality constraints G = FLOAT_DTYPE([[1000, 1500, 1200, 1350, 900, 1300, 0], [1, 1, 0, 0, 0, 0, -1]]) h = FLOAT_DTYPE([5000, 1]) # equality constraints A = FLOAT_DTYPE([[0, 0, 0, 0, 0, 0, 1]]) b = FLOAT_DTYPE([0]) # solver prob = cp.Problem(cp.Minimize(c.T@x), [G@x <= h, A@x == b]) score = prob.solve() # solution print("status:", prob.status) # optimal means solution found print("story points:", -prob.value) # optimal value for -<c, x> print("money:", G[0]@x.value) print("optimal x:", x.value) At this point I’m really curious about the results: status: optimal story points: 12.399999999999999 money: 4850.0 optimal x: [1. 0. 1. 1. 0. 1. 0.] We see thatx_1, x_3, x_4$and$x_6$got the job, providing$12.4$story points for a cost of$4850$. We want to thank all other candidates for their engagement and wish them success in their future endeavours. As an exercise, consider the following 2 scenarios: • How would the “shortcut” variant look like? • How can we encode the LP to give us the second-best solution? # Example 2: Preference-Based Partitioning In this example we want to illustrate how to model more complex behaviours: • Complex combinations of multiple thresholds will be required, so control variables are necessary • We want to encode “if and only if” relations, i.e. control variables are fully fixed and there are no$DC$s. • We have a large or unknown number of categories that must be coded as integers, not booleans Imagine the following scenario: 1. At our regional football club we are organizing a not-for-profit tournament. There are 11 distinct positions per team ($a$to$k$). People enter the tournament individually, providing their best, second-best and third-best prefered positions, and we must form the teams. To quantify the preferences, we give the first, second and third preference a score of$(1, \frac{2}{3}, \frac{1}{3})$, respectively 2. We are tasked with forming as many full teams as possible, leaving as little people behind as possible. Since this is a not-for-profit event we put fun above performance, i.e. we would like our partition to respect the participants’ preferences, and also to achieve the best possible relationship among every pair of participants within the same team 3. Furthermore, if there was any doubt that we are taking this way too seriously, we asked every participant to quantify their relationship with every other participant with a score in the$[-1, 1]$range,$1$being an extremely good relationship,$0$indifferent/unknown, and$-1$extremely bad. What could go wrong? 4. So far, an unexpectedly large amount of$N$people have enrolled. As for the pairwise relationships, we were able to gather all of them, totalling$M := \begin{pmatrix}11N\\2\end{pmatrix} = \frac{(11N)^2 - 11N}{2}$scores 5. Our objective is then to find a partition that maximizes the overall sum of preference scores, plus all interpersonal scores within each team We would really like to find the global maximum, but how could we go about it? In many aspects, this looks like a hardcore combinatorial problem with an extremely high number of combinations for a brute-force search. Can we do better than that? If we encode each participant as 11 boolean variables (one per position), and each pairwise relationship also as a boolean variable ($1$=active,$0$=inactive in all cases), we would have$11N + M$boolean variables, and we could (informally): 1. Ensure that each participant gets at most one position:$\sum_{i \in N_{pos}} x_i \leq 1$2. Ensure that only participant preferences are regarded:$\sum_{i \in N_{/pref}} x_i = 0$3. If and only if participants$C_i$and$C_j$are in the same team, activate the$(C_i, C_j)$relationship 4. Set the preference scores to the$c_P$partition, the relationship scores to the$c_R$partition and seek to maximize$c^T x$We see that there is hope, since we can already encode the objective and some of the constraints into an LP. But we haven’t seen yet how to encode “if and only if” relationships. Also, the relationship should only be active if both participants are in the same team. Furthermore, we didn’t enforce that teams have exactly one member per position. How could we achieve any of that through linear constraints? ### “If and only if both active” Relationship Let’s start with the simpler issue: the constraint$C_i + C_j - R_{ij} <= 1$ensures that if$C_i, C_j$are both active, then$R_{ij}$is also active. But otherwise, the variable is a$DC$, and we want it to be fixed to$0$instead. Changepoints and variable$R_{ij}$are the same, we just need to add an extra constraint to fill the gap:$C_iC_jR_{ij}$with$C_i + C_j - R_{ij} <= 1R_{ij}$with$-C_i - C_j + 2R_{ij} <= 000DC001DC010DC0111DC$Enforcing both constraints at the same time would ensure that$R_{ij}$is$1$if and only if$C_i, C_j$are both$1$. We can then use$R_{ij}$directly as an optimization variable to be multiplied with the corresponding relationship score. But we’re still far from done, we still need to address how will the teams get exactly one member per position. ### Modelling the Teams So far, our LP has no way of encoding how many members a team has, even less how many per position. We also don’t know how many teams we will end up having, so we can’t provide placeholders either. We need to re-model the whole problem before we can propose specific changepoints. A possible solution is to add a non-negative integer to each participant representing the team ID: We can use$0$as a “no team” catch-all term, and any value above as an arbitrary team ID. Consistency can be ensured as follows, for each participant: 1. The team ID is equal or greater than zero. 2. If the team ID is zero, the sum of all 11 boolean positions is zero. Otherwise, the sum is 1 (i.e. there is one position active) 3. The sum of all non-chosen positions is zero. This way, only chosen positions can be active Now that team ID and position states are consistent, we can handle relationships as follows, for each pair of participants: 1. Check if both are in the same non-zero team 2. Check if both have different positions 3. If both conditions hold, relationship must be active, otherwise inactive At this point we have discarded a majority of the infeasible connections, but we still aren’t enforcing teams of 11 members with unique positions. We can make progress by enforcing the following: 1. For each participant, enforce that they have either$11$or$0$active relationships In a way, we want participants to be either in a full team or in none. But still, duplicate positions may happen, and with our current model we have no way of keeping track of the team IDs. Luckily, we can exploit a key property of our model: transitivity. If the$(p, q)$and$(q, r)$relationships are active, this means that participants$p, q, r$are in the same team, so we must also enforce the$(p, r)$relationship to be active. But this can only happen if all 3 participants have distinct positions!. Even better: since we are enforcing that all participants in teams must have 11 active relationships, this property extends to all of them: by enforcing all 11 people to be related to each other (in a so-called clique), we will have guaranteed 11 distinct positions covered, because all possible triangles are present, and no single triangle can have repeated positions. Furthermore, we will also have guaranteed a unique ID for every team, because otherwise players from different teams would have to be connected. At this point we’re done with the higher-level modelling, and ready to convert our goals into LP constraints. Let’s identify the changepoints in bottom-up order: 1. Team IDs are feasible from zero upwards, all others are infeasible 2. There is a relevant changepoint when a participant’s team is zero and it moves to nonzero 3. If a participant has nonzero team, the sum of its 11 position vectors must be 1 (and 0 otherwise) 4. The sum of the non-chosen positions for a participant must be always zero 5. There is a relevant changepoint when two participants have the same team and they move to different teams 6. Same thing when two participants have the same position and move to a different one 7. When two participants have same team and different position they must have an active relationship variable, and inactive otherwise (combines the 2 prior changepoints) 8. Each participant must have either zero or 11 active relationships 9. All relationships must be transitive Now we can enforce the constraints, defining control variables whenever needed. Goal 1 is straightforward: for each participant’s team ID$t$integer variable, enforce: $$-t \leq 0$$ Goals 2 and 3 require us to define an if and only if relationship that is true only when team ID$t$is zero. We define an$x_0$control variable and 2 inequality constraints per participant as follows, with a sufficiently large integer K:$tx_0$with$-t - x_0 \leq -1x_0$with$t + Kx_0 \leq K\dotsDC02DC01DC001DC$This gives us goal 2. Then, for each participant, given$x_0$and the 11 (boolean) position variables$x_a, \dots, x_k$, we can enforce goal 3 via: $$x_0 + x_a + \dots + x_k = 1$$ That way, if$t=0$, no positions will be active; otherwise, exactly one position will be active. Goal 4 is also straightforward. Since we know ahead of time which positions did each participant choose, we isolate the discarded ones and enforce that they are all inactive: $$\sum_{i \in discarded} x_i = 0$$ Goal 5 requires an integer comparator variable$x_=$for each pair of participants. Since the difference between 2 team entries$\Delta_t$can be also a negative number, this is a bit more involved. Each integer comparator is a combination of 2 further auxiliary$x_\gt, x_\lt$variables, via 4 inequality and 1 equality constraints. All variables are boolean. Given a sufficiently large integer as$K$, we enforce$x_\gt$with the following 2 constraints:$\Delta_tx_\gt$with$\Delta_t - Kx_\gt \leq 0x_\gt$with$-\Delta_t + Kx_\gt \leq (K - 1)\dots1DC11DC0DC0-1DC0\dotsDC0$And$x_\lt$can be enforced analogously. Then, we enforce$x_=$as follows: $$x_\gt + x_\lt + x_= = 1$$ This way,$x_=$is$1$if and only if both$x_\gt = x_\lt = 0$, which happens exactly when$\Delta_t = 0$. Goal 6 requires to compare the 11 boolean variables between any 2 participants$\alpha$and$\beta$. We define a control variable$x_p$that is$1$if and only if both participants have different position. This is a variation of the integer comparator from goal 5 and also requires 2 auxiliary$x_\gt, x_\lt$variables. But now we define the difference between both positions as: $$\Delta_p = 2^0 \cdot \alpha_1 + 2^1 \cdot \alpha_2 + \dots a-2^0 \cdot \beta_1 - 2^1 \cdot \beta_2 - \dots$$ Since this is basically a binary encoding, we see that$\Delta_p$can only be zero if both participants have identical positions. So we can feed$\Delta_p$to an integer comparator like we did with$\Delta_t$, yielding our desired$x_p$. Note that in this case we enforce$x_p=1$when positions are different. This only requires a minor modification to the enforcer equality constraint. Instead of$x_\gt + x_\lt + x_p = 1$we just have to flip$x_p$, yielding$x_\gt + x_\lt + (1 - x_p) = 1$, or equivalently: $$x_\gt + x_\lt - x_p = 0$$ This enforces that$x_p$is zero when$\Delta_p$is zero, and 1 otherwise. Goal 7 makes use of$x_=$and$x_p$. The activity of the relationship$R$between 2 participants whenever their team ID is nonzero is heterogeneous:$x_=x_pR00001010$should not be allowed$111$It is not feasible to have 2 participants on the same (nonzero) team and position. Apart from that, the relationship is active if and only if both participants are in the same team and different positions. Let’s try to identify the relevant changing points: they can be exposed by the$\Gamma := 2x_= - x_p$operator:$x_=x_p\Gamma:= 2x_= - x_pR01-1000001111102$infeasible This looks much better now, because we can express it as inequalities: anything above 1 is infeasible, and the rest behaves like an if and only if. So now we can enforce$R$via just 2 inequality constraints, because we can create a$DC$/fixed/infeasible split with a single constraint:$\GammaR$with$\Gamma - R \leq 0R$with$KR -\Gamma \leq (K-1)-1DC00DC011DC2$infeasible$DC$Combining both inequality constraints gives us the desired behaviour for$R$. And importantly, it is not possible for any pair of participants to have same team and same position: that would imply$\Gamma = 2$, and there is no possible value of$R$that would support that. Hence,$\Gamma = 2$can’t be a feasible solution. But we must be careful: all participants with no team will end up with the same team ID and position (i.e. both$0$). With these constraints, we are forbidding that to happen, and this leads to an LP that has no solution at all! We must leave some breathing room, i.e. introduce the following extra requirement: A pair of participants is allowed to have the same team ID and position if and only if the team ID is zero. We can leverage our already defined$x_0$variable as the only changepoint we need. We just need to enforce the following rules: 1. Whenever both participants have$x_0=0$, the rules above should apply 2. If any or both have$x_0=1the rules shouldn’t apply This can be implemented as follows: \begin{aligned} \Gamma - R &\leq 0 + K(x_0+x_0’)\\ KR -\Gamma &\leq (K-1) + K(x_0+x_0’) \end{aligned} Note that with this modification the value ofR$is$DC$whenever there is any$x_0=1$. This is fine, since our next goal will handle those cases by enforcing$R=0$. In goal 8, we need to locate all$R_{ij}$variables for each participant$i$. Then we make use of the$x_0$variable that equals$1$if and only if the participant’s team ID is$0$, and encode the goal with the following constraint: $$11x_0 + \sum_{j} R_{ij} = 11$$ This way, if the participant has no team,$x_0 = 1$and all relationships will be inactive. If the participant has a team,$x_0=0$and exactly 11 relationships must be active. A literal implementation of Goal 9 would be by far the heaviest of all, since it would require two constraints for each pair of relationships, i.e.$2 \begin{pmatrix}M\\2\end{pmatrix} = M^2 - M$constraints. Given that$M$is already in$\mathcal{O}(N^2)$, this would yield an order of$\mathcal{O}(N^4)$constraints. This is in fact how the literature proposes to solve it. But wait, doesn’t goal 7 *already enforce transitivity? If participants$p, q$have an active$R$, and participants$q, r$as well, this implies: • Participants$p, q$and$q, r$are in the same team • Participants$p, q$and$q, r$have different positions • Also, importantly, it is not possible for any pair of participants to be in the same team and position (see goal 7) The first relationship is indeed transitive, so we are covered. The second relationship is a bit trickier, since “having a different position” is not transitive: if$p$is at position$a$, and both$q, r$are at position$b$, having the$(p, q), (q, r)$relationships active won’t imply that$(q, r)$is active as well. Luckily for us, this can’t happen since the constraints in goal 7 make it infeasible. If$(p, q)$and$(q, r)$have an active$R$, this must imply that all$p, q, r$are in the same team and have different positions. Hence,$(p, r)$will be active as well! Through detailed observation and handling of all cases we were able to reduce the overall size of the LP matrix from$\mathcal{O}(N^8)$to$\mathcal{O}(N^4)$. not bad! So it turned out that this was a bit more convoluted than expected, in fact this was a multi-state, multi-clique, transitivity-optimized version of the Maximum Edge Weight Clique (MEWC) problem. OK, at this point I’ll admit that our 3-step systematic didn’t quite replace all of the non-intuitive tweaks and quirks needed to encode this into an LP program, but still they were at the core of the design process, and hopefully removing the complexity there helps focusing all the brainwork on the big picture. A possible implementation would look as follows (complete script with tests and evaluation can be found here): import random from collections import defaultdict # import numpy as np import cvxpy as cp from cvxpy.settings import CPLEX import networkx as nx # ############################################################################ # # SYNTHETIC DATASET CREATION # ############################################################################ N = 61 NUM_POSITIONS = 5 POS_ENCODING = 2 ** np.arange(NUM_POSITIONS) # 1, 2, 4... PREFERENCE_SCORES = np.array([1, 2/3, 1/3]) * (NUM_POSITIONS - 1) MIN_NEIGHS_IF_NONZERO = 4 # at most N-1 IGNORED_PENALTY = 1 * (NUM_POSITIONS - 1) K = max(2 * N, POS_ENCODING.sum()) # K needs to be bigger than any of these 2 # Create graph with random relationships between -1 and 1 g = nx.complete_graph(N) for ori, dest in g.edges(): r = (random.random() * 2 - 1) ** 3 g.edges[ori, dest]["weight"] = r g_edge_idxs = np.array(g.edges) # shape (M, 2) # Create random preferences for all users prefs = np.zeros((N, NUM_POSITIONS)) p_idxs = np.array([random.sample(range(NUM_POSITIONS), len(PREFERENCE_SCORES)) for _ in range(N)]) for i, score in enumerate(PREFERENCE_SCORES): prefs[range(N), p_idxs[:, i]] = score # ############################################################################ # # VARIABLE PARTITION AND OPTIMIZATION OBJECTIVE # ############################################################################ # column partition in 3 sections: team IDs, indiv. vars, relationship vars ind_vars = NUM_POSITIONS + 1 # one per position + x0 rel_vars = 7 # [rel_score, x<, x>, x=, z<, z>, xp] num_rels = len(g.edges) num_pvars, num_relvars = N*ind_vars, num_rels*rel_vars beg1, beg2, beg3, num_vars = np.cumsum([0, N, num_pvars, num_relvars]) # indexes for column partitions t_idxs = np.arange(beg1, beg2) pos_idxs = np.array([np.arange(i, i+NUM_POSITIONS) for i in np.arange(beg2, beg3, ind_vars)]) x0_idxs = np.arange(beg2, beg3, ind_vars) + NUM_POSITIONS rel_idxs = np.arange(beg3, num_vars, rel_vars) + 0 xl_idxs = rel_idxs + 1 xg_idxs = rel_idxs + 2 xe_idxs = rel_idxs + 3 zl_idxs = rel_idxs + 4 zg_idxs = rel_idxs + 5 zp_idxs = rel_idxs + 6 # Create objective vector with 3 sections: team ID, preferences, relationships c = np.zeros(num_vars) for i, positions in enumerate(pos_idxs): c[positions] = prefs[i] rel_weights = [g.edges[idx]["weight"] for idx in g_edge_idxs] c[rel_idxs] = rel_weights c[x0_idxs] = -IGNORED_PENALTY # Create variables analogously to objective vector: teamID are int, rest bool x_int = cp.Variable(N, integer=True) x_bool = cp.Variable(num_pvars+num_relvars, boolean=True) x = cp.hstack([x_int, x_bool]) # ############################################################################ # # INEQUALITY CONSTRAINTS # ############################################################################ ineq_blocksizes = [N, # goal 1 N, N, # goal 2 num_rels, num_rels, num_rels, num_rels, # goal 5 num_rels, num_rels, num_rels, num_rels, # goal 6 num_rels, num_rels, # goal 7 N, N] # goal 8 ineq_begs = np.cumsum([0] + ineq_blocksizes) total_ineq = ineq_begs[-1] # G = np.zeros((total_ineq, num_vars)) h = np.zeros(total_ineq) # Goal 1: non-negativity of team IDs: -t <= 0 G[ineq_begs[0]:ineq_begs[1], t_idxs] = -np.eye(N) # Goal 2: x_0 iff t=0 for each n: (-t - x_0 <= -1), (t + K*x_0 <= K) G[ineq_begs[1]:ineq_begs[2], t_idxs] = -np.eye(N) G[ineq_begs[1]:ineq_begs[2], x0_idxs] = -np.eye(N) h[ineq_begs[1]:ineq_begs[2]] = -1 G[ineq_begs[2]:ineq_begs[3], t_idxs] = np.eye(N) G[ineq_begs[2]:ineq_begs[3], x0_idxs] = np.eye(N) * K h[ineq_begs[2]:ineq_begs[3]] = K # Goal 5 ineq: teamID integer comparator for each relationship: t_ori_idxs = t_idxs[g_edge_idxs[:, 0]] t_dest_idxs = t_idxs[g_edge_idxs[:, 1]] # xl constraints: (t1-t2) - K*xl <= 0 (t2-t1) + K*xl <= (K-1) G[ineq_begs[3]:ineq_begs[4]][range(num_rels), t_ori_idxs] = 1 G[ineq_begs[3]:ineq_begs[4]][range(num_rels), t_dest_idxs] = -1 G[ineq_begs[3]:ineq_begs[4]][range(num_rels), xl_idxs] = -K # G[ineq_begs[4]:ineq_begs[5]][range(num_rels), t_ori_idxs] = -1 G[ineq_begs[4]:ineq_begs[5]][range(num_rels), t_dest_idxs] = 1 G[ineq_begs[4]:ineq_begs[5]][range(num_rels), xl_idxs] = K h[ineq_begs[4]:ineq_begs[5]] = K - 1 # xg constraints: (t2-t1) - K*xg <= 0 (t1-t2) + K*xg <= (K-1) G[ineq_begs[5]:ineq_begs[6]][range(num_rels), t_ori_idxs] = -1 G[ineq_begs[5]:ineq_begs[6]][range(num_rels), t_dest_idxs] = 1 G[ineq_begs[5]:ineq_begs[6]][range(num_rels), xg_idxs] = -K # G[ineq_begs[6]:ineq_begs[7]][range(num_rels), t_ori_idxs] = 1 G[ineq_begs[6]:ineq_begs[7]][range(num_rels), t_dest_idxs] = -1 G[ineq_begs[6]:ineq_begs[7]][range(num_rels), xg_idxs] = K h[ineq_begs[6]:ineq_begs[7]] = K - 1 # Goal 6 ineq: binary-encoded position comparator for each relationship: # zl constraint 1: (bin(pos1)-bin(pos2)) - K*zl <= 0 # zl constraint 2: (bin(pos2)-bin(pos1)) + K*zl <= (K-1) for row_i, (ori, dest) in enumerate(g_edge_idxs): ori_positions, dest_positions = pos_idxs[ori], pos_idxs[dest] G[ineq_begs[7]+row_i][ori_positions] = POS_ENCODING G[ineq_begs[7]+row_i][dest_positions] = -POS_ENCODING G[ineq_begs[8]+row_i][ori_positions] = -POS_ENCODING G[ineq_begs[8]+row_i][dest_positions] = POS_ENCODING G[ineq_begs[7]:ineq_begs[8]][range(num_rels), zl_idxs] = -K G[ineq_begs[8]:ineq_begs[9]][range(num_rels), zl_idxs] = K h[ineq_begs[8]:ineq_begs[9]] = K - 1 # zg constraint 1: (bin(pos2)-bin(pos1)) - K*zl <= 0 # zg constraint 2: (bin(pos1)-bin(pos2)) + K*zl <= (K-1) for row_i, (ori, dest) in enumerate(g_edge_idxs): ori_positions, dest_positions = pos_idxs[ori], pos_idxs[dest] G[ineq_begs[9]+row_i][ori_positions] = -POS_ENCODING G[ineq_begs[9]+row_i][dest_positions] = POS_ENCODING G[ineq_begs[10]+row_i][ori_positions] = POS_ENCODING G[ineq_begs[10]+row_i][dest_positions] = -POS_ENCODING G[ineq_begs[9]:ineq_begs[10]][range(num_rels), zg_idxs] = -K G[ineq_begs[10]:ineq_begs[11]][range(num_rels), zg_idxs] = K h[ineq_begs[10]:ineq_begs[11]] = K - 1 # Goal 7 ineq: feasibility and state of relationships based on team ID and pos x0_ori_idxs = x0_idxs[g_edge_idxs[:, 0]] x0_dest_idxs = x0_idxs[g_edge_idxs[:, 1]] # Constraint 1 for each rel: (2xe-zp) - rel - K*x0 - K*x'0 <= 0 G[ineq_begs[11]:ineq_begs[12]][range(num_rels), xe_idxs] = 2 G[ineq_begs[11]:ineq_begs[12]][range(num_rels), zp_idxs] = -1 G[ineq_begs[11]:ineq_begs[12]][range(num_rels), rel_idxs] = -1 G[ineq_begs[11]:ineq_begs[12]][range(num_rels), x0_ori_idxs] = -K G[ineq_begs[11]:ineq_begs[12]][range(num_rels), x0_dest_idxs] = -K # Constraint 2 for each rel: K*rel -(2*xe-zp) - K*x0 - K*x'0 <= (K-1) G[ineq_begs[12]:ineq_begs[13]][range(num_rels), rel_idxs] = K G[ineq_begs[12]:ineq_begs[13]][range(num_rels), xe_idxs] = -2 G[ineq_begs[12]:ineq_begs[13]][range(num_rels), zp_idxs] = 1 G[ineq_begs[12]:ineq_begs[13]][range(num_rels), x0_ori_idxs] = -K G[ineq_begs[12]:ineq_begs[13]][range(num_rels), x0_dest_idxs] = -K h[ineq_begs[12]:ineq_begs[13]] = K - 1 # Goal 8: Enforce that each individual has either 0 relationships or >=NZ: # K*x0 + sum(rels) <= K (if x0==1, sum(rels) must be zero) G[ineq_begs[13]:ineq_begs[14]][range(N), x0_idxs] = K for i in range(N): # For individual i, find var indexes of all its relationships i_edge_idxs = np.where((g_edge_idxs == i).any(axis=1))[0] all_i_rels = rel_idxs[i_edge_idxs] # (N-1) # and set them G[ineq_begs[13]+i, all_i_rels] = 1 h[ineq_begs[13]:ineq_begs[14]] = K # -NZ*x0 - sum(rels) <= -NZ (if x0==0, sum(rels) must be >=NZ, otherwise DC) G[ineq_begs[14]:ineq_begs[15]][range(N), x0_idxs] = -MIN_NEIGHS_IF_NONZERO for i in range(N): # For individual i, find var indexes of all its relationships i_edge_idxs = np.where((g_edge_idxs == i).any(axis=1))[0] all_i_rels = rel_idxs[i_edge_idxs] # (N-1) # and set them G[ineq_begs[14]+i, all_i_rels] = -1 h[ineq_begs[14]:ineq_begs[15]] = -MIN_NEIGHS_IF_NONZERO # ############################################################################ # # EQUALITY CONSTRAINTS # ############################################################################ eq_blocksizes = [N, N, num_rels, num_rels] eq_begs = np.cumsum([0] + eq_blocksizes) total_eq = sum(eq_blocksizes) # A = np.zeros((total_eq, num_vars)) b = np.zeros(total_eq) # Goal 3: x_0 + sum(x_pos) = 1 for each n for i, row_i in enumerate(range(eq_begs[0], eq_begs[1])): A[row_i, x0_idxs[i]] = 1 A[row_i, pos_idxs[i]] = 1 b[eq_begs[0]:eq_begs[1]] = 1 # Goal 4: sum(discarded_positions) = 0 for each n for i, row_i in enumerate(range(eq_begs[1], eq_begs[2])): discarded_by_i = (prefs[i] == 0) A[row_i, pos_idxs[i]] = discarded_by_i # Goal 5 eq: enforce xe via xl+xg+xe = 1, once per relationship A[eq_begs[2]:eq_begs[3]][range(num_rels), xl_idxs] = 1 A[eq_begs[2]:eq_begs[3]][range(num_rels), xg_idxs] = 1 A[eq_begs[2]:eq_begs[3]][range(num_rels), xe_idxs] = 1 b[eq_begs[2]:eq_begs[3]] = 1 # Goal 6 eq: enforce zp via zl+zg-zp = 0, once per relationship A[eq_begs[3]:eq_begs[4]][range(num_rels), zl_idxs] = 1 A[eq_begs[3]:eq_begs[4]][range(num_rels), zg_idxs] = 1 A[eq_begs[3]:eq_begs[4]][range(num_rels), zp_idxs] = -1 breakpoint() # ############################################################################ # # SOLVER AND OPTIMIZATION # ############################################################################ prob = cp.Problem(cp.Maximize(c.T@x), [G@x <= h, A@x == b]) prob.solve(solver=CPLEX, verbose=True) # CPLEX worked best with no tweaking Exercises: • Would having an integer variable for each participant’s positions be more efficient than having several booleans? • Given a score of “quality” for each candidate, how could we factor it in so that the overall objective is 50% quality + 50% interpersonal score? • Assuming the transitivity optimization wasn’t possible, how would the regular transitivity constraints look like in a LP? I.e. if$(a, b)$and$(b, c)$, then$(a, c)$. • Could the inequality constraints form goals 5 and 6 be merged, resulting in half the constraints? if not, why not? # Further Considerations Finally we can consider ### Duality and variable vs. constraint tradeoff We have seen different ways of implementing the same relation, either with more variables and less constraints, or the converse. In this context, one relevant question is: which one should I prefer to get to my solution faster?. By the Strong Duality Theorem, we know that for each feasible linear program (called primal), there is an equivalent dual program that achieves the same optimum. And it turns out that the parameter matrix for the dual program is the transpose of the primal. Many modern ILP solvers are able to swap between primal and dual, and usually they identify which one is better. In the linked post, Michael Grant explains the point perfectly: Commercial solvers like Gurobi analyze the structure of a problem and decide whether the problem is best suited for solving in primal or dual form. As a general rule, users should not worry about this. In fact, it can often be counterproductive […]. So since the solver can end up transposing it anyway, it seems that reformulating the problem so that our matrix has less rows (less constraints) but more columns (more variables), or viceversa, should only be done if we end up with strictly less elements, or for the sake of formulation clarity. ### Connections with Graph Theory One interesting connection between this technique and Graph Theory is the problem of finding maximal weighted cliques in a graph, which, like ILP, is NP-complete in the general case. But due to its convexity, for many cases, an ILP formulation and solver finds the optimum in a reasonable amount of time. An example of this can be seen in constraint 3 of the extended MCPb formulation paper (online version can be found here) by Park, Lee and Park. The constraint$x_i + x_j - y_{ij} \leq 1$enforces that, if both$x$nodes are active, the$y\$ edge between them must also be active (note that this is a one-side implication, since an active edge with inactive nodes is not being constrained, but other constraints take care of that).
More generally, finding maximal cliques on densely connected k-partite graphs like Turán Graphs is a central problem for many signal processing applications, like compressing, denoising or object tracking (online version of the paper here).
A 13-4 Turán graph, from Wikipedia
### Connections with Machine Learning
The idea of achieving complex behaviour by linearly combining simpler, piece-wise rules is at the core of many powerful Machine Learning techniques.
Regarding neural networks, the DeepLearningBook by Goodfellow, Bengio and Courville summarizes the point well in its introduction:
The quintessential example of a deep learning model is the feedforward deep network or multilayer perceptron (MLP). […] The function is formed by composing many simpler functions.
More generally, in artificial neural networks, each layer performs a piecewise linear composition (followed then by a nonlinear transformation). And strongly related to this post, the idea that artificial neurons can implement arbitrary boolean functions was one of the first realizations in the field, stemming from the 1943 foundational paper by McCulloch and Pitts (online version can be found here). Much later (1989), Cybenko would formulate the Universal Approximation Theorem, which proves that artificial neural networks, given enough capacity, can approximate a very large family of functions. The following image provides a good intuition. In it, the authors show the expressivity of a deep NN with single-neuron hidden layers (check the paper for more details):
Source: ResNet with one-neuron hidden layers is a Universal Approximator, Lin and Jegelka, NeurIPS 2018). (online version can be found here).
Another tightly related concept is boosting, a meta-algorithm presented in Schapire’s 1990 paper (online version can be found here), which provided an affirmative answer to the 1988 question by Kearns and Valiant: “Can a set of weak learners create a single strong learner?". In this context, multiple weak predictors can be combined to produce a stronger one, thus reducing both variance and bias of the resulting composition. The linear combination of different thresholding mechanisms, like presented in this post, is a popular boosting strategy.
The power and simplicity of boosting proves to be very effective. Many current setups include it for different applications; e.g. the InstaBoost paper by Fang et al. (online version here) performs instance-based segmentation and inpainting, bringing forward the state of the art in several benchmarks.
But maybe the most powerful and compelling bridge that I’ve seen so far between Machine Learning and ILPs is the following paper, that got a spotlight at ICML 2021:
CombOptNet: Fit the Right NP-Hard Problem by Learning Integer Programming Constraints (Paulus et al) [pdf]
In it, the authors demonstrate the potential of neural networks to solve ILPs. They provide differentiable expressions for both the cost terms and the constraints, resulting in end-to-end trainable architectures that simultaneously extract the features from raw data and learn the constraints that define the combinatorial problem. The image below illustrates one very exciting application of this:
Example matchings predicted by CombOptNet (from the ICML 2021 paper by Primus et al.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9676123261451721, "perplexity": 2126.160217146759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363149.85/warc/CC-MAIN-20211205065810-20211205095810-00022.warc.gz"} |
https://www.physicsforums.com/threads/aint-no-proof-by-contradiction.77889/ | # Ain't No Proof By Contradiction
1. Jun 3, 2005
I recently saw something that said some mathematicians won't acknowledge proof by contradiction. What is the reason for that? Could somebody elaborate on this for me.
2. Jun 3, 2005
### Hurkyl
Staff Emeritus
I have no clue -- proof by contradiction is an essential mathematical tool.
Maybe it came from someone on yet another rant about how "obvious", say, treating 0 as a number leads to a contradiction, and since mathematicians wouldn't acknowledge his "proof", he leaped to that conclusion?
Last edited: Jun 3, 2005
3. Jun 3, 2005
### Cyrus
I think what they meant was that by finding a contradictory case, they can reject a proof, thus they won't acknowledge proof via contradition. It just means there is an illogical step in the proof that gives anwsers that are not experimentally observed. If your proof says something is true, but you do the calculation and get a different anwser, we can logically say that something is wrong in your proof, and we have provided sufficient contradictory evidence to reject it.
Last edited: Jun 3, 2005
4. Jun 3, 2005
### mathwonk
some people have trouble psychologically with an argument that does not even attempt to show any connection between the hypothesis and the conclusion. I call these "chicken little " proofs, and there are lots of them in my favorite subject differential topology: e.g. "if this theorem were not true then the sky would be falling, but the sky is obviously not falling, so the theorem must be true." yeah?? sooo?
e.g. if there is a never zero smooth vector field on a 2 - sphere, then there is also a smooth homotopy between the identity map and the antipodal map of the sphere, but that would imply by stokes theorem, and integrating the area form, that -4pi = +4pi, and that is false; so back there somewhere at the beginning of this argument, in fact there must be a zero of that vector field.
oh yeah??? well where is it??
that sort of thing bothers some people, i.e. proofs of existence of solutions that do not try to find them or even approximate them. but merely try to show that if there is no solution then dewey was actually elected president instead of truman.
Last edited: Jun 3, 2005
5. Jun 3, 2005
### robert Ihnot
This is about the Intuitionists and their rejection of the logic of the excluded middle, that is, the acceptance of the "Either A or not A" case.
The reason for this was their dislike of Cantor's transfinite and a demand for constructive proofs. Obviously they do not accept the Axiom of Choice, since, obviously no one can constructively make these choices.
Of course, this reduces the amount of math that can be proved.http://cs.wwc.edu/~aabyan/Logic/Book/book/node70.html [Broken]
Last edited by a moderator: May 2, 2017
6. Jun 3, 2005
### 1+1=1
Proof by contradiction? This useful technique assisted me in all of my proofs classes while in college. To me, using a proof by contradiction is great. You set the proof up for contradiction and soon the proof comes tumbling down...
7. Jun 4, 2005
### HackaB
It's funny that Brouwer was a constructivist, since the proof of his fixed point theorem is usually given as a proof by contradiction.
edit:
Yeah, I don't know what one would do without it. To me, some of the neatest proofs by contradiction are those where you can hypothesize a "least counterexample", and then proceed to construct a smaller one. Someone recently gave a proof like that here for Sylvester's line problem.
Last edited: Jun 4, 2005
8. Jun 4, 2005
### honestrosewater
I've only met intuitionistic logic in passing, but I thought they were more concerned about positive existence proofs, rejecting a proof that x exists because it is impossible for x to not exist. So I don't see why they would have a problem with proof by contradiction. And look what I found in robert Ihnot's link:
9. Jun 4, 2005
### robert Ihnot
HackaB: It's funny that Brouwer was a constructivist, since the proof of his fixed point theorem is usually given as a proof by contradiction.
This seems to be true. I have a reference which states for the fixed point theorem: "The case n = 3 was proved by E. J. Brouwer in 1909. Hadamard proved the general case in 1910, and Brouwer found a different proof in 1912. Since it must have an essentially non-constructive proof, it ran contrary to Brouwer's intuitionist ideals." http://www.absoluteastronomy.com/encyclopedia/B/Br/Brouwer_Fixed_Point_Theorem.htm [Broken]
Last edited by a moderator: May 2, 2017
10. Jun 4, 2005
"Intuitionists" sounds familiar, I'm pretty sure that's what it was talking about.
So do these people lose anything by thinking this way? Or can everything be proved in a different manner, with proof by contradiction just being a simple first choice in many cases?
11. Jun 4, 2005
### Icebreaker
How would you prove a theorem about the nonexistence of solutions to an equation, like Fermat's last theorem, without proof by contradiction? I'm not saying to look at Wiles' proof, I'm just saying that, hypothetically, how would it be done?
12. Jun 4, 2005
### honestrosewater
Last edited: Jun 4, 2005
13. Jun 4, 2005
### Hurkyl
Staff Emeritus
Actually, proof by contradiction is partially valid in intuitionist logic:
$$P \rightarrow (Q \wedge \neg Q) = \neg P$$
is an identity in any Heyting algebra. (The problem with proof by contradiction in general is that $P = \neg \neg P$ is not generally true)
(Heyting algebra is to intuitionist logic as Boolean algebra is to ordinary logic)
Actually... the axiom of choice is the theorem of choice in constructivist mathematics.
He proved his theorem before he switched -- IIRC, his theorem is not valid in general intuitionist logic.
14. Jun 4, 2005
### mathwonk
re brouwers mathematics: there is a difference between being able to give a non constructive argument and being satisfied with it. mathematicians are usually not like religious zealots. they are often willing to use methods they may not prefer.
I had a very sharp friend who kept struggling to perfect an almost finitistic proof of brouwers fixed point theorem. his idea was to diminish the role of the non intuitionistic part, or keep it confined to one final step in the argument.
i never understood why he did anything he did, but he definitely opened my eyes to distinctions that had never been visible before.
15. Jun 4, 2005
### robert Ihnot
The intutionists, as I had thought, did object to the Axiom of Choice:
Zermelo gave the standard argument that the axiom of choice implies the well-ordering principle. He argued that the axiom of choice was self-evident. The reactions to the proof came immediately. At the end of a note sent to Mathematische Annalen in December 1904, Borel states:
It seems to me that the objection against it is also valid for every reasoning where one assumes an arbitrary choice made an uncountable number of times, for such reasoning does not belong in mathematics.
As for the constuctionists: Constructive mathematics can mean many different things. The common thread is that some attention is paid to the distinction between the actual construction of a mathematical object and an indirect proof of its existence. The term was often used to indicate an avoidance of the axiom of choice---a principle that asserts the existence of a function with certain properties in situations in which it is particularly unclear how such a function could be constructed. It is also used to refer to the construction and analysis of algorithms, more-or-less ready to implement on a computer, in various branches of mathematics.http://www.math.fau.edu/Richman/HTML/CONSTRUC.HTM [Broken]
On the law of the excluded middle from the above reference: In the present context, the characteristic property is the rejection of the law of excluded middle. It is somewhat remarkable how this one metamathematical move embodies the essence of constructivism. Constructivism suggests rejection of the law of excluded middle because there is generally no computational basis for asserting "p or not p". For example, consider the statement, provable using the law of excluded middle, that some digit appears infinitely often in the decimal expansion of pi. Here the existence of an integer is claimed to be proved, but no method for its computation is indicated by the proof. It is less clear, but born out by experience and theoretical constructions such as recursive realizability, that theorems proved without the law of excluded middle automatically have a computational interpretation.
Last edited by a moderator: May 2, 2017
16. Jun 4, 2005
### mathwonk
actually theorems like the intermediate value theorem are to me something of a hoax. i.e. one claims such and such a continuous function has a "real" zero but one cannot find one. why not? because one cannot actually find most real numbers, due to the large (infinite) number of terms in their decimal expnasion.
so a more honest version of the theorem and entirely equivalent if you think about the meaning of "real numbers" and the proof of the IVT theorem, is to say directly that one can always find a rational number, or a finite decimal, at which the value of the function comes as close to zero as desired. that is really all that is proved. the rest is sleight of hand.
so in fact virginia there is no square root of 2, but there are arbitrarily close approximations. and if you then define the statement "there is a real square root of 2" to mean exactly the situation above, then you may say that sentence truthfully, but nonetheless tautologically.
i.e. it is really nonsense to claim that an infinite (Cauchy) sequence of rational numbers "is" a real solution to an equation, where the real solution is defined as the equivalence class of those rationals, in any sense other than that above.
i suspect this is where we lose the loyalty of most of our students when we started pretending this nonsense was the plain truth.
I.e. we should just admit that the language we use is a idealization intended to deal with the unfortunate state of affairs as they actually exist. Or perhaps more accurately historically, there is no satisfactory way to render all the points on a continuous line into computable "numbers".
hence many of us enjoy the axiomatic approach to real numbers, rather than the symbolic one, since it frees us entirely from any calculation at all.
[there chidren, i have proved there is a real solution to any equation of a certain kind but not only do i not display one, i do not even display any actual continuous functions, nor in fact do i ever show you a real number. my hands are entirely clean!]
I loved this sophistry as a young math major, but my students hate this sort of smoke and mirrors.
Last edited: Jun 4, 2005
17. Jun 4, 2005
### Hurkyl
Staff Emeritus
In constructivism, or at least some flavors, the axiom of choice can be proven. I think the basic idea is that you can tweak whatever method you used to construct a given set to produce a choice function on that set.
Or, alternatively, you simply can't create anything so complicated you can't construct a choice function for it.
I'll admit I'm not speaking from experience -- a friend of mine's field is nonstandard analysis, a topic so dependent on the axiom of choice that you can't even get to square-one without it. The whole thing is somewhat magical in how it works, so some would view results he quoted with suspicion (especially those who distrusted the axiom of choice). He was talking to a constructivist, and was certain he'd get that treatment, only to hear the guy state that the axiom of choice was a theorem, so everything was fine!
P.S. I can't tell how much of that mathwonk believes, and how much of that is him relating beliefs of others!
18. Jun 4, 2005
### mathwonk
for me math is not necessarily about belief, but includes the open consideration of logically equivalent options.
as i said, i myself liked the axiomatic approach (a real number is an element of a complete archimedean ordered field), but after trying for years to teach normal human beings, i.e. students who have still enough courage not to be cowed by what they are handed by their elders, I see other points of view as well.
I was able to remember how hard I worked as a student, to give up the desire to see a solution to a problem that had been "proved" to have one. I.e. it is an act of faith to accept a new meaning of the words one has always used, and to accept that a solution exists not because one is convinced of it, but because one has been force fed a new definition of the word "solution".
I think it would be useful in teaching, if we were more straightforward about how a concept changes when we move to an abstract treatment of it.
It is frustrating for meto talk about real numbers every year in a calculus class in which one never defines them. A comanion thread, has a vigorous discussion of whether or not 1 = .999... , showing clearly that many students who perhaps have studied calculus, would not know a real number if it asked them for an autograph.
does it make sense to claim to have convinced a student of the truth of a theorem, like IVT, that is true over the reals but not the rationals, if one has never explained the difference? And double that for todays students who think a real number is a decimal just long enough to fit on their calculator display.
astonishing as it may seem, every year i have students who reveal that they believe all real numbers are integers. yet they memorize the statement of the IVT and assert its correctness, because they were told it is so. Logical consistency plays no role at all in their allegiance.
so i guess i am finally saying the thread on 1 = .999..., as trivial as it seems, is actually addressing a key misunderstanding of todays calc students.
Last edited: Jun 4, 2005
19. Jun 5, 2005
### matt grime
Another thing Brouwer came to think ought to be part of proofs (I think this is intuitionist, but I'm not sure) is that a proof that A or B holds is only acceptable if one can prove explicitly A holds, or, if not, then we can prove B holds. I suspect this is related to the rejection of proof by contradiction since we can not show that not(not(A)and not(B)) is A or B.
20. Jun 5, 2005
### Hurkyl
Staff Emeritus
Right. We have $A \vee B \implies \neg(\neg A \wedge \neg B)$, but the other way we only have $\neg(\neg A \wedge \neg B) \implies \neg \neg(A \vee B)$
Here's a model in the plane that demonstrates these: (recall that values in a Heyting algebra, and thus intuitionistic logic, can be modelled as the open sets of some topological space)
A = left half-plane
B = right half-plane
A or B = plane minus y axis.
not A = right half-plane
not B = left half-plane
not A and not B = 0
not(not A and not B) = whole plane
not (A or B) = 0
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http://book.caltech.edu/bookforum/showthread.php?s=d8e33da378fe3d16c61207b2797d9ee0&t=1877 | LFD Book Forum Questions on Problem 2.24
#1
10-01-2012, 12:17 AM
mileschen Member Join Date: Sep 2012 Posts: 11
Questions on Problem 2.24
Though I have solved this problem, I still a little bit confusing.
(a) Eout. whether it is the test error Etest based on the test data set T, with size N, of a particular hypothesis g that's learnt from a particular training data set D (two points).
(b) Should the bias be computed based on the same test data set T? That is, bias = Ex[bias(x)] = 1/N * sum(bias(xi)) = 1/N * sum((g_x(xi) - f(xi))^2) for each xi in T, where g_x() is the average function.
(c) Should the var be computed based on the K data sets that learn the average function g_(x) and based on the test data set T? That is, var = Ex[var(x)] = 1/N * sum[1/k * sum((gk(xi) - g_x(xi))^2)].
for Eout, bias, and var, should the be computed based on the same test data set?
#2
10-01-2012, 06:10 AM
magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595
Re: Questions on Problem 2.24
(a) For this problem if you are given a linear hypothesis it should be possible to analytically compute . However, if you computed it on a test set T, it is fine.
(b) Yes. It is also true that Etest=bias+var. Why? (because we showed this for every x).
(c) The var is computed using the same data sets on which you learned and computed the average function. The average variance is computed over the distribution of the inputs. In the case you a test set, the average is taken over the test set. Just like bias(x), var(x) is also a function of x that captures how variable your prediction is at a point x. You take all your predictions on x learned from different data sets and compute the variance of those (just like you take the average of those to get the average function.
Remember that the only purpose of the test set or the input distribution P(x) is to compute an average over (x) of all these quantities. If you had a single test point as discussed in class, everything works there too.
Quote:
Originally Posted by mileschen Though I have solved this problem, I still a little bit confusing. (a) Eout. whether it is the test error Etest based on the test data set T, with size N, of a particular hypothesis g that's learnt from a particular training data set D (two points). (b) Should the bias be computed based on the same test data set T? That is, bias = Ex[bias(x)] = 1/N * sum(bias(xi)) = 1/N * sum((g_x(xi) - f(xi))^2) for each xi in T, where g_x() is the average function. (c) Should the var be computed based on the K data sets that learn the average function g_(x) and based on the test data set T? That is, var = Ex[var(x)] = 1/N * sum[1/k * sum((gk(xi) - g_x(xi))^2)]. for Eout, bias, and var, should the be computed based on the same test data set?
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#3
10-01-2012, 07:34 AM
mileschen Member Join Date: Sep 2012 Posts: 11
Re: Questions on Problem 2.24
I still have some questions.
var = Ex[var(x)], but var(x) = Ed[(gk(x) - g_(x))^x], where var(x) is computed based on the K data sets that learnt the average function g_(x). Then, how to compute var, which is a expected value of var(x)?
If var is computed on the same data sets that learnt the average function. Then, how to compute bias = Ex[bias(x)]? If still be computed in the same data set that learnt the average function?
#4
10-01-2012, 08:40 AM
magdon RPI Join Date: Aug 2009 Location: Troy, NY, USA. Posts: 595
Re: Questions on Problem 2.24
The point x has nothing to do with the data sets on which you learn. Fix any point x.
You can now compute M1=Ed[gk(x)].
You can also compute M2=Ed[gk(x)^2].
M1 and M2 are just two numbers which apply to the point x. Clearly M1 and M2 will change if you change x, so M1 and M2 are functions of x
Now, for example, if you have many x's (eg a test set) you can compute the average of and over those x's. This means you have to compute M1 and M2 for each of those x's. You can use the same learning data sets to do so.
Quote:
Originally Posted by mileschen I still have some questions. var = Ex[var(x)], but var(x) = Ed[(gk(x) - g_(x))^x], where var(x) is computed based on the K data sets that learnt the average function g_(x). Then, how to compute var, which is a expected value of var(x)? If var is computed on the same data sets that learnt the average function. Then, how to compute bias = Ex[bias(x)]? If still be computed in the same data set that learnt the average function?
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All times are GMT -7. The time now is 07:54 AM. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8180100917816162, "perplexity": 2843.7104944890852}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540481076.11/warc/CC-MAIN-20191205141605-20191205165605-00384.warc.gz"} |
http://math.tutorcircle.com/analytical-geometry/polar-coordinates.html | Sales Toll Free No: 1-855-666-7446
# Polar Coordinates
Top
Sub Topics In mathematics, the coordinate system plays a very important role, especially in geometry. It is a system of numbers, known as coordinates, which are utilized to uniquely position a point or a geometric element on the Euclidean space. Every object is positioned in space. The coordinate system is way to determine this position. The coordinates may have one or more tuples, i.e. they have a set of values representing an exact position of something. The order of these values is quite important. On the graphs and maps, you would have commonly seen a pair of numbers showing where a point is located. Here, the first number would indicate distance left or right and the second number denotes the distance up or down.Mainly, there are following types of coordinate systems:1) Number line2) Cartesian coordinate system3) Cylindrical coordinate system4) Polar coordinate system5) Spherical coordinate systemHere, we are going to discuss about polar coordinate system and its properties.
## System
The polar coordinate system is a type of coordinate system that is usually two-dimensional. In this coordinate system, each point on a plane is to be specified by the distance from a certain point, called the reference point as well as by an angle from a certain direction, called the reference direction. In polar coordinate system, the reference point is better known as the pole and the reference direction is said to be the polar axis.
Have a look at the following diagram of this system:
The polar coordinate is made of two elements: one is radial coordinate, while another is known as angular coordinate. The radial coordinate refers to the distance of point from the pole. This coordinate is simply termed as the radius. Whereas, the angular coordinate denotes the angle of given point from polar axis. This coordinate is also called polar angle or azimuth.
## Conversion
Let r and $\theta$ be the radial and angular coordinates in the polar coordinate system, then they can be converted into Cartesian coordinates using the following relations :
r cos $\theta$ and r sin $\theta$
where, $\theta$ is said to be the counterclockwise angle from the x-axis in Cartesian coordinate system. The values of r and $\theta$ can be calculated as below.
Squaring and adding above two equations, we get
$x^{2} + y^{2} = r^{2} cos^{2} \theta + r^{2} sin^{2} \theta$
$x^{2} + y^{2} = r^{2} (cos^{2} \theta + r^{2} sin^{2} \theta)$
$x^2+y^2 = r^{2}$
$r = \sqrt(x^2+y^2)$
Dividing second equation from first, we obtain
$\frac{y}{x}$ = $\frac{sin \theta}{cos \theta}$
$\frac{y}{x}$ = $tan \theta$
$\theta$ = $tan^{-1}$ $\frac{y}{x}$
## Cylindrical
When the polar coordinates are defined with respect to a cylinder, they are known as the cylindrical polar coordinates. Such coordinates are actually three dimensional coordinates.
The diagram of cylindrical polar coordinate system is shown below:
In this type of coordinate system, the axis of a right circular cylinder is considered as z-axis. From this axis, the perpendicular distance from the cylinder axis is denoted by radial coordinate and is denoted by r. Also, the angle of radius from horizontal axis is known as the azimuthal angle and is represented by symbol $\phi$. We denoted the cylindrical polar coordinates in the form of (r, $\phi$, z).
## Spherical
The polar coordinates can also be denoted with respect to a sphere. In this case, they are called the spherical polar coordinates. These coordinates are defined in three dimensional system.
Let us have a look at the diagram of spherical polar coordinate system below:
This coordinate system considers center of sphere as the origin of coordinate system. Radial distance is denoted by r. The angle of radial distance with vertical axis is considered as angular coordinate and is represented by $\theta$, while of radial distance angle from horizontal axis is called azimuthal angle and is denoted by $\phi$. The spherical polar coordinates has three tuples of the form (r, $\theta$, $\phi$).
## Grapher
In order to graph polar coordinates, we need to follow the steps written below:
Let us suppose the given polar coordinates are (r, $\theta$):
Step 1: Take a point (say O) as the center.
Step 2: From this point, a horizontal line (radial axis) is drawn.
Step 3: From this radial axis, the given azimuthal angle $\theta$ is inclined in the counterclockwise direction. If angle is negative, it should be inclined in clockwise direction.
Step 4: At this line, we denote given radial coordinate r. If r is negative, it should be shown in the reverse direction of this line.
Have a look at the following examples.
i) (1, $\frac{5 \pi}{4}$)
ii) (2, 3$\pi$)
iii) (1, $\frac{13 \pi}{4}$)
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https://enacademic.com/dic.nsf/enwiki/765584 | # Scale (map)
The scale of a map is defined as the ratio of a distance on the map to the corresponding distance on the ground.
If the region of the map is small enough for the curvature of the Earth to be neglected, then the scale may be taken as a constant ratio over the whole map. (A town plan would be an example). For maps covering larger areas, or the whole Earth, it is essential to use a map projection[1][2] from the sphere (or ellipsoid) to the plane. Such projections inevitably involve distortion and the scale can no longer be considered as constant. It is then necessary to introduce the concept of a variable point scale (or particular scale) which is defined as the ratio of the length of a small line element emanating from a point on the map to the length of the corresponding line element on the surface of the Earth. In general the point scale will vary with the position of the point and also the direction of the line element.
Tissot's indicatrix is often used to illustrate the variation of point scale. In the study of point scale it is convenient to define the projection formulae in such a way that the scale is unity, or nearly so, on some lines (or points) of the resulting map projection.
Clearly such a map projection must be comparable to the size of the Earth and, in order to represent it on a small sheet of paper, it must be scaled down by a constant ratio known as the representative fraction (RF) or principal scale. Thus we have to differentiate two uses of the word scale: the variable point scale inherent in the projection and the constant scale involved in the reduction to the printed (or screen) map.
## The terminology of scales
Map scales may be expressed in words (a lexical scale), as a ratio, or as a fraction. Examples are:
'one centimetre to one hundred metres' or 1:10,000 or 1/10,000
'one inch to one mile' or 1:63,360 or 1/63,360
'one centimetre to one thousand kilometres' or 1:100,000,000 or 1/100,000,000. (The ratio would usually be abbreviated to 1:100M).
In addition to the above many maps carry one or more (graphical) bar scales. For example some British maps presently (2009) use three bar scales for kilometres, miles and nautical miles.
A lexical scale on a recently published map, in a language known to the user, may be easier for a non-mathematician to visualise than a ratio: if the scale is an inch to two miles and he can see that two villages are about two inches apart on the map then it is easy to work out that they are about four miles apart on the ground.
On the other hand, a lexical scale may cause problems if it expressed in a language that the user does not understand or in obsolete or ill-defined units. On the other hand ratios and fractions may be more acceptable to the numerate user since they are immediately accessible in any language. For example a scale of one inch to a furlong (1:7920) will be understood by many older people in countries where Imperial units used to be taught in schools. But a scale of one pouce to one league may be about 1:144,000 but it depends on the cartographer's choice of the many possible definitions for a league, and only a minority of modern users will be familiar with the units used.
Maps are often described as small scale, typically for world maps or large regional maps, showing large areas of land on a small space, or large scale, showing smaller areas in more detail, typically for county maps or town plans. The town plan might be on a scale of 1:10,000 and the world map might be on a scale of 1:100,000,000. There is no hard and fast dividing line between "small" and "large" scales.
For maps covering large areas, a noticeable distortion may arise from the mapmaker's attempt to map the earth's curved surface onto a flat sheet of paper. The type of distortion will depend on the map projection used. In this case the true scale will vary over the area of the map, and the stated map scale will only be an approximation. This is discussed in great detail below.
## Large scale maps with curvature neglected
The region over which the earth can be regarded as flat depends on the accuracy of the survey measurements. If measured only to the nearest metre, then curvature is undetectable over a meridian distance of about 100 km and over an east-west line of about 80 km (at a latitude of 45 degrees). If surveyed to the nearest millimetre, then curvature is undetectable over a meridian distance of about 10 km and over an east-west line of about 8 km.[3] Thus a city plan of New York accurate to one metre or a building site plan accurate to one millimetre would both satisfy the above conditions for the neglect of curvature. They can be treated by plane surveying and mapped by scale drawings in which any two points at the same distance on the drawing are at the same distance on the ground. True ground distances are calculated by measuring the distance on the map and then multiplying by the inverse of the scale fraction or, equivalently, simply using dividers to transfer the separation between the points on the map to a bar scale on the map.
## Point scale (or particular scale)
It is known that a sphere (or ellipsoid) cannot be projected to the plane without distortion (as illustrated by the impossibility of smoothing an orange peel onto a flat surface). More formally it follows from the Theorema Egregium of Gauss. The only true representation of a sphere at constant scale is another sphere such as the schoolroom globe.
There is a limit to the practical size of such a globe and for detailed mapping we must use projections. The immediate corollary is that in any projection of the sphere to the plane the scale is variable: a constant separation on the map does not correspond to a constant separation on the ground. Graphical bar scales may be present on the map but they must be used with caution for they will be accurate on only some lines of the map. (This is discussed further in the examples in the following sections.) A good atlas will usually discuss scale variation in its preface.
Let P be a point at latitude ϕ and longitude λ on the sphere (or ellipsoid). Let Q be a neighbouring point and let α be the angle between the element PQ and the meridian at P: this angle is the azimuth angle of the element PQ. Let P' and Q' be corresponding points on the projection. The angle between the direction P'Q' and the projection of the meridian is the bearing β. In general $\alpha\ne\beta$. Comment: this precise distinction between azimuth (on the Earth's surface) and bearing (on the map) is not universally observed, many writers using the terms almost interchangeably.
Definition: the point scale at P is the ratio of the two distances P'Q' and PQ in the limit that Q approaches P. We write this as
$\mu(\lambda,\,\phi,\,\alpha)=\lim_{Q\to P}\frac{P'Q'}{PQ},$
where the notation indicates that the point scale is a function of the position of P and also the direction of the element PQ.
Definition: if P and Q lie on the same meridian (α = 0), the meridian scale is denoted by $h(\lambda,\,\phi)$ .
Definition: if P and Q lie on the same parallel (α = π / 2), the parallel scale is denoted by $k(\lambda,\,\phi)$.
Definition: if the point scale depends only on position, not on direction, we say that it is isotropic and conventionally denote its value in any direction by the parallel scale factor k(λ,φ).
Definition: A map projection is said to be conformal if the angle between two lines intersecting at a point P is the same as the angle between the projected lines at the projected point P'. A conformal map has an isotropic scale factor. Conversely an isotropic scale factor implies a conformal projection.
Isotropy of scale implies that small elements are stretched equally in all directions, that is the shape of a small element is preserved. This is the property of orthomorphism (from Greek 'right shape'). The qualification 'small' means that at some given accuracy of measurement no change can be detected in the scale factor over the element. Since conformal projections have an isotropic scale factor they have also been called orthomorphic projections. For example the Mercator projection is conformal since it is constructed to preserve angles and its scale factor is isotopic, a function of latitude only: Mercator does preserve shape in small regions.
Definition: on a conformal projection with an isotropic scale, points which have the same scale value may be joined to form the isoscale lines. These are not plotted on maps for end users but they feature in many of the standard texts. (See Snyder[1] pages 203—206.)
## The representative fraction (RF) or principal scale
There are two conventions used in setting down the equations of any given projection. For example, the equirectangular cylindrical projection may be written as
cartographers: x = aλ x = aφ
mathematicians: x = λ x = φ
Here we shall adopt the first of these conventions (following the usage in the surveys by Snyder). Clearly the above projection equations define positions on a huge cylinder wrapped around the Earth and then unrolled. We say that these coordinates define the projection map which must be distinguished logically from the actual printed (or viewed) maps. If the definition of point scale in the previous section is in terms of the projection map then we can expect the scale factors to be close to unity. For normal tangent cylindrical projections the scale along the equator is k=1 and in general the scale changes as we move off the equator. Analysis of scale on the projection map is an investigation of the change of k away from its true value of unity.
Actual printed maps are produced from the projection map by a constant scaling denoted by a ratio such as 1:100M (for whole world maps) or 1:10000 (for such as town plans). To avoid confusion in the use of the word 'scale' this constant scale fraction is called the representative fraction (RF) of the printed map and it is to be identified with the ratio printed on the map. The actual printed map coordinates for the equirectangular cylindrical projection are
printed map: x = (RF)aλ y = (RF)aφ
This convention allows a clear distinction of the intrinsic projection scaling and the reduction scaling.
From this point we ignore the RF and work with the projection map.
## Visualisation of point scale: the Tissot indicatrix
The Winkel tripel projection with Tissot's indicatrix of deformation
Consider a small circle on the surface of the Earth centred at a point P at latitude ϕ and longitude λ. Since the point scale varies with position and direction the projection of the circle on the projection will be distorted. Tissot proved that, as long as the distortion is not too great, the circle will become an ellipse on the projection. In general the dimension, shape and orientation of the ellipse will change over the projection. Superimposing these distortion ellipses on the map projection conveys the way in which the point scale is changing over the map. The distortion ellipse is known as Tissot's indicatrix. The example shown here is the Winkel tripel projection, the standard projection for world maps made by the National Geographic Society. The minimum distortion is on the central meridian at latitudes of 30 degrees (North and South). (Other examples[4][5]).
## Point scale for normal cylindrical projections of the sphere
The key to a quantitative understanding of scale is to consider an infinitesimal element on the sphere. The figure shows a point P at latitude ϕ and longitude λ on the sphere. The point Q is at latitude φ + δφ and longitude λ + δλ. The lines PK and MQ are arcs of meridians of length aδφ where a is the radius of the sphere and ϕ is in radian measure. The lines PM and KQ are arcs of parallel circles of length (acos ϕ)δλ withλ in radian measure. In deriving a point property of the projection at P it suffices to take an infinitesimal element PMQK of the surface: in the limit of Q approaching P such an element tends to an infinitesimally small planar rectangle.
Infinitesimal elements on the sphere and a normal cylindrical projection
Normal cylindrical projections of the sphere have x = aλ and y a function of latitude only. Therefore the infinitesimal element PMQK on the sphere projects to an infinitesimal element P'M'Q'K' which is an exact rectangle with a base δx = aδλ and height δy. By comparing the elements on sphere and projection we can immediately deduce expressions for the scale factors on parallels and meridians. (We defer the treatment of the scale in a general direction to a mathematical addendum to this page.)
parallel scale factor $\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$
meridian scale factor $\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\frac{y'(\phi)}{a}$
Note that the parallel scale factor k = sec ϕ is independent of the definition of y(ϕ) so it is the same for all normal cylindrical projections. It is useful to note that
at latitude 30 degrees the parallel scale is $k=\sec30^{\circ}=2/\sqrt{3}=1.15$
at latitude 45 degrees the parallel scale is $k=\sec45^{\circ}=\sqrt{2}=1.414$
at latitude 60 degrees the parallel scale is $k=\sec60^{\circ}=2$
at latitude 80 degrees the parallel scale is $k=\sec80^{\circ}=5.76$
at latitude 85 degrees the parallel scale is $k=\sec85^{\circ}=11.5$
The following examples illustrate three normal cylindrical projections and in each case the variation of scale with position and direction is illustrated by the use of Tissot's indicatrix.
## Three examples of normal cylindrical projection
### The equirectangular projection
The equidistant projection with Tissot's indicatrix of deformation
The equirectangular projection,[1][2][3] also known as the Plate Carrée (French for "flat square") or (somewhat misleadingly) the equidistant projection, is defined by
x = aλ, y = aϕ,
where a is the radius of the sphere, λ is the longitude from the central meridian of the projection (here taken as the Greenwich meridian at λ = 0) and ϕ is the latitude. Note that λ and ϕ are in radians (obtained by multiplying the degree measure by a factor of π/180). The longitude λ is in the range [ − π,π] and the latitude ϕ is in the range [ − π / 2,π / 2].
Since y'(ϕ) = 1 the previous section gives
parallel scale, $\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$ meridian scale $\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,1$
For the calculation of the point scale in an arbitrary direction see addendum.
The figure illustrates the Tissot indicatrix for this projection. On the equator h=k=1 and the circular elements are undistorted on projection. At higher latitudes the circles are distorted into an ellipse given by stretching in the parallel direction only: there is no distortion in the meridian direction. The ratio of the major axis to the minor axis is sec ϕ. Clearly the area of the ellipse increases by the same factor.
It is instructive to consider the use of bar scales that might appear on a printed version of this projection. The scale is true (k=1) on the equator so that multiplying its length on a printed map by the inverse of the RF (or principal scale) gives the actual circumference of the Earth. The bar scale on the map is also drawn at the true scale so that transferring a separation between two points on the equator to the bar scale will give the correct distance between those points. The same is true on the meridians. On a parallel other than the equator the scale is sec ϕ so when we transfer a separation from a parallel to the bar scale we must divide the bar scale distance by this factor to obtain the distance between the points when measured along the parallel (which is not the true distance along a great circle). On a line at a bearing of say 45 degrees ($\beta=45^{\circ}$) the scale is continuously varying with latitude and transferring a separation along the line to the bar scale does not give a distance related to the true distance in any simple way. (But see addendum). Even if we could work out a distance along this line of constant bearing its relevance is questionable since such a line on the projection corresponds to a complicated curve on the sphere. For these reasons bar scales on small scale maps must be used with extreme caution.
### Mercator projection
The Mercator projection with Tissot's indicatrix of deformation. (The distortion increases without limit at higher latitudes)
The Mercator projection maps the sphere to a rectangle (of infinite extent in the y-direction) by the equations[1][2][3]
$x = a\lambda\,$
$y = a\ln \left[\tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right]$
where a, $\lambda\,$ and $\phi \,$ are as in the previous example. Since y'(ϕ) = asec ϕ the scale factors are:
parallel scale
$k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$
meridian scale
$h\;=\;\frac{\delta y}{a\delta\phi\,}=\,\sec\phi$
In the mathematical addendum below we prove that the point scale in an arbitrary direction is also equal to sec ϕ so the scale is isotropic (same in all directions), its magnitude increasing with latitude as sec ϕ. In the Tissot diagram each infinitesimal circular element preserves its shape but is enlarged more and more as the latitude increases.
### Lambert's equal area projection
Lambert's normal cylindrical equal-area projection with Tissot's indicatrix of deformation
Lambert's equal area projection maps the sphere to a finite rectangle by the equations[1][2][3]
$x = a\lambda \qquad\qquad y = a\sin\phi$
where a, λ and ϕ are as in the previous example. Since y'(ϕ) = cos ϕ the scale factors are
parallel scale $\quad k\;=\;\frac{\delta x}{a\cos\phi\,\delta\lambda\,}=\,\sec\phi\qquad\qquad{}$
meridian scale $\quad h\;=\;\frac{\delta y}{a\delta\phi\,}=\,\cos\phi$
The calculation of the point scale in an arbitrary direction is given below.
The vertical and horizontal scales now compensate each other (hk=1) and in the Tissot diagram each infinitesimal circular element is distorted into an ellipse of the same area as the undistorted circles on the equator.
### Graphs of scale factors
The graph shows the variation of the scale factors for the above three examples. The top plot shows the isotropic Mercator scale function: the scale on the parallel is the same as the scale on the meridian. The other plots show the meridian scale factor for the Equirectangular projection (h=1) and for the Lambert equal area projection. These last two projections have a parallel scale identical to that of the Mercator plot. For the Lambert note that the parallel scale (as Mercator A) increases with latitude and the meridian scale (C) decreases with latitude in such a way that hk=1, guaranteeing area conservation.
## Scale variation on the Mercator projection
The Mercator point scale is unity on the equator because it is such that the auxiliary cylinder used in its construction is tangential to the Earth at the equator. For this reason the usual projection should be called a tangent projection. The scale varies with latitude as k = sec ϕ. Since sec ϕ tends to infinity as we approach the poles the Mercator map is grossly distorted at high latitudes and for this reason the projection is totally inappropriate for world maps (unless we are discussing navigation and rhumb lines). However, at a latitude of about 25 degrees the value of sec ϕ is about 1.1 so Mercator is accurate to within 10% in a strip of width 50 degrees centred on the equator. Narrower strips are better: a strip of width 16 degrees (centred on the equator) is accurate to within 1% or 1 part in 100.
A standard criterion for good large scale maps is that the accuracy should be within 4 parts in 10,000, or 0.04%, corresponding to k = 1.0004. Since sec ϕ attains this value at ϕ = 1.62 degrees (see figure below, red line). Therefore the tangent Mercator projection is highly accurate within a strip of width 3.24 degrees centred on the equator. This corresponds to north-south distance of about 360 km (220 mi). Within this strip Mercator is very good, highly accurate and shape preserving because it is conformal (angle preserving). These observations prompted the development of the transverse Mercator projections in which a meridian is treated 'like an equator' of the projection so that we obtain an accurate map within a narrow distance of that meridian. Such maps are good for countries aligned nearly north-south (like Great Britain) and a set of 60 such maps is used for the Universal Transverse Mercator (UTM). Note that in both these projections (which are based on various ellipsoids) the transformation equations for x and y and the expression for the scale factor are complicated functions of both latitude and longitude.
Scale variation near the equator for the tangent (red) and secant (green) Mercator projections.
## Secant, or modified, projections
The basic idea of a secant projection is that the sphere is projected to a cylinder which intersects the sphere at two parallels, say ϕ1 north and south. Clearly the scale is now true at these latitudes whereas parallels beneath these latitudes are contracted by the projection and their (parallel) scale factor must be less than one. The result is that deviation of the scale from unity is reduced over a wider range of latitudes.
As an example, one possible secant Mercator projection is defined by
$x = 0.9996a\lambda \qquad\qquad y = 0.9996a\ln \left(\tan \left(\frac{\pi}{4} + \frac{\phi}{2} \right) \right).$
The numeric multipliers do not alter the shape of the projection but it does mean that the scale factors are modified:
secant Mercator scale, $\quad k\;=0.9996\sec\phi.$
Thus
• the scale on the equator is 0.9996,
• the scale is k=1 at a latitude given by ϕ1 where sec ϕ1 = 1 / 0.9996 = 1.00004 so that ϕ1 = 1.62 degrees,
• k=1.0004 at a latitude ϕ2 given by sec ϕ2 = 1.0004 / 0.9996 = 1.0008 for which ϕ2 = 2.29 degrees. Therefore the projection has 1 < k < 1.0004 , that is an accuracy of 0.04%, over a wider strip of 4.58 degrees (compared with 3.24 degrees for the tangent form).
This is illustrated by the lower (green) curve in the figure of the previous section.
Such narrow zones of high accuracy are used in the UTM and the British OSGB projection, both of which are secant, transverse Mercator on the ellipsoid with the scale on the central meridian constant at k0 = 0.9996. The isoscale lines with k = 1 are slightly curved lines approximately 180 km east and west of the central meridian. The maximum value of the scale factor is 1.001 for UTM and 1.0007 for OSGB.
The lines of unit scale at latitude ϕ1 (north and south), where the cylindrical projection surface intersects the sphere, are the standard parallels of the secant projection.
Whilst a narrow band with | k − 1 | < 0.0004 is important for high accuracy mapping at a large scale, for world maps much wider spaced standard parallels are used to control the scale variation. Examples are
• Behrmann with standard parallels at 30N, 30S.
• Gall equal area with standard parallels at 45N, 45S.
Scale variation for the Lambert (green) and Gall (red) equal area projections.
The scale plots for the latter are shown below compared with the Lambert equal area scale factors. In the latter the equator is a single standard parallel and the parallel scale increases from k=1 to compensate the decrease in the meridian scale. For the Gall the parallel scale is reduced at the equator (to k=0.707) whilst the meridian scale is increased (to k=1.414). This gives rise to the gross distortion of shape in the Gall-Peters projection. (On the globe Africa is about as long as it is broad). Note that the meridian and parallel scales are both unity on the standard parallels.
Infinitesimal elements on the sphere and a normal cylindrical projection
For normal cylindrical projections the geometry of the infinitesimal elements gives
$\text{(a)}\quad \tan\alpha=\frac{a\cos\phi\,\delta\lambda}{a\,\delta\phi},$
$\text{(b)}\quad \tan\beta=\frac{\delta x}{\delta y} =\frac{a\delta \lambda}{\delta y}.$
The relationship between the angles β and α is
$\text{(c)}\quad \tan\beta=\frac{a\sec\phi}{y'(\phi)} \tan\alpha.\,$
For the Mercator projection y'(ϕ) = asec ϕ giving α = β: angles are preserved. (Hardly surprising since this is the relation used to derive Mercator). For the equidistant and Lambert projections we have y'(ϕ) = a and y'(ϕ) = acos ϕ respectively so the relationship between α and β depends upon the latitude ϕ. Denote the point scale at P when the infinitesimal element PQ makes an angle $\alpha \,$ with the meridian by μα. It is given by the ratio of distances:
$\mu_{\alpha}=\lim_{Q\to P}\frac{P'Q'}{PQ} = \lim_{Q\to P}\frac{\sqrt{\delta x^2 +\delta y^2}} {\sqrt{ a^2\, \delta\phi^2+a^2\cos^2\!\phi\, \delta\lambda^2}}.$
Setting δx = aδλ and substituting δϕ and δy from equations (a) and (b) respectively gives
$\mu_\alpha(\phi) = \sec\phi \left[\frac{\sin\alpha}{\sin\beta}\right].$
For the projections other than Mercator we must first calculate β from α and ϕ using equation (c), before we can find μα. For example the equirectangular projection has y' = a so that
$\tan\beta=\sec\phi \tan\alpha.\,$
If we consider a line of constant slope β on the projection both the corresponding value of α and the scale factor along the line are complicated functions of ϕ. There is no simple way of transferring a general finite separation to a bar scale and obtaining meaningful results.
## References
1. ^ a b c d e Snyder, John P. (1987). Map Projections - A Working Manual. U.S. Geological Survey Professional Paper 1395. United States Government Printing Office, Washington, D.C.. This paper can be downloaded from USGS pages. It gives full details of most projections, together with introductory sections, but it does not derive any of the projections from first principles. Derivation of all the formulae for the Mercator projections may be found in The Mercator Projections.
2. ^ a b c d Flattening the Earth: Two Thousand Years of Map Projections, John P. Snyder, 1993, pp. 5-8, ISBN 0-226-76747-7. This is a survey of virtually all known projections from antiquity to 1993.
3. ^ a b c d The Mercator Projections A pedagogical derivation of the formulae describing many of the variants of the Mercator projections. In particular normal and transverse projections on the sphere and ellipsoid along with their modified versions. It concludes with the Redfearn formulae used for the Transverse Mercator on the ellipsoid.
4. ^ Examples of Tissot's indicatrix. Some beautiful illustrations of the Tissot Indicatrix applied to a variety of projections other than normal cylindrical.
5. ^ Further examples of Tissot's indicatrix at Wikipedia commons.
Wikimedia Foundation. 2010.
### Look at other dictionaries:
• large-scale map — A map having a scale of 1:75,000 or larger. See also map … Military dictionary
• medium-scale map — A map having a scale larger than 1:600,000 and smaller than 1:75,000. See also map … Military dictionary
• small-scale map — A map having a scale smaller than 1:600,000. See also map … Military dictionary
• Map — /map/, n. Walter, c1140 1209?, Welsh ecclesiastic, poet, and satirist. Also, Mapes /mayps, may peez/. * * * I Graphic representation, drawn to scale and usually on a flat surface, of features usually geographic, geologic, or geopolitical of an… … Universalium
• map — mappable, adj. mapper, n. /map/, n., v., mapped, mapping. n. 1. a representation, usually on a flat surface, as of the features of an area of the earth or a portion of the heavens, showing them in their respective forms, sizes, and relationships… … Universalium
• MAP — See modified American plan. * * * I Graphic representation, drawn to scale and usually on a flat surface, of features usually geographic, geologic, or geopolitical of an area of the Earth or of any celestial body. Globes are maps represented on… … Universalium
• Map projection — A medieval depiction of the Ecumene (1482, Johannes Schnitzer, engraver), constructed after the coordinates in Ptolemy s Geography and using his second map projection A map projection is any method of representing the surface of a sphere or other … Wikipedia
• Map series — Not to be confused with atlas or map collection. Mairie de Loevenich (Germany), from the Topographic Survey of the Rhineland by Tranchot/Müffling, sheet 57 (published 1806/07) … Wikipedia
• Scale (ratio) — The concept of scale is applicable if a system is represented proportionally by another system. For example, for a scale model of an object, the ratio of corresponding lengths is a dimensionless scale, e.g. 1:25; this scale is larger than 1:50.In … Wikipedia
• scale — a measure describing the resolution of a map, architectural plan, or some similar document. A map, for example, might be described as 1:250 000 scale . In general, 1:n scale means that 1 unit of distance on the map or plan represents n of the… … Dictionary of units of measurement | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 32, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9156057834625244, "perplexity": 1031.0857095564643}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250598217.23/warc/CC-MAIN-20200120081337-20200120105337-00487.warc.gz"} |
https://hal.inria.fr/hal-01053454 | Skip to Main content Skip to Navigation
# A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation
Abstract : We consider the Monge-Kantorovich optimal transportation problem between two measures, one of which is a weighted sum of Diracs. This problem is traditionally solved using expensive geometric methods. It can also be reformulated as an elliptic partial differential equation known as the Monge-Ampere equation. However, existing numerical methods for this non-linear PDE require the measures to have finite density. We introduce a new formulation that couples the viscosity and Aleksandrov solution definitions and show that it is equivalent to the original problem. Moreover, we describe a local reformulation of the subgradient measure at the Diracs, which makes use of one-sided directional derivatives. This leads to a consistent, monotone discretisation of the equation. Computational results demonstrate the correctness of this scheme when methods designed for conventional viscosity solutions fail.
Document type :
Preprints, Working Papers, ...
Domain :
Complete list of metadatas
https://hal.inria.fr/hal-01053454
Contributor : Jean-David Benamou <>
Submitted on : Thursday, July 31, 2014 - 9:38:58 AM
Last modification on : Monday, December 14, 2020 - 5:00:07 PM
### Identifiers
• HAL Id : hal-01053454, version 1
• ARXIV : 1407.1300
### Citation
Jean-David Benamou, Brittany D. Froese. A viscosity framework for computing Pogorelov solutions of the Monge-Ampere equation. 2014. ⟨hal-01053454⟩
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https://fockphysics.wordpress.com/tag/notation/ | # Mathematical Notation: Summation
In response to David Wees’ post about summation notation, I’d like to suggest that the terseness of mathematical notation is a godsend when working on long calculations, but it should, perhaps, be collapsible for experts and expandable for beginners. What I mean is that while
$\sum\limits_{i=1}^{6} i^2$ or
$\sum\limits_{i\in1\ldots6} i^2$
may suffice for the expert, beginners may prefer
$\mathop{\Sigma\text{um}}\limits_{i\text{ from }1}^{\text{to }6} i^2$ or
$\mathop{\text{Sum}}\limits_{i\text{ from }1\text{ to }6} i^2$.
I hate the linear form that David Wees mentions,
Summation (i, 3, 6, i2) = 32 + 42 + 52 + 62 = 86,
for it loses the spatial memory aspect of the original summation convention. The real problem seems to be that this last expression serializes for computers well, but the other mathematics is hard to type. I would prefer “smaller bits” of mathematics, like Sum and Sequence:
Sum[Sequence[Lambda[i,i^2], 1..6]] or even Sum[(i->i^2)[1..6]]
It would be nice if computers would do f[A] if f[a] is defined for every a in A without some kind of function like Map or Apply. Here’s a longer version of the last expression:
Sum[Apply[Lambda[i,i^2],1..6]]
In fancy LaTeX form that might be:
$\sum (i\mapsto i^2)[1..6]$ or
$\sum \left[(i\mapsto i^2)[1..6]\right]$
if we want to make the operator precedence completely clear. Here, “1..6” is some Ruby-like syntactic sugar to mean the set (really: sequence) {1,2,3,4,5,6}. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 6, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9114459753036499, "perplexity": 2389.080090276748}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232255165.2/warc/CC-MAIN-20190519201521-20190519223521-00541.warc.gz"} |
https://www.frontporchmath.com/topics/algebra/algebraic-fractions/simplifying-algebraic-fractions-video/ | # Simplifying Algebraic Fractions
In this video, we look at how to simplify fractions containing variables. The example we will look at is $\tfrac{112ab}{7a}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.982159435749054, "perplexity": 682.1770969299857}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823817.62/warc/CC-MAIN-20181212091014-20181212112514-00332.warc.gz"} |
http://math.stackexchange.com/questions/233946/properties-of-continuous-bijection-fx-to-y/233947 | # Properties of continuous bijection $f:X\to Y$
Let $f:X\to Y$ be a continuous bijection.
If $Y$ is Hausdorff, then can we conclude that $X$ is also Hausdorff?
What I learned so far is that：
- If $X$ is compact, then $f(X)$ is also compact;
- If, $X$ is compact and $Y$ is Hausdorff, then $f^{-1}$ is continuous.
I've no idea how to approach the problem. Is there any counterexample for the statement above?
-
Yes! ${}{}{}{}{}$ – leo Nov 10 '12 at 2:58
Let $x_1, x_2 \in X$, $x_1 \ne x_2$. As $f$ is one-to-one, $f(x_1) \ne f(x_2)$ and as $Y$ is Hausdorff, there are disjoint open neighbourhoods $U_1 \ni f(x_1)$, $U_2 \ni f(x_2)$. This gives $x_1 \in f^{-1}(U_1)$, $x_2 \in f^{-1}(U_2)$. As $f$ is continuous, $f^{-1}(U_i)$ are open neighbourhoods of $x_i$, $i = 1,2$ moreover they are disjoint. So $X$ is Hausdorff.
Thank you for your answer! Why are $f^{-1}(U_i)$ disjoint? – Goku Nov 10 '12 at 3:09
As inverse images of disjoint sets are always: Suppose $x \in f^{-1}(U_1)\cap f^{-1}(U_2)$, then $f(x) \in U_1 \cap U_2$ ... – martini Nov 10 '12 at 3:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9915031790733337, "perplexity": 125.95898376463751}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-30/segments/1469257823072.2/warc/CC-MAIN-20160723071023-00005-ip-10-185-27-174.ec2.internal.warc.gz"} |
https://export.arxiv.org/abs/2004.04097 | physics.gen-ph
(what is this?)
# Title: Arithmetic loophole in Bell's theorem: An overlooked threat for entangled-state quantum cryptography
Authors: Marek Czachor
Abstract: Bell's theorem is supposed to exclude all local hidden-variable models of quantum correlations. However, an explicit counterexample shows that a new class of local realistic models, based on generalized arithmetic and calculus, can exactly reconstruct rotationally symmetric quantum probabilities typical of two-electron singlet states. The model is classical in the sense of Einstein, Podolsky, Rosen, and Bell: elements of reality exist and probabilities are computed by means of appropriate integrals over hidden variables. Probabilities have a Clauser-Horne product form typical of local realistic theories. However, neither the product nor the integral nor the representation of rotations are the usual ones. The integral has all the standard properties but only with respect to the arithmetic that defines the product. Certain formal transformations of integral expressions one finds in the usual proofs \`a la Bell do not work, so standard Bell-type inequalities cannot be proved. The system we consider is deterministic, local-realistic, rotationally invariant, observers have free will, detectors are perfect, so is free of all the canonical loopholes discussed in the literature. The model is quantum enough to fake quantum correlations, but classical enough to enable potential hacking into quantum encryptions certified by Bell's theorem.
Comments: Modified title Subjects: General Physics (physics.gen-ph) Cite as: arXiv:2004.04097 [physics.gen-ph] (or arXiv:2004.04097v4 [physics.gen-ph] for this version)
## Submission history
From: Marek Czachor [view email]
[v1] Mon, 6 Apr 2020 16:48:41 GMT (683kb,D)
[v2] Thu, 9 Apr 2020 09:08:01 GMT (686kb,D)
[v3] Mon, 20 Apr 2020 10:01:41 GMT (688kb,D)
[v4] Wed, 6 May 2020 12:10:53 GMT (691kb,D)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8900867104530334, "perplexity": 3502.008499925998}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347392141.7/warc/CC-MAIN-20200527044512-20200527074512-00410.warc.gz"} |
http://iqtutors.com.au/difference-of-two-squares.php | Difference of two squares (DOTS)
$a^2 - b^2 = (a + b)(a - b)$
The expressions $$a + b$$ and $$a - b$$ are called conjugates.
Conjugates are formed by changing the sign between two terms.
More examples:
ExpressionConjugate
$$x+y$$$$x-y$$
$$x+3$$$$x-3$$
$$x+\sqrt{5}$$$$x-\sqrt{5}$$
#### Factorisation of a difference of two squares (DOTS)
Factorise the following expressions:
#### Exercise #1
$\,\,\, 1) \quad x^2 - 1$ $\,\,\, 2) \quad x^2 - 36$ $\,\,\, 3) \quad x^2 - 64$ $\,\,\, 4) \quad x^2 - 16$ $\,\,\, 5) \quad x^2 - 25$
$\,\,\, 1) \quad x^2 - 1 = (x + 1)(x - 1)$ $\,\,\, 2) \quad x^2 - 36 = (x + 6)(x - 6)$ $\,\,\, 3) \quad x^2 - 64 = (x + 8)(x - 8)$ $\,\,\, 4) \quad x^2 - 16 = (x + 4)(x - 4)$ $\,\,\, 5) \quad x^2 - 25 = (x + 5)(x - 5)$
#### Exercise #2
$\,\,\, 1) \quad x^2 - 66$ $\,\,\, 2) \quad x^2 - 76$ $\,\,\, 3) \quad x^2 - 74$ $\,\,\, 4) \quad x^2 - 99$ $\,\,\, 5) \quad x^2 - 79$
$\,\,\, 1) \quad x^2 - 66 = (x + \sqrt{66})(x - \sqrt{66})$ $\,\,\, 2) \quad x^2 - 76 = (x + \sqrt{76})(x - \sqrt{76})$ $\,\,\, 3) \quad x^2 - 74 = (x + \sqrt{74})(x - \sqrt{74})$ $\,\,\, 4) \quad x^2 - 99 = (x + \sqrt{99})(x - \sqrt{99})$ $\,\,\, 5) \quad x^2 - 79 = (x + \sqrt{79})(x - \sqrt{79})$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9292413592338562, "perplexity": 575.9123599527707}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540544696.93/warc/CC-MAIN-20191212153724-20191212181724-00371.warc.gz"} |
https://www.esaral.com/q/the-sum-of-the-numerator-and-denominator-of-a-fraction-is-3-less-than-twice-the-denominator-68908/ | The sum of the numerator and denominator of a fraction is 3 less than twice the denominator.
Question:
The sum of the numerator and denominator of a fraction is 3 less than twice the denominator. If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Determine the fraction.
Solution:
Let the numerator and denominator of the fraction be $x$ and $y$ respectively. Then the fraction is $\frac{x}{y}$
The sum of the numerator and denominator of the fraction is 3 less than twice the denominator. Thus, we have
$x+y=2 y-3$
$\Rightarrow x+y-2 y+3=0$
$\Rightarrow x-y+3=0$
If the numerator and denominator are decreased by 1, the numerator becomes half the denominator. Thus, we have
$x-1=\frac{1}{2}(y-1)$
$\Rightarrow \frac{x-1}{y-1}=\frac{1}{2}$
$\Rightarrow 2(x-1)=y-1$
$\Rightarrow 2 x-2=y-1$
$\Rightarrow 2 x-y-1=0$
So, we have two equations
$x-y+3=0$
$2 x-y-5=0$
Here x and y are unknowns. We have to solve the above equations for x and y.
By using cross-multiplication, we have
$\frac{x}{(-1) \times(-1)-(-1) \times 3}=\frac{-y}{1 \times(-1)-2 \times 3}=\frac{1}{1 \times(-1)-2 \times(-1)}$
$\Rightarrow \frac{x}{1+3}=\frac{-y}{-1-6}=\frac{1}{-1+2}$
$\Rightarrow \frac{x}{4}=\frac{-y}{-7}=\frac{1}{1}$
$\Rightarrow \frac{x}{4}=\frac{y}{7}=1$
$\Rightarrow x=4, y=7$
Hence, the fraction is $\frac{4}{7}$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.999516487121582, "perplexity": 116.98813609390018}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320302622.39/warc/CC-MAIN-20220120190514-20220120220514-00200.warc.gz"} |
https://dlennon.org/page/20210320_annot_htf09 | ×
### Dustin Lennon
##### Applied Scientistdlennon.org
(206) 291-8893
notes elements of statistical learning machine learning
Annotations: Elements of Statistical Learning
My margin notes from reading Hastie, Tibshirani, and Friedman
Dustin Lennon
March 2021
https://dlennon.org/20210320_annot_htf09
March 2021
Abstract
#### Abstract
Mathematicians value brevity; details are left to the reader. As a lifetime consumer of mathematical ideas, I’ve often wondered if the authors have given too much credit to their audience. I find myself wanting to understand every step of a logical argument, and missing pieces give me pause. As such, I find it helpful to document, and hopefully elucidate, these points of confusion. Perhaps others will find this exercise useful.
Here, we covers one of my favorite machine learning books, “The Elements of Statistical Learning,” written by Hastie, Tibshirani, and Friedman. Specific references are to the 2009 print edition.
Chapter 2
#### Chapter 2
##### Expected Prediction Error: Linear Regression
Expected prediction error, or test error, is one of the key ideas in Chapter 2. In this context, HTF describes a bias variance tradeoff, and they work out the decomposition for both linear regression and nearest neighbor. For linear regression, this happens in 2.27 and 2.28, and the details are left to the reader as Exercise 2.5.
My confusion here started with the double expectations:
$\Evy{ \EvTb{ \left( y_0 - \yhat \right)^2 } }$
Recall that the goal is to estimate the prediction error at a new location, $x_0$. From 2.26, we could write,
$y_0 = \xbeta + \varepsilon_0$
with $x_0$ is known and $\beta$ an unknown constant. The only randomness enters through $\varepsilon_0$. We’ll associate the outer expectation, $\Evy{}$, with the above.
To make progress, let $\yhat$ be an estimator of $y_0$. As an estimator, it is a function of the data. As such, we require a sample, to be drawn from $\sspace$. For the linear regression problem, the estimator is
$\yhat = \xbhat$
The variability associated with the sample may be ascribed to the inner expectation, $\EvT{}$. A different sample from the population distribution would produce a different realization of $\bhat$.
Now we proceed by adding and subtracting $\xbeta$ and $\emean$ inside the original expression:
\begin{align*} \left( y_0 - \yhat \right)^2 & = \left( \color{red}{(y_0 - \xbeta)} + \color{green}{(\xbeta - \emean)} + \color{blue}{(\emean - \yhat)} \right)^2 \\ & = \color{red}{(y_0 - \xbeta)^2} \\ & \eqi + \color{green}{(\xbeta - \emean)^2} \\ & \eqi + \color{blue}{(\emean - \yhat)^2} \\ & \eqi + 2 \color{red}{(y_0 - \xbeta)} \color{green}{(\xbeta - \emean)} \\ & \eqi + 2 \color{red}{(y_0 - \xbeta)} \color{blue}{(\emean - \yhat)} \\ & \eqi + 2 \color{green}{(\xbeta - \emean)} \color{blue}{(\emean - \yhat)} \end{align*}
Consider the cross product terms above:
• The green terms are constant with respect to both $\Evy{}$ and $\EvT{}$.
• The red terms are constant with respect to $\EvT{}$ and have zero expectation under $\Evy{}$.
• The blue terms have zero expectation under $\EvT{}$.
Thus, all the cross product terms will be zero under the nested expectations, and, generally:
\begin{align} \Evy{ \EvTb{ \left( y_0 - \yhat \right)^2 } } & = \Evys{ \color{red}{(y_0 - \xbeta)^2} } + \color{green}{(\xbeta - \emean)^2} + \EvTs{ \color{blue}{(\emean - \yhat)^2} } \nonumber \\ & = \sigma^2 + \mbox{Bias(\yhat)}^2 + \VarOp_\sspace \yhat \label{decomposition} \end{align}
For the linear regression case, we can say more. In particular, the estimator $\yhat$ is unbiased; that is, $\EvT{\yhat} = \xbeta$. This follows from writing $\yhat = \xbhat = \xbeta + \underbrace{x_0^{\top} (X^{\top} X)^{-1} X^{\top}}_{l(X)^{\top}} \varepsilon$ and noting that
$\EvTs{l(X)^{\top} \varepsilon} = \Evb{ \Evs{ l(X)^{\top} \varepsilon | X } } = 0$
We can arrive at the final equation in 2.27 by simplifying the variance term above:
\begin{align*} \EvT{ (\yhat - \emean)^2 } & = \EvT{(\yhat - \xbeta)^2} \\ & = \EvT{(l(X)^{\top} \varepsilon)^2} \\ & = \EvTb{ \left( \sum_{i=1}^N l(X)_i \varepsilon_i \right)^2 } \\ & = \EvTb{ \sum_{i=1}^N l(X)_i^2 \varepsilon_i^2 } \\ & = \sigma^2 \EvTs{ x_0^{\top} (X^{\top} X)^{-1} x_0 } \end{align*}
noting from model assumptions that $\varepsilon$ and $X$ are independent as are $\varepsilon_i$ and $\varepsilon_j$.
##### Unbiased Estimators, Misspecified Models
Unfortunately, the unbiasedness of $\yhat$ leads to confusion later when discussing model complexity. In the latter context, low complexity models such as linear regression are said to have high bias and low variance.
The discrepancy seems to be explained by the following.
When we say that $\yhat$ is an unbiased estimator of $\xbeta$, we are tacitly assuming a correctly specified model, and our understanding is that the distribution of estimates based on new population samples would have a mean of $\xbeta$.
On the other hand, when we speak of unbiasedness in the context of model complexity it is with respect to a misspecified model. For low complexity models, we tend to underfit; for high complexity models, overfit. So, intuitively, when we claim that a low complexity model has high bias and low variance it is really a statement about the stability of predictions.
To be precise, linear regression is not unbiased for a misspecified model. Suppose
$y_0 = \xbeta + g(x_0) + \varepsilon_0 \\$
and
\begin{align*} \yhat & = \xbhat \\ & = x_0^{\top} (X^{\top} X)^{-1} X^{\top} y \\ & = x_0^{\top} (X^{\top} X)^{-1} X^{\top}(X \beta + g + \varepsilon) \\ & = \xbeta + \left(X^{\top} X\right)^{-1} X^{\top} \left( g + \varepsilon \right) \end{align*}
So,
\begin{align*} \emean & = \xbeta + x_0^{\top} \EvTs{ \left(X^{\top} X\right)^{-1} X^{\top} g } \end{align*}
and
\begin{align*} \mbox{Bias}(\yhat) & = \emean - (\xbeta + g_0) \\ & = x_0^{\top} \EvTs{ \left(X^{\top} X\right)^{-1} X^{\top} g } - g_0 \end{align*}
The claim in the context of model complexity, in terms of the bias variance tradeoff, is with respect to a misspecified model. And when the data is underfit by a low complexity model, we expect a stable prediction–low variance–that is consistently wrong, hence highly biased.
##### Expected Prediction Error: Nearest Neighbor
Here, we have
$y_0 = f(x_0) + \varepsilon_0$
As we revisit 2.46 and 2.47, any confusion is likely notational: the authors have collapsed the $\Evy{}$ and $\EvT{}$ into a single expectation operator.
Working as before–see equation $\ref{decomposition}$–we obtain 2.46.
\begin{align*} \Evy{ \EvT{ (y_0 - \yhat)^2 } } & = \sigma^2 + \mbox{Bias}^2\left( \nnfk \right) + \VarOp_\sspace \left( \nnfk \right) \end{align*}
To obtain 2.47, note that
$\begin{gather*} \nnfk = \nnkavg{y} \\ \EvT{ \nnfk } = \nnkfavg{x} \\ \end{gather*}$
As always, the bias is the difference between the expected value of the estimator and the true value, so $\mbox{Bias}\left(\nnfk\right) = \nnkfavg{x} - f(x_0)$.
The variance is the expectation of the squared difference between the estimator and its mean,
\begin{align*} \EvT{ \left( \nnfk - \EvT{ \nnfk } \right)^2 } & = \EvTb{ \left( \nnkavg{\varepsilon} \right)^2 } \\ & = \EvTb{ \frac{1}{k} \sum_{l=1}^k \varepsilon_{(l)}^2 } \\ & = \frac{\sigma^2}{k} \end{align*}
where we again use the fact that $\varepsilon_i$ and $\varepsilon_j$ are independent.
Combining the pieces yields 2.47. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998160004615784, "perplexity": 1382.075731542908}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487635724.52/warc/CC-MAIN-20210618043356-20210618073356-00124.warc.gz"} |
https://www.folgertkarsdorp.nl/tag/topics/ | ## Word2Vec: an introduction
[This is a practical tutorial about Word2Vec. If you don’t want to copy-paste all the code, you can download the corresponding IPython notebook here]
In the past two years, Word2Vec, developed by Mikolov et al. (2013), a tool for learning continuous word embeddings from raw text has gained a lot of traction. Word2Vec attempts to associate words with points in space. The spatial distance between words then describes the relation (similarity) between these words. Words that are spatially close, are similar. Words are represented by continuous vectors over x dimensions. This example shows the relation between a number of words where each word is represented by a vector of two dimensions:
import seaborn as sb
import numpy as np
words = ['queen', 'book', 'king', 'magazine', 'car', 'bike']
vectors = np.array([[0.1, 0.3], # queen
[-0.5, -0.1], # book
[0.2, 0.2], # king
[-0.3, -0.2], # magazine
[-0.5, 0.4], # car
[-0.45, 0.3]]) # bike
sb.plt.plot(vectors[:,0], vectors[:,1], 'o')
sb.plt.xlim(-0.6, 0.3)
sb.plt.ylim(-0.3, 0.5)
for word, x, y in zip(words, vectors[:,0], vectors[:,1]):
sb.plt.annotate(word, (x, y), size=12)
The displacement vector (the vector between two vectors) describes the relation between two words. This makes it possible to compare displacement vectors to find pairs of words that have a similar relation to each other. A famous example given in the original paper is the following analogy relation: queen : king :: woman : man which should be read as queen relates to king in the same way as woman relates to man. In algebraic formulation: v[queen] − v[king] = v[woman] − v[man]. This technique of analogical reasoning can be applied to e.g. question answering.
Word2Vec learns continuous word embeddings from plain text. But how? The model assumes the Distributional Hypothesis that words are characterized by words they hang out with. We can use that idea to estimate the probability of two words occurring near each other, e.g. what is the probability of the following words, given Cinderella, i.e $P(w|Cinderella)$?
import pandas as pd
s = pd.Series([0.1, 0.4, 0.01, 0.2, 0.05],
index=["pumpkin", "shoe", "tree", "prince", "luck"])
s.plot(kind='bar')
sb.plt.ylabel("$P(w|Cinderella)$")
#### Softmax Regression
Word2Vec is a very simple neural network with a single hidden layer. Have a look at the following picture. We’ll get into more details in a moment.
The model considers each word wo in turn along with a given context C (e.g. $w_O$ = Cinderella and $C$ = shoe). Now given this context, can we predict what $w_O$ should be? This is essentially a multiclass classification where we have as many labels as our vocabulary size $V$. Using softmax regression, we can compute a probability distribution $\hat{y}$ over the labels. The model attempts to minimize via Stochastic Gradient Descent the difference between the output distribution and the target distribution (which is a one-hot distribution which places all probability mass on the correct word). The difference between the two distribution is measured by the cross-entropy.
The neural network contains two matrics: $W$ and $W′$ of dimensions $V \times N$ and $N \times V$ respectively, where $V$ is the vocabulary size and $N$ the number of dimensions. Let’s make this all a little more concrete with a small example. Say we have a corpus containing the following documents:
sentences = [
'the king loves the queen',
'the queen loves the king',
'the dwarf hates the king',
'the queen hates the dwarf',
'the dwarf poisons the king',
'the dwarf poisons the queen']
We first transform these documents into bag-of-indices to enable easier computation:
from collections import defaultdict
def Vocabulary():
dictionary = defaultdict()
dictionary.default_factory = lambda: len(dictionary)
return dictionary
def docs2bow(docs, dictionary):
"""Transforms a list of strings into a list of lists where
each unique item is converted into a unique integer."""
for doc in docs:
yield [dictionary[word] for word in doc.split()]
vocabulary = Vocabulary()
sentences_bow = list(docs2bow(sentences, vocabulary))
We now construct the two matrices $W$ and $W′$:
import numpy as np
V, N = len(vocabulary), 3
WI = (np.random.random((V, N)) - 0.5) / N
WO = (np.random.random((N, V)) - 0.5) / N
Each row $i$ in $W$ corresponds to word $i$ and each column $j$ corresponds to the $j$-th dimension. Note that $W′$ isn’t simply the transpose of $W$ but a different matrix.
With the two matrices in place we continue with computing the posterior probability of an output word given some input word. Given an input word $w_I$, e.g. dwarf and its corresponding vector $W_I$, what is the probability that the output word $w_O$ is hates? Using the dot product $W_I \cdot W′^T_O$ we compute the distance between the input word dwarf and the output word hates:
print np.dot(WI[vocabulary['dwarf']],
WO.T[vocabulary['hates']])
which should print somethings like -0.0073. Now using softmax regression, we can compute the posterior probability P(w_O|w_I):
$$P(w_O|w_I) = y_i = \frac{exp(W_I \cdot W’^T_O)}{\sum^V_{j=1} exp(W_I \cdot W’^T_j)}$$
p = (np.exp(-np.dot(WI[vocabulary['dwarf']],
WO.T[vocabulary['hates']])) /
sum(np.exp(-np.dot(WI[vocabulary['dwarf']],
WO.T[vocabulary[w]]))
for w in vocabulary))
print p
which will print something like: 0.14.
#### Updating the hidden-to-output layer weights
Word2Vec attempts to associate words with points in space. These points in space are represented by the continuous embeddings of the words. All vectors are initialized as random points in space, so we need to learn better positions. The model does so by maximizing the equation above. The corresponding loss function which we try to minimize is $E=−\log P(w_O|w_I)$. First, let’s focus on how to update the hidden-to-output layer weights. Say the target output word is Cinderella. Given the aformentioned one-hot target distribution $t$, the error can be computed as $t_j−y_j=e_j$, where $t_j$ is 1 iff $w_j$ is the actual output word. So, the actual output word is Cinderella and we compute the posterior probability of $P(pumpkin | tree)$, the error will be $0 – P(pumpkin | tree)$, because pumpkin isn’t the actual ouput word.
To obtain the gradient on the hidden-to-output weights, we compute $e_j \cdot h_i$, where $h_i$ is a copy of the vector corresponding to the input word (only holds with a context of a single word). Finally, using stochastic gradient descent, with a learning rate $\nu$ we obtain the weight update equation for the hidden to output layer weights:
$$W’^{T (t)}_j = W’^{T (t-1)}_j – \nu \cdot e_j \cdot h_j$$
Assume the target word is king and the context or input word $C$ is queen. Given this input word we compute for each word in the vocabulary the posterior probability $P(word | queen)$. If the word is our target word, the error will be $1 – P(word | queen)$; otherwise $0 – P(word | queen)$. Finally, using stocastic gradient descent we update the hidden-to-output layer weights:
target_word = 'king'
input_word = 'queen'
learning_rate = 1.0
for word in vocabulary:
p_word_queen = (
np.exp(-np.dot(WO.T[vocabulary[word]],
WI[vocabulary[input_word]])) /
sum(np.exp(-np.dot(WO.T[vocabulary[w]],
WI[vocabulary[input_word]]))
for w in vocabulary))
t = 1 if word == target_word else 0
error = t - p_word_queen
WO.T[vocabulary[word]] = (
WO.T[vocabulary[word]] - learning_rate *
error * WI[vocabulary[input_word]])
print WO
#### Updating the input-to-hidden layer weights
Now that we have a way to update the hidden-to-output layer weights, we concentrate on updating the input-to-hidden layer weights. We need to backpropagate the prediction errors to the input-to-hidden weights. We first compute $EH$ which is an $N$ dimensional vector representing the sum of the hidden-to-output vectors for each word in the vocabulary weighted by their prediction error:
$$\sum^V_{j=1} e_j \cdot W’_{i,j} = {EH}_i$$
Again using the learning rate $\nu$ we update the weights using:
$$W^{(t)}_{w_I} = W^{(t-1)}_{w_I} – \nu \cdot EH$$
Let’s see how that works in Python:
WI[vocabulary[input_word]] = WI[vocabulary[input_word]] - learning_rate * WO.sum(1)
If we now would recompute the probability of each word given the input word queen, we see that the probability of king given queen has gone up:
for word in vocabulary:
p = (np.exp(-np.dot(WO.T[vocabulary[word]],
WI[vocabulary[input_word]])) /
sum(np.exp(-np.dot(WO.T[vocabulary[w]],
WI[vocabulary[input_word]]))
for w in vocabulary))
print word, p
#### Multi-word context
The model described above is the CBOW architecture of Word2Vec. However, we assumed that the context $C$ was only a single input word. This allowed us to simply copy the input vector to the hidden layer. If the context $C$ comprises multiple words, instead of copying the input vector we take the mean of their input vectors as our hidden layer:
$$h = \frac{1}{C} (W_1 + W_2 + \ldots + W_C)$$
The update functions remain the same except that for the update of the input vectors, we need to apply the update to each word in the contect $C$:
$$W^{(t)}_{w_I} = W^{(t-1)}_{w_I} – \frac{1}{C} \cdot \nu \cdot EH$$
Let’s see that in action. Again assume the target word is king. The context consists of two words: queen and loves.
target_word = 'king'
context = ['queen', 'loves']
We first take the average of the two context vectors:
h = (WI[vocabulary['queen']] + WI[vocabulary['loves']]) / 2
Then we apply the hidden-to-output layer update:
for word in vocabulary:
p = (np.exp(-np.dot(WO.T[vocabulary[word]], h)) /
sum(np.exp(-np.dot(WO.T[vocabulary[w]], h))
for w in vocabulary))
t = 1 if word == target_word else 0
error = t - p
WO.T[vocabulary[word]] = (
WO.T[vocabulary[word]] - learning_rate *
error * h)
print WO
Finally we update the vector of each input word in the context:
for word in context:
WI[vocabulary[word]] = (
WI[vocabulary[word]] - (1. / len(context)) *
learning_rate * WO.sum(1))
And that’s it! (Well… to really apply this learning algorithm to large collections of text, we need to add some optimization because our implementation will be incredibly slow. Fortunately the paper describes some really clever ways to speed up the computation such as hierarchical soft-max and negative sampling. These topics are however beyond the scope of this introduction. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8944724798202515, "perplexity": 1397.54864940967}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500076.87/warc/CC-MAIN-20230203221113-20230204011113-00279.warc.gz"} |
http://web.stanford.edu/class/ee263/quizzes/lin-fcts.html | $\newcommand{\ones}{\mathbf 1}$
If $f:\mathbf{R}^n \to \mathbf{R}^m$, with $f(x) = Ax$, then
Suppose $A \in \mathbf{R}^{10 \times 10}$ is tridiagonal, i.e., $a_{ij} = 0$ if $|i-j| > 1$, and $y=Ax$. Then $y_4$ does not depend on $x_1$.
• Correct! $y_i$ depends on $x_j$ if $a_{ij} \neq 0$.
• Incorrect.
