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https://math.stackexchange.com/questions/linked/208996 | 15 questions linked to/from half iterate of $x^2+c$
4k views
How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$?
How to get $f(x)$, if we know that $f(f(x))=x^2+x$? Is there an elementary function $f(x)$ that satisfies the equation?
17k views
Find $f(x)$ such that $f(f(x)) = x^2 - 2$
Find all $f(x)$ satisfying $f(f(x)) = x^2 - 2$. Presumably $f(x)$ is supposed to be a function from $\mathbb R$ to $\mathbb R$ with no further restrictions (we don't assume continuity, etc), but the ...
380 views
Find $f(f(\cdots f(x)))=p(x)$
$\newcommand{\nest}{\operatorname{nest}}$Let's define a function $\nest(f, x, k)$, which takes a function $f$, an input $x$, and a non-negative integer $k$, and calls $f$ on $x$ repeatedly ($k$ times)....
206 views
Find $f(x)$ satisfying $f(f(x))=x^x$
By inspection my attempts are always wrong. I really have no idea and given up. How to find $f(x)$ satisfying $f(f(x))=x^x$? My attempts: $f(x)=x^x$ $f(x)=x^{1/x}$ $f(x)=\frac{1}{x^x}$ My ...
241 views
How to solve the iterated function $f(f(x))=x^2+x$? [duplicate]
Any given setting for $f$ is acceptable. Iterated function
132 views
Find $f(0)$ if $f(f(x))=x^2-x+1$
Given $f : \mathbb{R} \to \mathbb{R}$ defined as $$f(f(x))=x^2-x+1$$ Find value of $f(0)$ I assumed $g(x)=f(f(x))$ we gave $$g(x)=(x-1)^2+(x-1)+1$$ Also $$g(x-1)=(x-1)^2-(x-1)+1$$ subtracting ...
330 views
find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that : $f(f(x))=x^2-2$ [duplicate]
This is a very hard functional equation. the problem is this : find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that : $f(f(x))=x^2-2$ to solve it i have no idea! can we solve it ...
178 views
Solving for best fit value $C$ in $\sqrt {\mathrm{Exp}_a^{[1/2]} (x) \cdot \mathrm{Exp}_b^{[1/2]} (x )} \sim\sim \mathrm{Exp}_C^{[1/2]} (x).$
Let $\mathrm{Exp}_t^{[y]} (x)$ denote the $y$ th iteration of the exponential function with base $t$ : $t^x.$ For example $\mathrm{Exp}_t^{[1]} (x) = t^x$. Let $\sim\sim$ denote best fit. Now as $x$...
185 views
Boundary of $x^2+x$ Julia set
How do you calculate an infinite fixed point for $f=x^2+x$, so that $f^{o n}(x)$ never repeats, and doesn't go to infinity and doesn't go to zero? Can such a sequence of points densely cover the ...
109 views
let $f(x)=x^2$, then $f(f(x))=x^4$, so $x^4$ is a continuous function from $\Bbb R$ to $\Bbb R$ which can be obtained as $f\circ f$ for a continuous $f\colon \Bbb R\to \Bbb R$. general example: for $f(... 1answer 89 views Easy looking functional equation. [closed] Find all functions that satisfy $$f (f (x))=x^2-3x+4$$ Any thoughts and approachs to find$f (x)$? 1answer 57 views Existance of a function satisfying given equation Examine the existence of a function$f:\mathbb R\to \mathbb R$such that:$\forall x \in \mathbb R\ :f(f(x))= x^2 +1$How to approach this type of problems? I study functional equation from a ... 1answer 86 views half composite function There is half derivative. Is there any definition of half composite function? Or is it possible to define half composite of a function? 1)$f^n=f \circ \cdots \circ f $($n$-times,$n \in \mathbb Z^{+...
The main idea is that if you have some well defined function $f$, is there a function $f^{\frac{1}{2}}$ such that $f^{\frac{1}{2}}\left(f^{\frac{1}{2}}\left(x\right)\right) = f\left(x\right)$. For ...
So I was playing around with function composition, and started wondering if there was a way to split up a function in to a repeated application of another function. Notation: Let $f^1 = f$ and \$f^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": 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": 0.9868249893188477, "perplexity": 340.59043205065535}, "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/1590347410284.51/warc/CC-MAIN-20200530165307-20200530195307-00353.warc.gz"} |
https://www.marefa.org/%D9%83%D8%B1%D8%A9_%D8%B1%D9%8A%D9%85%D8%A7%D9%86 | # كرة ريمان
The Riemann sphere can be visualized as the complex number plane wrapped around a sphere (by some form of stereographic projection – details are given below).
في الرياضيات ، كرة ريمان Riemann sphere ، على اسم الرياضي الشهير برنارد ريمان ، هي الطريقة الفريدة لإظهار السطح العقدي الممدد extended complex plane (السطح العقدي إضافة لنقطة في اللانهاية ) بحيث انه سيبدو من نقطة اللانهاية ممائلا لشكله عند أي عدد عقدي، بالذات بالنسبة للاستمرارية و الاشتقاقية. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ for infinity. With the Riemann model, the point "∞" is near to very large numbers, just as the point "0" is near to very small numbers.
The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as ${\displaystyle {\tfrac {1}{0}}=\infty }$ well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere.
In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds. In projective geometry, the sphere can be thought of as the complex projective line P1(C), the projective space of all complex lines in C2. As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics.
The extended complex plane is also called closed complex plane.
## فهرست
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## Extended complex numbers
The extended complex numbers consist of the complex numbers C together with ∞. The set of extended complex numbers may be written as C ∪ {∞}, and is often denoted by adding some decoration to the letter C, such as
${\displaystyle {\hat {\mathbb {C} }},\quad {\overline {\mathbb {C} }},\quad {\text{or}}\quad \mathbb {C} _{\infty }.}$
Geometrically, the set of extended complex numbers is referred to as the Riemann sphere (or extended complex plane).
### Arithmetic operations
Addition of complex numbers may be extended by defining, for z ∈ C,
${\displaystyle z+\infty =\infty }$
for any complex number z, and multiplication may be defined by
${\displaystyle z\times \infty =\infty }$
for all nonzero complex numbers z, with ∞ × ∞ = ∞. Note that ∞ – ∞ and 0 × ∞ are left undefined. Unlike the complex numbers, the extended complex numbers do not form a field, since ∞ does not have a multiplicative inverse. Nonetheless, it is customary to define division on C ∪ {∞} by
${\displaystyle {\frac {z}{0}}=\infty \quad {\text{and}}\quad {\frac {z}{\infty }}=0}$
for all nonzero complex numbers z, with /0 = ∞ and 0/ = 0. The quotients 0/0 and / are left undefined.
## ككرة
Stereographic projection of a complex number A onto a point α of the Riemann sphere
The Riemann sphere can be visualized as the unit sphere x2 + y2 + z2 = 1 in the three-dimensional real space R3. To this end, consider the stereographic projection from the unit sphere minus the point (0, 0, 1) onto the plane z = 0, which we identify with the complex plane by ζ = x + iy. In Cartesian coordinates (x, y, z) and spherical coordinates (θ, φ) on the sphere (with θ the zenith and φ the azimuth), the projection is
${\displaystyle \zeta ={\frac {x+iy}{1-z}}=\cot \left({\frac {1}{2}}\theta \right)\;e^{i\phi }.}$
Similarly, stereographic projection from (0, 0, −1) onto the plane z = 0, identified with another copy of the complex plane by ξ = xiy, is written
${\displaystyle \xi ={\frac {x-iy}{1+z}}=\tan \left({\frac {1}{2}}\theta \right)\;e^{-i\phi }.}$
## Automorphisms
A Möbius transformation acting on the sphere, and on the plane by stereographic projection
مقال رئيسي: Möbius transformation
The study of any mathematical object is aided by an understanding of its group of automorphisms, meaning the maps from the object to itself that preserve the essential structure of the object. In the case of the Riemann sphere, an automorphism is an invertible biholomorphic map from the Riemann sphere to itself. It turns out that the only such maps are the Möbius transformations. These are functions of the form
${\displaystyle f(\zeta )={\frac {a\zeta +b}{c\zeta +d}},}$
where a, b, c, and d are complex numbers such that adbc ≠ 0. Examples of Möbius transformations include dilations, rotations, translations, and complex inversion. In fact, any Möbius transformation can be written as a composition of these.
The Möbius transformations are homographies on the complex projective line. In projective coordinates, the transformation f can be written
${\displaystyle [\zeta ,\ 1]{\begin{pmatrix}a&c\\b&d\end{pmatrix}}\ =\ [a\zeta +b,\ c\zeta +d]\ =\ \left[{\tfrac {a\zeta +b}{c\zeta +d}},\ 1\right]\ =\ [f(\zeta ),\ 1].}$
## Applications
In complex analysis, a meromorphic function on the complex plane (or on any Riemann surface, for that matter) is a ratio f/g of two holomorphic functions f and g. As a map to the complex numbers, it is undefined wherever g is zero. However, it induces a holomorphic map (f, g) to the complex projective line that is well-defined even where g = 0. This construction is helpful in the study of holomorphic and meromorphic functions. For example, on a compact Riemann surface there are no non-constant holomorphic maps to the complex numbers, but holomorphic maps to the complex projective line are abundant.
The Riemann sphere has many uses in physics. In quantum mechanics, points on the complex projective line are natural values for photon polarization states, spin states of massive particles of spin 1/2, and 2-state particles in general (see also Quantum bit and Bloch sphere). The Riemann sphere has been suggested as a relativistic model for the celestial sphere.[1] In string theory, the worldsheets of strings are Riemann surfaces, and the Riemann sphere, being the simplest Riemann surface, plays a significant role. It is also important in twistor theory.
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## References
1. ^ R. Penrose (2007). The Road to Reality. Vintage books. pp. 428–430 (§18.5). ISBN 0-679-77631-1.
• Brown, James & Churchill, Ruel (1989). Complex Variables and Applications. New York: McGraw-Hill. ISBN 0-07-010905-2. Cite uses deprecated parameter |lastauthoramp= (help)
• Griffiths, Phillip & Harris, Joseph (1978). Principles of Algebraic Geometry. John Wiley & Sons. ISBN 0-471-32792-1. Cite uses deprecated parameter |lastauthoramp= (help)
• Penrose, Roger (2005). The Road to Reality. New York: Knopf. ISBN 0-679-45443-8.
• Rudin, Walter (1987). Real and Complex Analysis. New York: McGraw–Hill. ISBN 0-07-100276-6.
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https://cp3.irmp.ucl.ac.be/projects/madgraph/wiki/TopBSM?action=diff&version=2 | # Changes between Version 1 and Version 2 of TopBSM
Ignore:
Timestamp:
04/06/12 16:33:02 (8 years ago)
Comment:
--
### Legend:
Unmodified
v1 == Model for BSM physics studies in ttbar production ([http://arxiv.org/abs/0712.2355 arxiv:0712.2355]) Version 1.3 == The {{{topBSM}}} is a model implemented to study BSM effects in the ttbar invariant mass spectrum. This model includes the following possible resonances in the ttbar spectrum: The {{{ topBSM }}} is a model implemented to study BSM effects in the ttbar invariant mass spectrum. This model includes the following possible resonances in the ttbar spectrum: * spin-0, color singlet * spin-0, color octet * spin-2, RS model Note that this {{{topBSM}}} model can only be used for ttbar production. Any other final state might lead to inconsistencies in the evaluation of the diagrams. This model uses a special {{{param_card.dat}}} and {{{run_card.dat}}} that can be found [attachment:param_card.dat here] and [attachment:run_card.dat here]. Note that this {{{ topBSM }}} model can only be used for ttbar production. Any other final state might lead to inconsistencies in the evaluation of the diagrams. This model uses a special {{{ param_card.dat }}} and {{{ run_card.dat }}} that can be found [attachment:param_card.dat here] and [attachment:run_card.dat here]. === spin-0, color singlet === ([attachment:proc_card.dat proc_card.dat]: topBSM spin-0 color singlet proc_card.dat) The spin-0, color singlet particle, in the {{{topBSM}}} called {{{s0}}} (PDG code: 6000045), is a Higgs-like particle that couples only to top quarks. The production of the spin-0 is only through a top quark loop by gluon fusion. And its decay is directly to two top quarks with a branching ratio %$\textrm{BR}(s0\to t\bar{t})=1$%. It's coupling strength to the top quark is by default equal to the SM Higgs coupling to top quarks, ''i.e.'', %$im_t/v$%, but this can be changed in the {{{param_card.dat}}}. In the {{{param_card.dat}}} there are the following two lines: The spin-0, color singlet particle, in the {{{ topBSM }}} called {{{ s0 }}} (PDG code: 6000045), is a Higgs-like particle that couples only to top quarks. The production of the spin-0 is only through a top quark loop by gluon fusion. And its decay is directly to two top quarks with a branching ratio $\textrm{BR}(s0\to t\bar{t})=1$. It's coupling strength to the top quark is by default equal to the SM Higgs coupling to top quarks, ''i.e.'', %$im_t/v$%, but this can be changed in the {{{ param_card.dat }}}. In the {{{ param_card.dat }}} there are the following two lines: {{{ 1. 1.00000000e+00 # s0scalarf ,spin-0 scalar mult.fac. }}} These two values correspond to multiplication factors for the coupling strength, ''i.e.'', %$g_{s0tt}=$% {{{s0scalarf}}} %$i\frac{m_t}{v}+$% {{{s0axialf}}} %$\frac{m_t}{v}\gamma_5$%. Hence, the spin-0 can be a scalar or a pseudo-scalar or a mixed CP state by playing around with these two factors. Due to the loop in the production mechanism the coupling strength between the gluons and the {{{s0}}} depends on its momentum. Therefore it is important to set the flag {{{fixed_couplings}}} to false in the {{{run_card.dat}}}. (See above for a sample {{{run_card.dat}}}) The width is calculated automatically and is not read from the {{{param_card.dat}}} (this takes into account the values for {{{s0scalarf}}} and {{{s0axialf}}}). These two values correspond to multiplication factors for the coupling strength, ''i.e.'', %$g_{s0tt}=$% {{{ s0scalarf }}} %$i\frac{m_t}{v}+$% {{{ s0axialf }}} $\frac{m_t}{v}\gamma_5$. Hence, the spin-0 can be a scalar or a pseudo-scalar or a mixed CP state by playing around with these two factors. Due to the loop in the production mechanism the coupling strength between the gluons and the {{{ s0 }}} depends on its momentum. Therefore it is important to set the flag {{{ fixed_couplings }}} to false in the {{{ run_card.dat }}}. (See above for a sample {{{ run_card.dat }}}) The width is calculated automatically and is not read from the {{{ param_card.dat }}} (this takes into account the values for {{{ s0scalarf }}} and {{{ s0axialf }}}). === spin-0, color octet === ([attachment:proc_card_o0.dat proc_card_o0.dat]: topBSM spin-0 color octet proc_card.dat) The spin-0, color octet particle, in the {{{topBSM}}} called {{{o0}}} (PDG code: 6000046), is a scalar, colored particle that couples only to top quarks. The production of the spin-0 is only through a top quark loop by gluon fusion. And its decay is directly to two top quarks with a branching ratio %$\textrm{BR}(s0\to t\bar{t})=1$%. It's coupling strength to the top quark is by default equal to the SM Higgs coupling to top quarks, ''i.e.'', %$im_t/v$%, but this can be changed in the {{{param_card.dat}}}. In the {{{param_card.dat}}} there are the following two lines: The spin-0, color octet particle, in the {{{ topBSM }}} called {{{ o0 }}} (PDG code: 6000046), is a scalar, colored particle that couples only to top quarks. The production of the spin-0 is only through a top quark loop by gluon fusion. And its decay is directly to two top quarks with a branching ratio $\textrm{BR}(s0\to t\bar{t})=1$. It's coupling strength to the top quark is by default equal to the SM Higgs coupling to top quarks, ''i.e.'', %$im_t/v$%, but this can be changed in the {{{ param_card.dat }}}. In the {{{ param_card.dat }}} there are the following two lines: {{{ 3. 1.00000000e+00 # o0scalarf ,spin-0 scalar mult.fac. }}} These two values correspond to multiplication factors for the coupling strength, ''i.e.'', %$g_{o0tt}=$% {{{o0scalarf}}} %$i\frac{m_t}{v}+$% {{{o0axialf}}} %$\frac{m_t}{v}\gamma_5$%. Hence, the spin-0 can be a scalar or a pseudo-scalar or a mixed CP state by playing around with these two factors. Due to the loop in the production mechanism the coupling strength between the gluons and the {{{o0}}} depends on its momentum. Therefore it is important to set the flag {{{fixed_couplings}}} to false in the {{{run_card.dat}}}. (See above for a sample {{{run_card.dat}}}) The width is calculated automatically and is not read from the {{{param_card.dat}}} (this takes into account the values for {{{o0scalarf}}} and {{{o0axialf}}}). These two values correspond to multiplication factors for the coupling strength, ''i.e.'', %$g_{o0tt}=$% {{{ o0scalarf }}} %$i\frac{m_t}{v}+$% {{{ o0axialf }}} $\frac{m_t}{v}\gamma_5$. Hence, the spin-0 can be a scalar or a pseudo-scalar or a mixed CP state by playing around with these two factors. Due to the loop in the production mechanism the coupling strength between the gluons and the {{{ o0 }}} depends on its momentum. Therefore it is important to set the flag {{{ fixed_couplings }}} to false in the {{{ run_card.dat }}}. (See above for a sample {{{ run_card.dat }}}) The width is calculated automatically and is not read from the {{{ param_card.dat }}} (this takes into account the values for {{{ o0scalarf }}} and {{{ o0axialf }}}). === spin-1, color singlet === ([attachment:proc_card_S1.dat proc_card_S1.dat]: topBSM spin-1 color singlet proc_card.dat) The spin-1, color singlet particle in the {{{topBSM}}} is called {{{s1}}} (PDG code: 6000047). This spin-1 particle is a similar to the SM Z boson. Its mass and width have to be set in the {{{param_card.dat}}}. By default it has the same couplings as the SM Z boson (only couplings to fermions are implemented). By changing the multiplication factors in the {{{BLOCK MGUSER}}} in the {{{param_card.dat}}} the coupling strengths can be altered. The spin-1, color singlet particle in the {{{ topBSM }}} is called {{{ s1 }}} (PDG code: 6000047). This spin-1 particle is a similar to the SM Z boson. Its mass and width have to be set in the {{{ param_card.dat }}}. By default it has the same couplings as the SM Z boson (only couplings to fermions are implemented). By changing the multiplication factors in the {{{ BLOCK MGUSER }}} in the {{{ param_card.dat }}} the coupling strengths can be altered. === spin-1, color octet === ([attachment:proc_card_O1.dat proc_card_O1.dat]: topBSM spin-1 color octet proc_card.dat) The spin-1, color octet particle in the {{{topBSM}}} is called {{{o1}}} (PDG code: 6000048). This spin-1 particle is a similar to a heavy gluon. Its mass and width have to be set in the {{{param_card.dat}}}. By default it has the same couplings as the gluon (only couplings to quarks are implemented). By changing the multiplication factors in the {{{BLOCK MGUSER}}} in the {{{param_card.dat}}} the coupling strengths can be altered. The spin-1, color octet particle in the {{{ topBSM }}} is called {{{ o1 }}} (PDG code: 6000048). This spin-1 particle is a similar to a heavy gluon. Its mass and width have to be set in the {{{ param_card.dat }}}. By default it has the same couplings as the gluon (only couplings to quarks are implemented). By changing the multiplication factors in the {{{ BLOCK MGUSER }}} in the {{{ param_card.dat }}} the coupling strengths can be altered. === spin-2, ADD model === ([attachment:proc_card_ADD.dat proc_card_ADD.dat]: topBSM spin-2 ADD proc_card.dat) The spin-2 graviton particle of the large extra dimensions model (ADD) is called {{{s2}}} in the {{{topBSM}}} (PDG code: 6000049). Due to the large extra dimensions, the KK gravitons are almost degenerate in mass. Therefore in this model there is not a single resonance, but a very large number that contribute only together significantly. Effectively the denominator of the graviton propagator is calcelled by the sum over all the KK states. There is a cut-off scale {{{mstring}}} that you have to specify in the {{{param_card.dat}}}, as well as the number of extra dimensions (so far only implemented for 3 extra dimensions). The mass of the {{{s2}}} should be set equal to the cut-off scale, while the width is not used at all. Note that this cut-off scale is parameter in the model, this is '''not''' a cut on the ttbar invariant mass, and there will be [http://www.essaybank.com/ essay writing] events produced above this cut-off scale. For this model it is important that the couplings are calculated on an event-by-event basis, hence one should set the flag {{{fixed_couplings}}} in the {{{run_card.dat}}} to false. (For an example {{{run_card.dat}}} see above.) The spin-2 graviton particle of the large extra dimensions model (ADD) is called {{{ s2 }}} in the {{{ topBSM }}} (PDG code: 6000049). Due to the large extra dimensions, the KK gravitons are almost degenerate in mass. Therefore in this model there is not a single resonance, but a very large number that contribute only together significantly. Effectively the denominator of the graviton propagator is calcelled by the sum over all the KK states. There is a cut-off scale {{{ mstring }}} that you have to specify in the {{{ param_card.dat }}}, as well as the number of extra dimensions (so far only implemented for 3 extra dimensions). The mass of the {{{ s2 }}} should be set equal to the cut-off scale, while the width is not used at all. Note that this cut-off scale is parameter in the model, this is '''not''' a cut on the ttbar invariant mass, and there will be [http://www.essaybank.com/ essay writing] events produced above this cut-off scale. For this model it is important that the couplings are calculated on an event-by-event basis, hence one should set the flag {{{ fixed_couplings }}} in the {{{ run_card.dat }}} to false. (For an example {{{ run_card.dat }}} see above.) === spin-2, RS model === ([attachment:proc_card_RS.dat proc_card_RS.dat]: topBSM spin-2 RS proc_card.dat) In the RS model there are a number of KK resonances with their mass ratio's given by the zeros of the BesselJ function. The mass of the first resonance has to be given in the {{{param_card}}}, the others are calculated by the MadGraph code. Also the widths are calculated internally. Furthermore the ratio of %$\kappa/\bar{M}_{\textrm{planck}}$% also has to be specified in the {{{BLOCK MGUSER}}} to specify the size of the coupling. Note that the RS gravitons are implemented to couple only to quarks and gluons, but in the calculation of the widths, couplings to all SM particles are taken into account. Only the first 10 resonances are implemented, called {{{g1}}}, {{{g2}}},..., {{{g0}}} (PDG codes: 6000050...6000059) so setting the mass of the first resonance small and using a large value for the coupling strength should be used with care, because effects from higher resonances start getting more important in this part of the parameter space. In the RS model there are a number of KK resonances with their mass ratio's given by the zeros of the BesselJ function. The mass of the first resonance has to be given in the {{{ param_card }}}, the others are calculated by the MadGraph code. Also the widths are calculated internally. Furthermore the ratio of %$\kappa/\bar{M}_{\textrm{planck}}$% also has to be specified in the {{{ BLOCK MGUSER }}} to specify the size of the coupling. Note that the RS gravitons are implemented to couple only to quarks and gluons, but in the calculation of the widths, couplings to all SM particles are taken into account. Only the first 10 resonances are implemented, called {{{ g1 }}}, {{{ g2 }}},..., {{{ g0 }}} (PDG codes: 6000050...6000059) so setting the mass of the first resonance small and using a large value for the coupling strength should be used with care, because effects from higher resonances start getting more important in this part of the parameter space. -- Main.RikkertFrederix - 09 Dec 2008 | {"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": 0.8616478443145752, "perplexity": 2507.7475111317367}, "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/1575540531974.7/warc/CC-MAIN-20191211160056-20191211184056-00372.warc.gz"} |
https://math.msu.edu/seminars/TalkView.aspx?talk=9251 | ## Student Geometry/Topology II
• Hitesh Gakhar, MSU
• Dualities in Persistent (co)Homology
• 01/17/2018
• 4:10 PM - 5:00 PM
• C204A Wells Hall
For a filtered topological space, its persistent homology is a multi-set of half open real intervals known as barcode. Each bar represents the lifespan of a homology class. A fundamental principle is that the length of such a bar determines the significance of the corresponding class. In 2011, V. de Silva et al studied the relationships between (persistent) absolute homology, absolute cohomology, relative homology and relative cohomology. This talk will be a theoretical overview of that study.
## Contact
Department of Mathematics
Michigan State University | {"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": 0.9436988830566406, "perplexity": 3994.570043753531}, "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/1544376829140.81/warc/CC-MAIN-20181218102019-20181218124019-00383.warc.gz"} |
https://kar.kent.ac.uk/3155/ | # Primitive ideals and automorphism group of Uq+(B2)
Launois, Stephane (2007) Primitive ideals and automorphism group of Uq+(B2). Journal of Algebra and its Applications, 6 (1). pp. 21-47. ISSN 0219-4988.
PDF (Primitive Ideals and Automorphism Group ) | {"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": 0.9981454610824585, "perplexity": 3145.1520444350213}, "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-2020-05/segments/1579250606696.26/warc/CC-MAIN-20200122042145-20200122071145-00328.warc.gz"} |
https://brilliant.org/discussions/thread/playing-with-integrals-ineqaulitites-and-integra-j/ | ×
# Playing with Integrals: Ineqaulitites and Integrals 2
Although my previous post on this topic didn't cause a discussion I'll try to continue. If no one will be interested in problems from this post too, I'll switch to another theme.
Today I would like to present several inequalities directly involving integrals.
Problem 4. If $$f:[0,1]\rightarrow[0,1]$$ is a continuous function, prove the following inequality $\left(\int^1_0f(x)\,dx\right)^2\leq 2\int^1_0xf(x)\,dx$
Solution. To start our proof we'll consider the following function $F(t)=\left(\int^t_0f(x)\,dx\right)^2- 2\int^t_0xf(x)\,dx$ We know that $$F(0)=0$$, so to prove our inequality we simply need to prove taht $$F(t)$$ is a decreasing fucntion. $F^\prime(t)=2f(t)\int^t_0f(x)\,dx-2tf(t)=2f(t)\int^1_0(f(x)-1)\,dx$ However, we know that range of $$f(x)$$ is $$[0,1]$$, so $$f(x)-1\leq 0$$, subsequently by Theorem 1. $$\displaystyle\int^1_0(f(x)-1)\,dx\leq 0$$. So $F^\prime(t)=2f(t)\int^1_0(f(x)-1)\,dx\leq 0,$ which means that $$F(t)$$ is decreasing and $$F(1)\leq F(0)=0$$. Finaly we arrvie at the desired result $\left(\int^1_0f(x)\,dx\right)^2\leq 2\int^1_0xf(x)\,dx.$
Now try your own techniques to solve the following problems.
Problem 5. For continuous, differentiable function $$f:[0,1]\rightarrow\mathbb{R}$$, there exists $$a\in(0,1)$$ such that $$\displaystyle\int^a_0f(x)\,dx=0$$. Prove that
$\left|\int^1_0f(x)\,dx\right|\leq\frac{1-a}{2}\sup_{x\in(0,1)}|f^\prime(x)|$
Hint. For this problem you might consider the Mean Value Theorem
Problem 6. Let $$f:[0,1]\rightarrow\mathbb{R}$$ be a continuous function satisfying following property $\int^1_0 f(x)\,dx=\int^1_0 xf(x)\,dx=1$ Prove that $\int^1_0 f^2(x)\,dx\geq 4$
Hint. Try to search for a linear fucntion with the same proprties, also here $$f^2(x)=f(x)\cdot f(x)$$.
Note by Nicolae Sapoval
4 years, 2 months ago
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In problem 4, I think the statement should be $\left( \int_0^1 f(x) \: dx \right)^2 \le 2 \int_0^1 xf(x) \: dx.$ This is the only way the expression for $$F'(t)$$ makes sense. (Also, there exist counter-examples to the statement as is.)
- 4 years, 2 months ago
Yes, you're definitely right. Thanks for pointing that out!
- 4 years, 2 months ago
Please do continue. I like your posts very much and I am always a fan of inequalities and a newcommer at Calculus.Please do such another ones.Thanks.
- 4 years, 2 months ago | {"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": 0.9900376200675964, "perplexity": 1736.0400568423393}, "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-2018-13/segments/1521257648113.87/warc/CC-MAIN-20180323004957-20180323024957-00201.warc.gz"} |
https://www.math.ias.edu/seminars/abstract?event=94864 | # Collapsing hyperbolic structures: from rigidity to flexibility and back
Geometric Structures on 3-manifolds Topic: Collapsing hyperbolic structures: from rigidity to flexibility and back Speaker: Steve Kerckhoff Affiliation: Stanford University Date: Tuesday, April 5 Time/Room: 2:00pm - 3:00pm/S-101 Video Link: https://video.ias.edu/geostruct/2016/0405-Kerckhoff
This talk will be about some phenomena that occur as (singular) hyperbolic structures on 3-manifolds collapse to and transition through other geometric structures. Typically, the collapsed structures are much more flexible than the hyperbolic structures, leading to the question of which structures arise as limits of hyperbolic structures. | {"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": 0.9061042666435242, "perplexity": 4607.98916927448}, "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-2018-26/segments/1529267863407.58/warc/CC-MAIN-20180620011502-20180620031502-00637.warc.gz"} |
https://www.physicsforums.com/threads/reversing-the-order-of-integration.287745/ | # Reversing the order of integration
1. Jan 26, 2009
### s7b
Hi,
I want to reverse the order of this integration;
integral(from x=0 to x=3) integral(from y=sqrt(x/3) to y=1) of e^(y^3) dy dx
How do I change the limits of integration?
2. Jan 26, 2009
### NoMoreExams
Didn't I tell you what to do in the other thread though? Pretty sure you can move the threads you start (at least you could on the other boards I've been on) by going to Thread Tools
3. Jan 26, 2009
### s7b
Well I didn't know.
I also don't get how to draw the pictures.
4. Jan 26, 2009
### NoMoreExams
You are told what horizontal (i.e. x goes from and to i.e. between 0 and 3) and you are told what the vertical goes from (i.e. y goes from and to i.e. sqrt(x/3) to 1)
5. Jan 26, 2009
### s7b
I get that but don't you have to change the limits of integration when you reverse the order of integration?
6. Jan 26, 2009
### NoMoreExams
Yes, you said you don't know how to draw the picture, I told you what the picture would look like. Did you draw it? | {"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": 0.844733476638794, "perplexity": 1510.7137075478583}, "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/1521257649627.3/warc/CC-MAIN-20180324015136-20180324035136-00081.warc.gz"} |
https://byjus.com/question-answer/a-pump-is-used-to-lift-500-kg-of-water-from-a-depth-of-80-m-in-10-s/ | Question
A pump is used to lift $500\mathrm{kg}$ of water from a depth of $80\mathrm{m}$ in $10\mathrm{s}$. Calculate the power rating of the pump if its efficiency is $40%$
A
$100\mathrm{kW}$
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B
$40\mathrm{kW}$
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C
$16\mathrm{kW}$
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D
None of the above
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Solution
The correct option is A $100\mathrm{kW}$Step 1: Calculate the force exerted by the pumpGiven, mass of the water lifted is $500\mathrm{kg}$We know that $\mathrm{F}=\mathrm{m}g$, where $\mathrm{F}$ is the force exerted, $\mathrm{m}$ is the mass of the object and $g$ is the acceleration due to gravity. Assuming $g=10{\mathrm{ms}}^{-2}$,$\begin{array}{rcl}\mathrm{F}& =& 500×10\\ \therefore \mathrm{F}& =& 5000\mathrm{N}\end{array}$There the pump exerts $5000\mathrm{N}$ of force.Step 2: Calculate the work done by the pumpGiven, water is moved $80\mathrm{m}$ high.The formula for calculating work done is,$\mathrm{W}=\mathrm{F}·\mathrm{s}$Where, $\mathrm{W}$ is the work done, $\mathrm{F}$ is the force exerted and $\mathrm{s}$ is the displacement of the object.Thus, work done is,$\mathrm{W}=5000·80\phantom{\rule{0ex}{0ex}}\therefore \mathrm{W}=400000=4×{10}^{5}\mathrm{J}$Therefore, the pump does $4×{10}^{5}\mathrm{J}$ of work.Step 3: Calculate the power of the pumpGiven, the pump does the work in $10\mathrm{s}$The formula for power ($\mathrm{P}$) is given as,$\mathrm{P}=\frac{\mathrm{W}}{t}$Where, $t$ is the time taken to do the work.Thus,$\mathrm{P}=\frac{4×{10}^{5}}{10}=40\mathrm{kW}$Step 4: Calculate power ratingGiven, efficiency$=40%=0.4$We know that,$\mathrm{Power}\mathrm{rating}=\frac{\mathrm{Useful}\mathrm{power}}{\mathrm{efficieny}}$Thus, the power rating of the given pump is,$\mathrm{Power}\mathrm{rating}=\frac{40}{0.4}=100\mathrm{kW}$Hence, option A is correct.
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https://www.physicsforums.com/threads/subspaces-r-n-how-to-visualize.103680/ | # Subspaces, R^n How to visualize?
1. Dec 11, 2005
### bobby13
Hey,
I have no problems dealing with vectors in space, R^3. But I am having a lot of trouble with vectors in R^n. One of my basic questions is what is R^n. I mean doesn't the vector space already encompass everything? How do I visualize R^n vectors? Can you reccomend any good online tutorials that explain how to approach problems in the subspace units?
Any help would be great!!
Thanks!!!
2. Dec 11, 2005
### matt grime
Don't visualize it. Period. These are not spaces that have any physically visible manifestation really, so don't bother - it is unnecssary to do so, and unhelpful in the long run if you only attempt mathematics that is 'visualizable'.
R^n is just the set of n-tuples of numbers, and that is all you need.
3. Dec 11, 2005
### JasonRox
I can't even visualize R^3 half the time. :grumpy:
Look at the bright side of R^n, you'll never be asked to draw a graph.
4. Dec 11, 2005
### Hurkyl
Staff Emeritus
I dunno, I've seen people draw diagrams in much uglier spaces than than R^4. That's where the real fun begins.
5. Dec 12, 2005
### bobby13
Thnx! I think Im starting to get the hang of this stuff, after constantly and constantly reading over notes and trying examples. Quick question. In one example it says,
Let U = { X in R^n | AX = 0 (Zero Vector)}
Is U a subspace?
Am I correct in reading the question as, X is in R^n if and only if AX = 0. That is, when checking all three conditions, I should get the zero vector?
6. Dec 12, 2005
### hypermorphism
It is the definition of a set U. It says X is in U iff X is a vector in R^n such that AX=0. Now you have to determine whether U satisfies the axioms of a vector space.
7. Dec 12, 2005
### matt grime
that is how we all learned mathematics.
not quite, you should read it as X is in U if AX=0, where A is some fixed matrix, or that U is the subset of vectors x in R^n such that Ax=0. You need to show that U satisfies the axioms of a subspace, ie 0 is in U, and if x and y are in U that x+y is in U, and that if x is in U and t is a real number that tx is in U.
no, that doesn't follow at all
8. Dec 12, 2005
### JasonRox
Are you talking about the all the homogeneous solutions for the matrix A?
If that is the case, why not try yourself and see if it is a subspace. All you need to do is check if it is closed under addition and scalar multiplication.
In order words, show that if x,y is in U, then x+y is in U, and if k is any scalar, then kx is in U.
Note: If it has only one solution for the homegeneous system, then we know it is the trivial solution. The trivial solution is the zero vector itself, which creates a subspace all on it's own.
Last edited: Dec 12, 2005 | {"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": 0.8465360999107361, "perplexity": 690.5921286081982}, "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-2018-43/segments/1539583510749.37/warc/CC-MAIN-20181016093012-20181016114512-00128.warc.gz"} |
http://www.chegg.com/homework-help/questions-and-answers/356-cm-diameter-loop-rotated-uniform-electric-field-position-maximum-electric-flux--flux-p-q922098 | ## [HELP] Electric Field Uniform
A 35.6-cm-diameter loop is rotated in a uniform electric field until the position of maximum electric flux is found. The flux in this position is measured to be 6.91E+5 N-m2/C.
What is the magnitude of the electric field? | {"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": 0.9381398558616638, "perplexity": 558.8378284328015}, "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-2013-20/segments/1368704134547/warc/CC-MAIN-20130516113534-00042-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/34604/why-do-photons-add-mass-to-a-black-hole | # Why do photons add mass to a black hole?
Why do photons add mass to a black hole?
When photons are taken irreversibly into a black hole does the mass of the BH increase?
-
– Qmechanic Aug 20 '12 at 21:24
possible duplicate of If photons have no mass, how can they have momentum? – genneth Aug 20 '12 at 22:00
This is really just an expansion of Graham's answer.
It's a commonly made mistake that gravity, and therefore a black hole, is caused by matter. In fact the spacetime curvature is related to a quantity called the stress-energy tensor. This is usually represented by a matrix with ten independant values in it (it's a 4x4 matrix but it's symmetric so six of the elements in it are duplicated).
Only one of the elements in the matrix, $T_{00}$, depends directly on the mass, and actually that element gives the energy density, where mass is counted as energy using Einstein's equation $e = mc^2$.
So photons affect spacetime curvature because they contribute to the energy density even though they have no mass. Actually photons contribute to other elements of the matrix as well because they have a non-zero momentum and this too affects the spacetime curvature.
When photons are taken irreversibly into a black hole does the mass of the BH increase?
Yes, the mass of the black hole will increase by the photon energy divided by $c^2$.
Re your comment to Graham's question, yes, provided you add more energy than the black hole is radiating you will maintain or increase the black hole. You could add the energy using lots of low energy photons or a few high energy photons. It's the total energy added that matters.
-
The first part of your answer is a gem, but I was quite taken aback by the statement that you could measure a different $\Delta M$ from the gravitational field created and the inertia. Wasn't this equivalency the entire point of GR? I'm sitting here thinking "no way that's what he meant". My understanding would be completely wrong about everything if that's true. – AlanSE Aug 21 '12 at 14:02
@alanse: I didn't say you could measure a different inertia from gravitational mass, I said I wasn't sure if you would. Having said that, you're quite right of course because even though photons are massless they have a momentum. I was just suffering a temporary brain fade. Thanks for the injection of sanity :-) – John Rennie Aug 21 '12 at 14:27
I don't even understand how you can be unsure, if a photon hits the event horizon it will increase the mass of the BH. I don't understand where there is room for skepticism. Photons have no rest mass but they can impart mass onto other things. If this were about a particle of ordinary matter, a fully absorbed photon would impart mass through increased KE plus heat energy. In the case of a BH taken to be stationary, photon hitting at CM, there is no KE change, and 100% of the photon's energy adds to the BH mass. I'm baffled b/c you're the expert of the 2 of us! – AlanSE Aug 21 '12 at 14:36
You're quite correct of course. Put it down to old age and impending senility. – John Rennie Aug 21 '12 at 14:39
To expand on my "soft" understanding the photon does not hit the eh just the backup of material created by the eh. Does the same increase occur if the photon crosses the eh "un-interrupted" not yet quite far enough to prove GR mute when would the momentum effect the BH as soon as it passes the eh or is there a certain point were the energy becomes part of the BH. – Argus Aug 21 '12 at 19:10 | {"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": 0.903849720954895, "perplexity": 319.91307300840674}, "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-2014-15/segments/1397609533308.11/warc/CC-MAIN-20140416005213-00529-ip-10-147-4-33.ec2.internal.warc.gz"} |
https://www.aimsciences.org/journal/1930-8337/2014/8/1 | # American Institute of Mathematical Sciences
ISSN:
1930-8337
eISSN:
1930-8345
All Issues
## Inverse Problems & Imaging
February 2014 , Volume 8 , Issue 1
Select all articles
Export/Reference:
2014, 8(1): 1-22 doi: 10.3934/ipi.2014.8.1 +[Abstract](1271) +[PDF](481.0KB)
Abstract:
This paper concerns the reconstruction of an anisotropic conductivity tensor in an elliptic second-order equation from knowledge of the so-called power density functionals. This problem finds applications in several coupled-physics medical imaging modalities such as ultrasound modulated electrical impedance tomography and impedance-acoustic computerized tomography.
We consider the linearization of the nonlinear hybrid inverse problem. We find sufficient conditions for the linearized problem, a system of partial differential equations, to be elliptic and for the system to be injective. Such conditions are found to hold for a lesser number of measurements than those required in recently established explicit reconstruction procedures for the nonlinear problem.
2014, 8(1): 23-51 doi: 10.3934/ipi.2014.8.23 +[Abstract](1951) +[PDF](1474.3KB)
Abstract:
We apply an exterior approach" based on the coupling of a method of quasi-reversibility and of a level set method in order to recover a fixed obstacle immersed in a Stokes flow from boundary measurements. Concerning the method of quasi-reversibility, two new mixed formulations are introduced in order to solve the ill-posed Cauchy problems for the Stokes system by using some classical conforming finite elements. We provide some proofs for the convergence of the quasi-reversibility methods on the one hand and of the level set method on the other hand. Some numerical experiments in $2D$ show the efficiency of the two mixed formulations and of the exterior approach based on one of them.
2014, 8(1): 53-77 doi: 10.3934/ipi.2014.8.53 +[Abstract](1788) +[PDF](648.3KB)
Abstract:
This paper presents the Moreau envelope viewpoint for the L1/TV image denoising model. The main algorithmic difficulty for the numerical treatment of the L1/TV model lies in the non-differentiability of both the fidelity and regularization terms of the model. To overcome this difficulty, we propose five modified L1/TV models by replacing one or two non-differentiable functions in the L1/TV model with their corresponding Moreau envelopes. We prove that several existing approaches for the L1/TV model essentially solve some of the modified models, but not the original L1/TV model. Algorithms for the L1/TV model and its five variants are proposed under a unified framework based on fixed-point equations (via the proximity operator) which characterize the solutions of the models. Depending upon whether we smooth the regularization term or not, two different types of proximity algorithms are presented. The convergence rates of both types of the algorithms are improved significantly by exploring either the strategy of the Gauss-Seidel iteration, or the FISTA, or both. We compare the performance of various modified L1/TV models for the problem of impulse noise removal, and make recommendations based on our numerical experiments for using these models in applications.
2014, 8(1): 79-102 doi: 10.3934/ipi.2014.8.79 +[Abstract](1723) +[PDF](1252.2KB)
Abstract:
We present a new approach to solve the inverse source problem arising in Fluorescence Tomography (FT). In general, the solution is non-unique and the problem is severely ill-posed. It poses tremendous challenges in image reconstructions. In practice, the most widely used methods are based on Tikhonov-type regularizations, which minimize a cost function consisting of a regularization term and a data fitting term. We propose an alternative method which overcomes the major difficulties, namely the non-uniqueness of the solution and noisy data fitting, in two separate steps. First we find a particular solution called the orthogonal solution that satisfies the data fitting term. Then we add to it a correction function in the kernel space so that the final solution fulfills other regularity and physical requirements. The key ideas are that the correction function in the kernel has no impact on the data fitting, so that there is no parameter needed to balance the data fitting and additional constraints on the solution. Moreover, we use an efficient basis to represent the source function, and introduce a hybrid strategy combining spectral methods and finite element methods in the proposed algorithm. The resulting algorithm can dramatically increase the computation speed over the existing methods. Also the numerical evidence shows that it significantly improves the image resolution and robustness against noise.
2014, 8(1): 103-125 doi: 10.3934/ipi.2014.8.103 +[Abstract](1403) +[PDF](456.5KB)
Abstract:
We introduce a technique for recovering a sufficiently smooth function from its ray transforms over rotationally related curves in the unit disc of 2-dimensional Euclidean space. The method is based on a complexification of the underlying vector fields defining the initial transport and inversion formulae are then given in a unified form. The method is used to analyze the attenuated ray transform in the same setting.
2014, 8(1): 127-148 doi: 10.3934/ipi.2014.8.127 +[Abstract](2151) +[PDF](1571.8KB)
Abstract:
Electrical impedance tomography (EIT) is a non-invasive imaging modality in which the internal conductivity distribution is reconstructed based on boundary voltage measurements. In this work, we consider the application of EIT to non-destructive testing (NDT) of materials and, especially, crack detection. The main goal is to estimate the location, depth and orientation of a crack in three dimensions. We formulate the crack detection task as a shape estimation problem for boundaries imposed with Neumann zero boundary conditions. We propose an adaptive meshing algorithm that iteratively seeks the maximum a posteriori estimate for the shape of the crack. The approach is tested both numerically and experimentally. In all test cases, the EIT measurements are collected using a set of electrodes attached on only a single planar surface of the target -- this is often the only realizable configuration in NDT of large building structures, such as concrete walls. The results show that with the proposed computational method, it is possible to recover the position and size of the crack, even in cases where the background conductivity is inhomogeneous.
2014, 8(1): 149-172 doi: 10.3934/ipi.2014.8.149 +[Abstract](1630) +[PDF](451.8KB)
Abstract:
This article is devoted to the convergence analysis of a special family of iterative regularization methods for solving systems of ill--posed operator equations in Hilbert spaces, namely Kaczmarz-type methods. The analysis is focused on the Landweber--Kaczmarz (LK) explicit iteration and the iterated Tikhonov--Kaczmarz (iTK) implicit iteration. The corresponding symmetric versions of these iterative methods are also investigated (sLK and siTK). We prove convergence rates for the four methods above, extending and complementing the convergence analysis established originally in [22,13,12,8].
2014, 8(1): 173-197 doi: 10.3934/ipi.2014.8.173 +[Abstract](1581) +[PDF](557.4KB)
Abstract:
In bioluminescence tomography the location as well as the radiation intensity of a photon source (marked cell clusters) inside an organism have to be determined given the outside photon count. This inverse source problem is ill-posed: it suffers not only from strong instability but also from non-uniqueness. To cope with these difficulties the source is modeled as a linear combination of indicator functions of measurable domains leading to a nonlinear operator equation. The solution process is stabilized by a Tikhonov like functional which penalizes the perimeter of the domains. For the resulting minimization problem existence of a minimizer, stability, and regularization property are shown. Moreover, an approximate variational principle is developed based on the calculated domain derivatives which states that there exist smooth almost stationary points of the Tikhonov like functional near to any of its minimizers. This is a crucial property from a numerical point of view as it allows to approximate the searched-for domain by smooth domains. Based on the theoretical findings numerical schemes are proposed and tested for star-shaped sources in 2D: computational experiments illustrate performance and limitations of the considered approach.
2014, 8(1): 199-221 doi: 10.3934/ipi.2014.8.199 +[Abstract](1771) +[PDF](971.0KB)
Abstract:
We consider the inverse problem of finding sparse initial data from the sparsely sampled solutions of the heat equation. The initial data are assumed to be a sum of an unknown but finite number of Dirac delta functions at unknown locations. Point-wise values of the heat solution at only a few locations are used in an $l_1$ constrained optimization to find the initial data. A concept of domain of effective sensing is introduced to speed up the already fast Bregman iterative algorithm for $l_1$ optimization. Furthermore, an algorithm which successively adds new measurements at specially chosen locations is introduced. By comparing the solutions of the inverse problem obtained from different number of measurements, the algorithm decides where to add new measurements in order to improve the reconstruction of the sparse initial data.
2014, 8(1): 223-246 doi: 10.3934/ipi.2014.8.223 +[Abstract](1989) +[PDF](577.2KB)
Abstract:
In this paper, we presented an efficient algorithm to implement the regularization reconstruction of SPECT. Image reconstruction with priori assumptions is usually modeled as a constrained optimization problem. However, there is no efficient algorithm to solve it due to the large scale of the problem. In this paper, we used the superiorization of the expectation maximization (EM) iteration to implement the regularization reconstruction of SPECT. We first investigated the convergent conditions of the EM iteration in the presence of perturbations. Secondly, we designed the superiorized EM algorithm based on the convergent conditions, and then proposed a modified version of it. Furthermore, we gave two methods to generate desired perturbations for two special objective functions. Numerical experiments for SPECT image reconstruction were conducted to validate the performance of the proposed algorithms. The experiments show that the superiorized EM algorithms are more stable and robust for noised projection data and initial image than the classic EM algorithm, and outperform the classic EM algorithm in terms of mean square error and visual quality of the reconstructed images.
2014, 8(1): 247-257 doi: 10.3934/ipi.2014.8.247 +[Abstract](1366) +[PDF](314.1KB)
Abstract:
The equations of magneto-photoelasticity are derived for a nonhomogeneous background isotropic medium and for a variable gyration vector. They coincide with Aben's equations in the case of a homogeneous background medium and of a constant gyration vector. We obtain an explicit linearized formula for the fundamental solution under the assumption that variable coefficients of equations are sufficiently small. Then we consider the inverse problem of recovering the variable coefficients from the results of polarization measurements known for several values of the gyration vector. We demonstrate that the data can be easily transformed to a family of Fourier coefficients of the unknown function if the modulus of the gyration vector is agreed with the ray length.
2014, 8(1): 259-291 doi: 10.3934/ipi.2014.8.259 +[Abstract](1687) +[PDF](5178.8KB)
Abstract:
The grid method is one of the techniques available to measure in-plane displacement and strain components on a deformed material. A periodic grid is first transferred on the specimen surface, and images of the grid are compared before and after deformation. Windowed Fourier analysis-based techniques permit to estimate the in-plane displacement and strain maps. The aim of this article is to give a precise analysis of this estimation process. It is shown that the retrieved displacement and strain maps are actually a tight approximation of the convolution of the actual displacements and strains with the analysis window. The effect of digital image noise on the retrieved quantities is also characterized and it is proved that the resulting noise can be approximated by a stationary spatially correlated noise. These results are of utmost importance to enhance the metrological performance of the grid method, as shown in a separate article.
2014, 8(1): 293-320 doi: 10.3934/ipi.2014.8.293 +[Abstract](1779) +[PDF](3063.2KB)
Abstract:
Many effective models are available for segmentation of an image to extract all homogenous objects within it. For applications where segmentation of a single object identifiable by geometric constraints within an image is desired, much less work has been done for this purpose. This paper presents an improved selective segmentation model, without `balloon' force, combining geometrical constraints and local image intensity information around zero level set, aiming to overcome the weakness of getting spurious solutions by Badshah and Chen's model [8]. A key step in our new strategy is an adaptive local band selection algorithm. Numerical experiments show that the new model appears to be able to detect an object possessing highly complex and nonconvex features, and to produce desirable results in terms of segmentation quality and robustness.
2014, 8(1): 321-337 doi: 10.3934/ipi.2014.8.321 +[Abstract](1557) +[PDF](1015.8KB)
Abstract:
We propose an efficient nonlinear approximation scheme using the Polyharmonic Local Sine Transform (PHLST) of Saito and Remy combined with an algorithm to tile a given image automatically and adaptively according to its local smoothness and singularities. To measure such local smoothness, we introduce the so-called local Besov indices of an image, which is based on the pointwise modulus of smoothness of the image. Such an adaptive tiling of an image is important for image approximation using PHLST because PHLST stores the corner and boundary information of each tile and consequently it is wasteful to divide a smooth region of a given image into a set of smaller tiles. We demonstrate the superiority of the proposed algorithm using Antarctic remote sensing images over the PHLST using the uniform tiling. Analysis of such images including their efficient approximation and compression has gained its importance due to the global climate change.
2018 Impact Factor: 1.469 | {"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": 0.8543983101844788, "perplexity": 417.3364783670784}, "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/1575540482038.36/warc/CC-MAIN-20191205190939-20191205214939-00136.warc.gz"} |
https://socratic.org/questions/what-is-the-cube-root-of-88 | Algebra
Topics
# What is the cube root of 88?
Mar 6, 2018
$\sqrt[3]{88} = 2 \sqrt[3]{11}$ or decimal approximation to 25 decimal places that is 4.4479601811386310423307268
#### Explanation:
find a factor that is a cube number
$\sqrt[3]{8 \cdot 11}$
seperate the multiples using the radical law $\sqrt[n]{x y} = \sqrt[n]{x} \cdot \sqrt[n]{y}$
$\sqrt[3]{8} \cdot \sqrt[3]{11}$
The cube root of 8 is 2
$2 \cdot \sqrt[3]{11}$
$2 \sqrt[3]{11}$
Mar 6, 2018
See a see a solution process below
#### Explanation:
We can rewrite this expression as:
$\sqrt[3]{88} \implies \sqrt[3]{8 \cdot 11}$
We can then use this rule for radicals to simplify the expression:
$\sqrt[n]{\textcolor{red}{a} \cdot \textcolor{b l u e}{b}} = \sqrt[n]{\textcolor{red}{a}} \cdot \sqrt[n]{\textcolor{b l u e}{b}}$
$\sqrt[3]{\textcolor{red}{8} \cdot \textcolor{b l u e}{11}} \implies \sqrt[3]{\textcolor{red}{8}} \cdot \sqrt[3]{\textcolor{b l u e}{11}} \implies 2 \sqrt[3]{11}$
If you need an exact number: $\sqrt[3]{88} \implies 4.448$ rounded to the nearest thousandth.
##### Impact of this question
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https://link.springer.com/article/10.1007%2Fs40940-018-0082-9 | Glass Structures & Engineering
, Volume 4, Issue 1, pp 29–44
# Sensitivity study on climate induced internal pressure within cylindrical curved IGUs
• Minxi Bao
• Sam Gregson
SI: Challenging Glass paper
## Abstract
Insulated glass units (IGUs) are employed in modern buildings as a substitute for monolithic glass to reduce heat loss through windows. In the past decades, the pursuit of higher aesthetic design drives glazed products evolving from conventionally flat into more creative and dynamic curved shapes. The call for curved IGUs brings up a series of challenges, and one remarkable issue is the determination of load sharing and glass stress when evaluating its structural performance. Even though many national codes worldwide have established mature design approaches for load sharing of flat IGUs panels, such design approaches can scarcely be found for curved IGUs due to geometry complexity. In this paper, the author will use an FEA tool along with automatic iteration scripts to carry out a sensitivity study on the internal climatic pressure of cylindrically curved IGUs, considering a series of geometrical variables. The paper aims to evaluate the relationships between the internal pressure and the geometrical parameters of curved IGUs and normalize all these parameters into a dimensionless chart.
## Keywords
Curved IGUs Sensitivity Parametric study Geometrical impact
## 1 Introduction
Insulating glass units (IGUs) have been widely adopted in modern building envelopes for decades to improve thermal performance. The concept of the IGU was first invented and patented by Stetson (1865). The technology was then markedly improved over the century and generally emerged in United Kingdom in the early 1960s (Garvin and Marshall 1995). A typical double glazed IGU (shown in Fig. 1) consists of two glass panels separated by a spacer bar which is filled with desiccant in order to absorb the moisture condensation during service life. A primary sealant is applied between the spacer bar and glass panel, which provides an additional barrier against moisture infiltration. Secondary sealant is then filled outside the bottom of the spacer bar. The combination of edge sealants and spacer bar is usually referred as the edge-seal system, a crucial component in the IGUs that provides a gas and moisture barrier and structurally bonds glass panels together. Triple IGUs introduce an additional layer of glass and edge-seal system to provide higher insulation performance. For clarification, we are mainly focusing on double glazing units in this paper and the term “IGUs” used in this paper refers to double glazing units.
Contemporary architectural design is keen to integrate freeform aesthetics with thermal insulation for glass façades. Curved IGUs have been frequently proposed to deliver a more dynamic architectural language as well as retain satisfactory energy efficiency. However, there is a general lack of understanding of the structural behaviour of curved IGUs within the engineering industry, particularly the climatic induced internal pressures. Climatic loads are an inherent design problem with IGUs which cannot be solved by analytical equations for curved shapes. Climatic loads can lead to sealant failure, glass overstressing, and the panel deformation due to internal pressure change can lead to visible visual distortion as illustrated in Fig. 2.
As such, there is a demand for empirical charts or simple equations that express the relationships between geometry and the induced internal pressure loads to glass panels, so that designers can quickly specify reasonable glass build-up at the initial design stage.
## 2 Literature review
### 2.1 Internal pressure equilibrium in IGU cavity
In contrast to laminated glazing panels, one distinct step of IGU design is to specify the load share coefficient, i.e. how the loads are distributed to the two layers through the deformed air cavity. In addition, the phenomenon of climatic loads is a distinct feature of IGUs, whereby changes in the panel environment causes variation of internal pressures and loads the glass packages. The calculation of these phenomenon relies on knowledge of gas behaviour.
According to the ideal gas law, the relationship between the volume V occupied by a fixed amount of an ideal gas in moles n, its pressure P, and temperature T in Kelvin can be expressed as
\begin{aligned} pV=nRT \end{aligned}
(1)
where R is the gas constant, 8.314 J/mol. Thus, when a gas undergoes a change of state from state 0 to state 1 the volume V, pressure p and temperature T respect the following relationship
\begin{aligned} \frac{p_0 V_0 }{T_0 }=\frac{p_1 V_1 }{T_1 } \end{aligned}
(2)
Since the air or noble gas within the IGU cavity is hermetically sealed, it can be assumed to closely follow the ideal gas law. Load share between the glass packages is activated through the pressure change within the sealed gas. An installed IGU may encounter a number of environmental factors during its service life (Van-Den-Bergh et al. 2013), including (a) large temperature differences/thermal cycling, (b) atmospheric pressure fluctuation, (c) external wind loads/imposed loads. These all give rise to changes in the internal pressure of the sealed air, and thus impose load onto the glass packages. Quantifying the internal pressure change when the system reaches equilibrium is key to determining the loads on the two packages.
The cavity volume however is not constant. Internal pressure changes will cause deformation of the glass due to the glass bending behaviour and therefore changes in cavity volume. In summary, pressure change is a function of volume change, but volume change is in turn a function of internal pressure. It is this interrelation which complicates the calculation of internal pressure. What is problematic for the case of curved IGUs is that there is no analytic solution for volume change of the cavity due to changes in internal pressure.
Previous studies have outlined the workflow to find the pressure change at equilibrium state (Feldmeier 2003; Huveners et al. 2003; Vuolio 2003; Griffith and Marinov 2015). If the relationship between internal pressure and volume change is known, the climatic pressure can be calculated by expressing the volume change dV as a function of internal pressure change dp,
\begin{aligned} dV=f(dp) \end{aligned}
(3)
Substituting Eq. (3) into ideal gas law Eq. (2) we get the following equation, where dp is the only unknown variable to be solved
\begin{aligned} \frac{p_0 V_0 }{T_0 }=\frac{(p_0 +dp)(V_0 +f(dp))}{T_0 +dT} \end{aligned}
(4)
Another key concept relating to the gas pressure is the concept of the isochore pressure. In the context of IGUs the isochore pressure is defined as the internal pressure relative to external pressure caused by climatic changes, assuming that the cavity volume remains constant. For IGUs the climatic change which could occur subsequent to production are difference in altitude, difference in temperature and difference in external air pressure (BSI 2013; TRLV 2006) and is expressed as
\begin{aligned} p_{{ iso}} =c_H (H-H_p )+c_T (T-T_p )-(p-p_p )p_{{ iso}} \end{aligned}
(5)
where HT and p represent the altitude, temperature and ambient pressure at the installation site and $$H_{p},T_{p}$$ and $$p_{p}$$ represent the altitude, temperature and ambient pressure at the production site respectively; $$c_{H}$$ denotes coefficient for effect of altitude change on isochore pressure (0.12 kPa/m); $$c_{T}$$ is the coefficient for the effect of temperature change on isochore pressure (0.34 kPa/K).
Therefore, Eq. (4) can be transformed to:
\begin{aligned} (p_{{ atm}} +p_{{ iso}} )V_0 =(p_{{ atm}} +dp)(V_0 +f(dp)) \end{aligned}
(6)
where $$p_{{ atm}}$$ is atmostpheric pressure and dp represents the climatic load.
Volumetric stiffness $$k_{V}$$ is introduced to portray Eq. (3), i.e. the relation of volume change and pressure change (Griffith and Marinov 2015). Equation (6) can be converted to a quadratic equation, and internal pressure change dp can be solved.
\begin{aligned}&\left( p_{{ atm}} +p_{{ iso}} \right) V_0 =\left( p_{{ atm}} +dp\right) \left( V_0 +\frac{dp}{k_{V1} }+\frac{dp}{k_{V2} }\right) \end{aligned}
(7)
\begin{aligned}&dp=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a} \end{aligned}
(8)
where
\begin{aligned} a= & {} \left( \frac{1}{k_{V1} }+\frac{1}{k_{V2} }\right) ;\nonumber \\ b= & {} \left( V_0 +p_{{ atm}} \left( \frac{1}{k_{V1} }+\frac{1}{k_{V2} }\right) \right) ;c=-p_{{ iso}} V_0 \end{aligned}
where $$k_{v1}, k_{v2}$$ are defined as the volumetric stiffness of the two glass packages, and expressed as
\begin{aligned} k_{Vi} =\frac{dp}{dV_i } \end{aligned}
(9)
For curved panels it has been suggested that $$k_{V}$$ is solved by finite element analysis by applying a unit load to represent dp and calculating the volume change due to the displacement of the glass packages (Griffith and Marinov 2015). For flat panels some analytical equations have also been proposed.
### 2.2 Analytical solutions for flat IGUs
For flat IGUs, researchers have established analytical equations to express $$k_{v}$$. For small deformation, linear analytical solution can be derived from classical Timoshenko plate theory (1940)
\begin{aligned} dV= & {} \int _0^a \int _0^b w_{\left( x,y\right) } dxdy\nonumber \\= & {} \frac{16dp}{\pi ^{6}D} \int _0^a \int _0^b {\frac{\sin \frac{\pi x}{a}\sin \frac{\pi y}{b}}{\left( \frac{1}{a^{2}}+\frac{1}{b^{2}}\right) ^{2}}} dxdy\nonumber \\= & {} \frac{64a^{5}b^{5}}{\pi ^{8}D\left( a^{2}+b^{2}\right) }dp \end{aligned}
(10)
where $$w_{\left( {x,y} \right) }$$ is the deformation at any point of the plate. D is the bending stiffness of flat panel; a and b are the glass edge lengths;p is the applied pressure.
\begin{aligned} k_V =\frac{dp}{\left( \frac{64a^{5}b^{5}dp}{\pi ^{8}D\left( a^{2}+b^{2}\right) }\right) }=\frac{\pi ^{8}D\left( a^{2}+b^{2}\right) }{64a^{5}b^{5}} \end{aligned}
(11)
For those IGUs with larger dimensions and receiving higher level of loads, linear solution will lead to notable discrepancy from the actual bending behaviour because the membrane stress induced by large deformation is not considered. Non-linear analyses have been carried out however can be only solved via the aid from computational tools due to mathematical complexity. Feldmeier (2003) proposed a linear approximation which introduces a milestone parameter, the insulating glass unit factor:
\begin{aligned} \varphi =\frac{1}{1+(v_e +v_i )p_{{ atm}} /V_0 } \end{aligned}
(12)
where $$v_{e}$$ and $$v_{i}$$ are the volume per load for external and internal panel deformation respectively, in $$\hbox {m}^{3}/\hbox {kPa}$$.
Load share determination by insulating glass unit factor has been widely used as code of practice PrEN 16612 (BSI 2013). It is noted that though the nonlinearity in stress and deflection has been considered by introducing a parameter “Non-dimensional load”, the volume change, however, is still approximately by a linear expression:
\begin{aligned} V=k_5 A\frac{a^{4}}{h^{3}}\frac{F_d }{E} \end{aligned}
(13)
where $$k_{5}$$ is a geometry-related parameter, a is the shorter edge of the panel, h is the glass thickness, $$F_{d}$$ is the pressure load and E is the Young’s Modulus of glass.
Therefore, the volumetric stiffness is
\begin{aligned} k_V =\frac{h^{3}}{a^{4}}\frac{E}{Ak_5 }. \end{aligned}
(14)
### 2.3 Internal pressure load of cylindrically curved IGUs
Researchers have pointed out that for cylindrical IGUs, the climatic loads caused by the sealed gas is drastically greater than that of flat IGUs (Feldmeier 2003). Curved panels are stiffer due to geometric stiffness and hence yield less volume deformation. According to the reverse relationship, a lower volume change gives rises to a higher internal pressure change. It is worthy investigating correlations between internal pressure and different geometrical parameters of curved IGUs and evaluate the sensitivity degree of the internal pressure to each variable.
It has been observed that empirical or analytical solutions are not available for curved IGUs contrary to flat IGUs. The main reason is there is no accurate, general analytical expression to describe deformation of a curved plate. The overall deflection is attributed to both glass bending and membrane strain. The relative amounts of strain energy in these two behaviours varies with the plate geometry and is non-linear with respect to deflection. Therefore, a simple formula is not available to accurately depict the curved panel deformation. Previous researchers proposed empirical equations for calculating deflection of cylindrical shell within comparatively small ranges of the R / t and l / R ratios. The equations are calibrated by a dimension parameter $$\alpha$$ (changing in l / R), which is acquired from testing. It is claimed the results obtained from empirical charts do not differ from the laboratory measured data by more than approximately 20% (Roarks 1941). As for analytical solution, Batdorf (1947) pointed out the simplest method for cylinders or curved cylindrical panels is to represent radial displacement w by a trigonometric series expansion. Again, the expression is cumbersome and not in ease of solution.
\begin{aligned}&\nabla w^{8}+\frac{12Z^{2}}{b^{4}}\cdot \frac{\partial w^{4}}{\partial x^{4}}+k_x \frac{\pi ^{2}}{b^{2}}\nabla w^{4}\cdot \frac{\partial w^{2}}{\partial x^{2}}\nonumber \\&\quad +\,2k_S \frac{\pi ^{2}}{b^{2}}\cdot \nabla w^{4}\cdot \frac{\partial w^{2}}{\partial x\cdot \partial y}\nonumber \\&\quad +\,k_y \frac{\pi ^{2}}{b^{2}}\nabla w^{4}\cdot \frac{\partial w^{2}}{\partial y^{2}}=0 \end{aligned}
(15)
Among most of published documents (Feldmeier 2003; Huveners et al. 2003; Vuolio 2003; Griffith and Marinov 2015), FEM numerical method is believed to be most efficient way for the internal pressure investigations of curved IGUs. Nevertheless, although it is theoretically straightforward to investigate the stress/deformation behaviour of curved glass panels via numerical modelling, few commercially prevalent FEA software, especially those cost-efficient FEA software packages, can determine the internal pressure within a sealed air cavity at equilibrium state, namely pneumatic fluid analysis.
This paper introduces a computational-aided iteration process, which allows FEA software that does not support pneumatic fluid analysis to find the internal pressure change at equilibrium. A series of comparison charts are plotted to understand the geometrical impacts to the internal pressure of curved IGU. The study is particularly focused on climatic loads, as it has been found much more significant in curved IGUs compared to flat panels. The correlation between the internal pressure and the geometry of IGUs will be visualised in the charts to assist designers finding proper glass configurations at initial design stage. The charts are obtained from a series of parametric studies which is automated by scripts written in C# and within the parametric modelling software Grasshopper3D alongside FEM software STRAND7.
## 3 Methodology
### 3.1 Framework
The basis of the method employed in this paper generally follows the application of Eq. (7) and illustrated in Fig. 3. In order to obtain the volumetric stiffness, a variable of the governing equation, an FEM analysis is conducted, and displacements are integrated across the surface and used to calculate volume change and thereafter volume stiffness. Initially an arbitrary internal pressure is applied to obtain an initial approximation to the volumetric stiffness.
Direct application of Eq. (7) assumes that the volumetric stiffness is a constant and therefore independent of the internal pressure of the IGU. Due to geometric non-linearity of the panels the authors have found that assumption is inaccurate, we therefore employ an iterative procedure to update volume stiffness as the solution converges to the climatic pressure.
This process repeated until convergence as illustrated in the flow diagram below. In order to validate convergence within each iteration the current approximation of the climatic load is compared to the calculated internal pressure of the IGU for the given volume change. When the internal pressure applied in the FEM model and the climatic load calculated due to the volume change are within tolerance the procedure is terminated. Convergence tolerance is set to be 0.001 kPa.
### 3.2 FEM modelling description
Since STRAND 7 does not support gas element, nor pressure equilibrium analysis within a sealed cavity, the interactive action between two glass panels cannot be simulated. STRAND 7 is only used to analyse the bending behaviour of the curved glass panel at one iteration step. As plotted in Fig. 4 the glass deformation at each node will be integrated in Grasshopper script to obtain the overall volume change at the corresponding iteration step.
Glass panel is modelled with shell/plate element, first order QUAD 4. The thickness of the glass panel considered in the modelling is the effective thickness for laminated glass. The standard determination method of effective thickness is provided in prEN 16612 (BSI 2013). Element size is set to be 100 mm by 100 mm based on computational efficiency. Silicone joints are modelled by spring elements. Material properties employed in the modelling are listed in Table 1. It is worth noting that though silicone is a hyper-elastic material, the initial 6–7% tension/shear deformation follows an approximate linear stress–strain relation as observed in dow corning tensile test and shear curves (Dow Corning 2015). In this study, the resultant tension in silicone joint is subtle, i.e. less than 6% strain, therefore can be assumed to behave linearly.
### 3.3 Validation
In order to validate the methodology used in this study a comparison is made between the results of the proposed methodology for flat IGUs to the analytical equations which exist for flat IGUs. It is then assumed that the methodology can be employed to calculate the climatic loads of curved IGUs as no change in the algorithm is required to facilitate curved panels. Future work will extend the validation of the methodology via physical testing.
Table 2 below compares the results of the proposed methodology with the analytical results, based on the PrEN 16612, showing consistent results across the two approaches, validating the results of the methodology. It is assumed that the methodology can be extended to curved IGU panels with similar accuracy.
It is worth noting that in the derivation of the analytical method presented by PrEN 16612 are some simplifications and assumptions, therefore the analytical results shall not necessarily be taken as the exact answer.
Table 1
Material properties
Type
Young’s modulus E (MPa)
Poisson ratio v
Glass
70,000
0.22
Silicone
4
0.499
## 4 Sensitivity study
### 4.1 Geometry description
In the sensitivity study, the internal pressure changes due to climatic loads of curved IGUs are calculated by considering a series of variables. The geometric parameters include chord, length, radius, panel thickness, cavity width, and silicone bite as shown in Fig. 5.
Table 2
Comparison of analytical and FEA approaches for validation
Panel variables
Analytical results (PrEN 16612)
Algorithm results
Difference (%)
Length (mm)
Width (mm)
Cavity (mm)
Glass Thk. (mm)
1500
375
16
8
10.711
10.286
4.0
1500
750
16
8
2.448
2.381
2.7
1500
1500
16
8
0.451
0.453
0.4
1500
3000
16
8
0.178
0.177
0.6
1500
5000
16
8
0.132
0.131
0.8
1500
6000
16
8
0.124
0.123
0.8
1500
12,000
16
8
0.107
0.106
0.9
1500
375
16
24
15.805
15.662
0.9
1500
750
16
24
13.344
12.960
2.9
1500
1500
16
24
7.050
6.732
4.5
1500
3000
16
24
3.740
3.603
3.7
1500
5000
16
24
2.941
2.849
3.1
1500
6000
16
24
2.79
2.704
3.1
1500
12,000
16
24
2.470
2.399
2.9
1500
375
20
8
11.479
11.054
3.7
1500
750
20
8
2.948
2.864
2.8
1500
1500
20
8
0.560
0.568
1.4
1500
3000
20
8
0.222
0.221
0.5
1500
5000
20
8
0.165
0.163
1.2
1500
6000
20
8
0.155
0.153
1.3
1500
12,000
20
8
0.134
0.133
0.7
For the purposes of parametric study, the chord is kept constant at 1500 mm, and other parameters are changed within a realistic range. Each parameter is taken into account independently, with the other parameters staying consistent to the basic configuration. The variables are summarised in Table 3.
Table 3
Variable summary
Variables
Chord (mm)
Length (mm)
Thickness (mm)
Cavity width (mm)
Silicone bite (mm)
Symbol
c
l
R
t
s
b
Basic build-up
1500
1500
1500
8
16
10
Length
375–6000
750–10,000
Thickness
8–24
Cavity
12, 16, 20
Silicone
10–40
Characteristic climatic actions are set to be consistent to the recommended parameters given by TRLV technical note (2006), as shown in Table 4. It can be observed that climatic load in winter is a reverse version of summer, therefore either scenario can represent the magnitude of overall isochoric pressure. In this study, the climatic load is defined to occur during summer season.
Table 4
Action combination
Temperature change T (K)
Atmospheric pressure change $$dp_{{ atm}}$$ (kPa)
Altitude change dH (m)
Isochoric pressure $$p_{{ iso}}$$ (kPa)
Winter
$$-$$ 25
$$+$$ 4
$$-$$ 300
$$-$$ 16
Summer
$$+$$ 20
$$-$$ 2
$$+$$ 600
$$+$$ 16
### 4.3 Results and discussion
The results are presented in plots with internal pressure at equilibrium state as the dependent variable and the relevant parameter as the independent variable. Flat IGU and cylindrical curved IGU (also referred to here as “curved IGU”) will be compared in the same plot.
#### 4.3.1 Thickness
The effective glass pane thickness is varied between 8, 10, 12, 16, 20 and 24 mm. The calculated internal pressures varying with increasing thickness are presented in Fig. 6. As a comparison, two groups of data “flat IGU” and “asymptote” are plotted. The flat panel (R is $$\infty )$$ with same chord, length and climatic conditions is analyzed to obtain climatic pressures for each thickness. The asymptote obtained by assuming the glass panel is infinitely stiff and does not have any deformation when subjected to the climatic load, hence the overall cavity volume change is attributed to the silicone stretching only. It is assumed that this asymptote is the same for curved and flat geometries as the panel stiffness is infinite in both cases.
It can be observed that the internal pressure of the curved IGU rises mildly, varying from 4.25 to 5.88 kPa as the thickness itself increases by a multiple of three. In the contrast, flat panels generate a relative low internal pressure 0.435 kPa when $$\hbox {t} = 8$$ mm but grows rapidly to 4.75 kPa at $$\hbox {t} = 24$$ mm.
In parallel, to this comparison, we introduce “normalized internal pressure” $${r}({t}) = { dp}({x})$$/dp($${t}=8$$ mm). Figure 7 displays the normalized internal pressures of flat IGUs and curved IGUs. There is a strong contrast that r(t) of flat IGU increases by more than 1000%, whereas curved IGU increases by only 38% across the thickness range considered. The input information is summarised in Table 5.
According to Eq. (13), the volumetric stiffness is proportional to $$t^{3}$$, therefore the internal pressure of the flat panel is significantly sensitive to the thickness, whereas the stiffening effect in curved IGUs by increasing thickness is not as conspicuous as in flat panel. This is because the geometric stiffness due to curvature is governing the overall stiffness of the curved panel rather than plate bending. By comparing the equation of equilibrium for a flat plate and cylindrically curved plate given by Batdorf (1947), the additional term to account for the curvature effect is inversely proportional to t. but in flat panel, it is dictated by the flexural bending stiffness D, which is proportional to $$t^{3}$$. We can hence reason that, when the radius is increasing, the weight of curvature effect will be reduced, and the flexural stiffness becomes predominant, and the panel will be more sensitive to thickness variance.
Table 5
Input summary
Variables
Chord (mm)
Length (mm)
Thickness (mm)
Cavity width (mm)
Silicone bite (mm)
Symbol
c
l
R
t
s
b
Build-up
1500
1500
1500
8–24
16
10
Table 6
Input summary
Variables
Chord (mm)
Length (mm)
Thickness (mm)
Cavity width (mm)
Silicone bite (mm)
Symbol
c
l
R
t
s
b
Build-up
1500
1500
1500–6000
8
16
10
Table 7
Input summary
Variables
Chord (mm)
Length (mm)
Thickness (mm)
Cavity width (mm)
Silicone bite (mm)
Symbol
c
l
R
t
s
b
Build-up
1500
1500
1500
8
12, 16, 19.20
10
As has been described above, the relationship between the radius and internal pressure is essentially governed by the panel stiffness. Therefore, we plotted the internal pressure changing along increasing radius in Fig. 8. Since the chord of the panel is 1500 mm, the minimal radius of panel cannot be less than 750 mm. The curve is acting as a quasi-inverse function of radius, which is to say, it drops sharply initially and then levels off with larger radii. The input information is shown in Table 6.
#### 4.3.3 Cavity width
The most frequently used spacer bar widths in the market are considered in this study, namely 12 mm, 16 mm, 19 mm and 20 mm. The input information is shown in Table 7. As shown in Fig. 9, there is a subtle increase of internal pressure observed in both curved and flat IGU along cavity. Within the range, dp in curved IGUs is rising from 3.9 to 4.5 kPa and from 0.311 to 0.535 kPa in flat panels. By comparing the ratio in Fig. 10, flat panel displays 61% increase in contrast to 15% growth of curved IGU, therefore exhibits higher dependency of cavity width than curved IGU.
In order to understand the trends, we transform Eq. (8) to an expression that contains cavity width s and volumetric stiffness $$k_{V}$$. Since,
\begin{aligned} V_0 =A\cdot s \end{aligned}
(16)
where A is the area.
\begin{aligned} \frac{1}{k_{V,{ tot}} }=\frac{1}{k_{V1} }+\frac{1}{k_{V2} }=\frac{k_{V1} +k_{V2} }{k_{V1} k_{V2} } \end{aligned}
(17)
Substitute Eqs. (16) and (17) into (8), we have
\begin{aligned} dp\!=\!\frac{-\left( A\cdot s\!+\!p_{{ atm}} /k_{V,{ tot}} \right) \!+\!\sqrt{\left( A\cdot s\!+\!p_{{ atm}} /k_{V,{ tot}} \right) ^{2}\!-\!4k_{V,{ tot}} p_{{ iso}} }}{2/k_{V,{ tot}} }\nonumber \\ \end{aligned}
(18)
It can be found Eq. (18) is an increasing function, therefore the internal pressure is increasing with s.
#### 4.3.4 Silicone bite
Currently published analytical methods (BSI 2013) for flat IGUs have not included silicone stiffness. The glass edges are assumed to be rigidly fixed (referred as to “rigid fix”), which however, does not reflect the reality. Two panels of an IGU are held together by the silicone joint sealing around the spacer bar. In this parametric study, we examined the silicone bite from 10 to 40 mm, and compare against the analytical solution introduced in PrEN 16612 (BSI 2013) in Fig. 11. It shows that the internal pressure is relatively low, and the impact brought by silicone is negligible. This is because the silicone stiffness is far higher than the bending stiffness in flat IGUs, and the panel bending deformation contributes to the most of volume change and hence dominates the internal pressure.
On the contrary, Fig. 12 shows notable deviation in the internal pressure between rigid fix and silicone bond in curved IGUs. Rigid fix boundary condition gives 7.84 kPa as depicted by the asymptote, whereas 10 mm silicone bite leads to an internal pressure of 4.26 kPa, i.e. 45.7% lower than rigid fix. It ascends along with bigger silicone bite, namely higher silicone stiffness. We list the trends of silicone impacts to both flat IGUs and curved IGUs in Fig. 13 for a parallel comparison. What can be clearly seen is that the internal pressure of curved IGUs are significantly more susceptible to the size of silicone joint in contrast to flat IGUs. It can be explained also by the stiffness change of silicone. 10 mm deep silicone bite gives rise to a relatively lower stiffness and engaged more in the overall volume deformation. As it goes up, the less flexibility results in less volume change and subsequently higher internal pressure.
#### 4.3.5 Length
Panel length is varied from 375 to 6000 mm. Given the chord is unchanged, it can be deemed as a parametric study of aspect ratio. Both flat IGU and curved IGU exhibit strong dependency to the length. The nonlinear descending curves behave as an inverse function of l. For flat panels, the volumetric stiffness is expressed as Eq. (14) and there is an inverse relation to the bi-quadrate of short edge a. before aspect ratio reaches 1, the length is acting as the short edge a and thus leads to the rapid drop at the initial stage. Despite the lack of analytical expression of volumetric stiffness of curved IGUs, the similar curve pattern in Fig. 14 indicates that the overall stiffness of curved glass panel is predominantly driven by the longitudinal length l when the aspect ratio less than 1, i.e. l smaller than the chord c.
### 4.4 Summary
Based on the sensitivity studies carried out for each parameter, qualitative sensitivity degree is summarised to compare the difference between flat IGU and curved IGUs in Table 8.
Table 8
Sensitivity degree comparison
Curved IGU
Flat IGU
Silicone bite (b)
***
*
Length (l)
****
*****
****
N/A
Thickness (t)
Varying (less sensitive with smaller radius)
*****
Cavity width (s)
*
*
It’s found from Table 8 that changing cavity width exerts least impact among all five variables, therefore it will not be brought up for much discussion and assumed to be consistent in the following study.
It is worth noting the distinct influence of silicone bite in curved IGUs from flat IGUs. Therefore, when designing the internal pressure of curved IGUs, it’s not accurate to simply assume the panels are rigidly fixed at the edges as we normally do with flat IGUs.
To take into account the silicone stiffness into the overall volumetric stiffness $$k_{v, { tot}}$$, Eq. (17) can be written as
\begin{aligned} \frac{1}{k_{V,{ tot}} }=\frac{1}{k_{V1} }+\frac{1}{k_{V2} }+\frac{1}{k_{V3} } \end{aligned}
(19)
$$k_{{ Vs}}$$ denotes the volumetric stiffness of silicone. According to Eq. (9), $$k_{{ Vs}}$$ is a ratio of the applied pressure and the resultant volume change, and the latter is induced by the stretching of silicone bite regardless of panel deformation. Therefore, the volume change can be determined by calculating the cavity section area change due to a translational displacement of the glass panel as illustrated in Fig. 15.
Translational displacement x can be calculated by following Hook’s Law:
\begin{aligned} x=F/k=P\cdot a\cdot l/(E_s \cdot (2a+2l)\cdot b/s) \end{aligned}
(20)
where $$E_{S}$$ is the Young’s modulus of silicone, assumed as 4 MPa (Dow Corning 2015). s is the cavity width, c is the chord, b is the length of silicone bite, R is the radius and a denotes the arc length:
\begin{aligned} a=2R\cdot \arcsin \frac{c}{2R} \end{aligned}
(21)
The volume change can be expressed as:
\begin{aligned} dV_s =x\cdot c\cdot l \end{aligned}
(22)
Substitute Eqs. (20)–(22) into (9)
\begin{aligned} k_{{ Vs}} =\frac{\left( 2R\arcsin \frac{c}{2R}+l\right) E_S b}{scl^{2}R\arcsin \frac{c}{2R}} \end{aligned}
(23)
Since it can easily be concluded from Eq. (18) that internal pressure dp is a function of overall volumetric stiffness, the remarkable impact by silicone bite in curved IGUs can be explained. By observing Eq. (19), the total volumetric stiffness of a curved IGU can be deemed as an equivalent spring of a group of springs in series. Therefore, the equivalent spring stiffness is essentially dictated by the spring of the least stiffness. When R decreases, the curvature effect will go up and give rise to a greater volumetric stiffness of each glass panel, i.e. $$k_{v1}$$ and $$k_{v2}$$, which are much higher than the silicone stiffness. Therefore, silicone bite plays a remarkable role in curved IGUs. On the other hand, flat panels are relatively low, and hence $$k_{v1}$$ and $$k_{v2}$$ are smaller than $$k_{{ vs}}$$, then the overall stiffness is dominated by the bending stiffness of panels whereas silicone stiffness can be almost ignored.
## 5 Normalized internal pressure
Thus, for a curved IGUs with an arbitrary radius R, the internal pressure load due to climatic load will always fall within a range. The upper bound and lower bound of the range are defined as “infinite curvature effect” and “no curvature effect” respectively. The magnitude of the two bounds can be determined by simple hand calculation. When the panel hypothetically tends to be infinite stiff, ($$1/ k_{v1 }+1/ k_{v2})$$ will level off to zero, and according to Eq. (19) the volumetric stiffness $$k_{v,{ tot}}$$ equals to the silicone stiffness as calculated by Eq. (23). When there is no curvature effect, which means R is infinite, the panel is equivalent to flat panel, and the term ($$1/ k_{v1 }+1/ k_{v2})$$ can be obtained by substituting Eq. (14). For consistency consideration, the stiffness of silicone bite is considered in flat IGUs too.
Figures 16 and 17 show the internal pressure change in radius with and without taking into consideration of 10 mm silicone bite respectively. Upper bounds and lower bounds are plotted in the figures. Glass thickness, chord and length are fixed to be 8 mm, 1500 mm and 1500 mm respectively. The radius is ranging between 750 and 10 mm. It should be noted that 750 mm is the minimal radius that can be adopted for a chord 1500 mm. As has been discussed earlier, there is a big discrepancy due to the lack of silicone stiffness. If we compare the upper bounds of two figures, the resultant internal pressure by assuming rigid fixing is nearly twice of that with silicone bite and it’s far away from realistic values. In addition, the curvature effect is approaching infinite in Fig. 17 when the radius is reduced to 750 mm. However, it is not realistic either because the silicone joint that holding two panels together cannot provide rigid translational restraints but act as springs. This also explains that in Fig. 16, the internal pressure can’t reach 100% of $$t = \infty$$ even though the curvature has increased to the most.
With the purpose of expanding Fig. 16 for different radii and different lengths, a concept of “normalized internal pressure” $$r_{{ dp}}$$ is introduced to normalize the relations between internal pressure of curved IGUs and other impact parameters.
\begin{aligned} r_{{ dp}} =\frac{p_{{ upper}} -{ dp}}{p_{{ upper}} -p_{{ lower}} } \end{aligned}
(24)
where $$p_{{ upper}}$$ is the upper bound, and $$p_{{ lower}}$$ is the lower bound.
A dimensionless curvature parameter denoted as Z is employed as x axis, which is transformed from the curvature parameter introduced in Batdorf’s publication (1947)
\begin{aligned} Z=\left( \frac{Rt}{c^{2}}\right) \cdot 10^{3}. \end{aligned}
(25)
Within parameters explored in this study, designers may use Fig. 18 as a quick and easy indication to determine the magnitude of internal pressure due to climatic actions. The procedures are summarised below:
1. 1.
Calculate normalized curvature parameter Z using Eq. (25).
2. 2.
Based on normalized curvature parameters Z and the aspect ratio of the panel l / C, determine the corresponding normalized internal pressure $$r_{{ dp}}$$.
3. 3.
Calculate lower and upper bounds using Eqs. (23) and (14) respectively.
4. 4.
Calculate the magnitude of actual internal pressure dp using Eq. (24).
## 6 Conclusion
This paper reviews existing literature with respect to calculating the internal pressure change within IGUs due to climatic loads. The internal pressure can be mathematically expressed as a function is volumetric stiffness $$k_{V}$$. Since analytical expression of $$k_{V}$$ for cylindrically curved panels is unlikely to be solved, numerical iteration process is adopted to determine the internal pressure of curved IGUs. A sensitivity study on the geometric parameters including panel thickness, radius, length, cavity width and silicone bite depth is undertaken to identify the predominant impact factors. The correlations between the internal pressure and these parameters are plotted. The results between flat IGUs and curved IGUs are also compared and discussed. Following conclusions are drawn:
• For the same dimension, curved IGUs always generate higher internal pressure caused by the same climatic action than that of flat IGUs due to geometric stiffness of curved IGUs. Nevertheless, the sensitivity level to each parameter differs a lot between curved and flat IGUs.
• Silicone stiffness makes a significant difference in determining the internal pressure of curved IGUs but very minimal in flat IGUs. Therefore, it is necessary to consider the stiffness of silicone bite in the design to obtain accurate internal pressure.
• Internal pressure increases with increased glass thickness. However, the influence of thickness is varying with curvature. The higher the curvature is, the less sensitive the internal pressure to thickness, and vice versa. When the curvature is small enough and the panel is quasi-flat, the internal pressure will be approximately proportional to the thickness to the power three.
• Cavity width has a subtle influence for curved IGUs but has relatively higher weight in flat IGUs.
• Longitudinal length of has a big impact on internal pressure increase for both flat and curved IGUs, especially when the longitudinal length is less than the chord length.
• For an arbitrary radius, the resultant internal pressure always falls within a range. The upper and lower bounds of the range can be determined by hand calculation. The smaller the radius, the closer the internal pressure to the upper bounds, and vice versa.
• The concept of “normalized internal pressure” is proposed as a normalized expression of internal pressure in curved IGUs. This dimensionless parameter can be adopted for future empirical charts. In this paper, a dimensionless chart is drawn to depict the relations between the internal pressure and curved geometry.
• The dimensionless chart can be further expanded by considering different silicone bites /glass thickness combination, and therefore provide an empirical method for curved IGUs design.
## Notes
### Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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13. Vuolio, A.: Structural behaviour of glass structure in facades. Ph.D. Thesis, Technology Laboratory of Steel Structures (2003)Google Scholar | {"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": 0.8507910370826721, "perplexity": 2499.643963213304}, "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-2019-04/segments/1547583660258.36/warc/CC-MAIN-20190118172438-20190118194438-00232.warc.gz"} |
http://mathoverflow.net/questions/31538/non-abelian-class-field-theory-and-fundamental-groups?answertab=oldest | # Non-abelian class field theory and fundamental groups
Over the years, I've been somewhat in the habit of asking questions in this vein to experts in the Langlands programme.
As is well known, given an algebraic number field $K$, they propose to replace the reciprocity map $$A_K^\*/K^*\rightarrow Gal(K^{ab}/K)$$ of abelian class field theory by a correspondence between the $n$-dimensional representations $\rho$ of $Gal(\bar{K}/K)$ and certain automorphic representations $\pi_{\rho}$ of $GL_n(A_K)$. (We'll skip the Weil group business for this discussion.) Substantial arithmetic information is carried on either side by the $L$-functions, which are supposed to be equal.
This involves deep and beautiful mathematics whenever something can be proved, and there are many applications, such as the Sato-Tate conjecture or this recent paper of Chenevier and Clozel:
http://www.math.polytechnique.fr/~chenevier/articles/galoisQautodual2.pdf
(I mention this one because it is in some ways very close to the point of this question.)
However, there are elementary consequences of abelian class field theory that seem not to have obvious non-abelian analogues. The one I wish to mention today has to do with the fundamental group. Given a number field $K$ (assume it's totally imaginary to avoid some silly issues), how can we tell if it has non-trivial abelian unramified extensions? Class field theory says we can look at the class group, which is quite computable in principle, and even in practice for small discriminants. But now, suppose we go on to ask the non-abelian question: which number fields have $$\pi_1(Spec(O_K))=0?$$ That is to say, when does $K$ have no unramified extension at all, abelian or not? As far as I know, there is no easy answer to this question. Niranjan Ramachandran has pointed out that there are at least ten examples, $K=\mathbb{Q}$ (oops, that's real) and $K$ an imaginary quadratic field of class number one. I know of no others. Of course I would be happy to collect some more, if someone else has them lying around.
But the question I really wanted to ask is: Suppose we are in a Langlands paradise where everything reasonably conjectured by the programme is a theorem. Does this give a way to algorithmically (as we run over fields $K$) resolve this question as in the abelian case? Otherwise, is there a sensible refinement of the usual formulation that would subsume such applications?
I'm embarrassed to admit I hadn't followed the question mentioned by David Hansen (even after commenting on it). Thanks to David for pointing it out. Of course my main question still stands. I've changed the title following Andy Putman's suggestion. The original title evolved from a (humorously) provocative version that I normally use only among friends who already know I'm a Langlands fan: 'What is the Langlands programme good for?'
Regarding jnewton's very natural thought: in addition to other difficulties, one would also need to bound $n$.
Here is one more remark concerning jnewton's suggestion. Of course in the realm of classical holomorphic cusp forms, there are infinitely many of level one. More generally, it is shown in the paper
http://www.math.uchicago.edu/~swshin/Plancherel.pdf
that whenever $G$ is a split reductive group over $\mathbb{Q}$, there are infinitely many cuspidal automorphic representations that are unramified everywhere and belong to the discrete series at $\infty$. (I presume there are other results of this sort. This one I just happen to know from a talk last Fall.) According to Clozel's conjecture as you might find in
http://seven.ihes.fr/IHES/Scientifique/asie/textes/Clozel-juil06.pdf
(conjecture (2.1)), algebraic ones among them should correspond to motivic Galois representations (after we choose a representation of the dual group)*. I don't have the expertise to recognize algebraicity in such constructions, in addition to the danger that I'm misunderstanding something more elementary. But it seems to me quite a task to show directly that there are none corresponding to Artin representations. (The only case I could do myself is the classical one.)
Now, I would like very much to be corrected on all this. But such families do seem to indicate that a 'purely automorphic' approach to the the $\pi_1$ question is somewhat unlikely, at least within the current framework of the Langlands correspondence.
I suppose I'm sabotaging my own question.
*Note that in these situations, the Galois representations don't have to be unramified, since there is the choice of a coefficient field $\mathbb{Q}_p$. In general, they should only be crystalline at $p$.
Matthew: Since I didn't really expect a complete answer to my question, if you could write your extremely informative series of comments as an answer, I will accept it. (Barring the highly unlikely possibility that someone will write something better between the time you submit your answer and the time I look at it.)
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This is a fantastic question, but maybe it could have a more descriptive title? – Andy Putman Jul 12 '10 at 15:24
Minhyong, KConrad gives some further interesting examples of fields with no unramified extensions here: mathoverflow.net/questions/26491 – David Hansen Jul 12 '10 at 15:54
I guess the (strong) Artin conjecture tells you that given $L/K$ unramified, then non-trivial irreducible n-dimensional (complex) representations of $Gal(L/K)$ arise from cuspidal automorphic representations of $GL_n(\mathbb{A}_K)$ of level 1' (i.e. unramified at all finite places) and certain type at the archimedean places. So if we can rule out the existence of any of these, we can conclude that there's no non-trivial unramified $L$ - this may well be difficult to do though. – jnewton Jul 12 '10 at 17:08
@jnewton: It is indeed very difficult; the relevant automorphic representations cannot be counted by any naive application of the trace formula. This dichotomy is already apparent in classical modular forms - forms of weight $\geq 2$ and a given level can be counted precisely, but counting forms of weight $1$ is very difficult. – David Hansen Jul 12 '10 at 17:20
Does anyone know of an answer when we change "Langlands programme"'s role in the above to any other set of mostly-believed conjectures? – Dror Speiser Jul 12 '10 at 20:47
In response to Minhyong's request, I am reposting my comments above as an answer:
As James Newton commented, if $L/K$ is unramified, then an irreducible $n$-dimensional representation (over $\mathbb C$) of $Gal(L/K)$ will correspond, in the Langlands paradise, to a cuspidal automorphic representation of $GL_n(\mathbb A_K)$. The cuspidal automorphic representations that arise in this way are sometime (especially in the older literature) called "Galois type".
Thus one can (more or less --- there is the issue of irreducible vs. all reps. which I won't think about here) encode unramified extensions of $K$ whose Galois groups admit $n$-dimensional representations in terms of Galois type cuspidal automorphic representations $\pi$ of $GL_n(\mathbb A_K)$ that are unramified at every finite prime.
Now the question arises: how many such $\pi$ are there, and can one compute them?
Being of Galois type is (conjecturally, but we are in paradise!) purely a condition on $\pi_v$ for primes $v$ of $K$ lying over $\infty$, and in fact there are a finite number of prescribed representations of $GL_n(K_v)$ ($= GL_n(\mathbb R)$ or $GL_n(\mathbb C)$) which are allowed. (E.g. for $GL_n(K)$, the possibilities are limit of discrete series, corresponding to holomorphic weight one forms, or principal series with $\lambda = 1/4$, corresponding to Maass forms with eigenvalue of Laplacian equal to $1/4$.) For a given $n$ and $K$, these can be enumerated.
Now since we are asking that the "weight" (i.e. the collection of $\pi_v$ for $v|\infty$) be bounded (i.e. lie in a given finite set), and we are also asking that "level" be one (i.e. that there is no ramification at any finite prime), there are only a finite set of $\pi$ corresponding to irreducible everywhere unramified $n$-dimensional complex representations of $GL_n(Gal(\bar{K}/K)$. [Aside: Minhyong asked for a sketch of a proof of this; here goes: fixing the representation at infinity means that we are fixing a bunch of elliptic operators that the automorphic forms must satisfy. Fixing the level means that we are working on some particular quotient $X/\Gamma$ (here $X$ is the symmetric space attached to the real group in question, in our particular case $GL_n(K\otimes \mathbb R)$, and $\Gamma$ is the fixed level). This need not be compact (indeed won't be in our particular case), but the cuspidal condition (indeed, even the moderate growth condition that non-cuspidal automorphic forms are required to satisfy) means that we can pretend it is, since we explicitly rule out the possibility of extreme growth at infinity. So now we looking at sections of some bundle on a compact space satsifying a bunch of elliptic equations, and such a space of sections if finite dimensional. (The holomorphic modular forms case is the most familiar: in this case the elliptic equations are the Cauchy--Riemann equations. In the Maass form case, the corresponding fact is the finiteness of the eigenspaces of the Laplacian. These are good models for the general case.)]
To actually compute them (say for a fixed choice of $K$ and $n$) would be quite difficult (as David Hansen notes in his comment). The reason is that the relevant $\pi_v$ for $v|\infty$ are never discrete series (even when $n = 2$, and in any case, note that $GL_n(\mathbb R)$ never has discrete series if $n > 2$, and $GL_n(\mathbb C)$ never has discrete series when $n > 1$), and so standard applications of the trace formula to counting automorphic forms won't work.
Nevertheless, it seems that one might still be able to use the trace formula to analyze the situation, at least in principle. For example, Selberg used his original formulation of the trace formula for $SL_2(\mathbb R)/SL_2(\mathbb Z)$ to compute cuspidal Maass forms of level 1, and showed that the smallest eigenvalue $\lambda$ that occurs has $\lambda$ much greater than 1/4 (maybe closer to 90?).
And we all know that it is not hard to show that there are no holomorphic weight one forms of level one. So one can automorphically prove (modulo standard conjectures in the Maass form case) that there are no everywhere unramified two-dimensional complex representations of $Gal(\bar{\mathbb Q}/\mathbb Q)$. (This is of course an incredible battle, even in paradise, for a tiny portion of the information that Minkowski gives us, but is meant just to illustrate that this approach is not a priori ridiculous.)
What I don't see at all from this point of view is how to study all $n$ simultaneously. For example, one could imagine implementing this program and finding, for some $K$ and some $n$, maybe $n = 10^6$, that there are no unramified extensions $L/K$ with $L$ admitting an irrep. of dimension $\leq 10^6$. This doesn't rule out the possibility that there is a beautiful, everywhere unramified extension $L/K$ whose Galois group's lowest degree irrep. happens to be of enormous dimension.
The Langlands program seems to be intrinsically geared to thinking about linear representations of Galois groups, and to set the scene, you have to begin by choosing a linear group, which will then cut everything else down in a Procrustean manner. At least superficially (and this answer reflects just superficial thoughts about the question), it doesn't seem well adapted to questions related to the nature of $\pi_1(\mathcal O)$, where no a priori linear structure is given, or indeed expected.
[Added July 14, in response to Minhyong's question in the comments as to whether or not discrete series can convert into non-discrete series after applying some functoriality. The answer is essentially no, as I will now explain.
Added November 29, 2011: What follows is wrong; the answer seems rather to be yes. (See below for details.)
For an arithmetic geometer, one should think of an automorphic form on the adelic group $G(\mathbb A_K)$ as a morphism from the motivic Galois group (over the base number field $K$) to the $L$-group of $G$. (There are subtleties and caveats, of course, but they need not concern us here; all I will say about them is that automorphic forms can give rise to "motives" with non-integral $(p,q)$ in its Hodge decomposition, which necessitates enlarging the category of motives to an unknown larger category, whose hypothetical Tannakian group is called "the Langlands group".)
Now functoriality takes place when you have a map from the $L$-group of $G$ to the $L$-group of $H$; one can just compose this with a map from the motivic Galois group to the former, to obtain a map from the motivic Galois group to the latter. Functoriality is the assertion that the corresponding automorphic form on $H(\mathbb A_K)$ exists.
Now given an automorphic form $\pi$, its factors at the primes $v|\infty$ encode (via the local Langlands corresondence for $\mathbb R$ or $\mathbb C$) the Hodge numbers of the corresponding motive. One feature of discrete series is that (among other properties) they give rise to regular Hodge numbers, i.e. to sequences of $h^{p,q}$ with each $h^{p,q} \leq 1$. Now our original automorphic rep'n $\pi$ on $G(\mathbb A)$ corresponds to a motive whose Mumford--Tate group lies in the $L$-group of $G$, and if $\pi$ has discrete series components at primes above $\infty$, it has regular Hodge numbers at every place dividing $\infty$. If we then pass to a new motive by applying some map from the $L$-group of $G$ to the $L$-group of $H$, then concretely this corresponds to doing some kind of multilinear algebra on our motive, and the only way this can kill the property of having regular Hodge--Tate weights is if we do something like taking the diagonal map from the $L$-group of $G$ into its product with itself, and then embed the latter into the $L$-group of $H$. All such constructions will necessarily be a "reducible" rep'n of the $L$-group of $G$ in the $L$-group of $H$ (more precisely, the centralizer will be a non-trivial Levi), and the corresponding automorphic form won't be a cuspform.
But even if we destroy the property of having regular Hodge numbers, we typically still don't have an Artin motive. To get an Artin motive we have to have $h^{p,q} = 0$ unless $p = q = 0$, and to do this, we have to do even more destructive things, like map the $L$-group of $G$ into the $L$-group of $H$ via the trivial representation, or something similar. Again, this won't correspond to any kind of interesting automorphic forms, just those that correspond to (certain) sums of characters. So we can't produce interesting Galois type automorphic forms out of automorphic forms whose factors at primes above $\infty$ are discrete series.]
[Correction added Nov. 29, 2011: From the Galois/motivic point of view, we have an algebraic group (the Mumford--Tate group of some motive), with a representation (the particular motive), and the Mumford--Tate group contains a cocharacter whose eigenvalues are the Hodge numbers. Discrete series corresponds to the eigenspaces being one dimensional. We now apply some functoriality, which is essentially to say that we apply some multi-linear algebraic process to the representation. Now this can certainly produce eigenspaces for the cocharacter of multiplicity $> 1$. (E.g. the adjoint representation of $SL_3$ has a two-dimensional eigenspace.) So it seems that functoriality doesn't preserve being discrete series. It does preserve being tempered. And the remarks about not getting Artin motives still seem okay, since while the eigenspaces can become greater than $1$-dimensional, for all the eigenspaces to become trivial, we have to do something pretty destructive, like applying functoriality for the trivial representation.]
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But still, a portion of the Langlands programme is about the $\mathbb{C}$-algebraic completion of $\pi_1(O_K)$, and it would be nice if it could tell us whether or not the algebraic completion is trivial. – Minhyong Kim Jul 14 '10 at 5:09
I agree, but unfortunately it seems to really investigate it as a completion (!), passing through each possible dimension in turn. – Emerton Jul 14 '10 at 5:12
The Langlands Program grew out of Langlands' realization that Class Field Theory may be reformulated as an equivalence between 1-dim representations of the Galois group and automorphic forms on $GL_1.$ Thus "linear structure", in the sense described by Matt, is intrinsic to the LP. If you want something that does not involve it, a better choice would be a theory outputting some anabelian version of the Galois group (e.g. nilpotent completions have been studied). – Victor Protsak Jul 14 '10 at 5:48
Regarding the point that's been made about discrete series, let me just reveal my ignorance by asking the stupid question that gave me reason to worry. If you look at the unramified discrete series cusp forms of the sort constructed by Shin, is it impossible for it to transfer to a non-discrete one on some $GL_n$ corresponding to an Artin representation? – Minhyong Kim Jul 14 '10 at 6:11
To answer my own question, I guess the answer is 'yes', by what we know about unramified extensions of $\mathbb{Q}$. So the only reasonable question would be 'is it impossible for purely automorphic reasons' in a sense I hope is clear enough. – Minhyong Kim Jul 14 '10 at 10:38
I'm going to take the step of disagreeing slightly with the perspective offered in Professor Emerton's answer, in a bold attempt to prise away the green for myself.
The question is, what does the Langlands program have to say about the finiteness (or otherwise) of $\pi_1(\mathcal{O}_K)$? Let's first revisit what one knows by classical'' methods. Minkowski proved that if $K$ is an extension of degree $n$, then
$$\sqrt{|\Delta_K|} \ge \left(\frac{\pi}{4}\right)^{r_2} \frac{n^n}{n!},$$
which, asymptotically, implies that the root discriminant $\delta_K = |\Delta_K|^{1/n}$ is at least $e^2 \pi/4 - \epsilon$ for large $n$, and in particular, that $|\Delta_K| > 1$ if $n > 1$. This is purely a geometry of numbers argument.
What does the Langlands program have to say about this question? First, let me ask a related question: what does the Langlands program say about elliptic curves $E/\mathbb{Q}$ with good reduction everywhere? In this case, it implies (and is known, by Wiles!) that there exists a classical cuspidal modular form $$f \in S_2(\Gamma_0(1)).$$ The latter group vanishes, and so $E$ does not exist. Yet this latter fact seemingly requires an actual computation - namely, that $X_0(1)$ has genus zero. This could be tricky if one wants to replace $E$ by an abelian variety $A$ of (varying) dimension $g$, or replace $\mathbb{Q}$ by another field $F$. It turns out, however, that one can show that $f$ does not exist purely from the existence of the relevant functional equation - more on this later.
Let us return to our original question. Suppose we have a field $K$ unramified everywhere over $\mathbb{Q}$. It has been suggested that one should ponder the existence of algebraic automorphic forms $\pi$ for $\mathrm{GL}_n/\mathbb{Q}$ associated to irreducible Artin representations $\rho$ of $\mathrm{Gal}(K/\mathbb{Q})$. However, it is more natural to consider the regular representation. In this case, it is (of course) known by Hecke that $\zeta_K(s)$ has a meromorphic continuation that is entire away from $s = 1$, and, moreover, that $\zeta_K(s)$ satisfies a functional equation of a precise type. Now, purely given the existence of the functional equation for $\zeta_K(s)$, Odlyzko gives a formula(!) for $\log |\Delta_K|$ as some function of $s$ involving the roots of $\zeta_K$ (the function is (obviously) constant, but is not obviously constant). One then deduces a lower bound for $\delta_K$ of the form $2 \pi e^{\gamma} - \epsilon$, which is better than Minkowski's estimate. (Odlyzko's method can be refined to give better bounds.)
What is easy to miss in this argument is that role that the Langlands conjecture plays in this argument - in this case the theorem of Hecke - is already known! One might claim that this argument uses "more" than Langlands and ask for an argument that is purely algebraic and geometric (and here by "one", I mean Brian Conrad or Chevalley), but I think this is a little misguided. After all, I think it would be hard to prove that there does not exist any Maass form for $\mathrm{SL}_2(\mathbb{Z})$ of eigenvalue $1/4$ without using some analysis.
Can one argue similarly to see that there are no abelian varieties $A$ with good reduction everywhere? Indeed, Mestre gave such an argument (before Fontaine!). Namely, if $A$ is an abelian variety, and $L(A,s)$ is automorphic in the expected sense, then $A$ has conductor at least $10^{g}$. Moreover, this is close to optimal ($X_0(11)^g$ has conductor $11^g$). There are other arguments along these lines. Stephen Miller (and Fermigier independently) proves that there are no cusp forms for $\mathrm{SL}_n(\mathbb{Z})$ of "weight zero" (cohomological with the same infinitesimal character as the trivial representation) and "level one" for all $n$ in the range $2 \ldots 23$ - another generalization of the fact that $X_0(1)$ has genus zero - using Rankin--Selberg L-functions.