Let $x,y \in \mathbf{R}^3$. For $i = 1,2,3$, $y_i$ is the average of $x_1,\ldots,x_i$. Then we have $y = Ax$, with
Let $P \in \mathbf{R}^{m \times n}$ and $q \in \mathbf{R}^n$, where $P_{ij}$ is the price of good $j$ in country $i$, and $q_j$ is the quantity of good $j$ needed to produce some product. Then $(Pq)_i$ is
In the matrix $A\in {\mathbf R}^{m \times n}$, the entries in each row increase in size, moving from left to right. With $y=Ax$, this means
In the matrix $A\in {\mathbf R}^{n \times n}$, the significant entries in the matrix are all near the diagonal. With $y=Ax$, this means
The $j$th column of $AB$ is
The $i$th row of $AB$ is
The $(i,j)$ entry of $AB$ is | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9988382458686829, "perplexity": 172.2240002963884}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535924501.17/warc/CC-MAIN-20140909013109-00223-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/physics-i-should-be-able-to-do.41035/ | # Physics I should be able to do
• Start date
• #1
612
1
For some reason I can’t solve this:
10. [PSE6 1.P.039.] One cubic meter (1.00 m3) of aluminum has a mass of 2.70 103 kg, and 1.00 m3 of iron has a mass of 7.86 103 kg. Find the radius of a solid aluminum sphere that will balance a solid iron sphere of radius 3.10 cm on an equal-arm balance.
I keep getting $$r =(\frac{2.7*3.1^3}{7.86} )^{1/3}$$
Last edited:
Related Introductory Physics Homework Help News on Phys.org
• #2
jamesrc
Gold Member
476
1
Since it's an equal arm balance, the mass of the iron ball will equal the mass of the aluminum ball:
$$m_{\rm Fe} = m_{\rm Al}$$
$$\rho_{\rm Fe}V_{\rm Fe} = \rho_{\rm Al}V_{\rm Al}$$
$$\rho_{\rm Fe}\frac{4\pi r_{\rm Fe}^3}{3} = \rho_{\rm Al}\frac{4\pi r_{\rm Al}^3}{3}$$
Cancel out the common terms and solve for rFe in terms of the given parameters. If you follow that, you should find your mistakes.
• #3
612
1
i thought thats what i did
• #4
612
1
here are some other ones we arn't getting:
10.An auditorium measures 40.0 m 15.0 m 11.0 m. The density of air is 1.20 kg/m3.
(b) What is the weight of air in the room in pounds?
i found the volume to be: 233077ft^3 not sure what do form there.
11(b)Consider a solid disk of mass M=12.3kg and areal density =Dx3. Determine the value D if the radius of the disk is 0.25m.
I don’t even know where to start.
2. What level of precision is implied in an order-of-magnitude calculation?
i thought it was 3
Last edited:
• #5
382
2
Rearrange the equation by jamesrc :
$$r_{Al} =\sqrt[3]{\frac{\rho_{Fe}}{\rho_{Al}}r^3_{Fe}}$$
$$r_{Al}=\sqrt[3]{\frac{7.86103}{2.70103}(0.0310)^3}$$
= 0.0443 m
$$\rho=\frac{m}{v}$$
NOT
$$\rho=\frac{v}{m}$$
* Keep track of the units. Here, I used SI units.
• #6
382
2
JonF said:
here are some other ones we arn't getting:
10.An auditorium measures 40.0 m 15.0 m 11.0 m. The density of air is 1.20 kg/m3.
(b) What is the weight of air in the room in pounds?
i found the volume to be: 233077ft^3 not sure what do form there.
11(b)Consider a solid disk of mass M=12.3kg and areal density =Dx3. Determine the value D if the radius of the disk is 0.25m.
I don’t even know where to start.
10.(a) It is better to use the units given and then convert it lastly.
$$233077ft^3=6600m^3$$
Using the formula for density:
$$\rho_{air}=\frac{m_{air}}{v_{air}}$$
Then convert the mass to pound unit.
(b) I think it asks us to modify the formula for density a little bit so that it can be applied in the question. Since the disk is flat, i think we should use this :
$$\rho_{disk} = \frac{m_{disk}}{A_{disk}} = 3D$$
• #7
612
1
sorry for 11(b) it is dx^3.
so i get 12.3/pi(.25)^2 = dx^3 not sure where to go from there
but thank you guys i got all the rest
Last edited:
• #8
382
2
JonF,
Omit the hint given for 11(b). it looks useless now since the question has changed. what is x ? i am stuck as well.
• #9
612
1
Well the first question was
11. (a)Consider a two meterstick of mass M=18.2kg and linear density =Cx^4. Determine the value C.
So we did this: $$c*\int_0^2 x^4 dx= 18.2$$ and it was right.
This is the other question.
b) Consider a solid disk of mass M=12.3kg and areal density =Dx^3. Determine the value D if the radius of the disk is 0.25m.
• #10
382
2
$$m_{disk}=\rho_{disk}A_{disk}$$
See the attached image :
$$dm=Dx^3[\frac{1}{2}(d\theta)(r^2)]$$ ......(1)
$$x=rcos\theta$$.....(2)
(2)$$\rightarrow$$(1)
$$dm=\frac{D}{2}r^5cos^3\theta d\theta$$
$$\int_0^m dm = \int_0^{2\pi} \frac{D}{2}r^5 cos^3\theta d\theta$$
Hint :
1. $$cos^3\theta = cos\theta cos^2\theta$$
2.$$cos2\theta = 2cos^2\theta -1$$
3.$$cosAcosB = \frac{1}{2}[cos(A+B) + cos (A-B)]$$
• #11
612
1
uh you so lost me, but thanks for you help
• #12
HallsofIvy
Homework Helper
41,833
955
The very complicated formula Leong used was from saying that x, in polar coordinates, is given by x= r cos(θ). If the problem has said the density was
Dr3, that would be much simpler.
• #13
382
2
JonF,
I tried to integrate the equation and found that it equaled to zero!
I want to say sorry because you might think that i was such a *** person trying to teach others to do some problems when he himself didn't even try it first!
Since the disk areal density is given by $$Dx^3$$; then $$x \geq 0$$ because the density can never be negative.
So, my flawed equation is only valid for this range of $$\theta$$ : $$-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}$$.
You can integrate for these intervals:
1. $$[0,\frac{M}{4}]$$ and $$[0,\frac{\pi}{2}]$$ : For this value of $$\theta$$, the mass of the disk is just a quarter of the total mass since the mass of the disk is proportional to the area of the disk.
2.$$[0,\frac{M}{4}]$$ and $$[-\frac{\pi}{2},0,]$$ : For this value of $$\theta$$, the mass of the disk is just a quarter of the total mass since the mass of the disk is proportional to the area of the disk.
3.$$[0,\frac{M}{2}]$$ and $$[-\frac{\pi}{2},\frac{\pi}{2}]$$ : For this value of $$\theta$$, the mass of the disk is half of the total mass since the mass of the disk is proportional to the area of the disk.
All these intervals will yield the same results.
I choose option #1 to do this :
$$\int_0^\frac{M}{4} dm = \int_0^\frac{\pi}{2} \frac{D}{2}r^5 cos^3\theta d\theta$$
$$cos^3\theta = \frac{1}{4}[cos 3\theta + 3cos\theta]$$
$$[m]_0^\frac{M}{4}=\frac{D}{2}r^5\int_0^{\frac{\pi}{2}} \frac{1}{4}(cos3\theta +3cos\theta) d\theta$$
$$\frac{M}{4}=\frac{D}{2}r^5(\frac{1}{4})[\frac{sin3\theta}{3}+3sin\theta]_0^\frac{\pi}{2}$$
$$M=\frac{D}{2}r^5[\frac{8}{3}]$$
$$M=\frac{4}{3}Dr^5$$
$$D=\frac{3M}{4r^5}$$
$$D=\frac{3(12.3)}{4(0.25)^5}$$
$$=9.4X10^3$$ $$kg/m^5$$
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https://worldwidescience.org/topicpages/l/lorentz+symmetry+violation.html | #### Sample records for lorentz symmetry violation
1. Charged tensor matter fields and Lorentz symmetry violation via spontaneous symmetry breaking
International Nuclear Information System (INIS)
Colatto, L.P.; Penna, A.L.A.; Santos, W.C.
2003-10-01
We consider a model with a charged vector field along with a Cremmer-Scherk-Kalb-Ramond (CSKR) matter field coupled to a U(1) gauge potential. We obtain a natural Lorentz symmetry violation due to the local U(1) spontaneous symmetry breaking mechanism triggered by the imaginary part of the vector matter. The choice of the unitary gauge leads to the decoupling of the gauge-Kr sector from the Higgs-Kr sector. The excitation spectrum is carefully analyzed and the physical modes are identified. We propose an identification of the neutral massive spin-1 Higgs-like field with the massive Z' boson of the so-called mirror matter models. (author)
2. Search for Violations of Lorentz Invariance and CPT Symmetry in B-(s)(0) Mixing
NARCIS (Netherlands)
Aaij, R.; Beteta, C. Abellan; Adeva, B.; Adinolfi, M.; Ajaltouni, Z.; Akar, S.; Albrecht, J.; Alessio, F.; Alexander, M.; Ali, S.; Alkhazov, G.; Cartelle, P. Alvarez; Alves, A. A.; Amato, S.; Amerio, S.; Amhis, Y.; An, L.; Anderlini, L.; Andreassi, G.; Andreotti, M.; Andrews, J. E.; Appleby, R. B.; Gutierrez, O. Aquines; Archilli, F.; d'Argent, P.; Artamonov, A.; Artuso, M.; Aslanides, E.; Auriemma, G.; Baalouch, M.; Bachmann, S.; Back, J. J.; Badalov, A.; Baesso, C.; Baker, S.; Baldini, W.; Barlow, R. J.; Barschel, C.; Barsuk, S.; Barter, W.; Batozskaya, V.; Battista, V.; Beaucourt, L.; Beddow, J.; Bedeschi, F.; Bediaga, I.; Bel, L. J.; Onderwater, C. J. G.; Pellegrino, A.; Tolk, S.
2016-01-01
Violations of CPT symmetry and Lorentz invariance are searched for by studying interference effects in B-0 mixing and in B-s(0) mixing. Samples of B-0 -> J/psi K-S(0) and B-0(s) -> J/psi K+K- decays are recorded by the LHCb detector in proton-proton collisions at center-of-mass energies of 7 and 8
3. Lorentz Symmetry Violations from Matter-Gravity Couplings with Lunar Laser Ranging
Science.gov (United States)
Bourgoin, A.; Le Poncin-Lafitte, C.; Hees, A.; Bouquillon, S.; Francou, G.; Angonin, M.-C.
2017-11-01
The standard-model extension (SME) is an effective field theory framework aiming at parametrizing any violation to the Lorentz symmetry (LS) in all sectors of physics. In this Letter, we report the first direct experimental measurement of SME coefficients performed simultaneously within two sectors of the SME framework using lunar laser ranging observations. We consider the pure gravitational sector and the classical point-mass limit in the matter sector of the minimal SME. We report no deviation from general relativity and put new realistic stringent constraints on LS violations improving up to 3 orders of magnitude previous estimations.
4. Three questions on Lorentz violation
Energy Technology Data Exchange (ETDEWEB)
Iorio, Alfredo [Institute of Particle and Nuclear Physics, Charles University of Prague - V Holesovickach 2, 180 00 Prague 8 (Czech Republic); Department of Physics ' E. R. Caianiello' , University of Salerno and I.N.F.N. Naples, Gruppo Collegato di Salerno - Via Allende, 84081 Baronissi (Italy)
2007-05-15
We review the basics of the two most widely used approaches to Lorentz violation - the Standard Model Extension and Noncommutative Field Theory - and discuss in some detail the example of the modified spectrum of the synchrotron radiation. Motivated by touching upon such a fundamental issue as Lorentz symmetry, we ask three questions: What is behind the search for Lorentz violation? Is String Theory a physical theory? Is there an alternative to Supersymmetry?.
5. Search for violations of Lorentz invariance and $CPT$ symmetry in $B^0_{(s)}$ mixing
CERN Document Server
2016-06-15
Violations of $CPT$ symmetry and Lorentz invariance are searched for by studying interference effects in $B^0$ mixing and in $B^0_s$ mixing. Samples of $B^0\\to J/\\psi K^0_{\\mathrm{S}}$ and $B^0_s\\to J/\\psi K^+ K^-$ decays are recorded by the LHCb detector in proton--proton collisions at centre-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb$^{-1}$. No periodic variations of the particle-antiparticle mass differences are found, consistent with Lorentz invariance and $CPT$ symmetry. Results are expressed in terms of the Standard Model Extension parameter $\\Delta a_{\\mu}$ with precisions of $\\mathcal{O}(10^{-15})$ and $\\mathcal{O}(10^{-14})$ GeV for the $B^0$ and $B^0_s$ systems, respectively. With no assumption on Lorentz (non-)invariance, the $CPT$-violating parameter $z$ in the $B^0_s$ system is measured for the first time and found to be $\\mathcal{R}e(z) = -0.022 \\pm 0.033 \\pm 0.005$ and $\\mathcal{I}m(z) = 0.004 \\pm 0.011\\pm 0.002$, where the first uncertainti...
6. Search for Violations of Lorentz Invariance and CPT Symmetry in B_{(s)}^{0} Mixing.
Science.gov (United States)
2016-06-17
Violations of CPT symmetry and Lorentz invariance are searched for by studying interference effects in B^{0} mixing and in B_{s}^{0} mixing. Samples of B^{0}→J/ψK_{S}^{0} and B_{s}^{0}→J/ψK^{+}K^{-} decays are recorded by the LHCb detector in proton-proton collisions at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb^{-1}. No periodic variations of the particle-antiparticle mass differences are found, consistent with Lorentz invariance and CPT symmetry. Results are expressed in terms of the standard model extension parameter Δa_{μ} with precisions of O(10^{-15}) and O(10^{-14}) GeV for the B^{0} and B_{s}^{0} systems, respectively. With no assumption on Lorentz (non)invariance, the CPT-violating parameter z in the B_{s}^{0} system is measured for the first time and found to be Re(z)=-0.022±0.033±0.005 and Im(z)=0.004±0.011±0.002, where the first uncertainties are statistical and the second systematic.
7. Astroparticle tests of Lorentz symmetry
Energy Technology Data Exchange (ETDEWEB)
Diaz, Jorge [Karlsruhe Institute of Technology, Karlsruhe (Germany)
2016-07-01
Lorentz symmetry is a cornerstone of modern physics. As the spacetime symmetry of special relativity, Lorentz invariance is a basic component of the standard model of particle physics and general relativity, which to date constitute our most successful descriptions of nature. Deviations from exact symmetry would radically change our view of the universe and current experiments allow us to test the validity of this assumption. In this talk, I describe effects of Lorentz violation in cosmic rays and gamma rays that can be studied in current observatories.
8. Lorentz violation, gravitoelectromagnetic field and Bhabha scattering
Science.gov (United States)
Santos, A. F.; Khanna, Faqir C.
2018-01-01
Lorentz symmetry is a fundamental symmetry in the Standard Model (SM) and in General Relativity (GR). This symmetry holds true for all models at low energies. However, at energies near the Planck scale, it is conjectured that there may be a very small violation of Lorentz symmetry. The Standard Model Extension (SME) is a quantum field theory that includes a systematic description of Lorentz symmetry violations in all sectors of particle physics and gravity. In this paper, SME is considered to study the physical process of Bhabha Scattering in the Gravitoelectromagnetism (GEM) theory. GEM is an important formalism that is valid in a suitable approximation of general relativity. A new nonminimal coupling term that violates Lorentz symmetry is used in this paper. Differential cross-section for gravitational Bhabha scattering is calculated. The Lorentz violation contributions to this GEM scattering cross-section are small and are similar in magnitude to the case of the electromagnetic field.
9. Searching for Lorentz violation
International Nuclear Information System (INIS)
Allen, Roland E.; Yokoo, Seiichirou
2004-01-01
Astrophysical, terrestrial, and space-based searches for Lorentz violation are very briefly reviewed. Such searches are motivated by the fact that all superunified theories (and other theories that attempt to include quantum gravity) have some potential for observable violations of Lorentz invariance. Another motivation is the exquisite sensitivity of certain well-designed experiments and observations to particular forms of Lorentz violation. We also review some new predictions of a specific Lorentz-violating theory: If a fundamental energy m-bar c2 in this theory lies below the usual GZK cutoff E GZK , the cutoff is shifted to infinite energy; i.e., it no longer exists. On the other hand, if m-bar c2 lies above E GZK , there is a high-energy branch of the fermion dispersion relation which provides an alternative mechanism for super-GZK cosmic-ray protons
10. Constrained Gauge Fields from Spontaneous Lorentz Violation
CERN Document Server
Chkareuli, J L; Jejelava, J G; Nielsen, H B
2008-01-01
Spontaneous Lorentz violation realized through a nonlinear vector field constraint of the type $A_{\\mu}^{2}=M^{2}$ ($M$ is the proposed scale for Lorentz violation) is shown to generate massless vector Goldstone bosons, gauging the starting global internal symmetries in arbitrary relativistically invariant theories. The gauge invariance appears in essence as a necessary condition for these bosons not to be superfluously restricted in degrees of freedom, apart from the constraint due to which the true vacuum in a theory is chosen by the Lorentz violation. In the Abelian symmetry case the only possible theory proves to be QED with a massless vector Goldstone boson naturally associated with the photon, while the non-Abelian symmetry case results in a conventional Yang-Mills theory. These theories, both Abelian and non-Abelian, look essentially nonlinear and contain particular Lorentz (and $CPT$) violating couplings when expressed in terms of the pure Goldstone vector modes. However, they do not lead to physical ...
11. Lorentz violation naturalness revisited
Energy Technology Data Exchange (ETDEWEB)
Belenchia, Alessio; Gambassi, Andrea; Liberati, Stefano [SISSA - International School for Advanced Studies, via Bonomea 265, 34136 Trieste (Italy); INFN, Sezione di Trieste, via Valerio 2, 34127 Trieste (Italy)
2016-06-08
We revisit here the naturalness problem of Lorentz invariance violations on a simple toy model of a scalar field coupled to a fermion field via a Yukawa interaction. We first review some well-known results concerning the low-energy percolation of Lorentz violation from high energies, presenting some details of the analysis not explicitly discussed in the literature and discussing some previously unnoticed subtleties. We then show how a separation between the scale of validity of the effective field theory and that one of Lorentz invariance violations can hinder this low-energy percolation. While such protection mechanism was previously considered in the literature, we provide here a simple illustration of how it works and of its general features. Finally, we consider a case in which dissipation is present, showing that the dissipative behaviour does not percolate generically to lower mass dimension operators albeit dispersion does. Moreover, we show that a scale separation can protect from unsuppressed low-energy percolation also in this case.
12. Threshold analyses and Lorentz violation
International Nuclear Information System (INIS)
Lehnert, Ralf
2003-01-01
In the context of threshold investigations of Lorentz violation, we discuss the fundamental principle of coordinate independence, the role of an effective dynamical framework, and the conditions of positivity and causality. Our analysis excludes a variety of previously considered Lorentz-breaking parameters and opens an avenue for viable dispersion-relation investigations of Lorentz violation
13. Constrained gauge fields from spontaneous Lorentz violation
DEFF Research Database (Denmark)
Chkareuli, J. L.; Froggatt, C. D.; Jejelava, J. G.
2008-01-01
Spontaneous Lorentz violation realized through a nonlinear vector field constraint of the type AµAµ=M2 (M is the proposed scale for Lorentz violation) is shown to generate massless vector Goldstone bosons, gauging the starting global internal symmetries in arbitrary relativistically invariant...... theories. The gauge invariance appears in essence as a necessary condition for these bosons not to be superfluously restricted in degrees of freedom, apart from the constraint due to which the true vacuum in a theory is chosen by the Lorentz violation. In the Abelian symmetry case the only possible theory...... couplings when expressed in terms of the pure Goldstone vector modes. However, they do not lead to physical Lorentz violation due to the simultaneously generated gauge invariance. Udgivelsesdato: June 11...
14. Spacetime-varying couplings and Lorentz violation
International Nuclear Information System (INIS)
Kostelecky, V. Alan; Lehnert, Ralf; Perry, Malcolm J.
2003-01-01
Spacetime-varying coupling constants can be associated with violations of local Lorentz invariance and CPT symmetry. An analytical supergravity cosmology with a time-varying fine-structure constant provides an explicit example. Estimates are made for some experimental constraints
15. Symanzik–Becchi–Rouet–Stora lessons on renormalizable models with broken symmetry: The case of Lorentz violation
Energy Technology Data Exchange (ETDEWEB)
Del Cima, Oswaldo M.; Franco, Daniel H.T.; Piguet, Olivier, E-mail: [email protected]
2016-11-15
In this paper, we revisit the issue intensively studied in recent years on the generation of terms by radiative corrections in models with broken Lorentz symmetry. The algebraic perturbative method of handling the problem of renormalization of the theories with Lorentz symmetry breaking, is used. We hope to make clear the Symanzik's aphorism: “Whether you like it or not, you have to include in the lagrangian all counter terms consistent with locality and power-counting, unless otherwise constrained by Ward identities.”{sup 1}.
16. Lorentz Violation, Möller Scattering, and Finite Temperature
Directory of Open Access Journals (Sweden)
Alesandro F. Santos
2018-01-01
Full Text Available Lorentz and CPT symmetries may be violated in new physics that emerges at very high energy scale, that is, at the Planck scale. The differential cross section of the Möller scattering due to Lorentz violation at finite temperature is calculated. Lorentz-violating effects emerge from an interaction vertex due to a CPT-odd nonminimal coupling in the covariant derivative. The finite temperature effects are determined using the Thermo Field Dynamics (TFD formalism.
17. Statistical mechanics and Lorentz violation
International Nuclear Information System (INIS)
2004-01-01
The theory of statistical mechanics is studied in the presence of Lorentz-violating background fields. The analysis is performed using the Standard-Model Extension (SME) together with a Jaynesian formulation of statistical inference. Conventional laws of thermodynamics are obtained in the presence of a perturbed hamiltonian that contains the Lorentz-violating terms. As an example, properties of the nonrelativistic ideal gas are calculated in detail. To lowest order in Lorentz violation, the scalar thermodynamic variables are only corrected by a rotationally invariant combination of parameters that mimics a (frame dependent) effective mass. Spin-couplings can induce a temperature-independent polarization in the classical gas that is not present in the conventional case. Precision measurements in the residual expectation values of the magnetic moment of Fermi gases in the limit of high temperature may provide interesting limits on these parameters
18. Anomalous Lorentz and CPT violation
Science.gov (United States)
2018-01-01
If there exists Lorentz and CPT violation in nature, then it is crucial to discover and understand the underlying mechanism. In this contribution, we discuss one such mechanism which relies on four-dimensional chiral gauge theories defined over a spacetime manifold with topology ℛ3 × S 1 and periodic spin structure for the compact dimension. It can be shown that the effective gauge-field action contains a local Chern-Simons-like term which violates Lorentz and CPT invariance. For arbitrary Abelian U(1) gauge fields with trivial holonomies in the compact direction, this anomalous Lorentz and CPT violation has recently been established perturbatively with a Pauli-Villars-type regularization and nonperturbatively with a lattice regularization based on Ginsparg-Wilson fermions.
19. Sixth Meeting on CPT and Lorentz Symmetry
CERN Document Server
CPT and Lorentz Symmetry
2014-01-01
This book contains the Proceedings of the Sixth Meeting on CPT and Lorentz Symmetry, held at Indiana University in Bloomington on June 17–21, 2013. The Meeting focused on tests of these fundamental symmetries and on related theoretical issues, including scenarios for possible violations. Topics covered at the meeting include searches for CPT and Lorentz violations involving: accelerator and collider experiments; atomic, nuclear, and particle decays; birefringence, dispersion, and anisotropy in cosmological sources; clock-comparison measurements; electromagnetic resonant cavities and lasers; tests of the equivalence principle; gauge and Higgs particles; high-energy astrophysical observations; laboratory tests of gravity; matter interferometry; neutrino oscillations and propagation; oscillations and decays of neutral mesons; particle–antiparticle comparisons; post-newtonian gravity in the solar system and beyond; second- and third-generation particles; space-based missions; spectroscopy of hydrogen and ant...
20. Lorentz violation, gravitoelectromagnetism and Bhabha scattering at finite temperature
Science.gov (United States)
Santos, A. F.; Khanna, Faqir C.
2018-04-01
Gravitoelectromagnetism (GEM) is an approach for the gravitation field that is described using the formulation and terminology similar to that of electromagnetism. The Lorentz violation is considered in the formulation of GEM that is covariant in its form. In practice, such a small violation of the Lorentz symmetry may be expected in a unified theory at very high energy. In this paper, a non-minimal coupling term, which exhibits Lorentz violation, is added as a new term in the covariant form. The differential cross-section for Bhabha scattering in the GEM framework at finite temperature is calculated that includes Lorentz violation. The Thermo Field Dynamics (TFD) formalism is used to calculate the total differential cross-section at finite temperature. The contribution due to Lorentz violation is isolated from the total cross-section. It is found to be small in magnitude.
1. Strong binary pulsar constraints on Lorentz violation in gravity.
Science.gov (United States)
Yagi, Kent; Blas, Diego; Yunes, Nicolás; Barausse, Enrico
2014-04-25
Binary pulsars are excellent laboratories to test the building blocks of Einstein's theory of general relativity. One of these is Lorentz symmetry, which states that physical phenomena appear the same for all inertially moving observers. We study the effect of violations of Lorentz symmetry in the orbital evolution of binary pulsars and find that it induces a much more rapid decay of the binary's orbital period due to the emission of dipolar radiation. The absence of such behavior in recent observations allows us to place the most stringent constraints on Lorentz violation in gravity, thus verifying one of the cornerstones of Einstein's theory much more accurately than any previous gravitational observation.
2. Strong Binary Pulsar Constraints on Lorentz Violation in Gravity
CERN Document Server
Yagi, Kent; Yunes, Nicolas; Barausse, Enrico
2014-01-01
Binary pulsars are excellent laboratories to test the building blocks of Einstein's theory of General Relativity. One of these is Lorentz symmetry which states that physical phenomena appear the same for all inertially moving observers. We study the effect of violations of Lorentz symmetry in the orbital evolution of binary pulsars and find that it induces a much more rapid decay of the binary's orbital period due to the emission of dipolar radiation. The absence of such behavior in recent observations allows us to place the most stringent constraints on Lorentz violation in gravity, thus verifying one of the cornerstones of Einstein's theory much more accurately than any previous gravitational observation.
3. Lorentz and CPT violation in QED revisited: A missing analysis
Energy Technology Data Exchange (ETDEWEB)
Del Cima, Oswaldo M., E-mail: [email protected] [Universidade Federal Fluminense (UFF), Polo Universitario de Rio das Ostras, Rua Recife s/n, 28890-000, Rio das Ostras, RJ (Brazil); Fonseca, Jakson M., E-mail: [email protected] [Universidade Federal de Vicosa (UFV), Departamento de Fisica, Avenida Peter Henry Rolfs s/n, 36570-000, Vicosa, MG (Brazil); Franco, Daniel H.T., E-mail: [email protected] [Universidade Federal de Vicosa (UFV), Departamento de Fisica, Avenida Peter Henry Rolfs s/n, 36570-000, Vicosa, MG (Brazil); Piguet, Olivier, E-mail: [email protected] [Universidade Federal do Espirito Santo (UFES), Departamento de Fisica, Campus Universitario de Goiabeiras, 29060-900, Vitoria, ES (Brazil)
2010-05-03
We investigate the breakdown of Lorentz symmetry in QED by a CPT violating interaction term consisting of the coupling of an axial fermion current with a constant vector field b, in the framework of algebraic renormalization - a regularization-independent method. We show, to all orders in perturbation theory, that a CPT-odd and Lorentz violating Chern-Simons-like term, definitively, is not radiatively induced by the axial coupling of the fermions with the constant vector b.
4. Lorentz and CPT violation in QED revisited: A missing analysis
International Nuclear Information System (INIS)
Del Cima, Oswaldo M.; Fonseca, Jakson M.; Franco, Daniel H.T.; Piguet, Olivier
2010-01-01
We investigate the breakdown of Lorentz symmetry in QED by a CPT violating interaction term consisting of the coupling of an axial fermion current with a constant vector field b, in the framework of algebraic renormalization - a regularization-independent method. We show, to all orders in perturbation theory, that a CPT-odd and Lorentz violating Chern-Simons-like term, definitively, is not radiatively induced by the axial coupling of the fermions with the constant vector b.
5. Hiding Lorentz invariance violation with MOND
International Nuclear Information System (INIS)
Sanders, R. H.
2011-01-01
Horava-Lifshitz gravity is an attempt to construct a renormalizable theory of gravity by breaking the Lorentz invariance of the gravitational action at high energies. The underlying principle is that Lorentz invariance is an approximate symmetry and its violation by gravitational phenomena is somehow hidden to present limits of observational precision. Here I point out that a simple modification of the low-energy limit of Horava-Lifshitz gravity in its nonprojectable form can effectively camouflage the presence of a preferred frame in regions where the Newtonian gravitational field gradient is higher than cH 0 ; this modification results in the phenomenology of modified Newtonian dynamics (MOND) at lower accelerations. As a relativistic theory of MOND, this modified Horava-Lifshitz theory presents several advantages over its predecessors.
6. Quantum-gravity phenomenology, Lorentz symmetry, and the SME
International Nuclear Information System (INIS)
Lehnert, Ralf
2007-01-01
Violations of spacetime symmetries have recently been identified as promising signatures for physics underlying the Standard Model. The present talk gives an overview over various topics in this field: The motivations for spacetime-symmetry research, including some mechanisms for Lorentz breaking, are reviewed. An effective field theory called the Standard-Model Extension (SME) for the description of the resulting low-energy effects is introduced, and some experimental tests of Lorentz and CPT invariance are discussed
7. Lorentz and CPT violation in the Standard-Model Extension
Energy Technology Data Exchange (ETDEWEB)
Lehnert, Ralf, E-mail: [email protected] [Indiana University Center for Spacetime Symmetries (United States)
2013-03-15
Lorentz and CPT invariance are among the symmetries that can be investigated with ultrahigh precision in subatomic physics. Being spacetime symmetries, Lorentz and CPT invariance can be violated by minuscule amounts in many theoretical approaches to underlying physics that involve novel spacetime concepts, such as quantized versions of gravity. Regardless of the underlying mechanism, the low-energy effects of such violations are expected to be governed by effective field theory. This talk provides a survey of this idea and includes an overview of experimental efforts in the field.
8. Lorentz- and CPT-symmetry studies in subatomic physics
Energy Technology Data Exchange (ETDEWEB)
Lehnert, Ralf, E-mail: [email protected] [Leibniz Universität Hannover (Germany)
2016-12-15
Subatomic systems provide an exquisite test bench for spacetime symmetries. This work motivates such measurements, reviews the effective field theory test framework for the description of Lorentz and CPT violation, and employs this framework to study the phenomenology of spacetime-symmetry breaking in various subatomic systems.
9. Brane Lorentz symmetry from Lorentz breaking in the bulk
Energy Technology Data Exchange (ETDEWEB)
Bertolami, O [Departamento de Fisica, Instituto Superior Tecnico, Avenida Rovisco Pais 1, 1049-001 Lisbon (Portugal); Carvalho, C [Departamento de Fisica, Instituto Superior Tecnico, Avenida Rovisco Pais 1, 1049-001 Lisbon (Portugal)
2007-05-15
We propose the mechanism of spontaneous symmetry breaking of a bulk vector field as a way to generate the selection of bulk dimensions invisible to the standard model confined to the brane. By assigning a nonvanishing vacuum value to the vector field, a direction is singled out in the bulk vacuum, thus breaking the bulk Lorentz symmetry. We present the condition for induced Lorentz symmetry on the brane, as phenomenologically required.
10. Spin-dependent potentials, axion-like particles and Lorentz-symmetry violation. Beyond the Standard Model phenomenology at the low-energy frontier of physics
Energy Technology Data Exchange (ETDEWEB)
Cavalcanti Malta, Pedro
2017-06-27
It is well known that the Standard Model is not complete and many of the theories that seek to extend it predict new phenomena that may be accessible in low-energy settings. This thesis deals with some of these, namely, novel spin-dependent interparticle potentials, axion-like particles and Lorentz-symmetry violation. In Part I we discuss the spin-dependent potentials that arise due to the exchange of a topologically massive mediator, and also pursue a comparative study between spin-1/2 and spin-1 sources. In Part II we treat massive axion-like particles that may be copiously produced in core-collapse supernovae, thus leading to a non-standard flux of gamma rays. Using SN 1987A and the fact that after its observation no extra gamma-ray signal was detected, we are able to set robust limits on the parameter space of axion-like particles with masses in the 10 keV - 100 MeV range. Finally, in Part III we investigate the effects of Lorentz-breaking backgrounds in QED. We discuss two scenarios: a modification in the Maxwell sector via the Carroll-Field-Jackiw term and a new non-minimal coupling between electrons and photons. We are able to set upper limits on the coefficients of the backgrounds by using laboratory-based measurements.
11. Lorentz violations in multifractal spacetimes
Energy Technology Data Exchange (ETDEWEB)
Calcagni, Gianluca [Instituto de Estructura de la Materia, CSIC, Madrid (Spain)
2017-05-15
Using the recent observation of gravitational waves (GW) produced by a black-hole merger, we place a lower bound on the energy above which a multifractal spacetime would display an anomalous geometry and, in particular, violations of Lorentz invariance. In the so-called multifractional theory with q-derivatives, we show that the deformation of dispersion relations is much stronger than in generic quantum-gravity approaches (including loop quantum gravity) and, contrary to the latter, present observations on GWs can place very strong bounds on the characteristic scales at which spacetime deviates from standard Minkowski. The energy at which multifractal effects should become apparent is E{sub *} > 10{sup 14} GeV (thus improving previous bounds by 12 orders of magnitude) when the exponents in the measure are fixed to their central value 1 / 2. We also estimate, for the first time, the effect of logarithmic oscillations in the measure (corresponding to a discrete spacetime structure) and find that they do not change much the bounds obtained in their absence, unless the amplitude of the oscillations is fine tuned. This feature, unavailable in known quantum-gravity scenarios, may help the theory to avoid being ruled out by gamma-ray burst (GRB) observations, for which E{sub *} > 10{sup 17} GeV or greater. (orig.)
12. Lorentz violations and Euclidean signature metrics
International Nuclear Information System (INIS)
Barbero G, J. Fernando; Villasenor, Eduardo J.S.
2003-01-01
We show that the families of effective actions considered by Jacobson et al. to study Lorentz invariance violations contain a class of models that represent pure general relativity with a Euclidean signature. We also point out that some members of this family of actions preserve Lorentz invariance in a generalized sense
13. New bounds on isotropic Lorentz violation
International Nuclear Information System (INIS)
Carone, Christopher D.; Sher, Marc; Vanderhaeghen, Marc
2006-01-01
Violations of Lorentz invariance that appear via operators of dimension four or less are completely parametrized in the Standard Model Extension (SME). In the pure photonic sector of the SME, there are 19 dimensionless, Lorentz-violating parameters. Eighteen of these have experimental upper bounds ranging between 10 -11 and 10 -32 ; the remaining parameter, k-tilde tr , is isotropic and has a much weaker bound of order 10 -4 . In this Brief Report, we point out that k-tilde tr gives a significant contribution to the anomalous magnetic moment of the electron and find a new upper bound of order 10 -8 . With reasonable assumptions, we further show that this bound may be improved to 10 -14 by considering the renormalization of other Lorentz-violating parameters that are more tightly constrained. Using similar renormalization arguments, we also estimate bounds on Lorentz-violating parameters in the pure gluonic sector of QCD
14. k-essence explains a Lorentz violation experiment
International Nuclear Information System (INIS)
Li Miao; Pang Yi; Wang Yi
2009-01-01
Recently, a state of the art experiment shows evidence for Lorentz violation in the gravitational sector. To explain this experiment, we investigate a spontaneous Lorentz violation scenario with a generalized scalar field. We find that when the scalar field is nonminimally coupled to gravity, the Lorentz violation induces a deformation in the Newtonian potential along the direction of Lorentz violation.
15. Structural aspects of Lorentz-violating quantum field theory
Science.gov (United States)
Cambiaso, M.; Lehnert, R.; Potting, R.
2018-01-01
In the last couple of decades the Standard Model Extension has emerged as a fruitful framework to analyze the empirical and theoretical extent of the validity of cornerstones of modern particle physics, namely, of Special Relativity and of the discrete symmetries C, P and T (or some combinations of these). The Standard Model Extension allows to contrast high-precision experimental tests with posited alterations representing minute Lorentz and/or CPT violations. To date no violation of these symmetry principles has been observed in experiments, mostly prompted by the Standard-Model Extension. From the latter, bounds on the extent of departures from Lorentz and CPT symmetries can be obtained with ever increasing accuracy. These analyses have been mostly focused on tree-level processes. In this presentation I would like to comment on structural aspects of perturbative Lorentz violating quantum field theory. I will show that some insight coming from radiative corrections demands a careful reassessment of perturbation theory. Specifically I will argue that both the standard renormalization procedure as well as the Lehmann-Symanzik-Zimmermann reduction formalism need to be adapted given that the asymptotic single-particle states can receive quantum corrections from Lorentz-violating operators that are not present in the original Lagrangian.
16. Lorentz violation. Motivation and new constraints
International Nuclear Information System (INIS)
Liberati, S.; Maccione, L.
2009-09-01
We review the main theoretical motivations and observational constraints on Planck scale sup-pressed violations of Lorentz invariance. After introducing the problems related to the phenomenological study of quantum gravitational effects, we discuss the main theoretical frameworks within which possible departures from Lorentz invariance can be described. In particular, we focus on the framework of Effective Field Theory, describing several possible ways of including Lorentz violation therein and discussing their theoretical viability. We review the main low energy effects that are expected in this framework. We discuss the current observational constraints on such a framework, focusing on those achievable through high-energy astrophysics observations. In this context we present a summary of the most recent and strongest constraints on QED with Lorentz violating non-renormalizable operators. Finally, we discuss the present status of the field and its future perspectives. (orig.)
17. Lorentz violation. Motivation and new constraints
Energy Technology Data Exchange (ETDEWEB)
Liberati, S. [Scuola Internazionale Superiore di Studi Avanzati SISSA, Trieste (Italy); Istituto Nazionale di Fisica Nucleare INFN, Sezione di Trieste (Italy); Maccione, L. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2009-09-15
We review the main theoretical motivations and observational constraints on Planck scale sup-pressed violations of Lorentz invariance. After introducing the problems related to the phenomenological study of quantum gravitational effects, we discuss the main theoretical frameworks within which possible departures from Lorentz invariance can be described. In particular, we focus on the framework of Effective Field Theory, describing several possible ways of including Lorentz violation therein and discussing their theoretical viability. We review the main low energy effects that are expected in this framework. We discuss the current observational constraints on such a framework, focusing on those achievable through high-energy astrophysics observations. In this context we present a summary of the most recent and strongest constraints on QED with Lorentz violating non-renormalizable operators. Finally, we discuss the present status of the field and its future perspectives. (orig.)
18. Lorentz-violating theories in the standard model extension
Energy Technology Data Exchange (ETDEWEB)
Ferreira Junior, Manoel Messias [Universidade Federal do Maranhao (UFMA), Sao Luis, MA (Brazil)
2012-07-01
Full text: Lorentz-violating theories have been an issue of permanent interest in the latest years. Many of these investigations are developed under the theoretical framework of the Standard Model Extension (SME), a broad extension of the minimal Standard Model embracing Lorentz-violating (LV) terms, generated as vacuum expectation values of tensor quantities, in all sectors of interaction. In this talk, we comment on some general properties of the SME, concerning mainly the gauge and fermion sectors, focusing in new phenomena induced by Lorentz violation. The LV terms are usually separated in accordance with the behavior under discrete symmetries, being classified as CPT-odd or CPT-even, parity-even or parity-odd. We follow this classification scheme discussing some features and new properties of the CPT-even and CPT-odd parts of the gauge and fermion sectors. We finalize presenting some upper bounds imposed on the corresponding LV coefficients. (author)
19. Lorentz-violating electrodynamics and the cosmic microwave background.
Science.gov (United States)
Kostelecký, V Alan; Mewes, Matthew
2007-07-06
Possible Lorentz-violating effects in the cosmic microwave background are studied. We provide a systematic classification of renormalizable and nonrenormalizable operators for Lorentz violation in electrodynamics and use polarimetric observations to search for the associated violations.
20. Studying Lorentz-violating electromagnetic waves in confined media
International Nuclear Information System (INIS)
Viana, Davidson R.; Gomes, Andre H.; Fonseca, Jakson M.; Moura-Melo, Winder A.
2009-01-01
Full text. Planck energy scale is still far beyond current possibilities. A question of interest is whether the Lorentz symmetry remains valid at these extremely high energies, whose answer certainly would be useful whenever building grand unified theories, in which general relativity is consistently accommodated. Here, we study a reminiscent of this possible symmetry violation, incorporated in the body of the so-called Standard Model Extension (SME). More precisely, we deal with the pure (Abelian) gauge sector, so that we have a modified classical electromagnetism in (3+1) dimensions, whose Lagrangian include a term proportional to a (constant) background tensor that breaks the Lorentz symmetry, but respecting CPT. Our attention is devoted to the wave-like solutions constrained to propagate inside confined media, like waveguides and resonant cavities. Our preliminary findings indicate that Lorentz-breaking implies in modifications of the standard results which are proportional to the (very small) violating parameters, but could be largely enhanced by diminishing the size of the confined media. Under study is the case of a toroidal cavity where the electromagnetic field should respect the additional requirement of being single-valued in the (toroidal) angular variable. Perhaps, such an extra feature combined with the usual boundary conditions could lead us to large effects of this violation, somewhat similar to those predicted for CPT- and Lorentz-odd electromagnetic waves constrained to propagate along a hollow conductor waveguide. (author)
1. Lorentz-violating alternative to the Higgs mechanism?
International Nuclear Information System (INIS)
Alexandre, Jean; Mavromatos, Nick E.
2011-01-01
We consider a four-dimensional field-theory model with two massless fermions, coupled to an Abelian vector field without flavor mixing, and to another Abelian vector field with flavor mixing. Both Abelian vectors have a Lorentz-violating kinetic term, introducing a Lorentz-violation mass scale M, from which fermions and the flavor-mixing vector get their dynamical masses, whereas the vector coupled without flavor mixing remains massless. When the two coupling constants have similar values in order of magnitude, a mass hierarchy pattern emerges, in which one fermion is very light compared to the other, while the vector mass is of the order of the heavy fermion mass. The work presented here may be considered as a Lorentz-symmetry-violating alternative to the Higgs mechanism, in the sense that no scalar particle (fundamental or composite) is necessary for the generation of the vector-meson mass. However, the model is not realistic given that, as a result of Lorentz violation, the maximal (light-cone) speed seen by the fermions is smaller than that of the massless gauge boson (which equals the speed of light in vacuo) by an amount which is unacceptably large to be compatible with the current tests of Lorentz invariance, unless the gauge couplings assume unnaturally small values. Possible ways out of this phenomenological drawback are briefly discussed, postponing a detailed construction of more realistic models for future work.
2. Noncommutative gauge theory without Lorentz violation
International Nuclear Information System (INIS)
Carlson, Carl E.; Carone, Christopher D.; Zobin, Nahum
2002-01-01
The most popular noncommutative field theories are characterized by a matrix parameter θ μν that violates Lorentz invariance. We consider the simplest algebra in which the θ parameter is promoted to an operator and Lorentz invariance is preserved. This algebra arises through the contraction of a larger one for which explicit representations are already known. We formulate a star product and construct the gauge-invariant Lagrangian for Lorentz-conserving noncommutative QED. Three-photon vertices are absent in the theory, while a four-photon coupling exists and leads to a distinctive phenomenology
3. Constraining Lorentz Violation in Electroweak Physics
Science.gov (United States)
Lehnert, Ralf
2018-01-01
For practical reasons, the majority of past Lorentz tests has involved stable or quasistable particles, such as photons, neutrinos, electrons, protons, and neutrons. Similar efforts in the electroweak sector have only recently taken shape. Within this context, Lorentz-violation searches in the Standard-Model Extension’s Z-Boson sector will be discussed. It is argued that existing precision data on polarized electron-electron scattering can be employed to extract the first conservative two-sided limits on Lorentz breakdown in this sector at the level of 10-7.
4. Lorentz violation and black-hole thermodynamics
International Nuclear Information System (INIS)
Betschart, G.; Kant, E.; Klinkhamer, F.R.
2009-01-01
We consider nonstandard photons from nonbirefringent modified Maxwell theory and discuss their propagation in a fixed Schwarzschild spacetime background. This particular modification of Maxwell theory is Lorentz-violating and allows for maximal photon velocities differing from the causal speed c of the asymptotic background spacetime. In the limit of geometrical optics, light rays from modified Maxwell theory are found to propagate along null geodesics in an effective metric. We observe that not every Lorentz-violating theory with multiple maximal velocities different from the causal speed c modifies the notion of the event horizon, contrary to naive expectations. This result implies that not every Lorentz-violating theory with multiple maximal velocities necessarily leads to a contradiction with the generalized second law of thermodynamics.
5. Black holes in Lorentz-violating gravity theories
International Nuclear Information System (INIS)
Barausse, Enrico; Sotiriou, Thomas P
2013-01-01
Lorentz symmetry and the notion of light cones play a central role in the definition of horizons and the existence of black holes. Current observations provide strong indications that astrophysical black holes do exist in Nature. Here we explore what happens to the notion of a black hole in gravity theories where local Lorentz symmetry is violated, and discuss the relevant astrophysical implications. Einstein-aether theory and Hořava gravity are used as the theoretical background for addressing this question. We review earlier results about static, spherically symmetric black holes, which demonstrate that in Lorentz-violating theories there can be a new type of horizon and, hence, a new notion of black hole. We also present both known and new results on slowly rotating black holes in these theories, which provide insights on how generic these new horizons are. Finally, we discuss the differences between black holes in Lorentz-violating theories and in General Relativity, and assess to what extent they can be probed with present and future observations. (paper)
6. A Lorentz-Violating Alternative to Higgs Mechanism?
CERN Document Server
Alexandre, Jean
2011-01-01
We consider a four-dimensional field-theory model with two massless fermions, coupled to an Abelian vector field without flavour mixing, and to another Abelian vector field with flavour mixing. Both Abelian vectors have a Lorentz-violating kinetic term, introducing a Lorentz-violation mass scale $M$, from which fermions and the flavour-mixing vector get their dynamical masses, whereas the vector coupled without flavour mixing remains massless. When the two coupling constants have similar values in order of magnitude, a mass hierarchy pattern emerges, in which one fermion is very light compared to the other, whilst the vector mass is larger than the mass of the heavy fermion. The work presented here may be considered as a Lorentz-symmetry-Violating alternative to the Higgs mechanism, in the sense that no scalar particle (fundamental or composite) is necessary for the generation of the vector-meson mass. However, the model is not realistic given that, as a result of Lorentz Violation, the maximal (light-cone) s...
7. Lorentz Transformation from Symmetry of Reference Principle
International Nuclear Information System (INIS)
Petre, M.; Dima, M.; Dima, A.; Petre, C.; Precup, I.
2010-01-01
The Lorentz Transformation is traditionally derived requiring the Principle of Relativity and light-speed universality. While the latter can be relaxed, the Principle of Relativity is seen as core to the transformation. The present letter relaxes both statements to the weaker, Symmetry of Reference Principle. Thus the resulting Lorentz transformation and its consequences (time dilatation, length contraction) are, in turn, effects of how we manage space and time.
8. Vortices in superconductors from Lorentz violation
International Nuclear Information System (INIS)
Belich, H.; Orlando, M.T.D.; Costa-Soares, T.; Helayel-Neto, J.A.
2004-01-01
We start from a Lorentz non-invariant Abelian-Higgs model in 1+3 dimensions, and carry out its dimensional reduction to D = 1 + 2. The planar model resulting thereof is composed by a Maxwell-Chern-Simons-Proca gauge sector, a massive scalar sector, and a mixing term (involving the fixed background, v μ ) that realizes Lorentz violation for the reduced model. Vortex type solutions of the planar model are investigated in a superconducting environment . Our vortex solutions are electrically charged and exhibit a screened electric field. (author)
9. Lorentz violation and generalized uncertainty principle
Science.gov (United States)
Lambiase, Gaetano; Scardigli, Fabio
2018-04-01
Investigations on possible violation of Lorentz invariance have been widely pursued in the last decades, both from theoretical and experimental sides. A comprehensive framework to formulate the problem is the standard model extension (SME) proposed by A. Kostelecky, where violation of Lorentz invariance is encoded into specific coefficients. Here we present a procedure to link the deformation parameter β of the generalized uncertainty principle to the SME coefficients of the gravity sector. The idea is to compute the Hawking temperature of a black hole in two different ways. The first way involves the deformation parameter β , and therefore we get a deformed Hawking temperature containing the parameter β . The second way involves a deformed Schwarzschild metric containing the Lorentz violating terms s¯μ ν of the gravity sector of the SME. The comparison between the two different techniques yields a relation between β and s¯μ ν. In this way bounds on β transferred from s¯μ ν are improved by many orders of magnitude when compared with those derived in other gravitational frameworks. Also the opposite possibility of bounds transferred from β to s¯μ ν is briefly discussed.
10. BPS Lorentz-violating vortex solutions
International Nuclear Information System (INIS)
Casana, Rodolfo; Ferreira Junior, Manoel M.; Hora, E. da
2011-01-01
In this work, we deal with the construction of static Bogomol'nyi-Prasad-Sommerfield (BPS) rotationally symmetric configurations on the dimensional CPT-even Lorentz-breaking photonic sector of the Standard Model Extension (SME). The main objective of this presentation is to show the possibility of obtaining such BPS solutions, even in the presence of a Lorentz-violating background. A secondary objective is to analyze the effects of this background on such topologically non-trivial BPS configurations. In order to obtain these results, we deal with some specific components of Lorentz-violating field, handling with the static Euler-Lagrange equation of motion to gauge field, from which we fix temporal gauge (absence of electric field) as a proper gauge choice. Also, considering this equation, we consistently determine an interesting configuration (discarding non-interesting ones) to the Lorentz-breaking sector. Using this configuration and the standard rotationally symmetric vortex Ansatz (which describes the behaviors of Higgs and gauge fields via two profile functions, g(r) and a(r), respectively), we construct a rotationally symmetric expression to the energy density of the system. To obtain BPS solutions, we rewrite this expression in order to have static vortex solutions satisfying a set of first order differential equations (BPS ones). The existence of such solutions is strongly constrained by a relation between some parameters of the model, including the Lorentz-breaking one. Naturally, we show that the total energy of these BPS solutions is proportional to their magnetic flux, which is quantized according to their winding number. Using suitable boundary conditions (near the origin and asymptotically), we numerically integrate the BPS equations (by means of the shooting method). By this way, we obtain solutions for some physical quantities (Higgs field, magnetic field and energy density) for several values of the Lorentz-violating parameters. From these
11. Some impacts of Lorentz violation on cosmology
International Nuclear Information System (INIS)
Arianto; Zen, Freddy P.; Gunara, Bobby E.; Triyanta; Supardi
2007-01-01
The impact of Lorentz violation on the dynamics of a scalar field is investigated. In particular, we study the dynamics of a scalar field in the scalar-vector-tensor theory where the vector field is constrained to be unity and time like. By taking a generic form of the scalar field action, a generalized dynamical equation for the scalar-vector-tensor theory of gravity is obtained to describe the cosmological solutions. We present a class of exact solutions for an ordinary scalar field or phantom field corresponding to a power law coupling vector and the Hubble parameter. As the results, we find a constant equation of state in de Sitter space-time and power law expansion with the quadratic of coupling vector, while a dynamic equation of state is obtained for n > 2. Then, we consider the inflationary scenario based on the Lorentz violating scalar-vector-tensor theory of gravity with general power-law coupling vector and two typical potentials: inverse power-law and power-law potentials. In fact, both the coupling vector and the potential models affect the dynamics of the inflationary solutions. Finally, we use the dynamical system formalism to study the attractor behavior of a cosmological model containing a scalar field endowed with a quadratic coupling vector and a chaotic potential
12. Spontaneous Lorentz and diffeomorphism violation, massive modes, and gravity
International Nuclear Information System (INIS)
Bluhm, Robert; Fung Shuhong; Kostelecky, V. Alan
2008-01-01
Theories with spontaneous local Lorentz and diffeomorphism violation contain massless Nambu-Goldstone modes, which arise as field excitations in the minimum of the symmetry-breaking potential. If the shape of the potential also allows excitations above the minimum, then an alternative gravitational Higgs mechanism can occur in which massive modes involving the metric appear. The origin and basic properties of the massive modes are addressed in the general context involving an arbitrary tensor vacuum value. Special attention is given to the case of bumblebee models, which are gravitationally coupled vector theories with spontaneous local Lorentz and diffeomorphism violation. Mode expansions are presented in both local and spacetime frames, revealing the Nambu-Goldstone and massive modes via decomposition of the metric and bumblebee fields, and the associated symmetry properties and gauge fixing are discussed. The class of bumblebee models with kinetic terms of the Maxwell form is used as a focus for more detailed study. The nature of the associated conservation laws and the interpretation as a candidate alternative to Einstein-Maxwell theory are investigated. Explicit examples involving smooth and Lagrange-multiplier potentials are studied to illustrate features of the massive modes, including their origin, nature, dispersion laws, and effects on gravitational interactions. In the weak static limit, the massive mode and Lagrange-multiplier fields are found to modify the Newton and Coulomb potentials. The nature and implications of these modifications are examined.
13. Modelling Planck-scale Lorentz violation via analogue models
International Nuclear Information System (INIS)
Weinfurtner, Silke; Liberati, Stefano; Visser, Matt
2006-01-01
Astrophysical tests of Planck-suppressed Lorentz violations had been extensively studied in recent years and very stringent constraints have been obtained within the framework of effective field theory. There are however still some unresolved theoretical issues, in particular regarding the so called 'naturalness problem' - which arises when postulating that Planck suppressed Lorentz violations arise only from operators with mass dimension greater than four in the Lagrangian. In the work presented here we shall try to address this problem by looking at a condensed-matter analogue of the Lorentz violations considered in quantum gravity phenomenology. specifically, we investigate the class of two-component BECs subject to laserinduced transitions between the two components, and we show that this model is an example for Lorentz invariance violation due to ultraviolet physics. We shall show that such a model can be considered to be an explicit example high-energy Lorentz violations where the 'naturalness problem' does not arise
14. Bumpy black holes from spontaneous Lorentz violation
International Nuclear Information System (INIS)
Dubovsky, Sergei; Tinyakov, Peter; Zaldarriaga, Matias
2007-01-01
We consider black holes in Lorentz violating theories of massive gravity. We argue that in these theories black hole solutions are no longer universal and exhibit a large number of hairs. If they exist, these hairs probe the singularity inside the black hole providing a window into quantum gravity. The existence of these hairs can be tested by future gravitational wave observatories. We generically expect that the effects we discuss will be larger for the more massive black holes. In the simplest models the strength of the hairs is controlled by the same parameter that sets the mass of the graviton (tensor modes). Then the upper limit on this mass coming from the inferred gravitational radiation emitted by binary pulsars implies that hairs are likely to be suppressed for almost the entire mass range of the super-massive black holes in the centers of galaxies
15. Cosmic rays and the search for a Lorentz Invariance Violation
International Nuclear Information System (INIS)
Bietenholz, Wolfgang
2008-11-01
This is an introductory review about the on-going search for a signal of Lorentz Invariance Violation (LIV) in cosmic rays. We first summarise basic aspects of cosmic rays, focusing on rays of ultra high energy (UHECRs). We discuss the Greisen-Zatsepin-Kuz'min (GZK) energy cutoff for cosmic protons, which is predicted due to photopion production in the Cosmic Microwave Background (CMB). This is a process of modest energy in the proton rest frame. It can be investigated to a high precision in the laboratory, if Lorentz transformations apply even at factors γ ∝ O(10 11 ). For heavier nuclei the energy attenuation is even faster due to photo-disintegration, again if this process is Lorentz invariant. Hence the viability of Lorentz symmetry up to tremendous γ-factors - far beyond accelerator tests - is a central issue. Next we comment on conceptual aspects of Lorentz Invariance and the possibility of its spontaneous breaking. This could lead to slightly particle dependent ''Maximal Attainable Velocities''. We discuss their effect in decays, Cerenkov radiation, the GZK cutoff and neutrino oscillation in cosmic rays. We also review the search for LIV in cosmic γ-rays. For multi TeV γ-rays we possibly encounter another puzzle related to the transparency of the CMB, similar to the GZK cutoff, due to electron/positron creation and subsequent inverse Compton scattering. The photons emitted in a Gamma Ray Burst occur at lower energies, but their very long path provides access to information not far from the Planck scale. We discuss conceivable non-linear photon dispersions based on non-commutative geometry or effective approaches. No LIV has been observed so far. However, even extremely tiny LIV effects could change the predictions for cosmic ray physics drastically. An Appendix is devoted to the recent hypothesis by the Pierre Auger Collaboration, which identifies nearby Active Galactic Nuclei - or objects next to them - as probable UHECR sources. (orig.)
16. Cosmological constraints on Lorentz violating dark energy
CERN Document Server
Audren, B; Lesgourgues, J; Sibiryakov, S
2013-01-01
The role of Lorentz invariance as a fundamental symmetry of nature has been lately reconsidered in different approaches to quantum gravity. It is thus natural to study whether other puzzles of physics may be solved within these proposals. This may be the case for the cosmological constant problem. Indeed, it has been shown that breaking Lorentz invariance provides Lagrangians that can drive the current acceleration of the universe without experiencing large corrections from ultraviolet physics. In this work, we focus on the simplest model of this type, called ThetaCDM, and study its cosmological implications in detail. At the background level, this model cannot be distinguished from LambdaCDM. The differences appear at the level of perturbations. We show that in ThetaCDM, the spectrum of CMB anisotropies and matter fluctuations may be affected by a rescaling of the gravitational constant in the Poisson equation, by the presence of extra contributions to the anisotropic stress, and finally by the existence of ...
17. Cosmological constraints on Lorentz violating dark energy
Energy Technology Data Exchange (ETDEWEB)
Audren, B.; Lesgourgues, J. [FSB/ITP/LPPC, École Polytechnique Fédérale de Lausanne, CH-1015, Lausanne (Switzerland); Blas, D. [Theory Group, Physics Department, CERN, CH-1211 Geneva 23 (Switzerland); Sibiryakov, S., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, 117312 Moscow (Russian Federation)
2013-08-01
The role of Lorentz invariance as a fundamental symmetry of nature has been lately reconsidered in different approaches to quantum gravity. It is thus natural to study whether other puzzles of physics may be solved within these proposals. This may be the case for the cosmological constant problem. Indeed, it has been shown that breaking Lorentz invariance provides Lagrangians that can drive the current acceleration of the universe without experiencing large corrections from ultraviolet physics. In this work, we focus on the simplest model of this type, called ΘCDM, and study its cosmological implications in detail. At the background level, this model cannot be distinguished from ΛCDM. The differences appear at the level of perturbations. We show that in ΘCDM, the spectrum of CMB anisotropies and matter fluctuations may be affected by a rescaling of the gravitational constant in the Poisson equation, by the presence of extra contributions to the anisotropic stress, and finally by the existence of extra clustering degrees of freedom. To explore these modifications accurately, we modify the Boltzmann code class. We then use the parameter inference code Monte Python to confront ΘCDM with data from WMAP-7, SPT and WiggleZ. We obtain strong bounds on the parameters accounting for deviations from ΛCDM. In particular, we find that the discrepancy between the gravitational constants appearing in the Poisson and Friedmann equations is constrained at the level of 1.8%.
18. Imprints of supersymmetry in the Lorentz-symmetry breaking of Gauge Theories
Energy Technology Data Exchange (ETDEWEB)
Belich, H [Universidade Federal do Espirito Santo (UFES), Vitoria, ES (Brazil); Dias, G S; Leal, F J.L. [Instituto Federal de Educacao, Ciencia e Tecnologia do Espirito Santo (IFES), Vitoria, ES (Brazil); Durand, L G; Helayel-Neto, Jose Abdalla; Spalenza, W [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil); Grupo de Fisica Teorica Jose Leite Lopes (GFT-JLL), Petropolis, RJ (Brazil)
2011-07-01
Full text: The breaking of Lorentz symmetry that may take place at very high energies opens up a venue for the discussion of the interplay between the violations of supersymmetry and relativistic symmetry. Recently, there have appeared in the literature models which propose a residual (non-relativistic) supersymmetry after Lorentz symmetry has been broken in a Horava gravity scenario. We here propose an N=1-supersymmetric Abelian gauge model which realises the breaking of Lorentz invariance by means of a CPT-even term. Our attempt assumes the point of view that supersymmetry and Lorentz symmetry are broken down at the same scale. If this is the case, the fermionic sector of the supermultiplets that accomplish the breaking of the symmetries into consideration may give rise to condensates that play an important role in the photon and photino dispersion relations. Contemporarily, they may also point to a more fundamental origin for the (bosonic) tensors usually associated to the backgrounds that parametrize Lorentz-symmetry breaking. We also highlight that, by studying the the violation of Lorentz symmetry in connection with supersymmetry, we find out that the Myers-Pospelov Electrodynamics, proposed on the basis of an analysis of the set of dimension-five operators, naturally appears in the bosonic sector of our model. Also, as a result of the interconnection between the supersymmetry and Lorentz-symmetry breakings, the photino-photino and photon-photino mixings that correspond to the supersymmetric completion of the Myers-Pospelov purely photonic terms come out. Finally, we present some comments on the possible modifications the supersymmetric fermions may introduce in the dispersion relations for particles at (high) energies close to the scale where supersymmetry and Lorentz symmetry are broken. (author)
19. Imprints of supersymmetry in the Lorentz-symmetry breaking of Gauge Theories
International Nuclear Information System (INIS)
Belich, H.; Dias, G.S.; Leal, F.J.L.; Durand, L.G.; Helayel-Neto, Jose Abdalla; Spalenza, W.
2011-01-01
Full text: The breaking of Lorentz symmetry that may take place at very high energies opens up a venue for the discussion of the interplay between the violations of supersymmetry and relativistic symmetry. Recently, there have appeared in the literature models which propose a residual (non-relativistic) supersymmetry after Lorentz symmetry has been broken in a Horava gravity scenario. We here propose an N=1-supersymmetric Abelian gauge model which realises the breaking of Lorentz invariance by means of a CPT-even term. Our attempt assumes the point of view that supersymmetry and Lorentz symmetry are broken down at the same scale. If this is the case, the fermionic sector of the supermultiplets that accomplish the breaking of the symmetries into consideration may give rise to condensates that play an important role in the photon and photino dispersion relations. Contemporarily, they may also point to a more fundamental origin for the (bosonic) tensors usually associated to the backgrounds that parametrize Lorentz-symmetry breaking. We also highlight that, by studying the the violation of Lorentz symmetry in connection with supersymmetry, we find out that the Myers-Pospelov Electrodynamics, proposed on the basis of an analysis of the set of dimension-five operators, naturally appears in the bosonic sector of our model. Also, as a result of the interconnection between the supersymmetry and Lorentz-symmetry breakings, the photino-photino and photon-photino mixings that correspond to the supersymmetric completion of the Myers-Pospelov purely photonic terms come out. Finally, we present some comments on the possible modifications the supersymmetric fermions may introduce in the dispersion relations for particles at (high) energies close to the scale where supersymmetry and Lorentz symmetry are broken. (author)
20. Quantizations of D = 3 Lorentz symmetry
Energy Technology Data Exchange (ETDEWEB)
Lukierski, J. [University of Wroclaw, Institute for Theoretical Physics, Wroclaw (Poland); Tolstoy, V.N. [University of Wroclaw, Institute for Theoretical Physics, Wroclaw (Poland); Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow (Russian Federation)
2017-04-15
Using the isomorphism o(3; C) ≅ sl(2; C) we develop a new simple algebraic technique for complete classification of quantum deformations (the classical r-matrices) for real forms o(3) and o(2,1) of the complex Lie algebra o(3; C) in terms of real forms of sl(2; C): su(2), su(1,1) and sl(2; R). We prove that the D = 3 Lorentz symmetry o(2,1) ≅ su(1,1) ≅ sl(2; R) has three different Hopf-algebraic quantum deformations, which are expressed in the simplest way by two standard su(1,1) and sl(2; R) q-analogs and by simple Jordanian sl(2; R) twist deformation. These quantizations are presented in terms of the quantum Cartan-Weyl generators for the quantized algebras su(1,1) and sl(2; R) as well as in terms of quantum Cartesian generators for the quantized algebra o(2,1). Finally, some applications of the deformed D = 3 Lorentz symmetry are mentioned. (orig.)
1. New effects in the interaction between electromagnetic sources mediated by nonminimal Lorentz violating interactions
Energy Technology Data Exchange (ETDEWEB)
Borges, L.H.C.; Ferrari, A.F. [Universidade Federal do ABC, Centro de Ciencias Naturais e Humanas, Santo Andre, SP (Brazil); Barone, F.A. [Universidade Federal de Itajuba, IFQ, Itajuba, MG (Brazil)
2016-11-15
This paper is dedicated to the study of interactions between external sources for the electromagnetic field in the presence of Lorentz symmetry breaking. We focus on a higher derivative, Lorentz violating interaction that arises from a specific model that was argued to lead to interesting effects in the low energy phenomenology of light pseudoscalars interacting with photons. The kind of higher derivative Lorentz violating interaction we discuss are called nonminimal. They are usually expected to be relevant only at very high energies, but we argue they might also induce relevant effects in low energy phenomena. Indeed, we show that the Lorentz violating background considered by us leads to several phenomena that have no counterpart in Maxwell theory, such as nontrivial torques on isolated electric dipoles, as well as nontrivial forces and torques between line currents and point like charges, as well as among Dirac strings and other electromagnetic sources. (orig.)
2. Consistent Lorentz violation in flat and curved space
International Nuclear Information System (INIS)
Dvali, Gia; Pujolas, Oriol; Redi, Michele
2007-01-01
Motivated by the severity of the bounds on Lorentz violation in the presence of ordinary gravity, we study frameworks in which Lorentz violation does not affect the spacetime geometry. We show that there are at least two inequivalent classes of spontaneous Lorentz breaking that even in the presence of gravity result in Minkowski space. The first one generically corresponds to the condensation of tensor fields with tachyonic mass, which in turn is related to ghost condensation. In the second class, realized by the Dvali-Gabadadze-Porrati model or theories of massive gravitons, spontaneous Lorentz breaking is induced by the expectation value of sources. The generalization to de Sitter space is also discussed
3. Lorentz Violation in Warped Extra Dimensions
International Nuclear Information System (INIS)
Rizzo, Thomas G.
2011-01-01
Higher dimensional theories which address some of the problematic issues of the Standard Model(SM) naturally involve some form of D = 4 + n-dimensional Lorentz invariance violation (LIV). In such models the fundamental physics which leads to, e.g., field localization, orbifolding, the existence of brane terms and the compactification process all can introduce LIV in the higher dimensional theory while still preserving 4-d Lorentz invariance. In this paper, attempting to capture some of this physics, we extend our previous analysis of LIV in 5-d UED-type models to those with 5- d warped extra dimensions. To be specific, we employ the 5-d analog of the SM Extension of Kostelecky et al. which incorporates a complete set of operators arising from spontaneous LIV. We show that while the response of the bulk scalar, fermion and gauge fields to the addition of LIV operators in warped models is qualitatively similar to what happens in the flat 5-d UED case, the gravity sector of these models reacts very differently than in flat space. Specifically, we show that LIV in this warped case leads to a non-zero bulk mass for the 5-d graviton and so the would-be zero mode, which we identify as the usual 4-d graviton, must necessarily become massive. The origin of this mass term is the simultaneous existence of the constant non-zero AdS 5 curvature and the loss of general co-ordinate invariance via LIV in the 5-d theory. Thus warped 5-d models with LIV in the gravity sector are not phenomenologically viable.
4. Factoring the dispersion relation in the presence of Lorentz violation
International Nuclear Information System (INIS)
Colladay, Don; McDonald, Patrick; Mullins, David
2010-01-01
We produce an explicit formula for the dispersion relation for the Dirac equation in the standard model extension in the presence of Lorentz violation. Our expression is obtained using novel techniques which exploit the algebra of quaternions. The dispersion relation is found to conveniently factor in two special cases that each involve a mutually exclusive set of nonvanishing Lorentz-violating parameters. This suggests that a useful approach to studies of Lorentz-violating models is to split the parameter space into two separate pieces, each of which yields a simple, tractable dispersion relation that can be used for analysis.
5. Lorentz invariance violation in modified gravity
International Nuclear Information System (INIS)
Brax, Philippe
2012-01-01
We consider an environmentally dependent violation of Lorentz invariance in scalar-tensor models of modified gravity where General Relativity is retrieved locally thanks to a screening mechanism. We find that fermions have a modified dispersion relation and would go faster than light in an anisotropic and space-dependent way along the scalar field lines of force. Phenomenologically, these models are tightly restricted by the amount of Cerenkov radiation emitted by the superluminal particles, a constraint which is only satisfied by chameleons. Measuring the speed of neutrinos emitted radially from the surface of the earth and observed on the other side of the earth would probe the scalar field profile of modified gravity models in dense environments. We argue that the test of the equivalence principle provided by the Lunar ranging experiment implies that a deviation from the speed of light, for natural values of the coupling scale between the scalar field and fermions, would be below detectable levels, unless gravity is modified by camouflaged chameleons where the field normalisation is environmentally dependent.
6. ICECUBE NEUTRINOS AND LORENTZ INVARIANCE VIOLATION
Energy Technology Data Exchange (ETDEWEB)
Amelino-Camelia, Giovanni [Dipartimento di Fisica, Sapienza Università di Roma and INFN, Sez. Roma1, P.le A. Moro 2, I-00185 Roma (Italy); Guetta, D. [Osservatorio astronomico di Roma, v. Frascati 33, I-00040 Monte Porzio Catone (Italy); Piran, Tsvi [The Racah Institute for Physics, The Hebrew University of Jerusalem, Jerusalem 91904 (Israel)
2015-06-20
The IceCube neutrino telescope has found so far no evidence of gamma-ray burst (GRB) neutrinos. We here notice that these results assume the same travel times from source to telescope for neutrinos and photons, an assumption that is challenged by some much-studied pictures of spacetime quantization. We briefly review previous results suggesting that limits on quantum-spacetime effects obtained for photons might not be applicable to neutrinos, and we then observe that the outcome of GRB-neutrino searches could depend strongly on whether one allows for neutrinos to be affected by the minute effects of Lorentz invariance violation (LIV) predicted by some relevant quantum-spacetime models. We discuss some relevant issues using as an illustrative example three neutrinos that were detected by IceCube in good spatial coincidence with GRBs, but hours before the corresponding gamma rays. In general, this could happen if the earlier arrival reflects quantum-spacetime-induced LIV, but, as we stress, some consistency criteria must be enforced in order to properly test such a hypothesis. Our analysis sets the stage for future GRB-neutrino searches that could systematically test the possibility of quantum-spacetime-induced LIV.
7. Lorentz invariance violation in modified gravity
Energy Technology Data Exchange (ETDEWEB)
Brax, Philippe, E-mail: [email protected] [Institut de Physique Theorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/Yvette Cedex (France)
2012-06-06
We consider an environmentally dependent violation of Lorentz invariance in scalar-tensor models of modified gravity where General Relativity is retrieved locally thanks to a screening mechanism. We find that fermions have a modified dispersion relation and would go faster than light in an anisotropic and space-dependent way along the scalar field lines of force. Phenomenologically, these models are tightly restricted by the amount of Cerenkov radiation emitted by the superluminal particles, a constraint which is only satisfied by chameleons. Measuring the speed of neutrinos emitted radially from the surface of the earth and observed on the other side of the earth would probe the scalar field profile of modified gravity models in dense environments. We argue that the test of the equivalence principle provided by the Lunar ranging experiment implies that a deviation from the speed of light, for natural values of the coupling scale between the scalar field and fermions, would be below detectable levels, unless gravity is modified by camouflaged chameleons where the field normalisation is environmentally dependent.
8. Relativistic Anandan quantum phase and the Aharonov–Casher effect under Lorentz symmetry breaking effects in the cosmic string spacetime
Energy Technology Data Exchange (ETDEWEB)
Bakke, K., E-mail: [email protected] [Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900, João Pessoa-PB (Brazil); Furtado, C., E-mail: [email protected] [Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900, João Pessoa-PB (Brazil); Belich, H., E-mail: [email protected] [Departamento de Física e Química, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900, Vitória, ES (Brazil)
2016-09-15
From the modified Maxwell theory coupled to gravity, we establish a possible scenario of the violation of the Lorentz symmetry and write an effective metric for the cosmic string spacetime. Then, we investigate the arising of an analogue of the Anandan quantum phase for a relativistic Dirac neutral particle with a permanent magnetic dipole moment in the cosmic string spacetime under Lorentz symmetry breaking effects. Besides, we analyse the influence of the effects of the Lorentz symmetry violation and the topology of the defect on the Aharonov–Casher geometric quantum phase in the nonrelativistic limit.
9. Symmetry violating kaon decays
International Nuclear Information System (INIS)
Herczeg, P.
1979-01-01
An analysis of the muon number violating decay modes of the K-mesons is given. Subsequently, some new developments in the field of CP-violation are reviewed and the question of time-reversal invariance and the status of CPT-invariance are briefly considered. 42 references
10. Non-Abelian Gauge Theory in the Lorentz Violating Background
Science.gov (United States)
Ganai, Prince A.; Shah, Mushtaq B.; Syed, Masood; Ahmad, Owais
2018-03-01
In this paper, we will discuss a simple non-Abelian gauge theory in the broken Lorentz spacetime background. We will study the partial breaking of Lorentz symmetry down to its sub-group. We will use the formalism of very special relativity for analysing this non-Abelian gauge theory. Moreover, we will discuss the quantisation of this theory using the BRST symmetry. Also, we will analyse this theory in the maximal Abelian gauge.
11. Cosmic rays and the search for a Lorentz Invariance Violation
Energy Technology Data Exchange (ETDEWEB)
Bietenholz, Wolfgang, E-mail: [email protected]
2011-08-15
This is an introductory review about the ongoing search for a signal of Lorentz Invariance Violation (LIV) in cosmic rays. We first summarise basic aspects of cosmic rays, focusing on rays of ultrahigh energy (UHECRs). We discuss the Greisen-Zatsepin-Kuz'min (GZK) energy cutoff for cosmic protons, which is predicted due to photopion production in the Cosmic Microwave Background (CMB). This is a process of modest energy in the proton rest frame. It can be investigated to a high precision in the laboratory, if Lorentz transformations apply even at factors {gamma}{approx}O(10{sup 11}). For heavier nuclei, the energy attenuation is even faster due to photo-disintegration, again if this process is Lorentz invariant. Hence the viability of Lorentz symmetry up to tremendous {gamma}-factors-far beyond accelerator tests-is a central issue. Next, we comment on conceptual aspects of Lorentz Invariance and the possibility of its spontaneous breaking. This could lead to slightly particle dependent 'Maximal Attainable Velocities'. We discuss their effect in decays, Cerenkov radiation, the GZK cutoff and neutrino oscillation in cosmic rays. We also review the search for LIV in cosmic {gamma}-rays. For multi-TeV {gamma}-rays, we encounter another possible puzzle related to the transparency of the CMB, similar to the GZK cutoff, due to electron/positron creation and subsequent inverse Compton scattering. The photons emitted in a Gamma Ray Burst occur at lower energies, but their very long path provides access to information not that far from the Planck scale. We discuss conceivable nonlinear photon dispersions based on non-commutative geometry or effective approaches. No LIV has been observed so far. However, even extremely tiny LIV effects could change the predictions for cosmic ray physics drastically. An Appendix is devoted to the recent results by the Pierre Auger Collaboration, in particular the hypothesis that nearby Active Galactic Nuclei-or objects next to
12. Cosmic rays and the search for a Lorentz Invariance Violation
Energy Technology Data Exchange (ETDEWEB)
Bietenholz, Wolfgang [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany). John von Neumann-Inst. fuer Computing NIC
2008-11-15
This is an introductory review about the on-going search for a signal of Lorentz Invariance Violation (LIV) in cosmic rays. We first summarise basic aspects of cosmic rays, focusing on rays of ultra high energy (UHECRs). We discuss the Greisen-Zatsepin-Kuz'min (GZK) energy cutoff for cosmic protons, which is predicted due to photopion production in the Cosmic Microwave Background (CMB). This is a process of modest energy in the proton rest frame. It can be investigated to a high precision in the laboratory, if Lorentz transformations apply even at factors {gamma} {proportional_to} O(10{sup 11}). For heavier nuclei the energy attenuation is even faster due to photo-disintegration, again if this process is Lorentz invariant. Hence the viability of Lorentz symmetry up to tremendous {gamma}-factors - far beyond accelerator tests - is a central issue. Next we comment on conceptual aspects of Lorentz Invariance and the possibility of its spontaneous breaking. This could lead to slightly particle dependent ''Maximal Attainable Velocities''. We discuss their effect in decays, Cerenkov radiation, the GZK cutoff and neutrino oscillation in cosmic rays. We also review the search for LIV in cosmic {gamma}-rays. For multi TeV {gamma}-rays we possibly encounter another puzzle related to the transparency of the CMB, similar to the GZK cutoff, due to electron/positron creation and subsequent inverse Compton scattering. The photons emitted in a Gamma Ray Burst occur at lower energies, but their very long path provides access to information not far from the Planck scale. We discuss conceivable non-linear photon dispersions based on non-commutative geometry or effective approaches. No LIV has been observed so far. However, even extremely tiny LIV effects could change the predictions for cosmic ray physics drastically. An Appendix is devoted to the recent hypothesis by the Pierre Auger Collaboration, which identifies nearby Active Galactic Nuclei - or objects
13. The CTA Sensitivity to Lorentz-Violating Effects on the Gamma-Ray Horizon
CERN Document Server
Fairbairn, Malcolm; Ellis, John; Hinton, Jim; White, Richard
2014-01-01
The arrival of TeV-energy photons from distant galaxies is expected to be affected by their QED interaction with intergalactic radiation fields through electron-positron pair production. In theories where high-energy photons violate Lorentz symmetry, the kinematics of the process $\\gamma + \\gamma\\rightarrow e^+ + e^-$ is altered and the cross-section suppressed. Consequently, one would expect more of the highest-energy photons to arrive if QED is modified by Lorentz violation than if it is not. We estimate the sensitivity of Cherenkov Telescope Array (CTA) to changes in the $\\gamma$-ray horizon of the Universe due to Lorentz violation, and find that it should be competitive with other leading constraints.
14. Lorentz Violation of the Photon Sector in Field Theory Models
Directory of Open Access Journals (Sweden)
Lingli Zhou
2014-01-01
Full Text Available We compare the Lorentz violation terms of the pure photon sector between two field theory models, namely, the minimal standard model extension (SME and the standard model supplement (SMS. From the requirement of the identity of the intersection for the two models, we find that the free photon sector of the SMS can be a subset of the photon sector of the minimal SME. We not only obtain some relations between the SME parameters but also get some constraints on the SMS parameters from the SME parameters. The CPT-odd coefficients (kAFα of the SME are predicted to be zero. There are 15 degrees of freedom in the Lorentz violation matrix Δαβ of free photons of the SMS related with the same number of degrees of freedom in the tensor coefficients (kFαβμν, which are independent from each other in the minimal SME but are interrelated in the intersection of the SMS and the minimal SME. With the related degrees of freedom, we obtain the conservative constraints (2σ on the elements of the photon Lorentz violation matrix. The detailed structure of the photon Lorentz violation matrix suggests some applications to the Lorentz violation experiments for photons.
15. Generalizations of teleparallel gravity and local Lorentz symmetry
International Nuclear Information System (INIS)
Sotiriou, Thomas P.; Barrow, John D.; Li Baojiu
2011-01-01
We analyze the relation between teleparallelism and local Lorentz invariance. We show that generic modifications of the teleparallel equivalent to general relativity will not respect local Lorentz symmetry. We clarify the reasons for this and explain why the situation is different in general relativity. We give a prescription for constructing teleparallel equivalents for known theories. We also explicitly consider a recently proposed class of generalized teleparallel theories, called f(T) theories of gravity, and show why restoring local Lorentz symmetry in such theories cannot lead to sensible dynamics, even if one gives up teleparallelism.
16. Hadronic Lorentz violation in chiral perturbation theory including the coupling to external fields
Science.gov (United States)
Kamand, Rasha; Altschul, Brett; Schindler, Matthias R.
2018-05-01
If any violation of Lorentz symmetry exists in the hadron sector, its ultimate origins must lie at the quark level. We continue the analysis of how the theories at these two levels are connected, using chiral perturbation theory. Considering a 2-flavor quark theory, with dimension-4 operators that break Lorentz symmetry, we derive a low-energy theory of pions and nucleons that is invariant under local chiral transformations and includes the coupling to external fields. The pure meson and baryon sectors, as well as the couplings between them and the couplings to external electromagnetic and weak gauge fields, contain forms of Lorentz violation which depend on linear combinations of quark-level coefficients. In particular, at leading order the electromagnetic couplings depend on the very same combinations as appear in the free particle propagators. This means that observations of electromagnetic processes involving hadrons—such as vacuum Cerenkov radiation, which may be allowed in Lorentz-violating theories—can only reliably constrain certain particular combinations of quark coefficients.
17. Lorentz violation and black-hole thermodynamics: Compton scattering process
International Nuclear Information System (INIS)
Kant, E.; Klinkhamer, F.R.; Schreck, M.
2009-01-01
A Lorentz-noninvariant modification of quantum electrodynamics (QED) is considered, which has photons described by the nonbirefringent sector of modified Maxwell theory and electrons described by the standard Dirac theory. These photons and electrons are taken to propagate and interact in a Schwarzschild spacetime background. For appropriate Lorentz-violating parameters, the photons have an effective horizon lying outside the Schwarzschild horizon. A particular type of Compton scattering event, taking place between these two horizons (in the photonic ergoregion) and ultimately decreasing the mass of the black hole, is found to have a nonzero probability. These events perhaps allow for a violation of the generalized second law of thermodynamics in the Lorentz-noninvariant theory considered.
18. A Study of Gaugeon Formalism for QED in Lorentz Violating Background
Science.gov (United States)
Shah, Mushtaq B.; Ganai, Prince A.
2018-02-01
At the energy regimes close to Planck scales, the usual structure of Lorentz symmetry fails to address certain fundamental issues and eventually breaks down, thus paving the way for an alternative road map. It is thus argued that some subgroup of proper Lorentz group could stand consistent and might possibly help us to circumvent this problem. It is this subgroup that goes by the name of Very Special Relativity (VSR). Apart from violating rotational symmetry, VSR is believed to preserve the very tenets of special relativity. The gaugeon formalism due to type-I Yokoyama and type-II Izawa are found to be invariant under BRST symmetry. In this paper, we analyze the scope of this invariance in the scheme of VSR. Furthermore, we will obtain VSR modified Lagrangian density using path integral derivation. We will explore the consistency of VSR with regard to these theories.
19. Detecting Lorentz Violations with Gravitational Waves From Black Hole Binaries
Science.gov (United States)
Sotiriou, Thomas P.
2018-01-01
Gravitational wave observations have been used to test Lorentz symmetry by looking for dispersive effects that are caused by higher order corrections to the dispersion relation. In this Letter I argue on general grounds that, when such corrections are present, there will also be a scalar excitation. Hence, a smoking-gun observation of Lorentz symmetry breaking would be the direct detection of scalar waves that travel at a speed other than the speed of the standard gravitational wave polarizations or the speed of light. Interestingly, in known Lorentz-breaking gravity theories the difference between the speeds of scalar and tensor waves is virtually unconstrained, whereas the difference between the latter and the speed of light is already severely constrained by the coincident detection of gravitational waves and gamma rays from a binary neutron star merger.
20. Searches for Lorentz Violation in Top-Quark Production and Decay at Hadron Colliders
Energy Technology Data Exchange (ETDEWEB)
Whittington, Denver Wade [Indiana Univ., Bloomington, IN (United States)
2012-07-01
We present a first-of-its-kind confirmation that the most massive known elementary particle obeys the special theory of relativity. Lorentz symmetry is a fundamental aspect of special relativity which posits that the laws of physics are invariant regardless of the orientation and velocity of the reference frame in which they are measured. Because this symmetry is a fundamental tenet of physics, it is important to test its validity in all processes. We quantify violation of this symmetry using the Standard-Model Extension framework, which predicts the effects that Lorentz violation would have on elementary particles and their interactions. The top quark is the most massive known elementary particle and has remained inaccessible to tests of Lorentz invariance until now. This model predicts a dependence of the production cross section for top and antitop quark pairs on sidereal time as the orientation of the experiment in which these events are produced changes with the rotation of the Earth. Using data collected with the DØ detector at the Fermilab Tevatron Collider, we search for violation of Lorentz invariance in events involving the production of a $t\\bar{t}$ pair. Within the experimental precision, we find no evidence for such a violation and set upper limits on parameters describing its possible strength within the Standard-Model Extension. We also investigate the prospects for extending this analysis using the ATLAS detector at the Large Hadron Collider which, because of the higher rate of $t\\bar{t}$ events at that experiment, has the potential to improve the limits presented here.
1. Experimental Studies on the Lorentz Symmetry in Post-Newtonian Gravity with Pulsars
Directory of Open Access Journals (Sweden)
Lijing Shao
2016-12-01
Full Text Available Local Lorentz invariance (LLI is one of the most important fundamental symmetries in modern physics. While the possibility of LLI violation (LLIv was studied extensively in flat spacetime, its counterpart in gravitational interaction also deserves significant examination from experiments. In this contribution, I review several recent studies of LLI in post-Newtonian gravity, using powerful tools of pulsar timing. It shows that precision pulsar timing experiments hold a unique position to probe LLIv in post-Newtonian gravity.
2. CPT and Lorentz violation as signatures for Planck-scale physics
International Nuclear Information System (INIS)
Lehnert, Ralf
2009-01-01
In recent years, the breakdown of spacetime symmetries has been identified as a promising research field in the context of Planck-scale phenomenology. For example, various theoretical approaches to the quantum-gravity problem are known to accommodate minute violations of CPT invariance. This talk covers various topics within this research area. In particular, some mechanisms for spacetime-symmetry breaking as well as the Standard-Model Extension (SME) test framework will be reviewed; the connection between CPT and Lorentz invariance in quantum field theory will be exposed; and the a few experimental CPT tests with emphasis on matter-antimatter comparisons will be discussed.
3. Strong equivalence, Lorentz and CPT violation, anti-hydrogen spectroscopy and gamma-ray burst polarimetry
International Nuclear Information System (INIS)
Shore, Graham M.
2005-01-01
The strong equivalence principle, local Lorentz invariance and CPT symmetry are fundamental ingredients of the quantum field theories used to describe elementary particle physics. Nevertheless, each may be violated by simple modifications to the dynamics while apparently preserving the essential fundamental structure of quantum field theory itself. In this paper, we analyse the construction of strong equivalence, Lorentz and CPT violating Lagrangians for QED and review and propose some experimental tests in the fields of astrophysical polarimetry and precision atomic spectroscopy. In particular, modifications of the Maxwell action predict a birefringent rotation of the direction of linearly polarised radiation from synchrotron emission which may be studied using radio galaxies or, potentially, gamma-ray bursts. In the Dirac sector, changes in atomic energy levels are predicted which may be probed in precision spectroscopy of hydrogen and anti-hydrogen atoms, notably in the Doppler-free, two-photon 1s-2s and 2s-nd (n∼10) transitions
4. Why Cerenkov Radiation May Not Occur, Even When It Is Allowed by Lorentz-Violating Kinematics
Directory of Open Access Journals (Sweden)
Brett Altschul
2017-10-01
Full Text Available In a Lorentz-violating quantum field theory, the energy-momentum relations for the field quanta are typically modified. This affects the kinematics, and processes that are normally forbidden may become allowed. One reaction that clearly becomes kinematically possible when photons’ phase speeds are less than 1 is vacuum Cerenkov radiation. However, in spite of expectations, and in defiance of phase space estimates, a electromagnetic Chern–Simons theory with a timelike Lorentz violation coefficient does not feature any energy losses through Cerenkov emission. There is an unexpected cancelation, made possible by the existence of unstable long-wavelength modes of the field. The fact that the theory possesses a more limited form of gauge symmetry than conventional electrodynamics also plays a role.
5. Lorentz Invariance Violation and Modified Hawking Fermions Tunneling Radiation
Directory of Open Access Journals (Sweden)
Shu-Zheng Yang
2016-01-01
Full Text Available Recently the modified Dirac equation with Lorentz invariance violation has been proposed, which would be helpful to resolve some issues in quantum gravity theory and high energy physics. In this paper, the modified Dirac equation has been generalized in curved spacetime, and then fermion tunneling of black holes is researched under this correctional Dirac field theory. We also use semiclassical approximation method to get correctional Hamilton-Jacobi equation, so that the correctional Hawking temperature and correctional black hole’s entropy are derived.
6. Vacuum Cherenkov radiation for Lorentz-violating fermions
Science.gov (United States)
Schreck, M.
2017-11-01
The current work focuses on the process of vacuum Cherenkov radiation for Lorentz-violating fermions that are described by the minimal standard-model extension (SME). To date, most considerations of this important hypothetical process have been restricted to Lorentz-violating photons, as the necessary theoretical tools for the SME fermion sector have not been available. With their development in a very recent paper, we are now in a position to compute the decay rates based on a modified Dirac theory. Two realizations of the Cherenkov process are studied. In the first scenario, the spin projection of the incoming fermion is assumed to be conserved, and in the second, the spin projection is allowed to flip. The first type of process is shown to be still forbidden for the dimensionful a and b coefficients where there are strong indications that it is energetically disallowed for the H coefficients, as well. However, it is rendered possible for the dimensionless c , d , e , f , and g coefficients. For large initial fermion energies, the decay rates for the c and d coefficients were found to grow linearly with momentum and to be linearly suppressed by the smallness of the Lorentz-violating coefficient where for the e , f , and g coefficients this suppression is even quadratic. The decay rates vanish in the vicinity of the threshold, as expected. The decay including a fermion spin-flip plays a role for the spin-nondegenerate operators and it was found to occur for the dimensionful b and H coefficients as well as for the dimensionless d and g . The characteristics of this process differ much from the properties of the spin-conserving one, e.g., there is no threshold. Based on experimental data of ultra-high-energy cosmic rays, new constraints on Lorentz violation in the quark sector are obtained from the thresholds. However, it does not seem to be possible to derive bounds from the spin-flip decays. This work reveals the usefulness of the quantum field theoretic methods
7. Constraining spacetime nonmetricity with Lorentz-violation methods
Science.gov (United States)
Xiao, Zhi; Lehnert, Ralf; Snow, W. M.; Xu, Rui
2018-01-01
In this report, we will give the first constraints on in-matter nonmetricity. We will show how the effective-field-theory (EFT) toolbox developed for the study of Lorentz violation (LV) can be employed for investigations of the “effective LV” background caused by nonmetricity, a geometric object extending the notion of a Riemannian manifold. The idea is to probe for the effects of spacetime nonmetricity sourced by liquid 4He with polarized slow neutrons. We present the first constraints on isotropic and parity-odd nonmetricity components. Further constraints on anisotropic nonmetricity components within this EFT framework may be feasible with proper experimental techniques in the near future.
8. Prospects for testing Lorentz and CPT symmetry with antiprotons
Science.gov (United States)
Vargas, Arnaldo J.
2018-03-01
A brief overview of the prospects of testing Lorentz and CPT symmetry with antimatter experiments is presented. The models discussed are applicable to atomic spectroscopy experiments, Penning-trap experiments and gravitational tests. Comments about the sensitivity of the most recent antimatter experiments to the models reviewed here are included. This article is part of the Theo Murphy meeting issue Antiproton physics in the ELENA era'.
9. Prospects for testing Lorentz and CPT symmetry with antiprotons.
Science.gov (United States)
Vargas, Arnaldo J
2018-03-28
A brief overview of the prospects of testing Lorentz and CPT symmetry with antimatter experiments is presented. The models discussed are applicable to atomic spectroscopy experiments, Penning-trap experiments and gravitational tests. Comments about the sensitivity of the most recent antimatter experiments to the models reviewed here are included.This article is part of the Theo Murphy meeting issue 'Antiproton physics in the ELENA era'. © 2018 The Author(s).
10. CP violation and modular symmetries
International Nuclear Information System (INIS)
Dent, Thomas
2001-01-01
We reconsider the origin of CP violation in fundamental theory. Existing string models of spontaneous CP violation make ambiguous predictions, due to the arbitrariness of CP transformation and the apparent noninvariance of the results under duality. We find a modular CP invariance condition, applicable to any predictive model of spontaneous CP violation, which circumvents these problems; it strongly constrains CP violation by heterotic string moduli. The dilaton is also evaluated as a source of CP violation, but is likely experimentally excluded. We consider the prospects for explaining CP violation in strongly coupled strings and brane worlds
11. CP violation and modular symmetries
OpenAIRE
Dent, Thomas
2001-01-01
We reconsider the origin of CP violation in fundamental theory. Existing string models of spontaneous CP violation make ambiguous predictions, due to the arbitrariness of CP transformation and the apparent non-invariance of the results under duality. We find an unambiguous modular CP invariance condition, applicable to predictive models of spontaneous CP violation, which circumvents these problems; it strongly constrains CP violation by heterotic string moduli. The dilaton is also evaluated a...
12. Tests of Lorentz and CPT violation with MiniBooNE neutrino oscillation excesses
International Nuclear Information System (INIS)
Katori, Teppei
2014-01-01
Lorentz and CPT symmetry violaton is a predicted phenomenon of Planck–scale physics. Various types of data are analyzed to search for Lorentz violation under the Standard–Model Extension (SME) framework, including neutrino oscillation data. MiniBooNE is a short–baseline neutrino oscillation experiment at Fermilab. The measured excesses from MiniBooNE cannot be reconciled within the neutrino Standard Model (vSM); thus it might be a signal of new physics, such as Lorentz violation. We have analyzed the sidereal time dependence of MiniBooNE data for signals of the possible sidereal time dependence of the ocillation signals. we find that the v e appearance data prefer a sidereal time–independent solution, and the v-bar e appearance data slightly prefer a sidereal time–dependent solution, however, the statistical significance is not high to claim the discovery. Limits of order 10 −20 GeV are placed on combinations of SME coefficients
13. New test of Lorentz symmetry using ultrahigh-energy cosmic rays
Science.gov (United States)
Anchordoqui, Luis A.; Soriano, Jorge F.
2018-02-01
We propose an innovative test of Lorentz symmetry by observing pairs of simultaneous parallel extensive air showers produced by the fragments of ultrahigh-energy cosmic ray nuclei which disintegrated in collisions with solar photons. We show that the search for a cross-correlation of showers in arrival time and direction becomes background free for an angular scale ≲3 ° and a time window O (10 s ) . We also show that if the solar photo-disintegration probability of helium is O (10-5.5) then the hunt for spatiotemporal coincident showers could be within range of existing cosmic ray facilities, such as the Pierre Auger Observatory. We demonstrate that the actual observation of a few events can be used to constrain Lorentz violating dispersion relations of the nucleon.
14. Symmetry-violating kaon decays
International Nuclear Information System (INIS)
Herczeg, P.
1979-01-01
The content of this talk comprises two parts. In the first, an analysis of the muon number violating decay modes of the K-mesons is given. Subsequently, some new developments in the field of CP-violation are reviewed and the question of time-reversal invariance and the status of CPT-invariance are briefly considered. (auth)
15. Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation
International Nuclear Information System (INIS)
Mattingly, David M.; Maccione, Luca; Galaverni, Matteo; Liberati, Stefano; Sigl, Günter
2010-01-01
We study, within an effective field theory framework, O(E 2 M Pl 2 ) Planck-scale suppressed Lorentz invariance violation (LV) effects in the neutrino sector, whose size we parameterize by a dimensionless parameter η ν . We find deviations from predictions of Lorentz invariant physics in the cosmogenic neutrino spectrum. For positive O(1) coefficients no neutrino will survive above 10 19 eV. The existence of this cutoff generates a bump in the neutrino spectrum at energies of 10 17 eV. Although at present no constraint can be cast, as current experiments do not have enough sensitivity to detect ultra-high-energy neutrinos, we show that experiments in construction or being planned have the potential to cast limits as strong as η ν ∼ −4 on the neutrino LV parameter, depending on how LV is distributed among neutrino mass states. Constraints on η ν < 0 can in principle be obtained with this strategy, but they require a more detailed modeling of how LV affects the neutrino sector
16. Possible cosmogenic neutrino constraints on Planck-scale Lorentz violation
Energy Technology Data Exchange (ETDEWEB)
Mattingly, David M. [New Hamshire Univ., Durham, NH (United States); Maccione, Luca [DESY Hamburg (Germany). Theory Group; Galaverni, Matteo [INAF-IASF Bologna (Italy); Liberati, Stefano [INFN, Trieste (Italy); SISSA, Trieste (Italy); Sigl, Guenter [Hamburg Univ. (Germany). II. Inst. fuer Theoretische Physik
2009-11-15
We study, within an effective field theory framework, O(E{sup 2}/M{sup 2}{sub Pl}) Planck-scale suppressed Lorentz invariance violation (LV) effects in the neutrino sector, whose size we parameterize by a dimensionless parameter {eta}{sub {nu}}. We find deviations from predictions of Lorentz invariant physics in the cosmogenic neutrino spectrum. For positive O(1) coefficients no neutrino will survive above 10{sup 19} eV. The existence of this cutoff generates a bump in the neutrino spectrum at energies of 10{sup 17} eV. Although at present no constraint can be cast, as current experiments do not have enough sensitivity to detect ultra-high-energy neutrinos, we show that experiments in construction or being planned have the potential to cast limits as strong as {eta}{sub {nu}}
17. Extended hamiltonian formalism and Lorentz-violating lagrangians
Directory of Open Access Journals (Sweden)
2017-09-01
Full Text Available A new perspective on the classical mechanical formulation of particle trajectories in Lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant lagrangian and hamiltonian varieties is constructed. This approach enables calculation of trajectories using Hamilton's equations in momentum space and the Euler–Lagrange equations in velocity space away from certain singular points that arise in the theory. Singular points are naturally de-singularized by requiring the trajectories to be smooth functions of both velocity and momentum variables. In addition, it is possible to identify specific sheets of the dispersion relations that correspond to specific solutions for the lagrangian. Examples corresponding to bipartite Finsler functions are computed in detail. A direct connection between the lagrangians and the field-theoretic solutions to the Dirac equation is also established for a special case.
18. Violations of Lorentz invariance in the neutrino sector after OPERA
Energy Technology Data Exchange (ETDEWEB)
Maccione, Luca [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Liberati, Stefano [Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste (Italy); INFN, Sezione de Trieste (Italy); Mattingly, David M. [New Hampshire Univ., Durham (United States). Dept. of Physics
2011-10-15
The OPERA collaboration has recently reported that neutrinos travel faster than light. We review the theoretical situation of constraints on violations of Lorentz invariance, focusing in particular on the compatibility between the OPERA results with both previous constraints and recently obtained ones. We generalize to higher order operators the recent constraint provided by the absence of neutrino energy loss, via electron-positron pair production at OPERA energies, and show that no modi ed in vacuo dispersion relation within an effective field theory context is compatible with OPERA results. We conclude that the OPERA result is incompatible with current observations, at least without resorting to models beyond effective field theory, possibly with local environmental effects. (orig.)
19. Lorentz Invariance Violation effects on UHECR propagation: A geometrized approach
Science.gov (United States)
Torri, Marco Danilo Claudio; Bertini, Stefano; Giammarchi, Marco; Miramonti, Lino
2018-06-01
We explore the possibility to geometrize the interaction of massive fermions with the quantum structure of space-time, trying to create a theoretical background, in order to explain what some recent experimental results seem to implicate on the propagation of Ultra High Energy Cosmic Rays (UHECR). We will investigate part of the phenomenological implications of this approach on the predicted effect of the UHECR suppression, in fact recent evidences seem to involve the modification of the GZK cut-off phenomenon. The search for an effective theory, which can explain this physical effect, is based on Lorentz Invariance Violation (LIV), which is introduced via Modified Dispersion Relations (MDRs). Furthermore we illustrate that this perspective implies a more general geometry of space-time than the usual Riemannian one, indicating, for example, the opportunity to resort to Finsler theory.
20. Violations of Lorentz invariance in the neutrino sector after OPERA
International Nuclear Information System (INIS)
Maccione, Luca; Liberati, Stefano; Mattingly, David M.
2011-10-01
The OPERA collaboration has recently reported that neutrinos travel faster than light. We review the theoretical situation of constraints on violations of Lorentz invariance, focusing in particular on the compatibility between the OPERA results with both previous constraints and recently obtained ones. We generalize to higher order operators the recent constraint provided by the absence of neutrino energy loss, via electron-positron pair production at OPERA energies, and show that no modi ed in vacuo dispersion relation within an effective field theory context is compatible with OPERA results. We conclude that the OPERA result is incompatible with current observations, at least without resorting to models beyond effective field theory, possibly with local environmental effects. (orig.)
1. Constraints and stability in vector theories with spontaneous Lorentz violation
International Nuclear Information System (INIS)
Bluhm, Robert; Gagne, Nolan L.; Potting, Robertus; Vrublevskis, Arturs
2008-01-01
Vector theories with spontaneous Lorentz violation, known as bumblebee models, are examined in flat spacetime using a Hamiltonian constraint analysis. In some of these models, Nambu-Goldstone modes appear with properties similar to photons in electromagnetism. However, depending on the form of the theory, additional modes and constraints can appear that have no counterparts in electromagnetism. An examination of these constraints and additional degrees of freedom, including their nonlinear effects, is made for a variety of models with different kinetic and potential terms, and the results are compared with electromagnetism. The Hamiltonian constraint analysis also permits an investigation of the stability of these models. For certain bumblebee theories with a timelike vector, suitable restrictions of the initial-value solutions are identified that yield ghost-free models with a positive Hamiltonian. In each case, the restricted phase space is found to match that of electromagnetism in a nonlinear gauge
2. Extended hamiltonian formalism and Lorentz-violating lagrangians
Science.gov (United States)
2017-09-01
A new perspective on the classical mechanical formulation of particle trajectories in Lorentz-violating theories is presented. Using the extended hamiltonian formalism, a Legendre Transformation between the associated covariant lagrangian and hamiltonian varieties is constructed. This approach enables calculation of trajectories using Hamilton's equations in momentum space and the Euler-Lagrange equations in velocity space away from certain singular points that arise in the theory. Singular points are naturally de-singularized by requiring the trajectories to be smooth functions of both velocity and momentum variables. In addition, it is possible to identify specific sheets of the dispersion relations that correspond to specific solutions for the lagrangian. Examples corresponding to bipartite Finsler functions are computed in detail. A direct connection between the lagrangians and the field-theoretic solutions to the Dirac equation is also established for a special case.
3. Vacuum solutions of a gravity model with vector-induced spontaneous Lorentz symmetry breaking
International Nuclear Information System (INIS)
Bertolami, O.; Paramos, J.
2005-01-01
We study the vacuum solutions of a gravity model where Lorentz symmetry is spontaneously broken once a vector field acquires a vacuum expectation value. Results are presented for the purely radial Lorentz symmetry breaking (LSB), radial/temporal LSB and axial/temporal LSB. The purely radial LSB result corresponds to new black hole solutions. When possible, parametrized post-Newtonian parameters are computed and observational boundaries used to constrain the Lorentz symmetry breaking scale
4. Lorentz violating p-form gauge theories in superspace
Energy Technology Data Exchange (ETDEWEB)
Upadhyay, Sudhaker [Indian Institute of Technology Kharagpur, Centre for Theoretical Studies, Kharagpur (India); Shah, Mushtaq B.; Ganai, Prince A. [National Institute of Technology, Department of Physics, Srinagar, Kashmir (India)
2017-03-15
Very special relativity (VSR) keeps the main features of special relativity but breaks rotational invariance due to an intrinsic preferred direction. We study the VSR-modified extended BRST and anti-BRST symmetry of the Batalin-Vilkovisky (BV) actions corresponding to the p = 1, 2, 3-form gauge theories. Within the VSR framework, we discuss the extended BRST invariant and extended BRST and anti-BRST invariant superspace formulations for these BV actions. Here we observe that the VSR-modified extended BRST invariant BV actions corresponding to the p = 1, 2, 3-form gauge theories can be written in a manifestly covariant manner in a superspace with one Grassmann coordinate. Moreover, two Grassmann coordinates are required to describe the VSR-modified extended BRST and extended anti-BRST invariant BV actions in a superspace. These results are consistent with the Lorentz-invariant (special relativity) formulation. (orig.)
5. A perfectly conducting surface in electrodynamics with Lorentz symmetry breaking
Science.gov (United States)
Borges, L. H. C.; Barone, F. A.
2017-10-01
In this paper we consider a model which exhibits explicit Lorentz symmetry breaking due to the presence of a single background vector v^{μ } coupled to the gauge field. We investigate such a theory in the vicinity of a perfectly conducting plate for different configurations of v^{μ }. First we consider no restrictions on the components of the background vector and we treat it perturbatively up to second order. Next, we treat v^{μ } exactly for two special cases: the first one is when it has only components parallel to the plate, and the second one when it has a single component perpendicular to the plate. For all these configurations, the propagator for the gauge field and the interaction force between the plate and a point-like electric charge are computed. Surprisingly, it is shown that the image method is valid in our model and we argue that it is a non-trivial result. We show there arises a torque on the mirror with respect to its positioning in the background field when it interacts with a point-like charge. It is a new effect with no counterpart in theories with Lorentz symmetry in the presence of a perfect mirror.
6. A perfectly conducting surface in electrodynamics with Lorentz symmetry breaking
Energy Technology Data Exchange (ETDEWEB)
Borges, L.H.C. [UNESP, Campus de Guaratingueta, DFQ, Guaratingueta, SP (Brazil); Barone, F.A. [IFQ, Universidade Federal de Itajuba, Itajuba, MG (Brazil)
2017-10-15
In this paper we consider a model which exhibits explicit Lorentz symmetry breaking due to the presence of a single background vector v{sup μ} coupled to the gauge field. We investigate such a theory in the vicinity of a perfectly conducting plate for different configurations of v{sup μ}. First we consider no restrictions on the components of the background vector and we treat it perturbatively up to second order. Next, we treat v{sup μ} exactly for two special cases: the first one is when it has only components parallel to the plate, and the second one when it has a single component perpendicular to the plate. For all these configurations, the propagator for the gauge field and the interaction force between the plate and a point-like electric charge are computed. Surprisingly, it is shown that the image method is valid in our model and we argue that it is a non-trivial result. We show there arises a torque on the mirror with respect to its positioning in the background field when it interacts with a point-like charge. It is a new effect with no counterpart in theories with Lorentz symmetry in the presence of a perfect mirror. (orig.)
7. On the harmonic-type and linear-type confinement of a relativistic scalar particle yielded by Lorentz symmetry breaking effects
Energy Technology Data Exchange (ETDEWEB)
Bakke, K., E-mail: [email protected] [Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, 58051-900, João Pessoa-PB (Brazil); Belich, H., E-mail: [email protected] [Departamento de Física e Química, Universidade Federal do Espírito Santo, Av. Fernando Ferrari, 514, Goiabeiras, 29060-900, Vitória, ES (Brazil)
2016-10-15
Based on the Standard Model Extension, we investigate relativistic quantum effects on a scalar particle in backgrounds of the Lorentz symmetry violation defined by a tensor field. We show that harmonic-type and linear-type confining potentials can stem from Lorentz symmetry breaking effects, and thus, relativistic bound state solutions can be achieved. We first analyse a possible scenario of the violation of the Lorentz symmetry that gives rise to a harmonic-type potential. In the following, we analyse another possible scenario of the breaking of the Lorentz symmetry that induces both harmonic-type and linear-type confining potentials. In this second case, we also show that not all values of the parameter associated with the intensity of the electric field are permitted in the search for polynomial solutions to the radial equation, where the possible values of this parameter are determined by the quantum numbers of the system and the parameters associated with the violation of the Lorentz symmetry.
8. Lepton flavor violation and seesaw symmetries
Energy Technology Data Exchange (ETDEWEB)
Aristizabal Sierra, D., E-mail: [email protected] [Universite de Liege, IFPA, Department AGO (Belgium)
2013-03-15
When the standard model is extended with right-handed neutrinos the symmetries of the resulting Lagrangian are enlarged with a new global U(1){sub R} Abelian factor. In the context of minimal seesaw models we analyze the implications of a slightly broken U(1){sub R} symmetry on charged lepton flavor violating decays. We find, depending on the R-charge assignments, models where charged lepton flavor violating rates can be within measurable ranges. In particular, we show that in the resulting models due to the structure of the light neutrino mass matrix muon flavor violating decays are entirely determined by neutrino data (up to a normalization factor) and can be sizable in a wide right-handed neutrino mass range.
9. Cosmological evolution of interacting dark energy in Lorentz violation
International Nuclear Information System (INIS)
Zen, Freddy P.; Gunara, Bobby E.; Triyanta; Arianto; Purwanto, A.
2009-01-01
The cosmological evolution of an interacting scalar-field model in which the scalar field interacts with dark matter, radiation, and baryons via Lorentz violation is investigated. We propose a model of interaction through the effective coupling, anti β. Using dynamical system analysis, we study the linear dynamics of an interacting model and show that the dynamics of critical points are completely controlled by two parameters. Some results can be mentioned as follows. Firstly, the sequence of radiation, the dark matter, and the scalar-field dark energy exist and baryons are subdominant. Secondly, the model also allows for the possibility of having a universe in the phantom phase with constant potential. Thirdly, the effective gravitational constant varies with respect to time through anti β. In particular, we consider the simple case where anti β has a quadratic form and has a good agreement with the modified ΛCDM and quintessence models. Finally, we also calculate the first post-Newtonian parameters for our model. (orig.)
10. Generation of higher derivatives operators and electromagnetic wave propagation in a Lorentz-violation scenario
Energy Technology Data Exchange (ETDEWEB)
Borges, L.H.C., E-mail: [email protected] [Universidade Federal do ABC, Centro de Ciências Naturais e Humanas, Av. dos Estados, 5001, Santo André, SP, 09210-580 (Brazil); Dias, A.G., E-mail: [email protected] [Universidade Federal do ABC, Centro de Ciências Naturais e Humanas, Av. dos Estados, 5001, Santo André, SP, 09210-580 (Brazil); Ferrari, A.F., E-mail: [email protected] [Universidade Federal do ABC, Centro de Ciências Naturais e Humanas, Av. dos Estados, 5001, Santo André, SP, 09210-580 (Brazil); Nascimento, J.R., E-mail: [email protected] [Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa, Paraíba, 58051-970 (Brazil); Petrov, A.Yu., E-mail: [email protected] [Departamento de Física, Universidade Federal da Paraíba, Caixa Postal 5008, João Pessoa, Paraíba, 58051-970 (Brazil)
2016-05-10
We study the perturbative generation of higher-derivative Lorentz violating operators as quantum corrections to the photon effective action, originated from a specific Lorentz violation background, which has already been studied in connection with the physics of light pseudoscalars. We calculate the complete one loop effective action of the photon field through the proper-time method, using the zeta function regularization. This result can be used as a starting point to study possible effects of the Lorentz violating background we are considering in photon physics. As an example, we focus on the lowest order corrections and investigate whether they could influence the propagation of electromagnetic waves through the vacuum. We show, however, that no effects of the kind of Lorentz violation we consider can be detected in such a context, so that other aspects of photon physics have to be studied.
11. Violation of Particle Anti-particle Symmetry
CERN Multimedia
CERN. Geneva
2001-01-01
Symmetry is a fundamental concept which can be found in the whole range of human activities e. g. from arts to science. The beauty of a statues is often related to its symmetric form. In physics, all the laws are related to some sort of symmetry. Equally important is a small breakdown ofsymmetry. Even for the case of a statue, its beauty might be enhanced by introducing small distortions. In this course, we investigate the role symmetry in the world of elementary particles. Some symmetries found there are very similar to those which can be seen in our daily life, while others are more exotic and related to the quantum nature of the elementary particles. Our particular focus ismade on symmetry and its violation between the matter and anti-matter, known as CP violation. It is experimentally well established that particleand anti-particle behave a tiny bit differently in the world of elementary particles. We discuss how this would be explained and how we can extendour knowledge. Evolution of our universe is stro...
12. First test of Lorentz violation with a reactor-based antineutrino experiment
International Nuclear Information System (INIS)
Abe, Y.; Ishitsuka, M.; Konno, T.; Kuze, M.; Aberle, C.; Buck, C.; Hartmann, F.X.; Haser, J.; Kaether, F.; Lindner, M.; Reinhold, B.; Schwetz, T.; Wagner, S.; Watanabe, H.; Anjos, J.C. dos; Gama, R.; Lima, H.P.-Jr.; Pepe, I.M.; Bergevin, M.; Felde, J.; Maesano, C.N.; Bernstein, A.; Bowden, N.S.; Dazeley, S.; Erickson, A.; Keefer, G.; Bezerra, T.J.C.; Furuta, H.; Suekane, F.; Bezrukhov, L.; Lubsandorzhiev, B.K.; Yanovitch, E.; Blucher, E.; Conover, E.; Crum, K.; Strait, M.; Worcester, M.; Busenitz, J.; Goon, J.TM.; Habib, S.; Ostrovskiy, I.; Reichenbacher, J.; Stancu, I.; Sun, Y.; Cabrera, A.; Franco, D.; Kryn, D.; Obolensky, M.; Roncin, R.; Tonazzo, A.; Caden, E.; Damon, E.; Lane, C.E.; Maricic, J.; Miletic, T.; Milincic, R.; Perasso, S.; Smith, E.; Camilleri, L.; Carr, R.; Franke, A.J.; Shaevitz, M.H.; Toups, M.; Cerrada, M.; Crespo-Anadon, J.I.; Gil-Botella, I.; Lopez-Castano, J.M.; Novella, P.; Palomares, C.; Santorelli, R.; Chang, P.J.; Horton-Smith, G.A.; McKee, D.; Shrestha, D.; Chimenti, P.; Classen, T.; Collin, A.P.; Cucoanes, A.; Durand, V.; Fechner, M.; Fischer, V.; Hayakawa, T.; Lasserre, T.; Letourneau, A.; Lhuillier, D.; Mention, G.; Mueller, Th.A.; Perrin, P.; Sida, J.L.; Sinev, V.; Veyssiere, C.
2012-01-01
We present a search for Lorentz violation with 8249 candidate electron antineutrino events taken by the Double Chooz experiment in 227.9 live days of running. This analysis, featuring a search for a sidereal time dependence of the events, is the first test of Lorentz invariance using a reactor-based antineutrino source. No sidereal variation is present in the data and the disappearance results are consistent with sidereal time independent oscillations. Under the Standard-Model Extension, we set the first limits on 14 Lorentz violating coefficients associated with transitions between electron and tau flavor, and set two competitive limits associated with transitions between electron and muon flavor. (authors)
13. Spontaneous Lorentz violation and the long-range gravitational preferred-frame effect
International Nuclear Information System (INIS)
Graesser, Michael L.; Jenkins, Alejandro; Wise, Mark B.
2005-01-01
Lorentz-violating operators involving Standard Model fields are tightly constrained by experimental data. However, bounds are more model-independent for Lorentz violation appearing in purely gravitational couplings. The spontaneous breaking of Lorentz invariance by the vacuum expectation value of a vector field selects a universal rest frame. This affects the propagation of the graviton, leading to a modification of Newton's law of gravity. We compute the size of the long-range preferred-frame effect in terms of the coefficients of the two-derivative operators in the low-energy effective theory that involves only the graviton and the Goldstone bosons
14. Planck-scale Lorentz violation constrained by ultra-high-energy cosmic rays
Energy Technology Data Exchange (ETDEWEB)
Maccione, L. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Univ. Hamburg, II. Inst. fuer Theoretische Physik (Germany); Taylor, A.M. [Max-Planck-Inst. fuer Kernphysik, Heidelberg (Germany); Mattingly, D.M.; Liberati, S. [Scuola Internazionale Superiore di Studi Avanzati SISSA, Trieste (Italy); Istituto Nazionale di Fisica Nucleare INFN, Sezione di Trieste (Italy)
2009-09-15
We investigate the consequences of higher dimension Lorentz violating, CPT even kinetic operators that couple standard model fields to a non-zero vector field in an Effective Field Theory framework. Comparing the ultra-high energy cosmic ray spectrum reconstructed in the presence of such terms with data from the Pierre Auger observatory allows us to establish two sided bounds on the coefficients of the mass dimension five and six operators for the proton and pion. Our bounds imply that for both protons and pions, the energy scale of Lorentz symmetry breaking must be well above the Planck scale. In particular, the dimension five operators are constrained at the level of 10{sup -3}M{sup -1}{sub Planck}. The magnitude of the dimension six proton coefficient is bounded at the level of 10{sup -6}M{sup -2}{sub Planck} except in a narrow range where the pion and proton coefficients are both negative and nearly equal. In this small area, the magnitude of the dimension six proton coefficient must only be below 10{sup -3}M{sup -2}{sub Planck}. Constraints on the dimension six pion coefficient are found to be much weaker, but still below M{sup -2}{sub Planck}. (orig.)
15. Universal dynamics of spontaneous Lorentz violation and a new spin-dependent inverse-square law force
International Nuclear Information System (INIS)
Arkani-Hamed, Nima; Cheng, Hsin-Chia; Luty, Markus; Thaler, Jesse
2005-01-01
We study the universal low-energy dynamics associated with the spontaneous breaking of Lorentz invariance down to spatial rotations. The effective lagrangian for the associated Goldstone field can be uniquely determined by the non-linear realization of a broken time diffeomorphism symmetry, up to some overall mass scales. It has previously been shown that this symmetry breaking pattern gives rise to a Higgs phase of gravity, in which gravity is modified in the infrared. In this paper, we study the effects of direct couplings between the Goldstone boson and standard model fermions, which necessarily accompany Lorentz-violating terms in the theory. The leading interaction is the coupling to the axial vector current, which reduces to spin in the non-relativistic limit. A spin moving relative to the 'ether' rest frame will emit Goldstone Cerenkov radiation. The Goldstone also induces a long-range inverse-square law force between spin sources with a striking angular dependence, reflecting the underlying Goldstone shockwaves and providing a smoking gun for this theory. We discuss the regime of validity of the effective theory describing these phenomena, and the possibility of probing Lorentz violations through Goldstone boson signals in a way that is complementary to direct tests in some regions of parameter space
16. Charged Lifshitz black hole and probed Lorentz-violation fermions from holography
Energy Technology Data Exchange (ETDEWEB)
Luo, Cheng-Jian, E-mail: [email protected] [Department of Physics, Nanchang University, Nanchang, 330031 (China); Center for Relativistic Astrophysics and High Energy Physics, Nanchang University, Nanchang 330031 (China); Kuang, Xiao-Mei, E-mail: [email protected] [Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso (Chile); Shu, Fu-Wen, E-mail: [email protected] [Department of Physics, Nanchang University, Nanchang, 330031 (China); Center for Relativistic Astrophysics and High Energy Physics, Nanchang University, Nanchang 330031 (China)
2017-06-10
We analytically obtain a new charged Lifshitz solution by adding a non-relativistic Maxwell field in Hořava–Lifshitz gravity. The black hole exhibits an anisotropic scaling between space and time (Lifshitz scaling) in the UV limit, while in the IR limit, the Lorentz invariance is approximately recovered. We introduce the probed Lorentz-violation fermions into the background and holographically investigate the spectral properties of the dual fermionic operator. The Lorentz-violation of the fermions will enhance the peak and correspond larger fermi momentum, which compensates the non-relativistic bulk effect of the dynamical exponent (z). For a fixed z, when the Lorentz-violation of fermions increases to a critical value, the behavior of the low energy excitation goes from a non-Fermi liquid type to a Fermi liquid type, which implies a kind of phase transition.
17. Constraints on violation of Lorentz invariance from atmospheric showers initiated by multi-TeV photons
Energy Technology Data Exchange (ETDEWEB)
Rubtsov, Grigory; Satunin, Petr; Sibiryakov, Sergey, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect, 7a, 117312 Moscow (Russian Federation)
2017-05-01
Parameterizing hypothetical violation of Lorentz invariance at high energies using the framework of effective quantum field theory, we discuss its effect on the formation of atmospheric showers by very-high-energy gamma rays. In the scenario where Lorentz invariance violation leads to a decrease of the photon velocity with energy the formation of the showers is suppressed compared to the Lorentz invariant case. Absence of such suppression in the high-energy part of spectrum of the Crab nebula measured independently by HEGRA and H.E.S.S. collaborations is used to set lower bounds on the energy scale of Lorentz invariance violation. These bounds are competitive with the strongest existing constraints obtained from timing of variable astrophysical sources and the absorption of TeV photons on the extragalactic background light. They will be further improved by the next generation of multi-TeV gamma-ray observatories.
18. The flight of the bumblebee: solutions from a vector-induced spontaneous Lorentz symmetry breaking model
International Nuclear Information System (INIS)
Bertolami, Orfeu; Paramos, Jorge
2006-01-01
The vacuum solutions arising from a spontaneous breaking of Lorentz symmetry due to the acquisition of a vacuum expectation value by a vector field are derived. These include the purely radial Lorentz symmetry breaking (LSB), radial/temporal LSB and axial/temporal LSB scenarios. It is found that the purely radial LSB case gives rise to new black hole solutions. Whenever possible. Parametrized Post-Newtonian (PPN) parameters are computed and compared to observational bounds, in order to constrain the Lorentz symmetry breaking scale
19. Traces of Lorentz symmetry breaking in a hydrogen atom at ground state
Science.gov (United States)
Borges, L. H. C.; Barone, F. A.
2016-02-01
Some traces of a specific Lorentz symmetry breaking scenario in the ground state of the hydrogen atom are investigated. We use standard Rayleigh-Schrödinger perturbation theory in order to obtain the corrections to the ground state energy and the wave function. It is shown that an induced four-pole moment arises, due to the Lorentz symmetry breaking. The model considered is the one studied in Borges et al. (Eur Phys J C 74:2937, 2014), where the Lorentz symmetry is broken in the electromagnetic sector.
20. Traces of Lorentz symmetry breaking in a hydrogen atom at ground state
Energy Technology Data Exchange (ETDEWEB)
Borges, L.H.C. [Universidade Federal do ABC, Centro de Ciencias Naturais e Humanas, Santo Andre, SP (Brazil); Barone, F.A. [IFQ-Universidade Federal de Itajuba, Itajuba, MG (Brazil)
2016-02-15
Some traces of a specific Lorentz symmetry breaking scenario in the ground state of the hydrogen atom are investigated. We use standard Rayleigh-Schroedinger perturbation theory in order to obtain the corrections to the ground state energy and the wave function. It is shown that an induced four-pole moment arises, due to the Lorentz symmetry breaking. The model considered is the one studied in Borges et al. (Eur Phys J C 74:2937, 2014), where the Lorentz symmetry is broken in the electromagnetic sector. (orig.)
1. Traces of Lorentz symmetry breaking in a hydrogen atom at ground state
International Nuclear Information System (INIS)
Borges, L.H.C.; Barone, F.A.
2016-01-01
Some traces of a specific Lorentz symmetry breaking scenario in the ground state of the hydrogen atom are investigated. We use standard Rayleigh-Schroedinger perturbation theory in order to obtain the corrections to the ground state energy and the wave function. It is shown that an induced four-pole moment arises, due to the Lorentz symmetry breaking. The model considered is the one studied in Borges et al. (Eur Phys J C 74:2937, 2014), where the Lorentz symmetry is broken in the electromagnetic sector. (orig.)
2. Erratum (astro-ph/0510172) Robust Limits on Lorentz Violation from Gamma-Ray Bursts
CERN Document Server
AUTHOR|(CDS)2108556; Nanopoulos, D V; Sakharov, Alexander S; Sarkisyan-Grinbaum, E
2008-01-01
We correct the fitting formula used in refs. [1,2] to obtain a robust limit on a violation of Lorentz invariance that depends linearly on the photon energy. The correction leads to a slight increase of the limit on the scale of the violation, to M > 1.4 x 10^{16} GeV.
International Nuclear Information System (INIS)
Galvan Herrera, J.B.
1990-01-01
The left-right quiral symmetry is not conserved by the Standard model. A subgroup of the standard gauge group (SU(2) L ) breaks this symmetry in a explicit way. Moreover, the standard model, if there are theree or more matter generations, violates the CP discrete symmetry. This prediction has been experimentally demonstrated correct in the Kaon anti Kaon system. In this work some possible explanations to the CP violation parameter magnitude are researched. We have studied the variation of the Kobayashi-Maskawa matrix with the energy scale. To realize this work we have developed a general method to calculate the renormalization group equations of the Kobayashi-Maskawa matrix parameters. From these equations we could also calculate the renormalization group equation of the J parameter that characterizes the CP violation. This calculus has been applied in a concrete example: a typical supersymmetric model from superstring theories. This model can be seen like a natural extension of the supersymmetric standard model. This kind of models have a gauge group bigger that the standard one more particles and new terms of the Lagrangian. We have verified that such model provides us of a correct low energy fenomenology and, moreover other results, some particle spectrums have been developed. In the elaboration of this model some conditions, that the model has to respected to be compatible with the actual fenomenology, have been studied. The most interesting results of this thesis are the develop of a general method to calculate the renormalization group equations of the Kobayashi-Maskawa matrix parameters and the develop of a new mechanism of the radiative violation. This mechanism is related with the new terms of the Lagrangian. (Author)
4. Test of CPT and Lorentz symmetry in entangled neutral kaons with the KLOE experiment
International Nuclear Information System (INIS)
Babusci, D.; Balwierz-Pytko, I.; Bencivenni, G.; Bloise, C.; Bossi, F.; Branchini, P.; Budano, A.; Caldeira Balkeståhl, L.; Capon, G.; Ceradini, F.; Ciambrone, P.; Curciarello, F.; Czerwiński, E.; Danè, E.; De Leo, V.; De Lucia, E.; De Robertis, G.; De Santis, A.; De Simone, P.
2014-01-01
Neutral kaon pairs produced in ϕ decays in anti-symmetric entangled state can be exploited to search for violation of CPT symmetry and Lorentz invariance. We present an analysis of the CP-violating process ϕ→K S K L →π + π − π + π − based on 1.7 fb −1 of data collected by the KLOE experiment at the Frascati ϕ-factory DAΦNE. The data are used to perform a measurement of the CPT-violating parameters Δa μ for neutral kaons in the context of the Standard Model Extension framework. The parameters measured in the reference frame of the fixed stars are: Δa 0 =(−6.0±7.7 stat ±3.1 syst )×10 −18 GeV, Δa X =(0.9±1.5 stat ±0.6 syst )×10 −18 GeV, Δa Y =(−2.0±1.5 stat ±0.5 syst )×10 −18 GeV, Δa Z =(3.1±1.7 stat ±0.5 syst )×10 −18 GeV. These are presently the most precise measurements in the quark sector of the Standard Model Extension.
5. The energy-momentum spectrum in local field theories with broken Lorentz-symmetry
International Nuclear Information System (INIS)
Borchers, H.J.; Buchholz, D.
1984-05-01
Assuming locality of the observables and positivity of the energy it is shown that the joint spectrum of the energy-momentum operators has a Lorentz-invariant lower boundary in all superselection sectors. This result is of interest if the Lorentz-symmetry is (spontaneously) broken, such as in the charged sectors of quantum electrodynamics. (orig.)
6. Constraining Anisotropic Lorentz Violation via the Spectral-lag Transition of GRB 160625B
Energy Technology Data Exchange (ETDEWEB)
Wei, Jun-Jie; Wu, Xue-Feng; Shao, Lang [Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008 (China); Zhang, Bin-Bin [Instituto de Astrofísica de Andalucá (IAA-CSIC), P.O. Box 03004, E-18080 Granada (Spain); Mészáros, Peter [Department of Astronomy and Astrophysics, Pennsylvania State University, 525 Davey Laboratory, University Park, PA 16802 (United States); Kostelecký, V. Alan, E-mail: [email protected], E-mail: [email protected] [Physics Department, Indiana University, Bloomington, IN 47405 (United States)
2017-06-20
Violations of Lorentz invariance can lead to an energy-dependent vacuum dispersion of light, which results in arrival-time differences of photons with different energies arising from a given transient source. In this work, direction-dependent dispersion constraints are obtained on nonbirefringent Lorentz-violating effects using the observed spectral lags of the gamma-ray burst GRB 160625B. This burst has unusually large high-energy photon statistics, so we can obtain constraints from the true spectral time lags of bunches of high-energy photons rather than from the rough time lag of a single highest-energy photon. Also, GRB 160625B is the only burst to date having a well-defined transition from positive lags to negative lags, providing a unique opportunity to distinguish Lorentz-violating effects from any source-intrinsic time lag in the emission of photons of different energy bands. Our results place comparatively robust two-sided constraints on a variety of isotropic and anisotropic coefficients for Lorentz violation, including the first bounds on Lorentz-violating effects from operators of mass dimension 10 in the photon sector.
7. Effects of Lorentz violation through the γe → Wνe process in the Standard Model extension
International Nuclear Information System (INIS)
Aranda, J I; Ramírez-Zavaleta, F; Tututi, E S; Rosete, D A; Tlachino, F J; Toscano, J J
2014-01-01
Physics beyond the Fermi scale could show up through deviations of the gauge couplings predicted by the electroweak Yang–Mills sector. This possibility is explored in the context of the International Linear Collider through the helicity amplitudes for the γe → Wν e reaction to which the trilinear WWγ coupling contributes. The new physics effects on this vertex are parametrized in a model-independent fashion through an effective electroweak Yang–Mills sector, which is constructed by considering two essentially different sources of new physics. In one scenario, Lorentz violation will be considered exclusively as the source of new physics effects. This type of new physics is considered in an extension of the Standard Model (SM) that is known as the SM extension (SME), which is an effective field theory that contemplates CPT and Lorentz violation in a model-independent fashion. Any source of new physics that respects the Lorentz symmetry will be considered within the general context of the well-known conventional effective SM (CESM) extension. Both the SME and CESM descriptions include gauge invariant operators of dimension higher than 4, which, in general, transform as Lorentz tensors of rank higher than zero. In the former theory, observer Lorentz invariants are constructed by contracting these operators with constant Lorentz tensors, whereas in the latter the corresponding Lorentz invariant interactions are obtained contracting such operators with products of the metric tensor. In this work, we focus on a dimension 6 Lorentz 2-tensor, O αβ , which arises from an effective SU(2) L Yang–Mills sector. Contributions to the WWγ coupling arising from dimension 4 operators are ignored since they are strongly constrained. When these operators are contracted with a constant antisymmetric background tensor, b αβ , the corresponding observer invariant belongs to the SME, whereas if they are contracted with the metric tensor, g αβ , an effective interaction in
8. Maxwell-Chern-Simons vortices in a CPT-odd Lorentz-violating Higgs electrodynamics
International Nuclear Information System (INIS)
Casana, R.; Ferreira, M.M.; Hora, E. da; Neves, A.B.F.
2014-01-01
We study BPS vortices in a CPT-odd and Lorentz-violating Maxwell-Chern-Simons-Higgs (MCSH) electrodynamics attained from the dimensional reduction of the Carroll-Field-Jackiw-Higgs model. The Lorentz-violating parameter induces a pronounced behavior at origin (for the magnetic/electric fields and energy density) which is absent in the MCSH vortices. For some combination of the Lorentz-violating coefficients there always exists a sufficiently large winding number n 0 such that for all vertical stroke n vertical stroke ≥ vertical stroke n 0 vertical stroke the magnetic field flips sign, yielding two well-defined regions with opposite magnetic flux. However, the total magnetic flux remains quantized and proportional to the winding number. (orig.)
9. Tests of Lorentz violation in νμ→νe oscillations
International Nuclear Information System (INIS)
Auerbach, L.B.; Burman, R.L.; Donahue, J.B.; Garvey, G.T.; Louis, W.C.; Mills, G.B.; Sandberg, V.D.; White, D.H.; Caldwell, D.O.; Yellin, S.; Church, E.D.; McIlhany, K.L.; Strossman, W.H.; Cochran, A.K.; Fazely, A.R.; Gunasingha, R.; Imlay, R.L.; Metcalf, W.J.; Sung, M.; Katori, T.
2005-01-01
A recently developed standard-model extension (SME) formalism for neutrino oscillations that includes Lorentz and CPT violation is used to analyze the sidereal time variation of the neutrino event excess measured by the liquid scintillator neutrino detector (LSND) experiment. The LSND experiment, performed at Los Alamos National Laboratory, observed an excess, consistent with neutrino oscillations, of ν e in a beam of ν μ . It is determined that the LSND oscillation signal is consistent with no sidereal variation. However, there are several combinations of SME coefficients that describe the LSND data; both with and without sidereal variations. The scale of Lorentz and CPT violation extracted from the LSND data is of order 10 -19 GeV for the SME coefficients a L and Exc L . This solution for Lorentz and CPT violating neutrino oscillations may be tested by other short baseline neutrino oscillation experiments, such as the MiniBooNE experiment
10. Bounds on Cubic Lorentz-Violating Terms in the Fermionic Dispersion Relation
OpenAIRE
Bertolami, O.; Rosa, J. G.
2004-01-01
We study the recently proposed Lorentz-violating dispersion relation for fermions and show that it leads to two distinct cubic operators in the momentum. We compute the leading order terms that modify the non-relativistic equations of motion and use experimental results for the hyperfine transition in the ground state of the ${}^9\\textrm Be^+$ ion to bound the values of the Lorentz-violating parameters $\\eta_1$ and $\\eta_2$ for neutrons. The resulting bounds depend on the value of the Lorenz-...
11. A CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
International Nuclear Information System (INIS)
Ferreira Junior, Manoel; Casana, M.R.; Santos, Frederico E.P. dos; Silva, E.O.; Passos, E.
2013-01-01
Full text: The Standard Model Extension (SME) has been the usual framework for investigating signals of Lorentz violation in physical systems. It is the natural framework for studying properties of physical systems with Lorentz-violation since it includes Lorentz-violating terms in all sectors of the minimal standard model. The Lorentz-violating (LV) terms are generated as vacuum expectation values of tensors defined in a high energy scale. This framework has inspired a great deal of investigation in recent years. Such works encompass several distinct aspects involving fermion systems and radiative corrections, CPT- probing experiments, the electromagnetic CPT- and Lorentz-odd term, the 19 electromagnetic CPT-even coefficients. Recently, some studies involving higher dimensional operators have also been reported with great interest, including nonminimal interactions. These many contributions have elucidated the effects induced by Lorentz violation and served to set up stringent upper bounds on the LV coefficients. In the present work, we propose a new CPT-even, dimension-five, nonminimal coupling linking the fermionic and gauge fields in the context of the Dirac equation, involving the CPT-even tensor of the gauge term of the SME. By considering the nonrelativistic limit of the modified Dirac equation, we explicitly evaluate the new contributions to the nonrelativistic Hamiltonian. These new terms imply a direct correction on the anomalous magnetic moment, a kind of electrical Zeeman-like effect on the atomic spectrum, and a Rashba-like coupling term. These effects are then used to impose upper bounds on the magnitude of the non minimally coupled LV coefficients at the level of 1 part in 10 16 . (author)
12. A CPT-even and Lorentz-Violating nonminimal coupling in the Dirac equation
Energy Technology Data Exchange (ETDEWEB)
Ferreira Junior, Manoel; Casana, M.R.; Santos, Frederico E.P. dos; Silva, E.O. [UFMA, Sao Luis (Brazil); Passos, E. [UFCG, Campina Grande, PB (Brazil)
2013-07-01
Full text: The Standard Model Extension (SME) has been the usual framework for investigating signals of Lorentz violation in physical systems. It is the natural framework for studying properties of physical systems with Lorentz-violation since it includes Lorentz-violating terms in all sectors of the minimal standard model. The Lorentz-violating (LV) terms are generated as vacuum expectation values of tensors defined in a high energy scale. This framework has inspired a great deal of investigation in recent years. Such works encompass several distinct aspects involving fermion systems and radiative corrections, CPT- probing experiments, the electromagnetic CPT- and Lorentz-odd term, the 19 electromagnetic CPT-even coefficients. Recently, some studies involving higher dimensional operators have also been reported with great interest, including nonminimal interactions. These many contributions have elucidated the effects induced by Lorentz violation and served to set up stringent upper bounds on the LV coefficients. In the present work, we propose a new CPT-even, dimension-five, nonminimal coupling linking the fermionic and gauge fields in the context of the Dirac equation, involving the CPT-even tensor of the gauge term of the SME. By considering the nonrelativistic limit of the modified Dirac equation, we explicitly evaluate the new contributions to the nonrelativistic Hamiltonian. These new terms imply a direct correction on the anomalous magnetic moment, a kind of electrical Zeeman-like effect on the atomic spectrum, and a Rashba-like coupling term. These effects are then used to impose upper bounds on the magnitude of the non minimally coupled LV coefficients at the level of 1 part in 10{sub 16}. (author)
13. Test of CPT and Lorentz symmetry in entangled neutral kaons with the KLOE experiment
Energy Technology Data Exchange (ETDEWEB)
Babusci, D. [Laboratori Nazionali di Frascati dell' INFN, Frascati (Italy); Balwierz-Pytko, I. [Institute of Physics, Jagiellonian University, Cracow (Poland); Bencivenni, G.; Bloise, C.; Bossi, F. [Laboratori Nazionali di Frascati dell' INFN, Frascati (Italy); Branchini, P. [INFN Sezione di Roma Tre, Roma (Italy); Budano, A. [Dipartimento di Matematica e Fisica dell' Università “Roma Tre”, Roma (Italy); INFN Sezione di Roma Tre, Roma (Italy); Caldeira Balkeståhl, L. [Department of Physics and Astronomy, Uppsala University, Uppsala (Sweden); Capon, G. [Laboratori Nazionali di Frascati dell' INFN, Frascati (Italy); Ceradini, F. [Dipartimento di Matematica e Fisica dell' Università “Roma Tre”, Roma (Italy); INFN Sezione di Roma Tre, Roma (Italy); Ciambrone, P. [Laboratori Nazionali di Frascati dell' INFN, Frascati (Italy); Curciarello, F. [Dipartimento di Fisica e Scienze della Terra dell' Università di Messina, Messina (Italy); INFN Sezione di Catania, Catania (Italy); Czerwiński, E. [Institute of Physics, Jagiellonian University, Cracow (Poland); Danè, E. [Laboratori Nazionali di Frascati dell' INFN, Frascati (Italy); De Leo, V. [Dipartimento di Fisica e Scienze della Terra dell' Università di Messina, Messina (Italy); INFN Sezione di Catania, Catania (Italy); De Lucia, E. [Laboratori Nazionali di Frascati dell' INFN, Frascati (Italy); De Robertis, G. [INFN Sezione di Bari, Bari (Italy); De Santis, A., E-mail: [email protected] [Dipartimento di Fisica dell' Università “Sapienza”, Roma (Italy); INFN Sezione di Roma, Roma (Italy); De Simone, P. [Laboratori Nazionali di Frascati dell' INFN, Frascati (Italy); and others
2014-03-07
Neutral kaon pairs produced in ϕ decays in anti-symmetric entangled state can be exploited to search for violation of CPT symmetry and Lorentz invariance. We present an analysis of the CP-violating process ϕ→K{sub S}K{sub L}→π{sup +}π{sup −}π{sup +}π{sup −} based on 1.7 fb{sup −1} of data collected by the KLOE experiment at the Frascati ϕ-factory DAΦNE. The data are used to perform a measurement of the CPT-violating parameters Δa{sub μ} for neutral kaons in the context of the Standard Model Extension framework. The parameters measured in the reference frame of the fixed stars are: Δa{sub 0}=(−6.0±7.7{sub stat}±3.1{sub syst})×10{sup −18} GeV, Δa{sub X}=(0.9±1.5{sub stat}±0.6{sub syst})×10{sup −18} GeV, Δa{sub Y}=(−2.0±1.5{sub stat}±0.5{sub syst})×10{sup −18} GeV, Δa{sub Z}=(3.1±1.7{sub stat}±0.5{sub syst})×10{sup −18} GeV. These are presently the most precise measurements in the quark sector of the Standard Model Extension.
14. Hyperscaling violation and electroweak symmetry breaking
Energy Technology Data Exchange (ETDEWEB)
Elander, Daniel, E-mail: [email protected] [Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907-2036 (United States); Lawrance, Robert; Piai, Maurizio [Department of Physics, College of Science, Swansea University, Singleton Park, Swansea, Wales (United Kingdom)
2015-08-15
We consider a class of simplified models of dynamical electroweak symmetry breaking built in terms of their five-dimensional weakly-coupled gravity duals, in the spirit of bottom-up holography. The sigma-model consists of two abelian gauge bosons and one real, non-charged scalar field coupled to gravity in five dimensions. The scalar potential is a simple exponential function of the scalar field. The background metric resulting from solving the classical equations of motion exhibits hyperscaling violation, at least at asymptotically large values of the radial direction. We study the spectrum of scalar composite states of the putative dual field theory by fluctuating the sigma-model scalars and gravity, and discuss in which cases we find a parametrically light scalar state in the spectrum. We model the spontaneous breaking of the (weakly coupled) gauge symmetry to the diagonal subgroup by the choice of IR boundary conditions. We compute the mass spectrum of spin-1 states, and the precision electroweak parameter S as a function of the hyperscaling coefficient. We find a general bound on the mass of the lightest spin-1 resonance, by requiring that the indirect bounds on the precision parameters be satisfied, that implies that precision electroweak physics excludes the possibility of a techni-rho meson with mass lighter than several TeV.
15. Experimental tests of charge symmetry violation in parton distributions
International Nuclear Information System (INIS)
Londergan, J.T.; Murdock, D.P.; Thomas, A.W.
2005-01-01
Recently, a global phenomenological fit to high energy data has included charge symmetry breaking terms, leading to limits on the allowed magnitude of such effects. We discuss two possible experiments that could search for isospin violation in valence parton distributions. We show that, given the magnitude of charge symmetry violation consistent with existing global data, such experiments might expect to see effects at a level of several percent. Alternatively, such experiments could significantly decrease the upper limits on isospin violation in parton distributions
16. From symmetry violation to dynamics: The charm window
International Nuclear Information System (INIS)
Appel, J.A.
1997-12-01
C.S. Wu observed parity violation in the low energy process of nuclear decay. She was the first to observe this symmetry violation at any energy. Yet, her work taught us about the form and strengths of the couplings of the massive weak boson. Today, we use the same approach. We look for very much higher mass-scale interactions through symmetry violations in the decays of charm quark systems. These charm decays provide a unique window to new physics
17. Lorentz-violating vortex solutions in the CPT-even electrodynamics of the Standard Model Extension
International Nuclear Information System (INIS)
Casana, Rodolfo; Ferreira Junior, Manoel M.; Hora, E. da
2011-01-01
Full text: In this work, we investigate the formation of static rotationally symmetric solutions on the (1+3) dimensional CPT-even and Lorentz-violating photonic sector of the Standard Model Extension (SME). The main goal of this work is to show the possibility of obtaining these solutions, even in the presence of Lorentz-breaking fields. A secondary goal is to examine the effects of these fields on topologically non-trivial configurations. In order to obtain these results, we focus on specific components of Lorentz-violating background, dealing with static Euler-Lagrange equations, from which we fix temporal gauge (absence of electric field) as a proper gauge choice. We assume the usual rotationally symmetric Ansatz, inserting it in the Euler-Lagrange equations previously obtained. This Ansatz describes the Higgs and gauge fields via profile functions g(r) and a(r), respectively. From this Ansatz, we construct suitable boundary conditions near the origin. Also, we write the energy density in terms of these two profile functions, obtaining from it asymptotic boundary conditions. This set of conditions is used to numerically solve the Euler-Lagrange equations (by means of the shooting method). Finally, we plot solutions for some physical quantities (Higgs field, magnetic field and energy density) for several values of the Lorentz-violating parameters. From these plots, we discuss the influence of these coefficients on the topologically non-trivial rotationally symmetric configurations, focusing on the profiles of both magnetic field and energy density. (author)
18. Late-time acceleration and phantom divide line crossing with non-minimal coupling and Lorentz-invariance violation
International Nuclear Information System (INIS)
2008-01-01
We consider two alternative dark-energy models: a Lorentz-invariance preserving model with a non-minimally coupled scalar field and a Lorentz-invariance violating model with a minimally coupled scalar field. We study accelerated expansion and the dynamics of the equation of state parameter in these scenarios. While a minimally coupled scalar field does not have the capability to be a successful dark-energy candidate with line crossing of the cosmological constant, a non-minimally coupled scalar field in the presence of Lorentz invariance or a minimally coupled scalar field with Lorentz-invariance violation have this capability. In the latter case, accelerated expansion and phantom divide line crossing are the results of the interactive nature of this Lorentz-violating scenario. (orig.)
19. Essay on gravitation: The cosmological constant problem in brane-worlds and gravitational Lorentz violations
International Nuclear Information System (INIS)
Csaki, Csaba; Erlich, Joshua; Grojean, Christophe
2001-01-01
Brane worlds are theories with extra spatial dimensions in which ordinary matter is localized on a (3+1) dimensional submanifold. Such theories could have interesting consequences for particle physics and gravitational physics. In this essay we concentrate on the cosmological constant (CC) problem in the context of brane worlds. We show how extra-dimensional scenarios may violate Lorentz invariance in the gravity sector of the effective 4D theory, while particle physics remains unaffected. In such theories the usual no-go theorems for adjustment of the CC do not apply, and we indicate a possible explanation of the smallness of the CC. Lorentz violating effects would manifest themselves in gravitational waves travelling with a speed different from light, which can be searched for in gravitational wave experiments
20. Search for violation of Lorentz invariance in top quark pair production and decay
Czech Academy of Sciences Publication Activity Database
Abazov, V. M.; Abbott, B.; Acharya, B.S.; Kupčo, Alexander; Lokajíček, Miloš
2012-01-01
Roč. 108, č. 26 (2012), "261603-1"-"261603-7" ISSN 0031-9007 R&D Projects: GA MŠk LA08047; GA MŠk(CZ) LG12006 Institutional research plan: CEZ:AV0Z10100502 Keywords : D0 * violation Lorentz * pair productionl * decay * Batavia TEVATRON, Subject RIV: BF - Elementary Particles and High Energy Physics Impact factor: 7.943, year: 2012 http://prl.aps.org/abstract/PRL/v108/i26/e261603
1. Search for Violation of $CPT$ and Lorentz invariance in ${B_s^0}$ meson oscillations
CERN Document Server
2015-10-14
We present the first search for CPT-violating effects in the mixing of ${B_s^0}$ mesons using the full Run II data set with an integrated luminosity of 10.4 fb$^{-1}$ of proton-antiproton collisions collected using the D0 detector at the Fermilab Tevatron Collider. We measure the CPT-violating asymmetry in the decay $B_s^0 \\to \\mu^\\pm D_s^\\pm$ as a function of celestial direction and sidereal phase. We find no evidence for CPT-violating effects and place limits on the direction and magnitude of flavor-dependent CPT- and Lorentz-invariance violating coupling coefficients. We find 95\\% confidence intervals of $\\Delta a_{\\perp} < 1.2 \\times 10^{-12}$ GeV and $(-0.8 < \\Delta a_T - 0.396 \\Delta a_Z < 3.9) \\times 10^{-13}$ GeV.
2. A New Limit on Planck Scale Lorentz Violation from Gamma-ray Burst Polarization
Science.gov (United States)
Stecker, Floyd W.
2011-01-01
Constraints on possible Lorentz invariance violation (UV) to first order in E/M(sub Plank) for photons in the framework of effective field theory (EFT) are discussed, taking cosmological factors into account. Then. using the reported detection of polarized soft gamma-ray emission from the gamma-ray burst GRB041219a that is indicative' of an absence of vacuum birefringence, together with a very recent improved method for estimating the redshift of the burst, we derive constraints on the dimension 5 Lorentz violating modification to the Lagrangian of an effective local QFT for QED. Our new constraints are more than five orders of magnitude better than recent constraints from observations of the Crab Nebula.. We obtain the upper limit on the Lorentz violating dimension 5 EFT parameter absolute value of zeta of 2.4 x 10(exp -15), corresponding to a constraint on the dimension 5 standard model extension parameter. Kappa (sup 5) (sub (v)oo) much less than 4.2 X 10(exp -3)4 / GeV.
3. Lorentz invariance violation and electromagnetic field in an intrinsically anisotropic spacetime
Energy Technology Data Exchange (ETDEWEB)
Chang, Zhe [Chinese Academy of Sciences, Institute of High Energy Physics, Beijing (China); Chinese Academy of Sciences, Theoretical Physics Center for Science Facilities, Beijing (China); Wang, Sai [Chinese Academy of Sciences, Institute of High Energy Physics, Beijing (China)
2012-09-15
Recently, Kostelecky [V.A. Kostelecky, Phys. Lett. B 701, 137 (2011)] proposed that the spontaneous Lorentz invariance violation (sLIV) is related to Finsler geometry. Finsler spacetime is intrinsically anisotropic and naturally induces Lorentz invariance violation (LIV). In this paper, the electromagnetic field is investigated in locally Minkowski spacetime. The Lagrangian is presented explicitly for the electromagnetic field. It is compatible with the one in the standard model extension (SME). We show the Lorentz-violating Maxwell equations as well as the electromagnetic wave equation. The formal plane wave solution is obtained for the electromagnetic wave. The speed of light may depend on the direction of light and the lightcone may be enlarged or narrowed. The LIV effects could be viewed as influence from an anisotropic media on the electromagnetic wave. In addition, birefringence of light will not emerge at the leading order in this model. A constraint on the spacetime anisotropy is obtained from observations on gamma-ray bursts (GRBs). (orig.)
4. Ultra-large distance modification of gravity from Lorentz symmetry breaking at the Planck scale
International Nuclear Information System (INIS)
Gorbunov, Dmitry S.; Sibiryakov, Sergei M.
2005-01-01
We present an extension of the Randall-Sundrum model in which, due to spontaneous Lorentz symmetry breaking, graviton mixes with bulk vector fields and becomes quasilocalized. The masses of KK modes comprising the four-dimensional graviton are naturally exponentially small. This allows to push the Lorentz breaking scale to as high as a few tenth of the Planck mass. The model does not contain ghosts or tachyons and does not exhibit the van Dam-Veltman-Zakharov discontinuity. The gravitational attraction between static point masses becomes gradually weaker with increasing of separation and gets replaced by repulsion (antigravity) at exponentially large distances
5. Ultra-large distance modification of gravity from Lorentz symmetry breaking at the Planck scale
Energy Technology Data Exchange (ETDEWEB)
Gorbunov, Dmitry S. [Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary prospect, 7a, 117312 Moscow (Russian Federation); Sibiryakov, Sergei M. [Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary prospect, 7a, 117312 Moscow (Russian Federation)
2005-09-15
We present an extension of the Randall-Sundrum model in which, due to spontaneous Lorentz symmetry breaking, graviton mixes with bulk vector fields and becomes quasilocalized. The masses of KK modes comprising the four-dimensional graviton are naturally exponentially small. This allows to push the Lorentz breaking scale to as high as a few tenth of the Planck mass. The model does not contain ghosts or tachyons and does not exhibit the van Dam-Veltman-Zakharov discontinuity. The gravitational attraction between static point masses becomes gradually weaker with increasing of separation and gets replaced by repulsion (antigravity) at exponentially large distances.
6. Search for Violation of CPT and Lorentz Invariance in $B^0_s$ Meson Oscillations using the D0 Detector
Energy Technology Data Exchange (ETDEWEB)
Van Kooten, R. [Indiana U.
2017-01-01
A search is presented for CPT-violating effects in the mixing of $B^0_s$ mesons using the D0 detector at the Fermilab Tevatron Collider. The CPT-violating asymmetry in the decay $B^0_s \\rightarrow \\mu^{\\pm} D_s^{\\mp} X$ as a function of sidereal phase is measured. No evidence for CPT-violating effects is observed and limits are placed on CPT- and Lorentz-invariance violating coupling coefficients.
7. A Measurement of the muon neutrino charged current quasielastic interaction and a test of Lorentz violation with the MiniBooNE experiment
Energy Technology Data Exchange (ETDEWEB)
Katori, Teppei [Indiana Univ., Bloomington, IN (United States)
2008-12-01
The Mini-Booster neutrino experiment (MiniBooNE) at Fermi National Accelerator Laboratory (Fermilab) is designed to search for vμ → ve appearance neutrino oscillations. Muon neutrino charged-current quasi-elastic (CCQE) interactions (vμ + n → μ + p) make up roughly 40% of our data sample, and it is used to constrain the background and cross sections for the oscillation analysis. Using high-statistics MiniBooNE CCQE data, the muon-neutrino CCQE cross section is measured. The nuclear model is tuned precisely using the MiniBooNE data. The measured total cross section is σ = (1.058 ± 0.003 (stat) ± 0.111 (syst)) x 10-38 cm2 at the MiniBooNE muon neutrino beam energy (700-800 MeV). ve appearance candidate data is also used to search for Lorentz violation. Lorentz symmetry is one of the most fundamental symmetries in modern physics. Neutrino oscillations offer a new method to test it. We found that the MiniBooNE result is not well-described using Lorentz violation, however further investigation is required for a more conclusive result.
8. Exact Lorentz-violating all-loop ultraviolet divergences in scalar field theories
Energy Technology Data Exchange (ETDEWEB)
Carvalho, P.R.S. [Universidade Federal do Piaui, Departamento de Fisica, Teresina, PI (Brazil); Sena-Junior, M.I. [Universidade de Pernambuco, Escola Politecnica de Pernambuco, Recife, PE (Brazil); Universidade Federal de Alagoas, Instituto de Fisica, Maceio, AL (Brazil)
2017-11-15
In this work we evaluate analytically the ultraviolet divergences of Lorentz-violating massive O(N) λφ{sup 4} scalar field theories, which are exact in the Lorentz-violating mechanism, firstly explicitly at next-to-leading order and latter at any loop level through an induction procedure based on a theorem following from the exact approach, for computing the corresponding critical exponents. For attaining that goal, we employ three different and independent field-theoretic renormalization group methods. The results found for the critical exponents show that they are identical in the three distinct methods and equal to their Lorentz-invariant counterparts. Furthermore, we show that the results obtained here, based on the single concept of loop order of the referred terms of the corresponding β-function and anomalous dimensions, reduce to the ones obtained through the earlier non-exact approach based on a joint redefinition of the field and coupling constant of the theory, in the appropriate limit. (orig.)
9. Tests of local Lorentz invariance violation of gravity in the standard model extension with pulsars.
Science.gov (United States)
Shao, Lijing
2014-03-21
The standard model extension is an effective field theory introducing all possible Lorentz-violating (LV) operators to the standard model and general relativity (GR). In the pure-gravity sector of minimal standard model extension, nine coefficients describe dominant observable deviations from GR. We systematically implemented 27 tests from 13 pulsar systems to tightly constrain eight linear combinations of these coefficients with extensive Monte Carlo simulations. It constitutes the first detailed and systematic test of the pure-gravity sector of minimal standard model extension with the state-of-the-art pulsar observations. No deviation from GR was detected. The limits of LV coefficients are expressed in the canonical Sun-centered celestial-equatorial frame for the convenience of further studies. They are all improved by significant factors of tens to hundreds with existing ones. As a consequence, Einstein's equivalence principle is verified substantially further by pulsar experiments in terms of local Lorentz invariance in gravity.
10. Spontaneous breaking of Lorentz symmetry by ghost condensation in perturbative quantum gravity
Science.gov (United States)
Faizal, Mir
2011-10-01
In this paper, we will study the spontaneous breakdown of the Lorentz symmetry by ghost condensation in perturbative quantum gravity. Our analysis will be done in the Curci-Ferrari gauge. We will also analyse the modification of the BRST and anti-BRST transformations by the formation of this ghost condensate. It will be shown that even though the modified BRST and anti-BRST transformations are not nilpotent, their nilpotency is restored on-shell.
11. Dimensional reduction of a Lorentz and CPT-violating Maxwell-Chern-Simons model
Energy Technology Data Exchange (ETDEWEB)
Belich, H. Jr.; Helayel Neto, J.A. [Centro Brasileiro de Pesquisas Fisicas (CBPF), Rio de Janeiro, RJ (Brazil). Coordenacao de Teoria de Campos e Particulas; Grupo de Fisica Teorica Jose Leite Lopes, Petropolis, RJ (Brazil); E-mails: [email protected]; [email protected]; Ferreira, M.M. Jr. [Grupo de Fisica Teorica Jose Leite Lopes, Petropolis, RJ (Brazil); Maranhao Univ., Sao Luiz, MA (Brazil). Dept. de Fisica]. E-mail: [email protected]; Orlando, M.T.D. [Grupo de Fisica Teorica Jose Leite Lopes, Petropolis, RJ (Brazil); Espirito Santo Univ., Vitoria, ES (Brazil). Dept. de Fisica e Quimica; E-mail: [email protected]
2003-01-01
Taking as starting point a Lorentz and CPT non-invariant Chern-Simons-like model defined in 1+3 dimensions, we proceed realizing its dimensional to D = 1+2. One then obtains a new planar model, composed by the Maxwell-Chern-Simons (MCS) sector, a Klein-Gordon massless scalar field, and a coupling term that mixes the gauge field to the external vector, {nu}{sup {mu}}. In spite of breaking Lorentz invariance in the particle frame, this model may preserve the CPT symmetry for a single particular choice of {nu}{sup {mu}} . Analyzing the dispersion relations, one verifies that the reduced model exhibits stability, but the causality can be jeopardized by some modes. The unitary of the gauge sector is assured without any restriction , while the scalar sector is unitary only in the space-like case. (author)
12. Dimensional reduction of a Lorentz and CPT-violating Maxwell-Chern-Simons model
International Nuclear Information System (INIS)
Belich, H. Jr.; Helayel Neto, J.A.; Ferreira, M.M. Jr.; Maranhao Univ., Sao Luiz, MA; Orlando, M.T.D.; Espirito Santo Univ., Vitoria, ES
2003-01-01
Taking as starting point a Lorentz and CPT non-invariant Chern-Simons-like model defined in 1+3 dimensions, we proceed realizing its dimensional to D = 1+2. One then obtains a new planar model, composed by the Maxwell-Chern-Simons (MCS) sector, a Klein-Gordon massless scalar field, and a coupling term that mixes the gauge field to the external vector, ν μ . In spite of breaking Lorentz invariance in the particle frame, this model may preserve the CPT symmetry for a single particular choice of ν μ . Analyzing the dispersion relations, one verifies that the reduced model exhibits stability, but the causality can be jeopardized by some modes. The unitary of the gauge sector is assured without any restriction , while the scalar sector is unitary only in the space-like case. (author)
13. Quantization of Space-like States in Lorentz-Violating Theories
Science.gov (United States)
2018-01-01
Lorentz violation frequently induces modified dispersion relations that can yield space-like states that impede the standard quantization procedures. In certain cases, an extended Hamiltonian formalism can be used to define observer-covariant normalization factors for field expansions and phase space integrals. These factors extend the theory to include non-concordant frames in which there are negative-energy states. This formalism provides a rigorous way to quantize certain theories containing space-like states and allows for the consistent computation of Cherenkov radiation rates in arbitrary frames and avoids singular expressions.
14. Lorentz-violating Yang-Mills theory. Discussing the Chern-Simons-like term generation
Energy Technology Data Exchange (ETDEWEB)
Santos, Tiago R.S.; Sobreiro, Rodrigo F. [UFF-Universidade Federal Fluminense, Instituto de Fisica, Niteroi, RJ (Brazil)
2017-12-15
We analyze the Chern-Simons-like term generation in the CPT-odd Lorentz-violating Yang-Mills theory interacting with fermions. Moreover, we study the anomalies of this model as well as its quantum stability. The whole analysis is performed within the algebraic renormalization theory, which is independent of the renormalization scheme. In addition, all results are valid to all orders in perturbation theory. We find that the Chern-Simons-like term is not generated by radiative corrections, just like its Abelian version. Additionally, the model is also free of gauge anomalies and quantum stable. (orig.)
15. Classical kinematics and Finsler structures for nonminimal Lorentz-violating fermions
Energy Technology Data Exchange (ETDEWEB)
Schreck, M. [Indiana University, Indiana University Center for Spacetime Symmetries, Bloomington, IN (United States)
2015-05-15
In the current paper the Lagrangian of a classical, relativistic point particle is obtained whose conjugate momentum satisfies the dispersion relation of a quantum wave packet that is subject to Lorentz violation based on a particular coefficient of the nonminimal standard-model extension (SME). The properties of this Lagrangian are analyzed and two corresponding Finsler structures are obtained. One structure describes a scaled Euclidean geometry, whereas the other is neither a Riemann nor a Randers or Kropina structure. The results of the article provide some initial understanding of classical Lagrangians of the nonminimal SME fermion sector. (orig.)
16. Classical kinematics and Finsler structures for nonminimal Lorentz-violating fermions
International Nuclear Information System (INIS)
Schreck, M.
2015-01-01
In the current paper the Lagrangian of a classical, relativistic point particle is obtained whose conjugate momentum satisfies the dispersion relation of a quantum wave packet that is subject to Lorentz violation based on a particular coefficient of the nonminimal standard-model extension (SME). The properties of this Lagrangian are analyzed and two corresponding Finsler structures are obtained. One structure describes a scaled Euclidean geometry, whereas the other is neither a Riemann nor a Randers or Kropina structure. The results of the article provide some initial understanding of classical Lagrangians of the nonminimal SME fermion sector. (orig.)
17. Local effects of the quantum vacuum in Lorentz-violating electrodynamics
Science.gov (United States)
Martín-Ruiz, A.; Escobar, C. A.
2017-02-01
The Casimir effect is one of the most remarkable consequences of the nonzero vacuum energy predicted by quantum field theory. In this paper we use a local approach to study the Lorentz violation effects of the minimal standard model extension on the Casimir force between two parallel conducting plates in the vacuum. Using a perturbative method similar to that used for obtaining the Born series for the scattering amplitudes in quantum mechanics, we compute, at leading order in the Lorentz-violating coefficients, the relevant Green's function which satisfies given boundary conditions. The standard point-splitting technique allow us to express the vacuum expectation value of the stress-energy tensor in terms of the Green's function. We discuss its structure in the region between the plates. We compute the renormalized vacuum stress, which is obtained as the difference between the vacuum stress in the presence of the plates and that of the vacuum. The Casimir force is evaluated in an analytical fashion by two methods: by differentiating the renormalized global energy density and by computing the normal-normal component of the renormalized vacuum stress. We compute the local Casimir energy, which is found to diverge as approaching the plates, and we demonstrate that it does not contribute to the observable force.
18. Test of Lorentz and CPT violation with short baseline neutrino oscillation excesses
Energy Technology Data Exchange (ETDEWEB)
Aguilar-Arevalo, A.A. [Instituto de Ciencias Nucleares, Universidad Nacional Autonoma de Mexico, D.F. 04510 (Mexico); Anderson, C.E. [Yale University, New Haven, CT 06520 (United States); Bazarko, A.O. [Princeton University, Princeton, NJ 08544 (United States); Brice, S.J.; Brown, B.C. [Fermi National Accelerator Laboratory, Batavia, IL 60510 (United States); Bugel, L. [Massachusetts Institute of Technology, Cambridge, MA 02139 (United States); Cao, J. [University of Michigan, Ann Arbor, MI 48109 (United States); Coney, L. [Columbia University, New York, NY 10027 (United States); Conrad, J.M. [Massachusetts Institute of Technology, Cambridge, MA 02139 (United States); Cox, D.C. [Indiana University, Bloomington, IN 47405 (United States); Curioni, A. [Yale University, New Haven, CT 06520 (United States); Dharmapalan, R. [University of Alabama, Tuscaloosa, AL 35487 (United States); Djurcic, Z. [Argonne National Laboratory, Argonne, IL 60439 (United States); Finley, D.A. [Fermi National Accelerator Laboratory, Batavia, IL 60510 (United States); Fleming, B.T. [Yale University, New Haven, CT 06520 (United States); Ford, R.; Garcia, F.G. [Fermi National Accelerator Laboratory, Batavia, IL 60510 (United States); Garvey, G.T. [Los Alamos National Laboratory, Los Alamos, NM 87545 (United States); Grange, J. [University of Florida, Gainesville, FL 32611 (United States); Green, C. [Fermi National Accelerator Laboratory, Batavia, IL 60510 (United States); Los Alamos National Laboratory, Los Alamos, NM 87545 (United States); and others
2013-01-29
The sidereal time dependence of MiniBooNE {nu}{sub e} and {nu}{sup Macron }{sub e} appearance data is analyzed to search for evidence of Lorentz and CPT violation. An unbinned Kolmogorov-Smirnov (K-S) test shows both the {nu}{sub e} and {nu}{sup Macron }{sub e} appearance data are compatible with the null sidereal variation hypothesis to more than 5%. Using an unbinned likelihood fit with a Lorentz-violating oscillation model derived from the Standard Model Extension (SME) to describe any excess events over background, we find that the {nu}{sub e} appearance data prefer a sidereal time-independent solution, and the {nu}{sup Macron }{sub e} appearance data slightly prefer a sidereal time-dependent solution. Limits of order 10{sup -20} GeV are placed on combinations of SME coefficients. These limits give the best limits on certain SME coefficients for {nu}{sub {mu}}{yields}{nu}{sub e} and {nu}{sup Macron }{sub {mu}}{yields}{nu}{sup Macron }{sub e} oscillations. The fit values and limits of combinations of SME coefficients are provided.
19. Test of Lorentz and CPT violation with short baseline neutrino oscillation excesses
International Nuclear Information System (INIS)
Aguilar-Arevalo, A.A.; Anderson, C.E.; Bazarko, A.O.; Brice, S.J.; Brown, B.C.; Bugel, L.; Cao, J.; Coney, L.; Conrad, J.M.; Cox, D.C.; Curioni, A.; Dharmapalan, R.; Djurcic, Z.; Finley, D.A.; Fleming, B.T.; Ford, R.; Garcia, F.G.; Garvey, G.T.; Grange, J.; Green, C.
2013-01-01
The sidereal time dependence of MiniBooNE ν e and ν ¯ e appearance data is analyzed to search for evidence of Lorentz and CPT violation. An unbinned Kolmogorov–Smirnov (K–S) test shows both the ν e and ν ¯ e appearance data are compatible with the null sidereal variation hypothesis to more than 5%. Using an unbinned likelihood fit with a Lorentz-violating oscillation model derived from the Standard Model Extension (SME) to describe any excess events over background, we find that the ν e appearance data prefer a sidereal time-independent solution, and the ν ¯ e appearance data slightly prefer a sidereal time-dependent solution. Limits of order 10 −20 GeV are placed on combinations of SME coefficients. These limits give the best limits on certain SME coefficients for ν μ →ν e and ν ¯ μ →ν ¯ e oscillations. The fit values and limits of combinations of SME coefficients are provided.
20. Generation of geometrical phases and persistent spin currents in 1-dimensional rings by Lorentz-violating terms
Energy Technology Data Exchange (ETDEWEB)
Casana, R.; Ferreira, M.M., E-mail: [email protected]; Mouchrek-Santos, V.E.; Silva, Edilberto O.
2015-06-30
We have demonstrated that Lorentz-violating terms stemming from the fermion sector of the SME are able to generate geometrical phases on the wave function of electrons confined in 1-dimensional rings, as well as persistent spin currents, in the total absence of electromagnetic fields. We have explicitly evaluated the eigenenergies and eigenspinors of the electrons modified by the Lorentz-violating terms, using them to calculate the dynamic and the Aharonov–Anandan phases in the sequel. The total phase presents a pattern very similar to the Aharonov–Casher phase accumulated by electrons in rings under the action of the Rashba interaction. Finally, the persistent spin current were carried out and used to impose upper bounds on the Lorentz-violating parameters.
1. Aspects of quantum corrections in a Lorentz-violating extension of the abelian Higgs Model
Energy Technology Data Exchange (ETDEWEB)
Brito, L.C.T.; Fargnoli, H.G. [Universidade Federal de Lavras, MG (Brazil); Scarpelli, A.P. Baeta [Departamento de Policia Federal, Rio de Janeiro, RJ (Brazil)
2013-07-01
Full text: We have investigated new aspects related to the four-dimensional abelian gauge-Higgs model with the addition of the Carroll-Field-Jackiw term (CFJ). We have focused on one-loop quantum corrections to the photon and Higgs sectors and we have analyzed what kind of effects are induced at the quantum level by spontaneous gauge symmetry breaking due the presence of the CFJ term. We have shown that new finite and non-ambiguous Lorentz-breaking terms are induced in both sectors at second order in the background vector. Specifically in the pure gauge sector, a CPT-even aether term (free from ambiguities) is induced. A CPT-even term is also induced in the pure Higgs sector. Both terms have been mapped in the Standard Model Extension. Besides, aspects of the one-loop renormalization of the background vector dependent terms have been studied. The new divergences due the presence of the CFJ term were shown to be worked out by the renormalization condition which requires the vanishing of the vacuum expectation value of the Higgs field. So at one loop the CFJ term does not spoil the well known renormalizability of the model without Lorentz symmetry breaking terms. The calculations have been done within dimensional methods and in an arbitrary gauge choice. (author)
2. Lorentz invariance violation in the neutrino sector: a joint analysis from big bang nucleosynthesis and the cosmic microwave background
Science.gov (United States)
Dai, Wei-Ming; Guo, Zong-Kuan; Cai, Rong-Gen; Zhang, Yuan-Zhong
2017-06-01
We investigate constraints on Lorentz invariance violation in the neutrino sector from a joint analysis of big bang nucleosynthesis and the cosmic microwave background. The effect of Lorentz invariance violation during the epoch of big bang nucleosynthesis changes the predicted helium-4 abundance, which influences the power spectrum of the cosmic microwave background at the recombination epoch. In combination with the latest measurement of the primordial helium-4 abundance, the Planck 2015 data of the cosmic microwave background anisotropies give a strong constraint on the deformation parameter since adding the primordial helium measurement breaks the degeneracy between the deformation parameter and the physical dark matter density.
3. Lorentz invariance violation in the neutrino sector: a joint analysis from big bang nucleosynthesis and the cosmic microwave background
Energy Technology Data Exchange (ETDEWEB)
Dai, Wei-Ming; Cai, Rong-Gen [Chinese Academy of Sciences, CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, P.O. Box 2735, Beijing (China); University of Chinese Academy of Sciences, School of Physical Sciences, Beijing (China); Guo, Zong-Kuan [Chinese Academy of Sciences, CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, P.O. Box 2735, Beijing (China); University of Chinese Academy of Sciences, School of Astronomy and Space Science, Beijing (China); Zhang, Yuan-Zhong [Chinese Academy of Sciences, CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, P.O. Box 2735, Beijing (China)
2017-06-15
We investigate constraints on Lorentz invariance violation in the neutrino sector from a joint analysis of big bang nucleosynthesis and the cosmic microwave background. The effect of Lorentz invariance violation during the epoch of big bang nucleosynthesis changes the predicted helium-4 abundance, which influences the power spectrum of the cosmic microwave background at the recombination epoch. In combination with the latest measurement of the primordial helium-4 abundance, the Planck 2015 data of the cosmic microwave background anisotropies give a strong constraint on the deformation parameter since adding the primordial helium measurement breaks the degeneracy between the deformation parameter and the physical dark matter density. (orig.)
4. Lorentz invariance violation in the neutrino sector: a joint analysis from big bang nucleosynthesis and the cosmic microwave background
International Nuclear Information System (INIS)
Dai, Wei-Ming; Cai, Rong-Gen; Guo, Zong-Kuan; Zhang, Yuan-Zhong
2017-01-01
We investigate constraints on Lorentz invariance violation in the neutrino sector from a joint analysis of big bang nucleosynthesis and the cosmic microwave background. The effect of Lorentz invariance violation during the epoch of big bang nucleosynthesis changes the predicted helium-4 abundance, which influences the power spectrum of the cosmic microwave background at the recombination epoch. In combination with the latest measurement of the primordial helium-4 abundance, the Planck 2015 data of the cosmic microwave background anisotropies give a strong constraint on the deformation parameter since adding the primordial helium measurement breaks the degeneracy between the deformation parameter and the physical dark matter density. (orig.)
5. Overview: Parity Violation and Fundamental Symmetries
Science.gov (United States)
Carlini, Roger
2017-09-01
The fields of nuclear and particle physics have undertaken extensive programs of research to search for evidence of new phenomena via the precision measurement of observables that are well predicted within the standard model of electroweak interaction. It is already known that the standard model is incomplete as it does not include gravity and dark matter/energy and therefore likely the low energy approximation of a more complex theory. This talk will be an overview of the motivation, experimental methods and status of some of these efforts (past and future) related to precision in-direct searches that are complementary to the direct searches underway at the Large Hadron Collider. This abstract is for the invited talk associated with the Mini-symposium titled Electro-weak Physics and Fundamental Symmetries'' organized by Julie Roche.
6. Lorentz-violating contributions to the nuclear Schiff moment and nuclear EDM
Science.gov (United States)
Araujo, Jonas B.; Casana, Rodolfo; Ferreira, Manoel M.
2018-03-01
In the context of an atom endowed with nuclear electric dipole moments (EDM), we consider the effects on the Schiff moment of C P T -even Lorentz-violating (LV) terms that modify the Coulomb potential. First, we study the modifications on the Schiff moment when the nucleus interacts with the electronic cloud by means of a Coulomb potential altered only by the P -even LV components. Next, by supposing the existence of an additional intrinsic LV EDM generated by other LV sources, we assess the corrections to the Schiff moment when the interaction nucleus-electrons runs mediated by a Coulomb potential modified by both the P -odd and P -even LV components. We then use known estimates and EDM measurements to discuss upper bounds on the new Schiff moment components and the possibility of a nuclear EDM component ascribed to LV effects.
7. Constraints on spacetime anisotropy and Lorentz violation from the GRAAL experiment
Energy Technology Data Exchange (ETDEWEB)
Chang, Zhe [Chinese Academy of Sciences, Institute of High Energy Physics, Beijing (China); Chinese Academy of Sciences, Theoretical Physics Center for Science Facilities, Beijing (China); Wang, Sai [Chinese Academy of Sciences, Institute of High Energy Physics, Beijing (China)
2013-02-15
The GRAAL experiment could constrain the variations of the speed of light. The anisotropy of the speed of light may imply that the spacetime is anisotropic. Finsler geometry is a reasonable candidate to deal with the spacetime anisotropy. In this paper, the Lorentz invariance violation (LIV) of the photon sector is investigated in the locally Minkowski spacetime. The locally Minkowski spacetime is a class of flat Finsler spacetime and refers a metric with the anisotropic departure from the Minkowski one. The LIV matrices used to fit the experimental data are represented in terms of these metric deviations. The GRAAL experiment constrains the spacetime anisotropy to be less than 10{sup -14}. In addition, we find that the simplest Finslerian photon sector could be viewed as a geometric representation of the photon sector in the minimal standard model extension (SME). (orig.)
8. Effective potential in Lorentz-breaking field theory models
Energy Technology Data Exchange (ETDEWEB)
Baeta Scarpelli, A.P. [Centro Federal de Educacao Tecnologica, Nova Gameleira Belo Horizonte, MG (Brazil); Setor Tecnico-Cientifico, Departamento de Policia Federal, Belo Horizonte, MG (Brazil); Brito, L.C.T. [Universidade Federal de Lavras, Departamento de Fisica, Lavras, MG (Brazil); Felipe, J.C.C. [Universidade Federal de Lavras, Departamento de Fisica, Lavras, MG (Brazil); Universidade Federal dos Vales do Jequitinhonha e Mucuri, Instituto de Engenharia, Ciencia e Tecnologia, Veredas, Janauba, MG (Brazil); Nascimento, J.R.; Petrov, A.Yu. [Universidade Federal da Paraiba, Departamento de Fisica, Joao Pessoa, Paraiba (Brazil)
2017-12-15
We calculate explicitly the one-loop effective potential in different Lorentz-breaking field theory models. First, we consider a Yukawa-like theory and some examples of Lorentz-violating extensions of scalar QED. We observe, for the extended QED models, that the resulting effective potential converges to the known result in the limit in which Lorentz symmetry is restored. Besides, the one-loop corrections to the effective potential in all the cases we study depend on the background tensors responsible for the Lorentz-symmetry violation. This has consequences for physical quantities like, for example, in the induced mass due to the Coleman-Weinberg mechanism. (orig.)
9. Effective potential in Lorentz-breaking field theory models
International Nuclear Information System (INIS)
Baeta Scarpelli, A.P.; Brito, L.C.T.; Felipe, J.C.C.; Nascimento, J.R.; Petrov, A.Yu.
2017-01-01
We calculate explicitly the one-loop effective potential in different Lorentz-breaking field theory models. First, we consider a Yukawa-like theory and some examples of Lorentz-violating extensions of scalar QED. We observe, for the extended QED models, that the resulting effective potential converges to the known result in the limit in which Lorentz symmetry is restored. Besides, the one-loop corrections to the effective potential in all the cases we study depend on the background tensors responsible for the Lorentz-symmetry violation. This has consequences for physical quantities like, for example, in the induced mass due to the Coleman-Weinberg mechanism. (orig.)
10. R-symmetry violation in N=2 SUSY
International Nuclear Information System (INIS)
Volkov, G.G.; Maslikov, A.A.
1990-01-01
The present paper discusses the spontaneous R-symmetry violation in the N=2 SUSY SU(4)xU(1) model with soft SUSY breaking terms preserving finiteness. (In this case an invisible axion appears). In particular, the mechanism producting a light photino mass up to some GeV is suggested. In R-odd version of this model the mechanisms of enhancement of the neutrino decay is discussed. 10 refs.; 3 figs
11. CP symmetry violation. The search for its origin
International Nuclear Information System (INIS)
Cronin, J.W.
1981-01-01
The present experimental situation on detection of CP symmetry violation is presented. Interference between decays of long-lived (Ksub(L)sup(0)) and short-lived (Ksub(S)sup(0)) mesons into two charged pions serves a direct demonstration of the fact that the effect is caused by CP symmetry breaking. The time distribution of decays into π + π - when the 4-10 GeV Ksub(L) meson beam passes through a carbon regenerator is given as an example of the measurement accuracy. The measurements of the charge asymmetry in half-lepton channels of Ksub(L)→π +- l +- ν decay where l is an electron or a muon are discussed. It is noted that the presence of the charge asymmetry serves an indication of CP invariance violation and permits to carry out experimental differentiation between the matter and antimatter. Different theoretical assumptions on the nature of CP invariance violation are discussed. A list of experiments on search for CP, T and C invariance violation carried out in different laboratories of the world is given [ru
12. Constraints on relativity violations from gamma-ray bursts.
Science.gov (United States)
Kostelecký, V Alan; Mewes, Matthew
2013-05-17
Tiny violations of the Lorentz symmetry of relativity and the associated discrete CPT symmetry could emerge in a consistent theory of quantum gravity such as string theory. Recent evidence for linear polarization in gamma-ray bursts improves existing sensitivities to Lorentz and CPT violation involving photons by factors ranging from ten to a million.
13. Test of Lorentz symmetry with a 3He/129Xe clock-comparison experiment
International Nuclear Information System (INIS)
Gemmel, Claudia
2011-01-01
The minimal Standard Model Extension (SME) of Kostelecky and coworkers, which parametrizes the general treatment of CPT- and Lorentz invariance violation, predicts sidereal modulations of atomic transition frequencies as the Earth rotates relative to a Lorentz-violating background field. One method to search for these modulations is the so-called clock-comparison experiment, where the frequencies of co-located clocks are compared as they rotate with respect to the fixed stars. In this work an experiment is presented where polarized 3 He and 129 Xe gas samples in a glass cell serve as clocks, whose nuclear spin precession frequencies are detected with the help of highly sensitive SQUID sensors inside a magnetically shielded room. The unique feature of this experiment is the fact that the spins are precessing freely, with transverse relaxation times T * 2 of up to 4.4 h for 129 Xe and 14.1 h for 3 He. To be sensitive to Lorentz-violating effects, the influence of external magnetic fields is canceled via the weighted 3 He/ 129 Xe phase difference, ΔΦ=Φ he -(γ he )/(γ xe ) Φ xe . The Lorentz-violating SME parameters for the neutron, b n X and b n Y , are determined out of a χ 2 fit on the phase difference data of 7 spin precession measurements of 12 to 16 hours length. The piecewise defined fit model contains a sine and a cosine term to describe the sidereal modulation, as well as 7 offset terms, 7 linear terms and 7 . 2 exponential terms decreasing with T * 2,he and T * 2,xe , which are assigned to the respective measurement. The linear term in the weighted phase difference mainly arises from deviations of the gyromagnetic ratios from their literature values due to chemical shifts, while the exponential terms reflect the phase shifts resulting from demagnetization fields in the non-ideally spherical sample cell. The result of the χ 2 fit constrains the parameter b n perpendicular to =√((b n X ) 2 +(b n Y ) 2 ) to be -32 GeV at the 95% confidence level. This
14. Constraints on torsion from the bosonic sector of Lorentz violation and magnetogenesis data
International Nuclear Information System (INIS)
2011-01-01
A. Kostelecky et al. [Phys. Rev. Lett. 100 (2008) 111102], have shown that there is an exceptional sensitivity of spacetime torsion components by coupling it to fermions and constraining it to Lorentz violation. They obtain new constraints on torsion components down to the level of 10 -31 GeV. Yet more recently, L.C. Garcia de Andrade [Phys. Lett. B 468 (2011) 28] has shown that the photon sector of Lorentz violation (LV) Lagrangian leads to linear non-standard Maxwell equations where the magnetic field decays slower giving rise to a seed for galactic dynamos. In this paper bounds are placed on torsion based on the magnetogenesis or the origin of magnetic fields in the universe. On a coherence scale of 10 kpc, galactic magnetic fields of the order of some μG yield a torsion primordial field of the order of K 0 ∼10 -48 GeV. Just to give an idea of how tiny it is we mention that torsion limit in the Early universe yield K 0 ∼10 -31 GeV had been obtained by V. de Sabbata and C. Sivaram. Good limits were also obtained by B.R. Heckel et al. [Phys. Rev. D 78 (2008) 092006]. In our case the advantage from astro-particle physics point of view, is that a very small seed torsion field is enough to seed galactic dynamo. C. Sivaram limit is obtained from a massive photon electrodynamics [L.C. Garcia de Andrade, C. Sivaram, Ap. Space Sci. 209 (1993) 109] where a gauge invariant electrodynamics is used. Dynamo stars data are able to raise this value of torsion up to 10 -34 GeV at magnetar atmosphere. From these estimates one notices that they coincide with the ones obtained by A. Kostelecky et al., the difference being basically in the method. The ones here were obtained from magnetogenesis data while theirs were obtained from the Earth laboratory data from polarised electrons. Besides here one used the torsion derivatives while A. Kostelecky et al. uses the constant axial torsion tensor. Another fundamental distinction is that we use bosonic sector of the Lagrangian while
15. Lorentz violation bounds from torsion trace fermion sector and galaxy M51 data and chiral dynamos
Energy Technology Data Exchange (ETDEWEB)
Garcia de Andrade, L.C. [IF-UERJ, Departamento de Fisica Teorica, Rio de Janeiro, RJ (Brazil)
2017-06-15
Earlier we have computed a Lorentz violation (LV) bound for torsion terms via galactic dynamos and found bounds similar to the one obtained by Kostelecky et al. (Phys Rev Lett 100:111102, 2008) which is of the order of 10{sup -31} GeV. Their result was found making use of the axial torsion vector in terms of Dirac spinors and minimal torsion coupling in flat space-time of fermions. In this paper, a torsion dynamo equation obtained using the variation of the torsion trace and galaxy M51 data of 500 pc are used to place an upper bound of 10{sup -26} GeV in LV, which agrees with the one by Kostelecky and his group using an astrophysical framework background. Their lowest bound was obtained in earth laboratory using dual masers. One of the purposes of this paper is to apply the Faraday self-induction magnetic equation, recently extended to torsioned space-time, by the author to show that it lends support to physics in Riemann-Cartan space-time, in several distinct physical backgrounds. Backreaction magnetic effects are used to obtain the LV bounds. Previously Bamba et al. (JCAP 10:058, 2012) have used the torsion trace in their teleparallel investigation of the IGMF, with the argument that the torsion trace leads to less weaker effects than the other irreducible components of the torsion tensor. LV is computed in terms of a chiral-torsion-like current in the new dynamo equation analogous to the Dvornikov and Semikoz dynamo equation with chiral magnetic currents. Making use of the chiral-torsion dynamo equation we estimate the LV bounds in the early universe to be of the order of 10{sup -24} GeV, which was the order of the charged-lepton sector. Our main result is that it is possible to obtain more stringent bounds than the ones found in the fermion sector of astrophysics in the new revised 2017 data table for CPT and Lorentz violation by Kostelecky and Mewes. They found in several astrophysical backgrounds, orders of magnitude such as 10{sup -24} and 10{sup -23} Ge
16. Testing Lorentz invariance emergence in Ising Model using lattice Monte Carlo simulations
CERN Document Server
Stojku, Stefan
2017-01-01
All measurements performed so far at the observable energy scales show no violation of Lorentz invariance. However, it is yet impossible to check experimentally whether this symmetry holds at high energies such as the Planck scale. Recently, theories of gravitation with Lorentz violation, known as Horava-Lifshitz gravity [1, 2] have gained significant attention by treating Lorentz symmetry as an emergent phenomenon. A Lif-shitz type theory assumes an anisotropic scaling between space and time weighted by some critical exponent. In order for these theories to be viable candidates for quantum gravity description of the nature, Lorentz symmetry needs to be recovered at low energies.
17. The Search for Fundamental Symmetry Violation in Radium Nuclei
Science.gov (United States)
Dietrich, Matthew; Bishof, Michael; Bailey, Kevin; Greene, John; Mueller, Peter; O'Connor, Thomas; Lu, Zheng-Tian; Rabga, Tenzin; Ready, Roy; Singh, Jaideep
2017-09-01
Electric dipole moments (EDMs) are signatures of time-reversal, parity, and charge-parity (CP) violation, which makes them a sensitive probe of expected new physics beyond the Standard Model. Due to its large nuclear octupole deformation and high atomic mass, the radioactive Ra-225 isotope is a favorable EDM case; it is particularly sensitive to CP-violating interactions in the nuclear medium. We have developed a cold-atom approach of measuring the atomic EDM of atoms held stationary in an optical dipole trap, which we have used to place the only upper limit on the EDM of radium, |d(225Ra)|EDM, but also the first time the EDM of any octupole deformed species has been measured. We will present results on a new approach to spin detection that we expect to improve our EDM sensitivity by a factor of 20. Combined with upcoming improvements to our electric field generation, the next measurement should be competitive with the best neutron EDM result, in terms of sensitivity to CP-violating interactions. The Search for Fudamental Symmetry Violation in Radium Nuclei. This work is supported by the U.S. DOE, Office of Science, Office of Nuclear Physics, under Contract DE-AC02-06CH11357.
18. An application of Lorentz-invariance violation in black hole thermodynamics
Energy Technology Data Exchange (ETDEWEB)
Li, Guo-Ping; Zu, Xiao-Tao [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); Pu, Jin [University of Electronic Science and Technology of China, School of Physical Electronics, Chengdu (China); China West Normal University, College of Physics and Space Science, Nanchong (China); Jiang, Qing-Quan [China West Normal University, College of Physics and Space Science, Nanchong (China)
2017-10-15
In this paper, we have applied the Lorentz-invariance violation (LIV) class of dispersion relations (DRs) with the dimensionless parameter n = 2 and the ''sign of LIV'' η{sub +} = 1, to a phenomenological study of the effect of quantum gravity in a strong gravitational field. Specifically, we have studied the effect of the LIV-DR induced quantum gravity on the Schwarzschild black hole thermodynamics. The result shows that the effect of the LIV-DR induced quantum gravity speeds up the black hole evaporation, and its corresponding black hole entropy undergoes a leading logarithmic correction to the ''reduced Bekenstein-Hawking entropy'', and the ill-defined situations (i.e. the singularity problem and the critical problem) are naturally bypassed when the LIV-DR effect is present. Also, to put our results in a proper perspective, we have compared results with the earlier findings by another quantum-gravity candidate, i.e. the generalized uncertainty principle (GUP). Finally, we conclude from the inert remnants at the final stage of the black hole evaporation that, the GUP as a candidate for describing quantum gravity can always do as well as the LIV-DR by adjusting the model-dependent parameters, but in the same model-dependent parameters the LIV-DR acts as a more suitable candidate. (orig.)
19. String Quantum Gravity, Lorentz-Invariance Violation and Gamma-Ray Astronomy
CERN Document Server
Mavromatos, Nick E
2010-01-01
In the first part of the review, I discuss ways of obtaining Lorentz-Invariance-Violating (LIV) space-time foam in the modern context of string theory, involving brane world scenarios. The foamy structures are provided by lower-dimensional background brane defects in a D3-brane Universe, whose density is a free parameter to be constrained phenomenologically. Such constraining can be provided by high energy gamma-ray photon tests, including ultra-high energy/infrared photon-photon scattering. In the second part, I analyze the currently available data from MAGIC and FERMI Telescopes on delayed cosmic photon arrivals in this context. It is understood of course that conventional Astrophysics source effects, which currently are far from being understood, might be the dominant reason for the observed delayed arrivals. I also discuss how the stringent constraints from studies of synchrotron-radiation from distant Nebulae, absence of cosmic birefringence and non observation of ultra-high-energy cosmic photons can be ...
20. Black holes in multi-fractional and Lorentz-violating models
Energy Technology Data Exchange (ETDEWEB)
Calcagni, Gianluca [CSIC, Instituto de Estructura de la Materia, Madrid (Spain); Rodriguez Fernandez, David [Universidad de Oviedo, Department of Physics, Oviedo (Spain); Ronco, Michele [Universita di Roma ' ' La Sapienza' ' , Dipartimento di Fisica, Rome (Italy); INFN, Rome (Italy)
2017-05-15
We study static and radially symmetric black holes in the multi-fractional theories of gravity with q-derivatives and with weighted derivatives, frameworks where the spacetime dimension varies with the probed scale and geometry is characterized by at least one fundamental length l{sub *}. In the q-derivatives scenario, one finds a tiny shift of the event horizon. Schwarzschild black holes can present an additional ring singularity, not present in general relativity, whose radius is proportional to l{sub *}. In the multi-fractional theory with weighted derivatives, there is no such deformation, but non-trivial geometric features generate a cosmological-constant term, leading to a de Sitter-Schwarzschild black hole. For both scenarios, we compute the Hawking temperature and comment on the resulting black-hole thermodynamics. In the case with q-derivatives, black holes can be hotter than usual and possess an additional ring singularity, while in the case with weighted derivatives they have a de Sitter hair of purely geometric origin, which may lead to a solution of the cosmological constant problem similar to that in unimodular gravity. Finally, we compare our findings with other Lorentz-violating models. (orig.)
1. Lorentz invariance violation and chemical composition of ultra high energy cosmic rays
Energy Technology Data Exchange (ETDEWEB)
Saveliev, Andrey; Sigl, Guenter [Hamburg Univ. (Germany). II. Inst. fuer Theoretische Physik; Maccione, Luca [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany). Theory Group
2010-12-15
Motivated by experimental indications of a significant presence of heavy nuclei in the cosmic ray flux at ultra high energies (>or similar 10{sup 19} eV), we consider the effects of Planck scale suppressed Lorentz Invariance Violation (LIV) on the propagation of cosmic ray nuclei. In particular we focus on LIV effects on the photodisintegration of nuclei onto the background radiation fields. After a general discussion of the behavior of the relevant quantities, we apply our formalism to a simplified model where the LIV parameters of the various nuclei are assumed to kinematically result from a single LIV parameter for the constituent nucleons, {eta}, and we derive constraints on {eta}. Assuming a nucleus of a particular species to be actually present at 10{sup 20} eV the following constraints can be placed: -3 x 10{sup -2}
2. Lorentz invariance violation and simultaneous emission of electromagnetic and gravitational waves
Directory of Open Access Journals (Sweden)
E. Passos
2017-09-01
Full Text Available In this work, we compute some phenomenological bounds for the electromagnetic and massive gravitational high-derivative extensions supposing that it is possible to have an astrophysical process that generates simultaneously gravitational and electromagnetic waves. We present Lorentz invariance violating (LIV higher-order derivative models, following the Myers–Pospelov approach, to electrodynamics and massive gravitational waves. We compute the corrected equation of motion of these models, their dispersion relations and the velocities. The LIV parameters for the gravitational and electromagnetic sectors, ξg and ξγ, respectively, were also obtained for three different approaches: luminal photons, time delay of flight and the difference of graviton and photon velocities. These LIV parameters depend on the mass scales where the LIV-terms become relevant, M for the electromagnetic sector and M1 for the gravitational one. We obtain, using the values for M and M1 found in the literature, that ξg∼10−2, which is expected to be phenomenologically relevant and ξγ∼103, which cannot be suitable for an effective LIV theory. However, we show that ξγ can be interesting in a phenomenological point of view if M≫M1. Finally the relation between the variation of the velocities of the photon and the graviton in relation to the speed of light was calculated and resulted in Δvg/Δvγ≲1.82×10−3.
3. Black holes in multi-fractional and Lorentz-violating models
International Nuclear Information System (INIS)
Calcagni, Gianluca; Rodriguez Fernandez, David; Ronco, Michele
2017-01-01
We study static and radially symmetric black holes in the multi-fractional theories of gravity with q-derivatives and with weighted derivatives, frameworks where the spacetime dimension varies with the probed scale and geometry is characterized by at least one fundamental length l_*. In the q-derivatives scenario, one finds a tiny shift of the event horizon. Schwarzschild black holes can present an additional ring singularity, not present in general relativity, whose radius is proportional to l_*. In the multi-fractional theory with weighted derivatives, there is no such deformation, but non-trivial geometric features generate a cosmological-constant term, leading to a de Sitter-Schwarzschild black hole. For both scenarios, we compute the Hawking temperature and comment on the resulting black-hole thermodynamics. In the case with q-derivatives, black holes can be hotter than usual and possess an additional ring singularity, while in the case with weighted derivatives they have a de Sitter hair of purely geometric origin, which may lead to a solution of the cosmological constant problem similar to that in unimodular gravity. Finally, we compare our findings with other Lorentz-violating models. (orig.)
4. Searches for Lorentz violation in {sup 3}He/{sup 129}Xe clock comparison experiments
Energy Technology Data Exchange (ETDEWEB)
Allmendinger, F. [Universitaet Heidelberg, Physikalisches Institut (Germany); Burghoff, M. [Physikalisch-Technische Bundesanstalt (Germany); Heil, W., E-mail: [email protected]; Karpuk, S. [Johannes-Gutenberg Universitaet, Institut fuer Physik (Germany); Kilian, W.; Knappe-Grueneberg, S.; Mueller, W. [Physikalisch-Technische Bundesanstalt (Germany); Schmidt, U. [Universitaet Heidelberg, Physikalisches Institut (Germany); Schnabel, A.; Seifert, F. [Physikalisch-Technische Bundesanstalt (Germany); Sobolev, Yu [Johannes-Gutenberg Universitaet, Institut fuer Physik (Germany); Trahms, L. [Physikalisch-Technische Bundesanstalt (Germany); Tullney, K. [Johannes-Gutenberg Universitaet, Institut fuer Physik (Germany)
2013-03-15
We discuss the design and performance of a very sensitive low-field magnetometer based on the detection of free spin precession of gaseous, nuclear polarized {sup 3}He or {sup 129}Xe samples with a SQUID as magnetic flux detector. Characteristic spin precession times T{sub 2}{sup Asterisk-Operator} of up to 115 h were measured in low magnetic fields (about 1 {mu}T) and in the regime of motional narrowing. With the detection of the free precession of co-located {sup 3}He/{sup 129}Xe nuclear spins (clock comparison), the device can be used as ultra-sensitive probe for non-magnetic spin interactions, since the magnetic dipole interaction (Zeeman-term) drops out in the weighted frequency difference, i.e., {Delta}{omega} = {omega}{sub He} - {gamma}{sub He}/{gamma}{sub Xe}{center_dot}{omega}{sub Xe}. We report on searches for Lorentz violating signatures by monitoring the Larmor frequencies of co-located {sup 3}He/{sup 129}Xe spin samples as the laboratory reference frame rotates with respect to distant stars (sidereal modulation).
5. Black holes in multi-fractional and Lorentz-violating models.
Science.gov (United States)
Calcagni, Gianluca; Rodríguez Fernández, David; Ronco, Michele
2017-01-01
We study static and radially symmetric black holes in the multi-fractional theories of gravity with q -derivatives and with weighted derivatives, frameworks where the spacetime dimension varies with the probed scale and geometry is characterized by at least one fundamental length [Formula: see text]. In the q -derivatives scenario, one finds a tiny shift of the event horizon. Schwarzschild black holes can present an additional ring singularity, not present in general relativity, whose radius is proportional to [Formula: see text]. In the multi-fractional theory with weighted derivatives, there is no such deformation, but non-trivial geometric features generate a cosmological-constant term, leading to a de Sitter-Schwarzschild black hole. For both scenarios, we compute the Hawking temperature and comment on the resulting black-hole thermodynamics. In the case with q -derivatives, black holes can be hotter than usual and possess an additional ring singularity, while in the case with weighted derivatives they have a de Sitter hair of purely geometric origin, which may lead to a solution of the cosmological constant problem similar to that in unimodular gravity. Finally, we compare our findings with other Lorentz-violating models.
6. Time reversal symmetry violation in the YbF molecule
Energy Technology Data Exchange (ETDEWEB)
Sauer, B. E., E-mail: [email protected]; Devlin, J. A.; Hudson, J. J.; Kara, D. M.; Smallman, I. J.; Tarbutt, M. R.; Hinds, E. A. [Blackett Laboratory Imperial College London, Centre for Cold Matter (United Kingdom)
2013-03-15
We present a summary of the techniques used to test time reversal symmetry by measuring the permanent electric dipole moment of the YbF molecule. The results of a recent measurement (Hudson et al., Nature 473:493, 2011) are reported. We review some systematic effects which might mimic time reversal violation and describe how they are overcome. We then discuss improvements to the sensitivity of the apparatus, including both short term technical enhancements as well as a longer term goal to use laser cooled YbF in the experiment.
7. Constraints on Lorentz Invariance Violation from Fermi -Large Area Telescope Observations of Gamma-Ray Bursts
Science.gov (United States)
Vasileiou, V.; Jacholkowska, A.; Piron, F.; Bolmont, J.; Courturier, C.; Granot, J.; Stecker, Floyd William; Cohen-Tanugi, J.; Longo, F.
2013-01-01
We analyze the MeV/GeV emission from four bright Gamma-Ray Bursts (GRBs) observed by the Fermi-Large Area Telescope to produce robust, stringent constraints on a dependence of the speed of light in vacuo on the photon energy (vacuum dispersion), a form of Lorentz invariance violation (LIV) allowed by some Quantum Gravity (QG) theories. First, we use three different and complementary techniques to constrain the total degree of dispersion observed in the data. Additionally, using a maximally conservative set of assumptions on possible source-intrinsic spectral-evolution effects, we constrain any vacuum dispersion solely attributed to LIV. We then derive limits on the "QG energy scale" (the energy scale that LIV-inducing QG effects become important, E(sub QG)) and the coefficients of the Standard Model Extension. For the subluminal case (where high energy photons propagate more slowly than lower energy photons) and without taking into account any source-intrinsic dispersion, our most stringent limits (at 95% CL) are obtained from GRB 090510 and are E(sub QG,1) > 7.6 times the Planck energy (E(sub Pl)) and E(sub QG,2) > 1.3×10(exp 11) GeV for linear and quadratic leading order LIV-induced vacuum dispersion, respectively. These limits improve the latest constraints by Fermi and H.E.S.S. by a factor of approx. 2. Our results disfavor any class of models requiring E(sub QG,1) < or approx. E(sub Pl)
8. Tree-level equivalence between a Lorentz-violating extension of QED and its dual model in electron-electron scattering
Energy Technology Data Exchange (ETDEWEB)
Toniolo, Giuliano R.; Fargnoli, H.G.; Brito, L.C.T. [Universidade Federal de Lavras, Departamento de Fisica, Caixa Postal 3037, Lavras, Minas Gerais (Brazil); Scarpelli, A.P.B. [Setor Tecnico-Cientifico, Departamento de Policia Federal, Sao Paulo (Brazil)
2017-02-15
S-matrix amplitudes for the electron-electron scattering are calculated in order to verify the physical equivalence between two Lorentz-breaking dual models. We begin with an extended Quantum Electrodynamics which incorporates CPT-even Lorentz-violating kinetic and mass terms. Then, in a process of gauge embedding, its gauge-invariant dual model is obtained. The physical equivalence of the two models is established at tree level in the electron-electron scattering and the unpolarized cross section is calculated up to second order in the Lorentz-violating parameter. (orig.)
9. Tree-level equivalence between a Lorentz-violating extension of QED and its dual model in electron-electron scattering
International Nuclear Information System (INIS)
Toniolo, Giuliano R.; Fargnoli, H.G.; Brito, L.C.T.; Scarpelli, A.P.B.
2017-01-01
S-matrix amplitudes for the electron-electron scattering are calculated in order to verify the physical equivalence between two Lorentz-breaking dual models. We begin with an extended Quantum Electrodynamics which incorporates CPT-even Lorentz-violating kinetic and mass terms. Then, in a process of gauge embedding, its gauge-invariant dual model is obtained. The physical equivalence of the two models is established at tree level in the electron-electron scattering and the unpolarized cross section is calculated up to second order in the Lorentz-violating parameter. (orig.)
10. Lorentz-Symmetry Test at Planck-Scale Suppression With a Spin-Polarized 133Cs Cold Atom Clock.
Science.gov (United States)
Pihan-Le Bars, H; Guerlin, C; Lasseri, R-D; Ebran, J-P; Bailey, Q G; Bize, S; Khan, E; Wolf, P
2018-06-01
We present the results of a local Lorentz invariance (LLI) test performed with the 133 Cs cold atom clock FO2, hosted at SYRTE. Such a test, relating the frequency shift between 133 Cs hyperfine Zeeman substates with the Lorentz violating coefficients of the standard model extension (SME), has already been realized by Wolf et al. and led to state-of-the-art constraints on several SME proton coefficients. In this second analysis, we used an improved model, based on a second-order Lorentz transformation and a self-consistent relativistic mean field nuclear model, which enables us to extend the scope of the analysis from purely proton to both proton and neutron coefficients. We have also become sensitive to the isotropic coefficient , another SME coefficient that was not constrained by Wolf et al. The resulting limits on SME coefficients improve by up to 13 orders of magnitude the present maximal sensitivities for laboratory tests and reach the generally expected suppression scales at which signatures of Lorentz violation could appear.
11. Electromagnetic contribution to charge symmetry violation in parton distributions
Directory of Open Access Journals (Sweden)
X.G. Wang
2016-02-01
Full Text Available We report a calculation of the combined effect of photon radiation and quark mass differences on charge symmetry violation (CSV in the parton distribution functions of the nucleon. Following a recent suggestion of Martin and Ryskin, the initial photon distribution is calculated in terms of coherent radiation from the proton as a whole, while the effect of the quark mass difference is based on a recent lattice QCD simulation. The distributions are then evolved to a scale at which they can be compared with experiment by including both QCD and QED radiation. Overall, at a scale of 5 GeV2, the total CSV effect on the phenomenologically important difference between the d and u-quark distributions is some 20% larger than the value based on quark mass differences alone. In total these sources of CSV account for approximately 40% of the NuTeV anomaly.
12. Lorentz invariance violation and charge (non)conservation: A general theoretical frame for extensions of the Maxwell equations
International Nuclear Information System (INIS)
Laemmerzahl, Claus; Macias, Alfredo; Mueller, Holger
2005-01-01
All quantum gravity approaches lead to small modifications in the standard laws of physics which in most cases lead to violations of Lorentz invariance. One particular example is the extended standard model (SME). Here, a general phenomenological approach for extensions of the Maxwell equations is presented which turns out to be more general than the SME and which covers charge nonconservation (CNC), too. The new Lorentz invariance violating terms cannot be probed by optical experiments but need, instead, the exploration of the electromagnetic field created by a point charge or a magnetic dipole. Some scalar tensor theories and higher dimensional brane theories predict CNC in four dimensions and some models violating special relativity have been shown to be connected with CNC. Its relation to the Einstein Equivalence Principle has been discussed. Because of this upcoming interest, the experimental status of electric charge conservation is reviewed. Up to now there seem to exist no unique tests of charge conservation. CNC is related to the precession of polarization, to a modification of the 1/r-Coulomb potential, and to a time dependence of the fine structure constant. This gives the opportunity to describe a dedicated search for CNC
13. Soft CP violation and the global matter-antimatter symmetry of the universe
Science.gov (United States)
Senjanovic, G.; Stecker, F. W.
1980-01-01
Scenarios for baryon production are considered within the context of SU(5) and SO(10) grand unified theories where CP violation arises spontaneously. The spontaneous CP symmetry breaking then results in a matter-antimatter domain structure in the universe. Two possible, distinct types of theories of soft CP violation are defined. In the first type the CP nonconservation originates only from the breaking of SU(2) sub L X U(1) symmetry, and in the second type, even at the unification temperature scale, CP violation can emerge as a result of symmetry breaking by the vacuum expectation values of the superheavy Higgs sector scalars.
14. Anomalous Lorentz and CPT violation from a local Chern–Simons-like term in the effective gauge-field action
Directory of Open Access Journals (Sweden)
K.J.B. Ghosh
2018-01-01
Full Text Available We consider four-dimensional chiral gauge theories defined over a spacetime manifold with topology R3×S1 and periodic boundary conditions over the compact dimension. The effective gauge-field action is calculated for Abelian U(1 gauge fields Aμ(x which depend on all four spacetime coordinates (including the coordinate x4∈S1 of the compact dimension and have vanishing components A4(x (implying trivial holonomies in the 4-direction. Our calculation shows that the effective gauge-field action contains a local Chern–Simons-like term which violates Lorentz and CPT invariance. This result is established perturbatively with a generalized Pauli–Villars regularization and nonperturbatively with a lattice regularization based on Ginsparg–Wilson fermions.
15. Anomalous Lorentz and CPT violation from a local Chern-Simons-like term in the effective gauge-field action
Science.gov (United States)
Ghosh, K. J. B.; Klinkhamer, F. R.
2018-01-01
We consider four-dimensional chiral gauge theories defined over a spacetime manifold with topology R3 ×S1 and periodic boundary conditions over the compact dimension. The effective gauge-field action is calculated for Abelian U (1) gauge fields Aμ (x) which depend on all four spacetime coordinates (including the coordinate x4 ∈S1 of the compact dimension) and have vanishing components A4 (x) (implying trivial holonomies in the 4-direction). Our calculation shows that the effective gauge-field action contains a local Chern-Simons-like term which violates Lorentz and CPT invariance. This result is established perturbatively with a generalized Pauli-Villars regularization and nonperturbatively with a lattice regularization based on Ginsparg-Wilson fermions.
16. QED with minimal and nonminimal couplings: on the quantum generation of Lorentz violating terms in the pure photon sector
Energy Technology Data Exchange (ETDEWEB)
Gazzola, G.; Fargnoli, H.G.; Sampaio, Marcos; Nemes, M.C. [Universidade Federal de Minas Gerais (UFMG), Belo Horizonte, MG (Brazil); Scarpelli, A.P. Baeta [Departamento de Policia Federal (DPF), Sao Paulo, SP (Brazil). Setor Tecnico-Cientifico
2011-07-01
In this research we consider a modified version of quantum electrodynamics in four dimensions with the coupling between the photon and the fermion composed by two terms: a nonminimal and the minimal one. There are two interesting aspects in this model. First, gauge invariance is restored by the presence of the minimal coupling. Second, the quantum corrections will allow for the possibility of the generation of a Chern-Simons-like term. The fact that the model is gauge invariant allows for a more complete analysis on the value of both the coefficients of the hypothetical CPT odd and CPT even radiatively generated terms. A question that arises involves a possible violation of some Ward-Takahashi identity when radiative corrections are taken into account. In other words, is there an anomaly in the model? We show that, since conventional QED is gauge invariant, there is no room for a non transversal vacuum polarization tensor in the present model. This is study is to be presented in the following order: first we are to present the model; second we do an analysis on the generation of Lorentz violating terms in the pure gauge sector; third we carry out a calculation on gauge invariance grounds to fix the coefficients of the quantum corrections; and lastly the concluding comments. (author)
17. QED with minimal and nonminimal couplings: on the quantum generation of Lorentz violating terms in the pure photon sector
International Nuclear Information System (INIS)
Gazzola, G.; Fargnoli, H.G.; Sampaio, Marcos; Nemes, M.C.; Scarpelli, A.P. Baeta
2011-01-01
In this research we consider a modified version of quantum electrodynamics in four dimensions with the coupling between the photon and the fermion composed by two terms: a nonminimal and the minimal one. There are two interesting aspects in this model. First, gauge invariance is restored by the presence of the minimal coupling. Second, the quantum corrections will allow for the possibility of the generation of a Chern-Simons-like term. The fact that the model is gauge invariant allows for a more complete analysis on the value of both the coefficients of the hypothetical CPT odd and CPT even radiatively generated terms. A question that arises involves a possible violation of some Ward-Takahashi identity when radiative corrections are taken into account. In other words, is there an anomaly in the model? We show that, since conventional QED is gauge invariant, there is no room for a non transversal vacuum polarization tensor in the present model. This is study is to be presented in the following order: first we are to present the model; second we do an analysis on the generation of Lorentz violating terms in the pure gauge sector; third we carry out a calculation on gauge invariance grounds to fix the coefficients of the quantum corrections; and lastly the concluding comments. (author)
18. Planck-scale deformation of Lorentz symmetry as a solution to the UHECR and the TeV-$\\gamma$ paradoxes
CERN Document Server
Amelino-Camelia, G; Amelino-Camelia, Giovanni; Piran, Tsvi
2001-01-01
One of the most puzzling current experimental physics paradoxes is the arrival on Earth of Ultra High Energy Cosmic Rays with energies above the GZK threshold. The recent observation of 20TeV photons from Mk 501 is another somewhat similar paradox. Several models have been proposed for the UHECR paradox. No solution has yet been proposed for the TeV-$\\gamma$ paradox. Remarkably, the drastic assumption of a violation of ordinary Lorentz invariance would resolve both paradoxes. We present a formalism for the description of the type of Lorentz-invariance deformation (LID) that could be induced by non-trivial short-distance structure of space-time, and we show that this formalism is well suited for comparison of experimental data with LID predictions. We use the UHECR and TeV-$\\gamma$ data, as well as bounds on time-of-flight differences between photons of different energies, to constrain the LID parameter space. A model with only two parameters, an energy scale and a dimensionless parameter characterizing the fu...
19. Test of Lorentz symmetry with a {sup 3}He/{sup 129}Xe clock-comparison experiment
Energy Technology Data Exchange (ETDEWEB)
Gemmel, Claudia
2011-01-28
The minimal Standard Model Extension (SME) of Kostelecky and coworkers, which parametrizes the general treatment of CPT- and Lorentz invariance violation, predicts sidereal modulations of atomic transition frequencies as the Earth rotates relative to a Lorentz-violating background field. One method to search for these modulations is the so-called clock-comparison experiment, where the frequencies of co-located clocks are compared as they rotate with respect to the fixed stars. In this work an experiment is presented where polarized {sup 3}He and {sup 129}Xe gas samples in a glass cell serve as clocks, whose nuclear spin precession frequencies are detected with the help of highly sensitive SQUID sensors inside a magnetically shielded room. The unique feature of this experiment is the fact that the spins are precessing freely, with transverse relaxation times T{sup *}{sub 2} of up to 4.4 h for {sup 129}Xe and 14.1 h for {sup 3}He. To be sensitive to Lorentz-violating effects, the influence of external magnetic fields is canceled via the weighted {sup 3}He/{sup 129}Xe phase difference, {delta}{phi}={phi}{sub he}-({gamma}{sub he})/({gamma}{sub xe}) {phi}{sub xe}. The Lorentz-violating SME parameters for the neutron, b{sup n}{sub X} and b{sup n}{sub Y}, are determined out of a {chi}{sup 2} fit on the phase difference data of 7 spin precession measurements of 12 to 16 hours length. The piecewise defined fit model contains a sine and a cosine term to describe the sidereal modulation, as well as 7 offset terms, 7 linear terms and 7 . 2 exponential terms decreasing with T{sup *}{sub 2,he} and T{sup *}{sub 2,xe}, which are assigned to the respective measurement. The linear term in the weighted phase difference mainly arises from deviations of the gyromagnetic ratios from their literature values due to chemical shifts, while the exponential terms reflect the phase shifts resulting from demagnetization fields in the non-ideally spherical sample cell. The result of the {chi
20. Leptonic Dirac CP violation predictions from residual discrete symmetries
Directory of Open Access Journals (Sweden)
I. Girardi
2016-01-01
Full Text Available Assuming that the observed pattern of 3-neutrino mixing is related to the existence of a (lepton flavour symmetry, corresponding to a non-Abelian discrete symmetry group Gf, and that Gf is broken to specific residual symmetries Ge and Gν of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase δ of the neutrino mixing matrix U. The residual symmetries considered are: i Ge=Z2 and Gν=Zn, n>2 or Zn×Zm, n,m≥2; ii Ge=Zn, n>2 or Zn×Zm, n,m≥2 and Gν=Z2; iii Ge=Z2 and Gν=Z2; iv Ge is fully broken and Gν=Zn, n>2 or Zn×Zm, n,m≥2; and v Ge=Zn, n>2 or Zn×Zm, n,m≥2 and Gν is fully broken. For given Ge and Gν, the sum rules for cosδ thus derived are exact, within the approach employed, and are valid, in particular, for any Gf containing Ge and Gν as subgroups. We identify the cases when the value of cosδ cannot be determined, or cannot be uniquely determined, without making additional assumptions on unconstrained parameters. In a large class of cases considered the value of cosδ can be unambiguously predicted once the flavour symmetry Gf is fixed. We present predictions for cosδ in these cases for the flavour symmetry groups Gf=S4, A4, T′ and A5, requiring that the measured values of the 3-neutrino mixing parameters sin2θ12, sin2θ13 and sin2θ23, taking into account their respective 3σ uncertainties, are successfully reproduced.