Here is a final reason why one should expect some analysis. All of these arguments fundamentally require the discriminant of $K$ to be small. Any `algebraic'' method should be expected to work more generally. Yet consider the question of whether $\pi_1(\mathcal{O}_K)$ is finite when $\delta_K$ is large. The only known method for answering this question is producing an unramified extension $L/K$ such that the GS-inequality applies to $\pi_1(\mathcal{O}_L)$. The question on whether these groups are (always?) infinite when $\delta_K$ is sufficiently large is wildly open, and I know no good heuristics on this question.
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Dear FJ, This is a very nice post. – Emerton Jul 26 '10 at 18:18
For the neophytes, the GS-inequality is the Golod-Shararevich inequality. The paper by Mestre is Formules explicites et minorations de conducteurs de variétés algébriques, Compositio Math. 58 (1986), no. 2, 209--232. – Chandan Singh Dalawat Jul 27 '10 at 2:51
Very nice. So the gist of the argument is that refined lower bounds for discriminants or conductors are in fact consequences of analytic properties of $L$-functions? – Minhyong Kim Jul 27 '10 at 8:31
FJ: Sorry, I think I can accept only one answer. I do wish I could accept yours as well. – Minhyong Kim Jul 31 '10 at 9:04 | {"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": 0.9119622111320496, "perplexity": 316.65811023852154}, "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-2015-14/segments/1427131298660.78/warc/CC-MAIN-20150323172138-00175-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/835214/continuity-in-the-strong-vs-operator-topologies-on-a-compact-space | # Continuity in the Strong vs Operator topologies on a compact space
Suppose $X$ is a compact space, $H$ is a Hilbert space, and $f:X \rightarrow B(H)$ is continuous when $B(H)$ is given the strong topology. Does this imply that $f$ is continuous when $B(H)$ is given the operator norm topology?
My guess is that the answer is no, although I can't think of an easy counterexample.
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Let $H=\ell_2 (\mathbb{R} ) ,$ and let $T_n : \ell_2 \to \ell_2 ,$ be defined as follows: $$T_n \left(\sum_{j=1}^{\infty} x_j e_j \right) =x_n e_n ,$$ $$T\equiv 0 .$$ Let $$X=\{T\}\cup\bigcup_{n=1}^{\infty} \{T_n \}$$ and let $\tau_1$ denote the topology on $X$ induced from $(B(H), \tau )$ where $\tau$ denotes the strong topology on $B(H) .$ Let $\sigma$ denotes the operator norm topology on $B(H) ,$ and let $f:X\to B(H),$ be defined by $$f(A)=A .$$ Then $(X ,\tau_1 )$ is compact space and $f :(X, \tau_1 ) \to (B(H), \tau )$ is continuous but $f :(X, \tau_1 ) \to (B(H), \sigma )$ is not continuous.
It may help to notice that $X$ is homeomorphic to $\{1/n : n \ge 1 \} \cup \{0\}$, or to the one-point compactification $\mathbb{N} \cup \{\infty\}$. – Nate Eldredge Jun 15 '14 at 20:51 | {"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": 0.9995978474617004, "perplexity": 42.368182294785306}, "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/1469257827791.21/warc/CC-MAIN-20160723071027-00199-ip-10-185-27-174.ec2.internal.warc.gz"} |
http://brattahlid.tripod.com/sw3-42web.htm | The Disk
Opposing to the bulge, the disks of the spiral galaxies are rich in gas and dust clouds and in open clusters that behave as factories that produce young stars with a high metallic content. These stars belong to the so-called population I (contrarily to the older stars of the bulge and the halo, which belong to the population II). In the Milky Way, this disk extends for 80 000 light years. The thickness of the stellar disk diminishes from the centre towards the outskirts of the galaxies. However, the gaseous and dusty disk, which is larger than the stellar disk, is only about 700 light years thick close to the centre but 3000 light years thick at the exterior edges.
Composition
Neutral Hydrogen
One of the main components of the disk is the neutral hydrogen (1 proton and 1 electron), detected through its emissions at the 21-cm wavelength line, which take place when it changes from a high energy level to a low energy level. This material is mainly distributed along the spiral arms and in a ring located about 16.000 light years from the galactic centre.
Molecular Clouds
Other important components of the disk are the molecular clouds. They may reach dimensions of 3000 light years and, opposing to what succeeds with the neutral (atomic) hydrogen, they strongly prevail in the internal regions of the disk.
Inside them there may be found simple molecules as the oxydryl (OH), the carbon monoxide (CO) and other more complex, frequently organic, as the formic acid (HCOOH) or the methilic alcohol (CH3OH). The most frequent is the molecular hydrogen (H2).
It is thought that the dusty cocoons protect the gases from the ultraviolet radiation that dissociate the molecules. They also provide binders to which the gas atoms may be attached and be kept at comparatively low speeds. So, protected from the radiation and kept in low energy states, the hydrogen atoms have the chance to fuse themselves into the molecules that are found inside the clouds, under the dust and atomic gas layers.
On the other hand, the dust grains are thought to be composed by a solid core of silicon carbide surrounded by amorphous coal layers. They are also observed in these particles water ice, methane and carbon oxide layers.
M16: open cluster in Aquila (H. Frommert, C. Kronberg)
Molecular gas cloud inside the Aquila cluster
The Spiral Arms
The spiral arms are density waves (gravitational traps) that sweep the galaxy and force the matter that they intercept to softly keep the pace with the density wave. This phenomenon provides a slight increase of the matter density in the arms' region, which works as a stimulant for the production of new stars.
They are, for this reason, particularly rich in nebulae and open clusters containing high quantities of newborn stars. These stars, which frequently display large dimensions and an intense blue brightness, are precisely those who give the arms their known luminosity.
_ | {"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": 0.8532737493515015, "perplexity": 1459.8721133272334}, "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/1544376828697.80/warc/CC-MAIN-20181217161704-20181217183704-00119.warc.gz"} |
https://mathinsight.org/visualizing_two_dimensional_linear_system | # Math Insight
### Visualizing the solution to a two-dimensional system of linear ordinary differential equations
Below are two applets through which you can explore the solution a system of two linear ODEs, i.e., a system of the form \begin{align*} \diff{\vc{x}}{t} &= A \vc{x}\\ \vc{x}(0) &= \vc{x}_0, \end{align*} where $\vc{x}$ is a two-dimensional vector, $\vc{x}=(x,y)$, $A$ is a $2 \times 2$ matrix, and the initial condition is $\vc{x}_0=(x_0,y_0)$.
#### Interactive phase plane applet
The first applet shows the solution to $\diff{\vc{x}}{t} = A \vc{x}$, plotted both as functions as time and in the phase plane. The applet demonstrates how the phase plane represents the solution trajectory $(x(t),y(t))$ through time. It also illustrates the link between the solution and the eigenvalues and eigenvectors of $A$.
Control panel (Show)
Eigenvalues and eigenvectors (Show)
Equilibrium classification (Show)
Solution (Show)
A linear system with phase plane and versus time.
Illustration of the solution to a system of two linear ordinary differential equations. The system is of the form $\diff{\vc{x}}{t} = A\vc{x}$ with prescribed initial conditions $\vc{x}(0)=\vc{x}_0$, where $\vc{x}(t)=(x(t),y(t))$. The solution trajectory $(x(t),y(t))$ is plotted as a cyan curve on the phase plane in the left panel. In the right panel, the components of the solution $x(t)$ (top axes, solid cyan curve) and $y(t)$ (bottom axes, dashed cyan curve) are plotted versus time.
To visualize how the solution changes as a function of time in the phase plane, one can change the time $t$ with the slider in the right panel or press the play button (triangle) in the lower left of one of the panels to start the animation of $t$ increasing. The red points in both panel move with $t$ to correspond to the solution $(x(t),y(t))$.
Values of the matrix $A$ can be changed in the top control panel. The initial condition $\vc{x}(0)= (x_0,y_0)$ can be changed by dragging the cyan points in either panel or by entering numbers in the control panel.
If the eigenvalues of $A$ are real, then one can check the “show eigenvectors” box to show the directions of the eigenvectors of $A$ in the left phase plane. If the corresponding eigenvalue is not zero, arrows along the eigenvector indicates the direction the solution moves along the eigenvector direction. Checking the “show vector” box displays a vector from the origin to $(x(t),y(t))$, allowing one to track the direction of the solution even when the point $(x(t),y(t))$ moves out of view. Checking the “show decompositions” box, shows the decomposition of $(x(t),y(t))$ as a sum of components along the eigenvectors.
If you check the box “show eigenvalues”, then the phase plane plot shows an overlay of the eigenvalues, where the axes are reused to represent the real and imaginary axes of the complex plane. The eigenvalues appear as two points on this complex plane, and will be along the x-axis (the real axis) if the eigenvalues are real. If both eigenvalues are in the left half of the plane (which becomes shaded when the box is checked), then the equilibrium at the origin is stable.
The solution, eigenvalues, eigenvectors, and characterization of the equilibrium at the origin are shown in the sections at the bottom of the applet. These calculations depend on values of $A$ and initial condition chosen.
#### MIT cursor entry mathlet
The second applet, from the MIT Mathlet collections, is the Linear Phase Portraits: Cursor Entry Mathlet (distributed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license). In this applet, you specify the matrix by changing the trace and the determinant of the matrix $A$ (lower left), which determine the eigenvalues of the matrix $A$, and hence type of the system. The eigenvalues, however, don't fully determine the entries of the matrix. In the upper left, you can change two more quantities that determine the rotation and the asymmetry of the solutions in the phase plane. Combined with the eigenvalues, these quantities completely specify the entries of $A$.
LINEAR PHASE PORTRAITS: CURSOR ENTRY
The graphing window at right displays a few trajectories of the linear system x' = Ax. Below the window the name of the phase portrait is displayed, along with the matrix A and the eigenvalues of A.
To control the matrix one first sets the trace and the determinant by dragging the cursor over the diagram at bottom left or by grabbing the sliders below or to the left of that diagram. Select from among the matrices with given trace and determinant by dragging the cursor over the window at upper left, or by grabbing the sliders below and to the left of that window. The bottom slider conjugates the matrix A by a rotation matrix; the effect is to rotate the phase portrait. The left slider controls the "asymmetry" of A, half the difference of between its off-diagonal entries. When the eigenvalues are not real, the asymmetry is at least the imaginary part of the eigenvalue in absolute value, so the upper left window splits into two portions (corresponding to clockwise or counterclockwise spirals).
Depress the mousekey over the graphing window to display a trajectory through that point. The trajectory can be dragged by moving the cursor with the mousekey depressed. Releasing it will leave the trajectory in place. Click on [Clear] to clear all the trajectories.
© 2001 H. Hohn and H. Miller
Another applet that may be of interest is the Linear Phase Portraits: Matrix Entry Mathlet, also from the MIT Mathlet collections. | {"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": 0.8990577459335327, "perplexity": 469.23102426534507}, "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/1585370500331.13/warc/CC-MAIN-20200331053639-20200331083639-00049.warc.gz"} |
http://mathhelpforum.com/geometry/129410-sketch-locus.html | 1. ## Sketch the locus...
Not sure if this is the right forum, but here goes...
Sketch the locus of those points w with |w| = 3.
What is the locus...?
2. Originally Posted by jzellt
Not sure if this is the right forum, but here goes...
Sketch the locus of those points w with |w| = 3.
What is the locus...?
i suppose $w$ is a complex number here?
if so, say $w = x + iy$ and sketch $|x + iy| = \sqrt {x^2 + y^2} = 3$
if w isn't a complex number, the same principle will apply. you want the set of all points that are 3 units away from 0 (the origin). which would again yield the above graph. | {"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": 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": 0.933224618434906, "perplexity": 555.6257961409066}, "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-2013-48/segments/1386164805405/warc/CC-MAIN-20131204134645-00082-ip-10-33-133-15.ec2.internal.warc.gz"} |
https://physicscatalyst.com/article/omega-symbol-physics/ | # Meaning of Omega Symbol in physics
We use almost all greek alphabets to represent various physical quantities and units in physics. The purpose of this article is to look at the meaning of Omega Symbol in physics.
## Omega Symbol
• Omega is the 24th and last letter of the Greek alphabet.
• It stood for a long “o” sound in Ancient Greek. It still stands for “o” in Modern Greek, but there is no longer a difference between long and short vowels, so it sounds the same as Omicron.
We have both uppercase and lowercase letters in English Alphabets. Similar to this we also have both upper and lower case letters in greek alphabets. We will now look at what uppercase omega $\Omega$ and lowercase $\omega$ means in physics.
### Lowercase Omega $\omega$ symbol
Small or lowercase omega symbol $\omega$ is used to represent angular velocity in physics. In class 11 physics we study angular velocity while studying and analyzing problems related to circular motion or rotational motion. Sometimes lowercase omega symbol is also used to represents angular frequency.
### Uppercase Omega $\omega$
Capital or uppercase omega symbol $\Omega$ is used to represent the symbol for Ohms. Ohms is the unit of electrical resistance a quantity mostly used in electromagnetism in physics and also in engineering. | {"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": 0.8915380835533142, "perplexity": 843.3245185089452}, "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/1642320301475.82/warc/CC-MAIN-20220119155216-20220119185216-00284.warc.gz"} |
http://www.ams.org/joursearch/servlet/DoSearch?f1=msc&v1=58E20&jrnl=one&onejrnl=proc | # American Mathematical Society
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Matches for: msc=(58E20) AND publication=(proc) Sort order: Date Format: Standard display
Results: 1 to 30 of 53 found Go to page: 1 2
[1] Daoud Bshouty, Erik Lundberg and Allen Weitsman. A solution to Sheil-Small's harmonic mapping problem for polygons. Proc. Amer. Math. Soc. Abstract, references, and article information View Article: PDF [2] Jürgen Jost, Yi-Hu Yang and Kang Zuo. Harmonic maps and singularities of period mappings. Proc. Amer. Math. Soc. 143 (2015) 3351-3356. Abstract, references, and article information View Article: PDF [3] Shun Maeta. Polyharmonic maps of order $k$ with finite $L^p$ k-energy into Euclidean spaces. Proc. Amer. Math. Soc. 143 (2015) 2227-2234. Abstract, references, and article information View Article: PDF [4] Xiangrong Zhu. Bubble tree for approximate harmonic maps. Proc. Amer. Math. Soc. 142 (2014) 2849-2857. Abstract, references, and article information View Article: PDF [5] Faen Wu and Xinnuan Zhao. Non-existence of quadratic harmonic maps of $S^{4}$ into $S^{5}$ or $S^{6}$. Proc. Amer. Math. Soc. 141 (2013) 1083-1091. Abstract, references, and article information View Article: PDF [6] Shun Maeta. $k$-harmonic maps into a Riemannian manifold with constant sectional curvature. Proc. Amer. Math. Soc. 140 (2012) 1835-1847. MR 2869168. Abstract, references, and article information View Article: PDF [7] Kazuyuki Hasegawa. Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections. Proc. Amer. Math. Soc. 139 (2011) 309-317. MR 2729093. Abstract, references, and article information View Article: PDF [8] Tao Huang and Changyou Wang. Notes on the regularity of harmonic map systems. Proc. Amer. Math. Soc. 138 (2010) 2015-2023. MR 2596037. Abstract, references, and article information View Article: PDF [9] Gregory C. Verchota. Harmonic homeomorphisms of the closed disc to itself need be in $W^{1,p}$, $p<2$, but not $W^{1,2}$. Proc. Amer. Math. Soc. 135 (2007) 891-894. MR 2262887. Abstract, references, and article information View Article: PDF This article is available free of charge [10] Hsu Deliang. An approach to the regularity for stable-stationary harmonic maps. Proc. Amer. Math. Soc. 133 (2005) 2805-2812. MR 2146230. Abstract, references, and article information View Article: PDF This article is available free of charge [11] Daisuke Hirata. Blowup for $u_t = \Delta u + |\nabla u|^2 u$ from $\mathbb{R}^n$ into $\mathbb{R}^m$. Proc. Amer. Math. Soc. 133 (2005) 1823-1827. MR 2120283. Abstract, references, and article information View Article: PDF This article is available free of charge [12] Guowu Yao. Convergence of harmonic maps on the Poincaré disk. Proc. Amer. Math. Soc. 132 (2004) 2483-2493. MR 2052429. Abstract, references, and article information View Article: PDF This article is available free of charge [13] Andreas Gastel. On the harmonic Hopf construction. Proc. Amer. Math. Soc. 132 (2004) 607-615. MR 2022387. Abstract, references, and article information View Article: PDF This article is available free of charge [14] Guowu Yao. $\overline{\partial }$-energy integral and harmonic mappings. Proc. Amer. Math. Soc. 131 (2003) 2271-2277. MR 1963777. Abstract, references, and article information View Article: PDF This article is available free of charge [15] Lei Ni. A Bernstein type theorem for minimal volume preserving maps. Proc. Amer. Math. Soc. 130 (2002) 1207-1210. MR 1873798. Abstract, references, and article information View Article: PDF This article is available free of charge [16] Robert S. Strichartz. Harmonic mappings of the Sierpinski gasket to the circle. Proc. Amer. Math. Soc. 130 (2002) 805-817. MR 1866036. Abstract, references, and article information View Article: PDF This article is available free of charge [17] C. Greco. A bifurcation result for harmonic maps from an annulus to $S^2$ with not symmetric boundary data. Proc. Amer. Math. Soc. 129 (2001) 1199-1206. MR 1709752. Abstract, references, and article information View Article: PDF This article is available free of charge [18] Tom Y. H. Wan. A note on non-univalent harmonic maps between surfaces. Proc. Amer. Math. Soc. 129 (2001) 567-572. MR 1800239. Abstract, references, and article information View Article: PDF This article is available free of charge [19] I. Anic, V. Markovic and M. Mateljevic. Uniformly bounded maximal $\varphi$-disks, Bers space and harmonic maps. Proc. Amer. Math. Soc. 128 (2000) 2947-2956. MR 1664317. Abstract, references, and article information View Article: PDF This article is available free of charge [20] Harold Donnelly. Harmonic maps with noncontact boundary values. Proc. Amer. Math. Soc. 127 (1999) 1231-1241. MR 1473662. Abstract, references, and article information View Article: PDF This article is available free of charge [21] Kensho Takegoshi. A maximum principle for $P$-harmonic maps with $L^q$ finite energy. Proc. Amer. Math. Soc. 126 (1998) 3749-3753. MR 1469437. Abstract, references, and article information View Article: PDF This article is available free of charge [22] Luen-fai Tam. A note on harmonic forms on complete manifolds. Proc. Amer. Math. Soc. 126 (1998) 3097-3108. MR 1459152. Abstract, references, and article information View Article: PDF This article is available free of charge [23] Motoko Kotani. Harmonic 2-spheres with $r$ pairs of extra eigenfunctions. Proc. Amer. Math. Soc. 125 (1997) 2083-2092. MR 1372035. Abstract, references, and article information View Article: PDF This article is available free of charge [24] Seiichi Udagawa. Harmonic tori in quaternionic projective 3-spaces. Proc. Amer. Math. Soc. 125 (1997) 275-285. MR 1353402. Abstract, references, and article information View Article: PDF This article is available free of charge [25] Jingyi Chen. A boundary value problem for Hermitian harmonic maps and applications. Proc. Amer. Math. Soc. 124 (1996) 2853-2862. MR 1301014. Abstract, references, and article information View Article: PDF This article is available free of charge [26] Chiung-Jue Sung, Luen-fai Tam and Jiaping Wang. Bounded harmonic maps on a class of manifolds. Proc. Amer. Math. Soc. 124 (1996) 2241-2248. MR 1307567. Abstract, references, and article information View Article: PDF This article is available free of charge [27] Yunmei Chen and Livio Flaminio. Removability of the singular set of the heat flow of harmonic maps. Proc. Amer. Math. Soc. 124 (1996) 513-525. MR 1307502. Abstract, references, and article information View Article: PDF This article is available free of charge [28] Shiu-Yuen Cheng, Luen-Fai Tam and Tom Y.-H. Wan. Harmonic maps with finite total energy. Proc. Amer. Math. Soc. 124 (1996) 275-284. MR 1307503. Abstract, references, and article information View Article: PDF This article is available free of charge [29] James Eells and Paul Yiu. Polynomial harmonic morphisms between Euclidean spheres . Proc. Amer. Math. Soc. 123 (1995) 2921-2925. MR 1273489. Abstract, references, and article information View Article: PDF This article is available free of charge [30] Motoko Kotani. Connectedness of the space of minimal $2$-spheres in $S\sp {2m}(1)$ . Proc. Amer. Math. Soc. 120 (1994) 803-810. MR 1169040. Abstract, references, and article information View Article: PDF This article is available free of charge
Results: 1 to 30 of 53 found Go to page: 1 2
Comments: Email Webmaster © Copyright , American Mathematical Society Contact Us · Sitemap · Privacy Statement | {"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": 0.9287998676300049, "perplexity": 1968.7439618609458}, "config": {"markdown_headings": true, "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-2015-40/segments/1443737882743.52/warc/CC-MAIN-20151001221802-00188-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://tex.stackexchange.com/questions/154050/how-to-implement-book-style-notes-usage-fields | # How to implement book style notes/usage fields?
I would like to implement Note style fields like in here (excerpt from https://developer.android.com/training/graphics/opengl/environment.html#manifest for demonstration)
How can I do this in LaTeX ?
P.S. Please change the tags if I have chosen the wrong ones.
EDIT
I have came up with something more or less like this:
\vspace{1em}\hspace{1.5em}
\hbox{%
\vrule\hspace{.5em}\parbox{.9\textwidth}%
{
\textbf{Note:}
Text.....
}
}
\vspace{1em}
Which produces:
But I am having problems to define an environment with this.
What about something more similar to the one you are trying to copy, made with mdframed?
Code:
\documentclass{article}
\usepackage[framemethod=TikZ]{mdframed}
\usepackage{lipsum} % just for the example
\newmdenv[%
rightmargin = -10pt,
skipabove = 1.2\topskip,
rightline = false,
topline = false,
bottomline = false,
innertopmargin = 2pt,
innerbottommargin = 2pt,
linewidth = 2pt,
linecolor = cyan
]{leftlined}
\newenvironment{note}
{\leftlined\textbf{Note:}}
{\endleftlined}
\begin{document}
\noindent\lipsum[1]
\begin{note}
\lipsum[2]
\end{note}
\noindent\lipsum[1]
\end{document}
I've first defined a new mdframed environment called leftlined and then I've used it to define a new environment note that adds the text "Note:" at its beginning.
I have finally defined environment for this :
\newenvironment{note}[1]
{
\vspace{1em}\hspace{1.5em}
\hbox{%
\vrule\hspace{.5em}\parbox{.9\textwidth}%
{
\textbf{Note:}
#1
}}}
{\vspace{1em}}
To be used as follows:
\begin{note}
{"Lorem ipsum dolor sit amet, consectetur adipisicing elit,
sed do eiusmod tempor incididunt ut labore et dolore magna aliqua.
Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
nisi ut aliquip ex ea commodo consequat.
Duis aute irure dolor in reprehenderit in voluptate velit esse cillum
dolore eu fugiat nulla pariatur.
Excepteur sint occaecat cupidatat non proident, sunt in culpa qui
officia deserunt mollit anim id est laborum."}
\end{note}
Which would produce:
• It is probably fine but not "breakable" at a break page. A solution with mdframed is better in this regard. – pluton Jan 15 '14 at 3:24 | {"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": 0.8822939991950989, "perplexity": 1244.8012085276775}, "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-2020-05/segments/1579250592394.9/warc/CC-MAIN-20200118081234-20200118105234-00316.warc.gz"} |
http://mathhelpforum.com/algebra/17031-stumbled-upon-answer.html | # Math Help - Stumbled upon an answer....
1. ## Stumbled upon an answer....
Here's something I found interesting. "If $f(x) = 3x + 7$ and $g(x) = -2x + 5$, find the abscissa of a point which is on the graphs of both $f$ and $g$."
I really didn't know how to do it; the book certainly didn't spell it out anywhere. I started by graphing both equations, getting a rough idea of where the equations crossed, and plugged in a couple of points until I found the point $(-\frac{2}{5}, 5\frac{4}{5})$. Then I just happened to notice that earlier I had tried simplifying $3x + 7 = -2x + 5$, which simplifies to $5x = -2$, and that those numbers corresponded to the abscissa of the point in question. (Never mind that they also correspond to the numbers of the function $g(x)$).
I tried a few other simple linear equations and got the same results, so I feel like I've stumbled upon a consistent solution to the problem. Obviously I haven't discovered anything new, but I wonder, is this solution valid to all sets of two linear equations? If so, is there a named theorem for it?
2. Originally Posted by earachefl
Here's something I found interesting. "If $f(x) = 3x + 7$ and $g(x) = -2x + 5$, find the abscissa of a point which is on the graphs of both $f$ and $g$."
I really didn't know how to do it; the book certainly didn't spell it out anywhere. I started by graphing both equations, getting a rough idea of where the equations crossed, and plugged in a couple of points until I found the point $(-\frac{2}{5}, 5\frac{4}{5})$. Then I just happened to notice that earlier I had tried simplifying $3x + 7 = -2x + 5$, which simplifies to $5x = -2$, and that those numbers corresponded to the abscissa of the point in question. (Never mind that they also correspond to the numbers of the function $g(x)$).
I tried a few other simple linear equations and got the same results, so I feel like I've stumbled upon a consistent solution to the problem. Obviously I haven't discovered anything new, but I wonder, is this solution valid to all sets of two linear equations? If so, is there a named theorem for it?
It isn't so much a theorem as a process.
Consider the graphs of f(x) and g(x). Assume they are both linear functions and the slopes are different, so they cross.
Call the coordinates of the point where they cross (x, y). Then this same point (x, y) satisfies both:
y = f(x)
y = g(x)
So we have that
y = f(x) = g(x)
This is the step that you mentioned.
-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": 16, "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": 0.9045537114143372, "perplexity": 204.0128519223834}, "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-2015-32/segments/1438042989126.22/warc/CC-MAIN-20150728002309-00183-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://mathhelpforum.com/math-topics/159617-h3po4-naoh.html | 1. ## H3PO4+NaOH---> ???
I am stuck on a question for my homework. I know how to write the equation, but it says it forms 3 salts? All I'm getting is this: H3PO4+3NaOH--->3H20+Na3PO4.
This is the question:
H3PO4 can form three salts when it reacts with NaOH. Write the formula of each of these salts. Do not include physical states in your answers.
Salt with lowest molar mass:
Salt with intermediate molar mass:
Salt with higher molar mass:
I have no idea what to do?
2. There are 3 H in $H_3PO_4$. You can have reactions with one, two or three H atoms. You already have the one with 3. | {"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": 0.9555891752243042, "perplexity": 1299.732274669918}, "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-2013-48/segments/1387345775355/warc/CC-MAIN-20131218054935-00099-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://www.gradesaver.com/textbooks/science/physics/physics-principles-with-applications-7th-edition/chapter-15-the-laws-of-thermodynamics-questions-page-437/4 | ## Physics: Principles with Applications (7th Edition)
Published by Pearson
# Chapter 15 - The Laws of Thermodynamics - Questions: 4
#### Answer
Work is done on the gas and the internal energy increases.
#### Work Step by Step
We use the first law of thermodynamics. $$\Delta U = Q-W$$ If the gas is compressed adiabatically, Q=0. $$\Delta U = -W$$ The compression means that work was done on the gas, so the work done by the system, W, is negative. We see that the change in internal energy is a positive quantity, meaning the internal energy of the gas increased, so the temperature increased.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | {"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": 0.8125962615013123, "perplexity": 544.7999774038708}, "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-2017-17/segments/1492917119356.19/warc/CC-MAIN-20170423031159-00208-ip-10-145-167-34.ec2.internal.warc.gz"} |
http://www.met.reading.ac.uk/pplato2/h-flap/phys10_3.html | 1 Opening items
1.1 Module introduction
Classical physics, in the shape of Newtonian mechanics and Maxwell’s electromagnetism, reached its culmination at around the turn of the 19th century. The theory seemed complete, except for a little tidying up, but cracks were about to open up and cause the downfall of the whole structure. The unfolding of these events was dramatic and rapid. In the space of 30 years, classical physics was replaced by quantum physics as the fundamental theory of the world, with classical physics surviving only as a special case – admittedly, adequate for most everyday situations. This module traces part of the story of those hectic years; other modules begin the story and others carry it further. Here we pick up after the demonstration of particle–like behaviour of electromagnetic radiation and the publication of the de Broglie hypothesis, that matter should show wave–like behaviour. The resolution of this wave and particle dual behaviour for both matter and electromagnetic radiation is the door to quantum mechanics or wave mechanics and this story begins with the idea of a wavefunction and how this can be used.
Section 2 reviews some some basic quantum ideas, including de Broglie waves and Heisenberg uncertainty principle. Section 3 introduces the concept of a wavefunction and relates this to the probability distribution associated with the position of a particle.
The wavefunction of a free particle is then modelled, first as a complex travelling wave and then as a wave packet. The total energy, the momentum and the kinetic energy of the particle are then related to the angular wavenumber and angular frequency of the associated travelling wave. The discussion of the wave packet representation includes the significance of the phenomenon of dispersion and the relationship between the phase and group speeds of the packet.
Section 4 concerns the wavefunction of a particle which is confined in one dimension. The idea of a one-dimensional box is introduced, along with the stationary state wavefunctions and discrete energy levels relevant to such a system. Again, the development is guided closely by classical models of confined waves. The superposition of stationary states is briefly considered, as is the significance of transitions between stationary states.
Section 5 extends the discussion to the case of a particle which is confined in two or three dimensions, introducing the idea of degeneracy.
The overall strategy of the module is to develop quantum wave models by analogy with the classical wave models, rather than to adopt a wave equation approach from first principles. A more rigorous mathematical treatment of these topics – using the Schrödinger equation – is included elsewhere in FLAP.
Study comment Having read the introduction you may feel that you are already familiar with the material covered by this module and that you do not need to study it. If so, try the following Fast track questions. If not, proceed directly to the Subsection 1.3Ready to study? Subsection.
1.2 Fast track questions
Study comment Can you answer the following Fast track questions? If you answer the questions successfully you need only glance through the module before looking at the Subsection 6.1Module summary and the Subsection 6.2Achievements. If you are sure that you can meet each of these achievements, try the Subsection 6.3Exit test. If you have difficulty with only one or two of the questions you should follow the guidance given in the answers and read the relevant parts of the module. However, if you have difficulty with more than two of the Exit questions you are strongly advised to study the whole module.
Question F1
A particle state is represented by a wave packet which extends over a distance ∆x = 5.0 × 10−14 m. What is the uncertainty in the x–component of the particle’s momentum ∆px?
[Take h = 6.6 × 10−34 J s.]
The uncertainty ∆px, in the momentum of the particle is given by the Heisenberg uncertainty principle:
$\Delta x\,\Delta p_x \gtrsim \dfrac{h}{2\pi}$
Therefore:
$\Delta p_x \gtrsim \dfrac{h}{2\pi\Delta x} = \rm \dfrac{6.6\times10^{-34}\,J\,s}{2\pi\times5.0\times10^{-14}\,m} = 2.1\times10^{-21}\,kg\,m\,s^{-1}$
(∆px of half this value is also acceptable, using the more precise statement of the Heisenberg uncertainty principle: $\Delta x\,\Delta p_x \gtrsim \dfrac{h}{4\pi}$.)
Question F2
A particle is confined in one dimension between x = 0 and x = 5 × 10−14 m, by rigid impenetrable walls. Give an expression for the wavefunctions corresponding to the standing waves between the walls.
The wavefunction of the particle must be zero at the two end points, x = 0 and x = 5 × 10−14 m. Between these two points there may be any number n (a positive integer), of half wavelengths as shown in Figure 6. The expression for the spatial part of a wavefunction that satisfies these requirements is:
spatial wavefunctions: $\psi_n(x) = A_n\sin\left(\dfrac{n\pi x}{D}\right)$
where the length D is 5 ×10−14 m.
Question F3
What is the energy of the ground state of the particle in the previous question, if the mass of the particle is 9.1 × 10−31 kg?
Is this energy level degenerate?
The energy levels of a particle in this one–dimensional box are:
$E_n = \dfrac{n^2h^2}{8mD^2}$
In the ground state (n = 1) of the particle in the previous question:
$E_1 = \rm \dfrac{(6.6\times10^{-34}\,J\,s)^2}{8\times9.1\times10^{-31}\,kg\times(5.0\times10^{-14}\,m)^2} = 2.4\times10^{-11}\,J$
There is only one wavefunction (n = 1) with this energy and so the level is non-degenerate.
Study comment Having seen the Fast track questions you may feel that it would be wiser to follow the normal route through the module and to proceed directly to the following Ready to study? Subsection.
Alternatively, you may still be sufficiently comfortable with the material covered by the module to proceed directly to the Section 6Closing items.
Study comment In order to study this module you will need to be familiar with the following terms: de Broglie hypothesis, de Broglie wave, de Broglie wavelength, electron, energy, hydrogen atom (including the Bohr model), momentum, photon, Planck’s constant and quanta. You will need to be familiar with the basic characteristics of waves and wave packets, including; amplitude, frequency, wavelength, propagation speed, phase speed and group speed; and with wave phenomena such as diffraction, interference, and superposition; and to know the difference between a travelling wave and a standing wave. You also should be familiar with the differentiation and integration of simple functions of x, including exponentials and trigonometric functions. If you are uncertain about any of these terms then you can review them now by referring to the Glossary, which will also indicate where in FLAP they are developed. The following Ready to study questions will allow you to establish whether you need to review some of the topics before embarking on this module.
Question R1
A wave travelling along the x–axis consists of a moving disturbance that varies with time and position, which may be represented by a function of two variables y(x, t). The ‘shape’ of the wave at any particular time t = T is described by its wave profile at that time, y(x, T), which depends on the single variable x since T is a given constant in this case. The profile may be thought of as an instantaneous snapshot of the wave at the given time.
(a) At t = 0, the wave profile of a particular wave is given by y = Asin(kx), where k is a constant called the angular wavenumber of the wave. Write down the amplitude and wavelength of the wave.
(b) If the wave in part (a) has a frequency f, write down the general relationship between the wavelength, the frequency and the speed of the wave. If the angular frequency of the wave is ω = 2πf, write down the corresponding relationship between angular wavenumber, angular frequency and speed.
(a) For the wave profile y = Asin(kx), the maximum size of the displacement is referred to as the amplitude. So A is the amplitude. The wavelength is the smallest distance between points on the wave that have the same phase. The function sinθ is periodic, it repeats itself every time θ increases by 2π. Two points separated by a wavelength λ, say x and x + λ therefore must satisfy the condition k(x + λ) − kx = 2π. Therefore λ = 2π/k.
(b) The relationship between the wavelength λ, the frequency f, and the speed υ, of the wave is υ = fλ. In terms of angular wavenumber k and angular frequency ω, the speed is υ = ω/k.
Consult the Glossary for further information.
Question R2
At a particular time the transverse displacements due to two waves, acting at a common point, are y1 = 5sin(2x) and y2 = 2sin(2x). Use the principle of superposition to obtain the combined effect of the two waves at this time.
The principle of superposition states that the combined effect of the two waves acting at a point is their algebraic sum y = y2 + y2. In our case the effect is of a wave y = 7sin(2x).
Consult principle of superposition in the Glossary for further information.
Question R3
What is the de Broglie wavelength associated with a particle with momentum magnitude 5.0 × 10−21 kg m s−1?
[Take Planck’s constant, h, to have the value 6.6 × 10−34 J s]
The de Broglie hypothesis relates the momentum magnitude, p, of a particle and its de Broglie wavelength λdB, by λdB = h/p where h is Planck’s constant. In this case, therefore, the particle has a de Broglie wavelength:
$\lambda_{\rm dB} = \rm \dfrac{6.6\times10^{-34}\,J\,s}{5.0\times10^{-21}\,kg\,m\,s^{-1}} = 1.3\times10^{-13}\,m$
Consult the de Broglie hypothesis, de Broglie wavelength, and Planck’s constant in the Glossary for further information.
Question R4
Write down and evaluate the definite integral of the function f(x) = x3 between the limits x = 3 and x = 4.
The definite integral of the function f(x) between x = 3 and x = 4 is written $\displaystyle \int_3^4 f(x)\,dx$.
The value for our simple function is
$\displaystyle \int_3^4x^3\,dx = \left[\dfrac{x^4}{4}\right]_3^4 = \dfrac14(4^4-3^4) = 43.75$
Consult integration in the Glossary for further information.
Question R5
Give an expression for the indefinite integral $\int\sin(2x)\,dx$.
$\int\sin(2x)\,dx = -\frac12\cos(2x) + \text{constant of integration}$
2 A review of basic quantum physics
2.1 Particle–like behaviour of electromagnetic radiation
Max Planck’s i interpretation (1900–) of the distribution of light emitted by an idealized hot body, the so–called black–body spectrum, provided the first evidence that the interactions of radiation with matter were quantized.
If a body is heated to a high temperature it emits light over a wide continuous range of wavelengths. When the spectrum of this radiation is examined and the relative brightness of the emission from unit area of the surface is plotted as a function of wavelength, it is found that the wavelength for peak emission and the total radiated power are determined mainly by the temperature of the body, not by its material. In the case of an ideal emitter, a so–called black body, the spectrum would be entirely determined by the temperature. An explanation for the detailed shape of the black–body spectrum proved impossible using classical physics, which predicted that the brightness should increase without limit at high frequencies. Planck was able to resolve this by assuming a quantum model for the interaction of matter and radiation. In particular he assumed that the interaction involved the emission and absorption of quanta, (which later become known as photons) with energy hf, where f is the frequency of the radiation and h is Planck’s constant. The spectrum predicted on the basis of this assumption was in excellent agreement with the observed spectrum of black–body radiation. i
More direct evidence for photons was provided later through the photoelectric effect and the Compton effect. In the photoelectric effect, electrons are produced by shining ultraviolet radiation on to a metal surface. It is found that the maximum kinetic energy of the photoelectrons emitted from any particular surface depends only on the frequency of the incident radiation and not on the intensity of the radiation. In addition, there is a threshold frequency for a particular material and no photoelectrons are produced below this frequency, however intense the radiation. Classical wave models of electromagnetic radiation could not explain this and it was left to Einstein to interpret the observed behaviour in terms of a particle–like interaction between the radiation and the electrons in the material. In this interaction energy is absorbed from individual photons of discrete energy E given by:
photon energy E = hf(1)
This result became known as the Planck–Einstein formula.
Some 20 years later, a crucial series of experiments, involving the scattering of X–rays by different targets showed that the scattered X–rays had a slightly longer wavelength (or a lower frequency) than the incident radiation and also that the shift in wavelength depended on the scattering angle. These observations were inexplicable using classical wave ideas. However, Arthur H. Compton (1892–1962) gave a quantum interpretation of these results, which involved the photon having a momentum as well as the energy given by Equation 1. (The collision between the X–ray photon and an essentially free electron in the material could then be treated as a particle–particle collision, conserving energy and momentum in the usual way.) The required expression for the photon momentum magnitude was:
photon momentum p = E/c = h/λ(2)
This scattering phenomenon later became known as the Compton effect.
2.2 Wave–like behaviour of matter
The next stage in the evolution of quantum physics came when Louis de Broglie (1892–1987) suggested that since the particles (i.e. photons) which make up electromagnetic radiation can exhibit wave–like behaviour, perhaps the same is true of every other particle. This suggestion became known as the de Broglie hypothesis, and the wave associated with a particle, the de Broglie wave, was expected to have its de Broglie wavelength set by the magnitude of the momentum p of the particle, according to the expression:
de Broglie wavelength λdB = h/p(3)
The suggestion that particles of matter might exhibit wave–like behaviour implies that such particles might exhibit diffraction and interference. If so, the closely spaced planes of atoms in a crystalline solid might be used to diffract the de Broglie waves associated with an electron beam with particle energies of a few tens of electron volts. Such an experiment was carried out in 1927 by C. H. Davisson and L. H. Germer and they obtained a diffraction pattern in good agreement with de Broglie’s predicted wavelength. Subsequently, many other experiments have demonstrated that all particles, irrespective of charge, mass, shape or composition, produce a diffraction pattern which is consistent with the de Broglie hypothesis.
Figure 1 shows an instantaneous snapshot of the de Broglie wave for a particle with definite momentum px travelling along the x–direction.
The precise nature of de Broglie waves and the exact sense in which such waves are to be associated with particles was left unclear by de Broglie. However, subsequent work by others, notably Erwin Schrödinger (1887–1961), Werner Heisenberg (1901–1976) and Max Born (1882–1970), put de Broglie’s ideas onto a firmer mathematical footing and eventually brought about a complete revolution in physical thinking. Part of that revolution forms the main theme of this module, and we will come to it later. i
In the meantime, we will continue to use the term ‘de Broglie wave’ to describe the wave aspect of a particle, and we will summarize later work by saying that the de Broglie wave of a particle determines the relative likelihood of detecting the particle in any given region of space. In particular, continuing to use this somewhat over–simplified language, we can say that the probability of finding a particle in any small region of space is proportional to the square of the amplitude of the de Broglie wave in that region. In this sense the disturbance that constitutes a de Broglie wave may be thought of as a disturbance in the probability of finding the associated particle.
A simple one–dimensional de Broglie wave of fixed amplitude and wavelength, extending to infinity along the x–direction, corresponds to a particle whose momentum magnitude is perfectly known. Unfortunately, such a wave is not localized in space; its amplitude is the same everywhere, and so it conveys no information at all about the position of the particle. If we wish to produce a wave of finite extent, with some implied localization of the particle, then we must construct a wave packet by superposing (adding) waves, and arrange for this superposition to diminish sharply outside the expected range of particle positions ∆x. In discussing this process it is convenient to use as the variable the angular wavenumber k rather than the wavelength λ – the two are related by
angular wavenumber k = 2π/λ(4)
Question T1
At time t = 0, the instantaneous profiles of two de Broglie waves are ψ1(x) = A1cos(k1x) and ψ2(x) = A2cos(k2x). i These expressions show that the two are in phase at x = 0.
(a) Write down an expression for the profile of their superposition at t = 0 and give its value at x = 0.
(b) What are the values of x closest to zero for which ψ1(x) and ψ2(x) are exactly out of phase (in anti-phase) at t = 0? Express this value of x in terms of the difference in momentum of the corresponding particles.
(a) The superposed wave profile is:
ψ(x) = [ψ1(x) + ψ2(x)] = A1cos(k1x) + A2cos(k2x)
At x = 0 we have ψ(0) = [ψ1(0) + ψ2(0)] = A1 + A2.
(b) The waves are in anti–phase when the arguments of the two cosine functions differ by any odd multiple of π. The smallest values of x for which this occurs are when the arguments differ by π so that k1x = k2x ± π, thus, using p = hk/2π:
x = ±π/(k1k2) = ± h/[2(p1p2)]
The answer to Question T1 illustrates the principle that a more localized profile can be produced by superposing two other profiles, corresponding to two different angular wavenumbers, i.e. two different momenta, by using ψ(x) = [ψ1(x) + ψ2(x)] as a new wave profile.
Such a case is illustrated in Figure 2. The answer also shows that a position of constructive interference is separated from a position of destructive interference by a distance ∆x which is determined by the angular wavenumber difference ∆k. The two quantities ∆x and ∆k being inversely related: ∆k ≈ ±π/∆x.
We can go further than this, and use more than two waves in the superposition, as shown in Figure 3. This figure shows an example of the construction of a finite wave packet by the superposition of (in this case eight) waves of suitably chosen amplitudes and wavelengths. Although each contributing wave has a well defined wavelength and associated momentum, the resultant superposition does not.
Figure 4 A broad wave packet can be constructed from the superposition of waves with a narrow range of angular wavenumbers. Conversely a narrow wave packet requires a broad range of angular wavenumbers for its construction.
Figure 3 The localization of the wavefunction produced when eight waves with different amplitudes and wavelengths are added together using the principle of superposition.
The basic features arising from the mathematics are illustrated in Figure 4. It should be noted, in particular, that in order to localize the wave packet within a smaller and smaller region of space, there must be included in the superposition a wider and wider range of values of angular wavenumber for the contributing waves.
The greater the spread of angular wavenumbers ∆k, the narrower the width ∆x, of the corresponding wave packet.
Fourier analysis i quantifies this relationship in the simple expression:
xk ≈ 1(5)
Notice that Equation 5 is not given as an equality since, as you will see by looking at Figure 4, ∆x and ∆k are only approximate measures of spread and have not been defined precisely. This relationship is very important however, as it shows a trend which is always satisfied, irrespective of the shape of the wave packet.
In the context of a de Broglie wave packet, each of the superposed waves will have a different de Broglie wavelength,
de Broglie wavelength λdB = h/p(Eqn 3)
and hence a different associated particle momentum. A spread in angular wavenumber will therefore correspond to a spread in particle momentum. This implies that the wave packet corresponding to a particle whose position is known to within ∆x must be composed of de Broglie waves associated with particle momenta in the range
$\Delta p = \dfrac{h}{2\pi}\Delta k \approx \dfrac{h}{2\pi}\dfrac{1}{\Delta x}$
This leads to the Heisenberg uncertainty principle:
Heisenberg uncertainty principle $\Delta x\,\Delta p_x \gtrsim \dfrac{h}{2\pi}$(6) i
where ∆px represents the irreducible uncertainty in the x–component of the momentum of a particle that is known to be localized within ∆x.
Notice that we have replaced the approximation sign in Equation 5 by a ‘greater than or approximately equal to’ sign in Equation 6 to signify that in any experiment we can never obtain simultaneous information on position and momentum in a given direction to a precision which is better than the fundamental limit set by the wave nature of matter.
Question T2
In each of the two cases below, measurements are made simultaneously of position x, and x–component of momentum px. In each case the uncertainty ∆x, is given. Estimate the percentage uncertainty in the momentum due to the Heisenberg uncertainty principle. Take Planck’s constant h, as 6.6 × 10−34 J s.
(a) A bullet of mass 0.10 kg is travelling with a speed of 1.5 km s−1 and ∆x is 0.10 mm.
(b) A proton of mass 1.7 × 10−27 kg is travelling with a speed of 1.0 km s−1 for which ∆x is 1.0 × 10−10 m.
(a) For the bullet, the momentum is:
px = x = 0.10 kg × 1.5 × 103 m s−1 = 1.5 × 102 kg m s−1
The uncertainty in position is ∆x = 1.0 × 10−4 m. The Heisenberg uncertainty principle,
Heisenberg uncertainty principle $\Delta x\,\Delta p_x \gtrsim \dfrac{h}{2\pi}$(Eqn 6)
will give an estimate of ∆px as:
$\Delta p_x \gtrsim \dfrac{h}{2\pi\Delta x} = \rm \dfrac{6.6\times1^{-34}\,J\,s}{2\pi\times1.0\times10^{-4}\,m} = 1.1\times10^{-30}\,kg\,m\,s^{-1}$
The percentage uncertainty is:
$\dfrac{100\Delta p_x}{p_x}\% = \dfrac{100\times1.1\times10^{-30}}{1.5\times10^2}\% = 7.3\times10^{-31}\%$
(b) For the proton the momentum is:
px = mpυx = 1.7 × 10−27 kg × 1.0 × 103 m s−1 = 1.7 × 10−24 kg m s−1
For ∆x = 1.0 × 10−10 m:
$\Delta p_x \gtrsim \dfrac{h}{2\pi\Delta x} = \rm \dfrac{6.6\times10^{-34}\,J\,s}{2\pi\times1.0\times10^{-10}\,m} = 1.1\times10^{-24}\,kg\,m\,s^{-1}$
The percentage uncertainty is:
$\dfrac{100\Delta p_x}{p_x}\% = \dfrac{100\times1.1\times10^{-24}}{1.7\times10^{-24}}\% = 65\%$
We are now in a position to introduce the concept of a wavefunction and to begin the journey from the old de Broglie world to the new world of quantum mechanics. Our first task is to organize our vocabulary. So far, we have sometimes referred to electrons, protons, neutrons and other such entities rather loosely as ‘particles’. On other occasions we have had to accept that they exhibit wave–like behaviour. Neither the word ‘particle’ nor ‘wave’ conveys the full picture of their behaviour, so we will describe each of them simply as a quantum, indicating that they are neither waves nor particles but may exhibit features of either. i This is a new technical definition of the term. It is natural then to describe the study of the motion and the interaction of these quanta as quantum mechanics. This new world of quantum mechanics is a good deal more mathematical than the old world of de Broglie waves, but we will try to maintain the contact with de Broglie waves as long as we can.
3 The wavefunction for a free particle
The principles of quantum mechanics were introduced in a series of papers by Heisenberg, Schrödinger, Born and Pascual Jordan (1902–1980) in the years 1925–1926. Initially there were two quite distinct formulations of quantum mechanics, but they were soon shown to be mathematically equivalent and Schrödinger’s version (now known as wave mechanics) now provides the usual route of entry to the subject.
Central to Schrödinger’s approach is a mathematical quantity called the wavefunction. This quantity varies with time and position, just like a wave, and replaces the more primitive (and ill-defined) concept of a de Broglie wave. It is conventional to use the upper–case Greek letter Ψ (pronounced ‘psi’) to represent the wavefunction, so in a one–dimensional problem where its value depends on position x and time t, the wavefunction is usually written Ψ(x, t). The precise form of the wavefunction in any given situation is determined by solving a (complicated) equation known as the time-dependent Schrödinger equation. Learning how to formulate this equation to represent a given physical situation is an important skill, as is learning how to solve it, but we will not be concerned with either of those issues in this module. Instead, we will concentrate on the significance and physical interpretation of the wavefunctions that satisfy it.
In the conventional interpretation of quantum mechanics the wavefunction provides the most complete description of the behaviour of a system we can hope to have. This sounds rather grand, and it certainly has far reaching implications, but in practice we are often restricted to dealing with such simple systems that even their ‘complete description’ is fairly straightforward. In the case of a particle moving in one dimension, for instance, it usually boils down to knowing the energy and momentum, and something about the relative likelihood of finding the particle in various regions.
An important mathematical property of the wavefunction Ψ(x, t) that clearly distinguishes it from a de Broglie wave is that it is generally a complex quantity. That is to say, given particular values for the variables x and t, the corresponding value of the wavefunction will generally be complex number and may therefore be written in the form
Ψ(x, t) = a + ib i
where a and b are ordinary (real) numbers and i is a special algebraic quantity, usually referred to as the square root of −1, with the property
i2 = −1
Complex numbers are used in many parts of physics, but often only as a convenience. In quantum mechanics, however, they play an essential role and are unavoidable. In this particular module we will use them as little as possible, but even here we cannot avoid them completely and you will certainly need to know more about them if you intend to pursue the study of quantum mechanics.
For the moment, the one additional fact you need to know about any complex quantity is that it may always be associated with a unique real number called its modulus_of_a_complex_numbermodulus. The modulus of Ψ(x, t) = a + ib is written |Ψ(x, t)| and is defined in the following way
ifΨ(x, t) = a + ib
then |Ψ(x, t)| = (a2 + b2)1/2
In quantum mechanics the modulus of the wavefunction plays the role that we earlier (and over-simplistically) assigned to the square of the amplitude of a de Broglie wave. In other words, if the behaviour of a quantum is described by the wavefunction Ψ(x, t)
The probability of finding the particle within the small interval ∆x around the position x at time t is
∝ |Ψ(x, t)|2 ∆x(7) i
There are several points to note here:
• Since |Ψ(x, t)| is a real quantity (it doesn’t involve i), it must be the case that |Ψ(x, t)|2 is positive. It is therefore at least possible that |Ψ(x, t)|2 might represent a probability, since probabilities are represented by real numbers in the range 0 to 1 with 0 for no possibility and 1 for certainty.
• Since ∆x is taken to be very small, |Ψ(x, t)|2 can be thought of as the probability per unit length, or probability density around position x. i
• |Ψ(x, t)|2 ∆x is a statistical indicator of behaviour. Given a large number of experiments to measure the position of a particle, set up under identical conditions, it represents the fraction of those experiments that will indicate a particle in the range ∆x at a time t. The set up for the experiment is fixed but the results of individual experiments are not always the same.
• We may write an equality sign in place of the proportionality sign in Equation 6, provided we choose an appropriate scale of probability. For the moment, we do not need to be concerned with this refinement, but we will return to it in Subsection 4.2 when we discuss normalization.
Our task now is to introduce the appropriate wavefunctions for some simple situations and to determine how these wavefunctions may be used to obtain the characteristic properties of the associated particle – such as its position, momentum and kinetic energy. We begin with the case of a free particle.
3.1 A complex travelling wave to represent a free particle
If we are to represent a moving particle by a wave then it is reasonable to use a travelling wave and so we begin by reviewing how we represent a classical travelling wave, such as a transverse wave on a string. i
A classical travelling wave on a string
A wave travelling along a string is characterized by having an amplitude A, frequency f, angular frequency ω = 2πf, wavelength λ and angular wavenumber k = 2π/λ. i When such a wave propagates in the positive x–direction, the wave displacement y(x, t) can be represented by:
travelling wave y(x, t) = Acos(kxωt)(8) i
The wave represented by Equation 8 can be shown to be travelling in the positive x–direction, using the following argument. Consider the position of the wave at two times t = 0 and t = ∆t, where ∆t is very short compared to the period of the wave.
At t = 0: y(x, 0) = Acos(kx) and the wave has a maximum at x = 0. At t = ∆t: y(x, ∆t) = Acos(kxω ∆t) and the wave has a maximum when (kxω ∆t) = 0. This maximum occurs at x = ω ∆t/k.
We deduce that the maximum of the wave has moved a distance ∆x = ω ∆t/k in the time ∆t. It is apparent that the wave is travelling along the positive x–direction and that a point of fixed phase (e.g. a point of maximum displacement where kxωt = 0) advances with a phase speed υϕ given by:
wave phase speed: υϕ = ∆x/∆t = ω/k(9)
We could rewrite Equation 8 in terms of the wave phase speed as:
travelling wave y(x, t) = Acos[k(xυϕt)](10)
This expression for the travelling wave makes the role of the phase speed clear.
The travelling wave is shown in Figure 5.
Question T3
Obtain an expression similar to Equation 8,
travelling wave y(x, t) = Acos(kxωt)(Eqn 8)
but representing a wave travelling in the negative x–direction.
An expression for the wave travelling in the negative x–direction can be found by noting that at time ∆t the maximum displacement must be positioned at x = −ωt/k.
We can achieve this simply by replacing the negative sign in the argument of Equation 8,
travelling wave: y(x, t) = Acos(kxωt)(Eqn 8)
by a positive sign to obtain:
y(x, t) = Acos(kx + ωt)
A quantum travelling wave in one dimension
We can now write down the wavefunction of a freely moving quantum travelling in the +x–direction. (Remember, this is found by solving the time dependent Schrödinger equation, though we will not go into that in this module.)
free particle wavefunction Ψ(x, t) = Acos(kxωt) + iAsin(kxωt)(11) i
As you can see, it owes a great deal to the expression for a one–dimensional travelling wave (Equation 8),
travelling wave y(x, t) = Acos(kxωt)(Eqn 8)
but there is also a striking difference. This wavefunction involves i, the square root of −1, and is therefore intrinsically complex. We will examine the significance of this in a moment, but for the present let us exploit the similarities with a travelling wave.
As you might expect the total energy, the momentum and the kinetic energy of the quantum can all be expressed in terms of the parameters ω and k of the wavefunction, through Equations 1 and 3, as:
total energy:$E = hf = \dfrac{h\omega}{2\pi} = \hbar\omega$(12) i
momentum:$p_x = \dfrac{h}{\lambda_{\rm dB}} = \dfrac{hk}{2\pi} = \hbar k$(13)
kinetic energy:$E_{\rm kin} = \dfrac{m\upsilon_x^2}{2} = \dfrac{(m\upsilon_x)^2}{2m} = \dfrac{p_x^2}{2m} = \dfrac{\hbar^2k^2}{2m}$(14)
Where we have introduced the shorthand $\hbar$ for the commonly met quantity h/2π. Notice an important feature of quantum mechanics shown here – the properties of the quantum are derivable from the mathematical form of its wavefunction, in this case simply by inspection of the coefficients of x and t.
Now, let us see what information about the position of the quantum, can be derived from the wavefunction of Equation 11.
Using the general expression for the modulus of a complex quantity we see that in this case
|Ψ(x, t)| = [A2cos2(kxωt) + A2sin2(kxωt)]1/2
But sin2θ + cos2θ = 1 for all values of θ
So|Ψ(x, t)| = A
and|Ψ(x, t)|2 = A2(15)
Thus, the probability density is independent of x, and the likelihood of finding the quantum in a small region of fixed length ∆x is the same everywhere. You shouldn’t be surprised by this result: in the first place there is no reason why a freely moving quantum should be more likely to be found in one place than another; secondly this quantum has a well defined momentum magnitude so ∆p = 0 and it follows from the uncertainty principle that $\Delta x (=\hbar/\Delta p)$ will be undefined.
We might now be tempted to calculate the phase velocity from the wavefunction in Equation 11,
free particle wavefunction Ψ(x, t) = Acos(kxωt) + iAsin(kxωt)(Eqn 11)
When we do so we are in for a shock! By analogy with the wave on a string we find the phase speed of this wavefunction is:
phase speed: $\upsilon_\phi = \dfrac{\omega}{k} = \dfrac Ep = \dfrac{\frac12m\upsilon^2}{m\upsilon} = \dfrac{\upsilon}{2}$(16)
In Equation 16 we have used the usual expressions for the kinetic energy and momentum magnitude of a particle travelling with speed υ. We have arrived at the disturbing conclusion that the phase speed of the wavefunction is not the same as the velocity of the associated particle, but half this value! We will resolve this apparent paradox in the next subsection.
3.2 A travelling wave packet to represent a free particle
Just as we were able to construct a wave packet from de Broglie waves to represent a localized particle, so too we can produce a localized wavefunction by superposing free particle wavefunctions like that of Equation 11,
free particle wavefunction Ψ(x, t) = Acos(kxωt) + iAsin(kxωt)(Eqn 11)
If this quantum wave packet moves through space, its motion can represent the motion of the associated free particle. Of course, since the wave packet will involve a range of angular wavenumbers it will not describe a particle with precisely determined momentum but that, according to the uncertainty principle, is the price we must pay for having some idea where the particle is located.
When the phase speed of a wave depends on its wavelength or angular wavenumber, individual waves of a given k will move with different phase speeds whilst the wave packet itself will travel at the group speed. According to classical wave theory the group speed is the speed with which the energy is propagated and is given by the expression:
group speed: υg = /dk(17) i
We will soon use Equation 17 to investigate the group speed of our quantum wave packet. However, before we do that let us clarify the meaning of the equation by using it to investigate a packet of (hopefully familiar) electromagnetic waves.
Group speed of an electromagnetic wave packet in a vacuum
For an electromagnetic wave of frequency f and wavelength λ travelling in a vacuum:
c = fλ(18)
so the phase speed of a single electromagnetic wave is
phase speed: $\upsilon_\phi = \dfrac{\omega}{k} = \dfrac{2\pi f}{2\pi/\lambda} = f\lambda = c$(19)
The phase speed is constant, it is independent of k, and ω = kc. In a vacuum, an electromagnetic wave packet composed of many such waves with different values of k moves with an overall speed known as its group speed, given by
group speed: $\upsilon_{\rm g} = \dfrac{d\omega}{dk} = \dfrac{d}{dk}(ck) = c$(20)
Since the group speed is also c, the wave packet travels at the same speed as each of the constituent waves within it. The wave packet thus travels through a vacuum without change of shape.
In a medium the situation is rather different. If an electromagnetic wave packet travels through any material, other than vacuum, the phase speed of each constituent wave is reduced by the refractive index, which usually depends on k. In this case, the group speed and the various phase speeds differ and the wave packet changes its shape and spreads out as it propagates. This process is called dispersion. and the relation between ω and k is called the dispersion relation of the material.
Group speed of a quantum wave packet
To determine the group speed of the quantum wave packet representing a free particle we first need to determine its dispersion relation (i.e. the relationship between ω and k). We can do this by recognizing that for a free particle the total energy and the kinetic energy must be identical since it then follows from Equations 12 and 14,
total energy:$E = hf = \dfrac{h\omega}{2\pi} = \hbar\omega$(Eqn 12)
kinetic energy:$E_{\rm kin} = \dfrac{m\upsilon_x^2}{2} = \dfrac{(m\upsilon_x)^2}{2m} = \dfrac{p_x^2}{2m} = \dfrac{\hbar^2k^2}{2m}$(Eqn 14)
that:
quantum dispersion relation: $\omega = \dfrac{\hbar k^2}{2m}$(21)
so:
quantum group speed: $\upsilon_{\rm g} = \dfrac{d\omega}{dk} = \dfrac{d}{dk}\left(\dfrac{\hbar k^2}{2m}\right) = \dfrac{\hbar k}{m} = \dfrac{p_x}{m} = \upsilon_x$(22)
We thus find that for a quantum wave packet, the group speed with which the packet moves (i.e. speed with which the energy is transmitted) is equal to the speed of the associated particle. This resolves the problem we had at the end of Subsection 3.1, where we wrongly associated particle speed with phase speed rather than group speed.
Question T4
An electromagnetic wave packet moves through a material in which the dispersion relation is:
k = (1 + )
where a and b are positive constants. Obtain expressions for: (a) the phase speed and (b) the group speed of the wave packet.
Which of these two speeds is the greater?
(a) The phase speed is given by Equation 19,
phase speed: $\upsilon_\phi = \dfrac{\omega}{k} = \dfrac{2\pi f}{2\pi/\lambda} = f\lambda = c$(Eqn 19)
$\upsilon_\phi = \dfrac{\omega}{k} = \dfrac{\omega}{b\omega(1+a\omega)} = \dfrac{1}{b(1+a\omega)}$
(b) The group speed is given by Equation 17,
group speed: υg = /dk(Eqn 17)
$\upsilon_{\rm g} = \dfrac{d\omega}{dk} = \left(\dfrac{dk}{d\omega}\right)^{-1} = (b+2ab\omega)^{-1} = b(1+2a\omega)^{-1} = \dfrac{1}{b(1+2a\omega)}$
If a and b are positive constants, the denominator is larger in the second expression and so the phase speed is the greater.
4 The wavefunction for a particle confined in one dimension
We now turn our attention from freely moving particles (whether localized or not) to particles that are confined to a limited region of the x–axis. As with the travelling wave examples discussed earlier, we will approach the quantum wave problem through the familiar territory of classical waves, and we start by considering standing waves.
4.1 A particle confined in a one–dimensional box
A classical standing wave on a string
The simplest classical example of a one–dimensional standing wave is that of a vibrating string with fixed ends, such as occurs on the string of a musical instrument. For simplicity, we consider an elastic string which, in its rest state, is straight and taut with length D. It is fixed at each of its ends but may be made to vibrate at right angles to its length, if disturbed – for example, by plucking. A short while after being disturbed, a standing wave becomes established on the string and this wave can be analysed in terms of a superposition of oppositely directed travelling waves with various amplitudes; i this result can again be understood mathematically in terms of a Fourier superposition.
The significant difference between the free moving wave packet, discussed in Subsection 3.2, and the standing wave here, is that the wavelengths involved in the superposition can now have only certain particular values and do not form a continuous range.
The origin of this restriction on the contributing wavelengths is easy to explain. Since the end points of the string are fixed the displacement of the string must be zero at either end. These are known as the boundary conditions on the standing wave, and all the travelling waves that contribute to the superposition must obey them. In this case, the boundary conditions of zero displacement require that the standing wave can be expressed as a sum of travelling sine waves of definite wavelengths and amplitudes.
The particular wavelengths allowed are those for which an integer number of half wavelengths fits into the distance D, between the ends of the string.
Figure 6 shows ‘freeze-frame’ snapshots of some of these simplest standing waves that can arise. These elemental oscillations are known as the standing wave modes of the string. It is possible to excite a single mode, with a particular initial condition, but in general, several modes may be operating at the same time. When a single mode is excited, each point on the string oscillates at the same mode frequency but with an amplitude which depends on position along the string. Positions with zero amplitude are called node_in_a_standing_wavenodes and positions with maximum amplitude are called antinodes; the ends of the string are nodes. The distance between a node and the adjacent antinode is one quarter of the wavelength for that mode. If the string is excited into single mode oscillation it will continue in that mode until it is disturbed in some way, although energy losses due to friction usually damp the oscillation away, eventually.
✦ How many half wavelengths are contained in the distance D for the wave form in the bottom picture of Figure 6?
✧ There are two whole wavelengths shown. Therefore, four half wavelengths are contained between the end points.
The vertical displacements in any one of the one–dimensional standing wave modes, shown in Figure 6, may be represented mathematically as follows:
Standing wave mode n: yn(x, t) = Ansin(knx)cos(ωnt)(23)
The wavelength i of the mode n is determined by the condition:
$D = \dfrac{n\lambda_n}{2}$
for n = 1, 2, 3, 4, ... (i.e. n is a positive integer).
We can write the associated angular wavenumbers kn as:
$k_n = \dfrac{2\pi}{\lambda_n} = \dfrac{n\pi}{D}$(24)
so the modes become:
$y_n(x,\,t) = A_n\sin\left(\dfrac{n\pi x}{D}\right)\cos(\omega_n nt)$ for n = 1, 2, 3, ...(25)
For a given mode Equation 23 does not lead to a travelling wave. Each point at position x on the string undergoes oscillation at the same angular frequency ωn, but with an amplitude Ansin(knx), which depends on position. At an antinode the amplitude is An. Since all points on the string oscillate in phase at the same frequency, the time dependence of this stationary wave is the same at each point and for many purposes is of less interest than the spatial dependence of the mode.
However, if we ask what happens when two or more different modes operate simultaneously then we find that a more interesting time–dependence emerges. In particular, interference between two standing waves of different frequencies produces beats i at the difference frequency and the shape of the string is no longer a fixed sinusoid. Visually, the string appears to contain travelling waves which move back and forth along it.
A quantum wavefunction for one–dimensional confinement
For our quantum example, we will again consider the simplest case of one–dimensional confinement subject to appropriate boundary conditions. The wavefunction can then be completely specified in terms of a single position coordinate (x) and a time coordinate (t). The particle is not allowed outside of a finite range of x–values, say between x = 0 and x = D.
A realization of this situation could be a particle held in the space between two parallel infinite planes, separated by a distance D, measured along the x–axis. For example, we might locate one plane at x = 0 and the other at x = D, as shown in Figure 7a. This is usually called a one–dimensional box – it is actually a box in three–dimensional space, but the confinement is in one dimension only. The confinement of the particle means that the wavefunction must be zero everywhere outside the box, so there is no probability of finding the particle outside the box. It follows from this that we require the wavefunction to fall to zero at the boundary – in this case, at the walls of the box.
Thus the wavefunction must satisfy the boundary conditions:
Ψ(0, t) = 0 and Ψ(D, t) = 0 for all t
Before we consider this system in detail, let us examine whether the uncertainty principle has anything to say about this situation. When we make a mental picture of the situation we must be careful not to imply more information than we can legitimately claim from the Heisenberg uncertainty principle.
Question T5
Could we say that, in the realization of a one–dimensional box described above, the particle has only x–motion, i.e. that this is a one–dimensional problem, since the particle has no y– or z–components of velocity? Present your argument carefully. i
This claim is a valid possibility. It requires υy = 0 and υz = 0, so py = 0 and pz = 0. To have this perfect knowledge of py and pz implies that ∆py and ∆pz are both zero. From the Heisenberg uncertainty principle this is only tenable providing both ∆y and ∆z are undefined, so we are allowed no information about where the particle is to be found along either the y– or z–axes. We have admitted this from the start by having the space between the plates infinite in both y– and z–coordinates, so the statement of the question offers information which is consistent with the Heisenberg uncertainty principle.
Question T6
In the situation described in Question T5, could we claim that the particle is moving along the x-axis? Present your argument carefully. i
This claim offers information which is inconsistent with the Heisenberg uncertainty principle and so is not a valid possibility. If we know that the particle is moving along the x–axis then we are claiming not only that py = 0 and pz = 0 but also that y = 0 and z = 0, which requires perfect knowledge of the y– and z–coordinates and hence that ∆py and ∆pz are each zero. This is not possible, from the Heisenberg uncertainty principle.
The simplest wavefunctions describing a quantum confined between x = 0 and x = D, in a one–dimensional box may be written in the form:
confined wavefunction: i
${\it\Psi}_n(x,\,t) = \psi_n(x)[\cos(\omega_nt) + i\sin(\omega_nt)] = A_n\sin\left(\dfrac{n\pi x}{D}\right)[\cos(\omega_nt) + i\sin(\omega_nt)]$ n = 1, 2, 3, ...(26)
Note that these wavefunctions satisfy the boundary conditions, and that we can separate out the spatial and time dependencies, as we did for the classical wave. The part of the wavefunction that depends on position (x) is denoted by the lower case Greek letter ψ and is therefore written as ψn(x) in each case. This is called the spatial part of the wavefunction, or simply the spatial wavefunction. (Note the distinction between the wavefunction Ψn(x, t) and its spatial part ψn(x).) In this particular case:
spatial wavefunction:
$\psi_n(x) = A_n\sin(k_nx) = A_n\sin\left(\dfrac{n\pi x}{D}\right)$ n = 1, 2, 3, ...(27)
As for the classical case, the quantum wavefunctions consist of a set of modes for the system. The time–dependence is common to a particular mode and it is the spatial part of these wavefunctions which is of most interest. The spatial part of each of the first three wavefunctions, as given by Equation 27 is shown in Figure 7b. As in the classical case, if the system is excited into a particular single mode then it will continue with this same wavefunction indefinitely, unless it is perturbed in some way. These persisting single mode wavefunctions are called the stationary states of the system; they are the quantum equivalent of the modes of a classical system. Stationary states have the particular property that they correspond to a probability density |Ψn(x, t)|2 that is independent of time.
✦ Confirm that |Ψ(x, t)|2 is independent of time for these stationary states, and show that |Ψ(x, t)|2 = |ψ(x)|2.
✧ Applying the general expression for the modulus of a complex quantity to Equation 26,
${\it\Psi}_n(x,\,t) = \psi_n(x)[\cos(\omega_nt) + i\sin(\omega_nt)] = A_n\sin\left(\dfrac{n\pi x}{D}\right)[\cos(\omega_nt) + i\sin(\omega_nt)]$ n = 1, 2, 3, ...(Eqn 26)
$\lvert\,{\it\Psi}_n(x,\,t)\,\rvert^2 = A_n^2\sin^2\left(\dfrac{n\pi x}{D}\right)[\cos^2(\omega_nt) + i\sin^2(\omega_nt)] = A_n^2\sin^2\left(\dfrac{n\pi x}{D}\right) = \lvert\,\psi_n(x)\,\rvert^2$
The final modulus sign around ψn(x) is superfluous in this case since we have chosen An to be real, but in general An and hence ψn(x) may be complex, and the process of taking its modulus is then significant.
The time dependence of the nth wavefunction in Equation 26 is determined by the particle’s total energy, En through Equation 12:
total energy:$E = hf = \dfrac{h\omega}{2\pi} = \hbar\omega$(Eqn 12)
Equation 13 gives the momentum in the nth mode as:
momentum:$p_x = \dfrac{h}{\lambda_{\rm dB}} = \dfrac{hk}{2\pi} = \hbar k$(Eqn 13) i
Equation 14 gives the kinetic energy in nth mode as:
kinetic energy:$E_{\rm kin} = \dfrac{m\upsilon_x^2}{2} = \dfrac{(m\upsilon_x)^2}{2m} = \dfrac{p_x^2}{2m} = \dfrac{\hbar^2k^2}{2m}$(Eqn 14)
For this situation, there are no changes in potential energy in the box and so we may set the potential energy equal to zero everywhere in the box. This means that the kinetic energy is the same as the total energy En, which is:
The total energy: $E_n = \dfrac{\hbar^2k_n^2}{2m} = \dfrac{\hbar^2n^2\pi^2}{2mD^2} = \dfrac{n^2h^2}{8mD^2}$(28)
where n can take on any positive integer value (n ≥ 1). These allowed values of En are known as the energy levels of the system.
Figure 8 shows the first four such energy levels. The stationary state of lowest energy is called the ground state and the associated energy level is called the ground level. States with higher energy are called excited states and their energies are excited levels. Equation 28 represents a very important and characteristic feature of quantum mechanics. It shows that, if a particle is constrained (the constraint being represented by boundary conditions on the wavefunction) then its energy may not take on any arbitrary value. Only certain discrete energies determined by the integer n in Equation 28 are permitted. The energy is said to be quantized and n is referred to as a quantum number.
4.2 Comparison between the classical and quantum cases
We have now arrived at the predictions of quantum mechanics for a particle in a one–dimensional box. These predictions are in striking conflict with classical physics so we should take time to reflect on them and draw out just how fundamental the differences are. Before we draw together the conclusions of the quantum model, let us look briefly at how classical physics would model a particle trapped between these two impenetrable walls.
Question T7
Explain how classical physics could describe a particle with fixed kinetic energy, trapped in a one–dimensional box, given that there is no motion in the y– or z–directions. Comment on the speed and direction of the motion and whether the speed or energy is restricted in any way by the box.
We would probably have come to the conclusion that the particle was moving parallel to the x–axis and was continually rebounding elastically at each wall successively, with the x–component of momentum being reversed at each collision.
The particle would be pictured as moving back and forwards along a single line and it could move along this line with any speed, from zero upwards, and it could have any kinetic energy (or total energy), from zero upwards.
In contrast, the summary conclusions of our quantum treatment are as follows:
1
The system has available to it certain stationary states, or discrete states, each described by the appropriate wavefunction for that state. For the nth state:
${\it\Psi}_n(x,\,t) = \psi_n(x)[\cos(\omega_nt) + i\sin(\omega_nt)] = A_n\sin\left(\dfrac{n\pi x}{D}\right)[\cos(\omega_nt) + i\sin(\omega_nt)]$
2
Each stationary state has an associated definite discrete total energy:
$E_n = \dfrac{n^2h^2}{8mD^2}$
These are known as the energy levels of the system.
3$\vphantom{\dfrac{h^2}{8mD^2}}$
The system’s lowest energy is not zero but equal to $E_1 = \dfrac{h^2}{8mD^2}$.
4
For the stationary states the wavefunctions are complex standing waves, which do not travel and so imply no particular direction of motion for the associated particle in the box.
These conclusions differ at almost every point from the classical model!
✦ The energy becomes zero if we were to set n = 0 in the expression for En. Why does this give a result which is not physically sensible?
✧ If we put n = 0 in Equation 27,
$\psi_n(x) = A_n\sin(k_nx) = A_n\sin\left(\dfrac{n\pi x}{D}\right)$ n = 1, 2, 3, ...(Eqn 27)
then we shall have ψ = 0 everywhere which implies that the probability of finding the particle is zero at every point in the box. There is no particle in the box - it has already escaped!
These claims seem outrageous (particularly points 3 and 4), but before we are tempted to abandon this quantum model as being unrealistic, we should reflect on the fact that these discrete energy levels, which are a consequence of the confinement of the particle to a given region of space, are broadly similar what is observed to happen when an electron is confined within a hydrogen atom. Indeed, it was the discrete energy levels and the fixed transitions between them which were an integral part of the Bohr model of hydrogen and which were completely inexplicable using classical physics. Clearly, our one–dimensional box doesn’t look much like a hydrogen atom, but it shows some encouraging features as an atomic model. i
Question T8
In the state n = 1, as described by Equation 28,
The total energy:$E_n = \dfrac{\hbar^2k_n^2}{2m} = \dfrac{\hbar^2n^2\pi^2}{2mD^2} = \dfrac{n^2h^2}{8mD^2}$(Eqn 28)
the kinetic energy is known exactly. Since the motion is one–dimensional this appears to imply, by Equation 14,
kinetic energy:$E_{\rm kin} = \dfrac{m\upsilon_x^2}{2} = \dfrac{(m\upsilon_x)^2}{2m} = \dfrac{p_x^2}{2m} = \dfrac{\hbar^2k^2}{2m}$(Eqn 14)
that the momentum px is known exactly. We then appear to have the particle located in x to within ∆x = D but with the momentum px known exactly (∆px = 0). Are we claiming more information than is allowed by the Heisenberg uncertainty principle? Present your argument carefully.
If the particle has only x–motion and we know the kinetic energy (x2/2 = px2/(2m)) exactly, then we know px2 exactly. However, we do not know the direction of motion of the particle and so the sign of px is not known and ∆px is not zero.
This amounts to noting that the standing wave can be considered as a superposition of two travelling waves moving in opposite directions. We could also add, as suggested in the marginal note in Subsection 4.1, that we ought really to construct a wave packet in the box and this would require a superposition of wavefunctions corresponding to different momenta. Therefore, the statement of the question is consistent with the Heisenberg uncertainty principle and so is not invalid on this basis. DTH
Location of the particle in the one–dimensional box
Now we ask where the particle is most likely to be found within the box. From the general principle given in Equation 7, the probability of finding the particle in the interval between x and x + ∆x is proportional to |Ψ(x, t)|2x. Since we are dealing with a stationary state, this is proportional to the square of the spatial part of the wavefunction, |ψ(x)|2x.
✦ At which places is the particle most likely to be detected, when in its ground state?
✧ The probability of detection is proportional to |ψ1(x)|2. From Figure 7c, |ψ1(x)|2 has its maximum value at x = D/2, the mid-plane.
To find the actual numerical value of the probability near any point we must first of all normalize the wavefunction. This means that we must ensure that the total probability of finding the particle somewhere in the box is 1 (i.e. certainty). Mathematically, this implies
$\displaystyle \int_0^D \vert\,\psi_n(x)\,\rvert^2\,dx = A_n^2\int_0^D\sin^2\left(\dfrac{n\pi x}{D}\right)\,dx = 1$(29)
i.e.$\displaystyle \dfrac{A_n^2}{2}\int_0^D\left[1-\cos\left(\dfrac{2n\pi x}{D}\right)\right]\,dx = \dfrac{A_n^2}{2}\left[x - \left(\dfrac{D}{2n\pi x}\right)\sin\left(\dfrac{2n\pi x}{D}\right)\right]_0^D = A_n^2\dfrac{D}{2} = 1$
so that we must take $A_n = \sqrt{2/D\os}$ i and the normalized_wavefunctionnormalized spatial wavefunction for the state n is therefore:
normalized spatial wavefunction: $\psi_n(x) = \sqrt{\dfrac 2D}\sin\left(\dfrac{n\pi x}{D}\right)$(30) i
where n = 1, 2, 3, 4, ...
Question T9
What is the probability that a particle in the n = 5 state will be detected between x = 0 and x = +0.1D?
The probability, P5(0, +0.1D), that a particle in the n = 5 state will be detected between x = 0 and x = +0.1D is:
$\displaystyle P_5(0,\,+0.1D) = \int_0^{0.1D}\lvert\,\psi_5(x)\,\rvert^2\,dx$
where we must use the normalized wavefunction for state n = 5, which is:
$\psi_5(x) = \sqrt{\dfrac 2D}\sin\left(\dfrac{5\pi x}{D}\right)$
Thus the probability is:
$\displaystyle P_5(0,\,+0.1D) = \dfrac 2D \int_0^{0.1D}\lvert\,\sin^2\left(\dfrac{5\pi x}{D}\right)\,dx = \dfrac 2D\dfrac12\left[x - \dfrac{D}{10\pi}\sin\left(\dfrac{10\pi x}{D}\right)\right]_0^{0.1D}$
using the given integral.
However$\left[x - \dfrac{D}{10\pi}\sin\left(\dfrac{10\pi x}{D}\right)\right]_0^{0.1D} = \left[0.1D-\dfrac{D}{10\pi}\sin\left(\dfrac{10\pi\,0.1D}{D}\right)\right] - 0 = 0.1D$
since sinπ = 0. So, finally:
$\displaystyle P_5(0,\,+0.1D) = \dfrac 2D\dfrac120.1D = 0.1$
There is thus a 10% chance of finding the particle between x = 0 and x = +0.1D.
4.3 Energy changes or transitions in the one–dimensional box i
The quantum model for a particle in a one–dimensional box shows the existence of a set of stationary states of different energy. If the particle in one of these states is unperturbed, it will remain in the state indefinitely, in just the same way that a single standing wave mode on a string will continue with fixed energy indefinitely. If either system is perturbed, for example by gaining or losing energy, then the state of oscillation will change. In the classical case, other wave modes will become excited as the energy changes; in the quantum case, the system may make a transition between stationary states as the energy changes.
In Subsection 4.2 we noted that classically, when two standing wave modes are simultaneously operating on a string, the resulting disturbance has a time dependence at the difference frequency of the two modes. Visually, the string shows the presence of a disturbance which oscillates back and forth on the string, at this difference frequency. In the quantum case, the mixing of two stationary states with different characteristic angular frequencies ωn and ωm produces a superpositionsuperposition state which is no longer a stationary state and in which the probability density consequently depends on time. If we calculate the probability of finding the particle at a given position in the box we find that the probability density oscillates at the angular_frequencyangular beat frequency |ωnωm|. This corresponds to a real oscillation of the particle between the walls – rather as the classical picture suggested.
Since an electron is a charged particle, any oscillation that it exhibits should, according to classical physics, be accompanied by radiation. The oscillating electron would either be radiating away its energy (emission), or gaining energy (absorption) from an incoming electromagnetic wave. It is natural, therefore, to try to associate the emission and absorption of radiation by an atom with the presence of an electron in a superposition state, mixing two stationary states of the system. At some initial time the system is in one of these two states and at some later time it will be in the other state, having made a transition between the two states; in the interim the state is a non–stationary superposition state in which the proportions of the two stationary states are changing with time. The energy change may then be associated with a photon of frequency f = |ωnωm|/2π, and so corresponds to energy change
E = hf = h|ωnωm|/2π = |EnEm| i
Question T10
An electron is confined in the one–dimensional box in Figure 7a.
(a) Sketch the spatial wavefunction that describes the electron when its total energy is $E = \dfrac{9h^2}{8m_{\rm e}D^2}$.
(b) Where is the electron most likely to be detected when it has this energy?
(c) If the electron makes a transition from this state to the ground state, obtain an expression for the reduction of its energy.
(d) If this energy is given out as a quantum of radiation, what will be the frequency f of its associated radiation?
(a) The spatial wavefunction corresponding to E = 9h2/(8meD2) is given by Equation 27,
$\psi_n(x) = A_n\sin(k_nx) = A_n\sin\left(\dfrac{n\pi x}{D}\right)$ n = 1, 2, 3, ...(Eqn 27)
with n = 3 and m equal to the mass of the electron. Compare your sketch with Figure 7b (ψ3).
(b) The electron is most likely to be detected where |ψ(x)|2 is greatest. The three places can be seen in Figure 7c (ψ32).
(c) The ground state has energy E1 = h2/(8meD2). The loss of energy is therefore
$\Delta E = \dfrac{9h^2}{8m_{\rm e}D^2} - \dfrac{9h^2}{8m_{\rm e}D^2} = \dfrac{h^2}{m_{\rm e}D^2}$
(d) The radiation with frequency f has quanta with energy ∆E = hf.
Therefore$f = \dfrac{h}{m_{\rm e}D^2}$
5 A particle confined in two or three dimensions
The quantum treatment of a particle confined in one dimension has shown many encouraging features in relation to explaining the observed behaviour of an atom. The Heisenberg uncertainty principle explains why an atom is stable against collapse into the nucleus and the confined wave model predicts quantized energies of a confined electron and associates transitions between these with the emission or absorption of electromagnetic radiation. Now we need to extend the model to confinement in three dimensions.
5.1 Extension to two–dimensional confinement
We have seen from Section 4 that when a particle is confined in one dimension it may be found in certain quantum states labelled by a single quantum number, the integer n, and described by a wavefunction Ψn(x, t). Extending this idea to two dimensions means that the wavefunction must involve two spatial coordinates (e.g. x and y) as well as t. As usual, our deliberations will be guided by the classical analogue.
A classical standing wave on a square membrane
Consider a flexible two–dimensional surface, such as a taut square drum membrane stretched over and fixed to a square framework of side D (see Figure 9). Vibrations in this surface may be set up and there will be boundary conditions in both x– and y–directions, such that there can be no vibration along the perimeter of the square.
The displacement must be zero along both the lines y = 0 (the x–axis) and y = D for any value of x between 0 and D, and along both the lines x = 0 (the y–axis) and x = D for any value of y between 0 and D. These boundary conditions are complicated to write down, but they will affect the possible vibrations of the membrane in a way that is a simple extension of the one–dimensional case.
The standing waves that can be set up will involve an integer number of half wavelengths in each of the two independent spatial dimensions. Thus there will be two integers that define a particular mode of vibration of the drum, nx for the x–direction and ny for the y–direction. This may be most clearly seen by looking at the examples shown in Figure 9.
The integers nx and ny simply designate the number of half–wavelengths of the standing wave between the two boundaries along the relevant axis. The oscillations in the two dimensions are independent in the sense that any valid value of nx can be combined with any valid value of ny to produce a valid wave mode, labelled (nx, ny) on the membrane.
A quantum wavefunction for two–dimensional confinement
We need a two–dimensional box in which to confine our particle. A realization of this might involve using two pairs of parallel infinite planes, one pair separated by a distance D along the x–axis and the other pair separated by a distance D along the y–axis. The particle is completely unrestricted in z but is confined in x and y. Because we have not constrained the particle in z (∆z is undefined) we may legitimately claim that pz = 0, or ∆pz = 0; the particle can then be said to have no motion along z, or to have only motion along x and y. These two motions will be completely independent and each will have an associated kinetic energy, with the total kinetic energy given by the sum of the two independent kinetic energies. i
Treating the two contributions to the kinetic energy as being independent will require that we use two independent quantum numbers, nx and ny, to describe each stationary state. The energy levels of those states will then be given by an extension of Equation 28:
The total energy: $E_n = \dfrac{\hbar^2k_n^2}{2m} = \dfrac{\hbar^2n^2\pi^2}{2mD^2} = \dfrac{n^2h^2}{8mD^2}$(Eqn 28)
$E_{n_x,n_y} = \dfrac{h^2}{8mD^2}(n_x^2+n_y^2)$(31)
where nx = 1, 2, 3, 4, ... and ny = 1, 2, 3, 4, ...
The corresponding spatial wavefunctions now also require two quantum numbers to label them:
$\psi_{n_x,n_y}(x,\,y) = A_n\sin\left(\dfrac{n_x\pi x}{D}\right)\sin\left(\dfrac{n_y\pi y}{D}\right)$(32)
where nx = 1, 2, 3, 4, ... and ny = 1, 2, 3, 4, ...
Compare Equations 31 and 32 with Equations 28 and 27,
$\psi_n(x) = A_n\sin(k_nx) = A_n\sin\left(\dfrac{n\pi x}{D}\right)$ n = 1, 2, 3, ...(Eqn 27)
respectively. Equation 32 obviously satisfies the boundary conditions at x = 0 and x = D and also at y = 0 and y = D.
Shared energy levels – degeneracy
Equations 31 and 32,
$E_{n_x,n_y} = \dfrac{h^2}{8mD^2}(n_x^2+n_y^2)$(Eqn 31)
$\psi_{n_x,n_y}(x,\,y) = A_n\sin\left(\dfrac{n_x\pi x}{D}\right)\sin\left(\dfrac{n_y\pi y}{D}\right)$ n = 1, 2, 3, ...(Eqn 32)
also show a feature which did not occur for the one–dimensional case. Since the energy does not depend separately on nx and ny but on the special combination (nx2 + ny2), if we exchange the values of nx and ny we leave the energy unchanged. For example, the spatial wavefunctions ψ1,2 and ψ2,1 will correspond to the same energy level even though they are different functions. When different wavefunctions share the same energy level the wavefunctions are said to be degenerate and the system is said to exhibit degeneracy. In this case we see that the degeneracy arises as a consequence of the symmetry between the x and y variables. i
Table 1 Energies, expressed in units of h2/(8mD2), for a particle confined in two-dimensions.
nx = 1 nx = 2 nx = 3 nx = 4
ny = 1 2 5 10 17
ny = 2 5 8 13 20
ny = 3 10 13 18 25
ny = 4 17 20 25 32
Table 1 shows the energies, expressed in units of h2/(8mD2), for the states ψnx,ny for a range of values of nx and ny, you should be able to see several examples of degeneracy. Degeneracies usually arise in systems with symmetries – here, it is the symmetry between the x– and y–motion. If we were to break the symmetry, for example by allowing the box to have different dimensions in x and y, we would remove the degeneracy, since the energy depends on nx2 and ny2 separately but not on (nx2 + ny2).
Symmetry and symmetry breaking are important considerations in quantum mechanics.
5.2 Extension to three–dimensional confinement
The extension to three dimensions – where the complete specification of the wavefunction of the particle requires three spatial coordinates (x, y, z) – is now straightforward. The particle is confined to a cube of side D in which standing waves may be formed, as in Figure 10. The energy levels in this case are:
$E_{n_x,n_y,n_z} = \dfrac{h^2}{8mD^2}(n_x^2+n_y^2+n_z^2)$(33)
where nx = 1, 2, 3, 4, ...; ny = 1, 2, 3, 4, ...; nz = 1, 2, 3, 4, ...
The spatial wavefunctions are:
$\psi_{n_x,n_y,n_z}(x,\,y,\,z) = A_n\sin\left(\dfrac{n_x\pi x}{D}\right)\sin\left(\dfrac{n_y\pi y}{D}\right)\sin\left(\dfrac{n_z\pi z}{D}\right)$(34)
where nx = 1, 2, 3, 4, ...; ny = 1, 2, 3, 4, ...; nz = 1, 2, 3, 4, ...
In three dimensions, a far richer range of degeneracies becomes possible. First, we have the symmetry due to the three spatial coordinates; for each set of three quantum numbers there will be six permutations of these three which give the same energy. There are also now a whole range of accidental degeneracies, based on the accidental equality of (nx2+ ny2 + nz2) for two or more sets of three quantum numbers. For example, the state nx = 8, ny = 6, nz = 5, has the same energy as the state nx = 4, ny = 3, nz = 10. There are now many cases where several independent wavefunctions share a common energy level; the number of such wavefunctions sharing a given energy level is said to be the order of degeneracy of the energy level.
Question T11
(a) Write down an expression for the energy of the ground state of a particle confined in three dimensions. Explain whether or not this state is degenerate.
(b) What is the energy of the first excited state? Is this state degenerate and, if so, what is its order of degeneracy?
In the general case, the energy is given by:
$E_{n_x,n_y,n_z} = \dfrac{h^2}{8mD^2}(n_x^2+n_y^2+n_z^2)$
where nx, ny, nz are positive integers.
The corresponding spatial wavefunction is given by:
$\psi_{n_x,n_y,n_z}(x,\,y,\,z) = A\sin\left(\dfrac{n_x\pi x}{D}\right)\sin\left(\dfrac{n_y\pi y}{D}\right)\sin\left(\dfrac{n_z\pi z}{D}\right)$
(a) The ground (lowest energy) state is the one for which nx = ny = nz = 1. The energy is E1,1,1 = 3h2/(8mD2). There is only one state corresponding to this energy and so the state is not degenerate. Its spatial wavefunction is given by:
$\psi_{1,1,1}(x,\,y,\,z) = A\sin\left(\dfrac{\pi x}{D}\right)\sin\left(\dfrac{\pi y}{D}\right)\sin\left(\dfrac{\pi z}{D}\right)$
(b) The second lowest energy value may be obtained in three ways. The quantum numbers are:
(1) nx = 1, ny = 1, nz = 2;
or(2) nx = 1, ny = 2, nz = 1;
or(3) nx = 2, ny = 1, nz = 1.
Each gives rise to the energy 3h2/4mD2. There are three different spatial wavefunctions:
(1) $\psi_{1,1,2}(x,\,y,\,z) = A\sin\left(\dfrac{\pi x}{D}\right)\sin\left(\dfrac{\pi y}{D}\right)\sin\left(\dfrac{2\pi z}{D}\right)$ ;
(2) $\psi_{1,2,1}(x,\,y,\,z) = A\sin\left(\dfrac{\pi x}{D}\right)\sin\left(\dfrac{2\pi y}{D}\right)\sin\left(\dfrac{\pi z}{D}\right)$ ;
(3) $\psi_{2,1,1}(x,\,y,\,z) = A\sin\left(\dfrac{2\pi x}{D}\right)\sin\left(\dfrac{\pi y}{D}\right)\sin\left(\dfrac{\pi z}{D}\right)$ ;
This energy level has an order of degeneracy of 3.
6 Closing items
6.1 Module summary
1
Indications of the particle–like behaviour of electromagnetic radiation, include evidence for the photon as a particle with energy E = hf and momentum p = E/c = h/λ.
2
Indications of the wave–like behaviour of matter, as predicted by the de Broglie hypothesis (λdB = h/p), include a variety of particle diffraction experiments. The Heisenberg uncertainty principle states that the simultaneous uncertainties in the position and momentum of a particle obey the restriction
$\Delta x\,\Delta p_x \gtrsim \dfrac{h}{2\pi}$(Eqn 6)
3
The quantum mechanical wavefunction Ψ(x, t), of a particle moving in one dimension is a complex numbercomplex quantity found by solving the time–dependent Schrödinger equation. The probability of finding the particle within the small interval ∆x around the position x at time t being ∝ |Ψ(x, t)|2 ∆x, where |Ψ(x, t)|2 is the probability density at position x.
4
An unlocalized free particle may be represented by a wavefunction of the form
ψn(x, t) = Acos(kxωt) + i Asin(kxωt)
this describes a particle for which the total energy $E = \hbar\omega$, the momemtum $p_x = \hbar k$ and the kinetic energy $E_{\rm kin}= \hbar^2k^2/2m$.
5
The probability density for finding an unlocalized free particle at any value of x is constant and does not depend of x.
6
A localized free particle may be represented by a wavefunction constructed from a superposition of unlocalized free particle wavefunctions. Analysis of the dispersion relation for these wavefunctions shows that the superposition has a group speed that is equal to speed of the associated particle.
7
The quantum analogue to the confined wave on a stretched string is the particle confined in a one–dimensional box. The wavefunctions for this system consist of a set of stationary states for which the probability density is independent of time. The spatial wavefunctionspatial wavefunctions of these states must satisfy boundary conditions that require them to vanish at the edges of the box. The spatial wavefunctions form a discrete set and are given by
$\psi_n(x) = A_n\sin(k_nx) = A_n\sin\left(\dfrac{n\pi x}{D}\right)$ n = 1, 2, 3, ...(Eqn 27)
8
These spatial wavefunctions correspond to a restricted range of energy values for the particle. The allowed energy levels are
$E_n = \dfrac{n^2h^2}{8mD^2}$ n = 1, 2, 3, ...
where n is known as a quantum number. The ground state has a non–zero energy, which confirms that quantum mechanics forbids the idea of a confined particle at rest.
9
The stationary states imply no particular direction of motion for the confined particle and have a constant energy. If the system is perturbed and the energy does change, this can be associated with a transition between two stationary states. For an electron in a box, such a classical oscillation would be accompanied by the emission or absorption of electromagnetic radiation. This idea can be extended, in a modified way, to the quantum case.
10
A particle confined in two or three dimensions can be treated similarly, except that two or three independent quantum numbers are needed to specify the states and the energy levels. The energy levels are now:
$E_{n_x,n_y,n_z} = \dfrac{h^2}{8mD^2}(n_x^2+n_y^2+n_z^2)$(Eqn 33)
and the stationary state spatial wavefunctions are:
$\psi_{n_x,n_y,n_z}(x,\,y,\,z) = A_n\sin\left(\dfrac{n_x\pi x}{D}\right)\sin\left(\dfrac{n_y\pi y}{D}\right)\sin\left(\dfrac{n_z\pi z}{D}\right)$(Eqn 34)
The symmetry of the system produces some shared energy levels for several independent wavefunctions, an effect known as degeneracy, and some accidental degeneracies also occur.
Study comment As a final cautionary note we must admit that our approach in this module has been inspirational rather than rigorous. We have presented plausible arguments, based on analogies with classical wave ideas and have developed self–consistent models which throw some light on the quantum world. However, the classical models on which we have placed such reliance were themselves based on the rigorous solution of a wave equation, to which we have given scant reference here. The justification for the wavefunctions presented here also rests on the rigorous solution of an equation, the Schrödinger equation – but this must be left to other FLAP modules.
6.2 Achievements
Having completed this module, you should be able to:
A1
Define the terms that are emboldened and flagged in the margins of the module.
A2
State and use the relationship between a wave function Ψ(x, t) and the probability of detecting the particle to which it relates.
A3
Describe how a free particle may be represented either by a wavefunction consisting of a single complex travelling wave of definite wavelength, or by a wave packet constructed from such travelling waves and explain how these representations conform with the Heisenberg uncertainty principle.
A4
In the representation of a free particle by a single wavelength wavefunction, recall and use expressions for the total energy, the momentum and the kinetic energy of the particle in terms of the angular wavenumber and the wave angular frequency.
A5
Apply the Heisenberg uncertainty principle in simple cases.
A6
Write down the spatial wavefunctions for a particle constrained in a one–dimensional box.
A7$\vphantom{\dfrac{n^2h^2}{8mD^2}}$
Show that the energy levels for a particle constrained in a one–dimensional box are $E_n = \dfrac{n^2h^2}{8mD^2}$ and use this expression to discuss transitions between quantized energy levels.
A8
Explain what is meant by degeneracy of the energy levels of a particle confined in a two– or three–dimensional box. Determine orders of degeneracy in simple cases.
Study comment You may now wish to take the Exit test for this module which tests these Achievements. If you prefer to study the module further before taking this test then return to the topModule contents to review some of the topics.
6.3 Exit test
Study comment Having completed this module, you should be able to answer the following questions each of which tests one or more of the Achievements. i
Question E1 (A2)
A particle in a stationary state is represented by the spatial wavefunction ψ(x) = exp(−3x2).
(a) Is this spatial wavefunction normalized?
[You may use the following standard integral, $\int_{-\infty}^\infty \exp(-ax^2)\,dx = \sqrt{\pi/a\os}$.]
(b) If it is not, write down a normalized spatial wavefunction that represents the particle in the same state.
(c) What is the probability that the particle will be detected between x and x + ∆x (for small ∆x)?
If a particle in a stationary state is represented by a spatial wavefunction ψ(x), the probability that it will be detected between x and x + ∆x is |ψ(x)|2x.
The spatial wavefunction is normalized (‘to unity’) in quantum mechanics if
$\displaystyle \int_{-\infty}^\infty \lvert\,\phi(x)\,\rvert^2\,dx = 1$
If the integral is not equal to one, but to some other, finite, value I, then
$\dfrac{1}{\sqrt{I}}\psi(x)$
will be normalized. We may apply these facts to the spatial wavefunction ψ(x) = exp(−3x2), for which |ψ(x)|2 = exp(−6x2)
(a) $\displaystyle \int_{-\infty}^\infty \exp(-6x^2)dx = \sqrt{\pi/6\os}$ (i.e. a = 6 in the formula given). Since $I = \sqrt{\pi/6\os}$ is not equal to one, this spatial wavefunction is not normalized.
(b) The normalized spatial wavefunction in this case is therefore given by:
ψ(x) = (6/π)1/4exp(−3x2)
(c) The probability that the particle will be detected between x and x + ∆x is $\sqrt{6/\pi\os}\exp(-6x^2)\,\Delta x$.
Question E2 (A3, A4 and A5)
Explain why a particle represented by a wave packet may not simultaneously have both its position and momentum known to arbitrarily high precision. What is the relationship between ∆x and ∆px, the uncertainties in the measurements of position and momentum, respectively?
[Hint: No mathematical derivations are required.]
A wave packet is a superposition of waves using the principle of superposition. The wave packet is designed to have an appreciable value only over a finite range of values of position x. (Figure 4 shows examples of wave packets.) The fact that the wavefunction is negligibly small outside the chosen range, ensures that a position measurement will give a value within that range with high probability.
The functions superposed to form the wave packet may be, for instance, sine waves with a continuous range of values of the wave–number k. The mathematics of the superposition – Fourier analysis – determines the contributions of waves with each value of k necessary to produce a given wave packet, as in Figure 4. A general feature is that to produce a smaller value of the range ∆x, a larger range of values ∆k must be included in the superposition, so the two ranges vary inversely, with ∆x ∆k ≈ 1.
To make a more precise measurement of position, the value of ∆x must be made smaller. This then means that ∆k will have to become larger, as will the range of wavelengths ∆λ, since k = 2π/λ.
For a particle with momentum px, the wavelength is given by the de Broglie formula λdB = h/px. Therefore k is proportional to px: k = (2π/h)px and a wide spread of k means a wide spread in the momentum components of the wave packet: ∆k = (2π/h)∆px. The wider spread of values of px means that the measurement of px is less precise. The precision of the measurements of x and px therefore oppose one another; if we make x more precise by sharpening the wave packet, then px must be less precise, since more values of px contribute to the wave packet.
Substituting into ∆x ≈ 1/∆k with ∆k = (2π/h)∆px makes this relationship more quantitative:
$\Delta x \gtrsim \dfrac{h}{2\pi\Delta p_x}\quad\text{i.e.}\quad\Delta x\,\Delta p_x \gtrsim \dfrac{h}{2\pi}$
This is one statement of the Heisenberg uncertainty principle.
(Reread Subsections 2.1 and 2.2 if you had difficulty with this question.)
Question E3 (A7)
A helium nucleus has a mass of 6.6 × 10−27 kg. What is its minimum energy if it is confined in one dimension within a box of length 7.2 fm? (1 fm = 10−15 m) Explain the reasoning involved.
The energy levels of a particle in a one–dimensional box are: En = n2h2/(8mD2).
In this case the minimum energy corresponds to n = 1 and so:
$E_1 = \dfrac{h^2}{8mD^2} = \rm \dfrac{(6.6\times10^{-34}\,J\,s)^2}{8\times6.6\times10^{-27}\,kg\times(7.2\times10^{-15}\,m)^2} = 1.6\times10^{-13}\,J$
(Reread Subsections 4.1 and 4.2 if you had difficulty with this question.)
Question E4 (A4, A6 and A7)
A particle is confined in a one–dimensional box of dimension D.
(a) Write down the spatial wavefunction that is analogous to a standing wave that has five half wavelengths within the box.
(b) What is the momentum of a particle in the stationary state described by this spatial wavefunction?
(c) Show that the energy of the particle is 25h2/(8mD2) in this case.
The spatial wavefunction that has five half wavelengths within the box is shown in Figure 11. The wavelength is given by: D = 5λ/2 and nx = 5.
(a) The expression for this spatial wavefunction is
ψ5(x) = Asin(5πx D).
(b) Since λ = 2D/5 and p = h/λ it is tempting to say that px = 5h/(2D). But more caution is needed. Such certainty would conflict with the uncertainty principle in this case. In fact the localization of the particle means that the wavelength is not uniquely determined in this case, as was discussed in the answer to Question T8, but we can say px = ±5h/20.
(c) The energy is entirely kinetic and so:
$E_5 = \dfrac{p^2}{2m} = \dfrac{1}{2m}\left(\dfrac{5h}{2D}\right)^2 = \dfrac{25h^2}{8mD^2}$
Alternatively, the energy level for n = 5 is
$E_5 = \dfrac{n^2h^2}{8mD^2} = \dfrac{25h^2}{8mD^2}$
(Reread Subsections 4.1 and 4.2 if you had difficulty with this question.)
Question E5 (A8)
A particle confined to be within a cube of side D is in a stationary state with a spatial wavefunction:
$\psi_{n_x,n_y,n_z}(x,\,y,\,z) = A_n\sin\left(\dfrac{n_x\pi x}{D}\right)\sin\left(\dfrac{n_y\pi y}{D}\right)\sin\left(\dfrac{n_z\pi z}{D}\right)$
and with energy:
$E_{n_x,n_y,n_z} = \dfrac{h^2}{8mD^2}(n_x^2+n_y^2+n_z^2)$
(a) What does it mean to say that an energy level is degenerate?
(b) The energy level 7h2/(4mD2) is degenerate. Determine the order of degeneracy and give expressions for the corresponding wavefunctions.
(a) An energy level is degenerate if more than one wavefunction corresponds to the energy value.
(b) The energy value 7h2/4mD2, i.e. 14h2/8mD2 implies that (nx2 + ny2 + nz2) = 14.
One set of quantum numbers that satisfies this condition is:
(1) nx = 1, ny = 2, nz = 3.
The other combinations that also give the same energy are:
(2) nx = 1, ny = 3, nz = 2;
(3) nx = 2, ny = 1, nz = 3;
(4) nx = 2, ny = 3, nz = 1;
(5) nx = 3, ny = 1, nz = 2;
(6) nx = 3, ny = 2, nz = 1.
There are six combinations corresponding to six different wavefunctions, so the order of degeneracy is six. The six corresponding spatial wavefunctions are:
(1) $\psi_{1,2,3}(x,\,y,\,z) = A\sin\left(\dfrac{\pi x}{D}\right)\sin\left(\dfrac{2\pi y}{D}\right)\sin\left(\dfrac{3\pi z}{D}\right)$
(2) $\psi_{1,3,2}(x,\,y,\,z) = A\sin\left(\dfrac{\pi x}{D}\right)\sin\left(\dfrac{3\pi y}{D}\right)\sin\left(\dfrac{2\pi z}{D}\right)$
(3) $\psi_{2,1,3}(x,\,y,\,z) = A\sin\left(\dfrac{2\pi x}{D}\right)\sin\left(\dfrac{\pi y}{D}\right)\sin\left(\dfrac{3\pi z}{D}\right)$
(4) $\psi_{2,3,1}(x,\,y,\,z) = A\sin\left(\dfrac{2\pi x}{D}\right)\sin\left(\dfrac{3\pi y}{D}\right)\sin\left(\dfrac{\pi z}{D}\right)$
(5) $\psi_{3,1,2}(x,\,y,\,z) = A\sin\left(\dfrac{3\pi x}{D}\right)\sin\left(\dfrac{\pi y}{D}\right)\sin\left(\dfrac{2\pi z}{D}\right)$
(6) $\psi_{3,2,1}(x,\,y,\,z) = A\sin\left(\dfrac{3\pi x}{D}\right)\sin\left(\dfrac{2\pi y}{D}\right)\sin\left(\dfrac{\pi z}{D}\right)$
(Reread Subsections 5.1, and 5.2 if you have difficulty with this question.)
Study comment This is the final Exit test question. When you have completed the Exit test go back and try the Subsection 1.2Fast track questions if you have not already done so.
If you have completed both the Fast track questions and the Exit test, then you have finished the module and may leave it here. | {"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": 0.9243537783622742, "perplexity": 399.62717894574047}, "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-2018-05/segments/1516084888878.44/warc/CC-MAIN-20180120023744-20180120043744-00469.warc.gz"} |
https://rdrr.io/cran/crmPack/man/LogisticNormal-class.html | # Standard logistic model with bivariate normal prior
### Description
This is the usual logistic regression model with a bivariate normal prior on the intercept and slope.
### Details
The covariate is the natural logarithm of the dose x divided by the reference dose x^{*}:
logit[p(x)] = α + β \cdot \log(x/x^{*})
where p(x) is the probability of observing a DLT for a given dose x.
The prior is
(α, β) \sim Normal(μ, Σ)
The slots of this class contain the mean vector, the covariance and precision matrices of the bivariate normal distribution, as well as the reference dose.
### Slots
mean
the prior mean vector μ
cov
the prior covariance matrix Σ
prec
the prior precision matrix Σ^{-1}
refDose
the reference dose x^{*}
### Examples
1 2 3 model <- LogisticNormal(mean = c(-0.85, 1), cov = matrix(c(1, -0.5, -0.5, 1), nrow = 2), refDose = 50)
Want to suggest features or report bugs for rdrr.io? Use the GitHub issue tracker. Vote for new features on Trello. | {"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": 0.9013597965240479, "perplexity": 3521.8651644840447}, "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-2016-44/segments/1476988719139.8/warc/CC-MAIN-20161020183839-00494-ip-10-171-6-4.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-find-the-vertex-and-axis-of-symmetry-of-f-x-3x-2-12x-1 | Algebra
Topics
# How do you find the vertex and axis of symmetry of f(x)= 3x^2 + 12x + 1?
Mar 19, 2016
This is a quadratic equation of a parabola (the squared term gives it away)
$y = a {x}^{2} + b x + c$
the vertex is located where $x = - \frac{b}{2 a}$
this occurs where $x = - \frac{12}{2 \times 3}$ or at $x = - 2$
substitute into the equation to figure out the y coordinate of the vertex.
The axis of symmetry is the vertical line passing through the vertex which is $x = - 2$
##### Impact of this question
3318 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": 5, "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": 0.8227558732032776, "perplexity": 533.5647525422388}, "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-2022-21/segments/1652662534669.47/warc/CC-MAIN-20220520191810-20220520221810-00050.warc.gz"} |
https://analysisqual.wordpress.com/2011/05/22/september-2000/ | ## September 2000
Problem Statement: Use the open cover definition of compactness to show that any finite union of compact sets is compact.
Proof: Let $C_1,C_2,\dots,C_n$ be compact sets in $\mathbb{R}$. We wish to show that $C=C_1\cup C_2\cup\dots\cup C_n$ is a compact set.
Let $T=\bigcup\limits_{\alpha}T_{\alpha}$ be an open cover of $C$. Then $T$ is an open cover for $C_j$ for every $j\in[1,n]$. Since each $C_j$ is compact there is a finite subcover of $T$ which covers $C_j$ for each $j\in[1,n]$. So for each $C_j$ we may construct $T_{j}=T_{j_1}\cup T_{j_2}\cup\dots\cup T_{j_m}$, a finite subset of $T$ which covers $C_j$, for each $C_j$.
I claim that $T_C=\bigcup\limits_{j=1}^{n}T_j$ is a finite subcover of $C$.
First note that $T_C$ is a union of finitely many unions of open sets and so it is an open set. Now to verify that $T_C$ covers $C$. Let $x\in C$, then $x\in C_j$ for some $C_j$. Thus, by construction, $x\in T_j$ since $T_j$ is an open cover of $C_j$. Therefore, $x\in T_C$ and we may conclude that $T_C$ is a finite subcover of $T$.
Therefore, every open cover of $C$ has a finite subcover and so $C$ is compact.
$\Box$
Reflection: The key in this proof is that we had a finite union of compact sets. From there all we had to do was union up all of their finite subcovers. A finite union of a finite union is simply a finite union of open sets. | {"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": 34, "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": 0.9246275424957275, "perplexity": 35.42408759173757}, "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/1531676595531.70/warc/CC-MAIN-20180723071245-20180723091245-00181.warc.gz"} |
http://mathhelpforum.com/algebra/208060-prove-function-strictly-decreasing.html | # Thread: Prove function is strictly decreasing
1. ## Prove function is strictly decreasing
I've got functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ . $f$ is strictly increasing and $f\circ g$ is strictly decreasing I am asked to show $g$ is strictly decreasing. I already have a solution in front of me however I would like to know if this approach is correct.
Suppose $x_{1} < x_{2}$ then $f\left ( g\left ( x_{1} \right ) \right ) > f\left ( g\left ( x_{2} \right ) \right )$ $\Rightarrow g\left ( x_{1} \right ) > g\left ( x_{2} \right )$ since $f$ is strictly increasing.
therefore $g$ is strictly decreasing.
2. ## Re: Prove function is strictly decreasing
Yes, that's good. | {"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": 9, "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": 0.9899901747703552, "perplexity": 265.5384132739068}, "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-2017-09/segments/1487501170741.49/warc/CC-MAIN-20170219104610-00146-ip-10-171-10-108.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/algebra-1/chapter-5-linear-functions-5-3-slope-intercept-form-practice-and-problem-solving-exercises-page-311/54 | ## Algebra 1
Slope: $\frac{3}{2}$ $y$-intercept: $3$
We first must put the equation $2y-6=3x$ into slope-intercept form. To do this, first we add $6$ to both sides of the equation: $2y=3x+6$ Then we divide both sides of the equation by $2$: $y=\frac{3}{2}x+3$ The slope-intercept form of a linear equation of a non vertical line is $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept. We use this to pick out the slope and the $y$-intercept from the equation $y=\frac{3}{2}x+3$. Since $\frac{3}{2}$ takes the place of $m$ in the equation, it is the slope. Since $3$ takes the place of $b$ in the equation, it is the $y$-intercept. | {"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": 0.9839317798614502, "perplexity": 67.91581675344607}, "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-2018-51/segments/1544376823322.49/warc/CC-MAIN-20181210101954-20181210123454-00496.warc.gz"} |
https://wiki.seg.org/index.php?title=Dictionary:Absorption_coefficient&printable=yes | # Dictionary:Absorption coefficient
Other languages:
English • español
If the amplitude A is expressed as
${\displaystyle A=A_{0}e^{-\alpha x}\ }$
where x=distance, α is the absorption coefficient or attenuation factor. It usually varies linearly with frequency and is often expressed in dB/wavelength (typically 0.20 to 0.50 dB/λ). Distinction may or may not be made as to the reason for attenuation (i.e., absorption or some other mechanism). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "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": 0.9311102628707886, "perplexity": 2348.416858198609}, "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/1674764501407.6/warc/CC-MAIN-20230209045525-20230209075525-00170.warc.gz"} |
https://mathematics21.org/2013/07/05/partial-order-funcoids-reloids/?share=pinterest | # Partial order funcoids and reloids
Partial order funcoids and reloids formalize such things as “infinitely small” step rotating a circle counter-clockwise.
This is “locally” a partial order as every two nearby “small” sets (where we can define “small” for example as having the diameter (measuring along the circle) less than $\pi$) are ordered: which is before in the order of rotating the circle counter-clockwise and which is after.
The definition for partial order funcoid (and similarly partial order reloid) is a trivial generalization of the classical definition of partial order.
The endo-funcoid $f$ on a set $A$ is a partial order iff all of the following:
1. $f\sqsupseteq\mathrm{id}^{\mathsf{FCD}}_A$;
2. $f\sqcap f^{-1}\sqsubseteq\mathrm{id}^{\mathsf{FCD}}_A$;
3. $f\circ f\sqsubseteq f$.
This can also be defined for reloids entirely analogous to funcoids.
What are possible applications of partial order funcoids and partial order reloids? I yet don’t know.
## One comment
1. This “infinitely small counter-clockwise step” can be defined as the funcoid $f$ such that $X[f]Y$ iff for every $\epsilon>0$ there exists $\epsilon'\ge 0$ such that $\epsilon'<\epsilon$ and rotating the set $X$ $\epsilon'$ radians counter-clockwise produces a set which intersects with $Y$.
Exercise: Prove that the funcoid $f$ exists and that is a partial order funcoid. | {"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": 21, "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": 0.9899217486381531, "perplexity": 1004.2182307663717}, "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-2020-05/segments/1579251687725.76/warc/CC-MAIN-20200126043644-20200126073644-00194.warc.gz"} |
https://cs.stackexchange.com/questions/65212/min-cut-in-graph-with-demands-lower-bounds | # Min-cut in graph with demands/lower bounds
This week I read something about network flow from Algorithm Design. But I am confused about some concepts.
We say, if a graph G contains some nodes with demands, positive or negative, how to define the min cut for these graphs? If we still use the definition for simple graph, the value of max flow is not equal to the capacity of min cut. Or is it totally meaningless to talk about min-cut in a network with demands on some nodes.
Another thing is how to define the capacity of a min-cut for graph with lower bound on its edges. Still the sum of capacities of edges out of starting set? Or say, now we have a lower bound for this cut which is the sum of lower bounds of edges out of starting set. | {"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": 0.8419310450553894, "perplexity": 368.6065170317702}, "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/1575540543850.90/warc/CC-MAIN-20191212130009-20191212154009-00280.warc.gz"} |
http://stats.stackexchange.com/questions/26668/subset-models-in-auto-arima-function-in-forecast-package | # Subset models in auto.arima function in forecast package
I wanted to ask whether it was possible to use the auto.arima function to identify subset ARIMA models rather than those of pure lags? I have identified a model in Stata in subset lags that performs well and wanted to cross check this with the auto.arima() function but I can't seem to figure out if subset lags are supported.
-
No, auto.arima() does not allow for subset models. The FitAR package will automatically do subset AR modelling. I don't know of any R package that does automatic subset ARMA modelling. You can do manual subsetting, of course, with the arima() command in R by setting some coefficients to zero using the fixed argument -- see the help file for details. | {"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": 0.8442469835281372, "perplexity": 1404.6166712281674}, "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-2013-48/segments/1386164903523/warc/CC-MAIN-20131204134823-00068-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://link.springer.com/article/10.1023%2FB%3AJOCE.0000038345.99050.c0 | Journal of Oceanography
, Volume 60, Issue 3, pp 519–530
# Development of a Neural Network Algorithm for Retrieving Concentrations of Chlorophyll, Suspended Matter and Yellow Substance from Radiance Data of the Ocean Color and Temperature Scanner
• Akihiko Tanaka
• Motoaki Kishino
• Roland Doerffer
• Helmut Schiller
• Tomohiko Oishi
Article
DOI: 10.1023/B:JOCE.0000038345.99050.c0
Tanaka, A., Kishino, M., Doerffer, R. et al. Journal of Oceanography (2004) 60: 519. doi:10.1023/B:JOCE.0000038345.99050.c0
## Abstract
An algorithm is presented to retrieve the concentrations of chlorophyll a, suspended pariclulate matter and yellow substance from normalized water-leaving radiances of the Ocean Color and Temperature Sensor (OCTS) of the Advanced Earth Observing Satellite (ADEOS). It is based on a neural network (NN) algorithm, which is used for the rapid inversion of a radiative transfer procedure with the goal of retrieving not only the concentrations of chlorophyll a but also the two other components that determine the water-leaving radiance spectrum. The NN algorithm was tested using the NASA's SeaBAM (SeaWiFS Bio-Optical Mini-Workshop) test data set and applied to ADEOS/OCTS data of the Northwest Pacific in the region off Sanriku, Japan. The root-mean-square error between chlorophyll a concentrations derived from the SeaBAM reflectance data and the chlorophyll a measurements is 0.62. The retrieved chlorophyll a concentrations of the OCTS data were compared with the corresponding distribution obtained by the standard OCTS algorithm. The concentrations and distribution patterns from both algorithms match for open ocean areas. Since there are no standard OCTS products available for yellow substance and suspended matter and no in situ measurements available for validation, the result of the retrieval by the NN for these two variables could only be assessed by a general knowledge of their concentrations and distribution patterns.
Neural networkinverse modellingocean color algorithmchlorophyll asuspended matteryellow substanceADEOS/OCTS
© The Oceanographic Society of Japan 2004
## Authors and Affiliations
• Akihiko Tanaka
• 1
• Motoaki Kishino
• 2
• Roland Doerffer
• 3
• Helmut Schiller
• 3
• Tomohiko Oishi
• 4 | {"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": 0.8556005358695984, "perplexity": 4626.368663031391}, "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/1476988720153.61/warc/CC-MAIN-20161020183840-00081-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/49545-derivative-problem-print.html | # Derivative problem
• Sep 17th 2008, 06:34 PM
NotEinstein
Derivative problem
$sqrt((cot^2(x))+x^2)$
My attempt-
$sqrt(2cot(x) * (-csc^2x) +2x)
$
Here I believe I have two options.. not sure which one to go with if either..
$sqrt(2cot(x) * (-cot^2x - 1) +2x)
$
or
$sqrt(2cot(x) * (1/sin^2x) +2x) = sqrt(2cosx/sinx * (1/sin^2x)+2x)$
just trying to get it to simplest terms.
• Sep 17th 2008, 06:44 PM
skeeter
do you know how the chain rule works?
$\frac{d}{dx} \left[(\cot^2{x} + x^2)^{\frac{1}{2}}\right]$
$\frac{1}{2}(\cot^2{x} + x^2)^{-\frac{1}{2}} \cdot (-2\cot{x}\csc^2{x} + 2x)$
I'll leave the very little bit of possible simplification to you.
• Sep 17th 2008, 06:48 PM
javax
$
(\sqrt{\cot^2 x+x^2}) = \frac{(\cot^2 x)'+(x^2)'}{2\sqrt{\cot^2 x+x^2}} = \frac{-2\cot x \csc^2 x + 2x}{2\sqrt{\cot^2 x+x^2}} = \frac{x-\cot x \csc^2 x}{\sqrt{\cot^2 x+x^2}}
$
I don't think it can get any simplier.
• Sep 17th 2008, 06:49 PM
javax
Quote:
Originally Posted by skeeter
do you know how the chain rule works?
$\frac{d}{dx} \left[(\cot^2{x} + x^2)^{\frac{1}{2}}\right]$
$\frac{1}{2}(\cot^2{x} + x^2)^{-\frac{1}{2}} \cdot (-2\cot{x}\csc^2{x} + 2x)$
I'll leave the very little bit of possible simplification to you.
sorry I didn't see your post when I started to post (Lipssealed)
• Sep 17th 2008, 06:50 PM
NotEinstein
Quote:
Originally Posted by skeeter
do you know how the chain rule works?
$\frac{d}{dx} \left[(\cot^2{x} + x^2)^{\frac{1}{2}}\right]$
$\frac{1}{2}(\cot^2{x} + x^2)^{-\frac{1}{2}} \cdot (-2\cot{x}\csc^2{x} + 2x)$
I'll leave the very little bit of possible simplification to you.
just learned it yesterday.. guess this is good practice. only question. why after you took the derivative of the outermost term, did you put * (-2cotx ...)? i understand that you brought the 2 to the front using the power rule.. but wouldn't it stay positive?
EDIT: Nevermind, i understand now. Thank you
• Sep 17th 2008, 06:56 PM
skeeter
chain rule inside the chain rule ...
the derivative of $\cot^2{x}$ is
$2\cot{x} \cdot (-csc^2{x}) = -2\cot^2{x}\csc^2{x}$
• Sep 17th 2008, 07:39 PM
john doe
I think you made a mistake you squared the cot | {"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": 13, "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": 0.963111937046051, "perplexity": 3052.3607938291575}, "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-2017-51/segments/1512948511435.4/warc/CC-MAIN-20171210235516-20171211015516-00212.warc.gz"} |
https://physics.stackexchange.com/questions/363117/noethers-theorem-with-infinite-parameters | # Noether's theorem with infinite parameters
I'm trying to understand something regarding Noether's theorem - and with the given situation, my question isn't that much of a question, I'm rather just seeking confirmation whether I'm thinking right or not.
The situation:
Let $\mathcal L$ be a Lagrangian density, depending on some field $\phi$, and its first derivative. Noether's theorem (naively) says that if $\phi(x)\mapsto\phi(x)+\epsilon\delta\phi(x)$ is a specific infinitesimal deformation of the field (a more precise thing would be to say that this is a smooth 1-parameter family of finite deformations - and we're interested in behaviours under $d/d\epsilon|_{\epsilon=0}$), such that $\mathcal L$ changes by a divergence ($\delta\mathcal L=\partial_\mu K^\mu$), then the current $$j^\mu=\frac{\partial\mathcal L}{\partial(\partial_\mu\phi)}\delta\phi-K^\mu$$is conserved on-shell.
It is known that this can be recast in a different form, by making $\epsilon$ be a function instead of a parameter. Then the action won't be invariant in general, but the deformation of the action gives the same current $j^\mu$, and its conservation can be shown.
The problem is that this doesn't make sense, imo. To show this, consider the following, let $\epsilon(x)$ be the "infinitesimal" functional parameter, the variation is $$\phi(x)\mapsto\phi(x)+\epsilon(x)\delta\phi(x).$$ Let us define $\epsilon'(x)$ and $\epsilon$ as $\epsilon(x)=\epsilon\epsilon'(x)$, where here only $\epsilon$ is "infinitesimal". Now the variation has the form $$\phi(x)\mapsto\phi(x)+\epsilon\epsilon'(x)\delta\phi(x).$$ Now we redefine $\delta\phi(x)$ to $\delta\phi'(x)=\epsilon(x)\delta\phi(x)$, then the variation has the form $$\phi(x)\mapsto\phi(x)+\epsilon\delta\phi'(x).$$
This is literally the same form we had before we assumed $\epsilon$ is a function.
So this begs the question - what do we mean on an "infinite-parameter" variation? The problem is clearly caused by the fact, that if $\delta\phi(x)$ is specific, but reasonably arbitrary, then this still contains as many "free parameters" as the different possible values for $x$. Essentially, $\delta\phi(x)$ already contains infinite parameters.
The resolution:
Looking at specific examples, such as an $U(1)$ transformation of the free, massive, complex Klein-Gordon field, the finite transformation is $$\phi(x)\mapsto e^{i\epsilon}\phi(x).$$ Infinitesimally, this is $$\phi(x)\mapsto \phi(x)+i\epsilon\phi(x),$$ so $$\delta\phi(x)=i\phi(x).$$
Here we see, that $\delta\phi(x)$ depends on $x$ only through the unperturbed field $\phi(x)$ itself, so here the variation is truly 1-parameter.
If we do this for another archetypical example - spacetime translations - we get the same results.
The question:
Am I right in saying that the usual form of the Noether's theorem should be stated that we consider variations of the form $$\phi(x)\mapsto\phi(x)+\epsilon\delta\phi[\phi(x),\partial\phi(x)],$$ where $\delta\phi$ is a specific function of the field $\phi$, and possibly, its derivatives, but not the coordinates $x$?
Because only then does it makes any sense to me to discuss whether the variation has finite or infinite amount of parameters.
In Noether's first theorem, can the infinitesimal variation depend explicitly on the spacetime point $x^{\mu}$?
Example: Lagrangian formulation. Consider the Lagrangian $$L~:=~T-V ,\qquad T~:=~\frac{m}{2}\dot{q}^2,\qquad V~:=~\frac{\alpha}{q^2}. \tag{L1}$$ where $\alpha$ is a constant. It is easy to check that $$\delta q ~=~ \varepsilon (q-2t\dot{q}), \qquad \delta t ~=~0, \tag{L2}$$ is an infinitesimal quasi-symmetry $$\delta L ~=~\ldots ~=~\varepsilon\frac{dk^{0}}{dt}, \qquad k^0~:=~-2t L . \tag{L3}$$ Here $\varepsilon$ is a constant infinitesimal parameter. The bare Noether charge is $$Q^0~=~m\dot{q}( q-2t\dot{q}). \tag{L4}$$ The full Noether charge $$Q~:=~Q^0-k^0~=~mq\dot{q} -2t(T+V) \tag{L5}$$ is a conserved quantity.
Example: Hamiltonian formulation. Consider the Hamiltonian Lagrangian $$L_H~=~p\dot{q}-H, \qquad H~:=~\frac{p^2}{2m}+\frac{\alpha}{q^2}, \tag{H1}$$ where $\alpha$ is a constant. It is easy to check that $$\delta q ~=~ \varepsilon \left( q-\frac{2t}{m}p\right), \qquad \delta p ~=~- \varepsilon \left( p+\frac{4\alpha t}{q^3}\right), \qquad \delta t ~=~0, \tag{H2}$$ is an infinitesimal quasi-symmetry $$\delta L_H ~=~\ldots ~=~\varepsilon\frac{dk^{0}}{dt}, \qquad k^0~:=~2t\left(\frac{\alpha}{q^2} -\frac{p^2}{2m}\right) . \tag{H3}$$ Here $\varepsilon$ is a constant infinitesimal parameter. The bare Noether charge is $$Q^0~=~p\left( q-\frac{2t}{m}p\right). \tag{H4}$$ The full Noether charge $$Q~:=~Q^0-k^0~=~qp -2tH \tag{H5}$$ is a conserved quantity. This example is further discussed in this Phys.SE post. | {"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": 0.9894636869430542, "perplexity": 121.48759294232968}, "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-2019-43/segments/1570987823061.83/warc/CC-MAIN-20191022182744-20191022210244-00494.warc.gz"} |
http://www-old.newton.ac.uk/programmes/MAA/seminars/2005071209001.html | # MAA
## Seminar
### A common generalisation of the conjectures of Andre-Oort, Manin-Mumford and Mordell-Lang
Pink, R (ETH Zentrum)
Tuesday 12 July 2005, 09:00-10:00
Seminar Room 1, Newton Institute
#### Abstract
The formal similiarity between the Manin-Mumford and Mordell-Lang conjectures on the one hand, and the Andre-Oort conjecture on the other hand, suggests that a common generalization should exist for subvarieties of mixed Shimura varieties. We propose such a conjecture, explain why it implies all the stated conjectures, and explain its relation with existing results. | {"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": 0.8849184513092041, "perplexity": 1817.5935014453512}, "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-2015-18/segments/1429246649738.26/warc/CC-MAIN-20150417045729-00245-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://www.fz-juelich.de/iek/iek-8/EN/Expertise/Infrastructure/SAPHIR/Concept/Concept_node.html | Search
Institute of Energy and Climate Research
SAPHIR - Concept
The large atmospheric simulation chamber SAPHIR is a tool for the quantitative experimental investigation of tropospheric chemistry under natural condition and the evaluation of photo chemical models. The ultimate goal is the experimental verification of the chemistry of tropospheric trace compounds.
Evaluation of atmospheric photo chemistry
• What are the precursors and physical parameters determining the OH concentration?
• What is the influence of chemical and physical parameters on the lifetime of trace gases?
• Which factors control the formation of ozone and other secondary pollutants?
Methodology
Investigation of tropospheric chemistry under natural burdens of trace compounds. However, independent of time varying perturbations like small scale turbulent transport and unquantified local sources and sinks as they are inevitably present in the atmosphere near the ground. Specifically:
• De-coupling of chemical reactions and natural variability
• independent variation of trace gas concentrations in large range
• measuring the settling timing of radical species after pertubation
What is new?
• Simulation of tropospheric photo chemistry under natural conditions.
• direct measurement of the most reactive trace gas (OH) simultanously with a complete chemical and physical characterisation of the air
• direct comparison of field study and simulation experiment
• international platform for intercomparisons
Objectives
SAPHIR provides a platform for quantitative, experimental verification and improvement of photo chemical models at realistic concentrations and solar radiation. | {"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": 0.9399619102478027, "perplexity": 4420.404012880951}, "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-2020-40/segments/1600400213006.47/warc/CC-MAIN-20200924002749-20200924032749-00064.warc.gz"} |
https://didgaha.wordpress.com/2014/02/09/more-on-latex-in-wordpress/ | # More on Latex in WordPress.
While ago, I wrote a post on how to use a simple latex command into the WordPress. Unfortunately, this method is very limited and you can’t certainly include complex latex codes into your blog this way. Particularly, writing colorful texts, with equations, and numbering is a nightmare in the WordPress, unless you start writing in HTML.
Today, I came across this wonderful post on how to convert latex into something useable in WordPress. There is this free software written in Python called Latex2wp, which converts latex to HTML. I would like to thank the person who wrote this program. I have complied the example file included in the package, and copy and paste the HTML code into the WordPress editor. It is a rather long text, but you can see the part of it below. To appreciate the important of this software, just try to write first two lines below yourself, left alone equation numbering, etc.
Look at the document source to see how to strike out text, how to use different colors, and how to link to URLs with snapshot preview and how to link to URLs without snapshot preview.
There is a command which is ignored by pdflatex and which defines where to cut the post in the version displayed on the main page
Anything between the conditional declarations ifblog . . . fi is ignored by LaTeX and processed by latex2wp. Anything between iftex . . . fi is processed by LaTex and ignored by latex2wp.
This green sentence appears only in WordPress
This is useful if one, in desperation, wants to put pure HTML commands in the ifblog . . . fi scope.
Lemma 1 (Main) Let ${\cal F}$ be a total ramification of a compactifier, then
$\displaystyle \forall g \in {\cal F}. g^2 = \eta \ \ \ \ \ (1)$
…. | {"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": 2, "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": 0.9429930448532104, "perplexity": 1672.7704794958063}, "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-2017-26/segments/1498128321938.75/warc/CC-MAIN-20170627221726-20170628001726-00277.warc.gz"} |
http://math.stackexchange.com/questions/266535/a-problem-related-to-the-sdr?answertab=active | # A problem related to the SDR
Let $A_1, A_2, \ldots, A_{20}$ be twenty sets each of size $20$ such that $|A_i \cap A_j | \le 2$. Prove that they have a system of distinct representatives.
-
What is your progress so far? – rschwieb Dec 28 '12 at 16:26
Are you sure each of the $A_i$s have size 20, rather than their union has size 20? – Henning Makholm Dec 28 '12 at 16:37
## 1 Answer
Use Hall's Theorem. It says there exists a system of distinct representatives iff for any subset of the indices, $\{a_1,\ldots,a_k\}$, we have $|\displaystyle\cup_{j=1}^kA_{a_j}|\geq k$. It can be easily checked that your system satisfies this condition: For contradiction, suppose it does not. So, there exists a subset $\{b_1,\ldots,b_l\}$ such that $|\displaystyle\cup_{j=1}^l A_{b_j}|< l \leq 20$. But this is a contradiction since each subset in your system has size $20$.
I'm sure there are simpler solutions without using Hall's theorem for this particular problem, since the conditions are very strong. Comment: The problem is not correct if only the union of the subsets has size 20. A counter example is the system defined by $A_1=\{1,\ldots,20\}$, $A_i=\{1\}$ for i>1.
-
I almost understand your answer. But I see you didn't use the condition $|A_i \cap A_j| \le 2$ at all? – user53541 Jan 2 '13 at 16:15
That's right. The problem is still correct without that condition. I also have a simple way without using Hall's Theorem which proves it. Let me know if you want to know about that. – afshi7n Jan 2 '13 at 17:27
I guess that is a counting, right? Could you tell me your way.. – user53541 Jan 2 '13 at 18:21
Try to select a representative from each set, you will have enough elements in each set to avoid conflicts! – afshi7n Jan 3 '13 at 20:00 | {"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": 0.95770263671875, "perplexity": 184.8702983057622}, "config": {"markdown_headings": true, "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-2016-07/segments/1454701168011.94/warc/CC-MAIN-20160205193928-00328-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://waseda.pure.elsevier.com/en/publications/a-bayesian-inversion-framework-to-evaluate-parameter-and-predicti | # A Bayesian inversion framework to evaluate parameter and predictive inference of a simple soil respiration model in a cool-temperate forest in western Japan
Motomu Toda, Kazuki Doi, Masae I. Ishihara, Wakana A. Azuma, Masayuki Yokozawa
Research output: Contribution to journalArticlepeer-review
1 Citation (Scopus)
## Abstract
Careful modelling of soil carbon sequestration is essential to evaluate future terrestrial feedback to the earth climate system through atmosphere–surface carbon exchange. Few studies have evaluated, in bio- and geo-applications, parameter and predictive uncertainty of soil respiration models by considering the difference between observations and model predictions; i.e. residual error, which is assumed neither to be independent nor to be described by a normal (i.e. Gaussian) probability distribution with a mean of zero and constant variance. In this paper, we use 2-year observations of soil carbon flux from 2017 to 2018 (hereafter referred to as ‘long-term simulation’) obtained with two open-top chambers to estimate parameter and predictive uncertainty of a simple soil respiration model based on Bayesian statistics in a cool-temperate forest in western Japan. We also use a Gaussian innovative residual error model in which a generalised likelihood uncertainty estimation that accounts for correlated, heteroscedastic, non-normally distributed (i.e. non-Gaussian) residual error flexibly handles statistics varying in skewness and kurtosis. Results show that the effects of correlation and heteroscedasticity were eliminated adequately. Additionally, the posterior distribution of the residuals had a pattern intermediate to those of Gaussian and Laplacian (or double-exponential) distributions. Consequently, the predicted soil respiration rate, and range of uncertainty therein, well-matched the observational data. Furthermore, we compare results of parameter and predictive inference of the soil respiration model from the long-term simulation with those constrained of short-term simulations (i.e. 4-month subsets of the 2-year dataset) to determine the extent to which the approach used affects the estimation of parameter and predictive uncertainty. No significant difference in parameter estimates was found between the long-term simulation versus any of the short-term simulations, whereas short-term simulation analysis of the uncertainty at 50 %—i.e. between the lower (25 %) and upper (75 %) quartiles of the probability range—indicated distinctive variations in model parameters in summer when more vigorous activity of trees and organisms promotes carbon cycling between the atmosphere and ecosystem. Overall we demonstrate that the Bayesian inversion approach is useful as a means by which to evaluate effectively parameter and predictive uncertainty of a soil respiration model with precise representation of residual errors.
Original language English 108918 Ecological Modelling 418 https://doi.org/10.1016/j.ecolmodel.2019.108918 Published - 2020 Feb 15
## Keywords
• Bayesian statistics
• Data-model fusion
• DREAM algorithm
• Generalised likelihood
• Soil carbon flux
• Uncertainty
## ASJC Scopus subject areas
• Ecological Modelling | {"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": 0.861808180809021, "perplexity": 4063.8232159993736}, "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-2021-04/segments/1610703500028.5/warc/CC-MAIN-20210116044418-20210116074418-00276.warc.gz"} |
http://mphitchman.com/geometry/chapter-4.html | $\require{cancel}\newcommand{\nin}{} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&}$
We now have both the space ($\mathbb{C}^+$) and the transformations (Möbius transformations), and are just about ready to embark on non-Euclidean adventures. Before doing so, however, one more phrase needs defining: group of transformations. This phrase has a precise meaning. Not every collection of transformations is lucky enough to form a group. | {"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": 0.8658478260040283, "perplexity": 638.6142086847324}, "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-2018-22/segments/1526794866511.32/warc/CC-MAIN-20180524151157-20180524171157-00220.warc.gz"} |
https://exploringnumbertheory.wordpress.com/tag/chinese-remainder-theorem/page/2/ | # Introducing Carmichael numbers
This is an introduction to Carmichael numbers. We first discuss Carmichael numbers in the context of Fermat primality test and then discuss several basic properties. We also prove Korselt’s criterion, which gives a useful characterization of Carmichael numbers.
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Fermat Primality Test
Fermat’s little theorem states that if $p$ is a prime number, then $a^p \equiv a \ (\text{mod} \ p)$ for any integer $a$. Fermat primality test refers to the process of using Fermat little theorem to check the “prime vs. composite” status of an integer.
Suppose that we have a positive integer $n$ such that the “prime vs. composite” status is not known. If we can find an integer $a$ such that $a^n \not \equiv a \ (\text{mod} \ n)$, then we know for certain that the modulus $n$ is composite (or not prime). For example, let $n = \text{8,134,619}$. Note that $2^{8134619} \equiv 3024172 \ (\text{mod} \ 8134619)$. So we know right away that $n = \text{8,134,619}$ is not prime, even though we do not know what its prime factors are just from applying this test.
Given a positive integer $n$, whenever $a^n \not \equiv a \ (\text{mod} \ n)$, we say that $a$ is a Fermat witness for (the compositeness of) the integer $n$. Thus $2$ is a Fermat witness for $n = \text{8,134,619}$.
What if we try one value of $a$ and find that $a$ is not a witness for (the compositeness of) $n$? Then the test is inconclusive. The best we can say is that $n$ is probably prime. It makes sense to try more values of $a$. If all the values of $a$ we try are not witnesses for $n$ (i.e. $a^n \equiv a \ (\text{mod} \ n)$ for all the values of $a$ we try), then it “seems likely” that $n$ is prime. But if we actually declare that $n$ is prime, the decision could be wrong!
Take $n=\text{10,024,561}$. For several randomly chosen values of $a$, we have the following calculations:
$\displaystyle 5055996^{10024561} \equiv 5055996 \ (\text{mod} \ 10024561)$
$\displaystyle 4388786^{10024561} \equiv 4388786 \ (\text{mod} \ 10024561)$
$\displaystyle 4589768^{10024561} \equiv 4589768 \ (\text{mod} \ 10024561)$
$\displaystyle 146255^{10024561} \equiv 146255 \ (\text{mod} \ 10024561)$
$\displaystyle 6047524^{10024561} \equiv 6047524 \ (\text{mod} \ 10024561)$
The above calculations could certainly be taken as encouraging signs that $n=\text{10,024,561}$ is prime. With more values of $a$, we also find that $a^{10024561} \equiv a \ (\text{mod} \ 10024561)$. However, if we declare that $n=\text{10,024,561}$ is prime, it turns out to be a wrong conclusion.
In reality, $n=\text{10,024,561}$ is composite with $\text{10,024,561}=71 \cdot 271 \cdot 521$. Furthermore $a^{10024561} \equiv a \ (\text{mod} \ 10024561)$ for any integer $a$. So there are no witnesses for $n=\text{10,024,561}$. Any composite positive integer that has no Fermat witnesses is called a Carmichael number, in honor of Robert Carmichael who in 1910 found the smallest such number, which is 561.
Fermat primality test is always correct if the conclusion is that the integer being tested is a composite number (assuming there is no computational error). If the test says the number is composite, then it must be a composite number. In other words, there are no false negatives in using Fermat primality test as described above.
On the other hand, there can be false positives as a result of using Fermat primality test. If the conclusion is that the integer being tested is a prime number, it is possible that the conclusion is wrong. For a wrong conclusion, it could be that there exists a witness for the number being tested and that we have missed it. Or it could be that the number being tested is a Carmichael number. Though Carmichael numbers are rare but there are infinitely many of them. So we cannot ignore them entirely. For these reasons, Fermat primality test as described above is often not used. Instead, other extensions of the Fermat primality test are used.
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Carmichael Numbers
As indicated above, a Carmichael number is a positive composite integer that has no Fermat witnesses. Specifically, it is a positive composite integer that satisfies the conclusion of Fermat’s little theorem. In other words, a Carmichael number is a positive composite integer $n$ such that $a^n \equiv a \ (\text{mod} \ n)$ for any integer $a$.
Carmichael numbers are rare. A recent search found that there are $\text{20,138,200}$ Carmichael numbers between $1$ and $10^{21}$, about one in 50 trillion numbers (documented in this Wikipedia entry on Carmichael numbers). However it was proven by Alford, Granville and Pomerance in 1994 that there are infinitely many Carmichael numbers (paper).
The smallest Carmichael number is $561=3 \cdot 11 \cdot 17$. A small listing of Carmichael numbers can be found in this link, where the example of $n=\text{10,024,561}$ is found.
Carmichael numbers must be odd integers. To see this, suppose $n$ is a Carmichael number and is even. Let $a=-1$. By condition (1) of Theorem 1, we have $(-1)^n=1 \equiv -1 \ (\text{mod} \ n)$. On the other hand, $-1 \equiv n-1 \ (\text{mod} \ n)$. Thus $n-1 \equiv 1 \ (\text{mod} \ n)$. Thus we have $n \equiv 2 \ (\text{mod} \ n)$. It must be the case that $n=2$, contradicting the fact that $n$ is a composite number. So any Carmichael must be odd.
The following theorem provides more insight about Carmichael numbers. A positive integer $n$ is squarefree if its prime decomposition contains no repeated prime factors. In other words, the integer $n$ is squarefree means that if $\displaystyle n=p_1^{e_1} p_2^{e_2} \cdots p_t^{e_t}$ is the prime factorization of $n$, then all exponents $e_j=1$.
Theorem 1 (Korselt’s Criterion)
Let $n$ be a positive composite integer. Then the following conditions are equivalent.
1. The condition $a^n \equiv a \ (\text{mod} \ n)$ holds for any integer $a$.
2. The condition $a^{n-1} \equiv 1 \ (\text{mod} \ n)$ holds for any integer $a$ that is relatively prime to $n$.
3. The integer $n$ is squarefree and $p-1 \ \lvert \ (n-1)$ for any prime divisor $p$ of $n$.
Proof of Theorem 1
$1 \Longrightarrow 2$
Suppose that $a$ is relatively prime to the modulus $n$. Then let $b$ be the multiplicative inverse of $a$ modulo $n$, i.e., $ab \equiv 1 \ (\text{mod} \ n)$. By (1), we have $a^n \equiv a \ (\text{mod} \ n)$. Multiply both sides by the multiplicative inverse $b$, we have $a^{n-1} \equiv 1 \ (\text{mod} \ n)$.
$2 \Longrightarrow 3$
Let $\displaystyle n=p_1^{e_1} p_2^{e_2} \cdots p_t^{e_t}$ be the prime factorization of $n$ where $p_i \ne p_j$ for $i \ne j$ and each exponent $e_j \ge 1$. Since $n$ must be odd, each $p_j$ must be an odd prime.
We first show that each $e_j=1$, thus showing that $n$ is squarefree. To this end, for each $j$, let $a_j$ be a primitive root modulo $p_j^{e_j}$ (see Theorem 4 in the post Primitive roots of powers of odd primes). Consider the following system of linear congruence equations:
$x \equiv a_1 \ (\text{mod} \ p_1^{e_1})$
$x \equiv a_2 \ (\text{mod} \ p_2^{e_2})$
$\cdots$
$\cdots$
$\cdots$
$x \equiv a_t \ (\text{mod} \ p_t^{e_t})$
Since the moduli $p_j^{e_j}$ are pairwise relatively prime, this system must have a solution according to the Chinese Remainder Theorem (a proof is found here). Let $a$ one such solution. For each $j$, since $a_j$ is a primitive root modulo $p_j^{e_j}$, $a_j$ is relatively prime to $p_j^{e_j}$. Since $a \equiv a_j \ (\text{mod} \ p_j^{e_j})$, $a$ is relatively prime to $p_j^{e_j}$ for each $j$. Consequently, $a$ is relatively prime to $n$. By assumption (2), we have $a^{n-1} \equiv 1 \ (\text{mod} \ n)$.
Now fix a $j$ with $1 \le j \le t$. We show that $e_j=1$. Since $a^{n-1} \equiv 1 \ (\text{mod} \ n)$, $a^{n-1} \equiv 1 \ (\text{mod} \ p_j^{e_j})$. Since $a \equiv a_j \ (\text{mod} \ p_j^{e_j})$, we have $a_j^{n-1} \equiv 1 \ (\text{mod} \ p_j^{e_j})$. Note that the order of $a_j$ modulo $p_j^{e_j}$ is $\phi(p_j^{e_j})=p_j^{e_j-1}(p_j-1)$. Thus we have $p_j^{e_j-1}(p_j-1) \ \lvert \ (n-1)$. If $e_j>1$, then $p_j \ \lvert \ (n-1)$, which would mean that $p_j \ \lvert \ 1$. So it must be the case that $e_j=1$. It then follows that $(p_j-1) \ \lvert \ (n-1)$.
$3 \Longrightarrow 1$
Suppose that $n=p_1 p_2 \cdots p_t$ is a product of distinct prime numbers such that for each $j$, $(p_j-1) \ \lvert \ (n-1)$.
Let $a$ be any integer. First we show that $a^n \equiv a \ (\text{mod} \ p_j)$ for all $j$. It then follows that $a^n \equiv a \ (\text{mod} \ n)$.
Now fix a $j$ with $1 \le j \le t$. First consider the case that $a$ and $p_j$ are relatively prime. According to Fermat’s little theorem, $a^{p_j-1} \equiv 1 \ (\text{mod} \ p_j)$. Since $(p_j-1) \ \lvert \ (n-1)$, $a^{n-1} \equiv 1 \ (\text{mod} \ p_j)$. By the Chinese Remainder Theorem, it follows that $a^{n-1} \equiv 1 \ (\text{mod} \ n)$ and $a^n \equiv a \ (\text{mod} \ n)$. $\blacksquare$
Examples
With Korselt’s criterion, it is easy to verify Carmichael numbers as long as the numbers are factored. For example, the smallest Carmichael number is $561=3 \cdot 11 \cdot 17$. The number is obviously squarefree. furthermore $560$ is divisible by $2$, $10$ and $16$.
The number $\text{10,024,561}= 71 \cdot 271 \cdot 521$ is discussed above. We can also verify that this is a Carmichael number: $70 \ \lvert \ \text{10,024,560}$, $270 \ \lvert \ \text{10,024,560}$ and $520 \ \lvert \ \text{10,024,560}$.
Here’s three more Carmichael numbers (found here):
$\text{23,382,529} = 97 \cdot 193 \cdot 1249$
$\text{403,043,257} = 19 \cdot 37 \cdot 43 \cdot 67 \cdot 199$
$\text{154,037,320,009} = 23 \cdot 173 \cdot 1327 \cdot 29173$
We end the post by pointing out one more property of Carmichael numbers, that Carmichael numbers must have at least three distinct prime factors. To see this, suppose that $n=p \cdot q$ is a Carmichael number with two distinct prime factors $p$ and $q$. We can express $n-1$ as follows:
$n-1=pq-1=(p-1)q+q-1$
Since $n$ is Carmichael, $p-1 \ \lvert \ (n-1)$. So $n-1=(p-1)w$ for some integer $w$. Plugging this into the above equation, we see that $p-1 \ \lvert \ (q-1)$. By symmetry, we can also show that $q-1 \ \lvert \ (p-1)$. Thus $p=q$, a contradiction. So any Carmichael must have at least three prime factors.
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$\copyright \ 2013 \text{ by Dan Ma}$
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# The primitive root theorem
The primitive root theorem identifies all the positive integers for which primitive roots exist. The list of such integers is a restrictive list. This post along with two previous posts give a complete proof of this theorem using only elementary number theory. We prove the following theorem.
Main Theorem (The Primitive Root Theorem)
There exists a primitive root modulo $m$ if and only if $m=2$, $m=4$, $m=p^t$ or $m=2p^t$ where $p$ is an odd prime number and $t$ is a positive integer.
The theorem essentially gives a list of the moduli that have primitive roots. Any modulus outside this restrictive list does not have primitive roots. For example, any integer that is a product of two odd prime factors is not on this list and hence has no primitive roots. In the post Primitive roots of powers of odd primes, we show that the powers of an odd prime have primitive roots. In the post Primitive roots of twice the powers of odd primes, we show that the moduli that are twice the power of an odd prime have primitive roots. It is easy to verify that the moduli $2$ and $4$ have primitive roots. Thus the direction $\Longleftarrow$ of the primitive root theorem has been established. In this post we prove the direction $\Longrightarrow$, showing that if there exists a primitive root modulo $p$, then $p$ must be one of the moduli in the list stated in the theorem.
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LCM
The proof below makes use of the notion of the least common multiple. Let $a$ and $b$ be positive integers. The least common multiple of $a$ and $b$ is denoted by $\text{LCM}(a,b)$ and is defined as the least positive integer that is divisible by both $a$ and $b$. For example, $\text{LCM}(16,18)=144$. We can also express $\text{LCM}(a,b)$ as follows:
$\displaystyle \text{LCM}(a,b)=\frac{a \cdot b}{\text{GCD}(a,b)} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$
where $\text{GCD}(a,b)$ is the greatest common divisor of $a$ and $b$. The above formula reduces the calculation of LCM to that of calculating the GCD. To compute the LCM of two numbers, we can simply remove the common prime factors between the two numbers. When the number $a$ and $b$ are relatively prime, i.e., $\text{GCD}(a,b)=1$, we have $\displaystyle \text{LCM}(a,b)=a \cdot b$.
Another way to look at LCM is that it is the product of multiplying together the highest power of each prime number. For example, $48=2^4 \cdot 3$ and $18=2 \cdot 3^2$. Then $\text{LCM}(16,18)=2^4 \cdot 3^2=144$.
The least common divisor of the numbers $a_1,a_2,\cdots,a_n$ is denoted by $\text{LCM}(a_1,a_2,\cdots,a_n)$ and is defined similarly. It is the least positive integer that is divisible by all $a_j$. Since the product of all the numbers $a_j$ is one integer that is divisible by each $a_k$, we have:
$\displaystyle \text{LCM}(a_1,a_2,\cdots,a_n) \le a_1 \cdot a_2 \cdots a_n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (2)$
As in the case of two numbers, the LCM of more than two numbers can be thought as the product of multiplying together the highest power of each prime number. For example, the LCM of $48=2^4 \cdot 3$, $18=2 \cdot 3^2$ and $45=3^2 \cdot 5$ is $2^4 \cdot 3^2 \cdot 5=720$.
For a special case, there is a simple expression of LCM.
Lemma 1
Let $a_1,a_2,\cdots,a_n$ be positive integers.
Then $\displaystyle \text{LCM}(a_1,a_2,\cdots,a_n)=a_1 \cdot a_2 \cdots a_n$ if and only if the numbers $a_1,a_2,\cdots,a_n$ are pairwise relatively prime, i.e., $a_i$ and $a_j$ are relatively prime whenever $i \ne j$.
Proof of Lemma 1
$\Longleftarrow$
Suppose the numbers are pairwise relatively prime. Then there are no common prime factors in common between any two numbers on the list. Then multiplying together the highest power of each prime factor is the same as multiplying the individual numbers $a_1,a_2 \cdots,a_n$.
$\Longrightarrow$
Suppose $a_i$ and $a_j$ are not relatively prime for some $i \ne j$. As a result, $d=\text{GCD}(a_i,a_j)>1$. It follows that
$\displaystyle \text{LCM}(a_1,a_2,\cdots,a_n) \le \frac{a_1 \cdot a_2 \cdots a_n}{d} < a_1 \cdot a_2,\cdots a_n \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (3)$
To give a sense of why the above is true, let’s look at a simple case of $d=\text{GCD}(a_i,a_j)=p^u$ where $p$ is a prime number and $u \ge 1$. Assume that $a_i=a_1$, $a_j=a_2$ and $p^u$ is part of the prime factorization of $a_1$. Furthermore, note that $\displaystyle \text{LCM}(\frac{a_1}{p^u},a_2,\cdots,a_n)$ is identical to $\displaystyle \text{LCM}(a_1,a_2,\cdots,a_n)$. The following derivation confirms (3):
$\displaystyle \text{LCM}(a_1,a_2,\cdots,a_n)=\text{LCM}(\frac{a_1}{p^u},a_2,\cdots,a_n) \le \frac{a_1}{p^u} \cdot a_2 \cdots a_n < a_1 \cdot a_2,\cdots a_n$
With the above clarification, the lemma is established. $\blacksquare$
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Other Tools
We need to two more lemmas to help us prove the main theorem.
Lemma 2
Let $m$ and $n$ be positive integers that are relatively prime. Then $a \equiv b \ (\text{mod} \ m)$ and $a \equiv b \ (\text{mod} \ n)$ if and only if $a \equiv b \ (\text{mod} \ mn)$.
Lemma 3
The number $a$ is a primitive root modulo $m$ if and only if $\displaystyle a^{\frac{\phi(m)}{q}} \not \equiv 1 \ (\text{mod} \ m)$ for all prime divisors $q$ of $\phi(m)$.
Lemma 2 a version of the Chinese Remainder Theorem and is proved a previous post (see Theorem 2 in Primitive roots of twice the powers of odd primes or see Theorem 2 in Proving Chinese Remainder Theorem). Lemma 3 is also proved in a previous post (see Lemma 2 in More about checking for primitive roots).
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Breaking It Up Into Smaller Pieces
The proof of the direction $\Longrightarrow$ of the primitive root theorem is done in the following lemmas and theorems.
Lemma 4
Let $\displaystyle m=p_1^{e_1} p_2^{e_2} \cdots p_t^{e_t}$ be the prime factorization of the positive integer $m$. Let $a$ be a primitive root modulo $m$. Then the numbers $\phi(p_1^{e_1}), \phi(p_2^{e_2}),\cdots,\phi(p_t^{e_t})$ are pairwise relatively prime.
Proof of Lemma 4
Note that $a$ is relatively prime to $m$. So $a$ is relatively prime to each $p_j^{e_j}$. By Euler’s theorem, we have $\displaystyle a^{\phi(p_j^{e_j})} \equiv 1 \ (\text{mod} \ p_j^{e_j})$ for each $j$. Let $\displaystyle W=\text{LCM}(\phi(p_1^{e_1}),\phi(p_2^{e_2}),\cdots,\phi(p_t^{e_t}))$.
By definition of LCM, $\phi(p_j^{e_j}) \ \lvert \ W$ for each $j$. So $\displaystyle a^{W} \equiv 1 \ (\text{mod} \ p_j^{e_j})$ for each $j$. By the Chinese remainder theorem (Lemma 2 above), $\displaystyle a^{W} \equiv 1 \ (\text{mod} \ m)$. Since $a$ is a primitive root modulo $m$, it must be that $\phi(m) \le W$. Interestingly, we have:
$\displaystyle \phi(p_1^{e_1}) \phi(p_2^{e_2}) \cdots \phi(p_t^{e_t})=\phi(m) \le W \le \phi(p_1^{e_1}) \phi(p_2^{e_2}) \cdots \phi(p_t^{e_t})$
Thus $\displaystyle \text{LCM}(\phi(p_1^{e_1}),\phi(p_2^{e_2}),\cdots,\phi(p_t^{e_t}))=\phi(p_1^{e_1}) \phi(p_2^{e_2}) \cdots \phi(p_t^{e_t})$. By Lemma 1, the numbers $\phi(p_j^{e_j})$ are relatively prime. $\blacksquare$
The following theorems follow from Lemma 4. The main theorem is a corollary of these theorems.
Theorem 5
If there exists a primitive root modulo $m$, then $m$ cannot have two distinct prime divisors.
Proof of Theorem 5
Let $\displaystyle m=p_1^{e_1} p_2^{e_2} \cdots p_t^{e_t}$ be the prime factorization of $m$ where $t \ge 2$.
If $p_i$ and $p_j$ are odd prime with $i \ne j$, then $\phi(p_i^{e_i})=p_i^{e_i-1}(p_i-1)$ and $\phi(p_j^{e_j})=p_j^{e_j-1}(p_j-1)$ are both even and thus not relatively prime. If there exists a primitive root modulo $m$, $\phi(p_i^{e_i})$ and $\phi(p_j^{e_j})$ must be relatively prime (see Lemma 4). Since we assume that there exists a primitive root modulo $m$, $m$ cannot have two distinct odd prime divisors. $\blacksquare$
Theorem 6
Suppose that there exists a primitive root modulo $m$ and that $m$ has exactly one odd prime factor $p$. Then $m$ must be of the form $p^e$ or $2p^e$ where $e \ge 1$.
Proof of Theorem 6
By Theorem 5, the prime factorization of $m$ must be $m=2^{e_1} p^{e_2}$ where $e_1 \ge 0$ and $e_2 \ge 1$.
We claim that $e_1=0$ or $e_1=1$. Suppose $e_1 \ge 2$. Then $\phi(2^{e_1})=2^{e_1-1}$ and $\phi(p^{e_2})=p^{e_2-1}(p-1)$ are both even and thus not relatively prime. Lemma 4 tells us that there does not exist a primitive root modulo $m$. So if there exists a primitive root modulo $m$, then it must be the case that $e_1=0$ or $e_2=1$.
If $e_1=0$, then $m=p^{e_2}$. If $e_1=1$, then $m=2p^{e_2}$. $\blacksquare$
Lemma 7
Let $n=2^k$ where $k \ge 3$. Then $\displaystyle a^{\frac{\phi(n)}{2}} \equiv 1 \ (\text{mod} \ n)$ for any $a$ that is relatively prime to $n$.
Proof of Lemma 7
We prove this by induction on $k$. Let $k=3$. Then $n=8$ and $\displaystyle \frac{\phi(8)}{2}=2$. For any odd $a$ with $1 \le a <8$, it can be shown that $a^2 \equiv 1 \ (\text{mod} \ 8)$.
Suppose that the lemma holds for $k$ where $k \ge 3$. We show that it holds for $k+1$. Note that $\phi(2^k)=2^{k-1}$ and $\displaystyle \frac{\phi(2^k)}{2}=2^{k-2}$. Since the lemma holds for $k$, we have $\displaystyle v^{2^{k-2}} \equiv 1 \ (\text{mod} \ 2^k)$ for any $v$ that is relatively prime to $2^k$. We can translate this congruence into the equation $v^{2^{k-2}}=1+2^k y$ for some integer $y$.
Note that $\phi(2^{k+1})=2^{k}$ and $\displaystyle \frac{\phi(2^{k+1})}{2}=2^{k-1}$. It is also the case that $(v^{2^{k-2}})^2=v^{2^{k-1}}$. Thus we have:
$\displaystyle v^{2^{k-1}}=(1+2^k y)^2=1+2^{k+1} y+2^{2k} y^2=1+2^{k+1}(y+2^{k-1} y^2)$
The above derivation shows that $\displaystyle v^{\frac{\phi(2^{k+1})}{2}} \equiv 1 \ (\text{mod} \ 2^{k+1})$ for any $v$ that is relatively prime to $2^k$.
On the other hand, $a$ is relatively prime to $2^{k+1}$ if and only if $a$ is relatively prime to $2^k$. So $\displaystyle a^{\frac{\phi(2^{k+1})}{2}} \equiv 1 \ (\text{mod} \ 2^{k+1})$ for any $a$ that is relatively prime to $2^{k+1}$. Thus the lemma is established. $\blacksquare$
Theorem 8
Suppose that there exists a primitive root modulo $m$ and that $m=2^e$ where $e \ge 1$. Then $m=2^e$ where $e=1$ or $e=2$.
Proof of Theorem 8
Suppose $m=2^e$ where $e \ge 3$. By Lemma 7, $\displaystyle a^{\frac{\phi(m)}{2}} \equiv 1 \ (\text{mod} \ n)$ for any $a$ relatively prime to $m$. Since $2$ is the only prime divisor of $m$, by Lemma 3, there cannot be primitive root modulo $m$. Thus if there exists a primitive root modulo $m$ and that $m=2^e$ where $e \ge 1$, then the exponent $e$ can be at most $2$. $\blacksquare$
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Putting It All Together
We now put all the pieces together to prove the $\Longrightarrow$ of the Main Theorem. It is a matter of invoking the above theorems.
Proof of Main Theorem
Suppose that there exists a primitive root modulo $m$. Consider the following three cases about the modulus $m$.
1. $m$ has no odd prime divisor.
2. $m$ has exactly one odd prime divisor.
3. $m$ has at least two odd prime divisors.
$\text{ }$
Suppose Case 1 is true. Then $m=2^e$ where $e \ge 1$. By Theorem 8, $m=2$ or $m=4$.
Suppose Case 2 is true. Then Theorem 6 indicates that $m$ must be the power of an odd prime or twice the power of an odd prime.
Theorem 5 indicates that Case 3 is never true. Thus the direction $\Longrightarrow$ of the Main Theorem is proved. $\blacksquare$
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$\copyright \ 2013 \text{ by Dan Ma}$
# Primitive roots of twice the powers of odd primes
In a previous post, we show that there exist primitive roots modulo the power of an odd prime number (see Primitive roots of powers of odd primes). In this post we show that there exist primitive roots modulo two times the power of an odd prime number. Specifically we prove the following theorem.
Theorem 1
Let $p$ be an odd prime number. Let $j$ be any positive integer. Then there exist primitive roots modulo $p^j$.
We make use of the Chinese Remainder Theorem (CRT) in proving Theorem 1. We use the following version of CRT (also found in this post)
Theorem 2 (CRT)
Let $m$ and $n$ be positive integers that are relatively prime. Then $a \equiv b \ (\text{mod} \ m)$ and $a \equiv b \ (\text{mod} \ n)$ if and only if $a \equiv b \ (\text{mod} \ mn)$.
Proof of Theorem 2
$\Longrightarrow$
Suppose $a \equiv b \ (\text{mod} \ m)$ and $a \equiv b \ (\text{mod} \ n)$. Converting these into equations, we have $a=b+mx$ and $a=b+ny$ for some integers $x$ and $y$. It follows that $mx=ny$. This implies that $m \ \lvert \ ny$. Since $m$ and $n$ and relatively prime, $m \ \lvert \ y$ and $y=mt$ for some integer $t$. Now the equation $a=b+ny$ can be written as $a=b+mnt$, which implies that $a \equiv b \ (\text{mod} \ mn)$.
$\Longleftarrow$
Suppose $a \equiv b \ (\text{mod} \ mn)$. Then $a=b+mns$ for some integer $s$, which implies both congruences $a \equiv b \ (\text{mod} \ m)$ and $a \equiv b \ (\text{mod} \ n)$. $\blacksquare$
Proof of Theorem 1
Let $a$ be a primitive root modulo $p^j$ (shown to exist in the post Primitive roots of powers of odd primes). When $a$ is odd, we show that $a$ is a primitive root modulo $2p^j$. When $a$ is even, we show that $a+p^j$ is a primitive root modulo $2p^j$.
First the odd case. Since $a$ is a primitive root modulo $p^j$, $a^k \not \equiv 1 \ (\text{mod} \ p^j)$ for all positive $k<\phi(p^j)$. Since $a$ is odd, $a^k$ is odd for all integers $k \ge 1$. So $a^k \equiv 1 \ (\text{mod} \ 2)$ for all integers $k \ge 1$. By CRT (Theorem 2), $a^k \not \equiv 1 \ (\text{mod} \ 2p^j)$ for all positive $k<\phi(p^j)=\phi(2 p^j)$. This implies that $a$ is a primitive root modulo $2p^j$.
Now the even case. Note that $a+p^j$ is odd (even + odd is odd). It is also the case that $(a+p^j)^k$ is odd for all $k \ge 1$. Thus $(a+p^j)^k \equiv 1 \ (\text{mod} \ 2)$ for all $k \ge 1$.
In expanding $(a+p^j)^k$ using the binomial theorem, all terms except the first term $a^k$ is divisible by $p^j$. So $(a+p^j)^k \equiv a^k \ (\text{mod} \ p^j)$. Furthermore, $a^k \not \equiv 1 \ (\text{mod} \ p^j)$ for all positive $k<\phi(p^j)$ since $a$ is a primitive root modulo $p^j$. So $(a+p^j)^k \not \equiv 1 \ (\text{mod} \ p^j)$ for all positive $k<\phi(p^j)$.
By CRT (Theorem 2), we have $(a+p^j)^k \not \equiv 1 \ (\text{mod} \ 2p^j)$ for all positive $k<\phi(p^j)=\phi(2p^j)$. This implies that $a+p^j$ is a primitive root modulo $2p^j$. $\blacksquare$
The remainder of the proof of the primitive root theorem is found in The primitive root theorem.
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$\copyright \ 2013 \text{ by Dan Ma}$ | {"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": 467, "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": 0.9881381392478943, "perplexity": 61.68885699251573}, "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-2019-18/segments/1555578602767.67/warc/CC-MAIN-20190423114901-20190423140901-00090.warc.gz"} |
https://brilliant.org/problems/find-if-you-can/ | # How Does One Move?
A man is standing on a frictionless surface and there is no air resistance. If he wants to move, which of Newton's laws would he be using? Also Try my Easy Mechanics Set Easy Mechanics
× | {"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": 0.9204683303833008, "perplexity": 614.2806790992641}, "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-2021-17/segments/1618038088731.42/warc/CC-MAIN-20210416065116-20210416095116-00050.warc.gz"} |
http://twoqubits.wikidot.com/ex6-5 | Ex6 5
$\def\abs#1{|#1|}\def\i{\mathbf {i}}\def\ket#1{|{#1}\rangle}\def\bra#1{\langle{#1}|}\def\braket#1#2{\langle{#1}|{#2}\rangle}\def\tr{\mathord{\mbox{tr}}}\mathbf{Exercise\ 6.5}$
Show how to construct an efficient reversible circuit for every classical circuit along the lines of the construction of section 6.2.2 but without the assumption that $t$ is a power of $2$. Give the time and space bounds for your construction. | {"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": 0.965344250202179, "perplexity": 185.97843355527704}, "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/1505818687906.71/warc/CC-MAIN-20170921205832-20170921225832-00239.warc.gz"} |
https://www.toppr.com/guides/chemistry-formulas/chemical-reaction-formula/ | # Chemical Reaction Formula
## Chemical Reaction
Chemical Reaction Formula- it contains the reorganizing of atoms of the same or different elements to form new substances. Generally, we represent it with a chemical equation in which the reactants (substances that are broken apart) are written on the left and the products (a new substance that form) are written on the right. Furthermore, it the resulting product or the reactant are more than two then we separate them with a + sign. Moreover, we use an arrow (→) to separate the reactant side of the equation with the product side.
### Common Symbol
There are certain symbols that we use to distinguish between different types of reactions. These symbols are:
We use the ‘=’ symbol to denote a stoichiometric reaction.
The arrow ‘→’ symbol denotes a net forward reaction.
The right-left arrow symbol denotes a reaction in both directions.
Also, we use $$\rightleftharpoons$$ symbol to denote an equilibrium.
We can also state the physical state of chemical in parentheses after the chemical symbol, especially for ionic reactions. Furthermore, when stating the physical state the letter ‘s’ denotes solid, ‘l’ denotes a liquid, and ‘g’ symbolizes a gas, and ‘aq’ denotes an aqueous solution.
However, if the reaction requires energy then we indicate it above the arrow. We use the capital letter delta ($$\Delta$$) which we put on the reaction arrow to show that energy in the form of heat is added to the reaction. We use $$h\nu$$ if we add energy in the form of light. Moreover, we use other specific symbols for other types of energy or radiation.
### Balancing a chemical equation
The law of conservation of mass states that the amount of elements does not change in a chemical reaction, which means each side of the chemical equation must represent the same quantity of any particular element. In the same way, the chemical reaction conserves the charge. So, both sides of the equation must present a balanced equation.
We can balance the simple equation by inspection, i.e. by trial and error. In addition, another technique involves solving a system of linear equations. Furthermore, the balanced equation is written with the smallest whole-number coefficients. Most noteworthy, if there is no coefficient then the coefficient is 1.
For example: we can balance the burning of methane by putting a coefficient 1 before CH4:
1 CH4 + O2 → CO2 + H2O
As both sides of the arrow have one carbon atom so the equation is balanced.
### Ionic equations
These are chemical equations in which we can write electrolytes as distant ions. Moreover, we use ionic reactions for single and double displacement reactions that occur in aqueous solutions.
For example: The following precipitation reaction:
$$CaCl_{2} + 2AgNO_{3} \rightarrow CA(NO_{3})_{2} + 2AgCl\downarrow$$
Besides, the full ionic equation is:
$$Ca^{2+} + 2Cl^{-} + 2Ag^{+} + 2NO_{3}^{-} \rightarrow Ca^{2+} + 2NO_{3}^{-} +2AgCl \downarrow$$
However, in this reaction, the $$Ca^{2+}$$ and $$NO_{3}^{-}$$ ions remain in solution and are not part of the reaction. That means both these ions are identical in the reactant side and the product side of the chemical equation. This happens because these ions do not participate in this reaction, and are referred to as spectator ions. Also, a net ionic equation is the full ionic equation from which we have distant the spectator ions. Furthermore, the net ionic equation of the proceeding reaction is:
$$2Cl_{-} + 2Ag^{+} \rightarrow 2AgCl \downarrow$$
Or in reduced balanced form it is:
$$Cl_{-} + Ag^{+} \rightarrow AgCl \downarrow$$
## Solved Example For You
Question: Show the reaction of methane (CH4) and oxygen (O2) and their products?
Solution: When methane reacts with oxygen it produces carbon dioxide and water. The reaction is:
$$CH^{4} + 2O^{2} → CO^{2} + 2H^{2}O$$
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## Question Mark?
Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps. | {"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": 0.9075741171836853, "perplexity": 1326.319211124146}, "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/1634323585045.2/warc/CC-MAIN-20211016231019-20211017021019-00095.warc.gz"} |
https://proxieslive.com/if-intlimits_gamma-fzdzinbbb-r-does-it-imply-fzin-bbb-rforall-zingamma/ | # If $\int\limits_\gamma f(z)dz\in\Bbb R$ does it imply $f(z)\in \Bbb R~\forall z\in\gamma$?
If $$\int\limits_\gamma f(z)dz\in\Bbb R$$ does it imply $$f(z)\in \Bbb R~\forall z\in\gamma$$?
This is from a proof where $$\gamma$$ is a circle and we have $$1={1\over 2\pi}\int\limits_\gamma f(z)dz={1\over 2\pi}\int\limits_\gamma Re~f(z)dz$$ and I’m unsure how the second equality is justified. | {"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": 4, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9857470989227295, "perplexity": 79.98906492526925}, "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-2021-21/segments/1620243991904.6/warc/CC-MAIN-20210511060441-20210511090441-00182.warc.gz"} |
https://www.physicsforums.com/threads/find-the-vector-potentials.196948/ | # Find the vector potentials
1. Nov 8, 2007
### DieCommie
1. The problem statement, all variables and given/known data
A magnetic field of a long straight wire carrying a current I along the z-axis is given by the following expression:
$$\mathbf{B} = \frac{\mu_0I}{2\pi} \{\frac{-y}{x^2+y^2} \hat{x} + \frac{y}{x^2 + y^2} \hat{y} \}$$
Find two different potentials that will yield this field. Show explicitly that the curl of the difference between these two potentials vanishes.
2. Relevant equations
$$\nabla \times \mathbf{A} = \mathbf{B}$$
3. The attempt at a solution
I took a cross product to get this system...
$$\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y} = 0 \\$$
$$\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z} = \frac{-y}{x^2+y^2} \\$$
$$\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x} = \frac{y}{x^2+y^2} \\$$
I dont know what to do! Any ideas?
Last edited: Nov 8, 2007
2. Nov 11, 2007
### DieCommie
Maybe nobody knows how to do this problem...
But does anybody know how to, in general, find a vector 'A' given its cross product 'B'? | {"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": 0.8762885332107544, "perplexity": 645.7061896318677}, "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/1480698542217.43/warc/CC-MAIN-20161202170902-00049-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://calculator.academy/impact-force-calculator/ | # Impact Force Calculator
Enter the total deformation distance and the spring constant of the collision to determine the impact force of an object, such as a car crash.
## What is an impact force?
Impact force is a term used in physics, specifically in mechanics, to describe a force of a short period of time when objects collide. A high force applied to an object in a short amount of time results in a very quick acceleration. This fast acceleration is what cause much of the damage in collisions. Whether that is a car or a persons body part.
If this force as to be applied of a long period of time, the acceleration would be much smaller and the resulting damage would be much less. As a results, things like cars are designed to increase the amount of time the force is applied. In other words, cars are meant to crumple and deform because a slower acceleration, or in the case of a car crash, deceleration, is safer for a human body to endure.
## Impact Force Formula
The following formula is typically used to calculate and impact force.
W = 1/2 Fmax s = 1/2 k s2
• where
• W = work done (J, ft lb)
• Fmax = maximum force at the end of the deformation (N, lbf)
• s = deformation distance (m, ft)
• k = spring constant
As can be seen by the formula above, the work done is directly proportional to the spring constant and exponentially related to the deformation. For this reason cars are meant to deform as much as possible during crashes.
## How to calculate an impact force?
The following example is a step by step guide on how to calculate the impact force of a collision between two cars.
1. First, we must analyze the equation above to determine the missing variables and the correct formula. After re-arranging some variables, we find that the max force = k * s where k is the spring constant and s is the displacement.
2. We will assume for this problem that the car is colliding with a wall that has no deformation. In this case the deformation is equal to 1 meter.
3. Next, we need to determine the spring constant. The car is clearly not a traditional spring, but a front on collision can be studied very well to get an accurate representation of the spring constant of the car. For this example we will assume that the data collected by the car manufacturer shows a spring constant of 10,000 N/m.
4. Finally, and the information into the formula above and we find F = 10,000 N/m * 1 m = 10,000 N.
5. One last final step could be determine the resulting acceleration a human body feels during that time frame to determine the extent of injuries
More physics calculators | {"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": 0.8404489755630493, "perplexity": 386.76555552685966}, "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/1590347399830.24/warc/CC-MAIN-20200528170840-20200528200840-00394.warc.gz"} |
https://www.scribd.com/doc/150548979/Bash | # Bash Reference Manual
Reference Documentation for Bash Edition 4.1, for Bash Version 4.1. December 2009
Chet Ramey, Case Western Reserve University Brian Fox, Free Software Foundation
This text is a brief description of the features that are present in the Bash shell (version 4.1, 23 December 2009). This is Edition 4.1, last updated 23 December 2009, of The GNU Bash Reference Manual, for Bash, Version 4.1. Copyright c 1988–2009 Free Software Foundation, Inc. Permission is granted to make and distribute verbatim copies of this manual provided the copyright notice and this permission notice are preserved on all copies. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover texts being “A GNU Manual”, and with the Back-Cover Texts as in (a) below. A copy of the license is included in the section entitled “GNU Free Documentation License”. (a) The FSF’s Back-Cover Text is: You are free to copy and modify this GNU manual. Buying copies from GNU Press supports the FSF in developing GNU and promoting software freedom.” Published by the Free Software Foundation 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
i
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 1.2 What is Bash? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 What is a shell? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 3
Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Basic Shell Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1 Shell Syntax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.1 Shell Operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.1.2 Quoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2.1 Escape Character . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2.2 Single Quotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2.3 Double Quotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2.4 ANSI-C Quoting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3.1.2.5 Locale-Specific Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1.3 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Shell Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2.1 Simple Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.2 Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.3 Lists of Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2.4 Compound Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.4.1 Looping Constructs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2.4.2 Conditional Constructs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3.2.4.3 Grouping Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2.5 Coprocesses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 Shell Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.4 Shell Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.4.1 Positional Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4.2 Special Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.5 Shell Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5.1 Brace Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5.2 Tilde Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5.3 Shell Parameter Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5.4 Command Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.5.5 Arithmetic Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5.6 Process Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5.7 Word Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.5.8 Filename Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5.8.1 Pattern Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.5.9 Quote Removal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 Redirections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6.1 Redirecting Input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 The Directory Stack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Bash Conditional Expressions . . . . . . . . . . . . . . . . . . .ii Bash Reference Manual 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Command Search and Execution . . . . . . . . . . . . . . . . . .2 6. 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Signals . . . . . . . . . . .1 Simple Command Expansion . . . . . . . .8 Duplicating File Descriptors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . 4. . . . . . .6. . . . . . . . . . . . . . . .10 The Restricted Shell . . . .1 What is an Interactive Shell? . . . . . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . .9 Moving File Descriptors . . . . . . . . . . 6. . . 3. . . . . . . . . . . . . 3. . . . . . . 3. . . . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. .3. .7. . . . . . . . . . . . . . . .3. . . . . . . . . . . . . . . . . . . . . . . . .6. .7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Redirecting Output . . . . . . . .7 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Invoking Bash . . . . . . . . . . . . . . . . . . . . .2 Bourne Shell Variables . . . . . . . . . . . . . 4.2 The Shopt Builtin . . . . . . . . . . . . 6. . . . . . . . . . . . . . . . . .1 5. . . . . . . . . . . . . . . . .8. . . . . . 3. . . . .3 Command Execution Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interactive Shells . . . . . . . . . . . . . . 35 41 51 51 55 59 5 Shell Variables . . . . . . . . . . 27 27 27 28 28 28 28 29 29 29 29 30 30 31 32 32 33 4 Shell Builtin Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .7. . . . . . . . . . . . . . .9 Controlling the Prompt . . . . . . . . . . . . . . . . . . . . . . .6 Here Documents .4 Environment . . . . . . . . . 4. . . . . . . . . .11 Bash POSIX Mode .1 Directory Stack Builtins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 73 75 75 75 75 76 78 79 80 81 81 82 84 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 Opening File Descriptors for Reading and Writing . . . . . . . . . . . . . . . . . . . . . . . . 6. . . . . . . . . .5 Appending Standard Output and Standard Error . . . . . . . . 71 6. . . . . . . . . . . .3 Bourne Shell Builtins . . . . . . . . . . 3. . . . . . . . . . . . . . 6. . . . . . . . . . . . . .3 Interactive Shell Behavior . . . . . . . . . . . . . . . . . . . . . 3. . . . . . . . . . . .3. . . . . . . . . . . . . .7. . . . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3. . 3. . . . . . . . . . 61 5. . . . . . . . . . . . . 3. . . . . . . . . . 3.7 Here Strings . . . 6. . . . 6. . . . . . . . . .5 Shell Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 4. . . . . . . . . . . .6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modifying Shell Behavior . . .6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bash Startup Files . . . .4 Redirecting Standard Output and Standard Error . . . . . . . . . . .8 Shell Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 6. . . . . . . . . .7. . . . . . . . . . .6. . . . . . . . . . . . . 3. . . . . . . . . . . . .1 The Set Builtin . . . . . . . . . . . . . . . . . . 6. . . . .7 Executing Commands .6. . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . .3 Appending Redirected Output . . 3. . . . . . . . . . 35 4. . . . . . . . . . . . . . . . . .6. . . . . . . . 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Special Builtins . . . . . . . . . . . . .1 4. . . . . . . . . . . . . .6. . . . .6 Aliases . . . . . . . . . . . . . . . . 61 6 Bash Features . . . . . . 61 Bash Variables . . . . . . . . . . . . . 3. . . . . . .5 Exit Status . . . . . . . . . . . . . . . . . . . . . . .2 Is this Shell Interactive? . . . . . . . .7. . . . . . . . . . . . . Bash Builtin Commands . . . . . . 3. . . . . . . . . . . . . . 6. . . . . . . . . . . . . .
2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Readline vi Mode . . . 106 8. . 95 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compilers and Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 10. . . . . . . . . . . . . 121 9. . . . . . . 96 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3 Commands For Changing Text . . .3 Bash History Facilities . . . . . . . 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Programmable Completion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8. . . . . . . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 9 Using History Interactively . . . . . . . 93 8. . . . . . . . . . . . . . . . . . . . . . . . . .5 10. 121 121 123 123 124 125 10 Installing Bash . . . . . . . . . . . . . . . . . . . . . .1 Commands For Moving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2. . . . . .2 Introduction to Line Editing . . . . Sharing Defaults . . Optional Features . . . . . . 93 Readline Interaction . . . . . Operation Controls . . . . . . . . . . . . . . . . . . . 110 8. . . . . . . . . . . . . . . . . . . .6 10. . . . . . . . . . . . . . . . . . . . . .4 Readline Arguments . . . . . . . . . . . . 127 128 128 128 128 129 129 129 10. . . . . . . . . . . . . . . . . . . .7 10. . . . . . . . . . . . . . .1 Readline Bare Essentials . . . . . . . . . . 96 8. Bash History Builtins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 8. . . . . . . . . .4. 109 8. . . . . . . . .4 Bindable Readline Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Readline Movement Commands . . . . . . . . . . . . . . . . . . . 93 8. . . . . .1 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 8. . . . . . . . . . . . . . . 115 8. .3. . . . . . . .2 7. . . . . . . . . . . .1 9. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 8. . . .3. . . . .4. . .1 Event Designators . . . . . . . . . 95 8. . . . .4. . . . . . . . . . . . . . . . . . . . . . . .1 Readline Init File Syntax . . . . . . . . . . 90 Job Control Variables . . . . . . 94 8. . . . . . . . . . . . . . . . . . .2. . . . . . .2 Word Designators . . . . . . . . . . . . . .4. . . . . . . . . . . . . . . .3. . . . . . . . . . . . . . .3 Readline Init File . . . . . . . . . . .2 9. . . . . . . . . . . . . . . . . . . . . . . . 103 8. . . . . .3 Modifiers . . . . . .3 Readline Killing Commands . . . . . . . . . . . .iii 7 Job Control . . . . .2. . . . . . 92 8 Command Line Editing . . . . . . . . . . . . . . . . . Installation Names . . 114 8. . .3 Sample Init File . . . . . . . .2 10. . . . . . . . . 94 8. . . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 8. . . . . . . . . . 107 8. .8 Some Miscellaneous Commands . . . . . . . . . . .1 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. . . . . . . . . . . . . . . . .4 Killing And Yanking . . Compiling For Multiple Architectures . . . . . . . . . . . . . .7 Programmable Completion Builtins . . . . . . . . . . . . . . . . . . .4. . . . . . . . . . . . . . . . . . .3 Job Control Basics . . . . . . . . . . . . . . 127 Basic Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 Letting Readline Type For You . .5 Searching for Commands in the History . . . . . . . .3 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2 Commands For Manipulating The History . . .2 Conditional Init Constructs . . . . . . . . . . . . . . . . . . . . . . . . . . 110 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Job Control Builtins .1 8. . .5 Specifying Numeric Arguments . .3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifying the System Type . . . . . . 89 7. . 108 8. . . . .8 . . . . . . .2. . . . . . . . . History Expansion . . . . .7 Keyboard Macros . . . . . . . . . . . . . . . . . 9. . . . . . . . . . . . . . 9.
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Appendix A
Reporting Bugs . . . . . . . . . . . . . . . . . 135
Appendix B Major Differences From The Bourne Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
B.1 Implementation Differences From The SVR4.2 Shell . . . . . . . . . . 141
Appendix C GNU Free Documentation License . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Appendix D
D.1 D.2 D.3 D.4 D.5
Indexes . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
151 152 152 154 156
Index of Shell Builtin Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index of Shell Reserved Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parameter and Variable Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter 1: Introduction
1
1 Introduction
1.1 What is Bash?
Bash is the shell, or command language interpreter, for the gnu operating system. The name is an acronym for the ‘Bourne-Again SHell’, a pun on Stephen Bourne, the author of the direct ancestor of the current Unix shell sh, which appeared in the Seventh Edition Bell Labs Research version of Unix. Bash is largely compatible with sh and incorporates useful features from the Korn shell ksh and the C shell csh. It is intended to be a conformant implementation of the ieee posix Shell and Tools portion of the ieee posix specification (ieee Standard 1003.1). It offers functional improvements over sh for both interactive and programming use. While the gnu operating system provides other shells, including a version of csh, Bash is the default shell. Like other gnu software, Bash is quite portable. It currently runs on nearly every version of Unix and a few other operating systems − independently-supported ports exist for ms-dos, os/2, and Windows platforms.
1.2 What is a shell?
At its base, a shell is simply a macro processor that executes commands. The term macro processor means functionality where text and symbols are expanded to create larger expressions. A Unix shell is both a command interpreter and a programming language. As a command interpreter, the shell provides the user interface to the rich set of gnu utilities. The programming language features allow these utilities to be combined. Files containing commands can be created, and become commands themselves. These new commands have the same status as system commands in directories such as ‘/bin’, allowing users or groups to establish custom environments to automate their common tasks. Shells may be used interactively or non-interactively. In interactive mode, they accept input typed from the keyboard. When executing non-interactively, shells execute commands read from a file. A shell allows execution of gnu commands, both synchronously and asynchronously. The shell waits for synchronous commands to complete before accepting more input; asynchronous commands continue to execute in parallel with the shell while it reads and executes additional commands. The redirection constructs permit fine-grained control of the input and output of those commands. Moreover, the shell allows control over the contents of commands’ environments. Shells also provide a small set of built-in commands (builtins ) implementing functionality impossible or inconvenient to obtain via separate utilities. For example, cd, break, continue, and exec) cannot be implemented outside of the shell because they directly manipulate the shell itself. The history, getopts, kill, or pwd builtins, among others, could be implemented in separate utilities, but they are more convenient to use as builtin commands. All of the shell builtins are described in subsequent sections. While executing commands is essential, most of the power (and complexity) of shells is due to their embedded programming languages. Like any high-level language, the shell provides variables, flow control constructs, quoting, and functions.
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Shells offer features geared specifically for interactive use rather than to augment the programming language. These interactive features include job control, command line editing, command history and aliases. Each of these features is described in this manual.
A string of characters used to identify a file. POSIX blank builtin A family of open system standards based on Unix. and any processes descended from it. and underscores. The value is restricted to eight bits. numbers. Also referred to as an identifier. name A word consisting solely of letters.1 standard. ‘&&’. . operator process group A collection of related processes each having the same process group id. After expansion. Most reserved words introduce shell flow control constructs. ‘. Bash is primarily concerned with the Shell and Utilities portion of the posix 1003. ‘(’. separates words. exit status The value returned by a command to its caller. ‘<’.’.6 [Redirections]. ‘|&’. filename job job control A mechanism by which users can selectively stop (suspend) and restart (resume) execution of processes. the resulting fields are used as the command name and arguments. or ‘>’. A command that is implemented internally by the shell itself. A metacharacter is a blank or one of the following characters: ‘|’. so the maximum value is 255. metacharacter A character that. ‘&’. when unquoted. It is a newline or one of the following: ‘||’. page 26. ‘|’.. such as for and while. that are all in the same process group. ‘&’. A set of processes comprising a pipeline.’. A space or tab character. ‘. control operator A token that performs a control function. process group ID A unique identifier that represents a process group during its lifetime. Operators contain at least one unquoted metacharacter. or ‘)’. ‘)’. See Section 3.’. when executing a command. rather than by an executable program somewhere in the file system. reserved word A word that has a special meaning to the shell. for a list of redirection operators. ‘. A control operator or a redirection operator. field A unit of text that is the result of one of the shell expansions.Chapter 2: Definitions 3 2 Definitions These definitions are used throughout the remainder of this manual. ‘(’. Names are used as shell variable and function names. and beginning with a letter or underscore.
special builtin A shell builtin command that has been classified as special by the posix standard. token word A sequence of characters considered a single unit by the shell.4 Bash Reference Manual return status A synonym for exit status. Words may not include unquoted metacharacters. signal A mechanism by which a process may be notified by the kernel of an event occurring in the system. . A sequence of characters treated as a unit by the shell. It is either a word or an operator.
8 [Filename Expansion]. shell parameters. 6.7. 3. Alias expansion is performed by this step (see Section 6. Performs any necessary redirections (see Section 3. roughly speaking. page 26) and removes the redirection operators and their operands from the argument list.6 [Redirections].5. page 33). expands others.6 [Aliases]. which are a way to direct input and output from and to named files. This chapter briefly summarizes the shell’s ‘building blocks’: commands. The rules for evaluation and quoting are taken from the posix specification for the ‘standard’ Unix shell.1. page 6. the shell reads its input and divides the input into words and operators. and how the shell executes commands. page 29). shell expansions.Chapter 3: Basic Shell Features 5 3 Basic Shell Features Bash is an acronym for ‘Bourne-Again SHell’.2 [Shell Commands]. page 24) and commands and arguments. If the input indicates the beginning of a comment. breaking the expanded tokens into lists of filenames (see Section 3. .1 [Invoking Bash].1. Performs the various shell expansions (see Section 3.5 [Shell Expansions]. and the rest of that line. obeying the quoting rules described in Section 3. The shell then parses these tokens into commands and other constructs.1 Shell Operation The following is a brief description of the shell’s operation when it reads and executes a command. The Bourne shell is the traditional Unix shell originally written by Stephen Bourne. from a string supplied as an argument to the ‘-c’ invocation option (see Section 6. the shell ignores the comment symbol (‘#’). 4. control structures.7 [Executing Commands]. it proceeds through a sequence of operations. Otherwise. executes the specified command. shell functions. 3. Reads its input from a file (see Section 3. redirections. 2. Breaks the input into words and operators. All of the Bourne shell builtin commands are available in Bash. Basically. and makes that exit status available for further inspection or processing.8 [Shell Scripts]. page 7). waits for the command’s exit status. page 32).1 Shell Syntax When the shell reads input. Parses the tokens into simple and compound commands (see Section 3. 7. redirects input and output as needed. employing the quoting rules to select which meanings to assign various words and characters. 3.5 [Exit Status]. or from the user’s terminal. 5. page 79). Optionally waits for the command to complete and collects its exit status (see Section 3. These tokens are separated by metacharacters. Executes the command (see Section 3. page 71). removes the special meaning of certain words or characters. page 17).2 [Quoting]. the shell does the following: 1.
1 [Bash History Facilities]. The backslash retains its special meaning only when followed by one of the following characters: ‘\$’. and the backslash itself is not quoted. or newline. with the exception of ‘\$’.1. even when preceded by a backslash.6 Bash Reference Manual 3. Quoting can be used to disable special treatment for special characters. 3.5 [Shell Expansions]. and double quotes. ‘\’. ‘"’.1 Escape Character A non-quoted backslash ‘\’ is the Bash escape character.2. Backslash escape sequences. The special parameters ‘*’ and ‘@’ have special meaning when in double quotes (see Section 3. ‘!’.3 [Shell Parameter Expansion].2 Quoting Quoting is used to remove the special meaning of certain characters or words to the shell.1. and to prevent parameter expansion. ‘‘’. There are three quoting mechanisms: the escape character. must be quoted to prevent history expansion.2 Single Quotes Enclosing characters in single quotes (‘’’) preserves the literal value of each character within the quotes. The characters ‘\$’ and ‘‘’ retain their special meaning within double quotes (see Section 3. The word expands to string. with the exception of newline. backslashes that are followed by one of these characters are removed. with backslash-escaped characters replaced as specified by the ANSI C standard.2.1.3 [History Interaction]. if present. Within double quotes. it is removed from the input stream and effectively ignored). and. are decoded as follows: \a \b \e \E alert (bell) backspace an escape character (not ANSI C) . If a \newline pair appears. for more details concerning history expansion. usually ‘!’. Backslashes preceding characters without a special meaning are left unmodified. when history expansion is enabled. to prevent reserved words from being recognized as such. page 3) has special meaning to the shell and must be quoted if it is to represent itself.5.1.3 Double Quotes Enclosing characters in double quotes (‘"’) preserves the literal value of all characters within the quotes. page 121. It preserves the literal value of the next character that follows.2. page 123). Each of the shell metacharacters (see Chapter 2 [Definitions].1. If enabled.4 ANSI-C Quoting Words of the form \$’string’ are treated specially. The backslash preceding the ‘!’ is not removed. page 17). 3. page 19). the history expansion character. ‘‘’. ‘\’.2. the \newline is treated as a line continuation (that is. When the command history expansion facilities are being used (see Section 9. 3. See Section 9. 3. A double quote may be quoted within double quotes by preceding it with a backslash. history expansion will be performed unless an ‘!’ appearing in double quotes is escaped using a backslash. A single quote may not occur between single quotes. single quotes.
2 [The Shopt Builtin]. More complex shell commands are composed of simple commands arranged together in a variety of ways: in a pipeline in which the output of one command becomes the input of a second.1. See Section 6. An interactive shell without the interactive_comments option enabled does not allow comments. Some systems use the message catalog selected by the LC_MESSAGES shell variable. possibly adding a suffix of ‘. the replacement is double-quoted.3. as if the dollar sign had not been present. page 75. for a description of what makes a shell interactive.3 Comments In a non-interactive shell.1. or an interactive shell in which the interactive_comments option to the shopt builtin is enabled (see Section 4. Others create the name of the message catalog from the value of the TEXTDOMAIN shell variable. Still others use both variables in this fashion: TEXTDOMAINDIR/LC_MESSAGES/LC MESSAGES/TEXTDOMAIN. 3.3 [Interactive Shells]. If the current locale is C or POSIX. a word beginning with ‘#’ causes that word and all remaining characters on that line to be ignored.Chapter 3: Basic Shell Features 7 \f \n \r \t \v \\ \’ \" \nnn \xHH \cx form feed newline carriage return horizontal tab vertical tab backslash single quote double quote the eight-bit character whose value is the octal value nnn (one to three digits) the eight-bit character whose value is the hexadecimal value HH (one or two hex digits) a control-x character The expanded result is single-quoted. in a loop or conditional construct. 3. 3. If the string is translated and replaced. separated by spaces. The interactive_comments option is on by default in interactive shells.5 Locale-Specific Translation A double-quoted string preceded by a dollar sign (‘\$’) will cause the string to be translated according to the current locale.mo. you may need to set the TEXTDOMAINDIR variable to the location of the message catalog files. page 55).2.mo’.2 Shell Commands A simple shell command such as echo a b c consists of the command itself followed by arguments. If you use the TEXTDOMAIN variable. . or in some other grouping. the dollar sign is ignored.
or a newline. followed by ‘. The exit status of a pipeline is the exit status of the last command in the pipeline.’.3 [Command Execution Environment]. The ‘-p’ option changes the output format to that specified by posix. unless the pipefail option is enabled (see Section 4. Of these list operators. each command reads the previous command’s output.3. The return status (see Section 3. If pipefail is enabled. The TIMEFORMAT variable may be set to a format string that specifies how the timing information should be displayed. page 30). or ‘||’. The shell waits for all commands in the pipeline to terminate before returning a value.] The output of each command in the pipeline is connected via a pipe to the input of the next command. page 61. The statistics currently consist of elapsed (wall-clock) time and user and system time consumed by the command’s execution.2 Pipelines A pipeline is a sequence of simple commands separated by one of the control operators ‘|’ or ‘|&’. The format for a pipeline is [time [-p]] [!] command1 [ [| or |&] command2 . and optionally terminated by one of ‘. ‘&&’. and pipelines.5 [Exit Status]. terminated by one of the shell’s control operators (see Chapter 2 [Definitions].2. the standard error of command1 is connected to command2 ’s standard input through the pipe. .2. ‘&&’ and ‘||’ have equal precedence. the shell waits for all commands in the pipeline to complete. page 3). If the reserved word ‘!’ precedes the pipeline.’ and ‘&’. Each command in a pipeline is executed in its own subshell (see Section 3.2..1 [The Set Builtin]. It’s just a sequence of words separated by blanks.1 Simple Commands A simple command is the kind of command encountered most often. ‘&’. with the rest of the words being that command’s arguments. which have equal precedence. page 8). or 128+n if the command was terminated by signal n. it is shorthand for 2>&1 |. The use of time as a reserved word permits the timing of shell builtins. the pipeline’s return status is the value of the last (rightmost) command to exit with a non-zero status. See Section 5. for a description of the available formats. shell functions. The reserved word time causes timing statistics to be printed for the pipeline once it finishes.2 [Bash Variables].7. page 32) of a simple command is its exit status as provided by the posix 1003.’. If ‘|&’ is used. 3.. The first word generally specifies a command to be executed. This implicit redirection of the standard error is performed after any redirections specified by the command. page 51).1 waitpid function. That is. An external time command cannot time these easily. 3.2.3 [Lists].7. the exit status is the logical negation of the exit status as described above. ‘&’. If the pipeline is not executed asynchronously (see Section 3.3 Lists of Commands A list is a sequence of one or more pipelines separated by one of the operators ‘.8 Bash Reference Manual 3. This connection is performed before any redirections specified by the command. or zero if all commands exit successfully.
until The syntax of the until command is: until test-commands. The return status is the exit status of the last command executed in consequent-commands. An and list has the form command1 && command2 command2 is executed if. conditional commands. When job control is not active (see Chapter 7 [Job Control]. An or list has the form command1 || command2 command2 is executed if. Each construct begins with a reserved word or control operator and is terminated by a corresponding reserved word or operator. If a command is terminated by the control operator ‘&’. page 89). and and or lists are executed with left associativity. do consequent-commands.6 [Redirections]. the shell waits for each command to terminate in turn. the shell executes the command asynchronously in a subshell.’ are executed sequentially. 3. Commands separated by a ‘. and only if. it may be replaced with one or more newlines. the standard input for asynchronous commands. do consequent-commands.Chapter 3: Basic Shell Features 9 A sequence of one or more newlines may appear in a list to delimit commands. page 26) associated with a compound command apply to all commands within that compound command unless explicitly overridden. respectively. and mechanisms to group commands and execute them as a unit. and only if. The shell does not wait for the command to finish.4 Compound Commands Compound commands are the shell programming constructs.’ appears in the description of a command’s syntax.2. Any redirections (see Section 3. Note that wherever a ‘. 3. or zero if none was executed. The return status of and and or lists is the exit status of the last command executed in the list. done . while The syntax of the while command is: while test-commands. Bash provides looping constructs. command1 returns a non-zero exit status. This is known as executing the command in the background. done Execute consequent-commands as long as test-commands has an exit status which is not zero. and and or lists are sequences of one or more pipelines separated by the control operators ‘&&’ and ‘||’. The return status is the exit status of the last command executed. equivalent to a semicolon. and the return status is 0 (true). is redirected from /dev/null. in the absence of any explicit redirections.1 Looping Constructs Bash supports the following looping constructs. command1 returns an exit status of zero.4.2.
4.2 [Special Parameters]... page 35) may be used to control loop execution.2 Conditional Constructs if The syntax of the if command is: if test-commands. commands are executed and the arithmetic expression expr3 is evaluated. and the final command in the final if or elif clause has a non-zero exit status.. page 16). The return status is the exit status of the last command that executes. the consequent-commands list is executed. each elif list is executed in turn. it behaves as if it evaluates to 1.1 [Bourne Shell Builtins]. then more-consequents. for The syntax of the for command is: for name [ [in [words . esac case will selectively execute the command-list corresponding to the first pattern that matches word.5 [Shell Arithmetic]. with name bound to the current member.. The return status is the exit status of the last command executed. and the return status is zero.] ] .].4. page 78). the corresponding more-consequents is executed and the command completes. Each time expr2 evaluates to a non-zero value.. and execute commands once for each member in the resultant list. The return value is the exit status of the last command in list that is executed. The syntax of the case command is: case word in [ [(] pattern [| pattern].10 Bash Reference Manual Execute consequent-commands as long as test-commands has an exit status of zero. If ‘in words’ is not present.] [else alternate-consequents. The break and continue builtins (see Section 4. If the shell option nocasematch (see the description of case . as if ‘in "\$@"’ had been specified (see Section 3. If test-commands returns a non-zero status. If there are no items in the expansion of words. and if its exit status is zero.] fi The test-commands list is executed. An alternate form of the for command is also supported: for (( expr1 . do commands . ] do commands. then consequent-commands. If ‘else alternate-consequents’ is present. or zero if no condition tested true. the for command executes the commands once for each positional parameter that is set. 3. expr2 .. The return status is the exit status of the last command executed in consequent-commands. the arithmetic expression expr1 is evaluated according to the rules described below (see Section 6. no commands are executed. The arithmetic expression expr2 is then evaluated repeatedly until it evaluates to zero. then alternate-consequents is executed. done Expand words.. If any expression is omitted. and if its return status is zero. expr3 )) .2. done First. or false if any of the expressions is invalid.) command-list . or zero if none was executed. [elif more-test-commands.
parameter expansion..&’. each terminated by a ‘. or ‘.’ operator is used. The commands are executed after each selection until a break command is executed. and the ‘)’ operator terminates a pattern list.&’ in place of ‘.&’. Any other value read causes name to be set to null. If the line is empty. and arithmetic expansion. . command substitution. Otherwise.Chapter 3: Basic Shell Features 11 shopt in Section 4. at which point the select command completes. The ‘|’ is used to separate multiple patterns.. done The list of words following in is expanded.. If EOF is read. the match is performed without regard to the case of alphabetic characters. then the value of name is set to that word. the positional parameters are printed.. do commands. and quote removal before matching is attempted.2 [The Shopt Builtin]. Using ‘." If the ‘. the return status is the exit status of the command-list executed. The line read is saved in the variable REPLY. man | kangaroo ) echo -n "two". Here is an example using case in a script that could be used to describe one interesting feature of an animal: echo -n "Enter the name of an animal: " read ANIMAL echo -n "The \$ANIMAL has " case \$ANIMAL in horse | dog | cat) echo -n "four"... if any. If the ‘in words’ is omitted. each preceded by a number. *) echo -n "an unknown number of". Each pattern undergoes tilde expansion. ‘. The PS3 prompt is then displayed and a line is read from the standard input. The set of expanded words is printed on the standard error output stream.3. generating a list of items. parameter expansion. if any.’..’ causes the shell to test the patterns in the next clause. ‘. There may be an arbitrary number of case clauses.. the words and prompt are displayed again. Each clause must be terminated with ‘.&’.&’... The return status is zero if no pattern is matched. and execute any associated command-list on a successful match.].. It has almost the same syntax as the for command: select name [in words ..’.. The word undergoes tilde expansion. command substitution. as if ‘in "\$@"’ had been specified. If the line consists of a number corresponding to one of the displayed words.’ causes execution to continue with the command-list associated with the next clause. arithmetic expansion. select The select construct allows the easy generation of menus. esac echo " legs. no subsequent matches are attempted after the first pattern match. Using ‘. The first pattern that matches determines the command-list that is executed.&’ in place of ‘. the select command completes. page 55) is enabled. or ‘. A list of patterns and an associated command-list is known as a clause.
The return value is 0 if the string matches the pattern..3. page 76. is available. process substitution.5 [Shell Arithmetic]. the match is performed without regard to the case of alphabetic characters. ‘=~’. Word splitting and filename expansion are not performed on the words between the ‘[[’ and ‘]]’.12 Bash Reference Manual Here is an example that allows the user to pick a filename from the current directory.5. and quote removal are performed. When used with ‘[[’. page 78). page 55) is enabled. tilde expansion. The ‘<’ and ‘>’ operators sort lexicographically using the current locale.2 [Bash Builtins]. When the ‘==’ and ‘!=’ operators are used.3. the conditional expression’s return value is 2.2 [The Shopt Builtin]. When it is used. and 1 otherwise. do echo you picked \$fname \(\$REPLY\) break.2 [The Shopt Builtin]. If the value of the expression is non-zero. for a full description of the let builtin. page 55) is enabled. If the shell option nocasematch (see the description of shopt in Section 4. This is exactly equivalent to let "expression" See Section 4. Expressions are composed of the primaries described below in Section 6. The return value is 0 if the string matches (‘==’) or does not match (‘!=’)the pattern. parameter and variable expansion. done ((. An additional binary operator. Any part of the pattern may be quoted to force it to be matched as a string. [[. with the same precedence as ‘==’ and ‘!=’. select fname in *.. otherwise the return status is 1. and 1 otherwise. the string to the right of the operator is considered a pattern and matched according to the rules described below in Section 3. If the regular expression is syntactically incorrect.8. Conditional operators such as ‘-f’ must be unquoted to be recognized as primaries. arithmetic expansion. page 24. the return status is 0. the string to the right of the operator is considered an extended regular expression and matched accordingly (as in regex 3)). and displays the name and index of the file selected. Any part of the pattern may be quoted to force it to be matched as a string. Substrings matched by parenthesized subexpressions within the regular expression are saved in the . page 41.4 [Bash Conditional Expressions].. the match is performed without regard to the case of alphabetic characters. If the shell option nocasematch (see the description of shopt in Section 4.]] [[ expression ]] Return a status of 0 or 1 depending on the evaluation of the conditional expression expression.)) (( expression )) The arithmetic expression is evaluated according to the rules described below (see Section 6.1 [Pattern Matching]. command substitution..
there is a subtle difference between these two constructs due to historical reasons. {} { list. ! expression True if expression is false.7. so they must be separated from the list by blanks or other shell metacharacters. expression1 && expression2 True if both expression1 and expression2 are true.4.Chapter 3: Basic Shell Features 13 array variable BASH_REMATCH. and each of the commands in list to be executed in that subshell. Expressions may be combined using the following operators. () ( list ) Placing a list of commands between parentheses causes a subshell environment to be created (see Section 3. In addition to the creation of a subshell. } Placing a list of commands between curly braces causes the list to be executed in the current shell context. When commands are grouped. redirections may be applied to the entire command list. . Since the list is executed in a subshell. The exit status of both of these constructs is the exit status of list.3 [Command Execution Environment]. variable assignments do not remain in effect after the subshell completes. The parentheses are operators. page 30). The semicolon (or newline) following list is required. No subshell is created. This may be used to override the normal precedence of operators. The && and || operators do not evaluate expression2 if the value of expression1 is sufficient to determine the return value of the entire conditional expression. The element of BASH_REMATCH with index 0 is the portion of the string matching the entire regular expression. The braces are reserved words. and are recognized as separate tokens by the shell even if they are not separated from the list by whitespace.3 Grouping Commands Bash provides two ways to group a list of commands to be executed as a unit. the output of all the commands in the list may be redirected to a single stream.2. The element of BASH_REMATCH with index n is the portion of the string matching the nth parenthesized subexpression. 3. expression1 || expression2 True if either expression1 or expression2 is true. listed in decreasing order of precedence: ( expression ) Returns the value of expression. For example.
1 [Bourne Shell Builtins].14 Bash Reference Manual 3. Any redirections (see Section 3.2.3 Shell Functions Shell functions are a way to group commands for later execution using a single name for the group. NAME must not be supplied if command is a simple command (see Section 3. the parentheses are optional. the list of commands associated with that function name is executed. it is interpreted as the first word of the simple command. The format for a coprocess is: coproc [NAME] command [redirections] This creates a coprocess named NAME. otherwise. The file descriptors can be utilized as arguments to shell commands and redirections using standard word expansions. When the name of a shell function is used as a simple command name. Shell functions are executed in the current shell context. page 9). the shell creates an array variable (see Section 6. The standard output of command is connected via a pipe to a file descriptor in the executing shell. page 26). but may be any compound command listed above. The process id of the shell spawned to execute the coprocess is available as the value of the variable NAME PID. If the function reserved word is supplied.4 [Compound Commands]. The body of the function is the compound command compound-command (see Section 3. A coprocess is executed asynchronously in a subshell. page 35). . The return status of a coprocess is the exit status of command. page 26) associated with the shell function are performed when the function is executed.6 [Redirections]. 3. That command is usually a list enclosed between { and }. no new process is created to interpret them. the exit status of a function is the exit status of the last command executed in the body.5 Coprocesses A coprocess is a shell command preceded by the coproc reserved word. Functions are declared using this syntax: [ function ] name () compound-command [ redirections ] This defines a shell function named name.7 [Arrays]. with a two-way pipe established between the executing shell and the coprocess. When executed. and that file descriptor is assigned to NAME [0]. A function definition may be deleted using the ‘-f’ option to the unset builtin (see Section 4. The reserved word function is optional. page 80) named NAME in the context of the executing shell.1 [Simple Commands]. page 8).2. The exit status of a function definition is zero unless a syntax error occurs or a readonly function with the same name already exists. compound-command is executed whenever name is specified as the name of a command. This pipe is established before any redirections specified by the command (see Section 3. If NAME is not supplied.2.6 [Redirections]. the default name is COPROC. as if the command had been terminated with the ‘&’ control operator. and that file descriptor is assigned to NAME [1]. The wait builtin command may be used to wait for the coprocess to terminate. The standard input of command is connected via a pipe to a file descriptor in the executing shell. They are executed just like a "regular" command. When the coproc is executed.
the arguments to the function become the positional parameters during its execution (see Section 3. that is the function’s return status. Functions may be exported so that subshells automatically have them defined with the ‘-f’ option to the export builtin (see Section 4. Variables local to the function may be declared with the local builtin. If the builtin command return is executed in a function. it may be unset only by using the unset builtin command. if the extdebug shell option is enabled).1 [Positional Parameters]. When a function completes. Function names and definitions may be listed with the ‘-f’ option to the declare or typeset builtin commands (see Section 4. A variable has a value and zero or more attributes. a number. or one of the special characters listed below. the values of the positional parameters and the special parameter ‘#’ are restored to the values they had prior to the function’s execution. in the most common usage the curly braces that surround the body of the function must be separated from the body by blanks or newlines. Attributes are assigned using the declare builtin command (see the description of the declare builtin in Section 4. Any command associated with the RETURN trap is executed before execution resumes. Special parameter 0 is unchanged. All other aspects of the shell execution environment are identical between a function and its caller with these exceptions: the DEBUG and RETURN traps are not inherited unless the function has been given the trace attribute using the declare builtin or the -o functrace option has been enabled with the set builtin. page 35).4. These variables are visible only to the function and the commands it invokes. No limit is placed on the number of recursive calls. a ‘&’. Functions may be recursive. The ‘-F’ option to declare or typeset will list the function names only (and optionally the source file and line number.4 Shell Parameters A parameter is an entity that stores values. (in which case all functions inherit the DEBUG and RETURN traps). A variable is a parameter denoted by a name. The first element of the FUNCNAME variable is set to the name of the function while the function is executing. otherwise the function’s return status is the exit status of the last command executed before the return. and the ERR trap is not inherited unless the -o errtrace shell option has been enabled. When a function is executed. or a newline. page 41). page 35. A variable may be assigned to by a statement of the form .2 [Bash Builtins]. page 16). The null string is a valid value. Once a variable is set. Also. It can be a name. The special parameter ‘#’ that expands to the number of positional parameters is updated to reflect the change. Care should be taken in cases where this may cause a problem. when using the braces. page 41). This is because the braces are reserved words and are only recognized as such when they are separated from the command list by whitespace or another shell metacharacter. Note that shell functions and variables with the same name may result in multiple identically-named entries in the environment passed to the shell’s children. for the description of the trap builtin.Chapter 3: Basic Shell Features 15 Note that for historical reasons. the list must be terminated by a semicolon. 3. A parameter is set if it has been assigned a value.1 [Bourne Shell Builtins]. See Section 4.2 [Bash Builtins].1 [Bourne Shell Builtins]. If a numeric argument is given to return. the function completes and execution resumes with the next command after the function call.
and new values are appended to the array beginning at one greater than the array’s maximum index (for indexed arrays). or added as additional key-value pairs in an associative array. declare. it expands to a single word with the value of each parameter separated by the first character of the IFS special variable.4.5. If IFS is unset. Word splitting is not performed. "\$@" is equivalent to "\$1" "\$2" . All value s undergo tilde expansion. page 14).2 Special Parameters The shell treats several parameters specially. then value is evaluated as an arithmetic expression even if the \$((.7 [Arrays]. readonly. If IFS is null. That is. parameter and variable expansion. When a positional parameter consisting of more than a single digit is expanded. which is also evaluated.1 Positional Parameters A positional parameter is a parameter denoted by one or more digits..7 [Arrays]. assignment to them is not allowed.4.3 [Shell Functions].. the parameters are joined without intervening separators. or as \$N when N consists of a single digit. 3.5 [Arithmetic Expansion]. Positional parameter N may be referenced as \${N}. page 35). * Expands to the positional parameters. Positional parameters are assigned from the shell’s arguments when it is invoked. page 23). value is expanded and appended to the variable’s value. When ‘+=’ is applied to an array variable using compound assignment (see Section 6. where c is the first character of the value of the IFS variable. 3. page 80). other than the single digit 0. typeset. the ‘+=’ operator can be used to append to or add to the variable’s previous value.16 Bash Reference Manual name=[value] If value is not given. the expansion of the first parameter is joined with the beginning @ .)) expansion is not used (see Section 3.. The set and shift builtins are used to set and unset them (see Chapter 4 [Shell Builtin Commands]. and may be reassigned using the set builtin command. the parameters are separated by spaces. command substitution. and local builtin commands. When applied to a string-valued variable. page 80). These parameters may only be referenced. Filename expansion is not performed. the variable’s value is not unset (as it is when using ‘=’). each parameter expands to a separate word. with the exception of "\$@" as explained below. export. starting from one. and quote removal (detailed below). If the double-quoted expansion occurs within a word. If the variable has its integer attribute set. When the expansion occurs within double quotes. Positional parameters may not be assigned to with assignment statements. starting from one. In the context where an assignment statement is assigning a value to a shell variable or array index (see Section 6. value is evaluated as an arithmetic expression and added to the variable’s current value. The positional parameters are temporarily replaced when a shell function is executed (see Section 3. When the expansion occurs within double quotes. arithmetic expansion. it must be enclosed in braces. That is. Assignment statements may also appear as arguments to the alias.".. When ‘+=’ is applied to a variable for which the integer attribute has been set. the variable is assigned the null string... Expands to the positional parameters.. "\$*" is equivalent to "\$1c\$2c.