1. Implications of horizontal symmetries on baryon number violation in supersymmetric models
International Nuclear Information System (INIS)
Ben-Hamo, V.; Nir, Y.
1994-08-01
The smallness of the quark and lepton parameters and the hierarchy between them could be the result of selection rules due to a horizontal symmetry broken by a small parameter. The same selection rules apply to baryon number violating terms. Consequently, the problem of baryon number violation in supersymmetry may be solved naturally, without invoking any especially-designed extra symmetry. This mechanism is efficient enough even for low-scale flavor physics. Proton decay is likely to be dominated by the modes K + ν-bar i or K o μ + (e + ), and may proceed at observable rates. (authors). 15 refs
2. Current status and future prospect of space and time reversal symmetry violation on low energy neutron reactions
International Nuclear Information System (INIS)
Masuda, Yasuhiro
1993-01-01
In this report, the papers on symmetry violation under space reflection and time reversal and neutron spin, neutron spin rotation and P-violation, parity nonconservation in neutron capture reaction, some advantage of the search for CP-violation in neutron scattering, dynamic polarization of 139 La target, alexandrite laser for optical pumping, polarized 3 He system for T- and P-violation neutron experiments, control of neutron spin in T-violation neutron experiment, symmetry regarding time and space and angular distribution and angular correlation of radiation and particle beams, T-violation due to low temperature nuclear polarization and axion exploration using nuclear transition are collected. (K.I.)
3. Modified Adler sum rule and violation of charge symmetry
International Nuclear Information System (INIS)
Dominguez, C.A.; Moreno, H.; Zepeda, A.
The consequences of a once subtracted dispersion relation in the derivation of the Adler sum rule are investigated. It is shown that one can expect a breakdown of charge symmetry, of the isotriplet current hypothesis, and of scaling of the structure functions. These breakdowns are related to the possible presence of a non-zero subtraction function at asymptotic energies and arbitrary q 2 . Second class currents and PCAC relations are also discussed
4. A model of spontaneous CP violation and neutrino phenomenology with approximate LμLτ symmetry
International Nuclear Information System (INIS)
2013-01-01
We introduce a model where CP and Z 2 symmetry violate spontaneously. CP and Z 2 violate spontaneously through a singlet complex scalar S which obtains vacuum expectation value with phase S = Ve iα /2 and this is the only source of CP violation in this model. Low energy CP violation in the leptonic sector is connected to the large scale phase by three generations of left and right handed singlet fermions in the inverse see-saw like structure of model. We have considered approximate LμL τ symmetry to study neutrino phenomenology. Considering two mass square differences and three mixing angles including non zero θ 13 to their experimental 3σ limit, we have restricted the Lagrangian parameters for reasonably small value of L μ L τ symmetry breaking parameters. We have predicted the three masses, Dirac phase and two Majorana phases. We also evaluate CP violating parameter J CP , sum-mass and effective mass parameter involved in neutrino less double beta decay. (author)
5. A violation of CP symmetry in B meson decays
International Nuclear Information System (INIS)
Karyotakis, Y.; Monchenault, G.H. de
2002-01-01
This article reviews the issue of CP-violation and reports the most recent results about the observation of large CP asymmetries in the decay of neutral B-mesons. Some of the CP asymmetries in the neutral B-meson decay are expected to be large. CP-violation always involves quantum mechanical interference. This occurs for instance when there are 2 paths for a particle to decay into a given final state. The interference between the mixing-induced amplitude (B 0 → B-bar 0 → f) and the decay amplitude (B 0 → f) to a CP eigenstate f leads to a time dependent CP asymmetry that can be interpreted in terms of the angles of the unitary triangle (sin(2β)). The experimental challenge comes from the fact that B decays to some CP eigenstates have very small branching ratios and low efficiencies for complete reconstruction of the final state. It is therefore necessary to produce a very large number of B-mesons to perform a CP-measurement. To make the measurement possible, a new type of e + e - collider, called asymmetric B-factory has been designed. 2 asymmetric B-factories are operating in the world: PEP2 (Stanford, Usa) fitted with the Babar detector and KEK-B (Japan) which hosts Belle detector. The measurements given by Babar and Belle are in good agreement and can be combined. The average value is sin(2β) = 0.78 ± 0.08 and this value is in excellent agreement with the standard model predictions based on available experimental data. (A.C.)
6. Searches for discrete symmetries violation in ortho-positronium decay using the J-PET detector
Directory of Open Access Journals (Sweden)
Kamińska Daria
2015-12-01
Full Text Available In this paper, we present prospects for using the Jagiellonian positron emission tomograph (J-PET detector to search for discrete symmetries violations in a purely leptonic system of the positronium atom. We discuss tests of CP and CPT symmetries by means of ortho-positronium decays into three photons. No zero expectation values for chosen correlations between ortho-positronium spin and momentum vectors of photons would imply the existence of physics phenomena beyond the standard model. Previous measurements resulted in violation amplitude parameters for CP and CPT symmetries consistent with zero, with an uncertainty of about 10−3. The J-PET detector allows to determine those values with better precision, thanks to the unique time and angular resolution combined with a high geometrical acceptance. Achieving the aforementioned is possible because of the application of polymer scintillators instead of crystals as detectors of annihilation quanta.
7. Mass mixing, CP violation and left-right symmetry for heavy neutral mesons
International Nuclear Information System (INIS)
Ecker, G.; Grimus, W.
1985-01-01
We investigate M 0 - M-bar 0 mixing and CP violation in the minimal left-right symmetric gauge model with spontaneous P and CP violation. The dominant contributions to the mixing amplitude including QCD corrections are calculated explicitly for B 0 - B-bar 0 . While the amount of mixing is not much changed with respect to the standard model left-right symmetry can give rise to significantly larger CP violation in the B 0 sub(s) - B-bar 0 sub(s) system (up to two orders of magnitude for the dilepton charge asymmetry). Sizable CP violating effects require that the left-right contribution to the KsubLKsubS mass difference has the same sign as the standard model contribution. We also comment on D 0 - D-bar 0 mixing including a careful discussion of the standard model prediction. (Author)
8. Spontaneous Lorentz breaking at high energies
International Nuclear Information System (INIS)
Cheng, H.-C.; Luty, Markus A.; Mukohyama, Shinji; Thaler, Jesse
2006-01-01
Theories that spontaneously break Lorentz invariance also violate diffeomorphism symmetries, implying the existence of extra degrees of freedom and modifications of gravity. In the minimal model ('ghost condensation') with only a single extra degree of freedom at low energies, the scale of Lorentz violation cannot be larger than about M ∼ 100GeV due to an infrared instability in the gravity sector. We show that Lorentz symmetry can be broken at much higher scales in a non-minimal theory with additional degrees of freedom, in particular if Lorentz symmetry is broken by the vacuum expectation value of a vector field. This theory can be constructed by gauging ghost condensation, giving a systematic effective field theory description that allows us to estimate the size of all physical effects. We show that nonlinear effects become important for gravitational fields with strength Φ 1/2 ∼> g, where g is the gauge coupling, and we argue that the nonlinear dynamics is free from singularities. We then analyze the phenomenology of the model, including nonlinear dynamics and velocity-dependent effects. The strongest bounds on the gravitational sector come from either black hole accretion or direction-dependent gravitational forces, and imply that the scale of spontaneous Lorentz breaking is M ∼ 12 GeV, g 2 10 15 GeV). If the Lorentz breaking sector couples directly to matter, there is a spin-dependent inverse-square law force, which has a different angular dependence from the force mediated by the ghost condensate, providing a distinctive signature for this class of models
9. Charge symmetry violation in the electromagnetic form factors of the proton
International Nuclear Information System (INIS)
Shanahan, P.E.; Thomas, A.W.; Young, R.D.; Zanotti, J.M.; Pleiter, D.; Stueben, H.
2015-03-01
Experimental tests of QCD through its predictions for the strange-quark content of the proton have been drastically restricted by our lack of knowledge of the violation of charge symmetry (CSV). We find unexpectedly tiny CSV in the proton's electromagnetic form factors by performing the first extraction of these quantities based on an analysis of lattice QCD data. The resulting values are an order of magnitude smaller than current bounds on proton strangeness from parity violating electron-proton scattering experiments. This result paves the way for a new generation of experimental measurements of the proton's strange form factors to challenge the predictions of QCD.
10. Experimental searches for CP and CPT symmetries violation in the neutral kaons system
International Nuclear Information System (INIS)
Debu, P.
1996-01-01
The aim of this lecture is to give an overview of the experiments devoted to the study and research of CP, T and CPT symmetries invariance violations in the system of neutral K mesons. The discovery of K mesons has provided crucial informations for the elaboration of the standard model. However, the observation of CP violation has remained confined to the K system. The origin of the observed CP violation remains hypothetic. Its origin could be a complex phase in the mixing matrix of quarks. In the standard model of electroweak interactions, several evidences of the CP violation exist: the observed K neutral mesons (K L and K S ) are not proper states of CP and are due to CP violation in the K 0 - anti-K 0 mixture. On the other hand, the model predicts also a CP violation in decay amplitudes, named direct CP violation. Important experiments have been carried out for its demonstration. The K system is also the most precise test for CPT invariance. A description of the experiments in progress developed to improve the precision of these tests is given. The plan of the lecture is the following: after a recall of K 0 - anti-K 0 phenomenology, some important steps in the CP violation study are described. Then, the regeneration phenomenon is briefly described and two of the most recent measurements of the direct CP violation parameter are analysed. Finally, the CPT invariance tests are described with their parameters and the measurements in progress. A review of the principal results is given in conclusion with their improvements expected in a near future. (J.S.). 71 refs., 4 figs., 4 tabs
11. Lepton-flavour violation in a Pati-Salam model with gauged flavour symmetry
Energy Technology Data Exchange (ETDEWEB)
Feldmann, Thorsten; Luhn, Christoph; Moch, Paul [Theoretische Physik 1, Naturwissenschaftlich-Technische Fakultät,Universität Siegen, Walter-Flex-Straße 3, 57068 Siegen (Germany)
2016-11-11
Combining Pati-Salam (PS) and flavour symmetries in a renormalisable setup, we devise a scenario which produces realistic masses for the charged leptons. Flavour-symmetry breaking scalar fields in the adjoint representations of the PS gauge group are responsible for generating different flavour structures for up- and down-type quarks as well as for leptons. The model is characterised by new heavy fermions which mix with the Standard Model quarks and leptons. In particular, the partners for the third fermion generation induce sizeable sources of flavour violation. Focusing on the charged-lepton sector, we scrutinise the model with respect to its implications for lepton-flavour violating processes such as μ→eγ, μ→3e and muon conversion in nuclei.
12. Strange quark distribution and parton charge symmetry violation in a semi-inclusive process
International Nuclear Information System (INIS)
Kitagawa, Hisashi; Sakemi, Yasuhiro
2000-01-01
It is possible to observe a semi-inclusive reaction with tagged charged kaons using the RICH detector at DESY-HERA. Using the semi-inclusive process we study two kinds of parton properties in the nucleon. We study relations between cross sections and strange quark distributions, which are expected to be measured more precisely in such a process than in the process in which pions are tagged. We also investigate charge symmetry violation (CSV) in the nucleon, which appears in the region x ≤ 0.1. (author)
13. A phenomenological study of violation of CP and CPT symmetries in the neutral kaon system
International Nuclear Information System (INIS)
Kojima, Kazushi; Sugiyama, Wataru; Tsai, S.Y.
1996-01-01
A phenomenological study is given of the (possible) violation of CP and CPT symmetries in the K 0 -K-bar 0 system. Special attention is paid to the problem of phase ambiguity and phase convention. Mixing parameters and decay amplitudes are parametrized in a rephasing invariant way, and the well-known parameters η +- and η 00 describing 2π modes as well as various leptonic asymmetries are expressed in terms of these parameters. The parameters ε and Δ characterizing mixing between |K 0 > and |K-bar 0 > are treated with as little theoretical prejudice as possible. (author)
14. A strong astrophysical constraint on the violation of special relativity by quantum gravity.
Science.gov (United States)
Jacobson, T; Liberati, S; Mattingly, D
2003-08-28
Special relativity asserts that physical phenomena appear the same to all unaccelerated observers. This is called Lorentz symmetry and relates long wavelengths to short ones: if the symmetry is exact it implies that space-time must look the same at all length scales. Several approaches to quantum gravity, however, suggest that there may be a microscopic structure of space-time that leads to a violation of Lorentz symmetry. This might arise because of the discreteness or non-commutivity of space-time, or through the action of extra dimensions. Here we determine a very strong constraint on a type of Lorentz violation that produces a maximum electron speed less than the speed of light. We use the observation of 100-MeV synchrotron radiation from the Crab nebula to improve the previous limit by a factor of 40 million, ruling out this type of Lorentz violation, and thereby providing an important constraint on theories of quantum gravity.
15. Measurements of CPT Violation at LHCb
CERN Document Server
INSPIRE-00260865
2017-01-01
Recent measurements of CPT violation and Lorentz symmetry breaking in $B^0-\\bar{B}^0$ mixing and $B^0_s-\\bar{B}^0_s$ mixing, obtained from data taken by the LHCb experiment, are highlighted. The results are expressed in terms of the Standard-Model Extension (SME) coefficients, which incorporate both CPT and Lorentz violation. Due to the large boost of the $B$ mesons at LHCb, the SME coefficients can be determined with high precision. The bounds on these coefficients are improved significantly compared to previous measurements.
16. Search for violation of CPT and Lorentz invariance in B.sup.0./sup..sub.s./sub. meson oscillations
Czech Academy of Sciences Publication Activity Database
Abazov, V. M.; Abbott, B.; Acharya, B.S.; Kupčo, Alexander; Lokajíček, Miloš
2015-01-01
Roč. 115, č. 16 (2015), "161601-1"-"161601-8" ISSN 0031-9007 Institutional support: RVO:68378271 Keywords : Batavia TEVATRON Coll * DZERO * asymmetry * anti -p p * scattering * B/s0 anti -B/s0: mixing * CPT * violation * experimental results Subject RIV: BF - Elementary Particles and High Energy Physics Impact factor: 7.645, year: 2015
17. Induction of novel macroscopic properties by local symmetry violations in spin-spiral multiferroics
Science.gov (United States)
Meier, D.; Leo, N.; Becker, P.; Bohaty, L.; Ramesh, R.; Fiebig, M.
2011-03-01
Incommensurate (IC) structures are omnipresent in strongly correlated electron systems as high-TC superconductors, CMR manganites, as well as multiferroics. In each case they are origin of a pronounced symmetry reduction reflecting the complexity of the underlying microscopic interactions. Macroscopically, this can lead to new phases and possibilities to gain control of the host material. Here we report how the IC nature of a spin-spiral multiferroic induces new physical properties by renormalizing the relevant length scales of the system. Local symmetry violations directly manifest in the macroscopic response of the material and co-determine the multiferroic order giving rise to additional domain states. These usually hidden degrees of freedom become visible when non-homogenous fields are applied and condition for instance the second harmonic generation. Our study shows that incommensurabilities play a vital role in the discussion of the physical properties of multiferroics -- they represent a key ingredient for further enhancing the functionality of this class of materials. This work was supported by the DFG through the SFB 608. D.M. thanks the AvH for financial support.
18. Spontaneous Broken Local Conformal Symmetry and Dark Energy Candidate
International Nuclear Information System (INIS)
Liu, Lu-Xin
2013-01-01
The local conformal symmetry is spontaneously broken down to the Local Lorentz invariance symmetry through the approach of nonlinear realization. The resulting effective Lagrangian, in the unitary gauge, describes a cosmological vector field non-minimally coupling to the gravitational field. As a result of the Higgs mechanism, the vector field absorbs the dilaton and becomes massive, but with an independent energy scale. The Proca type vector field can be modelled as dark energy candidate. The possibility that it further triggers Lorentz symmetry violation is also pointed out
19. Searches for violation of the combined space reflection (P) and time reversal (T) symmetry in solid state experiments
International Nuclear Information System (INIS)
Sushkov, O.P.
2002-01-01
Full text: Electric dipole moment (EDM) of an elementary particle is a manifestation of the violation of the fundamental TP-symmetry. Because of the CRT-theorem TP-violation is related to CP-violation. Present experimental limitations on electron and neutron EDM as well as limitations on nuclear Schiff moments impose important constrains on physics beyond the standard model. Unfortunately the standard approaches for search of EDM in atomic, molecular, and neutron experiments are close to their sensitivity limit. There are novel suggestions for searches of the fundamental TP-violation in solid state experiments. Two groups lead by Lamoreaux (Los Alamos) and Hunter (Amherst college) are preparing these experiments. We calculate the expected effect. The improvement of sensitivity compared to the present level can reach 6-8 orders of magnitude!
20. A minimal spontaneous CP violation model with small neutrino mass and SU(2) x U(1) x Z3 symmetry
International Nuclear Information System (INIS)
Geng, C.Q.; Ng, J.N.
1988-04-01
It is shown that spontaneous CP violation and natural flavor conservation can occur in the SU(2) L x U(1) Y model based on two Higgs doublet and one Higgs singlet fields with a Z 3 discrete symmetry. Physical CP nonconservation is purely due to scalar-pseudoscalar mixings. In order for this to be a major source of CP violation a light spin-O boson of mass less than 10 GeV is required. The see-saw mechanism can be implemented to generate small neutrino masses. The model implies a relatively large electric dipole moment for charged leptons and small value for ε'/ε
1. Measurements of Direct CP Violation, CPT Symmetry, and Other Parameters in the Neutral Kaon System
Energy Technology Data Exchange (ETDEWEB)
Worcester, Elizabeth Turner [Univ. of Chicago, IL (United States)
2007-12-01
The authors present precision measurements of the direct CP violation parameter, Re(ϵ'/ϵ), the kaon parameters, Δm and τS, and the CPT tests, Φ± and ΔΦ, in neutral kaon decays. These results are based on the full dataset collected by the KTeV experiment at Fermi National Accelerator Laboratory during 1996, 1997, and 1999. This dataset contains ~ 15 million K → π0π0 decays and ~ 69 million K → π+π- decays. They describe significant improvements to the precision of these measurements relative to previous KTeV analyses. They find Re(ϵ'/ϵ = [19.2 ± 1.1(stat) ± 1.8(syst)] x 10-4, Δm = (5265 ± 10) x 106 hs-1, and τS = (89.62 ± 0.05) x 10-12 s. They measure Φ± = (44.09 ± 1.00)° and ΔΦ = (0.29 ± 0.31)°; these results are consistent with CPT symmetry.
2. Test of CPT and Lorentz invariance from muonium spectroscopy
NARCIS (Netherlands)
Hughes, V. W.; Perdekamp, M. Grosse; Kawall, D.; Liu, W.; Jungmann, K.; Putlitz, G. zu
2001-01-01
Following a suggestion of Kostelecky et al. we have evaluated a test of CPT and Lorentz invariance from the microwave spectroscopy of muonium. Hamiltonian terms beyond the standard model violating CPT and Lorentz invariance would contribute frequency shifts $\\delta\ 3. Left-right symmetry, mixing and CP violation in B0 - B-bar0 International Nuclear Information System (INIS) Ecker, G.; Grimus, W. 1986-01-01 We discuss B 0 - B-bar 0 mixing and CP violation in the minimal left-right symmetric model. While the amount of mixing is not much changed with respect to the standard model, left-right symmtery can give rise to significantly larger CP violation in the B 0 sub(s) - B-bar 0 sub(s) system. (Author) 4. Entropic information for travelling solitons in Lorentz and CPT breaking systems International Nuclear Information System (INIS) Correa, R.A.C.; Rocha, Roldão da; Souza Dutra, A. de 2015-01-01 In this work we group four research topics apparently disconnected, namely solitons, Lorentz symmetry breaking, supersymmetry, and entropy. Following a recent work (Gleiser and Stamatopoulos, 2012), we show that it is possible to construct in the context of travelling wave solutions a configurational entropy measure in functional space, from the field configurations. Thus, we investigate the existence and properties of travelling solitons in Lorentz and CPT breaking scenarios for a class of models with two interacting scalar fields. Here, we obtain a complete set of exact solutions for the model studied which display both double and single-kink configurations. In fact, such models are very important in applications that include Bloch branes, Skyrmions, Yang–Mills, Q-balls, oscillons and various superstring-motivated theories. We find that the so-called Configurational Entropy (CE) for travelling solitons shows that the best value of parameter responsible to break the Lorentz symmetry is one where the energy density is distributed equally around the origin. In this way, the information-theoretical measure of travelling solitons in Lorentz symmetry violation scenarios opens a new window to probe situations where the parameters responsible for breaking the symmetries are arbitrary. In this case, the CE selects the best value of the parameter in the model 5. CPT-symmetry studies with antihydrogen Energy Technology Data Exchange (ETDEWEB) Lehnert, Ralf, E-mail: [email protected] [Indiana University Center for Spacetime Symmetries (United States) 2012-05-15 Various approaches to physics beyond the Standard Model can lead to small violations of CPT invariance. Since CPT symmetry can be measured with ultra-high precision, CPT tests offer an interesting phenomenological avenue to search for underlying physics. We discuss this reasoning in more detail, comment on the connection between CPT and Lorentz invariance, and review how CPT breaking would affect the (anti)hydrogen spectrum. 6. Closeout Report - Search for Time Reversal Symmetry Violation with TREK at J-PARC Energy Technology Data Exchange (ETDEWEB) Kohl, Michael [Hampton Univ., VA (United States) 2015-04-15 academic positions. Two former graduate students of the group have graduated and received their PhD degrees in nuclear physics (Dr. Anusha Liyanage and Dr. Ozgur Ates). In particular, this award has enabled Dr. Kohl to pursue the TREK project (Time Reversal Experiment with Kaons) at J-PARC, which he has been leading and advancing as International Spokesperson. Originally proposed as a search for time reversal symmetry violation [6], the project has evolved into a precision test of lepton flavor universality in the Standard Model along with sensitive searches for physics beyond the Standard Model through a possible discovery of new particles such as a sterile neutrino or a neutral gauge boson from the hidden sector in the mass region up to 300 MeV/c2 [7]. Experiment TREK/E36, first proposed in 2010, has been mounted between November 2014 and April 2015, and commissioning with beam has been started in April 2015, with production running anticipated in early summer and late fall 2015. It uses the apparatus from the previous KEK/E-246 experiment with partial upgrades to measure the ratio of decay widths of leptonic two-body decays of the charged kaon to µν and eν, respectively, which is highly sensitive to the ratio of electromagnetic charged lepton couplings and possible new physics processes that could differentiate between μ and e, hence breaking lepton flavor universality of the Standard Model. Through the searches for neutral massive particles, TREK/E36 can severely constrain any new physics scenarios designed to explain the proton radius puzzle [12, 13]. 7. Are the invariance principles really truly Lorentz covariant? International Nuclear Information System (INIS) Arunasalam, V. 1994-02-01 It is shown that some sections of the invariance (or symmetry) principles such as the space reversal symmetry (or parity P) and time reversal symmetry T (of elementary particle and condensed matter physics, etc.) are not really truly Lorentz covariant. Indeed, I find that the Dirac-Wigner sense of Lorentz invariance is not in full compliance with the Einstein-Minkowski reguirements of the Lorentz covariance of all physical laws (i.e., the world space Mach principle) 8. Comment on self-inverse form of the Lorentz transformation International Nuclear Information System (INIS) Cook, R.J. 1979-01-01 It has been shown that the kinematic relations between two iertial reference frames in relative motion can be made symmetric by an appropriate orientation of the coordinate axes of the two frames. It follows from this symmetry and the principle of relativity that the transformation matrix, A, from one frame to the other, and its inverse, A -1 , are equal. This result, along with a limiting-velocity postulate, was used in a derivation of the Lorentz transformation. The present note points out that only two transformation laws are compatible with the symmetry condition A = A -1 . One of these is the Lorentz transformation and the other violates causality. Thus, if the limiting-velocity postulate is replaced by the requirement that causality be satisfied in all inertial frames, one arrives at a derivation of the Lorentz transformation based entirely on concepts which were known and widely accepted long before the advent of special relativity: the homogeneity and isotropy of space in all inertial frames, the principle of relativity, and the principle of causality 9. Bell's theorem, the measurement problem, Newton's self-gravitation and its connections to violations of the discrete symmetries C, P, T International Nuclear Information System (INIS) Hiesmayr, Beatrix C 2015-01-01 About 50 years ago John St. Bell published his famous Bell theorem that initiated a new field in physics. This contribution discusses how discrete symmetries relate to the big open questions of quantum mechanics, in particular:(i) how correlations stronger than those predicted by theories sharing randomness (Bell's theorem) relate to the violation of the CP symmetry and the P symmetry; and its relation to the security of quantum cryptography,(ii) how the measurement problem (“why do we observe no tables in superposition?”) can be polled in weakly decaying systems,(iii) how strongly and weakly interacting quantum systems are affected by Newton's self gravitation.These presented preliminary results show that the meson-antimeson systems and the hyperon- antihyperon systems are a unique laboratory to tackle deep fundamental questions and to contribute to the understand what impact the violation of discrete symmetries has. (paper) 10. Bell's theorem, the measurement problem, Newton's self-gravitation and its connections to violations of the discrete symmetries C, P, T Science.gov (United States) Hiesmayr, Beatrix C. 2015-07-01 About 50 years ago John St. Bell published his famous Bell theorem that initiated a new field in physics. This contribution discusses how discrete symmetries relate to the big open questions of quantum mechanics, in particular: (i) how correlations stronger than those predicted by theories sharing randomness (Bell's theorem) relate to the violation of the CP symmetry and the P symmetry; and its relation to the security of quantum cryptography, (ii) how the measurement problem (“why do we observe no tables in superposition?”) can be polled in weakly decaying systems, (iii) how strongly and weakly interacting quantum systems are affected by Newton's self gravitation. These presented preliminary results show that the meson-antimeson systems and the hyperon- antihyperon systems are a unique laboratory to tackle deep fundamental questions and to contribute to the understand what impact the violation of discrete symmetries has. 11. Strange and charge symmetry violating electromagnetic form factors of the nucleon International Nuclear Information System (INIS) Shanahan, P.E. 2016-01-01 We summarise recent work based on lattice QCD simulations of the electromagnetic form factors of the octet baryons from the CSSM/QCDSF/UKQCD collaborations. After an analysis of the simulation results using techniques to approach the infinite volume limit and the physical pseudoscalar masses at non-zero momentum transfer, the extrapolated proton and neutron form factors are found to be in excellent agreement with those extracted from experiment. Given the success of these calculations, we describe how the strange electromagnetic form factors may be estimated from these results under the same assumption of charge symmetry used in experimental determinations of those quantities. Motivated by the necessity of that assumption, we explore a method for determining the size of charge symmetry breaking effects using the same lattice results. (author) 12. Subtractions in the Adler sum rule and violation of charge symmetry International Nuclear Information System (INIS) Dominguez, C.A.; Moreno, H.; Zepeda, A. 1976-01-01 The consequences of a once-subtracted dispersion relation in the derivation of the Adler sum rule are investigated. It is shown that one can expect a breakdown of charge symmetry, of the isotriplet-current hypothesis, and of scaling of the structure functions. These breakdowns are related to the possible presence of a nonzero subtraction function at asymptotic energies and arbitrary q 2 . We also comment about second-class currents and PCAC (partial conservation of axial-vector current) relations 13. Fermion-number violation in regularizations that preserve fermion-number symmetry Science.gov (United States) Golterman, Maarten; Shamir, Yigal 2003-01-01 There exist both continuum and lattice regularizations of gauge theories with fermions which preserve chiral U(1) invariance (“fermion number”). Such regularizations necessarily break gauge invariance but, in a covariant gauge, one recovers gauge invariance to all orders in perturbation theory by including suitable counterterms. At the nonperturbative level, an apparent conflict then arises between the chiral U(1) symmetry of the regularized theory and the existence of ’t Hooft vertices in the renormalized theory. The only possible resolution of the paradox is that the chiral U(1) symmetry is broken spontaneously in the enlarged Hilbert space of the covariantly gauge-fixed theory. The corresponding Goldstone pole is unphysical. The theory must therefore be defined by introducing a small fermion-mass term that breaks explicitly the chiral U(1) invariance and is sent to zero after the infinite-volume limit has been taken. Using this careful definition (and a lattice regularization) for the calculation of correlation functions in the one-instanton sector, we show that the ’t Hooft vertices are recovered as expected. 14. Apparent violation of isospin symmetry in the 3H(3He,2H)4He reaction International Nuclear Information System (INIS) Rai, G.; Blyth, C.O.; England, J.B.A.; Farooq, A.; Karban, O.; Rawas, E.; Roman, S.; Vlastou, R. 1988-01-01 Angular distributions of the vector analyzing powers for the 3 H( 3 He, 2 H) 4 He reaction have been measured over the incident energy range 18--33 MeV. The measurements centered about 18 MeV display a deviation from the antisymmetric shape expected from isospin symmetry. Concentrating on the explanation of the 90 0 analyzing powers, we report the results of a distorted-wave Born approximation (DWBA) analysis which includes the direct and exchange processes and the spin-orbit potential. It is shown that the anomalous behavior of the 90 0 vector analyzing powers can be largely explained by the effect of a single F-wave potential resonance which leads to the magnification of the short-range differences between the 3 He and 3 H wave functions 15. Minimal flavour violation in the quark and lepton sector and the impact of extra dimensions on flavour changing neutral currents and electroweak symmetry breaking International Nuclear Information System (INIS) Weiler, A. 2007-01-01 We study flavor-changing decays of hadrons and leptons and an extra-dimensional approach to electroweak symmetry breaking. Specifically we study the framework of Minimal Flavour Violation (MFV) as an explanation of the flavour problem. We discuss the impact of a specific extra-dimensional model of the MFV class on flavour changing neutral currents. We derive model-independent upper bounds on rare decays. -We discuss the extension of the MFV framework from the quark to the lepton sector and show how baryogenesis through leptogenesis can be achieved and examine if possible correlations with charged lepton flavour violation exist. We discuss the dynamical breaking of the electroweak symmetry in extra dimensions by unifying gauge and Higgs fields and we show that realistic models are possible once the extra dimension is strongly curved. (orig.) 16. Minimal flavour violation in the quark and lepton sector and the impact of extra dimensions on flavour changing neutral currents and electroweak symmetry breaking Energy Technology Data Exchange (ETDEWEB) Weiler, A. 2007-01-16 We study flavor-changing decays of hadrons and leptons and an extra-dimensional approach to electroweak symmetry breaking. Specifically we study the framework of Minimal Flavour Violation (MFV) as an explanation of the flavour problem. We discuss the impact of a specific extra-dimensional model of the MFV class on flavour changing neutral currents. We derive model-independent upper bounds on rare decays. -We discuss the extension of the MFV framework from the quark to the lepton sector and show how baryogenesis through leptogenesis can be achieved and examine if possible correlations with charged lepton flavour violation exist. We discuss the dynamical breaking of the electroweak symmetry in extra dimensions by unifying gauge and Higgs fields and we show that realistic models are possible once the extra dimension is strongly curved. (orig.) 17. Lorentz covariant canonical symplectic algorithms for dynamics of charged particles Science.gov (United States) Wang, Yulei; Liu, Jian; Qin, Hong 2016-12-01 In this paper, the Lorentz covariance of algorithms is introduced. Under Lorentz transformation, both the form and performance of a Lorentz covariant algorithm are invariant. To acquire the advantages of symplectic algorithms and Lorentz covariance, a general procedure for constructing Lorentz covariant canonical symplectic algorithms (LCCSAs) is provided, based on which an explicit LCCSA for dynamics of relativistic charged particles is built. LCCSA possesses Lorentz invariance as well as long-term numerical accuracy and stability, due to the preservation of a discrete symplectic structure and the Lorentz symmetry of the system. For situations with time-dependent electromagnetic fields, which are difficult to handle in traditional construction procedures of symplectic algorithms, LCCSA provides a perfect explicit canonical symplectic solution by implementing the discretization in 4-spacetime. We also show that LCCSA has built-in energy-based adaptive time steps, which can optimize the computation performance when the Lorentz factor varies. 18. Lepton mixing and CP violation phase in the 3-3-1 model with neutral leptons based on T{sub 13} flavor symmetry Energy Technology Data Exchange (ETDEWEB) Vien, Vo Van, E-mail: [email protected] [Department of Physics, Tay Nguyen University, Le Duan, Buon Ma Thuot, DakLak (Viet Nam) 2015-08-15 We study a 3-3-1 model based on non-Abelian discrete symmetry group T{sub 13} which accommodates lepton mixing with non-zero θ{sub 13} and CP violation phase. The neutrinos get small masses and mixing with CP violation phase from S U(3) L antisextets which are all in triplets under T{sub 13}. If both breakings T{sub 13} → Z{sub 3} and Z{sub 3} → {Identity} are taken place in neutrino sector, and T{sub 13} is broken into Z{sub 3} in lepton sector, the realistic neutrino mixing form is obtained as a natural consequence of P{sub l} and T{sub 13} symmetries. The model predicts the lepton mixing with non-zero θ{sub 13}, and also gives a remarkable prediction of Dirac CP violation δ{sub CP} = 292.5∘ in the normal spectrum, and δ {sub CP} = 303.161∘ in the inverted spectrum which is still missing in the neutrino mixing matrix. There exist some regions of model parameters that can fit the experimental data in 2014 on neutrino masses and mixing without perturbation. (author) 19. Violation of CPT invariance in the early universe and leptogenesis/baryogenesis CERN Document Server Mavromatos, Nick E 2013-01-01 In this talk, I review some plausible scenarios entailing violation of CPT symmetry in the early Universe, due to space-time backgrounds which do not respect some of the assumptions for the validity of the CPT theorem (here considered will be Lorentz invariance and/or Unitarity). The key point in all these models is that the background induces different populations of fermions as compared to antifermions, and hence CPT Violation (CPTV), already in thermal equilibrium. Such populations may freeze out at various conditions depending on the details of the underlying microscopic model, thereby leading to leptogenesis and baryogenesis. Among the considered scenarios is a stringy one, in which the CPTV is associated with a cosmological background with torsion provided by the Kalb-Ramond antisymmetric tensor field (axion) of the string gravitational multiplet. We also discuss briefly (Lorentz Violating) CPTV models that go beyond the local effective lagrangian framework, such as a stochastic Finsler metric and D-par... 20. Improved test of Lorentz invariance in electrodynamics International Nuclear Information System (INIS) Wolf, Peter; Bize, Sebastien; Clairon, Andre; Santarelli, Giorgio; Tobar, Michael E.; Luiten, Andre N. 2004-01-01 We report new results of a test of Lorentz invariance based on the comparison of a cryogenic sapphire microwave resonator and a hydrogen-maser. The experimental results are shown together with an extensive analysis of systematic effects. Previously, this experiment has set the most stringent constraint on Kennedy-Thorndike type violations of Lorentz invariance. In this work we present new data and interpret our results in the general Lorentz violating extension of the standard model of particle physics (SME). Within the photon sector of the SME, our experiment is sensitive to seven SME parameters. We marginally improve present limits on four of these, and by a factor seven to ten on the other three 1. Angle and energy dependence of the superratio for π+ and π- elastic scattering from 3H and 3He: Evidence for charge-symmetry violation International Nuclear Information System (INIS) Pillai, C.; Barlow, D.B.; Berman, B.L.; Briscoe, W.J.; Mokhtari, A.; Nefkens, B.M.K.; Sadler, M.E. 1991-01-01 Data are presented on the energy and angle dependence of the charge-symmetry superratio R and simple ratios r 1 ' and r 2 ' for π ± elastic scattering from 3 H and 3 He. r 1 ' and r 2 ' were normalized with respect to π + d and π - d elastic scattering, which is assumed to have the ratio 1.0. The beam energies are T π =142, 180, and 220 MeV, and the scattering angle, θ L , ranges from 40 degree to 110 degree. In all cases measured it is found that R>1, r 1 ' congruent 1, and r 2 ' >1. These results provide substantial evidence for charge-symmetry violation. The angular distributions for π ± H and π ± 3 He elastic scattering also have been measured and comparisons are made with various model calculations 2. CP Violation course CERN Multimedia CERN. Geneva HR-RFA 2006-01-01 The lecture introduces the concepts and phenomena of matter-antimatter symmetry violation, so-called "CP" violation. The lecture is organized in four courses, the first being devoted to a historical overview and an introduction into fundamental discrete symmetries. The second course introduces the most compelling CP-violating phenomena, and presents the first experimental discovery of CP violation in the neutral kaon system. The third course discusses how CP violation is beautifully incorporated into the Standard Model of particle interactions, and how modern B-meson "factories" provide precise tests of this picture. Finally, the fourth and last course introduces CP violation and the genesis of our matter world. 3. Model for particle masses, flavor mixing, and CP violation, based on spontaneously broken discrete chiral symmetry as the origin of families International Nuclear Information System (INIS) Adler, S.L. 1999-01-01 We construct extensions of the standard model based on the hypothesis that Higgs bosons also exhibit a family structure and that the flavor weak eigenstates in the three families are distinguished by a discrete Z 6 chiral symmetry that is spontaneously broken by the Higgs sector. We study in detail at the tree level models with three Higgs doublets and with six Higgs doublets comprising two weakly coupled sets of three. In a leading approximation of S 3 cyclic permutation symmetry the three-Higgs-doublet model gives a open-quotes democraticclose quotes mass matrix of rank 1, while the six-Higgs-doublet model gives either a rank-1 mass matrix or, in the case when it spontaneously violates CP, a rank-2 mass matrix corresponding to nonzero second family masses. In both models, the CKM matrix is exactly unity in the leading approximation. Allowing small explicit violations of cyclic permutation symmetry generates small first family masses in the six-Higgs-doublet model, and first and second family masses in the three-Higgs-doublet model, and gives a nontrivial CKM matrix in which the mixings of the first and second family quarks are naturally larger than mixings involving the third family. Complete numerical fits are given for both models, flavor-changing neutral current constraints are discussed in detail, and the issues of unification of couplings and neutrino masses are addressed. On a technical level, our analysis uses the theory of circulant and retrocirculant matrices, the relevant parts of which are reviewed. copyright 1998 The American Physical Society 4. Phenomenologically viable Lorentz-violating quantum gravity. Science.gov (United States) Sotiriou, Thomas P; Visser, Matt; Weinfurtner, Silke 2009-06-26 Horava's "Lifschitz point gravity" has many desirable features, but in its original incarnation one is forced to accept a nonzero cosmological constant of the wrong sign to be compatible with observation. We develop an extension of Horava's model that abandons "detailed balance" and regains parity invariance, and in 3+1 dimensions exhibit all five marginal (renormalizable) and four relevant (super-renormalizable) operators, as determined by power counting. We also consider the classical limit of this theory, evaluate the Hamiltonian and supermomentum constraints, and extract the classical equations of motion in a form similar to the Arnowitt-Deser-Misner formulation of general relativity. This puts the model in a framework amenable to developing detailed precision tests. 5. Spontaneous CP violation from a quaternionic Kaluza-Klein theory International Nuclear Information System (INIS) Hanlon, B.E.; Joshi, G.C. 1991-01-01 Motivated by the isomorphism between the universal covering group of the six dimensional Lorentz group and the special linear group over the quaternions, a locally quaternionic covariant quantum mechanics is postulated to exist in six space-time dimensions. Compactifying onto the space-time M 4 x S 2 complex theory is retrieved on the four dimensional Minkowski space with the essential quaternionic nature confined to S 2 . Quaternionic spinors are introduced and a dimensionally reduced theory recovered which exhibits a CP violating effect via spontaneous symmetry breaking. 20 refs 6. Spontaneous violation of chiral symmetry in QCD vacuum is the origin of baryon masses and determines baryon magnetic moments and their other static properties International Nuclear Information System (INIS) Ioffe, B. L. 2009-01-01 A short review is presented of the spontaneous violation of chiral symmetry in QCD vacuum. It is demonstrated that this phenomenon is the origin of baryon masses in QCD. The value of nucleon mass is calculated, as well as the masses of hyperons and some baryonic resonances, and expressed mainly through the values of quark condensates - , q = u, d, s,-the vacuum expectation values (v.e.v.) of quark field. The concept of v.e.v. induced by external fields is introduced. It is demonstrated that such v.e.v. induced by static electromagnetic field results in quark condensate magnetic susceptibility, which plays the main role in determination of baryon magnetic moments. The magnetic moments of proton, neutron, and hyperons are calculated. The results of calculation of baryon octet β-decay constants are also presented. 7. Violations of Einstein's Relativity: Motivations, Theory, and Phenomenology International Nuclear Information System (INIS) Lehnert, Ralf 2011-01-01 One of the most difficult questions in present-day physics concerns a fundamental theory of space, time, and matter that incorporates a consistent quantum description of gravity. There are various theoretical approaches to such a quantum-gravity theory. Nevertheless, experimental progress is hampered in this research field because many models predict deviations from established physics that are suppressed by some power of the Planck scale, which currently appears to be immeasurably small. However, tests of relativity theory provide one promising avenue to overcome this phenomeno-logical obstacle: many models for underlying physics can accommodate a small breakdown of Lorentz symmetry, and numerous feasible Lorentz-symmetry tests have Planck reach. Such mild violations of Einstein's relativity have therefore become the focus of recent research efforts. This mini course provides a brief survey of the key ideas in this research field and is geared at both experimentalists and theorists. In particular, several theoretical mechanisms leading to deviations from relativity theory are presented; the standard theoretical framework for relativity violations at currently accessible energy scales (i.e., the SME) is reviewed, and various present and near-future experimental efforts within this field are discussed. 8. CP Violation International Nuclear Information System (INIS) Aleksan, R. 1993-06-01 The violation of the CP symmetry is a phenomenon, the origin of which is not yet well established and deserves a particular attention since it may be a fundamental property of Nature with very important consequences for the evolution of the universe. We propose in these lectures to have an overview of this phenomenon as we understand it so far. To this end, and after introducing the discrete space-time symmetries, we discuss the observation of the violation of the CP symmetry in the neutral kaon decays. We then derive the general formalism for any neutral system made of a particle and its antiparticle and discuss how CP violation is introduced. We show how this phenomenon is generated in the Standard Model of the electroweak interactions and what are the predictions that can be made. In particular we shall concentrate on the expected effects in the decays of mesons involving the b quark. We review the various possibilities for observing these effects, calculate their magnitudes and show how the consistency of the theory can be tested. Finally, we outline the experimental prospects for studying CP non conservation at an asymmetric B Factory to either verify the Standard Model mechanism or provide evidence for new physics. (author) 9. Study of charge-symmetry violation in. pi. /sup +/ and. pi. /sup -/ elastic scattering from /sup 3/H and /sup 3/He Energy Technology Data Exchange (ETDEWEB) Pillai, C.; Barlow, D.B.; Nefkens, B.M.K.; Berman, B.L.; Briscoe, W.J.; Mokhtari, A.; Petrov, A.M.; Sadler, M.E. 1988-06-30 New data on the charge-symmetric superratio R and the simple ratios r/sub 1/' and r/sub 2/' are reported for ..pi../sup +/ and ..pi../sup -/ elastic scattering from /sup 3/H and /sup 3/He at T/sub n/=142, 180, and 220 MeV at theta/sub ..pi../(lab) from 40/sup 0/ to 110/sup 0/; R=dsigma(..pi../sup +3/H)dsigma(..pi../sup -3/H)/dsigma(..pi../sup -3/He)dsigma(..pi../sup +3/ He), r/sub 1/'=dsigma(..pi../sup +3/H)dsigma(..pi../sup -/d)/dsigma(..pi../sup -3/He)dsigma(..pi../sup +/d), and r/sub 2/'=dsigma(..pi../sup -3/H)dsigma(..pi../sup +/d)/dsigma(..pi../sup +3/He)dsigma(..pi../sup -/d). We find that R > 1, r/sub 1/' approx. = 1, and r/sub 2/' > 1 at all energies and angles. These results are not accounted for by any of the available calculations which attempt to include only electromagnetic effects. This disagreement suggests that the violation of charge symmetry is due to a combination of strong and electromagnetic effects. 10. Lorentz and Poincaré invariance 100 years of relativity CERN Document Server Hsu Jong Ping 2001-01-01 This collection of papers provides a broad view of the development of Lorentz and Poincaré invariance and spacetime symmetry throughout the past 100 years. The issues explored in these papers include: (1) formulations of relativity theories in which the speed of light is not a universal constant but which are consistent with the four-dimensional symmetry of the Lorentz and Poincaré groups and with experimental results, (2) analyses and discussions by Reichenbach concerning the concepts of simultaneity and physical time from a philosophical point of view, and (3) results achieved by the union o 11. Traveling solitons in Lorentz and CPT breaking systems International Nuclear Information System (INIS) Souza Dutra, A. de; Correa, R. A. C. 2011-01-01 In this work we present a class of traveling solitons in Lorentz and CPT breaking systems. In the case of Lorentz violating scenarios, as far as we know, only static solitonic configurations were analyzed up to now in the literature. Here it is shown that it is possible to construct some traveling solitons which cannot be mapped into static configurations by means of Lorentz boosts due to explicit breaking. In fact, the traveling solutions cannot be reached from the static ones by using something similar to a Lorentz boost in those cases. Furthermore, in the model studied, a complete set of exact solutions is obtained. The solutions present a critical behavior controlled by the choice of an arbitrary integration constant. 12. Prospects for Lorentz and CPT tests with hydrogen and antihydrogen CERN Document Server Becker, Tobias Frederic 2017-01-01 As a summer student for 13 weeks in the ASACUSA-CUSP collaboration, under the supervision of Chloé Malbrunot, my project consisted in a first part on the theoretical treatment of Lorentz and CPT violation in hydrogen & antihydrogen in the framework of the Standard Model Extension SME and in second part on experimental measurements on a hydrogen beam. 13. Searching for CPT violation with cosmic microwave background data from WMAP and BOOMERANG. Science.gov (United States) Feng, Bo; Li, Mingzhe; Xia, Jun-Qing; Chen, Xuelei; Zhang, Xinmin 2006-06-09 We search for signatures of Lorentz and violations in the cosmic microwave background (CMB) temperature and polarization anisotropies by using the Wilkinson Microwave Anisotropy Probe (WMAP) and the 2003 flight of BOOMERANG (B03) data. We note that if the Lorentz and symmetries are broken by a Chern-Simons term in the effective Lagrangian, which couples the dual electromagnetic field strength tensor to an external four-vector, the polarization vectors of propagating CMB photons will get rotated. Using the WMAP data alone, one could put an interesting constraint on the size of such a term. Combined with the B03 data, we found that a nonzero rotation angle of the photons is mildly favored: [Formula: See Text]. 14. Testing Lorentz invariance in β decay Directory of Open Access Journals (Sweden) Sytema A. 2014-03-01 Experimentally we exploit the Gamow-Teller transition of polarized 20Na, where we can test the dependence of the β-decay rate on the spin orientation of 20Na. The polarization degree is measured using the β asymmetry, while the decay rate is measured by the γ yield. A change in the γ rate, when reversing the spin, implies Lorentz invariance violation. The decay rate should depend on sidereal time and the polarization direction relative to the rotation axis of the earth. The method of the measurement will be presented, together with the first results. 15. Does the relativity principle violate? International Nuclear Information System (INIS) Barashenkov, V.S. 1994-01-01 Theoretical and experimental data about a possible existence in Nature of some preferred reference frame with a violation of the principle of relativity are considered. The Einstein's and Lorentz's points of view are compared. Although some experiments are known which, in opinion of their authors, indicate the relativity principle violation persuasive evidences supporting this conclusion are absent for the present. The proposals of new experiments in this region, particularly with electron spin precession, are discussed. 55 refs., 4 figs 16. Charge symmetry at the partonic level Energy Technology Data Exchange (ETDEWEB) Londergan, J. T.; Peng, J. C.; Thomas, A. W. 2010-07-01 This review article discusses the experimental and theoretical status of partonic charge symmetry. It is shown how the partonic content of various structure functions gets redefined when the assumption of charge symmetry is relaxed. We review various theoretical and phenomenological models for charge symmetry violation in parton distribution functions. We summarize the current experimental upper limits on charge symmetry violation in parton distributions. A series of experiments are presented, which might reveal partonic charge symmetry violation, or alternatively might lower the current upper limits on parton charge symmetry violation. 17. Possible violations of the relativity theory International Nuclear Information System (INIS) Tiomno, J. 1985-01-01 A review of previous works of the author and collaborators on possible violations of the Theory of Relativity (SR) is made. It is shown that there is no contradiction of the predictions of the Lorentz Aether Theory, in the form presented in these papers, with existing experiments. Further experiments to detect these violations (or to confirm SR) are indicated. (Author) [pt 18. Quantum Space-Time Deformed Symmetries Versus Broken Symmetries CERN Document Server Amelino-Camelia, G 2002-01-01 Several recent studies have concerned the faith of classical symmetries in quantum space-time. In particular, it appears likely that quantum (discretized, noncommutative,...) versions of Minkowski space-time would not enjoy the classical Lorentz symmetries. I compare two interesting cases: the case in which the classical symmetries are "broken", i.e. at the quantum level some classical symmetries are lost, and the case in which the classical symmetries are "deformed", i.e. the quantum space-time has as many symmetries as its classical counterpart but the nature of these symmetries is affected by the space-time quantization procedure. While some general features, such as the emergence of deformed dispersion relations, characterize both the symmetry-breaking case and the symmetry-deformation case, the two scenarios are also characterized by sharp differences, even concerning the nature of the new effects predicted. I illustrate this point within an illustrative calculation concerning the role of space-time symm... 19. Dynamics on Lorentz manifolds CERN Document Server Adams, Scot 2001-01-01 Within the general framework of the dynamics of "large" groups on geometric spaces, the focus is on the types of groups that can act in complicated ways on Lorentz manifolds, and on the structure of the resulting manifolds and actions. This particular area of dynamics is an active one, and not all the results are in their final form. However, at this point, a great deal can be said about the particular Lie groups that come up in this context. It is impressive that, even assuming very weak recurrence of the action, the list of possible groups is quite restricted. For the most complicated of the 20. A new General Lorentz Transformation model International Nuclear Information System (INIS) Novakovic, Branko; Novakovic, Alen; Novakovic, Dario 2000-01-01 A new general structure of Lorentz Transformations, in the form of General Lorentz Transformation model (GLT-model), has been derived. This structure includes both Lorentz-Einstein and Galilean Transformations as its particular (special) realizations. Since the free parameters of GLT-model have been identified in a gravitational field, GLT-model can be employed both in Special and General Relativity. Consequently, the possibilities of an unification of Einstein's Special and General Theories of Relativity, as well as an unification of electromagnetic and gravitational fields are opened. If GLT-model is correct then there exist four new observation phenomena (a length and time neutrality, and a length dilation and a time contraction). Besides, the well-known phenomena (a length contraction, and a time dilation) are also the constituents of GLT-model. It means that there is a symmetry in GLT-model, where the center of this symmetry is represented by a length and a time neutrality. A time and a length neutrality in a gravitational field can be realized if the velocity of a moving system is equal to the free fall velocity. A time and a length neutrality include an observation of a particle mass neutrality. A special consideration has been devoted to a correlation between GLT-model and a limitation on particle velocities in order to investigate the possibility of a travel time reduction. It is found out that an observation of a particle speed faster then c=299 792 458 m/s, is possible in a gravitational field, if certain conditions are fulfilled 1. Study of the violation of the T and CP symmetries in the reactions Λb0 → Λ0 + a vector meson. Validation of the Front-end electronics for the PreShower detector of the LHCb experiment International Nuclear Information System (INIS) Conte, E. 2007-11-01 This thesis probes the beauty baryon physics in the framework of the LHCb experiment. The present study deals with the Λ b 0 → Λ 0 V decays where V is a vector meson such as J/Ψ(μ + μ - ), φ(K + K - ), ω(π + π - π0) or the ρ 0 - ω 0 (π + π - ) mixing. These processes allow to test independently the CP symmetry, which violation has not been observed yet in the baryonic sector, and the T symmetry, which experimental proofs are limited. Among the possible perspectives, a precise measurement of the Λ b 0 lifetime could contribute to the resolution of the raising theoretical-experimental puzzle. A phenomenological model of the Λ b 0 → Λ 0 V decays has been performed, from which branching ratios and angular distributions have been estimated. An advanced study of the reconstruction and the selection of these reactions by the LHCb apparatus shows that the channel Λ b 0 → Λ 0 J/Ψ is the dominant channel on both statistics and purity aspects. The Λ b 0 lifetime measure is the most imminent result; the constrains on asymmetries due to CP and T violation require several data taking years. Besides, an instrumental work has been achieved on the read-out electronics, called Front-End, of the experiment pre-shower. This contribution takes into account the validation of the prototype boards and the development of tools required by the qualification of the 100 production boards. (author) 2. Physics of the Lorentz Group Science.gov (United States) Başkal, Sibel 2015-11-01 This book explains the Lorentz mathematical group in a language familiar to physicists. While the three-dimensional rotation group is one of the standard mathematical tools in physics, the Lorentz group of the four-dimensional Minkowski space is still very strange to most present-day physicists. It plays an essential role in understanding particles moving at close to light speed and is becoming the essential language for quantum optics, classical optics, and information science. The book is based on papers and books published by the authors on the representations of the Lorentz group based on harmonic oscillators and their applications to high-energy physics and to Wigner functions applicable to quantum optics. It also covers the two-by-two representations of the Lorentz group applicable to ray optics, including cavity, multilayer and lens optics, as well as representations of the Lorentz group applicable to Stokes parameters and the Poincaré sphere on polarization optics. 3. Lorentz invariance on trial in the weak decay of polarized atoms Energy Technology Data Exchange (ETDEWEB) Mueller, Stefan E., E-mail: [email protected] [Kernfysisch Versneller Instituut (Netherlands) 2013-03-15 One of the most fundamental principles underlying our current understanding of nature is the invariance of the laws of physics under Lorentz transformations. Theories trying to unify the Standard Model with quantum gravity suggest that this invariance may be broken by the presence of Lorentz-violating background fields. Dedicated high-precision experiments at low energies could observe such suppressed signals from the Planck scale. At KVI, a test on Lorentz invariance of the weak interaction is performed searching for a dependence of the decay rate of spin-polarized nuclei on the orientation of their spin with respect to a fixed absolute galactical reference frame. An observation of such a dependence would imply a violation of Lorentz invariance. 4. Charge independence and charge symmetry Energy Technology Data Exchange (ETDEWEB) Miller, G A [Washington Univ., Seattle, WA (United States). Dept. of Physics; van Oers, W T.H. [Manitoba Univ., Winnipeg, MB (Canada). Dept. of Physics; [TRIUMF, Vancouver, BC (Canada) 1994-09-01 Charge independence and charge symmetry are approximate symmetries of nature, violated by the perturbing effects of the mass difference between up and down quarks and by electromagnetic interactions. The observations of the symmetry breaking effects in nuclear and particle physics and the implications of those effects are reviewed. (author). 145 refs., 3 tabs., 11 figs. 5. Charge independence and charge symmetry International Nuclear Information System (INIS) Miller, G.A. 1994-09-01 Charge independence and charge symmetry are approximate symmetries of nature, violated by the perturbing effects of the mass difference between up and down quarks and by electromagnetic interactions. The observations of the symmetry breaking effects in nuclear and particle physics and the implications of those effects are reviewed. (author). 145 refs., 3 tabs., 11 figs 6. Tests of CPT, Lorentz invariance and the WEP with antihydrogen International Nuclear Information System (INIS) Holzscheiter, M.H. 1999-01-01 Antihydrogen atoms, produced near rest, trapped in a magnetic well, and cooled to the lowest possible temperature (kinetic energy) could provide an extremely powerful tool for the search of violations of CPT and Lorentz invariance. Equally well, such a system could be used for searches of violations of the Weak Equivalence Principle (WEP) at high precision. The author describes his plans to form a significant number of cold, trapped antihydrogen atoms for comparative precision spectroscopy of hydrogen and antihydrogen and comment on possible first experiments 7. CPT symmetry tests with cold anti {rho} and antihydrogen Energy Technology Data Exchange (ETDEWEB) Yamazaki, Yasunori [RIKEN, Atomic Physics Laboratory, 2-1 Hirosawa, Wako, Saitama, 351-0198 (Japan); Ulmer, Stefan [RIKEN, Ulmer Initiative Research Unit, 2-1 Hirosawa, Wako, Saitama, 351-0198 (Japan) 2013-07-15 Precision comparisons of the properties of particles and their corresponding antiparticles are highly relevant because the Standard Model of elementary particle physics, a local, Lorentz-invariant field theory, is necessarily symmetric with respect to the combined CPT operation. This symmetry defines exact equality between the fundamental properties of particles and their anti-images. Any measured and confirmed violation constitutes a significant challenge to the Standard Model. Recent results of different CPT-tests are summarized, with emphasis to the high-precision measurement of the magnetic moment of the proton and the antiproton, as well as the precision investigation of antihydrogen ground state hyperfine splitting. (copyright 2013 by WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim) 8. Lorentz Covariance of Langevin Equation International Nuclear Information System (INIS) Koide, T.; Denicol, G.S.; Kodama, T. 2008-01-01 Relativistic covariance of a Langevin type equation is discussed. The requirement of Lorentz invariance generates an entanglement between the force and noise terms so that the noise itself should not be a covariant quantity. (author) 9. Parity violation in neutron resonances International Nuclear Information System (INIS) Mitchell, G.E.; Lowie, L.Y.; Bowman, J.D.; Knudson, J.; Crawford, B.E.; Delheij, P.P.J.; Haseyama, T.; Masaike, A.; Matsuda, Y.; Masuda, Y. 1997-01-01 The observation of very large parity violation in neutron resonances has led to a new approach to the study of symmetry breaking in nuclei. The origin of the enhancement of parity violation is discussed, as well as the new (statistical) analysis approach. The TRIPLE experimental system and analysis methods, their improvements are described. Sign correlation and results from recent parity violation experiments are presented and discussed. (author) 10. Self-similarity of high-pT hadron production in cumulative processes and violation of discrete symmetries at small scales (suggestion for experiment) International Nuclear Information System (INIS) Tokarev, M.V.; Zborovsky, I. 2009-01-01 The hypothesis of self-similarity of hadron production in relativistic heavy ion collisions for search for phase transition in a nuclear matter is discussed. It is offered to use the established features of z-scaling for revealing signatures of new physics in cumulative region. It is noted that selection of events on centrality in cumulative region could help to localize a position of a critical point. Change of parameters of the theory (a specific heat and fractal dimensions) near to a critical point is considered as a signature of new physics. The relation of the power asymptotic of ψ(z) at high z, anisotropy of momentum space due to spontaneous symmetry breaking, and discrete (C, P, T) symmetries is emphasized 11. Study of the breaking of the CP symmetry in the BABAR experiment; Etude de la violation de la symetrie CP dans l'experience BABAR Energy Technology Data Exchange (ETDEWEB) Ganjour, S 2007-09-15 This report summarizes my scientific activities from 1995 to 2007. During this period of time, my research work was related to the particle physics experiment BABAR. The BABAR experiment has been running since 1999 at the PEP-II e{sup +}e{sup -} asymmetric B-factory located at SLAC. This experiment searches for CP violation in the system of B mesons and tests the Standard Model through the measurements of the angles and the sides of the Unitarity Triangle. My research work is divided in five main topics: study of the BABAR magnet system and measurement of the magnetic field in the central tracking volume; project of the particle identification system based on aerogel counters for the forward region of the detector; conception of the magnetic shield and measurements of the fringe field in the region of photomultipliers of the DIRC (Detector of Internally Reflected Cherenkov light) system, the principal particle identification system of BABAR; development of the partial reconstruction technique of B mesons and study of the B{sup 0} {yields} D{sub s}{sup *} + D{sup *-} decays; measurement of CP violation in the B{sup 0} {yields} D{sup *{+-}}{pi}{sup {+-}} decays and constraint on the Unitary Triangle parameter sin(2{beta} + {gamma}) using these decays. (author) 12. R-parity violating supersymmetry CERN Document Server Barbier, R.; Besancon, M.; Chemtob, M.; Deandrea, A.; Dudas, E.; Fayet, Pierre; Lavignac, S.; Moreau, G.; Perez, E.; Sirois, Y. 2004-01-01 The possible appearance of R-parity violating couplings, and hence implicitly the question of lepton and baryon number conservation, has been emphasised since the early development of supersymmetric theories. The rich phenomenology implied by R-parity violation has now gained full attention in the search for supersymmetry. In this review, theoretical and phenomenological implications of R-parity violation in supersymmetric theories are discussed, in relation with particle and astroparticle physics. Fundamental aspects include the relation with continuous and discrete symmetries, up to more recent developments on the Abelian family symmetries and hierarchy of R-parity violating couplings. The question of the generation of the standard model neutrino masses and mixings is presented. The possible contributions of R-parity violating Yukawa couplings in processes involving virtual supersymmetric particles and the resulting constraints are reviewed. Finally, a survey of the direct production of supersymmetric parti... 13. Testing Lorentz invariance of dark matter with satellite galaxies Energy Technology Data Exchange (ETDEWEB) Bettoni, Dario [Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg (Germany); Nusser, Adi [Physics Department and the Asher Space Science Institute—Technion, Haifa 32000 (Israel); Blas, Diego; Sibiryakov, Sergey, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Theoretical Physics Department, CERN, CH-1211 Geneva 23 (Switzerland) 2017-05-01 We develop the framework for testing Lorentz invariance in the dark matter sector using galactic dynamics. We consider a Lorentz violating (LV) vector field acting on the dark matter component of a satellite galaxy orbiting in a host halo. We introduce a numerical model for the dynamics of satellites in a galactic halo and for a galaxy in a rich cluster to explore observational consequences of such an LV field. The orbital motion of a satellite excites a time dependent LV force which greatly affects its internal dynamics. Our analysis points out key observational signatures which serve as probes of LV forces. These include modifications to the line of sight velocity dispersion, mass profiles and shapes of satellites. With future data and a more detailed modeling these signatures can be exploited to constrain a new region of the parameter space describing the LV in the dark matter sector. 14. Are we observing Lorentz violation in gamma ray bursts? International Nuclear Information System (INIS) Pavlopoulos, Theodore G. 2005-01-01 From recent observations of gamma-ray bursts (GRBs), it appears that spectral time lags between higher-energy gamma rays photons and lower-energy photons vary with energy difference and time (distance) traveled. These lags appear to be smaller for the most luminous (close) bursts but larger for the fainter (farther away) bursts. From this observation, it has been suggested that it might be possible to determine the distance (L) these bursts have traveled from these time lags alone, without performing any red-shift measurements. These observed spreads (dispersion) of high-energy electromagnetic pulses of different energies with time contradict the special theory of relativity (STR). However, extended theories (ET) of the STR have been developed that contain a dispersive term, predicting the above observations. An example of such an ET is presented, allowing us to derive a relationship between time lags of gamma rays of different energies and distance L traveled from their origin. In addition, this theory predicts the origin of X-ray flashes 15. Gamma-Ray, Cosmic Ray and Neutrino Tests of Lorentz Invariance and Quantum Gravity Models Science.gov (United States) Stecker, Floyd 2011-01-01 High-energy astrophysics observations provide the best possibilities to detect a very small violation of Lorentz invariance such as may be related to the structure of space-time near the Planck scale of approximately 10(exp -35) m. I will discuss here the possible signatures of Lorentz invariance violation (LIV) from observations of the spectra, polarization, and timing of gamma-rays from active galactic nuclei and gamma-ray bursts. Other sensitive tests are provided by observations of the spectra of ultrahigh energy cosmic rays and neutrinos. Using the latest data from the Pierre Auger Observatory one can already derive an upper limit of 4.5 x 10(exp -23) to the amount of LIV of at a proton Lorentz factor of approximately 2 x 10(exp 11). This result has fundamental implications for quantum gravity models. I will also discuss the possibilities of using more sensitive space based detection techniques to improve searches for LIV in the future. 16. Transport properties of stochastic Lorentz models NARCIS (Netherlands) Beijeren, H. van Diffusion processes are considered for one-dimensional stochastic Lorentz models, consisting of randomly distributed fixed scatterers and one moving light particle. In waiting time Lorentz models the light particle makes instantaneous jumps between scatterers after a stochastically distributed 17. Constraints on CPT violation from Wilkinson Microwave Anisotropy Probe three year polarization data: A wavelet analysis International Nuclear Information System (INIS) Cabella, Paolo; Silk, Joseph; Natoli, Paolo 2007-01-01 We perform a wavelet analysis of the temperature and polarization maps of the cosmic microwave background (CMB) delivered by the Wilkinson Microwave Anisotropy Probe experiment in search for a parity-violating signal. Such a signal could be seeded by new physics beyond the standard model, for which the Lorentz and CPT symmetries may not hold. Under these circumstances, the linear polarization direction of a CMB photon may get rotated during its cosmological journey, a phenomenon also called cosmological birefringence. Recently, Feng et al. have analyzed a subset of the Wilkinson Microwave Anisotropy Probe and BOOMERanG 2003 angular power spectra of the CMB, deriving a constraint that mildly favors a nonzero rotation. By using wavelet transforms we set a tighter limit on the CMB photon rotation angle Δα=-2.5±3.0 (Δα=-2.5±6.0) at the one (two) σ level, consistent with a null detection 18. de Sitter group as a symmetry for optical decoherence International Nuclear Information System (INIS) Baskal, S; Kim, Y S 2006-01-01 Stokes parameters form a Minkowskian 4-vector under various optical transformations. As a consequence, the resulting two-by-two density matrix constitutes a representation of the Lorentz group. The associated Poincare sphere is a geometric representation of the Lorentz group. Since the Lorentz group preserves the determinant of the density matrix, it cannot accommodate the decoherence process through the decaying off-diagonal elements of the density matrix, which yields to an increase in the value of the determinant. It is noted that the O(3, 2) de Sitter group contains two Lorentz subgroups. The change in the determinant in one Lorentz group can be compensated by the other. It is thus possible to describe the decoherence process as a symmetry transformation in the O(3, 2) space. It is shown also that these two coupled Lorentz groups can serve as a concrete example of Feynman's rest of the universe 19. Study of CP Symmetry Violation in the Charmonium-K*(892) Channel By a Complete Time Dependent Angular Analysis (BaBar Experiment) Energy Technology Data Exchange (ETDEWEB) T' Jampens, Stephane; /Orsay 2006-09-18 This thesis presents the full-angular time-dependent analysis of the vector-vector channel B{sub d}{sup 0} {yields} J/{psi}(K{sub S}{sup 0}{pi}{sup 0})*{sup 0}. After a review of the CP violation in the B meson system, the phenomenology of the charmonium-K*(892) channels is exposed. The method for the measurement of the transversity amplitudes of the B {yields} J/{psi}K*(892), based on a pseudo-likelihood method, is then exposed. The results from a 81.9 fb{sup -1} of collected data by the BABAR detector at the {Upsilon}(4S) resonance peak are |A{sub 0}|{sup 2} = 0.565 {+-} 0.011 {+-} 0.004, |A{sub {parallel}}|{sup 2} = 0.206 {+-} 0.016 {+-} 0.007, |A{sub {perpendicular}}|{sup 2} = 0.228 {+-} 0.016 {+-} 0.007, {delta}{sub {parallel}} = -2.766 {+-} 0.105 {+-} 0.040 and {delta}{sub {perpendicular}} = 2.935 {+-} 0.067 {+-} 0.040. Note that ({delta}{sub {parallel}}, {delta}{sub {perpendicular}}) {yields} (-{delta}{sub {parallel}}, {pi} - {delta}{sub {perpendicular}}) is also a solution. The strong phases {delta}{sub {parallel}} and {delta}{sub {perpendicular}} are at {approx}> 3{sigma} from {+-}{pi}, signing the presence of final state interactions and the breakdown of the factorization hypothesis. The forward-backward analysis of the K{pi} mass spectrum revealed the presence of a coherent S-wave interfering with the K*(892). It is the first evidence of this wave in the K{pi} system coming from a B meson. The particularity of the B{sub d}{sup 0} {yields} J/{psi}(K{sub S}{sup 0}{pi}{sup 0})*{sup 0} channel is to have a time-dependent but also an angular distribution which allows to measure sin 2{beta} but also cos2{beta}. The results from an unbinned maximum likelihood fit are sin 2{beta} = -0.10 {+-} 0.57 {+-} 0.14 and cos 2{beta} = 3.32{sub -0.96}{sup +0.76} {+-} 0.27 with the transversity amplitudes fixed to the values given above. The other solution for the strong phases flips the sign of cos 2{beta}. Theoretical considerations based on the s-quark helicity 20. CP violation CERN Document Server 1989-01-01 Contents: CP Phenomenology: Introduction to CP Violation (C Jarlskog); CP-Violation in the K 0 -K 0 -System (K Kleinknecht); The Quark Mixing Matrix, Charm Decays and B Decays (S Stone); The Question of CP Noninvariance - As Seen through the Eyes of Neutral Beauty (I I Bigi et al.); In Search of CP Noninvariance in Heavy Quark Systems (L-L Chau); CP Violation at High Energy e + e - Colliders (J Bernabéu & M B Gavela); CP Violation in the Standard Model with Four Families (A Datta & E A Paschos); CP Effects When Neutrinos are their Own Antiparticles (B Kayser); On Spontaneous CP Violation Trigg 1. P and T violations in QED International Nuclear Information System (INIS) Pleitez, V. 1983-01-01 An abelian gauge theory with violation of P and T symmetries, is constructed other features of usual spinor quantum electrodynamics are maintained. The theory is applied to some scattering processes with polarized and unpolarized electrons. (Author) [pt 2. Symmetry and symmetry breaking International Nuclear Information System (INIS) Balian, R.; Lambert, D.; Brack, A.; Lachieze-Rey, M.; Emery, E.; Cohen-Tannoudji, G.; Sacquin, Y. 1999-01-01 The symmetry concept is a powerful tool for our understanding of the world. It allows a reduction of the volume of information needed to apprehend a subject thoroughly. Moreover this concept does not belong to a particular field, it is involved in the exact sciences but also in artistic matters. Living beings are characterized by a particular asymmetry: the chiral asymmetry. Although this asymmetry is visible in whole organisms, it seems it comes from some molecules that life always produce in one chirality. The weak interaction presents also the chiral asymmetry. The mass of particles comes from the breaking of a fundamental symmetry and the void could be defined as the medium showing as many symmetries as possible. The texts put together in this book show to a great extent how symmetry goes far beyond purely geometrical considerations. Different aspects of symmetry ideas are considered in the following fields: the states of matter, mathematics, biology, the laws of Nature, quantum physics, the universe, and the art of music. (A.C.) 3.$CPT$violation searches and prospects for LHCb CERN Document Server van Tilburg, Jeroen 2015-03-06 An overview of current experimental bounds on$CPT$violation in neutral meson mixing is given. New values for the$CPT$asymmetry in the$B^0$and$B_s^0$systems are deduced from BaBar, Belle and LHCb data. With dedicated analyses, LHCb will be able to further improve the bounds on$CPT$violation in the$D^0$,$B^0$and$B_s^0$systems. Since$CPT$violation implies violation of Lorentz invariance, the observed$CPT$asymmetry will exhibit sidereal- and boost-dependent variations. Such$CPT$-violating and Lorentz-violating effects are accommodated in the framework of the Standard-Model Extension (SME). The large boost of the neutral mesons produced at LHCb results in a high sensitivity to the corresponding SME coefficients. For the$B^0$and$B_s^0$systems, using existing LHCb data, we determine with high precision the SME coefficients that are not varying with sidereal time. With a full sidereal analysis, LHCb will be able to improve the existing SME bounds in the$D^0$,$B^0$and$B_s^0$systems by up t... 4. Lorentz covariant theory of gravitation International Nuclear Information System (INIS) Fagundes, H.V. 1974-12-01 An alternative method for the calculation of second order effects, like the secular shift of Mercury's perihelium is developed. This method uses the basic ideas of thirring combined with the more mathematical approach of Feyman. In the case of a static source, the treatment used is greatly simplified. Besides, Einstein-Infeld-Hoffmann's Lagrangian for a system of two particles and spin-orbit and spin-spin interactions of two particles with classical spin, ie, internal angular momentum in Moller's sense, are obtained from the Lorentz covariant theory 5. CP violation CERN Multimedia CERN. Geneva 1999-01-01 In the first two lectures, CP violation in the K system is pedagogically reviewed: its manifestations in the neutral K meson systems, in rare K meson decays and in decays of charged K mesons, and results from classical and current experiments, are discussed. In the third lecture, CP Violation in the B system and the forthcoming experimental tests will be discussed. 6. CP violation Indian Academy of Sciences (India) We have just entered a period during which we expect considerable progress toward understanding CP violation. Here we review what we have learnt so far, and what is to be expected in the near future. To do this we cover the foundation of CP violation at a level which can be understood by physicists who are not working ... 7. Uniformly bounded representations of the Lorentz groups International Nuclear Information System (INIS) Brega, A.O. 1982-01-01 For the Lorentz group G = SO/sub e/(n + 1, 1)(ngreater than or equal to 2) the author constructs a family of uniformly bounded representations by means of analytically continuing a certain normalization of the unitary principal series. The method the author uses relies on an analysis of various operators under a Mellin transform and extends earlier work of E.N. Wilson. In a series of papers Kunze and Stein initiated the theory of uniformly bounded representations of semisimple Lie groups; the starting point is the unitary principal series T(sigma,s) obtained in a certain subgroup M of G and a purely imaginary number s. From there Kunze and Stein constructed families of representations R(sigma,s) depending analytically on a parameter s in a domain D of C containing the imaginary axis which are unitarily equilvalent to T(sigma,s) for s contained in the set of imaginary numbers and whose operator norms are uniformly bounded for each s in D. In the case of the Lorentz groups SO/sub e/(n + 1, 1)(ngreater than or equal to2) and the trivial representation 1 of M, E.N. Wilson obtained such a family R(1,s) for the domain D = [s contained in the set of C: absolute value Re(s) Vertical Bar2]. For this domain D and for any representation sigma of M the author provides a family R(sigma,s) of uniformly bounded representations analytically continuing T(sigma,s), thereby generalizing Wilson's work. The author has also investigated certain symmetry properties of the representations R(sigma,s) under the action of the Weyl group. The trivial representation is Weyl group invariant and the family R(1,s) obtained by Wilson satisfies R(1,s) = R(1,-s) reflecting this. Obtained was the analogous result R(sigma,s) = R(sigma,-s) for some well known representations sigma that are Weyl group invariant. This involves the explicit computation of certain constants arising in the Fourier transforms of intertwining operators 8. B Factories and CP Violation International Nuclear Information System (INIS) Wilson, R. J. 2001-01-01 In this lecture, I will give an overview of the current and planned B meson facilities and the motivation for building them. The emphasis will be on the BaBar experiment at the PEP-II accelerator and on the primary physics motivation for these facilities: charge-parity symmetry violation. (Author) 11 refs 9. CP violating phenomena and theoretical results International Nuclear Information System (INIS) Grimus, W. 1987-01-01 An introduction to CP violating phenomena is given and the standard model and its most popular low energy extensions in this context are reviewed. The discussion comprises the minimal supersymmetric extension of the standard model, left-right symmetry, the standard model with more than one Higgs doublet and gauged horizontal symmetries. (Author) 10. Cosmological CP Violation CERN Document Server Tomaschitz, R 1994-01-01 Spinor fields are studied in infinite, topologically multiply connected Robertson-Walker cosmologies. Unitary spinor representations for the discrete covering groups of the spacelike slices are constructed. The spectral resolution of Dirac's equation is given in terms of horospherical elementary waves, on which the treatment of spin and energy is based in these cosmologies. The meaning of the energy and the particle-antiparticle concept is explained in the context of this varying cosmic background. Discrete symmetries, in particular inversions of the multiply connected spacelike slices, are studied. The violation of the unitarity of the parity operator, due to self-interference of P-reflected wave packets, is discussed. The violation of the CP and CPT invariance - already on the level of the free Dirac equation on this cosmological background - is pointed out. 11. CP violation International Nuclear Information System (INIS) Gilman, F.J. 1989-12-01 Predictions for CP violation in the three generation Standard Model are reviewed based on what is known about the Cabibbo-Kobayashi-Maskawa matrix. Application to the K and B meson systems are emphasized. 43 refs., 13 figs 12. Lorentz-invariant Bell's inequality International Nuclear Information System (INIS) Kim, Won Tae; Son, Edwin J. 2005-01-01 We study Bell's inequality in relation to the Einstein-Podolsky-Rosen paradox in the relativistic regime. For this purpose, a relativistically covariant analysis is used in the calculation of the Bell's inequality, which results in the maximally violated Bell's inequality in any reference frame 13. CP violation International Nuclear Information System (INIS) Quinn, H. 1995-12-01 In this talk the author briefly reviews the cosmological importance of CP violation and the status of calculations of baryogenisis in the context of the Standard Model. The author then turns to a discussion of Standard Model Predictions for CP violation in B decays, stressing the importance of multiple measurements to overconstrain the model parameters and thus search for indications of beyond-Standard-Model physics 14. Cosmology and CPT violating neutrinos Energy Technology Data Exchange (ETDEWEB) Barenboim, Gabriela; Salvado, Jordi [Universitat de Valencia-CSIC, Departament de Fisica Teorica y Instituto de Fisica Corpuscular, Burjassot (Spain) 2017-11-15 The combination charge conjugation-parity-time reversal (CPT) is a fundamental symmetry in our current understanding of nature. As such, testing CPT violation is a strongly motivated path to explore new physics. In this paper we study CPT violation in the neutrino sector, giving for the first time a bound, for a fundamental particle, in the CPT violating particle-antiparticle gravitational mass difference. We argue that cosmology is nowadays the only data sensitive to CPT violation for the neutrino-antineutrino mass splitting and we use the latest data release from Planck combined with the current baryonic-acoustic-oscillation measurement to perform a full cosmological analysis. To show the potential of the future experiments we also show the results for Euclid, a next generation large scale structure experiment. (orig.) 15. CP violating scalar Dark Matter Science.gov (United States) Cordero-Cid, A.; Hernández-Sánchez, J.; Keus, V.; King, S. F.; Moretti, S.; Rojas, D.; Sokołowska, D. 2016-12-01 We study an extension of the Standard Model (SM) in which two copies of the SM scalar SU(2) doublet which do not acquire a Vacuum Expectation Value (VEV), and hence are inert, are added to the scalar sector. We allow for CP-violation in the inert sector, where the lightest inert state is protected from decaying to SM particles through the conservation of a Z 2 symmetry. The lightest neutral particle from the inert sector, which has a mixed CP-charge due to CP-violation, is hence a Dark Matter (DM) candidate. We discuss the new regions of DM relic density opened up by CP-violation, and compare our results to the CP-conserving limit and the Inert Doublet Model (IDM). We constrain the parameter space of the CP-violating model using recent results from the Large Hadron Collider (LHC) and DM direct and indirect detection experiments. 16. Symmetry and bifurcations of momentum mappings International Nuclear Information System (INIS) Arms, J.M.; Marsden, J.E.; Moncrief, V. 1981-01-01 The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface. (orig.) 17. Symmetry and bifurcations of momentum mappings Science.gov (United States) Arms, Judith M.; Marsden, Jerrold E.; Moncrief, Vincent 1981-01-01 The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface. 18. Solution of the Lorentz-Dirac equation based on a new momentum expression CERN Document Server Yan, C C 1998-01-01 The Lorentz-Dirac equation is solved based on a new momentum expression given by p sup a =1/c sup 2 (u submu p supmu)u sup a +k du sup a /d tau. This new momentum expression is the form proposed by Barut modified to satisfy the condition imposed by Dirac. The solution turns out to be well behaved without violating causality or causing runaway. (author) 19. Nuclear probes of fundamental symmetries International Nuclear Information System (INIS) Adelberger, E.G. 1983-01-01 Nuclear experiments which probe the parity (P) and time-reversal (T) symmetries and lepton-number conservation are reviewed. The P-violating NN interaction, studied in the NN system and in light nuclei, provides an unique window on ΔS=0 hadronic weak processes. Results are in accord with expectations. Sensitive searches for T-violation via detailed balance, T-odd correlations in γ and β-decay, and a possible neutron electric dipole moment (EDM) are discussed. No T-violation is observed. The EDM limit is almost good enough to eliminate one of the leading theoretical explanations for CP violation. Experimental studies of double β-decay are reviewed. Although ββ nu nu decay has been convincingly detected in geochemical experiments there is no evidence for the lepton number violating ββ decay mode 20. A q-deformed Lorentz algebra International Nuclear Information System (INIS) Schmidke, W.B.; Wess, J.; Muenchen Univ.; Zumino, B.; Lawrence Berkeley Lab., CA 1991-01-01 We derive a q-deformed version of the Lorentz algebra by deformating the algebra SL(2, C). The method is based on linear representations of the algebra on the complex quantum spinor space. We find that the generators usually identified with SL q (2, C) generate SU q (2) only. Four additional generators are added which generate Lorentz boosts. The full algebra of all seven generators and their coproduct is presented. We show that in the limit q→1 the generators are those of the classical Lorentz algebra plus an additional U(1). Thus we have a deformation of SL(2, C)xU(1). (orig.) 1. Holography without translational symmetry CERN Document Server Vegh, David 2013-01-01 We propose massive gravity as a holographic framework for describing a class of strongly interacting quantum field theories with broken translational symmetry. Bulk gravitons are assumed to have a Lorentz-breaking mass term as a substitute for spatial inhomogeneities. This breaks momentum-conservation in the boundary field theory. At finite chemical potential, the gravity duals are charged black holes in asymptotically anti-de Sitter spacetime. The conductivity in these systems generally exhibits a Drude peak that approaches a delta function in the massless gravity limit. Furthermore, the optical conductivity shows an emergent scaling law:$|\\sigma(\\omega)| \\approx {A \\over \\omega^{\\alpha}} + B$. This result is consistent with that found earlier by Horowitz, Santos, and Tong who introduced an explicit inhomogeneous lattice into the system. 2. Lorentz invariance and the rotor Doppler shift experiments International Nuclear Information System (INIS) Rodrigues Junior, W.A.; Tiomno, J. 1984-01-01 It is shown that 'Rotor Doppler shift Experiments' provide a way to distinguish Einstein's Special Relativity (SR) from Lorentz's Aether Theory (LAT). Misconceptions in previous papers involving the Doppler shift experiments are examined. The theoretical and experimental data available on rotor Doppler shift experiments are analysed. Two models of SR violating theories are used to predict the output of a recently proposed experiment by Torr and Kolen. The first one corresponds to (strict) LAT and the other to an extended form of LAT Contrary to the first, the second theory leads to results in agreement with the preliminary experimental data of Torr et al indicating a breakdown both of SR and strict LAT. (Author) [pt 3. Lorentz invariance and the rotor Doppler shift experiments International Nuclear Information System (INIS) Rodrigues Junior, W.A.; Tiomno, J. 1984-01-01 It is shown that 'Rotor Doppler shift Experiments' provide a way to distinguish Einstein's Special Relativity (SR) from Lorentz's Aether Theory (LAT). Misconceptions in previous papers involving the Doppler shift experiments are examined. The theoretical and experimental data available on rotor Doppler shift experiments are analysed. Two models of SR violating theories are used to predict the output of a recently proposed experiment by Torr and Kolen. The first one corresponds to (strict) LAT and the other to an extended form of LAT. Contrary to the first, the second theory leads to results in agreement with the preliminary experimental data of Torr et al indicating a breakdown both of SR and strict LAT. (Author) [pt 4. Tests of Lorentz invariance using a microwave resonator International Nuclear Information System (INIS) Wolf, Peter; Bize, Sebastien; Clairon, Andre; Santarelli, Giorgio; Luiten, Andre N.; Tobar, Michael E. 2003-01-01 The frequencies of a cryogenic sapphire oscillator and a hydrogen maser are compared to set new constraints on a possible violation of Lorentz invariance. We determine the variation of the oscillator frequency as a function of its orientation (Michelson-Morley test) and of its velocity (Kennedy-Thorndike test) with respect to a preferred frame candidate. We constrain the corresponding parameters of the Mansouri and Sexl test theory to δ-β+1/2=(1.5±4.2)x10 -9 and β-α-1=(-3.1±6.9)x10 -7 which is of the same order as the best previous result for the former and represents a 30-fold improvement for the latter 5. CP violation International Nuclear Information System (INIS) Gronau, M. 1995-01-01 We review the present status of the Standard Model of CP violation, which is based on a complex phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix. So far CP violation has been observed only in K 0 -K 0 mixing, consistent with a sizable phase. The implications of future CP nonconserving measusrements in K and B decays are discussed within the model. Direct CP violation in K→2π may be observed in the near future, however this would not be a powerful test of the model. B decays provide a wide variety of CP asymmetry measurements, which can serve as precise tests of the Standard Model in cases where the asymmetry is cleanly related to phases of CKM matrix elements. Some of the most promising cases are discussed. ((orig.)) 6. A precision test of Lorentz invariance using room-temperature high-finesse optical resonators International Nuclear Information System (INIS) Eisele, Christian 2009-01-01 An apparatus for a test of a basic postulate of the theory of Special Relativity, the isotropy of the speed of light, has been developed. Deviations from the isotropy imply a violation of Lorentz invariance, a symmetry assumed by all established theories of the fundamental forces. Such a signal may provide a glimpse on physics beyond our current theories of the fundamental forces, the General Theory of Relativity and the Standard Modell of particle physics. Since long theoreticians try to unify General Relativity and the Standard Modell within one theory, a grand unified theory (GUT). So far they did not succeed, although promising candidate theories have been developed, e.g. string theories or loop quantum gravity. However, there are hints that Lorentz invariance might not be an exact symmetry of nature, but that deviations are to be expected. This is a strong motivation for tests of Lorentz invariance with increased sensitivity as the one presented within this thesis. We employ, for the first time for a test of the isotropy of the speed of light, monolithic optical resonators fabricated from a glass ceramic with ultra low expansion coefficient (ULE). By means of a monolithic Nd:YAG-laser (λ = 1064 nm) we measure the difference between the resonance frequencies of two orthogonally oriented resonators. The low thermal expansion coefficient reduces the influence of thermal fluctuations on the resonance frequencies, which are a function of the mirror spacing and the speed of light inside the resonators only. The complete optical setup has been put on top of active vibration isolation supports, which strongly damp mechanical vibrations. This improves the short-time stability of the resonators resonance frequencies. This technique is used for the first time in a Speed of Light Isotropy Test (SLIT) experiment. Furthermore, a system for the stabilization of the tilt of the optics breadboard is implemented, based on electromagnetic actuators. This stabilization is 7. A precision test of Lorentz invariance using room-temperature high-finesse optical resonators Energy Technology Data Exchange (ETDEWEB) Eisele, Christian 2009-10-28 An apparatus for a test of a basic postulate of the theory of Special Relativity, the isotropy of the speed of light, has been developed. Deviations from the isotropy imply a violation of Lorentz invariance, a symmetry assumed by all established theories of the fundamental forces. Such a signal may provide a glimpse on physics beyond our current theories of the fundamental forces, the General Theory of Relativity and the Standard Modell of particle physics. Since long theoreticians try to unify General Relativity and the Standard Modell within one theory, a grand unified theory (GUT). So far they did not succeed, although promising candidate theories have been developed, e.g. string theories or loop quantum gravity. However, there are hints that Lorentz invariance might not be an exact symmetry of nature, but that deviations are to be expected. This is a strong motivation for tests of Lorentz invariance with increased sensitivity as the one presented within this thesis. We employ, for the first time for a test of the isotropy of the speed of light, monolithic optical resonators fabricated from a glass ceramic with ultra low expansion coefficient (ULE). By means of a monolithic Nd:YAG-laser ({lambda} = 1064 nm) we measure the difference between the resonance frequencies of two orthogonally oriented resonators. The low thermal expansion coefficient reduces the influence of thermal fluctuations on the resonance frequencies, which are a function of the mirror spacing and the speed of light inside the resonators only. The complete optical setup has been put on top of active vibration isolation supports, which strongly damp mechanical vibrations. This improves the short-time stability of the resonators resonance frequencies. This technique is used for the first time in a Speed of Light Isotropy Test (SLIT) experiment. Furthermore, a system for the stabilization of the tilt of the optics breadboard is implemented, based on electromagnetic actuators. This stabilization is 8. Symmetry breaking in string theory International Nuclear Information System (INIS) Potting, R. 1998-01-01 A mechanism for a spontaneous breakdown of CPT symmetry appears in string theory, with possible implications for particle models. A realistic string theory might exhibit CPT violation at levels detectable in current or future experiments. A possible new mechanism for baryogenesis in the early Universe is also discussed 9. Effective lagrangian description on discrete gauge symmetries International Nuclear Information System (INIS) Banks, T. 1989-01-01 We exhibit a simple low-energy lagrangian which describes a system with a discrete remnant of a spontaneously broken continuous gauge symmetry. The lagrangian gives a simple description of the effects ascribed to such systems by Krauss and Wilczek: black holes carry discrete hair and interact with cosmic strings, and wormholes cannot lead to violation of discrete gauge symmetries. (orig.) 10. Reflections on symmetries at SPIN '94 International Nuclear Information System (INIS) Page, S.A. 1995-01-01 In my view, the parallel sessions on ''Symmetries'' were amongst the most stimulating sessions of this conference. Speakers reported on experimental tests of Charge Symmetry, Parity, and Time Reversal violation and their theoretical interpretation, spanning a wide range of energy scales and experimental techniques. I hope that this brief summary will whet the reader's appetite to explore the many contributed papers which follow 11. Can Lorentz-breaking fermionic condensates form in large N strongly-coupled Lattice Gauge Theories? OpenAIRE Tomboulis, E. T. 2010-01-01 The possibility of Lorentz symmetry breaking (LSB) has attracted considerable attention in recent years for a variety of reasons, including the attractive prospect of the graviton as a Goldstone boson. Though a number of effective field theory analyses of such phenomena have recently been given it remains an open question whether they can take place in an underlying UV complete theory. Here we consider the question of LSB in large N lattice gauge theories in the strong coupling limit. We appl... 12. Testing Lorentz invariance of dark matter CERN Document Server Blas, Diego; Sibiryakov, Sergey 2012-01-01 We study the possibility to constrain deviations from Lorentz invariance in dark matter (DM) with cosmological observations. Breaking of Lorentz invariance generically introduces new light gravitational degrees of freedom, which we represent through a dynamical timelike vector field. If DM does not obey Lorentz invariance, it couples to this vector field. We find that this coupling affects the inertial mass of small DM halos which no longer satisfy the equivalence principle. For large enough lumps of DM we identify a (chameleon) mechanism that restores the inertial mass to its standard value. As a consequence, the dynamics of gravitational clustering are modified. Two prominent effects are a scale dependent enhancement in the growth of large scale structure and a scale dependent bias between DM and baryon density perturbations. The comparison with the measured linear matter power spectrum in principle allows to bound the departure from Lorentz invariance of DM at the per cent level. 13. Testing Lorentz invariance of dark matter Energy Technology Data Exchange (ETDEWEB) Blas, Diego [Theory Group, Physics Department, CERN, CH-1211 Geneva 23 (Switzerland); Ivanov, Mikhail M.; Sibiryakov, Sergey, E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Faculty of Physics, Moscow State University, Vorobjevy Gory, 119991 Moscow (Russian Federation) 2012-10-01 We study the possibility to constrain deviations from Lorentz invariance in dark matter (DM) with cosmological observations. Breaking of Lorentz invariance generically introduces new light gravitational degrees of freedom, which we represent through a dynamical timelike vector field. If DM does not obey Lorentz invariance, it couples to this vector field. We find that this coupling affects the inertial mass of small DM halos which no longer satisfy the equivalence principle. For large enough lumps of DM we identify a (chameleon) mechanism that restores the inertial mass to its standard value. As a consequence, the dynamics of gravitational clustering are modified. Two prominent effects are a scale dependent enhancement in the growth of large scale structure and a scale dependent bias between DM and baryon density perturbations. The comparison with the measured linear matter power spectrum in principle allows to bound the departure from Lorentz invariance of DM at the per cent level. 14. The Scientific Correspondence of H A Lorentz CERN Document Server Kox, AJ 2008-01-01 Presents a selection of more than 400 letters from and to the Dutch physicist and Nobel Prize winner Hendrik Antoon Lorentz (1853-1928), covering the period from 1883 until a few months before his death. 15. On systems having Poincaré and Galileo symmetry International Nuclear Information System (INIS) Holland, Peter 2014-01-01 Using the wave equation in d≥1 space dimensions it is illustrated how dynamical equations may be simultaneously Poincaré and Galileo covariant with respect to different sets of independent variables. This provides a method to obtain dynamics-dependent representations of the kinematical symmetries. When the field is a displacement function both symmetries have a physical interpretation. For d=1 the Lorentz structure is utilized to reveal hitherto unnoticed features of the non-relativistic Chaplygin gas including a relativistic structure with a limiting case that exhibits the Carroll group, and field-dependent symmetries and associated Noether charges. The Lorentz transformations of the potentials naturally associated with the Chaplygin system are given. These results prompt the search for further symmetries and it is shown that the Chaplygin equations support a nonlinear superposition principle. A known spacetime mixing symmetry is shown to decompose into label-time and superposition symmetries. It is shown that a quantum mechanical system in a stationary state behaves as a Chaplygin gas. The extension to d>1 is used to illustrate how the physical significance of the dual symmetries is contingent on the context by showing that Maxwell’s equations exhibit an exact Galileo covariant formulation where Lorentz and gauge transformations are represented by field-dependent symmetries. A natural conceptual and formal framework is provided by the Lagrangian and Eulerian pictures of continuum mechanics 16. Discrete symmetries in the MSSM Energy Technology Data Exchange (ETDEWEB) Schieren, Roland 2010-12-02 The use of discrete symmetries, especially abelian ones, in physics beyond the standard model of particle physics is discussed. A method is developed how a general, abelian, discrete symmetry can be obtained via spontaneous symmetry breaking. In addition, anomalies are treated in the path integral approach with special attention to anomaly cancellation via the Green-Schwarz mechanism. All this is applied to the minimal supersymmetric standard model. A unique Z{sup R}{sub 4} symmetry is discovered which solves the {mu}-problem as well as problems with proton decay and allows to embed the standard model gauge group into a simple group, i.e. the Z{sup R}{sub 4} is compatible with grand unification. Also the flavor problem in the context of minimal flavor violation is addressed. Finally, a string theory model is presented which exhibits the mentioned Z{sup R}{sub 4} symmetry and other desirable features. (orig.) 17. Discrete symmetries in the MSSM International Nuclear Information System (INIS) Schieren, Roland 2010-01-01 The use of discrete symmetries, especially abelian ones, in physics beyond the standard model of particle physics is discussed. A method is developed how a general, abelian, discrete symmetry can be obtained via spontaneous symmetry breaking. In addition, anomalies are treated in the path integral approach with special attention to anomaly cancellation via the Green-Schwarz mechanism. All this is applied to the minimal supersymmetric standard model. A unique Z R 4 symmetry is discovered which solves the μ-problem as well as problems with proton decay and allows to embed the standard model gauge group into a simple group, i.e. the Z R 4 is compatible with grand unification. Also the flavor problem in the context of minimal flavor violation is addressed. Finally, a string theory model is presented which exhibits the mentioned Z R 4 symmetry and other desirable features. (orig.) 18. Fundamental symmetry studies at Los Alamos using epithermal neutrons International Nuclear Information System (INIS) Bowman, C.D.; Bowman, J.D.; Yuan, V.W. 1988-01-01 Fundamental symmetry studies using intense polarized beams of epithermal neutrons are underway at the LANSCE facility of the Los Alamos National Laboratory. Three classes of symmetry experiments can be explored: parity violation, and time reversal invariance violation for both parity-violating and parity-conserved observables. The experimental apparatus is described and performance illustrated with examples of recent measurements. Possible improvements in the facilities and prospective experiments are discussed. 15 refs., 10 figs 19. High Energy Astrophysics Tests of Lorentz Invariance and Quantum Gravity Models Science.gov (United States) Stecker, Floyd W. 2012-01-01 High energy astrophysics observations provide the best possibilities to detect a very small violation of Lorentz invariance such as may be related to the structure of space-time near the Planck scale of approx.10(exp -35) m. I will discuss the possible signatures of Lorentz invariance violation (LIV) that can be manifested by observing of the spectra, polarization, and timing of gamma-rays from active galactic nuclei and gamma-ray bursts. Other sensitive tests are provided by observations of the spectra of ultrahigh energy cosmic rays and neutrinos. Using the latest data from the Pierre Auger Observatory one can already derive an upper limit of 4.5 x 10(exp -23) on the fraction of LIV at a Lorentz factor of approx. 2 x 10(exp 11). This result has fundamental implications for quantum gravity models. I will also discuss the possibilities of using more sensitive space-based detection techniques to improve searches for LIV in the future. I will also discuss how the LIV formalism casts doubt on the OPERA superluminal neutrino claim. 20. Parity violation in the compound nucleus International Nuclear Information System (INIS) Mitchell, G. E.; Crawford, B. E.; Grossmann, C. A.; Lowie, L. Y.; Bowman, J. D.; Knudson, J.; Penttilae, S.; Seestrom, S. J.; Smith, D. A.; Yen, Yi-Fen; Yuan, V. W.; Delheij, P. P. J.; Haseyama, T.; Masaike, A.; Matsuda, Y.; Postma, H.; Roberson, N. R.; Sharapov, E. I.; Stephenson, S. L. 1999-01-01 Measurements have been performed on the helicity dependence of the neutron resonance cross section for many nuclei by our TRIPLE Collaboration. A large number of parity violations are observed. Generic enhancements amplify the signal for symmetry breaking and the stochastic properties of the compound nucleus permit the strength of the symmetry-breaking interaction to be determined without knowledge of the wave functions of individual states. A total of 15 nuclei have been analyzed with this statistical approach. The results are summarized 1. Compatibility of the Ampere and Lorentz force laws with the virtual-work concept International Nuclear Information System (INIS) Graneau, P. 1983-01-01 Whenever the reaction forces between parts of an electric circuit have to be calculated, as in the design of railguns, a choice has to be made between three available formulae which have evolved during the past 160 years. The first was Ampere's force law for the mechanical interaction between two current elements. Neumann then derived the virtual-work formula from what may be called the Ampere-Neumann electrodynamics. The last to be introduced was the Lorentz force law. This paper investigates whether both the Amperian and the Lorentzian forces are compatible with the virtual-work concept. The conclusion is that only Ampere's formula agrees in all cases with the virtual-work idea, but in special circumstances the Lorentz law will give the same result. After demonstrating how Ampere's law can be derived from the virtual-work formula, it is shown that for two closed circuits the relativistic component of the Lorentz force vanishes under the double integral around the two circuits. The remaining nonvanishing term is also present in the Ampere electrodynamics. This is not the case when considering the reaction forces between two parts of an isolated circuit. The Lorentz force is then, in general, not compatible with the virtual-work concept unless the circuit possesses a high degree of symmetry 2. Spinor Structure and Internal Symmetries Science.gov (United States) Varlamov, V. V. 2015-10-01 Spinor structure and internal symmetries are considered within one theoretical framework based on the generalized spin and abstract Hilbert space. Complex momentum is understood as a generating kernel of the underlying spinor structure. It is shown that tensor products of biquaternion algebras are associated with the each irreducible representation of the Lorentz group. Space-time discrete symmetries P, T and their combination PT are generated by the fundamental automorphisms of this algebraic background (Clifford algebras). Charge conjugation C is presented by a pseudoautomorphism of the complex Clifford algebra. This description of the operation C allows one to distinguish charged and neutral particles including particle-antiparticle interchange and truly neutral particles. Spin and charge multiplets, based on the interlocking representations of the Lorentz group, are introduced. A central point of the work is a correspondence between Wigner definition of elementary particle as an irreducible representation of the Poincaré group and SU(3)-description (quark scheme) of the particle as a vector of the supermultiplet (irreducible representation of SU(3)). This correspondence is realized on the ground of a spin-charge Hilbert space. Basic hadron supermultiplets of SU(3)-theory (baryon octet and two meson octets) are studied in this framework. It is shown that quark phenomenologies are naturally incorporated into presented scheme. The relationship between mass and spin is established. The introduced spin-mass formula and its combination with Gell-Mann-Okubo mass formula allows one to take a new look at the problem of mass spectrum of elementary particles. 3. Einstein-Yang-Mills-Lorentz black holes Energy Technology Data Exchange (ETDEWEB) Cembranos, Jose A.R.; Gigante Valcarcel, Jorge [Universidad Complutense de Madrid, Departamento de Fisica Teorica I, Madrid (Spain) 2017-12-15 Different black hole solutions of the coupled Einstein-Yang-Mills equations have been well known for a long time. They have attracted much attention from mathematicians and physicists since their discovery. In this work, we analyze black holes associated with the gauge Lorentz group. In particular, we study solutions which identify the gauge connection with the spin connection. This ansatz allows one to find exact solutions to the complete system of equations. By using this procedure, we show the equivalence between the Yang-Mills-Lorentz model in curved space-time and a particular set of extended gravitational theories. (orig.) 4. The Lorentz integral transform and its inversion International Nuclear Information System (INIS) Barnea, N.; Efros, V.D.; Leidemann, W.; Orlandini, G. 2010-01-01 The Lorentz integral transform method is briefly reviewed. The issue of the inversion of the transform, and in particular its ill-posedness, is addressed. It is pointed out that the mathematical term ill-posed is misleading and merely due to a historical misconception. In this connection standard regularization procedures for the solution of the integral transform problem are presented. In particular a recent one is considered in detail and critical comments on it are provided. In addition a general remark concerning the concept of the Lorentz integral transform as a method with a controlled resolution is made. (author) 5. Nonlinear Lorentz-invariant theory of gravitation International Nuclear Information System (INIS) Petry, W. 1976-01-01 A nonlinear Lorentz-invariant theory of gravitation and a Lorentz-invariant Hamiltonian for a particle with spin in the gravitational field are developed. The equations of motions are studied. The theory is applied to the three well known tests of General Relativity. In the special case of the red shift of spectral lines and of the deflection of light, the theory gives the same results as the General Theory of Relativity, whereas in the case of the perihelion of the Mercury, the theory gives 40,3'', in good agreement with experimental results of Dicke. (author) 6. Self-duality in generalized Lorentz superspaces International Nuclear Information System (INIS) Devchand, C.; Nuyts, J. 1996-12-01 We extend the notion of self-duality to spaces built from a set of representations of the Lorentz group with bosonic or fermionic behaviour, not having the traditional spin-one upper-bound of super Minkowski space. The generalized derivative vector fields on such superspace are assumed to form a superalgebra. Introducing corresponding gauge potentials and hence covariant derivatives and curvatures, we define generalized self-duality as the Lorentz covariant vanishing of certain irreducible parts of the curvatures. (author). 4 refs 7. Rotating optical cavity experiment testing Lorentz invariance at the 10-17 level International Nuclear Information System (INIS) Herrmann, S.; Senger, A.; Moehle, K.; Nagel, M.; Kovalchuk, E. V.; Peters, A. 2009-01-01 We present an improved laboratory test of Lorentz invariance in electrodynamics by testing the isotropy of the speed of light. Our measurement compares the resonance frequencies of two orthogonal optical resonators that are implemented in a single block of fused silica and are rotated continuously on a precision air bearing turntable. An analysis of data recorded over the course of one year sets a limit on an anisotropy of the speed of light of Δc/c∼1x10 -17 . This constitutes the most accurate laboratory test of the isotropy of c to date and allows to constrain parameters of a Lorentz violating extension of the standard model of particle physics down to a level of 10 -17 . 8. Theory prospective on leptonic CP violation International Nuclear Information System (INIS) Petcov, S.T. 2016-01-01 The phenomenology of 3-neutrino mixing, the current status of our knowledge about the 3-neutrino mixing parameters, including the absolute neutrino mass scale, and of the Dirac and Majorana CP violation in the lepton sector are reviewed. The problems of CP violation in neutrino oscillations and of determining the nature – Dirac or Majorana – of massive neutrinos are discussed. The seesaw mechanism of neutrino mass generation and the related leptogenesis scenario of generation of the baryon asymmetry of the Universe are considered. The results showing that the CP violation necessary for the generation of the baryon asymmetry of the Universe in leptogenesis can be due exclusively to the Dirac and/or Majorana CP-violating phase(s) in the neutrino mixing matrix U are briefly reviewed. The discrete symmetry approach to understanding the observed pattern of neutrino mixing and the related predictions for the leptonic Dirac CP violation are also reviewed. 9. Symmetry witnesses Science.gov (United States) Aniello, Paolo; Chruściński, Dariusz 2017-07-01 A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner’s theorem, the set of pure states—the rank-one projections—is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the sets of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable; i.e. reasoning in terms of quantum states, the sets of ‘uniform’ density operators of corresponding fixed rank are symmetry witnesses too. 10. CP violation in the lepton sector and implications for leptogenesis DEFF Research Database (Denmark) Hagedorn, C.; Mohapatra, R. N.; Molinaro, E. 2018-01-01 We review the current status of the data on neutrino masses and lepton mixing and the prospects for measuring the CP-violating phases in the lepton sector. The possible connection between low energy CP violation encoded in the Dirac and Majorana phases of the Pontecorvo-Maki-Nakagawa-Sakata mixing...... matrix and successful leptogenesis is emphasized in the context of seesaw extensions of the Standard Model with a flavor symmetry Gf (and CP symmetry).... 11. Reformulation of the symmetries of first-order general relativity Science.gov (United States) Montesinos, Merced; González, Diego; Celada, Mariano; Díaz, Bogar 2017-10-01 We report a new internal gauge symmetry of the n-dimensional Palatini action with cosmological term (n>3 ) that is the generalization of three-dimensional local translations. This symmetry is obtained through the direct application of the converse of Noether’s second theorem on the theory under consideration. We show that diffeomorphisms can be expressed as linear combinations of it and local Lorentz transformations with field-dependent parameters up to terms involving the variational derivatives of the action. As a result, the new internal symmetry together with local Lorentz transformations can be adopted as the fundamental gauge symmetries of general relativity. Although their gauge algebra is open in general, it allows us to recover, without resorting to the equations of motion, the very well-known Lie algebra satisfied by translations and Lorentz transformations in three dimensions. We also report the analog of the new gauge symmetry for the Holst action with cosmological term, finding that it explicitly depends on the Immirzi parameter. The same result concerning its relation to diffeomorphisms and the open character of the gauge algebra also hold in this case. Finally, we consider the non-minimal coupling of a scalar field to gravity in n dimensions and establish that the new gauge symmetry is affected by this matter field. Our results indicate that general relativity in dimension greater than three can be thought of as a gauge theory. 12. Electron correlation effects in the presence of non-symmetry dictated ... Indian Academy of Sciences (India) We numerically study the effect of non-symmetry dictated nodes (NSDN) on electron ... the absence of NSDN, attractive interaction between electrons give such an ... and the violation of parity effect, we first explain what are symmetry dictated. 13. Testing the Equivalence Principle and Lorentz Invariance with PeV Neutrinos from Blazar Flares. Science.gov (United States) Wang, Zi-Yi; Liu, Ruo-Yu; Wang, Xiang-Yu 2016-04-15 It was recently proposed that a giant flare of the blazar PKS B1424-418 at redshift z=1.522 is in association with a PeV-energy neutrino event detected by IceCube. Based on this association we here suggest that the flight time difference between the PeV neutrino and gamma-ray photons from blazar flares can be used to constrain the violations of equivalence principle and the Lorentz invariance for neutrinos. From the calculated Shapiro delay due to clusters or superclusters in the nearby universe, we find that violation of the equivalence principle for neutrinos and photons is constrained to an accuracy of at least 10^{-5}, which is 2 orders of magnitude tighter than the constraint placed by MeV neutrinos from supernova 1987A. Lorentz invariance violation (LIV) arises in various quantum-gravity theories, which predicts an energy-dependent velocity of propagation in vacuum for particles. We find that the association of the PeV neutrino with the gamma-ray outburst set limits on the energy scale of possible LIV to >0.01E_{pl} for linear LIV models and >6×10^{-8}E_{pl} for quadratic order LIV models, where E_{pl} is the Planck energy scale. These are the most stringent constraints on neutrino LIV for subluminal neutrinos. 