5 Shell Expansions Expansion is performed on the command line after it has been split into tokens. (A hyphen. not the subshell. (An underscore. and filename expansion. they are removed). Expands to the name of the shell or shell script. This is performed at the same time as parameter. In a () subshell. This is set at shell initialization. variable.8 [Shell Scripts]. and the expansion of the last parameter is joined with the last part of the original word. . if one is present.) At shell startup. # ? Expands to the number of positional parameters in decimal. Otherwise.) Expands to the current option flags as specified upon invocation. by the set builtin command. there is an additional expansion available: process substitution.. \$ ! 0 _ 3. set to the absolute pathname used to invoke the shell or shell script being executed as passed in the environment or argument list. tilde expansion. and arithmetic expansion and command substitution. expands to the last argument to the previous command. When there are no positional parameters. or those set by the shell itself (such as the ‘-i’ option). it expands to the process id of the invoking shell. If Bash is invoked with a file of commands (see Section 3. "\$@" and \$@ expand to nothing (i. Expands to the process id of the shell. When checking mail. Also set to the full pathname used to invoke each command executed and placed in the environment exported to that command. Expands to the exit status of the most recently executed foreground pipeline. \$0 is set to the name of that file. after expansion. There are seven kinds of expansion performed: • brace expansion • tilde expansion • parameter and variable expansion • command substitution • arithmetic expansion • word splitting • filename expansion The order of expansions is: brace expansion. then \$0 is set to the first argument after the string to be executed. page 71). Subsequently. word splitting. If Bash is started with the ‘-c’ option (see Section 6. parameter. and arithmetic expansion and command substitution (done in a left-to-right fashion). this parameter holds the name of the mail file. Expands to the process id of the most recently executed background (asynchronous) command. it is set to the filename used to invoke Bash. page 33). On systems that can support it.e. variable.Chapter 3: Basic Shell Features 17 part of the original word. as given by argument zero.1 [Invoking Bash].
To avoid conflicts with parameter expansion.dist.5. page 80). the string ‘\${’ is not considered eligible for brace expansion. The default increment is 1 or -1 as appropriate. expanding left to right.b}e ade ace abe A sequence expression takes the form {x.new. When characters are supplied.7 [Arrays]. Supplied integers may be prefixed with ‘0’ to force each term to have the same width. A { or ‘. and at least one unquoted comma or a valid sequence expression.bugs} or chown root /usr/{ucb/{ex.?*.5. other expansions expand a single word to a single word. is an integer. Patterns to be brace expanded take the form of an optional preamble. page 25) is performed. After all expansions.4.lib/{ex?. and incr.8 [Filename Expansion].y[. but the file names generated need not exist. an optional increment. Bash does not apply any syntactic interpretation to the context of the expansion or the text between the braces. left to right order is preserved. 3.9 [Quote Removal].’ may be quoted with a backslash to prevent its being considered part of a brace expression.1 Brace Expansion Brace expansion is a mechanism by which arbitrary strings may be generated. Note that both x and y must be of the same type.how_ex}} . page 24). inclusive. This mechanism is similar to filename expansion (see Section 3. To avoid conflicts with parameter expansion. When the increment is supplied. it is used as the difference between each term. and filename expansion can change the number of words of the expansion. Brace expansion is performed before any other expansions. the string ‘\${’ is not considered eligible for brace expansion. When integers are supplied. the expression expands to each number between x and y. and any characters special to other expansions are preserved in the result.. inclusive. Brace expansions may be nested. the shell attempts to force all generated terms to contain the same number of digits. This construct is typically used as shorthand when the common prefix of the strings to be generated is longer than in the above example: mkdir /usr/local/src/bash/{old. the expression expands to each character lexicographically between x and y.2 [Special Parameters]. The only exceptions to this are the expansions of "\$@" (see Section 3.edit}.c. zero-padding where necessary.5. The preamble is prefixed to each string contained within the braces. When either x or y begins with a zero. A correctly-formed brace expansion must contain unquoted opening and closing braces.. word splitting.18 Bash Reference Manual Only brace expansion. followed by an optional postscript. quote removal (see Section 3. The results of each expanded string are not sorted.incr]}. Any incorrectly formed brace expansion is left unchanged. page 16) and "\${name[@]}" (see Section 6. bash\$ echo a{d. and the postscript is then appended to each resulting string. where x and y are either integers or single characters. For example. It is strictly textual. followed by either a series of comma-separated strings or a seqeunce expression between a pair of braces.
In these cases. if it is set. or the tilde expansion fails. the tilde is replaced with the value of the HOME shell variable. the tilde-prefix is replaced with the home directory associated with the specified login name. Otherwise. ‘+’ is assumed. If the tilde-prefix is ‘~+’.5. which are optional but serve to protect the variable to be expanded from characters immediately following it which could be interpreted as part of the name. If the login name is invalid. all of the characters up to the first unquoted slash (or all characters. the value of the shell variable OLDPWD. and not within an embedded arithmetic expansion.5. The basic form of parameter expansion is \${parameter }. the matching ending brace is the first ‘}’ not escaped by a backslash or within a quoted string. as it would be displayed by the dirs builtin invoked with the characters following tilde in the tilde-prefix as an argument (see Section 6. one may use file names with tildes in assignments to PATH. if there is no unquoted slash) are considered a tilde-prefix. the characters in the tilde-prefix following the tilde are treated as a possible login name. If none of the characters in the tilde-prefix are quoted.8 [The Directory Stack]. page 81). If this login name is the null string. and the shell assigns the expanded value. and CDPATH. If HOME is unset. command substitution. consists of a number without a leading ‘+’ or ‘-’. If the tilde-prefix.Chapter 3: Basic Shell Features 19 3.2 Tilde Expansion If a word begins with an unquoted tilde character (‘~’).3 Shell Parameter Expansion The ‘\$’ character introduces parameter expansion. Each variable assignment is checked for unquoted tilde-prefixes immediately following a ‘:’ or the first ‘=’. or parameter expansion. If the characters following the tilde in the tilde-prefix consist of a number N. The value of parameter is substituted. The braces are required when parameter is a positional parameter with more . MAILPATH. the tilde-prefix is replaced with the corresponding element from the directory stack. sans the tilde. optionally prefixed by a ‘+’ or a ‘-’. When braces are used. command substitution. the word is left unchanged. the home directory of the user executing the shell is substituted instead. tilde expansion is also performed. The parameter name or symbol to be expanded may be enclosed in braces. is substituted. the value of the shell variable PWD replaces the tilde-prefix. If the tilde-prefix is ‘~-’. Consequently. The following table shows how Bash treats unquoted tilde-prefixes: ~ ~/foo ~fred/foo The subdirectory foo of the home directory of the user fred ~+/foo ~-/foo ~N ~+N ~-N ‘\$PWD/foo’ ‘\${OLDPWD-’~-’}/foo’ The string that would be displayed by ‘dirs +N’ The string that would be displayed by ‘dirs +N’ The string that would be displayed by ‘dirs -N’ The value of \$HOME ‘\$HOME/foo’ 3. or arithmetic expansion.
Positional parameters and special parameters may not be assigned to in this way. if it is not interactive. Bash uses the value of the variable formed from the rest of parameter as the name of the variable. Bash tests for a parameter that is unset or null. If parameter is ‘@’. \${parameter:offset} \${parameter:offset:length} Expands to up to length characters of parameter starting at the character specified by offset. the expansion of word is assigned to parameter. If length is omitted. Otherwise. If the first character of parameter is an exclamation point (!). Put another way. In each of the cases below. Omitting the colon results in a test only for a parameter that is unset. \${parameter:−word} If parameter is unset or null. page 78). When not performing substring expansion. nothing is substituted. word is subject to tilde expansion. the value of parameter is substituted. expands to the substring of parameter starting at the character specified by offset. and arithmetic expansion. exits. This is known as indirect expansion. parameter expansion. length must evaluate to a number greater than or equal to zero. The exceptions to this are the expansions of \${!prefix* } and \${!name [@]} described below. otherwise the expansion of word is substituted. the result is the length members of the array beginning with \${parameter[offset]}. If offset evaluates to a number less than zero. the expansion of word (or a message to that effect if word is not present) is written to the standard error and the shell. The value of parameter is then substituted. length and offset are arithmetic expressions (see Section 6. \${parameter:+word} If parameter is null or unset. The exclamation point must immediately follow the left brace in order to introduce indirection. command substitution. \${parameter:?word} If parameter is null or unset. the value of parameter is substituted. the value is used as an offset from the end of the value of parameter. this variable is then expanded and that value is used in the rest of the substitution. the result is length positional parameters beginning at offset. if the colon is omitted.5 [Shell Arithmetic]. or when parameter is followed by a character that is not to be interpreted as part of its name. rather than the value of parameter itself. the expansion of word is substituted. if the colon is included. Otherwise. the operator tests only for existence.20 Bash Reference Manual than one digit. a level of variable indirection is introduced. A negative offset is taken relative to one greater . using the form described below. \${parameter:=word} If parameter is unset or null. If parameter is an indexed array name subscripted by ‘@’ or ‘*’. the operator tests for both parameter ’s existence and that its value is not null. This is referred to as Substring Expansion.
page 24). in which case the indexing starts at 1 by default. If parameter is ‘@’ or ‘*’. \${parameter#word} \${parameter##word} The word is expanded to produce a pattern just as in filename expansion (see Section 3. then the result of the expansion is the value of parameter with the shortest matching pattern (the ‘%’ case) or the longest matching pattern (the ‘%%’ case) deleted. separated by the first character of the IFS special variable. the value substituted is the number of positional parameters. Substring expansion applied to an associative array produces undefined results. Substring indexing is zero-based unless the positional parameters are used. If parameter is an array variable subscripted with ‘@’ or ‘*’. \$@ is prefixed to the list. Note that a negative offset must be separated from the colon by at least one space to avoid being confused with the ‘:-’ expansion.Chapter 3: Basic Shell Features 21 than the maximum index of the specified array. If parameter is an array variable subscripted with ‘@’ or ‘*’. If the pattern matches a trailing portion of the expanded value of parameter. When ‘@’ is used and the expansion appears within double quotes. \${parameter%word} \${parameter%%word} The word is expanded to produce a pattern just as in filename expansion. the pattern removal operation is applied to each member of the array in turn. the value substituted is the number of elements in the array. the pattern removal operation is applied to each positional parameter in turn. expands to the list of array indices (keys) assigned in name. and the expansion is the resultant list. If parameter is ‘@’ or ‘*’. each variable name expands to a separate word. the pattern removal operation is applied to each positional parameter in turn. If parameter is an array name subscripted by ‘*’ or ‘@’. \${!prefix*} \${!prefix@} Expands to the names of variables whose names begin with prefix. If offset is 0. and the positional parameters are used. and the expansion is the resultant list.8 [Filename Expansion]. \${!name[@]} \${!name[*]} If name is an array variable. expands to 0 if name is set and null otherwise. then the result of the expansion is the expanded value of parameter with the shortest matching pattern (the ‘#’ case) or the longest matching pattern (the ‘##’ case) deleted. the pattern removal . If parameter is ‘*’ or ‘@’. and the expansion is the resultant list. If name is not an array. If the pattern matches the beginning of the expanded value of parameter. When ‘@’ is used and the expansion appears within double quotes. each key expands to a separate word. \${#parameter} The length in characters of the expanded value of parameter is substituted.5.
but they may be removed during word splitting. the substitution operation is applied to each member of the array in turn.’ expansions match and convert only the first character in the expanded value. If parameter is ‘@’ or ‘*’. the ‘. the substitution operation is applied to each positional parameter in turn. which matches every character. \${parameter/pattern/string} The pattern is expanded to produce a pattern just as in filename expansion.pattern} This expansion modifies the case of alphabetic characters in parameter. ‘‘’. Normally only the first match is replaced. the ‘^’ and ‘. it must match at the end of the expanded value of parameter.’ expansions convert each matched character in the expanded value. and the expansion is the resultant list. and the expansion is the resultant list. When the old-style backquote form of substitution is used. the case modification operation is applied to each member of the array in turn. all matches of pattern are replaced with string. If pattern begins with ‘%’.5. Command substitution occurs when a command is enclosed as follows: \$(command) or ‘command‘ Bash performs the expansion by executing command and replacing the command substitution with the standard output of the command.’ operator converts matching uppercase letters to lowercase.pattern} \${parameter. The first backquote not preceded by a . or ‘\’.. the case modification operation is applied to each positional parameter in turn. backslash retains its literal meaning except when followed by ‘\$’.22 Bash Reference Manual operation is applied to each member of the array in turn. it must match at the beginning of the expanded value of parameter. If parameter is an array variable subscripted with ‘@’ or ‘*’. The pattern is expanded to produce a pattern just as in filename expansion. Parameter is expanded and the longest match of pattern against its value is replaced with string. The ‘^^’ and ‘. and the expansion is the resultant list. and the expansion is the resultant list. If pattern begins with ‘/’. Embedded newlines are not deleted. If parameter is ‘@’ or ‘*’. with any trailing newlines deleted. matches of pattern are deleted and the / following pattern may be omitted. The ‘^’ operator converts lowercase letters matching pattern to uppercase. If pattern begins with ‘#’.. If pattern is omitted. \${parameter^pattern} \${parameter^^pattern} \${parameter. The command substitution \$(cat file) can be replaced by the equivalent but faster \$(< file). 3. and the expansion is the resultant list. it is treated like a ‘?’.4 Command Substitution Command substitution allows the output of a command to replace the command itself. If string is null. If parameter is an array variable subscripted with ‘@’ or ‘*’.
5. and splits the results of the other expansions into words on these characters. 3. If the expression is invalid. 3. then sequences of the whitespace characters space and tab are ignored at the beginning and end of the word. Arithmetic expansions may be nested. Bash prints a message indicating failure to the standard error and no substitution occurs. the default. command substitution.6 Process Substitution Process substitution is supported on systems that support named pipes (fifos) or the ‘/dev/fd’ method of naming open files. The name of this file is passed as an argument to the current command as the result of the expansion. writing to the file will provide input for list. escape the inner backquotes with backslashes. All tokens in the expression undergo parameter expansion. If IFS is unset. or its value is exactly <space><tab><newline>.5. 3. The shell treats each character of \$IFS as a delimiter.5. command substitution. process substitution is performed simultaneously with parameter and variable expansion.5 Arithmetic Expansion Arithmetic expansion allows the evaluation of an arithmetic expression and the substitution of the result. The evaluation is performed according to the rules listed below (see Section 6. and quote removal. all characters between the parentheses make up the command. and arithmetic expansion that did not occur within double quotes for word splitting. If IFS has a value other than the default.7 Word Splitting The shell scans the results of parameter expansion. then sequences of <space>. as long as the whitespace character is in the value of IFS (an IFS whitespace character). If the <(list) form is used. and <newline> at the beginning and end of the results of the previous expansions are ignored. <tab>. When using the \$(command) form. Command substitutions may be nested. If the >(list) form is used. the file passed as an argument should be read to obtain the output of list. Any character in IFS that is not IFS . The format for arithmetic expansion is: \$(( expression )) The expression is treated as if it were within double quotes. To nest when using the backquoted form.5 [Shell Arithmetic]. It takes the form of <(list) or >(list) The process list is run with its input or output connected to a fifo or some file in ‘/dev/fd’. otherwise the construct would be interpreted as a redirection. none are treated specially. Note that no space may appear between the < or > and the left parenthesis. If the substitution appears within double quotes.Chapter 3: Basic Shell Features 23 backslash terminates the command substitution. page 78). word splitting and filename expansion are not performed on the results. When available. command substitution. and any sequence of IFS characters not at the beginning or end serves to delimit words. but a double quote inside the parentheses is not treated specially. and arithmetic expansion.
5.3. page 55. If the nullglob option is set. Unquoted implicit null arguments. In other cases. the escaping backslash is discarded when matching. resulting from the expansion of parameters that have no values.’ are always ignored when GLOBIGNORE is set and not null. nullglob. make ‘.’ character is not treated specially. the ‘. Explicit null arguments ("" or ’’) are retained. an error message is printed and the command is not executed. If the shell option nocaseglob is enabled. so all other filenames beginning with a ‘. See the description of shopt in Section 4. The special pattern characters must be quoted if they are to be matched literally. 3. A sequence of IFS whitespace characters is also treated as a delimiter.1 [The Set Builtin]. failglob.8 Filename Expansion After word splitting. each matching filename that also matches one of the patterns in GLOBIGNORE is removed from the list of matches. along with any adjacent IFS whitespace characters.3. then the word is regarded as a pattern.5. are removed. the match is performed without regard to the case of alphabetic characters. If the value of IFS is null. However. The dotglob option is disabled when GLOBIGNORE is unset.’. When a pattern is used for filename expansion. unless the ‘-f’ option has been set (see Section 4. The nul character may not occur in a pattern. two adjacent ‘*’s used as a single pattern will match all files and zero or more directories and subdirectories. If the failglob shell option is set. setting GLOBIGNORE to a non-null value has the effect of enabling the dotglob shell option. no word splitting occurs. When matching a file name.1 Pattern Matching Any character that appears in a pattern. and ‘[’. Matches any single character. page 51). A backslash escapes the following character. unless the shell option dotglob is set. matches itself. a null argument results and is retained. no splitting is performed. If GLOBIGNORE is set. ? .. The GLOBIGNORE shell variable may be used to restrict the set of filenames matching a pattern. the character ‘.’ will match. The special pattern characters have the following meanings: * Matches any string. the word is left unchanged. ‘?’. delimits a field.8. the slash character must always be matched explicitly. and no matches are found. To get the old behavior of ignoring filenames beginning with a ‘. If followed by a ‘/’. Note that if no expansion occurs. other than the special pattern characters described below.’ and ‘.’ at the start of a filename or immediately following a slash must be matched explicitly. Bash scans each word for the characters ‘*’. and replaced with an alphabetically sorted list of file names matching the pattern. If no matching file names are found. for a description of the nocaseglob. If one of these characters appears. and the shell option nullglob is disabled. and dotglob options. the word is removed. If a parameter with no value is expanded within double quotes.*’ one of the patterns in GLOBIGNORE.24 Bash Reference Manual whitespace. two adjacent ‘*’s will match only directories and subdirectories. The filenames ‘.2 [The Shopt Builtin]. 3. including the null string. and ‘*’ is used in a filename expansion context. and no matches are found. When the globstar shell option is enabled.
A ‘−’ may be matched by including it as the first or last character in the set. !(pattern-list) Matches anything except one of the given patterns. Within ‘[’ and ‘]’. +(pattern-list) Matches one or more occurrences of the given patterns.Chapter 3: Basic Shell Features 25 [. .5. any character that sorts between those two characters. The word character class matches letters. 3. If the extglob shell option is enabled using the shopt builtin. ‘’’. To obtain the traditional interpretation of ranges in bracket expressions. where class is one of the following classes defined in the posix standard: alnum alpha ascii blank cntrl digit graph lower print punct space upper word xdigit A character class matches any character belonging to that class. character classes can be specified using the syntax [:class :].. an equivalence class can be specified using the syntax [=c =]. and ‘"’ that did not result from one of the above expansions are removed. using the current locale’s collating sequence and character set. all unquoted occurrences of the characters ‘\’. For example. and the character ‘_’. for example. Within ‘[’ and ‘]’. If the first character following the ‘[’ is a ‘!’ or a ‘^’ then any character not enclosed is matched. the syntax [. In the following description. a pattern-list is a list of one or more patterns separated by a ‘|’. Within ‘[’ and ‘]’.symbol . is matched. in the default C locale. digits.. @(pattern-list) Matches one of the given patterns. ‘[a-dx-z]’ is equivalent to ‘[abcdxyz]’. and in these locales ‘[a-dx-z]’ is typically not equivalent to ‘[abcdxyz]’. *(pattern-list) Matches zero or more occurrences of the given patterns. it might be equivalent to ‘[aBbCcDdxXyYz]’. Composite patterns may be formed using one or more of the following sub-patterns: ?(pattern-list) Matches zero or one occurrence of the given patterns. The sorting order of characters in range expressions is determined by the current locale and the value of the LC_COLLATE shell variable. if set. A ‘]’ may be matched by including it as the first character in the set.] matches the collating symbol symbol. you can force the use of the C locale by setting the LC_COLLATE or LC_ALL environment variable to the value ‘C’. A pair of characters separated by a hyphen denotes a range expression.] Matches any one of the enclosed characters. several extended pattern matching operators are recognized. inclusive. Many locales sort characters in dictionary order. which matches all characters with the same collation weight (as defined by the current locale) as the character c.9 Quote Removal After the preceding expansions.
For example. and the first character of the redirection operator is ‘<’. and port is an integer port number or service name. quote removal. In the following descriptions. Each redirection that may be preceded by a file descriptor number may instead be preceded by a word of the form {varname }. If it expands to more than one word. In this case. Redirection may also be used to open and close files for the current shell execution environment. tilde expansion.6 Redirections Before a command is executed. Bash reports an error. is subjected to brace expansion. for each redirection operator except >&. and word splitting.and <&-.is preceded by {varname }.or <&. file descriptor fd is duplicated. the redirection refers to the standard input (file descriptor 0). /dev/stdin File descriptor 0 is duplicated. the redirection refers to the standard output (file descriptor 1). The word following the redirection operator in the following descriptions. Bash handles several filenames specially when they are used in redirections. unless otherwise noted. If the first character of the redirection operator is ‘>’. parameter expansion. if the file descriptor number is omitted. Bash attempts to open a TCP connection to the corresponding socket. . /dev/tcp/host/port If host is a valid hostname or Internet address. /dev/stdout File descriptor 1 is duplicated. the shell will allocate a file descriptor greater than 10 and assign it to {varname }. The following redirection operators may precede or appear anywhere within a simple command or may follow a command. the value of varname defines the file descriptor to close. the command ls > dirlist 2>&1 directs both standard output (file descriptor 1) and standard error (file descriptor 2) to the file dirlist.26 Bash Reference Manual 3. its input and output may be redirected using a special notation interpreted by the shell. because the standard error was made a copy of the standard output before the standard output was redirected to dirlist. while the command ls 2>&1 > dirlist directs only the standard output to file dirlist. /dev/stderr File descriptor 2 is duplicated. filename expansion. If >&. Redirections are processed in the order they appear. Note that the order of redirections is significant. command substitution. from left to right. as described in the following table: /dev/fd/fd If fd is a valid integer. arithmetic expansion.
and the noclobber option to the set builtin has been enabled. The general format for redirecting input is: [n]<word 3. as they may conflict with file descriptors the shell uses internally.1 Redirecting Input Redirection of input causes the file whose name results from the expansion of word to be opened for reading on file descriptor n. the redirection is attempted even if the file named by word exists.6.6. or the standard input (file descriptor 0) if n is not specified. If the file does not exist it is created. There are two formats for redirecting standard output and standard error: &>word and >&word Of the two forms. Bash attempts to open a UDP connection to the corresponding socket. This is semantically equivalent to >word 2>&1 . Redirections using file descriptors greater than 9 should be used with care.Chapter 3: Basic Shell Features 27 /dev/udp/host/port If host is a valid hostname or Internet address. the first is preferred. or the redirection operator is ‘>’ and the noclobber option is not enabled. 3. or the standard output (file descriptor 1) if n is not specified. 3.6. If the redirection operator is ‘>|’. The general format for appending output is: [n]>>word 3. the redirection will fail if the file whose name results from the expansion of word exists and is a regular file.2 Redirecting Output Redirection of output causes the file whose name results from the expansion of word to be opened for writing on file descriptor n. or the standard output (file descriptor 1) if n is not specified. The general format for redirecting output is: [n]>[|]word If the redirection operator is ‘>’.6. and port is an integer port number or service name.4 Redirecting Standard Output and Standard Error This construct allows both the standard output (file descriptor 1) and the standard error output (file descriptor 2) to be redirected to the file whose name is the expansion of word. if it does exist it is truncated to zero size. A failure to open or create a file causes the redirection to fail. If the file does not exist it is created.3 Appending Redirected Output Redirection of output in this fashion causes the file whose name results from the expansion of word to be opened for appending on file descriptor n.
the format is: <<< word The word is expanded and supplied to the command on its standard input. command substitution.7 Here Strings A variant of here documents. The format for appending standard output and standard error is: &>>word This is semantically equivalent to >>word 2>&1 3.6. If n is not specified. If the digits in word do not specify a file descriptor open . or filename expansion is performed on word.8 Duplicating File Descriptors The redirection operator [n]<&word is used to duplicate input file descriptors. If word expands to one or more digits.6.6. 3.28 Bash Reference Manual 3. All of the lines read up to that point are then used as the standard input for a command. command substitution. the character sequence \newline is ignored. ‘\$’. the file descriptor denoted by n is made to be a copy of that file descriptor. file descriptor n is closed.6. all lines of the here-document are subjected to parameter expansion. then all leading tab characters are stripped from input lines and the line containing delimiter. arithmetic expansion. In the latter case. The format of here-documents is: <<[−]word here-document delimiter No parameter expansion. If word evaluates to ‘-’. and ‘\’ must be used to quote the characters ‘\’.5 Appending Standard Output and Standard Error This construct allows both the standard output (file descriptor 1) and the standard error output (file descriptor 2) to be appended to the file whose name is the expansion of word. a redirection error occurs. 3. If any characters in word are quoted. the standard input (file descriptor 0) is used.6 Here Documents This type of redirection instructs the shell to read input from the current source until a line containing only word (with no trailing blanks) is seen. This allows here-documents within shell scripts to be indented in a natural fashion. and arithmetic expansion. If the digits in word do not specify a file descriptor open for input. and the lines in the here-document are not expanded. The operator [n]>&word is used similarly to duplicate output file descriptors. and ‘‘’. the standard output (file descriptor 1) is used. If word is unquoted. If n is not specified. If the redirection operator is ‘<<-’. the delimiter is the result of quote removal on word.
If any words remain after expansion. If no command name results. As a special case. 3. or the standard input (file descriptor 0) if n is not specified.9 Moving File Descriptors The redirection operator [n]<&digitmoves the file descriptor digit to file descriptor n. and word does not expand to one or more digits. redirections are performed. the variables are added to the environment of the executed command and do not affect the current shell environment. If the file does not exist. parameter expansion.6. 4. The words that the parser has marked as variable assignments (those preceding the command name) and redirections are saved for later processing. The text after the ‘=’ in each variable assignment undergoes tilde expansion. and quote removal before being assigned to the variable. if n is omitted. 2. the redirection operator [n]>&digitmoves the file descriptor digit to file descriptor n. but do not affect the current shell environment. the standard output and standard error are redirected as described previously. 3. 3.7 Executing Commands 3. or on file descriptor 0 if n is not specified. an error occurs. command substitution. page 17). it is created. Redirections are performed as described above (see Section 3. 1. The words that are not variable assignments or redirections are expanded (see Section 3. the variable assignments affect the current shell environment. and the command exits with a non-zero status. 3. arithmetic expansion. the shell performs the following expansions. . If no command name results.Chapter 3: Basic Shell Features 29 for output.5 [Shell Expansions]. A redirection error causes the command to exit with a non-zero status.6 [Redirections].1 Simple Command Expansion When a simple command is executed. Similarly. a redirection error occurs. assignments. and redirections. If any of the assignments attempts to assign a value to a readonly variable. the first word is taken to be the name of the command and the remaining words are the arguments. digit is closed after being duplicated to n.7. or the standard output (file descriptor 1) if n is not specified. from left to right.10 Opening File Descriptors for Reading and Writing The redirection operator [n]<>word causes the file whose name is the expansion of word to be opened for both reading and writing on file descriptor n. Otherwise. page 26).6.
the exit status of the command is the exit status of the last command substitution performed. page 33. the shell searches for it in the list of shell builtins. or popd. as modified by redirections supplied to the exec builtin • the current working directory as set by cd. which consists of the following: • open files inherited by the shell at invocation. execution proceeds as described below. 1. Bash searches each element of \$PATH for a directory containing an executable file by that name.3 [Shell Functions]. If one of the expansions contained a command substitution. Argument 0 is set to the name given. If there were no command substitutions. and the remaining arguments to the command are set to the arguments supplied. and contains no slashes. If this execution fails because the file is not in executable format. pushd. the shell waits for the command to complete and collects its exit status. the command exits. 6. page 14. or inherited by the shell at invocation • the file creation mode mask as set by umask or inherited from the shell’s parent • current traps set by trap • shell parameters that are set by variable assignment or with set or inherited from the shell’s parent in the environment . If the name does not match a function. Otherwise. the command exits with a status of zero. If a match is found. that function is invoked as described in Section 3. 3. 4.7. If that function is not defined. If there exists a shell function by that name. the following actions are taken. If the command name contains no slashes.7.1 [Bourne Shell Builtins]. A full search of the directories in \$PATH is performed only if the command is not found in the hash table. 5. it is invoked with the original command and the original command’s arguments as its arguments. the shell searches for a defined shell function named command_not_found_handle. and the function’s exit status becomes the exit status of the shell. the shell executes the named program in a separate execution environment. that builtin is invoked. If that function exists. or if the command name contains one or more slashes. 2. If the command was not begun asynchronously. if any. and the file is not a directory. If the name is neither a shell function nor a builtin. 3. the shell prints an error message and returns an exit status of 127.8 [Shell Scripts]. Bash uses a hash table to remember the full pathnames of executable files to avoid multiple PATH searches (see the description of hash in Section 4. if it results in a simple command and an optional list of arguments. If the search is successful. it is assumed to be a shell script and the shell executes it as described in Section 3. the shell attempts to locate it. page 35).2 Command Search and Execution After a command has been split into words. 3. If the search is unsuccessful.30 Bash Reference Manual If there is a command name left after expansion.3 Command Execution Environment The shell has an execution environment.
7. Executed commands inherit the environment.Chapter 3: Basic Shell Features 31 • shell functions defined during execution or inherited from the shell’s parent in the environment • options enabled at invocation (either by default or with command-line arguments) or by set • options enabled by shopt (see Section 4. On invocation. Unless otherwise noted.3.6 [Aliases]. page 8).4 [Environment]. Bash clears the ‘-e’ option in such subshells. Builtin commands that are invoked as part of a pipeline are also executed in a subshell environment. it is invoked in a separate execution environment that consists of the following. The environment inherited by any executed command consists of the shell’s initial environment. and asynchronous commands are invoked in a subshell environment that is a duplicate of the shell environment. Otherwise. This is a list of name-value pairs. the new value becomes part of the environment. page 55) • shell aliases defined with alias (see Section 6. except that traps caught by the shell are reset to the values that the shell inherited from its parent at invocation. automatically marking it for export to child processes. 3. the default standard input for the command is the empty file ‘/dev/null’. and the value of \$PPID When a simple command other than a builtin or shell function is to be executed. and traps ignored by the shell are ignored A command invoked in this separate environment cannot affect the shell’s execution environment. The export and ‘declare -x’ commands allow parameters and functions to be added to and deleted from the environment. page 31) • traps caught by the shell are reset to the values inherited from the shell’s parent. whose values may be .2.7. Bash provides several ways to manipulate the environment. passed in the environment (see Section 3. Changes made to the subshell environment cannot affect the shell’s execution environment. the values are inherited from the shell. the shell scans its own environment and creates a parameter for each name found. plus any modifications and additions specified by redirections to the command • the current working directory • the file creation mode mask • shell variables and functions marked for export. Subshells spawned to execute command substitutions inherit the value of the ‘-e’ option from the parent shell.2 [The Shopt Builtin]. along with variables exported for the command. commands grouped with parentheses. • the shell’s open files. Command substitution. page 79) • various process ids. including those of background jobs (see Section 3.3 [Lists]. When not in posix mode. replacing the old. If the value of a parameter in the environment is modified. If a command is followed by a ‘&’ and job control is not active. the invoked command inherits the file descriptors of the calling shell as modified by redirections.4 Environment When a program is invoked it is given an array of strings called the environment. of the form name=value. the value of \$\$.
32 Bash Reference Manual modified in the shell. Exit statuses fall between 0 and 255.6 Signals When Bash is interactive. Under certain circumstances.7.4 [Shell Parameters]. so they may be used by the conditional and list constructs.7. If job control is in effect (see Chapter 7 [Job Control]. This seemingly counter-intuitive scheme is used so there is one well-defined way to indicate success and a variety of ways to indicate various failure modes. the shell may use values above 125 specially.1 [The Set Builtin]. and SIGTSTP.5 Exit Status The exit status of an executed command is the value returned by the waitpid system call or equivalent function. These assignment statements affect only the environment seen by that command. as explained below. If the ‘-k’ option is set (see Section 4. When job control is not in effect. a command which exits with a zero exit status has succeeded. not just those that precede the command name. the child process created to execute it returns a status of 127. If a command fails because of an error during expansion or redirection. The exit status is used by the Bash conditional commands (see Section 3. the exit status is greater than zero. When Bash receives a SIGINT. plus any additions via the export and ‘declare -x’ commands. and SIGINT is caught and handled (so that the wait builtin is interruptible).2. All builtins return an exit status of 2 to indicate incorrect usage. and SIGTSTP. Bash uses the value 128+N as the exit status. If a command is not found. When a command terminates on a fatal signal whose number is N. page 51).4. Non-builtin commands started by Bash have signal handlers set to the values inherited by the shell from its parent. in the absence of any traps. then all parameter assignments are placed in the environment for a command. it ignores SIGTERM (so that ‘kill 0’ does not kill an interactive shell). page 8). For the shell’s purposes. Bash ignores SIGTTIN.2 [Conditional Constructs]. the variable ‘\$_’ is set to the full path name of the command and passed to that command in its environment. All of the Bash builtins return an exit status of zero if they succeed and a non-zero status on failure. 3. A non-zero exit status indicates failure. page 15. asynchronous commands ignore SIGINT and SIGQUIT in addition to these inherited handlers. When Bash invokes an external command. If a command is found but is not executable. Exit statuses from shell builtins and compound commands are also limited to this range. The environment for any simple command or function may be augmented temporarily by prefixing it with parameter assignments. though. Bash ignores SIGQUIT. page 89). page 10) and some of the list constructs (see Section 3.2. it breaks out of any executing loops. 3. In all cases. less any pairs removed by the unset and ‘export -n’ commands. the shell will use special values to indicate specific failure modes. the return status is 126.3 [Lists]. as described in Section 3. SIGTTOU. Commands run as a result of command substitution ignore the keyboard-generated job control signals SIGTTIN. . SIGTTOU.3.
A shell script may be made executable by using the chmod command to turn on the execute bit. or some other interpreter and write the rest of the script file in that language. the remainder of the line specifies an interpreter for the program.2 [Job Control Builtins]. it should be removed from the jobs table with the disown builtin (see Section 7. an interactive shell resends the SIGHUP to all jobs. This subshell reinitializes itself. followed by the name of the script file. page 55). executing filename arguments is equivalent to executing bash filename arguments if filename is an executable shell script. and neither the ‘-c’ nor ‘-s’ option is supplied (see Section 6. In other words. Perl. followed by the rest of the arguments. awk.8 Shell Scripts A shell script is a text file containing shell commands. When Bash finds such a file while searching the \$PATH for a command. If the first line of a script begins with the two characters ‘#!’. Bash will perform this action on operating systems that do not handle it themselves. page 90) or marked to not receive SIGHUP using disown -h. Bash reads and executes commands from the file. If the huponexit shell option has been set with shopt (see Section 4. the positional parameters are unset. the trap will not be executed until the command completes. if any are given. the reception of a signal for which a trap has been set will cause the wait builtin to return immediately with an exit status greater than 128. Thus. If Bash is waiting for a command to complete and receives a signal for which a trap has been set. Before exiting. Stopped jobs are sent SIGCONT to ensure that they receive the SIGHUP. then exits. and looks in the directories in \$PATH if not found there. This mode of operation creates a non-interactive shell. If no additional arguments are supplied. Most versions of Unix make this a part of the operating system’s command execution mechanism. with the exception that the locations of commands remembered by the parent (see the description of hash in Section 4. Note that some older versions of Unix limit the interpreter name and argument to a maximum of 32 characters. When Bash runs a shell script. 3. running or stopped. To prevent the shell from sending the SIGHUP signal to a particular job. and the positional parameters are set to the remaining arguments. When Bash is waiting for an asynchronous command via the wait builtin. When such a file is used as the first non-option argument when invoking Bash. you can specify Bash. immediately after which the trap is executed. The shell first searches for the file in the current directory. Bash sends a SIGHUP to all jobs when an interactive login shell exits. it sets the special parameter 0 to the name of the file.Chapter 3: Basic Shell Features 33 The shell exits by default upon receipt of a SIGHUP. so that the effect is as if a new shell had been invoked to interpret the script. it spawns a subshell to execute it.2 [The Shopt Builtin]. rather than the name of the shell. . page 71). The arguments to the interpreter consist of a single optional argument following the interpreter name on the first line of the script file. page 35) are retained by the child.1 [Invoking Bash].3.1 [Bourne Shell Builtins].
since this ensures that Bash will be used to interpret the script. . even if it is executed under another shell.34 Bash Reference Manual Bash scripts often begin with #! /bin/bash (assuming that Bash has been installed in ‘/bin’).
1 Bourne Shell Builtins The following shell builtin commands are inherited from the Bourne Shell. (a period) . Otherwise the positional parameters are unchanged. filename [arguments] Read and execute commands from the filename argument in the current shell context. the command history (see Section 9. The return status is the exit status of the last command executed. and test builtins do not accept options and do not treat ‘--’ specially. page 121). The :. : (a colon) : [arguments] Do nothing beyond expanding arguments and performing redirections. The exit. . page 81). and the programmable completion facilities (see Section 8. they become the positional parameters when filename is executed. Unless otherwise noted. page 90). When Bash is not in posix mode. When the name of a builtin command is used as the first word of a simple command (see Section 3. the current directory is searched if filename is not found in \$PATH. break. or cannot be read. If filename is not found.2 [Job Control Builtins]. the PATH variable is used to find filename. the directory stack (see Section 6.7 [Programmable Completion Builtins]. or zero if no commands are executed. The return status is zero. page 117). false.2.1 [Directory Stack Builtins]. Other builtins that accept arguments but are not specified as accepting options interpret arguments beginning with ‘-’ as invalid options and require ‘--’ to prevent this interpretation. 4. let. break break [n] . Builtin commands are necessary to implement functionality impossible or inconvenient to obtain with separate utilities. as well as the builtin commands which are unique to or have been extended in Bash. the return status is non-zero. and shift builtins accept and process arguments beginning with ‘-’ without requiring ‘--’.2 [Bash History Builtins]. logout. continue.Chapter 4: Shell Builtin Commands 35 4 Shell Builtin Commands Builtin commands are contained within the shell itself. If any arguments are supplied. If filename does not contain a slash. each builtin command documented as accepting options preceded by ‘-’ accepts ‘--’ to signify the end of the options.1 [Simple Commands]. These commands are implemented as specified by the posix standard. page 8). without invoking another program. Many of the builtins have been extended by posix or Bash. Several builtin commands are described in other chapters: builtin commands which provide the Bash interface to the job control facilities (see Section 7. This builtin is equivalent to source.8. the shell executes the command directly. This section briefly describes the builtins which Bash inherits from the Bourne Shell. true.
and the directory change is successful. it is used as a search path. non-zero otherwise. CDPATH is not used. If ‘-a’ is supplied. eval eval [arguments] The arguments are concatenated together into a single command. otherwise the return status is non-zero. The return status is zero if the directory is successfully changed. redirections may be used to affect the current shell environment. continue continue [n] Resume the next iteration of an enclosing for. or select loop. exec exec [-cl] [-a name] [command [arguments]] If command is supplied. while. until. The ‘-c’ option causes command to be executed with an empty environment. If directory is ‘-’. the value of the HOME shell variable is used. the return status is zero. the execution of the nth enclosing loop is resumed. If there are no redirection errors. it replaces the shell without creating a new process. n must be greater than or equal to 1. If n is supplied. cd cd [-L|-P] [directory] Change the current working directory to directory. the shell places a dash at the beginning of the zeroth argument passed to command. while. Any trap on EXIT is executed before the shell terminates. exit exit [n] Exit the shell. If n is supplied. n must be greater than or equal to 1. If the shell variable CDPATH exists. the shell passes name as the zeroth argument to command. or select loop. the return status is zero. and its exit status returned as the exit status of eval. The return status is zero unless n is not greater than or equal to 1. If a non-empty directory name from CDPATH is used. If the ‘-l’ option is supplied. until. the absolute pathname of the new working directory is written to the standard output. symbolic links are followed by default or with the ‘-L’ option. export . The ‘-P’ option means to not follow symbolic links. If directory begins with a slash. which is then read and executed. If there are no arguments or only empty arguments. the nth enclosing loop is exited.36 Bash Reference Manual Exit from a for. or if ‘-’ is the first argument. If no command is specified. The return status is zero unless n is not greater than or equal to 1. the exit status is that of the last command executed. If n is omitted. it is equivalent to \$OLDPWD. returning a status of n to the shell’s parent. If directory is not given. This is what the login program does.
the name s refer to shell functions. getopts places that argument into the variable OPTARG. The colon (‘:’) and question mark (‘?’) may not be used as option characters. optstring contains the option characters to be recognized. which should be separated from it by white space. getopts places the next option in the shell variable name. If a variable name is followed by =value. The shell does not reset OPTIND automatically. a list of exported names is displayed. getopts parses those instead. the value of the variable is set to value. hash hash [-r] [-p filename] [-dt] [name] . The ‘-p’ option displays output in a form that may be reused as input. the option character found is placed in OPTARG and no diagnostic message is printed. otherwise the names refer to shell variables. OPTIND is set to the index of the first non-option argument. no error messages will be displayed. it must be manually reset between multiple calls to getopts within the same shell invocation if a new set of parameters is to be used. or ‘-f’ is supplied with a name that is not a shell function. If getopts is silent. the option is expected to have an argument. getopts can report errors in two ways. The ‘-n’ option means to no longer mark each name for export. then a colon (‘:’) is placed in name and OPTARG is set to the option character found. When the end of options is encountered. If getopts is silent. and getopts is not silent. getopts places ‘?’ into name and. even if the first character of optstring is not a colon. or if the ‘-p’ option is given. if not silent. If the first character of optstring is a colon. If a required argument is not found. If the variable OPTERR is set to 0. The return status is zero unless an invalid option is supplied. silent error reporting is used. getopts exits with a return value greater than zero. If the ‘-f’ option is supplied. and the index of the next argument to be processed into the variable OPTIND. and a diagnostic message is printed. one of the names is not a valid shell variable name. prints an error message and unsets OPTARG. a question mark (‘?’) is placed in name. When an option requires an argument. but if more arguments are given in args. If no names are supplied.Chapter 4: Shell Builtin Commands 37 export [-fn] [-p] [name[=value]] Mark each name to be passed to child processes in the environment. OPTIND is initialized to 1 each time the shell or a shell script is invoked. initializing name if it does not exist. If an invalid option is seen. Each time it is invoked. if a character is followed by a colon. In normal operation diagnostic messages are printed when invalid options or missing option arguments are encountered. OPTARG is unset. getopts normally parses the positional parameters. and name is set to ‘?’. getopts getopts optstring name [args] getopts is used by shell scripts to parse positional parameters.
or source.. information about remembered commands is printed. each name refers to a shell function. The ‘-a’ option means each name refers to an indexed array variable. The commands are found by searching through the directories listed in \$PATH. The ‘-r’ option causes the shell to forget all remembered locations. the return value is the exit status of the last command executed in the function. Any command associated with the RETURN trap is executed before execution resumes after the function or script. (or source) builtin. the value of the variable is set to value. The values of these names may not be changed by subsequent assignment. If the ‘-L’ option is supplied. If a variable name is followed by =value. The return status is zero unless an invalid option is supplied. pwd pwd [-LP] Print the absolute pathname of the current working directory. or the ‘-f’ option is supplied with a name that is not a shell function. The ‘-d’ option causes the shell to forget the remembered location of each name. the ‘-A’ option means each name refers to an associative array variable. If the ‘-P’ option is supplied. If the ‘-f’ option is supplied. This may also be used to terminate execution of a script being executed with the .38 Bash Reference Manual Remember the full pathnames of commands specified as name arguments. or if only ‘-l’ is supplied. The return status is zero unless a name is not found or an invalid option is supplied. The ‘-p’ option inhibits the path search. returning either n or the exit status of the last command executed within the script as the exit status of the script. The ‘-p’ option causes output to be displayed in a format that may be reused as input. the full pathname to which each name corresponds is printed.. return return [n] Cause a shell function to exit with the return value n. If no name arguments are given. The ‘-l’ option causes output to be displayed in a format that may be reused as input. the pathname printed will not contain symbolic links. so they need not be searched for on subsequent invocations. If no arguments are given. a list of all readonly names is printed. and filename is used as the location of name. readonly readonly [-aApf] [name[=value]] . one of the name arguments is not a valid shell variable or function name. the pathname printed may contain symbolic links. If the ‘-t’ option is supplied. The return status is non-zero if return is used outside a function and not during the execution of a script by . The return status is zero unless an error is encountered while determining the name of the current directory or an invalid option is supplied. If multiple name arguments are supplied with ‘-t’ the name is printed before the hashed full pathname. If n is not supplied. shift shift [n] . or if the ‘-p’ option is supplied. Mark each name as readonly.
. When the [ form is used. the result of the expression is the result of the binary test using the first and third arguments as operands.4 [Bash Conditional Expressions]. 1 argument The expression is true if and only if the argument is not null. test [ Evaluate a conditional expression expr. The positional parameters from n+1 . expr1 -o expr2 True if either expr1 or expr2 is true. the expression is false. \$#-n. it is assumed to be 1. Each operator and operand must be a separate argument. . \$# are renamed to \$1 . n must be a non-negative number less than or equal to \$#. 2 arguments If the first argument is ‘!’. The evaluation depends on the number of arguments. the expression is true if and only if the second argument is null. The test and [ builtins evaluate conditional expressions using a set of rules based on the number of arguments. test does not accept any options. the last argument to the command must be a ]. This may be used to override the normal precedence of operators. Returns the value of expr. Parameters represented by the numbers \$# to \$#-n+1 are unset. 3 arguments If the second argument is one of the binary conditional operators (see Section 6. page 76). Expressions may be combined using the following operators. nor does it accept and ignore an argument of ‘--’ as signifying the end of options. Expressions are composed of the primaries described below in Section 6. If the first argument is one of the unary conditional operators (see Section 6.4 [Bash Conditional Expressions]. If n is zero or greater than \$#. the positional parameters are not changed. the expression is true if the unary test is true. . 0 arguments The expression is false. non-zero otherwise. The return status is zero unless n is greater than \$# or less than zero.Chapter 4: Shell Builtin Commands 39 Shift the positional parameters to the left by n.4 [Bash Conditional Expressions]. If n is not supplied. If the . If the first argument is not a valid unary operator. see below. ! expr ( expr ) True if expr is false. page 76). . listed in decreasing order of precedence. expr1 -a expr2 True if both expr1 and expr2 are true. page 76. The ‘-a’ and ‘-o’ operators are considered binary operators when there are three arguments.
If a sigspec is DEBUG. The return status is zero. Refer to the description of the extdebug option to the shopt builtin (see Section 4.3. If the first argument is exactly ‘(’ and the third argument is exactly ‘)’. Otherwise.. times times Print out the user and system times used by the shell and its children. 4 arguments If the first argument is ‘!’. If arg is absent (and there is a single sigspec ) or equal to ‘-’. part of the test following the if or elif reserved words. case command. or source builtins finishes executing. page 55) for details of its effect on the DEBUG trap. or only ‘-p’ is given. each specified signal’s disposition is reset to the value it had when the shell was started. Each sigspec is either a signal name or a signal number.] The commands in arg are to be read and executed when the shell receives signal sigspec. the expression is parsed and evaluated according to precedence using the rules listed above. If a sigspec is RETURN. The ERR trap is not executed if the failed command is part of the command list immediately following an until or while keyword. every arithmetic for command.40 Bash Reference Manual first argument is ‘!’. part of a command executed in a && or || list. or if . the value is the negation of the two-argument test using the second and third arguments. the command arg is executed each time a shell function or a script executed with the . arg is executed when the shell exits. the command arg is executed before every simple command. and before the first command executes in a shell function. then the signal specified by each sigspec is ignored by the shell and commands it invokes. the result is the one-argument test of the second argument. If a sigspec is 0 or EXIT. subject to the following conditions. 5 or more arguments The expression is parsed and evaluated according to precedence using the rules listed above. the shell displays the trap commands associated with each sigspec. the result is the negation of the threeargument expression composed of the remaining arguments. the command arg is executed whenever a simple command has a non-zero exit status. Otherwise. the expression is false. If arg is the null string. Signal names are case insensitive and the SIG prefix is optional. trap prints the list of commands associated with each signal number in a form that may be reused as shell input. If no arguments are supplied. The ‘-l’ option causes the shell to print a list of signal names and their corresponding numbers.. If a sigspec is ERR. for command. trap trap [-lp] [arg] [sigspec .2 [The Shopt Builtin]. select command. If arg is not present and ‘-p’ has been supplied.
page 79. it is interpreted as a symbolic mode mask similar to that accepted by the chmod command. it is interpreted as an octal number. If no options are supplied. If mode is omitted. Thus. If the ‘-S’ option is supplied without a mode argument. unset unset [-fv] [name] Each variable or function name is removed. If the ‘-f’ option is given.Chapter 4: Shell Builtin Commands 41 the command’s return status is being inverted using !. and mode is omitted. If arguments are supplied. The return status is zero unless a name is readonly.2 Bash Builtin Commands This section describes builtin commands which are unique to or have been extended in Bash. umask umask [-p] [-S] [mode] Set the shell process’s file creation mask to mode. alias prints the list of aliases on the standard output in a form that allows them to be reused as input. If no value is given. Signals ignored upon entry to the shell cannot be trapped or reset. and non-zero otherwise. bind bind bind bind bind [-m [-m [-m [-m keymap] keymap] keymap] keymap] [-lpsvPSV] [-q function] [-u function] [-r keyseq] -f filename -x keyseq:shell-command . the name s refer to shell functions. each number of the umask is subtracted from 7. the current value of the mask is printed. These are the same conditions obeyed by the errexit option. each name refers to a shell variable. The return status is zero if the mode is successfully changed or if no mode argument is supplied. if not.. Some of these commands are specified in the posix standard. or the ‘-v’ option is given. Aliases are described in Section 6. Readonly variables and functions may not be unset.] Without arguments or with the ‘-p’ option. the output is in a form that may be reused as input. Note that when the mode is interpreted as an octal number. 4. and the function definition is removed. Trapped signals that are not being ignored are reset to their original values in a subshell or subshell environment when one is created. If the ‘-p’ option is supplied.6 [Aliases]. an alias is defined for each name whose value is given. If mode begins with a digit. the mask is printed in a symbolic format. alias alias [-p] [name[=value] .. a umask of 022 results in permissions of 755. the name and value of the alias is printed. The return status is zero unless a sigspec does not specify a valid signal.
subroutine name. In this case. If no other options are supplied with ‘-p’. caller caller [expr] Returns the context of any active subroutine call (a shell function or a script executed with the . When ‘-p’ is used with name arguments. declare will display the attributes and values of all variables having the attributes specified by the additional options.Chapter 4: Shell Builtin Commands 43 The return status is zero unless an invalid option is supplied or an error occurs. for example. The ‘-p’ option means to use a default value for PATH that is guaranteed to find all of the standard utilities. If either the ‘-V’ or ‘-v’ option is supplied. then display the values of variables instead. The current frame is frame 0. the ‘-V’ option produces a more verbose description. The return status is nonzero if shell-builtin is not a shell builtin command. Only shell builtin commands or commands found by searching the PATH are executed. declare will display . The ‘-v’ option causes a single word indicating the command or file name used to invoke command to be displayed. running ‘command ls’ within the function will execute the external command ls instead of calling the function recursively. command command [-pVv] command [arguments . and return its exit status. If no name s are given. retaining the functionality of the builtin within the function. This extra information may be used. a description of command is printed.. When ‘-p’ is supplied without name arguments.] Runs command with arguments ignoring any shell function named command. passing it args. This is useful when defining a shell function with the same name as a shell builtin. or source builtins).. declare declare [-aAfFilrtux] [-p] [name[=value] . caller displays the line number and source filename of the current subroutine call.. The return value is 0 unless the shell is not executing a subroutine call or expr does not correspond to a valid position in the call stack.. The ‘-p’ option will display the attributes and values of each name. and source file corresponding to that position in the current execution call stack.] Declare variables and give them attributes. additional options are ignored. and non-zero if not. the return status is zero if command is found. If a non-negative integer is supplied as expr. caller displays the line number. to print a stack trace. If there is a shell function named ls. The return status in this case is 127 if command cannot be found or an error occurred. Without expr. and the exit status of command otherwise. builtin builtin [shell-builtin [args]] Run a shell builtin.
With no other arguments. a list of shell builtins is printed. or no name arguments appear. If ‘-n’ is specified. The ‘-a’ option means to list each builtin with an indication of whether or not it is enabled. The ‘-d’ option will delete a builtin loaded with ‘-f’.. The return status is always 0. echo does not interpret ‘--’ to mean the end of options. type ‘enable -n test’. The ‘-f’ option means to load the new builtin command name from shared object filename. If the ‘-e’ option is given. Disabling a builtin allows a disk command which has the same name as a shell builtin to be executed without specifying a full pathname. the list consists of all enabled shell builtins. terminated with a newline. on systems that support dynamic loading. If the ‘-p’ option is supplied. Otherwise name s are enabled.] Output the arg s. the trailing newline is suppressed.. The xpg_ echo shell option may be used to dynamically determine whether or not echo expands these escape characters by default. the name s become disabled. For example. separated by spaces. If ‘-n’ is used. to use the test binary found via \$PATH instead of the shell builtin version. echo interprets the following escape sequences: \a \b \c \e \f \n \r \t \v \\ \0nnn \xHH enable enable [-a] [-dnps] [-f filename] [name . interpretation of the following backslash-escaped characters is enabled. even though the shell normally searches for builtins before disk commands.. even on systems where they are interpreted by default. alert (bell) backspace suppress further output escape form feed new line carriage return horizontal tab vertical tab backslash the eight-bit character whose value is the octal value nnn (zero to three octal digits) the eight-bit character whose value is the hexadecimal value HH (one or two hex digits) ..] Enable and disable builtin shell commands.Chapter 4: Shell Builtin Commands 45 echo [-neE] [arg . The ‘-E’ option disables the interpretation of these escape characters.
local can only be used within a function. it makes the variable name have a visible scope restricted to that function and its children.46 Bash Reference Manual If there are no options.. For each argument. mapfile mapfile [-n count] [-O origin] [-s count] [-t] [-u fd] [ -C callback] [-c quantum] [array] Read lines from the standard input into the indexed array variable array. Begin assigning to array at index origin. local local [option] name[=value] . help gives detailed help on all commands matching pattern. returning a status of n to the shell’s parent.4 [Special Builtins]. if supplied. Options.5 [Shell Arithmetic]. The default index is 0. The variable MAPFILE is the default array. If pattern is specified. if supplied. all lines are copied. page 78. If the last expression evaluates to 0. let let expression [expression] The let builtin allows arithmetic to be performed on shell variables. . The return status is zero unless a name is not a shell builtin or there is an error loading a new builtin from a shared object. Options. otherwise a list of the builtins is printed. If count is 0. The return status is zero unless local is used outside a function. have the following meanings: -n -O Copy at most count lines. If ‘-s’ is used with ‘-f’. logout logout [n] Exit a login shell. have the following meanings: -d -m -s Display a short description of each pattern Display the description of each pattern in a manpage-like format Display only a short usage synopsis for each pattern The return status is zero unless no command matches pattern. the new builtin becomes a special builtin (see Section 4. help help [-dms] [pattern] Display helpful information about builtin commands. The option can be any of the options accepted by declare. otherwise 0 is returned. a list of the shell builtins is displayed. a local variable named name is created.. page 59). or from file descriptor fd if the ‘-u’ option is supplied. or name is a readonly variable. and assigned value. Each expression is evaluated according to the rules given below in Section 6. an invalid name is supplied. let returns 1. The ‘-s’ option restricts enable to the posix special builtins.
then nothing is printed.] For each name. If the ‘-f’ option is used.. or nothing if ‘-t’ would not return ‘file’. (see Section 4. If the ‘-p’ option is used. readarray [-n count] [-O origin] [-s count] [-t] [-u fd] [ -C callback] [-c quantum] [array] Read lines from the standard input into the indexed array variable array. type returns all of the places that contain an executable named file. If the name is not found. The ‘-P’ option forces a path search for each name. type either returns the name of the disk file that would be executed. type prints a single word which is one of ‘alias’. typeset typeset [-afFrxi] [-p] [name[=value] . shell builtin. or shell reserved word. indicate how it would be interpreted if used as a command name. ulimit ulimit [-abcdefilmnpqrstuvxHST] [limit] ulimit provides control over the resources available to processes started by the shell. A synonym for mapfile.. however. on systems that allow such control. If a command is hashed. even if ‘-t’ would not return ‘file’.1 [Bourne Shell Builtins]. If the ‘-t’ option is used. ‘function’. as with the command builtin.. ‘builtin’.Chapter 4: Shell Builtin Commands 49 -u fd readarray Read input from file descriptor fd. if name is an alias. ‘-p’ and ‘-P’ print the hashed value. source source filename A synonym for . shell function. it has been deprecated in favor of the declare builtin command. and type returns a failure status. The return status is zero if all of the names are found. This includes aliases and functions.. disk file. non-zero if any are not found. or from file descriptor fd if the ‘-u’ option is supplied. type does not attempt to find shell functions. respectively. not necessarily the file that appears first in \$PATH. type type [-afptP] [name . page 35). If an option is given.] The typeset command is supplied for compatibility with the Korn shell. . it is interpreted as follows: -S Change and report the soft limit associated with a resource. If the ‘-a’ option is used. if and only if the ‘-p’ option is not also used. ‘file’ or ‘keyword’.
all aliases are removed. unalias unalias [-a] [name . The maximum real-time scheduling priority. both the hard and soft limits are set. A hard limit cannot be increased by a non-root user once it is set. ‘-p’. The maximum socket buffer size. the current soft limit.. The maximum number of file locks. the special limit values hard. The maximum number of open file descriptors (most systems do not allow this value to be set). Aliases are described in Section 6.50 Bash Reference Manual -H -a -b -c -d -e -f -i -l -m -n -p -q -r -s -t -u -v -x -T Change and report the hard limit associated with a resource. except for ‘-t’. If limit is given. The maximum number of threads. If no option is given. and no limit. .6 [Aliases]. Otherwise. The maximum amount of virtual memory available to the process. When setting new limits. soft. The maximum stack size. if neither ‘-H’ nor ‘-S’ is supplied. or an error occurs while setting a new limit. All current limits are reported. unless the ‘-H’ option is supplied. The maximum scheduling priority ("nice"). The return status is zero unless an invalid option or argument is supplied. page 79. which is in units of 512-byte blocks. If ‘-a’ is supplied. and unlimited stand for the current hard limit. respectively. The maximum resident set size (many systems do not honor this limit). and ‘-n’ and ‘-u’. ] Remove each name from the list of aliases. a soft limit may be increased up to the value of the hard limit. The maximum size that may be locked into memory. The maximum number of bytes in POSIX message queues. The maximum number of pending signals. The maximum size of files written by the shell and its children. which is in seconds. it is the new value of the specified resource. Values are in 1024-byte increments. The maximum amount of cpu time in seconds. The pipe buffer size. The maximum number of processes available to a single user. The maximum size of a process’s data segment. the current value of the soft limit for the specified resource is printed.. which are unscaled values. then ‘-f’ is assumed. The maximum size of core files created.
3 [Command Grouping].3 Modifying Shell Behavior 4. or one of the commands executed as part of a command list enclosed by braces (see Section 3. rather than before printing the next primary prompt. sorted according to the current locale. which may consist of a single simple command (see Section 3.. Exit immediately if a pipeline (see Section 3. A trap on ERR.2. a subshell command enclosed in parentheses (see Section 3. is executed before the shell exits.] set [+abefhkmnptuvxBCEHPT] [+o option] [argument . in a format that may be reused as input for setting or resetting the currently-set variables. only shell variables are listed.3 [Command Grouping].1 [Simple Commands].4.3.2 [Pipelines]. When options are supplied.4. Options. set set [--abefhkmnptuvxBCEHPT] [-o option] [argument .3 [Command Execution Environment]. if set. or if the command’s return status is being inverted with !.] If no options or arguments are supplied. -f -h Disable filename expansion (globbing). page 13) returns a non-zero status. page 30). The shell does not exit if the command that fails is part of the command list immediately following a while or until keyword. Read-only variables cannot be reset. .2. any command in a pipeline but the last. set displays the names and values of all shell variables and functions. This option is enabled by default. part of any command executed in a && or || list except the command following the final && or ||.Chapter 4: Shell Builtin Commands 51 4. Cause the status of terminated background jobs to be reported immediately. page 8).1 The Set Builtin This builtin is so complicated that it deserves its own section..7. In posix mode.. or to display the names and values of shell variables. page 13).2.. if specified. This option applies to the shell environment and each subshell environment separately (see Section 3. they set or unset shell attributes. Locate and remember (hash) commands as they are looked up for execution. set allows you to change the values of shell options and set the positional parameters.2. part of the test in an if statement. have the following meanings: -a -b -e Mark variables and function which are modified or created for export to the environment of subsequent commands. and may cause subshells to exit before executing all the commands in the subshell. page 8).
keyword monitor noclobber Same as -C. An interactive shell will not exit upon reading EOF. Same as -u. Same as -k. Currently ignored. Read commands but do not execute them. This option is ignored by interactive shells. Same as -b. Same as -f. Same as -e. as described in Section 9. braceexpand Same as -B. page 89).52 Bash Reference Manual -k All arguments in the form of assignment statements are placed in the environment for a command. page 121. This also affects the editing interface used for read -e. Job control is enabled (see Chapter 7 [Job Control]. Same as -T. Same as -h. errexit errtrace functrace ignoreeof . history Enable command history. Same as -E. not just those that precede the command name. -m -n -o option-name Set the option corresponding to option-name : allexport Same as -a. page 93). This option is on by default in interactive shells. noexec noglob nolog notify nounset Same as -n. this may be used to check a script for syntax errors. emacs Use an emacs-style line editing interface (see Chapter 8 [Command Line Editing].1 [Bash History Facilities]. hashall histexpand Same as -H. Same as -m.
Same as -P. Print a trace of simple commands.11 [Bash POSIX Mode]. This is intended to make Bash behave as a strict superset of that standard. The value of the PS4 variable is expanded and the resultant value is printed before the command and its expanded arguments. the effective user id is not reset. Same as -p. BASHOPTS. the return value of a pipeline is the value of the last (rightmost) command to exit with a non-zero status. Print shell input lines as they are read. If set. select commands. CDPATH and GLOBIGNORE variables. the \$BASH_ENV and \$ENV files are not processed. for commands. and the -p option is not supplied. The shell will perform brace expansion (see Section 3. Turning this option off causes the effective user and group ids to be set to the real user and group ids. case commands. and the SHELLOPTS. An error message will be written to the standard error. Turn on privileged mode. are ignored. these actions are taken and the effective user id is set to the real user id. If the -p option is supplied at startup. posix privileged verbose vi xtrace -p Same as -v.5. This also affects the editing interface used for read -e. and a noninteractive shell will exit. page 18). Same as -x. -t -u -v -x -B .Chapter 4: Shell Builtin Commands 53 onecmd physical pipefail Same as -t.1 [Brace Expansion]. If the shell is started with the effective user (group) id not equal to the real user (group) id. Use a vi-style line editing interface. Treat unset variables and parameters other than the special parameters ‘@’ or ‘*’ as an error when performing parameter expansion. page 84). if they appear in the environment. and arithmetic for commands and their arguments or associated word lists after they are expanded and before they are executed. In this mode. shell functions are not inherited from the environment. Exit after reading and executing one command. or zero if all commands in the pipeline exit successfully. This option is disabled by default. Change the behavior of Bash where the default operation differs from the posix standard to match the standard (see Section 6. This option is on by default.
Bash follows the logical chain of directories when performing commands which change the current directory. if ‘/usr/sys’ is a symbolic link to ‘/usr/local/sys’ then: \$ cd /usr/sys. pwd /usr/local -H -P -T If set. cause all remaining arguments to be assigned to the positional parameters. Signal the end of options.3 [History Interaction]. If set. any trap on ERR is inherited by shell functions. The ERR trap is normally not inherited in such cases. If set. The ‘-x’ and ‘-v’ options are turned off. The current set of options may be found in \$-. command substitutions. any trap on DEBUG and RETURN are inherited by shell functions. then the positional parameters are unset. .. ‘>&’. do not follow symbolic links when performing commands such as cd which change the current directory. The DEBUG and RETURN traps are normally not inherited in such cases.. The physical directory is used instead. \$2.. Otherwise. echo \$PWD /usr/sys \$ cd .. echo \$PWD /usr/local/sys \$ cd . in order. pwd /usr If set -P is on. to \$1. \$N. the positional parameters remain unchanged. By default.54 Bash Reference Manual -C -E Prevent output redirection using ‘>’. The special parameter # is set to N. The options can also be used upon invocation of the shell. and commands executed in a subshell environment. even if some of them begin with a ‘-’. The remaining N arguments are positional parameters and are assigned. This option is on by default for interactive shells. For example. page 123). . Enable ‘!’ style history substitution (see Section 9. and ‘<>’ from overwriting existing files. the positional parameters are set to the arguments. If there are no arguments. -- - Using ‘+’ rather than ‘-’ causes these options to be turned off. then: \$ cd /usr/sys. command substitutions. and commands executed in a subshell environment. If no arguments follow this option. The return status is always zero unless an invalid option is supplied. . .
nonzero otherwise. an argument to the cd builtin command that is not a directory is assumed to be the name of a variable whose value is the directory to change to. Unless otherwise noted. the return status is zero unless an optname is not a valid shell option. Suppresses normal output. The ‘-p’ option causes output to be displayed in a form that may be reused as input. a missing character.2 The Shopt Builtin This builtin allows you to change additional shell optional behavior. a command name that is the name of a directory is executed as if it were the argument to the cd command. Bash checks that a command found in the hash table exists before trying to execute it. With no options. The errors checked for are transposed characters. cdable_vars If this is set.. The list of shopt options is: autocd If set. checkhash . the corrected path is printed. minor errors in the spelling of a directory component in a cd command will be corrected. a normal path search is performed. the display is limited to those options which are set or unset. Other options have the following meanings: -s -u -q Enable (set) each optname. respectively. This option is only used by interactive shells. If this is set. -o If either ‘-s’ or ‘-u’ is used with no optname arguments. or with the ‘-p’ option.Chapter 4: Shell Builtin Commands 55 4. Disable (unset) each optname. Restricts the values of optname to be those defined for the ‘-o’ option to the set builtin (see Section 4. If a correction is found.] Toggle the values of variables controlling optional shell behavior. If multiple optname arguments are given with ‘-q’.3. shopt shopt [-pqsu] [-o] [optname . This option is only used by interactive shells. a list of all settable options is displayed. The return status when listing options is zero if all optnames are enabled. If a hashed command no longer exists. non-zero otherwise. page 51). with an indication of whether or not each is set.1 [The Set Builtin]. the shopt options are disabled (off) by default.. and the command proceeds. When setting or unsetting options. the return status is zero if all optnames are enabled.3. and a character too many. the return status indicates whether the optname is set or unset. cdspell If set.
If any jobs are running. behavior intended for use by debuggers is enabled: 1. a non-interactive shell will not exit if it cannot execute the file specified as an argument to the exec builtin command.2 [Bash Variables]. or source builtins). BASH_ARGC and BASH_ARGV are updated as described in their descriptions (see Section 5. and subshells invoked with ( command ) inherit the DEBUG and RETURN traps. page 61). Bash lists the status of any stopped and running jobs before exiting an interactive shell. if necessary. If the command run by the DEBUG trap returns a value of 2. Bash checks the window size after each command and. compat31 dirspell dotglob execfail expand_aliases If set.1 with respect to quoted arguments to the conditional command’s =~ operator. The shell always postpones exiting if any jobs are stopped. If this is set. Function tracing is enabled: command substitution.2 [Bash Builtins]. 4. 3. The ‘-F’ option to the declare builtin (see Section 4. Bash attempts spelling correction on directory names during word completion if the directory name initially supplied does not exist. 2. Bash includes filenames beginning with a ‘. If the command run by the DEBUG trap returns a non-zero value. 5. Bash attempts to save all lines of a multiple-line command in the same history entry.6 [Aliases]. An interactive shell does not exit if exec fails.’ in the results of filename expansion. page 89). cmdhist If set. This allows easy re-editing of multi-line commands. shell functions. and the shell is executing in a subroutine (a shell function or a shell script executed by the . Bash changes its behavior to that of version 3. the next command is skipped and not executed. If set. a call to return is simulated. this causes the exit to be deferred until a second exit is attempted without an intervening command (see Chapter 7 [Job Control].56 Bash Reference Manual checkjobs If set. If set. aliases are expanded as described below under Aliases. extdebug If set. checkwinsize If set. This option is enabled by default for interactive shells. updates the values of LINES and COLUMNS. . page 79. If set. page 41) displays the source file name and line number corresponding to each function name supplied as an argument. Section 6.
Error tracing is enabled: command substitution. shell functions. This option is enabled by default. histverify If set. and Readline is being used. page 24) are enabled. the extended pattern matching features described above (see Section 3. allowing further modification. only directories and subdirectories match. hostcomplete If set. huponexit If set.1 [Pattern Matching]. This option is enabled by default. histreedit If set. extglob extquote If set. the results of history substitution are not immediately passed to the shell parser.6 [Signals]. the pattern ‘**’ used in a filename expansion context will match a files and zero or more directories and subdirectories.5. Instead. Bash will send SIGHUP to all jobs when an interactive login shell exits (see Section 3. the history list is appended to the file named by the value of the HISTFILE variable when the shell exits.4. rather than overwriting the file. and Readline is being used. for a description of FIGNORE. If set. globstar If set.8.2 [Bash Variables]. page 32). If set.Chapter 4: Shell Builtin Commands 57 6. If set. \$’string’ and \$"string" quoting is performed within \${parameter} expansions enclosed in double quotes. patterns which fail to match filenames during filename expansion result in an expansion error. shell error messages are written in the standard gnu error message format. and subshells invoked with ( command ) inherit the ERROR trap.7. the suffixes specified by the FIGNORE shell variable cause words to be ignored when performing word completion even if the ignored words are the only possible completions. the resulting line is loaded into the Readline editing buffer.6 [Commands For Completion]. and Readline is being used. a user is given the opportunity to re-edit a failed history substitution. Bash will attempt to perform hostname completion when a word containing a ‘@’ is being completed (see Section 8. This option is enabled by default. histappend If set. page 61. failglob force_fignore If set. If the pattern is followed by a ‘/’. gnu_errfmt . See Section 5. page 110).
10 [The Restricted Shell]. lithist If enabled. restricted_shell The shell sets this option if it is started in restricted mode (see Section 6.58 Bash Reference Manual interactive_comments Allow a word beginning with ‘#’ to cause that word and all remaining characters on that line to be ignored in an interactive shell. If set. nullglob progcomp If set. page 82). The value may not be changed. and the cmdhist option is enabled. If set. mailwarn If set. This option is enabled by default. page 115) are enabled. This is not reset when the startup files are executed. arithmetic expansion. page 84). This option is enabled by default. page 71). The value may not be changed. multi-line commands are saved to the history with embedded newlines rather than using semicolon separators where possible.1 [Invoking Bash]. rather than themselves. and Readline is being used. the programmable completion facilities (see Section 8. no_empty_cmd_completion If set.6 [Programmable Completion]. prompt strings undergo parameter expansion. Bash allows filename patterns which match no files to expand to a null string. the message "The mail in mailfile has been read" is displayed. Bash matches filenames in a case-insensitive fashion when performing filename expansion. allowing the startup files to discover whether or not a shell is restricted. command substitution. login_shell The shell sets this option if it is started as a login shell (see Section 6. Bash matches patterns in a case-insensitive fashion when performing matching while executing case or [[ conditional commands. and a file that Bash is checking for mail has been accessed since the last time it was checked. This option is enabled by default. and quote removal after being expanded as described below (see Section 6. nocasematch If set. promptvars . nocaseglob If set.9 [Printing a Prompt]. Bash will not attempt to search the PATH for possible completions when completion is attempted on an empty line.
the shift builtin prints an error message when the shift count exceeds the number of positional parameters. The Bash posix mode is described in Section 6. the echo builtin expands backslash-escape sequences by default. Assignment statements preceding the command stay in effect in the shell environment after the command completes. the special builtins differ from other builtin commands in three respects: 1. Special builtins are found before shell functions during command lookup. 4. page 84. the source builtin uses the value of PATH to find the directory containing the file supplied as an argument. a non-interactive shell exits. xpg_echo If set. the return status is zero unless an optname is not a valid shell option. sourcepath If set. these builtins behave no differently than the rest of the Bash builtin commands. If a special builtin returns an error status. When Bash is executing in posix mode. the posix standard has classified several builtin commands as special. This option is enabled by default. These are the posix special builtins: break : . When Bash is not executing in posix mode.Chapter 4: Shell Builtin Commands 59 shift_verbose If this is set. The return status when listing options is zero if all optnames are enabled.4 Special Builtins For historical reasons. 2. 3.11 [Bash POSIX Mode]. continue eval exec exit export readonly return set shift trap unset . When setting or unsetting options. nonzero otherwise.
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the default for the cd builtin command. The value of this variable is also used by tilde expansion (see Section 3. A list of characters that separate fields. If this parameter is set to a filename and the MAILPATH variable is not set. CDPATH HOME A colon-separated list of directories used as a search path for the cd builtin command. or as an initial or trailing colon. A colon-separated list of directories in which the shell looks for commands. The default value is ‘\s-\v\\$ ’.Chapter 5: Shell Variables 61 5 Shell Variables This chapter describes the shell variables that Bash uses. See Section 6. page 82. page 92). A few variables used by Bash are described in different chapters: variables for controlling the job control facilities (see Section 7. for the complete list of escape sequences that are expanded before PS1 is displayed.1 Bourne Shell Variables Bash uses certain shell variables in the same way as the Bourne shell.9 [Printing a Prompt]. The value of the last option argument processed by the getopts builtin. Bash assigns a default value to the variable. A colon-separated list of filenames which the shell periodically checks for new mail. \$_ expands to the name of the current mail file. The default value is ‘> ’. When used in the text of the message.5. IFS MAIL MAILPATH OPTARG OPTIND PATH PS1 PS2 5. page 19). The current user’s home directory.3 [Job Control Variables]. The primary prompt string. The index of the last option argument processed by the getopts builtin.2 [Tilde Expansion]. Bash automatically assigns default values to a number of variables. A null directory name may appear as two adjacent colons. but other shells do not normally treat them specially. Each list entry can specify the message that is printed when new mail arrives in the mail file by separating the file name from the message with a ‘?’. The secondary prompt string. used when the shell splits words as part of expansion. Bash informs the user of the arrival of mail in the specified file. . In some cases.2 Bash Variables These variables are set or used by Bash. BASH The full pathname used to execute the current instance of Bash. A zero-length (null) directory name in the value of PATH indicates the current directory. 5.
This variable is readonly. When a subroutine is executed. BASH_COMMAND The command currently being executed or about to be executed. BASHPID BASH_ALIASES An associative array variable whose members correspond to the internal list of aliases as maintained by the alias builtin (see Section 4.2 [The Shopt Builtin]. page 35). page 55 for a description of the extdebug option to the shopt builtin). When a subroutine is executed. the parameters supplied are pushed onto BASH_ARGV.62 Bash Reference Manual BASHOPTS A colon-separated list of enabled shell options. Elements added to this array appear in the alias list.2 [Bash Startup Files]. page 55 for a description of the extdebug option to the shopt builtin).1 [Bourne Shell Builtins]. The number of parameters to the current subroutine (shell function or script executed with . the number of parameters passed is pushed onto BASH_ARGC. in which case it is the command executing at the time of the trap. such as subshells that do not require Bash to be re-initialized. The final parameter of the last subroutine call is at the top of the stack. BASH_EXECUTION_STRING The command argument to the ‘-c’ invocation option. BASH_CMDS An associative array variable whose members correspond to the internal hash table of commands as maintained by the hash builtin (see Section 4. . Expands to the process id of the current Bash process. BASH_ARGV An array variable containing all of the parameters in the current bash execution call stack. If this variable is in the environment when Bash starts up.3. page 55).3. The options appearing in BASHOPTS are those reported as ‘on’ by ‘shopt’.1 [Bourne Shell Builtins]. This differs from \$\$ under certain circumstances. unless the shell is executing a command as the result of a trap. The shell sets BASH_ARGV only when in extended debugging mode (see Section 4.2 [The Shopt Builtin].3.2 [The Shopt Builtin]. BASH_ARGC An array variable whose values are the number of parameters in each frame of the current bash execution call stack. its value is expanded and used as the name of a startup file to read before executing the script. The shell sets BASH_ARGC only when in extended debugging mode (see Section 4. Elements added to this array appear in the hash table. page 73. See Section 6. each shell option in the list will be enabled before reading any startup files. page 35). unsetting array elements cause commands to be removed from the hash table. the first parameter of the initial call is at the bottom. unsetting array elements cause aliases to be removed from the alias list. Each word in the list is a valid argument for the ‘-s’ option to the shopt builtin command (see Section 4. BASH_ENV If this variable is set when Bash is invoked to execute a shell script. or source) is at the top of the stack.
This variable is read-only. The corresponding source file name is \${BASH_SOURCE[\$i]}. BASH_VERSINFO A readonly array variable (see Section 6. . The element with index n is the portion of the string matching the nth parenthesized subexpression. The initial value is 0. \${BASH_LINENO[\$i]} is the line number in the source file where \${FUNCNAME[\$i]} was called (or \${BASH_ LINENO[\$i-1]} if referenced within another shell function). The file descriptor is closed when BASH_XTRACEFD is unset or assigned a new value. BASH_VERSINFO[3] The build version.7 [Arrays]. BASH_VERSION The version number of the current instance of Bash. BASH_VERSINFO[4] The release status (e. beta1 ). BASH_SUBSHELL Incremented by one each time a subshell or subshell environment is spawned.Chapter 5: Shell Variables 63 BASH_LINENO An array variable whose members are the line numbers in source files corresponding to each member of FUNCNAME. BASH_VERSINFO[2] The patch level. BASH_XTRACEFD If set to an integer corresponding to a valid file descriptor. The values assigned to the array members are as follows: BASH_VERSINFO[0] The major version number (the release ). Use LINENO to obtain the current line number. page 10). BASH_REMATCH An array variable whose members are assigned by the ‘=~’ binary operator to the [[ conditional command (see Section 3. page 80) whose members hold version information for this instance of Bash. Bash will write the trace output generated when ‘set -x’ is enabled to that file descriptor.2 [Conditional Constructs].g..2. BASH_SOURCE An array variable whose members are the source filenames corresponding to the elements in the FUNCNAME array variable. BASH_VERSINFO[1] The minor version number (the version).4. BASH_VERSINFO[5] The value of MACHTYPE. This allows tracing output to be separated from diagnostic and error messages. The element with index 0 is the portion of the string matching the entire regular expression.
page 115). This variable is available only in shell functions invoked by the programmable completion facilities (see Section 8. COMP_WORDBREAKS The set of characters that the Readline library treats as word separators when performing word completion. page 115). . for listing alternatives on partial word completion. COLUMNS COMP_CWORD An index into \${COMP_WORDS} of the word containing the current cursor position. for menu completion. Used by the select builtin command to determine the terminal width when printing selection lists.6 [Programmable Completion]. If COMP_WORDBREAKS is unset. ‘@’. Automatically set upon receipt of a SIGWINCH. COMP_KEY The key (or final key of a key sequence) used to invoke the current completion function. Note that setting BASH_XTRACEFD to 2 (the standard error file descriptor) and then unsetting it will result in the standard error being closed. for normal completion. or ‘%’. ‘!’.6 [Programmable Completion]. ‘?’. page 115). If the current cursor position is at the end of the current command. COMP_TYPE Set to an integer value corresponding to the type of completion attempted that caused a completion function to be called: TAB. page 115). even if it is subsequently reset. COMP_LINE The current command line.6 [Programmable Completion]. page 115). COMP_POINT The index of the current cursor position relative to the beginning of the current command. This variable is available only in shell functions and external commands invoked by the programmable completion facilities (see Section 8. it loses its special properties. This variable is available only in shell functions and external commands invoked by the programmable completion facilities (see Section 8. the value of this variable is equal to \${#COMP_LINE}. The line is split into words as Readline would split it. COMP_WORDS An array variable consisting of the individual words in the current command line.6 [Programmable Completion]. This variable is available only in shell functions invoked by the programmable completion facilities (see Section 8.6 [Programmable Completion]. to list completions if the word is not unmodified. This variable is available only in shell functions and external commands invoked by the programmable completion facilities (see Section 8. using COMP_ WORDBREAKS as described above. for listing completions after successive tabs.64 Bash Reference Manual Unsetting BASH_XTRACEFD or assigning it the empty string causes the trace output to be sent to the standard error.
Assignments to GROUPS have no effect and return an error status. The bottom-most element is "main". GROUPS An array variable containing the list of groups of which the current user is a member. it is removed from the list of matches. that is. and tokenization (see Section 9. If Bash finds this variable in the environment when the shell starts with value ‘t’. even if it is subsequently reset. even if it is subsequently reset. it loses its special properties. The numeric effective user id of the current user. Up to three characters which control history expansion. The editor used as a default by the ‘-e’ option to the fc builtin command. Assignment to this variable will not change the current directory. The second character is the character which signifies ‘quick substitution’ when seen as the first character on a line. it loses its special properties.6 [Programmable Completion]. A sample value is ‘. If DIRSTACK is unset. it loses its special properties. Directories appear in the stack in the order they are displayed by the dirs builtin. DIRSTACK An array variable containing the current contents of the directory stack. If GROUPS is unset. normally ‘!’. EMACS EUID FCEDIT FIGNORE FUNCNAME GLOBIGNORE histchars . A colon-separated list of suffixes to ignore when performing filename completion. If FUNCNAME is unset. A file name whose suffix matches one of the entries in FIGNORE is excluded from the list of matched file names. If a filename matched by a filename expansion pattern also matches one of the patterns in GLOBIGNORE. normally ‘^’. Assigning to members of this array variable may be used to modify directories already in the stack.3 [History Interaction]. page 123). the character which signifies the start of a history expansion. The history comment character causes history substitution to be skipped for the remaining words on the line. The element with index 0 is the name of any currentlyexecuting shell function. but the pushd and popd builtins must be used to add and remove directories. Assignments to FUNCNAME have no effect and return an error status. This variable is readonly. It does not necessarily cause the shell parser to treat the rest of the line as a comment. This variable exists only when a shell function is executing. usually ‘#’.o:~’ An array variable containing the names of all shell functions currently in the execution call stack. The first character is the history expansion character. The optional third character is the character which indicates that the remainder of the line is a comment when found as the first character of a word. page 115). quick substitution. even if it is subsequently reset.Chapter 5: Shell Variables 65 COMPREPLY An array variable from which Bash reads the possible completions generated by a shell function invoked by the programmable completion facility (see Section 8. it assumes that the shell is running in an emacs shell buffer and disables line editing. A colon-separated list of patterns defining the set of filenames to be ignored by filename expansion.
HISTFILE The name of the file to which the command history is saved. Combining these two patterns.66 Bash Reference Manual HISTCMD The history number. subject to the value of HISTIGNORE. by removing the oldest entries. and are added to the history regardless of the value of HISTIGNORE. The default value is 500. ‘&’ may be escaped using a backslash. all lines read by the shell parser are saved on the history list. If the list of values includes ‘ignorespace’. The default value is 500. time stamps are written to the history file so they may be preserved across shell sessions. or does not include a valid value. If HISTCMD is unset. A value of ‘erasedups’ causes all previous lines matching the current line to be removed from the history list before that line is saved. the backslash is removed before attempting a match. A pattern of ‘&’ is identical to ignoredups. Each pattern is tested against the line after the checks specified by HISTCONTROL are applied. Each pattern is anchored at the beginning of the line and must match the complete line (no implicit ‘*’ is appended). The second and subsequent lines of a multi-line compound command are not tested. HISTSIZE The maximum number of commands to remember on the history list. The second and subsequent lines of a multi-line compound command are not tested. its value is used as a format string for strftime to print the time stamp associated with each history entry displayed by the history builtin. The default value is ‘~/. HISTIGNORE subsumes the function of HISTCONTROL. Any value not in the above list is ignored.bash_history’. The history file is also truncated to this size after writing it when an interactive shell exits. . lines which begin with a space character are not saved in the history list. separating them with a colon. If HISTCONTROL is unset. This uses the history comment character to distinguish timestamps from other history lines. HISTFILESIZE The maximum number of lines contained in the history file. A value of ‘ignoreboth’ is shorthand for ‘ignorespace’ and ‘ignoredups’. HISTTIMEFORMAT If this variable is set and not null. it loses its special properties. even if it is subsequently reset. When this variable is assigned a value. and a pattern of ‘[ ]*’ is identical to ignorespace. HISTCONTROL A colon-separated list of values controlling how commands are saved on the history list. if necessary. In addition to the normal shell pattern matching characters. A value of ‘ignoredups’ causes lines which match the previous history entry to not be saved. or index in the history list. ‘&’ matches the previous history line. the history file is truncated. and are added to the history regardless of the value of HISTCONTROL. of the current command. If this variable is set. HISTIGNORE A colon-separated list of patterns used to decide which command lines should be saved on the history list. provides the functionality of ignoreboth. to contain no more than that number of lines.
5. If the variable does not exist. or does not name a readable file. the hostname list is cleared. LC_NUMERIC This variable determines the locale category used for number formatting. and collating sequences within filename expansion and pattern matching (see Section 3. but has no value. If the variable exists but does not have a numeric value (or has no value) then the default is 10. Used by the select builtin command to determine the column length for printing selection lists. HOSTNAME HOSTTYPE IGNOREEOF INPUTRC LANG LC_ALL LC_COLLATE The name of the Readline initialization file. and determines the behavior of range expressions. Used to determine the locale category for any category not specifically selected with a variable starting with LC_. equivalence classes.8 [Filename Expansion]. LC_MESSAGES This variable determines the locale used to translate double-quoted strings preceded by a ‘\$’ (see Section 3.5 [Locale Translation].Chapter 5: Shell Variables 67 HOSTFILE Contains the name of a file in the same format as ‘/etc/hosts’ that should be read when the shell needs to complete a hostname. page 7). in the standard gnu cpu-company-system format.2. This variable determines the collation order used when sorting the results of filename expansion. A string describing the machine Bash is running on.1. page 24). This is only in effect for interactive shells. . page 24). If HOSTFILE is set. This variable overrides the value of LANG and any other LC_ variable specifying a locale category. LINENO LINES MACHTYPE The line number in the script or shell function currently executing. Bash attempts to read ‘/etc/hosts’ to obtain the list of possible hostname completions. Bash adds the contents of the new file to the existing list. LC_CTYPE This variable determines the interpretation of characters and the behavior of character classes within filename expansion and pattern matching (see Section 3. overriding the default of ‘~/. The name of the current host.inputrc’. then EOF signifies the end of input to the shell. Automatically set upon receipt of a SIGWINCH. The list of possible hostname completions may be changed while the shell is running.5. Controls the action of the shell on receipt of an EOF character as the sole input. the next time hostname completion is attempted after the value is changed. When HOSTFILE is unset.8 [Filename Expansion]. the value denotes the number of consecutive EOF characters that can be read as the first character on an input line before the shell will exit. A string that fully describes the system type on which Bash is executing. If set.
The default is ‘+ ’. The default variable for the read builtin. If it is set while the shell is running.1 [The Set Builtin]. PROMPT_COMMAND If set. The previous working directory as set by the cd builtin.9 [Printing a Prompt]. a random integer between 0 and 32767 is generated. If this variable is not set. to indicate multiple levels of indirection. the value is used as the number of trailing directory components to retain when expanding the \w and \W prompt string escapes (see Section 6.11 [Bash POSIX Mode]. When it is time to check for mail. This variable is readonly. as if the ‘--posix’ invocation option had been supplied. the select command prompts with ‘#? ’ The value is the prompt printed before the command line is echoed when the ‘-x’ option is set (see Section 4. The current working directory as set by the cd builtin. as if the command set -o posix had been executed. A string describing the operating system Bash is running on. Assigning a value to this variable seeds the random number generator. Bash displays error messages generated by the getopts builtin command. or set to a value that is not a number greater than or equal to zero. If this variable is unset. the shell enters posix mode (see Section 6. The default is 60 seconds. OLDPWD OPTERR OSTYPE PIPESTATUS An array variable (see Section 6. PS3 PS4 The value of this variable is used as the prompt for the select command. page 84) before reading the startup files. PWD RANDOM REPLY .7 [Arrays].68 Bash Reference Manual MAILCHECK How often (in seconds) that the shell should check for mail in the files specified in the MAILPATH or MAIL variables. POSIXLY_CORRECT If this variable is in the environment when bash starts. PROMPT_DIRTRIM If set to a number greater than zero. as necessary. the value is interpreted as a command to execute before the printing of each primary prompt (\$PS1). page 82). page 80) containing a list of exit status values from the processes in the most-recently-executed foreground pipeline (which may contain only a single command). page 51). PPID The process id of the shell’s parent process. bash enables posix mode. the shell disables mail checking. If set to the value 1. Characters removed are replaced with an ellipsis. the shell does so before displaying the primary prompt.3. The first character of PS4 is replicated multiple times. Each time this parameter is referenced.
This variable is readonly. the value 3 is used. The value of this parameter is used as a format string specifying how the timing information for pipelines prefixed with the time reserved word should be displayed. At most three places after the decimal point may be specified.2 [Bash Builtins]. values of p greater than 3 are changed to 3. The elapsed time in seconds. If this variable is not set. A colon-separated list of enabled shell options. TMOUT If set to a value greater than zero.3. including minutes.FF s.2 [Conditional Constructs].1 [The Set Builtin]. Each word in the list is a valid argument for the ‘-o’ option to the set builtin command (see Section 4. TMOUT is treated as the default timeout for the read builtin (see Section 4.2. page 10) terminates if input does not arrive after TMOUT seconds when input is coming from a terminal. A value of 0 causes no decimal point or fraction to be output. The full pathname to the shell is kept in this environment variable. Bash assigns to it the full pathname of the current user’s login shell. The value of p determines whether or not the fraction is included. The optional p is a digit specifying the precision. A trailing newline is added when the format string is displayed. Bash acts as if it had the value \$’\nreal\t%3lR\nuser\t%3lU\nsys\t%3lS’ If the value is null.4. The number of CPU seconds spent in user mode. The select command (see Section 3. The ‘%’ character introduces an escape sequence that is expanded to a time value or other information. Assignment to this variable resets the count to the value assigned. and the expanded value becomes the value assigned plus the number of seconds since the assignment. The escape sequences and their meanings are as follows. page 51). no timing information is displayed. . each shell option in the list will be enabled before reading any startup files. page 41). the braces denote optional portions. computed as (%U + %S) / %R. The CPU percentage. SHELL SHELLOPTS SHLVL TIMEFORMAT Incremented by one each time a new instance of Bash is started.Chapter 5: Shell Variables 69 SECONDS This variable expands to the number of seconds since the shell was started. %% %[p][l]R %[p][l]U %[p][l]S %P A literal ‘%’. If this variable is in the environment when Bash starts up. the number of fractional digits after a decimal point. This is intended to be a count of how deeply your Bash shells are nested. If p is not specified. If it is not set when the shell starts. of the form MM mSS. The options appearing in SHELLOPTS are those reported as ‘on’ by ‘set -o’. The number of CPU seconds spent in system mode. The optional l specifies a longer format.
. the value is interpreted as the number of seconds to wait for input after issuing the primary prompt when the shell is interactive. TMPDIR UID If set. Bash uses its value as the name of a directory in which Bash creates temporary files for the shell’s use. Bash terminates after that number of seconds if input does not arrive. The numeric real user id of the current user.70 Bash Reference Manual In an interactive shell. This variable is readonly.
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6 Bash Features
This section describes features unique to Bash.
6.1 Invoking Bash
bash [long-opt] [-ir] [-abefhkmnptuvxdBCDHP] [-o option] [-O shopt_option] [argument ...] bash [long-opt] [-abefhkmnptuvxdBCDHP] [-o option] [-O shopt_option] -c string [argument ...] bash [long-opt] -s [-abefhkmnptuvxdBCDHP] [-o option] [-O shopt_option] [argument ...] In addition to the single-character shell command-line options (see Section 4.3.1 [The Set Builtin], page 51), there are several multi-character options that you can use. These options must appear on the command line before the single-character options to be recognized. --debugger Arrange for the debugger profile to be executed before the shell starts. Turns on extended debugging mode (see Section 4.3.2 [The Shopt Builtin], page 55 for a description of the extdebug option to the shopt builtin) and shell function tracing (see Section 4.3.1 [The Set Builtin], page 51 for a description of the -o functrace option). --dump-po-strings A list of all double-quoted strings preceded by ‘\$’ is printed on the standard output in the gnu gettext PO (portable object) file format. Equivalent to ‘-D’ except for the output format. --dump-strings Equivalent to ‘-D’. --help Display a usage message on standard output and exit successfully.
--init-file filename --rcfile filename Execute commands from filename (instead of ‘~/.bashrc’) in an interactive shell. --login Equivalent to ‘-l’.
--noediting Do not use the gnu Readline library (see Chapter 8 [Command Line Editing], page 93) to read command lines when the shell is interactive. --noprofile Don’t load the system-wide startup file ‘/etc/profile’ or any of the personal initialization files ‘~/.bash_profile’, ‘~/.bash_login’, or ‘~/.profile’ when Bash is invoked as a login shell. --norc Don’t read the ‘~/.bashrc’ initialization file in an interactive shell. This is on by default if the shell is invoked as sh.
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--posix
Change the behavior of Bash where the default operation differs from the posix standard to match the standard. This is intended to make Bash behave as a strict superset of that standard. See Section 6.11 [Bash POSIX Mode], page 84, for a description of the Bash posix mode.
--restricted Make the shell a restricted shell (see Section 6.10 [The Restricted Shell], page 84). --verbose Equivalent to ‘-v’. Print shell input lines as they’re read. --version Show version information for this instance of Bash on the standard output and exit successfully. There are several single-character options that may be supplied at invocation which are not available with the set builtin. -c string Read and execute commands from string after processing the options, then exit. Any remaining arguments are assigned to the positional parameters, starting with \$0. -i -l Force the shell to run interactively. Interactive shells are described in Section 6.3 [Interactive Shells], page 75. Make this shell act as if it had been directly invoked by login. When the shell is interactive, this is equivalent to starting a login shell with ‘exec -l bash’. When the shell is not interactive, the login shell startup files will be executed. ‘exec bash -l’ or ‘exec bash --login’ will replace the current shell with a Bash login shell. See Section 6.2 [Bash Startup Files], page 73, for a description of the special behavior of a login shell. Make the shell a restricted shell (see Section 6.10 [The Restricted Shell], page 84). If this option is present, or if no arguments remain after option processing, then commands are read from the standard input. This option allows the positional parameters to be set when invoking an interactive shell. A list of all double-quoted strings preceded by ‘\$’ is printed on the standard output. These are the strings that are subject to language translation when the current locale is not C or POSIX (see Section 3.1.2.5 [Locale Translation], page 7). This implies the ‘-n’ option; no commands will be executed.
-r -s
-D
[-+]O [shopt_option] shopt option is one of the shell options accepted by the shopt builtin (see Section 4.3.2 [The Shopt Builtin], page 55). If shopt option is present, ‘-O’ sets the value of that option; ‘+O’ unsets it. If shopt option is not supplied, the names and values of the shell options accepted by shopt are printed on the standard output. If the invocation option is ‘+O’, the output is displayed in a format that may be reused as input.
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--
A -- signals the end of options and disables further option processing. Any arguments after the -- are treated as filenames and arguments.
A login shell is one whose first character of argument zero is ‘-’, or one invoked with the ‘--login’ option. An interactive shell is one started without non-option arguments, unless ‘-s’ is specified, without specifying the ‘-c’ option, and whose input and output are both connected to terminals (as determined by isatty(3)), or one started with the ‘-i’ option. See Section 6.3 [Interactive Shells], page 75, for more information. If arguments remain after option processing, and neither the ‘-c’ nor the ‘-s’ option has been supplied, the first argument is assumed to be the name of a file containing shell commands (see Section 3.8 [Shell Scripts], page 33). When Bash is invoked in this fashion, \$0 is set to the name of the file, and the positional parameters are set to the remaining arguments. Bash reads and executes commands from this file, then exits. Bash’s exit status is the exit status of the last command executed in the script. If no commands are executed, the exit status is 0.
6.2 Bash Startup Files
This section describes how Bash executes its startup files. If any of the files exist but cannot be read, Bash reports an error. Tildes are expanded in file names as described above under Tilde Expansion (see Section 3.5.2 [Tilde Expansion], page 19). Interactive shells are described in Section 6.3 [Interactive Shells], page 75.
When Bash is invoked as an interactive login shell, or as a non-interactive shell with the ‘--login’ option, it first reads and executes commands from the file ‘/etc/profile’, if that file exists. After reading that file, it looks for ‘~/.bash_profile’, ‘~/.bash_login’, and ‘~/.profile’, in that order, and reads and executes commands from the first one that exists and is readable. The ‘--noprofile’ option may be used when the shell is started to inhibit this behavior. When a login shell exits, Bash reads and executes commands from the file ‘~/.bash_logout’, if it exists.
Invoked as an interactive non-login shell
When an interactive shell that is not a login shell is started, Bash reads and executes commands from ‘~/.bashrc’, if that file exists. This may be inhibited by using the ‘--norc’ option. The ‘--rcfile file’ option will force Bash to read and execute commands from file instead of ‘~/.bashrc’. So, typically, your ‘~/.bash_profile’ contains the line if [ -f ~/.bashrc ]; then . ~/.bashrc; fi after (or before) any login-specific initializations.
Invoked non-interactively
When Bash is started non-interactively, to run a shell script, for example, it looks for the variable BASH_ENV in the environment, expands its value if it appears there, and uses the
1 [Bash History Facilities]. it is unset in non-interactive shells.2 [Bash Startup Files]. It contains i when the shell is interactive. page 89) is enabled by default. An interactive shell generally reads from and writes to a user’s terminal. Startup files are read and executed as described in Section 6. and set in interactive shells.2 [Bash Variables]. 6. Job Control (see Chapter 7 [Job Control]. Command history (see Section 9. *) echo This shell is not interactive . page 73.3. without specifying the ‘-c’ option.3 Interactive Shells 6. or one started with the ‘-i’ option. Readline (see Chapter 8 [Command Line Editing]. Bash executes the value of the PROMPT_COMMAND variable as a command before printing the primary prompt. page 51). and expands and displays PS2 before reading the second and subsequent lines of a multi-line command. then echo This shell is not interactive else echo This shell is interactive fi 6. 6.1 What is an Interactive Shell? An interactive shell is one started without non-option arguments. page 123) are enabled by default. esac Alternatively. 2.. page 93) is used to read commands from the user’s terminal. \$PS1 (see Section 5. page 61). Bash . 3. Bash ignores the keyboard-generated job control signals SIGTTIN.3.Chapter 6: Bash Features 75 6. test the value of the ‘-’ special parameter. it changes its behavior in several ways. page 121) and history expansion (see Section 9.3. When job control is in effect. and SIGTSTP. Bash expands and displays PS1 before reading the first line of a command.1 [The Set Builtin]. and whose input and error output are both connected to terminals (as determined by isatty(3)). 5..3 Interactive Shell Behavior When the shell is running interactively. unless ‘-s’ is specified. For example: case "\$-" in *i*) echo This shell is interactive . 1. The ‘-s’ invocation option may be used to set the positional parameters when an interactive shell is started. SIGTTOU.3 [History Interaction].2 Is this Shell Interactive? To determine within a startup script whether or not Bash is running interactively. Bash inspects the value of the ignoreeof option to set -o instead of exiting immediately when it receives an EOF on its standard input when reading a command (see Section 4.3. 7. startup scripts may examine the variable PS1. Thus: if [ -z "\$PS1" ]. 4.
6 [Signals]. An interactive login shell sends a SIGHUP to all jobs on exit if the huponexit shell option has been enabled (see Section 3. 17. or 2. page 61). There are string operators and numeric comparison operators as well. or ‘/dev/stderr’. 21. If the file argument to one of the primaries is one of ‘/dev/stdin’. SIGINT is caught and handled ((see Section 3.5. Expansion errors due to references to unbound shell variables after ‘set -u’ has been enabled will not cause the shell to exit (see Section 4. will save the command history to the file named by \$HISTFILE when an interactive shell exits. Bash will check for mail periodically.3. The ‘<’ and ‘>’ operators sort lexicographically using the current locale.1 [The Set Builtin]. Redirection errors encountered by shell builtins will not cause the shell to exit. SIGINT will interrupt some shell builtins.6 [Signals]. When running in posix mode.76 Bash Reference Manual 8. Unary expressions are often used to examine the status of a file.2 [The Shopt Builtin]. -a file True if file exists.6 [Signals].2 [Bash Variables]. respectively. and MAILCHECK shell variables (see Section 5. 16. MAILPATH.7.3 [Shell Parameter Expansion]. The shell will not exit on expansion errors caused by var being unset or null in \${var:?word} expansions (see Section 3. then file descriptor N is checked. page 19). a special builtin returning an error status will not cause the shell to exit (see Section 6.3. rather than the link itself. page 32). The shell will check the value of the TMOUT variable and exit if a command is not read within the specified number of seconds after printing \$PS1 (see Section 5. 12. page 61). page 79) is performed by default.7. 19. page 51).11 [Bash POSIX Mode]. A failed exec will not cause the shell to exit (see Section 4. page 84).2 [Bash Variables].7. 9. 11. ‘/dev/stdout’. 15. The ‘-n’ invocation option is ignored. page 51). 6. page 35).6 [Aliases]. page 55).1 [The Set Builtin]. . 20.1 [Bourne Shell Builtins]. Bash ignores SIGTERM (see Section 3.4 Bash Conditional Expressions Conditional expressions are used by the [[ compound command and the test and [ builtin commands. In the absence of any traps. depending on the values of the MAIL. is checked. When used with ‘[[’. In the absence of any traps. If the file argument to one of the primaries is of the form ‘/dev/fd/N’. 1. Unless otherwise specified. Simple spelling correction for directory arguments to the cd builtin is enabled by default (see the description of the cdspell option to the shopt builtin in Section 4. 18. 10. page 32). file descriptor 0. 14. Parser syntax errors will not cause the shell to exit. Expressions may be unary or binary. Alias expansion (see Section 6. page 32).3. 13. and ‘set -n’ has no effect (see Section 4. primaries that operate on files follow symbolic links and operate on the target of the link.
True if file exists and is a character special file.1 [The Set Builtin]. page 51). file1 -ef file2 True if file1 and file2 refer to the same device and inode numbers. . -o optname True if shell option optname is enabled. True if file exists and is owned by the effective user id.3. True if file exists and its "sticky" bit is set. True if file exists and its set-group-id bit is set. -z string True if the length of string is zero. True if file exists and is a symbolic link. True if file exists and its set-user-id bit is set. True if file exists and is owned by the effective group id. True if file exists and is writable. or if file2 exists and file1 does not. The list of options appears in the description of the ‘-o’ option to the set builtin (see Section 4. True if file exists and is a regular file. True if file exists and is a directory. True if file exists. -n string string True if the length of string is non-zero. True if file exists and has a size greater than zero. True if file exists and has been modified since it was last read. True if file exists and is readable. or if file1 exists and file2 does not. True if file exists and is executable. file1 -ot file2 True if file1 is older than file2. True if file exists and is a named pipe (FIFO). True if file descriptor fd is open and refers to a terminal.Chapter 6: Bash Features 77 -b file -c file -d file -e file -f file -g file -h file -k file -p file -r file -s file -t fd -u file -w file -x file -O file -G file -L file -S file -N file True if file exists and is a block special file. True if file exists and is a symbolic link. True if file exists and is a socket. file1 -nt file2 True if file1 is newer (according to modification date) than file2.
Arg1 and arg2 may be positive or negative integers. The levels are listed in order of decreasing precedence. ‘-gt’. and values are the same as in the C language. The following list of operators is grouped into levels of equal-precedence operators. ‘-ne’. id++ id-. associativity. not equal to. respectively. Evaluation is done in fixed-width integers with no check for overflow.variable post-increment and post-decrement ++id --id variable pre-increment and pre-decrement -+ !~ ** */% +<< >> <= >= < > == != & ^ | && || unary minus and plus logical and bitwise negation exponentiation multiplication. ‘-lt’. or greater than or equal to arg2. remainder addition. 6. The operators and their precedence. string1 > string2 True if string1 sorts after string2 lexicographically.78 Bash Reference Manual string1 == string2 string1 = string2 True if the strings are equal. ‘-le’. though division by 0 is trapped and flagged as an error. greater than. arg1 OP arg2 OP is one of ‘-eq’. division. These arithmetic binary operators return true if arg1 is equal to. less than. less than or equal to. string1 != string2 True if the strings are not equal. as one of the shell expansions or by the let and the ‘-i’ option to the declare builtins. or ‘-ge’. string1 < string2 True if string1 sorts before string2 lexicographically.5 Shell Arithmetic The shell allows arithmetic expressions to be evaluated. subtraction left and right bitwise shifts comparison equality and inequality bitwise AND bitwise exclusive OR bitwise OR logical AND logical OR . ‘=’ should be used with the test command for posix conformance.
A leading ‘0x’ or ‘0X’ denotes hexadecimal. then the next command word following the alias is also checked for alias expansion. where base is a decimal number between 2 and 64 representing the arithmetic base. Otherwise. The digits greater than 9 are represented by the lowercase letters. parameter expansion is performed before the expression is evaluated.2 [The Shopt Builtin]. for instance. shell variables may also be referenced by name without using the parameter expansion syntax. The first word of each simple command. a shell function should be used (see Section 3. and removed with the unalias command. This means that one may alias ls to "ls -F". page 55). if unquoted. If arguments are needed.Chapter 6: Bash Features 79 expr ? expr : expr conditional operator = *= /= %= += -= <<= >>= &= ^= |= assignment expr1 . The characters ‘/’. and ‘_’. that word is replaced by the text of the alias. Within an expression. in that order. The first word of the replacement text is tested for aliases. lowercase and uppercase letters may be used interchangeably to represent numbers between 10 and 35. A shell variable that is null or unset evaluates to 0 when referenced by name without using the parameter expansion syntax. ‘=’ and any of the shell metacharacters or quoting characters listed above may not appear in an alias name. Aliases are not expanded when the shell is not interactive. If the last character of the alias value is a space or tab character. Aliases are created and listed with the alias command. as in csh. A shell variable need not have its integer attribute turned on to be used in an expression. 6. or when a variable which has been given the integer attribute using ‘declare -i’ is assigned a value. The value of a variable is evaluated as an arithmetic expression when it is referenced. including shell metacharacters. A null value evaluates to 0.6 Aliases Aliases allow a string to be substituted for a word when it is used as the first word of a simple command. expr2 comma Shell variables are allowed as operands. ‘‘’. the uppercase letters. and Bash does not try to recursively expand the replacement text. If so. The replacement text may contain any valid shell input. ‘@’. If base is less than or equal to 36. then base 10 is used.3. The shell maintains a list of aliases that may be set and unset with the alias and unalias builtin commands. but a word that is identical to an alias being expanded is not expanded a second time. If base # is omitted. Operators are evaluated in order of precedence. There is no mechanism for using arguments in the replacement text. unless the expand_aliases shell option is set using shopt (see Section 4. numbers take the form [base #]n. is checked to see if it has an alias. and n is a number in that base. Sub-expressions in parentheses are evaluated first and may override the precedence rules above. ‘\$’. Constants with a leading 0 are interpreted as octal numbers. . page 14).3 [Shell Functions].
because a function definition is itself a compound command. There is no maximum limit on the size of an array.. associative arrays use arbitrary strings. the subscript is ignored. This syntax is also accepted by the declare builtin. As a consequence. The braces are required to avoid conflicts with the shell’s filename expansion operators. Aliases are expanded when a command is read. When assigning to an associative array. an alias definition appearing on the same line as another command does not take effect until the next line of input is read. not when it is executed. To explicitly declare an array. Therefore. Associative arrays are created using declare -A name. aliases defined in a function are not available until after that function is executed. use declare -a name The syntax declare -a name[subscript] is also accepted. not when the function is executed. Any variable may be used as an indexed array. This behavior is also an issue when functions are executed. If the subscript is . page 78) and are zero-based. For almost every purpose. and do not use alias in compound commands. Aliases are expanded when a function definition is read. the subscript is required. Individual array elements may be assigned to using the name[subscript]=value syntax introduced above. Attributes may be specified for an array variable using the declare and readonly builtins. Indexing starts at zero. An indexed array is created automatically if any variable is assigned to using the syntax name[subscript]=value The subscript is treated as an arithmetic expression that must evaluate to a number greater than or equal to zero. When assigning to indexed arrays. Any element of an array may be referenced using \${name[subscript]}.5 [Shell Arithmetic]. To be safe. always put alias definitions on a separate line. if the optional subscript is supplied. Each attribute applies to all members of an array. Indexed arrays are referenced using integers (including arithmetic expressions (see Section 6.. shell functions are preferred over aliases. the declare builtin will explicitly declare an array. Bash always reads at least one complete line of input before executing any of the commands on that line. nor any requirement that members be indexed or assigned contiguously. that index is assigned to. Arrays are assigned to using compound assignments of the form name=(value1 . valuen) where each value is of the form [subscript]=string.80 Bash Reference Manual The rules concerning the definition and use of aliases are somewhat confusing. Indexed array assignments do not require the bracket and subscript.7 Arrays Bash provides one-dimensional indexed and associative array variables. 6. otherwise the index of the element assigned is the last index assigned to by the statement plus one. The commands following the alias definition on that line are not affected by the new alias.
8. Directories are added to the list with the pushd command. starting with zero. The read builtin accepts a ‘-a’ option to assign a list of words read from the standard input to an array. 6. unset name [subscript] destroys the array element at index subscript. 6. the expansion is the number of elements in the array. A subscript of ‘*’ or ‘@’ also removes the entire array. These subscripts differ only when the word appears within double quotes. The contents of the directory stack are also visible as the value of the DIRSTACK shell variable. Clears the directory stack by deleting all of the elements. . local. where name is an array. Displays the N th directory (counting from the right of the list printed by dirs when invoked without options). If the word is double-quoted. and \${name[@]} expands each element of name to a separate word. and can read values from the standard input into individual array elements. The null string is a valid value. unset name. If subscript is ‘@’ or ‘*’. An array variable is considered set if a subscript has been assigned a value.Chapter 6: Bash Features 81 ‘@’ or ‘*’. The dirs builtin displays the contents of the directory stack. the popd command removes directories from the list. and readonly builtins each accept a ‘-a’ option to specify an indexed array and a ‘-A’ option to specify an associative array.8 The Directory Stack The directory stack is a list of recently-visited directories. The unset builtin is used to destroy arrays. and the popd builtin removes specified directories from the stack and changes the current directory to the directory removed. \${#name[subscript]} expands to the length of \${name[subscript]}. The pushd builtin adds directories to the stack as it changes the current directory. and the expansion of the last parameter is joined with the last part of the original word. \${name[@]} expands to nothing. removes the entire array. \${name[*]} expands to a single word with the value of each array member separated by the first character of the IFS variable. The declare. Care must be taken to avoid unwanted side effects caused by filename expansion. Referencing an array variable without a subscript is equivalent to referencing with a subscript of 0. If the double-quoted expansion occurs within a word. When there are no array members. The set and declare builtins display array values in a way that allows them to be reused as input. the expansion of the first parameter is joined with the beginning part of the original word. This is analogous to the expansion of the special parameters ‘@’ and ‘*’.1 Directory Stack Builtins dirs dirs [+N | -N] [-clpv] Display the list of currently remembered directories. the word expands to all members of the array name. +N -N -c Displays the N th directory (counting from the left of the list printed by dirs when invoked without options). starting with zero.
the following table describes the special characters which can appear in the prompt variables: \a A bell character. Removes the N th directory (counting from the right of the list printed by dirs). popd is equivalent to popd +0. In addition. When no arguments are given. starting with zero. starting with zero) to the top of the list by rotating the stack.82 Bash Reference Manual -l -p -v popd Produces a longer listing.9 Controlling the Prompt The value of the variable PROMPT_COMMAND is examined just before Bash prints each primary prompt. -n +N Suppresses the normal change of directory when adding directories to the stack. Causes dirs to print the directory stack with one entry per line. pushd exchanges the top two directories. starting with zero. and cd to the new top directory. the default listing format uses a tilde to denote the home directory. Suppresses the normal change of directory when removing directories from the stack. Makes the current working directory be the top of the stack. i. . With no arguments. popd [+N | -N] [-n] Remove the top entry from the directory stack. and then executes the equivalent of ‘cd dir ’. prefixing each entry with its index in the stack. cds to dir. starting with zero) to the top of the list by rotating the stack. so that only the stack is manipulated. then the value is executed just as if it had been typed on the command line. Causes dirs to print the directory stack with one entry per line.e. so that only the stack is manipulated. -N dir 6. Brings the N th directory (counting from the left of the list printed by dirs. Removes the N th directory (counting from the left of the list printed by dirs). The elements are numbered from 0 starting at the first directory listed with dirs. If PROMPT_COMMAND is set and has a non-null value. Brings the N th directory (counting from the right of the list printed by dirs. +N -N -n pushd pushd [-n] [+N | -N | dir ] Save the current directory on the top of the directory stack and then cd to dir.. popd removes the top directory from the stack and performs a cd to the new top directory.
A carriage return. in 12-hour HH:MM:SS format. an empty format results in a locale-specific time representation. otherwise \$. The command number of this command. The basename of \$PWD. The time. The character whose ASCII code is the octal value nnn.’. The format is passed to strftime(3) and the result is inserted into the prompt string. while the command number is the position in the sequence of commands executed during the current shell session. The version of Bash (e.00. A backslash.g. which may include commands restored from the history file (see Section 9.. . the basename of \$0 (the portion following the final slash).g.. End a sequence of non-printing characters. The number of jobs currently managed by the shell. The command number and the history number are usually different: the history number of a command is its position in the history list. with \$HOME abbreviated with a tilde (uses the \$PROMPT_DIRTRIM variable). version + patchlevel (e.0) The current working directory. #. in 24-hour HH:MM format. in 12-hour am/pm format. The hostname.g. page 121). up to the first ‘. with \$HOME abbreviated with a tilde. The name of the shell.1 [Bash History Facilities]. The time. Begin a sequence of non-printing characters. 2. The history number of this command. The username of the current user.. in "Weekday Month Date" format (e.00) The release of Bash. "Tue May 26"). The basename of the shell’s terminal device name. in 24-hour HH:MM:SS format. \e \h \H \j \l \n \r \s \t \T \@ \A \u \v \V \w \W \! \# \\$ \nnn \\ \[ \] An escape character.Chapter 6: Bash Features 83 \d \D{format} The date. A newline. The braces are required. This could be used to embed a terminal control sequence into the prompt. The hostname. 2. If the effective uid is 0. The time. The time.
and quote removal. • Parsing the value of SHELLOPTS from the shell environment at startup. Bash will re-search \$PATH to find the new location. This is also available with ‘shopt -s checkhash’. • Specifying a filename containing a slash as an argument to the ‘-p’ option to the hash builtin command. it is expanded via parameter expansion. ENV. A restricted shell is used to set up an environment more controlled than the standard shell. • Using the enable builtin command to enable disabled shell builtins. the shell becomes restricted. ‘<>’. rbash turns off any restrictions in the shell spawned to execute the script. SIGTSTP. where signame is.8 [Shell Scripts]. When invoked as sh. When a command in the hash table no longer exists.84 Bash Reference Manual After the string is decoded. arithmetic expansion. When a command that is found to be a shell script is executed (see Section 3.2 [Bash Builtins]. The message printed by the job control code and builtins when a job exits with a non-zero status is ‘Done(status)’. These restrictions are enforced after any startup files are read. page 41). A restricted shell behaves identically to bash with the exception that the following are disallowed or not performed: • Changing directories with the cd builtin. command substitution. • Adding or deleting builtin commands with the ‘-f’ and ‘-d’ options to the enable builtin. ‘&>’. • Importing function definitions from the shell environment at startup. • Specifying a filename containing a slash as an argument to the . • Turning off restricted mode with ‘set +r’ or ‘set +o restricted’. or BASH_ENV variables. 6. 6. The message printed by the job control code and builtins when a job is stopped is ‘Stopped(signame )’. Bash enters posix mode after reading the startup files. or the ‘--restricted’ or ‘-r’ option is supplied at invocation. PATH. 2. • Setting or unsetting the values of the SHELL. . builtin command. ‘>|’. • Specifying command names containing slashes.10 The Restricted Shell If Bash is started with the name rbash. for example. ‘>&’. and ‘>>’ redirection operators. subject to the value of the promptvars shell option (see Section 4. • Specifying the ‘-p’ option to the command builtin. The following list is what’s changed when ‘posix mode’ is in effect: 1. page 33). • Redirecting output using the ‘>’.11 Bash POSIX Mode Starting Bash with the ‘--posix’ command-line option or executing ‘set -o posix’ while Bash is running will cause Bash to conform more closely to the posix standard by changing the behavior to match that specified by posix in areas where the Bash default differs. 3. • Using the exec builtin to replace the shell with another command.
This means that cd will fail if no valid directory name can be constructed from any of the entries in \$CDPATH. 5. 8. and may not start with a digit. 19. without the ‘SIG’ prefix. Assignment statements preceding shell function calls persist in the shell environment after the function returns. 7. Assignment statements preceding posix special builtins persist in the shell environment after the builtin completes. 23. they may not contain characters other than letters. for example. Declaring a function with an invalid name causes a fatal syntax error in non-interactive shells. 10. 12. If CDPATH is set. Function names must be valid shell names.sh_history’ (this is the default value of \$HISTFILE). which does not include an indication of whether the job is the current or previous job. 17. Non-interactive shells exit if filename in . and parameter expansion is performed on the values of PS1 and PS2 regardless of the setting of the promptvars option. 9. filename is not found. 15. The posix PS1 and PS2 expansions of ‘!’ to the history number and ‘!!’ to ‘!’ are enabled. If a posix special builtin returns an error status. A non-interactive shell exits with an error status if a variable assignment error occurs when no command name follows the assignment statements. Redirection operators do not perform filename expansion on the word in the redirection unless the shell is interactive. That is. 24. 14. 20. even if the a directory with the same name as the name given as an argument to cd exists in the current directory.Chapter 6: Bash Features 85 4. The default history file is ‘~/. and underscores. 21. Tilde expansion is only performed on assignments preceding a command name. as if a posix special builtin command had been executed. 6. digits. a non-interactive shell exits. The bg builtin uses the required format to describe each job placed in the background. Reserved words appearing in a context where reserved words are recognized do not undergo alias expansion. 22. 16. 11. Process substitution is not available. posix special builtins are found before shell functions during command lookup. when trying to assign a value to a readonly variable. The posix startup files are executed (\$ENV) rather than the normal Bash files. the cd builtin will not implicitly append the current directory to it. A non-interactive shell exits with an error status if the iteration variable in a for statement or the selection variable in a select statement is a readonly variable. rather than on all assignment statements on the line. Non-interactive shells exit if a syntax error in an arithmetic expansion results in an invalid expression. A variable assignment error occurs. 13. . and include things like passing incorrect options. variable assignment errors for assignments preceding the command name. and so on. redirection errors. Redirection operators do not perform word splitting on the word in the redirection. The fatal errors are those listed in the POSIX standard. 18. separated by spaces. The kill builtin does not accept signal names with a ‘SIG’ prefix. The output of ‘kill -l’ prints all the signal names on a single line.
When the xpg_echo option is enabled. it displays variable values without quotes. 38. When the pwd builtin is supplied the ‘-P’ option. The pwd builtin verifies that the value it prints is the same as the current directory.86 Bash Reference Manual 25. 32. the fc builtin does not include an indication of whether or not a history entry has been modified. 36. 42. 34. The . 31. The trap builtin doesn’t check the first argument for a possible signal specification and revert the signal handling to the original disposition if it is. If users want to reset the handler for a given signal to the original disposition. When not in posix mode. Each argument is displayed. 30. Bash clears the ‘-e’ option in such subshells. . 37. after escape characters are converted. it does not display them with a leading ‘alias ’ unless the ‘-p’ option is supplied. 27. When the set builtin is invoked without options. 28. unless they contain shell metacharacters. and source builtins do not search the current directory for the filename argument if it is not found by searching PATH. unless that argument consists solely of digits and is a valid signal number. The vi editing mode will invoke the vi editor directly when the ‘v’ command is run. 43. 41. Bash does not attempt to interpret any arguments to echo as options. The trap builtin displays signal names without the leading SIG. 26. When listing the history. The trap command is run once for each child that exits. The ulimit builtin uses a block size of 512 bytes for the ‘-c’ and ‘-f’ options. and the pathname constructed from \$PWD and the directory name supplied as an argument does not refer to an existing directory. When the cd builtin is invoked in logical mode. The arrival of SIGCHLD when a trap is set on SIGCHLD does not interrupt the wait builtin and cause it to return immediately. instead of checking \$VISUAL and \$EDITOR. When the alias builtin displays alias definitions. even if the result contains nonprinting characters. though the shell will attempt to execute such a file if it is the only so-named file found in \$PATH. it resets \$PWD to a pathname containing no symlinks. 35. The type and command builtins will not report a non-executable file as having been found. The export and readonly builtin commands display their output in the format required by posix. even in non-interactive shells. 39. 40. it does not display shell function names and definitions. 29. When the set builtin is invoked without options. even if it is not asked to check the file system with the ‘-P’ option. The default editor used by fc is ed. Alias expansion is always enabled. cd will fail instead of falling back to physical mode. Subshells spawned to execute command substitutions inherit the value of the ‘-e’ option from the parent shell. they should use ‘-’ as the first argument. 33.
2. page 129). Specifically: 1. rather than defaulting directly to ed.8 [Optional Features]. . Bash requires the xpg_echo option to be enabled for the echo builtin to be fully conformant.Chapter 6: Bash Features 87 There is other posix behavior that Bash does not implement by default even when in posix mode. The fc builtin checks \$EDITOR as a program to edit history entries if FCEDIT is unset. As noted above. by specifying the ‘--enable-strict-posix-default’ to configure when building (see Section 10. fc uses ed if EDITOR is unset. Bash can be configured to be posix-conformant by default.
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the fg command to continue it in the foreground. There are a number of ways to refer to a job in the shell. A user typically employs this facility via an interactive interface supplied jointly by the operating system kernel’s terminal driver and Bash. The user then manipulates the state of this job. or the kill command to kill it. The symbols ‘%%’ and ‘%+’ refer to the shell’s notion of the current job. When Bash starts a job asynchronously. If there is only a single job. using the bg command to continue it in the background. and the previous job with a ‘-’. . suspends the process.. how it works. The character ‘%’ introduces a job specification (jobspec ). Job number n may be referred to as ‘%n’.1 Job Control Basics Job control refers to the ability to selectively stop (suspend) the execution of processes and continue (resume) their execution at a later point. The previous job may be referenced using ‘%-’. Typing the suspend character (typically ‘^Z’. In output pertaining to jobs (e. the output of the jobs command). it prints a line that looks like: [1] 25647 indicating that this job is job number 1 and that the process id of the last process in the pipeline associated with this job is 25647.Chapter 7: Job Control 89 7 Job Control This chapter discusses what job control is. All of the processes in a single pipeline are members of the same job. A single ‘%’ (with no accompanying job specification) also refers to the current job. Only foreground processes are allowed to read from or. 7. and how Bash allows you to access its facilities. Background processes are those whose process group id differs from the terminal’s. and control to be returned to Bash. A ‘^Z’ takes effect immediately. The shell associates a job with each pipeline. It keeps a table of currently executing jobs. Control-Y) causes the process to be stopped when it attempts to read input from the terminal. ‘%+’ and ‘%-’ can both be used to refer to that job. the operating system maintains the notion of a current terminal process group id. If the operating system on which Bash is running supports job control. which is the last job stopped while it was in the foreground or started in the background. if the user so specifies with stty tostop. which may be listed with the jobs command. and has the additional side effect of causing pending output and typeahead to be discarded. unless caught. To facilitate the implementation of the user interface to job control. These processes are said to be in the foreground. Bash uses the job abstraction as the basis for job control. Control-Z) while a process is running causes that process to be stopped and returns control to Bash. Bash contains facilities to use it. Members of this process group (processes whose process group id is equal to the current terminal process group id) receive keyboard-generated signals such as SIGINT. which. the current job is always flagged with a ‘+’. write to the terminal. Typing the delayed suspend character (typically ‘^Y’. such processes are immune to keyboard-generated signals. Background processes which attempt to read from (write to when stty tostop is in effect) the terminal are sent a SIGTTIN (SIGTTOU) signal by the kernel’s terminal driver.g.
1 [The Set Builtin]. Bash does not print another warning..] Resume each suspended job jobspec in the background.3. jobspec does not specify a valid job or jobspec specifies a job that was started without job control. page 51). bringing job 1 from the background into the foreground. or non-zero if run when job control is disabled or. Bash reports an error. . Any trap on SIGCHLD is executed for each child process that exits.. fg fg [jobspec] Resume the job jobspec in the foreground and make it the current job. when run with job control enabled. refers to any job containing the string ‘ce’ in its command line. page 55). Using ‘%?ce’. jobs jobs [-lnprs] [jobspec] jobs -x command [arguments] The first form lists the active jobs. List only the process id of the job’s process group leader. If jobspec is not supplied. or using a substring that appears in its command line.2 Job Control Builtins bg bg [jobspec .3. Similarly.90 Bash Reference Manual A job may also be referred to using a prefix of the name used to start it. 7. as if it had been started with ‘&’. the current job is used. The return status is that of the command placed into the foreground. Bash waits until it is about to print a prompt before reporting changes in a job’s status so as to not interrupt any other output. Normally. and any stopped jobs are terminated. if the checkjobs option is enabled – see Section 4. The options have the following meanings: -l -n -p List process ids in addition to the normal information. If a second attempt to exit is made without an intervening command. (or running. Display information only about jobs that have changed status since the user was last notified of their status. The return status is zero unless it is run when job control is not enabled. or. on the other hand. Simply naming a job can be used to bring it into the foreground: ‘%1’ is a synonym for ‘fg %1’. ‘%ce’ refers to a stopped ce job. equivalent to ‘bg %1’ The shell learns immediately whenever a job changes state.2 [The Shopt Builtin]. The jobs command may then be used to inspect their status. and if the checkjobs option is enabled. If the ‘-b’ option to the set builtin is enabled. ‘%1 &’ resumes job 1 in the background. If jobspec is not supplied. the current job is used. If the prefix or substring matches more than one job. lists the jobs and their statuses. If an attempt to exit Bash is made while jobs are stopped. For example. when run with job control enabled. any jobspec was not found or specifies a job that was started without job control. the shell prints a warning message. Bash reports such changes immediately (see Section 4.
exit status is a number specifying a signal number or the exit status of a process terminated by a signal. all currently active child processes are waited for.. If any arguments are supplied when ‘-l’ is given. If jobspec is given. If the ‘-x’ option is supplied. If no jobspec is supplied. suspend suspend [-f] Suspend the execution of this shell until it receives a SIGCONT signal. Restrict output to stopped jobs. A login shell cannot be suspended. If sigspec and signum are not present. kill kill [-s sigspec] [-n signum] [-sigspec] jobspec or pid kill -l [exit_status] Send a signal specified by sigspec or signum to the process named by job specification jobspec or process id pid.. the ‘-f’ option can be used to override this and force the suspension. signum is a signal number. each jobspec is removed from the table of active jobs. the current job is used. the status of all jobs is listed. SIGTERM is used. If jobspec is not supplied. and neither the ‘-a’ nor ‘-r’ option is supplied. If no arguments are given. The return status is zero if at least one signal was successfully sent. all processes in the job are waited for. . but is marked so that SIGHUP is not sent to the job if the shell receives a SIGHUP. jobs replaces any jobspec found in command or arguments with the corresponding process group id..] Wait until the child process specified by each process id pid or job specification jobspec exits and return the exit status of the last command waited for. If neither jobspec nor pid specifies an active child process of the shell. and the return status is zero. the ‘-r’ option without a jobspec argument restricts operation to running jobs. passing it arguments.. the ‘-a’ option means to remove or mark all jobs. or non-zero if an error occurs or an invalid option is encountered. and the return status is zero. and executes command. If jobspec is not present. the return status is 127. wait wait [jobspec or pid . sigspec is either a case-insensitive signal name such as SIGINT (with or without the SIG prefix) or a signal number. The ‘-l’ option lists the signal names. the names of the signals corresponding to the arguments are listed. If a job spec is given.] Without options. the job is not removed from the table. output is restricted to information about that job.Chapter 7: Job Control 91 -r -s Restrict output to running jobs. disown disown [-ar] [-h] [jobspec . returning its exit status. If the ‘-h’ option is given.
is the command line used to start it. If this variable exists then single word simple commands without redirections are treated as candidates for resumption of an existing job.92 Bash Reference Manual When job control is not active. in this context. There is no ambiguity allowed. the kill and wait builtins do not accept jobspec arguments. the supplied string must be a prefix of a stopped job’s name. if set to ‘substring’.1 [Job Control Basics]. If set to any other value. 7. if there is more than one job beginning with the string typed. page 89). The name of a stopped job. the string supplied needs to match a substring of the name of a stopped job. The ‘substring’ value provides functionality analogous to the ‘%?’ job id (see Section 7. the string supplied must match the name of a stopped job exactly. They must be supplied process ids. . this provides functionality analogous to the ‘%’ job id.3 Job Control Variables auto_resume This variable controls how the shell interacts with the user and job control. If this variable is set to the value ‘exact’. then the most recently accessed job will be selected.
Notice how C-f moves forward a character. When you add text in the middle of a line. simply type them. The typed character appears where the cursor was. It is a loose convention that control keystrokes operate on characters while meta keystrokes operate on words. C-a C-e M-f M-b C-l Move to the start of the line. Move backward a word. and DEL. and then correct your mistake. you will notice that characters to the right of the cursor are ‘pushed over’ to make room for the text that you have inserted. You can undo all the way back to an empty line.2. For your convenience. C-d. . C-_ or C-x C-u Undo the last editing command. reprinting the current line at the top. Likewise. you can use your erase character to back up and delete the mistyped character. while M-f moves forward a word.2. the Backspace key be set to delete the character to the left of the cursor and the DEL key set to delete the character underneath the cursor. C-d Delete the character underneath the cursor.) 8. characters to the right of the cursor are ‘pulled back’ to fill in the blank space created by the removal of the text. when you delete text behind the cursor. you can move the cursor to the right with C-f. C-b C-f Move back one character. Move to the end of the line. Printing characters Insert the character into the line at the cursor. (Depending on your configuration. Move forward a word.1 Readline Bare Essentials In order to enter characters into the line. Move forward one character. If you mistype a character. Afterwards. Sometimes you may mistype a character. Clear the screen. where a word is composed of letters and digits. rather than the character to the left of the cursor.94 Bash Reference Manual 8. C-f. many other commands have been added in addition to C-b.2 Readline Movement Commands The above table describes the most basic keystrokes that you need in order to do editing of the input line. you can type C-b to move the cursor to the left. like C-d. A list of the bare essentials for editing the text of an input line follows. In that case. DEL or Backspace Delete the character to the left of the cursor. and then the cursor moves one space to the right. and not notice the error until you have typed several other characters. Here are some commands for moving more rapidly about the line.
then you can be sure that you can get the text back in a different (or the same) place later. to the start of the previous word. usually by yanking (re-inserting) it back into the line. 8. C-k M-d M-DEL Kill the text from the current cursor position to the end of the line. or. when you are typing another line. or. Yanking means to copy the mostrecently-killed text from the kill buffer. (‘Cut’ and ‘paste’ are more recent jargon for ‘kill’ and ‘yank’. if between words. Word boundaries are the same as those used by M-b. the text is saved in a kill-ring. if between words.2. The general way to pass numeric arguments to a command is to type meta digits before the command. C-y M-y Yank the most recently killed text back into the buffer at the cursor. Kill from the cursor the start of the current word. 8. Kill from the cursor to the end of the current word. Sometimes the argument acts as a repeat count. you get it all.3 Readline Killing Commands Killing text means to delete the text from the line. to the end of the next word. that command will act in a backward direction. but to save it away for later use. This is different than M-DEL because the word boundaries differ. If the first ‘digit’ typed is a minus sign (‘-’). C-w Here is how to yank the text back into the line. and yank the new top. Word boundaries are the same as those used by M-f. the text that you killed on a previously typed line is available to be yanked back later. If you pass a negative argument to a command which normally acts in a forward direction. There are two search modes: incremental and non-incremental. and then the command. so that when you yank it back. For example. The kill ring is not line specific.Chapter 8: Command Line Editing 95 8.1 [Bash History Facilities]. to kill text back to the start of the line. Kill from the cursor to the previous whitespace. which will delete the next ten characters on the input line.C-k’. You can only do this if the prior command is C-y or M-y.2. other times it is the sign of the argument that is significant. you can type the remainder of the digits. you might type ‘M-.5 Searching for Commands in the History Readline provides commands for searching through the command history (see Section 9. Once you have typed one meta digit to get the argument started.) If the description for a command says that it ‘kills’ text. When you use a kill command. to give the C-d command an argument of 10.4 Readline Arguments You can pass numeric arguments to Readline commands. page 121) for lines containing a specified string. Here is the list of commands for killing text.2. Rotate the kill-ring. Any number of consecutive kills save all of the killed text together. then the sign of the argument will be negative. you could type ‘M-1 0 C-d’. . For example.
converting them to a meta-prefixed key sequence. Readline will convert characters with the eighth bit set to an ascii key sequence by stripping the eighth bit and prefixing an ESC character. Readline will try to enable the application keypad when it is called. tilde expansion is performed when Readline attempts word completion. where the keystrokes are most similar to Emacs.98 Bash Reference Manual convert-meta If set to ‘on’. the meta key is used to send eight-bit characters. On many terminals. expand-tilde If set to ‘on’. horizontal-scroll-mode This variable can be set to either ‘on’ or ‘off’. The default is ‘on’. readline echoes a character corresponding to a signal generated from the keyboard. echo-control-characters When set to ‘on’. Completion characters will be inserted into the line as if they had been mapped to self-insert. The default is ‘off’. history-preserve-point If set to ‘on’. the history code attempts to place the point (the current cursor position) at the same location on each history line retrieved with previous-history or next-history. disable-completion If set to ‘On’. editing-mode The editing-mode variable controls which default set of key bindings is used. The default is ‘on’. If set to zero. This variable can be set to either ‘emacs’ or ‘vi’. Readline will try to enable any meta modifier key the terminal claims to support when it is called. The default is ‘off’. The default value is ‘on’. the number of entries in the history list is not limited. Readline starts up in Emacs editing mode. enable-keypad When set to ‘on’. history-size Set the maximum number of history entries saved in the history list. The default is ‘off’. The default is ‘off’. Readline will inhibit word completion. By default. Some systems need this to enable the arrow keys. on operating systems that indicate they support it. enable-meta-key When set to ‘on’. Setting it to ‘on’ means that the text of the lines being edited will scroll horizontally on a single screen line when they are longer than the width of the .
The name of . a colon. Readline will undo all changes to history lines before returning when accept-line is executed. Once you know the name of the command. a character denoting a file’s type is appended to the filename when listing possible completions. The default value is ‘off’. For instance. First you need to find the name of the command that you want to change. show-all-if-ambiguous This alters the default behavior of the completion functions. if any. so portions of the word following the cursor are not duplicated. It’s only active when performing completion in the middle of a word. If set to ‘on’. attempting completion when the cursor is after the ‘e’ in ‘Makefile’ will result in ‘Makefile’ rather than ‘Makefilefile’. The default is ‘off’. visible-stats If set to ‘on’. if this is enabled. and then the name of the command. If enabled. The default value is ‘off’. The following sections contain tables of the command name. Key Bindings The syntax for controlling key bindings in the init file is simple. readline does not insert characters from the completion that match characters after point in the word being completed. The default value is ‘off’. simply place on a line in the init file the name of the key you wish to bind the command to. The default is ‘off’. assuming there is a single possible completion. words which have more than one possible completion cause the matches to be listed immediately instead of ringing the bell. There can be no space between the key name and the colon – that will be interpreted as part of the key name. history lines may be modified and retain individual undo lists across calls to readline. Readline will display completions with matches sorted horizontally in alphabetical order. the default keybinding. rather than down the screen. and a short description of what the command does. The default is ‘off’. If set to ‘on’. By default. words which have more than one possible completion without any possible partial completion (the possible completions don’t share a common prefix) cause the matches to be listed immediately instead of ringing the bell. this alters the default completion behavior when inserting a single match into the line.100 Bash Reference Manual print-completions-horizontally If set to ‘on’. revert-all-at-newline If set to ‘on’. show-all-if-unmodified This alters the default behavior of the completion functions in a fashion similar to show-all-if-ambiguous. skip-completed-text If set to ‘on’.
A number of symbolic character names are recognized while processing this key binding syntax: DEL. by placing the key sequence in double quotes. RETURN. RUBOUT. as in the following example. and ‘ESC [ 1 1 ~’ is bound to insert the text ‘Function Key 1’. Some gnu Emacs style key escapes can be used. a second set of backslash escapes is available: . C-u is again bound to the function universal-argument (just as it was in the first example). and TAB. The bind -p command displays Readline function names and bindings in a format that can put directly into an initialization file. See Section 4. ‘C-x C-r’ is bound to the function re-read-init-file. SPC. NEWLINE. "keyseq": function-name or macro keyseq differs from keyname above in that strings denoting an entire key sequence can be specified. LFD. "\C-u": universal-argument "\C-x\C-r": re-read-init-file "\e[11~": "Function Key 1" In the above example. M-DEL is bound to the function backward-kill-word. The following gnu Emacs style escape sequences are available when specifying key sequences: \C\M\e \\ \" \’ control prefix meta prefix an escape character backslash ". ESC. readline allows keys to be bound to a string that is inserted when the key is pressed (a macro ). In addition to command names. page 41. SPACE. to insert the text ‘> output’ into the line). a double quotation mark ’.Chapter 8: Command Line Editing 101 the key can be expressed in different ways. RET. but the special character names are not recognized. ESCAPE. keyname : function-name or macro keyname is the name of a key spelled out in English. a single quote or apostrophe In addition to the gnu Emacs style escape sequences. For example: Control-u: universal-argument Meta-Rubout: backward-kill-word Control-o: "> output" In the above example.2 [Bash Builtins]. and C-o is bound to run the macro expressed on the right hand side (that is. depending on what you find most comfortable. C-u is bound to the function universalargument.
102 Bash Reference Manual \a \b \d \f \n \r \t \v \nnn \xHH alert (bell) backspace delete form feed newline carriage return horizontal tab vertical tab the eight-bit character whose value is the octal value nnn (one to three digits) the eight-bit character whose value is the hexadecimal value HH (one or two hex digits) When entering the text of a macro. Each program using the Readline library sets the application . mode The mode= form of the \$if directive is used to test whether Readline is in emacs or vi mode. for instance. the following binding will make ‘C-x \’ insert a single ‘\’ into the line: "\C-x\\": "\\" 8. perhaps to bind the key sequences output by the terminal’s function keys. Backslash will quote any other character in the macro text. term application The application construct is used to include application-specific settings. In the macro body. including ‘"’ and ‘’’. The text of the test extends to the end of the line. to set bindings in the emacsstandard and emacs-ctlx keymaps only if Readline is starting out in emacs mode.3. for instance. Unquoted text is assumed to be a function name. single or double quotes must be used to indicate a macro definition.2 Conditional Init Constructs Readline implements a facility similar in spirit to the conditional compilation features of the C preprocessor which allows key bindings and variable settings to be performed as the result of tests. \$if The \$if construct allows bindings to be made based on the editing mode. The term= form may be used to include terminal-specific key bindings. This allows sun to match both sun and sun-cmd. no characters are required to isolate it. For example. or the application using Readline. There are four parser directives used. the backslash escapes described above are expanded. the terminal being used. The word on the right side of the ‘=’ is tested against both the full name of the terminal and the portion of the terminal name before the first ‘-’. This may be used in conjunction with the ‘set keymap’ command.
This illustrates key binding. Commands in this branch of the \$if directive are executed if the test fails. the following command adds a key sequence that quotes the current or previous word in Bash: \$if Bash # Quote the current or previous word "\C-xq": "\eb\"\ef\"" \$endif \$endif \$else \$include This command.3. as seen in the previous example. This could be used to bind key sequences to functions useful for a specific program. . terminates an \$if command. and you can test for a particular value.Chapter 8: Command Line Editing 103 name. and conditional syntax. the following directive reads from ‘/etc/inputrc’: \$include /etc/inputrc 8. variable assignment.3 Sample Init File Here is an example of an inputrc file. This directive takes a single filename as an argument and reads commands and bindings from that file. For example. For instance.
include any systemwide bindings and variable # assignments from /etc/Inputrc \$include /etc/Inputrc # # Set various bindings for emacs mode. # Lines beginning with ’#’ are comments. and GDB.104 Bash Reference Manual # This file controls the behaviour of line input editing for # programs that use the GNU Readline library. Existing # programs include FTP. Bash. set editing-mode emacs \$if mode=emacs Meta-Control-h: # # Arrow keys # #"\M-OD": #"\M-OC": #"\M-OA": #"\M-OB": # # Arrow keys # "\M-[D": "\M-[C": "\M-[A": "\M-[B": # # Arrow keys # #"\M-\C-OD": #"\M-\C-OC": #"\M-\C-OA": #"\M-\C-OB": # # Arrow keys # #"\M-\C-[D": #"\M-\C-[C": backward-kill-word Text after the function name is ignored in keypad mode backward-char forward-char previous-history next-history in ANSI mode backward-char forward-char previous-history next-history in 8 bit keypad mode backward-char forward-char previous-history next-history in 8 bit ANSI mode backward-char forward-char . # # First. # # You can re-read the inputrc file with C-x C-r.
ask the user if he wants to see all of them set completion-query-items 150 .Chapter 8: Command Line Editing 105 #"\M-\C-[A": #"\M-\C-[B": C-q: quoted-insert \$endif previous-history next-history # An old-style binding. "\M-\C-v": "\C-a\C-k\$\C-y\M-\C-e\C-a\C-y=" \$endif # use a visible bell if one is available set bell-style visible # don’t strip characters to 7 bits when reading set input-meta on # allow iso-latin1 characters to be inserted rather # than converted to prefix-meta sequences set convert-meta off # display characters with the eighth bit set directly # rather than as meta-prefixed characters set output-meta on # if there are more than 150 possible completions for # a word. which is unbound "\C-xr": redraw-current-line # Edit variable on current line. # Macros that are convenient for shell interaction \$if Bash # edit the path "\C-xp": "PATH=\${PATH}\e\C-e\C-a\ef\C-f" # prepare to type a quoted word -# insert open and close double quotes # and move to just after the open quote "\C-x\"": "\"\"\C-b" # insert a backslash (testing backslash escapes # in sequences and macros) "\C-x\\": "\\" # Quote the current or previous word "\C-xq": "\eb\"\ef\"" # Add a binding to refresh the line. TAB: complete This happens to be the default.
4. point refers to the current cursor position. shell-forward-word () Move forward to the end of the next word. end-of-line (C-e) Move to the end of the line. backward-char (C-b) Move back a character. shell-backward-word () Move back to the start of the current or previous word. forward-char (C-f) Move forward a character.": yank-last-arg \$endif 8. Words are delimited by non-quoted shell metacharacters. (See Section 4. forward-word (M-f) Move forward to the end of the next word. backward-word (M-b) Move back to the start of the current or previous word. Words are delimited by non-quoted shell metacharacters. Words are composed of letters and digits. redraw-current-line () Refresh the current line. The text between the point and mark is referred to as the region. By default.106 Bash Reference Manual # For FTP \$if Ftp "\C-xg": "get \M-?" "\C-xt": "put \M-?" "\M-. for a more terse format. In the following descriptions.4 Bindable Readline Commands This section describes Readline commands that may be bound to key sequences.) Command names without an accompanying key sequence are unbound by default. clear-screen (C-l) Clear the screen and redraw the current line. this is unbound. Words are composed of letters and digits. You can list your key bindings by executing bind -P or. and mark refers to a cursor position saved by the set-mark command. leaving the current line at the top of the screen.2 [Bash Builtins]. page 41.1 Commands For Moving beginning-of-line (C-a) Move to the start of the current line. bind -p. 8. . suitable for an inputrc file.
next-history (C-n) Move ‘forward’ through the history list. fetching the previous command. If this line is non-empty. then restore the history line to its original state..e. this command is unbound. fetching the next command. reverse-search-history (C-r) Search backward starting at the current line and moving ‘up’ through the history as necessary. add it to the history list according to the setting of the HISTCONTROL and HISTIGNORE variables. A negative argument inserts the nth word from the end of the previous . beginning-of-history (M-<) Move to the first line in the history. With an argument n.2 Commands For Manipulating The History accept-line (Newline or Return) Accept the line regardless of where the cursor is. By default. This is an incremental search. end-of-history (M->) Move to the end of the input history. By default. forward-search-history (C-s) Search forward starting at the current line and moving ‘down’ through the the history as necessary. yank-nth-arg (M-C-y) Insert the first argument to the previous command (usually the second word on the previous line) at point.4. If this line is a modified history line. non-incremental-forward-search-history (M-n) Search forward starting at the current line and moving ‘down’ through the the history as necessary using a non-incremental search for a string supplied by the user. previous-history (C-p) Move ‘back’ through the history list. non-incremental-reverse-search-history (M-p) Search backward starting at the current line and moving ‘up’ through the history as necessary using a non-incremental search for a string supplied by the user. insert the nth word from the previous command (the words in the previous command begin with word 0). the line currently being entered. This is an incremental search. this command is unbound. history-search-forward () Search forward through the history for the string of characters between the start of the current line and the point. i. This is a non-incremental search.Chapter 8: Command Line Editing 107 8. history-search-backward () Search backward through the history for the string of characters between the start of the current line and the point. This is a non-incremental search.
and the last character typed was not bound to delete-char. backward-delete-char (Rubout) Delete the character behind the cursor. Successive calls to yank-last-arg move back through the history list. . lowercase the previous word. . unless the cursor is at the end of the line. With a negative argument. b. 8. moving the cursor forward as well. If point is at the beginning of the line. uppercase the previous word. forward-backward-delete-char () Delete the character under the cursor. !.. If the insertion point is at the end of the line. 1. By default. moving point past that word as well. upcase-word (M-u) Uppercase the current (or following) word. Negative arguments have no effect. then this transposes the last two characters of the line. for example. transpose-chars (C-t) Drag the character before the cursor forward over the character at the cursor. but do not move the cursor. then return eof. transpose-words (M-t) Drag the word before point past the word after point. or M-_) Insert last argument to the previous command (the last word of the previous history entry). quoted-insert (C-q or C-v) Add the next character typed to the line verbatim. this is not bound to a key. With an argument.) Insert yourself. there are no characters in the line. A. With a negative argument..3 Commands For Changing Text delete-char (C-d) Delete the character at point. yank-last-arg (M-. in which case the character behind the cursor is deleted. The history expansion facilities are used to extract the last argument. the argument is extracted as if the ‘!n’ history expansion had been specified.4. inserting the last argument of each line in turn. behave exactly like yank-nth-arg. as if the ‘!\$’ history expansion had been specified. but do not move the cursor.108 Bash Reference Manual command. A numeric argument means to kill the characters instead of deleting them. self-insert (a. If the insertion point is at the end of the line. this transposes the last two words on the line. Once the argument n is computed. This is how to insert key sequences like C-q. downcase-word (M-l) Lowercase the current (or following) word.
4 Killing And Yanking kill-line (C-k) Kill the text from point to the end of the line. By default. . With a negative argument. but do not move the cursor. unix-filename-rubout () Kill the word behind point. capitalize the previous word. shell-kill-word () Kill from point to the end of the current word. Word boundaries are the same as backward-word. Word boundaries are the same as shell-backwardword. Each call to readline() starts in insert mode. to the end of the next word. backward-kill-word (M-DEL) Kill the word behind point. backward-kill-line (C-x Rubout) Kill backward to the beginning of the line. 8. this command is unbound.Chapter 8: Command Line Editing 109 capitalize-word (M-c) Capitalize the current (or following) word. this is unbound. By default. This command affects only emacs mode. The killed text is saved on the kill-ring. Characters bound to backwarddelete-char replace the character before point with a space. this is unbound. delete-horizontal-space () Delete all spaces and tabs around point. Word boundaries are the same as forward-word. overwrite-mode () Toggle overwrite mode. or if between words. vi mode does overwrite differently. to the end of the next word. switches to insert mode. kill-word (M-d) Kill from point to the end of the current word. switches to overwrite mode. With an explicit positive numeric argument. no matter where point is. With an explicit non-positive numeric argument. using white space as a word boundary. or if between words. Word boundaries are the same as shell-forward-word. using white space and the slash character as the word boundaries. backward-kill-word () Kill the word behind point. unix-line-discard (C-u) Kill backward from the cursor to the beginning of the current line. kill-whole-line () Kill all characters on the current line. characters bound to self-insert replace the text at point rather than pushing the text to the right. By default. In overwrite mode. unix-word-rubout (C-w) Kill the word behind point. The killed text is saved on the kill-ring.4.
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kill-region () Kill the text in the current region. By default, this command is unbound. copy-region-as-kill () Copy the text in the region to the kill buffer, so it can be yanked right away. By default, this command is unbound. copy-backward-word () Copy the word before point to the kill buffer. The word boundaries are the same as backward-word. By default, this command is unbound. copy-forward-word () Copy the word following point to the kill buffer. The word boundaries are the same as forward-word. By default, this command is unbound. yank (C-y) Yank the top of the kill ring into the buffer at point. yank-pop (M-y) Rotate the kill-ring, and yank the new top. You can only do this if the prior command is yank or yank-pop.
8.4.5 Specifying Numeric Arguments
digit-argument (M-0, M-1, ... M--) Add this digit to the argument already accumulating, or start a new argument. M-- starts a negative argument. universal-argument () This is another way to specify an argument. If this command is followed by one or more digits, optionally with a leading minus sign, those digits define the argument. If the command is followed by digits, executing universal-argument again ends the numeric argument, but is otherwise ignored. As a special case, if this command is immediately followed by a character that is neither a digit or minus sign, the argument count for the next command is multiplied by four. The argument count is initially one, so executing this function the first time makes the argument count four, a second time makes the argument count sixteen, and so on. By default, this is not bound to a key.
8.4.6 Letting Readline Type For You
complete (TAB) Attempt to perform completion on the text before point. The actual completion performed is application-specific. Bash attempts completion treating the text as a variable (if the text begins with ‘\$’), username (if the text begins with ‘~’), hostname (if the text begins with ‘@’), or command (including aliases and functions) in turn. If none of these produces a match, filename completion is attempted. possible-completions (M-?) List the possible completions of the text before point.
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insert-completions (M-*) Insert all completions of the text before point that would have been generated by possible-completions. menu-complete () Similar to complete, but replaces the word to be completed with a single match from the list of possible completions. Repeated execution of menu-complete steps through the list of possible completions, inserting each match in turn. At the end of the list of completions, the bell is rung (subject to the setting of bell-style) and the original text is restored. An argument of n moves n positions forward in the list of matches; a negative argument may be used to move backward through the list. This command is intended to be bound to TAB, but is unbound by default. menu-complete-backward () Identical to menu-complete, but moves backward through the list of possible completions, as if menu-complete had been given a negative argument. delete-char-or-list () Deletes the character under the cursor if not at the beginning or end of the line (like delete-char). If at the end of the line, behaves identically to possiblecompletions. This command is unbound by default. complete-filename (M-/) Attempt filename completion on the text before point. possible-filename-completions (C-x /) List the possible completions of the text before point, treating it as a filename. complete-username (M-~) Attempt completion on the text before point, treating it as a username. possible-username-completions (C-x ~) List the possible completions of the text before point, treating it as a username. complete-variable (M-\$) Attempt completion on the text before point, treating it as a shell variable. possible-variable-completions (C-x \$) List the possible completions of the text before point, treating it as a shell variable. complete-hostname (M-@) Attempt completion on the text before point, treating it as a hostname. possible-hostname-completions (C-x @) List the possible completions of the text before point, treating it as a hostname. complete-command (M-!) Attempt completion on the text before point, treating it as a command name. Command completion attempts to match the text against aliases, reserved words, shell functions, shell builtins, and finally executable filenames, in that order.
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possible-command-completions (C-x !) List the possible completions of the text before point, treating it as a command name. dynamic-complete-history (M-TAB) Attempt completion on the text before point, comparing the text against lines from the history list for possible completion matches. dabbrev-expand () Attempt menu completion on the text before point, comparing the text against lines from the history list for possible completion matches. complete-into-braces (M-{) Perform filename completion and insert the list of possible completions enclosed within braces so the list is available to the shell (see Section 3.5.1 [Brace Expansion], page 18).
8.4.7 Keyboard Macros
start-kbd-macro (C-x () Begin saving the characters typed into the current keyboard macro. end-kbd-macro (C-x )) Stop saving the characters typed into the current keyboard macro and save the definition. call-last-kbd-macro (C-x e) Re-execute the last keyboard macro defined, by making the characters in the macro appear as if typed at the keyboard.
8.4.8 Some Miscellaneous Commands
re-read-init-file (C-x C-r) Read in the contents of the inputrc file, and incorporate any bindings or variable assignments found there. abort (C-g) Abort the current editing command and ring the terminal’s bell (subject to the setting of bell-style). do-uppercase-version (M-a, M-b, M-x, ...) If the metafied character x is lowercase, run the command that is bound to the corresponding uppercase character. prefix-meta (ESC) Metafy the next character typed. This is for keyboards without a meta key. Typing ‘ESC f’ is equivalent to typing M-f. undo (C-_ or C-x C-u) Incremental undo, separately remembered for each line. revert-line (M-r) Undo all changes made to this line. This is like executing the undo command enough times to get back to the beginning.
8. display-shell-version (C-x C-v) Display version information about the current instance of Bash. . magic-space () Perform history expansion on the current line and insert a space (see Section 9. history-expand-line (M-^) Perform history expansion on the current line. This performs alias and history expansion as well as all of the shell word expansions (see Section 3. and the line is redrawn. a ‘*’ is appended before pathname expansion. glob-expand-word (C-x *) The word before point is treated as a pattern for pathname expansion.6 [Aliases]. page 123). or M-_) A synonym for yank-last-arg. edit-and-execute-command (C-xC-e) Invoke an editor on the current command line. \$EDITOR. Any argument is ignored. replacing the word. The Readline vi mode behaves as specified in the posix 1003. and the list of matching file names is inserted. page 79). and execute the result as shell commands.3 [History Interaction]. Bash attempts to invoke \$VISUAL. shell-expand-line (M-C-e) Expand the line as the shell does. a ‘*’ is appended before pathname expansion. If a numeric argument is supplied. with an asterisk implicitly appended. If a numeric argument is supplied. and emacs as the editor. glob-list-expansions (C-x g) The list of expansions that would have been generated by glob-expand-word is displayed. in that order. insert-last-argument (M-. operate-and-get-next (C-o) Accept the current line for execution and fetch the next line relative to the current line from the history for editing. history-and-alias-expand-line () Perform history and alias expansion on the current line.2 standard. This pattern is used to generate a list of matching file names for possible completions.5 Readline vi Mode While the Readline library does not have a full set of vi editing functions. it does contain enough to allow simple editing of the line. page 17). alias-expand-line () Perform alias expansion on the current line (see Section 6.5 [Shell Expansions].114 Bash Reference Manual glob-complete-word (M-g) The word before point is treated as a pattern for pathname expansion.
any compspec defined with the ‘-D’ option to complete is used as the default. page 17). for a description of FIGNORE. the actions specified by the compspec are used.5 [Shell Expansions]. After these matches have been generated.2 [Bash Variables]. COMP_KEY. you are already placed in ‘insertion’ mode. page 110) is performed. and COMP_TYPE variables are assigned values as described above (see Section 5. as described above (see Section 3. First. it is used to generate the list of matching words. and so forth. command substitution. Only matches which are prefixed by the word being completed are returned.6 [Commands For Completion]. The results are split using the rules described above (see Section 3. the command name is identified. First. If no compspec is found for the full pathname.7 [Word Splitting].1 [The Set Builtin].4.2 [Bash Variables]. and the matching words become the possible completions. See Section 5. If those searches do not result in a compspec. page 61). page 51). Shell quoting is honored. but the FIGNORE shell variable is used. Pressing ESC switches you into ‘command’ mode. where you can edit the text of the line with the standard vi movement keys. any shell function or command specified with the ‘-F’ and ‘-C’ options is invoked. The string is first split using the characters in the IFS special variable as delimiters. the string specified as the argument to the ‘-W’ option is considered.Chapter 8: Command Line Editing 115 In order to switch interactively between emacs and vi editing modes. When the command or function is invoked. an attempt is made to find a compspec for the portion following the final slash. tilde expansion. parameter and variable expansion. Once a compspec has been found. When you enter a line in vi mode. 8. Next. The GLOBIGNORE shell variable is not used to filter the matches. If a shell function is being invoked. If the command word is a full pathname. and arithmetic expansion. move to previous history lines with ‘k’ and subsequent lines with ‘j’. the shell variable FIGNORE is used to filter the matches. Any completions specified by a filename expansion pattern to the ‘-G’ option are generated next. The Readline default is emacs mode.6 Programmable Completion When word completion is attempted for an argument to a command for which a completion specification (a compspec ) has been defined using the complete builtin (see Section 8. Each word is then expanded using brace expansion. If the command word is the empty string (completion attempted at the beginning of an empty line). If a compspec is not found. page 61. COMP_POINT. The results of the expansion are prefix-matched against the word being completed. If a compspec has been defined for that command. page 23).5. When the function or command .7 [Programmable Completion Builtins]. the default Bash completion described above (see Section 8. a compspec for the full pathname is searched for first.3. The words generated by the pattern need not match the word being completed. page 117). the COMP_WORDS and COMP_CWORD variables are also set. the programmable completion facilities are invoked. as if you had typed an ‘i’. When the ‘-f’ or ‘-d’ option is used for filename or directory name completion. any compspec defined with the ‘-E’ option to complete is used. the COMP_ LINE. the compspec is used to generate the list of possible completions for the word. use the ‘set -o emacs’ and ‘set -o vi’ commands (see Section 4.
When using the ‘-F’ or ‘-C’ options. and write the matches to the standard output. all completion specifications. completion attempted on a blank line.sh" >/dev/null 2>&1 && return 124 } complete -D -F _completion_loader 8. existing completion specifications are printed in a way that allows them to be reused as input. The matches will be generated in the same way as if the programmable completion code had generated them directly from a completion specification with the same flags. or if no options are supplied... The ‘-E’ option indicates that the remaining options and actions should apply to “empty” command completion. If word is specified. while available. completion attempted on a command for which no completion has previously been defined. the following default completion function would load completions dynamically: _completion_loader() { . The return value is true unless an invalid option is supplied. if no name s are supplied.] complete -pr [-DE] [name . the various shell variables set by the programmable completion facilities. If the ‘-p’ option is supplied. which may be any option accepted by the complete builtin with the exception of ‘-p’ and ‘-r’. each kept in a file corresponding to the name of the command.] Specify how arguments to each name should be completed. compgen compgen [option] [word] Generate possible completion matches for word according to the options. only those completions matching word will be displayed. assuming that there is a library of compspecs. For instance. or no matches were generated. The ‘-D’ option indicates that the remaining options and actions should apply to the “default” command completion.d/\$1. The ‘-r’ option removes a completion specification for each name. that is. will not have useful values. complete complete [-abcdefgjksuv] [-o comp-option] [-DE] [-A action] [G globpat] [-W wordlist] [-F function] [-C command] [-X filterpat] [-P prefix] [-S suffix] name [name . rather than being loaded all at once.. "/etc/bash_completion..7 Programmable Completion Builtins Two builtin commands are available to manipulate the programmable completion facilities. that is. . or.Chapter 8: Command Line Editing 117 completions to be built dynamically as completion is attempted.
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The process of applying these completion specifications when word completion is attempted is described above (see Section 8.6 [Programmable Completion], page 115). The ‘-D’ option takes precedence over ‘-E’. Other options, if specified, have the following meanings. The arguments to the ‘-G’, ‘-W’, and ‘-X’ options (and, if necessary, the ‘-P’ and ‘-S’ options) should be quoted to protect them from expansion before the complete builtin is invoked. -o comp-option The comp-option controls several aspects of the compspec’s behavior beyond the simple generation of completions. comp-option may be one of: bashdefault Perform the rest of the default Bash completions if the compspec generates no matches. default dirnames filenames Tell Readline that the compspec generates filenames, so it can perform any filename-specific processing (like adding a slash to directory names quoting special characters, or suppressing trailing spaces). This option is intended to be used with shell functions specified with ‘-F’. nospace plusdirs Tell Readline not to append a space (the default) to words completed at the end of the line. After any matches defined by the compspec are generated, directory name completion is attempted and any matches are added to the results of the other actions. Use Readline’s default filename completion if the compspec generates no matches. Perform directory name completion if the compspec generates no matches.
-A action The action may be one of the following to generate a list of possible completions: alias arrayvar binding builtin command directory Directory names. May also be specified as ‘-d’. Alias names. May also be specified as ‘-a’. Array variable names. Readline key binding names (see Section 8.4 [Bindable Readline Commands], page 106). Names of shell builtin commands. May also be specified as ‘-b’. Command names. May also be specified as ‘-c’.
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disabled enabled export file function group helptopic
Names of disabled shell builtins. Names of enabled shell builtins. Names of exported shell variables. May also be specified as ‘-e’. File names. May also be specified as ‘-f’. Names of shell functions. Group names. May also be specified as ‘-g’. Help topics as accepted by the help builtin (see Section 4.2 [Bash Builtins], page 41).
hostname
Hostnames, as taken from the file specified by the HOSTFILE shell variable (see Section 5.2 [Bash Variables], page 61). Job names, if job control is active. May also be specified as ‘-j’. Shell reserved words. May also be specified as ‘-k’. Names of running jobs, if job control is active. Service names. May also be specified as ‘-s’. Valid arguments for the ‘-o’ option to the set builtin (see Section 4.3.1 [The Set Builtin], page 51). Shell option names as accepted by the shopt builtin (see Section 4.2 [Bash Builtins], page 41). Signal names. Names of stopped jobs, if job control is active. User names. May also be specified as ‘-u’. Names of all shell variables. May also be specified as ‘-v’.
job keyword running service setopt shopt signal stopped user variable -G globpat
The filename expansion pattern globpat is expanded to generate the possible completions. -W wordlist The wordlist is split using the characters in the IFS special variable as delimiters, and each resultant word is expanded. The possible completions are the members of the resultant list which match the word being completed. -C command command is executed in a subshell environment, and its output is used as the possible completions.
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-F function The shell function function is executed in the current shell environment. When it finishes, the possible completions are retrieved from the value of the COMPREPLY array variable. -X filterpat filterpat is a pattern as used for filename expansion. It is applied to the list of possible completions generated by the preceding options and arguments, and each completion matching filterpat is removed from the list. A leading ‘!’ in filterpat negates the pattern; in this case, any completion not matching filterpat is removed. -P prefix prefix is added at the beginning of each possible completion after all other options have been applied. -S suffix suffix is appended to each possible completion after all other options have been applied. The return value is true unless an invalid option is supplied, an option other than ‘-p’ or ‘-r’ is supplied without a name argument, an attempt is made to remove a completion specification for a name for which no specification exists, or an error occurs adding a completion specification. compopt compopt [-o option] [-DE] [+o option] [name] Modify completion options for each name according to the options, or for the currently-execution completion if no name s are supplied. If no options are given, display the completion options for each name or the current completion. The possible values of option are those valid for the complete builtin described above. The ‘-D’ option indicates that the remaining options should apply to the “default” command completion; that is, completion attempted on a command for which no completion has previously been defined. The ‘-E’ option indicates that the remaining options should apply to “empty” command completion; that is, completion attempted on a blank line. The ‘-D’ option takes precedence over ‘-E’. The return value is true unless an invalid option is supplied, an attempt is made to modify the options for a name for which no completion specification exists, or an output error occurs.
2 Bash History Builtins Bash provides two builtin commands which manipulate the history list and history file. no truncation is performed. The shell stores each command in the history list prior to parameter and variable expansion but after history expansion is performed. search commands are available in each editing mode that provide access to the history list (see Section 8. The text of the last \$HISTSIZE commands (default 500) is saved. causes the shell to attempt to save each line of a multi-line command in the same history entry. the lines are appended to the history file. The history builtin may be used to display or modify the history list and manipulate the history file. The HISTCONTROL and HISTIGNORE variables may be set to cause the shell to save only a subset of the commands entered.Chapter 9: Using History Interactively 121 9 Using History Interactively This chapter describes how to use the gnu History Library interactively. the history is not saved. For information on using the gnu History Library in other programs. When the history file is read. subject to the values of the shell variables HISTIGNORE and HISTCONTROL. see the gnu Readline Library Manual.1 Bash History Facilities When the ‘-o history’ option to the set builtin is enabled (see Section 4. The lithist shell option causes the shell to save the command with embedded newlines instead of semicolons. The file named by the value of HISTFILE is truncated. to contain no more than the number of lines specified by the value of the HISTFILESIZE variable. marked with the history comment character. After saving the history.bash_history’). page 41.3. The shopt builtin is used to set these options. The shell allows control over which commands are saved on the history list. if enabled. It should be considered a user’s guide. otherwise the history file is overwritten. the history file is truncated to contain no more than \$HISTFILESIZE lines. the shell provides access to the command history. If HISTFILE is unset. If HISTFILESIZE is not set. page 51). if necessary. the history is initialized from the file named by the HISTFILE variable (default ‘~/. the time stamp information associated with each history entry is written to the history file. If the histappend shell option is set (see Section 4. When using command-line editing. lines beginning with the history comment character followed immediately by a digit are interpreted as timestamps for the previous history line. The builtin command fc may be used to list or edit and re-execute a portion of the history list.2 [Bash Builtins].1 [The Set Builtin]. When an interactive shell exits. from a user’s standpoint. the list of commands previously typed. fc . See Section 4. 9. page 41). page 107).2 [Bash Builtins]. When the shell starts up. for a description of shopt. or if the history file is unwritable. If the HISTTIMEFORMAT is set. adding semicolons where necessary to preserve syntactic correctness.2 [Commands For History]. the last \$HISTSIZE lines are copied from the history list to the file named by \$HISTFILE.4. The value of the HISTSIZE shell variable is used as the number of commands to save in a history list. 9. The cmdhist shell option.
In the first form. The ‘-r’ flag reverses the order of the listing. or vi if neither is set. so that typing ‘r cc’ runs the last command beginning with cc and typing ‘r’ re-executes the last command (see Section 6. Otherwise. the commands are listed on standard output. it is used as a format string for strftime to display the time stamp associated with each displayed history entry. command is re-executed after each instance of pat in the selected command is replaced by rep. If last is not specified it is set to first. have the following meanings: -c Clear the history list. If the shell variable HISTTIMEFORMAT is set and not null. These are lines appended to the history file since the beginning of the current Bash session. In the second form. history history history history history history [n] -c -d offset [-anrw] [filename] -ps arg With no options. the value of the following variable expansion is used: \${FCEDIT:-\${EDITOR:-vi}}. the editor given by ename is invoked on a file containing those commands. If ename is not given. When editing is complete. Lines prefixed with a ‘*’ have been modified. -a -n Append the new history lines (history lines entered since the beginning of the current Bash session) to the history file. The ‘-n’ flag suppresses the command numbers when listing. An argument of n lists only the last n lines. No intervening blank is printed between the formatted time stamp and the history line. the edited commands are echoed and executed. .122 Bash Reference Manual fc [-e ename] [-lnr] [first] [last] fc -s [pat=rep] [command] Fix Command. Both first and last may be specified as a string (to locate the most recent command beginning with that string) or as a number (an index into the history list. If first is not specified it is set to the previous command for editing and −16 for listing. a range of commands from first to last is selected from the history list. Options. display the history list with line numbers. A useful alias to use with the fc command is r=’fc -s’. offset should be specified as it appears when the history is displayed. if supplied. This may be combined with the other options to replace the history list completely. If the ‘-l’ flag is given. where a negative number is used as an offset from the current command number). This says to use the value of the FCEDIT variable if set. or the value of the EDITOR variable if that is set. Append the history lines not already read from the history file to the current history list.6 [Aliases]. -d offset Delete the history entry at position offset. page 79).
the end of the line. that is. ‘\$’. This may be shortened to !\$. Refer to command line n. except the 0th. Refer to the most recent command starting with string. The word matched by the most recent ‘?string?’ search. ‘=’ or ‘(’ (when the extglob shell option is enabled using the shopt builtin). A ‘:’ separates the event specification from the word designator. For example.2 Word Designators Word designators are used to select desired words from the event. ‘*’. replacing string1 with string2. The first argument. Words are inserted into the current line separated by single spaces. ‘-y’ abbreviates ‘0-y’. Refer to the command n lines back. Equivalent to !!:s/string1/string2/. When you type this. designates the last argument of the preceding command. designates the second argument of the most recent command starting with the letters fi. For many applications. Words are numbered from the beginning of the line. !! !!:\$ !fi:2 designates the preceding command. . It may be omitted if the word designator begins with a ‘^’. !# The entire command line typed so far. It is not an error to use ‘*’ if there is just one word in the event. ^string1^string2^ Quick Substitution. the empty string is returned in that case. All of the words. with the first word being denoted by 0 (zero). Refer to the previous command. or ‘%’. word 1. This is a synonym for ‘!-1’.124 Bash Reference Manual ! Start a history substitution. except when followed by a space. ‘-’. this is the command word. This is a synonym for ‘1-\$’. 9. The nth word. Repeat the last command. The last argument. The trailing ‘?’ may be omitted if the string is followed immediately by a newline. Here are the word designators: 0 (zero) n ^ \$ % x-y * The 0th word. tab. the preceding command is repeated in toto.3. A range of words. !n !-n !! !string !?string[?] Refer to the most recent command containing string.
3. Any delimiter may be used in place of ‘/’. it is replaced by old. but break into words at spaces. tabs. 9. you can add a sequence of one or more of the following modifiers. but omits the last word. Used in conjunction with ‘s’. & g a G Repeat the previous substitution.Chapter 9: Using History Interactively 125 x* x- Abbreviates ‘x-\$’ Abbreviates ‘x-\$’ like ‘x*’. Cause changes to be applied over the entire event line. Remove all but the trailing suffix. Remove all leading pathname components. and newlines. . If a word designator is supplied without an event specification. as in gs/old/new/. Quote the substituted words as with ‘q’. leaving the tail. The delimiter may be quoted in old and new with a single backslash. The final delimiter is optional if it is the last character on the input line. Print the new command but do not execute it. leaving only the head. or with ‘&’. A single backslash will quote the ‘&’. each preceded by a ‘:’. the previous command is used as the event. leaving the basename. Quote the substituted words. Apply the following ‘s’ modifier once to each word in the event. If ‘&’ appears in new.3 Modifiers After the optional word designator. Remove a trailing suffix of the form ‘.suffix’. h t r e p q x s/old/new/ Substitute new for the first occurrence of old in the event line. Remove a trailing pathname component. escaping further substitutions.
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To find out more about the options and arguments that the configure script understands. you might need to type ‘sh . it prints messages telling which features it is checking for. it creates a shell script named config. ‘doc’. This will also install the manual pages and Info file. Optionally. and several others).cache’ contains results you don’t want to keep. please try to figure out how configure could check whether or not to do them. The configure shell script attempts to guess correct values for various system-dependent variables used during compilation. It uses those values to create a ‘Makefile’ in each directory of the package (the top directory. you may remove or edit it. 4. 3. The simplest way to compile Bash is: 1. You can remove the program binaries and object files from the source code directory by typing ‘make clean’. and mail diffs or instructions to bash-maintainers@gnu. Type ‘make install’ to install bash and bashbug. While running. . type bash-2. os/2. and ‘support’ directories.1 Basic Installation These are installation instructions for Bash./configure --help at the Bash prompt in your Bash source directory.status that you can run in the future to recreate the current configuration. a file ‘config. nearly every version of Unix.in’ is used to create configure by a program called Autoconf. make sure you are using Autoconf version 2. Finally. The file ‘configure. If you do this. type ‘make distclean’.h’ file containing system-dependent definitions. Running configure takes some time.Chapter 10: Installing Bash 127 10 Installing Bash This chapter provides basic instructions for installing Bash on the various supported platforms.org so they can be considered for the next release. To also remove the files that configure created (so you can compile Bash for a different kind of computer). It also creates a ‘config. and Windows platforms. If you’re using csh on an old version of System V. and several non-Unix systems such as BeOS and Interix. The distribution supports the gnu operating systems. If you need to do unusual things to compile Bash.cache’ that saves the results of its tests to speed up reconfiguring.log’ containing compiler output (useful mainly for debugging configure). and a file ‘config. each directory under ‘lib’. 10./configure’ to configure Bash for your system. cd to the directory containing the source code and type ‘.50 or newer. If at some point ‘config. You only need ‘configure. type ‘make tests’ to run the Bash test suite.in’ if you want to change it or regenerate configure using a newer version of Autoconf./configure’ instead to prevent csh from trying to execute configure itself.04\$ . the ‘builtins’. Other independent ports exist for ms-dos. 2. Type ‘make’ to compile Bash and build the bashbug bug reporting script.
Alternatively. Usually configure can figure that out. by placing the object files for each architecture in their own directory. or by specifying a value for the DESTDIR ‘make’ variable when running ‘make install’./configure On systems that have the env program. If you have to use a make that does not supports the VPATH variable. Here’s an example that creates a build directory in the current directory from a source directory ‘/usr/gnu/src/bash-2. you must use a version of make that supports the VPATH variable.’. such as GNU make. To do this. you can do that on the command line like this: CC=c89 CFLAGS=-O2 LIBS=-lposix ./configure The configuration process uses GCC to build Bash if it is available. ‘make install’ will install into ‘/usr/local/bin’. but if it prints a message saying it can not guess the host type.0/support/mkclone -s /usr/gnu/src/bash-2. You can specify separate installation prefixes for architecture-specific files and architecture-independent files.0 . ‘/usr/local/man’.3 Compiling For Multiple Architectures You can compile Bash for more than one kind of computer at the same time. You can specify an installation prefix other than ‘/usr/local’ by giving configure the option ‘--prefix=PATH’. 10. cd to the directory where you want the object files and executables to go and run the configure script from the source directory. Documentation and other data files will still use the regular prefix.0’: bash /usr/gnu/src/bash-2. If you give configure the option ‘--exec-prefix=PATH’.2 Compilers and Options Some systems require unusual options for compilation or linking that the configure script does not know about. but need to determine by the type of host Bash will run on. you can do it like this: env CPPFLAGS=-I/usr/local/include LDFLAGS=-s . ‘make install’ will use PATH as the prefix for installing programs and libraries.4 Installation Names By default. 10. use ‘make distclean’ before reconfiguring for another architecture. The mkclone script requires Bash. so you must have already built Bash for at least one architecture before you can create build directories for other architectures. if your system supports symbolic links. etc..128 Bash Reference Manual 10. give it the ‘--host=TYPE’ . 10. You may need to supply the ‘--srcdir=PATH’ argument to tell configure where the source files are. You can give configure initial values for variables by setting them in the environment. configure automatically checks for the source code in the directory that configure is in and in ‘. After you have installed Bash for one architecture. you can use the ‘support/mkclone’ script to create a build tree which has symbolic links back to each file in the source directory.5 Specifying the System Type There may be some features configure can not figure out automatically. you can compile Bash for one architecture at a time in the source code directory. Using a Bourne-compatible shell.
--help --quiet --silent -q Print a summary of the options to configure. where package is something like ‘bash-malloc’ or ‘purify’.. 10. Or.g.7 Operation Controls configure recognizes the following options to control how it operates. not widely used. --with-afs Define if you are using the Andrew File System from Transarc.site that gives default values for variables like CC. See the file ‘support/config. Usually configure can determine that directory automatically. ‘TYPE’ can either be a short name for the system type.cache’. configure also accepts some other. you can set the CONFIG_SITE environment variable to the location of the site script. Set file to ‘/dev/null’ to disable caching. --version Print the version of Autoconf used to generate the configure script. cache_ file. and exit. Here is a complete list of the ‘--enable-’ and ‘--with-’ options that the Bash configure recognizes. 10. where feature indicates an optional part of Bash. for debugging configure. or a canonical name with three fields: ‘CPU-COMPANY-SYSTEM’ (e. --with-bash-malloc Use the Bash version of malloc in the directory ‘lib/malloc’./config. use ‘--disable-feature’. Do not print messages saying which checks are being made. There are also several ‘--with-package’ options.6 Sharing Defaults If you want to set default values for configure scripts to share. such as ‘sun4’.2’). but an older version originally derived . boilerplate options. 10. then ‘PREFIX/etc/config. but not all configure scripts do.8 Optional Features The Bash configure has a number of ‘--enable-feature’ options. A warning: the Bash configure looks for a site script. and exit.sub’ for the possible values of each field.Chapter 10: Installing Bash 129 option. --cache-file=file Use and save the results of the tests in file instead of ‘. ‘configure --help’ prints the complete list. --srcdir=dir Look for the Bash source code in directory dir. and prefix.site’ if it exists. To configure Bash without a feature that is enabled by default. use ‘--without-package’. ‘i386-unknown-freebsd4. This is not the same malloc that appears in gnu libc. you can create a site shell script called config.site’ if it exists. configure looks for ‘PREFIX/share/config. To turn off the default use of a package.
for details of the builtin and enable builtin commands. --enable-disabled-builtins Allow builtin commands to be invoked via ‘builtin xxx’ even after xxx has been disabled using ‘enable -n xxx’.5.7 [Arrays].2. page 18.2 [Pipelines]. (see Section 3. page 80). for a complete description.b}c → bac bbc ). page 10). for example. --enable-brace-expansion Include csh-like brace expansion ( b{a.2. --enable-casemod-attributes Include support for case-modifying attributes in the declare builtin and assignment statements.3 [History Interaction]. page 10). will have their values converted to uppercase upon assignment. page 8).2 [Pipelines]. --enable-cond-command Include support for the [[ conditional command. page 41.4. --enable-debugger Include support for the bash debugger (distributed separately). .1 [Brace Expansion]. --enable-array-variables Include support for one-dimensional array shell variables (see Section 6. --enable-bang-history Include support for csh-like history substitution (see Section 9. popd. and dirs builtins (see Section 6. See Section 3.2. Variables with the uppercase attribute.2 [Conditional Constructs].2.4.2 [Conditional Constructs].2 [Bash Builtins].Chapter 10: Installing Bash 131 --enable-arith-for-command Include support for the alternate form of the for command that behaves like the C language for statement (see Section 3. page 81). page 9). --enable-coprocesses Include support for coprocesses and the coproc reserved word (see Section 3. --enable-directory-stack Include support for a csh-like directory stack and the pushd.2. --enable-command-timing Include support for recognizing time as a reserved word and for displaying timing statistics for the pipeline following time (see Section 3. See Section 4. (see Section 3.8 [The Directory Stack]. This allows pipelines as well as shell builtins and functions to be timed. --enable-casemod-expansion Include support for case-modifying word expansions. page 123). page 8).1 [Looping Constructs].4. --enable-cond-regexp Include support for matching POSIX regular expressions using the ‘=~’ binary operator in the [[ conditional command.
page 89). this option has no effect.. when called as rbash. \$PS3. --enable-help-builtin Include the help builtin.2 [Bash Builtins]. --enable-job-control This enables the job control features (see Chapter 7 [Job Control].5. and \$PS4 prompt strings.4. page 10).10 [The Restricted Shell].2.6 [Redirections].132 Bash Reference Manual --enable-dparen-arithmetic Include support for the ((. See Section 6. --enable-extended-glob-default Set the default value of the extglob shell option described above under Section 4. enters a restricted mode. if the operating system supports them. page 41). for a complete list of prompt string escape sequences. . page 93). --enable-prompt-string-decoding Turn on the interpretation of a number of backslash-escaped characters in the \$PS1. --enable-progcomp Enable the programmable completion facilities (see Section 8. page 55 to be enabled. --enable-history Include command history and the fc and history builtin commands (see Section 9. for a description of restricted mode. --enable-multibyte This enables support for multibyte characters if the operating system provides the necessary support.6 [Programmable Completion].2 [The Shopt Builtin].2 [Conditional Constructs]. page 84. page 23) if the operating system provides the necessary support.1 [Pattern Matching]. --enable-extended-glob Include support for the extended pattern matching features described above under Section 3. Bash. page 26).1 [Bash History Facilities]. which displays help on shell builtins and variables (see Section 4. --enable-net-redirections This enables the special handling of filenames of the form /dev/tcp/host/port and /dev/udp/host/port when used in redirections (see Section 3. page 82.5. --enable-readline Include support for command-line editing and history with the Bash version of the Readline library (see Chapter 8 [Command Line Editing]. page 24. See Section 6..6 [Process Substitution].9 [Printing a Prompt]. --enable-restricted Include support for a restricted shell. page 121). \$PS2. page 115). If this is enabled. --enable-process-substitution This enables process substitution (see Section 3.8.3.)) command (see Section 3. If Readline is not enabled.
page 10). which makes the Bash echo behave more like the version specified in the Single Unix Specification. This aids in translating the text to different languages. for a description of the escape sequences that echo recognizes. You may need to disable this if your compiler cannot handle very long string literals. This sets the default value of the xpg_echo shell option to on. --enable-strict-posix-default Make Bash posix-conformant by default (see Section 6. which allows the generation of simple menus (see Section 3.11 [Bash POSIX Mode]. --enable-usg-echo-default A synonym for --enable-xpg-echo-default. Read the comments associated with each definition for more information about its effect. Some of these are not meant to be changed.2. page 84). The file ‘config-top. --enable-xpg-echo-default Make the echo builtin expand backslash-escaped characters by default. --enable-single-help-strings Store the text displayed by the help builtin as a single string for each help topic.4. version 3. --enable-separate-helpfiles Use external files for the documentation displayed by the help builtin instead of storing the text internally.2 [Conditional Constructs].h’ contains C Preprocessor ‘#define’ statements for options which are not settable from configure. without requiring the ‘-e’ option. page 41. See Section 4. beware of the consequences if you do.Chapter 10: Installing Bash 133 --enable-select Include the select builtin. .2 [Bash Builtins].
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All bug reports should include: • The version number of Bash. Once you have determined that a bug actually exists.bash. you are encouraged to mail that as well! Suggestions and ‘philosophical’ bug reports may be mailed to [email protected] or posted to the Usenet newsgroup gnu. If you have a fix.gnu.bug. and that it appears in the latest version of Bash.ramey@case. • A description of the bug behaviour.Appendix A: Reporting Bugs 135 Appendix A Reporting Bugs Please report all bugs you find in Bash.org/pub/gnu/bash/. • The hardware and operating system. Please send all reports concerning this manual to chet. • The compiler used to compile Bash. But first.edu. . • A short script or ‘recipe’ which exercises the bug and may be used to reproduce it. The latest version of Bash is always available for FTP from ftp://ftp. bashbug inserts the first three items automatically into the template it provides for filing a bug report. you should make sure that it really is a bug. use the bashbug command to submit a bug report.
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• Bash provides a programmable word completion mechanism (see Section 8. • Bash implements csh-like history expansion (see Section 9. • Bash includes the select compound command. The display of the timing statistics may be controlled with the TIMEFORMAT variable. page 93) and the bind builtin.. page 8). page 115). which allows the generation of simple menus (see Section 3. • Bash has one-dimensional array variables (see Section 6. A number of these differences are explained in greater depth in previous sections. .1. The Bash history list maintains timestamp information and uses the value of the HISTTIMEFORMAT variable to display it.4.2. even where the posix specification differs from traditional sh behavior (see Section 6. page 80). redirection. and compopt.11 [Bash POSIX Mode].3 [History Interaction]. and builtin commands complete. this section quickly details the differences of significance.. parameter and variable expansion. page 71). is supported (see Section 3.2 [Pipelines]. and ‘--dump-po-strings’ invocation options list the translatable strings found in a script (see Section 3. This section uses the version of sh included in SVR4. Bash uses the posix standard as the specification of how these features are to be implemented. • Bash supports the \$".2. • Bash has multi-character invocation options (see Section 6. expr3 )) arithmetic for command.2 (the last version of the historical Bourne shell) as the baseline reference.5 [Locale Translation]. similar to the C language (see Section 3. Very useful when an if statement needs to act only if a test fails.6 [Programmable Completion]. page 121) and the history and fc builtins to manipulate it. • Bash has command history (see Section 9. page 10). expr2 . There are some differences between the traditional Bourne shell and Bash. • Bash has the time reserved word and command timing (see Section 3.1 [Looping Constructs]. • Bash implements the for (( expr1 .2 [Conditional Constructs].1 [Invoking Bash].Appendix B: Major Differences From The Bourne Shell 137 Appendix B Major Differences From The Bourne Shell Bash implements essentially the same grammar. ‘--dump-strings’. page 84).7 [Arrays]." quoting syntax to do locale-specific translation of the characters between the double quotes.1 [Bash History Facilities]. Bash provides a number of built-in array variables. and quoting as the Bourne Shell. page 8). • Bash has command-line editing (see Chapter 8 [Command Line Editing]. page 9). The ‘-D’.. • The \$’. Several of the Bash builtins take options to act on arrays. • Bash implements the ! keyword to negate the return value of a pipeline (see Section 3.2 [Pipelines].4 [ANSI-C Quoting].’ quoting syntax. compgen. The Bash ‘-o pipefail’ option to set will cause a pipeline to return a failure status if any command fails..1.2.2. and the appropriate variable expansions and assignment syntax to use them. page 7). page 123).2. which expands ANSI-C backslash-escaped characters in the text between the single quotes. to manipulate it. page 6).4. • Bash is posix-conformant.2.
5.5. page 10). • The posix \$() form of command substitution is implemented (see Section 3. • Bash includes the posix pattern removal ‘%’. which makes conditional testing part of the shell grammar (see Section 3.5. page 19). page 18) and tilde expansion (see Section 3. EUID.4.138 Bash Reference Manual • Bash includes the [[ compound command. • Bash supports the ‘+=’ assignment operator.2. page 23). . is available (see Section 3. • Bash automatically assigns variables that provide information about the current user (UID. page 19). which returns the length of \${xx}. The Bourne shell does not normally do this unless the variables are explicitly marked using the export command. including optional regular expression matching.7 [Word Splitting]. • Bash provides shell arithmetic.4 [Command Substitution]. ‘#’.5. which appends to the value of the variable named on the left hand side.5. See Section 5.4.5. the current host (HOSTTYPE. the (( compound command (see Section 3.5.6 [Process Substitution]. and the instance of Bash that is running (BASH. which expands to the substring of var’s value of length length. which matches pattern and replaces it with replacement in the value of var.3 [Shell Parameter Expansion]. page 19). beginning at offset. and GROUPS). • Bash has indirect variable expansion using \${!word} (see Section 3. • Bash provides optional case-insensitive matching for the case and [[ constructs.2 [Tilde Expansion].2. page 10). MACHTYPE.3 [Shell Parameter Expansion]. ‘%%’ and ‘##’ expansions to remove leading or trailing substrings from variable values (see Section 3. • The expansion \${var/[/]pattern[/replacement]}. and preferred to the Bourne shell’s ‘‘ (which is also implemented for backwards compatibility). not all words (see Section 3.5. is available (see Section 3. • The expansion \${var:offset[:length]}. • The expansion \${#xx}. page 19). and BASH_VERSINFO). page 23). is present (see Section 3. • The expansion \${!prefix}* expansion.2 [Bash Variables]. is supported (see Section 3. BASH_VERSION. • The IFS variable is used to split only the results of expansion. page 61.3 [Shell Parameter Expansion].5.2 [Conditional Constructs]. page 19).3 [Shell Parameter Expansion]. which expands to the names of all shell variables whose names begin with prefix. and arithmetic expansion (see Section 6. • Bash can expand positional parameters beyond \$9 using \${num}.3 [Shell Parameter Expansion]. and HOSTNAME).1 [Brace Expansion]. page 78).5. OSTYPE.2 [Conditional Constructs]. This closes a longstanding shell security hole.3 [Shell Parameter Expansion].6 [Aliases]. page 22). page 19). page 19). for details. • Bash includes brace expansion (see Section 3. page 79).5 [Shell Arithmetic]. • Bash implements command aliases and the alias and unalias builtins (see Section 6. • Bash has process substitution (see Section 3. • Variables present in the shell’s initial environment are automatically exported to child processes.5.
• The Bash cd and pwd builtins (see Section 4. • Bash implements extended pattern matching features when the extglob shell option is enabled (see Section 3. allowing a file to be opened for both reading and writing.1 [Bourne Shell Builtins]. and the ‘&>’ redirection operator. • Bash functions are permitted to have local variables using the local builtin. page 26). • The Bash exec builtin takes additional options that allow users to control the contents of the environment passed to the executed command. • Bash treats a number of filenames specially when they are used in redirection operators (see Section 3.5. page 41). equivalence classes. page 26). and collating symbols (see Section 3. The ‘>|’ redirection operator may be used to override noclobber. for directing standard output and standard error to the same file (see Section 3. even builtins and functions (see Section 3. page 24).2 [Bash Builtins]. • Bash implements the ‘[n]<&word’ and ‘[n]>&word’ redirection operators. In sh. • It is possible to have a variable and a function with the same name. and provides access to that builtin’s functionality within the function via the builtin and command builtins (see Section 4.6 [Redirections]. .1 [The Set Builtin].5. page 35). page 24). page 31).6 [Redirections].1 [Bourne Shell Builtins].2 [Bash Builtins]. all variable assignments preceding commands are global unless the command is executed from the file system. and what the zeroth argument to the command is to be (see Section 4. page 26).2 [Bash Builtins]. • Individual builtins may be enabled or disabled using the enable builtin (see Section 4. • Bash includes the ‘<<<’ redirection operator.3. which move one file descriptor to another. allowing a string to be used as the standard input to a command.2 [Bash Builtins].8. page 41). • Variable assignments preceding commands affect only that command. • The command builtin allows selective disabling of functions when command lookup is performed (see Section 4. including character classes. sh does not separate the two name spaces. • The noclobber option is available to avoid overwriting existing files with output redirection (see Section 4. page 35) each take ‘-L’ and ‘-P’ options to switch between logical and physical modes.1 [Pattern Matching]. • Bash performs filename expansion on filenames specified as operands to input and output redirection operators (see Section 3. • Bash contains the ‘<>’ redirection operator.3 [Shell Functions]. • Bash allows a function to override a builtin with the same name.6 [Redirections]. page 51).6 [Redirections]. page 26). page 41).4 [Environment]. • Bash can open network connections to arbitrary machines and services with the redirection operators (see Section 3. page 14). and thus useful recursive functions may be written (see Section 4.7. page 41).8 [Filename Expansion]. • Shell functions may be exported to children via the environment using export -f (see Section 3.Appendix B: Major Differences From The Bourne Shell 139 • Bash implements the full set of posix filename expansion operators.
The disown builtin can remove a job from the internal shell job table (see Section 7. it does not suffer from many of the limitations of the SVR4. and dirs builtins to manipulate it (see Section 6. The trap builtin (see Section 4. The ERR trap is not inherited by shell functions unless the -o errtrace option to the set builtin is enabled.9 [Printing a Prompt].2 shell has two privilege-related builtins (mldmode and priv) not present in Bash. Bash also makes the directory stack visible as the value of the DIRSTACK shell variable. The SVR4. page 81). page 35).2 sh uses a TIMEOUT variable like Bash uses TMOUT.2 Shell Since Bash is a completely new implementation. Bash implements a csh-like directory stack. The SVR4. page 35) allows a RETURN pseudo-signal specification. . The Bash restricted mode is more useful (see Section 6. More features unique to Bash may be found in Chapter 6 [Bash Features]. The SVR4. or source returns. B. • Bash does not allow unbalanced quotes. Commands specified with an RETURN trap are executed before execution resumes after a shell function or a shell script executed with . by using the system() C library function call).2 shell.2 [Bash Builtins].2 shell will silently insert a needed closing quote at EOF under certain circumstances.Appendix B: Major Differences From The Bourne Shell 141 • • • • • • • • • • • are executed after a simple command fails.g. page 71.. popd. For instance: • Bash does not fork a subshell when redirecting into or out of a shell control structure such as an if or while statement. and provides the pushd.8 [The Directory Stack]. page 90) or suppress the sending of SIGHUP to a job when the shell exits as the result of a SIGHUP. with a few exceptions.10 [The Restricted Shell]. The Bash umask builtin permits a ‘-p’ option to cause the output to be displayed in the form of a umask command that may be reused as input (see Section 4.1 [Bourne Shell Builtins]. similar to EXIT and DEBUG.1 [Bourne Shell Builtins]. Bash does not use the SHACCT variable or perform shell accounting. The RETURN trap is not inherited by shell functions unless the function has been given the trace attribute or the functrace option has been enabled using the shopt builtin. This can be the cause of some hardto-find errors. the SVR4. If the shell is started from a process with SIGSEGV blocked (e. page 41).2 shell restricted mode is too limited. The Bash type builtin is more extensive and gives more information about the names it finds (see Section 4.1 Implementation Differences From The SVR4. • The SVR4. page 82). it misbehaves badly. Bash interprets special backslash-escaped characters in the prompt strings when interactive (see Section 6.2 shell uses a baroque memory management scheme based on trapping SIGSEGV. page 84).2 [Job Control Builtins]. Bash does not have the stop or newgrp builtins. Bash includes a number of features to support a separate debugger for shell scripts.
as enumerated in the posix standard. or SIGCHLD. PATH. • The SVR4.2 shell does not allow the IFS. • The SVR4. some versions of the shell dump core if the second argument begins with a ‘-’.142 Bash Reference Manual • In a questionable attempt at security. commonly 100. • The SVR4.2 shell exits a script if any builtin fails. • Bash allows multiple option arguments when it is invoked (-x -v). the SVR4. . • The SVR4. • The SVR4. Bash exits a script only if one of the posix special builtins fails. PS1. MAILCHECK. or PS2 variables to be unset.2 shell treats ‘^’ as the undocumented equivalent of ‘|’.2 shell does not allow users to trap SIGSEGV. the SVR4.2 shell. and only for certain failures. In fact.2 shell behaves differently when invoked as jsh (it turns on job control). will alter its real and effective uid and gid if they are less than some magic threshold value. SIGALRM. when invoked without the ‘-p’ option.2 shell allows only one option argument (-xv). This can lead to unexpected results.
VERBATIM COPYING . The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. An image format is not Transparent if used for any substantial amount of text. that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor. The Document may contain zero Invariant Sections. LaTEX input format. The “Title Page” means. the title page itself. A copy that is not “Transparent” is called “Opaque”. such as “Acknowledgements”. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. plus such following pages as are needed to hold. The “Cover Texts” are certain short passages of text that are listed. “Endorsements”. represented in a format whose specification is available to the general public. For works in formats which do not have any title page as such.144 Bash Reference Manual under this License. or absence of markup. but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License. These Warranty Disclaimers are considered to be included by reference in this License. SGML or XML for which the DTD and/or processing tools are not generally available. “Dedications”. SGML or XML using a publicly available DTD. Texinfo input format. and standard-conforming simple HTML. has been arranged to thwart or discourage subsequent modification by readers is not Transparent. preceding the beginning of the body of the text.) To “Preserve the Title” of such a section when you modify the Document means that it remains a section “Entitled XYZ” according to this definition. A copy made in an otherwise Transparent file format whose markup. or “History”. A “Transparent” copy of the Document means a machine-readable copy. “Title Page” means the text near the most prominent appearance of the work’s title. A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. PostScript or PDF designed for human modification. and the machine-generated HTML. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors. and a Back-Cover Text may be at most 25 words. (Here XYZ stands for a specific section name mentioned below. in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words. Examples of suitable formats for Transparent copies include plain ascii without markup. PostScript or PDF produced by some word processors for output purposes only. legibly. If the Document does not identify any Invariant Sections then there are none. and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. The “publisher” means any person or entity that distributes copies of the Document to the public. XCF and JPG. for a printed book. Examples of transparent image formats include PNG. 2. the material this License requires to appear in the title page. as Front-Cover Texts or Back-Cover Texts.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 J A alias . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 45 36 36 36 36 S set . . . . . . . . . . . . . . . . . 81 disown . . . . . . . . . . . shopt . . . . . . . . 82 47 82 38 D declare . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 complete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pushd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pwd . . . . . . . . . . . . . . . . . . . . . 46 C caller . 43 cd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 . . . . . . . exit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 fg . . . . . . . . . . . . . . . . . . . . . . 46 logout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . builtin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 M mapfile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 help . . . . source . . . . . . . . . . . printf . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 dirs . . . . . . . . . . . . . . 46 P popd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 G getopts . . . . . . . . . . . . . . . . . . 43 compgen . . . . . . . . export . . . . . . 46 local . . . . 36 command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 R read . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 49 38 38 E echo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 [ [ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exec . . . . . . . . . . . readarray . . . . 37 : : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . readonly . return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 history . . . . . . . .1 Index of Shell Builtin Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . enable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 compopt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bind . . . . . . . . . . . . 90 B bg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 continue . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 L let . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 38 55 49 91 F fc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 41 35 43 K kill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . suspend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Appendix D: Indexes 151 Appendix D Indexes D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eval . 35 H hash . . . . . .
.. . . . . . . . . . . . umask . . . . . . . . . . . .. . . . trap . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 14 ] ]] . . . . . . . . . . . . . . . . . . .. . . . . . 11 C case . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 17 \$ \$ . . . . . . . . 49 41 50 41 W wait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . 17 # # . . . . . . . . . . . 10 for . . . . . . . . . . . . . . . . . . . . . . .. . . . 39 40 40 49 49 U ulimit . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 } } . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 10 [ [[ . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . type . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 F fi . . . . . . 91 D. . . . . . . . . . . . . . . . . . . . . . . . .. . .. .. . . . . . . . . . . . . . . . . . . . . . . 13 S select . . . . . . . . . . . . 10 in . . .. . . . . . . . . . . . typeset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 T then . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .152 Bash Reference Manual T test . . . . . . . . . . . . . . . 10 time . . . . . . . . . . . . . . . .. . . . . . . . . . 10 esac . . . . . . .. . . . . . . . . . . . 9 D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 9 E elif . . 8 else . . . . . . . . . . . . . . . . . . . . 10 W while . .. . . . . unset . 8 D do . . 17 * * . 9 done . . .3 Parameter and Variable Index ! ! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 { { . . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 U until . . . . . . . . . . . 12 I if . . . . . . . . . . . . . .. 10 function . . . . . . . . . . . . . . . . . .2 Index of Shell Reserved Words ! !. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . unalias . . . . . . . . 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. .. . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . times . . . . .. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . history-preserve-point . . . . . . completion-prefix-display-length . . BASH_EXECUTION_STRING . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 GROUPS . . . . . . . . . . . . . . . . HISTFILESIZE . . . . . . . . . . . . . . . . . . . . . 98 @ @ . . . . . . . . . . COMP_TYPE . . . . . . . . . . . . . . . . BASH_ENV . . . . . . . . . . . 61 62 62 62 62 62 62 62 63 63 63 63 63 63 63 62 62 97 97 G GLOBIGNORE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COMP_CWORD . . . . . . . . . . . . . . . . . . . . . . . . . . COMP_WORDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HISTCMD . . . . . . . . BASH_VERSINFO . . . . . . . . . . . . . . . . . . . input-meta . . . . . . . . . . . HOSTTYPE . . . . . . HOSTFILE . . . . . BASH_XTRACEFD . . . . . . BASH_COMMAND . . . . . . . . isearch-terminators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HISTIGNORE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 ? ? . . . . . . . . . . . . . . . 92 B BASH . . . . . . . . . 65 FIGNORE . . . . . . . . . . . . . . . . . . . . . . . . . . . . BASH_LINENO . . . . . . . . . . . . . . . . . . . . . . . . . 61 64 97 64 64 64 64 64 64 64 97 I IFS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 disable-completion . . . . . . . . . . . . . . . . . . . . . . . . . . COLUMNS . . expand-tilde . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HISTFILE . . . . . . . . . . . . . . BASH_VERSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BASH_CMDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 65 A auto_resume . history-size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 convert-meta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COMP_KEY . . . . . HOSTNAME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 H histchars . . . . . . . . . . . 17 completion-query-items . . . . . . . . . . . . . . . BASH_SOURCE . . . . . . . . . . . . . . . . . . . 97 COMPREPLY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Appendix D: Indexes 153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 0 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 F FCEDIT . . . . . . comment-begin . . . . . . . . . . . . . . . . . . . . . EUID . . . . . . . BASH_ARGV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 65 98 65 98 _ . . COMP_LINE . . . . BASH_REMATCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . COMP_POINT . . . . . . . . . horizontal-scroll-mode . . . . . . . . . 16 E editing-mode . . . . . . . . . . . . . . . . 65 66 66 66 66 66 98 98 66 66 61 98 67 67 67 C CDPATH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BASHPID . . . . . . . BASH_SUBSHELL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . bell-style . . . . . . . . . . . . INPUTRC . . . . . . . . . . . . . . . . . . . . . . . . . . 61 67 99 67 99 K keymap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 FUNCNAME . . . . . . . . . . . . . . . . . . bind-tty-special-chars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EMACS . BASHOPTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BASH_ALIASES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HISTTIMEFORMAT . . . . . . . . . . . . . . . . . HOME . . . . . . . . . . 17 D DIRSTACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 . . . . . . . . . . . COMP_WORDBREAKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BASH_ARGC . . . . . . . . . . . . . . enable-keypad . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HISTCONTROL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . IGNOREEOF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HISTSIZE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OPTIND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 SHELLOPTS . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 P page-completions . . . . . . . . . . . . . . . . 67 61 68 61 99 99 99 99 S SECONDS . . . . . . . . . . . . . . match-hidden-files . . . . . . . . . . . . . . . . . . . . . . . . . . . . output-meta . . . . . . . . . . . . . . . . . . . . . . . . . . complete-command (M-!) . . . . . . . . . . . . . . . . beginning-of-history (M-<) . . . . . . . . . . . . . 68 61 68 61 68 99 T TEXTDOMAIN . . . . . . . . . . . . . OPTARG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 skip-completed-text . . . . . . . . . . . . . . . . . . . . . . . . . . LC_MESSAGES . . . . . . . . . . . . . . . . . . . . . . . . . . PIPESTATUS . . . . . . . . . . . . meta-flag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . character-search-backward (M-C-]) . . LINES . . . . . . . . . . . backward-word (M-b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . backward-kill-line (C-x Rubout) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . backward-delete-char (Rubout) . . . . . . . . . . . . . . . . . . . . . . backward-kill-word (M-DEL) . . . . . . . . . . . . . . . . . . . . . . . . . 106 C call-last-kbd-macro (C-x e) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LC_CTYPE . . LC_NUMERIC . . . . . . . . . . . . . . . . . .154 Bash Reference Manual L LANG . 99 61 68 68 68 68 U UID . . . . . . . . . . . . . . . . . . . . . . . . . . . . complete-hostname (M-@) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 show-all-if-ambiguous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 SHELL . . . complete (TAB) . . . . . . . . . . . . . . . . . . . . . . LC_COLLATE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4 Function Index A abort (C-g) . . . . . . . . . 100 D. . . . . 69 TMOUT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS2 . 69 SHLVL . . mark-symlinked-directories . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 revert-all-at-newline . . . . . . 7. . . . . . . . . . . . . . . . . 107 alias-expand-line () . . . . . . . . MAILPATH . . . . . . . . . . . . . . . . . . . 112 accept-line (Newline or Return) . . . . complete-username (M-~) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PATH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 O OLDPWD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 61 61 68 68 68 R RANDOM . . . . . . . . . . . . . . . . . . . 112 109 113 113 106 110 111 111 111 112 111 B backward-char (C-b) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . clear-screen (C-l) . 70 V visible-stats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PWD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 67 67 67 67 67 67 67 PROMPT_DIRTRIM . . . . . . . . MAIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . backward-kill-word () . . . . . . . . . . . . . . . . 68 REPLY . . . . . . . . . 7 TIMEFORMAT . . . . . . . complete-into-braces (M-{) . . . . . . . . . . . . . . . . . . . . . . . . . . MAILCHECK . . PPID . . . . . . . . . . . . . . . . . . . . LC_ALL . . . . . . . . . . . . . . . . . . . . . . PROMPT_COMMAND . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 M MACHTYPE . . . . . . . . . . . PS4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 108 109 109 109 106 107 . . . . . . . . . . . . . . . . . . 100 show-all-if-unmodified . . . . . . capitalize-word (M-c) . . . complete-filename (M-/) . . . . 69 TMPDIR . . . . . . . . . . . . . . OSTYPE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OPTERR . . . . . . . . . . . 7 TEXTDOMAINDIR . . . . . . . . . . . . . . . . . . . . . . . . . POSIXLY_CORRECT . . . . . . . . . . . . . . . mark-modified-lines . . . . . . . . . . 114 beginning-of-line (C-a) . . . . . . . . . . . . . character-search (C-]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LINENO . . . . . . . . . .
. delete-char-or-list () . . . . . . . . . . . . . . . . . . . . .. . . . . . . . skip-csi-sequence () . . . . . . . . . . . . end-kbd-macro (C-x )) . . . . . . possible-hostname-completions (C-x @) . . . . . . . . . . . . . 107 non-incremental-forward-search-history (M-n) . . . . . . . . . . . . . . . . . . . . 114 114 107 107 S self-insert (a. . . . . . .. . . or M-_) . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . !. digit-argument (M-0. . . . . . . . . . . . . . . . . . . . . . M-1. . . . . . . A. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . delete-horizontal-space () . . . . . kill-whole-line () . . . . . . . . . . . . . . . delete-char (C-d) . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dump-macros () . . . . . . . 108 R re-read-init-file (C-x C-r) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . copy-forward-word () . . . . . . . . . . . . . . shell-expand-line (M-C-e) . . history-search-forward () . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . redraw-current-line () . . . . . . 114 overwrite-mode () . . . . . . . . . 113 insert-completions (M-*) . . . . . . . previous-history (C-p) . . forward-char (C-f) . . . . . . .. . . . . . . . . . . 1. . . . . . . . 111 menu-complete-backward () . . . . . . . . . . . . . . forward-search-history (C-s) . . . . . do-uppercase-version (M-a. . . . 112 106 107 112 H history-and-alias-expand-line () . . . . . . . . 109 P possible-command-completions (C-x !) . . . . . . . 114 glob-expand-word (C-x *) . . . . . . . . . . . . . . . . . M-b. . . . . . 108 113 106 114 106 109 113 112 I insert-comment (M-#) . . . . . . . . . . .. . . . . . . . . . . . . . . . .. . possible-completions (M-?) . . . .. . . . . . . . . . . . . . . . . M-x. shell-kill-word () . 108 106 107 106 G glob-complete-word (M-g) . . . . . . . . . . . . . .. 107 non-incremental-reverse-search-history (M-p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 110 109 109 D dabbrev-expand () . . . . . . . copy-region-as-kill () . . . . . . . . . . . . . . . . . . . . . . . . . . . possible-variable-completions (C-x \$) . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . dump-functions () . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . .) . . . . . . .Appendix D: Indexes 155 complete-variable (M-\$) . . .... . . . . . . . . . . . . . . . . . . . . . . history-expand-line (M-^) . . . . . . . . . . . . . . 111 N next-history (C-n) .. . . . .. . . . . . . . . 111 110 110 110 K kill-line (C-k) . . . . . . . . . . forward-word (M-f) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .. . . start-kbd-macro (C-x () . . . . . . . . . revert-line (M-r) . . . . . . . . . . . . . . . . . 114 menu-complete () . . . . . . . . . exchange-point-and-mark (C-x C-x) . . . . . . . . 112 108 111 109 110 114 112 108 113 113 113 112 M magic-space () . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . possible-username-completions (C-x ~) . . . . . . . prefix-meta (ESC) . . . . . . . . . downcase-word (M-l) . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . history-search-backward () . . . . . . . . . . . . . . . . . . . possible-filename-completions (C-x /) . . . . . . . . . . . . .. . . shell-backward-word () .. . . . . . . . . . . . . . . . . . . . . 114 . . . . kill-word (M-d) .. . .. . kill-region () . . . . . . . . . . . . . . . . . 114 112 107 106 113 operate-and-get-next (C-o) . . . . . . . . . . . . . . . . . . . . . .. . . . set-mark (C-@) . . . . . . . . . . .. . . . . .. shell-forward-word () . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . end-of-history (M->) . . . . . . . dump-variables () . . .. . . . . . . . . . . . . . . . .. display-shell-version (C-x C-v) . . . . . . . .. . . . .) .. . . . . . 107 O E edit-and-execute-command (C-xC-e) . . . . . . . . . . . . . . . . 111 insert-last-argument (M-. . . dynamic-complete-history (M-TAB) . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . 114 glob-list-expansions (C-x g) . . . . . end-of-line (C-e) . . . . . . . . . . . . . . copy-backward-word () . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . b. . . . M--) . . . . . . reverse-search-history (C-r) . 112 110 111 111 111 111 112 107 F forward-backward-delete-char () . . . . . . . . . . . 114 Q quoted-insert (C-q or C-v) . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Bourne shell . . readline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . yank-last-arg (M-. . . . . . . . . . . . looping . . . . . . . . . . . . . . . . . . . . . grouping . . . . . . . . . . . . . shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 unix-filename-rubout () . . . . . . . . . . . . . . . . . . . . . . . 19 expansion. . . . 108 U undo (C-_ or C-x C-u) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . history events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73. . . . . . yank-nth-arg (M-C-y) . . . . . . . . . . . . . . .5 Concept Index A alias expansion . . . 8 commands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 filename . . . . . . . . . . . . . . . . . . . 76 B background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 unix-line-discard (C-u) . . . . . . . History. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 command search . . shell . . . . . . 94 environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 upcase-word (M-u) . . 14 F field . . . . . . 108 transpose-words (M-t) . . . . . . . . . . . . . . . . . . . . . conditional . . . . . . . . . . . . . . . . . . . . . . . . . . 5 brace expansion . . . 29 command history . . . . . . . . . . . . . . . . 8 commands. . 117 configuration . . . arithmetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . yank-pop (M-y) . 94 command execution . . . . . . . . . . 17 expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 event designators . . . 31 evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 exit status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 coprocess . . 89 functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shell . . . . pathname . . . . . . . . . . . . . . . . . . . . . . . . . . . history list . 7 commands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 command expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tilde . . . . . . . . . . . . . . . . . . 24 expansion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Y yank (C-y) . . . . . . . . . . . . . . . . 96 installation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 commands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 commands. . . shell . . . . . . . . . . . . . . . . 75 internationalization . . . . . . . . . . . . . . . . . . . . lists . . . . . . . . . . . . . . . . . . . . 110 108 107 110 D. . . . . . . . . . . . . . 113 transpose-chars (C-t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .156 Bash Reference Manual T tilde-expand (M-&) . . . . . . . . . . . . . . . . . . . . . . . arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . how to use . . conditional . . . . . . . . . . . . . 3 C command editing . . . parameter . . . . . . . . . . . . . . . . . . . . 9 commands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 execution environment . . . . . . . . . . . . . . . . . . . . . 30 command substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 universal-argument () . . . . . 81 . . . . . . . . . . . . . . . . . . . . . 24 expansion. . brace . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . arithmetic expansion . . . . 7 completion builtins . . . . . arrays . . . . . . . 127 control operator . . . . . . . . . . . . . . . 32 expansion . . . . . . . . . . . . . . . . . . . . . . . . . . 109 unix-word-rubout (C-w) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 builtin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 expansion. . . . . . . . . . . . . . . . . . . . . 93 interactive shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 initialization file. . . . . . . . . . . . . . . . . . . . 79 78 23 78 80 E editing command lines . . . . . . . 127 Bash installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 123 123 121 120 I identifier . . . . 14 H history builtins . . . . . 7 D directory stack . . . . . . . . . . . 10 commands. . . . . . . . . . . . . . . . . . . . . . . . . . . 24 foreground . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . simple . 18 expansion. 13 commands. . . 19 expressions. . . . . . . . . . . . . arithmetic . . . . . . . . . . or M-_) . . . . . . . . . . . . . . . . . . . . . . . history expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . readline . 127 interaction. . . . . . . . . . . . . . . . . . . 78 expressions. . . . . . . . . . . . . . . . . . . . . . arithmetic evaluation . . . . . . . . 22 command timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . compound . . . . . . . . . . . . . . . . . . . pipelines . 89 Bash configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 filename expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . arithmetic . . . . . . . . . . . . . . filename . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 14 shell script . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 . . . . . . . . . . . . . . . . . . . . how to use . . . . . . . . . . . . . . 19 token . . . . . . . . . . . . . . . . . . . . . . . shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 W word . . . . . . . . . . . . . . . . . . . . . 15 parameters. . . . . 3 process group ID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 V variable. . . . . 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 shell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 shell function . . . . . . . 7 login shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 job control . . . . . . . . . . . . . . . 115 prompting . . 92 redirection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ANSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 native languages . . . . . . . . . . . 94 O operator. . . . . . . 6 K kill ring . . . . . 3 POSIX Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . shell . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 metacharacter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 R Readline. . . . . . . . . . . . . . . . . . . . . . . . . . readline . . . . . . . . . . . . 73 suspending jobs . . . . . special . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 quoting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 return status . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 parameters. . . . . . . . . .Appendix D: Indexes 157 J job . . . . . . . . . . . . . . . . . . . . . . . 4 word splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 M matching. . . 95 killing text . . 24 pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 shell variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 signal handling . . . . . . . . . . . . . . 7 notation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 N name . . . 23 Y yanking text . 3 T tilde expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . readline . . . . . . . . . . . . . . 3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 pattern matching . . . . . . 3 restricted shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 S shell arithmetic . . . . . . . . . . . . . . . . . . 59 startup files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 L localization . . . . . . . . . . . . . . . . interactive . . . . 7 P parameter expansion . . . . . . . . . . . . . . . . . . 84 process group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 special builtin . . . . . . . 19 parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 programmable completion . . . . . . . . . . . . . 8 POSIX . . . . . . . . 26 reserved word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 variables. . . . . . . . . . . . . . . . . . . 16 pathname expansion . 89 Q quoting . 4 translation. . . . . . . . . . . . . . . 3 process substitution . . . . . . native languages . . . . . . . . . positional . . . . . . . . . . . . . . . . . . . . . . . pattern . . . | {"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": 0.8740200996398926, "perplexity": 522.6468098654248}, "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-2017-30/segments/1500549426639.7/warc/CC-MAIN-20170726222036-20170727002036-00008.warc.gz"} |
https://brilliant.org/problems/a-problem-by-ritesh-manna-2/ | # A problem by Ritesh Manna
Level pending
ABC is a triangle.The angle opposite to the side BC is double the angle opposite to the side AC.If AC=4 units and AB=5 units then find BC (in units).