14. Classroom Experiment to Verify the Lorentz Force Indian Academy of Sciences (India) Home; Journals; Resonance – Journal of Science Education; Volume 8; Issue 3. Classroom Experiment to Verify the Lorentz Force. Somnath Basu Anindita Bose Sumit Kumar Sinha Pankaj Vishe S Chatterjee. Classroom Volume 8 Issue 3 March 2003 pp 81-86 ... 15. Lorentz Spengler's descriptions of chitons (Mollusca: Polyplacophora) NARCIS (Netherlands) Kaas, P.; Knudsen, J. 1992-01-01 The present paper deals with an important Danish paper on the Polyplacophora, published in 1797 by Lorentz Spengler: Udförlig Beskrivelse over det mangeskallede Konkylie-Slaegt, af Linnaeus kaldet Chiton; med endeel nye Arter og Varieteter. -Skrivter af Naturhistorie-Selskabet, 4e Bind, Ie Hefte, 16. Characterisation of Embeddings in Lorentz Spaces Czech Academy of Sciences Publication Activity Database Gogatishvili, Amiran; Johansson, M.; Okpoti, C.A.; Persson, L. E. 2007-01-01 Roč. 76, č. 1 (2007), s. 69-92 ISSN 0004-9727 R&D Projects: GA ČR GA201/05/2033 Institutional research plan: CEZ:AV0Z10190503 Keywords : non-increasing rearrangement * Lorentz spaces * weights Subject RIV: BA - General Mathematics Impact factor: 0.297, year: 2007 17. Mirror symmetry CERN Document Server Voisin, Claire 1999-01-01 This is the English translation of Professor Voisin's book reflecting the discovery of the mirror symmetry phenomenon. The first chapter is devoted to the geometry of Calabi-Yau manifolds, and the second describes, as motivation, the ideas from quantum field theory that led to the discovery of mirror symmetry. The other chapters deal with more specialized aspects of the subject: the work of Candelas, de la Ossa, Greene, and Parkes, based on the fact that under the mirror symmetry hypothesis, the variation of Hodge structure of a Calabi-Yau threefold determines the Gromov-Witten invariants of its mirror; Batyrev's construction, which exhibits the mirror symmetry phenomenon between hypersurfaces of toric Fano varieties, after a combinatorial classification of the latter; the mathematical construction of the Gromov-Witten potential, and the proof of its crucial property (that it satisfies the WDVV equation), which makes it possible to construct a flat connection underlying a variation of Hodge structure in the ... 18. Discrete symmetries and their stringy origin International Nuclear Information System (INIS) Mayorga Pena, Damian Kaloni 2014-05-01 Discrete symmetries have proven to be very useful in controlling the phenomenology of theories beyond the standard model. In this work we explore how these symmetries emerge from string compactifications. Our approach is twofold: On the one hand, we consider the heterotic string on orbifold backgrounds. In this case the discrete symmetries can be derived from the orbifold conformal field theory, and it can be shown that they are in close relation with the orbifold geometry. We devote special attention to R-symmetries, which arise from discrete remnants of the Lorentz group in compact space. Further we discuss the physical implications of these symmetries both in the heterotic mini-landscape and in newly constructed models based on the Z 2 x Z 4 orbifold. In both cases we observe that the discrete symmetries favor particular locations in the orbifold where the particles of standard model should live. On the other hand we consider a class of F-theory models exhibiting an SU(5) gauge group, times additional U(1) symmetries. In this case, the smooth compactification background does not permit us to track the discrete symmetries as transparently as in orbifold models. Hence, we follow a different approach and search for discrete subgroups emerging after the U(1)s are broken. We observe that in this approach it is possible to obtain the standard Z 2 matter parity of the MSSM. 19. Val L. Fitch, the CP Violation, and Antimatter Science.gov (United States) dropdown arrow Site Map A-Z Index Menu Synopsis Val L. Fitch, the CP Violation, and Antimatter Resources ) 'to verify a fundamental tenet of physics, known as CP [charge-parity] symmetry, by showing that two into two pi mesons. Cronin and Fitch had found an example of CP violation. The discovery's 20. Covariant Renormalizable Modified and Massive Gravity Theories on (Non) Commutative Tangent Lorentz Bundles CERN Document Server Vacaru, Sergiu I 2014-01-01 The fundamental field equations in modified gravity (including general relativity; massive and bimetric theories; Ho\\vrava-Lifshits, HL; Einstein--Finsler gravity extensions etc) posses an important decoupling property with respect to nonholonomic frames with 2 (or 3) +2+2+... spacetime decompositions. This allows us to construct exact solutions with generic off--diagonal metrics depending on all spacetime coordinates via generating and integration functions containing (un-) broken symmetry parameters. Such nonholonomic configurations/ models have a nice ultraviolet behavior and seem to be ghost free and (super) renormalizable in a sense of covariant and/or massive modifications of HL gravity. The apparent noncommutativity and breaking of Lorentz invariance by quantum effects can be encoded into fibers of noncommutative tangent Lorentz bundles for corresponding "partner" anisotropically induced theories. We show how the constructions can be extended to include conjectured covariant reonormalizable models with... 1. The Lorentz Theory of Electrons and Einstein's Theory of Relativity Science.gov (United States) Goldberg, Stanley 1969-01-01 Traces the development of Lorentz's theory of electrons as applied to the problem of the electrodynamics of moving bodies. Presents evidence that the principle of relativity did not play an important role in Lorentz's theory, and that though Lorentz eventually acknowledged Einstein's work, he was unwilling to completely embrace the Einstein… 2. CP violation experiment at Fermilab International Nuclear Information System (INIS) Hsiung, Yee B. 1990-07-01 The E731 experiment at Fermilab has searched for ''direct'' CP violation in K 0 → ππ, which is parametrized by var-epsilon '/var-epsilon. For the first time, in 20% of the data set, all four modes of the K L,S → π + π - (π 0 π 0 ) were collected simultaneously, providing a great check on the systematic uncertainty. The result is Re(var-epsilon '/var-epsilon) = -0.0004 ± 0.0014 (stat) ± 0.0006(syst), which provides no evidence for ''direct'' CP violation. The CPT symmetry has also been tested by measuring the phase difference Δφ = φ 00 - φ ± between the two CP violating parameters η 00 and η ± . We fine Δφ = -0.3 degrees ± 2.4 degree(stat) ± 1.2 degree(syst). Using this together with the world average φ ± , we fine that the phase of the K 0 -bar K 0 mixing parameter var-epsilon is 44.5 degree ± 1.5 degree. Both of these results agree well with the predictions of CPT symmetry. 17 refs., 10 figs 3. Observational Aspects of Symmetries of the Neutral B Meson System CERN Document Server Fidecaro, Maria; Ruf, Thomas 2015-01-01 We revisit various results, which have been obtained by the BABAR and Belle Collaborations over the last twelve years, concerning symmetry properties of the Hamiltonian, which governs the time evolution and the decay of neutral B mesons. We find that those measurements, which established CP violation in B meson decay, 13 years ago, had as well established T (time-reversal) symmetry violation. They also confirmed CPT symmetry in the decay (T$_{CPT}$= 0) and symmetry with respect to time-reversal ($\\epsilon$= 0) and to CPT ($\\delta$= 0) in the$B^0 \\bar{B}^0oscillation. 4. The nucleon- nucleon interaction and symmetries International Nuclear Information System (INIS) Van Oers, W.T.H. 1992-11-01 With the advent of the possibility to study nucleon-nucleon scattering at medium energies, its extension to investigate fundamental symmetries was recognized early on. It was precisely the introduction of rotational invariance, parity conservation, time reversal invariance, and isotopic spin conversation that led to the description of the N - N scattering matrix in terms of five complex amplitudes: one set of five for proton-proton scattering and one set of five for neutron-proton scattering, or alternatively, one set for the isotopic spin state ι=ο and the other for the isotopic spin state ι=1. Clearly, if one or more of the above constraints are removed, there are additional amplitudes that need to be considered. To be meaningful, experiment requires observables that are particularly sensitive to the violation of a conservation law or symmetry principle. During the last decade a series of precision experiments has been performed to measure charge- symmetry breaking in n - p elastic scattering (corresponding to isotopic spin non-conservation), and to measure parity violation in p-p scattering. For a particle-anti-particle system,like the pp or λλ system one can raise the question of CP violation in a system other than the neutral kaon system may become possible in the near future through pp →λλ and pp→ ≡ ≡. A description is given of the ongoing efforts to measure charge symmetry breaking, parity violation and CP violation.(author). 42 refs., 6 figs 5. On the origin of neutrino flavour symmetry International Nuclear Information System (INIS) King, Stephen F.; Luhn, Christoph 2009-01-01 We study classes of models which are based on some discrete family symmetry which is completely broken such that the observed neutrino flavour symmetry emerges indirectly as an accidental symmetry. For such 'indirect' models we discuss the D-term flavon vacuum alignments which are required for such an accidental flavour symmetry consistent with tri-bimaximal lepton mixing to emerge. We identify large classes of suitable discrete family symmetries, namely the Δ(3n 2 ) and Δ(6n 2 ) groups, together with other examples such as Z 7 x Z 3 . In such indirect models the implementation of the type I see-saw mechanism is straightforward using constrained sequential dominance. However the accidental neutrino flavour symmetry may be easily violated, for example leading to a large reactor angle, while maintaining accurately the tri-bimaximal solar and atmospheric predictions. 6. CP Violation and B Physics International Nuclear Information System (INIS) Quinn, Helen R 2001-01-01 These lectures provide a basic overview of topics related to the study of CP Violation in B decays. In the first lecture, I review the basics of discrete symmetries in field theories, the quantum mechanics of neutral but flavor-non-trivial mesons, and the classification of three types of CP violation [1]. The actual second lecture which I gave will be separately published as it is my Dirac award lecture and is focused on the separate topic of strong CP Violation. In Lecture 2 here, I cover the Standard Model predictions for neutral B decays, and in particular discuss some channels of interest for CP Violation studies. Lecture 3 reviews the various tools and techniques used to deal with the hadronic physics effects. In Lecture 4, I briefly review the present and planned experiments that can study B decays. I cannot teach all the details of this subject in this short course, so my approach is instead to try to give students a grasp of the relevant concepts and an overview of the available tools. The level of these lectures is introductory. I will provide some references to more detailed treatments and current literature, but this is not a review article so I do not attempt to give complete references to all related literature. By now there are some excellent textbooks that cover this subject in great detail [1]. I refer students to these for more details and for more complete references to the original literature 7. B physics and CP violation International Nuclear Information System (INIS) Quinn, H. 2002-01-01 These lectures provide a basic overview of topics related to the study of CP Violation in B decays. In the first lecture, I review the basics of discrete symmetries in field theories, the quantum mechanics of neutral but flavor-non-trivial mesons, and the classification of three types of CP violation. The actual second lecture which I gave will be separately published as it is my Dirac award lecture and is focussed on the separate topic of strong CP Violation. In Lecture 2 here, I cover the Standard Model predictions for neutral B decays, and in particular discuss some channels of interest for CP Violation studies. Lecture 3 reviews the various tools and techniques used to deal with the hadronic physics effects. In Lecture 4, I briefly review the present and planned experiments that can study B decays. I cannot teach all the details of this subject in this short course, so my approach is instead to try to give students a grasp of the relevant concepts and an overview of the available tools. The level of these lectures is introductory. I will provide some references to more detailed treatments and current literature, but this is not a review article so I do not attempt to give complete references to all related literature. By now there are some excellent textbooks that cover this subject in great detail. I refer students to these for more details and for more complete references to the original literature. (author) 8. Problems of CP-violation in early unification theories International Nuclear Information System (INIS) Liparteliani, A.G.; Monich, V.A.; Volkov, G.G. 1985-01-01 The present work studies possible mechanisms of P and CP-violation in the frames of an approach based on early unification of fundamental local symmetries, i.e., Pati-Salam four-colour symmetry, extended weak isotopic symmetry and that of quark-lepton generations. The work also studies the influence of generations mixing on the rates of rare processes in each of 3 classes of interactions 9. Is CP a gauge symmetry? International Nuclear Information System (INIS) Choi, K.; Kaplan, D.B.; Nelson, A.E. 1993-01-01 Conventional solutions to the strong CP problem all require the existence of global symmetries. However, quantum gravity may destroy global symmetries, making it hard to understand why the electric dipole moment of the neutron (EDMN) is so small. We suggest here that CP is actually a discrete gauge symmetry, and is therefore not violated by quantum gravity. We show that four-dimensional CP can arise as a discrete gauge symmetry in theories with dimensional compactification, if the original number of Minkowski dimensions equals 8k+1, 8k+2 or 8k+3, and if there are certain restrictions on the gauge group; these conditions are met by superstrings. CP may then be broken spontaneously below 10 9 GeV, explaining the observed CP violation in the kaon system without inducing a large EDMN. We discuss the phenomenology of such models, as well as the peculiar properties of cosmic 'SP strings' which could be produced at the compactification scale. Such strings have the curious property that a particle carried around the string is turned into its CP conjugate. A single CP string renders four-dimensional space-time nonorientable. (orig.) 10. The geometric role of symmetry breaking in gravity International Nuclear Information System (INIS) Wise, Derek K 2012-01-01 In gravity, breaking symmetry from a group G to a group H plays the role of describing geometry in relation to the geometry of the homogeneous space G/H. The deep reason for this is Cartan's 'method of equivalence,' giving, in particular, an exact correspondence between metrics and Cartan connections. I argue that broken symmetry is thus implicit in any gravity theory, for purely geometric reasons. As an application, I explain how this kind of thinking gives a new approach to Hamiltonian gravity in which an observer field spontaneously breaks Lorentz symmetry and gives a Cartan connection on space. 11. Neutrality of the lorentz transformations in SRT International Nuclear Information System (INIS) Hamdan, N.; Baza, S. 2005-01-01 The special theory of Relativity (SRT), gives us two results, the dilation of time and the contraction of the Length, which have been refuted by many scientists. The solution to these kinematical effects has driven researchers to develop new methods. One of these methods is using the physical law equations and apply the principle of relativity to them. With this approach, we reformulated the SRT in a simple manner which has dynamical applications without using the Lorentz transformations (LT) and its kinematical effects. We obtained the results which require the invariant of Maxwell's field equations under the LT in a way different to that of Einsterin. In the present paper, we get the LT from the Lorentz force. In contrast to Einstein's LT with its kinematical effects, the LT produced in this paper is simply a neutral transformation. Containing no physical significance, i.e. LT and its kinematical effects do not explain any physical phenomenon. (author) 12. Minimal Flavour Violation and Beyond CERN Document Server Isidori, Gino 2012-01-01 We review the formulation of the Minimal Flavour Violation (MFV) hypothesis in the quark sector, as well as some "variations on a theme" based on smaller flavour symmetry groups and/or less minimal breaking terms. We also review how these hypotheses can be tested in B decays and by means of other flavour-physics observables. The phenomenological consequences of MFV are discussed both in general terms, employing a general effective theory approach, and in the specific context of the Minimal Supersymmetric extension of the SM. 13. Symmetry and bifurcations of momentum mappings Energy Technology Data Exchange (ETDEWEB) Arms, J.M.; Marsden, J.E.; Moncrief, V. 1981-01-01 The zero set of a momentum mapping is shown to have a singularity at each point with symmetry. The zero set is diffeomorphic to the product of a manifold and the zero set of a homogeneous quadratic function. The proof uses the Kuranishi theory of deformations. Among the applications, it is shown that the set of all solutions of the Yang-Mills equations on a Lorentz manifold has a singularity at any solution with symmetry, in the sense of a pure gauge symmetry. Similarly, the set of solutions of Einstein's equations has a singularity at any solution that has spacelike Killing fields, provided the spacetime has a compact Cauchy surface. 14. Electromagnetic reactions of few-body systems with the Lorentz integral transform method International Nuclear Information System (INIS) Leidemann, W. 2007-01-01 Various electromagnetic few-body break-up reactions into the many-body continuum are calculated microscopically with the Lorentz integral transform (LIT) method. For three- and four-body nuclei the nuclear Hamiltonian includes two- and three-nucleon forces, while semirealistic interactions are used in case of six- and seven-body systems. Comparisons with experimental data are discussed. In addition various interesting aspects of the 4 He photodisintegration are studied: investigation of a tetrahedrical symmetry of 4 He and a test of non-local nuclear force models via the induced two-body currents 15. BOOK REVIEW: Symmetry Breaking Science.gov (United States) Ryder, L. H. 2005-11-01 One of the most fruitful and enduring advances in theoretical physics during the last half century has been the development of the role played by symmetries. One needs only to consider SU(3) and the classification of elementary particles, the Yang Mills enlargement of Maxwell's electrodynamics to the symmetry group SU(2), and indeed the tremendous activity surrounding the discovery of parity violation in the weak interactions in the late 1950s. This last example is one of a broken symmetry, though the symmetry in question is a discrete one. It was clear to Gell-Mann, who first clarified the role of SU(3) in particle physics, that this symmetry was not exact. If it had been, it would have been much easier to discover; for example, the proton, neutron, Σ, Λ and Ξ particles would all have had the same mass. For many years the SU(3) symmetry breaking was assigned a mathematical form, but the importance of this formulation fell away when the quark model began to be taken seriously; the reason the SU(3) symmetry was not exact was simply that the (three, in those days) quarks had different masses. At the same time, and in a different context, symmetry breaking of a different type was being investigated. This went by the name of spontaneous symmetry breaking' and its characteristic was that the ground state of a given system was not invariant under the symmetry transformation, though the interactions (the Hamiltonian, in effect) was. A classic example is ferromagnetism. In a ferromagnet the atomic spins are aligned in one direction only—this is the ground state of the system. It is clearly not invariant under a rotation, for that would change the ground state into a (similar but) different one, with the spins aligned in a different direction; this is the phenomenon of a degenerate vacuum. The contribution of the spin interaction, s1.s2, to the Hamiltonian, however, is actually invariant under rotations. As Coleman remarked, a little man living in a ferromagnet would 16. Parity violations in electron-nucleon scattering and the SU(2)sub(L)xSU(2)sub(R)xU(1)sub(L+R) electroweak symmetry International Nuclear Information System (INIS) Rajpoot, S. 1981-07-01 The SU(2)sub(L) x SU(2)sub(R) x U(1)sub(L+R) model of electroweak interactions is described with the most general gauge couplings gsub(L), gsub(R) and gsub(L+R). The case in which neutrino neutral current interactions are identical to the standard SU(2)sub(L) x U(1)sub(L+R) model is discussed in detail. It is shown that with the weak angle lying in the experimental range sin 2 thetaSUB(w)=0.23+-0.015 and 1 2 /gsub(R) 2 <3 it is possible to explain the amount of parity violation observed at SLAC and at the same time predict values of the ''weak charge'' in bismuth to lie in the range admitted by the controversal data from different experiments. (author) 17. CP and other gauge symmetries in string theory International Nuclear Information System (INIS) Dine, M.; Leigh, R.G.; MacIntire, D.A. 1992-01-01 We argue that CP is a gauge symmetry in string theory. As a consequence, CP cannot be explicitly broken either perturbatively or nonperturbatively; there can be no nonperturbative CP-violating parameters. String theory is thus an example of a theory where all θ angles arise due to spontaneous CP violation, and are in principle calculable 18. Symmetry, Symmetry Breaking and Topology Directory of Open Access Journals (Sweden) Siddhartha Sen 2010-07-01 Full Text Available The ground state of a system with symmetry can be described by a group G. This symmetry group G can be discrete or continuous. Thus for a crystal G is a finite group while for the vacuum state of a grand unified theory G is a continuous Lie group. The ground state symmetry described by G can change spontaneously from G to one of its subgroups H as the external parameters of the system are modified. Such a macroscopic change of the ground state symmetry of a system from G to H correspond to a “phase transition”. Such phase transitions have been extensively studied within a framework due to Landau. A vast range of systems can be described using Landau’s approach, however there are also systems where the framework does not work. Recently there has been growing interest in looking at such non-Landau type of phase transitions. For instance there are several “quantum phase transitions” that are not of the Landau type. In this short review we first describe a refined version of Landau’s approach in which topological ideas are used together with group theory. The combined use of group theory and topological arguments allows us to determine selection rule which forbid transitions from G to certain of its subgroups. We end by making a few brief remarks about non-Landau type of phase transition. 19. Novel characteristics of energy spectrum for 3D Dirac oscillator analyzed via Lorentz covariant deformed algebra. Science.gov (United States) Betrouche, Malika; Maamache, Mustapha; Choi, Jeong Ryeol 2013-11-14 We investigate the Lorentz-covariant deformed algebra for Dirac oscillator problem, which is a generalization of Kempf deformed algebra in 3 + 1 dimension of space-time, where Lorentz symmetry are preserved. The energy spectrum of the system is analyzed by taking advantage of the corresponding wave functions with explicit spin state. We obtained entirely new results from our development based on Kempf algebra in comparison to the studies carried out with the non-Lorentz-covariant deformed one. A novel result of this research is that the quantized relativistic energy of the system in the presence of minimal length cannot grow indefinitely as quantum number n increases, but converges to a finite value, where c is the speed of light and β is a parameter that determines the scale of noncommutativity in space. If we consider the fact that the energy levels of ordinary oscillator is equally spaced, which leads to monotonic growth of quantized energy with the increment of n, this result is very interesting. The physical meaning of this consequence is discussed in detail. 20. Universe symmetries International Nuclear Information System (INIS) Souriau, J.M. 1984-01-01 The sky uniformity can be noticed in studying the repartition of objects far enough. The sky isotropy description uses space rotations. The group theory elements will allow to give a meaning at the same time precise and general to the word a ''symmetry''. Universe models are reviewed, which must have both of the following qualities: - conformity with the physic known laws; - rigorous symmetry following one of the permitted groups. Each of the models foresees that universe evolution obeys an evolution equation. Expansion and big-bang theory are recalled. Is universe an open or closed space. Universe is also electrically neutral. That leads to a work hypothesis: the existing matter is not given data of universe but it appeared by evolution from nothing. Problem of matter and antimatter is then raised up together with its place in universe [fr 1. CP properties of symmetry-constrained two-Higgs-doublet models CERN Document Server Ferreira, P M; Nachtmann, O; Silva, Joao P 2010-01-01 The two-Higgs-doublet model can be constrained by imposing Higgs-family symmetries and/or generalized CP symmetries. It is known that there are only six independent classes of such symmetry-constrained models. We study the CP properties of all cases in the bilinear formalism. An exact symmetry implies CP conservation. We show that soft breaking of the symmetry can lead to spontaneous CP violation (CPV) in three of the classes. 2. Concerning the equivalence of Lorentz's and Einstein's theories International Nuclear Information System (INIS) Clube, S.V.M. 1978-01-01 A clear distinction is drawn between derivations of the Lorentz transformations by Lorentz and Einstein. The choice as to which derivation is correct is still open to experimental test. Possible reasons are given for preferring the Lorentz derivation in terms of a material aether, and the role of covariance in physical theory is considered to be heuristic rather than fundamental. The existence of a material aether also permits one to question the fundamental role of fields in modern theory 3. Lorentz invariance with an invariant energy scale. Science.gov (United States) Magueijo, João; Smolin, Lee 2002-05-13 We propose a modification of special relativity in which a physical energy, which may be the Planck energy, joins the speed of light as an invariant, in spite of a complete relativity of inertial frames and agreement with Einstein's theory at low energies. This is accomplished by a nonlinear modification of the action of the Lorentz group on momentum space, generated by adding a dilatation to each boost in such a way that the Planck energy remains invariant. The associated algebra has unmodified structure constants. We also discuss the resulting modifications of field theory and suggest a modification of the equivalence principle which determines how the new theory is embedded in general relativity. 4. Mixed Lorentz boostedZ^{0}'s$CERN Document Server Kjaer, N J 2001-01-01 A novel technique is proposed to study systematic errors on jet reconstruction in W physics measurements at LEP2 with high statistical precision. The method is based on the emulation of W pair events using Mixed Lorentz Boosted Z0 events. The scope and merits of the method and its statistical accuracy are discussed in the context of the DELPHI W mass measurement in the fully hadronic channel. The numbers presented are preliminary in the sense that they do not constitute the final DELPHI systematic errors. 5. Studies on representation of the Lorentz group and gauge theory International Nuclear Information System (INIS) Hanitriarivo, R. 2002-01-01 This work is focused on studies about the representation of the Lorentz group and gauge theory. The mathematical tools required for the different studies are presented, as well as for the representation of the Lorentz group and for the gauge theory. Representation of the Lorentz group gives the possible types of fields and wave functions that describe particles: fermions are described by spinors and bosons are described by scalar or vector. Each of these entities (spinors, scalars, vectors) are characterized by their behavior under the action of Lorentz transformations.Gauge theory is used to describe the interactions between particles. [fr 6. From anomalies of finite symmetries to heterotic GUTs Science.gov (United States) Vaudrevange, Patrick K. S. 2017-11-01 We review the role of finite symmetries for particle physics with special emphasis on discrete anomalies and on their possible origin from extra dimensions. Then, we apply our knowledge on finite symmetries to the problematic proton decay operators of various mass-dimensions, focusing on ℤ4R , i.e. a special R-symmetry of order 4. We show that this ℤ4R symmetry can naturally originate from extra dimensions as a discrete remnant of higher-dimensional Lorentz symmetry. Finally, in order to obtain a unified picture from the heterotic string theory we discuss grand unified theories (GUTs) in extra dimensions compactified on ℤ2 × ℤ2 orbifolds and show how proton decay operators can be suppressed in a certain class of orbifolds. 7. A model for the origin and mechanisms of CP violation International Nuclear Information System (INIS) Wu, Y. 1995-01-01 In this talk I will show that the two-Higgs doublet model with vacuum CP violation and approximate global U(1) family symmetries may provide one of the simplest and attractive models for understanding the origin and mechanisms of CP violation. It is shown that the mechanism of spontaneous symmetry breaking provides not only a mechanism for generating masses of the bosons and fermions, but also a mechanism for creating CP-phases of the bosons and fermions, so that CP violation occurs, after spontaneous symmetry breaking, in all possible ways from a single CP phase of the vacuum and is generally classified into four types of CP-violating mechanism. A new type of CP-violating mechanism in the charged Higgs boson interactions of the fermions is emphasized and can provide a consistent description for both established and reported CP-, P-, and T-violating phenomena. Of particular importance is the new source of CP violation for charged Higgs boson interactions that lead to the value of ε'/ε as large as 10 -3 independent of the CKM phase. copyright 1995 American Institute of Physics 8. B-L violating supersymmetric couplings International Nuclear Information System (INIS) Ramond, P. 1983-01-01 We consider two problems: one is the possible effect of the breaking of Peccei-Quinn symmetry on the inflationary universe scenario; the other is the remark that even the minimal supersymmetric SU 5 theory contains B-L violating couplings which give rise to neutrino masses and family-diagonal proton decay. However the strength of these couplings is limited by the gauge hierarchy 9. Neutrino properties and fundamental symmetries International Nuclear Information System (INIS) Bowles, T.J. 1996-01-01 This is the final report of a three-year, Laboratory-Directed Research and Development (LDRD) project at the Los Alamos National Laboratory (LANL). There are two components to this work. The first is a development of a new detection scheme for neutrinos. The observed deficit of neutrinos from the Sun may be due to either a lack of understanding of physical processes in the Sun or may be due to neutrinos oscillating from one type to another during their transit from the Sun to the Earth. The Sudbury Neutrino Observatory (SNO) is designed to use a water Cerenkov detector employing one thousand tonnes of heavy water to resolve this question. The ability to distinguish muon and tau neutrinos from electron neutrinos is crucial in order to carry out a model-independent test of neutrino oscillations. We describe a developmental exploration of a novel technique to do this using 3 He proportional counters. Such a method offers considerable advantages over the initially proposed method of using Cerenkov light from capture on NaCl in the SNO. The second component of this work is an exploration of optimal detector geometry for a time-reversal invariance experiment. The question of why time moves only in the forward direction is one of the most puzzling problems in modern physics. We know from particle physics measurements of the decay of kaons that there is a charge-parity symmetry that is violated in nature, implying time-reversal invariance violation. Yet, we do not understand the origin of the violation of this symmetry. To promote such an understanding, we are developing concepts and prototype apparatus for a new, highly sensitive technique to search for time-reversal-invariance violation in the beta decay of the free neutron. The optimized detector geometry is seven times more sensitive than that in previous experiments. 15 refs 10. Introduction to Ives' 'Derivation of the Lorentz transformations' International Nuclear Information System (INIS) Ruderfer, M. 1979-01-01 Lorentz ether theory is elevated on a par with special relativity by Ives' derivation of the Lorentz transformations. The two theories combined then demand the physical existence of a relativistic ether. This is supported by the still unfolding hierarchy of matter. Cogent implications for physical theory follow. (Auth.) 11. Vortex deformation and reduction of the Lorentz force International Nuclear Information System (INIS) Vuorio, M. 1977-01-01 A vortex of an extreme II-type superconductor is considered in the presence of a transport current. The equivalence of Magnus and Lorentz forces in a static vortex is discussed and the effect of vortex deformation is included in calculating corrections to the conventional expression of the Lorentz force. (author) 12. Lorentz Invariant Spectrum of Minimal Chiral Schwinger Model Science.gov (United States) Kim, Yong-Wan; Kim, Seung-Kook; Kim, Won-Tae; Park, Young-Jai; Kim, Kee Yong; Kim, Yongduk We study the Lorentz transformation of the minimal chiral Schwinger model in terms of the alternative action. We automatically obtain a chiral constraint, which is equivalent to the frame constraint introduced by McCabe, in order to solve the frame problem in phase space. As a result we obtain the Lorentz invariant spectrum in any moving frame by choosing a frame parameter. 13. New paradigm for baryon and lepton number violation International Nuclear Information System (INIS) Fileviez Pérez, Pavel 2015-01-01 The possible discovery of proton decay, neutron–antineutron oscillation, neutrinoless double beta decay in low energy experiments, and exotic signals related to the violation of the baryon and lepton numbers at collider experiments will change our understanding of the conservation of fundamental symmetries in nature. In this review we discuss the rare processes due to the existence of baryon and lepton number violating interactions. The simplest grand unified theories and the neutrino mass generation mechanisms are discussed. The theories where the baryon and lepton numbers are defined as local gauge symmetries spontaneously broken at the low scale are discussed in detail. The simplest supersymmetric gauge theory which predicts the existence of lepton number violating processes at the low scale is investigated. The main goal of this review is to discuss the main implications of baryon and lepton number violation in physics beyond the Standard Model. 14. Lie-isotopic generalization of the Poincare symmetry: Classical formulation International Nuclear Information System (INIS) Santilli, R.M. 1991-03-01 This paper is devoted to the origin and methodology of the several phenomenological predictions of deviations from Einstein's Special Relativity and related Lorentz symmetry in the behaviour of the lifetime of unstable hadrons at different speeds, that exist in the literature since the early '60's. After reviewing the background phenomenological literature, we outline the Lie-isotopic symmetry of the emerging deformations of the Minkowski metric introduced in a preceding paper, and extend the results to the construction of the full Poincare-isotopic symmetry. The local isomorphism of the Poincare-isotopic symmetry with the conventional symmetry is proved for all possible topology-preserving deformations of the Minkowski metric. In this way we establish that the phenomenological predictions of deviations recalled earlier must be specifically referred to Einstein's Special Relativity, but they cannot be referred to the Lorentz (or to the Poincare) symmetry which remains exact. Particular attention is devoted to the proof of the compatibility of the exact validity of the Special Relativity for the center-of-mass trajectory of a hadron in a particle accelerator, with conceivable deviations from the same relativity in the interior structural problem. For completeness, the analysis is complemented with a few remarks on the gravitational profile. First, we review the pioneering Lie-isotopic generalization of Einstein's Gravitation worked out by Gasperini, which possesses precisely a locally Lorentz-isotopic structure. We then restrict this theory to the interior gravitational problem in order to achieve compatibility with the particle setting. The paper concludes with a review of the need to finally conduct direct experimental measures of the lifetime of unstable hadrons at different speeds, in order to finally resolve whether Einsteins's Special and General Relativities are locally valid in the interior of hadrons, or structurally more general relativities must be worked 15. CP violation in B decay OpenAIRE Yamamoto, Hitoshi 2001-01-01 We review the physics of CP violation in B decays. After introducing the CKM matrix and how it causes CP violation, we cover three types of CP violation that can occur in B decays: CP violation in mixing, CP violation by mixing-decay interference, and CP violation in decay. 16. The nucleon- nucleon interaction and symmetries Energy Technology Data Exchange (ETDEWEB) Van Oers, W T.H. 1992-11-01 With the advent of the possibility to study nucleon-nucleon scattering at medium energies, its extension to investigate fundamental symmetries was recognized early on. It was precisely the introduction of rotational invariance, parity conservation, time reversal invariance, and isotopic spin conversation that led to the description of the N - N scattering matrix in terms of five complex amplitudes: one set of five for proton-proton scattering and one set of five for neutron-proton scattering, or alternatively, one set for the isotopic spin state {iota}={omicron} and the other for the isotopic spin state {iota}=1. Clearly, if one or more of the above constraints are removed, there are additional amplitudes that need to be considered. To be meaningful, experiment requires observables that are particularly sensitive to the violation of a conservation law or symmetry principle. During the last decade a series of precision experiments has been performed to measure charge- symmetry breaking in n - p elastic scattering (corresponding to isotopic spin non-conservation), and to measure parity violation in p-p scattering. For a particle-anti-particle system,like the pp or {lambda}{lambda} system one can raise the question of CP violation in a system other than the neutral kaon system may become possible in the near future through pp {yields}{lambda}{lambda} and pp{yields} {identical_to} {identical_to}. A description is given of the ongoing efforts to measure charge symmetry breaking, parity violation and CP violation.(author). 42 refs., 6 figs. 17. Minimal flavour violation and neutrino masses without R-parity DEFF Research Database (Denmark) Arcadi, G.; Di Luzio, L.; Nardecchia, M. 2012-01-01 symmetry breaking all the couplings of the superpotential including the R-parity violating ones. If R-parity violation is responsible for neutrino masses, our setup can be seen as an extension of MFV to the lepton sector. We analyze two patterns based on the non-abelian flavour symmetries SU(3)(4) circle...... times SU(4) and SU(3)(5). In the former case the total lepton number and the lepton flavour number are broken together, while in the latter the lepton number can be broken independently by an abelian spurion, so that visible effects and peculiar correlations can be envisaged in flavour changing charged... 18. Study of the violation of the T and CP symmetries in the reactions {lambda}{sub b}{sup 0} {yields} {lambda}{sup 0} + a vector meson. Validation of the Front-end electronics for the PreShower detector of the LHCb experiment; Recherche de la violation des symetries CP et T dans les reactions {lambda}{sub b}{sup 0} {yields} {lambda}{sup 0} + un meson vecteur. Validation de l'architecture de lecteur des canaux du detecteur de pied de gerbe de l'experience LHCb Energy Technology Data Exchange (ETDEWEB) Conte, E 2007-11-15 This thesis probes the beauty baryon physics in the framework of the LHCb experiment. The present study deals with the {lambda}{sub b}{sup 0} {yields} {lambda}{sup 0}V decays where V is a vector meson such as J/{psi}({mu}{sup +}{mu}{sup -}), {phi}(K{sup +}K{sup -}), {omega}({pi}{sup +}{pi}{sup -}{pi}0) or the {rho}{sup 0} - {omega}{sup 0}({pi}{sup +}{pi}{sup -}) mixing. These processes allow to test independently the CP symmetry, which violation has not been observed yet in the baryonic sector, and the T symmetry, which experimental proofs are limited. Among the possible perspectives, a precise measurement of the {lambda}{sub b}{sup 0} lifetime could contribute to the resolution of the raising theoretical-experimental puzzle. A phenomenological model of the {lambda}{sub b}{sup 0} {yields} {lambda}{sup 0}V decays has been performed, from which branching ratios and angular distributions have been estimated. An advanced study of the reconstruction and the selection of these reactions by the LHCb apparatus shows that the channel {lambda}{sub b}{sup 0} {yields} {lambda}{sup 0}J/{psi} is the dominant channel on both statistics and purity aspects. The {lambda}{sub b}{sup 0} lifetime measure is the most imminent result; the constrains on asymmetries due to CP and T violation require several data taking years. Besides, an instrumental work has been achieved on the read-out electronics, called Front-End, of the experiment pre-shower. This contribution takes into account the validation of the prototype boards and the development of tools required by the qualification of the 100 production boards. (author) 19. Probing Fundamental Symmetries: Questioning the Very Basics of Conservation Laws Science.gov (United States) Mohanmurthy, Prajwal 2017-09-01 Is the Lorentz-CPT symmetry, a core component of the standard model, valid? To what extent are the CP and T symmetries broken in the strong sector? What are we doing about the existing strong-CP problem? Do neutrons oscillate (like neutral kaons) or break the (Baryon - Lepton) number conservation? In this presentation, we will go over some of the experiments probing fundamental symmetries trying to answer the above questions. I will, very briefly, introduce the CompEx & nEx experiments probing the Lorentz symmetry in the electromagnetic (EM) sector, the nEDM experiment probing CP and T symmetries in the strong sector, NStar experiment searching for neutron oscillations, MASS & BDX experiments searching for axion like particles & dark matter. We will then briefly touch upon the highlights of these experiments and focus on the path we are taking towards answering those questions while also connecting the dots [experiments] with CEU. PM would like to acknowledge support from SERI SNSF Grant 2015.0594. 20. Relativistic transformation law of quantum fields: A slight generalization consistent with the equivalence of all Lorentz frames International Nuclear Information System (INIS) Ingraham, R.L. 1985-01-01 The well-known relativistic transformation law of quantum fields satisfies the relativity principle, which asserts the complete equivalence of all Lorentz (inertial) frames as far as physical measurements go. We point out a slight generalization which is allowed by the relativity principle, but violates a further, tacit assumption usually made in connection with it but which is actually logically independent of it and subject to a feasible experimental test. The interest of the generalization is that it permits the incorporation of an ultraviolet cutoff in a simple, direct way which avoids the usual difficulties 1. Some symmetries in nuclei International Nuclear Information System (INIS) Henley, E.M. 1981-09-01 Internal and space-time symmetries are discussed in this group of lectures. The first of the lectures deals with an internal symmetry, or rather two related symmetries called charge independence and charge symmetry. The next two discuss space-time symmetries which also hold approximately, but are broken only by the weak forces; that is, these symmetries hold for both the hadronic and electromagnetic forces 2. Two-Higgs-doublet models with Minimal Flavour Violation International Nuclear Information System (INIS) Carlucci, Maria Valentina 2010-01-01 The tree-level flavour-changing neutral currents in the two-Higgs-doublet models can be suppressed by protecting the breaking of either flavour or flavour-blind symmetries, but only the first choice, implemented by the application of the Minimal Flavour Violation hypothesis, is stable under quantum corrections. Moreover, a two-Higgs-doublet model with Minimal Flavour Violation enriched with flavour-blind phases can explain the anomalies recently found in the ΔF = 2 transitions, namely the large CP-violating phase in B s mixing and the tension between ε K and S ψKS . 3. Discrete symmetries with neutral mesons Science.gov (United States) Bernabéu, José 2018-01-01 Symmetries, and Symmetry Breakings, in the Laws of Physics play a crucial role in Fundamental Science. Parity and Charge Conjugation Violations prompted the consideration of Chiral Fields in the construction of the Standard Model, whereas CP-Violation needed at least three families of Quarks leading to Flavour Physics. In this Lecture I discuss the Conceptual Basis and the present experimental results for a Direct Evidence of Separate Reversal-in-Time T, CP and CPT Genuine Asymmetries in Decaying Particles like Neutral Meson Transitions, using Quantum Entanglement and the Decay as a Filtering Measurement. The eight transitions associated to the Flavour-CP eigenstate decay products of entangled neutral mesons have demonstrated with impressive significance a separate evidence of TRV and CPV in Bd-physics, whereas a CPTV asymmetry shows a 2σ effect interpreted as an upper limit. Novel CPTV observables are discussed for K physics at KLOE-2, including the difference between the semileptonic asymmetries from KL and KS, the ratios of double decay rate Intensities to Flavour-CP eigenstate decay products and the ω-effect. Their observation would lead to a change of paradigm beyond Quantum Field Theory, however there is nothing in Quantum Mechanics forbidding CPTV. 4. Models of dynamical R-parity violation Energy Technology Data Exchange (ETDEWEB) Csáki, Csaba; Kuflik, Eric [Department of Physics, LEPP, Cornell University, Ithaca, NY 14853 (United States); Slone, Oren; Volansky, Tomer [Raymond and Beverly Sackler School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978 (Israel) 2015-06-08 The presence of R-parity violating interactions may relieve the tension between existing LHC constraints and natural supersymmetry. In this paper we lay down the theoretical framework and explore models of dynamical R-parity violation in which the breaking of R-parity is communicated to the visible sector by heavy messenger fields. We find that R-parity violation is often dominated by non-holomorphic operators that have so far been largely ignored, and might require a modification of the existing searches at the LHC. The dynamical origin implies that the effects of such operators are suppressed by the ratio of either the light fermion masses or the supersymmetry breaking scale to the mediation scale, thereby providing a natural explanation for the smallness of R-parity violation. We consider various scenarios, classified by whether R-parity violation, flavor breaking and/or supersymmetry breaking are mediated by the same messenger fields. The most compact case, corresponding to a deformation of the so called flavor mediation scenario, allows for the mediation of supersymmetry breaking, R-parity breaking, and flavor symmetry breaking in a unified manner. 5. Rotation associated with product of two Lorentz transformations International Nuclear Information System (INIS) Van Wyk, C.B. 1984-01-01 In the usual presentation of the Lorentz transformation there is an almost complete absence of the use of products of these transformations. One of the reasons for this appears to be the large amount of calculation involved when multi-plying the 4X4 matrices of the vector representation of the Lorentz transformation. In the article this problem is partly cleared up by using the coordinate free two-component spinor representation of rotations and Lorentz transformations. It is also shown that the theory derived in the article can be applied to Thomas precission in a very simple and direct way 6. CP violation outside the standard model phenomenology for pedestrians International Nuclear Information System (INIS) Lipkin, H.J. 1993-01-01 So far the only experimental evidence for CP violation is the 1964 discovery of K L →2π where the two mass eigenstates produced by neutral meson mixing both decay into the same CP eigenstate. This result is described by two parameters ε and ε'. Today ε ∼ its 1964 value, ε' data are still inconclusive and there is no new evidence for CP violation. One might expect to observe similar phenomena in other systems and also direct CP violation as charge asymmetries between decays of charge conjugate hadrons H ± → f ± . Why is it so hard to find CP violation? How can B Physics help? Does CP lead beyond the standard model? The author presents a pedestrian symmetry approach which exhibits the difficulties and future possibilities of these two types of CP-violation experiments, neutral meson mixing and direct charge asymmetry: what may work, what doesn't work and why 7. Is CP violation maximal International Nuclear Information System (INIS) Gronau, M. 1984-01-01 Two ambiguities are noted in the definition of the concept of maximal CP violation. The phase convention ambiguity is overcome by introducing a CP violating phase in the quark mixing matrix U which is invariant under rephasing transformations. The second ambiguity, related to the parametrization of U, is resolved by finding a single empirically viable definition of maximal CP violation when assuming that U does not single out one generation. Considerable improvement in the calculation of nonleptonic weak amplitudes is required to test the conjecture of maximal CP violation. 21 references 8. Symmetries of Ginsparg-Wilson chiral fermions International Nuclear Information System (INIS) Mandula, Jeffrey E. 2009-01-01 The group structure of the variant chiral symmetry discovered by Luescher in the Ginsparg-Wilson description of lattice chiral fermions is analyzed. It is shown that the group contains an infinite number of linearly independent symmetry generators, and the Lie algebra is given explicitly. CP is an automorphism of this extended chiral group, and the CP transformation properties of the symmetry generators are found. The group has an infinite-parameter invariant subgroup, and the factor group, whose elements are its cosets, is isomorphic to the continuum chiral symmetry group. Features of the currents associated with these symmetries are discussed, including the fact that some different, noncommuting symmetry generators lead to the same Noether current. These are universal features of lattice chiral fermions based on the Ginsparg-Wilson relation; they occur in the overlap, domain-wall, and perfect-action formulations. In a solvable example, free overlap fermions, these noncanonical elements of lattice chiral symmetry are related to complex energy singularities that violate reflection positivity and impede continuation to Minkowski space. 9. Discrete R symmetries for the MSSM and its singlet extensions CERN Document Server Lee, Hyun Min; Ratz, Michael; Ross, Graham G; Schieren, Roland; Schmidt-Hoberg, Kai; Vaudrevange, Patrick K S 2011-01-01 We determine the anomaly free discrete R symmetries, consistent with the MSSM, that commute with SU(5) and suppress the$\\mu$parameter and nucleon decay. We show that the order M of such$Z_M^R$symmetries has to divide 24 and identify 5 viable symmetries. The simplest possibility is a$Z_4^R$symmetry which commutes with SO(10). We present a string-derived model with this$Z_4^R$symmetry and the exact MSSM spectrum below the GUT scale; in this model$Z_4^R$originates from the Lorentz symmetry of compactified dimensions. We extend the discussion to include the singlet extensions of the MSSM and find$Z_4^R$and$Z_8^R$are the only possible symmetries capable of solving the$\\mu$problem in the NMSSM. We also show that a singlet extension of the MSSM based on a$Z_{24}^R$symmetry can provide a simultaneous solution to the$\\mu$and strong CP problem with the axion coupling in the favoured window. 10. Chiral symmetry and chiral-symmetry breaking International Nuclear Information System (INIS) Peskin, M.E. 1982-12-01 These lectures concern the dynamics of fermions in strong interaction with gauge fields. Systems of fermions coupled by gauge forces have a very rich structure of global symmetries, which are called chiral symmetries. These lectures will focus on the realization of chiral symmetries and the causes and consequences of thier spontaneous breaking. A brief introduction to the basic formalism and concepts of chiral symmetry breaking is given, then some explicit calculations of chiral symmetry breaking in gauge theories are given, treating first parity-invariant and then chiral models. These calculations are meant to be illustrative rather than accurate; they make use of unjustified mathematical approximations which serve to make the physics more clear. Some formal constraints on chiral symmetry breaking are discussed which illuminate and extend the results of our more explicit analysis. Finally, a brief review of the phenomenological theory of chiral symmetry breaking is presented, and some applications of this theory to problems in weak-interaction physics are discussed 11. Charge symmetry breaking in parton distribution functions from lattice QCD Energy Technology Data Exchange (ETDEWEB) Horsley, R.; Zanotti, J.M. [Edinburgh Univ. (United Kingdom). School of Physics and Astronomy; Nakamura, Y. [Regensburg Univ. (Germany). Inst. fuer Theoretische Physik; Tsukuba Univ., Ibaraki (Japan). Center for Computational Sciences; Pleiter, D. [Deutsches Elektronen-Synchrotron (DESY), Zeuthen (Germany); Rakow, P.E.L. [Liverpool Univ. (United Kingdom). Theoretical Physics Div.; Schierholz, G. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Stueben, H. [Konrad-Zuse-Zentrum fuer Informationstechnik Berlin (Germany); Thomas, A.W.; Young, R.D. [Adelaide Univ. SA (Australia). School of Physics and Chemistry; Winter, F. [Edinburgh Univ. (United Kingdom). School of Physics and Astronomy; Regensburg Univ. (Germany). Inst. fuer Theoretische Physik 2010-12-15 By determining the quark momentum fractions of the octet baryons from N{sub f}=2+1 lattice simulations, we are able to predict the degree of charge symmetry violation in the parton distribution functions of the nucleon. This is of importance, not only as a probe of our understanding of the non-perturbative structure of the proton but also because such a violation constrains the accuracy of global ts to parton distribution functions and hence the accuracy with which, for example, cross sections at the LHC can be predicted. A violation of charge symmetry may also be critical in cases where symmetries are used to guide the search for physics beyond the Standard Model. (orig.) 12. Charge symmetry breaking in parton distribution functions from lattice QCD International Nuclear Information System (INIS) Horsley, R.; Zanotti, J.M.; Rakow, P.E.L.; Stueben, H.; Thomas, A.W.; Young, R.D.; Winter, F.; Regensburg Univ. 2010-12-01 By determining the quark momentum fractions of the octet baryons from N f =2+1 lattice simulations, we are able to predict the degree of charge symmetry violation in the parton distribution functions of the nucleon. This is of importance, not only as a probe of our understanding of the non-perturbative structure of the proton but also because such a violation constrains the accuracy of global ts to parton distribution functions and hence the accuracy with which, for example, cross sections at the LHC can be predicted. A violation of charge symmetry may also be critical in cases where symmetries are used to guide the search for physics beyond the Standard Model. (orig.) 13. Direct Lorentz force compensation flowmeter for electrolytes Energy Technology Data Exchange (ETDEWEB) Vasilyan, S., E-mail: [email protected]; Froehlich, Th. [Institute of Process Measurement and Sensor Technology, Ilmenau University of Technology, 98684 Ilmenau (Germany) 2014-12-01 A simplified method of contactless Lorentz force (LF) measurements for flow meters on electrolytes is described and realized. Modification and comparative representation are discussed against recently well-developed methods. Based on the catapult effect, that current carrying conductor experiences a repulsive force in a magnetic field, we demonstrate force measurement method of LF velocimetry applications by commonly known “electromagnetic force” compensation principle. Measurement approach through zero point stability is considered to minimize mechanical influences and avoid gravimetric uncertainties. Here, the current carrying wires are static fixed in the vicinity of magnet system at zero point stable position, while occurring deflection of magnets by electrolyte flow is compensated by external applied current within wires. Measurements performed by developed servo-system which drives control loop by means of optical position sensor for simplified (i) single wire and (ii) coil-like extended compensation schemes. Guided by experiments on electrolyte flow, we demonstrate the applicability of adopted principle for conductivities ranging from 2 to 20 S/m. Further improvements are discussed in agreement with the parameters of demonstration setup, straightforward theory, and experimental results. We argue that this method is potentially suitable for: (a) applications with higher conductivity like molten metal (order of 10{sup 6 }S/m) assuming spatial configuration of setup and (b) for lower range of conductivity (below 1 S/m) while this is strongly subject to stiffness of system and noise mainly mechanical and thermal radiations. 14. Lorentz and CPT invariances and the Einstein-Podolsky-Rosen correlations International Nuclear Information System (INIS) Beauregard, O.C. de 1984-01-01 This paper shows that there is no conflict between Einstein-Podolsky-Rosen (EPR) correlation and the new 1925 - 55 ''microrelativity principle'' stating the Lorentz and CPT invariance of physical law at the microlevel. The CPT invariance concept is a perfectly legal heir of the 1876 Loschmidt T-invariance concept. Therefore, the EPR-paradox can be understood as synthetizing two earlier ''paradoxes'': the wavelike probability calculus, and the T- or CPT-symmetry of elementary physical processes. The CPT-invariance can be summarized as the basic requirement of second quantization, that particle emission and antiparticle absorption are mathematically equivalent. The phenomenology displays causality as arrowless at the microlevel. The relativistic S-matrix scheme displays the CPT invariance of causality concept at the microlevel. In order to strengthen the point that the Lorentz and CPT invariant schemes of relativistic quantum mechanics do contain the full formalization of the EPR correlation, the covariant calculations pertaining to the subject are presented. The formalization of the EPR correlation and its interpretation are contained in the existing relativistic quantum mechanics. (Kato, T.) 15. Implications of Lorentz covariance for the guidance equation in two-slit quantum interference International Nuclear Information System (INIS) Holland, Peter; Philippidis, Chris 2003-01-01 It is known that Lorentz covariance fixes uniquely the current and the associated guidance law in the trajectory interpretation of quantum mechanics for spin-(1/2) particles. In the nonrelativistic domain this implies a guidance law for the electron which differs by an additional spin-dependent term from that originally proposed by de Broglie and Bohm. In this paper, we explore some of the implications of the modified guidance law. We bring out a property of mutual dependence in the particle coordinates that arises in product states, and show that the quantum potential has scalar and vector components, which implies the particle is subject to a Lorentz-like force. The conditions for the classical limit and the limit of negligible spin are given, and the empirical sufficiency of the model is demonstrated. We then present a series of calculations of the trajectories based on two-dimensional Gaussian wave packets which illustrate how the additional spin-dependent term plays a significant role in structuring both the individual trajectories and the ensemble. The single packet corresponds to quantum inertial motion. The distinct features encountered when the wave function is a product or a superposition are explored, and the trajectories that model the two-slit experiment are given. The latter paths exhibit several new characteristics compared with the original de Broglie-Bohm ones, such as crossing of the axis of symmetry 16. Dynamics and control of Lorentz-augmented spacecraft relative motion CERN Document Server Yan, Ye; Yang, Yueneng 2017-01-01 This book develops a dynamical model of the orbital motion of Lorentz spacecraft in both unperturbed and J2-perturbed environments. It explicitly discusses three kinds of typical space missions involving relative orbital control: spacecraft hovering, rendezvous, and formation flying. Subsequently, it puts forward designs for both open-loop and closed-loop control schemes propelled or augmented by the geomagnetic Lorentz force. These control schemes are entirely novel and represent a significantly departure from previous approaches. 17. Measurement of the Lorentz-FitzGerald body contraction Science.gov (United States) Rafelski, Johann 2018-02-01 A complete foundational discussion of acceleration in the context of Special Relativity (SR) is presented. Acceleration allows the measurement of a Lorentz-FitzGerald body contraction created. It is argued that in the back scattering of a probing laser beam from a relativistic flying electron cloud mirror generated by an ultra-intense laser pulse, a first measurement of a Lorentz-FitzGerald body contraction is feasible. 18. The BTZ black hole as a Lorentz-flat geometry Energy Technology Data Exchange (ETDEWEB) Alvarez, Pedro D., E-mail: [email protected] [Rudolf Peierls Centre for Theoretical Physics, University of Oxford (United Kingdom); Pais, Pablo, E-mail: [email protected] [Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia (Chile); Universidad Andrés Bello, Av. República 440, Santiago (Chile); Rodríguez, Eduardo, E-mail: [email protected] [Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Concepción (Chile); Salgado-Rebolledo, Patricio, E-mail: [email protected] [Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia (Chile); Departamento de Física, Universidad de Concepción, Casilla 160-C, Concepción (Chile); Physique Théorique et Mathématique, Université Libre de Bruxelles and International Solvay Institutes, Campus Plaine C.P. 231, B-1050 Bruxelles (Belgium); Zanelli, Jorge, E-mail: [email protected] [Centro de Estudios Científicos (CECs), Av. Arturo Prat 514, Valdivia (Chile); Universidad Andrés Bello, Av. República 440, Santiago (Chile) 2014-11-10 It is shown that 2+1 dimensional anti-de Sitter spacetimes are Lorentz-flat. This means, in particular, that any simply-connected patch of the BTZ black hole solution can be endowed with a Lorentz connection that is locally pure gauge. The result can be naturally extended to a wider class of black hole geometries and point particles in three-dimensional spacetime. 19. Symmetries of cosmological Cauchy horizons International Nuclear Information System (INIS) Moncrief, V.; Isenberg, J. 1983-01-01 We consider analytic vacuum and electrovacuum spacetimes which contain a compact null hypersurface ruled by closed null generators. We prove that each such spacetime has a non-trivial Killing symmetry. We distinguish two classes of null surfaces, degenerate and non-degenerate ones, characterized by the zero or non-zero value of a constant analogous to the ''surface gravity'' of stationary black holes. We show that the non-degenerate null surfaces are always Cauchy heizons across which the Killing fields change from spacelike (in the globally hyperbolic regions) to timelike (in the acausal, analytic extensions). For the special case of a null surface diffeomorphic to T 3 we characterize the degenerate vacuum solutions completely. These consists of an infinite dimensional family of ''plane wave'' spacetimes which are entirely foliated by compact null surfaces. Previous work by one of us has shown that, when one dimensional Killing symmetries are allowed, then infinite dimensional families of non-degenerate, vacuum solutions exist. We recall these results for the case of Cauchy horizons diffeomorphic to T 3 and prove the generality of the previously constructed non-degenerate solutions. We briefly discuss the possibility of removing the assumptions of closed generators and analyticity and proving an appropriate generalization of our main results. Such a generalization would provide strong support for the cosmic censorship conjecture by showing that causality violating, cosmological solutions of Einstein's equations are essentially an artefact of symmetry. (orig.) 20. Symmetries and nuclei International Nuclear Information System (INIS) Henley, E.M. 1987-01-01 Nuclei are very useful for testing symmetries, and for studies of symmetry breaking. This thesis is illustrated for two improper space-time transformations, parity and time-reversal and for one internal symmetry: charge symmetry and independence. Recent progress and present interest is reviewed. 23 refs., 8 figs., 2 tabs 1. Hendrik Antoon Lorentz: his role in physics and society Science.gov (United States) Berends, Frits 2009-04-01 Hendrik Antoon Lorentz (1853-1928) was appointed in 1878 to a chair of theoretical physics at the University of Leiden, one of the first of such chairs in the world. A few years later Heike Kamerlingh Onnes became his experimental colleague, after vehement discussions in the faculty. Lorentz strongly supported Kamerlingh Onnes then, and proved subsequently to be an ideal colleague. With Lorentz's electron theory the classical theory of electromagnetism obtained its final form, at the time often called the Maxwell-Lorentz theory. In this theory the Zeeman effect could be explained: the first glimpse of the electron. The Nobel Prize followed in 1902. The Lorentz transformation, established in 1904, preceded the special theory of relativity. Later on, Lorentz played a much admired role in the debate on the new developments in physics, in particular as chairman of a series of Solvay conferences. Gradually his stature outside of physics grew, both nationally as chairman of the Zuiderzee committee and internationally as president of the International Commission on Intellectual Cooperation of the League of Nations. At his funeral the overwhelming tribute was the recognition of his unique greatness. Einstein said about him 'He meant more to me personally than anyone else I have met on my life's journey'. 2. Hendrik Antoon Lorentz: his role in physics and society Energy Technology Data Exchange (ETDEWEB) Berends, Frits [Emeritus Theoretical Physics, Leiden University (Netherlands) 2009-04-22 Hendrik Antoon Lorentz (1853-1928) was appointed in 1878 to a chair of theoretical physics at the University of Leiden, one of the first of such chairs in the world. A few years later Heike Kamerlingh Onnes became his experimental colleague, after vehement discussions in the faculty. Lorentz strongly supported Kamerlingh Onnes then, and proved subsequently to be an ideal colleague. With Lorentz's electron theory the classical theory of electromagnetism obtained its final form, at the time often called the Maxwell-Lorentz theory. In this theory the Zeeman effect could be explained: the first glimpse of the electron. The Nobel Prize followed in 1902. The Lorentz transformation, established in 1904, preceded the special theory of relativity. Later on, Lorentz played a much admired role in the debate on the new developments in physics, in particular as chairman of a series of Solvay conferences. Gradually his stature outside of physics grew, both nationally as chairman of the Zuiderzee committee and internationally as president of the International Commission on Intellectual Cooperation of the League of Nations. At his funeral the overwhelming tribute was the recognition of his unique greatness. Einstein said about him 'He meant more to me personally than anyone else I have met on my life's journey'. 3. Hendrik Antoon Lorentz: his role in physics and society International Nuclear Information System (INIS) Berends, Frits 2009-01-01 Hendrik Antoon Lorentz (1853-1928) was appointed in 1878 to a chair of theoretical physics at the University of Leiden, one of the first of such chairs in the world. A few years later Heike Kamerlingh Onnes became his experimental colleague, after vehement discussions in the faculty. Lorentz strongly supported Kamerlingh Onnes then, and proved subsequently to be an ideal colleague. With Lorentz's electron theory the classical theory of electromagnetism obtained its final form, at the time often called the Maxwell-Lorentz theory. In this theory the Zeeman effect could be explained: the first glimpse of the electron. The Nobel Prize followed in 1902. The Lorentz transformation, established in 1904, preceded the special theory of relativity. Later on, Lorentz played a much admired role in the debate on the new developments in physics, in particular as chairman of a series of Solvay conferences. Gradually his stature outside of physics grew, both nationally as chairman of the Zuiderzee committee and internationally as president of the International Commission on Intellectual Cooperation of the League of Nations. At his funeral the overwhelming tribute was the recognition of his unique greatness. Einstein said about him 'He meant more to me personally than anyone else I have met on my life's journey'. 4. The Lorentz-Dirac equation in light of quantum theory International Nuclear Information System (INIS) Nikishov, A.I. 1996-01-01 To high accuracy, an electron in ultrarelativistic motion 'sees' an external field in its rest frame as a crossed field (E=H, E·H=0). In this case, quantum expressions allow the introduction of a local intensity of the radiation, which determines the radiative term of the force of radiative reaction. For γ=(1-v2)-1/2>> 1 this term is much larger than the mass term, i.e., the term with xd3do. Under these conditions, the reduced Lorentz-Dirac equation, which is obtained from the full Lorentz-Dirac equation by eliminating the terms xd3do and xe on the right side using the equation of motion without taking into account the force of radiative reaction, is equivalent to good accuracy to the original Lorentz-Dirac equation. Exact solutions to the reduced Lorentz-Dirac equation are obtained for a constant field and the field of a plane wave. For γ∼1 a local expression for the radiative term cannot be obtained quantitatively from the quantum expressions. In this case the mass (Lorentz-Dirac) terms in the original and reduced Lorentz-Dirac equations are not small compared to the radiative term. The predictions of these equations, which depend appreciably on the mass terms, are therefore less reliable 5. Up sector of minimal flavor violation: top quark properties and direct D meson CP violation Energy Technology Data Exchange (ETDEWEB) Bai, Yang; Berger, Joshua; Hewett, JoAnne L.; Li, Ye 2013-07-01 Minimal Flavor Violation in the up-type quark sector leads to particularly interesting phenomenology due to the interplay of flavor physics in the charm sector and collider physics from flavor changing processes in the top sector. We study the most general operators that can affect top quark properties and D meson decays in this scenario, concentrating on two CP violating operators for detailed studies. The consequences of these effective operators on charm and top flavor changing processes are generically small, but can be enhanced if there exists a light flavor mediator that is a Standard Model gauge singlet scalar and transforms under the flavor symmetry group. This flavor mediator can satisfy the current experimental bounds with a mass as low as tens of GeV and explain observed D-meson direct CP violation. Additionally, the model predicts a non-trivial branching fraction for a top quark decay that would mimic a dijet resonance. 6. Flavored dark matter beyond Minimal Flavor Violation CERN Document Server Agrawal, Prateek; Gemmler, Katrin 2014-10-13 We study the interplay of flavor and dark matter phenomenology for models of flavored dark matter interacting with quarks. We allow an arbitrary flavor structure in the coupling of dark matter with quarks. This coupling is assumed to be the only new source of violation of the Standard Model flavor symmetry extended by a$U(3)_\\chi$associated with the dark matter. We call this ansatz Dark Minimal Flavor Violation (DMFV) and highlight its various implications, including an unbroken discrete symmetry that can stabilize the dark matter. As an illustration we study a Dirac fermionic dark matter$\\chi$which transforms as triplet under$U(3)_\\chi$, and is a singlet under the Standard Model. The dark matter couples to right-handed down-type quarks via a colored scalar mediator$\\phi$with a coupling$\\lambda$. We identify a number of "flavor-safe" scenarios for the structure of$\\lambda$which are beyond Minimal Flavor Violation. For dark matter and collider phenomenology we focus on the well-motivated case of$b$-... 7. Flavored dark matter beyond Minimal Flavor Violation International Nuclear Information System (INIS) Agrawal, Prateek; Blanke, Monika; Gemmler, Katrin 2014-01-01 We study the interplay of flavor and dark matter phenomenology for models of flavored dark matter interacting with quarks. We allow an arbitrary flavor structure in the coupling of dark matter with quarks. This coupling is assumed to be the only new source of violation of the Standard Model flavor symmetry extended by a U(3) χ associated with the dark matter. We call this ansatz Dark Minimal Flavor Violation (DMFV) and highlight its various implications, including an unbroken discrete symmetry that can stabilize the dark matter. As an illustration we study a Dirac fermionic dark matter χ which transforms as triplet under U(3) χ , and is a singlet under the Standard Model. The dark matter couples to right-handed down-type quarks via a colored scalar mediator with a coupling. We identify a number of ''flavor-safe'' scenarios for the structure of which are beyond Minimal Flavor Violation. Also, for dark matter and collider phenomenology we focus on the well-motivated case of b-flavored dark matter. Furthermore, the combined flavor and dark matter constraints on the parameter space of turn out to be interesting intersections of the individual ones. LHC constraints on simplified models of squarks and sbottoms can be adapted to our case, and monojet searches can be relevant if the spectrum is compressed 8. Asymmetric transmission in planar chiral split-ring metamaterials: Microscopic Lorentz-theory approach DEFF Research Database (Denmark) Novitsky, Andrey; Galynsky, Vladimir M.; Zhukovsky, Sergei 2012-01-01 The electronic Lorentz theory is employed to explain the optical properties of planar split-ring metamaterials. Starting from the dynamics of individual free carriers, the electromagnetic response of an individual split-ring meta-atom is determined, and the effective permittivity tensor...... of the metamaterial is calculated for normal incidence of light. Whenever the split ring lacks in-plane mirror symmetry, the corresponding permittivity tensor has a crystallographic structure of an elliptically dichroic medium, and the metamaterial exhibits optical properties of planar chiral structures. Its...... transmission spectra are different for right-handed versus left-handed circular polarization of the incident wave, so the structure changes its transmittance when the direction of incidence is reversed. The magnitude of this change is shown to be related to the geometric parameters of the split ring... 9. Discrete symmetries: A broken look at QCD International Nuclear Information System (INIS) Goldman, T. 1996-01-01 The alphabet soup of discrete symmetries is briefly surveyed with a view towards those which can be tested at LISS and two particularly interesting cases are called out. A LISS experiment may be able to distinguish CP violation that is not due to the QCD θ term. The elements of a model of parity violation in proton-nucleon scattering, which is consistent with lower energy LAMPF and ANL results, are reviewed in the light of new information on diquarks and the proton spin fraction carried by quarks. The prediction that the parity violating total cross section asymmetry should be large at LISS energies is confirmed. The results of such an experiment can be used both to obtain new information about the diquark substructure of the nucleon and to provide bounds on new right-chiral weak interactions 10. A unique$Z_4^R$symmetry for the MSSM CERN Document Server Lee, Hyun Min; Ratz, Michael; Ross, Graham G; Schieren, Roland; Schmidt-Hoberg, Kai; Vaudrevange, Patrick K S 2011-01-01 We consider the possible anomaly free Abelian discrete symmetries of the MSSM that forbid the mu-term at perturbative order. Allowing for anomaly cancellation via the Green-Schwarz mechanism we identify discrete R-symmetries as the only possibility and prove that there is a unique Z_4^R symmetry that commutes with SO(10). We argue that non-perturbative effects will generate a mu-term of electroweak order thus solving the mu-problem. The non-perturbative effects break the Z_4^R symmetry leaving an exact Z_2 matter parity. As a result dimension four baryon- and lepton-number violating operators are absent while, at the non-perturbative level, dimension five baryon- and lepton-number violating operators get induced but are highly suppressed so that the nucleon decay rate is well within present bounds. 11. Renormalisation group improved leptogenesis in family symmetry models International Nuclear Information System (INIS) Cooper, Iain K.; King, Stephen F.; Luhn, Christoph 2012-01-01 We study renormalisation group (RG) corrections relevant for leptogenesis in the case of family symmetry models such as the Altarelli-Feruglio A 4 model of tri-bimaximal lepton mixing or its extension to tri-maximal mixing. Such corrections are particularly relevant since in large classes of family symmetry models, to leading order, the CP violating parameters of leptogenesis would be identically zero at the family symmetry breaking scale, due to the form dominance property. We find that RG corrections violate form dominance and enable such models to yield viable leptogenesis at the scale of right-handed neutrino masses. More generally, the results of this paper show that RG corrections to leptogenesis cannot be ignored for any family symmetry model involving sizeable neutrino and τ Yukawa couplings. 12. Isospin symmetry violation, meson production and η-nucleus ... Indian Academy of Sciences (India) The experimental details and results obtained so far are presented ... to the large mass of η-meson (547 MeV), this S11 resonance is very close to η-N ... 3He particles were detected at the normal focal plane whereas tritons were measured. 13. Isospin symmetry violation, meson production and η-nucleus ... Indian Academy of Sciences (India) The experiment was perfomed at the cooler synchrotron accelerator. COSY, Jülich at several beam energies close to the corresponding production threshold. We also have ongoing programmes on -nucleus final-state interaction studies via + 6Li → 7Be + reactions, high resolution search for dibaryonic resonances ... 14. Weak interaction models with spontaneously broken left-right symmetry International Nuclear Information System (INIS) Mohapatra, R.H. 1978-01-01 The present status of weak interaction models with spontaneously broken left-right symmetry is reviewed. The theoretical basis for asymptotic parity conservation, manifest left-right symmetry in charged current weak interactions, natural parity conservation in neutral currents and CP-violation in the context of SU(2)/sub L/ circled x SU (2)/sub R/ circled x U(1) models are outlined in detail. Various directions for further research in the theoretical and experimental side are indicated 15. CP violation and supersymmetry-breaking in superstring models International Nuclear Information System (INIS) Dent, T.E. 2000-09-01 In this thesis I discuss aspects of the phenomenology of heterotic string, theory, using low-energy effective supergravity models. I investigate the origin of CP violation, the implications for low-energy physics of the modular invariance of the theory, supersymmetry-breaking via gaugino condensation in a hidden sector, and the interplay between these topics. I review the theory of CP violation and the problem of CP violation in supersymmetry phenomenology. In a scenario where the origin of CP violation lies in the compactification of the extra dimensions of string theory, I present simple models which include a duality symmetry acting on the compactification modulus and on observable fields. I show how the structure of the theory affects CP-violating observables, and discuss the effect of such a symmetry on low-energy physics in general. I present a detailed investigation of supersymmetry-breaking by gaugino condensation in supergravity, in particular as applied to the stabilisation of string moduli. For hidden sectors with or without matter I calculate corrections to the usual formulae for the scalar potential and soft supersymmetry-breaking terms. I discuss the phenomenological implications of these corrections and show that they may affect the value of the compactification modulus. and consequently the prospects for predictions of CP violation in string models. (author) 16. Quantum-gravity-motivated Lorentz-symmetry tests with laser interferometers International Nuclear Information System (INIS) Amelino-Camelia, Giovanni; Laemmerzahl, Claus 2004-01-01 We consider the implications for laser interferometry of the quantum-gravity-motivated modifications in the laws of particle propagation, which are presently being considered in attempts to explain puzzling observations of ultra-high-energy cosmic rays. We show that there are interferometric set-ups in which the Planck-scale effect on propagation leads to a characteristic signature. A naive estimate is encouraging with respect to the possibility of achieving Planck-scale sensitivity, but we also point out some severe technological challenges which would have to be overcome in order to achieve this sensitivity 17. The symmetry of man. Science.gov (United States) Ermolenko, Alexander E; Perepada, Elena A 2007-01-01 The paper contains a description of basic regularities in the manifestation of symmetry of human structural organization and its ontogenetic and phylogenetic development. A concept of macrobiocrystalloid with inherent complex symmetry is proposed for the description of the human organism in its integrity. The symmetry can be characterized as two-plane radial (quadrilateral), where the planar symmetry is predominant while the layout of organs of radial symmetry is subordinated to it. Out of the two planes of symmetry (sagittal and horizontal), the sagittal plane is predominant. The symmetry of the chromosome, of the embrio at the early stages of cell cleavage as well as of some organs and systems in their phylogenetic development is described. An hypothesis is postulated that the two-plane symmetry is formed by two mechanisms: a) the impact of morphogenetic fields of the whole crystalloid organism during embriogenesis and, b) genetic mechanisms of the development of chromosomes having two-plane symmetry. 18. Phenomenology of CP violation International Nuclear Information System (INIS) Ecker, G. 1987-01-01 A short survey of the theoretical status of CP violation is presented. The Standart Model is confronted with the present experimental situation. Possible future tests of our notions of CP violation are discussed, concentrating on rare K decays. Other promising reactions such as B decays are briefly reviewed. Among alternative models of CP violation, multi-Higgs extensions of the Standart Model, left-right symmetric gauge theories and minimal SUSY models are discussed. Finally, the relevance of generalized CP invariance is emphasized. 64 refs., 7 figs. (Author) 19. Errors and violations International Nuclear Information System (INIS) Reason, J. 1988-01-01 This paper is in three parts. The first part summarizes the human failures responsible for the Chernobyl disaster and argues that, in considering the human contribution to power plant emergencies, it is necessary to distinguish between: errors and violations; and active and latent failures. The second part presents empirical evidence, drawn from driver behavior, which suggest that errors and violations have different psychological origins. The concluding part outlines a resident pathogen view of accident causation, and seeks to identify the various system pathways along which errors and violations may be propagated 20. Non-commutative phase space and its space-time symmetry International Nuclear Information System (INIS) Li Kang; Dulat Sayipjamal 2010-01-01 First a description of 2+1 dimensional non-commutative (NC) phase space is presented, and then we find that in this formulation the generalized Bopp's shift has a symmetric representation and one can easily and straightforwardly define the star product on NC phase space. Then we define non-commutative Lorentz transformations both on NC space and NC phase space. We also discuss the Poincare symmetry. Finally we point out that our NC phase space formulation and the NC Lorentz transformations are applicable to any even dimensional NC space and NC phase space. (authors) 1. LORENTZ PHASE IMAGING AND IN-SITU LORENTZ MICROSCOPY OF PATTERNED CO-ARRAYS International Nuclear Information System (INIS) VOLKOV, V.V.; ZHU, Y. 2003-01-01 Understanding magnetic structures and properties of patterned and ordinary magnetic films at nanometer length-scale is the area of immense technological and fundamental scientific importance. The key feature to such success is the ability to achieve visual quantitative information on domain configurations with a maximum ''magnetic'' resolution. Several methods have been developed to meet these demands (Kerr and Faraday effects, differential phase contrast microscopy, magnetic force microscopy, SEMPA etc.). In particular, the modern off-axis electron holography allows retrieval of the electron-wave phase shifts down to 2π/N (with typical N = 10-20, approaching in the limit N ∼ 100) in TEM equipped with field emission gun, which is already successfully employed for studies of magnetic materials at nanometer scale. However, it remains technically demanding, sensitive to noise and needs highly coherent electron sources. As possible alternative we developed a new method of Lorentz phase microscopy [1,2] based on the Fourier solution [3] of magnetic transport-of-intensity (MTIE) equation. This approach has certain advantages, since it is less sensitive to noise and does not need high coherence of the source required by the holography. In addition, it can be realized in any TEM without basic hardware changes. Our approach considers the electron-wave refraction in magnetic materials (magnetic refraction) and became possible due to general progress in understanding of noninterferometric phase retrieval [4-6] dealing with optical refraction. This approach can also be treated as further development of Fresnel microscopy, used so far for imaging of in-situ magnetization process in magnetic materials studied by TEM. Figs. 1-3 show some examples of what kind information can be retrieved from the conventional Fresnel images using the new approach. Most of these results can be compared with electron-holographic data. Using this approach we can shed more light on fine details of 2. Violation of the Appelquist-Carazzone decoupling in a nonsupersymmetric grand unified theory International Nuclear Information System (INIS) Chankowski, Piotr H.; Wagner, Jakub 2008-01-01 We point out that in nonsupersymmetric grand unified theories, in which the SU(5) gauge symmetry is broken down to the standard model gauge group by a 24 Higgs multiplet the Appelquist-Carazzone decoupling is violated. This is because the SU(2) L Higgs triplet contained in the 24 acquires a dimension-full coupling to the SU(2) L Higgs doublets which is proportional to the grand unified symmetry breaking vacuum expectation value. As a result, at one-loop heavy gauge and Higgs fields contribution to tadpoles generates a vacuum expectation value of the triplet which is not suppressed for V→∞ and violates the custodial symmetry 3. In-depth Study on Cylinder Wake Controlled by Lorentz Force International Nuclear Information System (INIS) Zhang Hui; Fan Bao-Chun; Chen Zhi-Hua 2011-01-01 The underlying mechanisms of the electromagnetic control of cylinder wake are investigated and discussed. The effects of Lorentz force are found to be composed of two parts, one is its direct action on the cylinder (the wall Lorentz force) and the other is applied to the fluid (called the field Lorentz force) near the cylinder surface. Our results show that the wall Lorentz force can generate thrust and reduce the drag; the field Lorentz force increases the drag. However, the cylinder drag is dominated by the wall Lorentz force. In addition, the field Lorentz force above the upper surface decreases the lift, while the upper wall Lorentz force increases it. The total lift is dominated by the upper wall Lorentz force. (fundamental areas of phenomenology(including applications)) 4. Violation of causality in f(T) gravity Energy Technology Data Exchange (ETDEWEB) Otalora, G. [Pontificia Universidad Catolica de Valparaiso, Instituto de Fisica, Valparaiso (Chile); Reboucas, M.J. [Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ (Brazil) 2017-11-15 In the standard formulation, the f(T) field equations are not invariant under local Lorentz transformations, and thus the theory does not inherit the causal structure of special relativity. Actually, even locally violation of causality can occur in this formulation of f(T) gravity. A locally Lorentz covariant f(T) gravity theory has been devised recently, and this local causality problem seems to have been overcome. The non-locality question, however, is left open. If gravitation is to be described by this covariant f(T) gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its field equations allow homogeneous Goedel-type solutions, which necessarily leads to violation of causality on non-local scale. Here, to look into the potentialities and difficulties of the covariant f(T) theories, we examine whether they admit Goedel-type solutions. We take a combination of a perfect fluid with electromagnetic plus a scalar field as source, and determine a general Goedel-type solution, which contains special solutions in which the essential parameter of Goedel-type geometries, m{sup 2}, defines any class of homogeneous Goedel-type geometries. We show that solutions of the trigonometric and linear classes (m{sup 2} < 0 and m = 0) are permitted only for the combined matter sources with an electromagnetic field matter component. We extended to the context of covariant f(T) gravity a theorem which ensures that any perfect-fluid homogeneous Goedel-type solution defines the same set of Goedel tetrads h{sub A}{sup μ} up to a Lorentz transformation. We also showed that the single massless scalar field generates Goedel-type solution with no closed time-like curves. Even though the covariant f(T) gravity restores Lorentz covariance of the field equations and the local validity of the causality principle, the bare existence of the Goedel-type solutions makes apparent that the covariant formulation of f(T) gravity 5. Pittsburgh PLI Violations Report Data.gov (United States) Allegheny County / City of Pittsburgh / Western PA Regional Data Center — Report containing Department of Permits, Licenses, and Inspections violation notices that have been issued by the City after October 15, 2015 6. On the ether-like Lorentz-breaking actions International Nuclear Information System (INIS) Petrov, A.Yu; Nascimento, J.R.; Gomes, M.; Silva, A. J. da 2011-01-01 We demonstrate the generation of the CPT-even, ether-like Lorentz-breaking actions for the scalar and electro-magnetic fields via their appropriate Lorentz-breaking coupling to spinor fields in three, four and five space-time dimensions. Besides, we show that the ether-like terms for the spinor field also can be generated as a consequence of the same couplings. The key result which will be presented here is the finiteness of the ether-like term for the electromagnetic field not only in three and five space-time dimensions where it is natural due to known effects of the dimensional regularization but also in four space-time dimensions. Moreover, we present the calculation of the last result within different calculational schemes and conclude that the result for the four-dimensional ether-like term for the electromagnetic field essentially depending on the calculation scheme, similarly to the result for the Carroll-Field-Jackiw (CFJ) term which probably signalizes a possibility for arising of a new anomaly. Also we discuss the dispersion relations in the theories with ether-like Lorentz-breaking terms which allows to discuss the consistency of the Lorentz-breaking modified theories for different (space-like or time-like) Lorentz-breaking vectors and find the tree-level effective (Breit) potential for fermion scattering and the one-loop effective potential corresponding to the action of the scalar field. (author) 7. Geometry of Majorana neutrino and new symmetries CERN Document Server Volkov, G G 2006-01-01 Experimental observation of Majorana fermion matter gives a new impetus to the understanding of the Lorentz symmetry and its extension, the geometrical properties of the ambient space-time structure, matter--antimatter symmetry and some new ways to understand the baryo-genesis problem in cosmology. Based on the primordial Majorana fermion matter assumption, we discuss a possibility to solve the baryo-genesis problem through the the Majorana-Diraco genesis in which we have a chance to understand creation of Q(em) charge and its conservation in our D=1+3 Universe after the Big Bang. In the Majorana-Diraco genesis approach there appears a possibility to check the proton and electron non-stability on the very low energy scale. In particle physics and in our space-time geometry, the Majorana nature of the neutrino can be related to new types of symmetries which are lying beyond the binary Cartan-Killing-Lie algebras/superalgebras. This can just support a conjecture about the non-completeness of the SM in terms of ... 8. Parity and time-reversal violation in nuclei and atoms International Nuclear Information System (INIS) Adelberger, E.G. 1986-01-01 Two topics are briefly reviewed: the parity (P)-violating NN interaction and the time-reversal (T) and P-violating electric moments (EDM's) of atoms. The ΔI = 1 P-violating NN amplitude dominated by weak π +- exchange is found to be appreciably smaller than bag-model predictions. This may be a dynamical symmetry of flavor-conserving hadronic weak processes reminiscent of the ΔI = 1/2 rule in flavor-changing decays. General principles of experimental searches for atomic EDM's are discussed. Atomic EDM's are sensitive to electronic or nuclear EDM's and to a P-and-T-violating electron-quark interaction. Even though the experimental precision is still ∼10 4 times worse than counting statistics, the recent results have reached a sensitivity to nuclear EDM's which rivals that of the neutron EDM data. Further significant improvements can be expected 9. Recent Results on T and CP Violation at BABAR Energy Technology Data Exchange (ETDEWEB) Perez Perez, Alejandro [Istituto Nazionale di Fisica Nucleare (INFN), Pisa (Italy). 2015-02-06 CP-violation (CPV) and Time-reversal violation (TRV) are intimately related through the CPT theorem: if one of these discrete symmetries is violated the other one has to be violated in such a way to conserve CPT. Although CPV in the B0B0-bar system has been established by the B-factories, implying indirectly TRV, there is still no direct evidence of TRV. We report on the observation of TRV in the B-meson system performed with a dataset of 468 × 106 BB-bar pairs produced in Υ(4S) decays collected by the BABAR detector at the PEP-II asymmetric-energy e+e- collider at the SLAC National Accelerator Laboratory. We also report on other CPV measurements recently performed on the B-meson system 10. LHC experimental sensitivity to CP violating gtt couplings CERN Document Server Sjölin, J 2003-01-01 The level of CP violation in pp to tt+X induced by the standard model is known to be below the experimental sensitivity by many orders of magnitude. However, in some effective theories, it is plausible that new CP violating physics could reveal itself as additional non- renormalizable terms in the Lagrangian. Since these should respect the symmetries of the low-energy gauge interaction, violate CP and generate the correct event topology, the set of allowed terms is highly restricted. This analysis gives an estimate of the expected experimental sensitivity to the lowest order effective CP violating gtt interaction term beyond the standard model using simulated data from the ATLAS detector at the LHC. (36 refs). 11. CP violation in the lepton sector with Majorana neutrinos International Nuclear Information System (INIS) Aguila, F. del 1995-01-01 We study CP violation in the lepton sector in extended models with right-handed neutrinos, without and with left-right symmetry, and with arbitrary mass terms. We find the conditions which must be satisfied by the neutrino and charged lepton mass matrices for CP conservation. These constraints, which are independent of the choice of weak basis, are proven to be also sufficient in simple cases. This invariant formulation makes apparent the necessary requirements for CP violation, as well as the size of CP violating effects. As an example, we show that CP violation can be much larger in left-right symmetric models than in models with only additional right-handed neutrinos, i.e., without right-handed currents. (orig.) 12. CP violation and B0-(B0)-bar mixing International Nuclear Information System (INIS) Aleksan, R. 1996-01-01 The status of CP violation and B 0 -(B 0 )-bar mixing is given and the subsequent constraints in the framework of the Standard Model are discussed. Recent result on CP violation in the kaon system and related topics are reviewed, including the status of T violation and the tests of the CPT symmetry. The results on B 0 -(B 0 )-bar mixing are presented followed by the studies on B d 0 -(B d 0 )-bar and B s 0 -(B s 0 )-bar oscillations. Finally, the prospects of progress on understanding CP violation are discussed in framework of the new projects expected to produce results at the turn of the century. (author) 13. Realizing total reciprocity violation in the phase for photon scattering. Science.gov (United States) Deák, László; Bottyán, László; Fülöp, Tamás; Merkel, Dániel Géza; Nagy, Dénes Lajos; Sajti, Szilárd; Schulze, Kai Sven; Spiering, Hartmut; Uschmann, Ingo; Wille, Hans-Christian 2017-02-22 Reciprocity is when wave or quantum scattering satisfies a symmetry property, connecting a scattering process with the reversed one. While reciprocity involves the interchange of source and detector, it is fundamentally different from rotational invariance, and is a generalization of time reversal invariance, occurring in absorptive media as well. Due to its presence at diverse areas of physics, it admits a wide variety of applications. For polarization dependent scatterings, reciprocity is often violated, but violation in the phase of the scattering amplitude is much harder to experimentally observe than violation in magnitude. Enabled by the advantageous properties of nuclear resonance scattering of synchrotron radiation, we have measured maximal, i.e., 180-degree, reciprocity violation in the phase. For accessing phase information, we introduced a new version of stroboscopic detection. The scattering setting was devised based on a generalized reciprocity theorem that opens the way to construct new types of reciprocity related devices. 14. Dark Matter and observable lepton flavour violation International Nuclear Information System (INIS) Heurtier, Lucien; Univ. Libre de Bruxelles; Teresi, Daniele 2016-07-01 Seesaw models with leptonic symmetries allow right-handed (RH) neutrino masses at the electroweak scale, or even lower, at the same time having large Yukawa couplings with the Standard Model leptons, thus yielding observable effects at current or near-future lepton-flavour-violation (LFV) experiments. These models have been previously considered also in connection to low-scale leptogenesis, but the combination of observable LFV and successful leptogenesis has appeared to be difficult to achieve unless the leptonic symmetry is embedded into a larger one. In this paper, instead, we follow a different route and consider a possible connection between large LFV rates and Dark Matter (DM). We present a model in which the same leptonic symmetry responsible for the large Yukawa couplings guarantees the stability of the DM candidate, identified as the lightest of the RH neutrinos. The spontaneous breaking of this symmetry, caused by a Majoron-like field, also provides a mechanism to produce the observed relic density via the decays of the latter. The phenomenological implications of the model are discussed, finding that large LFV rates, observable in the near-future μ→e conversion experiments, require the DM mass to be in the keV range. Moreover, the active-neutrino coupling to the Majoron-like scalar field could be probed in future detections of supernova neutrino bursts. 15. The Symmetry behind Extended Flavour Democracy and Large Leptonic Mixing CERN Document Server Silva-Marcos, Joaquim I 2002-01-01 We show that there is a minimal discrete symmetry which leads to the extended flavour democracy scenario constraining the Dirac neutrino, the charged lepton and the Majorana neutrino mass term ($M_R$) to be all proportional to the democratic matrix, with all elements equal. In particular, this discrete symmetry forbids other large contributions to$M_R$, such as a term proportional to the unit matrix, which would normally be allowed by a$S_{3L}\\times S_{3R}\$ permutation symmetry. This feature is crucial in order to obtain large leptonic mixing, without violating 't Hooft's, naturalness principle.
16. Special Relativity in Week One: 3) Introducing the Lorentz Contraction
Science.gov (United States)
Huggins, Elisha
2011-05-01
This is the third of four articles on teaching special relativity in the first week of an introductory physics course.1,2 With Einstein's second postulate that the speed of light is the same to all observers, we could use the light pulse clock to introduce time dilation. But we had difficulty introducing the Lorentz contraction until we saw the movie "Time Dilation, an Experiment with Mu-Mesons" by David Frisch and James Smith.3,4 The movie demonstrates that time dilation and the Lorentz contraction are essentially two sides of the same coin. Here we take the muon's point of view for a more intuitive understanding of the Lorentz contraction, and use the results of the movie to provide an insight into the way we interpret experimental results involving special relativity.
17. Comment on 'Lorentz transformations with arbitrary line of motion'
International Nuclear Information System (INIS)
Tjiang, Paulus C; Sutanto, Sylvia H
2007-01-01
A short comment regarding the derivation of Lorentz transformation proposed by Iyer and Prabhu (2007 Eur. J. Phys. 11 183-90) is given. It is shown that the proposed derivation is similar to that appearing in the standard textbooks of classical mechanics, electrodynamics and the theory of relativity. In fact, those textbooks also provide an elegant form of the Lorentz matrix for the (3+1)-dimensional case, which Iyer and Prabhu claim to be difficult to attain because of its algebraic complexity. We also provide the derivation of the (3+1)-dimensional version of the Lorentz matrix using a method analogous to that proposed by Iyer and Prabhu, and show that the result is completely equivalent to the (3+1)-dimensional version appearing in the textbooks. (letters and comments)
18. Convexity and concavity constants in Lorentz and Marcinkiewicz spaces
Science.gov (United States)
Kaminska, Anna; Parrish, Anca M.
2008-07-01
We provide here the formulas for the q-convexity and q-concavity constants for function and sequence Lorentz spaces associated to either decreasing or increasing weights. It yields also the formula for the q-convexity constants in function and sequence Marcinkiewicz spaces. In this paper we extent and enhance the results from [G.J.O. Jameson, The q-concavity constants of Lorentz sequence spaces and related inequalities, Math. Z. 227 (1998) 129-142] and [A. Kaminska, A.M. Parrish, The q-concavity and q-convexity constants in Lorentz spaces, in: Banach Spaces and Their Applications in Analysis, Conference in Honor of Nigel Kalton, May 2006, Walter de Gruyter, Berlin, 2007, pp. 357-373].
19. Origin of family symmetries
International Nuclear Information System (INIS)
Nilles, Hans Peter
2012-04-01
Discrete (family) symmetries might play an important role in models of elementary particle physics. We discuss the origin of such symmetries in the framework of consistent ultraviolet completions of the standard model in field and string theory. The symmetries can arise due to special geometrical properties of extra compact dimensions and the localization of fields in this geometrical landscape. We also comment on anomaly constraints for discrete symmetries.
20. Origin of family symmetries
Energy Technology Data Exchange (ETDEWEB)
Nilles, Hans Peter [Bonn Univ. (Germany). Bethe Center for Theoretical Physics; Bonn Univ. (Germany). Physikalisches Inst.; Ratz, Michael [Technische Univ. Muenchen, Garching (Germany). Physik-Department; Vaudrevange, Patrick K.S. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2012-04-15
Discrete (family) symmetries might play an important role in models of elementary particle physics. We discuss the origin of such symmetries in the framework of consistent ultraviolet completions of the standard model in field and string theory. The symmetries can arise due to special geometrical properties of extra compact dimensions and the localization of fields in this geometrical landscape. We also comment on anomaly constraints for discrete symmetries.
1. Symmetry, asymmetry and dissymmetry
International Nuclear Information System (INIS)
Wackenheim, A.; Zollner, G.
1987-01-01
The authors discuss the concept of symmetry and defect of symmetry in radiological imaging and recall the definition of asymmetry (congenital or constitutional) and dissymmetry (acquired). They then describe a rule designed for the cognitive method of automatic evaluation of shape recognition data and propose the use of reversal symmetry [fr
2. Symmetry and electromagnetism
International Nuclear Information System (INIS)
Fuentes Cobas, L.E.; Font Hernandez, R.
1993-01-01
An analytical treatment of electrostatic and magnetostatic field symmetry, as a function of charge and current distribution symmetry, is proposed. The Newmann Principle, related to the cause-effect symmetry relation, is presented and applied to the characterization of simple configurations. (Author) 5 refs
3. Weak C* Hopf Symmetry
OpenAIRE
Rehren, K. -H.
1996-01-01
Weak C* Hopf algebras can act as global symmetries in low-dimensional quantum field theories, when braid group statistics prevents group symmetries. Possibilities to construct field algebras with weak C* Hopf symmetry from a given theory of local observables are discussed.
4. Gauge symmetry breaking
International Nuclear Information System (INIS)
Weinberg, S.
1976-01-01
The problem of how gauge symmetries of the weak interactions get broken is discussed. Some reasons why such a heirarchy of gauge symmetry breaking is needed, the reason gauge heirarchies do not seem to arise in theories of a given and related type, and the implications of theories with dynamical symmetry breaking, which can exhibit a gauge hierarchy
5. Synchronizing and controlling hyperchaos in complex Lorentz-Haken systems
Energy Technology Data Exchange (ETDEWEB)
Jinqing, Fang [Academia Sinica, Beijing, BJ (China). Inst. of Atomic Energy
1995-03-01
Synchronizing hyperchaos is realized by the drive-response relationship in the complex Lorentz-Haken system and its higher-order cascading systems for the first time. Controlling hyperchaos is achieved by the intermittent proportional feedback to all of the drive (master) system variables. The complex Lorentz-Haken system describes the detuned single-mode laser and is taken as a typical example of hyperchaotic synchronization to clarify our ideas and results. The ideas and concepts could be extended to some nonlinear dynamical systems and have prospects for potential applications, for example. to laser, electronics, plasma, cryptography, communication, chemical and biological systems and so on. (8 figs., 2 tabs.).
6. Synchronizing and controlling hyperchaos in complex Lorentz-Haken systems
International Nuclear Information System (INIS)
Fang Jinqing
1995-03-01
Synchronizing hyperchaos is realized by the drive-response relationship in the complex Lorentz-Haken system and its higher-order cascading systems for the first time. Controlling hyperchaos is achieved by the intermittent proportional feedback to all of the drive (master) system variables. The complex Lorentz-Haken system describes the detuned single-mode laser and is taken as a typical example of hyperchaotic synchronization to clarify our ideas and results. The ideas and concepts could be extended to some nonlinear dynamical systems and have prospects for potential applications, for example. to laser, electronics, plasma, cryptography, communication, chemical and biological systems and so on. (8 figs., 2 tabs.)
7. Complex scaling and residual flavour symmetry in the neutrino mass ...
Probir Roy
2017-10-09
Oct 9, 2017 ... Leptonic Dirac CP violation must be maximal while atmospheric neutrino mixing need not be exactly maximal. Each of the two Majorana phases, to be probed by the search for 0νββ decay, has to be zero or π and a normal neutrino mass hierarchy is allowed. Keywords. Neutrinos; residual flavour symmetry; ...
8. Asymmetry in Nature-Discrete Symmetries in Particle Physics and ...
Home; Journals; Resonance – Journal of Science Education; Volume 7; Issue 3. Asymmetry in Nature - Discrete Symmetries in Particle Physics and their Violation - Background and ... Theoretical Studies, Indian Institute of Science, Bangalore 560012, India. Indian Institute of Technology, Chennai. Aligarh Muslim University.
9. The minimal extension of the Standard Model with S3 symmetry
International Nuclear Information System (INIS)
Lee, C.E.; Lin, C.; Yang, Y.W.
1991-01-01
In this paper the two Higgs-doublet extension of the standard electroweak model with S 3 symmetry is presented. The flavour changing neutral Higgs interaction are automatically absent. A permutation symmetry breaking scheme is discussed. The correction to the Bjorken's approximation and the CP-violation factor J are given within this scheme
10. String constraints on discrete symmetries in MSSM type II quivers
Energy Technology Data Exchange (ETDEWEB)
Anastasopoulos, Pascal [Technische Univ. Wien (Austria). Inst. fur Theor. Phys.; Cvetic, Mirjam [Univ. of Pennsylvania, Philadelphia PA (United States). Dept. of Physics and Astronomy; Univ. of Maribor (Slovenia). Center for Applied Mathematics and Theoretical Physics; Richter, Robert [Hamburg Univ. (Germany). 2. Inst. fuer Theoretische Physik; Vaudrevange, Patrick K.S. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany)
2012-11-15
We study the presence of discrete gauge symmetries in D-brane semirealistic compactifications. After establishing the constraints on the transformation behaviour of the chiral matter for the presence of a discrete gauge symmetry we perform a systematic search for discrete gauge symmetries within semi-realistic D-brane realizations, based on four D-brane stacks, of the MSSM and the MSSM with three right-handed neutrinos. The systematic search reveals that Proton hexality, a discrete symmetry which ensures the absence of R-parity violating terms as well as the absence of dangerous dimension 5 proton decay operators, is only rarely realized. Moreover, none of the semi-realistic local D-brane configurations exhibit any family dependent discrete gauge symmetry.
11. Symmetry in running.
Science.gov (United States)
Raibert, M H
1986-03-14
Symmetry plays a key role in simplifying the control of legged robots and in giving them the ability to run and balance. The symmetries studied describe motion of the body and legs in terms of even and odd functions of time. A legged system running with these symmetries travels with a fixed forward speed and a stable upright posture. The symmetries used for controlling legged robots may help in elucidating the legged behavior of animals. Measurements of running in the cat and human show that the feet and body sometimes move as predicted by the even and odd symmetry functions.
12. Symmetries of Chimera States
Science.gov (United States)
Kemeth, Felix P.; Haugland, Sindre W.; Krischer, Katharina
2018-05-01
Symmetry broken states arise naturally in oscillatory networks. In this Letter, we investigate chaotic attractors in an ensemble of four mean-coupled Stuart-Landau oscillators with two oscillators being synchronized. We report that these states with partially broken symmetry, so-called chimera states, have different setwise symmetries in the incoherent oscillators, and in particular, some are and some are not invariant under a permutation symmetry on average. This allows for a classification of different chimera states in small networks. We conclude our report with a discussion of related states in spatially extended systems, which seem to inherit the symmetry properties of their counterparts in small networks.
13. Parastatistics and gauge symmetries
International Nuclear Information System (INIS)
Govorkov, A.B.
1982-01-01
A possible formulation of gauge symmetries in the Green parafield theory is analysed and the SO(3) gauge symmetry is shown to be on a distinct status. The Greenberg paraquark hypothesis turns out to be not equivalent to the hypothesis of quark colour SU(3)sub(c) symmetry. Specific features of the gauge SO(3) symmetry are discussed, and a possible scheme where it is an exact subgroup of the broken SU(3)sub(c) symmetry is proposed. The direct formulation of the gauge principle for the parafield represented by quaternions is also discussed
14. Family symmetries in F-theory GUTs
CERN Document Server
King, S F; Ross, G G
2010-01-01
We discuss F-theory SU(5) GUTs in which some or all of the quark and lepton families are assigned to different curves and family symmetry enforces a leading order rank one structure of the Yukawa matrices. We consider two possibilities for the suppression of baryon and lepton number violation. The first is based on Flipped SU(5) with gauge group SU(5)\\times U(1)_\\chi \\times SU(4)_{\\perp} in which U(1)_{\\chi} plays the role of a generalised matter parity. We present an example which, after imposing a Z_2 monodromy, has a U(1)_{\\perp}^2 family symmetry. Even in the absence of flux, spontaneous breaking of the family symmetry leads to viable quark, charged lepton and neutrino masses and mixing. The second possibility has an R-parity associated with the symmetry of the underlying compactification manifold and the flux. We construct an example of a model with viable masses and mixing angles based on the gauge group SU(5)\\times SU(5)_{\\perp} with a U(1)_{\\perp}^3 family symmetry after imposing a Z_2 monodromy.
15. Translational spacetime symmetries in gravitational theories
International Nuclear Information System (INIS)
Petti, R J
2006-01-01
How to include spacetime translations in fibre bundle gauge theories has been a subject of controversy, because spacetime symmetries are not internal symmetries of the bundle structure group. The standard method for including affine symmetry in differential geometry is to define a Cartan connection on an affine bundle over spacetime. This is equivalent to (1) defining an affine connection on the affine bundle, (2) defining a zero section on the associated affine vector bundle and (3) using the affine connection and the zero section to define an 'associated solder form', whose lift to a tensorial form on the frame bundle becomes the solder form. The zero section reduces the affine bundle to a linear bundle and splits the affine connection into translational and homogeneous parts; however, it violates translational equivariance/gauge symmetry. This is the natural geometric framework for Einstein-Cartan theory as an affine theory of gravitation. The last section discusses some alternative approaches that claim to preserve translational gauge symmetry
16. Translational spacetime symmetries in gravitational theories
Energy Technology Data Exchange (ETDEWEB)
Petti, R J [MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760 (United States)
2006-02-07
How to include spacetime translations in fibre bundle gauge theories has been a subject of controversy, because spacetime symmetries are not internal symmetries of the bundle structure group. The standard method for including affine symmetry in differential geometry is to define a Cartan connection on an affine bundle over spacetime. This is equivalent to (1) defining an affine connection on the affine bundle, (2) defining a zero section on the associated affine vector bundle and (3) using the affine connection and the zero section to define an 'associated solder form', whose lift to a tensorial form on the frame bundle becomes the solder form. The zero section reduces the affine bundle to a linear bundle and splits the affine connection into translational and homogeneous parts; however, it violates translational equivariance/gauge symmetry. This is the natural geometric framework for Einstein-Cartan theory as an affine theory of gravitation. The last section discusses some alternative approaches that claim to preserve translational gauge symmetry.
17. Symmetry in social exchange and health
Science.gov (United States)
Siegrist, Johannes
2005-10-01
Symmetry is a relevant concept in sociological theories of exchange. It is rooted in the evolutionary old norm of social reciprocity and is particularly important in social contracts. Symmetry breaking through violation of the norm of reciprocity generates strain in micro-social systems and, above all, in victims of non-symmetric exchange. In this contribution, adverse healthconsequences of symmetry breaking in contractual social exchange are analysed, with a main focus on the employment contract. Scientific evidence is derived from prospective epidemiological studies testing the model of effort-reward imbalance at work. Overall, a twofold elevated risk of incident disease is observed in employed men and women who are exposed to non-symmetric exchange. Health risks include coronary heart disease, depression and alcohol dependence, among others. Preliminary results suggest similar effects on health produced by symmetry breaking in other types of social relationships (e.g. partnership, parental roles). These findings underline the importance of symmetry in contractual social exchange for health and well-being.
18. CP violations in the Universe
Science.gov (United States)
Auriemma, Giulio
2003-12-01
The origin of the asymmetry between matter and antimatter that is evident in our part of the Universe is one of the open questions in cosmology, because the CPT symmetry between matter and antimatter seems to be absolutely conserved at microscopic level. We repeat here the classical proofs which exclude the viability of a Universe baryon symmetric on the average, or the observed asymmetry as an initial conditions. The current understanding is that the asymmetry should have been dynamically generated before nucleosynthesis, by B, C, and CP violating processes, acting out of thermodynamical equilibrium, as suggested by Sakharov in the 70's. The physical realizations of these conditions would be possible, in principle, also in the framework of the Standard Model of elementary particles, but the present limits on the mass of the higgs particle exclude this possibility. Finally we present the model of baryogenesis through leptogenesis, which is allowed by a minimal extension of the Standard Model, which has the appeal of being testable in future long-baseline neutrino oscillation experiments.
19. Generalized global symmetries
International Nuclear Information System (INIS)
Gaiotto, Davide; Kapustin, Anton; Seiberg, Nathan; Willett, Brian
2015-01-01
A q-form global symmetry is a global symmetry for which the charged operators are of space-time dimension q; e.g. Wilson lines, surface defects, etc., and the charged excitations have q spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries (q=0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a subgroup). They can also have ’t Hooft anomalies, which prevent us from gauging them, but lead to ’t Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.
20. Internal space-time symmetries of massive and massless particles and their unification
International Nuclear Information System (INIS)
Kim, Y.S.
2001-01-01
It is noted that the internal space-time symmetries of relativistic particles are dictated by Wigner's little groups. The symmetry of massive particles is like the three-dimensional rotation group, while the symmetry of massless particles is locally isomorphic to the two-dimensional Euclidean group. It is noted also that, while the rotational degree of freedom for a massless particle leads to its helicity, the two translational degrees of freedom correspond to its gauge degrees of freedom. It is shown that the E(2)-like symmetry of of massless particles can be obtained as an infinite-momentum and/or zero-mass limit of the O(3)-like symmetry of massive particles. This mechanism is illustrated in terms of a sphere elongating into a cylinder. In this way, the helicity degree of freedom remains invariant under the Lorentz boost, but the transverse rotational degrees of freedom become contracted into the gauge degree of freedom
1. Probing CPT violation with CMB polarization measurements
Energy Technology Data Exchange (ETDEWEB)
Xia Junqing, E-mail: [email protected] [Scuola Internazionale Superiore di Studi Avanzati, Via Beirut 2-4, I-34014 Trieste (Italy); Li Hong; Zhang Xinmin [Institute of High Energy Physics, Chinese Academy of Science, P.O. Box 918-4, Beijing 100049 (China); Theoretical Physics Center for Science Facilities (TPCSF), Chinese Academy of Science (China)
2010-04-12
The electrodynamics modified by the Chern-Simons term L{sub cs}approxp{sub m}uA{sub n}uF-tilde{sup m}u{sup n}u with a non-vanishing p{sub m}u violates the Charge-Parity-Time Reversal symmetry (CPT) and rotates the linear polarizations of the propagating Cosmic Microwave Background (CMB) photons. In this Letter we measure the rotation angle DELTAalpha by performing a global analysis on the current CMB polarization measurements from the five-year Wilkinson Microwave Anisotropy Probe (WMAP5), BOOMERanG 2003 (B03), BICEP and QUaD using a Markov Chain Monte Carlo method. Neglecting the systematic errors of these experiments, we find that the results from WMAP5, B03 and BICEP all are consistent and their combination gives DELTAalpha=-2.62+-0.87deg (68% C.L.), indicating a 3sigma detection of the CPT violation. The QUaD data alone gives DELTAalpha=0.59+-0.42deg (68% C.L.) which has an opposite sign for the central value and smaller error bar compared to that obtained from WMAP5, B03 and BICEP. When combining all the polarization data together, we find DELTAalpha=0.09+-0.36deg (68% C.L.) which significantly improves the previous constraint on DELTAalpha and test the validity of the fundamental CPT symmetry at a higher level.
2. Four-dimensional aether-like Lorentz-breaking QED revisited and problem of ambiguities
Energy Technology Data Exchange (ETDEWEB)
Baeta Scarpelli, A.P. [Setor Tecnico-Cientifico, Departamento de Policia Federal, Rua Hugo D' Antola, 95, Lapa, Sao Paulo (Brazil); Mariz, T. [Universidade Federal de Alagoas, Instituto de Fisica, Maceio, Alagoas (Brazil); Nascimento, J.R.; Petrov, A.Yu. [Universidade Federal da Paraiba, Departamento de Fisica, Caixa Postal 5008, Joao Pessoa, Paraiba (Brazil)
2013-08-15
In this paper, we consider the perturbative generation of the CPT-even aether-like Lorentz-breaking term in the extended Lorentz-breaking QED within different approaches and discuss its ambiguities. (orig.)
3. Four-dimensional aether-like Lorentz-breaking QED revisited and problem of ambiguities
International Nuclear Information System (INIS)
Baeta Scarpelli, A.P.; Mariz, T.; Nascimento, J.R.; Petrov, A.Yu.
2013-01-01
In this paper, we consider the perturbative generation of the CPT-even aether-like Lorentz-breaking term in the extended Lorentz-breaking QED within different approaches and discuss its ambiguities. (orig.)
4. CP violation in atoms
International Nuclear Information System (INIS)
Barr, S.M.
1992-01-01
Electric dipole moments of large atoms are an excellent tool to search for CP violation beyond the Standard Model. These tell us about the electron EDM but also about CP-violating electron-nucleon dimension-6 operators that arise from Higgs-exchange. Rapid strides are being made in searches for atomic EDMs. Limits on the electron EDM approaching the values which would be expected from Higgs-exchange mediated CP violation have been achieved. It is pointed out that in this same kind of model if tan β is large the effects in atoms of the dimension-6 e - n operators may outweigh the effect of the electron EDM. (author) 21 refs
5. Symmetry and symmetry breaking in quantum mechanics
International Nuclear Information System (INIS)
Chomaz, Philippe
1998-01-01
In the world of infinitely small, the world of atoms, nuclei and particles, the quantum mechanics enforces its laws. The discovery of Quanta, this unbelievable castration of the Possible in grains of matter and radiation, in discrete energy levels compels us of thinking the Single to comprehend the Universal. Quantum Numbers, magic Numbers and Numbers sign the wave. The matter is vibration. To describe the music of the world one needs keys, measures, notes, rules and partition: one needs quantum mechanics. The particles reduce themselves not in material points as the scholars of the past centuries thought, but they must be conceived throughout the space, in the accomplishment of shapes of volumes. When Einstein asked himself whether God plays dice, there was no doubt among its contemporaries that if He exists He is a geometer. In a Nature reduced to Geometry, the symmetries assume their role in servicing the Harmony. The symmetries allow ordering the energy levels to make them understandable. They impose there geometrical rules to the matter waves, giving them properties which sometimes astonish us. Hidden symmetries, internal symmetries and newly conceived symmetries have to be adopted subsequently to the observation of some order in this world of Quanta. In turn, the symmetries provide new observables which open new spaces of observation
6. Noncommutativity and unitarity violation in gauge boson scattering
International Nuclear Information System (INIS)
Hewett, J. L.; Petriello, F. J.; Rizzo, T. G.
2002-01-01
We examine the unitarity properties of spontaneously broken noncommutative gauge theories. We find that the symmetry breaking mechanism in the noncommutative standard model of Chaichian et al. leads to an unavoidable violation of tree-level unitarity in gauge boson scattering at high energies. We then study a variety of simplified spontaneously broken noncommutative theories and isolate the source of this unitarity violation. Given the group theoretic restrictions endemic to noncommutative model building, we conclude that it is difficult to build a noncommutative standard model under the Weyl-Moyal approach that preserves unitarity
7. Quantum walks, deformed relativity and Hopf algebra symmetries.
Science.gov (United States)
Bisio, Alessandro; D'Ariano, Giacomo Mauro; Perinotti, Paolo
2016-05-28
We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras. © 2016 The Author(s).
8. Topics in CP violation
International Nuclear Information System (INIS)
Quinn, H.R.
1993-02-01
Given the varied backgrounds of the members of this audience this talk will be a grab bag of topics related to the general theme of CP Violation. I do not have time to dwell in detail on any of them. First, for the astronomers and astrophysicists among you, I want to begin by reviewing the experimental status of evidence for CP violation in particle processes. There is only one system where this has been observed, and that is in the decays of neutral K mesons
9. Topics in CP violation
Science.gov (United States)
Quinn, H. R.
1993-02-01
Given the varied backgrounds of the members of this audience this talk will be a grab bag of topics related to the general theme of CP Violation. I do not have time to dwell in detail on any of them. First, for the astronomers and astrophysicists among you, I want to begin by reviewing the experimental status of evidence for CP violation in particle processes. There is only one system where this has been observed, and that is in the decays of neutral K mesons.
10. Direct observation of rectified motion of vortices by Lorentz microscopy
We have investigated the vortex dynamics for the `ratchet' operation in a niobium superconductor via a direct imaging of Lorentz microscopy. We directly observe one-directional selective motion of field-gradient-driven vortices along fabricated channels. This results from the rectification of vortices in a spatially asymmetric ...
11. On Einstein's kinematics and his derivation of Lorentz transformation equations
International Nuclear Information System (INIS)
Gulati, Shobha; Gulati, S.P.
1981-01-01
Recently the present authors have claimed that Einstein's historic derivation of 1905 of Lorentz transformation equations is a 'howler' - a correct result achieved through some incorrect steps. In the present contribution, this howler is fully resolved. Incidently, Einstein's kinematical considerations are found to be void of any new definitional elements or conventionality as unjustifiably claimed by Einstein and some other scientists. (author)
12. On the Lorentz degree of a product of polynomials
KAUST Repository
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence of a result of Barnard et al. (1991) on polynomials with nonnegative coefficients.
13. The scientific correspondence of H. A. Lorentz: Volume I
NARCIS (Netherlands)
Kox, A.J.
2008-01-01
This book presents a selection of 434 carefully annotated letters from and to the Dutch physicist and Nobel Prize winner Hendrik Antoon Lorentz (1853-1928), covering the period from 1883 until a few months before his death in February 1928. Most of these letters are of a scientific nature, with the
14. Lorentz force actuation of a heated atomic force microscope cantilever.
Science.gov (United States)
Lee, Byeonghee; Prater, Craig B; King, William P
2012-02-10
We report Lorentz force-induced actuation of a silicon microcantilever having an integrated resistive heater. Oscillating current through the cantilever interacts with the magnetic field around a NdFeB permanent magnet and induces a Lorentz force that deflects the cantilever. The same current induces cantilever heating. With AC currents as low as 0.2 mA, the cantilever can be oscillated as much as 80 nm at resonance with a DC temperature rise of less than 5 °C. By comparison, the AC temperature variation leads to a thermomechanical oscillation that is about 1000 times smaller than the Lorentz deflection at the cantilever resonance. The cantilever position in the nonuniform magnetic field affects the Lorentz force-induced deflection, with the magnetic field parallel to the cantilever having the largest effect on cantilever actuation. We demonstrate how the cantilever actuation can be used for imaging, and for measuring the local material softening temperature by sensing the contact resonance shift.
15. Anomalous current in periodic Lorentz gases with infinite horizon
Energy Technology Data Exchange (ETDEWEB)
Dolgopyat, Dmitrii I [University of Maryland, College Park (United States); Chernov, Nikolai I [University of Alabama at Birmingham, Birmingham, Alabama (United States)
2009-08-31
Electric current is studied in a two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, that is, free flights between collisions are bounded, the resulting current J is proportional to the voltage difference E, that is, J=1/2 D*E+o(||E||), where D* is the diffusion matrix of a Lorentz particle moving freely without an electric field (see a mathematical proof). This formula agrees with Ohm's classical law and the Einstein relation. Here the more difficult model with an infinite horizon is investigated. It is found that infinite corridors between scatterers allow the particles (electrons) to move faster, resulting in an abnormal current (causing 'superconductivity'). More precisely, the current is now given by J=1/2 DE| log||E|| | + O(||E||), where D is the 'superdiffusion' matrix of a Lorentz particle moving freely without an electric field. This means that Ohm's law fails in this regime, but the Einstein relation (suitably interpreted) still holds. New results are also obtained for the infinite-horizon Lorentz gas without external fields, complementing recent studies by Szasz and Varju. Bibliography: 31 titles.
16. Anomalous current in periodic Lorentz gases with infinite horizon
International Nuclear Information System (INIS)
Dolgopyat, Dmitrii I; Chernov, Nikolai I
2009-01-01
Electric current is studied in a two-dimensional periodic Lorentz gas in the presence of a weak homogeneous electric field. When the horizon is finite, that is, free flights between collisions are bounded, the resulting current J is proportional to the voltage difference E, that is, J=1/2 D*E+o(||E||), where D* is the diffusion matrix of a Lorentz particle moving freely without an electric field (see a mathematical proof). This formula agrees with Ohm's classical law and the Einstein relation. Here the more difficult model with an infinite horizon is investigated. It is found that infinite corridors between scatterers allow the particles (electrons) to move faster, resulting in an abnormal current (causing 'superconductivity'). More precisely, the current is now given by J=1/2 DE| log||E|| | + O(||E||), where D is the 'superdiffusion' matrix of a Lorentz particle moving freely without an electric field. This means that Ohm's law fails in this regime, but the Einstein relation (suitably interpreted) still holds. New results are also obtained for the infinite-horizon Lorentz gas without external fields, complementing recent studies by Szasz and Varju. Bibliography: 31 titles.
17. On the Lorentz degree of a product of polynomials
KAUST Repository
2015-01-01
In this note, we negatively answer two questions of T. Erdélyi (1991, 2010) on possible lower bounds on the Lorentz degree of product of two polynomials. We show that the correctness of one question for degree two polynomials is a direct consequence
18. A note on Lorentz transformation and pseudo-rapidity distributions
International Nuclear Information System (INIS)
Hama, Y.
1980-07-01
It is shown that although rapidity and pseudo-rapidity are almost equivalent variables, their difference may in pratice become quite remarkable. Non Lorentz invariance of pseudo-rapidity distributions may cause appearance of strange effects at first sight, such as deformation of a perfectly symmetric particle distribution into an asymmetric one when going to another frame. (Author) [pt
19. Adaptive compensation of Lorentz force detuning in superconducting RF cavities
Energy Technology Data Exchange (ETDEWEB)
Pischalnikov, Yuriy [Fermilab; Schappert, Warren [Fermilab
2011-11-01
The Lorentz force can dynamically detune pulsed Superconducting RF cavities and considerable additional RF power can be required to maintain the accelerating gradient if no effort is made to compensate. Fermilab has developed an adaptive compensation system for cavities in the Horizontal Test Stand, in the SRF Accelerator Test Facility, and for the proposed Project X.
20. What Governs Lorentz Factors of Jet Components in Blazars? Xinwu ...
Abstract. We use a sample of radio-loud Active Galactic Nuclei. (AGNs) with measured black hole masses to explore the jet formation mechanisms in these sources. We find a significant correlation between black hole mass and the bulk Lorentz factor of the jet components for this sample, while no significant correlation is ...
1. What Governs Lorentz Factors of Jet Components in Blazars?
We use a sample of radio-loud Active Galactic Nuclei (AGNs) with measured black hole masses to explore the jet formation mechanisms in these sources. We find a significant correlation between black hole mass and the bulk Lorentz factor of the jet components for this sample, while no significant correlation is present ...
2. Helicity and evanescent waves. [Energy transport velocity, helicity, Lorentz transformation
Energy Technology Data Exchange (ETDEWEB)
Agudin, J L; Platzeck, A M [La Plata Univ. Nacional (Argentina); Albano, J R [Instituto de Astronomia y Fisica del Espacio, Buenos Aires, Argentina
1978-02-20
It is shown that the projection of the angular momentum of a circularly polarized electromagnetic evanescent wave along the mean velocity of energy transport (=helicity) can be reverted by a Lorentz transformation, in spite of the fact that this velocity is c.
3. Special Relativity in Week One: 3) Introducing the Lorentz Contraction
Science.gov (United States)
Huggins, Elisha
2011-01-01
This is the third of four articles on teaching special relativity in the first week of an introductory physics course. With Einstein's second postulate that the speed of light is the same to all observers, we could use the light pulse clock to introduce time dilation. But we had difficulty introducing the Lorentz contraction until we saw the movie…
4. Neutrino mixing: from the broken μ-τ symmetry to the broken Friedberg–Lee symmetry
International Nuclear Information System (INIS)
Xing, Zhizhong
2007-01-01
I argue that the observed flavor structures of leptons and quarks might imply the existence of certain flavor symmetries. The latter should be a good starting point to build realistic models towards deeper understanding of the fermion mass spectra and flavor mixing patterns. The μ-τ permutation symmetry serves for such an example to interpret the almost maximal atmospheric neutrino mixing angle (θ 23 ~ 45°) and the strongly suppressed CHOOZ neutrino mixing angle (θ 13 < 10°). In this talk I like to highlight a new kind of flavor symmetry, the Friedberg–Lee symmetry, for the effective Majorana neutrino mass operator. Luo and I have shown that this symmetry can be broken in an oblique way, such that the lightest neutrino remains massless but an experimentally-favored neutrino mixing pattern is achievable. We get a novel prediction for θ 13 in the CP-conserving case: sinθ 13 = tanθ 12 |(1 - tanθ 23 )/(1 + tanθ 23 )|. Our scenario can simply be generalized to accommodate CP violation and be combined with the seesaw mechanism. Finally I stress the importance of probing possible effects of μ-τ symmetry breaking either in terrestrial neutrino oscillation experiments or with ultrahigh-energy cosmic neutrino telescopes. (author)
5. Symmetries in nature
International Nuclear Information System (INIS)
Mainzer, K.
1988-01-01
Symmetry, disymmetry, chirality etc. are well-known topics in chemistry. But they cannot only be found on the molecular level of matter. Atoms and elementary particles in physics are also characterized by particular symmetry groups. Even living organisms and populations on the macroscopic level have functional properties of symmetry. The whole physical, chemical, and biological evolution seems to be regulated by the emergence of new symmetries and the breaking down of old ones. One is reminded of Heisenberg's famous statement: 'Die letzte Wurzel der Erscheinungen ist also nicht die Materie, sondern das mathematische Gesetz, die Symmetrie, die mathematische Form' (Wandlungen in den Grundlagen der Naturwissenschaften, 1959). Historically the belief in symmetry and simplicity of nature has a long philosophical tradition from the Pythagoreans, Plato and Greek astronomers to Kepler and modern scientists. Today, 'symmetries in nature' is a common topic of mathematics, physics, chemistry, and biology. A lot of Nobel prizes were given in honour of inquiries concerning symmetries in nature. The fascination of symmetries is not only motivated by science, but by art and religion too. Therefore 'symmetris in nature' is an interdisciplinary topic which may help to overcome C.P. Snow's 'Two Cultures' of natural sciences and humanities. (author) 17 refs., 21 figs
6. Symmetries in nature
Energy Technology Data Exchange (ETDEWEB)
Mainzer, K
1988-05-01
Symmetry, disymmetry, chirality etc. are well-known topics in chemistry. But they cannot only be found on the molecular level of matter. Atoms and elementary particles in physics are also characterized by particular symmetry groups. Even living organisms and populations on the macroscopic level have functional properties of symmetry. The whole physical, chemical, and biological evolution seems to be regulated by the emergence of new symmetries and the breaking down of old ones. One is reminded of Heisenberg's famous statement: 'Die letzte Wurzel der Erscheinungen ist also nicht die Materie, sondern das mathematische Gesetz, die Symmetrie, die mathematische Form' (Wandlungen in den Grundlagen der Naturwissenschaften, 1959). Historically the belief in symmetry and simplicity of nature has a long philosophical tradition from the Pythagoreans, Plato and Greek astronomers to Kepler and modern scientists. Today, 'symmetries in nature' is a common topic of mathematics, physics, chemistry, and biology. A lot of Nobel prizes were given in honour of inquiries concerning symmetries in nature. The fascination of symmetries is not only motivated by science, but by art and religion too. Therefore 'symmetris in nature' is an interdisciplinary topic which may help to overcome C.P. Snow's 'Two Cultures' of natural sciences and humanities. (author) 17 refs., 21 figs.
7. Symmetries in nuclei
International Nuclear Information System (INIS)
Arima, A.
2003-01-01
(1) There are symmetries in nature, and the concept of symmetry has been used in art and architecture. The symmetry is evaluated high in the European culture. In China, the symmetry is broken in the paintings but it is valued in the architecture. In Japan, however, the symmetry has been broken everywhere. The serious and interesting question is why these differences happens? (2) In this lecture, I reviewed from the very beginning the importance of the rotational symmetry in quantum mechanics. I am sorry to be too fundamental for specialists of nuclear physics. But for people who do not use these theories, I think that you could understand the mathematical aspects of quantum mechanics and the relation between the angular momentum and the rotational symmetry. (3) To the specialists of nuclear physics, I talked about my idea as follows: dynamical treatment of collective motions in nuclei by IBM, especially the meaning of the degeneracy observed in the rotation bands top of γ vibration and β vibration, and the origin of pseudo-spin symmetry. Namely, if there is a symmetry, a degeneracy occurs. Conversely, if there is a degeneracy, there must be a symmetry. I discussed some details of the observed evidence and this correspondence is my strong belief in physics. (author)
8. Masses, flavor mix and CP violation
International Nuclear Information System (INIS)
Chaussard, L.
2004-06-01
The author describes the relationships between masses, mixing of flavors and CP violation. This document is divided into 4 chapters: 1) fermions' masses, 2) mixing of flavors and CP violation, 3) beauty physics and 4) neutrino physics. In chapter 1 an attempt is made to explain what is behind the concepts of lepton mass and quark mass. As for neutrinos, the only neutral fermion, Dirac's and Majorana's views are exposed as well as their consequences. Fermion flavors are mixed in the process of mass generation and this mix is responsible for the breaking of CP and T symmetries. In chapter 2 the author shows how the analysis of particle oscillations from neutral mesons (K 0 , D 0 , B d 0 and B s 0 ) and from neutrinos can shed light on CP violation. Chapter 3 is dedicated to the contribution of beauty physics to the determination of the unitary triangle, through the oscillations of beauty mesons. In chapter 4 the author reviews the experimental results obtained recently concerning neutrino mass and neutrino oscillations and draws some perspectives on future neutrino experiments. (A.C.)
9. 48 CFR 2803.104-10 - Violations or possible violations.
Science.gov (United States)
2010-10-01
... General IMPROPER BUSINESS PRACTICES AND PERSONAL CONFLICTS OF INTEREST Safeguards 2803.104-10 Violations... action to be taken. The types of actions that would normally be taken when a violation has occurred that...
10. Beautiful CP violation
International Nuclear Information System (INIS)
Dunietz, I.
1997-01-01
CP violation is observed to date only in K 0 decays and is parameterizable by a single quantity ε. Because it is one of the least understood phenomena in the Standard Model and holds a clue to baryogenesis, it must be investigated further. Highly specialized searches in K 0 decays are possible. Effects in B decays are much larger. In addition to the traditional B d → J/ψK S , π + π - asymmetries, CP violation could be searched for in already existing inclusive B data samples. The rapid B s --anti B s oscillations cancel in untagged B s data samples, which therefore allow feasibility studies for the observation of CP violation and the extraction of CKM elements with present vertex detectors. The favored method for the extraction of the CKM angle γ is shown to be unfeasible and a solution is presented involving striking direct CP violation in charged B decays. Novel methods for determining the B s mixing parameter Δm are described without the traditional requirement of flavor-specific final states
11. Scaling violation in QCD
International Nuclear Information System (INIS)
Furmanski, W.
1981-08-01
The effects of scaling violation in QCD are discussed in the perturbative scheme, based on the factorization of mass singularities in the light-like gauge. Some recent applications including the next-to-leading corrections are presented (large psub(T) scattering, numerical analysis of the leptoproduction data). A proposal is made for extending the method on the higher twist sector. (author)
12. Lepton flavor violation
International Nuclear Information System (INIS)
Cooper, M.D. Brooks, M.; Hogan, G.E.
1997-01-01
The connection of rare decays to supersymmetric grand unification is highlighted, and a review of the status of rare decay experiments is given. Plans for future investigations of processes that violate lepton flavor are discussed. A new result from the MEGA experiment, a search for μ + → e + γ, is reported to be B.R. -11 with 90% confidence
13. Electron scattering violates parity
CERN Multimedia
2004-01-01
Parity violation has been observed in collisions between electrons at the Stanford Linear Accelerator Center (SLAC) in the US. The resuls, which are in agreement with the Stanford Model of particle physics, also provide a new measurement of the weak charge of the electron (½ page)
14. Parity violating electron scattering
International Nuclear Information System (INIS)
McKeown, R.D.
1990-01-01
Previous measurements of parity violation in electron scattering are reviewed with particular emphasis on experimental techniques. Significant progress in the attainment of higher precision is evident in these efforts. These pioneering experiments provide a basis for consideration of a future program of such measurements. In this paper some future plans and possibilities in this field are discussed
15. From physical symmetries to emergent gauge symmetries
International Nuclear Information System (INIS)
Barceló, Carlos; Carballo-Rubio, Raúl; Di Filippo, Francesco; Garay, Luis J.
2016-01-01
Gauge symmetries indicate redundancies in the description of the relevant degrees of freedom of a given field theory and restrict the nature of observable quantities. One of the problems faced by emergent theories of relativistic fields is to understand how gauge symmetries can show up in systems that contain no trace of these symmetries at a more fundamental level. In this paper we start a systematic study aimed to establish a satisfactory mathematical and physical picture of this issue, dealing first with abelian field theories. We discuss how the trivialization, due to the decoupling and lack of excitation of some degrees of freedom, of the Noether currents associated with physical symmetries leads to emergent gauge symmetries in specific situations. An example of a relativistic field theory of a vector field is worked out in detail in order to make explicit how this mechanism works and to clarify the physics behind it. The interplay of these ideas with well-known results of importance to the emergent gravity program, such as the Weinberg-Witten theorem, are discussed.
16. The Symmetry of Multiferroics
OpenAIRE
Harris, A. Brooks
2006-01-01
This paper represents a detailed instruction manual for constructing the Landau expansion for magnetoelectric coupling in incommensurate ferroelectric magnets. The first step is to describe the magnetic ordering in terms of symmetry adapted coordinates which serve as complex valued magnetic order parameters whose transformation properties are displayed. In so doing we use the previously proposed technique to exploit inversion symmetry, since this symmetry had been universally overlooked. Havi...
17. Leptonic CP violation theory
DEFF Research Database (Denmark)
Hagedorn, C.
2017-01-01
I summarize the status of theoretical predictions for the yet to be measured leptonic CP phases, the Dirac phase δ and the two Majorana phases α and β. I discuss different approaches based on: (a) a flavor symmetry without and with corrections, (b) different types of sum rules and (c) flavor and CP...... symmetries. I show their predictive power with examples. In addition, I present scenarios in which low and high energy CP phases are connected so that predictions for the CP phases α, β and δ become correlated to the sign of the baryon asymmetry YB of the Universe that is generated via leptogenesis....
18. CP violation in K decays
International Nuclear Information System (INIS)
Gilman, F.J.
1989-05-01
Recent theoretical and experimental progress on the manifestation of CP violation in K decays, and toward understanding whether CP violation originates in a phase, or phases, in the weak mixing matrix of quarks is reviewed. 23 refs., 10 figs
19. Approximate and renormgroup symmetries
Energy Technology Data Exchange (ETDEWEB)
Ibragimov, Nail H. [Blekinge Institute of Technology, Karlskrona (Sweden). Dept. of Mathematics Science; Kovalev, Vladimir F. [Russian Academy of Sciences, Moscow (Russian Federation). Inst. of Mathematical Modeling
2009-07-01
''Approximate and Renormgroup Symmetries'' deals with approximate transformation groups, symmetries of integro-differential equations and renormgroup symmetries. It includes a concise and self-contained introduction to basic concepts and methods of Lie group analysis, and provides an easy-to-follow introduction to the theory of approximate transformation groups and symmetries of integro-differential equations. The book is designed for specialists in nonlinear physics - mathematicians and non-mathematicians - interested in methods of applied group analysis for investigating nonlinear problems in physical science and engineering. (orig.)
20. Approximate and renormgroup symmetries
International Nuclear Information System (INIS)
Ibragimov, Nail H.; Kovalev, Vladimir F.
2009-01-01
''Approximate and Renormgroup Symmetries'' deals with approximate transformation groups, symmetries of integro-differential equations and renormgroup symmetries. It includes a concise and self-contained introduction to basic concepts and methods of Lie group analysis, and provides an easy-to-follow introduction to the theory of approximate transformation groups and symmetries of integro-differential equations. The book is designed for specialists in nonlinear physics - mathematicians and non-mathematicians - interested in methods of applied group analysis for investigating nonlinear problems in physical science and engineering. (orig.) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9183012843132019, "perplexity": 1841.5308617651317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027313428.28/warc/CC-MAIN-20190817143039-20190817165039-00451.warc.gz"} |
https://www.physicsforums.com/search/373266/ | # Search results
So it is the probability density right? Cause that is what I want. If it isn't then how different would it be? student
Can you explain to me again, why what I have drawn is not the wavefunction but the probability density? I think it might be the wavefunction.....I am still confused. I think it should be the continuation of exponential decay when the wave emerges from the barrier for E < V and for E > V, I...
So something like this where the amplitude of the transmitted wave is reduced....its intensity is reduced.
Like this? Also, just wana clarify something, I know I am probably being pedantic but anyways...Is 50 % transmission is the same as R = T = 0.5? Cause the question asks for the probability density sketches for 50% transmission. I think 50% transmission is the same as R = T = 0.5...
5. ### Quantum mechanics in DIRAC notation
This is the quantum harmonic oscillator where \hat a\dagger and \hat a are the step down and step up (SOMETIMES CALLED LADDER) operators according to your definitions. Correction: I think you meant to say \hat N = \hat a\dagger \hat a
So would the probability density look like this? The transmitted wave and reflected waves have a reduced amplitude...i.e. they are 1/2 of the original amplitude (incident amplitude). This is the plot of the probability density. Does it make sense? Looking forward to hearing from you soon...
Consider a Quantum Mechanical particle approaching a barrier (potential) of height V_0 and width a. What will the sketch of the probability density look like if there is a 50% chance of reflection and a 50% chance of transmission? Can you explain why cause after reading Griffith' s Quantum...
8. ### How to graph in Latex ?
How do I plot graphs in LaTeX? Example sin(x) to begin with. :frown: Also, how do I insert pictures in LaTeX? Example, simple circuit diagrams. student :confused:
9. ### Diode output waveform
Can you explain to me step by step how I would go about reasoning as to what it should look like?
10. ### Diode output waveform
I have done clamping, clipping, half wave rectifier and full wave rectifier circuits. For the diagram, I think it is a cos wave with amplitude 10 V but have no idea why. I am genuinely lost here. student
11. ### Diode output waveform
Sketch the output voltage as a function of time. The AC voltage source is V_{o}cos(\omega)t with V_{o} = 10V and \omega = 2000rad/sec. I have posted a diode circuit question in the attachment Ok, I think it should be a sine curve with a 10 V amplitude but am not too sure about the period...
12. ### Fourier integral / transform ? What is it really?
If i multiple both sides by exp(-ik'x) the LHS gives exp(-ik'x-(x/2a)^2). I' m not sure what to do with this to simplify it further. Do i have to try to complete the square in this exponential now?
13. ### Fourier integral / transform ? What is it really?
Any suggestions guys?
14. ### Calculation of Excitation potentialfrom I-V curve
Plot distance along x axis versus peak number. calculate the slope using the method of least squares. Plot distance from 0th peak versus peak number. Calculate the slope as before. The ratio of the slopes gives the excitation potential. There is another method using the current just before...
15. ### Fourier integral / transform ? What is it really?
Find phi(k) I need help with this question as far as what am I looking for and how do I use a Fourier transform cause I think I need one. student
16. ### Data Analysis web site for experiments ?
Is there a site where I could see what people have done in various experiments? For example, I will be doing the verlocity of light experiment and would like to know what kinds of data analysis people have done so I can get some ideas as to what I could do with my data. Is there any such...
17. ### Particle in a BOX - what are allowed momenta?
I know about that, but it depends on what the size of the box is. The only reason things worked out nicely to one expression was due to the fact that the box was centered at the origin and was of size L rather than what I have which is 2L. I think there was an expression relating energy to...
18. ### Franck Hertz questions
I' m not very knowledgable on this issue....perhaps someone else can help ...I say excitation energy is the difference between peaks and subtract the potential at the location of the first peak to get the contact potential...WAIT for SOMEONE ELSE to back this up. student1938
19. ### Particle in a BOX - what are allowed momenta?
Particle in a BOX -- what are allowed momenta? Ok I am trying to come up with the first five eigenfunctions for the particle in a box of size 2L. Now, I gave the appropriate initial conditions and get as a solution phi(x) = bcos(kx) + asin(kx). I said that phi(-L) = phi(L) = 0 which game me...
20. ### Lagrange' s eequations of a suspended set of rods
got it..thanks for all the help
21. ### Lagrange' s eequations of a suspended set of rods
Anything,,,I just wana finish this off please.
22. ### Lagrange' s eequations of a suspended set of rods
Or shoul it be the same?
23. ### Lagrange' s eequations of a suspended set of rods
So for the side rods, then it should just be l/2*omega ^ 2 right?
24. ### Lagrange' s eequations of a suspended set of rods
oh did you use v = omega X r where r = l and the angle is 90 degrees.....now that makes sense. So then T = (1/2)*m*(l*omega)^2 and I can now use Lagrange' s Eqs right?
25. ### Lagrange' s eequations of a suspended set of rods
Then it should just be l/t but then wht happens to t?
26. ### Lagrange' s eequations of a suspended set of rods
the speed should just be d(lcos(theta))/dt right? I think thta lcos(theta) is the horizontal distance and we are dividing by time here to get speed right? | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9077634811401367, "perplexity": 863.804507852714}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514575596.77/warc/CC-MAIN-20190922160018-20190922182018-00207.warc.gz"} |
http://mathhelpforum.com/pre-calculus/185669-verifying-identity-print.html | # Verifying that this is an identity?
• August 5th 2011, 12:58 PM
explodingtoenails
Verifying that this is an identity?
Trig and I are not very good friends...
$\displaystyle \frac{1 + tan^2x}{csc^2x} = tan^2x$
This is what I tried:
$\displaystyle \frac{1 + (sin^2x/cos^2x)}{(1/sin^2x)} = tan^2x$
$\displaystyle \frac{cos^2x + sin^2x}{cos^2x} * \frac{sin^2x}{1} = tan^2x$
$(sin^2x)(sin^2x) = tan^2x$
I don't know why it's not equaling up. D:
• August 5th 2011, 01:03 PM
TheChaz
Re: Verifying that this is an identity?
Quote:
Originally Posted by explodingtoenails
Trig and I are not very good friends...
$\displaystyle \frac{1 + tan^2x}{csc^2x} = tan^2x$
This is what I tried:
$\displaystyle \frac{1 + (sin^2x/cos^2x)}{(1/sin^2x)} = tan^2x$
$\displaystyle \frac{cos^2x + sin^2x}{cos^2x} * \frac{sin^2x}{1} = tan^2x$
...
I don't know why it's not equaling up. D:
From here, the top left is ONE.
Then you have (sin/cos)^2
• August 5th 2011, 01:06 PM
explodingtoenails
Re: Verifying that this is an identity?
Thank you so much dude. I didn't know that cos^2x + sin^2x = 1.
• August 5th 2011, 01:42 PM
TheChaz
Re: Verifying that this is an identity?
Quote:
Originally Posted by explodingtoenails
Thank you so much dude. I didn't know that cos^2(x) + sin^2(x) = 1.
That's kind of important! From it, many (most?) identities are derived.
• August 15th 2011, 07:55 PM
Drexel28
Re: Verifying that this is an identity?
Quote:
Originally Posted by TheChaz
That's kind of important! From it, many (most?) identities are derived.
The most important identity really. By this I mean that as rings $\mathbb{R}[x,y]/(x^2-y^2-1)\cong \mathbb{R}[\cos(\theta),\sin(\theta)]$. In other words, if you take the ring $\mathbb{R}[x,y]$ and impose the conditions that $x^2+y^2=1$ then you really just have the ring of trigonometric polynomials $\mathbb{R}[\cos(\theta),\sin(\theta)]$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 11, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8393840789794922, "perplexity": 2358.1198904966463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414119649133.8/warc/CC-MAIN-20141024030049-00169-ip-10-16-133-185.ec2.internal.warc.gz"} |
https://www.zakmhammedi.com/publication/mhammedi-pac-bayes-2019/ | # PAC-Bayes Un-Expected Bernstein Inequality
### Abstract
We present a new PAC-Bayesian generalization bound. Standard bounds contain a square-root $L_n \mathrm{KL}/n$ complexity term which dominates unless $L_n$, the empirical error of the learning algorithm’s randomized predictions, vanishes. We manage to replace $L_n$ by a term which vanishes in many more situations, essentially whenever the employed learning algorithm is sufficiently stable on the dataset at hand. Our new bound consistently beats state-of-the-art bounds both on a toy example and on UCI datasets (with large enough $n$). Theoretically, unlike existing bounds, our new bound can be expected to converge to 0 faster whenever a Bernstein/Tsybakov condition holds, thus connecting PAC-Bayesian generalization and excess risk bounds—for the latter it has long been known that faster convergence can be obtained under Bernstein conditions. Our main technical tool is a new concentration inequality which is like Bernstein’s but with $X^2$ taken outside its expectation.
Publication
Advances in Neural Information Processing Systems 32
##### Zak Mhammedi
###### Postdoctoral Associate
My research interests include the theory of online learning and statistical generalization. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8988288640975952, "perplexity": 1128.3232392381624}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500154.33/warc/CC-MAIN-20230204205328-20230204235328-00126.warc.gz"} |
http://tijarohonline.blogspot.com/2010/07/bias-of-transistor.html | ## Thursday, July 1, 2010
### Bias of Transistor
Bipolar transistor amplifiers must be properly biased to operate correctly. In circuits made with individual devices (discrete circuits), biasing networks consisting of resistors are commonly employed. Much more elaborate biasing arrangements are used in integrated circuits, for example, bandgap voltage references and current mirrors.
The operating point of a device, also known as bias point, quiescent point, or Q-point, is the point on the output characteristics that shows the DC collector–emitter voltage (Vce) and the collector current (Ic) with no input signal applied. The term is normally used in connection with devices such as transistors.
### Signal requirements for Class A amplifiers
For analog circuit operation, the Q-point is placed so the transistor stays in active mode (does not shift to operation in the saturation region or cut-off region) when input is applied. For digital operation, the Q-point is placed so the transistor does the contrary - switches from "on" to "off" state. Often, Q-point is established near the center of active region of transistor characteristic to allow similar signal swings in positive and negative directions. Q-point should be stable. In particular, it should be insensitive to variations in transistor parameters (for example, should not shift if transistor is replaced by another of the same type), variations in temperature, variations in power supply voltage and so forth. The circuit must be practical: easily implemented and cost-effective.
### Thermal considerations
At constant current, the voltage across the emitter–base junction VBE of a bipolar transistor decreases 2 mV (silicon) and 1.8mV (germanium) for each 1°C rise in temperature (reference being 25°C). By the Ebers–Moll model, if the base–emitter voltage VBE is held constant and the temperature rises, the current through the base–emitter diode IB will increase, and thus the collector current IC will also increase. Depending on the bias point, the power dissipated in the transistor may also increase, which will further increase its temperature and exacerbate the problem. This deleterious positive feedback results in thermal runaway.[1] There are several approaches to mitigate bipolar transistor thermal runaway. For example,
• Negative feedback can be built into the biasing circuit so that increased collector current leads to decreased base current. Hence, the increasing collector current throttles its source.
• Heat sinks can be used that carry away extra heat and prevent the base–emitter temperature from rising.
• The transistor can be biased so that its collector is normally less than half of the power supply voltage, which implies that collector–emitter power dissipation is at its maximum value. Runaway is then impossible because increasing collector current leads to a decrease in dissipated power; this notion is known as the half-voltage principle.
The circuits below primarily demonstrate the use of negative feedback to prevent thermal runaway.
## Types of bias circuit for Class A amplifiers
The following discussion treats five common biasing circuits used with Class A bipolar transistor amplifiers:
1. Fixed bias
2. Collector-to-base bias
3. Fixed bias with emitter resistor
4. Voltage divider bias
5. Emitter bias
### Fixed bias (base bias)
Fixed bias (Base bias)
This form of biasing is also called base bias. In the example image on the right, the single power source (for example, a battery) is used for both collector and base of transistor, although separate batteries can also be used.
In the given circuit,
VCC = IBRB + Vbe
Therefore,
IB = (VCC - Vbe)/RB
For a given transistor, Vbe does not vary significantly during use. As VCC is of fixed value, on selection of RB, the base current IB is fixed. Therefore this type is called fixed bias type of circuit.
Also for given circuit,
VCC = ICRC + Vce
Therefore,
Vce = VCC - ICRC
The common-emitter current gain of a transistor is an important parameter in circuit design, and is specified on the data sheet for a particular transistor. It is denoted as β on this page.
Because
IC = βIB
we can obtain IC as well. In this manner, operating point given as (VCE,IC) can be set for given transistor.
Merits:
• It is simple to shift the operating point anywhere in the active region by merely changing the base resistor (RB).
• A very small number of components are required.
Demerits:
• The collector current does not remain constant with variation in temperature or power supply voltage. Therefore the operating point is unstable.
• Changes in Vbe will change IB and thus cause RE to change. This in turn will alter the gain of the stage.
• When the transistor is replaced with another one, considerable change in the value of β can be expected. Due to this change the operating point will shift.
• For small-signal transistors (e.g., not power transistors) with relatively high values of β (i.e., between 100 and 200), this configuration will be prone to thermal runaway. In particular, the stability factor, which is a measure of the change in collector current with changes in reverse saturation current, is approximately β+1. To ensure absolute stability of the amplifier, a stability factor of less than 25 is preferred, and so small-signal transistors have large stability factors.[citation needed]
Usage:
Due to the above inherent drawbacks, fixed bias is rarely used in linear circuits (i.e., those circuits which use the transistor as a current source). Instead, it is often used in circuits where transistor is used as a switch. However, one application of fixed bias is to achieve crude automatic gain control in the transistor by feeding the base resistor from a DC signal derived from the AC output of a later stage.
### Collector-to-base bias
Collector-to-base bias
This configuration employs negative feedback to prevent thermal runaway and stabilize the operating point. In this form of biasing, the base resistor RB is connected to the collector instead of connecting it to the DC source VCC. So any thermal runaway will induce a voltage drop across the RC resistor that will throttle the transistor's base current.
From Kirchhoff's voltage law, the voltage $V_{\text{R}_{\text{b}}}$ across the base resistor Rb is
$V_{\text{R}_{\text{b}}} = V_{\text{cc}} \, - \, \mathord{\overbrace{(I_{\text{c}} + I_{\text{b}}) R_{\text{c}}}^{\text{Voltage drop across } R_{\text{c}}}} \, - \, \mathord{\overbrace{V_{\text{be}}}^{\text{Voltage at base}}}.$
By the Ebers–Moll model, Ic = βIb, and so
$V_{\text{R}_{\text{b}}} = V_{\text{cc}} - (\overbrace{\beta I_{\text{b}}}^{I_{\text{c}}} + I_{\text{b}}) R_{\text{c}} - V_{\text{be}} = V_{\text{cc}} - I_{\text{b}} (\beta + 1) R_{\text{c}} - V_{\text{be}}.$
From Ohm's law, the base current $I_{\text{b}} = V_{\text{R}_{\text{b}}} / R_{\text{b}}$, and so
$\overbrace{I_{\text{b}} R_{\text{b}}}^{V_{\text{R}_{\text{b}}}} = V_{\text{cc}} - I_{\text{b}} (\beta + 1) R_{\text{c}} - V_{\text{be}}.$
Hence, the base current Ib is
$I_{\text{b}} = \frac{ V_{\text{cc}} - V_{\text{be}} }{ R_{\text{b}} + ( \beta + 1 ) R_{\text{c}} }$
If Vbe is held constant and temperature increases, then the collector current Ic increases. However, a larger Ic causes the voltage drop across resistor Rc to increase, which in turn reduces the voltage $V_{\text{R}_{\text{b}}}$ across the base resistor Rb. A lower base-resistor voltage drop reduces the base current Ib, which results in less collector current Ic. Because an increase in collector current with temperature is opposed, the operating point is kept stable.
Merits:
• Circuit stabilizes the operating point against variations in temperature and β (ie. replacement of transistor)
Demerits:
• In this circuit, to keep Ic independent of β, the following condition must be met:
$I_{\text{c}} = \beta I_{\text{b}} = \frac { \beta (V_{\text{cc}} - V_{\text{be}})}{R_{\text{b}} + R_{\text{c}} + \beta R_{\text{c}}} \approx \frac{(V_{\text{cc}} - V_{\text{be}})}{R_{\text{c}}}$
which is the case when
$\beta R_{\text{c}} \gg R_{\text{b}}.$
• As β-value is fixed (and generally unknown) for a given transistor, this relation can be satisfied either by keeping Rc fairly large or making Rb very low.
• If Rc is large, a high Vcc is necessary, which increases cost as well as precautions necessary while handling.
• If Rb is low, the reverse bias of the collector–base region is small, which limits the range of collector voltage swing that leaves the transistor in active mode.
• The resistor Rb causes an AC feedback, reducing the voltage gain of the amplifier. This undesirable effect is a trade-off for greater Q-point stability.
Usage: The feedback also decreases the input impedance of the amplifier as seen from the base, which can be advantageous. Due to the gain reduction from feedback, this biasing form is used only when the trade-off for stability is warranted.
### Fixed bias with emitter resistor
Fixed bias with emitter resistor
The fixed bias circuit is modified by attaching an external resistor to the emitter. This resistor introduces negative feedback that stabilizes the Q-point. From Kirchhoff's voltage law, the voltage across the base resistor is
VRb = VCC - IeRe - Vbe.
From Ohm's law, the base current is
Ib = VRb / Rb.
The way feedback controls the bias point is as follows. If Vbe is held constant and temperature increases, emitter current increases. However, a larger Ie increases the emitter voltage Ve = IeRe, which in turn reduces the voltage VRb across the base resistor. A lower base-resistor voltage drop reduces the base current, which results in less collector current because Ic = ß IB. Collector current and emitter current are related by Ic = α Ie with α ≈ 1, so increase in emitter current with temperature is opposed, and operating point is kept stable.
Similarly, if the transistor is replaced by another, there may be a change in IC (corresponding to change in β-value, for example). By similar process as above, the change is negated and operating point kept stable.
For the given circuit,
IB = (VCC - Vbe)/(RB + (β+1)RE).
Merits:
The circuit has the tendency to stabilize operating point against changes in temperature and β-value.
Demerits:
• In this circuit, to keep IC independent of β the following condition must be met:
$I_C = \beta I_B = \frac { \beta (V_{CC} - V_{be})}{R_B+ ( \beta+1) R_E} \approx \frac {(V_{CC} - V_{be})}{R_E}$
which is approximately the case if
( β + 1 )RE >> RB.
• As β-value is fixed for a given transistor, this relation can be satisfied either by keeping RE very large, or making RB very low.
• If RE is of large value, high VCC is necessary. This increases cost as well as precautions necessary while handling.
• If RB is low, a separate low voltage supply should be used in the base circuit. Using two supplies of different voltages is impractical.
• In addition to the above, RE causes ac feedback which reduces the voltage gain of the amplifier.
Usage:
The feedback also increases the input impedance of the amplifier when seen from the base, which can be advantageous. Due to the above disadvantages, this type of biasing circuit is used only with careful consideration of the trade-offs involved.
### Voltage divider bias
Voltage divider bias
The voltage divider is formed using external resistors R1 and R2. The voltage across R2 forward biases the emitter junction. By proper selection of resistors R1 and R2, the operating point of the transistor can be made independent of β. In this circuit, the voltage divider holds the base voltage fixed independent of base current provided the divider current is large compared to the base current. However, even with a fixed base voltage, collector current varies with temperature (for example) so an emitter resistor is added to stabilize the Q-point, similar to the above circuits with emitter resistor.
In this circuit the base voltage is given by:
$V_B = \$ voltage across $R_2 \$ $= V_{cc} \frac{R_2}{(R_1+R_2)} - I_B \frac{R_1 R_2}{(R_1+R_2)}$
$\approx V_{cc} \frac{R_2}{(R_1+R_2)}$ provided $I_B << I_2 = V_B / R_2 \$.
Also $V_B = V_{be} + I_ER_E \$
For the given circuit,
$I_B =\frac { \frac {V_{CC}}{1+R_1/R_2} - V_{be} } {( \beta + 1)R_E + R_1 \parallel R_2 } .$
Merits:
• Unlike above circuits, only one dc supply is necessary.
• Operating point is almost independent of β variation.
• Operating point stabilized against shift in temperature.
Demerits:
• In this circuit, to keep IC independent of β the following condition must be met:
$I_C = \beta I_B = \beta \frac { \frac {V_{CC}}{1+R_1/R_2} - V_{be} } {( \beta + 1)R_E + R_1 \parallel R_2 } \approx \frac { \frac {V_{CC}}{1+R_1/R_2}- V_{be}} {R_E} ,$
which is approximately the case if
$( \beta + 1 ) R_E >> R_1 \parallel R_2$
where R1 || R2 denotes the equivalent resistance of R1 and R2 connected in parallel.
• As β-value is fixed for a given transistor, this relation can be satisfied either by keeping RE fairly large, or making R1||R2 very low.
• If RE is of large value, high VCC is necessary. This increases cost as well as precautions necessary while handling.
• If R1 || R2 is low, either R1 is low, or R2 is low, or both are low. A low R1 raises VB closer to VC, reducing the available swing in collector voltage, and limiting how large RC can be made without driving the transistor out of active mode. A low R2 lowers Vbe, reducing the allowed collector current. Lowering both resistor values draws more current from the power supply and lowers the input resistance of the amplifier as seen from the base.
• AC as well as DC feedback is caused by RE, which reduces the AC voltage gain of the amplifier. A method to avoid AC feedback while retaining DC feedback is discussed below.
Usage:
The circuit's stability and merits as above make it widely used for linear circuits.
#### Voltage divider with AC bypass capacitor
Voltage divider with capacitor
The standard voltage divider circuit discussed above faces a drawback - AC feedback caused by resistor RE reduces the gain. This can be avoided by placing a capacitor (CE) in parallel with RE, as shown in circuit diagram.
This capacitor is usually chosen to have a low enough reactance at the signal frequencies of interest such that RE is essentially shorted at AC, thus grounding the emitter. Feedback is therefore only present at DC to stabilize the operating point, in which case any AC advantages of feedback are lost.
Of course, this idea can be used to shunt only a portion of RE, thereby retaining some AC feedback.
### Emitter bias
Emitter bias
When a split supply (dual power supply) is available, this biasing circuit is the most effective, and provides zero bias voltage at the emitter or collector for load. The negative supply VEE is used to forward-bias the emitter junction through RE. The positive supply VCC is used to reverse-bias the collector junction. Only two resistors are necessary for the common collector stage and four resistors for the common emitter or common base stage.
We know that,
VB - VE = Vbe
If RB is small enough, base voltage will be approximately zero. Therefore emitter current is,
IE = (VEE - Vbe)/RE
The operating point is independent of β if RE >> RB
Merit:
Good stability of operating point similar to voltage divider bias.
Demerit:
This type can only be used when a split (dual) power supply is available.
## Class B and AB amplifiers
### Signal requirements
Class B and AB amplifiers employ 2 active devices to cover the complete 360 deg of input signal flow. Each transistor is therefore biased to perform over approximately 180 deg of the input signal. Class B bias is when the collector current Ic with no signal is just conducting (about 1 % of maximum possible value). Class AB bias is when the collector current Ic is about 1/4 of maximum possible value. The class AB push–pull output amplifier circuit below could be the basis for a moderate-power audio amplifier.
A practical amplifier circuit
Q3 is a common emitter stage that provides amplification of the signal and the DC bias current through D1 and D2 to generate a bias voltage for the output devices. The output pair are arranged in Class AB push–pull, also called a complementary pair. The diodes D1 and D2 provide a small amount of constant voltage bias for the output pair, just biasing them into the conducting state so that crossover distortion is minimized. That is, the diodes push the output stage into class-AB mode (assuming that the base-emitter drop of the output transistors is reduced by heat dissipation).
This design automatically stabilizes its operating point, since overall feedback internally operates from DC up through the audio range and beyond. The use of fixed diode bias requires the diodes to be both electrically and thermally matched to the output transistors. If the output transistors conduct too much, they can easily overheat and destroy themselves, as the full current from the power supply is not limited at this stage.
A common solution to help stabilize the output device operating point is to include some emitter resistors, typically an ohm or so. Calculating the values of the circuit's resistors and capacitors is done based on the components employed and the intended use of the amplifier. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 19, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8616302609443665, "perplexity": 2160.7966985864355}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657129409.8/warc/CC-MAIN-20140914011209-00134-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"} |
http://ptsymmetry.net/?tag=paolo-amore | ## Non-Hermitian oscillators with $$T_d$$ symmetry
Paolo Amore, Francisco M. Fernández, Javier Garcia
We analyse some PT-symmetric oscillators with $$T_d$$ symmetry that depend on a potential parameter $$g$$. We calculate the eigenvalues and eigenfunctions for each irreducible representation and for a range of values of $$g$$. Pairs of eigenvalues coalesce at exceptional points $$g_c$$; their magnitude roughly decreasing with the magnitude of the eigenvalues. It is difficult to estimate whether there is a phase transition at a nonzero value of g as conjectured in earlier papers. Group theory and perturbation theory enable one to predict whether a given space-time symmetry leads to real eigenvalues for sufficiently small nonzero values of $$g$$.
http://arxiv.org/abs/1409.2672
Quantum Physics (quant-ph)
## Is space-time symmetry a suitable generalization of parity-time symmetry?
Paolo Amore, Francisco M. Fernández, Javier Garcia
We discuss space-time symmetric Hamiltonian operators of the form $$H=H_{0}+igH^{\prime}$$, where $$H_{0}$$ is Hermitian and $$g$$ real. $$H_0$$ is invariant under the unitary operations of a point group $$G$$ while $$H^\prime$$ is invariant under transformation by elements of a subgroup $$G^\prime$$ of $$G$$. If $$G$$ exhibits irreducible representations of dimension greater than unity, then it is possible that $$H$$ has complex eigenvalues for sufficiently small nonzero values of $$g$$. In the particular case that $$H$$ is parity-time symmetric then it appears to exhibit real eigenvalues for all $$0<g<g_c$$, where $$g_{c}$$ is the exceptional point closest to the origin. Point-group symmetry and perturbation theory enable one to predict whether $$H$$ may exhibit real or complex eigenvalues for $$g>0$$. We illustrate the main theoretical results and conclusions of this paper by means of two- and three-dimensional Hamiltonians exhibiting a variety of different point-group symmetries.
http://arxiv.org/abs/1405.5234
Quantum Physics (quant-ph)
## $$\mathcal{PT}$$-symmetric strings
Paolo Amore, Francisco M. Fernández, Javier Garcia, German Gutierrez
We study both analytically and numerically the spectrum of inhomogeneous strings with $$\mathcal{PT}$$-symmetric density. We discuss an exactly solvable model of $$\mathcal{PT}$$-symmetric string which is isospectral to the uniform string; for more general strings, we calculate exactly the sum rules $$Z(p) \equiv \sum_{n=1}^\infty 1/E_n^p$$, with $$p=1,2,\dots$$ and find explicit expressions which can be used to obtain bounds on the lowest eigenvalue. A detailed numerical calculation is carried out for two non-solvable models depending on a parameter, obtaining precise estimates of the critical values where pair of real eigenvalues become complex.
http://arxiv.org/abs/1306.1419
Mathematical Physics (math-ph)
## Comment on: `Numerical estimates of the spectrum for anharmonic PT symmetric potentials’ [Phys. Scr. \textbf{85} (2012) 065005]
Paolo Amore, Francisco M Fernández
We show that the authors of the commented paper draw their conclusions from the eigenvalues of truncated Hamiltonian matrices that do not converge as the matrix dimension increases. In one of the studied examples the authors missed the real positive eigenvalues that already converge towards the exact eigenvalues of the non-Hermitian operator and focused their attention on the complex ones that do not. We also show that the authors misread Bender’s argument about the eigenvalues of the harmonic oscillator with boundary conditions in the complex-x plane (Rep. Prog. Phys. 70 (2007) 947).
http://arxiv.org/abs/1209.6357
Quantum Physics (quant-ph) | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9704554677009583, "perplexity": 515.892160931494}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370490497.6/warc/CC-MAIN-20200328074047-20200328104047-00391.warc.gz"} |
https://www.physicsforums.com/threads/rotational-kinetic-energy-of-bicycle-wheels.220348/ | # Rotational Kinetic Energy of bicycle wheels
1. ### rugbygirl
5
A bicycle has wheels of radius 0.33 m. Each wheel has a rotational inertia of 0.082 kg* m2 about its axle. The total mass of the bicycle including the wheels and the rider is 74 kg. When coasting at constant speed, what fraction of the total kinetic energy of the bicycle (including rider) is the rotational kinetic energy of the wheels?
I thought this: Rotational KE = (1/2)Iw^2
=(1/2)(second bold number)w^2
Linear KE= (1/2)mv^2
= (1/2)(third bold number)(radius*w)^2 (i.e. plug in r*w for v)
Total KE is equal to Rotational KE + Linear KE
(1/2)Iw^2/ (some # * w^2)
2. ### luben
71
i think you are correct
eventually one takes away the w^2 in both numerator and denominator, then gets a result independent of w
3. ### rugbygirl
5
i keep getting .041/4.07 and that is not right
4. ### Cyrus
How many wheels does a bicycle have?
Please post in the HW forums. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9386594891548157, "perplexity": 4247.119922065276}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398457127.55/warc/CC-MAIN-20151124205417-00121-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://clumath343s15s2.wikidot.com/clulinearalgebra343s2015s2definitions | Clulinearalgebra343s2015s2definitions
A linear equation in the variables $x_1, x_2, ... , x_n$ is an equation that can be written in the form
(1)
$$a_1 x_1 + a_2 x_2 + ... + a_n x_n = b$$
A system of linear equations is a collection of linear equations involving the same variables.
A solution of the system is a list of numbers that makes each equation true.
The set of all solutions is the solution set.
Two systems are equivalent if they have the same solution set.
#### Linear Combination
Let $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$ be vectors in $\mathbb{R}^n$.
Let $c_1, c_2, ..., c_p$ be scalars.
Then
(2)
\begin{align} y=c_1\bar{v}_1 + c_2\bar{v}_2 + ... + c_p \bar{v}_p \end{align}
is the linear combination of $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$
#### Span
Let $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p \in \mathbb{R}^n$. The **span** of $\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$ is the set of all linear combinations of$\bar{v}_1 , \bar{v}_2, ... , \bar{v}_p$
page revision: 0, last edited: 04 Feb 2015 22:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 1, "x-ck12": 0, "texerror": 0, "math_score": 0.9673251509666443, "perplexity": 655.4579555009007}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160923.61/warc/CC-MAIN-20180925024239-20180925044639-00109.warc.gz"} |
http://mathhelpforum.com/calculus/111811-derivative-problem-e-x.html | # Math Help - derivative problem with e^x
1. ## derivative problem with e^x
y= e ^ 5- (2/x) ............ whats the derivative? for some reason i cant find the answer!!
thanks!
2. Originally Posted by I"love"mymathteacher
y= e ^ 5- (2/x) ............ whats the derivative? for some reason i cant find the answer!!
thanks!
$y=e^{5-\frac{2}{x}}$
Use the chain rule. Whenever a composite function looks complicated like this, just introduce a variable. Let $u=5-\frac{2}{x}$. So now $y=e^u$. So now use the chain rule $\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$. I'll let you try to finish this, we aren't really suppose to just give answers around here. Reply with what you think the answer is, if you're still stuck, I will help you finish it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 4, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9916287064552307, "perplexity": 885.6077958059772}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398451872.11/warc/CC-MAIN-20151124205411-00136-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/51222/is-there-an-example-of-a-sigma-algebra-that-is-not-a-topology | Is there an example of a sigma algebra that is not a topology?
Is there an example of a sigma algebra that is not a topology? If this is not the case, is it possible to prove that all sigma algebras are topologies?
-
The Borel sets in $[0,1]$, obviously. It contains all points but not all subsets. – t.b. Jul 13 '11 at 14:37
@Claudia: Didn't you pose exactly the same question here yesterday? Maybe I'm going mad... – Stefan Walter Jul 13 '11 at 14:45
@Stefan: I had the same déjà-vu, don't worry about your mental health :) I see only now that Joel made more or less the same point as I did in a comment – t.b. Jul 13 '11 at 14:50
I do not recall topologies being $\sigma$-algebras at all... They are not usually closed under complements nor countable intersections. – Asaf Karagila Jul 13 '11 at 14:58
See also mathoverflow.net/questions/70137/… same question and basically the same answer. – plusepsilon.de Jul 13 '11 at 15:17
If a $\sigma$-algebra contains all one-point sets then the topology it generates is the discrete topology, so the only thing we need to ensure is that we have a $\sigma$-algebra in which not every set is measurable.
The very first example that comes to mind is already an example. As there are non-Borel sets in $[0,1]$ the Borel sets are not the discrete topology, but the smallest topology that contains the Borel sets is the discrete topology by the above paragraph.
(I'm ignoring choice issues here, as I lack the expertise).
A similar argument works for the Lebesgue $\sigma$-algebra.
In a positive direction: Every $\sigma$-algebra on a countable set is a topology.
To sum up:
• On finite or countable sets every $\sigma$-algebra is a topology.
• In view of Pete's answer below, on every uncountable set there is a $\sigma$-algebra that isn't a topology, namely the countable-cocountable $\sigma$-algebra. This example is probably the optimal one in terms of simplicity.
For the sake of completeness:
• Pete L. Clark's answer in this thread uses choice in the very weak form "the countable union of countably many countable sets is countable", as noted by Nate Eldredge.
• Andrés Caicedo argues beautifully in this MO-thread why some choice is necessary to ensure that there exist non-Borel sets.
• If I interpret François G. Dorais's answer here correctly, there are models of ZF with the property that the $\sigma$-algebra generated by the singleton subsets of a set is equal to the power set. (Please correct me if this is too naïve a rendering)
• Asaf Karagila adds in a comment below: "[...] an even freakier statement is given in Jech's The Axiom of Choice, Ch. 5, Exercise 14. There is an extension of every transitive model, with the same $\aleph$-cardinals, and for every $\alpha$ there is a set $X$ which is a countable union of countable sets, and $P(X)$ can be partitioned into $\aleph_{\alpha}$ nonempty sets."
• Jay adds: As John B. S. Haldane might have said, "Set theory without the axiom of choice is not only queerer than we suppose, but queerer than we can suppose."
-
@Asaf: No I mean that I don't know how to construct a non-Borel set without assuming at least some choice. And I mean that I ignore choice issues in general. – t.b. Jul 13 '11 at 16:32
I have no choice but to agree with Andres... :-) – Asaf Karagila Jul 13 '11 at 16:45
@Asaf: I guess so :) You might also be interested in François G. Dorais's answer here. By the way, I have nothing against asking about and investigating choice, but I'm a lazy person and stick to good ol' ZFC - don't try to tell an old man to change his ways... Foundational issues are interesting but more a hobby of mine than anything else. – t.b. Jul 13 '11 at 17:17
I saw this question before, an even freakier statement is given in Jech's The Axiom of Choice, Ch. 5, Exercise 14. There is an extension of every transitive model, with the same $\aleph$-cardinals, and for every $\alpha$ there is a set $X$ which is a countable union of countable sets, and $P(X)$ can be partitioned into $\aleph_\alpha$ nonempty sets. Creepy, right? :-) – Asaf Karagila Jul 13 '11 at 17:37
As John B. S. Haldane might have said, "Set theory without the axiom of choice is not only queerer than we suppose, but queerer than we can suppose." – Jay Jul 14 '11 at 0:31
Let $S$ be any uncountable set, and let $\mathcal{A}$ be the collection of all subsets of $S$ which are either countable or have countable complement.
This collection is evidently closed under complementation. If I have a countable union of elements of $\mathcal{A}$, all of which are countable, then the union is countable. Otherwise, at least one element is cocountable, hence so is the union. A similar argument works for intersections. So $\mathcal{A}$ is a $\sigma$-algebra.
It is not a topology because it contains all the singleton sets but not all subsets of $S$ -- in particular, it contains no set which is uncountable with uncountable complement.
-
Ah very nice! That's much simpler than the Borel sets. I guess I let myself get carried away... – t.b. Jul 14 '11 at 1:23
Of course, this argument is also using the axiom of choice (a countable union of countable sets is countable). – Nate Eldredge Jul 14 '11 at 3:52
@Theo: If the space is $T_1$ then every countable set is $\Sigma^0_2$ and every cocountable set is $\Pi^0_2$, therefore the $\sigma$-algebra denoted by Pete as $\mathcal A$ is a sub-algebra of $\Delta^0_3$. It is simpler than Borel sets, but the Borel sets tend to include them. (I know both you as well Pete are aware of that, this is a side remark for the less familiar reader :-)) – Asaf Karagila Jul 14 '11 at 22:08
@Nate: Yes, I am assuming the axiom of choice. I am (like most mathematicians?) in fact always assuming the Axiom of Choice, although (unlike many mathematicians?) I agree that it is nice to know what happens in the absence of AC. But really, how far does one get in measure theory without the "fact" that countable unions of countable sets are countable? (This is not a rhetorical question.) – Pete L. Clark Jul 15 '11 at 1:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9379573464393616, "perplexity": 328.5030033694437}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-32/segments/1438042982502.13/warc/CC-MAIN-20150728002302-00111-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/86835-finding-definite-integral-f-x-but-not-given-function-print.html | # Finding definite integral of f(x) but not given a function?
• May 1st 2009, 02:24 PM
ceb0196
Finding definite integral of f(x) but not given a function?
I think my prof is trying to trip me up. Here is the question(sorry, I don't know the codes to make it in the right form):
Suppose: integrand with b=6 and a=1 of f(x) dx=9, integrand with b=4 and a=6 of f(x) dx=4, then find integrand with b=4 and a=1
does that make any sense? How do I solve this without being given a function??
• May 1st 2009, 02:27 PM
mr fantastic
Quote:
Originally Posted by ceb0196
I think my prof is trying to trip me up. Here is the question(sorry, I don't know the codes to make it in the right form):
Suppose: integrand with b=6 and a=1 of f(x) dx=9, integrand with b=4 and a=6 of f(x) dx=4, then find integrand with b=4 and a=1
does that make any sense? How do I solve this without being given a function??
You're given $\int_1^6 f(x) \, dx = 9$ and $\int_4^6 f(x) \, dx = 4$.
Note that $\int_1^6 f(x) \, dx = \int_1^4 f(x) \, dx + \int_4^6 f(x) \, dx \Rightarrow \int_1^4 f(x) \, dx = \int_1^6 f(x) \, dx - \int_4^6 f(x) \, dx$.
• May 1st 2009, 02:32 PM
Plato
Is this your question: $\int_1^6 f = 9\;\& \;\int_6^4 f = 4\; \Rightarrow \;\int_1^4 f = ?$?
If so, notice that $\int_4^6 f = - 4\;\& \;\int_1^4 f + \int_4^6 f = \int_1^6 f$.
• May 1st 2009, 02:35 PM
ceb0196
Yes, that is the question. Thanks to both for the start; I will try to work it out!
• May 1st 2009, 02:52 PM
ceb0196
I just can't figure out what I am missing. How do I solve without having a function that I can plug F(b) and F(a) into??
• May 1st 2009, 02:58 PM
mr fantastic
Quote:
Originally Posted by ceb0196
I just can't figure out what I am missing. How do I solve without having a function that I can plug F(b) and F(a) into??
Read posts #2 and #3 again. Read them carefully. Where is your trouble in plugging in the value of things that you know and solving for the thing that you don't know?
• May 1st 2009, 03:14 PM
ceb0196
Is it 13? It can't possibly be that simple...
• May 1st 2009, 03:26 PM
Plato
Quote:
Originally Posted by ceb0196
Is it 13? It can't possibly be that simple...
Oh yes, it is indeed that simple!
• May 1st 2009, 03:36 PM
ceb0196
Oh my goodness gracious. Thanks to all of you for helping me fumble through the last week of this class! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9474145770072937, "perplexity": 1394.1093732635875}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1430460196625.80/warc/CC-MAIN-20150501060316-00058-ip-10-235-10-82.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/20745/why-electrons-are-relativistic-in-graphene-and-non-relativistic-in-vacuum/20747 | # Why electrons are relativistic in Graphene and non relativistic in vacuum?
If a free region in space has a potential difference of one volt, an electron in this region will acquire kinetic energy of 1 eV. Its speed will be much smaller than the speed of light hence it will be a non relativistic electron.
On the other hand conduction electrons in graphene are relativistic for the same potential difference.
Question is how come that when the electrons are in vacuum they are non relativistic, and when they are inside Graphene they are relativistic (for the same potential difference)?
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I think you assume from the first sentence $v\approx \sqrt{\frac{2 U_{eV}}{m_e}}<<c_0$ But your second sentence is a little bit misleading "conduction electrons in graphene are relativistic for the same potential difference". What do you mean? Electron mobility? Drift velocity? They are nonrelativistic. Or are you asking for the electron movement in a molecule? Well for Carbon the Schrödinger equation is a good approximation, you do not need the Dirac equation. You have to consider relativity for s electrons for heavy elements with high charge with e.g. ZORA - zeroth order relativistic approxi – Alex1167623 Feb 26 '12 at 0:10
Please do not answer questions if you are not familiar with the field. A quick google immediately brings up a wealth of information about the statement. In this case, there is no connection to the actual speed of light and the statement is a purely formal one regarding the equation of motion for quasiparticles. – genneth Feb 26 '12 at 2:32
@genneth You are right that this should not be an answer, but rather a comment on the given question to improve it. Pushed the wrong button :-(. But I am dissapointed too that a proper answer was not given by you, nor a reference. Hans de Vries finally clearified it, thnx. – Alex1167623 Feb 29 '12 at 9:33
As far as I understand, electrons in graphene are not relativistic, although quasiparticles in graphene are indeed described by the massless Dirac equation. However, for graphene, the speed velocity in this equation is replaced by the Fermi velocity, which is much smaller.
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@Revo: In my book, a particle is relativistic if its velocity is comparable to the velocity of light. If you use a different definition, please give me a reference to a reliable source (if you just alluded in your comment that the eigenvalues of velocity projections for a Dirac particle are always +-c due to Zitterbewegung, this does not seem to be relevant to your question). The velocity of the quasiparticles in graphene is always comparable to the "velocity of light" in the massless Dirac equation for graphene, but that "velocity of light" is not the genuine velocity of light. – akhmeteli Feb 9 '12 at 22:14
@Revo: No. I believe a particle is relativistic when its velocity is comparable to the velocity of light in vacuum. In most cases the velocity of speed in media is comparable to that in vacuum, so the clarification about vacuum is usually omitted. I agree that some exotic media may present exceptions. That does not mean that the particle that you describe must be described by a relativistic equation. It just so happens that quasiparticles in graphene can be described satisfactorily (to some extent) by an equation looking exactly like the massless Dirac equation with lesser "velocity of light" – akhmeteli Feb 10 '12 at 19:35
@Revo: I have two problems with your reasoning. While I agree that the standard Dirac equation is a relativistic equation and that it correctly describes a relativistic spin one half particle, that does not mean that if a particle is correctly described by the Dirac equation, it is necessarily relativistic, because the Dirac equation correctly describes slow particles as well. The above reasoning is correct, however, for the standard MASSLESS Dirac equation, as such an equation does not describe correctly slow particles. The other problem is described in another comment. – akhmeteli Feb 11 '12 at 1:19
@Revo: The other problem is as follows. The massless Dirac equation used for quasiparticles in graphene is not the standard massless Dirac particles for the following reasons. While it looks exactly like the standard massless Dirac equation, the speed constant in this equation is much less than the velocity of speed in vacuum, so it only describes particles that are slow compared to the velocity of light in vacuum. Furthermore, the equation is not relativistic in the sense that it is not invariant under Lorentz transforms, it is only correct in the frame of reference of the graphene lattice. – akhmeteli Feb 11 '12 at 1:30
@Revo: you are mistaken about the link between the Dirac equation and relativity. The Dirac equation correctly describes a single particle relativistically, but does not have to. One can use it to do other things. The statement "electrons in graphene are relativistic" is a purely formal statement about the lack of a rest mass for the quasiparticles. – genneth Feb 26 '12 at 2:30
The statement that in graphene the "conduction electrons are massless" is because the energy levels (bands) are proportional to their momenta.
So the $E = \sqrt{p^2+m^2}$ relation of a free electron becomes $E\propto p$ in graphene.
Massless particles travel all at the same speed because of the $E\propto p$ relation but this characteristic velocity in graphene is far below c though, only 0.3% of the speed of light.
The reason that the relation $E\propto p$ leads to a characteristic speed is due to the quantum mechanical wave character. $E$ is proportional to the phase changes in time, $p$ is proportional to the phase changes in space and therefor $p/E$ is proportional to the velocity. In the case that $E\propto p$ there is a characteristic velocity $v$ independent of the energy level.
The most striking aspect of graphene is that its electronic energy levels, or “bands,” produce conduction electrons whose energies are directly proportional to their momentum. This is the energy-momentum relationship exhibited by photons, which are massless particles of light. Electrons and other particles of matter normally have energies that depend on the square of their momentum.
When the bands are plotted in three dimensions, the photonlike energy-momentum relationship appears as an inverted cone, called a Dirac cone. This unusual relationship causes conduction electrons to behave as though they were massless, like photons, so that all of them travel at roughly the same speed (about 0.3 percent of the speed of light). This uniformity leads to a conductivity greater than copper.
Hans
- | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9003373384475708, "perplexity": 359.21548256571936}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500832738.80/warc/CC-MAIN-20140820021352-00009-ip-10-180-136-8.ec2.internal.warc.gz"} |
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[1] Anastasia Chavez and Nicole Yamzon. The Dehn--Sommerville relations and the Catalan matroid. Proc. Amer. Math. Soc. 145 (2017) 4041-4047. Abstract, references, and article information View Article: PDF [2] Nina Amenta, Jesús A. De Loera and Pablo Soberón. Helly's theorem: New variations and applications. Contemporary Mathematics 685 (2017) 55-95. Book volume table of contents View Article: PDF [3] Vincent Pilaud and Christian Stump. Vertex barycenter of generalized associahedra. Proc. Amer. Math. Soc. 143 (2015) 2623-2636. Abstract, references, and article information View Article: PDF [4] A. A. Ayzenberg. Substitutions of polytopes and of simplicial complexes, and multigraded betti numbers. Trans. Moscow Math. Soc. 74 (2013) 175-202. Abstract, references, and article information View Article: PDF [5] Jesús A. De Loera and Edward D. Kim. Combinatorics and geometry of transportation polytopes: An update. Contemporary Mathematics 625 (2014) 37-76. Book volume table of contents View Article: PDF [6] Thomas Tradler and Ronald Umble. Tensor products of $A_\infty$-algebras with homotopy inner products. Trans. Amer. Math. Soc. 365 (2013) 5153-5198. Abstract, references, and article information View Article: PDF [7] Julien Paupert. A simple method to compute volumes of even-dimensional Coxeter polyhedra. Contemporary Mathematics 590 (2013) 167-175. Book volume table of contents View Article: PDF [8] Satoshi Murai. $h$-vectors of simplicial cell balls. Trans. Amer. Math. Soc. 365 (2013) 1533-1550. Abstract, references, and article information View Article: PDF [9] Gábor Hetyei. The short toric polynomial. Trans. Amer. Math. Soc. 365 (2013) 1441-1468. Abstract, references, and article information View Article: PDF [10] Douglas Bowman and Alon Regev. Counting equivalence classes of vertex pairs modulo the dihedral action on the associahedron. Proc. Amer. Math. Soc. 141 (2013) 779-789. Abstract, references, and article information View Article: PDF [11] Lukas Katthän. On homology spheres with few minimal non-faces. Proc. Amer. Math. Soc. 140 (2012) 2489-2500. Abstract, references, and article information View Article: PDF [12] Olga Holtz, Amos Ron and Zhiqiang Xu. Hierarchical zonotopal spaces. Trans. Amer. Math. Soc. 364 (2012) 745-766. MR 2846351. Abstract, references, and article information View Article: PDF [13] Fiammetta Battaglia. Betti numbers of the geometric spaces associated to nonrational simple convex polytopes. Proc. Amer. Math. Soc. 139 (2011) 2309-2315. MR 2784795. Abstract, references, and article information View Article: PDF This article is available free of charge [14] Alberto Corso and Uwe Nagel. Monomial and toric ideals associated to Ferrers graphs. Trans. Amer. Math. Soc. 361 (2009) 1371-1395. MR 2457403. Abstract, references, and article information View Article: PDF This article is available free of charge [15] David L. Donoho and Jared Tanner. Counting faces of randomly projected polytopes when the projection radically lowers dimension. J. Amer. Math. Soc. 22 (2009) 1-53. MR 2449053. Abstract, references, and article information View Article: PDF This article is available free of charge [16] Toshiyuki Akita. A formula for the Euler characteristics of even dimensional triangulated manifolds. Proc. Amer. Math. Soc. 136 (2008) 2571-2573. MR 2390528. Abstract, references, and article information View Article: PDF This article is available free of charge [17] V. A. Zalgaller. Symmetric geodesics on cubes: Algorithms for finding them. St. Petersburg Math. J. 16 (2005) 423-436. MR 2068348. Abstract, references, and article information View Article: PDF This article is available free of charge [18] Chuanming Zong. What is known about unit cubes. Bull. Amer. Math. Soc. 42 (2005) 181-211. MR 2133310. Abstract, references, and article information View Article: PDF [19] Günter M. Ziegler. Projected products of polygons. Electron. Res. Announc. Amer. Math. Soc. 10 (2004) 122-134. MR 2119033. Abstract, references, and article information View Article: PDF [20] Margaret M. Bayer and Richard Ehrenborg. The toric $h$-vectors of partially ordered sets. Trans. Amer. Math. Soc. 352 (2000) 4515-4531. MR 1779486. Abstract, references, and article information View Article: PDF This article is available free of charge [21] Masaaki Wada, Yasushi Yamashita and Han Yoshida. An inequality for polyhedra and ideal triangulations of cusped hyperbolic 3-manifolds. Proc. Amer. Math. Soc. 124 (1996) 3905-3911. MR 1346992. Abstract, references, and article information View Article: PDF This article is available free of charge [22] A. J. W. Duijvestijn. The number of polyhedral (3-connected planar) graphs. Math. Comp. 65 (1996) 1289-1293. MR 1348044. Abstract, references, and article information View Article: PDF This article is available free of charge [23] Mark Purtill. Andr\'e permutations, lexicographic shellability and the $cd$-index of a convex polytope . Trans. Amer. Math. Soc. 338 (1993) 77-104. MR 1094560. Abstract, references, and article information View Article: PDF This article is available free of charge
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http://mathhelpforum.com/algebra/23952-dividing-fractions.html | # Math Help - Dividing fractions
1. ## Dividing fractions
when I divide two fractions example
[1/2] / [3/5] I know that I have to invert and then multiply
[1/2]*[5/3] so my answer will be 5/6
My question is why do I invert then multiply inorder to get the correct answer, Why cannot I just say that it is 3/10, why do we need to invert and multiply, why is this the rule that we must invert then multiply?
2. Originally Posted by schinb64
when I divide two fractions example
[1/2] / [3/5] I know that I have to invert and then multiply
[1/2]*[5/3] so my answer will be 5/6 My question is why do I invert then multiply inorder to get the correct answer?
Lets take the following equation as an example. I'm not entirely sure how to explain using words.
$\frac{4}{1} \div \frac{2}{1} = \frac{4}{2} = \frac{2}{1}$ I'm sure you understand that.
$\frac{4}{1} \times \frac{1}{2} = \frac{4}{2} = \frac{2}{1}$
Does this make sense? I'm not too good a teacher.
3. Originally Posted by schinb64
when I divide two fractions example
[1/2] / [3/5] I know that I have to invert and then multiply
[1/2]*[5/3] so my answer will be 5/6 My question is why do I invert then multiply inorder to get the correct answer?
$\frac{ \frac{1}{2} }{ \frac{3}{5} }$
We want to clear out the fractions in the numerator and denominator, so we need to multiply top and bottom by the LCM of 2 and 5, which is 10:
$= \frac{ \frac{1}{2} }{ \frac{3}{5} } \cdot \frac{2 \cdot 5}{2 \cdot 5}$
$= \frac{ \frac{1}{2} \cdot 2 \cdot 5 }{ \frac{3}{5} \cdot 2 \cdot 5 }$
$= \frac{1 \cdot 5}{3 \cdot 2} = \frac{1}{2} \times \frac{5}{3}$
This is, by far, the harder method to do the problem with, but it is the way to prove that the shortcut you learned is correct.
-Dan
4. Originally Posted by schinb64
when I divide two fractions example
[1/2] / [3/5] I know that I have to invert and then multiply
[1/2]*[5/3] so my answer will be 5/6
My question is why do I invert then multiply inorder to get the correct answer, Why cannot I just say that it is 3/10, why do we need to invert and multiply, why is this the rule that we must invert then multiply?
Originally Posted by topsquark
$\frac{ \frac{1}{2} }{ \frac{3}{5} }$
We want to clear out the fractions in the numerator and denominator, so we need to multiply top and bottom by the LCM of 2 and 5, which is 10:
$= \frac{ \frac{1}{2} }{ \frac{3}{5} } \cdot \frac{2 \cdot 5}{2 \cdot 5}$
$= \frac{ \frac{1}{2} \cdot 2 \cdot 5 }{ \frac{3}{5} \cdot 2 \cdot 5 }$
$= \frac{1 \cdot 5}{3 \cdot 2} = \frac{1}{2} \times \frac{5}{3}$
This is, by far, the harder method to do the problem with, but it is the way to prove that the shortcut you learned is correct.
-Dan
You guys are good because when I was in 6th and asked why questions my teacher said," don't ever ask why in math. Some old guy said so that's why." In the future my teachers said they didn't know why. I just accepted that it was a rule some old guy that made up the rules and we follow them.
5. Originally Posted by Itachi888Uchiha
You guys are good because when I was in 6th and asked why questions my teacher said," don't ever ask why in math. Some old guy said so that's why." In the future my teachers said they didn't know why. I just accepted that it was a rule some old guy that made up the rules and we follow them.
haha, interesting. well, the rule is, dividing by a fraction is the same as multiplying by its inverse, and this is not just because some old guy said so. it makes sense when you think about it. dividing something by two is the same as making two halves of it. so dividing by 2 is the same as multiplying by 1/2. etc
6. Originally Posted by Itachi888Uchiha
You guys are good because when I was in 6th and asked why questions my teacher said," don't ever ask why in math. Some old guy said so that's why." In the future my teachers said they didn't know why. I just accepted that it was a rule some old guy that made up the rules and we follow them.
Always ask "why?" It's the way you learn this stuff, as opposed to merely being able to do it.
-Dan | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 10, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.884868323802948, "perplexity": 408.1290836244471}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461864121714.34/warc/CC-MAIN-20160428172201-00133-ip-10-239-7-51.ec2.internal.warc.gz"} |
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## Increase font in Message list (Messages app)
(Topic created: 01-08-2023 11:17 PM)
Asteroid
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I'm using [Samsung] Messages on an A53 5G phone. I can change the text OK in each chat by pinching to zoom. BUT the text size on the opening (Messages) screen is too small for me, and pinching doesn't work. Is there a way to increase it?
### 1 Solution
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Solution
Asteroid
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Thanks for the tips, all. I was hoping to increase this font without increasing the systemwide size, but it seems that's not possible. It's helpful to know that.
3 Replies
Supernova
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You can adjust this in the phone's settings. Go to Display; then Screen Zoom.
There isn't a separate way for *just* Messages. This zoom setting applies to all apps on your phone.
Galaxy
Options
Solution
Asteroid
Options
Thanks for the tips, all. I was hoping to increase this font without increasing the systemwide size, but it seems that's not possible. It's helpful to know that. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8807027339935303, "perplexity": 4468.4558438435915}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943746.73/warc/CC-MAIN-20230321193811-20230321223811-00451.warc.gz"} |
http://tetration.org/Tetration/index.html | Tetration.org What Lies Beyond Exponentiation?
# Tetration
Tetration is defined as iterated exponentiation but while exponentiation is essential to a large body of mathematics, little is known about tetration due to its chaotic properties. The standard notation for tetration is $^{1}a=a, ^{2}a=a^a, ^{3}a=a^{a^a},$ and so on. Mathematicians have been researching tetration since at least the time of Euler but it is only at the end of the twentieth century that the combination of advances in dynamical systems and access to powerful computers is making real progress possible.
The big question in tetration research is how can tetration be extended to complex numbers. How do you compute numbers like $^{.5}2$, and $^{\pi i}e$ ? This web site will show how to compute these and other problems. Two important related concepts to tetration are continuously iterated functions and the Ackermann function.
Tetration by Escape Tetration by Period
# Continuous iteration of functions
The study of the iterations of a complex function $f(z)$ are known as complex dynamics, where $f^0(z)\equiv z$, $f^1(z) \equiv f(z)$, $f^2(z) \equiv f(f(z)), \ldots$. The meaning of $f^n(z)$, when $n \in \mathbb{N}$ is clear. It is when $f(z)$ is viewed in higher dimensional dynamics with $n \in \mathbb{R}$ or $n \in \mathbb{C}$ that $f^n(z)$ as a continuously iterationed functions can represent any possible system in physics. There are two ways in physics to represent arbitrary dynamical systems, as partial differential equations and as a continuously iterationed function. Partial differential equations are known as flows while discretely iterationed functions are known as maps. The continuous iteration of functions is known as Lie groups when the dynamical system is not chaotic.
Consider when $f(z)\equiv a^z$, then $f(1)= a^1=a$, $f^2(1)= a^a=^{2}a$, and $f^3(1)= a^{a^a}=^{3}a, \ldots$ . Therefore $f^n(1)= ^{n}a$. So tetration for complex numbers can be easily defined if complex functions can be continuosly iterated, $n,z \in \mathbb{C}$ for $f^n(z)$.
# The Ackermann function
The Ackermann function consists of addition, multiplication, exponentiation, tetration, pentation and so on.
Arithmetic Standard Ackermann Knuth Conway Addition a+b ack(a,b,0) Multiplication a*b ack(a,b,1) Exponentiation ab ack(a,b,2) a↑b a→b→1 Tetration ba ack(a,b,3) a↑↑b a→b→2 Pentation ba ack(a,b,4) a↑↑↑b a→b→3 Hexation ack(a,b,5) a↑↑↑↑b a→b→4 ... ... ...
# Uniting Two Kinds of Science - The Old and the New
In 1986 Stephen Wolfram pointed out that the problem of extending tetration to the complex numbers was actually part of the much larger and more important problem of unifying the discreet representation of chaotic systems in mathematics with the continuous representation of chaotic systems in physics, of unifying maps from iterated functions and flows from PDEs. He maintained that the duality prevented the derivation of mathematical solutions for continuous chaotic systems as are found in physics. My understanding of Wolfram’s position is that he was interested in the possibility that there might be a principle at play even deeper than the Principle of Equivalence and that it might be possible to formulate a single mathematical approach to dynamics encompassing iterated functions, cellular automata, PDEs, and recurrence equations. Wolfram suggested at if tetration could be defined for complex numbers then those results might be generalized to unify discrete maps and continuous flows.
## How can one extend recursive function definitions to continuous numbers?
How can one extend recursive function definitions to continuous numbers? What is the continuous analog of the Ackermann function? - Stephen Wolfram
Let $f(z) \equiv a \rightarrow z \rightarrow k$.
#### Theorem. When $f^n(z)$ where $n \in \mathbb{C}$ is defined, then $a \rightarrow b \rightarrow k+1$ where $a,b \in \mathbb{C}$.
\begin{eqnarray} f(1) &=& a \rightarrow 1 \rightarrow k = a\\ f^2(1) &=& f(a) = a \rightarrow a \rightarrow k = a \rightarrow 2 \rightarrow k+1\\ f^3(1) &=& f(a \rightarrow a \rightarrow k) = a \rightarrow (a \rightarrow a \rightarrow k) \rightarrow k = a \rightarrow 3 \rightarrow k+1\\ f^n(1) &=& a \rightarrow n \rightarrow k+1 \\ \end{eqnarray}
Therefore, when $f^n(z)$ where $n \in \mathbb{C}$ is defined, then $a \rightarrow b \rightarrow k+1$ where $a,b \in \mathbb{C}$ is defined. $\bullet$
### An Equivalent Problem
So for tetration and the Ackermann function in general, the problem of extending them to the complex numbers can be simply reduced to the problem of continuous iteration of functions. The problem of continuous iteration of functions is solved by taking the Taylor series $f^t(z)=\sum_{j=1}^\infty D^j f^t(0) z^j$ .
## The Derivatives of Iterated Functions
Consider the holomorphic function $f(z): \mathbb{C} \rightarrow \mathbb{C}$ and its iterates $f^{\;\:t}(z), t \in \mathbb{N}$. The standard convention of using a coordinate translation to set a fixed point at zero is invoked, $f(0)\equiv 0$, giving $f(z)=\sum_{n=1}^{\infty} \frac{f_n}{n!} z^n$ for $0\leq |z|< R$ for some positive $R$. Note that $f(z)$ is the exponential generating function of the sequence $f_0, f_1, \ldots ,f_\infty$, where $f_0=0$ and $f_1$ will be written as $\lambda$. The expression $f_j^k$ denotes $(D^j f(z))^k |_{z=0}$ .
Note: The symbol $t$ for time assumes $t \in \mathbb{N}$, that time is discrete. This allows the variable $n$ to be used solely in the context of differentiation in this paper. Beginning with the second derivitive each component will be expressed in a general form using summations and referred to here as Schroeder summations.
### The First Derivative
The first derivative of a function at its fixed point $Df(0)=f_1$ is often represented by $\lambda$ and referred to as the multiplier or the Lyapunov characteristic number; its logarithm is known as the Lyapunov exponent. Let $g(z)=f^{t-1}(z)$, then \begin{eqnarray*} Df(g(z))&=&f'(g(z))g'(z)\\ &=&f'(f^{t-1}(z))Df^{t-1}(z)\\ &=&\prod^{t-1}_{k_1=0}f'(f^{t-k_1-1}(z))\\ \end{eqnarray*} \begin{eqnarray} Df^t(0)&=&f'(0)^t\nonumber\\ &=&f_1^t = \lambda^t \label{eq:TheFirstDerivative} \end{eqnarray}
### The Second Derivative
\begin{eqnarray*} D^2f(g(z))&=&f''(g(z))g'(z)^2+f'(g(z))g''(z)\\ &=&f''(f^{t-1}(z))(Df^{t-1}(z))^2+f'(f^{t-1}(z))D^2f^{t-1}(z) \end{eqnarray*} Setting $g(z) = f^{t-1}(z)$ results in \begin{eqnarray} D^2f^t(0)&=& f_2 \lambda^{2t-2}+\lambda D^2f^{t-1}(0)\nonumber \end{eqnarray} When $\lambda \neq 0$, a recurrence equation is formed that is solved as a summation. \begin{eqnarray} D^2f^t(0)&=&f_2\lambda^{2t-2}+\lambda D^2f^{t-1}(0)\nonumber\\ &=&\lambda^0f_2 \lambda^{2t-2}\nonumber\\ &&+\lambda^1f_2 \lambda^{2t-4}\nonumber\\ &&+\cdots\nonumber\\ &&+\lambda^{t-2}f_2 \lambda^2\nonumber\\ &&+\lambda^{t-1}f_2 \lambda^0\nonumber\\ &=&f_2\sum_{k_1=0}^{t-1}\lambda^{2t-k_1-2} \label{eq:TheSecondDerivative} \end{eqnarray}
### The Third Derivative
Continuing on with the third derivative, \begin{eqnarray} D^3f(g(z))&=&f'''(g(z))g'(z)^3+3f''(g(z))g'(z)g''(z)+f'(g(z))g'''(z)\nonumber\\ &=&f'''(f^{t-1}(z))(Df^{t-1}(z))^3\nonumber\\ &&+3f''(f^{t-1}(z))Df^{t-1}(z)D^2f^{t-1}(z)\nonumber\\ &&+f'(f^{t-1}(z))D^3f^{t-1}(z)\nonumber \end{eqnarray} \begin{eqnarray} D^3f^t(0)&=&f_3\lambda^{3t-3}+3 f_2^2\sum_{k_1=0}^{t-1}\lambda^{3t-k_1-5} +\lambda D^3f^{t-1}(0) \nonumber\\ &=&f_3\sum_{k_1=0}^{t-1}\lambda^{3t-2k_1-3} +3f_2^2 \sum_{k_1=0}^{t-1} \sum_{k_2=0}^{t-k_1-2} \lambda^{3t-2k_1-k_2-5} \label{eq:TheThirdDerivative} \end{eqnarray}
Note that the index $k_1$ from the second derivative is renamed $k_2$ in the final summation of the third derivative. A certain amount of renumbering is unavoidable in order to use a simple index scheme.
## Iterated Functions
Putting the pieces together and setting the fixed point at $f_0$ gives,
\begin{eqnarray} f^t(z)&=&\sum_{j=0}^\infty D^j f^t(0) z^j \\ &=&f_0+\lambda^t (z-f_0)+( f_2\sum_{k_1=0}^{t-1}\lambda^{2t-k_1-2}) (z-f_0)^2+ (f_3\sum_{k_1=0}^{t-1}\lambda^{3t-2k_1-3} +3f_2^2 \sum_{k_1=0}^{t-1} \sum_{k_2=0}^{t-k_1-2} \lambda^{3t-2k_1-k_2-5}) (z-f_0)^3+ \ldots \end{eqnarray}
So far we have covered a decent amount of algebra, but still $t \in \mathbb{N}$. The equation $f^t(z)$, $t \in \mathbb{N}$ is important because it is convergent when $f(z)$ is convergent. A number of different attempts have been made to extend tetration to complex numbers, but have failed because they couldn't show convergence.
### Hyperbolic Fixed Points
When $\lambda$ is neither zero nor a root of unity $\lambda^t \neq 1, t \in \mathbb{N}$, then the nested summations simplify to
\begin{eqnarray} f^t(z)=f_0 &+& \lambda ^t (z-f_0)+\frac{\lambda ^{-1+t} \left(-1+\lambda ^t\right) f_2}{2 (-1+\lambda )} (z-f_0)^2 \\ & + & \frac{1}{6} \left(\frac{3 \lambda ^{-2+t} \left(-1+\lambda ^t\right) \left(-\lambda +\lambda ^t\right) f_2^2}{(-1+\lambda )^2 (1+\lambda )}+\frac{\lambda ^{-1+t} \left(-1+\lambda ^{2 t}\right) f_3}{-1+\lambda ^2}\right) (z-f_0)^3+\ldots \end{eqnarray}
#### Hyperbolic Tetration
Let $a_0$ be a limit point for $f(z)=a^z$, so that $a^{a_0}=a_0$. Also $a_1=\lambda$. This results in a definition for tetration of complex points for all except the set of points with rationally neutral fixed points. For the real numbers $a=e^{e^{-1}}\approx 1.44467, a=e^{-e}\approx 0.065988$ have rationally neutral fixed points while $a=1$ is a superattractor. All other real values of $a$ are defined by hyperbolic tetration.
\begin{eqnarray} {}^t a = a_o & + & \lambda ^t\left(1-a_o\right)+\frac{\lambda ^{-1+t} \left(-1+\lambda ^t\right) \text{Log}\left(a_o\right){}^2}{2 (-1+\lambda )}\left(1-a_o\right){}^2 \\ & + &\frac{1}{6}\text{ }\left(\frac{3 \lambda ^{-2+t} \left(-1+\lambda ^t\right) \left(-\lambda +\lambda ^t\right)\text{ }\text{Log}\left(a_o\right){}^4}{(-1+\lambda )^2 (1+\lambda )}+\frac{\lambda ^{-1+t} \left(-1+\lambda ^t\right) \left(1+\lambda ^t\right)\text{ }\text{Log}\left(a_o\right){}^3}{(-1+\lambda ) (1+\lambda )}\right)\left(1-a_o\right){}^3+\ldots \end{eqnarray}
##### An Example of Hyperbolic Tetration
Consider a question posed at the begining of this article: what is ${}^{.5}2$. The function $f(z)=2^z$ has an infinite number of fixed points that are nearly periodic, including fixed points at $a_0=0.824679+1.56743 i$, $a_0=3.51524 + 10.8801 i$, and $a_0=4.36143 + 20.0872 i$. The fractal below is the Julia set of the function $f(z)=2^z$ .
For limit point $a_0=0.824679+1.56743 i, {}^{.5}2=1.824-0.745596 i$, for limit point $a_0=3.51524 + 10.8801 i, {}^{.5}2=-171.818+199.332 i$, and for limit point $a_0=4.36143 + 20.0872 i, {}^{.5}2=-1178.18+1829.92 i$. Therefore ${}^{.5}2=\{\ldots,1.824-0.745596 i,-171.818+199.332 i,-1178.18+1829.92 i,\ldots\}$.
### Parabolic Neutral Fixed Points
$f^t(z)=z+\frac{1}{2}t f_2 (z-f_0)^2 +\frac{1}{12}(3(t^2-t)f_2^2+2tf_3)(z-f_0)^3+\ldots$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 3, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9953742027282715, "perplexity": 1018.2822353091858}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-26/segments/1498128319636.73/warc/CC-MAIN-20170622161445-20170622181445-00500.warc.gz"} |
https://www.physicsforums.com/threads/absolute-convergence-help.543375/ | # Absolute convergence help
1. Oct 23, 2011
### miglo
1. The problem statement, all variables and given/known data
$$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sqrt{n}+1}{n+1}$$
2. Relevant equations
absolute convergence test
3. The attempt at a solution
by book says that the series converges because $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges
but they dont show how the absolute value of the original series converges, and ive tried showing it myself but i keep getting divergence
i know that as n grows larger and larger the behavior of $$\frac{\sqrt{n}+1}{n+1}$$ is similar to that of $$\frac{\sqrt{n}}{n}$$ so i tried using limit comparison and direct comparison with $\frac{1}{n}$ but i keep getting divergence
i tried the integral test but i kept getting divergence also
ive been trying this for far too long so any help would be greatly appreciated
2. Oct 23, 2011
### LCKurtz
You are correct that the positive term series diverges.
3. Oct 23, 2011
### miglo
but my book says that the original series converges by the absolute convergence test
so wouldnt that mean that $$\sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1}$$ converges also? or is this an error in the book?
4. Oct 23, 2011
### LCKurtz
The series is not absolutely convergent. It may be convergent with the alternating signs in which case it would be called "conditionally convergent". (I didn't check that). But the positive term series you are asking about is definitely divergent. You know it is because you correctly checked it.
5. Oct 23, 2011
### miglo
well then ill just check to see if it convergences by the alternating series test
thanks a lot! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9797552227973938, "perplexity": 489.9967986480084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647649.70/warc/CC-MAIN-20180321121805-20180321141805-00615.warc.gz"} |
https://rank1neet.com/8-4-redox-reactions-and-electrode-processes/ | # 8.4 Redox Reactions And Electrode Processes
Zn(s) + Cu2+ (aq) → Zn2+ (aq) + Cu(s)
The experiment corresponding to the above reaction , can also be observed if zinc rod is dipped in copper sulphate solution.
The redox reaction takes place and during the reaction, zinc is oxidised to zinc ions and copper ions are reduced to metallic copper due to direct transfer of electrons from zinc to copper ion. During this reaction heat is also evolved. Now we modify the experiment in such a manner that for the same redox reaction transfer of electrons takes place indirectly.
This necessitates the separation of zinc metal from copper sulphate solution. We take copper sulphate solution in a beaker and put a copper strip or rod in it. We also take zinc sulphate solution in another beaker and put a zinc rod or strip in it.
Now reaction takes place in either of the beakers and at the interface of the metal and its salt solution in each beaker both the reduced and oxidized forms of the same species are present. These represent the species in the reduction and oxidation half reactions.
A redox couple is defined as having together the oxidised and reduced forms of a substance taking part in an oxidation or reduction half reaction.
This is represented by separating the oxidised form from the reduced form by a vertical line or a slash representing an interface (e.g. solid/solution). For example in this experiment the two redox couples are represented as Zn2+/Zn and Cu2+/Cu. In both cases, oxidised form is put before the reduced form.
Now we put the beaker containing copper sulphate solution and the beaker containing zinc sulphate solution side by side (Figure).
We connect solutions in two beakers by a salt bridge (a U-tube containing a solution of potassium chloride or ammonium nitrate usually solidified by boiling with agar agar and later cooling to a jelly like substance).
This provides an electric contact between the two solutions without allowing them to mix with each other. The zinc and copper rods are connected by a metallic wire with a provision for an ammeter and a switch. The set-up as shown in Figure is known as Daniell cell.
When the switch is in the off position, no reaction takes place in either of the beakers and no current flows through the metallic wire. As soon as the switch is in the on position, we make the following observations:
The transfer of electrons now does not take place directly from Zn to Cu2+ but through the metallic wire connecting the two rods as is apparent from the arrow which indicates the flow of current.
The electricity from solution in one beaker to solution in the other beaker flows by the migration of ions through the salt bridge. We know that the flow of current is possible only if there is a potential difference between the copper and zinc rods known as electrodes here.
The potential associated with each electrode is known as electrode potential. If the concentration of each species taking part in the electrode reaction is unity (if any gas appears in the electrode reaction, it is confined to 1 atmospheric pressure) and further the reaction is carried out at 298K, then the potential of each electrode is said to be the Standard Electrode Potential.
By convention, the standard electrode potential (E- of hydrogen electrode is 0.00 volts. The electrode potential value for each electrode process is a measure of the relative tendency of the active species in the process to remain in the oxidised/reduced form.
A negative E- means that the redox couple is a stronger reducing agent than the H+/H2 couple. A positive E- means that the redox couple is a weaker reducing agent than the H+/H2 couple.
The standard electrode potentials are very important and we can get a lot of other useful information from them. You will learn more about electrode reactions and cells in Class XII. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8109395503997803, "perplexity": 955.3319775792215}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600401641638.83/warc/CC-MAIN-20200929091913-20200929121913-00781.warc.gz"} |
https://socratic.org/questions/how-do-you-factor-w-2-4w-3 | Algebra
Questions Topics
# How do you factor w^2+4w+3?
Then teach the underlying concepts
Don't copy without citing sources
preview
?
#### Explanation
Explain in detail...
#### Explanation:
I want someone to double check my answer
1
Mar 7, 2018
$\left(w + 3\right) \left(w + 1\right)$
#### Explanation:
In this case, since $a = 1$ (in $a {x}^{2} + b x + c$), simply look for two numbers whose sum is $4$ and whose product is $3$. You can easily see that the two numbers are $1$ and $3$.
Then, just write it down: $\left(w + 3\right) \left(w + 1\right)$
Check: try expanding the brackets using FOIL.
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##### Impact of this question
16 views around the world | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 8, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8118396401405334, "perplexity": 4210.433982980205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257650262.65/warc/CC-MAIN-20180324112821-20180324132821-00440.warc.gz"} |
https://math.stackexchange.com/questions/39142/a-local-diffeomorphism-of-euclidean-space-that-is-not-a-diffeomorphism | A local diffeomorphism of Euclidean space that is not a diffeomorphism
Could someone give me an example of a local diffeomorphism from $\mathbb{R}^p$ to $\mathbb{R}^p$ (function of class say $C^k$ with an invertible differential map in each point) that is not a diffeomorphism..
in the real line (1 dim case) that would mean a function with a continuous non null derivative on an open $V$ of $\mathbb{R}$ that is not bijective which does not make sense thus any local diffeomorphism on the real line is a diffeo..
Could one give me a counterexample in a higher dimension?
• Take the complex exponential $e^z$ as a function from $R^2$ to $R^2$. For every horizontal strip [iy+ i(y+2Pi) ) (i.e., including iy, but not i(y+2Pi)) there is an inverse--a logz --, but there is no global inverse. – gary May 15 '11 at 1:37
• there is no inverse around zero – yoyo May 15 '11 at 2:27
• Correction: that should be the strip [iy, i(y+2Pi)) – gary May 15 '11 at 2:38
• @yoyo: I assume you mean the target zero , right? Otherwise, the inverse function guarantees the existence of a local diffeomorphism at each point, including 0 in the domain, since d/dz($e^z$)=1 at z=0; same for all other points' or using the $R^2$ version of the inverse function theorem. – gary May 15 '11 at 22:36
• You are wrong about the one dimensional case: the exponential function $\mathbb R\to \mathbb R: x\mapsto e^x$ is a non surjective local diffeomorphism. – Georges Elencwajg Sep 1 '14 at 20:09
Consider $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ defined by $f(x,y)=(e^x \cos y,e^x \sin y)$
$Df(x,y)$ is always invertible because $\det Df(x,y)=e^{2x}$ but clearly $f$ is not one to one. It is periodic with period $2\pi$.
In higher dimensions, one can use $f(x)=(e^{x_1}\cos x_2, e^{x_1}\sin x_2, x_3, \dots, x_n)$. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.840181827545166, "perplexity": 390.6688792781842}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027330962.67/warc/CC-MAIN-20190826022215-20190826044215-00275.warc.gz"} |
https://www.arxiv-vanity.com/papers/1408.5326/ | # Tracy-Widom asymptotics for a random polymer model with gamma-distributed weights
Neil O’Connell
University of Warwick
Janosch Ortmann
University of Toronto
###### Abstract
We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Seppäläinen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.
## 1 Introduction
Denote by the set of ‘paths’ of the form , where , as shown in Figure 1. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8180020451545715, "perplexity": 378.0126804218462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057225.57/warc/CC-MAIN-20210921161350-20210921191350-00529.warc.gz"} |
http://m-phi.blogspot.com/2013/07/the-newman-objection-to-ramsey-sentence.html | ## Wednesday, 3 July 2013
### The Newman Objection to Ramsey-Sentence Structuralism, II
I posted on this topic a couple of months ago. This is a quick follow up. The Newman Objection to Ramsey-sentence (= Carnapian) structuralism can be given quickly, if we first use two definitions and a lemma. Below, $L$ is an interpreted language with a well-defined O/T distinction; $\Theta$ in $L$ is a finitely axiomatized theory. $\Re(\Theta)$ is then the Ramsey-sentence of $\Theta$. (The details here can become very, very messy; but it simply muddies the water to go into all that.)
Definition 1. Ramsey-sentence structuralism (Carnap)
The synthetic content of $\Theta$ = $\Re(\Theta)$.
Definition 2. Constructive empiricism (van Fraassen)
The content of a theory $\Theta$ that we may accept is: $\Theta$ is empirically adequate.
Ramsey-sentence Lemma
$\Re(\Theta)$ is true iff $\Theta$ has an empirically correct model $\mathcal{M}$ satisfying a cardinality condition on the entities it thinks are unobservables.
Then the objection can be put like this:
Newman Objection
Ramsey-sentence structuralism $\approx$ Constructive empiricism
The argument for this is: let us suppose that Ramsey-sentence structuralism is true and Constructive empiricsm is true. Then, because of the Lemma, it follows that accepting $\Re(\Theta)$ $\approx$ accepting $\Theta$'s empirical adequacy (that is, the sole difference is the cardinality commitment). QED.
Like all philosophical arguments, the technical regimentation is not 100% tight, and many possible queries can arise (in my view, these are: the precise scheme of formalization for theories; the precise definition of "empirically adequate"; and the range of the second-order quantifiers). Even so, under various clarifications, precisifications, modifications of definitions, etc., the conclusion remains more or less invariant. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9361308813095093, "perplexity": 2746.025024635042}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593438.33/warc/CC-MAIN-20180722174538-20180722194538-00332.warc.gz"} |
http://math.stackexchange.com/questions/383293/how-to-interpret-rank-of-a-matrix-intuitively | # How to interpret “rank” of a matrix intuitively?
What is the physical interpretation of "rank" of a matrix ? Why is it called "rank" ?
-
The rank is the dimension of the image of the matrix. A 3x3 matrix with rank 2 sends all vectors in 3-dimensional space into a 2-dimensional subset of 3-dimensional space.
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Regarding the importance of the "rank" of a matrix, and help to understand what this is, intuitively, you'll find some wonderful answers here, to a previous post at math.se:
A wonderful website for origins of mathematical terms is found here:
Click on "[R]"...scroll down to:
"Rank (of a matrix or determinant)":
RANK (of a determinant or matrix) was coined by F. G. Frobenius, who used the German word Rang in his paper "Uber homogene totale Differentialgleichungen," J. reine angew. Math. Vol. 86 (1879) p.1. This is according to C. C. MacDuffee, The Theory of Matrices, Springer (1933). Frobenius was defining the rank of a determinant but the term travelled.
In English, rank (of a matrix) is found in the monograph "Quadratic forms and their classification by means of invariant factors", by T. J. Bromwich, Cambridge UP, 1906. This citation was provided by Rod Gow, who writes that it is possible that an earlier book c. 1900 by G. B. Mathews, a revision of R. F. Scott's 1880 book on determinants, contains the word.
Rank is also found in 1907 in Introduction to Higher Algebra by Maxime Bôcher: where it is defined in the same way as in Frobenius:
Definition 3. A matrix is said to be of rank r if it contains at least one r-rowed determinant which is not zero, while all determinants of order higher than r which the matrix may contain are zero. A matrix is said to be of rank 0 if all its elements are 0. ... For brevity, we shall speak also of the rank of a determinant, meaning thereby the rank of the matrix of the determinant.