× | {"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": 0.8838914036750793, "perplexity": 4004.147533908572}, "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-2017-13/segments/1490218189031.88/warc/CC-MAIN-20170322212949-00152-ip-10-233-31-227.ec2.internal.warc.gz"} |
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## anonymous one year ago How do I do this? What is the answer? Solve for x: -3|2x + 6| = -12 1)x = 1 and x = 5 2)x = -1 and x = -5 3)x = -9 and x = 3 4)No solution Delete Cancel Submit
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1. Kash_TheSmartGuy
• one year ago
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First, distribute:$-3\left| 2x+6 \right|=-12$$6x+18=-12$$6x+(18-18)=(-12)-18$$6x=-30$$\frac{ 6x }{ 6 }=\frac{ -30 }{ 6 }$$x=-5$I would go for B.
2. Kash_TheSmartGuy
• one year ago
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You get it, right?
3. anonymous
• one year ago
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are you sure? I need to get this question right. And yes I get it but I just want to make sure that the answer is b.
4. Kash_TheSmartGuy
• one year ago
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@Michele_Laino
5. anonymous
• one year ago
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?
6. anonymous
• one year ago
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is it right?
7. Michele_Laino
• one year ago
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We can rewrite your equation as follows: $\left| {2x + 6} \right| = 4$ I have divided both sides by -3
8. Michele_Laino
• one year ago
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now we have to consider these 2 cases: $\Large \left\{ \begin{gathered} 2x + 6 \geqslant 0 \hfill \\ 2x + 6 = 4 \hfill \\ \end{gathered} \right.\; \cup \,\left\{ {\begin{array}{*{20}{c}} {2x + 6 < 0} \\ { - \left( {2x + 6} \right) = 4} \end{array}} \right.$
9. Michele_Laino
• one year ago
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2
the solutions of your equations are given by the solutions of those system of inequality above
10. Michele_Laino
• one year ago
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equation*
11. Michele_Laino
• one year ago
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for example I solve the first system: I get this: $\Large \left\{ \begin{gathered} 2x + 6 \geqslant 0 \hfill \\ 2x + 6 = 4 \hfill \\ \end{gathered} \right. \Rightarrow \left\{ {\begin{array}{*{20}{c}} {x \geqslant - 3} \\ {x = - 1} \end{array}} \right.$ am I right?
12. Michele_Laino
• one year ago
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2
|dw:1435429357830:dw|
13. Michele_Laino
• one year ago
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|dw:1435429414248:dw|
14. Michele_Laino
• one year ago
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since x=-1, belongs to the interval (-3, +infinity), then x=-1 is a solution of your original equation
15. Michele_Laino
• one year ago
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now, please do the same with the second system: $\Large \left\{ {\begin{array}{*{20}{c}} {2x + 6 < 0} \\ { - \left( {2x + 6} \right) = 4} \end{array}} \right.$
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Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy. | {"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": 1.000006079673767, "perplexity": 2775.4709268455335}, "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-2016-44/segments/1476988719468.5/warc/CC-MAIN-20161020183839-00550-ip-10-171-6-4.ec2.internal.warc.gz"} |
http://papers.nips.cc/paper/7122-improving-the-expected-improvement-algorithm | NIPS Proceedingsβ
Improving the Expected Improvement Algorithm
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Abstract
The expected improvement (EI) algorithm is a popular strategy for information collection in optimization under uncertainty. The algorithm is widely known to be too greedy, but nevertheless enjoys wide use due to its simplicity and ability to handle uncertainty and noise in a coherent decision theoretic framework. To provide rigorous insight into EI, we study its properties in a simple setting of Bayesian optimization where the domain consists of a finite grid of points. This is the so-called best-arm identification problem, where the goal is to allocate measurement effort wisely to confidently identify the best arm using a small number of measurements. In this framework, one can show formally that EI is far from optimal. To overcome this shortcoming, we introduce a simple modification of the expected improvement algorithm. Surprisingly, this simple change results in an algorithm that is asymptotically optimal for Gaussian best-arm identification problems, and provably outperforms standard EI by an order of magnitude. | {"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": 0.9318603277206421, "perplexity": 398.913077153425}, "config": {"markdown_headings": false, "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-2019-13/segments/1552912203947.59/warc/CC-MAIN-20190325112917-20190325134917-00322.warc.gz"} |
https://ukdiss.com/examples/features-of-cyrene.php | Disclaimer: This dissertation has been written by a student and is not an example of our professional work, which you can see examples of here.
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# Presence of a Doublet in the IR Spectrum
Info: 19522 words (78 pages) Dissertation
Published: 16th Dec 2019
Tagged: Sciences
1 INTRODUCTION
1.1 Aims of the work
This work examines some features of Cyrene, in particular, the investigation of the presence of a doublet in the IR spectrum in the absorption region of the carbonyl area. A preliminary study of Cyrene-water interaction was done through different technique in order to better understand the characteristic of this system in order to suggest a possible industrial application.
This work can be divided in two main themes:
Preliminary spectroscopic analysis of the Cyrene spectra: were analysed in particular the vibrational zone of the Carbonyl in order to explain the presence of a double band in this region. Many technique were used like mid IR in different temperature and Raman, and the study of the spectra was extended to the C-H vibrational zone, as well as abinitio calculation. Different models were presented in order to justify the experimental observation.
Analysis of Cyrene-water interaction: The system was investigated by 1HNMR and 13CNMR to analyse the behaviour of the Cyrene in solution. The ratio between the amount of geminal diol, Cyrene and free H2O in different solution concentration were calculate. Preliminary analysis like DSC, DLS, and SAXS, were performed, in order to characterise the solution behaviour, and possible application of such system was also suggested.
1.2 Green Chemistry
With the continuous increase in population and the consequent growing demand for consumer products, pharmaceuticals and personal care products, it has become essential to ensure economic and social development are sustainable.[1]{Gas-Phase, 2015 #1} Industrial process in fact require the use of solvents , is estimate an use of 20 millions of tons per year, many of these solvent are also volatile or toxic.[2]
The dispersion of these organic volatile compounds in the atmosphere is one of the major cause of pollution as well as the practice to burn solvent at end of life cycle. Many of them in fact contains nitrogen and sulfur heteroatoms that are known to lead to atmospheric pollution. With the purpose to find safer and more environmentally friendly solvents, chemists are developing new chemicals with high performance but low environmental impact using the green chemistry principles developed by Paul Anastas and John Warner.[3]Table 1.1
Table 1.1 shows the 12 principles of Green Chemistry
Prevention It is better to prevent waste than to treat or clean up waste after it has been created. Atom Economy Synthetic methods should be designed to maximize the incorporation of all materials used in the process into the final product. Less Hazardous Chemical Syntheses Wherever practicable, synthetic methods should be designed to use and generate substances that possess little or no toxicity to human health and the environment. Designing Safer Chemicals Chemical products should be designed to affect their desired function while minimizing their toxicity. Safer Solvents and Auxiliaries The use of auxiliary substances (e.g., solvents, separation agents, etc.) should be made unnecessary wherever possible and innocuous when used. Design for Energy Efficiency Energy requirements of chemical processes should be recognized for their environmental and economic impacts and should be minimized. If possible, synthetic methods should be conducted at ambient temperature and pressure. Use of Renewable Feedstocks A raw material or feedstock should be renewable rather than depleting whenever technically and economically practicable. Reduce Derivatives Unnecessary derivatization (use of blocking groups, protection/ deprotection, temporary modification of physical/chemical processes) should be minimized or avoided if possible, because such steps require additional reagents and can generate waste. Catalysis Catalytic reagents (as selective as possible) are superior to stoichiometric reagents. Design for Degradation Chemical products should be designed so that at the end of their function they break down into innocuous degradation products and do not persist in the environment. Real-time analysis for Pollution Prevention Analytical methodologies need to be further developed to allow for real-time, in-process monitoring and control prior to the formation of hazardous substances. Inherently Safer Chemistry for Accident Prevention Substances and the form of a substance used in a chemical process should be chosen to minimize the potential for chemical accidents, including releases, explosions, and fires.
Among these principle used to suggest the choice, the develop and the use of more eco-friendly solvent there are parameter like toxicity or volatility and many others, but in general to determinate the eco sustainability of a solvent we have to consider the balance between its effectiveness in the considered chemical process, and the environmental effect through all the life cycle of the solvent. For this reason the interest of green chemistry for new alternative solvent such water, supercritical fluids, ionic liquids, is growing. Recent discoveries in green chemistry include, the development of sustainable solvents, which are low polluting, ease recyclable and produced from renewable sources. Many of these solvents maintain or enhance the performance compared to traditional solvents commonly used in industrial processes.[3]
Among these new solvents there is the dihydrolevoglucosenone (Cyrene), which already has shown promise as an alternative dipolar aprotic (Figure 1). [1] Cyrenes renewable nature, lack of mutagenicity, and easily of disposable makes it particularly attractive as solvent.[1]
1.3 Solvent properties
Solvents are use in order to assist in separation and transporting of materials in chemical process. It can stabilize an intermediate of reaction shifting the equilibrium, or simple through its chemical physical properties, like hydrogen bonding ability or boiling point, influence the yield of the process. Normally the solvent are used to dissolve chemicals, separate the reagents promoting their contact, the reason of this capability is in the formation of favourable interaction between solvent and solute, but both thermodynamic and kinetic consideration have to be take in account.[4]
The choice of the correct solvent is then decisive to promote a specific reaction or influence a chemical physical process. Of course it can not exist a green solvent that cover all the possible range of application, and its choice, is done in base of the necessity of a specific industrial process, normally is followed the rule “ like dissolve like”.
Molecules with a permanent dipole are call Dipolar, while the term Apolar should be used rarely to describe a solvent , because phenomena like polarizability have to be take in account.5[5]
We can distinguish also between aprotic solvent, whose molecular structure does not have a dissociable H+ hydrogen atom, and protic solvents that are capable to yield an H+.
Obviously, intermolecular solute/solvent interactions are of highly complicated nature and difficult to determine quantitatively.6[6]
Is well known that the position and intensity of absorption bands in Uv/vis, IR, NMR, and ESR spectroscopy, are solvent-dependent,7[7] these medium effects could depend from many causes , for instance the differential solvation of reactants and products rates of chemical reactions or the different physical absorption of electromagnetic radiation. The degree of this differential solvation depends mainly on the intermolecular forces between solute and surrounding solvent molecules.8[8],9
Nowadays are available many parameters to describe the principal characteristic of solvent. A general distinction between solvent it could be done measuring their polarity, even if could be better define it as solvation power of a solvent. In fact It depends on the balance of several factor like electrostatic, inductive, dispersive, charge-transfer and hydrogen bonding forces.5,10
Various physical and chemical properties like dielectric constant, electron pair acceptor and donor ability, and the ability to stabilize charge separation in an indicator dye, were used in order to describe correctly the polarity of solvents.5
Chemists have tried to understand solvent effects on chemical reactions in terms of the so-called solvent polarity. Hildebrand’s solubility parameter (δ) that provides a numerical estimate of the degree of interaction between materials, and can be a good indication of solubility, it is normally express in the ENT scale were 0.0 is for tetramethylsilane and 1.0 for water.
1.4 Cyrene as solvent
Dihydrolevoglucosenone (Cyrene) is a new dipolar aprotic solvent, (figure 1.1) It can be synthesised in two simple steps from biomass, thus ensuring a low environmental
Figure 1.1. Structure of Cyrene
footprint, as well as improving its economic viability.1The pure compound is obtained by simple hydrogenation of the levoglucosenone (Scheme 1.1). 11
Scheme 1.1. the figure shows the steps for the synthesis of Cyrene from Biomass
As told before numerous parameter are used to characterize the property of a solvent. Among these the most important for Cyrene are reported in table 1.2. . The Kamlet–Abboud–Taft parameters for Cyrene indicate that this solvent is aprotic, the hydrogen bond donor (HBD) ability α is in fact equal to zero. Cyrene shows a similar π* value ,corresponding a measure of general polarity or polarizability (excluding hydrogen bonding effects), to those of highly dipolar aprotic solvents, but with a slightly lower β value which is an indicator of hydrogen bond accepting ability.1 In table 1.2. are also reported the value of Hildebrand solubility parameter, which define a solvent affinity to solve a solute. A solute is soluble in solvents which have a similar solubility parameter.12
It is possible, locate a solvent in the ‘‘Hansen space’’, a threedimensional representation of dispersion (δd), polar (δp) and hydrogen bonding (δh) interactions. Two solvents that are close in the Hansen space, exhibit similar solubilising properties. Considering this parameter the closest solvent in the Hansen space is NMP (N-Methyl-2-pyrrolidone). 1 Although Cyrene has a high boiling point (about 2030-2100C) which makes it less suitable to be removed by evaporation from the reaction environments, thanks to its characteristics , shown above, it is an interesting dipolar aprotic solvent.1
Table 1.2. Kamlet-Abboud-Taft parameter for Cyrene
Cyrene NMP Cycloehanone 0,333 0,355 0,281 α 0,00 0,00 0,00 β O,61 0,75 0,58 Π* 0,93 0,90 0,71 δD/MPa0,5 18,8 18,0 17,8 δp/MPa0,5 10,6 12,3 6,3 δH/MPa0,5 6,9 7,2 5,1
Cyrene shows the unique spectroscopic features previously not reported in the published literature for this molecule. This study has been complicated by not only the unusual behaviour of this molecule, but also from the limited literature available exploring its properties. As you can see from figure 1.1, the structural characteristics of Cyrene include a carbonyl group and an acetal group in a completely asymmetric cyclic molecule (it belongs in fact to C1 group),13,14 one of consequence of this characteristic is the possible presence in IR or Raman spectra of bands resulting from resonance phenomena. One of these bands could be due to the presence of a Fermi resonance.
1.5 Fermi resonance
One of the assumptions that was immediately investigated, in order to justify the presence of a doublet band in the vibrational region of the carbonyl group in the Cyrene spectra, was the possible presence of a Fermi resonance.
The IR spectroscopy is an analytical technique which based on the interaction between the electromagnetic radiation and matter. The IR spectroscopy is a type of Vibration spectroscopy technique, in fact when an organic molecule is invested by an Infrared radiation whose frequency is comprised between 10,000 and 100 cm-1, the energy transferred from the radiation is converted in vibrational energy.
In a spectrometer the molecule is irradiated with a whole range of infrared frequencies but they are capable of absorbing radiation energy only at certain specific frequencies which match the natural vibrational frequencies of the molecule, namely in the infrared region of the electromagnetic spectrum.
Although the infrared spectra of polyatomic molecules involve vibrational transitions along with rotational transitions, the peaks in the infrared spectrum could be assigned as a fundamental vibrational transition only , especially in the case of low-resolution spectra of polyatomic molecules, were, the rotational fine structure is lost because to the intermolecular interactions. In a real IR spectra, the motion of nuclei cannot be isolated from the motion of other closely surrounding nuclei in the molecule, and for this reason, the characteristic absorption of a particular functional group in a molecule can be assigned only to a range of characteristic frequencies in the infrared spectrum.
Hence, in the low-resolution IR spectrum the frequencies bands, in first approximation, is due to a vibrational transition only. 15
The vibrational spectroscopy has as objective the analysis of molecular vibrational transitions between different states. To do that is necessary to know the manner in which the energy levels are organized, and which the allowed transitions are in a considerate system. In order to analyze the vibrations of a molecule, it is helpful to determine the number of degrees of freedom available to vibration.
“To represent the movement in the space of a molecule consisting of N atoms, we need to use 3N coordinates. These coordinate represents the 3N degrees of freedom necessary to describe the translational motion, rotational and vibrational mode in a molecule. To describe the translational motion of a molecule, that could be identifies with the movement of his center of gravity in the space, we need three coordinates. To describe its rotational motion and the orientation in the space, we need to use two or three coordinates, depending on whether the molecule is linear or nonlinear. The vibration motion concerns the reciprocal displacement of the atoms in the molecule. The degrees of vibrational freedom are equal to the difference between the total degrees of freedom and the sum of those rotational and translational 3N- (3 + 2) or 3N- (3 + 3), depending on whether the molecule is linear or not. These also are the normal modes of vibration of the molecule. The normal coordinates correspond to the actual vibrational modes that the molecule will undergo.”16
Often it can be helpful to use another coordinate system to describe the vibrational motion of a molecule, called internal coordinates, that is comprised of bond and angles.15
In a molecule, it is possible distinguish between different type of vibration movements; the Vibration stretching indicated with the letter ν, in which we have a modification in the length of the bonds , can be of two types, symmetrical and asymmetric. In the symmetrical Stretching the bonds are longer or shorter at the same time, while in the asymmetric stretching a bond is extended, an another is shortened at the same time ;
The vibration bending (δ) is another type of vibration in which we have a change the bond angles. We can distinguish four species of bending in which are possible different movements of the atoms relative to the plane that them it contains: 15
1. Rocking
2. Scissoring
3. Twisting
4. Wagging
The vibrations of a molecule set up a potential called force field,and the simplest force field model is the harmonic oscillator.
To describe the energetic vibrational state of a molecule we need to solve the Schrödinger equation for the system considered.
For instance, if one considers the simplest force field model, the harmonic oscillator, defined as a diatomic molecule. (see figure 1.2)
Figure1.2
In which the form of the potential energy it could be wrote as a series of Taylor we can write:17
V(x) =V(0)+ (
dVdx
)₀ x +
12
(
d2 vdx2
)₀ x2
x= r-re
Where re is the bond length at a minimum of potential energy of a molecule. The first two terms are equal to 0 and the remaining term could be interpreted as a force constant. Then the harmonic approximation for the potential energy will be:
V(R)=
12
Kx2 K= (
d2 vdx2
)₀
And the curve of the potential energy is described from a parabola. The introduction of this form of potential energy in the Schrödinger equation give 18
-ħ22μ d2ψdx2
+
12
Kx2 = E
Where
μ =
m1 m2 m1+m2
and the solution of the Schrödinger equation (converting in wave number cm-1) is: 19
G(v)=
v+12
ν̃
Where
ν̃ =
12πc Kμ 12
=
ω2πc
ω=
Kμ 12
v is the vibrational quantum number and can have only integer values 0,1,2,3… where the state at
v=0
corresponde to the ground-state of vibrational energy.
From this expression we can see that the energetic level of a harmonic oscillator are uniformly separate from one to other. In reality, the harmonic oscillator model is an approximation, the real molecules do not obey at Hooke’s law. It follows that the power of recall nuclei is not proportional to their displacement, especially in large amplitudes of vibration. Moreover this model does not consider the breaking of the bond and the repulsion between the nuclei. A model more appropriate use the equation of Morse, an empirical expression to describe the form of the potential energy for the molecular vibrations, that is:20,21
V(x)= De
1De-ax
2
where De is the dissociation energy and a is a constant for a particular molecule.
When this equation is used to solve the Schrödinger equation, the shape of the vibrational energy levels became:
G(ν)=
ν+12
ν̃ ̶
ν+122
ν̃ xe
Where
xe =
a2ħ2μω
=
ν̃ 4De
and Xe represent an anharmonicity constant for bond stretching vibration, and ν̃ is the oscillation frequency. One result of the use of this form of potential is, when v increase, the different vibrational levels become closely together. ( figure 1.3 )22
Figure 1.3 shows the trend of the Morse potential compared with the one for the pure harmonic model
For non-diatomic molecules where are possible different vibration mode, in fact in diatomic molecule only the stretching in the direction of the bond is possible, it is convenient define 3N-6 displacement coordinates Ri, that is the minimum number necessary to describe the changes of nuclear configuration. It must be underline that the real bonds in the molecules are not elastic and it is always present a certain grade of anharmonicity. One of the effects of the anharmonicity is that normal vibrations are mixed, and the energy levels are perturbed from their harmonic position by anharmonicity. Taking into account the anharmonicity, the potential energy of such system is given by: 23
V=
12!
∑i,jfi,j RiRj
+
13! ∑i,j,kfi,j,k RiRjRk
+
14! ∑i,j,k,lfi,j,k,l RiRjRkRl
+…….
The first term in the potential is the quadratic term of the harmonic force field. The second and third terms correspond to cubic and quartic contributions to the force field. The solution of the Schrödinger equation for a molecule in which all vibrations are not degenerative gives:
∑iG(Vi)
=
∑iν̃ νi+12
+
∑i≤jxij νi+12 νj+12+
…….
Not all the vibration are active in IR. For the harmonic model, it was found that the quantum number changes by one, but also the dipole moment of the molecule must change in the course of an allowed transition. Therefore, a particular vibrational mode of a polyatomic molecule will be active in the infrared, if the molecule’s dipole change in a normal mode vibration.
Classically the dipole moment of a molecule is determined as follows:
μx
=
∑iqixi
μy
=
∑iqixy
μz
=
∑iqixz
where all atoms are considered to have a partial charge, and atoms that are chemically equivalent are assumed to have the same charge. The displacement of the atoms in a normal mode of vibration is then considered if there is a change in the dipole moment of the molecule. Therefore the symmetry of molecules have a fundamental importance in the description of the spectroscopic phenomena, because the possible IR and also Raman active bands in the spectra depends from the symmetry of the molecule.24Nevertheless to understand a Fermi resonance phenomena, whose study is object of this thesis, is not necessary to deepen in to the description of group theory;
Therefore, we need only to remember few elementary principles.
For all the allowed transition, the vector dipolar moment must be:
μ =
∫ψf* μ̂ ψi
d
τ
and ∆ν = ±1
where
ψf*
is the wave function of a final state of transition and
ψi
is the wave function of the initial state of the transition and the vibrational quantum number have to change of one unity.
1.6 Selection rules and types of transitions
The general selection rules states that a vibrational transition is active in the IR if there is a variation of the electric dipole moment of the molecule during vibration and that the vibrational transitions are permitted for ∆ν= ± 1,± 2, ± 3, …
The fundamental transitions are those that take place between the ground state and the vibrational first excited level (v=0 ⟶ν = 1). The corresponding band is called fundamental band and is usually a strong.
The transition from the ground state to the excited state ν= 2 is called second harmonic, the transition between the ground state and the excited state ν = 3 is the third harmonic, and so on, the corresponding bands are called first overtone, second overtone, etc.
The intensity of these bands is much lower than the fundamental bands and becomes weaker with the increasing quantum number of the excited arrival level.
Another type of transition is that which occurs from one excited vibrational state to another excited vibrational state with higher energy. This type of transition is called “hot transition” and the corresponding band is called “hot band”.
Until now we have considered that the IR spectra are due to vibration of molecule, but the molecular spectra are complicated by the presence of rotation and degenerated levels. Given the symmetry of molecules and the anharmonicity of bonds, and the other phenomena already discussed, in the IR spectra are present other kind of bands. 25
These bands are called combination bands and derive from the transitions between a fundamental state and a state where more normal modes are excited. Even in this case, the resulting bands are significantly weaker than the related fundamental bands.
As shown before, one of the effects of the anharmonicity is that normal vibrations are mixed, and the energy levels are perturbed from their harmonic position by anharmonicity. All the perturbation is rigidly restricted by symmetry, in fact only levels with the same symmetry can perturb each other.
In the polyatomic molecule it may happen that two vibrational levels belonging to different vibrations or at combination of vibrations with the same symmetry and nearly the same energy may give rise to an unusually large perturbation.
If such perturbation, involve a fundamental band and an overtone or a combination of bands, is called Fermi resonance in honor of the physicist that discovered this phenomena for the first time in the spectra of the CO2 molecule. The Fermi resonance is a kind of anharmonic resonance and is named of type one if involve a fundamental and an overtone band, and of type two if involve a fundamental and a combination of bands. The quantum mechanics effect of this interaction is that the two energy levels repel each other and the upper one in energy is pushed at highest energy and the lower in energy is pushed down.26
The macroscopic effect of this interaction is shown in the IR spectra (but also in the Raman spectra) as a doublet band in a region where only one band is expected. Normally the fundamental band has the stronger intensity and is at highest energy where the other peak due to an overtone or combination of bands is smaller and it is at lower energy. Is usually convenient investigate about the nature of this interaction using the Perturbation theory.27
The magnitude of perturbation between two energy levels in fact depends on the value of the matrix element Wi,j of perturbation function W:
Wi,j =
∫ψi0
Ŵ
ψj0*
d
τ
Where the
ψi0
and
ψj0
are the zero approximation eigenfunctions of the two vibrational level that perturb each other.28
The magnitude of the shift between levels can be obtained applying the first order perturbation theory. For instance, considering the case of two levels with unperturbed energies
Ei0
and
Ej0
, and ∆E is the shift of the levels, is possible write the follow secular determinant
Ei0-∆EWWEj0-∆E
Solving the determinant we have:
∆E=
(Ei0+Ej0)/2
±
4W2 +δ212
/2
Where δ is the energy of separation
Ei0
Ej0
of the unperturbed level. If δ is large compared with 2W, namely, when the levels are well separated, the equation became:
∆E≈
(Ei0+Ej0)/2
±
δ2+W2δ
In the case of the Fermi resonance the wave function
ψi
and
ψj
of perturbed state are a linear combination of the wave function
ψi0
and
ψj0
ψi=aψi0-bψj0
ψj0
=b
ψi0+aψj0
Where a and b are numerical coefficients.29
It should be stressed that in many cases , as for instance in the C02 spectra , one of the effects of a strong mixing of the vibrational levels is that , the Raman spectrum may shows bands with the similar intensity of IR bands involved in Fermi resonance.30
In the light of the above considerations, it is possible, in principle, obtain from the IR spectra all the information necessary to calculate the Fermi coupling constant that can give us the proof of the existence in the our spectra of the Fermi resonance that involve the fundamental vibration of the carbonyl.
Chapter 2
A first look to the spectroscopic properties of Cyrene
Chapter 2
2.1 Characterisation of the sample and some spectroscopic considerations about Cyrene
Initial analysis was used to verify the structure of the Cyrene molecule and ascertain the purity of the substance provided by the chemical manufacturer (CIRCA). 1H NMR and 13C NMR analysis, were performed at room temperature on the double distilled Cyrene in CDCl3 as solvent. As we can see from the figures 2.1 and 2.2 there are no presence of signals which encourages us to think that the samples do not conform to the specifications and declarations of the company provided in an attachment with the sample.(See appendix)
Figure 2.1. 1HNMR of Dihydrolevoglucosenone in CDCl3
Figure 2.2. 13CNMR of Cyrene in CDCl3
In the 1HNMR exist a direct relation between the area of peaks and the numbers of protons, so is possible by integration give a quantitative analysis. In the 13CNMR were the 13C nuclei do not have uniform relaxation time and as consequence, the relative integrated area of the peaks not give the correct number of C nuclei. However from the 13 CNMR spectra is possible recognize the carbons atoms that are not bonded to the H atoms. These in fact show peak with small intensity, because the relaxation time of a C not bonded to H atom is longer. For this reason is possible recognise from the intensity of the peaks carbonyl groups. The presence in the molecule of C equivalent atoms is responsible for the different number of peak in the spectra, and the number of C atoms present in the molecule. Also the chemical shift of 13C are related to the Hybridization and with the electronegativity of the substituent.1
The 13CNMR in figure 2.2 shows only signal directly related with the molecular structure of Cyrene, in particular is possible see the single signal for the carbonyl (a), and the six signal expected from the asymmetric structure of the Cyrene. The GC-FID and GC-MS analysis of Cyrene indicated that the organic impurities were determined to be lower than 2% in weight. (Figure2.3)
Figure 2.3. GC-MS of Cyrene
Simultaneously, FT-IR spectra of the double distilled pure Cyrene demonstrated the presence of a characteristic double peak in the absorption region of the carbonyl at 1742cm-1 and 1725cm-1 (Figure 2.4).
Figure 2.4. FT-IR of pure Cyrene at room temperature
The presence of this double peak was consistently found in all the samples examined and the relative intensity of these peaks were constant. However, based on the molecular structure, a single intense signal would be expected in this region instead of a doublet. As is known, the infrared absorption band arising from the C=O stretching vibration has been studied more extensively than any other and a significant amount of information is known of the factors influencing its frequency and intensity.2
The frequency of the carbonyl absorption is determinate almost wholly by the nature of its immediate chemical environment and the structure of the rest of the molecule is of little importance. Unless, in the molecule are present phenomena like dimerization or keto–enol tautomerism or some resonance effect.3
For this reason it is possible to obtain a considerable amount of data studying the nature, intensity and position of this band. 2 It is also known that the frequency of the carbonyl group is influenced by the different chemical environment. For instance, its position depend by the nature of the solvent in which it is situated .4-7
The presence of two peaks in the carbonyl region of the IR spectra, as reported in literature,2,8,9can be attributed to a wide range of causes, including the opening of the molecule with the resultant formation of rotational isomers, the fast opening and closing of the ring bone (like in sugars),10 the presence of carbonyls with two different chemical environment (caused for example by molecular aggregation),2 complex formation, the presence of two different molecular configurations,8,9athe formation of a dimer or another molecular oligomer, or lastly a dual band may be the result of a some kind of coupling (for instance a Fermi resonance).14
The first step in order to give an explanation of this phenomenon, was to check for the presence of other inorganic impurities in the sample. In fact due to the nature of Cyrene synthesis processes, a small quantity of Pd could form complex through the carbonyl, causing the splitting of the relative band.9b
It was therefore obtained from the company other Cyrene (double distilled) samples, made using different methods. In which a very small concentration of Pd (few ppm as declared by CIRCA, but not quantified) was present. It was verified that in crude Cyrene and twice distilled samples the IR spectra had the same doublet peak with the same intensity ratio. To confirm that the carbonyl splitting did not depend on the presence of impurities from the production methods, the FTIR spectra was performed on a pure Cyrene sample (0.019511 mol) in which 0.0218 g of PdCl2 (1.229×10-4 mol) was deliberately added. The IR spectra exhibited the same shape for the carbonyl band with ratio intensity, suggesting that the nature of this phenomenon is due to other causes.
The presence of additional impurities such as water were considered as a potential reason for the presence of a double carbonyl. Therefore, a Karl Fischer’s titration was carried out on the Cyrene and an average of 0.33% in weight of water was found to be present in the samples. ( Figure 2.5)
Figure 2.5. shows the report for the KF tritation
The Karl Fischer reaction for water determination follows the equation:
H2O + I2 + [RNH]+ SO3CH3+ 2 RN [RNH]+ SO4CH3 + 2 [RNH]+ I
Eq 2.0
where NR is a base, in which CH3OH is normally used as solvent. In this reaction one mole of I2 is consumed for each mole of H2O. Normally aldehydes and ketones gives problems in the KF titration because they form acetals and ketals respectively with conventional KF reagents.
Eq.2.1
As we can see from the Eq. 2.1 the reaction forms water, which is also titrated, as result is shown an erroneously high water content in the samples. With aldehydes another side reaction, can also occur:
Eq.2.2
In the bisulfite addition, is consumed water and it leads to an erroneously low water content. 11 For all these reason In the titration of Cyrene we used HYDRANAL® a apposite solvent CH3OH free with a specific base that do not give side reactions.
At the same time, the IR measurements were repeated and no OH peak related to the presence of water was found in the FT-IR of any samples.
Figure 2.6 shows a particular spectra in the region of the carbonyl absorption were there are two distinct bands clearly present for the carbonyl group, albeit partially overlapping. It must be noted that the second derivative of the IR spectra, for the pure liquid Cyrene in this region, shows that the first signal (at a higher energy) is due to a single contribution, while the second is formed by several signals.
The deconvolution of the Cyrene spectra using OPUS indicated the possible presence of several peaks composing both the vibrational bands for the carbonyl, this result could be due at an instrumental artefact as consequence of the magnification of the spectra.
Figure 2.6. FT-IR spectra for Cyrene in the region of the carbonyl
The distance between the maximum between the two peaks is about 17 cm-1. The presence of these two peaks and their difference in energy is consistent with the assumption that the carbonyl could undergo an intense intermolecular hydrogen bonding interactions .3
Another possible explanation is the presence of two different chemical environments for the carbonyl through the formation of an “ordered” structure in Cyrene. For example, the formation of dimers, trimers or other oligomers, some of these possible structures are shown in figure (Figure 2.7).
Figure 2.7. Possible oligomers of Cyrene
Spectroscopic studies of molecular structure are normally extended from the conventional mid-IR range to the far-IR or near IR.13
For instance, at low-frequency molecular vibrations provide unique fingerprints, this part of the spectra is highly sensitivity to intra- and intermolecular interactions. The study of this region is normally used to investigate the conformation of many systems, including proteins.14,15
Gas-phase mid-IR spectroscopy combined with theoretical calculations have demonstrated great potential in the successful structural analysis of molecules of biological interest.12–17
In the case of Cyrene, study of the spectra in the mid-IR region was limited, due to the high density of vibrational states and its low symmetry, so the use of additional techniques to resolve individual mid-IR bands is required.22 Furthermore, the mid-IR data quite often allow only assignment of families of structures.17 Albeit with these limitation, an explanation of the spectroscopic behaviour of Cyrene is postulated.
The use of IR and Raman spectroscopy is historically proven to be a fundamental tool for the analysis of ordered systems in which intra and inter molecular interactions between functional groups of a molecule have a direct impact on its position of the bands and their intensity.23
One of the methods to verify the presence of such structures is through FTIR or Raman analysis on pure Cyrene at different temperatures, comparing to the spectra obtained with the one in gas phase (fig.2.9). In presence of ordered structures like dimer or other oligomers, the relative height of the IR signals for the carbonyl band, should vary with changes in the temperature. 4-5-6 -7 (14,15,24,25)
It was therefore examined the FTIR spectrum of the pure Cyrene in Gas phase.(fig2.8-2.9)
Figure 2.8. The IR spectrum in the absorption region of the carbonyl in the gas phase.
Figure 2.8.
Figure 2.9. FT-IR spectra for Cyrene in gas phase
The presence of only one peak relative to the absorption of carbonyl was found at higher energy. In fact, the IR spectra of condensate phases, demonstrate a displacement in the position of peaks at lower energy if compared with the gas phase spectrum. In literature it could be found that the magnitude of this shift is related to the nature of the force that interact between the molecules of the condensate phase. It can be seen that the distance from the maximum of the peaks, between the gas and the liquid phase in the carbonyl region of absorption, is about 17-18 cm-1 . At this point the IR spectra of pure liquid Cyrene at different temperature was performed.
In the presence of dimers, trimers or other oligomers, it is expected to see an increment in intensity of one of the peak in the doublet. The signal related to the monomer form (normally is the one at higher energy) and a decrease of the peaks related to the formation of oligomers. This behaviour could be explained with the progressive breaking of hydrogen bonds between molecules leading to an increase in the proportion of monomer form in the solvent.
From a series of DRIFTs IR spectra carried out at 30 °C up to 200 °C (Figure 2.10) it was shown that the relative heights of the two peaks in the carbonyl absorption region do not significantly change and the relative distance between peaks remains constant.
Figure 2.10. Behaviour of the carbonyl peaks at different temperature (DRIFT): in red the peak of Cyrene in gas phase.
This result was unexpected and is in contrast with the hypothesis of the presence of structures like dimers or other oligomers. Although, in principle this phenomenon may be justified by the existence of strong hydrogen bonding interactions or with a diffuse hydrogen bonding net between the molecules. In fact is well known that when a strong hydrogen interaction exists between molecules, the intensity of the peaks in the IR at different temperature may not change drastically in a wide range of temperature. At this stage, some brief considerations on the hydrogen bonding were considered.
According to the structure of the Cyrene, the functional group that are able to form inter molecular Hydrogen bonding between the oxygen of the carbonyl and the oxygens of acetal, which are both able to interact with the hydrogens present in another Cyrene molecules. But is the oxygen of the carbonyl due the polarity of its bond that is expected to be involved in much stronger intermolecular hydrogen bonding, even if the typical bond strength that take place between the carbonyl and an alkyl hydrogen is in the range between -1,9 and -5,5 KJ/mol . 27(26,27) it could give a significant contribute to the stability of possible oligomers.
It should also be noted that from thermogram (Figure 2.11.) of Cyrene there is no evidence of degradation and a mass loss is exhibited at 125 oC. Also the Derivative of the TGA confirms the purity of the molecule, showing the expected behaviour during the evaporation of the sample with the increasing of temperature. Have to be stressed that no visible quantity of water was noticed. It was therefore possible to exclude the presence of a double band due to the degradation of Cyrene with the temperature.
Figure 2.11. Thermogram of pure Cyrene
2.2 Fermi resonance or not?
As previously discussed the spectroscopic splitting of a band can be attribute to a wide range of causes.
One of the assumptions that was immediately investigated, was the possible presence of a Fermi resonance involving the carbonyl of the Cyrene molecule. The spectroscopic behaviour of Cyrene in different solvents was investigated. At this point is useful give a brief description of the spectroscopic behaviour of Cyrene in different solvents, that is solvent is summarised as follows:
The band at 1725 cm-1 in polar solvents increases in intensity and widens, while observing only a limited displacement in the carbonyl frequency from 1725 cm-1 at about 1723 cm-1, while in less polar solvents, the intensity of the band is reduced (Figures 2.12-2.14). The most intense band at about 1743 cm-1, decreasing in intensity in the polar solvents and shows a slight blue shift of just 1-2cm-1 (1745cm-1 ) .
Figure 2.12. Trend of the carbonyl peaks in Toluene and aqueous solution
Figure 2.13. Trend of carbonyl peaks in water and D2O
Figure 2.14. Behaviour of Cyrene in Toluene
The IR spectra of pure Cyrene and Cyrene in different solvent, does not demonstrate conventional characteristics for a carbonyl. 2,28
In fact, the relative insensitivity of the first band of the carbonyl to the different chemical environment, while in contrast the second show a larger shift in the spectra, suggest that the carbonyl of Cyrene in not free from each other’s interaction. A free carbonyl, that is not involved in formation of oligomers or which does not undergo inter or intra molecular hydrogen bonding shows a strongest shift in the characteristic frequency of vibration in different solvents. The spectroscopic behaviour of Cyrene could indicate a strong hydrogen interaction between Cyrene molecules. Such interactions may lead to the presence of a specific molecular organization in liquid Cyrene in which there are two different chemical environment for the carbonyl. However, another possibility, for this behaviour is dependent on the presence of a Fermi resonance.15 (29)
In general, Fermi resonance results in the splitting of two vibrational bands that have similar energy and same symmetry in both IR and Raman spectroscopies. The two interacting bands are a fundamental vibration and an overtone or combination band.16(30)
The wave functions for the two resonant bands mix according to the harmonic oscillator model, the result is a shift in frequency and a change in intensity in their IR and Raman spectrum, and, two intense bands are observed, instead of the expected strong band for the fundamental and a weak band for the overtone or combination band.17(31)
If two vibrational mode have same symmetry and similar energies, mixing occurs and the resulting modes can be described by a linear combination of the two interacting modes.32 The effect of this interaction is to increase the splitting between their energy levels, as consequence the distance between the peaks of the interacting vibrational modes increase in the IR spectra.33 The mixing of the two states also tend to equalize the intensities of the bands involved in the Fermi resonance, and a weak overtone or combination band show significant intensity while the fundamental band tends to decrease its intensity. Because the coupling bands have nearly the same frequency, the interaction could be affected if one band undergoes a frequency shift, for instance due to deuteration or a solvent effect, while the other does not, as it happens in the case of Cyrene. This shift in the vibration frequency lead to decoupling the two bands.34,35
It is essential to emphasize that the relative intensities components of a Fermi resonance multiplet can varies significantly in frequency with temperature if one of the resonant band change in frequency. (e.g. the C=O fundamental).18-19(36,37)
When bands are broad, Fermi resonance perturbation of localised levels cannot be applied. This broadening can be the result of a number of things, including intermolecular interaction and shortened excited state lifetimes.
In the case of Cyrene, the bands are sufficiently separated and not excessively wide however, the profiles of the bands are different as seen by fitting using the program OPUS.
In particular, the band at 1745 cm-1 approximately has a Lorentzian profile, while the band at 1725 cm-1 has a mixed Gaussian-Lorentzian profile. It was not possible to perform a satisfactory fit for the bands related to the carbonyl in different solvent, due to the presence of abnormal bands whose position changes with the concentration of the solvent. This behaviour probably is due to the presence in the peak of overtone or combination bands that changes position and width in different solvents.
It had been seen that the IR spectra of Cyrene and twice distilled Cyrene in different solvent, does not demonstrate conventional characteristics for a carbonyl. It was noted from the IR spectra in various solvent the ratio between the intensities of the bands change significantly. See Figure 2.10-2.14
Although this behaviour can be explained by the effect of the solvent, 20-21 (38,39) for instance the intensity of the carbonyl increase in solvent with proton donor , due to the change in the dipolar moment of C=O , or with the formation of carbonyl of structure with the solvent , as a dimer or other oligomers . The ratio between the intensities of the two peaks in the Cyrene spectra, are unaffected by the temperature, thus could excluding the formation of a dimer or the existence of two different molecular configurations with different energy. 14,15, 24,25
This behaviour, could indicate a strong hydrogen interaction with the other molecules of Cyrene or with the solvent. In fact in the case of two different conformer form, with the increasing of temperature, we should see a stronger different in the relative intensity of the bands with a corresponding significant change in the shape of bands in the fingerprint region.40
However, this was not observed in the IR spectra of gas and condensate phase.41,42?
The simplest comparison in the literature suggesting a Fermi resonance is present in cyclopentanone.22-23-24(43,44,45)? However, the cyclopentanone did not show a clear Fermi resonance behaviour with the solvent variation method. The other suggestion postulated by the authors was the presence of a dimer, which could also explain the doublet present in the IR spectrum for this molecule. 22-24(43-45)
This literature reported methodology was therefore applied to Cyrene. To determine if the hypothesis of a Fermi resonance is correct, polarized and not polarized Raman Spectra for pure liquid Cyrene was obtained, as well as second series of IR spectra conducted different temperature using a specific cell for liquids (although one of the window was cracked). Although Raman polarized that not polarized spectra shows that the relative intensities of the peaks of carbonyl, in liquid Cyrene at room temperature, do not change substantially (figure 2.10), as expected in the presence of Fermi Resonance.22-23(44-45) The ratio between the carbonyl bands, in this second set of IR spectra recorder at different temperature, change with the temperature is in contrast to previous experiments (figure 2.15-2.16). 22-25(43-46)
Figure 2.15.The IR spectra at different temperature.
Figure 2.16. IR spectra of Cyrene at different temperature
Figure 2.17. Raman spectra of pure liquid Cyrene at room temperature
Figure 2.18. Polarised Raman spectra of pure liquid Cyrene at room temperature
Given the lack of reproducibility in the IR at different temperatures, it was necessary to repeat this experiment using another instrumentation. A third series of analysis at elevated temperatures were made using the REACT FTIR in the range of 25-140 0 C. figure2.19-2.20
Figure 2.19
Figure 2.20 n
Unfortunately the instruments resolution was only 4 cm-1. As observed in figure xxx, the relatively intensity of peaks does not change significantly. Once correct the baseline for the third series of IR spectra temperature it was possible calculate the value for the total integral of the carbonyl peaks from 25 °C to the one at 140 °C. The variation of the area of the peak at highest energy is compensate with the decrement of the one at lower energy, and the value of the total integral remain constant, which is indicative of an absence of resonance phenomena.
Nevertheless, given the variability of result in different series of IR spectra at temperature, others methods have been utilised to disprove the existence of Fermi resonance. These includes, the “solvent variation method”, which is a semi empirical method that is valid for large molecules, but not for small molecules such as CO2.5 (14)
This method provides the calculation from the IR spectrum of the value of Fermi constant coupling for the bands by an interaction in various solvents. If this value does not change in different solvents, this provides evidence to support a Fermi resonance.14 -26
The data of ratio between the intensity of the two peak in the carbonyl region and the relative distance between the maximum intensity of the peaks in different solvents are necessary to calculate the Fermi coupling constant using is method (figure 2.21, tablet 2, ).14 (9)
The coupling constant was calculated using the following equation:
R= {∆ +(∆2-4W2)1/2}/{∆- (∆2-4W2)1/2}
Were R is the ratio between the integrated intensity of peaks, W is the Fermi resonance coupling constant and ∆ is the difference of frequency between the peaks at the maximum of intensity.
Figure 2.21. The trend of Fermi’s coupling constant
As shown in the figure 1.21 the value of Fermi coupling constant change strongly, if the presence of a Fermi resonance was confirmed it should remain almost constant.
Tablet 2. Data collecting from IR spectra and using for the calculation
Others tests that are used to prove a fermi resonance are proposed in the literature include the isotope substitution method, which as has been demonstrated as a simple method that can easily exclude both ,the presence of Fermi resonance as well as the presence of two different molecular conformations as responsible for this oddity in the IR spectra.8,49
However, in the case of Cyrene this technique would be technically difficult and particularly expensive. Although the variation solvent method cannot be considered as definitive proof to deny coupling phenomena, the experimental data obtained indicates that the Fermi resonance is not involved the carbonyl of Cyrene molecule. The last remaining hypothesis that can explain the doublet band in this region of the spectra, is the formation of oligomers between Cyrene molecules and will be discussed in the next section.
2.3 Which molecular organization for Cyrene?
The important role of the hydrogen bond.
This section discusses the experimental data suggesting that Cyrene is organized in a complex bulk, in which two different “organizational” environments are present for the carbonyl, thus explaining both the spectroscopic oddity of the doublet band and other unusual physical characteristics of density and boiling point.
As told in the introduction The Kamlet–Abboud–Taft parameters for Cyrene indicate that this solvent is aprotic (Table 1.2), with a similar π* value ,that corresponding to a measure of its dipolarity, to those of highly dipolar aprotic solvents. It shows also a slightly lower β value which is an indicator of hydrogen bond accepting ability.1(50)
From this data and from the structure of Cyrene it should be expected that the principal force interact between Cyrene molecules is the hydrogen bonding. In order to define the strength of this interaction, it was performed an wide series of IR of Cyrene in different solvents and simultaneously it was check the IR spectra of liquid Cyrene compared with the one in gas phase in the C-H region.
Figure 2.22
Figure 2.22. reports the spectra related to the liquid Cyrene (blue) and the spectra of Cyrene in gas phase (red). Comparing the peak at 2900 cm-1 in liquid phase with the one in gas phase, shows an increase in intensity compared with the signal at 2985 cm-1. This signal remains in in a similar position for both the liquid and gas phase Cyrene. It is be clear signs of interaction of a specific hydrogen bond between the Cyrene molecules in the liquid state. As told before, once excluded resonance phenomena, the appearance in liquid phase of a second peak (shown as a doublet in the carbonyl zone) is probably due to structuring (not necessarily a dimer), which involves specifically the C-H and C = O group.
It is well known that some weak hydrogen bonding has an effect on the no shift or in same case,51as for the Cyrene , a blue shift relative of C-H vibrational mode in the IR spectra.
Due the nature of both C-H and C=O it is possible for the existence of a weak hydrogen bonding interaction,51 leading to an extended hydrogen bonding interaction between molecules, thus explaining the high value of both density and boiling point. This non shift of C-H peak passing from the gas phase to a condensate form, is another unique aspect of the Cyrene spectra. In Cyrene may exist a very particular case of hydrogen bonding, namely an anti-hydrogen bonding.52
Over the last 20 years, a large number of studies on hydrogen bonding had been produced. Recently hydrogen bonding interaction with peculiar characteristic blue shift or non-conventional hydrogen bonding between two different H atoms has been reported.53-52
The classic hydrogen bonding is described as a non-covalent and attractive interaction between a proton donor X-H (electron acceptor) and a proton acceptor Y that can be involve the same or different molecules.63 According at this description, in a conventional hydrogen bonding the hydrogen atom is bonded to an electronegative group or atom and when this is involve in hydrogen bonding through the interaction with a proton acceptor Y (electron donator), the distance between X and, became smaller than the sum of Van Der Waals radii of the atoms X and Y. In the same time the distance X-H increase and this bond elongation can be visualized in the IR spectra as a red shift in the frequency of the fundamental vibration of X-H , also it is generally observed an increase of its intensity.
The formation of this interaction lead to a decrease of the magnetic shield of proton involved in the hydrogen bonding, that is visible in a shift at lower field in NMR, it was found in fact a correlation between the experimental hydrogen bond distance and strength an NMR chemical shift.64-69
This shift is proportional to the number of hydrogen bonds between molecules.69
At the same time the electron density flows from a proton acceptor to the donor. 70
The electron density is in fact is transferred from the proton acceptor (lone pair) to the σ* antibonding orbital of the X-H bond, causing the weakening and the elongation of this bond and the decrease of X-H stretch frequency. 71
However, the Hydrogen bonding is a kind of interaction that is far to be completely understood. In theory the hydrogen bonding could be explained as an attractive force between partial electric charges with different polarity. In fact, the electronegative atom X attracts the electron density from the partially positively charged hydrogen, and this attracts a lone pair electron from Y, but it true only for weak hydrogen bonds. The hydrogen bonding shows several type of contribution, due to electrostatic and polarization effect and charge transfer, especially when a delocalization effect or dispersion forces plays an important role.72,73
The wide nature of these effects is reflected by a wide range of shifting and intensity in the IR spectra. For instance the shift in the IR can go from few cm-1 to more than 100 cm-1 in same case, with energy interactions between molecule in the order of 15-40 Kcal/mol for strong hydrogen bonding, and from 4-15 Kcal/mol in the case of a moderate or weak hydrogen bonding interaction. This wide variability in the strength, depends strongly from the angle and the distance between the groups involved in this type of interaction.
How was already told the red shift and the consequent increasing in intensity of the spectral band involved in the hydrogen bonding , is a spectroscopic characteristic of this kind of interaction. However, there are other situation in which the stretching vibration of X-H is shifted to higher frequency (blue shift). When that happens we are in front of a non-conventional type of hydrogen bonding called Blue shifting or improper or anti hydrogen bonding. This peculiar type of interaction is typically observed in the cases in which the X-H bond elongating hyper conjugative n(Y)⟶σ*(X-H) interaction is relatively weak. 74According to the theory when the interaction is weak and the hybrid orbital of X in X-H bond undergo a change of both polarization and hybridization, we have a shortening of X-H bond that cause a blue shift in the spectra.
Over the years many model were proposed in order to explain this phenomena, but its real nature is object of debates in the scientific community.
As we saw before with the formation of a conventional the hydrogen bonding the electron density flows from lone pairs of the proton acceptor to Y to the σ* antibonding molecular orbital of the proton donor X-H with an increasing of electron density on it. As consequence we have an increasing of polarity of X-H bond (a larger negative charge on the proton donor and an increasing of positive charge on the proton acceptor). This is the cause of the weakening of A-H bond and its red shift in the IR stretching vibrational mode as well as an increase of the total dipole moment of the system.
For the Blue shifting hydrogen bonding and others improper hydrogen bonding, it was proposed more than a mechanism, we will report only one as example:75,76
The flow of electron density from the proton acceptor to the antibonding molecular orbital of proton donor, like in a conventional hydrogen bonding, is followed by the shortening of X-H bond.77,78,79, 80
“The bond length X–H is controlled by a balance of two main and opposite effects. One of them is the hyperconjugative interaction
n(X)→ σ*(X-H) that is responsible for elongation of the X–H bond, whereas the other is the increase in the s character of the hybrid molecular orbital at X, accompanying the polarization of the X–H bond. If the latter effect dominates, the X–H bond contracts and ν(X–H) undergoes a blue shift. If the former dominates, the red shift of ν(X–H) occurs.” 106,107
However, as told above this effect is due to a balance of a variety of effects including electrostatic , polarization charge transfer , dispersion , steric repulsion force ; an examples is the case of C–H—–O bond present in the methane-water complex , were without the dispersion contribution , the complex would not exist,81,82,83
Another example is the hydrogen bonding between fragments with opposite charge where the dipole –dipole term is not predominant as shown not all hydrogen bonds are just electrostatic in nature.
However the balance between this varieties of effect could explain the not movement of the C-H band in the IR spectra. A major part of a class of nonconventional hydrogen bonds belongs to the C–H——B bonds, in particularly at the C–H——-O bonds. The influence of this kind of interaction it is shown to be very important in complex systems .84,85A first description of the hydrogen bonding between the C-H group with an oxygen was proposed by Suton but only in recent times, its central role to explain the structure of proteins their stability and folding was stressed.86-89
In particular it was observed as this kind of weak hydrogen bonding interaction was responsible to the behavior of a wide range of biological molecules. Although The energy of this bond is just a bit stronger than the van der Waals bonds ,90( about 2-4 Kcal/mol ), the C-H….O=C role in the proteins , is fundamental , even if it represent only the 6-10% of the total of the hydrogen bonding , its angular preference , and its numbers , are responsible for many specific interaction between biological molecules. At the contrary the role of this bonds is secondary in the aggregate and crystal91
It was shown that the strength of this bond depend strongly on the nature of the proton donor acidity and from the proton acceptor basicity. In fact the strength of the C–H—B hydrogen bonds and their stretching modes n(C–H) strongly depend on the nature of the proton donor92 as C(sp)-H > C(sp2)-H > C(sp3)-H. Therefore ,the importance of this weak interaction do not have to be underestimated , for instance Dougill and Jeffrey used C–H—O=C ‘‘polarization bonding’’ for the explanation of the anomalously high melting point of dimethyloxalate. In the same way, the presence of this hydrogen bonding could explain the chemical-physical property of the Cyrene.
2.4 Which kind of organization we should expect from Cyrene? And which kind of oligomer is possible for this solvent?
Although the existence a such weak interaction has been demonstrated, the behaviour of Cyrene in various solvents, suggests the presence of an intermolecular oligomerization involving the carbonyl. To confirm the existence of such a structure a series of experiment were undertaken, including: FTIR of Cyrene in very dilute solution with CCl4, comparison between calculated and experimental value of ΔH of vaporization, comparison between calculated and experimental IR spectra in gas phases, and calculated IR spectra for some dimer form of Cyrene.
FTIR of twice distilled Cyrene in CCl4 (in proportion of 1/200 mol), indicates a strong band for the carbonyl of Cyrene at 1749cm-1 (figure 2.23).
Figure 2.23
Two smaller and very wide bands at 1697 cm-1 and at 1650 cm-1 attributable to combination bands are also present. Although these bands could due to the presence of oligomer, the most plausible explanation is that they are related with a characteristic vibration of Cyrene. These are also present in the Cyrene gas phase and because their intensity does not increasing with temperature they cannot been attributed to hot bands. Probably the same bands are responsible for the complication in the fitting of the bands using OPUS.
Another proof supporting the self-assembly of Cyrene molecules come from the determination of the ΔH of vaporization of the pure Cyrene through the classical Clausius-Clapeyron experiment.
Fig.2.24
Figure 2.24
As we can see from the figure, plotting the experimental data of the natural logarithm of the pressure lnP against the temperature 1/T we have obtained a line which slope was calculated from the classical relation:
lnP1P2=ΔHvapR1T2-1T1
Slope=-ΔHvapR
The value obtained for the enthalpy of vaporization of pure Cyrene was 67.142 KJ/mol. This value was compared with the one calculated using ACD/labs that shows a value of ΔHvap=46.83 ±3 KJ/mol at boiling point. Fig2.25
Figure 2.25
The experimental value of the entropy of vaporization ΔSvap at boiling point using the relation was calculated through:
ΔSvap=ΔHvapT
The result of ΔSvap was 137 J/molK. The
ΔHvap
, typically has an error of greater than 10%, the very high value of entropy could be an indication of molecular interaction. In fact following the Trouton’s rule positive deviation from the theoretical value of the entropy of vaporization are related with the presence of hydrogen bonding interaction between molecule.93-96
In addition, the calculations of IR spectra of Cyrene in gas phase were compared with the experimental data. Figure 2.26
Figure 2.26
It became apparent the remarkable similarity of the calculate IR spectra of Cyrene in gas phase, with the experimental one, in particular in the finger print region. The calculations based on IR spectra for several dimers demonstrated that it was possible notice the splitting of the carbonyl band.
Figure 2.27
As observed in figure 2.27, the conformation of this dimer present in solution, two different chemical environment for the carbonyl, that could explain both the splitting of the carbonyl band and the relative insensibility from the chemical environment of one of the peaks. Although is possible shown many other oligomers models, that could explain this behaviour, they require more accurate calculation and a deeper spectroscopic study that for obvious reason cannot be object of this study. However, the following calculation, gives a reasonable explanation of the experimental spectroscopic characteristic of Cyrene, strongly suggesting the existence an oligomer structures responsible for its chemical physic characteristic.
Figure 2.28
Figure 2.28 note the splitting of the band for the carbonyl
The model above presented shows the clear spitting of the carbonyl band and the simultaneous increase of the antisymmetric stretching in the C-H zone, exactly as observed in the experimental IR.
Hence, as shown in the previous paragraph, crosschecking the experimental data with all the calculation, it was reasonably exclude the possible existence of Fermi resonance involving the carbonyl and at the same time then attributed the second peak in the carbonyl region to the presence of an oligomer form of Cyrene that conventionally was attributed to a dimer.
The probable existence of these dimer forms in the pure liquid Cyrene, leads to the question if is possible or not the presence of a long range order in the solution, as shown for instance for the liquid crystals or ionic liquid solvents. To answer this question a new set of experiment using a Polarized Optical Microscope and a light scattering spectroscopy were performed.
The idea that Cyrene could shows some pattern like liquid crystal comes from the fact that its molecular structure shows a typical amphiphilicity and an anisotropy that is necessary for the formation of such ordered structures. see Figure 2.29
Figure 2.29
In fact, the anisotropic interactions when sufficiently intense favouring a common alignment of the molecules. As is possible to see from the calculation reported in figure, that an amphiphilic structure with a significant dipole moment passing through the plane between the carbonyl and the alpha carbon. In other molecules like 4-pentyl-4′-cyanobiphenyl (5CB), shown in Figure 2.30 the presence a large electric dipole moment aligned along the long axis molecular that is responsible for the strong anisotropy. 97
The 5CB molecule is particularly important because it is chemically very stable and has a liquid crystalline phase (nematic phase) at room temperature. 98 (figure 2.30)
Figure 2.30
2.5 Polarized Optical Microscopy
Polarized optical microscopy usually represents the first useful technique to investigate the presence of liquid crystalline face as well as the miscibility between two different compounds. The natural light oscillates on different frequencies that correspond to the colours of the spectrum, these oscillations occur on all planes perpendicular to the direction of light wave propagation. When the natural light pass through a polarizing filter, this selects the waves according to the plane of their vibration. The light that results, remaining composed of waves at various frequencies, but vibrates only on one of these plans. The polarized light microscope is equipped with a device that filters the white light through two different polarizing filters. The first one is situated between the light source and the sample, the second is located between the sample and the objective. The two polarizers are oriented relative to one another, so that, if the first lets through only the vertical oscillations, the second that is rotated by 90 ° compared to the first, let pass only the horizontal oscillations. If a sample is not optically active, the vertically polarized light passed through the first filter would be completely obscured by the second. Many samples like liquid crystal have the characteristic to be birefringent. The birefringent material deviate the polarized light passed through the first filter and divided it into two radiations that oscillate at different frequency with a plane rotated by 90 ° to one another. In this way the analyser filter is achieved by a radiation in which still present an horizontal component, and the light that pass through appear coloured, because the phenomena of interference between the two different frequencies of light.
Using this technique at room temperature, no useful images were derived from investigation, thing that exclude the presence of visible typical patters due to the existence of nematic or smectic phases typical of liquid crystal.99
Although many liquid crystals materials may not always be in a liquid-crystal phase, like for instance the thermotropic liquid crystals, it was possible exclude the existence of this behaviour in the range of temperature from -70 to 25 oC, performing some DSC experiment.
2.6 DSC analysis of pure Cyrene
In order to investigate the chemical physical behaviour of Cyrene, light polarised microscopy and DSC and MDSC were used to investigate these solvent samples.
The Differential Scanning Calorimetry is a technique that measures the heat flow and the temperature associated with transitions of the material subjected to heating in a controlled atmosphere. 100
This analysis technique allows obtaining a large number of information like for instance transition temperatures, the degree of crystallinity, purity, the specific heat, and the endothermic or exothermic transitions peaks. Inside the apparatus, it is put two aluminium capsules, thermally isolated between them, one containing a material that have to be characterized and the other acts as reference. A computer adjusts the flow of heat, between two crucibles in which are positioned the two capsules in order to equilibrate their temperature. When, the temperature varies as effect of some transitions that takes place in the material, the crucible containing the sample generates a flow of heat endothermic or exothermic, and the computer records these flow changes and adjusts the heater trying to maintain it constant respect to the reference. Normally the analysis, are performed in an inert atmosphere, (usually nitrogen) in order to avoid possible oxidation of the sample, which could distort the detection of the transition phenomena.100
The choice of the type of capsule, hermetic or not, its size and material, as well as the method of analysis are decisive for the correct interpretation of the experimental data. We have use for the analysis samples of 13-14 mg and aluminium pan with hermetic lid provided from TA instrument. (Fig 2.31)
.
Figure 2.31. TA Instruments Q2000 using for the DSC analysis
The transition from amorphous solid to crystalline solid is an exothermic process and in the DSC curve is shown as an upwards peak. When the temperature rises and the sample reaches its melting temperature, is shown a peak downwards in the thermogram related with this endothermic process. The glass transition is a kinetic phenomenon that is measured in the heating run. The measured value of the glass transition depends on many factor like the cooling rate, experimental conditions, and from the thermomechanical history of the sample. The lower the cooling rate, the lower is the resulting glass transition temperature.
The consequence is that the value of its temperature depends on the measurement conditions and cannot be precisely defined. In many cases, an enthalpy relaxation peak is observed that overlaps the glass transition. This depends on the history of the sample. Sometimes the physical aging below the glass transition leads to enthalpy relaxation.100-103
The glass transition appears as a step in the DSC curve and shows a change of the specific heat capacity, Cp, from the solid to the liquid phase.
Cold crystallization is an exothermic crystallization process. It is normally observed when a heating sample that has previously been cooled very quickly, has had no time to crystallize. Below the glass transition, molecular mobility is severely restricted and cold crystallization does not occur. Melting is the transition process from the solid to the liquid state. It is an endothermic process and occurs at a defined temperature for pure substances. This characteristic is used for the identification of a specific substance or for the determination of its grade of purity. During this process temperature remains constant and the heat required to change of state is known as the latent heat of melting. 104
In this work, the differential scanning calorimetry was used to observe the phenomena of crystallization and melting of Cyrene not reported in literature.
Figure 2.32
The figure 2.32. show only the last heating cycle in which is possible notice the possible amorphous behaviour of pure Cyrene. Although is not possible see the peak of the glass transition, that should be at lower temperature than 70o C , that is the working limit temperature of the instrument, it is evident the presence of an exothermic peak at -64,5oC that correspond to the cold crystallization of the samples , it results formed by at list two separate peaks . The fact that in not evident the glass transition temperature further complicate the analysis because a wide physical properties change at glass transition temperature , as for instance the specific heat capacity Cp , or the dielectric constant. As we can see from the figure the melting peak consists in two peaks, whose relative intensity depend from the thermal history of the samples. The general behaviour is in line with an amorphous material in which are present more than one form of possible crystals. Cyrene could shows a polymorphism in the experimental condition. See figure 2.33
Figure2.33
From the figure is possible notice the absence of cold crystallization wen a lower cooling rate is performed and no other peaks relative at other phase transition of were observed. 105
From the data above presented, it could be take it as almost certain, that Cyrene does not shows a long range order like in liquid crystal ,in fact the correspondent transition peak are absent. This lack of long range order in Cyrene solution is also shown in the SAXS experiment reported in the follow chapter.
Chapter 3
Cyrene and water solution
3.1 The Water Molecule and its liquid structure
In the molecule of water, the hydrogen atoms are bonded to the oxygen atom through two of the four molecular orbitals with sp3 hybridization. the overall geometry of the molecule is tetrahedral , with two of the sp3 orbital remaining not involved in any bond. Because this, the geometry is slightly distorted ,and the angles between bonds that in a pure tetrahedral molecule should have a value of 109.47° is redused at 104.5 ° because of the greater space occupied by the molecular orbitals containing electronic doublets. As well known , the oxygen atom has a higher electronegativity than the hydrogen atom and the O-H bonds in the water molecule is polar, with a negative partial charge on the oxygen atom and a positive partial charge on the hydrogen atoms. This structure and its polar character are responsible for the particular chemical physical characteristics of water ,In fact properties like high melting and boiling points, , high specific heat and negative thermal expansion coefficient , can be explained in terms of hydrogen bonding interaction. 1
We have to remember that hydrogen generally shows a remarkable directionality . In fact hydrogen bonding in water shows strong directionality due to the fact that oxygen atoms with its electron pair orbitals behave as hydrogen bond acceptor wile the hydrogen atoms act as hydrogen bond donor. The formation of an hydrogen bonding through the interaction between ,an hydrogen atom of a O-H bond of a molecule ,with the oxygen atom of another water molecule , generally shows a OH⋅⋅⋅O bond angle of 180° . In liquid water, the Hydrogen bond energy is about 1.32 kcal/mole,2but its value depend from many factor as for instance , from the orientation and the position of other hydrogen bonded molecules, their bond length and angle , and temperature. 3 A wide range of techniques were used in order to describe the real structure of water,4 for instance using infrared and Raman spectroscopy or X-ray and neutron scattering through which It was noticed that the structure of liquid water is dynamic, 5,6and its properties are determined by a fast rearrangement of the molecules , this is due to the fact that broken Hydrogen bonds are unstable , and they can survive only for very short times. Many studies sugest that In liquid water a short range structure is present , in which at least 4 molecules are assembled in tetrahedral shape, but the apparent number of molecules appear larger in the second coordination shell.7,8,9
During the years many type of models were proposed in order to discover the real structure of the liquid water , untill now no one alone is capable to explain estensively the property of water .The principal models that describe the water structure are , the continuum model , mixture model, and statistical thermodynamic model . The continuum model , describe the hydrogen bonding interaction in liquid water like a continuous net in which the interaction never been broken, in which the prevalent structure of water depends from the temperature , because the consecutive change of hydrogen bond length and energy.
In the mixture models instead structure water consists of two different zone , in one is present an ordered ice-like structure and in a second a less order structure. the Statistical Thermodynamic Model instead,10,11 water is composed of small, quasi spherical clusters, in which non-bonded water molecules filling the space between clusters. Other water molecules with three two or one hydrogen bonding can be found on the periphery of these clusters. The latest studies sustein the suggestion that in liquid water are present two different region , one with a low density in which a tetrahedral shape structure is dominate , and the other one that shows an high density in which interstitial water is present forming less ordered structures.11
3.2 Cyrene –water system
Cyrene has a carbonyl functional group , and like other ketones or aldehydes , in the presence of water forms a geminal diol , which is in equilibrium with the ketone form according to the following general reaction (Figure 3.1).
Figure 3.1. Equilibrium between Keto-diol form of Cyrene in water
It has been demonstrated that, in the presence of an acid or a base, water react rapidly to the carbonyl group of aldehydes and ketones establishing a reversible equilibrium with a hydrate form. It were proposed two different mechanism different for the hydration reaction in both basic and acid solution. 12 Figure3.2, 3.3
Hydration base catalysed reaction mechanism
Figure 3.2
Both water and hydroxide ion are nucleophiles ,but because HO is much more nucleophilic than H2O, even at low concentrations reacts with the carbonyl much faster than H2O. 13
Here the OH act as catalyst facilitating the establishment of the equilibrium, in fact although OH is consumed in the first step of the reaction , it is regenerated in the second step, without affect the equilibrium between the ketone and its hydrate form.
Hydratation acid catalysed Reation
Figure 3.3.
Generallyhydrates form of ketones are present only in a small fraction in their mixture of water solutions. Can be cited for instance the case of the formaldehyde solution in which the hydrate form is dominant , while in the case of the acetone solution the hydrate form is very small. However the hydrates of some ketone are stable enough to be present in high concentration , even if as told before generally the hydrate form of aldehyde are more stable than the germinal diol of many ketones, in both cases ,the reaction is normally very fast . However several methods have been designed in order to calculate its rate. Like for instance the UV.14-20
.
Furthermore through 1HNMR analysis it is possible in principle determinate the ratio between the ketone form and the hydrate , and calculate the value of the dissociation constant.
Because the reversibility of the reaction and the relative instability of many hydrates form of ketones , the isolation of the germinal diol is difficult. In fact removing water from the reaction in order to isolate the germinal diol form can convert the hydrate form in the ketone. For these reason the isolation of the germinal diol of Cyrene is very difficult. Only few hydrate form are stable enough to be isolated , for instance the germinal diol of the tricloroacetaldehyde , it is due to the strong stabilization effect caused by the interaction between the hydrogen of diol and the Cl atoms.
3.2 general factors affecting the geminal diol equilibrium
As we said In most cases the resulting geminal diol is unstable and cannot be easily isolated even if for instance , we have seen that in solution like formaldehyde hydrate form is predominant. The equilibrium position in these kind of reactions, is influenced by both steric hindrance and the electronic effects of substituent groups close to the carbonyl. In the case of the acetaldehyde the stability of its hydrate form could be explained by the fact that it has a weaker π component of its carbonyl bond if compared with the others aldehyde or ketones . Also the small steric hindrance of the hydrogen atoms, has an important role to favour the addition reaction of water. If we compare the small hydration of the acetone, we can say that this is due to the stabilizing effect of the electron-methyl groups on the carbonyl double bond. In fact , electron donating alkyl groups stabilized the partial positive charge on the carbonyl carbon and decreases the amount of geminal diol product, also its steric hindrance obstacle the addiction of water. The alkyl groups, therefore, both for electronic and steric effects, increase the stability of the carbonyl respect to the germinal diol form , making the addition of water unfavourable. To the contrary , as shown for the tricloroacetaldehyde, the addition of strong electron attractor groups in alpha destabilizes the carbonyl and tends to form stable geminal diols. However the order of reactivity of the cyclic ketones suggests that the release of internal strain due to the formation of hydrate is an important factor in that influence the rate of hydration. Cyclopropanone, for instance , exists only in its hydrate form and its O-C-O angle measure 1100.21
In order to describe the behaviour of water-Cyrene system a wide range of experiment were performed. A series of FTIR spectra of Cyrene solution in water and D2O at different dilution were done . Figure 3.4
Figure 3.4.
Fig trend of the bands in aqueous solution in the carbonyl zone : in light blue Cyrene 100% ,in blue Cyrene 80% , in green Cyrene 70%, in purple Cyrene 65%, in red Cyrene 50% .
In the figure is possible to see the trand of the carbonyl bands with the growing dilution, all the bands were normalised on the first peak . In particulr it is possible underline that with the addition of water the relative intensity of the carbonyl bands change strongly . The first peak decrease and the second one increase becaming wider as effect of the formation of hydrogen bonding between water and Cyrene molecule. Have to be stressed that the second band is due to the formation of a wide range of hydrate forms involving the C=O . It was not possible do a sotisfactory fitting because the complessity of the behaviour with the dilution. Nevertheless, because it can not be osserved in the spectra an isosbestic point , the formation of the dublet involve more than two species, conventionally monomer and hydrates form of carbonyl. Other part of the spectra were studied, for instance the fingerprint zone , but because its complessity due to the contemporary presence in the solution of the geminal diol form , it was not possible to find a clear indication of the possible structure or interaction of bouth, Cyrene and geminal diol form . In order to better investigate this behaviour it should be necessary to use Raman spectroscopy or far infrared, as well as the correct band assignment. Although above the dilution of 50% in weigth , bouth peaks were covered by the broad band of water making difficult the interpretation of the spectra, It was possible follow the trend of the carbonyl doublet studing the solution of Cyrene in D2O. (see figure 3.5)
Figure3.5. trend of the carbonyl bands of Cyrene in D2O at different dilution . In red 100% Cyrene, in blue Cyrene 65% in weight
In particular the ratio between the bands increase until the loss of signal with the increasing of dilution. It was always possible distinguee two bands, the second one about 1725 cm-1 wider and higher then the first one. In order to determinate the composition of Cyrene , and germinal diol in solution it was crucial the investigation by NMR in various concentration in both , D2O and water solution. (Figure 3.6.)
Figure 3.6. 13CNMR of 10% in Weight of Cyrene in D2O
Figure2.6. highlights a small peak relative to the carbonyl, and there is an intense peak for the geminal diol, a clear sign that the equilibria at this concentration of Cyrene in D2O has moved to the diol form. It was therefore possible by integrating peaks of the 1HNMR for the D2O -Cyrene system , to calculate the relative ratio of moles of diol for each mole of Cyrene present in solutions at different concentrations of D2O (fig 3.7). Using this method, it was possible to show the relative molar ratios of Cyrene, Geminal Diol (OD)2 and free D2O as a function of initial weight percentage Cyrene in D2O (table 3.1).
Figure 3.7. 1HNMR of 65%Wt Cyrene in D2O
In figure are shown the peak relative at Cyrene and Its hydrate form
Table 3.1. Moles of geminol diol and free H2O per every mole of Cyrene present in different water solutions
These data were used to determine the general trend and behaviour of Cyrene in D2O plotting the experimental data (Figure 3.8.). The same analysis was then conducted for Cyrene in H2O (Table 3.2).
Figure 3.8. the figure shows the moles of germinal diol and free D2O in different Cyrene-water solution
Table 3.2. Moles of germinal diol and free H2O per mole of Cyrene present in different concentration of Cyrene-water solution
Figure 3.9.
As can be seen from figure 2.9., the behaviour of Cyrene in H2O looks very different and exhibits a maximum, which is not present in that obtained for the Cyrene-D2O system. In fact, if the behaviour of the geminal diol is compared, two different systems are obtained for each mole of Cyrene (figure3.10).
Figure 3.10. Comparison between the plotted data from the solutions of Cyrene in D2O and H2O in different weight concentration
Figure 2.10 .highlights two contrasting trends, which can be explained by the fact that the D2O forms the more intense hydrogen bonds with the Cyrene that would shift the equilibrium toward the formation of the diol. It should also be stressed that the formation of the geminal diol and its relative equilibrium with the keto form has important consequences on the behaviour of Cyrene in aqueous solution. In fact, thanks to the presence of two hydroxyl groups in place of the carbonyl, the Cyrene is able to interact in a more complex way with its chemical environment, forming more easily interactions with other molecules present in solution through hydrogen bonds. The equilibrium between the ketonic form and the diol form, can be used in theory, to increase the solvating ability of the Cyrene, in all those situations in which the formation of hydrogen bonds favours the dissolution of the substances in solution, for example in the solubilisation of biomass. This aspect of the research is actually object of an article under writing, but it not will be included in this thesis. Here it will be reported only few experimental data as example.
Other 1HNMR at different temperature were performed in order to understand the trend of the equilibrium between ketone form and the germinal diol form with the changing of temperature. Fig 3.11-3.12
figure 3.11. 1HNMR of 65%Wt Cyrene in H2O at different temperature
figure 3.12. 1HNMR of 65% Cyrene in D2O at different temperature
both the figures relative to the 1HNMR experiment in H2O and D2O show that there is an increment of the Ketone form when the temperature increase. It is well known that the hydration ration equilibrium is very sensitive to temperature.22
Although the collected data in theory are enough to determinate the constant of equilibrium for this reaction , this could be done only by thermodynamic way , due to the rapidity of the reaction. However as shown from the curve in figure 2.10 , the kinetic of Cyrene in water has a very complex behaviour , and could be very difficult to determinate because it have to involve the formation of clathrate, as is shown clearly from its (non linear) dependence of the association constant Kh of germinal diol-Cyrene with the concentration of water.23 a
Kd=Cyrene KetoneGeminal Diol =1Kh
eq.3.1
3.3 Density and viscosity measurements
It was performed measurement of the density in all the concentration range of Cyrene-water and Cyrene D2O solutions . Figure 3.13
Figure 3.13. shows a comparison between the trend of the density in the Cyrene solution of D2O and water
the figure point out the strong deviation from the ideality in behaviour in both the solvent . In particular we have to stress the presence of a maximum in density that should correspond whit the formation of same kind of structure ( probably clathrate) at 80-85%wtg concentration. This trend , in the current state of my knowledge , is not been reported in literature for any common green solvent utilised . Notice that in D2O the same maximum is shifted at lower concentration value , may be due to the solvent effect , related at the stronger hydrogen bonding interaction present in D2O.23b
the same position of the maximum was found for the peak shown in the viscosity graph of
Cyrene-water solution. figure 3.14
Figure 3.14 show the uncommon data of viscosity values.
Table 3.3
Table2.3 in table are reported the average of value for the viscosity data
SAXS and DLS experiment Experiment
SAXS analyses for aqueous solutions of Cyrene and in D2O were obtained in an attempt to observe the eventual formation of structures in these solvents (Figure 3.15.).
Figure 3.15. SAXS for Cyrene in D2O
In light blue pure Cyrene, in violet 95wgt% Cyrene, in green 65wgt% Cyrene , in red 35wgt% Cyrene, in blue 5Cyrene wgt%
From the figure is possible see the presence of a intense peak related with the normal distribution of molecule of Cyrene in solution in different concentration . This scattering peak shows structure with an average of dimensional size of 5A0 , compatible with a single free molecule of Cyrene. At low concentrations of Cyrene in D2O, the figure x shows a peak where typically a flat line would be expected, this result is constant with the possible formation a structure between the Cyrene and D2O. However, it is easy to hypothesise the formation of complex interactions between Cyrene and water, with the consequent formation of structures.(Figure 3.16.)
Figure 3.16. A tentative model of how the Cyrene/geminal diol aggregates in water (in very dilute solution).
Dynamic light scattering experiment were made in order to confirm the existence of structure in very diluted Cyrene water solution. Figure 3.17
Figure 3.17
At this extreme dilution as we can see from the previous experiment , only geminal form of Cyrene id predominant , so the possible presence of structure is imputable to the interaction between this form and water . As we can see from the figure a clear peak shows a size distribution around 2 nm that is in accordance with the result of SAXS experiment . This result is compatible with the presence of structure of about 20-30 Ao . Nevertheless , more experiment have to be done to confirm this result , due to the not optimal poly dispersion of the sample .
3.4 DSC experiment of Cyrene water solution
In the determination of the method for the DSC analysis for Cyrene , it was chosen the program with the heating –cooling rate that allow to have the better data in the large range of concentration of Cyrene- water solution. This because the physical behaviour of Cyrene-water change drastically with the temperature. For all the data reported was used the following method:
1)equilibrate at 20°C,
2)Ramp 5°C/min to -70°C,
3)Ramp 15°C/min to 20°C (x 2 times)
The method above allow to have useful data from very diluted solution to about 45%wgt of Cyrene-water solution , reducing at the same time the super cooling effect of water. Figure 2.18
Figure 3.18 Same example of the DSC in the range of 50%wgt concentration
From the DSC thermogram was possible , using the function onset available in the for the instrument, calculate the value of the melting point depression .figure 3.19
Figure 3.19 Example of onset calculation using the specific function available in the software of the TA Instruments DSC Q2000
The same value were compared with the one obtained by using a fibre optic (SAIREM miniflow) for samples at different dilution , that were first frozen using liquid nitrogen . figure2.20
Figure3.20
fig the trend of melting and freezing point obtained using different method
Is possible to see as the value obtained using DSC calculation from the melting point , are really close to the one observed using the optic fiber during the melting of frozen solution .
It is possible to see that the value of freezing point are shifted at lower temperature, due to the super cooling effect of the water. From all the experimental data is possible observe the presence of an eutectic point close to -550C.
Have to be report ,that during the melting of the Cyrene-water solution at certain concentration, was possible observe the existence of a uncommon phenomena like the formation of a kind of ice with a consistence similar to the toothpaste , with a very high viscosity. Figure3.21
Figure 3.21. shows the consistency of high Cyrene water ice.
This phenomena is probably related with the formation of an amorphous system , according with the DSC thermogram , in which is not possible observe any peak relative at freezing or melting .
At this point it was attempt the calculation of the molecular weight of Cyrene using the melting point depression instead to the freezing point depression , obtained by the DSC thermogram recorded in very dilute Cyrene-water concentration. The data of the freezing point depression of Cyrene –water solution were not used because the thermograms shows a supercooling peak for the water at very low concentration.24-26
It was used the follow equation:
ΔT = iKfm Eq.3.2
Where , ΔT is the difference in temperature, Kf is the cryoscopic constant of water = 1.86 °Ckg/mol
And i is the van ‘t Hoff factor in this case i=1, and m= molality of the solution.
The result were not useful , probably because the contemporary presence of Cyrene molecules and its germinal diol form, and because their interactions with water, that make the system not ideal. However it was possible using the DSC data perform the calculation of the activity of water in the system.
Calculation of the activity of water
Using the follow equation and the melting point DSC data it was calculate the value of the activity of water.27
Eq.3.3
lnaw=-ΔHfR (Tf-T)TfT+ΔCfR(Tf-T)T-lnTfT
Eq.3.3
Table 3.4
Figure3.22
As we have seen , Cyrene shows an complex behaviour in water, but theoretically it is possible thanks to the collected data, relate the coefficient of activity and melting point depression , to a specific structure, as well as the fraction of water bonded with Cyrene . Certainly many other experiment are necessary in order to confirm the data above reported and clarify the nature of Cyrene water interaction.28-30
[1] J. Sherwood, M. De bruyn, A. Constantinou, L. Moity, C. R. McElroy, T. J. Farmer, T. Duncan, W. Raverty, A. J. Hunt and J. H. Clark, Chemical Communications 2014, 50, 9650-9652.
[2] J. H. Clark, T. J. Farmer, A. J. Hunt and J. Sherwood, International journal of molecular sciences 2015, 16, 17101-17159.
[3] P. T. Anastas and J. C. Warner, Green chemistry: Theory and practice 1998, 29-56.
[4] F. M. Kerton and R. Marriott, Alternative solvents for green chemistry, Royal Society of chemistry, 2013, p.
[5] F. M. Kerton, Alternative solvents for green chemistry, Royal Society of chemistry, 2013, ch.1.3
[6] C. Reichardt, Chemical Reviews 1994, 94, 2319-2358.
[7] R. C., Solvents and Solvent Effects in Organic Chemistry, VCH Publishers Weinheim, 1988, p.
[8] A. Buckingham, P. Fowler and J. M. Hutson, Chemical Reviews 1988, 88, 963-988.
1)Dihydrolevoglucosenone (Cyrene) as a bio-based alternative for dipolar aprotic solvents† James Sherwood,a Mario De bruyn,a Andri Constantinou,a Laurianne Moity,a C. Rob McElroy,a Thomas J. Farmer,a Tony Duncan,b Warwick Raverty,b Andrew J. Hunta and James H. Clark*a
2)Opportunities for Bio-Based Solvents Created as Petrochemical and Fuel Products Transition towards Renewable Resources
Clark, James H ; Farmer, Thomas J ; Hunt, Andrew J ; Sherwood, James
International journal of molecular sciences, 2015, Vol.16(8), pp.17101-59
3)*Anastas, P. T.; Warner, J. C. Green Chemistry: Theory and Practice, Oxford University Press: New York, 1998, p.30. By permission of Oxford University Press.
4)Alternative Solvents for Green
Chemistry Francesca M. Kerton RSC, 2009
5)Alternative Solvents for Green
Chemistry Francesca M. Kerton RSC, 2009 ch.1.3
6)Chem. Rev. 1594, 94, 231S2358 2319 Solvatochromic Dyes as Solvent Polarity Indicators Christian Reichardt
7)Reichardt, C. Solvents and Solvent Effects in Organic Chemistry, 2nd ed.; VCH Publishers: Weinheim, 1988
8)Buckingham, A. D.; Fowler, P. W.; Hutson, J. M. Chem. Rev.
9) Huyskens, P. L., Luck, W. A. P., Zeegers-Huyskens, T., Eds.
10) D. J. Adams, P. J. Dyson and S. J. Taverner, Chemistry in Alternative
Reaction Media, John Wiley & Sons Ltd, Chichester, 2004.
11)G. R. Court, C. H. Lawrence, W. D. Raverty and A. J. Duncan, US Pat.,
2012/0111714 A1, 2012.
12)Further property of ionic liquids: Hildebrand solubility parameter from new molecular thermodynamic model M.M. Alavianmehr a , S.M. Hosseini a, ⁎, A.A. Mohsenipour c , J. Moghadasi
13). J. P.lowe, in ,Quantum Chemistry ,Academic Press , second edition ,Chapter13 , pp. 429
14). B. Schrader ,in ,Infrared and Raman spectroscopy methods and applications, VCH , first edition,1995, pp.39-41
15) Mueller fundamentals of quantum chemistry KLUWER ACADEMIC PUBLISHERS 2001)
16) herzberg, molecular spectra and molecular structure vol 2 1988
17) D.Griffiths, introduction of quantum mechanics
18) herzberg ,molecular spectra and molecular structure vol 1 pag 75, 1988
19) High resolution spectroscopy second edition wiley pag 172
20) Banwell fundamentals of molecular spectroscopy pag 77 1983
21) hollas high resolution spectroscopy pag165
22) (from Wikipedia) draw by Mark Somoza March 26, 2006
23) High resolution spectroscopy second edition wiley pag 246
24) see for instance the symmetry of the molecule of water Banwell fundaments of molecular spectroscopy pag 94
25) see the examples of water, influence of rotation on the spectra of polyatomic molecule. Banwell, Fundamentals of molecular spectroscopy 1983, pag 95-97)
26) hollas , high resolution spectroscopy second edition wiley 1998
27) Herzberg molecular spectra and molecular structure vol 1 1988 ,pag14
28) Herzberg molecular spectra and molecular structure vol 2 pag 216 1988)
29) Hollas High resolution spectroscopy second edition wiley Pag 248)
30)Hollas High resolution spectroscopy second edition wiley Pag 249
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Related Services | {"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": 0.8449575901031494, "perplexity": 1609.596737465891}, "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-2021-17/segments/1618038464146.56/warc/CC-MAIN-20210418013444-20210418043444-00229.warc.gz"} |
http://tex.stackexchange.com/questions/88138/xetex-mathspec-math-environment-font?answertab=oldest | # XeTeX/mathspec math environment font
How does one set the font for text inside a math environment? If I use \setmathsfont(Latin){FontName}, I can change the font for individual letters in the math environment, but not for functions such as \sin. (These aren't controlled by the set "Symbols" either.)
Here's a MWE:
\documentclass[a4paper,12pt]{article}
\usepackage[MnSymbol]{mathspec}
\usepackage[no-sscript]{xltxtra}
\defaultfontfeatures{Mapping=tex-text}
\setmainfont{Times}
\setmathsfont(Digits){Times}
\setmathsfont(Latin){Times}
%\setmathsfont(Symbols){Times}
\begin{document}
Some text digits 1, 2, 3 and some math digits $1, 2, 3$.
Some italic text \textit{text}, and some math text $text$.
But, we have \textit{sin x} and $\sin x$. Or, \textit{arctan x} and $\arctan x$.
\end{document}
-
The math-font system is a rather complex system, but I think the question is answered quite well in this related thread: how to select math font in document. – aagaard Dec 25 '12 at 11:41
You have to do \setmathrm; here I use TeX Gyre Termes, but what font you're using is irrelevant.
\documentclass[a4paper,12pt]{article}
\usepackage{mathspec}
\defaultfontfeatures{Ligatures=TeX}
\setmainfont{TeX Gyre Termes}
\setmathsfont(Digits){TeX Gyre Termes}
\setmathsfont(Latin){TeX Gyre Termes}
\setmathrm{TeX Gyre Termes}
\begin{document}
Some text digits 1, 2, 3 and some math digits $1, 2, 3$.
Some italic text \textit{text}, and some math text $text$.
Also $\sin x$ and $\arctan x$.
\end{document}
If you have a recent and updated TeX Live, you can also use the new TG Termes Math font with unicode-math:
\documentclass[a4paper,12pt]{article}
\usepackage{unicode-math}
\defaultfontfeatures{Ligatures=TeX}
\setmainfont{TeX Gyre Termes}
\setmathfont{TG Termes Math}
\begin{document}
Some text digits 1, 2, 3 and some math digits $1, 2, 3$.
Some italic text \textit{text}, and some math text $text$.
Also $\sin x$ and $\arctan x$.
\end{document}
This is actually much better than mathspec, because the math font has the right parameters for being used in formulas.
-
Thanks, \setmathrm does the trick. – Conic Dec 25 '12 at 17:11 | {"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": 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": 0.9980959892272949, "perplexity": 4677.060157259946}, "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-2015-11/segments/1424936464303.77/warc/CC-MAIN-20150226074104-00114-ip-10-28-5-156.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/block-sliding-down-frictionless-ramp-while-being-pulled-with-constant-force.188756/ | # Homework Help: Block sliding down frictionless ramp while being pulled with constant force?
1. Oct 3, 2007
### soliel021
1. The problem statement, all variables and given/known data
A block of ice slides down a frictionless ramp at theta = 60 degrees while an ice worker pulls on the block (via a rope) with a constant force of 50.0 N. The block slides a distance d = 0.8m along the ramp and its kinetic energy increases by 128 J.
(a) Calculate the work done on the block by the gravitational force.
(b) Calculate the work done on the block by the normal force.
(c) Calculate the percent increase in its kinetic energy if it slid down the ramp the same distance with no rope attached.
2. Relevant equations
I have no idea, that's my problem!
3. The attempt at a solution
I don't even know where to start!
2. Oct 3, 2007
### Sourabh N
Do you know the formulas for work and concept of free body diagram(FBD, in short) ?
Last edited: Oct 3, 2007
3. Oct 3, 2007
### soliel021
no and yes...
I understand the concept of free body diagrams, but there are so many formulas that I really need help choosing which ones to use for this specific problem.
4. Oct 3, 2007
### Sourabh N
OK. Draw a free body diagram.
For I and II : W = F.r
You know F from FBD for both cases and r is given to you. Use your "intelligence" to find the angle between F and r and calculate the dot product in each case.
Last edited: Oct 3, 2007
5. Oct 3, 2007
### soliel021
This may be a really stupid question, but what does "r" stand for? Is it displacement?
6. Oct 3, 2007
### jesus1987
r really should be s or distance i think...
7. Oct 3, 2007
### jesus1987
If it has a constant force then that must mean that F=ma is constant, therefore its acceleration isnt changing, as this would alter the force, so how is the block of ice gaining Ek due to a change in velocity?
8. Oct 3, 2007
### soliel021
yeah, I don't get it either! and don't I need to know the mass of the block of ice?? I'm so confused...
9. Oct 3, 2007
### learningphysics
If there is an acceleration, then velocity is changing.
10. Oct 3, 2007
### learningphysics
you don't need the mass.
Net work done by all forces = change in kinetic energy.
There are 3 forces to consider here. Gravity, the normal force, and the 50N force.
What is the work done by the normal force?
What is the work done by the 50N force?
$$W_{gravity} + W_{normal} + W_{50} = 128$$
you can solve for Wgravity using this equation. so the idea is to solve part b) before part a). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9505361318588257, "perplexity": 1182.8179628318658}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583517495.99/warc/CC-MAIN-20181023220444-20181024001944-00129.warc.gz"} |
https://www.physicsforums.com/threads/harmonic-oscillator.245481/ | # Harmonic Oscillator
1. Jul 17, 2008
Griffiths solves the Schrodinger equation using ladder operators, and he then says that there must be a "lowest rung," or $$\psi_{0}$$, such that a_$$\psi_{0}$$ = 0. I'm guessing this also means that E = 0 for a_$$\psi_{0}$$, which is why it wouldn't be allowable.
Because a_$$\psi_{0}$$ is (mathematically, at least) a solution to the Schrodinger equation, it corresponds to the energy ($$E_{0}$$ - $$\hbar\omega$$), which equals zero. Doesn't this mean that $$E_{0}$$ = $$\hbar\omega$$?
But then this conflicts with Griffith's statement that $$E_{0}$$ = $$\frac{1}{2}\hbar\omega$$. HELP!
2. Jul 17, 2008
### CompuChip
Watch out! The statement $a_- \psi_0 = 0$ just says that when you apply the ladder operator to this state, you get nothing, i.e. the vacuum. Physicists have a tendency to use "natural" notation, and you will often see things like $a_- |0\rangle = 0$. Note that here the $|0\rangle$ on the left hand side is a state (wavevector) and the 0 on the right hand side is the zero vector, which is not a state. Griffiths is just kind enough to call it $\psi_0$ instead of $|0\rangle$ so you don't get completely confused
You say that $a_- \psi_0$ is a solution to the Schrodinger equation: that's right. But it isn't a physical solution. It's just the solution $\psi = 0$. The energy of $\psi_0$ is $\frac12 \hbar \omega$ (formula 2.61) which can be seen by applying the Hamiltonian in the form (2.56) to this state and using that $a_-$ annihilates it.
Hope that makes it more clear.
3. Jul 18, 2008
Thanks, CompuChip! | {"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": 0.8712671995162964, "perplexity": 301.9935659409108}, "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-2017-04/segments/1484560281353.56/warc/CC-MAIN-20170116095121-00149-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://en.wikipedia.org/wiki/Carnot_Cycle | Carnot cycle
(Redirected from Carnot Cycle)
The Carnot cycle is a theoretical thermodynamic cycle proposed by Nicolas Léonard Sadi Carnot in 1824 and expanded upon by others in the 1830s and 1840s. It provides an upper limit on the efficiency that any classical thermodynamic engine can achieve during the conversion of heat into work, or conversely, the efficiency of a refrigeration system in creating a temperature difference (e.g. refrigeration) by the application of work to the system. It is not an actual thermodynamic cycle but is a theoretical construct.
Every single thermodynamic system exists in a particular state. When a system is taken through a series of different states and finally returned to its initial state, a thermodynamic cycle is said to have occurred. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine. A system undergoing a Carnot cycle is called a Carnot heat engine, although such a "perfect" engine is only a theoretical construct and cannot be built in practice.[1] However, a microscopic Carnot heat engine has been designed and run.[2]
Essentially, there are two systems at temperatures Th and Tc (hot and cold respectively), which are so large that their temperatures are practically unaffected by a single cycle. As such, they are called "heat reservoirs". Since the cycle is reversible, there is no generation of entropy during the cycle; entropy is conserved. During the cycle, an arbitrary amount of entropy ΔS is extracted from the hot reservoir, and deposited in the cold reservoir. Since there is no volume change in either reservoir, they do no work, and during the cycle, an amount of energy ThΔS is extracted from the hot reservoir and a smaller amount of energy TcΔS is deposited in the cold reservoir. The difference in the two energies (Th-TcS is equal to the work done by the engine.
Stages
Figure 1: A Carnot cycle illustrated on a PV diagram to illustrate the work done.
The Carnot cycle when acting as a heat engine consists of the following steps:
1. Reversible isothermal expansion of the gas at the "hot" temperature, T1 (isothermal heat addition or absorption). During this step (1 to 2 on Figure 1, A to B in Figure 2) the gas is allowed to expand and it does work on the surroundings. The temperature of the gas does not change during the process, and thus the expansion is isothermal. The gas expansion is propelled by absorption of heat energy Q1 from the high temperature reservoir and results in an increase of entropy of the gas in the amount ${\displaystyle \Delta S_{1}=Q_{1}/T_{1}}$.
2. Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step (2 to 3 on Figure 1, B to C in Figure 2) the mechanisms of the engine are assumed to be thermally insulated, thus they neither gain nor lose heat (an adiabatic process). The gas continues to expand, doing work on the surroundings, and losing an amount of internal energy equal to the work that leaves the system. The gas expansion causes it to cool to the "cold" temperature, T2. The entropy remains unchanged.
3. Reversible isothermal compression of the gas at the "cold" temperature, T2. (isothermal heat rejection) (3 to 4 on Figure 1, C to D on Figure 2) Now the surroundings do work on the gas, causing an amount of heat energy Q2 to leave the system to the low temperature reservoir and the entropy of the system decreases in the amount ${\displaystyle \Delta S_{2}=Q_{2}/T_{2}}$. (This is the same amount of entropy absorbed in step 1, as can be seen from the Clausius inequality.)
4. Isentropic compression of the gas (isentropic work input). (4 to 1 on Figure 1, D to A on Figure 2) Once again the mechanisms of the engine are assumed to be thermally insulated, and frictionless, hence reversible. During this step, the surroundings do work on the gas, increasing its internal energy and compressing it, causing the temperature to rise to T1 due solely to the work added to the system, but the entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.
In this case,
${\displaystyle \Delta S_{1}=\Delta S_{2}}$,
or,
${\displaystyle Q_{1}/T_{1}=Q_{2}/T_{2}}$.
This is true as ${\displaystyle Q_{2}}$ and ${\displaystyle T_{2}}$ are both lower and in fact are in the same ratio as ${\displaystyle Q_{1}/T_{1}}$.
The pressure-volume graph
When the Carnot cycle is plotted on a pressure volume diagram, the isothermal stages follow the isotherm lines for the working fluid, adiabatic stages move between isotherms and the area bounded by the complete cycle path represents the total work that can be done during one cycle.
Properties and significance
The temperature-entropy diagram
Figure 2: A Carnot cycle acting as a heat engine, illustrated on a temperature-entropy diagram. The cycle takes place between a hot reservoir at temperature TH and a cold reservoir at temperature TC. The vertical axis is temperature, the horizontal axis is entropy.
A generalized thermodynamic cycle taking place between a hot reservoir at temperature TH and a cold reservoir at temperature TC. By the second law of thermodynamics, the cycle cannot extend outside the temperature band from TC to TH. The area in red QC is the amount of energy exchanged between the system and the cold reservoir. The area in white W is the amount of work energy exchanged by the system with its surroundings. The amount of heat exchanged with the hot reservoir is the sum of the two. If the system is behaving as an engine, the process moves clockwise around the loop, and moves counter-clockwise if it is behaving as a refrigerator. The efficiency to the cycle is the ratio of the white area (work) divided by the sum of the white and red areas (heat absorbed from the hot reservoir).
The behaviour of a Carnot engine or refrigerator is best understood by using a temperature-entropy diagram (TS diagram), in which the thermodynamic state is specified by a point on a graph with entropy (S) as the horizontal axis and temperature (T) as the vertical axis. For a simple closed system (control mass analysis), any point on the graph will represent a particular state of the system. A thermodynamic process will consist of a curve connecting an initial state (A) and a final state (B). The area under the curve will be:
${\displaystyle Q=\int _{A}^{B}T\,dS\quad \quad (1)}$
which is the amount of thermal energy transferred in the process. If the process moves to greater entropy, the area under the curve will be the amount of heat absorbed by the system in that process. If the process moves towards lesser entropy, it will be the amount of heat removed. For any cyclic process, there will be an upper portion of the cycle and a lower portion. For a clockwise cycle, the area under the upper portion will be the thermal energy absorbed during the cycle, while the area under the lower portion will be the thermal energy removed during the cycle. The area inside the cycle will then be the difference between the two, but since the internal energy of the system must have returned to its initial value, this difference must be the amount of work done by the system over the cycle. Referring to figure 1, mathematically, for a reversible process we may write the amount of work done over a cyclic process as:
${\displaystyle W=\oint PdV=\oint (dQ-dU)=\oint (TdS-dU)=\oint TdS-\oint dU=\oint TdS\quad \quad \quad \quad (2)}$
Since dU is an exact differential, its integral over any closed loop is zero and it follows that the area inside the loop on a T-S diagram is equal to the total work performed if the loop is traversed in a clockwise direction, and is equal to the total work done on the system as the loop is traversed in a counterclockwise direction.
A Carnot cycle taking place between a hot reservoir at temperature TH and a cold reservoir at temperature TC.
The Carnot cycle
A visualization of the Carnot cycle
Evaluation of the above integral is particularly simple for the Carnot cycle. The amount of energy transferred as work is
${\displaystyle W=\oint PdV=\oint TdS=(T_{H}-T_{C})(S_{B}-S_{A})}$
The total amount of thermal energy transferred from the hot reservoir to the system will be
${\displaystyle Q_{H}=T_{H}(S_{B}-S_{A})\,}$
and the total amount of thermal energy transferred from the system to the cold reservoir will be
${\displaystyle Q_{C}=T_{C}(S_{B}-S_{A})\,}$
The efficiency ${\displaystyle \eta }$ is defined to be:
${\displaystyle \eta ={\frac {W}{Q_{H}}}=1-{\frac {T_{C}}{T_{H}}}\quad \quad \quad \quad \quad \quad \quad \quad \quad (3)}$
where
${\displaystyle W}$ is the work done by the system (energy exiting the system as work),
${\displaystyle Q_{C}}$ is the heat taken from the system (heat energy leaving the system),
${\displaystyle Q_{H}}$ is the heat put into the system (heat energy entering the system),
${\displaystyle T_{C}}$ is the absolute temperature of the cold reservoir, and
${\displaystyle T_{H}}$ is the absolute temperature of the hot reservoir.
${\displaystyle S_{B}}$ is the maximum system entropy
${\displaystyle S_{A}}$ is the minimum system entropy
This definition of efficiency makes sense for a heat engine, since it is the fraction of the heat energy extracted from the hot reservoir and converted to mechanical work. A Rankine cycle is usually the practical approximation.
The Reversed Carnot cycle
The Carnot heat-engine cycle described is a totally reversible cycle. That is, all the processes that comprise it can be reversed, in which case it becomes the Carnot refrigeration cycle. This time, the cycle remains exactly the same except that the directions of any heat and work interactions are reversed. Heat is absorbed from the low-temperature reservoir, heat is rejected to a high-temperature reservoir, and a work input is required to accomplish all this. The P-V diagram of the reversed Carnot cycle is the same as for the Carnot cycle except that the directions of the processes are reversed.[3]
Carnot's theorem
It can be seen from the above diagram, that for any cycle operating between temperatures ${\displaystyle T_{H}}$ and ${\displaystyle T_{C}}$, none can exceed the efficiency of a Carnot cycle.
A real engine (left) compared to the Carnot cycle (right). The entropy of a real material changes with temperature. This change is indicated by the curve on a T-S diagram. For this figure, the curve indicates a vapor-liquid equilibrium (See Rankine cycle). Irreversible systems and losses of energy (for example, work due to friction and heat losses) prevent the ideal from taking place at every step.
Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between those same reservoirs. Thus, Equation 3 gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient. Rearranging the right side of the equation gives what may be a more easily understood form of the equation. Namely that the theoretical maximum efficiency of a heat engine equals the difference in temperature between the hot and cold reservoir divided by the absolute temperature of the hot reservoir. To find the absolute temperature in kelvins, add 273.15 degrees to the Celsius temperature. Looking at this formula an interesting fact becomes apparent. Lowering the temperature of the cold reservoir will have more effect on the ceiling efficiency of a heat engine than raising the temperature of the hot reservoir by the same amount. In the real world, this may be difficult to achieve since the cold reservoir is often an existing ambient temperature.
In other words, maximum efficiency is achieved if and only if no new entropy is created in the cycle, which would be the case if e.g. friction leads to dissipation of work into heat. In that case the cycle is not reversible and the Clausius theorem becomes an inequality rather than an equality. Otherwise, since entropy is a state function, the required dumping of heat into the environment to dispose of excess entropy leads to a (minimal) reduction in efficiency. So Equation 3 gives the efficiency of any reversible heat engine.
Corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient. Rearranging the right side of the equation gives what may be a more easily understood form of the equation. Namely that the theoretical maximum efficiency of a heat engine equals the difference in temperature between the hot and cold reservoir divided by the absolute temperature of the hot reservoir. To find the absolute temperature in kelvin, add 273.15 degrees to the Celsius temperature. Looking at this formula an interesting fact becomes apparent. Lowering the temperature of the cold reservoir will have more effect on the ceiling efficiency of a heat engine than raising the temperature of the hot reservoir by the same amount. In the real world, this may be difficult to achieve since the cold reservoir is often an existing ambient temperature. In other words, maximum efficiency is achieved if and only if no new entropy is created in the cycle.[clarification needed] Otherwise, since entropy is a state function, the required dumping of heat into the environment to dispose of excess entropy leads to a reduction in efficiency. This relation transforms Carnot's inequality into an exact equality that applies to an arbitrary heat engine coupled to two heat reservoirs and operating at an arbitrary rate.
In mesoscopic heat engines, work per cycle of operation fluctuates due to thermal noise. For the case when work and heat fluctuations are counted, there is exact equality that relates average of exponents of work performed by any heat engine and the heat transfer from the hotter heat bath.[4].
Efficiency of real heat engines
Carnot realized that in reality it is not possible to build a thermodynamically reversible engine, so real heat engines are even less efficient than indicated by Equation 3. In addition, real engines that operate along this cycle are rare. Nevertheless, Equation 3 is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.
Although Carnot's cycle is an idealisation, the expression of Carnot efficiency is still useful. Consider the average temperatures,
${\displaystyle \langle T_{H}\rangle ={\frac {1}{\Delta S}}\int _{Q_{in}}TdS}$
${\displaystyle \langle T_{C}\rangle ={\frac {1}{\Delta S}}\int _{Q_{out}}TdS}$
at which heat is input and output, respectively. Replace TH and TC in Equation (3) by 〈TH〉 and 〈TC〉 respectively.