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This is a nice explanation. :-) – Babak S. May 6 '13 at 14:51
@amWhy: I totally agree with Babak! +1 – Amzoti May 7 '13 at 0:27
Thanks, @Amzoti (and Babak!) – amWhy May 7 '13 at 0:28 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8109520077705383, "perplexity": 278.7953865131759}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783393442.26/warc/CC-MAIN-20160624154953-00009-ip-10-164-35-72.ec2.internal.warc.gz"} |
http://clay6.com/qa/34252/which-of-the-following-quantities-is-zero-on-an-average-for-the-molecules-o | Browse Questions
Which of the following quantities is zero on an average for the molecules of an ideal gas in equilibrium ?
$(a)\;kinetic\;energy\qquad(b)\;momentum\qquad(c)\;density\qquad(d)\;speed$
Explanation :
At equilibrium, the momentum is conserved and hence zero.
edited Aug 9, 2014 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.98504638671875, "perplexity": 975.538995948185}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988721405.66/warc/CC-MAIN-20161020183841-00305-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/question-why-this.100434/ | # Question: Why this ?
1. Nov 18, 2005
Hi
How are you friends ?
I want to know why when I let fall tow things that have a different mass, why they reach the ground in the same time?
2. Nov 18, 2005
### mcah5
Newtons F = m*a and the approximation F = m*g, so a = g
Although I guess if you really want to be anal, the heavier object hits very slightly first because the earth is accelerating up to the heavier object. So two earth sized masses would fall toward each other faster than a ping pong ball and the earth.
3. Nov 18, 2005
### vaishakh
The equation of gravitational law shows it
F = GMm/r^2 where F is force applied due to gravitation by earth, G is a constant, M is the mass of the earth, m is the mass of the thrown object and r is the radius of the earth and from the law distancebetween the centre of two objects.
F = ma
therefore a = F/m = which here is g.
thus g is equal for all objects. may it be anything. now what mcah5 explained is the gravitational force applied by object on earth. the heavy celestial bodies apply gavitational force on earth due to which the gravitational force on earth due to object becomes negligible.
4. Nov 18, 2005
### DaveC426913
OK, now that the 'how' has been answered with equations, perhaps we can answer the 'why'? i.e a more intuitive answer.
I respond to your question with another question:
Tell me why you think they wouldn't they hit the ground at the same time. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8816916942596436, "perplexity": 770.9733437912977}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718423.28/warc/CC-MAIN-20161020183838-00131-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://tex.stackexchange.com/tags/texlive/new | # Tag Info
1
You need to add the .cls to some folder that your tex distribution can find, for compiling. You can for example make a folder in your home directory and register it with tex. Then you can easily keep your folder of tex goodies when you wipe your machine. See this Q & A for details. Secondly, you need to make LyX aware of the package you added. See this ...
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There is no need for installing texlive as root. It makes sense to set the owner of the texlive to the user: sudo chown -R <username>.users /usr/local/texlive Then you can run all <command>-sys as a default user.
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It turns out to be easier than expected. While the documentation is out of date, the solution is easy. install texlive-lang-hebrew and texlive-lang-other install the culmus-latex as per instructions (I used version 0.7, e.g., http://sourceforge.net/projects/ivritex/files/culmus-latex/culmus-latex-0.7/culmus-latex-0.7-r1.tar.gz/download) After that ...
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I solved the problem by something very simple: In Options -> Configure Texmaker I typed the full path for pdflatex's and lualatex's and biber's binary. I got the full path by opening the terminal and typing "which [pdflatex|lualatex|biber]". It works now. However, I don't understand WHY providing the full path to Texmaker is necessary. In Terminal, I can ...
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You can use \hypersetup just before \tableofcontents within a group: \documentclass[12pt]{book} \usepackage{blindtext} \usepackage{xcolor} \definecolor{darkgreen}{RGB}{41,159,49} \usepackage[colorlinks=true, urlcolor=blue, linkcolor=darkgreen, citecolor=darkgreen]{hyperref} \begin{document} {\hypersetup{linkcolor =black} \tableofcontents } ...
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If you want to create a TeX Live package, you need to do the following steps. I assume here that you are not shipping any binaries, but only files in a the texmf hierarchy. Also assume that the package is called foo: prepare a TDS tree of the package, say in foo, that is, you have foo/tex, foo/fonts etc. write a tlpobj file in foo/tlpkg/tlpobj/foo.tlpobj ...
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The following compiles also fine in my Ubuntu machine: \includegraphics[width=\paperwidth, height=\headheight, baby=0.25]{heads/test} Seems that Fedora has a more up to date version of graphicx package (and/or of some other related package) that does not contain the bug. (Obviously, "baby" does not belong in the keyval options for includegraphics, but the ...
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If you can not remove this (rather pointless) definition as it is being auto-generated, put \newcommand\mathplus{+} in your document preamble so that \mathplus expands to +.
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Many places on the web linux users will find the advise: add blah blah to PATH in ~/.bashrc. In general this is not a good idea, because only when ~/.bashrc has been executed, is the PATH change visible to programmes. If you are a command line jockey (like me) and open every program through a terminal, you will not see the difference. However, if you ...
4
\rule has never been robust in LaTeX. You can use \protect\rule or you can use \usepackage{fixltx2e} which updates various things, including making \rule robust. Perhaps your previous document was using that package.
1
It looks like the .sty file is available here. http://people.hss.caltech.edu/~kcb/LaTeX.shtml If you run kpsewhich -var-value=TEXMFHOME In terminal, it should give you path to a folder called texmf/. If you put the .sty file in a subdirectory of that file such as texmf/tex/latex/commonstuff/ then you should be able to run texhash [path to texmf folder] ...
1
Try the command kpsewhich latex.ltx to locate your TeX installation. On my system it gives the location /usr/share/texlive/texmf-dist/tex/latex/base/latex.ltx
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You have most likely forgotten to enter the installation path into the file "/etc/environment" Enter the full path, which is most likely "/usr/local/texlive/2014/bin/x86_64-linux/" into the line "PATH:" Note: You need to edit the file with sudo rights and entries are separated with a colon ":"
2
Issue has been solved via update to package bigfoot. Thanks everyone!
1
I recently encountered a very similar error message in Windows 7 MiKTeX 2.9. The solution was to simply run udpmap.exe as administrator. The program is located in: C:\Program Files\MiKTeX 2.9\miktex\bin\x64\
2
What I can see from the terminal output is the following: No file MSP430.aux. This might be the first time you're compiling MSP430.tex or have erased MSP430.aux prior to compiling. This might be automated if you have \nofiles as part of your preamble. (/usr/share/texmf/tex/latex/tipa/t3cmr.fd) A font definition file loaded as a result of ...
0
Configure -> Manage Repositories and using a http-server solved my problem.
Top 50 recent answers are included | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9201475977897644, "perplexity": 3933.540910880378}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931008720.43/warc/CC-MAIN-20141125155648-00099-ip-10-235-23-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/the-soccer-ball-problem-in-relativity.44360/ | # The Soccer Ball Problem in Relativity
1. Sep 23, 2004
### marcus
This recent paper describes the socalled Soccerball problem in quantizing relativity.
Franz Hinterleitner
Canonical DSR
http://arxiv.org/gr-qc/0409087 [Broken]
It makes an attempt to solve the soccerball problem, also known as "the problem of macroscopic objects".
Here is how the problem originated.
Relativity is probably going to be quantized (there's been a fair amount of progress towards it and some interesting results about removing singularities.)
Nobody can say by what approach but ANY approach to quantum gravity, especially if it is quantizing spacetime geometry, is likely to have the Planck scale playing an essential role.
This probably means that the Planck length will have to look the same to all observers, or to a large class of observers. Not only the speed of light is invariant, in other words. Besides having an invariant speed we may also have to allow for another invariant quantity, an invariant length perhaps, or an invariant energy.
Energy and length invariants amount to much the same thing because of the relation of wavelength and frequency to energy.
So there have been appearing these various proposed multi-special relativity frameworks. And there's a widely shared expectation that whatever eventually turns out to be workable as a quantum theory of gravity is going to have some kind of DSR (double-invariant-scale SR) or multiple invariant scale SR as its flat limit.
that is the limiting case where matter is sparse enough and gravity weak enough so that space is not noticeably curved----the flat limit is our everyday reality.
So even the large distances that gammaray bursts travel to come to us are approximable not by the flat space of ordinary SR but more likely by the flat space of some DSR.
This, interestingly enough, appears to be testable!
But meanwhile there is a theoretical problem. When SR is modified to give it another invariant scale there turns out to be a limit on momentum, or atleast on momentum density The momentum limit is the Planck momentum and it is very reasonable when applied to microscopic particles.
But it would not do as a limit on the momentum of macroscopic objects. By kicking a soccer ball one can give it more than the planck momentum.
Hinterleitner has contrived to make the limit be one on how much momentum can be concentrated in a small space. So soccerballs, because by planck standards they are not very dense, can have all the momentum they want.
here is Hinterleitner abstract:
"For a certain example of a "doubly special relativity theory" the modified space-time Lorentz transformations are obtained from momentum space transformations by using canonical methods. In the sequel an energy-momentum dependent space-time metric is constructed, which is essentially invariant under the modified Lorentz transformations. By associating such a metric to every Planck volume in space and the energy-momentum contained in it, a solution of the problem of macroscopic bodies in doubly special relativity is suggested."
The Soccerball Problem was mentioned in several recent papers on multi-special relativity (by Smolin, Kowalski-Glikman, Livine, Girelli, Oriti and others). I first remember reading about it in a paper of Rovelli some time back, but dont remember the title.
Last edited by a moderator: May 1, 2017
2. Sep 23, 2004
### skywolf
what about black holes, i mean what would their density, would that mean that you cant give a black hole momentum? or am i completely misunderstanding this subject?
3. Sep 23, 2004
### marcus
the first reaction would be not to worry about black holes---at least not about macroscopic black holes, because
the role intended for DSR is that of a "flat limit" of quantum geometry.
DSR is just a variation on the Minkowski space of special relativity.
So it would not be used to model black holes, or any situation where there was perceptible curvature
but that only seems to dispose of the problem
the fact is I cannot resolve it at this point. Because it seems to me that this paper is envisaging at once a standard flat space, like that of SR, a Minkowski space slightly modified in how the Lorentz group works on it, and at the same time allowing for massive particles, perhaps even very small black holes, to exist in it. How could these exist in the space without deforming it.
So maybe this DSR is only good as an approximation for very mundane everyday situations where there are not high (Planckian) density objects around to make the geometry non-flat. Maybe it it only works as an approximation in very mundane circumstances where the angles of a triangle always add up to 180 degrees and all that. And it is only the formulas of this version of DSR that are set up so it looks like near-planckian densities could be plugged in.
I will look at the article some more. Have you tried it, skywolf? It seems
clearer than usual in its writing, so probably more help than I can be right now.
Similar Discussions: The Soccer Ball Problem in Relativity | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8228925466537476, "perplexity": 773.0342884461336}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818693240.90/warc/CC-MAIN-20170925182814-20170925202814-00039.warc.gz"} |
https://www.proofwiki.org/wiki/Definition:Unit_(One) | # Definition:Unit (One)
## Definition
A numerical quantity whose cardinality corresponds to the number $1$ (one) is called a unit.
In the words of Euclid:
An unit is that of which each of the things that exist is called one.
### One of Naturally Ordered Semigroup
Let $\struct {S, \circ, \preceq}$ be a naturally ordered semigroup.
Let $S^*$ be the zero complement of $S$.
By Zero Complement is Not Empty, $S^*$ is not empty.
Therefore, by Naturally Ordered Semigroup Axiom $\text {NO} 4$: Existence of Distinct Elements, $\struct {S^*, \circ, \preceq}$ has a smallest element for $\preceq$.
This smallest element is called one and denoted $1$.
## Also known as
In older writings, a unit is often rendered as an unit; the rules of grammar have since evolved. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9863449335098267, "perplexity": 1611.897998642425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652663016853.88/warc/CC-MAIN-20220528123744-20220528153744-00420.warc.gz"} |
https://www.intechopen.com/chapters/16724 | Open access peer-reviewed chapter
# Composite Materials under Extreme Radiation and Temperature Environments of the Next Generation Nuclear Reactors
By Nikolaos Simos
Submitted: November 14th 2010Reviewed: March 23rd 2011Published: July 20th 2011
DOI: 10.5772/21488
## 1. Introduction
In the nuclear energy renaissance, driven by fission reactor concepts utilizing very high temperatures and fast neutron spectra, materials with enhanced performance that exceeds are expected to play a central role. With the operating temperatures of the Generation III reactors bringing the classical reactor materials close to their performance limits there is an urgent need to develop and qualify new alloys and composites. Efforts have been focused on the intricate relations and the high demands placed on materials at the anticipated extreme states within the next generation fusion and fission reactors which combine high radiation fluxes, elevated temperatures and aggressive environments. While nuclear reactors have been in operation for several decades, the structural materials associated with the next generation options need to endure much higher temperatures (1200oC), higher neutron doses (tens of displacements per atom, dpa), and extremely corrosive environments, which are beyond the experience on materials accumulated to-date. The most important consideration is the performance and reliability of structural materials for both in-core and out-of-core functions.
While there exists a great body of nuclear materials research and operating experience/performance from fission reactors where epithermal and thermal neutrons interact with materials and alter their physio-mechanical properties, a process that is well understood by now, there are no operating or even experimental facilities that will facilitate the extreme conditions of flux and temperature anticipated and thus provide insights into the behaviour of these well understood materials. Materials, however, still need to be developed and their interaction and damage potential or lifetime to be quantified for the next generation nuclear energy. Based on material development advances, composites, and in particular ceramic composites, seem to inherently possess properties suitable for key functions within the operating envelope of both fission and fusion reactors. In advanced fission reactors composite materials are being designed in an effort to extend the life and improve the reliability of fuel rod cladding as well as structural materials. Composites are being considered for use as core internals in the next generation of gas-cooled reactors. Further, next-generation plasma-fusion reactors, such as the International Thermonuclear Experimental Reactor (ITER) will rely on the capabilities of advanced composites to safely withstand extremely high neutron fluxes while providing superior thermal shock resistance. In addition it will be required by the composite to possess and maintain under severe neutron irradiation extremely high thermal conductivity to enable the flow of the anticipated extreme thermal heat loads generated in the core. The first wall and blanket surrounding the core in the fusion reactor are the two elements where composites are considered leading candidates.
Composites of special interest to both fission and fusion next generation nuclear reactors are carbon-fiber (C/C) and silicon carbide fiber (SiCf/SiC), and more recently, C/SiC composites. These are continuous fiber-reinforced materials of either carbon or silicon carbide fibers infiltrated with a similar matrix. During the last two decades a number of studies have been conducted to address the feasibility and response of the two composites to different radiation environments of fission and fusion reactors and identify their limitations. While these composite structures have a significant advantage over materials used in the same reactor applications (i.e. nuclear graphite, BeO and metal alloys) because of the physical and mechanical properties they possess, they also experience limitations that require quantification. Carbon-fiber composites for example while they can have customized architecture to enhance desired properties, such as thermal conductivity, they too may experience anisotropic dimensional changes and be susceptible to irradiation-induced degradation. SiCf/SiC composites, on the other hand exhibit good fracture resistance and low induced activity due to the irradiation stability of the SiC crystal but their technology is less mature. Critical issues such as cost, fabrication and joining as well as uncertainties due to lack of experience data in performance/survivability and lifetime in the combined extremes of high temperature and high fast neutron fluxes require further evaluation to qualify and quantify their performance.
During the last three decades and driven primarily by the fusion reactor needs (i.e. ITER) a extensive array of neutron-irradiation experiments at high temperatures have been conducted using available test reactors while the technology in composites was maturing. Facilities such as the High Flux Isotope Reactor (HFIR) at Oak Ridge National Laboratory, Japan Materials Testing Reactor (JTMR) as well as test reactors in Europe have been used to irradiate newly developed alloys and composites to anticipated fluence levels in the next generation fusion and fission reactors. As mentioned above, the understanding of the behaviour of reactor materials through studies and experience from the operating low energy neutron reactors may not necessarily transfer to the new materials under a different neutron spectrum. While the peak of the energy spectrum in the fission reactors to-date is ~1 MeV, neutrons in fast-spectrum reactors of the next generation are of several MeV while neutrons from fusion reactions will have energies of ~14 MeV. Based purely on modelling it is anticipated that the induced damage in the microstructure of materials will be similar to the one corresponding to that induced by less than 1 MeV neutrons. While the assumption may be generally correct, there is greater uncertainty with composites which do not form a perfectly oriented structure and are subject to the effects of dissimilar response between the fibre structure and the matrix. In addition, interaction with higher energy neutrons will generate more hydrogen and helium (as a result of nuclear interactions and transmutation products). Helium trapped within basal graphite planes and irradiation-induced defects will form bubbles and degrade the microstructure of the constituents. Therefore understanding how these composite structures behave in the fast neutron environment and what the degradation rate of their key properties (thermal conductivity, expansion, strength, etc.) is important and assessment by means of extrapolation from available data will be risky.
To observe the effects of high energy irradiating particles on composite structures, irradiation studies have been launched using the Brookhaven National Laboratory (BNL) proton linear accelerator and the target station at the isotope production facility (BLIP). The tuneable, ~25 kW (~110 μA peak) accelerator can accelerate protons to energies up to 200 MeV. Irradiation damage studies on graphite, carbon composites and new alloys have been performed in different phases using the proton beam directly on the materials and reaching fluences of ~1021 protons/cm2. Results on the finding following proton irradiations on carbon composites, graphite and composite-like structures are presented in subsequent sections. In a different irradiation mode where the isotope targets completely absorb the 116 MeV protons, a mostly isotropic fast neutron is generated downstream of the isotope targets from the spallation process. This neutron spectrum with mean energy of ~9 MeV is utilized for the irradiation of composites and new alloys. While peak fluences are of the order of 1019 – 1020 neutron/cm2 for each yearly proton beam run (much lower than the ~1021 -1022 n/cm2) expected in the fusion reactor, still the interaction of these new composites with predominantly fast neutrons is expected to provide useful indicators of their stability and resilience to damage. Neutron irradiation studies at BNL BLIP have been completed recently for a number of super alloys and nano-structured coatings (Al2O3, TiO2) on various substrates. Nanostructured coatings, along with AlBeMet, an aluminium-beryllium metal matrix composite, and structures made of fusion-bonded dissimilar materials constitute a special class of composites which are discussed in a subsequent section.
## 2. Composites and extreme environments
The development of advanced materials to be used in next generation reactors (Zinkle, 2004) has been driven by the need to endure the extreme environment consisting of prolonged, highly damaging radiation fluxes (tens of dpa), extreme temperatures (above 1000º C and up to 1200º C) and high stress conditions, which together can push well-understood and widely used materials beyond their limit. In fusion and fission reactor applications there are certain key properties that the materials which are playing a pivotal role (i.e. first wall and blanket in a fusion reactor) must maintain. These include low activation, structural integrity, dimensional stability, thermal conductivity and inherent ability to absorb thermal shock.
### 2.1. C/C composites
Carbon fibre reinforced composites (Cf/C) are an attractive choice for use in the extreme environment of the next generation reactors because of some key properties they inherently possess, namely enhanced strength as compared to nuclear graphite, thermal shock resistance because of their unique structure, extremely low thermal expansion, enhanced thermal conductivity due to the presence and directionality of fibres and low neutron activation. Due to their attractive properties carbon-carbon composites have enjoyed widespread use in advanced technologies which have led to the maturity of their technology and fabrication. A wide variety of architectures of the fibre/matrix have been developed as well as fabrication techniques. Most widely used architectures are the two-dimensional (2D Cf/C) and three-dimensions (3D Cf/C) forms. Shown in Figures 1 and 2 are sections of the three-dimensional architecture of the composite (FSI 3D Cf/C) indicating the orderly fibre bundle (thickness of ~265μm) and matrix arrangement.
While carbon composites exhibit enhanced properties when compared to graphite, radiation-induced damage from neutrons or other energetic particles such as protons is far less well understood. To the contrary, nuclear graphite has been extensively studied for radiation-induced degradation for almost sixty (60) years and so the degradation of the key properties as a function of the neutron fluence such as thermal conductivity, dimensional stability and strength has been established thus leading to limitation thresholds for its use in more extreme environments (Gittus, 1975; Maruyama & Harayama 1992; Nikolaenko et al. 1999). Key findings from these studies on graphite are the anisotropic dimensional changes that take place at higher radiation doses and most importantly the degradation of the thermal conductivity. Within the last two decades a body of experimental research work on irradiation damage of carbon-carbon composites has been reported prompted primarily by the need to identify higher performance, low neutron activation for the first wall of fusion reactors such as the International Thermonuclear Experimental Reactor (ITER). Of primary interest in these reported studies (Burchell, 1992, 1994; Burchell et al. 1996; Barabash, et al., 1998) are neutron irradiation induced dimensional changes, thermal conductivity and mechanical properties.
#### 2.1.1. Shock resistance
One of the important attributes of C/C composites in the fusion reactor environment is their inherent ability to absorb thermal shock. In an effort to quantify the ability of the C/C composite to absorb thermal shock and so be used as the material of choice in a number of high power accelerator applications including accelerator targets for the Long Baseline Neutrino Experiment, Neutrino Factory (LBNE), beam collimating elements for the Large Hadron Collider (LHC) or energetic beam absorbers, experiments have been performed using intense pulses of energetic protons. In these experiments (Simos et al., 2005) performed using the 24 GeV proton beam at the Accelerating Gradient Synchrotron (AGS) at BNL the shock performance of FMI 3D C/C composite targets (16-cm long, 1-cm diameter rods) was measured and compared to that of ATJ graphite. Shown in Figure 4 is the shock test arrangement where 3D C/C composite and ATG graphite targets are instrumented with fibre-optic strain gauges mounted on the surface of the target and measuring extremely fast axial strain transients in the target resulting from its interaction with the 24 GeV proton beam. The response from the intense (3.0 e+12 protons) and focussed (0.3mm x 0.7mm) proton pulses on the ATJ graphite and 3D C/C targets is shown in Figure 5 where the two are compared. It is evident from the comparison that the carbon-carbon composite shows a much lower response to the shock induced by the beam while radial reverberations indicated by the high frequency cycles within each axial cycle are damped out as a result of the impedance interfaces (fibre/matrix) and the voids that are present as shown in Figure 3.
Due the potential implications and applications of high shock resistance in the carbon-carbon composites which stem from the “effective” low thermal expansion coefficient specific studies (Hereil, 1997) have focussed on experimentally verifying the compressive wave velocities in a plate-impact configuration by wave decomposition. As observed in (Simos et al., 2006; Hereil et al, 1997) the problem of shock in materials such as C/C remains very complex due to the anisotropy and the fibre-matrix interfaces as well as the response to dynamic loads of wave propagation in the individual components. In such materials, understanding the behaviour at the mesoscale is important for modelling and implementation of these composites in large-scale designs.
For high power accelerators, while the thermal shock resistance is superior to graphite and so is the retention of thermal conductivity, degradation as a result of energetic proton irradiation can be a serious limiting factor along with the dimensional stability required of critical elements such as primary beam intercepting accelerator target and collimators. Irradiation damage studies have been conducted in recent years ( Simos et al, 2006a , 2006b, 2008) using the BNL 200 MeV Linac beam at the isotope production facility (BLIP). The main objective was to assess the proton-induced damage at energies higher than the thermal and fast neutrons these composite structures have been exposed in test reactors like HFIR and JMTR and qualify the differences stemming from the irradiating species (protons vs. neutrons) and energies (neutron energies a few MEV and proton energies up to 200 MeV).
The dimensional changes of irradiated 3D C/C composite are shown in Figure 11. For the 3D architecture the material exhibits a negative CTE in all directions. As seen in Figure 11 for the un-irradiated samples, there is accelerated shrinkage >600º C attributed to the influence of the matrix within the fibre structure. Reversal from shrinkage to growth at these temperatures was observed in previous studies (Burchell, 1994). Measurements of thermal conductivity of the irradiated 3D C/C and IG-43 graphite samples following proton irradiation revealed that thermal conductivity reduced by a factor of three (3) for the 3D C/C for 0.25 dpa fluence and by a factor of six (6) for IG-43 and similar fluence. To address the significantly higher irradiation damage from energetic protons a new irradiation experiment has been initiated where the carbon composite will be exposed to a neutron flux at the BNL BLIP facility which results from the spallation of protons with upstream isotope targets. The goal is to compare the damage at similar fluences of energetic protons and energetic neutrons.
### 2.2. SiCf/SiC
Silicon carbide fibre reinforced composites (SiCf/SiC), along with the C/C composites, considered as prime candidates for the first wall and blanket structural material in fusion reactors. Due to their low activation and irradiation stability these composites have a clear advantage over the C/C composites especially when the neutron doses are expected to be high as it is the case of fusion reactors where tens of dpa over the lifetime will be accumulated. Its inherent stability stems from the isotropic dimensional change of the cubic SiC crystal (Bonal et al. 2009) which tends to saturate at modest irradiation levels. It also exhibits good fracture resistance and excellent mechanical properties at high temperatures. While the carbon composite technology and manufacturing is more mature than that of the SiC composites, which currently have limited structural applicability outside nuclear reactors. Significant progress has been made in recent years to both eliminate issues of early grades of the composite associated with poor irradiation performance (Snead et al., 1992) and reduce cost through adoption of novel fabrication techniques (Katoh et al., 2010a) such as the nano-infiltrated transient-eutectic (NITE) process (Bonal et al., 2009). For nuclear grade SiCf/SiC composites the costly chemical vapour infiltration technique (CVI) is used.
An experimental study on the effects of high temperatures and proton or fast neutron irradiation has been initiated at Brookhaven National Laboratory. The goal is to subject SiCf/SiC composite to similar irradiation fields that the carbon composites have been exposed to and make comparison in the irradiation-induced damage. As discussed in the previous section, significant damage was observed in C/C composites at levels far below the observed limits in neutron irradiation environments. Shown in Figure 12 is an optical image of SiCf/SiC composite section showing the fibre bundle thickness (~128μm) and the SiC matrix thickness (~343μm). Also shown in Figure 12 is the distribution of voids within the architecture. Figures 13 and 14 are SEM images of the fibre/matrix interfaces and of the individual fibres.
To assess the effect of high temperature on the SiCf/SiC in terms of dimensional changes, structure and density composite samples were brought to 1000º C in atmosphere and the changes were made with precise instruments at the BNL isotope facility. Dimensional changes were more pronounced along the fibres (shrinkage of ~1%) while in the direction normal to the fibres they were of the order of 0.09%. Density reduction of ~0.8% was also observed (ρrt = 2.4324 g/cc) following the annealing of the sample for one hr at 1000º C. Shown if Figure 17 is dimensional changes obtained for temperatures up to 610º C and are compared with those of 3D C/C. As seen in Figure 17 during the first thermal cycle there is an adjustment in the both structures except more pronounced in SiCf/SiC which expands with increasing temperature in contrast to C/C which shrinks for the selected temperature range. Based on the stabilized thermal expansion, the thermal expansion coefficient (CTE) in the range of 200-600º C was estimated as 3.7 10-6/K. Experimental results (Zhang, 2006) on carbon fibre reinforced SiC (via CVI method) up to 1400º C showed values similar average values in the 200-600º C but with dramatic fluctuations above 800º C. Following the planned proton irradiation of the SiCf/SiC and the fast neutron irradiation using the spallation process at the isotope production facility at BNL the effects on the physio-mechanical properties will be studied and compared with the C/C composites.
### 2.3. Composite-like structures
A number of material structures not adhering to the classical definition of composites involving a matrix with fibre-reinforcement, i.e. C/C, C/SiC, SiCf/SiC, etc., can still be considered composites, or more appropriately composite-like with potential applications in the next generation fusion and fission nuclear reactors. These can be based on (a) the embedment of particles of one material into the lattice of another thus maintaining the individual characteristics, (b) the bonding of dissimilar materials using solid state reaction of chemical vapour deposition with the help of an interface layer, and (c) on deposition of nano-structured coatings on substrates to either enhance the properties of the combined structure or protect the substrate. Because of their potential for use in nuclear reactors, some of these composite-like structures have been studied for radiation damage and extreme temperatures.
#### 2.3.1. AlBeMet
AlBeMet, while by metallurgical definition may be considered an alloy, is in fact a composite of aluminium and beryllium consisting typically of ~62% commercially pure beryllium and 38% of commercially pure aluminium by weight. The two metals involved in forming AlBeMet do not fully mix but instead the beryllium particles are embedded in a pure aluminium lattice. The powder is produced by a gas atomization process yielding a fine beryllium structure. The two granular forms are mixed at temperatures just below the melting points of the two metals and a pressure that prompt the particles to form a stable bond. The result is a non-typical composite with some very appealing thermo-physical properties since it combines the workability of aluminium and, for the most part, the hardness of beryllium. Interest in this special composite has been increased in recent years primarily for use in special components of particle accelerator systems and in particular in the accelerator target envelope characterized by high-radiation, high temperature and thermal shock conditions. The combination of low-Z, good thermal conductivity (210 W/m-K) and low electrical resistivity (3.5e-6 Ohm-cm), combined with its workability and hardness, make it very attractive for special components such as magnetic horns and targets. Unknown was its radiation resistance and dimensional stability which are key parameters for potential applications in nuclear reactor systems. The effects of radiation on the physical and mechanical properties of this unique composite have been studied using direct energetic protons and secondary fast neutrons of the BNL accelerator complex through a series of irradiation experiments. Peak proton fluences of 3.0 1020 p/cm2 at 140 MeV using the 200 MeV proton beam at BNL BLIP and, through a different study, fast neutron fluences of ~1019 n/cm2 were achieved in these accelerator-based irradiation experiments. The arrangement of specially designed test samples of beryllium (similar to AlBeMet samples) in the irradiation space intercepting the proton beam is shown in Figure 18. The numbered tensile specimens (dog-bone) are 42mm long and 1.5mm thick and have a strain gauge length of 6mm. The matching specimens are used for post-irradiation analysis of thermal expansion, electrical and thermal conductivity.
Post-irradiation studies revealed that AlBeMet is dimensionally stable following irradiation and that it resists embrittlement and degradation even at high proton fluences where materials such as graphite and carbon composites have shown to undergo serious degradation. As indicated above, the proton fluences received may be representative of a much more severe irradiation condition when correlated to either thermal or fast neutrons. Depicted in Figure 19 is the thermal expansion of both AlBeMet and beryllium for peak fluence of 1.2 1020 p/cm2 and the measured CTE of AlBeMet as a function of average proton fluence. The effects of proton irradiation on the stress-strain relation of AlBeMet and its comparison with beryllium are shown in Figure 20. While it is confirmed that the ultimate tensile strength of beryllium is higher than that of AlBeMet, the AlBeMet appears to increase its strength following irradiation (as expected in all metals due to the pinning of dislocations) but without loss of the ductility anticipated to accompany the induced hardening. Further tests are planned where AlBeMet will be exposed to higher proton and accelerator-produced fast neutron fluences to explore its mechanical behaviour.
#### 2.3.2. Bonded dissimilar materials
Bonding of dissimilar materials to create a “composite-like” structure and ensuring its ability to maintain the integrity of the interfaces under extreme temperature conditions and high radiation fluences is an important challenge. Applications of such composites can be seen in nuclear fuel elements, fusion reactor plasma facing components where high-Z materials such as tungsten with higher erosion resistance will protect low Z materials like carbon or beryllium and particle accelerator targets where the variation of the material atomic number Z from the centre of the intercepted beam is important for both particle-mass interaction and heat removal from the hot central part ( Simos et al., 2006b ). In all applications and due to the dissimilarity of the thermal expansion in the bonded material structure, high stresses can develop at the interfaces leading to micro-cracking or even separation. Such condition can dramatically reduce a key property of the composite layer that controls the primary function such as heat transfer across the interfaces through thermal conduction. Shown in Figure 21 is a schematic of a TRISO-coated particle and pebble bed fuel sphere for Generation-IV Very High Temperature Reactor (VHTR). Maintaining the integrity of the interfaces between the various layers around the fuel kernel under high temperatures and extreme radiation fluxes is crucial. The dimensional changes occurring as a result of the two simultaneous effects do not necessarily coincide in terms of direction (growth or shrinkage) and thus a better understanding of the interface mechanics under such conditions is required.
There is a significant need in the nuclear industry and in particular in the first wall of fusion reactors, of graphite/metals bonding to form a coating or cladding on the low Z materials (i.e. graphite) and reduce the erosion rate. Direct graphite-metal junctions (Brossa et al., 1992) for use in the first wall of fusion reactors are extremely sensitive to thermal cycling due to differences in thermal expansion so techniques have been developed and applied with the introduction of interface agents (such as silver) that will prevent the formation of fragile components and metal dissolution. Thus brazing between stainless steel and graphite, Mo and graphite as well as W and graphite has been produced using vacuum plasma spray (VPS) and chemical vapour deposition (CVD) techniques. To study the resilience to thermal shock, graphite/Mo or graphite/W bonding was achieved by using an intermediate layer of Mo, V, or Mo-Ti and applying a solid state reaction bonding technique (Fukatomi et al., 1985).
To assess the effect of proton irradiation bombarding the muon target at J-PARC facility where a 3 GeV, 333 μA proton beam is intercepted by a graphite target at a rate of 25 Hz an experimental study was initiated at Brookhaven National Laboratory. The study consisted of an irradiation phase using the 200 MeV proton beam of the BNL Linac and of a post-irradiation analysis to observe the degradation of the target-like composite structure that was made for the study. Shown in Figure 22 is a test specimen consisting of three materials (copper and titanium alloy Ti-6Al-4V) and graphite (IG-43) and two interfaces (graphite to titanium and copper to titanium alloy). To form the two interfaces the silver brazing technique in vacuum was applied. Two types of geometry in the 42mm long (4mm x 4mm cross sectional area) specimens was used, one at 45º (as shown) and one with normal or 900 interfaces. Shown in the SEM image of Figure 22 and prior to irradiation is the achieved bonding/interface (extremely faint) between copper and titanium alloy for the 900 interface.
The study demonstrated the serious effects that energetic protons at fluences above ~5.0 1020 protons/cm2 have on graphite and thus its interface or bonding with metals. In the fusion reactor, however, composite structures that involve graphite with metal cladding or coating will be exposed to higher fluences of 14 MeV fast neutrons. While for such neutron energies it is anticipated that the cross section of nuclear interaction is similar to that of the 200 MeV protons, confirmatory investigations are necessary. It should be emphasized that nuclear graphite exposed to higher fluences of thermal neutrons (< 1 MeV) than the ones achieved with the 200 MeV protons has shown much greater resilience to irradiation damage. Therefore, to address the potentially different response of such graphite to metal bonding when fast neutrons are the irradiating species (as in the fusion reactor) a radiation damage experiment using the spallation-produced fast neutrons at the BNL isotope facility has been launched. During this irradiation phase fast neutron fluences of ~2.0e+19 n/cm2 and dominated by energies between 1 MeV and 30 MeV will be achieved. While these levels are far below the fusion reactor fluences anticipated for the graphite/metal composite structures, a comparison between proton and neutron irradiation at similar exposure levels can be made.
#### 2.3.3. Nanostructured coatings
While nano-structured coatings on metal substrates form a unique class of materials with a wide range of applications, the combined coating/substrate structure can be also characterized as a composite. These structures exhibit similar behaviour at the interface between the substrate and the coating as fibre-reinforced matrices stemming from the mismatch of thermal expansion coefficient which leads to elevated interface stress fields at high temperatures. With recent advances in the techniques and application of nanocoatings on base materials such as thermal spray deposition (Tsakalakos, 2009), interest has increased for their potential use in nuclear reactor systems and in particular in plasma-facing components of fusion reactors.
In fusion reactor environments where the low Z materials of the plasma-facing wall (carbon or beryllium) require protection from erosion, coatings based on tungsten and its alloys have been explored (Koch, 2007). In such setting, however, extreme radiation damage presents an additional challenge for these relatively untested structures which alters the physio-mechanics of the interaction between the substrate and the coating due to the fact that the rate of change of the thermal expansion (CTE) as a function of the radiation fluence may differ significantly between the distinct materials. Therefore, the potential for micro-cracking and even separation between the substrate and the coating under a combination of extreme temperatures and radiation fluxes requires experimental investigation.
Experimental studies focussing on the radiation and extreme temperature effects on alumina (Al2O3) and titania (TiO2) nano-structured coatings applied on Ti-6Al-4V and 4130 steel alloy substrates were launched to assess their susceptibility. Specifically, 200 μm-thick coatings consisting of 87% Al2O3 and 13% TiO2 (grid-blasted) and 600 μm-thick Al2O3 (thermally sprayed) on Ti-6Al-4V substrates were used along with 600μm-thick Al2O3 and 600 μm-thick amorphous Fe coating on alloy steel 4130 substrates. The behaviour of the interface of the between the substrate and the coating was evaluated for temperatures reaching 1200oC. In addition, the radiation damage from the spallation-based radiation field at BNL BLIP using 116 MeV protons. During irradiation, nano-coated samples received a neutron fluence of ~2.0e+19 n/cm2 with mean energy of 9 MeV. Combined with the neutron fluence, the coatings received a secondary proton fluence of ~3.2e+15 p/cm2 of 23 MeV mean energy, a photon fluence of ~3.0 e+19 γ/cm2 of 1 MeV mean energy and ~2.4e+16 e/cm2 of 1 MeV mean energy. Of primary interest was the effect of irradiation on the thermo-mechanical behaviour of the structures. Shown in Figure 25 are changes that occur at the interface of 600μm-thick Al2O3 on Ti-6Al-4V substrate from room temperature, to 900º C and 1200º C. Demonstrated is the resilience of the composite structure despite the high interface stresses which result in shear failure planes in the substrate (middle image). At higher temperatures (1200º C) the substrate material begins to re-arrange across the shear failure plane. The effect of extreme temperatures on the 600μm-thick Al2O3 layer deposited on alloy steel 4130 substrate is quite different as shown in Figure 26. At elevated temperatures (>600º C) an inter-metallic layer begins to form at the interface eventually leading to complete separation of the nano-structured alumina layer at 1200º C.
The effects of irradiation on the thermal expansion of the coated samples were studied and are depicted in Figures 27 and 28 where they are compared with their un-irradiated counterparts. Observed in the un-irradiated case is that the coating and substrate adjust at certain temperatures to accommodate for the dissimilarities in thermal expansion coefficient. The adjustment which first occurs at a low temperature (~150º C) re-occurs at a higher temperature in a subsequent thermal cycle. This behaviour in which the two dissimilar materials adjust at a particular temperature is very similar to what has been observed near phase transitions in bcc metals such as tungsten. The radiation effects on the stress-strain behaviour based on specially-designed and irradiated specimens with the fluences mentioned above are currently being evaluated.
## 3. Conclusion
With the development of the new generation composites such as C/C, and SiC/SiC as well as special bonds and coatings rapidly advancing and, in the process, performance in extreme environments is better understood and quantified, there is high degree of confidence that these material structures will be able to support the needs of the next generation reactors. A multitude of efforts world-wide have been aiding in closing the knowledge gap on these very promising materials during the last three decades while, with the adoption of novel processing techniques, have made their fabrication at a large-scale feasible. However, due to the harshness of the nuclear environment of the future reactors, consisting of a combination of extremes in temperature and radiation flux, further work is necessary to qualify these composites since the available data are the results of small-scale experimental efforts. The extensive experience on the constituents of these composites from fission reactors may not necessarily provide a good basis to assess the performance of the integrated composite in the elevated nuclear environments. As some of the irradiation experiments using more energetic particles than the thermal neutrons from fission reactors on materials with well understood behaviour (i.e. graphite) showed is that irradiation-induced damage may occur at a faster rate at much lower thresholds. This emphasizes the need to understand and quantify the performance of both the constituents and the final composite structure under prolonged exposure to higher energy neutrons that make up the flux in the next generation fusion and fission reactors.
Irradiation damage studies to-date focussing on the next generation composites and using the available facilities have shown that it is feasible with these new material forms to achieve the performance required through extrapolation. However, the actual conditions in the fusion and next generation fission reactors are expected to be more severe in terms of flux, fluence and temperature. These may result in a much greater spectrum of changes in the physio-mechanical properties of these materials especially in hydrogen and helium formation. The knowledge of the behaviour of these promising composite materials at these levels, either extrapolated or acquired through tests simulating the anticipated conditions, will still be at a small scale. For application in the large-scale of the fusion or fission reactor environment the small-scale must be extrapolated to the realistic size of the components. Therefore a better handle of the scaling must be achieved with the development and implementation of numerical codes. Because of the variability within their structure the numerical models need to consider spatial variation of the properties. Code benchmarking efforts focussed in the specifically on the prediction of the response of these composites will be necessary. Important attributes that make these composites attractive, such as shock resistance, need special attention and further experimental work due to the enormous complexity of the problem associated with fibre-reinforced composites. The effect of irradiation on the degradation of the physical and mechanical properties that control the response to shock absorption, for example, down to the interface between the fibre and the matrix need to be understood so the performance of the bulk composite can be assessed.
## Acknowledgments
The author gratefully acknowledges the input of Dr. H. Ludewig and Dr. H. Kirk for discussions on the subject and for reviewing and commenting on the manuscript, Dr. L. Snead for providing SiC/SiC samples for the study and Prof. T. Tsakalakos for providing the nanostructured coating samples and for discussions on the subject. The help of A. Kandasamy, Dr. A. Stein and Dr. J. Warren for facilitating the optical microscopy and SEM images is much appreciated and acknowledged.
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Nikolaos Simos (July 20th 2011). Composite Materials under Extreme Radiation and Temperature Environments of the Next Generation Nuclear Reactors, Metal, Ceramic and Polymeric Composites for Various Uses, John Cuppoletti, IntechOpen, DOI: 10.5772/21488. Available from:
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https://physics.stackexchange.com/questions/270535/how-to-explain-implicit-time-dependence-to-someone?noredirect=1 | # How to explain implicit time dependence to someone? [duplicate]
I am trying to explain what implicit time dependence is and how it differs from explicit time dependence, but I'm unsure how "sound" my explanation is. Here is what I said:
Suppose I have a function $T(x,y,z,t)$ which is a temperature field of a room. If $T$ has explicit time-dependence, then even though I may stay at a fixed point, $(x,y,z)$, in the room, the value of $T$ at that point can change over time, the temperature may increase or decrease over time. Now, suppose for simplicity that the temperature at each point in the room doesn't change in time, i.e. $T$ has no explicit time dependence such that $T=T(x,y,z)$.
Then, suppose that I start walking around the room, now the temperature at each point remains fixed, but now my position $(x(t),y(t),z(t))$ is time dependent, i.e. it changes in time due to me walking around the room. As such, the temperature function $T$, that I use to measure the temperature at each point, is said to have implicit time dependence since its value changes over time, not due to the value at each given point changing over time, but because the position, $(x(t),y(t),z(t))$, at which it is evaluated, is changing over time.
Hence, for explicit time dependence, we have that $\frac{\partial T}{\partial t}\neq 0$, since even if my position remains fixed, the temperature at that point is changing over time. For implicit time dependence (and no explicit time dependence), we have that $\frac{\partial T}{\partial t}= 0$, since the value of the temperature at each point is fixed in time, however, $\frac{dT}{dt}\neq 0$, since my position is changing over time and so the temperature will implicitly change over time due to my movement around the room.
Would this be a correct explanation at all, or would something else be better?
• – ACuriousMind Jul 29 '16 at 10:59
• @ACuriousMind I read through this post before writing my own. I wasn't sure whether I fully understood the accepted answer given, so I wanted to check that my intuitive understanding is correct, if I were to explain it to someone?! – user35305 Jul 29 '16 at 11:08
• Also related: physics.stackexchange.com/q/264463/2451 – Qmechanic Jul 29 '16 at 14:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8724548816680908, "perplexity": 188.45633108372755}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-34/segments/1596439737168.97/warc/CC-MAIN-20200810175614-20200810205614-00508.warc.gz"} |
http://mathhelpforum.com/calculus/38291-solved-couple-double-integrals.html | Math Help - [SOLVED] A couple of double integrals
1. [SOLVED] A couple of double integrals
EDIT: Solved all of them.
I guess I don't know how to solve any of the following, because I keep getting the wrong answer all the time. It's mostly the boundaries that I'm having trouble with.
1) $\int{\int_D{x^3}}dxdy$ where $D= \left\{(x,y)\in R^2; 1\leq x^2 + 9y^2 \leq 9; x \geq 3y\right\}$
2) $\int{\int_D{x}}dxdy$ where $D= \left\{(x,y)\in R^2; 1\leq x^2 + 2xy + 4y^2 \leq 1; x \geq 0; y \geq 0\right\}$
3) $\int{\int_D{\frac{dxdy}{(1+x^2-y^2)^2}}}dxdy$ where $D$ is the triangle with corners in $(0,0), (1, -1)$ and $(3,1)$.
So here's what I tried to do:
1)
$\left\{\begin{array}{l}u= x\\v = 3y\end{array}\right.$
$|J(u,v)| = \frac{1}{3}$
$1 \leq u^2 + v^2 \leq 9, u \geq v$
$\left\{\begin{array}{c}u= rcos\varphi\\v = rsin\varphi\end{array}\right.$
$|J(r, \varphi)| = r$
So the limits become $1 < r < 3$ and $\frac{-3\pi}{4}< \varphi < \frac{\pi}{4}$
2. Originally Posted by Spec
I guess I don't know how to solve any of the following, because I keep getting the wrong answer all the time. It's mostly the boundaries that I'm having trouble with.
1) $\int{\int_D{x^3}}dxdy$ where $D= \left\{(x,y)\in R^2; 1\leq x^2 + 9y^2 \leq 9; x \geq 3y\right\}$
[snip]
So here's what I tried to do:
1)
$\left\{\begin{array}{l}u= x\\v = \frac{y}{3}\end{array}\right.$
$|J(x,y)| = \frac{1}{3}$
Boundaries?
$1 \leq u^2 + v^2 \leq 9$
$\left\{\begin{array}{c}u= rcos\varphi\\v = rsin\varphi\end{array}\right.$
$|J(r, \varphi)| = r$
Boundaries?
$\int{\int_D{r^3cos^3\varphi}}\cdot |J(u,v)| dxdy = \int{\int_D{\frac{r^4cos^3\varphi}{3}}}dxdy$
[snip]
I have time for one:
After your transformation to uv-coordinates (which was a good move by the way), you have $\frac{1}{3} \int \int_{D'} u^3 \, du \, dv$ where $D' = \left\{(u ,v) \in R^2; 1 \leq u^2 + v^2 \leq 9; u \geq 9v \Rightarrow v \leq \frac{u}{9}\right\}$.
Drawing a diagram that shows D' will be a big help for you.
Switch to polar coordinates and you have $\frac{1}{3} \int_{r = 1}^{r = 3} \int_{\varphi = \alpha}^{\varphi = \beta} r^4 \cos^3\varphi \, d\varphi \, dr$ where $\alpha = \tan^{-1} \left( \frac{1}{9} \right) - \pi$ and $\beta = \tan^{-1} \left( \frac{1}{9} \right)$.
3. I'm not getting the right answer by using that. It's supposed to be $\frac{121\sqrt{2}}{9}$, but I'm getting something way more complicated.
I would appreciate some help with the other integrals as well.
4. Originally Posted by Spec
I'm not getting the right answer by using that. It's supposed to be $\frac{121\sqrt{2}}{9}$, but I'm getting something way more complicated.
I would appreciate some help with the other integrals as well.
There's an error with the Jacobian. It's meant to be $\left| \begin{array}{cc}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} \right|$
$= \left| \begin{array}{cc}
1 & 0 \\
0 & 3 \end{array} \right| = 3$
.
I can't see any other error. But I get $\frac{5929 \sqrt{82}}{1681}$.
Maybe someone else can see what I can't (if there's anything to see).
If I have time I'll take a look at the others. I would have thought that the second one is similar to the first in many ways ......
5. Originally Posted by Spec
[snip]
3) $\int{\int_D{\frac{dxdy}{(1+x^2-y^2)^2}}}dxdy$ where $D$ is the triangle with corners in $(0,0), (1, -1)$ and $(3,1)$.
[snip]
The triangular region D is bounded by the lines $y = -x, ~ y = \frac{x}{3}\,$ and $y = x - 2$.
Note that $1 + x^2 - y^2 = 1 + (x - y)(x + y)$. This suggests making the transformation
u = x - y .... (1)
v = x + y .... (2)
(1) + (2): $x = \frac{u+v}{2}$.
(2) - (1): $y = \frac{-u+v}{2}$.
The Jacobian of the transformation is therefore $J(u, v) = \left| \begin{array}{cc}
\frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\
& \\
\frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{array} \right|$
$= \left| \begin{array}{cc}
\frac{1}{2} & \frac{1}{2} \\
& \\
-\frac{1}{2} & \frac{1}{2} \end{array} \right| = \frac{1}{2}$
.
The triangular region D transforms to a simple triangular region D' bounded by the following lines:
$y = - x \Rightarrow x + y = 0 \Rightarrow {\color{red}v = 0}$.
$y = x - 2 \Rightarrow x - y = 2 \Rightarrow {\color{red}u = 2}$.
$y = \frac{x}{3} \Rightarrow \frac{-u + v}{2} = \frac{u + v}{6} \Rightarrow {\color{red}v = 2u}$.
So the integral becomes $\frac{1}{2} \int_{u=0}^{u=2} \int_{v = 0}^{v = 2u} \frac{1}{(1 + uv)^2} \, dv \, du$ and it should be blue sky from here.
6. Originally Posted by mr fantastic
$\alpha = \tan^{-1} \left( \frac{1}{9} \right) - \pi$
$\beta = \tan^{-1} \left( \frac{1}{9} \right)$
Why are they $\tan^{-1} \left( \frac{1}{9} \right)$ ?
7. Originally Posted by wingless
Why are they $\tan^{-1} \left( \frac{1}{9} \right)$ ?
In transforming from xy-coordinates to uv-coordinates, the line x = 3y becomes u = 3(3v) = 9v => v = u/9, which has a gradient of 1/9 ......
The transformation is done to facilitate a switch to polar coordinates.
8. Originally Posted by Spec
I'm not getting the right answer by using that. It's supposed to be $\frac{121\sqrt{2}}{9}$, but I'm getting something way more complicated.
I would appreciate some help with the other integrals as well.
I got $\frac{121\sqrt{2}}{9}$. Here's my approach:
$\int{\int_D{x^3}}dxdy$ where $D= \left\{(x,y)\in R^2; 1\leq x^2 + 9y^2 \leq 9; x \geq 3y\right\}$
Let $x = r \cos \theta$ and $y = \frac{1}{3}r \sin \theta$. Then
$D'=1\leq r^2 \leq 9;~\cos\theta \geq \sin\theta$.
$\int\int_D x^3~dx~dy = \int\int_{D'}r^3\cos^3 \theta \frac{r}{3}~dr~d\theta$.
From $1\leq r^2 \leq 9,~1\leq r \leq 3$.
From $\cos\theta \geq \sin\theta,~-\frac{3\pi}{4}\leq \theta \leq \frac{\pi}{4}$.
$\int^{\frac{\pi}{4}}_{-\frac{3\pi}{4}}\int_1^ 3\frac{1}{3}r^4\cos^3 \theta ~dr~d\theta = \frac{121 \sqrt{2}}{9}$.
9. Originally Posted by wingless
I got $\frac{121\sqrt{2}}{9}$. Here's my approach:
$\int{\int_D{x^3}}dxdy$ where $D= \left\{(x,y)\in R^2; 1\leq x^2 + 9y^2 \leq 9; x \geq 3y\right\}$
Let $x = r \cos \theta$ and $y = \frac{1}{3}r \sin \theta$. Then
$D'=1\leq r^2 \leq 9;~\cos\theta \geq \sin\theta$.
$\int\int_D x^3~dx~dy = \int\int_{D'}r^3\cos^3 \theta \frac{r}{3}~dr~d\theta$.
From $1\leq r^2 \leq 9,~1\leq r \leq 3$.
From $\cos\theta \geq \sin\theta,~-\frac{3\pi}{4}\leq \theta \leq \frac{\pi}{4}$.
$\int^{\frac{\pi}{4}}_{-\frac{3\pi}{4}}\int_1^ 3\frac{1}{3}r^4\cos^3 \theta ~dr~d\theta = \frac{121 \sqrt{2}}{9}$.
The one thing I didn't check - Spec's transformation from xy-coordinates to uv-coordinates. I used the original transformation (see the quote in post #2) without checking it
Looking for my mistake, I see that it should have been v = 3y, NOT v = y/3. Then I notice Spec has edited the question to reflect this - s/he obviously realised the same thing (*ahem* it's good form to include the reason for an edit, by the way .....)
My mistake for not checking right from the start I knew it would be something simple.
Moral of the lesson - don't assume that even the simplest things are done correctly.
10. Originally Posted by mr fantastic
Moral of the lesson - don't assume that even the simplest things are done correctly.
That's so true | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 73, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.94693922996521, "perplexity": 518.5561974306603}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394678704953/warc/CC-MAIN-20140313024504-00037-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://www.scottmarlowe.com/?tag=/negative+energy | # Scott Marlowe
### Author of the Alchemancer and Assassin Without a Name fantasy series
It's believed that when the Big Bang occurred some 14 billion years ago that equal amounts of positive and negative energy were produced. But where positive energy is readily observable in the world around us, negative energy is not. This is because negative energy does not exist in non-vacuum conditions. Introduce a vacuum, however, and you'll not only find negative energy present, but also that positive energy seems to have gone missing.
Stephen Hawking uses the following analogy to describe the existence of negative energy: Imagine a man with a shovel digs a hole. As dirt is shoveled out of the hole, it is made into a pile on the surface. Once the man is finished, he is left with a hole and a mound that perfectly balance each other out. Such is the case with positive and negative energy.
Hawking's analogy applies to the creation of our universe as well, which supposedly was created from nothing. Since the Big Bang produced equal amounts of positive and negative energy, something was created from nothing; take the sum of those two amounts and you're left with 0.
# So what exactly is negative energy?
It's not antimatter, which has positive energy. For example, an electron with a positive charge, or positron, is considered anti-matter.
It's also not dark energy, which is thought to make up 68.3% of the universe's mass (on a mass-energy equivalence basis).
Negative energy is perhaps something stranger. Described as "the inherent fluctuations in energy that exists in any energy or magnetic field," negative energy is referred to as a form of "exotic matter." As a concept, negative energy was first proposed in 1928 by British physicist Paul Adrien Maurice Didec. Negative energy was a component of his formula, the Dirac equation, which held that quantum states of positive and negative energy were in balance with one another. In a non-vacuum environment (like here inside our planet's atmosphere), negative energy is not observable. However, inside a vacuum, negative energy is present while positive energy is not.
In 1948, the Dutch physicist Hendrick Casimir predicted that a small attractive force could exist between two uncharged, parallel plates in a vacuum. Should the plates be resting extremely close to one another, negative energy is produced since the number of electromagnetic waves between the two plates becomes less than that of surrounding space. In essence, a negative state of energy becomes present when the wavelengths of particles in a certain region of space are less than what may normally be measured.
Negative energy has been produced in a lab via what's called the Casimir effect. This phenomenon revolves around the idea that vacuum, contrary to its portrayal in classical physics, isn't empty. According to quantum theory, vacuum is full of electromagnetic fluctuations. Distorting these fluctuations can create negative energy.
An interesting phenomenon observed within a negative energy vacuum is that light actually travels faster than it does in a normal vacuum. This has fueled interest in the application of negative energy fields to theories involving FTL travel. Further interest in this area has been fueled by experiments demonstrating that negative energy can distort space-time. In regions of extreme space-time curvature, the existence of negative energy may someday allow for the creation of sustainable wormholes.
It is theorized that negative energy that 'falls back into' a black hole has the effect of lowering the black hole's total mass. In effect, as more and more negative energy falls back into the black hole it will diminish over time, evaporating until nothing is left. This is contrary to the belief that black holes gain in mass indefinitely (or at least until the entire universe has been swallowed).
# Conclusion
Negative energy is only one of an assortment of strange phenomenon of which we still do not know enough about. I have my own theories on negative energy, however, and how it might fit into our universe. More precisely, I have ideas on how it fits into the world of my fiction.
Different forms of energy (made-up or not) have already played a major role in some of my writing. For example, in The Alchemancer series, elemental energy plays a substantial role in the story. Negative energy has also made an appearance, though in somewhat subtle fashion as in when Aaron turns his encorder on Ensel Rhe to discover that the eslar's life force signature is negative. Negative energy will play a larger role moving forward into the next book, The Inversion Solution.
Also, negative energy is coming into play in the Assassin Without a Name series. Look for its specific mention in The Goddard Affair, set for release on June 4, 2014. It's already made an appearance in Night of Zealotry, the 3rd assassin tale in the series, when the Jakaree activated their mysterious black energy machine.
Negative energy, as well as other exotic, similar types, will continue to play a role in my fiction. I hope you enjoyed this brief primer. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8019454479217529, "perplexity": 712.4628399080061}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039747215.81/warc/CC-MAIN-20181121052254-20181121074254-00173.warc.gz"} |
https://en.m.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/def_cqp | # Fractals/Iterations in the complex plane/def cqp
Definitions
## InternalEdit
Internal addresses describe the combinatorial structure of the Mandelbrot set.[1]
• is not constant within hyperbolic component. Example : internal address of -1 is 1->2 and internal address of 0.9999 is 1[2]
• of hyperbolic component is defined as a internal address of it's center
# AngleEdit
## Types of angleEdit
Principal branch or complex number argument
external angle internal angle plain angle
parameter plane ${\displaystyle arg(\Phi _{M}(c))\,}$ ${\displaystyle arg(\rho _{n}(c))\,}$ ${\displaystyle arg(c)\,}$
dynamic plane ${\displaystyle arg(\Phi _{c}(z))\,}$ ${\displaystyle arg(z)\,}$
where :
• ${\displaystyle \rho _{n}(c)}$ is a multiplier map
• ${\displaystyle \Phi (c)\,}$ is a Boettcher function
### externalEdit
The external angle is a angle of point of set's exterior. It is the same on all points on the external ray
### internalEdit
The internal angle[3] is an angle of point of component's interior
• it is a rational number and proper fraction measured in turns
• it is the same for all point on the internal ray
• in a contact point ( root point ) it agrees with the rotation number
• root point has internal angle 0
${\displaystyle \alpha ={\frac {p}{q}}\in \mathbb {Q} }$
### plainEdit
The plain angle is an agle of complex point = it's argument [4]
• turns
• degrees
## Number typesEdit
Angle ( for example external angle in turns ) can be used in different number types
Examples :
the external arguments of the rays landing at z = −0.15255 + 1.03294i are :[5]
${\displaystyle (\theta _{20}^{-},\theta _{20}^{+})=(0.{\overline {00110011001100110100}},0.{\overline {00110011001101000011}})}$
where :
${\displaystyle \theta _{20}^{-}=0.{\overline {00110011001100110100}}_{2}=0.{\overline {20000095367522590181913549340772}}_{10}={\frac {209716}{1048575}}={\frac {209716}{2^{20}-1}}}$
# CurvesEdit
Types:
• topology:
• closed versus open
• simple versus not simple
• other properities:
• invariant
• critical
Description[6]
• plane curve = it lies in a plane.
• closed = it starts and ends at the same place.
• simple = it never crosses itself.
## closedEdit
Closed curves are curves whose ends are joined. Closed curves do not have end points.
• Simple Closed Curve : A connected curve that does not cross itself and ends at the same point where it begins. It divides the plane into exactly two regions ( Jordan curve theorem ). Examples of simple closed curves are ellipse, circle and polygons.[7]
• complex Closed Curve ( not simple = non-simple ) It divides the plane into more than two regions. Example : Lemniscates.
"non-self-intersecting continuous closed curve in plane" = "image of a continuous injective function from the circle to the plane"
### CircleEdit
#### Unit circleEdit
Unit circle ${\displaystyle \partial D\,}$ is a boundary of unit disk[8]
${\displaystyle \partial D=\left\{w:abs(w)=1\right\}}$
where coordinates of ${\displaystyle w\,}$ point of unit circle in exponential form are :
${\displaystyle w=e^{i*t}\,}$
## Critical curvesEdit
Diagrams of critical polynomials are called critical curves.[9]
These curves create skeleton of bifurcation diagram.[10] (the dark lines[11])
## Escape linesEdit
"If the escape radius is equal to 2 the contour lines have a contact point (c= -2) and cannot be considered as equipotential lines" [12]
## InvariantEdit
Types:
• topological
• shift invariants
examples :
• curve is invariant for the map f ( evolution function ) if images of every point from the curve stay on that curve
## IsocurvesEdit
### Equipotential linesEdit
Equipotential lines = Isocurves of complex potential
"If the escape radius is greater than 2 the contour lines are equipotential lines" [13]
## Jordan curveEdit
Illustration of the Jordan curve theorem. The Jordan curve (drawn in black) divides the plane into an "inside" region (light blue) and an "outside" region (pink).
Jordan curve = a simple closed curve that divides the plane into an "interior" region bounded by the curve and an "exterior" region containing all of the nearby and far away exterior points[14]
## LaminationEdit
Lamination of the unit disk is a closed collection of chords in the unit disc, which can intersect only in an endpoint of each on the boundary circle[15][16]
It is a model of Mandelbrot or Julia set.
A lamination, L, is a union of leaves and the unit circle which satisfies :[17]
• leaves do not cross (although they may share endpoints) and
• L is a closed set.
## LeafEdit
Chords = leaves = arcs
A leaf on the unit disc is a path connecting two points on the unit circle. [18]
## Open curveEdit
Curve which is not closed. Examples : line, ray.
## RayEdit
Rays are :
• invariant curves
• dynamic or parameter
• external or internal
### Internal rayEdit
Dynamic internal ( blue segment) and external ( red ray) rays
Internal rays are :
• dynamic ( on dynamic plane , inside filled Julia set )
• parameter ( on parameter plane , inside Mandelbrot set )
### SpiderEdit
A spider S is a collection of disjoint simple curves called legs [19]( extended rays = external + internal ray) in the complex plane connecting each of the post-critical points to infnity [20]
See :
## VeinEdit
"A vein in the Mandelbrot set is a continuous, injective arc inside in the Mandelbrot set"
"The principal vein ${\displaystyle v_{p/q}}$ is the vein joining ${\displaystyle c_{p/q}}$ to the main cardioid" (Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems. A dissertation by Giulio Tiozzo )
# DiscretizationEdit
discretization[21] and its reverse [22]
# DynamicsEdit
## symbolicEdit
"Symbolic dynamics encodes :
• a dynamical system ${\displaystyle f:X\to X}$ by a shift map on a space of sequences over finite alphabet using Markov paritition of the space ${\displaystyle X}$
• the points of space ${\displaystyle X}$ by their itineraries with respect to the paritition " ( Volodymyr Nekrashevych - Symbolic dynamics and self-similar groups
# equationEdit
## differentialEdit
differential equations
• exact analytic solutions.
• approximated solution
• use perturbation theory to approximate the solutions
# FunctionEdit
## DerivativeEdit
Derivative of Iterated function (map)
### Derivative with respect to cEdit
On parameter plane :
• ${\displaystyle c}$ is a variable
• ${\displaystyle z_{0}=0}$ is constant
${\displaystyle {\frac {d}{dc}}f_{c}^{(p)}(z_{0})=z'_{p}\,}$
This derivative can be found by iteration starting with
${\displaystyle z_{0}=0\,}$
${\displaystyle z'_{0}=1\,}$
and then
${\displaystyle z_{p}=z_{p-1}^{2}+c\,}$
${\displaystyle z'_{p}=2\cdot z_{p-1}\cdot z'_{p-1}+1\,}$
This can be verified by using the chain rule for the derivative.
• Maxima CAS function :
dcfn(p, z, c) :=
if p=0 then 1
else 2*fn(p-1,z,c)*dcfn(p-1, z, c)+1;
Example values :
${\displaystyle z_{0}=0\qquad \qquad z'_{0}=1\,}$
${\displaystyle z_{1}=c\qquad \qquad z'_{1}=1\,}$
${\displaystyle z_{2}=c^{2}+c\qquad z'_{2}=2c+1\,}$
### Derivative with respect to zEdit
${\displaystyle z'_{n}\,}$ is first derivative with respect to c.
This derivative can be found by iteration starting with
${\displaystyle z'_{0}=1\,}$
and then :
${\displaystyle z'_{n}=2*z_{n-1}*z'_{n-1}\,}$
## GermEdit
Germ [26] of the function f in the neighborhood of point z is a set of the functions g which are indistinguishable in that neighborhood
${\displaystyle [f]_{z}=\{g:g\sim _{z}f\}.}$
See :
## mapEdit
• differences between map and the function [27]
• Iterated function = map[28]
• an evolution function[29] of the discrete nonlinear dynamical system[30]
${\displaystyle z_{n+1}=f_{c}(z_{n})\,}$
is called map ${\displaystyle f_{c}}$ :
${\displaystyle f_{c}:z\to z^{2}+c.\,}$
## typesEdit
### PolynomialEdit
#### CriticalEdit
Critical polynomial :
${\displaystyle Q_{n}=f_{c}^{n}(z_{cr})=f_{c}^{n}(0)\,}$
so
${\displaystyle Q_{1}=f_{c}^{1}(0)=c\,}$
${\displaystyle Q_{2}=f_{c}^{2}(0)=c^{2}+c\,}$
${\displaystyle Q_{3}=f_{c}^{3}(0)=(c^{2}+c)^{2}+c\,}$
These polynomials are used for finding :
• centers of period n Mandelbrot set components. Centers are roots of n-th critical polynomials ${\displaystyle centers=\{c:f_{c}^{n}(z_{cr})=0\}\,}$ ( points where critical curve Qn croses x axis )
• Misiurewicz points ${\displaystyle M_{n,k}=\{c:f_{c}^{k}(z_{cr})=f_{c}^{k+n}(z_{cr})\}\,}$
#### post-critically finiteEdit
a post-critically finite polynomial = all critical points have finite orbit
### ResurgentEdit
"resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the origin. Loosely speaking, these functions resurrect, or surge up - in a slightly different guise, as it were - at their singularities"
J. Écalle, 1980[31]
# glitchesEdit
glitches = Incorrect parts of renders[32] using perturbation techique
# InvariantsEdit
sth is invariant with respect to the transformation = non modified, steady
Topological methods for the analysis of dynamical systems
Invariants type
• metric invariants
• dynamical invariants,
• topological invariants.
## dynamicalEdit
Dynamical invariants = invariants of the dynamical system
• periodic points
• fixed point
• invariant curve
• periodic ray
• external
• internal
Dynamical Invariants Derived from Recurrence Plots[33]
# IntervalEdit
a partition of an interval into subintervals
• Markov paritition[34]
# ItineraryEdit
${\displaystyle S(x)}$ is an itinerary of point x under the map f relative to the paritirtion.
It is a right-infinite sequence of zeros and ones [35]
${\displaystyle S(x)=s_{1}s_{2}s_{3}...s_{n}}$
where
Examples :
For rotation map ${\displaystyle R_{p/q}}$ and invariant interval ${\displaystyle I}$ ( circle ) :
${\displaystyle I=(0,1]}$
one can compute ${\displaystyle x_{c}}$ :
${\displaystyle x_{c}=1-{\frac {p}{q}}}$
and split interval into 2 subintervals ( lower circle paritition):
${\displaystyle I_{0}=(0,x_{c}]}$
${\displaystyle I_{1}=(x_{c},1]}$
then compute s according to it's relation with critical point :
${\displaystyle s_{n}={\begin{cases}0:x_{n}x_{c}\end{cases}}}$
Itinerary can be converted[36] to point ${\displaystyle x\in [0,1]}$
${\displaystyle \gamma (S_{n})=0.s_{1}s_{2}s_{3}...s_{n}=\sum _{n=0}^{n-1}{\frac {s_{n}}{2^{n}}}=x_{n}}$
# MagnitudeEdit
magnitude of the point ( complex number in 2D case) = it's distance from the origin
# MapEdit
description
## typesEdit
• The map f is hyperbolic if every critical orbit converges to a periodic orbit.[37]
#### FormsEdit
##### c form : ${\displaystyle z^{2}+c}$ Edit
• math notation : ${\displaystyle f_{c}(z)=z^{2}+c\,}$
• Maxima CAS function :
f(z,c):=z*z+c;
(%i1) z:zx+zy*%i;
(%o1) %i*zy+zx
(%i2) c:cx+cy*%i;
(%o2) %i*cy+cx
(%i3) f:z^2+c;
(%o3) (%i*zy+zx)^2+%i*cy+cx
(%i4) realpart(f);
(%o4) -zy^2+zx^2+cx
(%i5) imagpart(f);
(%o5) 2*zx*zy+cy
• math notation
${\displaystyle \ f_{c}^{(0)}(z)=z=z_{0}}$
${\displaystyle \ f_{c}^{(1)}(z)=f_{c}(z)=z_{1}}$
...
${\displaystyle \ f_{c}^{(p)}(z)=f_{c}(f_{c}^{(p-1)}(z))}$
or with subscripts :
${\displaystyle \ z_{p}=f_{c}^{(p)}(z_{0})}$
• Maxima CAS function :
fn(p, z, c) :=
if p=0 then z
elseif p=1 then f(z,c)
else f(fn(p-1, z, c),c);
zp:fn(p, z, c);
##### lambda form : ${\displaystyle z^{2}+\lambda z}$ Edit
More description Maxima CAS code ( here m not lambda is used ) :
(%i2) z:zx+zy*%i;
(%o2) %i*zy+zx
(%i3) m:mx+my*%i;
(%o3) %i*my+mx
(%i4) f:m*z+z^2;
(%o4) (%i*zy+zx)^2+(%i*my+mx)*(%i*zy+zx)
(%i5) realpart(f);
(%o5) -zy^2-my*zy+zx^2+mx*zx
(%i6) imagpart(f);
(%o6) 2*zx*zy+mx*zy+my*zx
##### Switching between formsEdit
Start from :
• internal angle ${\displaystyle \theta ={\frac {p}{q}}}$
Multiplier of fixed point :
${\displaystyle \lambda =re^{2\pi \theta i}}$
When one wants change from lambda to c :[39]
${\displaystyle c=c(\lambda )={\frac {\lambda }{2}}\left(1-{\frac {\lambda }{2}}\right)={\frac {\lambda }{2}}-{\frac {\lambda ^{2}}{4}}}$
or from c to lambda :
${\displaystyle \lambda =\lambda (c)=1\pm {\sqrt {1-4c}}}$
Example values :
${\displaystyle \theta }$ r c fixed point alfa ${\displaystyle z_{c}}$ ${\displaystyle \lambda }$ fixed point ${\displaystyle z_{\lambda }}$
1/1 1.0 0.25 0.5 1.0 0
1/2 1.0 -0.75 -0.5 -1.0 0
1/3 1.0 0.64951905283833*i-0.125 0.43301270189222*i-0.25 0.86602540378444*i-0.5 0
1/4 1.0 0.5*i+0.25 0.5*i i 0
1/5 1.0 0.32858194507446*i+0.35676274578121 0.47552825814758*i+0.15450849718747 0.95105651629515*i+0.30901699437495 0
1/6 1.0 0.21650635094611*i+0.375 0.43301270189222*i+0.25 0.86602540378444*i+0.5 0
1/7 1.0 0.14718376318856*i+0.36737513441845 0.39091574123401*i+0.31174490092937 0.78183148246803*i+0.62348980185873 0
1/8 1.0 0.10355339059327*i+0.35355339059327 0.35355339059327*i+0.35355339059327 0.70710678118655*i+0.70710678118655 0
1/9 1.0 0.075191866590218*i+0.33961017714276 0.32139380484327*i+0.38302222155949 0.64278760968654*i+0.76604444311898 0
1/10 1.0 0.056128497072448*i+0.32725424859374 0.29389262614624*i+0.40450849718747 0.58778525229247*i+0.80901699437495
One can easily compute parameter c as a point c inside main cardioid of Mandelbrot set :
${\displaystyle c=c_{x}+c_{y}*i}$
of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this c / cpp code by Wolf Jung[40]
double InternalAngleInTurns;
double t = InternalAngleInTurns *2*M_PI; // from turns to radians
double Cx, Cy; /* C = Cx+Cy*i */
// main cardioid
or this Maxima CAS code :
/* conformal map from circle to cardioid ( boundary
of period 1 component of Mandelbrot set */
F(w):=w/2-w*w/4;
/*
circle D={w:abs(w)=1 } where w=l(t,r)
t is angle in turns ; 1 turn = 360 degree = 2*Pi radians
*/
ToCircle(t,r):=r*%e^(%i*t*2*%pi);
(
[w],
/* point of unit circle w:l(internalAngle,internalRadius); */
w:ToCircle(angle,radius), /* point of circle */
float(rectform(F(w))) /* point on boundary of period 1 component of Mandelbrot set */
)$compile(all)$
/* ---------- global constants & var ---------------------------*/
Numerator :1;
DenominatorMax :10;
/* --------- main -------------- */
for Denominator:1 thru DenominatorMax step 1 do
(
InternalAngle: Numerator/Denominator,
display(Denominator),
display(c),
/* compute fixed point */
alfa:float(rectform((1-sqrt(1-4*c))/2)), /* alfa fixed point */
display(alfa)
)$ ### Circle mapEdit Circle map [41] • irrational rotation[42] ### Doubling mapEdit definition [43] C function ( using GMP library) : // rop = (2*op ) mod 1 void mpq_doubling(mpq_t rop, const mpq_t op) { mpz_t n; // numerator mpz_t d; // denominator mpz_inits(n, d, NULL); // mpq_get_num (n, op); // mpq_get_den (d, op); // n = (n * 2 ) % d mpz_mul_ui(n, n, 2); mpz_mod( n, n, d); // output mpq_set_num(rop, n); mpq_set_den(rop, d); mpz_clears(n, d, NULL); } • Maxima CAS function using numerator and denominator as an input doubling_map(n,d):=mod(2*n,d)/d$
or using rational number as an input
DoublingMap(r):=
block([d,n],
n:ratnumer(r),
d:ratdenom(r),
mod(2*n,d)/d)\$
• Common Lisp function
(defun doubling-map (ratio-angle)
" period doubling map = The dyadic transformation (also known as the dyadic map,
bit shift map, 2x mod 1 map, Bernoulli map, doubling map or sawtooth map "
(let* ((n (numerator ratio-angle))
(d (denominator ratio-angle)))
(setq n (mod (* n 2) d)) ; (2 * n) modulo d
(/ n d))) ; result = n/d
-- by Claude Heiland-Allen
-- type Q = Rational
double :: Q -> Q
double p
| q >= 1 = q - 1
| otherwise = q
where q = 2 * p
• C++
// mndcombi.cpp by Wolf Jung (C) 2010.
// http://mndynamics.com/indexp.html
// n is a numerator
// d is a denominator
// f = n/d is a rational fraction ( angle in turns )
// twice is doubling map = (2*f) mod 1
// n and d are changed ( Arguments passed to function by reference)
void twice(unsigned long long int &n, unsigned long long int &d)
{ if (n >= d) return;
if (!(d & 1)) { d >>= 1; if (n >= d) n -= d; return; }
unsigned long long int large = 1LL;
large <<= 63; //avoid overflow:
if (n < large) { n <<= 1; if (n >= d) n -= d; return; }
n -= large;
n <<= 1;
large -= (d - large);
n += large;
}
#### Inverse function of doubling mapEdit
Every angle α ∈ R/Z measured in turns has :
In Maxima CAS :
InvDoublingMap(r):= [r/2, (r+1)/2];
Note that difference between these 2 preimages
${\displaystyle {\frac {\alpha }{2}}-{\frac {\alpha +1}{2}}={\frac {1}{2}}}$
is half a turn = 180 degrees = Pi radians.
${\displaystyle \alpha }$ ${\displaystyle d^{1}(\alpha )}$ ${\displaystyle d^{-1}(\alpha )}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle {\frac {1}{1}}}$ ${\displaystyle \left\{{\frac {1}{4}},{\frac {3}{4}}\right\}}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle {\frac {2}{3}}}$ ${\displaystyle \left\{{\frac {1}{6}},{\frac {4}{6}}\right\}}$ ${\displaystyle {\frac {1}{4}}}$ ${\displaystyle {\frac {1}{2}}}$ ${\displaystyle \left\{{\frac {1}{8}},{\frac {5}{8}}\right\}}$ ${\displaystyle {\frac {1}{5}}}$ ${\displaystyle {\frac {2}{5}}}$ ${\displaystyle \left\{{\frac {1}{10}},{\frac {6}{10}}\right\}}$ ${\displaystyle {\frac {1}{6}}}$ ${\displaystyle {\frac {1}{3}}}$ ${\displaystyle \left\{{\frac {1}{12}},{\frac {7}{12}}\right\}}$ ${\displaystyle {\frac {1}{7}}}$ ${\displaystyle {\frac {2}{7}}}$ ${\displaystyle \left\{{\frac {1}{14}},{\frac {4}{7}}\right\}}$
### First return mapEdit
definition [46]
"In contrast to a phase portrait, the return map is a discrete description of the underlying dynamics. .... A return map (plot) is generated by plotting one return value of the time series against the previous one "[47]
"If x is a periodic point of period p for f and U is a neighborhood of x, the composition ${\displaystyle f^{\circ p}\,}$ maps U to another neighborhood V of x. This locally defined map is the return map for x." ( W P Thurston : On the geometry and dynamics of Iterated rational maps)
"The first return map S → S is the map defined by sending each x0 ∈ S to the point of S where the orbit of x0 under the system first returns to S." [48]
"way to obtain a discrete time system from a continuous time system, called the method of Poincar´e sections Poincar´e sections take us from : continuous time dynamical systems on (n + 1)-dimensional spaces to discrete time dynamical systems on n-dimensional spaces"[49]
### Multiplier mapEdit
Multiplier map ${\displaystyle \lambda }$ gives an explicit uniformization of hyperbolic component ${\displaystyle \mathrm {H} }$ by the unit disk ${\displaystyle \mathbb {D} }$ :
${\displaystyle \lambda :\mathrm {H} \to \mathbb {D} }$
Multiplier map is a conformal isomorphism.[50]
### Rotation mapEdit
Rotation map ${\displaystyle R}$ describes counterclockwise rotation of point ${\displaystyle \theta }$ thru ${\displaystyle {\frac {p}{q}}}$ turns on the unit circle :
${\displaystyle R_{\frac {p}{q}}(\theta )=\theta +{\frac {p}{q}}}$
It is used for computing :
### Shift mapEdit
names :
• bit shift map ( because it shifts the bit ) = if the value of an iterate is written in binary notation, the next iterate is obtained by shifting the binary point one bit to the right, and if the bit to the left of the new binary point is a "one", replacing it with a zero.
• 2x mod 1 map ( because it is math description of it's action )
Shift map (one-sided binary left shift ) acts on one-sided infinite sequence of binary numbers by
${\displaystyle \sigma (b_{1},b_{2},b_{3},\ldots )=(b_{2},b_{3},b_{4},\ldots )}$
It just drops first digit of the sequence.
${\displaystyle \sigma ^{2}(S)=\sigma (\sigma (S))}$
${\displaystyle \sigma ^{k}(b_{1}b_{2}\ldots )=b_{k+1}b_{k+2}\ldots }$
If we treat sequence as a binary fraction :
${\displaystyle x=0.b_{1},b_{2},b_{3},\ldots }$
then shift map = the dyadic transformation = dyadic map = bit shift map= 2x mod 1 map = Bernoulli map = doubling map = sawtooth map
${\displaystyle \sigma (x)=2x{\bmod {1}}}$
and "shifting N places left is the same as multiplying by 2 to the power N (written as 2N)"[51] ( operator << )
# MultiplierEdit
Multiplier of periodic z-point : [52]
Math notation :
${\displaystyle \lambda _{c}(z)={\frac {df_{c}^{(p)}(z)}{dz}}\,}$
Maxima CAS function for computing multiplier of periodic cycle :
m(p):=diff(fn(p,z,c),z,1);
where p is a period. It takes period as an input, not z point.
period ${\displaystyle f^{p}(z)\,}$ ${\displaystyle \lambda _{c}(z)\,}$
1 ${\displaystyle z^{2}+c\,}$ ${\displaystyle 2z\,}$
2 ${\displaystyle z^{4}+2cz^{2}+c^{2}+c}$ ${\displaystyle 4z^{3}+4cz}$
3 ${\displaystyle z^{8}+4cz^{6}+6c^{2}z^{4}+2cz^{4}+4c^{3}z^{2}+4c^{2}z^{2}+c^{4}+2c^{3}+c^{2}+c}$ ${\displaystyle 8z^{7}+24cz^{5}+24c^{2}z^{3}+8cz^{3}+8c^{3}z+8c^{2}z}$
It is used to :
• compute stability index of periodic orbit ( periodic point) = ${\displaystyle |\lambda |=r}$ ( where r is a n internal radius
• multiplier map
# NumberEdit
## Rotation numberEdit
The rotation number[53][54][55][56] of the disk ( component) attached to the main cardioid of the Mandelbrot set is a proper, positive rational number p/q in lowest terms where :
• q is a period of attached disk ( child period ) = the period of the attractive cycles of the Julia sets in the attached disk
• p descibes fc action on the cycle : fc turns clockwise around z0 jumping, in each iteration, p points of the cycle [57]
Features :
• in a contact point ( root point ) it agrees with the internal angle
• the rotation numbers are ordered clockwise along the boundary of the componant
• " For parameters c in the p/q-limb, the filled Julia set Kc has q components at the fixed point αc . These are permuted cyclically by the quadratic polynomial fc(z), going p steps counterclockwise " Wolf Jung
def[58]
# OrbitEdit
Orbit is a sequence of points = trajectory
## CriticalEdit
Forward orbit[59] of a critical point[60][61] is called a critical orbit. Critical orbits are very important because every attracting periodic orbit[62] attracts a critical point, so studying the critical orbits helps us understand the dynamics in the Fatou set.[63][64] [65]
${\displaystyle z_{0}=z_{cr}=0\,}$
${\displaystyle z_{1}=f_{c}(z_{0})=c\,}$
${\displaystyle z_{2}=f_{c}(z_{1})=c^{2}+c\,}$
${\displaystyle z_{3}=f_{c}(z_{2})=(c^{2}+c)^{2}+c\,}$
${\displaystyle ...\,}$
This orbit falls into an attracting periodic cycle.
Code :
"https://github.com/conanite/rainbow/blob/master/src/arc/rainbow/spiral.arc
This software is copyright (c) Conan Dalton 2008. Permission to use it is granted under the Perl Foundations's Artistic License 2.0.
This software uses javacc which is copyright (c) its authors
"
(def plot (plt c)
(with (z 0+0i
n 0
repeats 0)
(while (and (small z) (< n 10000) (< repeats 1000))
(assign n (+ n 1)
z (+ c (* z z))
repeats (if (apply plt (complex-parts z))
(+ repeats 1)
0)))))
Here are images:
## InverseEdit
Inverse = Backward
# ParameterEdit
Parameter
• point of parameter plane : " is renormalizable if restriction of some of its iterate gives a polinomial-like map of the same or lower degree. " [68]
• parameter of the function
• Markov
# PeriodEdit
The smallest positive integer value p for which this equality
${\displaystyle f^{p}(z_{0})=z_{0}}$
holds is the period[69] of the orbit.[70]
${\displaystyle z_{0}}$ is a point of periodic orbit ( limit cycle ) ${\displaystyle \{z_{0},\dots ,z_{p-1}\}}$ .
More is here
# PlaneEdit
Planes [73]
Douady’s principle : “sow in dynamical plane and reap in parameter space”.
## Dynamic planeEdit
• z-plane for fc(z)= z^2 + c
• z-plane for fm(z)= z^2 + m*z
## Parameter planeEdit
See :[74]
Types of the parameter plane :
• c-plane ( standard plane )
• exponential plane ( map) [75][76]
• flatten' the cardiod ( unroll ) [77][78] = "A region along the cardioid is continuously blown up and stretched out, so that the respective segment of the cardioid becomes a line segment. .." ( Figure 4.22 on pages 204-205 of The Science Of Fractal Images)[79]
• transformations [80]
# PointsEdit
## Band-mergingEdit
the band-merging points are Misiurewicz points[81]
## BiaccessibleEdit
If there exist two distinct external rays landing at point we say that it is a biaccessible point. [82]
## CenterEdit
### Nucleus or center of hyperbolic componentEdit
A center of a hyperbolic component H is a parameter ${\displaystyle c_{0}\in H\,}$ ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [83]
Synonyms :
• Nucleus of a Mu-Atom [84]
How to find center/s ?
### Center of Siegel DiscEdit
Center of Siegel disc is a irrationally indifferent periodic point.
Mane's theorem :
"... appart from its center, a Siegel disk cannot contain any periodic point, critical point, nor any iterated preimage of a critical or periodic point. On the other hand it can contain an iterated image of a critical point." [85]
## CriticalEdit
A critical point[86] of ${\displaystyle f_{c}\,}$ is a point ${\displaystyle z_{cr}\,}$ in the dynamical plane such that the derivative vanishes:
${\displaystyle f_{c}'(z_{cr})=0.\,}$
Since
${\displaystyle f_{c}'(z)={\frac {d}{dz}}f_{c}(z)=2z}$
implies
${\displaystyle z_{cr}=0\,}$
we see that the only (finite) critical point of ${\displaystyle f_{c}\,}$ is the point ${\displaystyle z_{cr}=0\,}$ .
${\displaystyle z_{0}}$ is an initial point for Mandelbrot set iteration.[87]
## CutEdit
The "neck" of this eight-like figure is a cut-point.
Cut points in the San Marco Basilica Julia set. Biaccessible points = landing points for 2 external rays
Cut point k of set S is a point for which set S-k is dissconected ( consist of 2 or more sets).[88] This name is used in a topology.
Examples :
• root points of Mandelbrot set
• Misiurewicz points of boundary of Mandelbrot set
• cut points of Julia sets ( in case of Siegel disc critical point is a cut point )
These points are landing points of 2 or more external rays.
Point which is a landing point of 2 external rays is called biaccesible
Cut ray is a ray which converges to landing point of another ray. [89] Cut rays can be used to construct puzzles.
Cut angle is an angle of cut ray.
## fixedEdit
Periodic point when period = 1
## FeigenbaumEdit
Self similarity in the Mandelbrot set shown by zooming in on a round feature while panning in the negative-x direction. The display center pans from (−1, 0) to (−1.31, 0) while the view magnifies from 0.5 × 0.5 to 0.12 × 0.12 to approximate the Feigenbaum ratio${\displaystyle \delta }$ .
The Feigenbaum Point[90] is a :
• point c of parameter plane
• is the limit of the period doubling cascade of bifurcations
• an infinitely renormalizable parameter of bounded type
• boundary point between chaotic ( -2 < c < MF ) and periodic region ( MF< c < 1/4)[91]
${\displaystyle MF^{(n)}({\tfrac {p}{q}})=c}$
Generalized Feigenbaum points are :
• the limit of the period-q cascade of bifurcations
• landing points of parameter ray or rays with irrational angles
Examples :
• ${\displaystyle MF^{(0)}=MF^{(1)}({\tfrac {1}{2}})=c=-1.401155}$
• -.1528+1.0397i)
The Mandelbrot set is conjectured to be self- similar around generalized Feigenbaum points[92] when the magnification increases by 4.6692 (the Feigenbaum Constant) and period is doubled each time[93]
n Period = 2^n Bifurcation parameter = cn Ratio ${\displaystyle ={\dfrac {c_{n-1}-c_{n-2}}{c_{n}-c_{n-1}}}}$
1 2 -0.75 N/A
2 4 -1.25 N/A
3 8 -1.3680989 4.2337
4 16 -1.3940462 4.5515
5 32 -1.3996312 4.6458
6 64 -1.4008287 4.6639
7 128 -1.4010853 4.6682
8 256 -1.4011402 4.6689
9 512 -1.401151982029
10 1024 -1.401154502237
infinity -1.4011551890 ...
Bifurcation parameter is a root point of period = 2^n component. This series converges to the Feigenbaum point c = −1.401155
The ratio in the last column converges to the first Feigenbaum constant.
" a "Feigenbaum point" (an infinitely renormalizable parameter of bounded type, such as the famous Feigenbaum value which is the limit of the period-2 cascade of bifurcations), then Milnor's hairiness conjecture, proved by Lyubich, states that rescalings of the Mandelbrot set converge to the entire complex plane. So there is certainly a lot of thickness near such a point, although again this may not be what you are looking for. It may also prove computationally intensive to produce accurate pictures near such points, because the usual algorithms will end up doing the maximum number of iterations for almost all points in the picture." Lasse Rempe-Gillen[94]
## infinityEdit
The point at infinity [95]" is a superattracting fixed point, but more importantly its immediate basin of attraction - that is, the component of the basin containing the fixed point itself - is completely invariant (invariant under forward and backwards iteration). This is the case for all polynomials (of degree at least two), and is one of the reasons that studying polynomials is easier than studying general rational maps (where e.g. the Julia set - where the dynamics is chaotic - may in fact be the whole Riemann sphere). The basin of infinity supports foliations into "external rays" and "equipotentials", and this allows one to study the Julia set. This idea was introduced by Douady and Hubbard, and is the basis of the famous "Yoccoz puzzle"." Lasse Rempe-Gillen[96]
## MisiurewiczEdit
Misiurewicz point[97] = " parameters where the critical orbit is pre-periodic.
Examples are:
• band-merging points of chaotic bands (the separator of the chaotic bands Bi−1 and Bi )[98]
• the branch points
• tips in the Mandelbrot set ( tips of the midgets ) [99]
Characteristic Misiurewicz pointof the chaotic band of the Mandelbrot set is :[100]
• the most prominent and visible Misiurewicz point of a chaotic band
• have the same period as the band
• have the same period as the gene of the band
## Myrberg-FeigenbaumEdit
MF = the Myrberg-Feigenbaum point is the different name for the Feigenbaum Point.
## Parabolic pointEdit
parabolic points : this occurs when two singular points coallesce in a double singular point (parabolic point)[101]
## PeriodicEdit
Point z has period p under f if :
${\displaystyle z:\ f^{p}(z)=z}$
## PinchingEdit
"Pinching points are found as the common landing points of external rays, with exactly one ray landing between two consecutive branches. They are used to cut M or K into well-defined components, and to build topological models for these sets in a combinatorial way. " ( definition from Wolf Jung program Mandel )
See for examples :
• period 2 = Mandel, demo 2 page 3.
• period 3 = Mandel, demo 2 page 5 [102]
## post-criticalEdit
A post-critical point is a point
${\displaystyle z=f(f(f(...(z_{cr}))))}$
where ${\displaystyle z_{cr}}$ is a critical point. [103]
## precriticalEdit
precritical points, i.e., the preimages of 0
## rootEdit
The root point :
• has a rotational number 0
• it is a biaccesible point ( landing point of 2 external rays )
## singularEdit
the singular points of a dynamical system
In complex analysis there are four classes of singularities:
• Isolated singularities: Suppose the function f is not defined at a, although it does have values defined on U \ {a}.
• The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U \ {a}. The function g is a continuous replacement for the function f.
• The point a is a pole or non-essential singularity of f if there exists a holomorphic function g defined on U with g(a) nonzero, and a natural number n such that f(z) = g(z) / (za)n for all z in U \ {a}. The least such number n is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with n increased by 1 (except if n is 0 so that the singularity is removable).
• The point a is an essential singularity of f if it is neither a removable singularity nor a pole. The point a is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.
• Branch points are generally the result of a multi-valued function, such as ${\displaystyle {\sqrt {z}}}$ or ${\displaystyle \log(z)}$ being defined within a certain limited domain so that the function can be made single-valued within the domain. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. When the cut is genuinely required, the function will have distinctly different values on each side of the branch cut. The shape of the branch cut is a matter of choice, however, it must connect two different branch points (like ${\displaystyle z=0}$ and ${\displaystyle z=\infty }$ for ${\displaystyle \log(z)}$ ) which are fixed in place.
# PortraitEdit
## orbit portraitEdit
### typesEdit
There are two types of orbit portraits: primitive and satellite. [104]If ${\displaystyle v}$ is the valence of an orbit portrait ${\displaystyle {\mathcal {P}}}$ and ${\displaystyle r}$ is the recurrent ray period, then these two types may be characterized as follows:
• Primitive orbit portraits have ${\displaystyle r=1}$ and ${\displaystyle v=2}$ . Every ray in the portrait is mapped to itself by ${\displaystyle f^{n}}$ . Each ${\displaystyle A_{j}}$ is a pair of angles, each in a distinct orbit of the doubling map. In this case, ${\displaystyle r_{\mathcal {P}}}$ is the base point of a baby Mandelbrot set in parameter space.
• Satellite ( non-primitive ) orbit portraits have ${\displaystyle r=v\geq 2}$ . In this case, all of the angles make up a single orbit under the doubling map. Additionally, ${\displaystyle r_{\mathcal {P}}}$ is the base point of a parabolic bifurcation in parameter space.
# Processes and phenomenonaEdit
## Contraction and dilatationEdit
• the contraction z → z/2
• the dilatation z → 2z.
## Implosion and explosionEdit
Explosion (above) and implosion ( below)
Implosion is :
• the process of sudden change of quality fuatures of the object, like collapsing (or being squeezed in)
• the opposite of explosion
Example : parabolic implosion in complex dynamics, when filled Julia for complex quadratic polynomial set looses all its interior ( when c goes from 0 along internal ray 0 thru parabolic point c=1/4 and along extrnal ray 0 = when c goes from interior , crosses the bounday to the exterior of Mandelbrot set)[105]
Explosion is a :
• is a sudden change of quality fuatures of the object in an extreme manner,
• the opposite of implosion
Example : in exponential dynamics when λ> 1/e , the Julia set of ${\displaystyle E_{\lambda }(z)=\lambda e^{z}}$ is the entire plane.[106]
## TuningEdit
• definition[107]
• examples
Conformal radius of Siegel Disk [108][109]
Escape radius ( ER ) or bailout value is a radius of circle target set used in bailout test
Minimal Escape Radius should be grater or equal to 2 :
${\displaystyle ER=max(2,|c|)\,}$
Better estimation is :[110][111]
${\displaystyle ER={\frac {1}{2}}+{\sqrt {{\frac {1}{4}}+|c|}}}$
• radius of inner circle, where inner circle with center at fixed point is the biggest circle inside Siegel Disc.
• minimal distance between center of Siel Disc and critical orbit
• absolute value of multiplier ${\displaystyle r=|\lambda |}$
# SequencesEdit
A sequence is an ordered list of objects (or events).[112]
A series is the sum of the terms of a sequence of numbers.[113] Some times these names are not used as in above definitions.
## OrbitEdit
Orbit can be:
• forward = sequence of points
• backward ( inverse )
• tree in case of multivalued function
• sequence
# SetEdit
definition[114]
## ComponentEdit
### Components of parameter planeEdit
mu-atom , ball, bud, bulb, decoration, lake and lakelet.[115]
#### IslandsEdit
Names :
• mini Mandelbrot set
• 'baby'-Mandelbrot set
• island mu-molecules = embedded copy of the Mandelbrot Set[116]
• Bug
• Island
• Mandelbrotie
• Midget
List of islands :
#### Primary and satelliteEdit
"Hyperbolic components come in two kinds, primitive and satellite, depending on the local properties of their roots." [117]
def [118]
#### Hyperbolic component of Mandelbrot setEdit
Boundaries of hyperbolic components of Mandelbrot set
Domain is an open connected subset of a complex plane.
"A hyperbolic component H of Mandelbrot set is a maximal domain (of parameter plane) on which ${\displaystyle f_{c}\,}$ has an attracting periodic orbit.
A center of a H is a parameter ${\displaystyle c_{0}\in H\,}$ ( or point of parameter plane ) such that the corresponding periodic orbit has multiplier= 0." [119]
A hyperbolic component is narrow if it contains no component of equal or lesser period in its wake [120]
features of hyperbolic component
• period
• islandhood ( shape = cardiod or circle )
• lower and upper external angle of rays landing on it's root
• center (
• root
• orientation
• size
#### LimbEdit
13/34 limb and wake on the left image
p/q limb is a part of Mandelbrot set contained inside p/q wake
#### WakeEdit
Wakes of Mandelbrot Set to Period 10
Wake is the region of parameter plane enclosed by two external rays landing on the same root point on the boundary of main cardioid ( period 1 hyperbolic component).
Angles of the external rays that land on the root point one can find by :
### Components of dynamical planeEdit
In case of Siegel disc critical orbit is a boundary of component containing Siegel Disc.
## DomainEdit
Domain in mathematical analysis it is an open connected set
### Jordan domainEdit
"A Jordan domain[121] J is the the homeomorphic image of a closed disk in E2. The image of the boundary circle is a Jordan curve, which by the Jordan Curve Theorem separates the plane into two open domains, one bounded, the other not, such that the curve is the boundary of each." [122]
Lea-Fatu flower
## Planar setEdit
a non-separating planar set is a set whose complement in the plane is connected.[123]
Sepal
## Target setEdit
How target set is changing along internal ray 0
### Elliptic caseEdit
Target set in elliptic case = inner circle
For the elliptic dynamics, when there is a Siegel disc, the target set is an inner circle
### Hyperbolic caseEdit
Infinity is allways hyperbolic attractor for forward iteration of polynomials. Target set here is an exterior of any shape containing all point of Julia set ( and it's interior). There are also other hyperbolic attractors.
In case of forward iteration target set ${\displaystyle T\,}$ is an arbitrary set on dynamical plane containing infinity and not containing points of filled Julia set.
For escape time algorithms target set determines the shape of level sets and curves. It does not do it for other methods.
#### Exterior of circleEdit
This is typical target set. It is exterior of circle with center at origin ${\displaystyle z=0\,}$ and radius =ER :
${\displaystyle T_{ER}=\{z:abs(z)>ER\}\,}$
Circle of radius=ER centered at the origin is : ${\displaystyle \{z:abs(z)=ER\}\,}$
#### Exterior of squareEdit
Here target set is exterior of square of side length ${\displaystyle s\,}$ centered at origin
${\displaystyle T_{s}=\{z:abs(re(z))>s~~{\mbox{or}}~~abs(im(z))>s\}\,}$
### Parabolic case : petalEdit
trap in parabolic case
In the parabolic case target set shoul be iside petal
## TrapEdit
Trap is an another name of the target set. It is a set which captures any orbit tending to point inside the trap ( fixed / periodic point ).
# TestEdit
## Bailout test or escaping testEdit
Two sets after bailout test: escaping white and non-escaping black
Distance to fixed point for various types of dynamics
It is used to check if point z on dynamical plane is escaping to infinity or not.[124] It allows to find 2 sets :
• escaping points ( it should be also the whole basing of attraction to infinity)[125]
• not escaping points ( it should be the complement of basing of attraction to infinity)
In practice for given IterationMax and Escape Radius :
• some pixels from set of not escaping points may contain points that escape after more iterations then IterationMax ( increase IterMax )
• some pixels from escaping set may contain points from thin filaments not choosed by maping from integer to world ( use DEM )
If ${\displaystyle z_{n}}$ is in the target set ${\displaystyle T\,}$ then ${\displaystyle z_{0}}$ is escaping to infinity ( bailouts ) after n forward iterations ( steps).[126]
The output of test can be :
• boolean ( yes/no)
• integer : integer number (value of the last iteration)
# TreeEdit
## Farey treeEdit
Farey tree = Farey sequence as a tree
## Hubbard treeEdit
"Hubbard trees are finite planar trees, equipped with self-maps, which classify postcritically finite polynomials as holomorphic dynamical systems on the complex plane." [127] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 170, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8282327055931091, "perplexity": 2953.704871914602}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542250.48/warc/CC-MAIN-20161202170902-00132-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://mathematics.huji.ac.il/event/ntag-amnon-yekutieli-bgu-derived-category-sheaves-commutative-dg-rings?ref_tid=3830 | # NT&AG: Amnon Yekutieli (BGU), "The Derived Category of Sheaves of Commutative DG Rings"
Abstract: In modern algebraic geometry we encounter the notion of derived intersection of subschemes. This is a sophisticated way to encode what happens when two subschemes Y_1 and Y_2 of a given scheme X intersect non-transversely. The classical intersection multiplicity can be extracted from the derived intersection. When the ambient scheme X is affine, it is not too hard to describe the derived intersection, by taking flat DG ring resolutions of the structure sheaves of the subschemes Y_1 or Y_2. This also works when the scheme X is quasi-projective. However, derived intersections in more general schemes X could only be treated using the very difficult homotopical methods of derived algebraic geometry. Several months ago I discovered a "cheap" way to construct flat resolutions of sheaves of rings. The resolutions are by semi-pseudo-free sheaves of DG rings. The main advantage is that the geometry does not change: all the action takes place on the original topological space X. Using semi-pseudo-free resolutions it is possible to produce derived intersections as above. It is also possible to get a direct presentation of the cotangent complex of a scheme (at least in characteristic 0). Presumably the derived adic completion of Shaul, so far studied only in the affine case, could be globalized using our our methods. Lastly, the semi-pseudo-free resolutions give rise to a congruence on the category of sheaves of commutative DG rings on the space X, that we call relative quasi-homotopy. The functor from the homotopy category to the derived category turns out to be a faithful right Ore localization. This fact gives tight control on the derived category. It should be noted that in this situation there does not seem to exist a Quillen model structure, so the traditional approaches would fail. In the talk I will explain the various ideas listed above. More details can be found in the eprint arxiv:1608.04265.
## Date:
Mon, 12/12/2016 - 14:00 to 15:00
## Location:
Ros Building, 70A | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9191389679908752, "perplexity": 448.9885264144757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315750.62/warc/CC-MAIN-20190821022901-20190821044901-00055.warc.gz"} |
https://brahma.tcs.tifr.res.in/events/approximate-colouring-using-semi-definite-programming?page=1&mini=2021-08 | # Approximate Colouring using Semi Definite Programming
Speaker:
## Time:
Monday, 14 November 2011, 16:00 to 17:00
## Venue:
• A-212 (STCS Seminar Room)
Determining the chromatic number of a graph is known to be NP-hard. We will consider the problem of coloring a $k$-colorable graph on n vertices with $n^{\alpha}$ colors (where $\alpha < 1$). We will give an algorithm for this problem using Semi Definite Programming (a generalization of LP ). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9888318777084351, "perplexity": 636.3188458275353}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588246.79/warc/CC-MAIN-20211028003812-20211028033812-00350.warc.gz"} |
https://engineering.stackexchange.com/questions/21123/is-it-possible-to-calculate-the-shear-wave-velocity-in-lab-using-ultrasonic-test | # Is it possible to calculate the shear wave velocity in lab using ultrasonic test?
I have an ultrasonic testing equipment that can only measure the direct and semi-direct ( (a) and (b) in the image below):
I can measure compression wave velocities fairly easily, but I would like to know if it is possible to measure the shear wave velocity.
Is it possible to measure the shear wave velocity in a lab setting using this? What should the distance between the two sensors be to provide accurate measurments?
Well, nothing is impossible. But, lets see what you are trying to do:
This diagram is simplified; in reality, there are 3 different P (compression) waves and 2 different S (shear) waves.
Measurement of the first P wave is very easy because it depends on the first arrival time (it is the fastest wave).
But to measure the S wave, you first need to isolate the noise, the reflections caused by P waves, the refraction and then determine if you have the correct S wave or not.
Doing so is very hard.
However, you can use specific transducers called shear wave transducer that will generate shear waves directly so you can measure them. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9624540209770203, "perplexity": 520.5247885021563}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400241093.64/warc/CC-MAIN-20200926102645-20200926132645-00168.warc.gz"} |
http://www.insight-things.com/why-you-can-multiply-means-and-expected-values | Maybe you had to multiply means or expected values already. If you know, for instance, how often people go shopping on average and how much money they spent on their shopping tours on average then you could multiply both to obtain the average amount spent. In this post I will explain why multiplying means and expected values is a valid operation.
As in “Why you can add means and expected values”, my reasoning will be based on expected values. You way want to read that post first to gain an understanding of the key ideas which I will use straight forward in this post. Otherwise, let’s start with the definition of our expected value for two multiplied random variables $X$ and $Y$:
$\displaystyle \mathrm{E}\big[X \times Y\big]=\sum\limits_{x} \sum\limits_{y} x \times y \times p_{x,y}(x,y)$
Above you see that we add all possible products resulting when $X$ and $Y$ take specific values. The probability for each combination has been stated by $p_{x,y}(x,y)$. Now we assume – and this is important! – that $X$ and $Y$ are independent. Under the condition of independence $p_{x,y}(x,y)$ can be expanded to $p_{x}(x)\times p_{y}(y)$. Wait until you spot the implications on our calculation!
$\displaystyle \mathrm{E}\big[X Y\big]=\sum\limits_{x} \sum\limits_{y} x \times y \times p_{x}(x) \times p_{y}(y)$
$\displaystyle \mathrm{E}\big[X Y\big]=\sum\limits_{x} x \times p_{x}(x) \times \sum\limits_{y} y \times p_{y}(y)$
With our simplification we were able to take $x$ and $p_{x}(x)$ out of the inner sum, because both measures are constant in terms of $y$. Have you recognized that the inner sum, which iterates for all values for $y$, now resembles the expected value $\mathrm{E}[Y]$?
$\displaystyle \mathrm{E}\big[X Y\big]=\sum\limits_{x} x \times p_{x}(x) \times \mathrm{E}\big[Y\big]$
$\mathrm{E}[Y]$ is constant in terms of $x$ and, therefore, can be taken in front of the remaining sum:
$\displaystyle \mathrm{E}\big[X Y\big]= \mathrm{E}\big[Y\big] \times \sum\limits_{x} x \times p_{x}(x)$
Surprise! 😀 The sum gives the expected value of $Y$.
$\displaystyle \mathrm{E}\big[X Y\big]= \mathrm{E}\big[Y\big] \times \mathrm{E}\big[X\big]$
Et voilà! The above line is what we wanted to show in the first place. Keep in mind that you could replace all the sum signs by integrals to show that you can also multiply continous random variables. Further, multiplications works also for means as they are approximations of the expected value of a distribution. So, it is valid to state $\overline{xy}=\overline{x}\times \overline{y}$, if both measures are independent! | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 24, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9286667108535767, "perplexity": 233.00221471776536}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000414.26/warc/CC-MAIN-20190626174622-20190626200622-00387.warc.gz"} |
http://qubitrot.org/problems/9cmclr8bhnvoyu91cw5oro60av01gc | # Frankel 1.2 Problems
## (1)
The real projective plane $$\mathbb{R}P^2$$ is defined the be the set of all lines in $$\mathbb{R}^3$$ that pass through the origin. We must show that this forms a 2-dimensional manifold. We can cover this set with three subsets:
\begin{align}U_x &:= \text{all lines which do not lie on the yz-plane}, \\ U_ y &:= \text{all lines which do not lie on the xz-plane}, \\ U_z &:= \text{all lines which do not lie on the xy-plane}.\end{align}
For each of these we can define a coordinate system by associating each line is the set with a point in the plane. That is, a map $$\phi_i(U_i) : \mathbb{R}P^2 \to \mathbb{R}^2, \text { for } i = x,y,z$$. To do this for $$U_z$$, chose any point $$(x,y,z)$$ other than the origin on the line and construct
$$\phi_z(U_z) = \left(\frac{x}{z},\frac{y}{z}\right).$$
So a line in $$U_z$$ gets mapped to a point in $$\mathbb{R}^2$$ which is the intersection of that line and the $$z=1$$ plane. The other coordinate patches are defined similarity.