For the Carnot cycle, or its equivalent, the average value 〈TH〉 will equal the highest temperature available, namely TH, and 〈TC〉 the lowest, namely TC. For other less efficient cycles, 〈TH〉 will be lower than TH, and 〈TC〉 will be higher than TC. This can help illustrate, for example, why a reheater or a regenerator can improve the thermal efficiency of steam power plants—and why the thermal efficiency of combined-cycle power plants (which incorporate gas turbines operating at even higher temperatures) exceeds that of conventional steam plants. The first prototype of the diesel engine was based on the Carnot cycle. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 44, "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": 0.8577083349227905, "perplexity": 305.6758581035992}, "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-2017-17/segments/1492917122886.86/warc/CC-MAIN-20170423031202-00297-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://physics.stackexchange.com/questions/184742/reduction-of-nambu-goto-action-to-true-degrees-of-freedom?noredirect=1 | # Reduction of Nambu-Goto action to true degrees of freedom
I) First consider the point particle
$$S=m\int\sqrt{-\dot{X}^2}d\tau.$$
If you choose the static gauge $$\tau=X^0$$ and replace it in the action you get
$$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau.$$
So now, you have an equivalent action with true degrees of freedom only. In fact, you can do the same with the light-cone gauge $$\tau=X^+$$ and obtain
$$S=m\int\sqrt{2\dot{X}^--\dot{X}^I\dot{X}^I}d\tau.$$
II) Is it possible to do the same for the string? I have found no reference doing this. I have been trying replacing the conditions of the light-cone gauge. I think that the answer could be
$$\mathcal{L}\sim \dot{X}^I\dot{X}^I -{X^I}' {X^I}'.\tag{12.81}$$
This is eq.(12.81) in the 2nd edition of Zwiebach's book.
Additional Information: Basically you have to show that the gauge conditions $$n\cdot X=2\beta \alpha'n\cdot p \tau ,\tag{1}$$
$$n\cdot p=\frac{2\pi}{\beta}n\cdot \mathcal{P},\tag{2}$$
(where $$\beta=2$$ for open string and $$\beta=1$$ for closed string) imply $$\dot{X^2}= -X'^2\quad\text{and}\quad \dot{X}\cdot X'=0.$$
And with these relations you can easily reduce the action.
For the open string you can show that condition (2) implys $$\dot{X}\cdot X'=0.$$
This can be done using the boundary conditions of the open string.$$^1$$
But for the closed string I have found no way to show that. In Zwiebach page 180 there is an idea of this for open and closed strings.$$^2$$
To sum up, my question is if one can deduce the same for the closed string. What would be the procedure?
$$^1$$See Sundermayer, Constrained Dynamics page 218, or Hansen-Regge-Teitelboim, Constrained Hamiltonian Systems page 58.
$$^2$$ But in the case of closed string the full $$\sigma=0$$ line is constructed by requiring that at each point its tangent be orthogonal to $$X'$$. So it is like he is imposing $$\dot{X}\cdot X'=0$$, not deducing it.
• It is possible. It's known as light cone quantization. – Prahar Mitra May 18 '15 at 22:30
Here is an outline of the reduction from the Nambu-Goto (NG) action to the light-cone (LC) formulation from a Hamiltonian perspective:
1. The starting point is the Hamiltonian formulation of the NG string, cf. e.g. this Phys.SE post. The Hamiltonian is of the form "Lagrange multipliers times constraints"$$^1$$ $$H~:=~\int_0^{\ell}\! d\sigma~{\cal H}, \qquad {\cal H}~=~\lambda^{\alpha} \chi_{\alpha}, \qquad \alpha~\in~\{0,1\}.\tag{1}$$
2. The two first class constraints read $$\chi_0~:=~P\cdot X^{\prime}~\approx~0, \qquad \chi_1~:=~\frac{P^2}{2T_0}+\frac{T_0}{2}(X^{\prime})^2~\approx~0.\tag{2}$$ The two first class constraints $$\chi_{\alpha}$$ originate from (and generate) the world-sheet (WS) reparametrization invariance of the Nambu-Goto action.
3. The corresponding Hamiltonian Lagrangian then reads$$^1$$ $$L_H~:=~\int_0^{\ell}\! d\sigma~{\cal L}_H, \qquad {\cal L}_H ~:=~ P\cdot \dot{X}-{\cal H} . \tag{3}$$ The Euler-Lagrange eqs. for the Hamiltonian Lagrangian (3) are Hamilton's eqs. $$\dot{X}^{\mu}~\approx~\lambda^0X^{\mu\prime}+ \frac{\lambda^1}{T_0}P^{\mu},\qquad \dot{P}^{\mu}~\approx~\left(\lambda^0 P^{\mu}+ T_0\lambda^1 X^{\mu\prime}\right)^{\prime},\tag{4}$$ together with the two constraints (2).
4. We use light-cone (LC) coordinates $$X^{\pm}~:=~\frac{X^0\pm X^1}{\sqrt{2}} ,\tag{5}$$ and Minkowski metric $$\eta_{\mu\nu} ~dX^{\mu}~ dX^{\nu} ~=~ -(dX^0)^2 + (dX^1)^2 + (dX^{\perp})^2$$ $$~=~ -2dX^+dX^- + (dX^{\perp})^2,\tag{6}$$ in the target space (TS). [The $$\perp$$ symbol denotes transversal coordinates.]
5. We fix the gauge symmetry by imposing two corresponding gauge-fixing conditions $$X^+(\tau,\sigma)~=~f(p^+(\tau))\tau\qquad\text{and}\qquad P^+(\tau,\sigma)~=~p^+(\tau), \tag{7}$$ i.e. the LC gauge, where $$f>0$$ is some positive function, usually taken to be a linear or constant function. We have dropped a conventional multiplicative constant in the condition (7b) for simplicity.
6. In the LC gauge (7), the Hamiltonian Lagrangian density (3) becomes $${\cal L}_H ~=~P\cdot \dot{X} -\lambda^0\left\{-X^{-\prime} p^+ +\chi^{\perp}_0 \right\} -\lambda^1\chi_1$$ $$~\sim~P\cdot \dot{X} - \lambda^{0 \prime} X^- p^+ -\lambda^0\chi^{\perp}_0 -\lambda^1 \left\{-\frac{p^+ P^-}{T_0} + \chi^{\perp}_1\right\}. \tag{8}$$ The $$\sim$$ symbol means here equal modulo total derivative terms. Also we have introduced the hopefully natural short-hand notation $$\chi^{\perp}_0~:=~P^{\perp}\cdot X^{\perp\prime}, \qquad \chi^{\perp}_1~:=~\frac{(P^{\perp})^2}{2T_0}+\frac{T_0}{2}(X^{\perp\prime})^2.\tag{9}$$
7. Next decompose the two coordinates $$X^-$$ and $$P^-$$ in mean-value coordinates $$x^-(\tau)~:=~ \frac{1}{\ell}\int_0^{\ell}\! d\sigma~X^-(\tau,\sigma),\qquad p^-(\tau)~:=~ \frac{1}{\ell}\int_0^{\ell}\! d\sigma~P^-(\tau,\sigma),\tag{10}$$ and coordinates $$Y^-~:=~X^- - x^-,\qquad R^-~:=~P^- - p^-,\tag{11}$$ with zero means.
8. The kinetic term in the Hamiltonian Lagrangian (8) simplifies to $$\int_0^{\ell}\! d\sigma~P\cdot \dot{X} ~=~\int_0^{\ell}\!d\sigma\left\{\left(-f(p^+)-\tau\frac{df(p^+)}{d\tau}\right) P^- -p^+\dot{X}^- + P^{\perp}\cdot \dot{X}^{\perp}\right\}$$ $$~=~- \ell f(p^+)p^- - \ell p^+\dot{x}^- +\int_0^{\ell}\!d\sigma~ P^{\perp}\cdot \dot{X}^{\perp}.\tag{12}$$ In the last equality of (12) we have dropped the term $$\frac{df(p^+)}{d\tau}$$, because $$p^+$$ is a constant of motion, since $$x^-$$ is a cyclic coordinate (in the Hamiltonian sense), cf. eq. (16) below.
9. Note that in the Hamiltonian Lagrangian (8) the variables $$Y^-$$ and $$R^-$$ only appear in one place each inside $$X^-$$ and $$P^-$$, respectively. Integration over $$Y^-$$ and $$R^-$$ implies that $$\lambda^{0 \prime\prime}~\approx~ 0 \quad\text{and}\quad \lambda^{1 \prime}~\approx~ 0,\tag{13}$$ respectively. To understand the extra derivatives in eq. (13), see e.g. this Phys.SE post for the analogous argument for the Polyakov string action. Pertinent boundary/periodicity conditions then implies that $$\lambda^{0 \prime}~\approx~ 0.\tag{14}$$ So both the two Lagrange multipliers $$\lambda^0$$ and $$\lambda^1$$ are global constants.
10. Integration over the constant $$\lambda^1$$-mode imposes the LC energy condition (2b) $$p^- ~\approx~ \frac{T_0}{p^+\ell}\int_0^{\ell}\! d\sigma ~\chi^{\perp}_1,\qquad \chi^{\perp}_1~:=~\frac{(P^{\perp})^2}{2T_0}+\frac{T_0}{2}(X^{\perp\prime})^2.\tag{15}$$ Then $$p^-$$ is no longer an independent variable. The LC Hamiltonian reads $$H^{LC}~:=~f(p^+)\ell p^-+\lambda^0 \int_0^{\ell}\! d\sigma~\chi^{\perp}_0 ~=~\int_0^{\ell}\! d\sigma~{\cal H}^{LC}, \tag{16}$$ $${\cal H}^{LC}~:=~\lambda^0\chi^{\perp}_0 +T_0\frac{f(p^+)}{p^+} \chi^{\perp}_1.\tag{17}$$ It is tempting to introduced the (possibly confusing) short-hand notation $$\lambda^1~:=~T_0\frac{f(p^+)}{p^+}~>~0 \tag{18}$$ in eq. (17), so that the LC Hamiltonian (17) superficially has the same form as the original Hamiltonian (1). The $$\lambda^1$$ in eq. (18) should not be confused with the Lagrange multiplier zero-mode $$\lambda^1$$, which at this stage has been integrated out.
11. The gauge-fixed LC Hamiltonian Lagrangian then becomes $$L^{LC}_H~=~ - \ell p^+\dot{x}^- +\int_0^{\ell}\!d\sigma~ P^{\perp}\cdot \dot{X}^{\perp} - H^{LC},\tag{19}$$ where the LC Hamiltonian $$H^{LC}$$ is given in eq. (16). The remaining dynamical variables are the transversal variables $$X^{\perp}$$ and $$P^{\perp}$$ and the zero-modes $$x^-$$ and $$p^+$$. The non-zero fundamental Poisson brackets read $$\{x^-, p^+\}_{PB}~=~-\frac{1}{\ell}, \qquad \{X^{\mu\perp}(\sigma), P^{\perp}_{\nu}(\sigma^{\prime})\}_{PB}~=~\delta^{\mu}_{\nu}\delta(\sigma-\sigma^{\prime}).\tag{20}$$ This answers OP's question about the true degrees of freedom. On top of that there is a zero-mode Lagrange multiplier $$\lambda^0$$, to be discussed below.
12. Now let's discuss the role of the zero-mode Lagrange multiplier $$\lambda^0$$. For the open string with free ends, we should impose Neumann boundary conditions $$\frac{\partial{\cal L}_H}{\partial X^{\prime}_{\mu}}~=~0\quad\text{for}\quad \sigma~\in~\{0,\ell\}, \tag{21}$$ which for the coordinate $$\mu=+$$ imply that the constant mode $$\lambda^0=0$$ must vanish (if $$p^+\neq 0$$).
13. For the remainder of this answer, we will discuss the closed string. Integration over the constant mode Lagrange multiplier $$\lambda^0$$ imposes the so-called level-matching constraint/condition (LMC) $$\int_0^{\ell}\!d\sigma~\chi^{\perp}_0~\approx~0, \qquad \chi^{\perp}_0~:=~P^{\perp}\cdot X^{\perp\prime}. \tag{22}$$ Conversely, an integration over a particular transversal string mode would assign a (quantum) average value to $$\lambda^0$$. However, to keep a nice clean classical picture, cf. eoms (23) below, we prefer to postpone the integrations over a particular transversal string mode and the zero-mode $$\lambda^0$$ for later.
14. The LC Hamilton's eqs. read $$\dot{X}^{\perp}~\approx~\lambda^0X^{\perp\prime}+ \frac{\lambda^1}{T_0}P^{\perp},\qquad \dot{P}^{\perp}~\approx~\lambda^0P^{\perp\prime}+ T_0\lambda^1X^{\perp\prime\prime},\tag{23}$$ where $$\lambda^1$$ is given by eq. (18). Eliminating the transversal momenta $$P^{\perp}$$ yields $$\ddot{X}^{\perp}-2\lambda^0 \dot{X}^{\perp\prime} + (\lambda^0+\lambda^1)(\lambda^0-\lambda^1) X^{\perp\prime\prime}~\approx~0. \tag{24}$$
15. Let's introduce new WS coordinates $$\sigma^{\pm}~:=~ \sigma \pm \lambda^{\pm}\tau ~\equiv~(\sigma + \lambda^0\tau) \pm \lambda^1\tau, \qquad \lambda^{\pm}~:=~\lambda^1\pm \lambda^0, \tag{25}$$ along the characteristics of the PDE (24). The LC Hamilton's eqs. (23) become $$P^{\perp}~\approx~T_0(\partial_+ - \partial_-)X^{\perp},\qquad (\partial_+ - \partial_-)P^{\perp}~\approx~T_0(\partial_+ + \partial_-)^2X^{\perp}.\tag{26}$$ The eom (24) factorizes $$\partial_+\partial_-X^{\perp}~\approx~0, \tag{27}$$ with full solution being a sum of a left- and a right-mover $$X^{\perp}~\approx~ X^{\perp}_L(\sigma^+)+X^{\perp}_R(\sigma^-). \tag{28}$$ Periodicity conditions impose further conditions on the left- and right-movers, cf. Refs. 1-5.
16. On-shell, the LC Hamiltonian (16) becomes $$H^{LC}~\approx~T_0\int_0^{\ell}\! d\sigma\left[\lambda^+(\partial_+X_L^{\perp})^2 + \lambda^-(\partial_-X_R^{\perp})^2 \right]$$ $$~=~T_0\int_0^{\ell}\! d\sigma\left[ \lambda^0 \left\{(\partial_+X_L^{\perp})^2-(\partial_-X_R^{\perp})^2\right\} + \lambda^1 \left\{(\partial_+X_L^{\perp}\}^2+(\partial_-X_R^{\perp})^2\right\}\right],\tag{29}$$ where $$\lambda^1$$ is given by eq. (18). Notice that the implicit dependence of $$\lambda^0$$ in eq. (29) always appears in the combination $$\sigma + \lambda^0\tau$$ [due to the new WS coordinates $$\sigma^{\pm}$$, cf. eq. (25)]. Since the string is $$\ell$$-periodic, we can shift the $$\sigma$$-integration in LC Hamiltonian (29) to get rid of the implicit $$\lambda^0$$-dependence. Thus the $$\lambda^0$$ in front of the LMC (22) is the only actual $$\lambda^0$$-dependence, as it should be. Integration over $$\lambda^0$$ enforces the LMC (22).
17. Finally, let's return to OP's question. Classically, the orthogonally condition $$\dot{X}\cdot X^{\prime}~\approx~0,\tag{30}$$ that OP asks about, is equivalent to picking the zero-mode $$\lambda^0=0$$ to be zero, cf. eqs. (2a) and (4a). This is what happens in the open string. In the closed string, we are supposed to integrate over $$\lambda^0$$. However, we can get away with working in a $$\lambda^0=0$$ "gauge" if we additionally impose the level-matching constraint (22) by hand. This latter approach is often taken in string theory textbooks.
References:
1. B. Zwiebach, A first course in String Theory, 2nd edition, 2009.
2. J. Polchinski, String Theory, Vol. 1, 1998.
3. R. Blumenhagen, D. Lust and S. Theisen, Basic Concepts of String Theory, 2012.
4. K. Sundermeyer, Constrained Dynamics, Lecture Notes in Physics 169, 1982.
5. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994.
--
$$^1$$ Here is a proof that our starting point in this answer, the Hamiltonian Lagrangian (3), describes the NG string, at least classically. If we integrate out the $$P^{\mu}$$ momenta in the Hamiltonian Lagrangian (3), we get the Lagrangian density$$^2$$
$${\cal L}~=~T_0\frac{\left(\dot{X}-\lambda^0 X^{\prime}\right)^2}{2\lambda^1} -\frac{T_0\lambda^1}{2} (X^{\prime})^2.\tag{i}$$
Integrating out next the auxiliary variables $$\lambda^0$$ leads to $$\left. {\cal L}\right|_{\lambda_0} ~=~-\frac{T_0{\cal L}_{(1)}}{2(X^{\prime})^2\lambda^1} -\frac{T_0\lambda^1}{2}(X^{\prime})^2,\tag{ii}$$ where $${\cal L}_{(1)}~:=~-\det\left(\partial_{\alpha} X\cdot \partial_{\beta} X\right)_{\alpha\beta} ~=~(\dot{X}\cdot X^{\prime})^2-\dot{X}^2(X^{\prime})^2~\geq~ 0.\tag{iii}$$
Finally integrating out
$$\lambda^1~>~0 \tag{iv}$$
then yields the standard NG Lagrangian density
$${\cal L}_{NG}~:=~-T_0\sqrt{{\cal L}_{(1)}}. \tag{v}$$
[We have assumed that the auxiliary variable $$\lambda^1>0$$ is positive (iv) to get rid of a negative square root branch. It is imperative that the negative square root branch is not present. After Wick rotation, it would lead to an unstable exponentially growing (rather than an exponentially suppressed) Boltzmann factor. ] Conversely, if we Legendre transform the Lagrangian density (i), we get back the Hamiltonian density (1), cf. my Phys.SE here. Moreover, we should mention the well-known fact that if we integrate out the full WS metric $$h_{\alpha\beta}$$ in the Polyakov Lagrangian density
$${\cal L}_P~=~-\frac{T_0}{2} \sqrt{-h} h^{\alpha\beta} \partial_{\alpha}X \cdot\partial_{\beta}X ~=~\frac{T_0}{2} \left\{\frac{\left(h_{\sigma\sigma}\dot{X}- h_{\tau\sigma}X^{\prime}\right)^2}{\sqrt{-h}h_{\sigma\sigma}} - \frac{ \sqrt{-h}}{h_{\sigma\sigma}}(X^{\prime})^2 \right\}, \tag{vi}$$
then we get the standard NG Lagrangian density (v), cf. this Phys.SE post. [We choose the branch for the square root $$\sqrt{-h}$$ that has the same sign as $$h_{\sigma\sigma}$$ in order to make the kinetic term $$\dot{X}^2$$ positive definite.] Conversely, the Polyakov Lagrangian density (vi) can be derived from the Polyakov (P) De Donder-Weyl (DDW) Lagrangian density
\begin{align}{\cal L}_{P,DDW}~=~&P^{\alpha} \cdot \partial_{\alpha}X +\frac{h_{\alpha\beta}P^{\alpha}\cdot P^{\beta}}{2T_0\sqrt{-h}} \cr ~=~& P^{\tau}\cdot \dot{X} +P^{\sigma}\cdot X^{\prime}+ \frac{(P^{\sigma} + \lambda^0 P^{\tau})^2}{2T_0 \lambda^1} - \frac{\lambda^1}{2T_0} (P^{\tau})^2\end{align} \tag{vii}
by integrating out the polymomenta $$P^{\alpha}=(P^{\tau};P^{\sigma})$$. See also e.g. my Phys.SE answer here. In the second equality of eq. (vii), we have identified
$$\lambda^0~=~\frac{h_{\tau\sigma}}{h_{\sigma\sigma}} ~=~-\frac{h^{\tau\sigma}}{h^{\tau\tau}},$$ $$\lambda^1~=~\frac{\sqrt{-h}}{h_{\sigma\sigma}} ~=~ \frac{-1}{\sqrt{-h}h^{\tau\tau}}~\geq~0 \quad\Leftrightarrow\quad h~:=~\det\left(h_{\alpha\beta}\right)_{\alpha\beta}~=~-\left(\lambda^1 h_{\sigma\sigma}\right)^2~\leq~0 . \tag{viii}$$
Similarly, in the Lagrangian picture, the Polyakov Lagrangian density (vi) is equal to the Lagrangian density (i) under the identification (viii). The point is that only 2 out of the 3 degrees of freedom in the WS metric $$h_{\alpha\beta}$$ enter the Polyakov action due to Weyl symmetry at the classical level. Therefore the WS metric $$h_{\alpha\beta}$$ can be replaced with only 2 variables $$\lambda^0$$ and $$\lambda^1$$. The correspondence (vi) $$\leftrightarrow$$ (i) establishes a more refined equivalence between the Polyakov and the Nambu-Goto Lagrangian formulations than just integrating out the full WS metric $$h_{\alpha\beta}$$ by brute force.
Finally, if we only integrate out
$$P^{\sigma}~\approx~-T_0\lambda^1X^{\prime}-\lambda^0P^{\tau}\tag{ix}$$
in the Polyakov De Donder-Weyl Lagrangian density (vii) but keep the $$P^{\tau}\equiv P$$ variable, then the Polyakov De Donder-Weyl Lagrangian density (vii) becomes the Hamiltonian Lagrangian density (3), i.e. our starting point in this answer. This shows that the Nambu-Goto and the Polyakov Hamiltonian formulations are equivalent, cf. this Phys.SE post.
$$^2$$ The Gaussian integration over the auxiliary variable $$\lambda^0\equiv \lambda^0_M$$ looks naively unstable in Minkowski signature. One should Wick rotate $$\tau_E=i\tau_M$$ to Euclidean signature to get a Lagrangian density $$-{\cal L}_M={\cal L}_E>0$$ bounded from below with $$-i\lambda^0_M=\lambda^0_E\in\mathbb{R}$$. In other words, the product $$\lambda^0_M\tau_M=\lambda^0_E\tau_E$$ should remain invariant under Wick rotation. | {"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": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 137, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9793290495872498, "perplexity": 409.0228529854503}, "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/1614178381989.92/warc/CC-MAIN-20210308052217-20210308082217-00401.warc.gz"} |
https://math.stackexchange.com/questions/2717434/pseudonorm-well-definedness/2717630 | # Pseudonorm Well Definedness
The question is as follows (Its not very long - most of it is just definitions!):
A nonnegative real-valued function $\| \cdot \|$ defined on a vector space $X$ is called a pseudonorm if $\|x + y\| \le \|x\| + \|y\|$ and $\|\alpha x\| = |\alpha|\; \|x\|$. Define $x \cong y$, provided $\|x - y\| = 0$. Define $X/\cong$ to be the set of equivalence classes of X under $\cong$ and for $x \in X$ define $[x]$ to be the equivalence class of $x$. Show that $X/\cong$ is a normed vector space if we define $\alpha[x] +\beta[y]$ to be the equivalence class of $\alpha x + \beta y$ and define $\|[x]\|=\|x\|$.
I think there's a bunch of stuff to do here:
• Show linearity of classes is well defined
• Show classes, so defined, form a vector space
• Show the norm is well defined
• Show the norm, so defined, satisfies the usual properties
My questions are:
1. Are these the things to do in the problem?
2. If so, I can do the last 3, but I'm getting mixed up on the first -- arguing that linearity of classes is well-defined. Usually I start with 2 elements that are in the same class and show the operation defined on one is equal to operation defined on the other. But here we have a binary operation - so I'm confused about how this argument should proceed. It occurred to me that can show $\alpha[x]+\beta[y] = [\alpha x + \beta y]$ as sets but I'm not sure this is sufficient.
• Use $\le$ in the definition. fixed. – GEdgar Apr 1 '18 at 14:15
• Try this to start ... $[0]$ is a vector subspace, and every $[x]$ is a coset of $[0]$. – GEdgar Apr 1 '18 at 14:18
• @GEdgar can you elaborate? Part of my problem is I'm having trouble with the logic of the argument and what I need to show. I can see that your hint will show equality as sets -- but is that enough? – yoshi Apr 1 '18 at 14:47
Let $x,y,x',y' \in X$ such that $[x] = [x']$ and $[y] = [y']$. To verify that addition and scalar multiplication is well defined, we wish to show that $[\alpha x + \beta y] = [\alpha x' + \beta y']$ for any scalars $\alpha, \beta$.
We have
$$\|\alpha x + \beta y - (\alpha x' + \beta y')\| \le |\alpha|\|x-x'\| + |\beta|\|y-y'\| = 0$$
because $\|x-x'\| = \|y-y'\| = 0$ by assumption. Therefore $[\alpha x + \beta y] = [\alpha x' + \beta y']$ so we can define
$$[x] + [y] = [x+y]$$ $$\alpha[x] = [\alpha x]$$
It is similar for the norm. Assume that $[x] = [x']$ and we wish to show that $\|x\| = \|x'\|$.
We have
$$\|x\| \le \|x-x'\| + \|x'\| = \|x'\|$$ $$\|x'\| \le \|x'-x\| + \|x\| = \|x\|$$
so $\|x\| = \|x'\|$. Therefore, we can define $\|[x]\| = \|x\|$.
Now it should be straightforward to check that the operations and the norm satisfy the required axioms. | {"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": 0.9729969501495361, "perplexity": 162.03021121539908}, "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-2020-05/segments/1579251728207.68/warc/CC-MAIN-20200127205148-20200127235148-00287.warc.gz"} |
http://mathoverflow.net/questions/49272/pull-back-connection | pull-back connection
I have a question related to the definition of the pull-back connection, more specifically about its uniqueness or the canonical way to induce it.
The definition that one finds in general goes along the following lines: let $F:P\rightarrow B$ be map between differentiable manifolds, let $\pi:E\rightarrow B$ be a vector bundle and $\nabla$ a connection on $E$. Then the connection $F^*\nabla$ is uniquely determined by the following relation
$(F^*\nabla)_X(F^* s):=F^*(\nabla_{df(X)} s)$.
This should uniquely determine the connection right?
Let us start with the most trivial case, when $B$ is a point. Then $E$ is a vector space and the pull-back of $E$ is a trivial vector space over $P$. A connection on $E\rightarrow pt$ is an endomorphism of $E$ (which trivially satisfies the Leibniz relation). Could anybody explain how does that canonically induce a connection on $P\times E$, presumably the trivial connection $d$ if one starts with the zero endomorphism of $E$? Leibniz relation does not suffice. One could say let us make a convention here. But the more general question is how does one define $(F^*\nabla)_X$ when $X\in\ker{dF}$, in general?
-
What is $F^*s$? – Deane Yang Dec 13 '10 at 16:06
I don't know how to define $F^*\nabla$ functorially (i.e., without using local co-ordinates and/or trivializations). I suggest first working this out using local co-ordinates and trivializations. – Deane Yang Dec 13 '10 at 16:17
I take it back. Your equation is right. But I still recommend playing around with it using local co-ordinates and trivialization. It's a rather confusing equation (at least for me). – Deane Yang Dec 13 '10 at 16:26
From the right-hand side of your equation, if $dF(X) = 0$ then $(F^*\nabla)_X = 0$ as well (note it should be $dF$, not $df$). – Matt Noonan Dec 13 '10 at 16:44
Over a pt, connections vanish (since vector fields & 1-forms are 0). In general "$dF(X)$" makes no sense as a vector field on $B$. View connections as additive maps from sections of $E$ to sections of $E \otimes \Omega^1_B$ over varying opens in $B$. Local sections of $F^{\ast}(E)$ are function-linear combinations of $F$-pullbacks of local sections of $E$, so the pullback rule (using pullback of 1-forms and of local sections) and Leibniz yield uniqueness. Construction with local coords gives local existence (& yields d when $B$ is pt and $E$ trivial), so by uniqueness get global existence. – BCnrd Dec 13 '10 at 17:15
Hi, I have seen the equation you gave as a definition many times. For example, I think it is also used in a corresponding Wikipedia article. Nevertheless, as you correctly pointed out, it does not give a reasonable/unique description. A better formulation/definition of the pullback connection can be found for example in Milnor's and Stasheff's book 'Characteristic classes' on p. 292, Lemma 3 and its proof (definition by universal property/commutative diagram; proof: computation in local coordinates; it's the precise version of what the equation you gave tries to capture). I hope this helps more or less.
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The commutative diagram in Milnor and Stasheff is exactly the same equation, only written in a different language. As others pointed out, it does define the pull-back connection uniquely but you can not use it directly for arbitrary sections of the pull-back bundle. – Sergei Ivanov Dec 14 '10 at 9:03
Here's my summary of the situation:
1) First, observe that the space of local sections of the pullback bundle is generated by the space of sections of the original bundle composed with the map $F$. (This is better stated using sheaf language)
2) So, using the Leibniz rule, it suffices to define the pullback connection on a section obtained by composing a section of the original bundle with $F$.
3) The formula given above accomplishes this. It is worth noting that in this formula you should view $X$ as a single vector and not as a vector field.
-
As an aside, I stumbled onto this, because the use of a pullback connection is implicit when analyzing variations of geodesics and Jacobi fields on a Riemannian manifold. I've never seen this discussed explicitly in any textbook or paper, but it's actually quite necessary to make all the arguments rigorous. – Deane Yang Dec 13 '10 at 19:06
Dear Deane: Aha, I didn't think of the interpretation with $X$ as a single vector (since the notation $\nabla_X$ is usually used with $X$ a vector field). OK, that then makes sense, but working that pointwise then requires a separate argument to verify the smoothness aspect (i.e., recognizing how to bypass the single-vector formulation after all). If we use the 1-form formulation then we work with local smooth sections throughout, so smoothness "comes along for the ride", and the single-vector interpretation can be inferred afterwards. Well, six of one, half dozen of the other. :) – BCnrd Dec 13 '10 at 19:41
Dear BCnrd, could you clarify what smoothness aspect you are referring to? – Deane Yang Dec 13 '10 at 20:16
Dear Deane: I just mean that if we make a pointwise construction then we may need to do more work (an explicit calculations over a suitable small open domain, or something else) to verify that its output is smooth in local coordinates, whereas if we make construction in terms of local sections of bundles (or in other words, from the viewpoint of sheaves) then the smoothness of the output of the connection operator drops out automatically from the framework. – BCnrd Dec 13 '10 at 23:50
Dear BCnrd, thanks for the clarification. But I don't think there's any additional work required to verify smoothness. Smoothness of the pull-back connection follows directly from the smoothness of the original connection, smoothness of $F$, linear pointwise dependence on the tangent vector, and the Leibniz formula. – Deane Yang Dec 14 '10 at 0:01
See 19.12.6 (page 246) in this book.
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To Matt: The relation $(F^*\nabla)_X=0$ does not satisfy Leibniz relation. Meanwhile I found out the answer to my question (a friend clarified it for me).
It turns out that there is an isomorphism $\Gamma(P;\pi^*E)\simeq \Gamma(P;\mathbb{C})\otimes \Gamma(B;E)$ where the tensor product is over $\Gamma(B;\mathbb{C})$. It is obviously true when $E$ is a vector space. Now use Leibniz relation $\nabla_X(f\otimes s)=X(f)\otimes s+f\otimes\nabla_Xs$ to extend the connection from $\Gamma(B;E)$ to $\Gamma (P;\pi^*E)$. In the trivial case one gets indeed that $\nabla_X(f\otimes s)=X(f)\otimes s$.
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@Unknown: see my comment above. The equation you wrote down is only meaningful for pull-backs of sections $s$. If you multiply $F^*s$ by an arbitrary function $h$, it will no longer be the pull back of a section over $B$ if $h$ is not constant along $F^{-1}B$. In other words, you were trying to force Leibniz rule somewhere it has no business being. The given expression is, indeed, enough to specify the pull-back connection, but it is not so simple as plug-and-play. – Willie Wong Dec 13 '10 at 17:15
Willie i didn't say it was easy:) In the background lurks the isomorphism $\Gamma(P;\pi^*E)\simeq \Gamma(P;\mathbb{C})\otimes \Gamma(B;E)$ which is by not obvious(at least not to me). But Leibniz relation does make sense and it does define the connection everywhere and it also explains the trivial connection on the trivial bundle. – user11538 Dec 13 '10 at 17:29
Dear unknown: the isom. you're using is false in complex-analytic and algebraic cases (unsure in $C^{\infty$-case, but seems silly to use a defn that only works there when there's a simple procedure applicable in all cases). Think more locally (it's good for you!): use local existence and global uniqueness to get global existence. What you propose with "global" tensors works locally with no hard work; then use uniqueness to globalize. Your comment to Matt is unclear since your original statement of the pullback-relation with global vector fields makes no sense ($dF(X)$ makes no sense on $B$). – BCnrd Dec 13 '10 at 18:26
Dear BCnrd, Thank you for your interesting comments but... 1) I was only interested in the differentiable case. I am not sure what you mean by a connection in the analytic or algebraic cases. (an analytic(algebraic) splitting of the tangent bundle?) so I am really satisfied with the isomorphism of smooth sections i was talking about. By the way, it's not mine. 2)I admit the comment to Matt lacked details, probably thinking that it was clear $X$ should be thought as a tangent vector and not a vector field (as I wrote); (retrospectively that was really not the main issue of the question); – user11538 Dec 13 '10 at 20:30
Dear unknown: later in life you'll want to use complex manifolds or complex algebraic varieties. (Connections remain all about making vector fields act as "directional deriv." operators on sections of the bundle, as in diff'ble case.) Then no bump functions, so cannot work entirely so "globally" as above. That's why I outlined an alternative to exploit your preferred calculations on a local level, coupled with global uniqueness (which can be proved by local calculations!) to infer global existence. It really is easier that way (e.g., no non-obvious isoms needed). You'll appreciate it later. – BCnrd Dec 14 '10 at 0:01
Just to put what others have said into an explicit formula, note that any section of $F^*E$ is of the form
$\sum_j \varphi_j F^*s_j$,
for certain $\varphi_1, \ldots, \varphi_n \in C^{\infty}(P)$, and $s_1, \ldots, s_n \in \Gamma^{\infty}(E)$. Then in the notation of the question (with $X \in T_pP$ a single tangent vector), at $p \in P$ we have by the Leibniz rule and the defining relation of $F^*\nabla$,
$(F^*\nabla)_X \left(\sum_j \varphi_j F^*s_j\right) = \sum_j \left( d\varphi_j(X) F^*s_j + \varphi_j (F^*\nabla)_X F^*s_j \right) = \sum_j \left( d\varphi_j(X) F^*s_j + \varphi_j F^*(\nabla_{dF(X)}s_j) \right)$
(with everything evaluated at $p$ where appropriate). This gives back $F^*\nabla = d$ in your example where $B$ is a point and you take the zero endomorphism of $E$.
- | {"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": 0.9542781114578247, "perplexity": 424.6771476909002}, "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/1438042982013.25/warc/CC-MAIN-20150728002302-00069-ip-10-236-191-2.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/139942/reference-for-studying-the-method-of-stationary-phase/141332 | Reference for studying the method of stationary phase
I would like to learn about the stationary phase method, as part of the theory of Fourier Integral Operators.
From what I can see so far, Hormander's book "The Analysis of Linear Partial Differential Operators I" seems to be the standard reference, but I was wondering whether somebody has some positive experience with alternative textbooks that I could use to supplement Hormander's exposition, in case it gets too dense.
Many thanks for your suggestions !
- | {"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": 0.8825797438621521, "perplexity": 253.49565959246635}, "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-2014-10/segments/1393999679238/warc/CC-MAIN-20140305060759-00008-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mpmath.org/doc/current/functions/index.html | # Mathematical functions¶
Mpmath implements the standard functions from Python’s math and cmath modules, for both real and complex numbers and with arbitrary precision. Many other functions are also available in mpmath, including commonly-used variants of standard functions (such as the alternative trigonometric functions sec, csc, cot), but also a large number of “special functions” such as the gamma function, the Riemann zeta function, error functions, Bessel functions, etc. | {"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": 0.9610435366630554, "perplexity": 668.6694302117546}, "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/1490218189686.31/warc/CC-MAIN-20170322212949-00227-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://davidthemathstutor.com.au/2020/01/22/rate-of-change/ | # Rate of Change
We are all familiar with many physical rates of change. The rate of change of distance is called velocity. If distance is being measured in meters and time is being measured in seconds, the rate of change of distance (velocity) is measured in units meters per second (m/sec). Water filling a bucket can be measured in liters. If time is measured in minutes, then the rate of change of the amount of water in the bucket has units liters per minute (ltr/min). Or we cold measure the height of the water in the bucket in centimeters. The rate of change would be in centimeters per minute (cm/min).
Graphically, the rate of change of a function indicates how fast it increases or decreases as you move along the x-axis. From your study of linear equations, the standard form of an equation of a line is y = mx + c, where m is the gradient or slope of the line. For example, the line y = 2x + 5 has a gradient of 2 which means that it rises 2 units for every unit you move to the right along the x-axis. If this was the equation of the distance of a particle moving from some reference point, where the distance (y-axis) was measured in centimeters and the time (x-axis) was measured in seconds, the velocity of the particle would be 2 cm/sec which is the same as the gradient of the graph. However, the gradient (velocity) is constant since the gradient is 2 anywhere on the graph. All linear graphs have a constant gradient (rate of change). What about non-linear graphs?
Look at the graph of a non-linear function below:
You would say that the function is increasing (positive rate of change) up to about x = -0.6, decreases (negative rate of change) between about -0.6 and 0.6, and increases after 0.6. The rate of change is different depending on where you are on the graph. For many physical problems that have been modelled with an equation, we want to know what the rate of change is at different values of x. A very common problem to solve is to find where the rate of change is zero. The solution to this would find the maximum and/or minimum points because these are the points where the rate of change goes from positive to negative or vice versa.
What does this have to do with calculus? The mathematical term for the rate of change of a function is the derivative of the function and finding the derivative of a function will be the first thing I will define in my next post. Finding the derivative of a function is an operation in calculus, and this is usually the first topic developed in a calculus subject. | {"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": 0.9518662095069885, "perplexity": 134.52492762544463}, "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-2020-45/segments/1603107874340.10/warc/CC-MAIN-20201020221156-20201021011156-00346.warc.gz"} |
https://gaurish4math.wordpress.com/tag/ussr-olympiad-book/ | # Arithmetic & Geometry
Standard
After a month of the unexpected break from mathematics, I will resume the regular weekly blog posts. It’s a kind of relaunch of this blog, and I will begin with the discussion of an arithmetic problem with a geometric solution. This is problem 103 from The USSR Olympiad Problem Book:
Prove that
$\displaystyle{t_1+t_2+\ldots + t_n = \left\lfloor\frac{n}{1}\right\rfloor + \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor + \ldots + \left\lfloor\frac{n}{n}\right\rfloor}$
where $t_k$ is the number of divisors of the natural number $k$.
One can solve this problem by using the principle of mathematical induction or using the fact that the number of integers of the sequence $1,2,3,\ldots , n$ which are divisible by any chosen integer $k$ is equal to $\left\lfloor \frac{n}{k}\right\rfloor$. The second approach of counting suggests an elegant geometric solution.
Consider the equilateral hyperbola $\displaystyle{y = \frac{k}{x}}$ , (of which we shall take only the branches in the first quadrant):
We note all the points in the first quadrant which have integer coordinates as the intersection point of the dotted lines. Now, if an integer $x$ is a divisor of the integer $k$, then the point $(x,y)$ is a point on the graph of the hyperbola $xy=k$. Conversely, if the hyperbola $xy=k$ contains an integer point, then the x-coordinate is a divisor of $k$. Hence the number of integers $t_k$ is equal to the number of the integer points lying on the hyperbola $xy=k$. The number $t_k$ is thus equal to the number of absciassas of integer points lying on the hyperbola $xy=k$. Now, we make use of the fact that the hyperbola $xy=n$ lies “farther out” in the quadrant than do $xy=1, xy=2, xy=3, \ldots, xy=n-1$. The following implication hold:
The sum $t_1+t_2\ldots + t_n$ is equal to the number of integer points lying under or on the hyperbola $xy=n$. Each such point will lie on a hyperbola $xy=k$, where $k\leq n$. The number of integer points with abscissa $k$ located under the hyperbola is equal to the integer part of the length of the segment $\overline{AB}$ [in figure above $k=3$]. That is $\left\lfloor\frac{n}{k}\right\rfloor$, since $\displaystyle{|\overline{AB}|=\frac{n}{k}}$, i.e. ordinate of point $A$ on hyperbola $xy=n$ for abscissa $k$. Thus, we obtain
$\displaystyle{t_1+t_2+\ldots + t_n = \left\lfloor\frac{n}{1}\right\rfloor + \left\lfloor\frac{n}{2}\right\rfloor + \left\lfloor\frac{n}{3}\right\rfloor + \ldots + \left\lfloor\frac{n}{n}\right\rfloor}$
Caution: Excess of anything is harmful, even mathematics.
$\displaystyle{\sum_{k=2}^{n} \frac{1}{k}}$ can never be an integer for any value of $n$.
Firstly, observe that among the natural numbers from 2 to $n$ there is exactly one natural number which has the highest power of 2 as its divisor. Now, while summing up the reciprocals of these natural numbers we will get a fraction as the answer. In that fraction, the denominator will be an even number since it’s the least common multiple of all numbers from 2 to $n$. And the numerator will be an odd number since it’s the sum of $(n-2)$ even numbers with one odd number (corresponding to the reciprocal of the number with the highest power of 2 as the factor). Since under no circumstances an even number can completely divide an odd number, denominator can’t be a factor of the numerator. Hence the fraction can’t be reduced to an integer and the sum can never be an integer. | {"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": 37, "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": 0.9369010925292969, "perplexity": 165.25473294139152}, "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/1531676594790.48/warc/CC-MAIN-20180723012644-20180723032644-00149.warc.gz"} |
https://kerodon.net/tag/00AL | # Kerodon
$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
Construction 2.2.8.9 (The Pith of a $2$-Category). Let $\operatorname{\mathcal{C}}$ be a $2$-category. We define a new $2$-category $\operatorname{Pith}(\operatorname{\mathcal{C}})$ as follows:
• The objects of $\operatorname{Pith}(\operatorname{\mathcal{C}})$ are the objects of $\operatorname{\mathcal{C}}$.
• For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the category $\underline{\operatorname{Hom}}_{\operatorname{Pith}(\operatorname{\mathcal{C}})}( X, Y)$ is the core $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)^{\simeq }$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ (see Construction 1.2.4.4).
• The composition law, associativity constraints, and unit constraints of $\operatorname{Pith}(\operatorname{\mathcal{C}})$ are given by restricting the composition law, associativity constraints, and unit constraints of $\operatorname{\mathcal{C}}$.
Then $\operatorname{Pith}(\operatorname{\mathcal{C}})$ is a $(2,1)$-category which we will refer to as the pith of $\operatorname{\mathcal{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": 0.9945703148841858, "perplexity": 350.4734335758098}, "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-2022-21/segments/1652662558030.43/warc/CC-MAIN-20220523132100-20220523162100-00324.warc.gz"} |
https://link.springer.com/article/10.1007/s10953-014-0257-1 | # Density, Viscosity and Surface Tension of Binary Mixtures of 1-Butyl-1-Methylpyrrolidinium Tricyanomethanide with Benzothiophene
## Abstract
The effects of temperature and composition on the density and viscosity of pure benzothiophene and ionic liquid (IL), and those of the binary mixtures containing the IL 1-butyl-1-methylpyrrolidynium tricyanomethanide ([BMPYR][TCM] + benzothiophene), are reported at six temperatures (308.15, 318.15, 328.15, 338.15, 348.15 and 358.15) K and ambient pressure. The temperature dependences of the density and viscosity were represented by an empirical second-order polynomial and by the Vogel–Fucher–Tammann equation, respectively. The density and viscosity variations with compositions were described by polynomials. Excess molar volumes and viscosity deviations were calculated and correlated by Redlich–Kister polynomial expansions. The surface tensions of benzothiophene, pure IL and binary mixtures of ([BMPYR][TCM] + benzothiophene) were measured at atmospheric pressure at four temperatures (308.15, 318.15, 328.15 and 338.15) K. The surface tension deviations were calculated and correlated by a Redlich–Kister polynomial expansion. The temperature dependence of the interfacial tension was used to evaluate the surface entropy, the surface enthalpy, the critical temperature, the surface energy and the parachor for pure IL. These measurements have been provided to complete information of the influence of temperature and composition on physicochemical properties for the selected IL, which was chosen as a possible new entrainer in the separation of sulfur compounds from fuels. A qualitative analysis on these quantities in terms of molecular interactions is reported. The obtained results indicate that IL interactions with benzothiophene are strongly dependent on packing effects and hydrogen bonding of this IL with the polar solvent.
## Introduction
New international regulations require the removal of low level sulfur compounds such as thiophene, benzothiophene, methyldibenzothiophenes, 4,6-dibenzothiophenethiols, thioethers, and disulfides from fuels. From an industrial point of view these are new challenges to decrease the sulfur content in diesel fuel in the USA and Europe [1, 2]. The total sulfur content in European gasoline and diesel fuels must be at a maximum concentration of 10 ppm [2]. Thus, the emission of sulfur from petrol and diesel oils, which is linked to acid rain phenomena, plays a crucial role in pollution problems of large conglomerates. The hydrodesulfurization (HDS) process, the established method used in industrial technologies to remove organic sulfur compounds from fuels, cannot achieve these low sulfur targets and uses higher temperature, higher pressure, larger reactor volumes and more active catalysts [3]. Therefore, the easy liquid–liquid equilibrium (LLE) extraction process is proposed for deep desulfurization with ionic liquids (ILs) [47]. ILs are not volatile, are nonflammable and show excellent solvation capacity mainly via hydrogen bonding. Great effort has been made to design and synthesize novel ILs to match potential applications such as media for extraction processes [414]. ILs that consist of a short alkane chain with polar groups such as oxygen, or nitryle, or hydroxyl substituent in cation and cyano-subgroups in the anion, are expected to be good entrainers in many separation processes. The application of ILs in the desulfurization process has already been reported in the literature [714]. It is therefore a challenge to design ILs that incorporate progressively larger extraction selectivity, while maintaining viscosity, density and surface tension convenient for a new technology.
Several recent attempts have focused on the design and synthesis of ILs with high selectivity for the separation of sulfur compounds from alkanes. The 1-alkylpyrrolidinium-based ILs with different anions [14] have been recently studied in our laboratory in ternary LLE (IL + thiophene + heptane) mixtures at T = 298.15 K. The highest selectivity (S max = 133.4) with high solute distribution ratio (β = 3.47) was found for 1-butyl-1-methylpyrrolidinium tricyanomethanide [BMPYR][TCM] [14]. However, larger extraction parameters are presented by 1-ethyl-3-methylimidazolium tricyanomethanide, [EMIM][TCM] [13]. Promising results in ternary LLE measurements were obtained also with 1-ethyl-3-methylimidazolium bis{(trifluoromethyl)sulfonyl}imide, [EMIM][NTf2] [11], and 1-butyl-1-methylpyrrolidinium tetracyanoborate, [BMPYR][TCB] [14].
The current work represents a continuation of our systematic study on desulfurization of fuels. We have just reported experimental ternary LLE data for three ILs, which we expected to show high selectivity for the extraction of thiophene: 1-butyl-1-methylpyrrolidinium trifluoromethanesulfonate, [BMPYR][CF3SO3], 1-butyl-1-methylpyrrolidinium tricyanomethanide, [BMPYR][TCM], and 1-hexyl-3-methylimidazolium tetracyanoborate, [HMIM][TCB] [14]. Therefore, we thought it quite possible that the incorporation of the pyrrolidinium cation and the tricyanomethanide anion would improve the extraction activity of benzothiophene. The ternary systems {IL (1) + thiophene or benzothiophene (2) + heptane (3)} were measured at T = 308.15 K and ambient pressure. The [BMPYR][TCM] was found to show high selectivity in the desulfurization process. Recently, very good results for the extraction of sulfur compounds from model mixtures of real fuels were also obtained with tricyanomethanide-based, [TCM], ILs [7]. The extraction of thiophene and dibenzothiophene (about 95 wt%) was reported for their simultaneous separation from alkanes with pyridine-based and imidazolium-based ILs [7].
Liquid–liquid extraction, such as extraction of benzothiophene from petrol and diesel oils, is greatly affected by viscosity and liquid surface tension. Knowledge about the physicochemical properties such as density, viscosity, or surface tension and thermodynamic surface properties is necessary in order to design any process involving ILs on an industrial scale [15]. The excess functions calculated from viscosity and density of binary systems play a very important role in the understanding of molecular interactions that exist in the bulk of liquids and on the surface. Recently, we published new data for N-octylisochinolinium bis{(trifluoromethyl)sulfony}imide, [OiQuin][NTf2], for possible extraction of 2-phenylethanol from the aqueous phase [16], and for 1-alkyl-cyanopyridinium-based ILs [17] as well as [EMIM][TCM] [18], for possible extraction of sulfur compounds from fuels. As in all hydrogen-bonded liquids, the structural organization of constituents makes ILs behave as very viscous fluids. The high viscosity of ILs is widely known and usually destroys the mass transport of extractants in new IL-entrainers and limits their generalized use for a variety of applications. Studies of physicochemical properties, besides helping in deciding limits of increase in temperature for the desired applications, are expected to reflect the molecular interactions in binary systems.
Liquid surface tension as an equilibrium thermodynamic property is important for engineering aspects related to use of the IL as an entrainer. Surface tension of a liquid is related to the intermolecular interaction potential energy and the liquid interfacial microstructure. Knowledge of the impact of temperature on the surface tensions of fluids is essential for most industrial applications. In recent years, measurements of the experimental surface tension data has been focused mainly on imidazolium-based ILs, which are air and moisture stable, and their binary solutions with alcohols and water [19, 20]. The surface tensions of ILs are usually lower than that of water (71.98 mN·m−1 at T = 293.15 K, 0.1 MPa, [21]); for example, for 1-ethyl-3-methylimidazolium tricyanomethanide it is 50.94 mN·m−1 at T = 298.15 K [18].
The current work represents a continuation of our systematic study on desulfurization of fuels and physicochemical properties of ILs. We report an experimental investigation of the density, viscosity and surface tension for the pure IL, [BMPYR][TCM], and benzothiophene, as well as of binary mixtures containing ([BMPYR][TCM] + benzothiophene) as a function of temperature and composition at ambient pressure.
Using the quasi-linear variation of surface tension with temperature observed for the pure IL, the surface thermodynamic properties, such as surface entropy, surface enthalpy, surface energy, the critical temperature and parachor were determined. The data obtained were analyzed to determine the effect of temperature on fundamental physicochemical and thermodynamic properties.
## Experimental Section
### Materials
The sample of 1-butyl-1-methylpyrrolidinium tricyanomethanide, [BMPYR][TCM], was from Iolitec (≥0.98 mass fraction), M w = 232.32 g·mol−1, CAS No. 878027-72-6. The sample was dried for several days at 300 K under reduced pressure to remove volatile impurities and trace water and was then stored in a desiccator under an inert atmosphere. Benzothiophene, Sigma Aldrich Chemie GmbH (≥0.99 mass fraction), CAS No. 95-15-8, was stored over freshly activated molecular sieves (4Å, Union Carbide).
The structure of [BMPYR][TCM] is shown in Scheme 1. Physical properties: density, ρ, dynamic viscosity, η, and surface tension, σ, of pure IL and benzothiophene, together with the literature data, are listed in Table 1 [2224].
The water content was analyzed by Karl Fischer titration (method TitroLine KF). The sample of IL, or solvent, was dissolved in methanol and titrated in steps of 0.0025 cm3. The error in the water content is ±10 ppm by mass for the 3 cm3 of injected IL. The water content in solvents used was less than 350 ppm by mass.
### Density Measurements
The densities of the chemicals used and their mixtures were measured using an Anton Paar GmbH 4500 vibrating-tube densimeter (Graz, Austria) thermostatted over the (308.15–358.15) K temperature range. The temperature was controlled with two integrated Pt 100 platinum thermometers providing good precision of (±0.01 K). The densimeter includes an automatic correction for the viscosity of the sample. The apparatus is precise to within 1 × 10−5 g·cm−3, and the overall uncertainty of the measurements was estimated to be better than 5 × 10−5 g·cm−3. The calibration of the densimeter was performed at atmospheric pressure using doubly distilled and degassed water {CAS: 77-32-18-5; Anton Paar GmbH, liquid density standard, density, 0.99820 ± 0.00002 g·cm−3 (293.15 K); literature density 0.9982323 g·cm−3 (293.15 K, KNOVEL DIPPR); conductivity, κ = 8 μS}, specially-purified benzene (CAS: 71-43-2; standard CHE USC 11; CHEMIPAN, Poland, 0.9999 in mass fraction), and dried air. The data are similar to the literature data of different ILs [14, 1618].
### Viscosity Measurements
Viscosity measurements were carried out in an Anton Paar BmbH AMVn (Graz, Austria) programmable rheometer, with a nominal uncertainty of ±0.1 % and reproducibility <0.05 % for viscosities from 2.54 to 370 mPa·s. Temperature was controlled internally with a precision of ± 0.01 K in the range from 308.15 to 358.15 K. The diameter of the capillary was 1.8 mm for viscosities from 2.5 to 70 mPa·s. The diameter of the balls was 1.5 mm.
### Surface Tension Measurements
The surface tension measurements were made with a Tensiometer (KSV Sigma 701 System Finland) using a Du-Noüy ring taking into account the Zuidema Waters correction. Measurements were performed using the ring method that is widely used [25] since the studies of Harkins and Jordan [26] that improved the accuracy and established tables of correction factors based on the work of Freund and Freund [27]. The force acting on the balance was recorded with respect to time. The maximum value of the downward force was used to calculate the surface tension. All measurements were repeated three to five times. The equipment has both a control and a mechanic unit that are connected to a PC-controlled instrument for the precise measurement of a liquid with an uncertainty of ±0.04 mN·m−1. Temperature was maintained at the desired value within ±0.1 K.
## Results and Discussion
### Effect of Temperature and Composition on Density and Viscosity
The experimental data of density, ρ, and dynamic viscosity, η, as a function of mole fraction, x 1, of the {[BMPYR][TCM] (1) + benzothiophene (2)} system at different temperatures are listed in Table 2.
Fit parameters with R 2 = 1 for the empirical correlation (see Eqs. 1 and 2) of the density as a function of temperature (a 0, a 1and a 2) and concentration (b i ), for pure substances and for mixtures, are listed in Tables 1S and 2S in the supplementary material (SM), respectively:
$$\rho = a_{2} T^{2} + a_{1} T + a_{0}$$
(1)
$$\rho = b_{4} x_{1}^{4} + b_{3} x_{1}^{3} + b_{2} x_{1}^{2} + b_{1} x_{1} + b_{0}$$
(2)
The density of [BMPYR][TCM] is lower than that of benzothiophene, but the viscosity is almost ten times higher. The densities of the IL range in values from 1.00076 g·cm−3 at T = 308.15 K (ρ = 1.00066 g·cm−3, extrapolated value from [22]) to 0.97100 g·cm−3 at T = 358.15 K, and of benzothiophene from 1.15081 g·cm−3 at T = 308.15 K to 1.10592 g·cm−3 at T = 358.15 K. Immiscibility in the binary solutions of {[BMPYR][TCM] (1) + benzothiophene (2)} was observed in our ternary LLE measurements [13, 28]. The data presented in this work do not cover the compositions at the immiscibility gap (see Figs. 1, 2).
The viscosity decreases with increasing benzothiophene content. The dynamic viscosity of the pure IL and the mixtures as a function of temperature, through the whole composition range, was correlated by the well-known Vogel–Fulcher–Tammann, VFT equation [2931],
$$\eta = CT^{0.5} \exp \left(\frac{D}{{T - T_{0} }}\right)$$
(3)
The fit parameters, determined empirically, are in general C, D and T 0 when a linear relation is observed between logarithmic value of ηT 0.5 and (TT 0)−1. For the best correlation of the experimental curves, the value of T 0 = 118.01 K (T g,1 = 178.01 K [32] −60 K) was used in the calculations. A single value of the parameter T 0 was used for different concentrations. Figure 3 depicts the dynamic viscosity as a function of temperature. The temperature dependence of viscosity becomes distinctly nonlinear, especially at low benzothiophene content. The parameters C and D from Eq. 3 change smoothly with composition for the system, as shown in Table 3S in the supplementary material.
The composition dependence of viscosity was described by the following polynomial:
$$\eta = c_{3} x_{1}^{3} + c_{2} x_{1}^{2} + c_{1} x_{1}^{{}} + c_{0}$$
(4)
The parameters of the correlation are listed in Table 4S in the supplementary material and the calculated lines are shown in Fig. 4. The dynamic viscosity of the IL changes from 20.56 mPa·s at T = 308.15 K to 6.47 mPa·s at T = 358.15 K, and for benzothiophene from 2.94 mPa·s at T = 308.15 K to 1.14 mPa·s at T = 358.15 K. The values of viscosity presented in this work are higher than that reported for [EMIM][TCM] [18], which was suggested as a very good entrainer for the extraction of sulfur compounds from alkanes. ILs exhibit high viscosities that are usually higher than those for molecular organic solvents. Both density and viscosity decrease with an increase of temperature.
The values of excess molar volumes, V E m , of the mixtures formed from two polar compounds are the result of a number of effects which may contribute terms differing in sign. Disruption of H-bonds in the IL molecules makes a positive contribution, but specific interaction between two dissimilar molecules makes negative contributions to V E m . The free-volume effect, which depends on differences in the characteristic pressures and temperatures of the components (described by Flory formalism [33]), makes a negative contribution. Packing effects or conformational changes of the molecules in the mixtures are more difficult to categorize. However, interstitial accommodation and the effect of the condensation give further negative contributions.
Experimental excess molar volume V E m data of {[BMPYR][TCM] (1) + benzothiophene (2)} are listed in Table 2. The data were correlated by the well-known polynomial Redlich–Kister equation (Eq. 5):
$$V_{m}^{\text {E}} = x_{1} (x_{1} - 1)\sum\limits_{i = 0}^{i = 3} {A_{i} (1 - 2x_{1} )}^{i - 1}$$
(5)
$$\sigma_{V} = \left[ \left\{ \sum\limits_{i = 1}^{n} {(V_{m}^{{\text {E}({\text{exp}}.)}} - V_{m}^{{\text {E} ( {\text{calc}} . )}} )} /(n - k) \right\} \right]^{1/2}$$
(6)
where x 1 is the mole fraction of the IL and V E m is the molar excess volume. The values of the parameters (A i ) were determined using the least-squares method. The fit parameters are summarized in Table 5S in the supplementary material, along with the corresponding standard deviations, σ V , for the correlations (Eq. 6), where n is the number of experimental points and k is the number of coefficients. The values of V E m , as well as the Redlich–Kister fits, are plotted in Fig. 5 as a function of the mole fraction. The V E m values exhibit negative deviations from ideality over the entire composition range. The graph also shows the unsymmetrical variation of these excess molar volumes with composition. The minimum of V E m is close to −1.8067 cm3·mol−1, at mole fraction x 1 = 0.3233 (at T = 308.15 K) and is shifted to lower values of mole fraction of the IL. The values of V E m decrease as the temperature increases. The strength of interactions between the IL and benzothiophene is at its highest and most negative at the higher temperature. This has to be the result of a more efficient packing effect rather than due to interactions at higher temperature.
The values of the excess dynamic viscosity, Δη, are listed in Table 2. These values were correlated with the following Redlich–Kister equation:
$$\varDelta \eta = x_{1} (x_{1} - 1)\sum\limits_{i = 0}^{i = 3} {B_{i} (1 - 2x_{1} )}^{i - 1}$$
(7)
$$\sigma_{\Delta \eta } \, = \,\left[ { \left\{ \sum\limits_{{{{i}} = 1}}^{{n}} {(\Delta \eta^{{{ \exp } }} - \Delta\eta^{{{\text{calc}} }} )} /({{n}} - {{k}}) \right\} } \right]^{1/2}$$
(8)
The parameters are listed in Table 6S in the supplementary material. Figure 6 shows the positive values of the excess dynamic viscosity for this binary system with Δη max minimally shifted to a lower IL mole fraction.
### Effect of Temperature and Composition on the Surface Tension
The values of surface tension, σ, of [BMPYR][TCM] at different temperatures (308.15 K to 338.15 K) are listed in Table 3. Within the present study, the surface tension of [BMPYR][TCM] at T = 308.15 K is 48.04 mN·m−1. This value is much higher than those for other, mainly imidazolium, ILs [19], but is very similar to the tricyanamide-based IL [EMIM][TCM] measured by us (49.91 mN·m−1 at T = 298.15 K) [18]. The surface tension is much higher for the IL than for benzothiophene and decreases with increasing concentration of benzothiophene, implying that the benzothiophene molecules tend to adsorb at the air–solution interface due to it hydrophobicity. The surface tension decreases with an increase of temperature, which is typical for organic solvents.
The correlation of the surface tension as a function of temperature and composition was represented with the equations:
$$\sigma = d_{1} T + d_{0}$$
(9)
$$\sigma = e_{3} x_{1}^{3} + e_{2} x_{1}^{2} + e_{1} x_{1} + e_{0}$$
(10)
The obtained parameters are shown in Tables 7S and 8S in the supplementary material for temperature and composition dependences, respectively. The surface tension decreases with an increase of temperature and of benzothiophene content in the binary mixtures (see Figs. 7, 8).
The absence of a breakpoint in this mixture confirms the special interactions observed in the LLE in its ternary system [1114]. These properties cannot be deduced using phase equilibrium data only. A regularly increasing value of the solution surface tension indicates that the two compounds, the IL and benzothiophene, are present at the gas/liquid interface. The [BMPYR][TCM] IL is a complex molecule, in which the Columbic forces, hydrogen bonds and van der Waals forces all are present in the interaction between the cation and anion, as well as between the dissimilar molecules in the solution, with the hydrogen bonds being probably the most important forces in the IL at higher mole fractions. This can be explained by the high capacity of benzothiophene to form π-π interactions, making possible an easy accommodation of benzothiophene into the IL’s structure. On the other hand benzothiophene, in comparison with alcohols, is not a substance forming associates between similar molecules, and thus theoretically fewer molecules are free to interact with the IL in the solution and adsorb on the air–liquid surface. The surface tension of the {[BMPYR][TCM] + benzothiophene} solutions present formally similar patterns to those measured earlier [EMIM][TCM] [18]. According to our results, the regular decrease of the surface tension observed with decreasing IL mole fraction confirms that this behavior can be explained by strong interaction (IL + benzothiophene) within the investigated mole fraction region (see Fig. 8).
For the better understanding the results of this work, the surface tension deviation (Δσ) was calculated according to the equation:
$$\Delta \sigma = \sigma - \sum\limits_{i = 0}^{2} {x_{i} \sigma_{i} }$$
(11)
where x i and Δσ i are the mole fraction and surface tension deviation of component i, respectively. The surface tension deviations were correlated by means of the Redlich–Kister equation in the following form:
$$\Delta \sigma = x_{1} (x_{1} - 1)\sum\limits_{i = 0}^{i = 3} {C_{i} (1 - 2x_{1} )}^{i - 1}$$
(12)
where x i and Δσ i are the mole fraction and surface tension deviation of component i, respectively. The surface tension deviations at different temperatures are listed in Table 3. The values of parameters C i /(mN·m−1) have been determined using the least-squares method:
$$\sigma_{\Delta \sigma } = \left[ \left\{ \sum\limits_{i = 1}^{n} {(\Delta \sigma^{{{\text{exp}}}} - \Delta \sigma^{{{\text{calc}}}} )} /(n - k) \right\} \right]^{1/2}$$
(13)
The standard deviation, σ Δσ , is given by the formula (Eq. 13) where n is the number of experimental points and k is the number of coefficients. The parameters and standard deviations σ Δσ are listed in Table 9S in the supplementary material. The values of Δσ i are positive for all compositions of {[BMPYR][TCM] (1) + benzothiophene (22)} over the measured composition range as can be seen in Fig. 9. The maximum value of Δσ i is 5.36 N·m−1 and shifts to a lower mole fraction of the IL, x 1 = 0.3233 at T = 388.15 K. Values of Δσ i increase with an increase of temperature. This is similar to observations for [EMIM][TCM] [18], but opposite to that observed for (IL + an alcohol) binary mixtures [3436]. Changes with temperature may be attributed to diminishing of the hydrogen bonding between cation and anion in the IL, and then a new distribution of interactions exists at the surface and in the bulk region.
The measurements of the surface tension as a function of temperature provide the possibility of calculating the surface thermodynamic functions in the measured temperature range (308.15–338.15) K. The surface entropy (S σ) and the surface enthalpy (H σ) were calculated from the following equations [37, 38]:
$$S^{\sigma } = - \frac{\text{d}\sigma }{\text{d}T}$$
(14)
$$H^{\sigma } = \sigma - T\left( {\frac{\text{d}\sigma }{\text{d}T}} \right)$$
(15)
The thermodynamic functions for [BMPYR][TCM] at T = 308.15 K are listed in Table 4. The surface entropy is quite high {S σ = (55.00 ± 0.05) × 10−6 N·m−1·K−1 at T = 308.15 K}, but lower than that for [EMIM][TCM] {S σ = (10.61 ± 0.08) × 10−5 N·m−1·K−1 at T = 298.15 K [18] }, and higher than those for many ionic liquids [16, 17, 19, 36]. The lower is the surface entropy, the lower is the surface organization of the solution. The lower value of entropy of the IL shows that the partial molar entropy of the IL decreases at the contact between the IL and air in the surface region. The surface enthalpy {H σ = (64.98 ± 0.05) × 10−3 N·m−1 at T = 308.15 K} is lower than those observed for [EMIM][TCM] (T = 298.15 K) [18] and other ionic liquids [16, 17, 19, 36].
Because of the negligible vapour pressure of the IL, the critical temperature, (T c) can be estimated from the measurements of surface tension as a function of temperature according to following two formulae:
$$\sigma \left( {\frac{M}{\rho }} \right)^{{{\raise0.7ex\hbox{2} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{3}}}} = K\left( {T_{\text{c}}^{\text{E}} - T} \right)$$
(16)
$$\sigma = E^{\sigma } \left( {1 - \frac{T}{{T_{c}^{\text{G}} }}} \right)^{{{\raise0.7ex\hbox{{11}} \!\mathord{\left/ {\vphantom {{11} 9}}\right.\kern-0pt} \!\lower0.7ex\hbox{9}}}}$$
(17)
The critical temperature may be calculated from the Eötvös equation, (Eq. 16) [39], where K is a constant, ρ/(g·cm−3) is the density, M/(g·mol−1) is the molar mass, T/(K) is the temperature of the measured surface tension σ/(N·m−1), and $$T_{\text{c}}^{\text{E}}$$ /(K) is the Eötvös critical temperature. The critical temperature can be also calculated from the alternative van der Waals–Guggenheim equation (Eq. 17) for traditional organic liquids [38, 40], where E σ is the total surface energy of the IL, which equals the surface enthalpy as long as there is negligible volume change due to thermal expansion at temperatures well removed from the Guggenheim critical temperature $$T_{\text{c}}^{\text{G}}$$ /(K). The critical temperatures in this work, calculated from (Eqs. 16 and 17) and the total surface energy of the IL, are presented in Table 4. The two obtained values of the critical temperatures differ slightly from each other, ($$T_{\text{c}}^{\text{E}}$$ /(K) = 1646 and $$T_{\text{c}}^{\text{G}}$$ /(K) = 1377), and are higher than those of other ILs [16, 17, 19, 36]. The total surface energy of the IL is equal to 65.46 ± 0.05 mN·m−1 at T = 308.15 K, which is twice as large as that for 1-butyl-3-cyanopyridinium bis{(trifluoromethyl)sulfonyl}imide, [BCN3Py][NTf2] [17], and similar to [EMIM][TCM] (T = 298.15 K) [18]. According to the corresponding states correlations, in both equations (Eqs. 16 and 17) the surface tension becomes null at the critical temperature [40].
Using the definition of parachor (Eq. 18) and the measured density in a range of temperature (308.15 to 338.15) K, the parachor was calculated and the values are listed in Table 5.
$$P = \frac{{M\sigma^{1/4}}}{\rho }$$
(18)
The obtained value, 616.18 at T = 308.15 K (mN·m−1)1/4·cm3·mol−1, is similar to many values published earlier for other ILs [16, 17, 19, 36].
## Conclusions
The density, viscosity and surface tension of 1-butyl-1-methylpyrrolidinium tricyanomethanide, [BMPYR][TCM], were measured. The consequences of adding different amounts of benzothiophene and increasing the temperature were investigated. Through density, viscosity and surface tension measurements, it is established that both the increase in temperature and addition of benzothiophene lead to decreases in Coulombic, hydrogen bonding and van der Waals interactions and hence to structural disorder in the ionic liquid.
Negative deviations in the range of measured mole fraction were observed for the excess molar volumes, V E m , and positive deviations were observed for both the excess dynamic viscosity, Δη, and surface tension deviation, ∆σ. The results show that addition of benzothiophene increases the density but decreases the viscosity and surface tension of the mixture, which results in a loss of structural order at the interface and in the bulk of the IL. The molecular interpretation of the possible interactions for similar and dissimilar molecules, together with the packing effects, were discussed for the measured properties presented here.
The [BMPYR][TCM] IL presents surface tension value of the same order as observed for conventional ionic liquids and much higher than those reported for common organic solvents. The thermodynamic functions of the surface, such as surface entropy and enthalpy, were found to be similar to, or lower than those reported for other ILs. The molecular interpretation of the possible cross-hydrogen bonding between the IL and benzothiophene is difficult because of the immiscibility gap at low IL mole fractions. Thus, the negative excess molar volume data may be interpreted in terms of the packing effect. The packing effects or conformational changes of the molecules play a decisive role in the formation of associates in the solution and of the gas/liquid interface. Values of the parachor derived from the temperature dependence of the surface tension values are believed to be accurate enough for engineering calculations. The comparison made with [EMIM][TCM] [18], measured earlier, shows that the nature of the anion is the dominant factor in determining the extraction process, while changing the cation plays a minor role. Both ILs are similar in their nonaggregation behavior with thiophene or benzothiophene and their surface thermodynamic functions.
The results of the correlations with the second order polynomials, Redlich–Kister equation, and VFT equation for density, viscosity, excess molar volumes, viscosity deviation, surface tension and the surface tension deviation were presented, each with very low standard deviations.
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## Acknowledgments
This work has been supported by the project of National Science Center 011/01/B/ST5/00800.
## Author information
Authors
### Corresponding author
Correspondence to Urszula Domańska.
## Electronic supplementary material
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Domańska, U., Królikowska, M. & Walczak, K. Density, Viscosity and Surface Tension of Binary Mixtures of 1-Butyl-1-Methylpyrrolidinium Tricyanomethanide with Benzothiophene. J Solution Chem 43, 1929–1946 (2014). https://doi.org/10.1007/s10953-014-0257-1
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• DOI: https://doi.org/10.1007/s10953-014-0257-1
### Keywords
• ([BMPYR][TCM] + benzothiophene)
• Experimental density
• Dynamic viscosity
• Surface tension
• Molecular interactions
• Thermodynamics | {"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": 0.8876605033874512, "perplexity": 3351.943634711314}, "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-2022-27/segments/1656103033816.0/warc/CC-MAIN-20220624213908-20220625003908-00729.warc.gz"} |
http://mathoverflow.net/questions/129479/a-question-of-galois-cohomology | # a question of Galois cohomology
Let $R$ be a complete DVR with algebraically closed residue field $k$ and fractional field $K$ , $PGL(2)$ the automorphic group of projective line over $\overline K$.
My question is:
When $H^{1}(Gal(\overline K/K), PGL(2))=0$ ? Is this group trivial if $Char(K) \neq 2$ ?
-
A more informative title wouldn't hurt, I think. – Kestutis Cesnavicius May 3 '13 at 2:53
The claimed triviality holds (the nonabelian cohomology set is not a group though), and I don't think you need $Char(K) \neq 2$. To argue this, I will use the long exact nonabelian cohomology sequence of the central extension $1 \rightarrow \mathbf{G}_m \rightarrow GL_2 \rightarrow PGL_2 \rightarrow 1$, a segment of which reads $H^1(K, GL_2) \rightarrow H^1(K, PGL_2) \rightarrow H^2(K, \mathbf{G}_m)$. Firstly, $H^1(K, GL_2)$ is the one-point set because it classifies rank 2 vector bundles over $Spec(K)$, of which there is only the trivial one. Secondly, $K$ is a $C_1$ field by a theorem of Lang (see Serre "Galois cohomology", p. 80, II.3.3 c)), hence is of $dim \le 1$, so its Brauer group $H^2(K, \mathbf{G}_m)$ vanishes (loc. cit. for more details).
The same argument shows that $H^1(K, PGL_n)$ is the one-point set for any $n$ and any $C_1$ field $K$.
Alternatively this cohomology group classifies principal homogeneous spaces for $PGl_2$ and these correspond precisely to rank 4 Azumaya algebras $A$ by descent theory. But you can lift idempotent elements in $\overline A$ over the residue field $k$ to idempotent elements in $A$ over $R$. Consequently any such algebra has non identity idempotents in it and so must be a 2×2 ring of matrices over $R$. The characteristic of $k$ or $K$ doesn't matter. – Ray Hoobler May 4 '13 at 21:06 | {"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": 0.9327566027641296, "perplexity": 291.38950521621405}, "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-2014-23/segments/1405997894260.71/warc/CC-MAIN-20140722025814-00039-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://nuit-blanche.blogspot.com/2016/03/provable-non-convex-phase-retrieval.html | ## Friday, March 18, 2016
### Provable Non-convex Phase Retrieval with Outliers: Median Truncated Wirtinger Flow
A phase transition for a phase retrieval problem:
Provable Non-convex Phase Retrieval with Outliers: Median Truncated Wirtinger Flow by Huishuai Zhang, Yuejie Chi, Yingbin Liang
Solving systems of quadratic equations is a central problem in machine learning and signal processing. One important example is phase retrieval, which aims to recover a signal from only magnitudes of its linear measurements. This paper focuses on the situation when the measurements are corrupted by arbitrary outliers, for which the recently developed non-convex gradient descent Wirtinger flow (WF) and truncated Wirtinger flow (TWF) algorithms likely fail. We develop a novel median-TWF algorithm that exploits robustness of sample median to resist arbitrary outliers in the initialization and the gradient update in each iteration. We show that such a non-convex algorithm provably recovers the signal from a near-optimal number of measurements composed of i.i.d. Gaussian entries, up to a logarithmic factor, even when a constant portion of the measurements are corrupted by arbitrary outliers. We further show that median-TWF is also robust when measurements are corrupted by both arbitrary outliers and bounded noise. Our analysis of performance guarantee is accomplished by development of non-trivial concentration measures of median-related quantities, which may be of independent interest. We further provide numerical experiments to demonstrate the effectiveness of the approach.
Join the CompressiveSensing subreddit or the Google+ Community or the Facebook page and post there !
Liked this entry ? subscribe to Nuit Blanche's feed, there's more where that came from. You can also subscribe to Nuit Blanche by Email, explore the Big Picture in Compressive Sensing or the Matrix Factorization Jungle and join the conversations on compressive sensing, advanced matrix factorization and calibration issues on Linkedin. | {"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": 0.8819085955619812, "perplexity": 1210.6837148910092}, "config": {"markdown_headings": true, "markdown_code": false, "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-2018-13/segments/1521257647612.53/warc/CC-MAIN-20180321102234-20180321122234-00220.warc.gz"} |
http://tex.stackexchange.com/questions/7055/local-and-global-cross-references | # Local and global cross-references
Sections are referenced in a lot of distinct places:
In table of contents. Managed by \tableofcontents In minitocs. Managed by minitoc package In headers. Managed by fancyhdr package Like a leader section. Managed by \section In text. Managed by \ref In tocs and minitocs, because of each section appears below its chapter (wits some indentation) it's not necessary to prefix the section number with the chapter number. The label scope is local; so I need a local renderization. Example:
1. Chapter One ------------------1
1. First section -----------1
2. Second section ----------3
2. Chapter Two -----------------5
1. First section ...........5
This means that sections 1.1 and 2.1 must be rendered as section 1.
When we use \section command we know what chapter we are in: it's showed in the headers of te page. So I want also local scope:
1. First section
bla bla
2. Second section
bla bla
In headers I want global scope:
1. Chapter One 1.1. First section
In text we can make a crossreference to whatever point of text. So we need global scope:
See section 1.1 (first section in first chapter)
This is the problem: The default behavior give as global scope every where. But if I redefine \thesection to local scope, then I redefine everywhere.
This is my code:
\documentclass[catalan]{book}
\usepackage{fancyhdr}
\usepackage{nameref}
\usepackage{minitoc}
\usepackage{hyperref}
\makeindex
\dominitoc
%\renewcommand{\thesection}{\arabic{section}}
\renewcommand{\chaptermark}[1]{\markboth{\thechapter\ #1}{}} \renewcommand{\sectionmark}[1]{\markboth{\thechapter.\thesection\ \ #1}}
\pagestyle{fancy}
\begin{document} \tableofcontents \chapter{One}
\minitoc[e]
\section{First section}
\label{sec:A}
aaaaaa \newpage bbbbbbbb \newpage
ccccccccccc
\chapter{Two} \minitoc[e]
\section{Second section}
\label{sec:B}
See sections \ref{sec:A} and \ref{sec:B}
\end{document}
-
I might be able to help, but I don't really understand what you want. What part of the document is incorrect if you uncomment \renewcommand{\thesection}{\arabic{section}} in your example? – Ian Thompson Dec 13 '10 at 17:09
If I uncomment \renewcommand{\thesection}{\arabic{section}} I obtain always local scope. But I want global scope in headers and in-text cross-references. I only want local scope in minitocs and in the begining of a section or chapter – meren Dec 13 '10 at 17:22
@meren. I suspect that the reason you are struggling here is that most people would regard what you want to do as being rather confusing. Referencing 'Section 1' makes sense in a document with no chapters, but I've never seen this done in a larger document. – Joseph Wright Dec 25 '10 at 9:24
Use \renewcommand{\thesection}{\arabic{section}} to get the numbers beside the section titles right. Changing this otherwise is difficult.
Use \renewcommand{\sectionmark}[1]{\markright{\thechapter.\thesection\ #1}} for the headers.
For the references you can use e.g. zref to set up a "fullsec" reference. But I would find it confusing. And what if you have a subsection 2.1? I would put a \label behind the chapter too and then write something like see section 1 in chapter 2.
-
Well. But what about minitoc? – meren Dec 14 '10 at 15:15
And to have "see section 1 in chapter 2" how can I obtain the chapter number? – meren Dec 14 '10 at 15:17
Like you optain the section number: With \label after the \chapter and \ref. – Ulrike Fischer Dec 14 '10 at 15:27
But then I've to know the chapter of each reference! But I only want to know the label. – meren Dec 14 '10 at 20:28
Originally I posted a confused question, how-can-i-get-the-chapter-number-like-a-prefix-of-all-references. That question was the genesis of this one.
The answer posted by TH. there is the good answer to the current question
-
Here the link to that question: tex.stackexchange.com/q/6254/2975 – Martin Scharrer Jan 31 '11 at 15:15
I guess you mean "posted by TH."? – Hendrik Vogt Jan 31 '11 at 15:16
@Hendrik: Heh. I'm not really attached to that period. I just needed a third character when I registered. – TH. Feb 1 '11 at 9:13
@TH: I know. I just wanted to avoid further misunderstandings. – Hendrik Vogt Feb 1 '11 at 10:52 | {"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": 0.8745613694190979, "perplexity": 2785.2023761609166}, "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-2016-22/segments/1464051299749.12/warc/CC-MAIN-20160524005459-00035-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://socratic.org/questions/how-do-you-find-nth-term-rule-for-1-3-9-27 | Precalculus
Topics
How do you find nth term rule for 1,3,9,27,...?
Jan 31, 2017
The n-th term of this sequence appears to be ${3}^{n - 1}$, $n \ge 1$.
Explanation:
These are powers of $3$ ordered from ${3}^{0} = 1$ to ${3}^{a}$ (for an integer $a \ge 1$). However, the first convenient value for $n$ is $1$, not $0$ (imagine saying the 0th term of a sequence). Because of that, since the first term is actually ${3}^{0}$, we need to start from the first term ($n = 1$) being
${3}^{1 - 1}$. The next is ${3}^{2 - 1}$, ${3}^{3 - 1}$ ... ${3}^{n - 1}$.
Impact of this question
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https://www.math.ias.edu/seminars/abstract?event=47980 | # Geometric structure and the local Langlands conjecture
Joint IAS/Princeton University Number Theory Seminar Topic: Geometric structure and the local Langlands conjecture Speaker: Paul Baum Affiliation: Pennsylvania State University Date: Thursday, May 1 Time/Room: 4:30pm - 5:30pm/Fine 214, Princeton University
Let $G$ be a connected split reductive $p$-adic group. Examples are $\mathrm{GL}(n,F)$, $\mathrm{SL}(n, F )$, $\mathrm{SO}(n, F)$, $\mathrm{Sp}(2n, F )$, $\mathrm{PGL}(n, F )$ where $n$ can be any positive integer and $F$ can be any finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers. The smooth dual of $G$ is the set of equivalence classes of smooth irreducible representations of $G$. The representations are on vector spaces over the complex numbers. In the smooth dual there are subsets known as the Bernstein components, and the smooth dual is the disjoint union of the Bernstein components. This talk will explain a conjecture due to Aubert-Baum-Plymen-Solleveld (ABPS) which says that each Bernstein component is a complex affine variety. These affine varieties are explicitly identified as certain extended quotients. The infinitesimal character of Bernstein and the L-packets which appear in the local Langlands conjecture are then described from this point of view. Granted a mild restriction on the residual characteristic of the field $F$ over which $G$ is defined, ABPS has been proved for any Bernstein component in the principal series of $G$. A corollary is that the local Langlands conjecture is valid throughout the principal series of $G$. The above is joint work with Anne-Marie Aubert, Roger Plymen, and Maarten Solleveld. | {"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": 0.9256784319877625, "perplexity": 301.9046450451115}, "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-2018-39/segments/1537267156613.38/warc/CC-MAIN-20180920195131-20180920215531-00384.warc.gz"} |
https://stacks.math.columbia.edu/tag/01XJ | # The Stacks Project
## Tag 01XJ
Lemma 29.4.5. Let $f : X \to S$ be a morphism of schemes. Assume that $f$ is quasi-separated and quasi-compact.
1. For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the higher direct images $R^pf_*\mathcal{F}$ are quasi-coherent on $S$.
2. If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$ such that $R^pf_*\mathcal{F} = 0$ for all $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}$ on $X$.
3. In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$ such that for every morphism of schemes $S' \to S$ we have $R^p(f')_*\mathcal{F}' = 0$ for $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}'$ on $X'$. Here $f' : X' = S' \times_S X \to S'$ is the base change of $f$.
Proof. We first prove (1). Note that under the hypotheses of the lemma the sheaf $R^0f_*\mathcal{F} = f_*\mathcal{F}$ is quasi-coherent by Schemes, Lemma 25.24.1. Using Cohomology, Lemma 20.8.4 we see that forming higher direct images commutes with restriction to open subschemes. Since being quasi-coherent is local on $S$ we may assume $S$ is affine.
Assume $S$ is affine and $f$ quasi-compact and separated. Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$. We will prove this case of (1) by induction on $t$. If $t = 1$ then the morphism $f$ is affine by Morphisms, Lemma 28.11.12 and (1) follows from Lemma 29.2.3. If $t > 1$ write $X = U \cup V$ with $V$ affine open and $U = U_1 \cup \ldots \cup U_{t - 1}$ a union of $t - 1$ open affines. Note that in this case $U \cap V = (U_1 \cap V) \cup \ldots (U_{t - 1} \cap V)$ is also a union of $t - 1$ affine open subschemes, see Schemes, Lemma 25.21.8. We will apply the relative Mayer-Vietoris sequence $$0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots$$ see Cohomology, Lemma 20.9.3. By induction we see that $R^pa_*\mathcal{F}$, $R^pb_*\mathcal{F}$ and $R^pc_*\mathcal{F}$ are all quasi-coherent. This implies that each of the sheaves $R^pf_*\mathcal{F}$ is quasi-coherent since it sits in the middle of a short exact sequence with a cokernel of a map between quasi-coherent sheaves on the left and a kernel of a map between quasi-coherent sheaves on the right. Using the results on quasi-coherent sheaves in Schemes, Section 25.24 we see conclude $R^pf_*\mathcal{F}$ is quasi-coherent.
Assume $S$ is affine and $f$ quasi-compact and quasi-separated. Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$. We will prove (1) by induction on $t$. In case $t = 1$ the morphism $f$ is separated and we are back in the previous case (see previous paragraph). If $t > 1$ write $X = U \cup V$ with $V$ affine open and $U$ a union of $t - 1$ open affines. Note that in this case $U \cap V$ is an open subscheme of an affine scheme and hence separated (see Schemes, Lemma 25.21.6). We will apply the relative Mayer-Vietoris sequence $$0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots$$ see Cohomology, Lemma 20.9.3. By induction and the result of the previous paragraph we see that $R^pa_*\mathcal{F}$, $R^pb_*\mathcal{F}$ and $R^pc_*\mathcal{F}$ are quasi-coherent. As in the previous paragraph this implies each of sheaves $R^pf_*\mathcal{F}$ is quasi-coherent.
Next, we prove (3) and a fortiori (2). Choose a finite affine open covering $S = \bigcup_{j = 1, \ldots m} S_j$. For each $i$ choose a finite affine open covering $f^{-1}(S_j) = \bigcup_{i = 1, \ldots t_j} U_{ji}$. Let $$d_j = \max\nolimits_{I \subset \{1, \ldots, t_j\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_{ji})\right)$$ be the integer found in Lemma 29.4.4. We claim that $n(X, S, f) = \max d_j$ works.
Namely, let $S' \to S$ be a morphism of schemes and let $\mathcal{F}'$ be a quasi-coherent sheaf on $X' = S' \times_S X$. We want to show that $R^pf'_*\mathcal{F}' = 0$ for $p \geq n(X, S, f)$. Since this question is local on $S'$ we may assume that $S'$ is affine and maps into $S_j$ for some $j$. Then $X' = S' \times_{S_j} f^{-1}(S_j)$ is covered by the open affines $S' \times_{S_j} U_{ji}$, $i = 1, \ldots t_j$ and the intersections $$\bigcap\nolimits_{i \in I} S' \times_{S_j} U_{ji} = S' \times_{S_j} \bigcap\nolimits_{i \in I} U_{ji}$$ are covered by the same number of affines as before the base change. Applying Lemma 29.4.4 we get $H^p(X', \mathcal{F}') = 0$. By the first part of the proof we already know that each $R^qf'_*\mathcal{F}'$ is quasi-coherent hence has vanishing higher cohomology groups on our affine scheme $S'$, thus we see that $H^0(S', R^pf'_*\mathcal{F}') = H^p(X', \mathcal{F}') = 0$ by Cohomology, Lemma 20.14.6. Since $R^pf'_*\mathcal{F}'$ is quasi-coherent we conclude that $R^pf'_*\mathcal{F}' = 0$. $\square$
The code snippet corresponding to this tag is a part of the file coherent.tex and is located in lines 679–695 (see updates for more information).
\begin{lemma}
\label{lemma-quasi-coherence-higher-direct-images}
Let $f : X \to S$ be a morphism of schemes.
Assume that $f$ is quasi-separated and quasi-compact.
\begin{enumerate}
\item For any quasi-coherent $\mathcal{O}_X$-module $\mathcal{F}$ the
higher direct images $R^pf_*\mathcal{F}$ are quasi-coherent on $S$.
\item If $S$ is quasi-compact, there exists an integer $n = n(X, S, f)$
such that $R^pf_*\mathcal{F} = 0$ for all $p \geq n$ and any
quasi-coherent sheaf $\mathcal{F}$ on $X$.
\item In fact, if $S$ is quasi-compact we can find $n = n(X, S, f)$
such that for every
morphism of schemes $S' \to S$ we have $R^p(f')_*\mathcal{F}' = 0$
for $p \geq n$ and any quasi-coherent sheaf $\mathcal{F}'$
on $X'$. Here $f' : X' = S' \times_S X \to S'$ is the base change of $f$.
\end{enumerate}
\end{lemma}
\begin{proof}
We first prove (1). Note that under the hypotheses of the lemma the sheaf
$R^0f_*\mathcal{F} = f_*\mathcal{F}$ is quasi-coherent by
Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
Using
Cohomology, Lemma \ref{cohomology-lemma-localize-higher-direct-images}
we see that forming higher direct images commutes with restriction
to open subschemes. Since being quasi-coherent is local on $S$ we
may assume $S$ is affine.
\medskip\noindent
Assume $S$ is affine and $f$ quasi-compact and separated.
Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$.
We will prove this case of (1) by induction on $t$.
If $t = 1$ then the morphism $f$ is affine by
Morphisms, Lemma \ref{morphisms-lemma-morphism-affines-affine}
and (1) follows from
Lemma \ref{lemma-relative-affine-vanishing}.
If $t > 1$ write $X = U \cup V$ with $V$ affine open and
$U = U_1 \cup \ldots \cup U_{t - 1}$ a union of $t - 1$ open affines.
Note that in this case
$U \cap V = (U_1 \cap V) \cup \ldots (U_{t - 1} \cap V)$
is also a union of $t - 1$ affine open subschemes, see
Schemes, Lemma \ref{schemes-lemma-characterize-separated}.
We will apply the relative Mayer-Vietoris sequence
$$0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots$$
see Cohomology, Lemma \ref{cohomology-lemma-relative-mayer-vietoris}.
By induction we see that
$R^pa_*\mathcal{F}$, $R^pb_*\mathcal{F}$ and $R^pc_*\mathcal{F}$
are all quasi-coherent. This implies that each of the sheaves
$R^pf_*\mathcal{F}$ is quasi-coherent since it sits in the middle of a short
exact sequence with a cokernel of a map between quasi-coherent sheaves
on the left and a kernel of a map between quasi-coherent sheaves on the right.
Using the results on quasi-coherent sheaves in
Schemes, Section \ref{schemes-section-quasi-coherent} we see
conclude $R^pf_*\mathcal{F}$ is quasi-coherent.
\medskip\noindent
Assume $S$ is affine and $f$ quasi-compact and quasi-separated.
Let $t \geq 1$ be the minimal number of affine opens needed to cover $X$.
We will prove (1) by induction on $t$.
In case $t = 1$ the morphism $f$ is separated and we are back
in the previous case (see previous paragraph).
If $t > 1$ write $X = U \cup V$ with $V$ affine open and
$U$ a union of $t - 1$ open affines.
Note that in this case $U \cap V$ is an open subscheme of an affine
scheme and hence separated (see
Schemes, Lemma \ref{schemes-lemma-affine-separated}).
We will apply the relative Mayer-Vietoris sequence
$$0 \to f_*\mathcal{F} \to a_*(\mathcal{F}|_U) \oplus b_*(\mathcal{F}|_V) \to c_*(\mathcal{F}|_{U \cap V}) \to R^1f_*\mathcal{F} \to \ldots$$
see Cohomology, Lemma \ref{cohomology-lemma-relative-mayer-vietoris}.
By induction and the result of the previous paragraph we see that
$R^pa_*\mathcal{F}$, $R^pb_*\mathcal{F}$ and $R^pc_*\mathcal{F}$
are quasi-coherent. As in the previous paragraph this implies each of
sheaves $R^pf_*\mathcal{F}$ is quasi-coherent.
\medskip\noindent
Next, we prove (3) and a fortiori (2). Choose a finite affine open
covering $S = \bigcup_{j = 1, \ldots m} S_j$. For each $i$ choose
a finite affine open covering
$f^{-1}(S_j) = \bigcup_{i = 1, \ldots t_j} U_{ji}$.
Let
$$d_j = \max\nolimits_{I \subset \{1, \ldots, t_j\}} \left(|I| + t(\bigcap\nolimits_{i \in I} U_{ji})\right)$$
be the integer found in
Lemma \ref{lemma-vanishing-nr-affines-quasi-separated}.
We claim that $n(X, S, f) = \max d_j$ works.
\medskip\noindent
Namely, let $S' \to S$ be a morphism of schemes and let
$\mathcal{F}'$ be a quasi-coherent sheaf on $X' = S' \times_S X$.
We want to show that $R^pf'_*\mathcal{F}' = 0$ for $p \geq n(X, S, f)$.
Since this question is local on $S'$ we may assume that $S'$ is affine
and maps into $S_j$ for some $j$. Then $X' = S' \times_{S_j} f^{-1}(S_j)$
is covered by the open affines $S' \times_{S_j} U_{ji}$, $i = 1, \ldots t_j$
and the intersections
$$\bigcap\nolimits_{i \in I} S' \times_{S_j} U_{ji} = S' \times_{S_j} \bigcap\nolimits_{i \in I} U_{ji}$$
are covered by the same number of affines as before the base change.
Applying
Lemma \ref{lemma-vanishing-nr-affines-quasi-separated}
we get $H^p(X', \mathcal{F}') = 0$. By the first part of the proof
we already know that each $R^qf'_*\mathcal{F}'$ is quasi-coherent
hence has vanishing higher cohomology groups on our affine scheme $S'$,
thus we see that $H^0(S', R^pf'_*\mathcal{F}') = H^p(X', \mathcal{F}') = 0$
by Cohomology, Lemma \ref{cohomology-lemma-apply-Leray}.
Since $R^pf'_*\mathcal{F}'$ is quasi-coherent
we conclude that $R^pf'_*\mathcal{F}' = 0$.
\end{proof}
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http://www.personal.psu.edu/t20/papers/massint/ | # Mass problems and intuitionism
Stephen G. Simpson1
Department of Mathematics
First draft: July 25, 2007
This draft: April 28, 2008
Notre Dame Journal of Formal Logic, 49, 2008, 127-136
Pennsylvania State University
### Abstract:
Let be the lattice of Muchnik degrees of nonempty subsets of . The lattice has been studied extensively in previous publications. In this note we prove that the lattice is not Brouwerian.
# Introduction
Definition 1 Let denote the set of natural numbers, . Let denote the Baire space, . Following Medvedev [27] and Rogers [32, §13.7] we define a mass problem to be an arbitrary subset of . For mass problems and we say that is Medvedev reducible or strongly reducible to , abbreviated , if there exists a partial recursive functional such that for all . We say that is Muchnik reducible or weakly reducible to , abbreviated , if for all there exists such that is Turing reducible to . Clearly Medvedev reducibility implies Muchnik reducibility, but the converse does not hold.
Definition 2 A Medvedev degree or degree of difficulty or strong degree is an equivalence class of mass problems under mutual Medvedev reducibility. A Muchnik degree or weak degree is an equivalence class of mass problems under mutual Muchnik reducibility. We write the Medvedev degree of . We write the Muchnik degree of . Let be the set of Medvedev degrees, partially ordered by Medvedev reducibility. There is a natural embedding of the Turing degrees into given by . Let be the set of Muchnik degrees, partially ordered by Muchnik reducibility. There is a natural embedding of the Turing degrees into given by . Here is the singleton set whose only element is .
Definition 3 Let be a lattice. For we define to be the unique minimum such that . Note that may or may not exist in . Following Birkhoff [8,9] (first two editions) and McKinsey/Tarski [25] we say that is Brouwerian if exists in for all and has a top element. It is known (see Birkhoff [9, §IX.12] [10, §II.11] or McKinsey/Tarski [25] or Rasiowa/Sikorski [31, §I.12]) that if is Brouwerian then is distributive and has a bottom element and for all in the sublattice
is again Brouwerian.
Remark 1 Given a Brouwerian lattice , we may view as a model of first-order intuitionistic propositional calculus. Namely, for we define , , as above, and where is the top element of . We may also define if and only if in . There is a completeness theorem (see Tarski [52] or McKinsey/Tarski [24,25,26] or Rasiowa/Sikorski [31, §IX.3] or Rasiowa [30, § XI.8]) saying that a first-order propositional formula is intuitionistically provable if and only if it evaluates identically to the bottom element in all Brouwerian lattices.
Remark 2 Brouwerian lattices have also been studied under other names and with other notation and terminology. A pseudo-Boolean algebra is a lattice such that the dual of is Brouwerian; see Rasiowa/Sikorski [31] and Rasiowa [30]. Pseudo-Boolean algebras are also known as Heyting algebras; see Balbes/Dwinger [2, Chapter IX], Fourman/Scott [18], and Grätzer [19]. Brouwerian lattices are also known as Brouwer algebras; see Sorbi [48,49], Sorbi/Terwijn [51], and Terwijn [53,54,55,56,57]. Remarkably, the so-called Brouwerian lattices of Birkhoff [10] (third edition) are dual to those of Birkhoff [8,9] (first two editions). We adhere to the terminology of Birkhoff [8,9].
Remark 3 It is known that and are Brouwerian lattices. There is a natural homomorphism of onto given by . This homomorphism preserves the binary lattice operations and and the top and bottom elements, but it does not preserve the binary if-then operation .
Remark 4 The relationship between mass problems and intuitionism has a considerable history. Indeed, it seems fair to say that the entire subject of mass problems originated from intuitionistic considerations. The impetus came from Kolmogorov 1932 [22,23] who informally proposed to view Heyting's intuitionistic propositional calculus [20] as a calculus of problems'' (Aufgabenrechnung''). This idea amounts to what is now known as the BHK or Brouwer/Heyting/Kolmogorov interpretation of the intuitionistic propositional connectives; see Troelstra/van Dalen [59, §§1.3.1 and 1.5.3]. Elaborating Kolmogorov's idea, Medvedev 1955 [27] introduced and noted that is a Brouwerian lattice. Later Muchnik 1963 [28] introduced and noted that is a Brouwerian lattice. Some further papers in this line are Skvortsova [47], Sorbi [48,49,50], Sorbi/Terwijn [51], and Terwijn [54,53,55,56,57].
Definition 4 Let denote the Cantor space, . Following Simpson [40] let be the sublattice of consisting of the Medvedev degrees of nonempty subsets of , and let be the sublattice of consisting of the Muchnik degrees of nonempty subsets of .
Remark 5 The lattices and are mathematically rich and have been studied extensively. See Alfeld [1], Binns [3,4,5,6], Binns/Simpson [7], Cenzer/Hinman [11], Cole/Simpson [13], Kjos-Hanssen/Simpson [21], Simpson [34,35,37,38,39,40,41,42,43,45,44], Simpson/Slaman [46], and Terwijn [54]. It is known that contains not only the recursively enumerable Turing degrees [42] but also many specific, natural Muchnik degrees which arise from foundationally interesting topics. Among these foundationally interesting topics are algorithmic randomness [40,42], reverse mathematics [36,40,41,43], almost everywhere domination [43], hyperarithmeticity [13], diagonal nonrecursiveness [40,42], subrecursive hierarchies [21,40], resource-bounded computational complexity [21,40], and Kolmogorov complexity [21]. Recently Simpson [44] has applied and to prove a new theorem in symbolic dynamics.
Remark 6 It is known that and are distributive lattices with top and bottom elements. Moreover, the natural lattice homomorphism of onto restricts to a natural lattice homomorphism of onto preserving top and bottom elements.
Remark 7 In view of Remarks 3, 4, 5 and 6, it is natural to ask whether and are Brouwerian lattices. The purpose of this note is to show that is not a Brouwerian lattice. Letting denote the top element of , we shall produce a family of Muchnik degrees such that does not exist in . In other words, does not exist in .
Remark 8 It remains open whether is a Brouwerian lattice. Terwijn [54] has shown that the dual of is not a Brouwerian lattice. It remains open whether the dual of is a Brouwerian lattice.
# Proof that is not Brouwerian
In this section we prove that the lattice is not Brouwerian.
Definition 5 For we write to mean that is Turing reducible to , i.e., is computable relative to the Turing oracle . We write the Turing jump of . In particular the halting problem the Turing jump of . We use standard recursion-theoretic notation from Rogers [32]. We say that is majorized by if for all .
We begin with four well known lemmas.
Lemma 1 Given we can find such that is .
Proof.Since , it follows by Post's Theorem (see for instance [32, §14.5, Theorem VIII]) that is . From this it follows that the singleton set is . Let be a recursive predicate such that our is the unique such that holds. Let where is defined by the least such that holds. It is easy to verify that and is .
Lemma 2 If is and is nonrecursive, then is not majorized by any recursive function.
Proof.This lemma is equivalent to, for instance, [40, Theorem 4.15].
Lemma 3 For all nonempty sets we have .
Proof.This lemma is a restatement of the well known Kleene Basis Theorem. Namely, every nonempty subset of contains an element which is . See for instance the proof of [42, Lemma 5.3].
Lemma 4 Let be nonempty such that no element of is recursive. Then we can find such that and .
Proof.By Lemma 3 it suffices to find such that and . To construct we may proceed as in the proof of Lemma 5 below. The construction is easier than in Lemma 5, because we can ignore .
Lemma 5 Let be nonempty . Let be such that and . Then we can find such that and and .
Proof.We adapt the technique of Posner/Robinson [29].
Let be a recursive tree such that paths through . By Lemmas 1 and 2 we may safely assume that is not majorized by any recursive function.
For integers and strings we write
where the least such that either or . Note that the mapping is recursive and monotonic, i.e., implies . Moreover, for all we have if and only if . Here we are writing
In order to prove Lemma 5, we shall inductively define an increasing sequence of strings , . We shall then let . In presenting the construction, we shall identify strings with their Gödel numbers.
Stage . Let the empty string.
Stage . Assume that has been defined. The definition of will be given in a finite number of substages.
Substage . Let .
Substage . Assume that has been defined. Let the least such that either
and
or
and .
Note that exists, because otherwise would be majorized by the recursive function least such that and . If (1) holds with let . If (2) holds with let . This completes our description of the construction.
We claim that, within each stage , (2) holds for some . Otherwise, we would have infinite increasing sequences of strings
and
with for all . Moreover, these sequences would be recursive relative to , namely where least such that (1) holds. Thus, letting , we would have and . Thus , a contradiction. This proves our claim.
From the previous claim it follows that is defined for all . By construction, the sequence , is recursive relative to . Moreover, is recursive relative to , because for all we have .
Finally let . Clearly .
We claim that the sequence is . Namely, given , we may use and as oracles to compute as follows. We begin with . Given we use the oracle to compute . Then, using the oracle , we ask whether there exists such that and . If so, we compute the least such . If not, we use the oracle to compute . This proves our claim.
From the previous claim it follows that . Hence .
We claim that . To see this, let be such that is a total function. Consider what happened at stage of the construction. Consider the least such that (2) holds, i.e., . Since (2) holds, there does not exist such that and . In particular, letting be an initial segment of such that and , we have . Hence . This proves our claim.
From the two previous claims, it follows that . The proof of Lemma 5 is now finished.
Remark 9 By a similar argument we can prove the following. Let be . Let be of hyperimmune Turing degree such that . Let be such that . Then we can find such that and and .
Lemma 6 Let be nonempty . Let be . Then
Proof.This is Simpson's Embedding Lemma. See [42, Lemma 3.3] or [45].
We are now ready to prove our main result.
Theorem 1 is not Brouwerian.
Proof.Let be the set of completions of Peano Arithmetic. Recall from Simpson [40] that the top element of . By Lemma 4 let be such that and . Let
and note that . By Lemmas 1 and 6 we have .
It is well known (see for instance [40, Remark 3.9]) that is a complete lattice. This means that for all the least upper bound and the greatest lower bound exist in . Therefore, within , let
and note that in . In other words, in .
We claim that . Otherwise, let where is nonempty . Since , we have for all . Since , it follows that . By Lemma 5 let be such that and and . Let
and note that . By Lemmas 1 and 6 we have . By Lemma 3 we have , hence contradicting the definition of . This proves our claim.
Because it follows that does not exist in . Thus is not Brouwerian.
Remark 10 The same proof shows that for all in we can find in such that does not exist in . On the other hand, we know at least a few nontrivial instances where exists in . For example, letting be the Muchnik degree of the set of 1-random reals, Theorem 8.12 of Simpson [40] tells us that in and exists in . In fact, in is equal to in , which is equal to . We do not know any instances of where exists in and both and are in .
## Bibliography
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Bjørn Kjos-Hanssen and Stephen G. Simpson.
Mass problems and Kolmogorov complexity.
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Zur Deutung der intuitionistischen Logik.
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On the interpretation of intuitionistic logic.
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Translation of [22] with commentary and additional references.
24
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29
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Degrees joining to .
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30
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An Algebraic Approach to Non-Classical Logics.
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The Mathematics of Metamathematics.
Number 41 in Polska Akademia Nauk, Monografie Matematyczne. Panstwowe Wydawnictwo Naukowe, 1963.
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32
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Theory of Recursive Functions and Effective Computability.
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XIX + 482 pages.
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S. G. Simpson, editor.
Reverse Mathematics 2001.
Number 21 in Lecture Notes in Logic. Association for Symbolic Logic, 2005.
X + 401 pages.
34
Stephen G. Simpson.
FOM: natural r.e. degrees; Pi01 classes.
FOM e-mail list [16], 13 August 1999.
35
Stephen G. Simpson.
FOM: priority arguments; Kleene-r.e. degrees; Pi01 classes.
FOM e-mail list [16], 16 August 1999.
36
Stephen G. Simpson.
Subsystems of Second Order Arithmetic.
Perspectives in Mathematical Logic. Springer-Verlag, 1999.
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37
Stephen G. Simpson.
Medvedev degrees of nonempty Pi01 subsets of 2omega.
COMP-THY e-mail list [14], 9 June 2000.
38
Stephen G. Simpson.
Mass problems.
24 May 2004.
Preprint, 24 pages, submitted for publication.
39
Stephen G. Simpson.
FOM: natural r.e. degrees.
FOM e-mail list [16], 27 February 2005.
40
Stephen G. Simpson.
Mass problems and randomness.
Bulletin of Symbolic Logic, 11:1-27, 2005.
41
Stephen G. Simpson.
sets and models of .
In [33], pages 352-378, 2005.
42
Stephen G. Simpson.
An extension of the recursively enumerable Turing degrees.
Journal of the London Mathematical Society, 75:287-297, 2007.
43
Stephen G. Simpson.
Mass problems and almost everywhere domination.
Mathematical Logic Quarterly, 53:483-492, 2007.
44
Stephen G. Simpson.
Medvedev degrees of 2-dimensional subshifts of finite type.
Ergodic Theory and Dynamical Systems, 2008.
Preprint, 8 pages, 1 May 2007, accepted for publication.
45
Stephen G. Simpson.
Some fundamental issues concerning degrees of unsolvability.
In [12], pages 313-332, 2008.
46
Stephen G. Simpson and Theodore A. Slaman.
Medvedev degrees of subsets of .
July 2001.
Preprint, 4 pages, in preparation, to appear.
47
Elena Z. Skvortsova.
A faithful interpretation of the intuitionistic propositional calculus by means of an initial segment of the Medvedev lattice.
Sibirskii Matematicheskii Zhurnal, 29:171-178, 1988.
In Russian.
48
Andrea Sorbi.
Some remarks on the algebraic structure of the Medvedev lattice.
Journal of Symbolic Logic, 55:831-853, 1990.
49
Andrea Sorbi.
Embedding Brouwer algebras in the Medvedev lattice.
Notre Dame Journal of Formal Logic, 32:266-275, 1991.
50
Andrea Sorbi.
The Medvedev lattice of degrees of difficulty.
In [15], pages 289-312, 1996.