To show that $$\mathbb{R}P^2$$ forms a manifold, we must show that for any line existing in more than one coordinate patch, the coordinates in one patch can be differentiably expressed in terms of the coordinates in another patch.
$$\phi_x(\phi_z^{-1}(a,b)) = \phi_x(x,y,1) = \left(\frac{y}{x},\frac{1}{x}\right).$$
This is $$C^\infty$$ over $$U_z \cap U_x$$ ($$x \neq 0$$ in $$U_x$$). Likewise for all other patch relationships. Therefore, the real projective plane is a 2-dimensional manifold.
## (2)
The real projective space $$\mathbb{R}P^3$$ is a 3-dimensional manifold. A point $$p = (x^1,x^2,x^3,x^4)$$ is identified with $$(\lambda x^1, \lambda x^2, \lambda x^3,\lambda x^4)$$ for all $$\lambda \neq 0 \in \mathbb{R}$$. Following the example of $$\mathbb{R}P^2$$, we cover $$\mathbb{R}P^3$$ with the 4 coordinate patches
$$U_i = \{(\lambda x^1, \lambda x^2, \lambda x^3,\lambda x^4) \, | \, x^i \neq 0\}, \,\,\, i = 1,2,3,4.$$
Then the coordinate maps $$\phi_i : U_i \to \mathbb{R}^3$$ are clear, e.g.
$$\phi_1(U_1) = \left(\frac{x^2}{x^1}, \frac{x^3}{x^1}, \frac{x^4}{x^1}\right).$$
Thus,
$$\phi_2(\phi_1^{-1}(a,b,c,d)) = \phi_2(1,x^2,x^3,x^4) = \left(\frac{1}{x^2}, \frac{x^3}{x^2}, \frac{x^4}{x^2}\right),$$
which is $$C^\infty$$. Hence $$\mathbb{R}P^3$$ is a 3-dimensional manifold. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.991561770439148, "perplexity": 293.27140469415946}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267867055.95/warc/CC-MAIN-20180624195735-20180624215735-00059.warc.gz"} |
http://cms.math.ca/cjm/kw/Lipschitz | Canadian Mathematical Society www.cms.math.ca
location: Publications → journals
Search results
Search: All articles in the CJM digital archive with keyword Lipschitz
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1. CJM Online first
Ostrovskii, Mikhail; Randrianantoanina, Beata
Metric spaces admitting low-distortion embeddings into all $n$-dimensional Banach spaces For a fixed $K\gg 1$ and $n\in\mathbb{N}$, $n\gg 1$, we study metric spaces which admit embeddings with distortion $\le K$ into each $n$-dimensional Banach space. Classical examples include spaces embeddable into $\log n$-dimensional Euclidean spaces, and equilateral spaces. We prove that good embeddability properties are preserved under the operation of metric composition of metric spaces. In particular, we prove that $n$-point ultrametrics can be embedded with uniformly bounded distortions into arbitrary Banach spaces of dimension $\log n$. The main result of the paper is a new example of a family of finite metric spaces which are not metric compositions of classical examples and which do embed with uniformly bounded distortion into any Banach space of dimension $n$. This partially answers a question of G. Schechtman. Keywords:basis constant, bilipschitz embedding, diamond graph, distortion, equilateral set, ultrametricCategories:46B85, 05C12, 30L05, 46B15, 52A21
2. CJM 2012 (vol 65 pp. 702)
Taylor, Michael
Regularity of Standing Waves on Lipschitz Domains We analyze the regularity of standing wave solutions to nonlinear Schrödinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in Hölder spaces. Keywords:standing waves, elliptic regularity, Lipschitz domainCategories:35J25, 35J65
3. CJM 2006 (vol 58 pp. 64)
Filippakis, Michael; Gasiński, Leszek; Papageorgiou, Nikolaos S.
Multiplicity Results for Nonlinear Neumann Problems In this paper we study nonlinear elliptic problems of Neumann type driven by the $p$-Laplac\-ian differential operator. We look for situations guaranteeing the existence of multiple solutions. First we study problems which are strongly resonant at infinity at the first (zero) eigenvalue. We prove five multiplicity results, four for problems with nonsmooth potential and one for problems with a $C^1$-potential. In the last part, for nonsmooth problems in which the potential eventually exhibits a strict super-$p$-growth under a symmetry condition, we prove the existence of infinitely many pairs of nontrivial solutions. Our approach is variational based on the critical point theory for nonsmooth functionals. Also we present some results concerning the first two elements of the spectrum of the negative $p$-Laplacian with Neumann boundary condition. Keywords:Nonsmooth critical point theory, locally Lipschitz function,, Clarke subdifferential, Neumann problem, strong resonance,, second deformation theorem, nonsmooth symmetric mountain pass theorem,, $p$-LaplacianCategories:35J20, 35J60, 35J85
4. CJM 2004 (vol 56 pp. 655)
Tao, Xiangxing; Wang, Henggeng
On the Neumann Problem for the Schrödinger Equations with Singular Potentials in Lipschitz Domains We consider the Neumann problem for the Schr\"odinger equations $-\Delta u+Vu=0$, with singular nonnegative potentials $V$ belonging to the reverse H\"older class $\B_n$, in a connected Lipschitz domain $\Omega\subset\mathbf{R}^n$. Given boundary data $g$ in $H^p$ or $L^p$ for $1-\epsilon Keywords:Neumann problem, Schrödinger equation, Lipschitz, domain, reverse Hölder class,$H^p\$ spaceCategories:42B20, 35J10
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© Canadian Mathematical Society, 2015 : https://cms.math.ca/ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8723766803741455, "perplexity": 1304.0115596177627}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398447769.81/warc/CC-MAIN-20151124205407-00177-ip-10-71-132-137.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/varation-of-parameters-fun-diff-eq-question-where-do-i-go-next.111140/ | # Varation of Parameters fun diff EQ question, where do i go next?
1. Feb 17, 2006
### mr_coffee
Varation of Parameters fun!! diff EQ question, where do i go next?
This is my first attempt at doing Variation of parameters, didn't go to bad, things cancled out pretty well but now i'm almost done but i'm stuck!
The problem says: Find the solution of y''+15y'+56y = 54*e^(-5t), with y(0) = 8, and y'(0) = 2, y = ?
Here is my work:
http://img157.imageshack.us/img157/6798/s2ly.jpg [Broken]
http://img157.imageshack.us/img157/6389/s26za.jpg [Broken]
If you don't follow me or see an error right off the bat, please let me know! Thanks!
Last edited by a moderator: May 2, 2017
2. Feb 18, 2006
### HallsofIvy
Staff Emeritus
Now integrate u1= -54e3t and u2'= 54e2t to find u1 and u2 of course! Once you know those put them into "u1e-8t+ u2e-7t to get a "specific solution" and add that to the general solution of the homogenous equation.
3. Feb 18, 2006
### mr_coffee
ahh thanks Ivey, i'm alittle confused on what u mean when u said add it to the homogenous equation, the orginal equation wasn't homogenous was set equal to 54e^(-5t), are you saying add it to the r^2+15r+56 = 0? Thanks here is what i got now:
http://img140.imageshack.us/img140/848/lastscan1ia.jpg [Broken]
Last edited by a moderator: May 2, 2017
4. Feb 18, 2006
### HallsofIvy
Staff Emeritus
Well, that's pretty much the whole idea! If $y(t)= C_1e^{-8t}+ C_2e^{-7t}$ is the general solution to y"+ 15y'+ 56y= 0 and $y(t)= 9e^{-5t}$ satisfies y"+ 15y'+ 56y= 54e-5t, then
The general solution to y"+ 15y'+56y= 54e-5t is
$y(t)= C_1e^{-8t}+ C_2e^{-7t}+ 9e^{-5t}$.
5. Feb 18, 2006
### mr_coffee
Thanks again Ivey!
I don't know if its right because webhw is still down, yay!
but this is what I got:
http://img114.imageshack.us/img114/6418/lastscan3jq.jpg [Broken]
Last edited by a moderator: May 2, 2017
6. Feb 18, 2006
### assyrian_77
I personally think it's easier to solve it like this: First, as usual, you solve the homogeneous equation which is basically setting the right hand side (RHS) to zero, i.e. $y"+ 15y'+56y=0$. This will give you $y_{h}(t)=C_1e^{-8t}+ C_2e^{-7t}$. Next you solve the "special" or "particular" equation by making an ansatz (educated guess) depending on what expression you have on the RHS. In this case I would choose the ansatz $y_{p}(t)=Ae^{-5t}$. Differentiate this and insert into the original equation gives you a value on A. The total solution is simply $y=y_{h}+y_{p}$. And then use your initial conditions to get the values of $C_1$ and $C_2$.
Sound complicated, but that's how I used to solve these things.
And yes, I too got $C_1=-40$ and $C_2=39$.
7. Feb 19, 2006
### mr_coffee
i like ur method alot better! But how did you make an educational guess of $y_{p}(t)=Ae^{-5t}$ ? I understand why u would guess an e^(t) but how did u konw it should be e^(-5t)? Thanks!
8. Feb 19, 2006
### Benny
You can find out more about that method if you look up 'method of undetermined coefficients.' I find that reduction of order is generally faster because you don't need to memorise specific forms for certain situations.
9. Feb 19, 2006
### assyrian_77
I made the ansatz $y_{p}(t)=Ae^{-5t}$ because the the RHS of the original equation had the exponential term $e^{-5t}$. More generally, I could have chosen the ansatz $y_{p}(t)=Ae^{-5t}+Be^{+5t}$ but the second term is redundant.
Similar Discussions: Varation of Parameters fun diff EQ question, where do i go next? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8710132837295532, "perplexity": 1697.677134212852}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608617.80/warc/CC-MAIN-20170525233846-20170526013846-00360.warc.gz"} |
https://www.solidot.org/news/view?id=145273 | ## Graph Isomorphism for \$(H_1,H_2)\$-free Graphs: An Almost Complete Dichotomy. (arXiv:1811.12252v1 [cs.DM])
We consider the Graph Isomorphism problem for classes of graphs characterized by two forbidden induced subgraphs \$H_1\$ and \$H_2\$. By combining old and new results, Schweitzer settled the computational complexity of this problem restricted to \$(H_1,H_2)\$-free graphs for all but a finite number of pairs \$(H_1,H_2)\$, but without explicitly giving the number of open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. By combining previously known results for Graph Isomorphism with known results for boundedness of clique-width, we reduce the number of open cases to 14. By proving a number of new results we then further reduce this number to seven. By exploiting the strong relationship between Graph Isomorphism and clique-width, we also prove that the class of \$(\mbox{gem},P_1+2P_2)\$-free graphs has unbounded clique-width. This reduces the number of open cases for boundedness of clique-width for \$(H_1,H_2)\$-free grap 查看全文>> | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8534355163574219, "perplexity": 735.4654504731302}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247518425.87/warc/CC-MAIN-20190222135147-20190222161147-00176.warc.gz"} |
http://www.r-bloggers.com/not-all-proportion-data-are-binomial-outcomes/ | # Not all proportion data are binomial outcomes
March 24, 2013
By
(This article was first published on Are you cereal? » R, and kindly contributed to R-bloggers)
It really is trivial. Not every proportion is frequency. There are things that have values bounded between 0 and 1 and yet they are neither probabilities, nor frequencies. Why do I even bother to write this? Because some kinds of proportions should be treated as unbounded continuous variables, and should be analyzed using appropriate statistical machinery (e.g. assuming normal error structure). This may not be entirely clear after reading the chapter in Michael Crawley's The R Book (2007) that deals with proportions (Chapter 16: "Proportion data") and that focuses exclusively on the proportions which are frequencies.
Proportion is frequency when we count numbers of binary outcomes of a bernoulli-distributed random process (e.g. coin toss). If one is a frequentist he can say that the proportion (or frequency) of heads in the total number of flips is equal to the bias of the coin, or he can directly link the frequency to the probability that the coin is equal. Coin tosses are a dull example, so here are other kinds of data in which proportions are frequencies and which follow the same distribution: percentage mortalities, infection rates of diseases, proportions of patients responding to treatments, sex ratios and so on (examples taken from Crawley, 2007).
These data should be modeled with the assumption of binomial error structure, for example by using logistic regression. Here is an example of such data (black dots; the data are artificial) and model (red line):
Proportion is not frequency when we use the proportion to standardize and relativize continuous data. For example, length of a male leg covers lower proportion of the total body height than length of a female leg. Or: Percentages of weight gains or losses after a medical treatment. Or: Proportional decrease of population of an endangered species resulting from proportional destruction of an area of a rain forest. And so on.
Interestingly, these proportions can sometimes have interpretable negative values (e.g. negative percentage weight loss is weight gain). Also, it is not as clear as in the previous case what error structure should we assume. I would guess that in most cases it would be the distribution of the original, non-proportional and "non-standardized" variable.
Here is an example of proportional weight loss of patients (black dots; the data are artificial) after a drug treatment. In this case normal linear regression model is fitted:
As I've said, it is quite trivial. However, do let me know if I am trivially mistaken here. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8208412528038025, "perplexity": 1036.3224944734638}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931007797.72/warc/CC-MAIN-20141125155647-00243-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/136548/ring-of-witt-vectors-and-tensor-product-of-fields | # Ring of Witt Vectors and Tensor product of Fields
Let $p > 2$ be a prime, and let $\textbf{F}_{p} = \textbf{Z}/p\textbf{Z}$. Let $k_{1}$ be a finite field over $\textbf{F}_{p}$, and let $k$ be a perfect field of characteristic $p$. Then we have ring isomorphism $k_{1} \otimes_{\textbf{F}_{p}} k \cong \oplus_{i=1}^{n} l_{i}$ where $l_{i}$ are finite extensions of $k$.
Question: How do we prove that $W(k_{1}) \otimes_{\textbf{Z}_{p}} W(k) \cong \oplus_{i=1}^{n} W(l_{i})$, where $W(k)$ denote the ring of Witt vectors of $k$?
Any suggestions or comments would be greatly appreciated.
-
It is not true in general that when $k$ is finite you will always have $k\subset k_1$ or $k_1\subset k$. – Kevin Ventullo Jul 12 '13 at 20:04
@KevinVentullo: Oops, sorry. I was being stupid. But the above result is still true in that case. I edited my question. Thanks anyway! – david Jul 12 '13 at 20:32
A family of special cases, including the ones of interest to you, of an earlier, more general question was addressed by @WilberdvanderKallen. – L Spice Jan 5 at 3:06 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9445659518241882, "perplexity": 141.93862921700529}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701163512.72/warc/CC-MAIN-20160205193923-00130-ip-10-236-182-209.ec2.internal.warc.gz"} |
http://www.computer.org/csdl/trans/tc/2003/05/t0633-abs.html | Subscribe
Issue No.05 - May (2003 vol.52)
pp: 633-644
ABSTRACT
<p><b>Abstract</b>—A statistical multiplexer is a basic model used in the design and the dimensioning of communication networks. The multiplexer model consists of a single server queue with constant service time and a more or less complicated arrival process. The aim is to determine the packet loss probability as a function of the capacity of the buffer. In this paper, we show how rational approximation techniques may be applied to compute the packet loss efficiently. The approach is based on the knowledge of a limited number of sample values, together with the decay rate of the probability distribution function. A strategy is proposed where the sample points are chosen automatically. The accuracy of the approach is validated by comparison with both analytical results obtained using a matrix-analytic method and simulation results.</p>
INDEX TERMS
Statistical multiplexing, Markovian arrival process, matrix-analytic methods, Newton-Padé approximation.
CITATION
Annie Cuyt, R.B. Lenin, Gert Willems, Chris Blondia, Peter Rousseeuw, "Computing Packet Loss Probabilities in Multiplexer Models Using Rational Approximation", IEEE Transactions on Computers, vol.52, no. 5, pp. 633-644, May 2003, doi:10.1109/TC.2003.1197129 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8983585834503174, "perplexity": 774.3725722268274}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-06/segments/1422115926735.70/warc/CC-MAIN-20150124161206-00042-ip-10-180-212-252.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/different-positive-negative-error-values.450200/ | # Different positive/negative error values
1. Nov 21, 2010
### zhermes
If you have some distribution, the standard deviation is defined as symmetric about the mean; what measure do people use for different positive and negative error values?
2. Nov 22, 2010
### St41n
What do you mean? Why can't you use standard deviation?
3. Nov 22, 2010
### HallsofIvy
Staff Emeritus
What is it, exactly, that you are trying to measure?
Similar Discussions: Different positive/negative error values | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8969783186912537, "perplexity": 3922.4697838598795}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988718296.19/warc/CC-MAIN-20161020183838-00025-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://thuviencokhi.com/news/Kien-thuc-co-khi-co-ban/Inverted-Pendulum-System-Modeling-265/ | # Inverted Pendulum: System Modeling
Thứ tư - 07/05/2014 19:26 | Đã xem: 423
## Dynamic Systems
Dynamic systems are systems that change or evolve in time according to a fixed rule. For many physical systems, this rule can be stated as a set of first-order differential equations:
(1)
In the above equation, is the state vector, a set of variables representing the configuration of the system at time . For instance in a simple mechanical mass-spring-damper system, the two state variables could be the position and velocity of the mass. is the vector of control inputs at time , representing the externally applied "forces" on the system, and is a possibly nonlinear function giving the time derivative (rate of change) of the state vector, for a particular state, input, and time.
The state at any future time, , may be determined exactly given knowledge of the initial state, , and the time history of the inputs, , between and by integrating Eq.(1). Though the state variables themselves are not unique, there is a minimum number of state variables, , required in a given system for the above to hold true. is referred to as the system order and determines the dimensionality of the state-space. The system order usually corresponds to the number of independent energy storage elements in the system.
The relationship given in Eq.(1) is very general and can be used to describe a wide variety of different systems; unfortunately, it may be very difficult to analyze. There are two common simplifications which make the problem more tractable. First, if the function, , does not depend explicitly on time, i.e. , then the system is said to betime invariant. This is often a very reasonable assumption, since the underlying physical laws themselves do not typically depend on time. For time invariant systems, the parameters or coefficients of the function, , are constant. The control input, however, may still be time dependent, .
The second common assumption concerns the linearity of the system. In reality, nearly every physical system is nonlinear. In other words, is typically some complicated function of the state and inputs. These nonlinearities arise in many different ways, one of the most common in control systems being "saturation" in which an element of the system reaches a hard physical limit to its operation. Fortunately, over a sufficiently small operating range (think tangent line near a curve), the dynamics of most systems are approximately linear, that is .
Until the advent of digital computers (and to a large extent thereafter), it was only practical to analyze linear time invariant (LTI) systems. Consequently, most of the results of control theory are based on these assumptions. Fortunately, as we shall see, these results have proven to be remarkably effective and many significant engineering challenges have been solved using LTI techniques. In fact, the true power of feedback control systems are that they work (are robust) in the presence of the unavoidable modeling uncertainty.
## State-Space Representation
For continuous linear time invariant (LTI) systems, the standard state-space representation is given below:
(2)
(3)
where is the vector of state variables (nx1), is the time derivative of state vector (nx1), is the input or control vector (px1), is the output vector (qx1), is the system matrix (nxn), is the input matrix (nxp), is the output matrix (qxn), is the feedforward matrix (qxp).
The output equation, Eq.(3), is necessary because often there are state variables which are not directly observed or are otherwise not of interest. The output matrix, , is used to specify which state variables (or combinations thereof) are available for use by the controller. Also often there is no direct feedforward in which case is the zero matrix.
The state-space representation, also referred to as the time-domain representation, can easily handle multi-input/multi-output (MIMO) systems, systems with non-zero initial conditions, and nonlinear systems via Eq.(1). Consequently, the state-space representation is used extensively in "modern" control theory.
## Transfer Function Representation
LTI systems have the extremely important property that if the input to the system is sinusoidal, then the output will also be sinusoidal at the same frequency but in general with different magnitude and phase. These magnitude and phase differences as a function of frequency are known as the frequency response of the system.
Using the Laplace transform, it is possible to convert a system's time-domain representation into a frequency-domain output/input representation, known as the transfer function. In so doing, it also transforms the governing differential equation into an algebraic equation which is often easier to analyze.
The Laplace transform of a time domain function, , is defined below:
(4)
where the parameter is a complex frequency variable. It is very rare in practice that you will have to directly evaluate a Laplace transform (though you should certainly know how). It is much more common to look up the transform of the function you are interested in in a table such as the one found here: Laplace Transform Table
The Laplace transform of the nth derivative of a function is particularly important:
(5)
Frequency-domain methods are most often used for analyzing LTI single-input/single-output (SISO) systems, e.g. those governed by a constant coefficient differential equation as follows:
(6)
The Laplace transform of this equation is given below:
(7)
where and are the Laplace Transforms of and respectively. Note that when finding transfer functions, we always assume that the each of the initial conditions, , etc. is zero. The transfer function from input to output is therefore:
(8)
It is useful to factor the numerator and denominator of the transfer function into the so called zero-pole-gain form:
(9)
The zeros of the transfer function, , are the roots of the numerator polynomial, i.e. the values of s such that . The poles of the transfer function, , are the roots of the denominator polynomial, i.e. the values of s such that . Both the zeros and poles may be complex valued (have both real and imaginary parts). The system Gain is .
Note that we can also determine the transfer function directly form the state-space representation as follows:
(10)
## Mechanical Systems
Newton's laws of motion form the basis for analyzing mechanical systems. Newton’s second law, Eq. (11), states that the sum of the forces acting on a body equals its mass times acceleration. Newton's third law, for our purposes, states that if two bodies are connected, then they experience the same magnitude force acting in opposite directions.
(11)
When applying this equation, it is best to construct a free body diagram (FBD) of the sysetm showing all applied forces.
## Example: Mass-Spring-Damper System
The free body diagram for this system is shown below. The spring force is proportional to the displacement of the mass, , and the viscous damping force is proportional to the velocity of the mass, . Both forces oppose the motion of the mass and are therefore shown in the negative -direction. Note also, that corresponds to the position of the mass when the spring is unstretched.
Now we proceed by summing the forces and applying Newton’s second law, Eq. (11), in each direction of the problem. In this case, there are no forces acting in the -direction; however, in the -direction we have:
(12)
This equation, known as the governing equation, completely characterizes the dynamic state of the system. Later, we will see how to use this to calculate the response of the system to any external input, , as well as analyze system properties such as stability and performance.
To determine the state-space representation of the mass-spring-damper system, we must reduce the second order governing equation to a set of two first order differential equations. To this end, we choose the position and velocity as our state variables.
(13)
Note also that these state variables correspond to the potential energy in the spring and the kinetic energy of the mass respectively. Often when choosing state variables it is helpful to consider the independent energy storage elements in the system.
The state equation in this case is as follows:
(14)
If, for instance, we are interested in controlling the position of the mass, then the output equation is as follows:
(15)
## Entering State-Space Models into MATLAB
Now we will show you how to enter the equations derived above into a m-file for MATLAB. Let's assign numerical values to each of the variables.
m mass 1.0 kgk spring constant 1.0 N/mb damping constant 0.2 Ns/mF input force 1.0 N
Create a new m-file and enter the following commands.
m = 1;k = 1;b = 0.2;F = 1;A = [0 1; -k/m -b/m];B = [0 1/m]';C = [1 0];D = [0];sys = ss(A,B,C,D)
sys = a = x1 x2 x1 0 1 x2 -1 -0.2 b = u1 x1 0 x2 1 c = x1 x2 y1 1 0 d = u1 y1 0 Continuous-time state-space model.
The Laplace transform for this system assuming zero initial conditions is
(16)
and therefore the transfer function from force input to displacement output is
(17)
## Entering Transfer Function Models into MATLAB
Now we will show how to enter the transfer function derived above into MATLAB. Enter the following commands into the m-file in which you defined the system parameters.
s = tf('s');sys = 1/(m*s^2+b*s+k)
sys = 1 --------------- s^2 + 0.2 s + 1 Continuous-time transfer function.
Note that we have used the symbolic s variable here to define our transfer function model. We recommend using this method most of the time; however, in some circumstances, for instance in older versions of MATLAB or when interfacing with SIMULINK, you may need to define the transfer function model using the numerator and denominator polynomial coefficients directly. In these cases, use the following commands:
num = [1];den = [m b k];sys = tf(num,den)
sys = 1 --------------- s^2 + 0.2 s + 1 Continuous-time transfer function.
## Electrical Systems
Like Newton’s laws in mechanical systems, Kirchoff’s circuit laws are the basic analytical tool in electrical systems. Kirchoff’s current law (KCL) states that the sum of the electrical currents entering and exiting a node in a circuit must be equal. Kirchoff’s voltage law (KVL) states that the sum of voltage differences around any closed loop in the circuit is zero. When applying KVL, the source voltages are typically taken as positive and the load voltages taken as negative.
## Example: RLC Circuit
We will now consider a simple series combination of three passive electrical elements: a resistor, an inductor, and a capacitor, known as an RLC Circuit.
Since this circuit is a single loop, each node only has one input and output; therefore, application of KCL simply shows that the current is the same throughout the circuit at any given time, . Now applying KVL around the loop and using the sign conventions indicated in the diagram, we arrive at the following governing equation.
(18)
We note that that the governing equation for the RLC circuit has an analogous form to the mass-spring-damper mechanical system. In particular, they are both second order systems where the charge (integral of current) corresponds to displacement, the inductance to mass, the resistance to viscous damping, and the inverse capacitance to the spring stiffness. These analogies and others like them turn out to be quite useful conceptually in understanding the behavior of dynamical systems.
The state-space representation is found by choosing the charge and current as the state variables.
(19)
where,
(20)
The state equation is therefore:
(21)
We choose the current as ouput as follows:
(22)
The transfer function representation may be found by taking the Laplace transform as we did for the mass-spring-damper or from the state-space equation as follows:
(23)
(24)
The RLC state-space and transfer fcuntion models can be entered into MATLAB using the same procedure as discussed for the mass-spring-damper system above.
## System Identification
In this section, we have seen how to model systems using basic physical principles; however, often this is not possible either because the parameters of the system are uncertain, or the underlying processes are simply not known. In these cases, we must rely on experimental measurements and statistical techniques to develop a system model, a process known as system identification.
System identification may be performed using either time-domain or frequency-domain data, see the Introduction: System Identification page.
Also refer to MATLAB’s System Identification Toolbox for more information on this subject.
## System Conversions
Most operations in MATLAB can be performed on either the transfer function, the state-space model, or the zero-pole-gain form. Furthermore, it is simple to transfer between these if the other form of representation is required. If you need to learn how to convert from one representation to the other, see the Introduction: System Conversions page.
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Mọi thắc mắc hay ý kiến xin gửi vào mục Liên hệ hoặc gửi qua Email: [email protected] | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9185543656349182, "perplexity": 939.6580603353008}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818688158.27/warc/CC-MAIN-20170922022225-20170922042225-00055.warc.gz"} |
http://mathhelpforum.com/math-topics/128423-magnitude-acceleration-help.html | # Math Help - Magnitude of acceleration help
1. ## Magnitude of acceleration help
i am having trouble with following problem and need some help in solving it, it will be much appreciated, thank you in advance
If an astronaut throws a tool with a force of 16.0 , what is the magnitude of the acceleration of the astronaut during the throw? Assume that the total mass of the astronaut after she throws the tool is 80.0 .
thanks again
2. Originally Posted by elexis10
i am having trouble with following problem and need some help in solving it, it will be much appreciated, thank you in advance
If an astronaut throws a tool with a force of 16.0 , what is the magnitude of the acceleration of the astronaut during the throw? Assume that the total mass of the astronaut after she throws the tool is 80.0 .
thanks again
reaction force is 16 N ... use Newton's 2nd, a = F/m
3. thank you very much...i got the correct answer | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9525113701820374, "perplexity": 364.32183133039337}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657134620.5/warc/CC-MAIN-20140914011214-00187-ip-10-234-18-248.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/sig-figs-still-mastering-them.187917/ | # Sig Figs, Still Mastering Them
1. Sep 29, 2007
### mircobot
I know this seem pretty ridiculous to ask to such a large forum, but for my physics lab I have found a spring constant to have a value of 0.1368 N/cm with an uncertainty of 5x10^-4. How do I express the spring constant?
So far I think it is (13.68 +/- 0.05)x10^-2 N/cm but the scientific notation is wrong on the 0.1368 value.
Also, on a side note. For my lab I made a graph Force vs Delta X of Spring. We used a nylon sheath and put mass on the end of it and recorded the distance stretched. The graph shows two linear relationships. One linear slope that starts up with a very small slop, and then suddenly jerks to the right to another linear relationship with a larger slope. My professor has already told me that it has nothing to do with the yield or ultimate strength of the nylon sheath. Any ideas to help me out?
Last edited: Sep 29, 2007
2. Sep 29, 2007
### EricVT
I don't think there is anything wrong with simply writing k = 0.1368 +/- .0005 N/cm. That's how I would personally write it, but I know how labs can be pretty strict on these sorts of procedures.
3. Sep 29, 2007
### mgb_phys
probably 1.368 (+/- 0.005)x10^-1 but i would prefer evict's as clearer
Similar Discussions: Sig Figs, Still Mastering Them | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8490014672279358, "perplexity": 1461.512780918455}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189239.16/warc/CC-MAIN-20170322212949-00504-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://testbook.com/question-answer/select-the-option-that-is-related-to-the-third-ter--6368ccf5d19cf0867107cffc | # Select the option that is related to the third term in the same way as the second term is related to the first term.12 : 72 :: 6 : ?
This question was previously asked in
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1. 28
2. 18
3. 64
4. 46
5. Not attempted
Option 2 : 18
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## Detailed Solution
The logic follows here is,
Let (1st number : 2nd number)
(1st number)2 ÷ 2 = (2nd number)
Now by following the same logic we have;
• 12 : 72
⇒ (12)2 ÷ 2 = 144 ÷ 2 = 72 = 2nd number
Similarly,
• 6 : ?
⇒ (6)2 ÷ 2 = 36 ÷ 2 = 18 = 2nd number
Hence, the correct answer is "18". | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9148128032684326, "perplexity": 3392.8789327661843}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296949642.35/warc/CC-MAIN-20230331113819-20230331143819-00352.warc.gz"} |
https://mathhelpboards.com/threads/problem-of-the-week-11-june-11th-2012.1203/ | # Problem of the Week #11 - June 11th, 2012
Status
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• Moderator
• #1
#### Chris L T521
##### Well-known member
Staff member
Jan 26, 2012
995
Thanks to those who participated in last week's POTW!! Here's this week's problem.
-----
Problem: Let $G=\{a+b\sqrt{2} : a,b\in\mathbb{Q}\}$ and let $\displaystyle H=\left\{\begin{bmatrix} a & 2b \\ b & a\end{bmatrix} : a,b\in\mathbb{Q}\right\}$ be two groups. Show that $G$ and $H$ are isomorphic as groups under addition; i.e. find a bijective map $\varphi:G\rightarrow H$ such that for any $x,y\in G$, $\varphi(x+y) = \varphi(x) + \varphi(y)$, where $\varphi(x),\varphi(y)\in H$.
Are $G$ and $H$ isomorphic under multiplication? If yes, prove it. If not, provide a counterexample.
-----
Last edited:
• Moderator
• #2
#### Chris L T521
##### Well-known member
Staff member
Jan 26, 2012
995
This problem was correctly answered by Sudharaka. You can find his solution below.
$$\mbox{Let, }\varphi:G\rightarrow H\mbox{ such that, }\varphi:a+b\sqrt{2}\mapsto\begin{bmatrix} a & 2b \\ b & a\end{bmatrix}\mbox{ where }a,\,b\in\mathbb{Q}$$
First we shall show that $$\varphi$$ is a well defined, bijective function.
Take any two elements; $$a_{1}+b_{1}\sqrt{2},\,a_{2}+b_{2}\sqrt{2}\in G$$ such that, $$a_{1}+b_{1}\sqrt{2}=a_{2}+b_{2}\sqrt{2}\mbox{ where }a_{1},\,a_{2},\,b_{1},\,b_{2}\in\mathbb{Q}\,.$$
$\Rightarrow (a_{1}-a_{2})+(b_{1}-b_{2})\sqrt{2}=0$
Since, $$a_{1}-a_{2},\,b_{1}-b_{2}\in\mathbb{Q}$$ it can be easily shown that,
$a_{1}-a_{2}=b_{1}-b_{2}=0$
$\Rightarrow a_{1}=a_{2}\mbox{ and }b_{1}=b_{2}$
$\therefore\begin{bmatrix} a_{1} & 2b_{1} \\ b_{1} & a_{1}\end{bmatrix}=\begin{bmatrix} a_{2} & 2b_{2} \\ b_{2} & a_{2}\end{bmatrix}$
$\mbox{That is }\varphi\mbox{ is a well defined function.}~~~~~~~~~~(1)$
Take any two elements; $$a_{1}+b_{1}\sqrt{2},\,a_{2}+b_{2}\sqrt{2}\in G$$ such that, $$\varphi(a_{1}+b_{1}\sqrt{2})=\varphi(a_{2}+b_{2}\sqrt{2})\,.$$ Then,
$\begin{bmatrix} a_{1} & 2b_{1} \\ b_{1} & a_{1}\end{bmatrix}=\begin{bmatrix} a_{2} & 2b_{2} \\ b_{2} & a_{2}\end{bmatrix}$
$\Rightarrow a_{1}=a_{2}\mbox{ and }b_{1}=b_{2}$
$\mbox{That is }\varphi\mbox{ is injective.}~~~~~~~~~~(2)$
Take any $$\begin{bmatrix} a & 2b \\ b & a\end{bmatrix}\in H\,.$$ Then there exist $$a+b\sqrt{2}\in G$$ such that,
$\varphi(a+b\sqrt{2})=\begin{bmatrix} a & 2b \\ b & a\end{bmatrix}$
$\mbox{Therefore }\varphi\mbox{ is surjective.}~~~~~~~~~~(3)$
Take any $$x=a_{1}+b_{1}\sqrt{2},\,y=a_{2}+b_{2}\sqrt{2}\in G$$ and consider $$\varphi(x+y)\,.$$
\begin{eqnarray}
\varphi(x+y)&=&\varphi\left((a_{2}+a_{2})+(b_{1}+b_{2})\sqrt{2}\right)\\
&=&\begin{bmatrix} a_{1}+a_{2} & 2(b_{1}+b_{2}) \\ b_{1}+b_{2} & a_{1}+a_{2}\end{bmatrix}\\
&=&\begin{bmatrix} a_{1} & 2b_{1} \\ b_{1} & a_{1}\end{bmatrix}+\begin{bmatrix} a_{2} & 2b_{2} \\ b_{2} & a_{2}\end{bmatrix}\\
&=&\varphi(x)+\varphi(y)
\end{eqnarray}
$\therefore\varphi(x+y)=\varphi(x)+\varphi(y)\, \forall\,x,\,y\in G~~~~~~~~~~~~(4)$
By (1), (2), (3) and (4);
$(G,+)\cong(H,+)$
We shall show that $$G$$ and $$H$$ are isomorphic under multiplication with respect to the same function, $$\varphi\,.$$ Consider, $$\varphi(x\cdot y)$$ where $$x=a_{1}+b_{1}\sqrt{2},\,y=a_{2}+b_{2}\sqrt{2}\in G$$
\begin{eqnarray}
\varphi(xy)&=&\varphi\left((a_{1}+b_{1}\sqrt{2})(a_{2}+b_{2}\sqrt{2})\right)\\
&=&\varphi\left((a_{1}a_{2}+2b_{1}b_{2})+(a_{1}b_{2}+b_{1}a_{2})\sqrt{2}\right)\\
&=&\begin{bmatrix} a_{1}a_{2}+2b_{1}b_{2} & 2(a_{1}b_{2}+b_{1}a_{2}) \\ a_{1}b_{2}+b_{1}a_{2} & a_{1}a_{2}+2b_{1}b_{2}\end{bmatrix}\\
&=&\begin{bmatrix} a_{1} & 2b_{1} \\ b_{1} & a_{1}\end{bmatrix}\begin{bmatrix} a_{2} & 2b_{2} \\ b_{2} & a_{2}\end{bmatrix}\\
&=&\varphi(x).\varphi(y)
\end{eqnarray}
$\therefore\varphi(x\cdot y)=\varphi(x)\cdot\varphi(y)\, \forall\,x,\,y\in G$
Since we have already shown that $$\varphi$$ is a bijective function,
$\therefore (G,\,\cdot)\cong(H,\,\cdot)$
Q.E.D.
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http://master-studios.net/cellular-automaton-used-as-dynamic-model-for-spiking-neuronal-nets-part-1-termination/ | # Cellular Automaton used as dynamic model for Spiking Neural Nets (Part I: Termination)
## Cellular Automaton
Cellular Automaton (CA) is a discret time model used in many areas and also know for heavily discussed by Stephan Wolfram in „A new kind of science“. They are best known for „Game of life“. Such a model consists of a regular grid which changes by each discret timestep. The changes are defined by a set of rules which change the state of each cell. Each rules takes the surrounding states of the cells into account. Here an example:
(souce: wikipedia)
So let’s define the model:
We have a set of cells and each cell has neighour cells. To generalize the above mentioned grid we use a graph $$G$$ which is a grid graph with edges $$E$$ and nodes/vertices $$V$$. Further we have a set of k states $$\sigma \in \Sigma$$ and $$|\Sigma| = k$$ each node $$v \in V$$ has such an initial state $$i : V \rightarrow I \subseteq \Sigma$$ (at this point the definition is kind of a „relaxed“ graph coloring). Further the system changes it’s state by each discret timetemp $$t$$. So we have a transition function $$\phi : \Sigma^n \rightarrow \Sigma$$ which changes the state of a vertex $$v \in V$$ based on the neighbour nodes‘ states given to $$\phi$$ via an n-tuple. $$\phi_{t_i}$$ determines the state at timestep $$t_i$$. Further for the timestep $$t_0$$ we have the initial states via $$i()$$. For each further timestep $$t_i$$ the states of all vertices is calculated via $$\phi()$$ synchronously and takes the states from the timestep $$t_{i-1}$$
When running such a CA it can: 1) reach a fixed state which does not change anymore, hence it is deterministic: $$\phi_{t_{i-1}} = \phi_{t_i}$$ 2) reach a state beginning from that timestep $$t_{i-d}$$ the state will cyclic, repeatly occure s.t. $$\exists v \in V: \phi(v)_{t_{i-d}} = \phi(v)_{t_i} = \phi(v)_{t_{i+d}}$$. In that cycle this equation holds for every timestep for the specific $$v$$. Since there can be many v but the cycle size $$d$$ is different it is not possible to define a termination criteria based on the states. It is hence necessary to define an abortion criteria.
## The first problem: Termination
To reach an abortion cirteria we define one further parameter. First $$l : \mathbb{N} \rightarrow \mathbb{N}$$ which takes the timestep $$t_i$$ as parameter and gives the number of cells which are „live“ (e.g. which has to be taken into account and are „not dead cells“).
$$\frac{(l(t) – l(t – 1))^{2}}{|V| + t^2} < \xi$$
$$|V|$$ gives the number of vertices in the grid graph $$G$$. To give an intuition of this formular (it very easy I think). The dividend first calculates the difference between two states and gives the square product of this difference. First of all this ensures that the difference is positive and second, like the mean squared error, some drastically changes in the cellular automaton has higher impact on the result. It did not check if it will give better results if we increase the potency number. Further the divisor: At the beginning $$t$$ will be 1 (since we can not calculate the change in the first state $$t=0$$) and to that we add the number of vertex (cells) we have at all. This polynom which adds the overall number is necessary to keep the result of the formular „low“. Otherwise we get a strange scale. Think about the situation where we only have $$t^2$$ at the divisor. It will end up at $$t=1$$ with the dividend itself an maybe will give a large number. The $$t^2$$ is added to ensure that the whole result of the formular is getting lower and lower for sure to reach the criteria. The $$\xi$$ is now a threshold which will be defined to reach the abortion.
I already implemented this abortion criteria at the game of life. See it on github: https://github.com/mrqc/spiking-neural-networks-with-cellular-automaton. with $$\xi = 0.1$$.
I will start now to put up a model of Spiking Neural Networks (SNN) which can transfer a spike train via such a CA model.
Notes: There exists a concept called CoDi, which is a cellular automaton model for spiking neural networks. My first thoughts about this was one year ago and I did not know that something like this exists. I will have a look at it. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9085102677345276, "perplexity": 445.2521100578903}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891813187.88/warc/CC-MAIN-20180221004620-20180221024620-00427.warc.gz"} |
http://mathhelpforum.com/number-theory/273561-how-can-beh-7-divides-y.html | # Thread: how can beh 7 divides y
1. ## how can beh 7 divides y
show: if 7 | 2x - 3y and 7 | 28x+35y then 7|y
(2x - 3y = 28 + 35 y) / 7
only 28 and 35 is divisible by 7 which is 4 and 5 quotient respectively... how come 7 divides y?
2. ## Re: how can beh 7 divides y
7 | 28x + 35y = 7(4x + 5y) doesn't provide us with any useful information...this is true for all integral values of x and y.
7 | 2x - 3y let x = 5 and y = 1...7 does not divide 1.
3. ## Re: how can beh 7 divides y
show: if $7 | 2x - 3y$ and $7 | 4x + 5y$ then $7|y$
4. ## Re: how can beh 7 divides y
Thanks Idea. That makes a doable problem.
$Given\ x,\ y \in \mathbb Z,\ 7\ |\ 2x - 3y,\ and\ 7\ |\ 4x + 5y.$
Spoiler:
$\therefore \exists\ m,\ n \in \mathbb Z\ such\ that\ 7m = 2x - 3y\ and\ 7n = 4x + 5y.$
$7m = 2x - 3y \implies x = 0.5(7m + 3y).$
$\therefore 7n = 4x + 5y \implies 7n = 14m + 11y \implies y = \dfrac{7(n - 2m)}{11} = \dfrac{7}{11} * (n - 2m).$
$But\ y,\ (n - 2m) \in \mathbb Z \implies \exists\ k \in \mathbb Z\ such\ that\ 11k = (n - 2m) \implies$
$y = \dfrac{7}{11} * 11k = 7k$
$THUS\ 7\ |\ y.$
5. ## Re: how can beh 7 divides y
Note that if $7|a$ then $7|2a$. If also $7|b$, then $7|(2a-b)$.
6. ## Re: how can beh 7 divides y
Thank you sir Jeff | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 3, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9080790281295776, "perplexity": 2253.0610898693685}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463608617.80/warc/CC-MAIN-20170525233846-20170526013846-00528.warc.gz"} |
https://www.physicsforums.com/threads/solving-d-e-with-laplace.102877/ | # Homework Help: Solving D.E with Laplace
1. Dec 5, 2005
I have
y'' + 2y' - 3y = Dirac(t)
I use the laplacetransformation and get
s^2Y + 2sY - 3Y = 1
Y = 1/4 * 1/(s-1) - 1/4 * 1/(s+3) (skipped some steps)
I try to use inverselaplace and recieve
y = 1/4*exp(t) * H(t) - 1/4 * exp(-3t)*H(t)
Where H(t) is the heaviside function.
the correct answear should be 1/4*exp(t) *( H(t) -1) - 1/4 * exp(-3t)*H(t).
What am I doing wrong. They ask for "the limited solution" does that mean anything particulairy?
Thnx for any help
2. Dec 5, 2005
### Tide
You left out the initial values when you performed the Laplace transform on the original DE.
3. Dec 5, 2005
There were no initial values if you mean like y(0) och y'(0). But I think I understand it anyhow. the key lies in the "limited solution". A limited solution cant eb allowed to grow infinite, and therefor we have to compensate for the 1/4 exp(t)
4. Dec 5, 2005
### saltydog
Hello guys. Are you referring to the following IVP:
$$y^{''}+2y^{'}-3y=\delta_0(t);\quad y(0)=0\quad y^{'}(0)=0^-$$
(You'll need initial conditions to solve via Laplace transforms.)
It's a bit awkward that one initial condition is given "prior" to t=0. Perhaps this is the "limiting" case the author refers to. Anyway, I don't get all those Heaviside's. I get one and it has to be inserted manually:
$$y(t)=H(t)\left(\frac{e^t}{4}-\frac{e^{-3t}}{4}\right)$$
Last edited: Dec 5, 2005 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9214469790458679, "perplexity": 3314.1141253057617}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376827137.61/warc/CC-MAIN-20181215222234-20181216004234-00377.warc.gz"} |
https://www.physicsforums.com/threads/magnetic-dipole-moment-does-not-include-the-permeability-of-free-space.78238/ | # Magnetic Dipole Moment does not include the permeability of free space?
1. Jun 7, 2005
### H_man
Magnetic Dipole Moment does not include the permeability of free space???
Hi,
The formula for a magnetic field in a current loop involves the permeability of free space.
But the formula for the magnetic dipole moment which seems to represent the flux through a current loop does not incorporate the permeability of free space.
Whats happening here?
2. Jun 7, 2005
### Meir Achuz
It's just a question of where you put the constant mu0.
It's not usually put in the mag mom. Then mu0/4pi is put into the Eq. for B, just as it is for B due to currents.
The mag mom is NOT the flux through a current loop.
3. Jun 7, 2005
### Crosson
Anything that you measure involving the magnetic moment also involves a magnetics field, and the permeability is included in that magnetic fielc.
4. Jun 7, 2005
### H_man
Thanks for that, though its still not entirely clear.
What exactly is the Magnetic Dipole Moment a measure of then?
I ask this as I know in the Stern Gerlach exp. the Ag atoms experience a torque due to the dipole moments of the electrons being acted upon by the magnetic field into which it enters. And I originally assumed that the dipole moment was a way of stating the magentic field strength of the current loop....
5. Jun 7, 2005
### Crosson
It is a measure of how strong the interaction between that dipole and an external magnetic field would be.
6. Jun 7, 2005
### Gokul43201
Staff Emeritus
Use the analogue of the electric dipole moment. This is simply the charge times a separation. You do not see $\epsilon_0$ anywhere, do you ?
Likewise, there is no reason for the magnetic dipole moment to have $\mu_0$ in it. This is by definition.
Last edited: Jun 7, 2005
7. Jun 8, 2005
### H_man
Thanks guys, its all much clearer now.
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https://www.maths.ox.ac.uk/node/33743 | # Irreducible polynomials over finite fields produced by composition of quadratics
Heath-Brown, D
Micheli, G
1 January 2019
## Journal:
REVISTA MATEMATICA IBEROAMERICANA
## Last Updated:
2020-03-25T10:12:24.69+00:00
3
35
10.4171/RMI/1072
847-855
## abstract:
© European Mathematical Society For a set S of quadratic polynomials over a finite field, let C be the (infinite) set of arbitrary compositions of elements in S. In this paper we show that there are examples with arbitrarily large S such that every polynomial in C is irreducible. As a second result, when #S > 1, we give an algorithm to determine whether all the elements in C are irreducible, using only O(#S(log q)3q1/2) operations.
1028195
Submitted
Journal Article | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8036674857139587, "perplexity": 811.595865844192}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585371813538.73/warc/CC-MAIN-20200408104113-20200408134613-00481.warc.gz"} |
https://cs.stackexchange.com/tags/recurrence-relation/new | Tag Info
1
Your recurrence isn't really defined for $n = 1$. This suggests using a different value for the lower bound in the integral. The formula given by Akra–Bazzi doesn't actually depend on the exact value of the lower bound that you choose – it can at most affect the big Theta constant or, in your case, some lower order term. If you choose any lower bound which ...
1
The guess $O(n^2)$ also works: $$T(n) \leq 2c\lfloor n/2\rfloor^2 + n \leq \frac{c}{2} n^2 + n \leq cn^2,$$ as long as $c \geq 2$. The author did not guess the answer. Presumably the author already knew the answer. In practice, for most recurrences you would encounter one of the following methods will work: Open up the recurrence (also known as ...
2
To guess simple recurrences most useful step is to study basic intuition beyond so-called master theorem that looks at any recursion as if it is implicit tree. Your case is admissible for it and thus easy: on each step you have half of task (peek one branch down in tree), and $O(n)$ work. Multiply tree height and amount of work on each step and you will get ...
1
The transformation: You define $S(m) = T(2^m)$ which is absolutely fine. $T(m) = T(m^{1/2}) + m$, so $T(2^m) = T(2^{m/2}) + 2^m$. Therefore $S(m) = T(2^m) = T(2^{m/2}) + 2^m = S(m/2) + 2^m$. That's the mistake you made, the last term is $2^m$ and not $m$. Try $n = 2^{1024}$: $T(2^{1024}) = T(2^{512}) + 2^{1024} = T(2^{256}) + 2^{512} + 2^{1024}$ and ...
-4
Complexity of above recurrence is O(m) which is O(log m) and now n = 2 ^m so m = log n and hence complexity is O(log log n).
2
You can solve this using the master theorem, whose proof uses a recursion tree argument. Define $S(n) = T(2^n)$. The main observation is that $\sqrt[3]{2^n} = 2^{n/3}$, and so $$S(n) = 9S(n/3) + O(1).$$ The solution is $S(n) = \Theta(n^2)$, and so $$T(n) = S(\log n) = \Theta(\log^2 n).$$ You can also open the recurrence directly, obtaining T(n) \...
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http://swmath.org/software/9344 | # QPSI
[QPSI]. A MAPLE package for the determination of quasi-polynomial symmetries and invariants We present the quasi-polynomial symmetries and invariants (QPSI) MAPLE package for the systematic determination of quasi-polynomial symmetries, invariants and invariant tensor fields for dynamical systems. A brief survey of the theoretical results obtained by the authors used in the package is given.
## Keywords for this software
Anything in here will be replaced on browsers that support the canvas element
## References in zbMATH (referenced in 3 articles )
Showing results 1 to 3 of 3.
Sorted by year (citations) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9901061058044434, "perplexity": 2379.493982326719}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171281.53/warc/CC-MAIN-20170219104611-00050-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://proofwiki.org/wiki/Primitive_of_Power_of_Sine_of_a_x_by_Power_of_Cosine_of_a_x | Primitive of Power of Sine of a x by Power of Cosine of a x
$\displaystyle \int \sin^m a x \cos^n a x \rd x = \frac {-\sin^{m - 1} a x \cos^{n + 1} a x} {a \paren {m + n} } + \frac {m - 1} {m + n} \int \sin^{m - 2} a x \cos^n a x \rd x + C$
$\displaystyle \int \sin^m a x \cos^n a x \rd x = \frac {\sin^{m + 1} a x \cos^{n - 1} a x} {a \paren {m + n} } + \frac {n - 1} {m + n} \int \sin^m a x \cos^{n - 2} a x \rd x + C$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9961982369422913, "perplexity": 95.83152122248207}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347432521.57/warc/CC-MAIN-20200603081823-20200603111823-00263.warc.gz"} |
http://www.scottaaronson.com/blog/?p=327 | ## Floating in Platonic heaven
In the comments section of my last post, Jack in Danville writes:
I may have misunderstood [an offhand comment about the “irrelevance” of the Continuum Hypothesis] … Intuitively I’ve thought the Continuum Hypothesis describes an aspect of the real world.
I know we’ve touched on similar topics before, but something tells me many of you are hungerin’ for a metamathematical foodfight, and Jack’s perplexity seemed as good a pretext as any for starting a new thread.
So, Jack: this is a Deep Question, but let me try to summarize my view in a few paragraphs.
It’s easy to imagine a “physical process” whose outcome could depend on whether Goldbach’s Conjecture is true or false. (For example, a computer program that tests even numbers successively and halts if it finds one that’s not a sum of two primes.) Likewise for P versus NP, the Riemann Hypothesis, and even considerably more abstract questions.
But can you imagine a “physical process” whose outcome could depend on whether there’s a set larger than the set of integers but smaller than the set of real numbers? If so, what would it look like?
I submit that the key distinction is between
1. questions that are ultimately about Turing machines and finite sets of integers (even if they’re not phrased that way), and
2. questions that aren’t.
We need to assume that we have a “direct intuition” about integers and finite processes, which precedes formal reasoning — since without such an intuition, we couldn’t even do formal reasoning in the first place. By contrast, for me the great lesson of Gödel and Cohen’s independence results is that we don’t have a similar intuition about transfinite sets, even if we sometimes fool ourselves into thinking we do. Sure, we might say we’re talking about arbitrary subsets of real numbers, but on closer inspection, it turns out we’re just talking about consequences of the ZFC axioms, and those axioms will happily admit models with intermediate cardinalities and other models without them, the same way the axioms of group theory admit both abelian and non-abelian groups. (Incidentally, Gödel’s models of ZFC+CH and Cohen’s models of ZFC+not(CH) both involve only countably many elements, which makes the notion that they’re telling us about some external reality even harder to understand.)
Of course, everything I’ve said is consistent with the possibility that there’s a “truth” about CH floating in Platonic heaven, or even that a plausible axiom system other than ZFC could prove or disprove CH (which was Gödel’s hope). But the “truth” of CH is not going to have consequences for human beings or the physical universe independent of its provability, in the same way that the truth of P=NP could conceivably have consequences for us even if we weren’t able to prove or disprove it.
For mathematicians, this distinction between “CH-like questions” and “Goldbach/Riemann/Pvs.NP-like questions” is a cringingly obvious one, probably even too obvious to point out. But I’ve seen so many people argue about Platonism versus formalism as if this distinction didn’t exist — as if one can’t be a Platonist about integers but a formalist about transfinite sets — that I think it’s worth hammering home.
To summarize, Kronecker had it backwards. Man and Woman deal with the integers; all else is the province of God.
### 91 Responses to “Floating in Platonic heaven”
1. Luca Says:
Amen!
2. Moodworves Says:
I’m having trouble imagining a process who’s outcome is dependent on the P vs. NP question. Certainly if P=NP, and the proof/algorithm is easy enough to find, that could influence a process, but is there a process that gives the answer to the P vs. NP question? In other words, is the P vs. NP question reducible a halting problem?
3. komponisto Says:
Let Man and Woman deal with the integers; all else is the province of God.
This sounds dangerously close to a quote of Errett Bishop:
“Classical mathematics concerns itself with operations that can be carried out by God.. Mathematics belongs to man, not to God… When a man proves a positive integer to exist, he should show how to find it. If God has mathematics of his own that needs to be done, let him do it himself”.
Such a view is of course Wrong with a capital W, for several reasons:
1. God doesn’t exist, so if we don’t do it, no one will.
2. It implicitly assumes Platonism: that when mathematicians talk about transfinite sets, they aren’t just talking about consequences of the ZFC axioms in the first place.
3. We don’t know enough about physics to know whether transfinite sets “correspond to reality” even in the naive sense that everyone always assumes in these discussions.
4. Assuming we did have this kind of knowledge, we would need to express it in the form of a formal physical theory, which would then necessarily make use of these mathematical concepts.
4. Scott Says:
Moodworves, P=NP is reducible to the halting problem with an oracle for the halting problem. To put it differently, P=NP means that there exists a Turing machine M and integers c,k such that for all SAT instances φ of size n, M decides φ after at most cnk steps. Ignoring the details, this is tantamount to saying that there exists an integer x such that for integers y, some computable predicate A(x,y) is true. Which is still a statement about integers (just like Goldbach’s Conjecture), the only difference being that now there are two quantifiers instead of one.
5. Scott Says:
komponisto, two quick responses:
1. My view is very different from the view of Bishop you quoted. Unlike him, I have no problem at all accepting nonconstructive existence proofs. (Of course constructive proofs often yield deeper insights, better algorithms, etc., but those are added bonuses.) More generally, I’d never, ever advocate throwing away interesting math to uphold a philosophical principle. I’m not talking about discarding anything we can prove; I’m talking about how to deal with statements we know we can’t prove.
2. If you think a saying like “all else is the province of God” presupposes God’s existence, you’re reading way too much into it! Think of the legal concept “act of God” (meaning “not an act of anyone you can sue”).
6. Moodworves Says:
Ah, thanks Scott!
Correct me if I’m wrong, but it seems that the second quantifier makes the P vs. NP question fundamentally different from Goldbach’s Conjecture, in that it is possible that it doesn’t affect the behavior of any computable process. Since our universe (as far as we understand it) doesn’t have any halting oracles, it could some day be relegated to Plato’s Math Heaven (where it can sit around with the Continuum Hypothesis). …That’s a strange thought considering the importance a positive or negative result would have.
7. Liron Says:
Thanks Scott, I was really wondering about what’s real and what isn’t, but now it’s obvious in retrospect.
8. csrster Says:
Scott, shh! You were on your way to a Templeton Prize nomination there for a couple of hours.
9. asdf Says:
Scott, I saw your paper about whether P=NP might be independent of ZFC. ZFC seems pretty artificial and bogus to me (powerset function, ok; iterating it through the entire transfinite list of ordinals: wtf?). And maybe that independence is unlikely.
But I wonder if anyone has considered whether P=NP might be independent of first order Peano arithmetic. I’m thinking of the graph minor theorem (a generalization of Kruskal’s tree theorem) as an example of that sort of thing. It doesn’t seem so outrageous (at least to a neophyte like me) that a theorem about an infinite set of graphs is independent of PA, since the obvious well-orderings on these graphs have order type larger than epsilon-0 which is the largest ordinal that “fits” in PA, so some inductions over those graphs may not fit in PA. And the graph minor theorem proves nonconstructively(!!) that certain problems are solvable in polynomial time.
Anyway in an earlier comment thread someone suggested it would be amusing if P=NP but the fastest algorithm for SAT was O(n^10000) or something like that. If P vs NP is independent of PA, it might be natural for the situation to be a lot worse, e.g. P=NP (provable in 2nd order arithmetic) but the fastest algorithm for SAT is O(n^B(B(k+7))) where k is the number of states of the smallest TM that solves an np-complete problem (in exponential time) and B is the (uncomputably fast-growing) busy beaver function.
Anyway the above is just raving bogosity, no real mathematical thought behind it, but if it turns out to be true, there’s probably someone out there in Platonic heaven laughing at us….
10. asdf Says:
Also I wonder if you know of Woodin’s argument that CH is false, based on infinitary logic. And Paul Cohen was not a platonist but nonetheless once said he thought CH was obviously false, because the powerset operation was so much more powerful than diagonalization.
I’ve been trying to make up a joke but all I have is the beginning, so sorry if that gets you ready to hear something funny, but the funny part doesn’t come. It begins: Kurt Godel died in 1978 and went immediately to heaven, where like all new arrivals, he was given wings and a harp and asked if he had any questions. The first thing he wanted to know was whether the continuum hypothesis is true. …
I’m not sure where to go from there. I’ve thought of a couple directions. Maybe someone can think of a suitable conclusion. I realized after a while that it’s sort of similar to a very famous joke about Wolfgang Pauli and the fine structure constant, that you all probably know.
11. asdf Says:
Darn, I wish there was a way to edit these comments. Anyway I had forgotten to ask in the first one, whether anyone had thought of what natural well-orderings there might be on P-time Turing machines and that sort of thing, and what the corresponding ordinals would be. Is that a completely bogus thing to want to know?
12. Pascal Koiran Says:
Maybe one reason why no “natural” example has been found yet is that one would have to reason outside of ZFC to obtain such an example.
Consider for instance diophantine equations. Since their solvability is Turing-undecidable (Matiyasevich) there must exist specific equations whose solvability is undecidable in ZFC (assuming of course that ZFC is consistent, or you can prove anything and its negation).
But the moment you exhibit such an equation (call it E) you know that E is, in fact, unsolvable! Indeed, simple proofs of solvability (namely, an integral solution) exist for all solvable equations.
We therefore have a proof (in ZFC) of non-solvability for E.
So the original proof that E is ZFC-undecidable must have been obtained outside of ZFC. QED
Possibly, this argument breaks down if one looks for a “natural example” which is higher up in the arithmetic hierarchy.
13. Job Says:
If P=NP, then i find it unintuitive that NPC problems would require n^c where c is very large. Why would it be n^10000 for example? All the relevant data can be scanned in n^2, so it would either be of the form 2^f(n) due to some requirement that a portion of the 2^n subsets be analyzed, or of the form n^c where c
14. Job Says:
…where c is small (~3).
15. asdf Says:
Job, if c were small it would be computable and someone would have solved the problem by now ;-). That’s what makes me chuckle over the idea that it’s enormous, uncomputably large in some parameter having to do with the problem class.
Pascal, where would something like the twin primes conjecture come in? It might be independent but the independence doesn’t imply either its truth or its falsity.
16. Andy Says:
I agree with Scott’s general message, although I think as we go higher in the arithmetic hierarchy we should be correspondingly less confident about our intuitions.
The one point I would emphasize is that in asking how ‘concrete’ or ‘physically meaningful’ a statement S is, we should look, not at its syntactic, ‘apparent’ content (cardinalities of the sets it concerns, level of existential-universal alternation, etc.), but at its ‘essential’ content: the syntactic content of the least-complex statement S’ provably equivalent to S under ZFC (though we have latitude to decide what syntactic resources count as ‘complex’), or at least the least-complex known equivalent.
Scott already affirmed this, when he described P vs NP as being reducible to a halting problem relative to a HALT oracle. A naive translation would’ve placed P vs NP one level higher in the arithmetic hierarchy (corresponding to an attempt to ‘guess’ which NP machine accepted a hard language), but Cook’s theorem allows us to remove that quantifier–we know a SAT checker is the best guess.
Similarly, research in set theory has shown that CH is provably equivalent to lower-complexity statements (i.e. ones with fewer alternations). Bill G. discussed one example: CH is equivalent to
‘there exists a coloring of the reals with countably many colors, without a monochromatic solution to x + y = w + z in distinct w, x, y, z.’
Can we even rule out that CH might be provably equivalent to a statement involving only natural numbers, albeit high in the arithmetic hierarchy?
17. Andy Says:
Bill’s discussion:
http://weblog.fortnow.com/2007/11/it-was-stupid-question-or.html
18. Scott Says:
asdf, a few responses:
1. If you go further into my survey, you’ll see that there’s a whole subfield (which Razborov, Krajicek, Pudlak, and others have been involved in) which tries to prove P=NP independent of weak fragments of PA. For example, it’s now known that Resolution and various other proof systems too weak to prove things like the Pigeonhole Principle can’t prove circuit lower bounds. Alas, these techniques are nowhere near being able to handle anything as rich as PA — and the great irony is that if they were, then we’d probably understand enough about proof complexity to be able to prove NP≠coNP (which implies P≠NP)!
2. I don’t understand Woodin’s argument for 2Aleph0=Aleph2, though I know he has such an argument. My own favorite argument for not(CH) is Freiling’s: in ZFC+CH, it’s possible to assign a countable subset S(x)⊂[0,1] to every real number x∈[0,1], so that for every (x,y) pair, either y∈S(x) or x∈S(y). That seems incredibly counterintuitive: how could a set that’s countable for every x possibly be “dense” enough that the union of it with its flipped version would cover the entire unit square? Whereas in ZFC+not(CH) this isn’t possible. (See Lecture 2 of the Democritus series.)
3. Like ZFC, your joke about Gödel has multiple consistent extensions. Maybe he ends up starving himself because he thinks God is going to poison him?
19. david Says:
Pascal, how do you draw the conclusion that “there must exist specific equations whose solvability is undecidable in ZFC” from “solving diophantine equations is undecidable”? This only means that there is no general algorithm to solve the question, but any specific equation may be proven not to have any solutions in ZFC, only that the same proof method won’t work for all of them.
20. david Says:
Ok I get it, we can enumarate all statements provable in ZFC and see if one of these says that equation has no solution.
21. Scott Says:
I removed the word “Let” from the sentence “Let Man and Woman deal with the integers…”, just to eliminate a whiff of prescriptive dogmatism.
22. Pascal Koiran Says:
>Ok I get it, we can enumarate all statements provable
>in ZFC and see if one of these says that equation
> has no solution.
That’s correct.
23. david Says:
Anyway it is true that we can exhibit specific equations whose solvability is unprovable in ZFC, once we fix encodings.
We can write a program P which, given a diophantine equation E, enumerates all proofs in ZFC and checks if one of them gives a solution of E, or proves there is none, and if so, outputs the answer.
We want to find an E such that P(E) never halts (which implies E is unsolvable). But the proof of non-computability of the halting problem tells us just how to do this.
Given an integer i, we can write a program Q(i) that does the following:
1. Write a diophantine equation E(i) such that E(i) has a zero if and only if the I-th Turing machine halts on input i.
2. Compute P(E(i)); if it halts, we know i does not halt on input i (this is where we use that ZFC is consistent).
Now if we let i = code of program Q, it follows that P(E(Q)) does not halt, so e(Q) is unprovable in ZFC.
So, it seems there is an algorithm to explicitly find an expression (namely E(Q)) such that E is undecidable in ZFC, and hence unsolvable.
We can also prove in ZFC that if E is solvable, there is a proof for this (compute the value of E on the purported solution).
I guess (I’m not entirely sure) inside ZFC we can carry out all these steps, but we have to assume ZFC is consistent in order to draw the conclusion that P(E(i)) halts implies Mi(i) doesn’t. Am I right?
24. Pascal Koiran Says:
David, at first glance I am happy with your argument; the only thing that worries me is that the conclusion that you reach is in contradiction with mine…
I am afraid that I don’t know enough set theory to be sure who’s right.
We need a wise, fair and knowledgable referee in this case:
Scott, please weigh in!
25. Scott Says:
Pascal: Yes, I believe it’s possible to write down an explicit Diophantine equation that has a solution iff ZFC is inconsistent (and hence, whose solvability is independent of ZFC). This is non-obvious but should follow from the Matiyasevich/MRDP Theorem. I don’t know if that Diophantine equation will need variables in the exponents or not. Does anyone who actually knows want to enlighten us?
26. Pascal Koiran Says:
Scott, I am beginning to believe it too. The equation E would “simulate” a Turing machine that enumerates all proofs in ZFC and halts when it finds a contradiction.
So indeed, Solvable(E) is independent of ZFC (assuming that ZFC is consistent).
However, the statement E’:
Solvable(E) “ZFC inconsistent”
would presumably be provable in ZFC (because the Matiyasevich-based argument can presumably be formalized within ZFC).
So the next challenge would be to exhibit a diophantine equation E such that the associated statement E’ is independent of ZFC!
27. zzz Says:
CH just says sets are either computably enumerable |Z|, or uncomputable |R|. Never really understood what the fuss is.
28. Pascal Koiran Says:
Also I think you would not need variables in the exponents of E: Matiyasevich showed that those don’t buy you any additional power (and the conversion from these “exponential diophantine equations” to ordinary diophantine equations is effective!)
29. Pascal Koiran Says:
One last comment for tonight: in my definition of E’ an equivalence sign got eaten by wordpress.
Solvable(E) equivalent to “ZFC inconsistent”.
30. Sam Nead Says:
I have a question which I hope belongs in this thread: Suppose that T is a Turing machine and N is an input. It is possible that there is a proof that T(N) halts and perhaps there is a proof that T(N) does not halt. Of course, a formal proof assumes some collection of axioms. So, can there be a pair (T, N) which provably halts if we assume ZFC and provably does not halt if we assume ZF+notC?
Can the workings of a Turing machine depend on the axioms we assume? And if so, what could this possibly mean?
31. Joseph Hertzlinger Says:
I’ve been trying to make up a joke but all I have is the beginning, so sorry if that gets you ready to hear something funny, but the funny part doesn’t come. It begins: Kurt Godel died in 1978 and went immediately to heaven, where like all new arrivals, he was given wings and a harp and asked if he had any questions. The first thing he wanted to know was whether the continuum hypothesis is true. …
By analogy with the Pauli joke, God would hand Godel a paper with an elegant philosophical argument answering the Continuum Question. Godel would leaf through it, point to page epsilon 0, and say “There’s a mistake right over here…”
32. asdf Says:
Sam Nead, I think there is a TM like you are asking for. Someone more knowledgeable should confirm/unconfirm this, but I believe that the MRDP theorem implies that one can construct a diophantine equation system that has a solution (set of integers) iff AC is true. So the TM would just start enumerating sets of integers and checking whether they were a solution to that diophantine system, halting if a solution is fonud.
33. Scott Says:
Sam and asdf: No, a given Turing machine either halts or doesn’t halt; it makes no difference what axioms you assume! This is an absolutely crucial point, and is a huge part of what I was trying to get across in this post.
You might ask, what makes me so sure? Well, suppose you want to believe that a given Turing machine M has “indeterminate” behavior — i.e., that there’s no objective fact about whether M halts or not, separate from what can be proved about the question in various formal systems like ZFC. Then why on earth would you suppose there’s an objective fact about whether ZFC proves M halts? After all, the existence of a proof just corresponds to the halting of another Turing machine. So you see that there’s an infinite regress: if the question “does M halt?” is meaningless in the absence of a proof one way or the other, then question “is there a proof?” is equally meaningless.
Let’s consider two concrete examples:
1. A Turing machine that searches for inconsistencies in ZFC will run forever, assuming ZFC is consistent. Of course ZFC can’t prove it runs forever, but that’s just because ZFC is consistent (i.e., because it does run forever)!
2. On the other hand, I now claim that assuming ZF+Con(ZF) is consistent, there’s no Turing machine that can be proved in ZF to halt iff AC is true.
Proof: Suppose such a machine M existed. Then
ZF |= “If M halts, then AC is true.”
This implies that if M halts, then ZF proves AC. But we know from Cohen that if ZF is consistent then ZF doesn’t prove AC. Hence M doesn’t halt.
Now, the whole argument above can be formalized in the system ZF+Con(ZF). Hence ZF+Con(ZF) proves that M doesn’t halt. This means that ZF+Con(ZF) proves not(AC). But we know that ZF+Con(ZF) doesn’t prove not(AC), assuming ZF+Con(ZF) is consistent. (I can’t remember who showed this, but it follows from the fact that large cardinal axioms don’t decide AC.) So M can’t exist, QED.
(Question for experts: can one weaken the assumption in the above result, from ZF+Con(ZF) is consistent to ZF is consistent?)
So asdf: no, the MRDP theorem can’t possibly yield a diophantine system that has a solution iff AC is true. What you might have been thinking is this: the MRDP theorem does yield a diophantine system that has a solution iff ZF proves AC. But that’s a different question, and assuming ZF is consistent we know the answer to it (no) — hence the diophantine system in question simply won’t have a solution.
34. asdf Says:
Scott, doh, thanks, that was a good explanation of my silly error. What I was actually thinking may have been somewhat different:
Is there a Turing machine T such that
1) T halts on every input
2) That T halts on every input is provable in ZFC, but it is not provable in ZF in the absence of AC.
I believe the answer to the above is yes.
35. wolfgang Says:
> So, can there be a pair (T, N) which provably halts if we assume ZFC and provably does not halt if we assume ZF+notC?
Now this is a really stupid question:
What if N (the input) is simply “we assume ZFC” or “we assume ZF+notC” in whatever encoding and T (the Turing machine) simply checks if it is one or the other?
36. wolfgang Says:
Sorry, please disregard my previous comment, it makes no sense.
37. Jack in Danville Says:
I get it! There are only enumerably transfinite quantities in the physical world. That’s a profound statement. It follows there are ultimate Planckian units of time and volume, the points of the physical world; and geodesics have a dimension orthogonal to the dimension of direction.
38. Gil Kalai Says:
“For mathematicians, this distinction between “CH-like questions” and “Goldbach/Riemann/Pvs.NP-like questions” is a cringingly obvious one, probably even too obvious to point out. But I’ve seen so many people argue about Platonism versus formalism as if this distinction didn’t exist — as if one can’t be a Platonist about integers but a formalist about transfinite sets — that I think it’s worth hammering home.”
I must admit that I do not see this distinction as obvious or clear, in fact, I do not see it. I am not sure it is meaningful for practicing mathematics. For both kinds of questions some intuition emerges and occasionally the intuition is incorrect.
39. Scott Says:
It follows there are ultimate Planckian units of time and volume, the points of the physical world; and geodesics have a dimension orthogonal to the dimension of direction.
Huh? I don’t understand what you’re talking about (what’s the “dimension of direction”?). I was only arguing for an implication in the other direction: it follows from the fact that we’re finite creatures who can only ever engage in finite chains of reasoning (and in particular, from the Church-Turing Thesis), that CH can have no effect on us separate from its provability.
40. Scott Says:
Is there a Turing machine T such that
1) T halts on every input
2) That T halts on every input is provable in ZFC, but it is not provable in ZF in the absence of AC.
Interesting question! I’m pretty sure a generalization of the argument from my previous comment would rule this out. I just landed in Seattle and am way too tired to think, but let me get back to you about it later (unless someone wants to beat me to the punch).
41. asdf Says:
Joseph Hetzinger, I like your conclusion of the joke. Maybe Woodin’s argument could even be incorporated into it somehow.
Gil and Scott, I thought the difference between a Goldbach-type question and a CH type question was the number of alternating quantifiers. That would make the twin prime conjecture (that there are infinitely many p such that p and p+2 are both prime) a CH-type question: it doesn’t involve any uncountable sets, but it is not necessarily decidable by a Turing machine as either true or false.
Re being a platonist about the integers but a formalist about transfinite sets: well, what are the integers anyway? They are described (up to isomorphism) by the Peano axioms in second-order logic, but believing those axioms would impute some kind of existence to every subset of the integers, and there are uncountably many such subsets…
42. Scott Says:
asdf: No, the difference between “Goldbach-type questions” and “CH-type questions” has nothing to do with the number of quantifiers. The difference is that in the one case the quantifiers range over integers, while in the other they range over transfinite sets.
And no, I don’t agree that the Peano axioms “impute some kind of existence” to every subset of integers. After all, the key point about PA is that it only involves quantification over integers, and not over sets of integers.
43. asdf Says:
Arggh, I misspelled Joseph Hertzlinger’s name above, I was thinking of someone else after scrolling down. My apologies.
Scott, I don’t see how to convert your argument about ZF vs ZFC into one where we’re only discussing provability of whether the machine halts. But, I’m not very good at this subject, as you can surely tell.
BTW, here is an interesting article by Doron Zeilberger, who doesn’t believe in any infinite sets of any kind, i.e. he thinks there are really only finitely many integers and there is a largest one, and shows how to develop calculus from there: pdf link.
44. asdf Says:
PA (first order Peano arithmetic) quantifies only over integers, but because of that, it has nonstandard models. Goodstein’s theorem is a fairly straightforward theorem about integers that is unprovable in PA (but provable in PA+CON(PA) if I understand it right). The classical Peano axioms contain an induction axiom which says (from Wikipedia “Peano axioms”): If K is a set such that:
* 0 is in K, and
* for every natural number n, if n is in K, then S(n) is in K,
then K contains every natural number. This is where second-order logic comes in: that axiom quantifies over all sets of integers. The result is that the Peano axioms have only one model. However, since it’s second-order logic, the completeness and compactness theorems of first-order logic don’t hold, so there are sentences about the unique model of the Peano axioms that are true but are not theorems. I don’t think we can say first-order PA describes the platonic integers, since PA has (as you put it) multiple consistent extensions.
45. cody Says:
asdf, havent you read Scott’s biggest number essay?
…Doren must have discovered that 83 is the largest integer.
46. cody Says:
i am not intimate enough with ZFC to know why we all have such confidence (faith?) in its consistency. so, as a physicist, id like to admit that lack of simultaneity is still a very non-intuitive result for me to cope with… and so my question is, why are Gödel’s incompleteness theorems so well accepted when ZFC is not provably consistent?
also, in regards to the last post, mathematicians seem to be (on average), the most demanding, least accepting of conjecture, group of individuals so far established, (if not possible, thanks Cauchy), so its hard to imagine you guys biting bullets at all. which is intended as a compliment, not criticism.
47. John Sidles Says:
I’d like to thank asdf for providing the link to Doron Zeilberger’s ultrafinitist manifesto … this essay was a lot of fun to read!
Zeilberger’s observation that “There are many ways to divide mathematics into two-culture dichotomies” was for me an especially enjoyable starting point.
Zeilberger’s essay divides mathematics into an (old-fashioned) culture of the continuous versus a (new-fangled) culture of the discrete. The essay then argues that the discrete side of the dichotomy contains all of the truth of the continuous side, with none of the bedeviling transfinite conundrums.
But is this really the case? Zeilberger’s essay notes the ubiquity of interval arithmetic in theorem-proving on the discrete side. He asserts that this arithmetic is governed by “obvious rules” … leaving the reader to assume that these rules have no philosophical depth.
But are all the implications of interval arithmetic’s seeming simple rules really obvious? Definitely not! Until quite recently, for example, it definitely was *not* obvious that computing the cube of an interval matrix is NP-hard.
Zeilberger’s essay thus confronts us with an unexpectedly difficult dichotomous choice between two mathematical paradises: Cantor’s transfinite paradise—which has the flaw that seeming truths in set-theory are undecidable—versus Zeilberger’s “ultrafinite paradise”—which the flaw that even the simplest mathematical questions have answers that are NP-hard to compute.
Since my own mathematical philosophy is unitarian, it is for me an article of faith that all mathematical paradises are fundamentally the same paradise.
It follows, therefore, that the transfinite obstruction to uniquely choosing set-theory axioms must be identical to the ultrafinite obstruction of proving P≠NP.
Having demonstrated philosophically that this transfinite / ultrafinite equivalence must exist, I will leave the details of actually proving it to mathematicians who are wiser than myself! 🙂
48. Jack in Danville Says:
Huh?
Doh! Well I thought (and bought) you were arguing there are physically no transfinite sets of cardinality greater than Aleph-null.
That would apply to points in physical space. If the collection of points in a line segment, or any finite path, cannot have a higher cardinality, I cannot see how the set can have a cardinality of merely Aleph-null, so the points in a finite path, or a finite volume of space, must be finite.
(If I haven’t already gotten into trouble, surely this is where I do.)
Finite points in finite space requires a smallest unit of space (a Planck volume?). Any path in spacetime, for instance a geodesic, would consist of a series of these teeny-tiny volumes strung together. Hence as well as having length the path would have a circumference (an additional dimension perpendicular to the dimension of length).
49. Pascal Koiran Says:
Even if there is a smallest unit of length, quantum mechanics could provide the continuous with a victory over the discrete: amplitudes are complex numbers, and I’ve never heard of a “smallest amplitude.”
Have the physicists ever proposed such a thing ?
50. Bram Cohen Says:
Scott, given that ZFC is consistent, doesn’t that mean that every diophantine equations with no solutions qualifies as one having solutions iff ZFC is inconsistent?
51. Scott Says:
Bram: What you want, and what Matiyasevich/MDRP gives you, is a Diophantine equation that can be proved in ZFC to have solutions iff ZFC is inconsistent.
52. Scott Says:
Pascal: Plenty of people have speculated about “QM with discrete amplitudes,” but no one has proposed such a theory that makes any sense. The fundamental problem is that the discrete subgroups of the unitary group all seem to be “trivial” (e.g., they don’t allow entanglement) or “unphysical” (e.g. the Clifford group).
53. Scott Says:
Well I thought (and bought) you were arguing there are physically no transfinite sets of cardinality greater than Aleph-null.
Jack: Sets are mathematical objects; I’m not even sure what it would mean for them to “physically exist.” For me the question is not what exists; it what we ever need to invoke to explain our experiences. Because we’re finite beings, who live for finite amounts of time and discriminate between observations with finite precision, all our knowledge and reasoning can be expressed as finite strings of bits. Goldbach’s Conjecture and the Riemann Hypothesis both make predictions about what the outcomes of certain operations on finite strings of bits are going to be, whereas CH makes no such prediction. That’s the key difference between them as I see it.
54. Scott Says:
my question is, why are Gödel’s incompleteness theorems so well accepted when ZFC is not provably consistent?
Cody: I wouldn’t say Gödel’s theorems require us to “assume” ZFC is consistent. They say either there are such-and-such limits on what ZFC can prove, or else ZFC is inconsistent — in which case it can prove anything, but who cares?
55. John Sidles Says:
Pascal: lattice gauge theory has all the ingredients you require — space is discrete, the values of the gauge fields are (or can be chosen to be) discrete too, and the resulting theory is well-posed mathematically, efficient algorithmically, and can be directly linked to experiment.
This article by Kenneth Wilson is a wonderful account of how all these ideas were worked-out.
From a fundamental physics point of view, however, this discretizing leads nowhere — by design! — because the whole point is to devise a lattice theory such that the discreteness parameter disappears from the final predictions.
This is yet another example of the ubiquity of “duality” in physics and mathematics … in which the main point of Discipline “A” commonly appears as a small parameter or unwanted side-effect of Discipline “B”.
My own interest in the Continuum Hypothesis chiefly resides in trying to guess what other problems it might be dual to. I am a little bit surprised that no one else is posting about this point of view!
56. komponisto Says:
Scott:
Goldbach’s Conjecture and the Riemann Hypothesis both make predictions about what the outcomes of certain operations on finite strings of bits are going to be, whereas CH makes no such prediction. That’s the key difference between them as I see it.
But if you’re a formalist, to ask about CH is just to ask whether there is a proof of CH in ZFC — and then we’re right back in Turing Machine Land.
So my question for you, Scott, is: why aren’t you a formalist?
57. asdf Says:
A formalist might believe that ZFC is not the best formalization of set theory for doing math in. They might prefer some other axioms instead. And then the question of whether CH is a theorem is back in play.
58. Scott Says:
komponisto: I’m reluctant to buy into any sort of -ism without being sure of what I’m getting. So for example, could a formalist believe P≠NP, even supposing the question were proved independent of ZFC? If not, then I am not a formalist.
59. komponisto Says:
Along the lines of asdf’s comment, I suspect that if the P vs. NP question were to be proved independent of ZFC, there would be a movement to revisit ZFC’s status as the “official” axiom system of mathematics. (Indeed, there was/is such a movement with CH, but it hasn’t really taken off, I suppose because CH is not seen as a particularly urgent question by the larger mathematical community.)
Like Platonists, formalists can have preferences among axiom systems; the difference is that formalists don’t attribute “incorrectness” to the systems they’re less interested in.
60. John Sidles Says:
Komponisto, please correct me if I’m wrong, but if the P vs. NP question were proved to be independent of ZFC, wouldn’t that immediately imply P≠NP?
On the following grounds. One proof that P = NP would be a concrete algorithm in P that solved NP-complete problems. So if P≠NP is independent of ZFC, then no such proof exists, and hence, no such algorithm exists.
Probably this point is already clear to most people … or else my own understanding of this implication is simply wrong.
61. komponisto Says:
John: My understanding from reading Scott’s paper on this topic is that there might be such an algorithm, but it might be impossible to prove that it works.
By the way, I thought your earlier comment was right on the mark.
62. komponisto Says:
Let’s try that second link again.
63. asdf Says:
CH used to be an urgent issue. It stopped being urgent when Cohen proved its independence.
I don’t think it’s possible to prove P!=NP is independent of ZFC. I.e. it might be independent, but (within ZFC) there can be no proof of this. The reason is that P!=NP is a statement about the standard integers, and these are the same in every model of ZFC, unlike the situation with CH. Maybe Scott’s article says more about this. I should re-read it now that I know a little bit more logic than I did the last time.
64. asdf Says:
No wait, what I said above makes no sense. If the standard integers are the same in every model of ZFC, then obviously a first-order statement about them can’t be independent. Can somebody straighten me out?
65. Scott Says:
Live from STOC 2008:
asdf: It’s perfectly conceivable (even if astronomically unlikely) that P≠NP could be proved independent of ZFC, despite being about standard integers. Consis(ZFC) is also about standard integers, but we know it’s independent of ZFC.
John: If Goldbach’s Conjecture were proved independent of ZFC, that would immediately imply Goldbach’s Conjecture. However, the same is not true for P≠NP. The difference is that if Goldbach is false, then there’s necessarily a proof it’s false; but if P=NP, then there’s not necessarily a proof (as komponisto says, there could be a polytime algorithm for SAT, but no way to prove its efficiency or correctness).
66. Job Says:
If “is P!=NP?” is a particular instance of a problem L and “is P=NP?” is an instance of a problem L’, then would L be in NP? What about L’?
Formally i don’t know what L’s input is, but it seems plausible that given a proof that P!=NP, it can be quickly verified. Is that probably the case?
67. Job Says:
In more detail, to avoid asking a blurry question, suppose L is the problem:
Given two complexity classes A and B, are they different?
And L’ would be the complement.
68. Scott Says:
Job, “complexity class” is itself a blurry notion.
69. John Sidles Says:
Konponisto and Scott, thank you both for your clarifying replies, which helped my understanding a lot … Scott’s survey article was a very great help too. Tricky stuff, this set theory!
70. mitchell porter Says:
Scott: can you imagine a “physical process” whose outcome could depend on whether there’s a set larger than the set of integers but smaller than the set of real numbers?
Naively it seems possible that the subset structure of the continuum might have ‘detectable’ implications for real analysis, and hence for continuum-based physics. I’d ask the gurus of FOM, such as Harvey Friedman, some of whom have worked on the practical implications of large cardinal axioms.
71. Walt Says:
As far as I understand your argument, Scott, it advances two basic claims. The first claim is that there exist statements whose truth has no impact on what theorems are true about integers and Turing machines. It’s not obvious that this is true. Fortunately, it really is true, and the Continuum Hypothesis is an example. Forcing cannot change the truth of any statement about the integers, so any statement proven independent by forcing cannot have any consequences for the integers. But this is only one of the two main ways to prove statements independent of ZFC. The other main way is to prove that the statement implies the consistency of ZFC, which means such a statement makes predictions about integers as well as the reals. The types of things Woodin talks about are of this type.
The other claim is that there are no physical processes that can’t be modelled as Turing machines. I have to admit everything I know about this claim I learned from your blog, but my sense is that has the statement of a plausible conjecture, rather than an established fact.
72. JerboaKolinowski Says:
Hi Scott,
I think I can imagine a physical process whose outcome depended on the existence of a set larger than the integers but smaller than the reals. However, my admitted lack of mathematical sophistication may make this easier for me than for some!
In my limited understanding, the independence of CH from ZFC means that in speaking of the “existence” such a set we must be using the word “existence” as a gloss for “existence under some set of axioms which is not ZFC”, and so my imagined physical process depends on there being some interesting set of axioms under which we are prepared to say that there either is or is not a set greater than the integers and less than the reals.
The physical process I imagine, then, is just some mathematician writing down a proof under this as-yet-undreamed-of axiom set, where the (non)existence of the set in question is a result or dependency of the proof (or, if you like, a machine check of this proof). Because I am a mathematical unsophisticate, I don’t have to try very hard to imagine this. In particular, I don’t feel the need to specify the axiom set – I just imagine the mathematician in the act of writing and leave the details to her 😉
In this respect, the “existence” of the set seems in a position no different from the “existence” of other mathematical objects: it “exists” in the same way we would say that the solution to a problem “exists”. Naturally some problems (or axiom sets) seem more important or fundamental to us than others, and in those cases we’re more tempted towards a platonic viewpoint, perhaps.
73. david Says:
Sidles: Even if a proof that P=NP gives a concrete algorithm in P to solve SAT, this doesn’t mean that it is provable in ZFC that it runs in polynomial time.
So the question may be independent of ZFC and still this need not imply that P=NP.
74. John Sidles Says:
Thank you David .. and more generally, many thanks to *everyone* who is contributing to this very enjoyable topic … I’ve learned a lot, and I’m sure many other folks have too. Again, I especially commend Scott’s survey Is P Versus NP Formally Independent? as a starting point for further reading.
75. John Sidles Says:
Folks following this thread might enjoy reading the numerous good research ideas on the wiki Vision Nuggets for Theoretical Computer Science.
Included on the wiki are Scott’s nugget entitled Efficient computation in the physical world (?) and Avi Wigderson’s nugget entitled P != NP as a law of nature.
76. Raoul Ohio Says:
There are about 37k Google hits for “joke about Wolfgang Pauli and the fine structure constant” (Turns out I remember it). Is this a great universe, or what?
77. RM Says:
But can you imagine a “physical process” whose outcome could depend on whether there’s a set larger than the set of integers but smaller than the set of real numbers? If so, what would it look like?
Well, venturing into wild speculation here, such a process might look something like the Casimir Effect. Imagine a similar phenomenon (which I’ll call the Rimisac Effect just to make it clear that I am not claiming that the Casimir Effect forces us to believe the continuum exists, and to allow me to bend to physics a bit) in which two plates in empty space are drawn together by something akin to vacuum modes. There are an infinite number of such modes both between the plates and outside them, but the infinity within is countable while the infinity without is uncountable.
Now suppose we know from experiments with exciting modes that these modes are energetically degenerate: the energy density of a cavity doesn’t depend on which modes are excitied, only on the number of excitations, and our theory says that this fact should carry over to the vacuum limit (no excitations). Thus we find that the energy density in any countably infinite set of vacuum modes is the same, but less than it would be for an uncountable set. Add in some theorem that no discrete model of the universe (satisfying some reasonable constraints) can give rise to uncountably infite modes outside the plates, and the Rimisac experiment may well give an empirical measurement of whether the continuum “exists” in this universe.
As for the Continuum Hypothesis, let us imagine that we can modify the Rimisac experiment with some sort of metacavity structure that sets the countable and uncountable vacua in a carefully balanced tension such that theorists predict will either result in erratic jumps between the two types or else equilibrate to an intermediate infinity if such a thing “exists”. Thus in this hypothetical scenario the Continuum Hypothesis could be equivalent to the claim that there can exist cavities with Rimisic energy densities greater than that of countable-mode cavities and less than that of the continuuous vacuum. I make no claims about the relevance to actual physics, only that this seems to be a conceptual cartoon of what it might “look like” for the CH to be relevant to the physical world.
78. Job Says:
I suppose complexity classes like P, NP, etc are undecidable languages. We can’t have a TM decide the elements of P or NP.
In addition P_k and NP_k also seem to be undecidable. P_k being the class containing languages whose TMs complete in n^k time, similarly for NP_k.
If P and NP are “probably” different, where would they start diverging at the beginning? Or maybe would P_1 = NP_1, P_2 = NP_2 but then P_3 != NP_3 and so on?
The complexity classes P_(ksb) and NP_(ksb) would be decidable. These being the classes containing languages whose TMs complete in n^k time on inputs less than s in length and where the TM definition takes no more than b bits. If P_(ksb) is not equal to NP_(ksb) for some given values of k, s and b, then does this imply that P != NP?
If P != NP, then must there be values of k, s and b such that P_(ksb) and NP_(ksb) are different? In other words, can a brute force attempt at settling P vs NP succeed?
79. Job Says:
I apologize in advance if the above makes no sense, sometimes i write things without thinking them through and usually regret it.
80. Jonathan Vos Post Says:
RM’s “Rimisac Effect” is clever but does not, I think, answer the question.
Because of particle-wave duality and Bohr’s principle of complementarity, the wave description of Casimir effect is dual to a photon description. I have not seen a good theoretical nor experimental description of photons with infinitesimal energy, for either of several definitions of infinitesimal.
Note that Cantor did believe that his hierarchy of infinity applied to an atomistic model of physical reality, and he said that the mind (or soul) was made of infinitesimal particles of a higher order of infinity than the particles of matter. But he never said how this could be tested, and nobody believed him.
81. Walt Says:
I just read Scott’s survey, and it is good. It also makes me realize 80 percent of my comment was superfluous…
82. Yury Says:
I’d like to answer the question Scott asked in comment 33:
can one weaken the assumption in the above result, from ZF+Con(ZF) is consistent to ZF is consistent?
Yes, it suffices to assume that ZF is consistent.
The truth value of the formula “M halts” depends only on the set of natural numbers, \omega, in the model. If two models have the same natural numbers then either M halts in both models, or M halts in neither of them (that is, “M halts” is an absolute formula). In particular, “M halts” relativized to the constructible universe L is true if and only if “M halts” is true in V.
Now we choose a model T of ZF in which AC is false, we get that
T |= AC is false, T |= AC^L is true
If ZF implied that “AC holds iff M halts”, then we would get that
T |= M doesn’t halt
T |= (M halts)^L
we would get a contradiction.
In general, when we prove independence results by forcing or by considering L, we don’t change truth values of arithmetic formulas.
83. Darran Says:
This is the first time I’ve heard somebody else say this, but it’s kind of what I’ve always thought as well. That is to say I think questions like P=NP and the Riemann hypothesis have real answers “out there in the heavens above”. However I’m a formalist about stuff like the Continuum hypothesis.
Things like the Continuum hypothesis have always struck me as artefacts of the language of set theory. Sets are such a basic concept that they’re approaching the point of concepts left undefined and can easily lead to self-contradiction if they aren’t limited in their scope. However if you limited them too much they don’t really correspond to our intuitive understanding of a set. Hence we end up with axioms like ZFC which are purposefully vague about exactly when something is a set. Which leads to a question like CH having no answer since it asks how many subsets of the Naturals there are. Definitely something you can’t answer if you haven’t said exactly what a set is.
Hence CH seems to be a language thing.
Constrast this with P=NP or Cauchy’s integral theorem, very definite statements about well defined objects.
84. Job Says:
Scott, if i’m getting annoying let me know and i’ll stop spamming but i have a question on a variant of P vs NP, namely:
Is P “exactly” equal to NP?
In other words, given that a solution to problem P can be verified in exactly n^k time and g(n) space, can it also be solved in exactly n^k time and g(n) space?
Do we know the answer to this already? I was thinking about that question and the following problem doesn’t seem to be verifiable as quickly as it is solvable:
Given an array of n numbers, identify a sequence of n or less steps that can sort the array.
It requires at least nlogn operations to solve but can be verified in linear time, isn’t that right? But this isn’t a yes/no problem anyway. Do we know that P isn’t “exactly” equal to NP?
85. Scott Says:
Job, that’s actually an excellent question, and it turns out that we do know the answer to it. Paul, Pippenger, Szemeredi and Trotter showed in 1983 that DTIME(n)≠NTIME(n); that is, there are problems solvable in nondeterministic linear time but not deterministic linear time, on reasonable models such as multi-tape Turing machines. (Their lower bound on the deterministic time needed to solve these problems is n times an extremely slow-growing function of n, basically iterated log.)
However, with all such results you need to be careful in defining the model. Regarding your sorting example, there are several issues:
(1) Even verifying a sort will require n*log(n) time, if we measure time by the number of bit operations. This is because, if each number in the array has at least n possible values (as is needed for the n*log(n) lower bound to hold), then the numbers will take log(n) bits each to specify, hence even reading them all will take n*log(n) time.
(2) If we instead adopt the comparison model (where the only allowed operation is to compare two numbers, but each comparison takes unit time), then if we’re going to be consistent and apply the same rules to the witness, it again will take n*log(n) time to verify a sort (by a generalization of the standard proof that sorting takes n*log(n) time in the comparison model).
(3) If we consider more powerful models — which involve “unit-cost comparisons” but also bit operations — then the n*log(n) lower bound for sorting breaks down (in many such models it’s actually known to be false).
So, I don’t know whether there’s any reasonable model for which sorting yields a separation between DTIME(n) and NTIME(n).
86. Job Says:
How cool, that’s really interesting.
87. sirix Says:
Scott, I generally agree with your article. However, today I’ve learned about Whitehead Problem (see wikipedia). I am stunned that it is undecidable.
I’m not saying that it contradicts your logic or anything, but still, I’m stunned, and it does change my thinking about set theory and stuff a bit.
88. Joe Shipman Says:
No arithmetical statement can depend on CH, but the ontology of fundamental physical theories involves sets that are not only uncountable, but several levels of infinity up from the integers. Itamar Pitowsky showed that the EPR paradox could be resolved if a certain kind of nonmeasurable set exists, which allows the physical weirdness to be explained by a Banach-Tarski-like mathematical weirdness; his model (later elaborated by Stanley Gudder) involved iterated integrals of (necessarily nonmeasurable) functions giving different answers with the order of integration corresponding to the order noncommuting observables were measured. Pitowsky and Gudder USED CH to build their models.
In my thesis (“Cardinal Conditions for Strong Fubini Theorems”, October 1990 Transactions of the AMS) I showed that some such trans-ZFC assumption was necessary, because it is consistent with ZFC that iterated integrals (for non-negative functions to avoid trivial counterexamples) always match when they exist.
My favorite candidate for an axiom that settles CH is the existence of a countably additive measure on ALL (not just the measurable) subsets of the reals. This axiom (known as RVM for “real-valued measurable cardinal”) is equiconsistent with a measurable cardinal, entails the continuum being very large (having many “weakly inaccessible cardinals” below it), and has a certain intuitive plausibility. (It also implies the strong Fubini theorems I alluded to above — iterated integrals must agree when they exist.)
The Banach-Tarski phenomenon shows that such a measure could not be rotationally invariant. But a possible alternative history for physics could have had Riemann not die young and discover General Relativity quite early, before the quantum theory destroyed the intuition of continuous space.
In this alternate timeline, Banach-Tarski might been discovered shortly thereafter and the assumption that was sacrificed as unphysical could have been the isotropy of space rather than its continuity, since General Relativity already requires anisotropy and assumes continuity, and so RVM could have come to be accepted as an axiom. By the time quantum mechanics had been discovered, RVM would have been shown so fruitful (it proves Con(ZFC) for example!) that it would be retained as a mathematical axiom and CH would be considered settled.
89. Scott Says:
Yury #82: Thanks very much for sharing! If I’m not mistaken, your argument should also answer in the negative the question from comment #40 (which I just realized I never answered).
90. John Sidles Says:
Joe Shipman sez: “The Banach-Tarski phenomenon shows …”
Joe, have you ever read White Light? It’s one of the very few quasi-comedic novels ever written about the CF … which is what earned it a place in our UW QSE Group’s library of subversive literature.
Perhaps I’ll transcribe some excerpts from White Light in the next few days … it includes quite a few quotations from Cantor’s own writings on the physical and meta-physical implications of CH.
91. Yury Says:
Scott, yes, the argument also gives a negative answer to the question in post #40. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8454605937004089, "perplexity": 839.147234890403}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463613796.37/warc/CC-MAIN-20170530051312-20170530071312-00181.warc.gz"} |
http://mathhelpforum.com/trigonometry/114594-inverse.html | # Math Help - inverse
1. ## inverse
If sin^-1(a)=%pi/6 and cos^-1(%pi/4)=b, then the exact value of sin(%pi*a+b) is:
i think you get rid of the inverse by going
sin %pi/6 = a so then a = 1/2
cosb = %pi/4
but im not sure how you solve for b or get the value the question is asking for.....
2. Originally Posted by samtheman17
If sin^-1(a)=%pi/6 and cos^-1(%pi/4)=b, then the exact value of sin(%pi*a+b) is:
i think you get rid of the inverse by going
sin %pi/6 = a so then a = 1/2
cosb = %pi/4
but im not sure how you solve for b or get the value the question is asking for.....
Good start ,
$\sin^{-1} a=\frac{\pi}{6}$
$a=\frac{1}{2}$
$\cos^{-1} \frac{\pi}{4}=b$
$\cos b=\frac{\pi}{4}$
$=\sin (\pi\cdot a+b)=\sin (\frac{\pi}{2}+b)$
$=\sin \frac{\pi}{2}\cos b+\cos \frac{\pi}{2}\sin b$
$
=\cos b=\frac{\pi}{4}
$
3. ahh okay thanks,
but then how do you put it into the form of
sin(%pi*a + b)?
i think you have to use
sin(a+b)=cos(a)sin(b)+sin(a)cos(b)
but i can't seem to get it right.....
4. Originally Posted by samtheman17
ahh okay thanks,
but then how do you put it into the form of
sin(%pi*a + b)?
i think you have to use
sin(a+b)=cos(a)sin(b)+sin(a)cos(b)
but i can't seem to get it right.....
$\sin (\frac{1}{2}\pi+b)$ ... substitute the a with 1/2
after that , that's what i did , expand using the sin formulas . | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9864386916160583, "perplexity": 1954.4886823922745}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500830094.68/warc/CC-MAIN-20140820021350-00209-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/112952-tangent.html | Math Help - Tangent
1. Tangent
Determine the equation of the tangent to y=18/(x+2)^2 at the point (1,2).
I found the derivative of the function and then I substituted 1 into the Mt.
Then I substituted (1,2) into y=Mtx + b to find the y intercept.
I didn't get the right answer with the textbook. Is mine right or theirs?
f'(x)=-36 / (x+2) ^3
Mt= -36/27
y=-36/27x - 1.5
4x+3y-10=0
2. I got the book's answer. I was doing the same thing as you up until you found "b". Something went different after that. Here's what I did:
y = 18/(x+2)^2
y=mx+b
m = dy/dx = (x+2^2)(0) - (18(2(x+2)(1))/(x+2)^4
= 0-36(x+2)/(x+2)^4
= -36/(x+2)^3
= -36/27
m = -4/3
Substitute m into y=mx+b
y = (-4/3)x+b
Substitute (1,2) to find b
2 = (-4/3)(1) + b
b= 10/3
Into the equation:
y=mx+b
y= -4/3x+10/3
Multiply by 3 to get standard form:
3y=-4x+ 10
Set equal to zero:
$
4x+3y-10 = 0
$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 1, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8330147862434387, "perplexity": 3756.487038575229}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701148428.26/warc/CC-MAIN-20160205193908-00024-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.nature.com/articles/456182a?error=cookies_not_supported&code=dc3e9d95-d6ca-4525-a57b-d6951f12556c | Quantum physics
# Swift control of a single spin
## Article metrics
For now, quantum information processing systems remain a dream. Step by step, however, progress towards that goal is being made, with one promising route involving a novel means of manipulating electron spin.
The basic quantity of magnetic recording — the working principle of a computer's hard disk — is an electron's spin. Although the technology for magnetic recording is reaching recording densities as high as 1 terabit per square inch (ref. 1), storing a single bit of information still involves around 105 electron spins. Future quantum information processing systems that use the electron's spin as a unit of quantum information2,3 — a quantum bit or qubit — will require the qubit to be stored in a single spin and manipulated on a timescale in which the coherence of the spin is preserved. Press et al.4 (page 218 of this issue) report that, using ultrafast laser pulses, they have controlled and observed the spin of a single electron in a semiconductor during the spin's coherence time.
The quantum state of a qubit that is based on an electron's spin can be described by a vector, known as a Bloch vector, in a sphere (a Bloch sphere), as shown in Figure 1. An arbitrary single-qubit gate operation is expressed in terms of the rotation of the Bloch vector. In general, the process is split into three rotation steps, called Euler rotations — for example, two rotations about the x axis and one about the z axis. But how can spin-state rotations with arbitrary angles about the two axes be achieved? The rotation about the z axis is implemented by applying a static magnetic field along this axis. This field induces the energy separation between the spin-up and spin-down states, known as Zeeman splitting, and the spin state precesses about the z axis with an angular frequency that is proportional to the amplitude of the field — a phenomenon known as Larmor precession. The rotation about the x axis is achieved by applying an oscillating (microwave) driving field that is resonant with the energy separation between the spin-up and spin-down states. This technique is called electron spin resonance (ESR). The coherent interaction between the spin state and the oscillating field results in the periodic rotation of the spin state about the x axis, and is called Rabi oscillation.
Although the rotation of a single electron spin has been successfully demonstrated using ESR5,6, the time required to achieve rotation with this technique is rather long — typically longer than a few nanoseconds. For quantum information processing to remain effective, rotation must be performed within a timescale that is much shorter than the spin's decoherence time.
One way of increasing the spin-state rotation is to use ultrafast laser pulses. In place of the microwave field used in ESR, a circularly polarized laser pulse is applied along the x axis (Fig. 1). The laser induces transitions between the two spin states through intermediate excited states in a process called stimulated Raman adiabatic passage (STIRAP)7,8. The effect of STIRAP is to rotate the spin about the x axis, as in ESR.
Another challenge for spin-based quantum information processing is to establish a technique for observing the spin state of a single electron. Until now, optical5,9,10 and electrical6,11 methods have done the job. Optical pumping5,10 in particular, which is a standard technique in atomic systems12, can be a sensitive tool for single-spin detection. In this method, one observes the emission (or absorption) of a photon following spin-selective optical excitation by a pumping laser. A further advantage of this technique is that the same laser can be used to initialize the spin state.
Press et al.4 have exploited ultrafast laser control and optical pumping to manipulate a single electron spin in a semiconductor using charged quantum dots. Quantum dots are nanometre-sized, artificially fabricated semiconductor structures in which electrons are confined in all three dimensions. The ground state of a neutral quantum dot has no net spin because it forms a singlet state with equal numbers of spin-up and spin-down electrons. When an extra electron, or an electron 'hole', is added to a neutral quantum dot, it will acquire a net charge and a spin. This can be achieved, for instance, by introducing impurities in the semiconductor, a method known as doping.
The above techniques can be used in combination with an optical microscope to optically control and observe the spin state in a single quantum dot. In Press and colleagues' experiment4, the spin state was initialized to the spin-up state by optical pumping. The spin state was then rotated about the x axis by a laser pulse through STIRAP. By changing the intensity of the rotating laser pulse, the authors obtained a remarkable result: they observed up to six and a half rotations (periodic Rabi oscillations) of a single spin state.
In a second experiment, the authors observed an effect known as Ramsey interference. The initialized spin-up state was first rotated by 90° about the x axis, and then rotated by an arbitrary angle about the z axis using Larmor precession. After a certain period, it was rotated back by −90° about the x axis. In terms of Euler rotations, the total process corresponds to performing a rotation about the y axis. The spin state projected onto the z axis was then observed using optical pumping and photon detection. The result was the observation of Ramsey fringes of interference with amplitudes that decay within about 200 picoseconds. This experiment is the first clear proof-of-principle demonstration of complete control of the single-spin state using an ultrafast laser.
There is no experiment that doesn't have a few 'buts'. First, the measurement of the spin state was obtained from a large, time-averaged ensemble of events, not from a single-shot measurement, a feat that has already been achieved using an electrical method11. Second, the amplitude of the Rabi oscillation fell with increasing number of rotations because of incoherent processes induced by the laser rotating the spin. Decoherence in the Ramsey fringes also seems to occur quite rapidly; the short coherence time is attributed mainly to the continuous optical pumping, and could be made longer in future experiments. If the spin coherence time in quantum dots is extended to a few microseconds13, 105 single-qubit gate operations could occur within this time4.
We have reached a stage at which we can manipulate and observe a single electron spin: albeit not perfectly, we have obtained arbitrary single-qubit gates of spins. The next step will be to realize scalable two-qubit gates, which, together with the single-qubit gates, can form a universal set for quantum computing2,14. Another challenge is to interface electron-spin-based qubits with other qubits, such as photons or nuclear spins, so that we can use appropriate qubits for different tasks, such as processing, communicating and storing quantum information.
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Edamatsu, K. Swift control of a single spin. Nature 456, 182–183 (2008) doi:10.1038/456182a | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8156449794769287, "perplexity": 806.7900714602696}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496672313.95/warc/CC-MAIN-20191123005913-20191123034913-00445.warc.gz"} |
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