51
Andrea Sorbi and Sebastiaan A. Terwijn.
Intermediate logics and factors of the Medvedev lattice.
Annals of Pure and Applied Logic, 2008.
Preprint, 22 pages, 20 June 2006, to appear.
52
Alfred Tarski.
Der Aussagenkalkül und die Topologie.
Fundamenta Mathematicae, 31:103-134, 1938.
53
Sebastiaan A. Terwijn.
Constructive logic and the Medvedev lattice.
Notre Dame Journal of Formal Logic, 47:73-82, 2006.
54
Sebastiaan A. Terwijn.
The Medvedev lattice of computably closed sets.
Archive for Mathematical Logic, 45:179-190, 2006.
55
Sebastiaan A. Terwijn.
Constructive Logic and Computational Lattices.
Habilitationsschrift, Universität Wien, 2007.
VI + 110 pages.
56
Sebastiaan A. Terwijn.
Kripke models, distributive lattices, and Medvedev degrees.
Studia Logica, 85:327-340, 2007.
57
Sebastiaan A. Terwijn.
On the structure of the Medvedev lattice.
Journal of Symbolic Logic, 73:543-558, 2008.
58
V. M. Tikhomirov, editor.
Selected Works of A. N. Kolmogorov, Volume I, Mathematics and Mechanics.
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XIX + 551 pages.
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Anne S. Troelstra and Dirk van Dalen.
Constructivism in Mathematics, an Introduction.
Studies in Logic and the Foundations of Mathematics. North-Holland, 1988.
Volume I, XX + 342 + XIV pages.
Mass problems and intuitionism
This document was generated using the LaTeX2HTML translator Version 2002-2-1 (1.71)
Copyright © 1993, 1994, 1995, 1996, Nikos Drakos, Computer Based Learning Unit, University of Leeds.
Copyright © 1997, 1998, 1999, Ross Moore, Mathematics Department, Macquarie University, Sydney.
The command line arguments were:
latex2html -split 0 massint
The translation was initiated by Stephen G Simpson on 2008-04-28
#### Footnotes
... Simpson1
The author thanks Sebastiaan Terwijn for helpful correspondence. The author's research was partially supported by the United States National Science Foundation, grants DMS-0600823 and DMS-0652637, and by the Cada and Susan Grove Mathematics Enhancement Endowment at the Pennsylvania State University.
Stephen G Simpson 2008-04-28 | {"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": 0.9807436466217041, "perplexity": 2306.271587831947}, "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-2014-23/segments/1406510273676.54/warc/CC-MAIN-20140728011753-00414-ip-10-146-231-18.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/sums-substituting-two-sums.284210/ | # Homework Help: Sums: Substituting two sums
1. Jan 11, 2009
### Niles
1. The problem statement, all variables and given/known data
Hi all.
Lets assume that we know the following:
$$\sum\limits_{n = - \infty }^\infty {\varepsilon _n (t)} \exp ( - i\omega _n t) = a_0 + \sum\limits_{n = 1}^\infty {\varepsilon _n (t)} \exp ( - i\omega _n t),$$
where a0 is the contribution for n=0. Now I have an expression for a function f given by the following:
$$f = \sum\limits_{n = - \infty }^\infty {\varepsilon _n (t)} \exp ( - i\omega _n t)\frac{1}{{Z(\omega _n )}}.$$
Am I allowed to write f as this?:
$$f = a_0 \frac{1}{{Z(\omega _0 )}} + \sum\limits_{n = 1}^\infty {\varepsilon _n (t)} \exp ( - i\omega _n t)\frac{1}{{Z(\omega _n )}},$$
i.e. substitute the sum? Personally, I think yes, but I am a little unsure, which is why I thought it would be best to check here. Thanks in advance.
Best regards,
Niles.
2. Jan 11, 2009
### HallsofIvy
This appears to be saying that
$$\sum\limites_{n=-\infty}^{-1}\varepsilon_n(t)}\exp(-i\omega_n t)= 0[/itex] But that does NOT mean that [tex]\sum\limits_{n = -\infty}^{-1} {\varepsilon _n (t)} \exp ( - i\omega _n t)\frac{1}{{Z(\omega _n )}}= 0$$
since the $Z(\omega_n)$ term may change the contribution of each indvidual term in the sum.
3. Jan 11, 2009
### Niles
Aw man, I made an error in my first post. Very stupid of me, because now I made you look at something which was not my intention; I am very sorry for that.
What I meant to write is the following (I'm still really sorry, I should have taken a closer look before posting):
Lets assume that we know the following:
$$\sum\limits_{n = - \infty }^\infty {\varepsilon _n (t)} \exp ( - i\omega _n t) = a_0 + \sum\limits_{n = 1}^\infty {n^2} \exp ( - i\omega _n t).$$
Now we look at an expression for a function f given by the following:
$$f = \sum\limits_{n = - \infty }^\infty {\varepsilon _n (t)} \exp ( - i\omega _n t)\frac{1}{{Z(\omega _n )}}.$$
Am I allowed to write f as:
$$f = a_0 \frac{1}{{Z(\omega _0 )}} + \sum\limits_{n = 1}^\infty {n^2} \exp ( - i\omega _n t)\frac{1}{{Z(\omega _n )}}.$$
I have double-checked, and the formulas are correct now. Sorry, again.
Best regards,
Niles.
4. Jan 11, 2009
### HallsofIvy
And the answer is still "No". Because $Z(\omega_n)$ may be different for different n, it can completely change the sum.
5. Jan 11, 2009
### Niles
I had actually hoped that you would answer "Yes", because the "No" means that my calculations are wrong. But the quantity $Z(\omega_n)$ does depend on n, so it is different for different n.
I will have to think about this. I guess I made a wrong assumption along the way.. I do that sometimes (as we can also see from this thread ).
Thanks for helping, I really do appreciate 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": 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": 0.9226993322372437, "perplexity": 521.0544140151111}, "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-2018-43/segments/1539583514708.24/warc/CC-MAIN-20181022050544-20181022072044-00424.warc.gz"} |
https://electronics.stackexchange.com/questions/326419/%CE%99nput-referred-noise-of-an-amplifier | # Ιnput referred noise of an amplifier
I have a problem with a low noise amplifier because I can't find in ADS(Advance design system) software how to integrating under the are of a measured curve in a specific frequency range.Can someone help me how I can achieve that?
Figure 1 shows a paper result and how the noise measured and Figure 2,3 shows my results.As we can see the measured values for the noise is approximately the same but I would like to integrate under the are of a measured curve from 10Hz to 98kHz to find the total input referred noise and I don't know how I can do it.Also, I had searched a lot on the internet and I didn't find something useful.Can someone help me? | {"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": 0.9660290479660034, "perplexity": 479.51651737793924}, "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/1623487611445.13/warc/CC-MAIN-20210614043833-20210614073833-00246.warc.gz"} |
https://proofwiki.org/wiki/Category:Definitions/Ordered_Integral_Domains | # Category:Definitions/Ordered Integral Domains
This category contains definitions related to Ordered Integral Domains.
Related results can be found in Category:Ordered Integral Domains.
### Definition 1
An ordered integral domain is an integral domain $\struct {D, +, \times}$ which has a strict positivity property $P$:
$(\text P 1)$ $:$ Closure under Ring Addition: $\ds \forall a, b \in D:$ $\ds \map P a \land \map P b \implies \map P {a + b}$ $(\text P 2)$ $:$ Closure under Ring Product: $\ds \forall a, b \in D:$ $\ds \map P a \land \map P b \implies \map P {a \times b}$ $(\text P 3)$ $:$ Trichotomy Law: $\ds \forall a \in D:$ $\ds \paren {\map P a} \lor \paren {\map P {-a} } \lor \paren {a = 0_D}$ For $\text P 3$, exactly one condition applies for all $a \in D$.
### Definition 2
An ordered integral domain is an ordered ring $\struct {D, +, \times, \le}$ which is also an integral domain.
That is, it is an integral domain with an ordering $\le$ compatible with the ring structure of $\struct {D, +, \times}$:
$(\text {OID} 1)$ $:$ $\le$ is compatible with ring addition: $\ds \forall a, b, c \in D:$ $\ds a \le b$ $\ds \implies$ $\ds \paren {a + c} \le \paren {b + c}$ $(\text {OID} 2)$ $:$ Strict positivity is closed under ring product: $\ds \forall a, b \in D:$ $\ds 0_D \le a, 0_D \le b$ $\ds \implies$ $\ds 0_D \le a \times b$
An ordered integral domain can be denoted:
$\struct {D, +, \times \le}$
where $\le$ is the total ordering induced by the strict positivity property.
## Subcategories
This category has only the following subcategory.
## Pages in category "Definitions/Ordered Integral Domains"
The following 11 pages are in this category, out of 11 total. | {"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": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9987816214561462, "perplexity": 666.1208510151888}, "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-2021-39/segments/1631780057787.63/warc/CC-MAIN-20210925232725-20210926022725-00399.warc.gz"} |
https://ez.analog.com/amplifiers/f/q-a/536651/problems-with-simulation-of-a-common-source-with-current-mirror-load | # Problems with simulation of a common source with current mirror load
I'm trying to perform an ac analysis on the circuit with LTSpice and level 1 parameters. From the theory I should get a pole at frequency 1/(RL*(Cgd+Cl)) and a zero in gm1/Cgd. The gain should be gm1*RL, where RL=[(lambda1*I)^(-1)] || [(lambda2*I)^(-1)].
(W/L)1=20/1; I=77.5uA; kn=110uA/V^2; lambda1=0.04V^-1; lambda2=0.05V^-1; RL=143kOhm; Cl=100fF and then Cl=27pF
Now, the questions are:
1) is Cgd=CGDO * W with W=Weff?
2) as I have CGDO=220pF/m and a transistor with W=20um I should have Cgd=4.4 fF but from computer simulation Cgd=26.5fF (as I get the zero at 22GHz and gm1 is 584uS). I don't understand why... is the Cgd of 4.4fF wrong? Is the Weff so much different from W?
3) If I use Cgd=26.5fF and I apply the Miller effect, the pole I calculate with Cl=100fF is close to the one I get from computer simulation. If I try to increase the Cl up to 27pF this is not true. In fact I need to increase the value of Cgd to get a pole frequency closer to the one of LTSpice. But this is wrong, because the zero stays always at 22GHz and Cgd doesn't change (as well as gm1)
So, generally speaking, I do not know how to explain the behavior of the simulation and I can't "chek" the hand calculations given by the theory.
How can I solve the problem?
SC con specchio analisi in frequenza.asc | {"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": 0.9237534999847412, "perplexity": 1948.6854627714185}, "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/1606141201836.36/warc/CC-MAIN-20201129153900-20201129183900-00091.warc.gz"} |
https://it.mathworks.com/help/pde/ug/pde.transientthermalresults.html | # TransientThermalResults
Transient thermal solution and derived quantities
## Description
A `TransientThermalResults` object contains the temperature and gradient values in a form convenient for plotting and postprocessing.
The temperature and its gradient are calculated at the nodes of the triangular or tetrahedral mesh generated by `generateMesh`. Temperature values at the nodes appear in the `Temperature` property. The solution times appear in the `SolutionTimes` property. The three components of the temperature gradient at the nodes appear in the `XGradients`, `YGradients`, and `ZGradients` properties. You can extract solution and gradient values for specified time indices from `Temperature`, `XGradients`, `YGradients`, and `ZGradients`.
To interpolate the temperature or its gradient to a custom grid (for example, specified by `meshgrid`), use `interpolateTemperature` or `evaluateTemperatureGradient`.
To evaluate heat flux of a thermal solution at nodal or arbitrary spatial locations, use `evaluateHeatFlux`. To evaluate integrated heat flow rate normal to a specified boundary, use `evaluateHeatRate`.
## Creation
Solve a transient thermal problem using the `solve` function. This function returns a transient thermal solution as a `TransientThermalResults` object.
expand all
## All Transient Thermal Models
Finite element mesh, returned as an FEMesh Properties object.
Temperature values at nodes, returned as a vector or matrix.
Data Types: `double`
Solution times, returned as a real vector. `SolutionTimes` is the same as the `tlist` input to `solve`.
Data Types: `double`
## Non-Axisymmetric Models
x-component of the temperature gradient at nodes, returned as a vector or matrix.
Data Types: `double`
y-component of the temperature gradient at nodes, returned as a vector or matrix.
Data Types: `double`
z-component of the temperature gradient at nodes, returned as a vector or matrix.
Data Types: `double`
## Axisymmetric Models
r-component of the temperature gradient at nodes, returned as a vector or matrix.
Data Types: `double`
z-component of the temperature gradient at nodes, returned as a vector or matrix.
Data Types: `double`
## Object Functions
`evaluateHeatFlux` Evaluate heat flux of a thermal solution at nodal or arbitrary spatial locations `evaluateHeatRate` Evaluate integrated heat flow rate normal to specified boundary `evaluateTemperatureGradient` Evaluate temperature gradient of a thermal solution at arbitrary spatial locations `interpolateTemperature` Interpolate temperature in a thermal result at arbitrary spatial locations
## Examples
collapse all
Solve a 2-D transient thermal problem.
Create a transient thermal model for this problem.
`thermalmodel = createpde('thermal','transient');`
Create the geometry and include it in the model.
```SQ1 = [3; 4; 0; 3; 3; 0; 0; 0; 3; 3]; D1 = [2; 4; 0.5; 1.5; 2.5; 1.5; 1.5; 0.5; 1.5; 2.5]; gd = [SQ1 D1]; sf = 'SQ1+D1'; ns = char('SQ1','D1'); ns = ns'; dl = decsg(gd,sf,ns); geometryFromEdges(thermalmodel,dl); pdegplot(thermalmodel,'EdgeLabels','on','FaceLabels','on') xlim([-1.5 4.5]) ylim([-0.5 3.5]) axis equal```
For the square region, assign these thermal properties:
• Thermal conductivity is $10\text{\hspace{0.17em}}\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$
• Mass density is $2\text{\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$
• Specific heat is $0.1\text{\hspace{0.17em}}\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$
```thermalProperties(thermalmodel,'ThermalConductivity',10, ... 'MassDensity',2, ... 'SpecificHeat',0.1, ... 'Face',1);```
For the diamond region, assign these thermal properties:
• Thermal conductivity is $2\text{\hspace{0.17em}}\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$
• Mass density is $1\text{\hspace{0.17em}}\mathrm{kg}/{\mathrm{m}}^{3}$
• Specific heat is $0.1\text{\hspace{0.17em}}\mathrm{J}/\left({\mathrm{kg}\cdot }^{\circ }\mathrm{C}\right)$
```thermalProperties(thermalmodel,'ThermalConductivity',2, ... 'MassDensity',1, ... 'SpecificHeat',0.1, ... 'Face',2);```
Assume that the diamond-shaped region is a heat source with a density of $4\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{m}}^{2}$.
`internalHeatSource(thermalmodel,4,'Face',2);`
Apply a constant temperature of $0{\phantom{\rule{0.16666666666666666em}{0ex}}}^{\circ }C$ to the sides of the square plate.
`thermalBC(thermalmodel,'Temperature',0,'Edge',[1 2 7 8]);`
Set the initial temperature to 0 °C.
`thermalIC(thermalmodel,0);`
Generate the mesh.
`generateMesh(thermalmodel);`
The dynamics for this problem are very fast. The temperature reaches a steady state in about 0.1 seconds. To capture the interesting part of the dynamics, set the solution time to `logspace(-2,-1,10)`. This command returns 10 logarithmically spaced solution times between 0.01 and 0.1.
`tlist = logspace(-2,-1,10);`
Solve the equation.
`thermalresults = solve(thermalmodel,tlist)`
```thermalresults = TransientThermalResults with properties: Temperature: [1481x10 double] SolutionTimes: [1x10 double] XGradients: [1481x10 double] YGradients: [1481x10 double] ZGradients: [] Mesh: [1x1 FEMesh] ```
Plot the solution with isothermal lines by using a contour plot.
```T = thermalresults.Temperature; pdeplot(thermalmodel,'XYData',T(:,10),'Contour','on','ColorMap','hot')```
Analyze heat transfer in a rod with a circular cross-section and internal heat generation by simplifying a 3-D axisymmetric model to a 2-D model.
Create a transient thermal model for solving an axisymmetric problem.
`thermalmodel = createpde('thermal','transient-axisymmetric');`
The 2-D model is a rectangular strip whose x-dimension extends from the axis of symmetry to the outer surface and whose y-dimension extends over the actual length of the rod (from `-`1.5 m to 1.5 m). Create the geometry by specifying the coordinates of its four corners. For axisymmetric models, the toolbox assumes that the axis of rotation is the vertical axis passing through r = 0.
`g = decsg([3 4 0 0 .2 .2 -1.5 1.5 1.5 -1.5]');`
Include the geometry in the model.
`geometryFromEdges(thermalmodel,g);`
Plot the geometry with the edge labels.
```figure pdegplot(thermalmodel,'EdgeLabels','on') axis equal```
The rod is composed of a material with these thermal properties.
```k = 40; % thermal conductivity, W/(m*C) rho = 7800; % density, kg/m^3 cp = 500; % specific heat, W*s/(kg*C) q = 20000; % heat source, W/m^3```
Specify the thermal conductivity, mass density, and specific heat of the material.
```thermalProperties(thermalmodel,'ThermalConductivity',k,... 'MassDensity',rho,... 'SpecificHeat',cp);```
Specify internal heat source and boundary conditions.
`internalHeatSource(thermalmodel,q);`
Define the boundary conditions. There is no heat transferred in the direction normal to the axis of symmetry (edge 1). You do not need to change the default boundary condition for this edge. Edge 2 is kept at a constant temperature T = 100 °C.
`thermalBC(thermalmodel,'Edge',2,'Temperature',100);`
Specify the convection boundary condition on the outer boundary (edge 3). The surrounding temperature at the outer boundary is 100 °C, and the heat transfer coefficient is $50\text{\hspace{0.17em}}\mathrm{W}/\left(\mathrm{m}{\cdot }^{\circ }\mathrm{C}\right)$.
```thermalBC(thermalmodel,'Edge',3,... 'ConvectionCoefficient',50,... 'AmbientTemperature',100);```
The heat flux at the bottom of the rod (edge 4) is $5000\text{\hspace{0.17em}}\mathrm{W}/{\mathrm{m}}^{2}$.
`thermalBC(thermalmodel,'Edge',4,'HeatFlux',5000);`
Specify that the Initial temperature in the rod is zero.
`thermalIC(thermalmodel,0);`
Generate the mesh.
`generateMesh(thermalmodel);`
Compute the transient solution for solution times from `t = 0` to `t = 50000` seconds.
```tfinal = 50000; tlist = 0:100:tfinal; result = solve(thermalmodel,tlist)```
```result = TransientThermalResults with properties: Temperature: [259x501 double] SolutionTimes: [1x501 double] RGradients: [259x501 double] ZGradients: [259x501 double] Mesh: [1x1 FEMesh] ```
Plot the temperature distribution at `t = 50000` seconds.
```T = result.Temperature; figure pdeplot(thermalmodel,'XYData',T(:,end),'Contour','on') axis equal title(sprintf('Transient Temperature at Final Time (%g seconds)',tfinal))```
Introduced in R2017a
Get trial now | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 10, "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": 0.9033968448638916, "perplexity": 1437.4489737867725}, "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-2020-45/segments/1603107909746.93/warc/CC-MAIN-20201030063319-20201030093319-00467.warc.gz"} |
https://www.physicsforums.com/threads/weight-normal-force-of-a-block.237969/ | # Weight/Normal Force of a block
• Start date
• #1
AnkhUNC
91
0
[SOLVED] Weight/Normal Force of a block
## Homework Statement
If P = 1.98, M = 1, Theta = 45 degrees what is the weight of m in Newtons?
http://img261.imageshack.us/img261/3339/fig450ph7.gif [Broken]
## The Attempt at a Solution
I'm apparently doing this wrong. I was trying to solve for the normal force given that N = Mg/cos theta but this is just 9.8 so I'm doing something wrong. Any advice would be appreciated.
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• #2
Mentor
45,461
1,947
Any further description of the problem? Is the block sliding down the incline?
• #3
AnkhUNC
91
0
P (1.98) is such that the block is not moving.
Thats all I'm given :(
• #4
Mentor
45,461
1,947
Start by identifying the forces acting on each mass. Draw a free body diagram for each.
What can you say about the net force on the block?
• #5
Mentor
45,461
1,947
I was trying to solve for the normal force given that N = Mg/cos theta ...
How did you deduce this?
• #6
AnkhUNC
91
0
N cosθ – mg = ma(y). a(y) = 0 so N = Mg/cos theta?
Forces on triangle block are W (Mass pointed down), P (->), Normal force up mgcos(theta) opposite the angle of the block?
Small block is w pointed straight down and Normal force at an angle opposite (90d?)
• #7
Mentor
45,461
1,947
N cosθ – mg = ma(y). a(y) = 0 so N = Mg/cos theta?
Good. But don't mix M and m: N = mg/cos(theta).
Forces on triangle block are W (Mass pointed down), P (->), Normal force up mgcos(theta) opposite the angle of the block?
Only worry about horizontal forces on the triangle.
Small block is w pointed straight down and Normal force at an angle opposite (90d?)
OK
Keep going. Apply Newton's 2nd law to the horizontal direction and see what you can deduce. Hint: What net horizontal force acts on the block? On both masses?
• #8
AnkhUNC
91
0
P = (M + m) g tanθ?
But this gives me m = .2020408163 so * 9.8 just = 1.98 again which is wrong.
N = Nx i + Ny j = N sinθ i + N cosθ j
–Nsinθ = –mgtanθ
P – mg tanθ = Max
N sinθ = mg tanθ = max
P = (M + m) g tanθ
I guess I'm just having a hard time seeing what I need to get out of this.
F = Mgcos(45) - mgsin(45)? That doesn't even look right...
• #9
Mentor
45,461
1,947
P = (M + m) g tanθ?
No. Just apply Newton's 2nd law to M + m. (Where did you get the g tanθ?)
After you do that, apply it to the block alone.
• #10
AnkhUNC
91
0
Ah its equal to 1! Sweet! I don't know if I did it right or not though. I did N = Mg/cos(45) to find N then N = mg/sin(45) so m = 1!
• #11
Mentor
45,461
1,947
P = (M + m) g tanθ?
Actually, I take it back. This seems correct to me. (I just didn't see how you got. You gave your conclusion first. )
But your data doesn't seem OK.
But this gives me m = .2020408163 so * 9.8 just = 1.98 again which is wrong.
I don't see how you deduced this value for m.
• #12
Mentor
45,461
1,947
Ah its equal to 1! Sweet! I don't know if I did it right or not though. I did N = Mg/cos(45) to find N then N = mg/sin(45) so m = 1!
Where did you get N = Mg/cos(45) = mg/sin(45)? I thought we had established that N = mg/cos(45).
• #13
AnkhUNC
91
0
I was just trying to find a value for N without having m. Like I said I got the right answer but I'm sure I did it wrong
• #14
Mentor
45,461
1,947
What makes you think you got the right answer?
• #15
Homework Helper
25,838
256
N cosθ – mg = ma(y). a(y) = 0 so N = Mg/cos theta?
Hi AnkhUNC!
waaah … you've left out the y-component of the poor little friction force.
When you draw a diagram, you should always mark in all the forces
You need to take components in the normal direction, so that the friction component will be 0.
Then N = ?
• #16
Mentor
45,461
1,947
..the poor little friction force.
Don't cry, tiny-tim! Now I'm starting to cry...
I would assume, lacking any statement to the contrary, that the surfaces are frictionless.
• #17
AnkhUNC
91
0
The answer checked as right but like I said I was sure I did it wrong. If its not moving then N is going to be equal to P?
Any websites you can recommend to help with these types of problems? :(
• #18
Mentor
45,461
1,947
How about the site you're on right now?
Just attack it systematically, as I suggest in post #7. (I assume the surfaces are frictionless, correct?)
(I'd say that given the data you supplied, there is no correct answer. But you can solve for m symbolically in terms of P and M.)
• #19
AnkhUNC
91
0
Yeah there isn't a frictional force. I think the problem was just to get me thinking about the process and I spent way too much time on it :P
• #20
Mentor
45,461
1,947
Just do it step by step. Analyze the forces and apply Newton's 2nd law:
(1) To the block (vertical direction)
(2) To the block (horizontal direction)
(3) To the entire system (horizontal direction)
• #21
Homework Helper
25,838
256
Don't cry, tiny-tim! Now I'm starting to cry...
I would assume, lacking any statement to the contrary, that the surfaces are frictionless.
Yes, looking at the diagram again, I think you're right!
Presumably the whole system is accelerating, but the little block is not moving relative to the big block, so resolving vertically was correct after all.
oh … I'm so much more cheerful now!
Thanks, Doc Al!
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429 | {"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": 0.8673970699310303, "perplexity": 1790.0585100363774}, "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/1674764499541.63/warc/CC-MAIN-20230128090359-20230128120359-00169.warc.gz"} |
https://projecteuclid.org/euclid.die/1399395747 | Differential and Integral Equations
Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball
Abstract
The normalized or game-theoretic $p$-Laplacian operator given by $$-\Delta_p^Nu:=-\frac{1}{p}|\nabla u|^{2-p}\Delta_p(u)$$ for $p\in(1,\infty)$ with $\Delta_pu=\rm{div}(|\nabla u|^{p-2}\nabla u)$ has no apparent variational structure. Showing the existence of a first (positive) eigenvalue of this fully nonlinear operator requires heavy machinery as in [6]. If it is restricted to the class of radial functions, however, the normalized $p$-Laplacian transforms into a linear Sturm-Liouville operator. We investigate radial eigenfunctions to this operator under homogeneous Dirichlet boundary conditions and come up with an explicit complete orthonormal system of Bessel functions in a suitably weighted $L^2$-space. This allows us to give a Fourier-series representation for radial solutions to the corresponding evolution equation $u_t-\Delta_p^Nu=0$.
Article information
Source
Differential Integral Equations, Volume 27, Number 7/8 (2014), 659-670.
Dates
First available in Project Euclid: 6 May 2014
Kawohl, Bernd; Krӧmer, Stefan; Kurtz, Jannis. Radial eigenfunctions for the game-theoretic $p$-Laplacian on a ball. Differential Integral Equations 27 (2014), no. 7/8, 659--670. https://projecteuclid.org/euclid.die/1399395747 | {"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": 0.9807616472244263, "perplexity": 1052.0777916474694}, "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-2019-04/segments/1547583897417.81/warc/CC-MAIN-20190123044447-20190123070447-00346.warc.gz"} |
https://scicomp.stackexchange.com/questions/7273/is-stabilization-of-energy-equation-needed-when-momentum-equation-needs-it/7276 | # Is stabilization of energy equation needed when momentum equation needs it?
When SUPG/PSPG stabilization is added to momentum equation of flow problem, is needed stabilization for energy equation also? I would guess that when stabilization for velocity works fine so one gets velocity without spurious wiggles and assuming that thermal diffusivity is large enough then answer is no. How are these two conditions quantified in the language of dimensionless quantites?
$$\mathrm{Pe}_h = h v / K$$
where $h$ is mesh size, $v$ is the magnitude of velocity, and $K$ is diffusivity. It is analogous to cell Reynolds number for the momentum equation and is small when "thermal diffusivity is large compared to advection". It is common common in macro-scale fluid dynamics that thermal diffusivity $K$ is very small. If the cell Péclet is much larger than 1, you need stabilization. | {"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": 0.9776221513748169, "perplexity": 744.2919784623465}, "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-2021-21/segments/1620243988831.77/warc/CC-MAIN-20210508001259-20210508031259-00023.warc.gz"} |
https://www.physicsforums.com/threads/div-g_ab-0.239558/ | # Div G_ab = 0
• #1
716
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## Main Question or Discussion Point
What physical meaning can be ascribed to the non-divergence of the Einstein tensor? I find it counterintuitive since I associate divergence with field sources (like the electrical field of a proton) and obviously a gravitational field has a source. Is there a parallel with Newton's formulation of gravity that might be instructive?
Last edited:
Related Special and General Relativity News on Phys.org
• #2
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What physical meaning can be ascribed to the non-divergence of the Einstein tensor? I find it counterintuitive since I associate divergence with field sources (like the electrical field of a proton) and obviously a gravitational field has a source. Is there a parallel with Newton's formulation of gravity that might be instructive?
Since the Einstein tensor is proportional to the stress-energy-momentum tensor, T it means that energy and momentum is conserved since div T = 0. This holds true even when the cosmological constant is non-zero.
Pete
• #3
haushofer
2,267
621
What physical meaning can be ascribed to the non-divergence of the Einstein tensor? I find it counterintuitive since I associate divergence with field sources (like the electrical field of a proton) and obviously a gravitational field has a source. Is there a parallel with Newton's formulation of gravity that might be instructive?
You can also see them as constraints on the equations of motion. Intuïtively you can understand them as follows: in general relativity one wants to solve for the metric tensor, which is symmetric and thus can have 10 independent entries ( n*(n+1)/2 ). However, one is free to choose the coordinates, and this gives some freedom in your equations ( which can be seen as a gauge-freedom ). The consequences of this can be calculated to be the Bianchi-identities, which give that the Einstein tensor is divergence-free. Note that this is an identity rather than a symmetry; the description of physics doesn't depend on your coordinates.
A parallel with Newton's formulation is a little tricky; but you have to be aware of the fact that the Einstein tensor already contains second order derivatives of the metric, just like Poisson's equation is a second order differential equation of the classical gravitational field ! A sensible parallel would appear to me that due to constraints on the gravitational field the third order divergence of the classical gravitational field would be zero.
• #4
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So Einstein created the stress-energy tensor first, then made $$G_{ab}$$ to match it?
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https://hpi.de/friedrich/news/2015/two-papers-nominated-for-best-paper-award-at-gecco-2015.html | Prof. Dr. Tobias Friedrich
12.06.2015
# Two papers nominated for best paper award of the Genetic and Evolutionary Computation Conference (GECCO 2015)
### Robustness of Ant Colony Optimization to Noise
Tobias Friedrich, Timo Kötzing, Martin S. Krejca, Andrew M. Sutton
Track: ACO
Abstract: Recently Ant Colony Optimization (ACO) algorithms have been proven to be efficient in uncertain environments, such as noisy or dynamically changing fitness functions. Most of these analyses focus on combinatorial problems, such as path finding.
We analyze an ACO algorithm in a setting where we try to optimize the simple OneMax test function, but with ad- ditive posterior noise sampled from a Gaussian distribu- tion. Without noise the classical (μ + 1)-EA outperforms any ACO algorithm, with smaller μ being better; however, with large noise, the (μ + 1)-EA fails, even for high val- ues of μ (which are known to help against small noise). In this paper we show that ACO is able to deal with arbitrarily large noise in a graceful manner, that is, as long as the evaporation factor ρ is small enough dependent on the parameter σ2 of the noise and the dimension n of the search space (ρ = o(1/(n(n + σ log n)2 log n))), optimization will be successful.
### Improved Runtime Bounds for the (1+1) EA on Random 3-CNF Formulas Based on Fitness-Distance Correlation
Benjamin Doerr, Frank Neumann, Andrew M. Sutton
Track: Theory
Abstract: With this paper, we contribute to the theoretical understanding of randomized search heuristics by investigating their behavior on random 3-SAT instances. We improve the results for the (1+1) EA obtained by Sutton and Neumann [PPSN 2014, 942--951] in three ways. First, we reduce the upper bound by a linear factor and prove that the (1+1) EA obtains optimal solutions in time O(n log n) with high probability on asymptotically almost all high-density satisfiable 3-CNF formulas. Second, we extend the range of densities for which this bound holds to satisfiable formulas of at least logarithmic density. Finally, we complement these mathematical results with numerical experiments that summarize the behavior of the (1+1) EA on formulas along the density spectrum, and suggest that the implicit constants hidden in our bounds are low.
Our proofs are based on analyzing the run of the algorithm by establishing a fitness-distance correlation. This approach might be of independent interest and we are optimistic that it is useful for the analysis of randomized search heuristics in various other settings. To our knowledge, this is the first time that fitness-distance correlation is explicitly used to rigorously prove a performance statement for an evolutionary algorithm. | {"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": 0.9361720085144043, "perplexity": 1252.4957938075252}, "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/1480698541839.36/warc/CC-MAIN-20161202170901-00097-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://plainmath.net/34938/given-that-loga-5-approx0-65-and-loga-3-approx0-44-evaluate-e | # Given that \log_a(5)\approx0.65 and \log_a(3)\approx0.44, evaluate e
Given that ${\mathrm{log}}_{a}\left(5\right)\approx 0.65$ and ${\mathrm{log}}_{a}\left(3\right)\approx 0.44$, evaluate each of the following. Hint: use the properties of logarithms to rewrite the given logarithm in terms of the logarithms of 5 and 3
a)${\mathrm{log}}_{a}\left(0.6\right)$
b)${\mathrm{log}}_{a}\left(\sqrt{3}\right)$
c)${\mathrm{log}}_{a}\left(15\right)$
d)${\mathrm{log}}_{a}\left(25\right)$
e)${\mathrm{log}}_{a}\left(75\right)$
f)${\mathrm{log}}_{a}\left(1.8\right)$
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jlo2niT
Step1
${\mathrm{log}}_{a}\left(5\right)\approx 0.65{\mathrm{log}}_{a}\left(3\right)\approx 0.44$
properties of logarithms are
1.$\mathrm{log}\left(ab\right)={\mathrm{log}}_{c}\left(a\right)+{\mathrm{log}}_{c}\left(b\right)$
2.${\mathrm{log}}_{c}\left(\frac{a}{b}\right)={\mathrm{log}}_{c}\left(a\right)-{\mathrm{log}}_{c}\left(b\right)$
3.${\mathrm{log}}_{c}\left({a}^{b}\right)=b{\mathrm{log}}_{c}a$
Step2
${\mathrm{log}}_{a}\left(0.6\right)={\mathrm{log}}_{a}\left(\frac{3}{5}\right)$
${\mathrm{log}}_{a}\left(0.6\right)={\mathrm{log}}_{a}\left(3\right)-{\mathrm{log}}_{a}\left(5\right)$
${\mathrm{log}}_{a}\left(0.6\right)=0.44-0.65$
${\mathrm{log}}_{a}\left(0.6\right)=-0.21$
Step3
${\mathrm{log}}_{a}\left(\sqrt{3}\right)={\mathrm{log}}_{a}\left({3}^{\frac{1}{2}}\right)$
${\mathrm{log}}_{a}\left(\sqrt{3}\right)=\frac{1}{2}{\mathrm{log}}_{a}\left(3\right)$
${\mathrm{log}}_{a}\left(\sqrt{3}\right)=\frac{1}{2}\cdot 0.44$
${\mathrm{log}}_{a}\left(\sqrt{3}\right)=0.22$
Step4
${\mathrm{log}}_{a}\left(15\right)={\mathrm{log}}_{a}\left(5\cdot 3\right)$
${\mathrm{log}}_{a}\left(15\right)={\mathrm{log}}_{a}\left(5\right)+{\mathrm{log}}_{a}\left(3\right)$
${\mathrm{log}}_{a}\left(15\right)=0.65+0.44$
${\mathrm{log}}_{a}\left(15\right)=1.09$
Step5
${\mathrm{log}}_{a}\left(25\right)={\mathrm{log}}_{a}\left({5}^{2}\right)$
${\mathrm{log}}_{a}\left(25\right)=2{\mathrm{log}}_{a}\left(5\right)$
${\mathrm{log}}_{a}\left(25\right)=2\cdot 0.65$
${\mathrm{log}}_{a}\left(25\right)=1.3$
Step6
${\mathrm{log}}_{a}\left(75\right)={\mathrm{log}}_{a}\left(25\cdot 3\right)$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 46, "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": 0.9229051470756531, "perplexity": 1529.086283615926}, "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-2022-33/segments/1659882571692.3/warc/CC-MAIN-20220812105810-20220812135810-00744.warc.gz"} |
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2335282/?tool=pubmed | • We are sorry, but NCBI web applications do not support your browser and may not function properly. More information
BMC Bioinformatics. 2008; 9: 117.
Published online 2008 February 25.
PMCID: PMC2335282
# A mixture model approach to sample size estimation in two-sample comparative microarray experiments
## Abstract
### Background
Choosing the appropriate sample size is an important step in the design of a microarray experiment, and recently methods have been proposed that estimate sample sizes for control of the False Discovery Rate (FDR). Many of these methods require knowledge of the distribution of effect sizes among the differentially expressed genes. If this distribution can be determined then accurate sample size requirements can be calculated.
### Results
We present a mixture model approach to estimating the distribution of effect sizes in data from two-sample comparative studies. Specifically, we present a novel, closed form, algorithm for estimating the noncentrality parameters in the test statistic distributions of differentially expressed genes. We then show how our model can be used to estimate sample sizes that control the FDR together with other statistical measures like average power or the false nondiscovery rate. Method performance is evaluated through a comparison with existing methods for sample size estimation, and is found to be very good.
### Conclusion
A novel method for estimating the appropriate sample size for a two-sample comparative microarray study is presented. The method is shown to perform very well when compared to existing methods.
## Background
One of the most frequently used experimental setups for microarrays is the two-sample comparative study, i.e. a study that compares expression levels in samples from two different experimental conditions. In the case of replicated two-sample comparisons statistical tests may be used to assess the significance of the measured differential expression. A natural test statistic for doing so is the t-statistic (see e.g. [1]), which will be our focus here. In the context of two-sample comparisons it is also convenient to introduce the concept of 'effect size'. In this paper effect size is taken to mean: the difference between two conditions in a gene's mean expression level, divided by the common standard deviation of the expression level measurements.
In an ordinary microarray experiment thousands of genes are measured simultaneously. Performing a statistical test for each gene leads to a multiple hypothesis testing problem, and a strategy is thus needed to control the number of false positives among the tests. A successful approach to this has been to control the false discovery rate (FDR) [2], or FDR-variations like the positive false discovery rate (pFDR) [3].
To obtain the wanted results from an experiment it is important that an appropriate sample size, i.e. number of biological replicates, is used. A goal can, for example, be set in terms of a specified FDR and average power, and a sample size chosen so that the goal may be achieved [4].
In the last few years many methods have been suggested that can help estimate the needed sample size. Some early approaches [5,6] relied on simulation to see the effect of sample size on the FDR. Later work established explicit relationships between sample size and FDR. A common feature of the more recent methods is that they require knowledge of the distribution of effect sizes in the experiment to be run. In lack of this distribution there are two alternatives. The first alternative is simply specifying the distribution to be used. The choice may correspond to specific patterns of differential expression that one finds interesting, or it can be based on prior knowledge of how effect sizes are distributed. Many of the available methods discuss sample size estimates for specified distributions [7-11]. The second alternative is estimating the needed distribution from a pilot data set. Ferreira and Zwinderman [12] discuss one such approach. Assuming that the probability density functions for the test statistics are symmetric and belonging to a location family, they obtain the wanted distribution using a deconvolution estimator. One should note that, for the sample sizes often used in a microarray experiment, the noncentral density functions for t-statistics depart from these assumptions. Hu et al. [13] and Pounds and Cheng [14] discuss two different approaches. Both methods recognize that test statistics for differentially regulated genes are noncentrally distributed, and aim to estimate the corresponding noncentrality parameters. From the noncentrality parameters, effect sizes can be found. Hu et al. consider, as we do, t-statistics and estimate the noncentrality parameters by fitting a 3-component mixture model to the observed statistics of pilot data. Pounds and Cheng consider F-statistics and estimate a noncentrality parameter for each observation. They then rescale the estimates according to a Bayesian q-value interpretation [15]. A last approach that needs mention is that of Pawitan et al. [16], which fits a mixture model to observed t-statistics using a likelihood-based criterion. The approach is not explored as a sample size estimation method in the paper by Pawitan et al., but it can be put to this use.
In this article we introduce a mixture model approach to estimating the underlying distribution of noncentrality parameters, and thus also effect sizes, for t-statistics observed in pilot data. The number of mixture components used is not restricted, and we present a novel, closed form, algorithm for estimating the model parameters. We then demonstrate how this model can be used for sample size estimation. By examining the relationships between FDR and average power, and between FDR and the false nondiscovery rate (FNR), we are able to choose sample sizes that control these measures in pairs. To validate our model and sample size estimates, we test its performance on simulated data. We include the estimates made by the methods of Hu et al., Pounds and Cheng and Pawitan et al.
## Results and Discussion
### Notation, assumptions and test statistics
Throughout this text tν (λ) represents the probability density function (pdf) of a t-distributed random variable with ν degrees of freedom and noncentrality parameter λ. A central t pdf, tν (0), can also be written tν. A tν (λ) evaluated at x is written tν (x; λ).
Assume gene expression measurements can be made in a pilot study. For a particular gene we denote the n1 measurements from condition 1 and the n2 from condition 2 by X1i, (i = 1, ..., n1) and X2j, (j = 1, ..., n2). Let (μ1, $σ12$) and (μ2, $σ22$) be expectation and variance for each X1i and X2j, respectively. For simplicity we focus in this paper on the case where $σ12=σ22=σ2$. As is common in microarray data analysis, the X1is and X2js are assumed to be normally distributed random variables.
Measured expression levels are often transformed before normality is assumed.
A statistic frequently used to detect differential expression in this setting, is the t-statistic. Two versions of t-statistics can be used, depending on the experimental setup. In the first setup, measurements for each condition are made separately. Inference is based on the two-sample statistic
$T1=n1n2(n1+n2)−1(X¯1−X¯2)(n1+n2−2)−1[(n1−1)S12+(n2−1)S22],$
(1)
where $X¯k=nk−1∑i=1nkXki$ and $Sk2=(nk−1)−1∑i=1nk(Xki−X¯k)2$. Under the null hypothesis H0 : μ1 = μ2, T1 has a $tn1+n2−2$ pdf. If, however, H0 is not true, and there is a difference μ1 - μ2 = ξ, the pdf of the statistic is a $tn1+n2−2(ξσ−1n1n2(n1+n2)−1$. This setup includes comparing measurements from single color arrays and two-color array reference designs. In a second kind of experiment, measurements are paired. In the case of, for example, n two-color slides that compare the two conditions directly, then n1 = n2 = n, and the statistic used is
$T2=d¯Sd2/n,$
(2)
where $d¯=n−1∑i=1n(X1i−X2i)$ and $Sd2=(n−1)−1∑i=1n(di−d¯)2$. Now, under H0 : μ1 = μ2, T2 has as a tn - 1 pdf. If μ1 - μ2 = ξ, however, the pdf of T2 is a tn - 1 (ξ σ-1 $n$).
In both experimental setups we note that the pdf of t-statistics for truly unregulated genes is tν (0). For truly regulated genes the pdf is tν (δ), with δ ≠ 0 reflecting the gene's level of differential expression. We also note that this δ is proportional to the gene's effect size, ξ/σ . The δs can be considered realizations of some underlying random variable Δ, distributed as h(δ). Under our assumptions the observed t-scores should thus be modelled as a mixture of tν (δ)-distributions, with the h(δ) as its mixing distribution. The h(δ) is not directly observed and must be estimated.
In the following, the t-statistics calculated in an experiment are assumed to be independent. This assumption, and the assumption that $σ12=σ22=σ2$, may not hold in the microarray setting. In the Testing section we examine cases where these assumptions are not satisfied to see how results are affected.
### Algorithm for estimating effect sizes
Let yj, j = 1, ..., m denote observed t-statistics for the m genes of an experiment, having Yjs as corresponding random variables. Let f(y) be their pdf. Our mixture model can then generally be stated as
f(y; h) = ∫ tν (y; δ) dh(δ),
where h(δ) is any probability measure, discrete or continuous. To estimate h(δ) we want to find a probability measure that maximizes the likelihood, L(h), of our observations, where
$L(h)=∏i=1mf(yi;h).$
It has been shown [17] that to solve this maximization problem, when L(h) is bounded, it is sufficient to consider only discrete probability measures with m or fewer points of support. Motivated by this we choose h(δ) to be a discrete probability measure, and aim to fit a mixture model of the form
$f(y)=∑i=0gπitν(y;δi)=π0tν(y;0)+∑i=1gπitν(y;δi).$
(3)
The h(δ) is thus a distribution where Pr(δ = δi) = πi, (i = 0, ..., g), and $∑i=0gπi=1$. The second form of f(y) in (3) is due to knowing that δ = 0 for unregulated genes.
We now aim to find the parameters of a model like (3) with a fixed number of components g + 1. It is clear that finding these parameters can be formulated as a missing data problem, which suggests the use of the EM-algorithm [18]. Although this approach has been discussed in earlier work [13,16], a closed form EM-algorithm that solves the problem has not been available until now. The main difficulty with constructing the algorithm is the lack of a closed form expression for the noncentral t pdf. In the remainder of this section we show how the needed algorithm can be obtained.
As is usual with EM, random component-label vectors Z1, ..., Zm are introduced that define the origin of Y1, ..., Ym. These have Zij = (Zj)i equal to one or zero according to whether or not Yj belongs to the ith component. A Zj is distributed according to
$Pr(Zj=zj)=π0z0jπ1z1j⋯πgzgj,$
where the zj is a realized value of the random Zj.
We proceed by recognizing the fact that a noncentral t pdf of is itself a mixture. The cumulative distribution of a variable distributed according to tν (δ) is (see [19])
$Fν(y;δ)=12ν2Γ(ν2)∫0∞vν−1e−v2212π∫−∞yvνe−12(s−δ)2dsdv.$
Differentiating Fν (y) with respect to y, and substituting $v=νu$, yields
$Fν(y;δ)=∫0∞u2πe−12(y−δu)2u2ν2ν2−1Γ(ν2)(νu)ν−1e−νu22du.$
(4)
This form of the noncentral t pdf can be identified as a mixture of normal $N(δu,1u2)$ distributions, with a scaled χν mixing distribution for the random variable U. Based on the characterization in (4) we can introduce a new set of missing data uij, (i = 0, ..., g; j = 1, ..., m) that are realizations of Uij, and defined so that (Yj|uij, zij = 1) follows a $N(δuij,1uij2)$ distribution. Restating the model in this form, as a mixture of mixtures, is a vital step in finding the closed form algorithm.
The yjs augmented by the zijs and uijs form the complete-data set. The complete-data log-likelihood may be written
$logLc(π,δ)=∑i=0g∑j=1mzijlog(πifc(yj|uij,zij=1)gc(uij)),$
(5)
where π = (π1, ..., πg), δ = (δ1, ..., δg) and fc(y|u, z) and gc(u) are the above-mentioned normal and scaled χν distribution, respectively. The E-step of the EM-algorithm requires the expectation of Lc in (5) conditional on the data. Combining (4) with (5) we find that we need the expectations
E(Zij|yj) and E(Uij|yj, zij = 1).
Calculating the first expectation is straightforward (see e.g. [20]) and is found to be
$E(Zij|yj)=πitν(yj;δi)∑i=0gπitν(yj;δi).$
(6)
Calculating the second expectation is harder, but by using Bayes theorem we find that it can be stated as (now omitting indices for clarity)
$E(Uij|yj,zij=1)=∫0∞ufc(u|y,z)du$
(7)
$=∫0∞ufc(y|u,z)gc(u)∫0∞fc(y|u′,z)gc(u′)du′du$
(8)
where fc(u|y, z) is the pdf of (Uij|yj, zij = 1). Note that gc(u|z) = gc(u) since Uij and Zij are independent.
The integral in (8) must now be evaluated. To do this, we note that the denominator of the integrand is itself an integral and that it does not depend on the integrating variable u. In effect, (8) is thus the ratio of two integrals. After a substitution of variables in both integrals, $(w=uy2+ν)$ and $(w=u′y2+ν)$, we find that this ratio can be rewritten in terms of H h-functions as
$E(Uij|yj,zij=1)=ν+1yj2+νHhv+1(−yjδiyj2+v)Hhv(−yjδiyj2+v),$
(9)
where $Hhk(x)=∫0∞wkk!e−12(w+x)2$dw for an integer k ≥ 0. The properties of the H hk-functions are discussed in [21]. A particularly nice property is that it satisfies the recurrence relation
(k + 1) H hk + 1 (x) = - x H hk (x) + H hk - 1 (x).
With easily calculated $Hh−1(x)=e−12x2$, $Hh0(x)=∫x∞e−12u2du$, we have a convenient way of computing (9).
The M-step requires maximizing the Lc with respect to π and δ. This is accomplished by simple differentiation and yields maximizers
$π^i=1m∑j=1mzij$
(10)
$δ^i=∑j=1mzijyjuij∑j=1mzij.$
(11)
Equations (6) and (9) – (11) constitute the backbone of the needed closed form EM-algorithm to fit a mixture model like (3) with g + 1 components. Parameter estimates are updated according to the scheme
$i.)uij(k)=Eπ^(k)δ^(k)(Uij|yj,zij=1),zij(k)=Eπ^(k)δ^(k)(Zij|yj)ii.)π^i(k+1)=1m∑j=1mzij(k),δ^i(k)=∑j=1mzij(k)yjuij(k)∑j=1mzij(k).$
On convergence, the estimated π and δ are used as the parameters of a h(δ) with g + 1 point masses. As discussed above we fix δ0 = 0.
An issue that has received much attention is estimating the proportion of true null hypotheses when many hypothesis tests are performed simultaneously (e.g. [3,22,23]). In the microarray context this amounts to estimating the proportion of truly unregulated genes among all genes considered. Referring to (3) we see that this quantity enters our model as π0. To draw on the extensive work on π0-estimation, we suggest using a known conservative π0-estimate to guide the choice of some model parameters. This is discussed below. In our implementation we use the convex decreasing density estimate proposed in [24], but a different estimate may be input.
Assessing the appropriate number of components in a mixture model is a difficult problem that is not completely resolved. An often adopted strategy is using measures like the Akaike Information Criterion (AIC) [25] or the Bayesian Information Criterion (BIC) [26]. We find from simulation that, in our setting, these criteria seem to underestimate the needed number of components (refer to the Testing section for some of the simulation results). This is possibly due to the large proportion of unregulated genes often found in microarray data. With relatively few regulated genes, the gain in likelihood from fitting additional components is, for these criteria, not enough to justify the additional parameters used. In our implementation we use g = log2((1 - $π^0$) m), where $π^0$ is the above-mentioned estimate. This choice is motivated by experience, but has proven itself adequate in our numerical studies. It also reflects the fact that a single component should provide sufficient explanation for the unregulated genes, while the remaining g components explain the regulated ones. A different g may be specified by users of the sample size method.
A complication that could arise when fitting the mixture model is that one or more of the ${δi}i=1g$, could be assigned to δ = 0, or very close to it, thus affecting the fit of the central component. To avoid this, we define a small neighbourhood around δ0 = 0 from which all g remaining components are excluded. A δi, i ≠ 0, that tries crossing into this neighbourhood while fitting the model, is simply halted at the neighbourhood boundary. The boundary is determined by finding the smallest $δ˜$ for which it is possible to tell apart the tν (0) distribution from a tν ($δ˜$) one, based on samples of sizes $π^0m$ and (1 - $π^0$) m/g, respectively. The latter sample size assumes regulated genes to be evenly distributed among their components. The samples are taken as evenly spaced points within the 0.05 and 0.95 quantiles of the two distributions. The criterion used to check if the two samples originated from the same distribution is a two-sample Kolmogorov-Smirnov test with a significance level of 0.05. The rationale behind this criteron is that, for the g components associated with regulated genes, we only want to allow those that with reasonable certainty can be distinguished from the central component. Again, the $π^0$ used is the estimate discussed above.
Another difficulty related to fitting a mixture model is that the optimal fit is not unique, something which might cause convergence problems. The difficulty is due to the fact that permuting mixture components does not change the likelihood function. In our implementation we do not have any constraints that resolve this problem. We did, however, track the updates of our mixture components in a number of test runs and did not see this problem occur.
In summary, our approach provides estimates, ${δ^i}i=0g$ and ${π^i}i=0g$, of the noncentrality parameters in the data and a set of weights. Together these quantities make up an estimate of the distribution h(δ). As seen in the section on test-statistics a δ is proportional to the effect size, ξ/σ. No estimates or assumptions are thus made on the numerical size of the means or variances in the data. We only estimate a set of mean shift to variance ratios.
### Algorithm for estimating sample sizes for FDR control
An important issue in experimental design is estimating the sample size required for the experiment to succeed. We now outline how to choose sample sizes that control FDR together with other measures, and how the model discussed above can be used for this purpose.
Table Table11 summarizes the possible outcomes of m hypothesis tests. All table elements except m are random variables. From Table Table11 the much used positive false discovery rate (pFDR) [3] can be defined as E(R0/MR|MR > 0). In [3] it is also proven that, for a given significance region and under assumed independence for the tests, we have
Outcomes of m hypothesis tests
pFDR = Pr(H0 true|H0 rejected),
(12)
and that, in terms of p-values p and a chosen p-value cutoff α, (12) can be rewritten as
$pFDR(α)=π0αPr(p<α).$
(13)
Equation (13) is important in sample size estimation as it provides the relationship between pFDR, significance region and sample size. Sample size will determine Pr(p <α) since the shape of the t-distributions depends on n1, n2. (An explanation is provided at the end this section). The application of (13) to sample size estimation was first discussed by Hu et al. [13].
Using (13), and a fitted mixture model, one can estimate the sample size that achieves a specified α and pFDR. The remaining issue is choosing an appropriate α and pFDR. The pFDR is an easily understood measure, and its size can be set directly by users of the sample size estimation method. How to pick α, on the other hand, is not as clear. One solution to this is to restate (13) as relationships between α, pFDR and other statistical measures. Hu et al. present one such relationship. They suggest picking the α by specifying the expected number of hypotheses to be rejected, E(MR). Their idea is substituting Pr(p <α) = E(MR)/m in (13) to get
$α=pFDRπ0E(MR)m.$
(14)
In words, instead of specifying an α, one can specify E(MR). This way of obtaining α, however, has a shortcoming. It provides little direct information to the user about the experiment's ability to recognize regulated genes. In our view, a more informative way to choose α would be to let the user specify quantities such as average power or the false nondiscovery rate (FNR). We now discuss how this can be accomplished.
Power is defined as the probability of rejecting a hypothesis that is false. In the microarray multiple hypothesis setting, average power controls the proportion of regulated genes that is correctly identified. Setting α through an intuitively appealing measure as average power would thus be helpful. A relationship between α and average power can be found by rewriting the denominator of the right side in (13) as
Pr(p <α) = Pr(p <α|H0 true)π0 + Pr(p <α|H0 false)(1 - π0).
Recognizing Pr(p <α|H0 false) as average power, (13) can be inverted to find
(15)
Combining (15) and (13) one can now find sample sizes that achieve a specified pFDR and average power, with no need of specifying α.
Another interesting measure to control is the false nondiscovery rate (FNR), the expected proportion of false negatives among the hypotheses not rejected. In other words, the FNR controls the proportion of regulated genes erroneously accepted as unregulated. We use a version of the FNR discussed in [15] called the pFNR = E(A1/MA|MA > 0), which, under the same assumptions as for the pFDR, can be stated as
pFNR = Pr(H0 false|H0 accepted).
Rewriting this probability in terms of pFDR and α yields
$α=1−pFDR−π01−pFNR−pFDR.$
(16)
Again, specifying a pFNR will correspond to a specific choice of α. The pFNR approach to setting α could be interesting to use. One should note, however, that in the microarray setting this measure can sometimes be hard to apply. The reason is the potentially large MA, due to a high proportion of unregulated genes. A large MA makes pFNR numerically small, and a reasonable size may be hard to set.
Having chosen a pFDR and α (from average power or pFNR), we need to find a sample size that solves (13). To do this, we express Pr(p <α) via our mixture model (3) as
$Pr(p<α)=∫y>|y0|f(y;h)dy,$
(17)
where y0 is the t-score corresponding to a p-value cutoff α for testing the null hypothesis. Since f(y) is a weighted sum of tν (δ)s, the integrals needed to be taken are simply quantiles of (noncentral) t-distributions. Sample size affects (17) because all tν (δ)s to be integrated are on the form $ts1(n)(ξiσi−1s2(n))$, with s1(n) and s2(n) dependent on n1 and n2. (Refer to pdfs of (1) and (2)). If the effect sizes ${ξiσi−1}i=0g$ and the weights ${πi}i=0g$ were known we could evaluate (17) for all s1(n), s2(n).
These parameters can, however, be found from the fitted model. To obtain the effect sizes we can equate $δi=ξiσi−1s2(n˜),i=(0,...,g)$, where ñ is the sample size used in the fit. The weights are estimated directly. Having expressed Pr(p <α) in terms of sample size we aim to find one that solves (13). This problem can be reformulated as finding the root of the function
$S(n)=pFDR∑i=0g(πi∫y>|y0|ts1(n)(ξiσi−1s2(n))dy)−π0α.$
For finding the root of S(n) we implement a bisection method.
### Testing
To evaluate our approach we implemented the EM-algorithm and the sample size estimation method described above. We then ran tests on simulated data sets. For reasons discussed above we focused on controlling average power, along with the pFDR, in our sample size estimates.
When using simulated data sets it is possible to calculate the true sample size needed to achieve a given combination of pFDR and average power. To evaluate the performance of our method we compared our estimates to the true values. For comparison with existing approaches we also included the estimates made by the methods of Hu et al. [13], Pounds and Cheng [14] and Pawitan et al. [16].
#### Use of existing methods
In their paper Hu et al. discuss three different mixture models. In our comparison we used their truncated normal model, as this seemed to be the favored one. To produce sample size estimates using this model one needs to input the wanted pFDR and E(MR). As we wished to control pFDR along with average power, we calculated the E(MR) corresponding to each choice of pFDR and average power as E(MR) = average power·m(1 - π0)/(1 – pFDR). All tests were run with default parameters as found in the source code. In the implementation of Hu et al., however, three parameters (diffthres, iternum, gridsize) were missing default values. Reasonable choices (0.01,10,100) were kindly provided by the authors.
For the method of Pounds and Cheng one needs to input quantities called anticipated false discovery ratio (aFDR) and anticipated average power. For the estimates presented here we used the corresponding pFDR and average power combination. As Pounds and Cheng work with F-statistics, the t-statistics calculated in our tests were transformed accordingly. (If T follows a tν distribution, then T2 is distributed as F1,ν). In their implementation Pounds and Cheng set an upper limit, nmax, on the sample size estimates, and replace all estimates above nmax with nmax itself. The default value of nmax = 50 was replaced with nmax = 1000 in our tests. The reason for this was that sample size estimates of more than 50 could, and did, occur in the tests. Using a low nmax would then affect the comparison to the other methods. Apart from nmax all tests were run with default parameters as found in the source code.
In [16] Pawitan et al. discuss a method for fitting a mixture model to t-statistics. The fitted model is used to make an estimate of the proportion of unregulated genes, π0. The use of their method for sample size estimation is mentioned, but is not further explored or tested. The only input needed to fit a model is the number of mixture components. In our tests, this number was determined using the AIC, as suggested by Pawitan et al. in their paper. In the implementation of Pawitan et al. the assignment of non-central components close to the central one is not restricted. A preliminary test run using their unadjusted model fit showed that sample size estimates were greatly deteriorated by this. In our tests we therefore adjusted their fitted model by collapsing all non-central components within a given threshold (|δ| < 0.75) into the central one. Our model adjustment corresponds to the π0-estimation procedure used in the implementation of Pawitan et al. For our test runs we used our procedures to produce sample size estimates based on the models fitted by this method.
#### Test procedure and results
For the test results presented here we used m = 10000 genes, and n1 = n2 = 5 measurements per group. For the proportion of unregulated genes we examined the cases of π0 = 0.7 and π0 = 0.9.
In a first set of tests we considered sample size estimates in the case of normally distributed measurements and equal variances. In this setting we simulated data with and without a correlation structure. The true distribution of noncentrality parameters for the m1 = (1 - π0)m regulated genes was generated in the following way: A random sample was drawn from a N(1,0.52) and a N(-1,0.52) distribution. Both samples were of size m1/2, and together they made up the ${ξk}k=1m1$ for the regulated genes in data set. The measurement variances were set to $σ2=σ12=σ22=0.52$. The noncentrality parameters of the regulated genes were then calculated as ${ξkσ−1n1n2(n1+n2)−1}k=1m1$ (Refer to discussion of (1)). Based on our experience with microarray data analysis the above choices seem plausible for log-transformed microarray measurements. Since the true noncentrality parameters are all known, the true sample size needed to achieve a particular pFDR and average power can be calculated.
Correlation was introduced using a block correlation structure with block size 50. The reasoning behind such a structure was discussed in [27]. All genes, regulated and unregulated, were randomly assigned to their blocks. A correlation matrix for each group was then generated by first sampling random values from a uniform U(-1, 1) distribution into the off-diagonal elements of a symmetric matrix. Then, using the iteration procedure described in [28], we iterated to find the positive semidefinite matrix with unit diagonal that was closest to our randomly generated one.
Using the above approach we simulated data. For each test case we generated 50 data sets and made sample size estimates based on these. In an initial test run we wanted to evaluate our choice of using a larger number of mixture components, g, than what is suggested by the AIC or BIC criterion. To do so, two models were fitted to each simulated data set using our algorithm. One model had our chosen number of mixture components, the other had the number indicated by the AIC. Sample size estimates were then produced for both models. The reason for comparing with the AIC instead of the BIC is that the BIC, in this setting, will favor even fewer components than the AIC. In this initial run only uncorrelated data were used. The sample size estimation results are listed in Table Table2.2. The average number of components chosen by the AIC and our method were, respectively, 6.1 and 11.8 for π0 = 0.7 and 5.0 and 10.1 for π0 = 0.9. Based on our findings we concluded that there may be an advantage to using more components than suggested by the AIC, and we used this larger number of components in the remaining tests. We then turned our attention to comparing the different approaches to sample size estimation. Uncorrelated and correlated data were generated and sample size estimates were produced using all four methods. The results are found in the upper half of Table Table3.3. In general it seems that our estimates are close to their true values. Results are slightly better when there is no correlation between genes. As was to be expected, accuracy decreases, and standard deviation increases, with increasing power. This is probably related to the difficult problem of describing the distribution of noncentrality parameters well near the point of no regulation, i.e. close to δ = 0. The estimates of Hu et al. seem largely to be further from the true value than our estimates, and to be more conservative, but have lower standard deviation. The deviation from the true value is particularly high in estimates for high power. The estimates of Pounds and Cheng seem to deviate from the true value, be more conservative than our estimates and have higher standard deviation. The conservativeness of the estimates of Pounds and Cheng is seen from their own numerical tests as well, in which the estimated actual power exceeds the desired power. The estimates of Pawitan et al. appear to be better than those of Hu et al. and Pounds and Cheng, but still seem to be further from the true value than our estimates, and to have higher standard deviation. For high power there is a tendency of underestimating the needed sample size using this method.
Evaluating the number of mixture components.
Evaluating sample size estimates from different methods.
Tests with normally distributed measurements were also run, in which the ${ξk}k=1m1$ were drawn from gamma distributions and where variances differed according to the model discussed below. Results were similar to those discussed above (not shown).
In a second set of tests we wanted to simulate data from a model having the characteristics of a true microarray experiment. We also wanted to see how sample size estimates were affected if the assumptions of normality and equal variances did not hold. To accomplish this we based our simulation on the Swirl data set, which is included in the limma software package [29], and on a model for gene expression level measurements discussed by Rocke and Durbin [30]. The model of Rocke and Durbin states that
w = α + μeη + ,
(18)
where w is the intensity measurement, μ is the expression level, α is the background level and and η are error terms distributed as N (0, $σε2$) and N (0, $ση2$) respectively. Using the estimation method discussed in their paper we estimated the parameters of (18) for the Swirl data set. The estimated parameters $(α^,σ^ε,σ^η)$ were, for the mutant: (394.12, 150.84, 0.18), and for the wild-type: (612.99, 291.40, 0.19). To generate a set of log-ratios representative of the same data we performed a significance analysis as outlined in the limma user's guide. Using a cutoff level of 0.10 for the FDR-adjusted p-values, we obtained a set of 280 log-ratios for genes likely to be regulated. Log-ratios for the regulated genes in our tests were sampled from this set with replacement. The true expression levels were generated by sampling from the background-corrected mean intensities of the genes in the mutant data set. To simulate microarray data for two conditions the following procedure was used: A set of log-ratios, and the true expression levels for one condition, were sampled. The true expression levels for the other condition were then calculated. Using the above-mentioned model, with their respective sets of parameters, measurements were simulated for both conditions and then log-transformed. To introduce correlation in this setting we added a random effect, γ, to the log-transformed measurements for each correlated block of genes. The γ was drawn from a N (0, $0.5ση2$) distribution. The block size was again assumed to be 50, and the genes were assigned randomly to each block. The true sample size requirements were in this case estimated by repeatedly drawing data sets from the given model and calculating their average power and FDR on a fine grid of cutoff values for the t-statistics. A direct calculation is possible since the regulated genes are known.
After generating a model as described above we again simulated data with and without a correlation structure. For each test setting we sampled 50 data sets and made sample size estimates from the data using all four methods. The results are summarized in the lower half of Table Table3.3. For the methods of Hu et al. and Pounds and Cheng the trend is the same as in the first set of tests. Our method seems to slightly overestimate the needed sample size, while the method of Pawitan et al. now interestingly provides the estimates closest to the true value. The standard deviations for the estimates of Pawitan et al. are still somewhat higher than ours.
Note that, since the implementations of Hu et al. and Pounds and Cheng support only sample size estimates based on two-sample t-statistics (1), all tests listed are based on this statistic. Our implementation supports both types, and tests were also run to check the case of one-sample t-statistics (2). The results were similar to those of the two-sample t-statistics (not shown).
## Conclusion
We have in this article discussed a mixture model approach to estimating the distribution of noncentrality parameters, and thus effect sizes, among regulated genes in a two-sample comparative microarray study. The model can be fitted to t-statistics calculated from pilot data for the study. We have also illustrated how the model can be used to estimate sample sizes that control the pFDR along with other statistical measures like average power or pFNR. In the microarray setting our results will often also be approximately valid when using the FDR and FNR instead of the pFDR and pFNR. This is due to, referring to Table Table1,1, that one frequently will have Pr(MR) ≈ 1 and Pr(MA) ≈ 1 in this setting. Sample size estimation methods like the one presented are useful in the planning of any large scale microarray experiment.
We examined the accuracy of our sample size estimates by performing a series of numerical studies. The conclusions are that our estimates are reasonably accurate, and have low variance, for moderate cutoffs in the error measures used. For stringent cutoffs we see a larger variance and a somewhat lowered accuracy. Overall our method seems to provide better results than the available sample size estimation methods of Hu et al. and Pounds and Cheng. We have also evaluated a method by Pawitan et al. for fitting a mixture model and its use in sample size estimation. The results using the method are good, and our tests suggest that optimizing the method of Pawitan et al. for use in sample size estimation could be interesting.
The decreased accuracy for stringent cutoffs found in the estimates is probably due to the difficult task of describing the distribution of regulated genes well near the point of no regulation, that is near δ = 0. It is important that the characterization of such genes is precise, but also that it does not affect the estimated distribution of the unregulated genes. Our solution in this article was to introduce a small neighbourhood around δ = 0 in which no other components are fitted. Better ways of differentiating between the two distributions close to 0 could be a subject of further study.
In our tests we also checked how correlation among the genes would affect the sample size estimates. We found that the estimates were only moderately affected. Nevertheless, we believe that estimation methods that incorporate correlation among genes is an important topic for future studies.
For the microarray setting the use of a moderated t-statistic, as discussed in [31], is often more appropriate than the ordinary t-statistic. For our method to be directly applicable to this type of statistic one would need to know that moderated t-statistics for unregulated genes follow a central t-distribution, and one would need the distribution's degrees of freedom. In [31] this distributional result is shown to hold, with augmented degrees of freedom for the central t-distribution. One would also need to know that moderated t-statistics for a regulated gene, with some given degree of regulation, follow a noncentral t-distribution, and one would need the distribution's degrees of freedom and noncentrality parameter. For this second result we are not aware of any findings. If this second result is shown to hold, then our method can be applied to moderated t-statistics as well. Testing the performance of our algorithm with moderated t-statistics, even without the second result, could also be interesting.
In our work we focus on average power as the control measure used along with the pFDR. One should note that having an average power of 0.9 means being able to correctly identify 90% of the regulated genes. In many experiments achieving this may not be interesting. One example is studies that aim only at identifying the small set of marker genes that best distinguishes one sample from the other. Other examples are experiments that, when comparing a treated and untreated tissue sample, only are interested in the most heavily affected regulatory pathways. In both these examples a power well below 0.9 could suffice. Our estimates are in this case particularly useful since they seem to be both accurate, and have low variance, for moderate power cutoffs with a low pFDR.
Although our main goal in this paper was fitting a model to be used in sample size estimation we would like to emphasize that the fitted model itself does provide some interesting information. One example is a direct estimate of π0, the proportion of unregulated genes, which we often found to be better than the one made by existing methods.
Other subjects for future work include a speed-up of the algorithm. The convergence of a straightforward EM-algorithm is known to be rather slow, and methods that improve on it will be implemented. Another issue that may be investigated further is picking the number of model components. As already mentioned, using a few more components than suggested by the traditionally used information criteria would often improve sample size estimates substantially. A different approach might be needed to choose the number of components in this setting.
The methods discussed in this article are applicable to any two-sample comparative multiple hypothesis testing situation where t-tests are used, and not only to problems in the microarray setting.
## Availability
An implementation of the methods discussed in this paper for the R environment is available from the authors upon request.
## Authors' contributions
HM and TSJ developed the statistical algorithms, AMB provided the biological insight. TSJ implemented the methods, ran the tests and drafted the manuscript. HM and AMB revised the manuscript. All authors read and approved the final manuscript.
## Acknowledgements
This work was supported by grants NFR 143250/140 and NFR 151991/S10 from the biotechnology and the functional genomics (FUGE) programs of the Norwegian Research Council (AMB, TSJ). Financial support was also given by the cross-disciplinary project "BIOEMIT – Prediction and modification in functional genomics: combining bioinfomatical, bioethical, biomedical, and biotechnlogical research" at the Norwegian University of Science and Technology (HM). The authors would like to thank Mette Langaas for reading the manuscript and making suggestions for improvements. We would also like to thank two anonymous reviewers for their careful reading and helpful advice.
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PubMed citations for these articles | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 62, "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": 0.8814708590507507, "perplexity": 1107.546864021923}, "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-2013-48/segments/1386163040002/warc/CC-MAIN-20131204131720-00031-ip-10-33-133-15.ec2.internal.warc.gz"} |
http://www.aanda.org/articles/aa/abs/2005/26/aa2060-04/aa2060-04.html | Free access
Issue A&A Volume 437, Number 2, July II 2005 501 - 515 Interstellar and circumstellar matter http://dx.doi.org/10.1051/0004-6361:20042060
A&A 437, 501-515 (2005)
DOI: 10.1051/0004-6361:20042060
H CO and CH OH abundances in the envelopes around low-mass protostars
J. K. Jørgensen1, F. L. Schöier2 and E. F. van Dishoeck1
1 Leiden Observatory, PO Box 9513, 2300 RA Leiden, The Netherlands
e-mail: [email protected]
2 Stockholm Observatory, AlbaNova, 106 91 Stockholm, Sweden
(Received 23 September 2004 / Accepted 22 March 2005)
Abstract
This paper presents the third in a series of single-dish studies of molecular abundances in the envelopes around a large sample of 18 low-mass pre- and protostellar objects. It focuses on typical grain mantle products and organic molecules, including H2CO, CH3OH and CH3CN. With a few exceptions, all H2CO lines can be fit by constant abundances of throughout the envelopes if ortho- and para lines are considered independently. The current observational dataset does not require a large H2CO abundance enhancement in the inner warm regions, but this can also not be ruled out. Through comparison of the H2CO abundances of the entire sample, the H2CO ortho-para ratio is constrained to be consistent with thermalization on grains at temperatures of 10-15 K. The H2CO abundances can be related to the empirical chemical network established on the basis of our previously reported survey of other species and is found to be closely correlated with that of the nitrogen-bearing molecules. These correlations reflect the freeze-out of molecules at low temperatures and high densities, with the constant H2CO abundance being a measure of the size of the freeze-out zone. An improved fit to the data is obtained with a "drop" abundance structure in which the abundance is typically a few 10-10 when the temperature is lower than the evaporation temperature and the density high enough so that the timescale for depletion is less than the lifetime of the core. The location of the freeze-out zone is constrained from CO observations. Outside the freeze-out zone, the H2CO abundance is typically a few . The observations show that the CH3OH lines are significantly broader than the H2CO lines, indicating that they probe kinematically distinct regions. CH3OH is moreover only detected toward a handful of sources and CH3CN toward only one, NGC 1333-IRAS2. For NGC 1333-IRAS2, CH3OH and CH3CN abundance enhancements of two-three orders of magnitude at temperatures higher than 90 K are derived. In contrast, the NGC 1333-IRAS4A and IRAS4B CH3OH data are fitted with a constant abundance and an abundance enhancement at a lower temperature of 30 K, respectively. This is consistent with a scenario where CH3OH probes the action of compact outflows on the envelopes, which is further supported by comparison to high frequency, high excitation CS J=10-9 and HDO line profiles which uniquely probe warm, dense gas. The extent to which the outflow dominates the abundance enhancements compared with the passively heated inner envelope depends on the filling factors of the two components in the observing beam.
Key words: stars: formation -- ISM: molecules -- ISM: abundances -- radiative transfer -- astrochemistry | {"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": 0.8233556151390076, "perplexity": 2086.467319476047}, "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-2016-50/segments/1480698542686.84/warc/CC-MAIN-20161202170902-00485-ip-10-31-129-80.ec2.internal.warc.gz"} |
http://www.ck12.org/physical-science/Energy-Conversion-in-Physical-Science/lesson/Energy-Conversion-MS-PS/ | <img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Energy Conversion
## Different forms of energy—such as electrical, chemical, and thermal energy—often change to other forms of energy.
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Energy Conversion
Sari and Daniel are spending a stormy Saturday afternoon with cartons of hot popcorn and a spellbinding movie. They are obviously too focused on the movie to wonder where all the energy comes from to power their weekend entertainment. They’ll give it some thought halfway through the movie when the storm causes the power to go out!
### Changing Energy
Watching movies, eating hot popcorn, and many other activities depend on electrical energy. Most electrical energy comes from the burning of fossil fuels, which contain stored chemical energy. When fossil fuels are burned, the chemical energy changes to thermal energy and the thermal energy is then used to generate electrical energy. These are all examples of energy conversion. Energy conversion is the process in which one kind of energy changes into another kind. When energy changes in this way, the energy isn’t used up or lost. The same amount of energy exists after the conversion as before. Energy conversion obeys the law of conservation of energy, which states that energy cannot be created or destroyed.
### How Energy Changes Form
Besides electrical, chemical, and thermal energy, some other forms of energy include mechanical and sound energy. Any of these forms of energy can change into any other form. Often, one form of energy changes into two or more different forms. For example, the popcorn machine below changes electrical energy to thermal energy. The thermal energy, in turn, changes to both mechanical energy and sound energy. You can read the Figure below how these changes happen.
1. The popcorn machine changes electrical to thermal energy, which heats the popcorn. 2. The heat causes the popcorn to pop. You can see that the popping corn has mechanical energy (energy of movement). It overflows the pot and falls into the pile of popcorn at the bottom of the machine. 3. The popping corn also has energy. That's why it makes popping sounds.
### Kinetic-Potential Energy Changes
Mechanical energy commonly changes between kinetic and potential energy. Kinetic energy is the energy of moving objects. Potential energy is energy that is stored in objects, typically because of their position or shape. Kinetic energy can be used to change the position or shape of an object, giving it potential energy. Potential energy gives the object the potential to move. If it does, the potential energy changes back to kinetic energy.
That’s what happened to Sari. After she and Daniel left the theater, the storm cleared and they went for a swim. That’s Sari in the Figure below coming down the water slide. When she was at the top of the slide, she had potential energy. Why? She had the potential to slide into the water because of the pull of gravity. As she moved down the slide, her potential energy changed to kinetic energy. By the time she reached the water, all the potential energy had changed to kinetic energy.
Q: How could Sari regain her potential energy?
A: Sari could climb up the steps to the top of the slide. It takes kinetic energy to climb the steps, and this energy would be stored in Sari as she climbed. By the time she got to the top of the slide, she would have the same amount of potential energy as before.
Q: Can you think of other fun examples of energy changing between kinetic and potential energy?
A: Playground equipment such as swings, slides, and trampolines involve these changes.
### Summary
• Energy conversion is the process in which energy changes from one form or type to another. Energy is always conserved in energy conversions.
• Different forms of energy—such as electrical, chemical, and thermal energy—often change to other forms of energy.
• Mechanical energy commonly changes back and forth between kinetic and potential energy.
### Review
1. Define energy conversion.
2. Relate energy conversion to the law of conservation of energy.
3. Describe an original example of energy changing from one form to two other forms.
4. Explain how energy changes back and forth between kinetic and potential energy when you jump on a trampoline. Include a sketch to help explain the energy conversions.
### Notes/Highlights Having trouble? Report an issue.
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### Vocabulary Language: English
energy conversion
Process in which energy changes from one type or form to another.
### Explore More
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Please wait... | {"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": 0.8595771193504333, "perplexity": 1082.1880002279527}, "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-2017-09/segments/1487501172902.42/warc/CC-MAIN-20170219104612-00442-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/59848/convex-subsets-of-a-group | # Convex subsets of a group
Assume that $(G,+)$ is an Abelian topological group (maybe locally compact, if necessary) and assume that $V$ is an open connected neighbourhood of zero. Does there exist an open "convex" neighborhood of zer0 $W$ such that $W \subset V$?
A subset $A$ of an Abelian group $G$ we call convex if for each $x\in G$ the condition $2x\in A+A$ implies that $x \in A$. (It is a generalization of notion of convex set in a real linear space).
Thanks.
-
If $V=G$ then $G$ is such a neighborhood of zero.
If $(G,+)$ is the $\mathbb{Z}/2\mathbb{Z}$ with the discrete topology and $V = \{\operatorname{zero}\}$,
then there is no such neighborhood of zero.
(Since $[1]+[1] = \operatorname{zero} \;$ and $\; [1]\not\in V \:$ .)
Therefore whether or not there is such a neighborhood
of zero depends on things which you did not specify.
-
Thanks. I have not experience in topological groups. If I assume that group is "locally convex", that is it has a basis in zero consisting of convex open subsets, then the answer will be true (assumption that $V$ is connected in this case is unnecessary). Maybe you know another additional conditions on $G$ or on $V$ under which my question has positive answer. – Richard Aug 26 '11 at 10:46
If the topology is generated by a family of semi-norms that satisfy $||x||_i \leq ||x+x||_i$ for all $x$ and $i$, then the group is locally convex. – Ricky Demer Aug 26 '11 at 22:35 | {"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": 0.9351570010185242, "perplexity": 138.22371985688355}, "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/1440645195983.63/warc/CC-MAIN-20150827031315-00015-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.popflock.com/learn?s=ADM_mass | ADM Mass
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ADM Mass
Richard Arnowitt, Stanley Deser and Charles Misner at the ADM-50: A Celebration of Current GR Innovation conference held in November 2009[1] to honor the 50th anniversary of their paper.
The ADM formalism (named for its authors Richard Arnowitt, Stanley Deser and Charles W. Misner) is a Hamiltonian formulation of general relativity that plays an important role in canonical quantum gravity and numerical relativity. It was first published in 1959.[2]
The comprehensive review of the formalism that the authors published in 1962[3] has been reprinted in the journal General Relativity and Gravitation,[4] while the original papers can be found in the archives of Physical Review.[2][5]
## Overview
The formalism supposes that spacetime is foliated into a family of spacelike surfaces ${\displaystyle \Sigma _{t}}$, labeled by their time coordinate ${\displaystyle t}$, and with coordinates on each slice given by ${\displaystyle x^{i}}$. The dynamic variables of this theory are taken to be the metric tensor of three dimensional spatial slices ${\displaystyle \gamma _{ij}(t,x^{k})}$ and their conjugate momenta ${\displaystyle \pi ^{ij}(t,x^{k})}$. Using these variables it is possible to define a Hamiltonian, and thereby write the equations of motion for general relativity in the form of Hamilton's equations.
In addition to the twelve variables ${\displaystyle \gamma _{ij}}$ and ${\displaystyle \pi ^{ij}}$, there are four Lagrange multipliers: the lapse function, ${\displaystyle N}$, and components of shift vector field, ${\displaystyle N_{i}}$. These describe how each of the "leaves" ${\displaystyle \Sigma _{t}}$ of the foliation of spacetime are welded together. The equations of motion for these variables can be freely specified; this freedom corresponds to the freedom to specify how to lay out the coordinate system in space and time.
## Notation
Most references adopt notation in which four dimensional tensors are written in abstract index notation, and that Greek indices are spacetime indices taking values (0, 1, 2, 3) and Latin indices are spatial indices taking values (1, 2, 3). In the derivation here, a superscript (4) is prepended to quantities that typically have both a three-dimensional and a four-dimensional version, such as the metric tensor for three-dimensional slices ${\displaystyle g_{ij}}$ and the metric tensor for the full four-dimensional spacetime ${\displaystyle {^{(4)}}g_{\mu \nu }}$.
The text here uses Einstein notation in which summation over repeated indices is assumed.
Two types of derivatives are used: Partial derivatives are denoted either by the operator ${\displaystyle \partial _{i}}$ or by subscripts preceded by a comma. Covariant derivatives are denoted either by the operator ${\displaystyle \nabla _{i}}$ or by subscripts preceded by a semicolon.
The absolute value of the determinant of the matrix of metric tensor coefficients is represented by ${\displaystyle g}$ (with no indices). Other tensor symbols written without indices represent the trace of the corresponding tensor such as ${\displaystyle \pi =g^{ij}\pi _{ij}}$.
## Derivation
### Lagrangian formulation
The starting point for the ADM formulation is the Lagrangian
${\displaystyle {\mathcal {L}}={^{(4)}R}{\sqrt {^{(4)}g}},}$
which is a product of the square root of the determinant of the four-dimensional metric tensor for the full spacetime and its Ricci scalar. This is the Lagrangian from the Einstein-Hilbert action.
The desired outcome of the derivation is to define an embedding of three-dimensional spatial slices in the four-dimensional spacetime. The metric of the three-dimensional slices
${\displaystyle g_{ij}={^{(4)}}g_{ij}}$
will be the generalized coordinates for a Hamiltonian formulation. The conjugate momenta can then be computed as
${\displaystyle \pi ^{ij}={\sqrt {^{(4)}g}}\left({^{(4)}}\Gamma _{pq}^{0}-g_{pq}{^{(4)}}\Gamma _{rs}^{0}g^{rs}\right)g^{ip}g^{jq},}$
using standard techniques and definitions. The symbols ${\displaystyle {^{(4)}}\Gamma _{ij}^{0}}$ are Christoffel symbols associated with the metric of the full four-dimensional spacetime. The lapse
${\displaystyle N=\left(-{^{(4)}g^{00}}\right)^{-1/2}}$
and the shift vector
${\displaystyle N_{i}={^{(4)}g_{0i}}}$
are the remaining elements of the four-metric tensor.
Having identified the quantities for the formulation, the next step is to rewrite the Lagrangian in terms of these variables. The new expression for the Lagrangian
${\displaystyle {\mathcal {L}}=-g_{ij}\partial _{t}\pi ^{ij}-NH-N_{i}P^{i}-2\partial _{i}\left(\pi ^{ij}N_{j}-{\frac {1}{2}}\pi N^{i}+\nabla ^{i}N{\sqrt {g}}\right)}$
is conveniently written in terms of the two new quantities
${\displaystyle H=-{\sqrt {g}}\left[^{(3)}R+g^{-1}\left({\frac {1}{2}}\pi ^{2}-\pi ^{ij}\pi _{ij}\right)\right]}$
and
${\displaystyle P^{i}=-2\pi ^{ij}{}_{;j},}$
which are known as the Hamiltonian constraint and the momentum constraint respectively. Note also that the lapse and the shift appear in the Hamiltonian as Lagrange multipliers.
### Equations of motion
Although the variables in the Lagrangian represent the metric tensor on three-dimensional spaces embedded in the four-dimensional spacetime, it is possible and desirable to use the usual procedures from Lagrangian mechanics to derive "equations of motion" that describe the time evolution of both the metric ${\displaystyle g_{ij}}$ and its conjugate momentum ${\displaystyle \pi ^{ij}}$. The result
${\displaystyle \partial _{t}g_{ij}={\frac {2N}{\sqrt {g}}}\left(\pi _{ij}-{\tfrac {1}{2}}\pi g_{ij}\right)+N_{i;j}+N_{j;i}}$
and
{\displaystyle {\begin{aligned}\partial _{t}\pi ^{ij}=&-N{\sqrt {g}}\left(R^{ij}-{\tfrac {1}{2}}Rg^{ij}\right)+{\frac {N}{2{\sqrt {g}}}}g^{ij}\left(\pi ^{mn}\pi _{mn}-{\tfrac {1}{2}}\pi ^{2}\right)-{\frac {2N}{\sqrt {g}}}\left(\pi ^{in}{\pi _{n}}^{j}-{\tfrac {1}{2}}\pi \pi ^{ij}\right)\\&-{\sqrt {g}}\left(\nabla ^{i}\nabla ^{j}N-g^{ij}\nabla ^{n}\nabla _{n}N\right)+\nabla _{n}\left(\pi ^{ij}N^{n}\right)-{N^{i}}_{;n}\pi ^{nj}-{N^{j}}_{;n}\pi ^{ni}\end{aligned}}}
is a non-linear set of partial differential equations.
Taking variations with respect to the lapse and shift provide constraint equations
${\displaystyle H=0}$
and
${\displaystyle P^{i}=0,}$
and the lapse and shift themselves can be freely specified, reflecting the fact that coordinate systems can be freely specified in both space and time.
## Applications
### Application to quantum gravity
Using the ADM formulation, it is possible to attempt to construct a quantum theory of gravity in the same way that one constructs the Schrödinger equation corresponding to a given Hamiltonian in quantum mechanics. That is, replace the canonical momenta ${\displaystyle \pi ^{ij}(t,x^{k})}$ and the spatial metric functions by linear functional differential operators
${\displaystyle {\hat {g}}_{ij}(t,x^{k})\mapsto g_{ij}(t,x^{k}),}$
${\displaystyle {\hat {\pi }}^{ij}(t,x^{k})\mapsto -i{\frac {\delta }{\delta g_{ij}(t,x^{k})}}.}$
More precisely, the replacing of classical variables by operators is restricted by commutation relations. The hats represents operators in quantum theory. This leads to the Wheeler-DeWitt equation.
### Application to numerical solutions of the Einstein equations
There are relatively few known exact solutions to the Einstein field equations. In order to find other solutions, there is an active field of study known as numerical relativity in which supercomputers are used to find approximate solutions to the equations. In order to construct such solutions numerically, most researchers start with a formulation of the Einstein equations closely related to the ADM formulation. The most common approaches start with an initial value problem based on the ADM formalism.
In Hamiltonian formulations, the basic point is replacement of set of second order equations by another first order set of equations. We may get this second set of equations by Hamiltonian formulation in an easy way. Of course this is very useful for numerical physics, because the reduction of order of differential equations must[clarification needed] be done, if we want to prepare equations for a computer.
## ADM energy and mass
ADM energy is a special way to define the energy in general relativity, which is only applicable to some special geometries of spacetime that asymptotically approach a well-defined metric tensor at infinity - for example a spacetime that asymptotically approaches Minkowski space. The ADM energy in these cases is defined as a function of the deviation of the metric tensor from its prescribed asymptotic form. In other words, the ADM energy is computed as the strength of the gravitational field at infinity.
If the required asymptotic form is time-independent (such as the Minkowski space itself), then it respects the time-translational symmetry. Noether's theorem then implies that the ADM energy is conserved. According to general relativity, the conservation law for the total energy does not hold in more general, time-dependent backgrounds - for example, it is completely violated in physical cosmology. Cosmic inflation in particular is able to produce energy (and mass) from "nothing" because the vacuum energy density is roughly constant, but the volume of the Universe grows exponentially.
## Application to modified gravity
By using the ADM decomposition and introducing extra auxiliary fields, in 2009 Deruelle et al. found a method to find the Gibbons-Hawking-York boundary term for modified gravity theories "whose Lagrangian is an arbitrary function of the Riemann tensor".[6]
## Notes
1. ^ ADM-50: A Celebration of Current GR Innovation
2. ^ a b Arnowitt, R.; Deser, S.; Misner, C. (1959). "Dynamical Structure and Definition of Energy in General Relativity" (PDF). Physical Review. 116 (5): 1322-1330. Bibcode:1959PhRv..116.1322A. doi:10.1103/PhysRev.116.1322.
3. ^ Chapter 7 (pp. 227-265) of Louis Witten (ed.), Gravitation: An introduction to current research, Wiley: New York, 1962.
4. ^ Arnowitt, R.; Deser, S.; Misner, C. (2008). "Republication of: The dynamics of general relativity". General Relativity and Gravitation. 40 (9): 1997-2027. arXiv:gr-qc/0405109. Bibcode:2008GReGr..40.1997A. doi:10.1007/s10714-008-0661-1.
5. ^ The papers are:
6. ^ Deruelle, Nathalie; Sasaki, Misao; Sendouda, Yuuiti; Yamauchi, Daisuke (2010). "Hamiltonian formulation of f(Riemann) theories of gravity". Progress of Theoretical Physics. 123 (1): 169-185. arXiv:0908.0679. Bibcode:2010PThPh.123..169D. doi:10.1143/PTP.123.169.
## References
This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 34, "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": 0.9641780257225037, "perplexity": 632.6687514732118}, "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-2019-43/segments/1570986705411.60/warc/CC-MAIN-20191020081806-20191020105306-00306.warc.gz"} |
https://mathoverflow.net/questions/19755/problems-concerning-subspace-of-m-nc/19764 | # problems concerning subspace of M_n(C)
let M_n(c) denote the n times n matrices over the complex number field. N be a subspace of
M_n(C).
1. if all the matrices in N are non-invertible , what is the maximum of the dimension of N can be?
2. if all the matrices in N commute with each other , what is the maximum of the dimension of N can be?
3. if all the matrices in N are nilpotent , what is the maximum of the dimension of N can be?
4. if all the non-zero matrices in N are invertible , what is the maximum of the dimension of N can be?
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Could you provide some motivation or context to allay the coming worries that this is a homework problem? – Pete L. Clark Mar 29 '10 at 19:05
Also, why don't you tell us what you have tried already? For most of these, there are some fairly obvious lower bounds. The question is whether one can do better. – Pete L. Clark Mar 29 '10 at 19:18
You might have better luck posting this kind of question on artofproblemsolving.com. I say this not because the question is inappropriate for MO but because I know there are a lot of strong problem-solvers there who like to think about this kind of question, although a few of them are here... – Qiaochu Yuan Mar 29 '10 at 20:06
for problem 4 , if the base field is C , the answer is 1. But if the base field is R, the answer may be greater than 1 ,right? – zhaoliang Mar 29 '10 at 20:52
Sure. One can reduce to subalgebras without loss of generality since the product and inverse of invertible matrices is invertible, and then you're just looking at a division algebra over R with a finite-dimensional representation. For that see en.wikipedia.org/wiki/… . – Qiaochu Yuan Mar 29 '10 at 21:29
1. "Non-invertible" means rank $\leq n-1$, and thus the upper bound $n\left(n-1\right)$ follows from the Theorem in paragraph 8.3 in Victor Prasolov's "Problems and theorems in linear algebra" (see here for a DVI file and here for a PDF version). (Scroll to page 58.) The reference given there is Flanders H., On spaces of linear transformations with bound rank, J. London Math. Soc. 37 (1962), pp. 10-16.
2. We can WLOG assume that our subspace $N$ is actually a subalgebra of $\mathrm{M}_n\left(\mathbb C\right)$ (because otherwise, we can replace it by the subalgebra it generates, and it will still have the property that any two of its elements commute), so the question is how large a commutative subalgebra of $\mathrm{M}_n\left(\mathbb C\right)$ can get. This has been solved by I. Schur (see the 2 links in that topic).
4. Here the maximal dimension is $1$, and Petya has told why.
As for 3., I can prove the upper bound $\frac{n^2}{2}$ (strangely enough, for $\mathbb C$ only), but unfortunately there is room between it and the lower bound $\frac{n\left(n-1\right)}{2}$.
-
great thanks! but what if the field is R , the real number field ,not C ? – zhaoliang Mar 29 '10 at 20:35
Interestingly enough, I can solve 3. over R, but not over C. – darij grinberg Mar 29 '10 at 20:47
For 3, it's a theorem of Gerstenhaber that $n(n−1)/2$ is best possible and this holds only for conjugates of the strictly upper triangular matrices - jstor.org/pss/2372773 – dke Mar 29 '10 at 21:00
Thanks for the link,dke (I had never seen the original article): I was writing my answer when you posted it! Did you notice that Gerstenhaber's second sentence (which states "the main purpose of this paper") is completely false, because he forgot to say that his k matrices are linearly independant ! – Georges Elencwajg Mar 29 '10 at 21:45
For problem 4 over the real field, the answer is the Radon-Hurwitz function at $n$. See for instance Petrovic, "On nonsingular matrices and Bott periodicity." The Radon-Hurwitz function is defined to be $\rho(n)=8a+2^b$, where the largest power of 2 dividing $n$ is $2^{4a+b}$, $a\geq 0$, $0\leq b\leq 3$.
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thank you Stanley, but I dont know how to open ps.gz files. would you please check the file? – zhaoliang Mar 30 '10 at 5:15
oh, I visited the webpage and downloaded the pdf version of the article. that's nice of you. now I have found the answers to all the four problems. thank everyone, thank MO! – zhaoliang Mar 30 '10 at 5:22
By the way, are you professor Stanley, the author of the <<enumeration combinatorics>>? – zhaoliang Mar 30 '10 at 5:24
zhaoliang, to read a file called filename.ps.gz on Linux or Unix, type "gunzip filename.ps.gz". This should give you a file called "filename.ps", which you can view in the usual way (e.g. by typing "gv filename.ps") or convert to pdf if you prefer (e.g. by typing "ps2pdf filename.ps"). Don't know what you do on other operating systems. – Tom Leinster Mar 30 '10 at 7:46
Ah, but I see Douglas has edited the link, so you don't need my instructions after all. – Tom Leinster Mar 30 '10 at 7:48
Dear zhaoliang, here is the answer (from Gerstenhaber's thesis) to question 3.
a) The maximal dimension of a space of $n$ times $n$ nilpotent matrices is $\frac {n(n-1)}{2}$.
b) The subspaces of that dimension are exactly: the space of strictly upper triangular matrices and its conjugates.
Here is a fairly modern related article in the bibliography of which you will find the original references : http://www.win.tue.nl/~jdraisma/publications/NilpotentSubspacesv14.pdf
If you understand mathematical Portuguese (which is easy), here
http://ptmat.fc.ul.pt/~pedro/tese.pdf
is an interesting thesis attacking this kind of problem both with algebraic geometry and combinatorics: a combination that should warm the heart of many a MathOverflower...
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thank you , Elencwajg. I am a chinese student in Peking university. so Portuguese ......^_^ the first pdf file is nice for me. – zhaoliang Mar 29 '10 at 21:41
also in problem 4, if we change the base field to R, the real number field , wether the dimension of N can be larger – zhaoliang Mar 29 '10 at 21:55
Dear zhaoliang, I am very sorry for the poor formulation of my remark about Portuguese. What I meant was that Portuguese is an Indo-European language whose scientific vocabulary has much in common with that of English. Moreover it is a Romance language and so even closer to French, Spanish, Italian, Romanian and my remark was addressed to the non-negligible part of us who have some knowledge of one of those. This may not apply to you but I must tell you my sincere admiration for your command of English, a language from a linguistic family completely different to your mother tongue (mandarin?) – Georges Elencwajg Mar 29 '10 at 22:04
thank you,Elencwajg.In fact, there are lots of fallacies in my spelling and grammer~^_^~ I re-read the second file, it seems I can guess the meaning of the text by its math formula. But this one still used some algebraic geometry, huh? I saw Zariski topology. – zhaoliang Mar 29 '10 at 22:44
no,it's not a homework. Do you think a homework can be so hard as these ones? :P.
I proposed them by myself. of course I don't konw whether others have solved them. They are very 'natrul' and easy to formulate, but I find it quite hard to deal with them.
for I, I guess it is n(n-1), the A satisfy Ax=0 for a given x not 0 is ok
for II, I guess it is n(n-1)/2, uptriangle with zero engenvalues will suffice
for III and IV I have no idea.
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When you say "for II" do you mean "for 3"? – Jonas Meyer Mar 29 '10 at 19:39
Thanks, that's better. (I think you have mixed up II and III in your answer.) I have to run now and am not a matrix algebra expert anyway, but I'll tell you my guesses: I. the same as yours. II. n (keyword: Cartan subalgebra). III. the same as yours. IV: 1. (Hint: det = 0 is a hypersurface in affine $n^2$-space which intersects your subspace at the origin.) – Pete L. Clark Mar 29 '10 at 19:42
Answer in 4 is one. Hint: look at the expression det(A+tB) and think when it equals to 0. – Petya Mar 29 '10 at 19:48
You can do better than n for II (if n is at least 4). It came up at a recent question that you can have a subalgebra of dimension 1 + the floor of (n/2)^2 by considering scalars + 2-by-2 block strictly upper triangular matrices. When you only require subspace rather than subalgebra, perhaps you can do better. – Jonas Meyer Mar 29 '10 at 19:49
oh, dke ,I didnt catch you. skew-symmetric matrices do not commute in general. – zhaoliang Mar 29 '10 at 20:10 | {"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": 0.8915942907333374, "perplexity": 772.0758523989545}, "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-2015-40/segments/1443737936627.82/warc/CC-MAIN-20151001221856-00036-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://www.lessonplanet.com/teachers/galaxy-distances-and-mixed-fractions | # Galaxy Distances and Mixed Fractions
In this galaxies and fractions worksheet, students solve 8 problems involving the distances of galaxies from each other by using mixed fractions to solve each problem. They use the megaparsec as the unit of measurement.
Concepts
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https://www.physicsforums.com/threads/photon-electron-collision-problem.204822/ | Photon-Electron collision problem
1. Dec 16, 2007
Rainbow
Let us consider a free electron in space, which is initially at rest. Now let us consider a photon of frequency f, which collides with our free electron giving all its energy to it. This energy will manifest itself as the K.E. of the electron after the collision. Therefore, we can write
hf=(1/2)mv^2 where, h: Planck's Constant
m: Mass of the electron
v: velocity of the electron after the collision
Also, whole of the momentum of the photon will also be transfered to the electron
hf/c=mv
Solving the two equations, we get v=0 or v=2c(which defies special relativity).
Now, v=0 cannot be the solution, as the energy in the electron has to manifest itself in some or the other form, and the only form is K.E. (if I'm not mistaken).
The solution v=2c is not compatible with the fact that c is the ultimate speed.
So, what's the problem?
2. Dec 16, 2007
George Jones
Staff Emeritus
Newtonian concepts don't work in this situation. Relativistic energy and momentum conservation must be used, and these concepts forbid your situation. On its own, a free electron cannot absorb a photon.
3. Dec 16, 2007
Rainbow
Thank you for the help.
4. Dec 16, 2007
Staff: Mentor
This is also the reason why when an electron and a positron anihlate they always produce at least two photons. With just one the 4-momentum wouldn't be conserved.
Similar Discussions: Photon-Electron collision problem | {"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": 0.8170924782752991, "perplexity": 1073.2793674953023}, "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-2017-51/segments/1512948592972.60/warc/CC-MAIN-20171217035328-20171217061328-00249.warc.gz"} |
https://math.stackexchange.com/questions/1431166/affine-parameterization-of-null-geodesics | # Affine parameterization of null geodesics
How does one find an affine parameter for a null geodesic? I found this advice on planetmath.org:
Take s as an arbitrary parameter;
Set $$u^\mu=\frac{dx^\mu}{ds}$$ Then $$u^\mu \nabla_\mu u^\nu = f(s)u^\nu$$
If the RHS is zero, s is affine; if not, s is not affine. Does this make sense? It strikes me that to find an affine parameter all you need is the equations of the curve $x^\mu (s)$ and their derivatives $u^\mu$. Is this true? I was wondering if we need to know the metric tensor $g$ also? I thought one needs to write the geodesic equation $$\frac{d^2 x^\mu}{ds^2}+\Gamma^\mu_{\nu \sigma}\frac{dx^\nu}{ds}\frac{dx^\sigma}{ds}=0$$ and check if its RHS is zero or not. But then we also need to know the $\Gamma$s or we need to know the metric tensor $g$ if we rewrite the above geodesic equation in terms of $g$ instead of $\Gamma$s.
What you are missing is that $\nabla_\mu$ is the covariant derivative (aka Levi-Civita connection) associated to the metric $g$. It already encodes in it the connection coefficients. So hidden in the notation $\nabla_\mu$ is the knowledge on the metric you need.
That said, if you know that $x^\mu(s)$ is a geodesic curve, and you know the tangent vectors $u^\mu(s)$, you can actually find an affine re-parametrisation without too much of the underlying geometry. What you are looking for is a function $s = s(t)$ such that $x^\mu(s(t))$ is affinely parametrized by $t$. If you consider its inverse function $t = t(s)$, and let $v^\mu$ be the tangent vector relative to the parameter $t$, you can by chain rule reason that
$$v^\mu(t(s)) t'(s) = u^\mu(s)$$
Inserting this into the equation you have
$$t'(s)^2 (v^\mu\nabla_\mu v^\nu)(t(s)) + t''(s) v^\mu(t(s)) = f(s) t'(s) v^\mu(t(s))$$
By assumption $t$ is an affine parametrisation, so the transport of $v$ by itself vanishes. So you end up having to solve the second order ODE
$$t'' = t' f \implies \log (t') ' = f \implies t' = \exp \int f \implies t = \int \exp \int f$$
So... while the geodesic equation itself must encode geometric data (the connection coefficients), the algorithm for finding an affine parameter when given an arbitrarily parametrized geodesic curve only involves solving an ODE and the geometric data can be conveniently forgotten.
• I'm a bit confused, I thought that if a curve is parametrized as $u\mapsto \gamma(u)$, then $u$ is said to be an affine parameter if $g_{\gamma(u)}(\dot{\gamma}(u),\dot{\gamma}(u))=\mathrm{constant}$. At least this is the case in the Riemannian setting. In the Lorentzian setting, if we have a null curve $\gamma$ then $g_{\gamma(u)}(\dot{\gamma}(u),\dot{\gamma}(u))=0$. So by this definition any parametrization is affine. – Aerinmund Fagelson Jan 31 '18 at 12:03
• @AerinmundFagelson: this is one place where your Riemannian "definition" fails. In Riemannian geometry, the condition $g(\dot{\gamma}, \dot{\gamma})$ being constant along a geodesic is equivalent to $\nabla_{\dot{\gamma}} \dot{\gamma} = 0$. This equivalence is false for null curves in Lorentzian geometry. Hence affine parametrization in general pseudo-Riemannian geometry uses the more precise definition $\nabla_{\dot{\gamma}} \dot{\gamma} = 0$. (For non-affine parametrizations, you allow $\nabla_{\dot{\gamma}} \dot{\gamma} \propto \dot{\gamma}$ to define geodesics.) – Willie Wong Jan 31 '18 at 17:06 | {"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": 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": 0.9734602570533752, "perplexity": 174.00508113744326}, "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/1563195525374.43/warc/CC-MAIN-20190717181736-20190717203736-00412.warc.gz"} |
http://tex.stackexchange.com/questions/163535/how-do-i-put-a-label-inside-a-feynman-diagram-circle | # How do I put a label inside a Feynman diagram circle?
I have this simple Feynman diagram with a circle:
I would like to put the number "1" in the middle of that circle.
I've tried these:
\fmfiv{lab=1}{c}
\fmfiv{lab=1}{.5[nw,se])
\fmfiv{lab=1}{(.4999w, .5h)}
\fmfiv{lab=1}{(.5001w, .5h)}
but whatever constants I put, latex refuses to put it INSIDE the circle.
The latex:
\documentclass[24pt]{article}
\usepackage{amsmath,amsfonts,epsf}
\usepackage{amssymb}
\usepackage[pdftex]{graphicx}
\usepackage{grffile}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{2ptcorrection1}
\begin{fmfgraph*}(220,12)
\fmfpen{thick}
\fmfleft{i1}
\fmfright{o1}
\fmf{plain}{i1,v1}
\fmf{dashes, left=1, tension=0.3}{v1,v2}
\fmf{plain, right=1, tension=0.3}{v1,v2}
\fmf{plain}{v2,o1}
\end{fmfgraph*}
\end{fmffile}
\end{document}
btw, the coordinates are: "c" for center, sw, se, ne, nw.
Update: thank you hftf for the answer.
-
I was able to make the label appear to be in the center of the circle by using a phantom (invisible) edge across the circle and setting the label.dist to 0 to place the label in the middle of that edge.
\documentclass{minimal}
\usepackage{grffile}
\usepackage{feynmp-auto}
\begin{document}
\begin{fmffile}{2ptcorrection1}
\begin{fmfgraph*}(220,12)
\fmfpen{thick}
\fmfleft{i1}
\fmfright{o1}
\fmf{plain}{i1,v1}
\fmf{dashes, left=1, tension=0.3}{v1,v2}
\fmf{plain, right=1, tension=0.3}{v1,v2}
\fmf{plain}{v2,o1}
% This is where the magic happens:
\fmf{phantom,label.dist=0,label=1}{v1,v2}
\end{fmfgraph*}
\end{fmffile}
\end{document}
And this is what it looks like:
- | {"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": 0.8952832221984863, "perplexity": 2047.9347331308938}, "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-2015-22/segments/1432207926828.58/warc/CC-MAIN-20150521113206-00260-ip-10-180-206-219.ec2.internal.warc.gz"} |
http://mathhelpforum.com/calculus/97986-mean-value-theorem.html | # Math Help - Mean Value Theorem
1. ## Mean Value Theorem
I'm having trouble with this question on mean value theorem
1. x+(96/x) on interval [6,16]
the problem with this equation is i get 0/(10) and when i check the answer is 4sq.rt(6) I'm using the f(b)-f(a)/b-a so what am i'm missing?
2. Originally Posted by goldenroll
I'm having trouble with this question on mean value theorem
1. x+(96/x) on interval [6,16]
the problem with this equation is i get 0/(10) and when i check the answer is 4sq.rt(6) I'm using the f(b)-f(a)/b-a so what am i'm missing?
$1-\frac{96}{x^2}=\frac{\left(16+\frac{96}{16}\right)-\left(6+\frac{96}{6}\right)}{16-6}$
Solve for x.
Join Date: Feb 2009
Posts: 25
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3. that comes out to (22-22)/10? because the answer in the book says it come out to 4sq.rt6,
4. Originally Posted by goldenroll
that comes out to (22-22)/10? because the answer in the book says it come out to 4sq.rt6,
Exactly. But you have only done the right side of the equation...
Thanks: 0
Hint hint...
5. hahaha, I guess we just found out were I got lost at. when you say I have only done the right side of the equation what do you mean.
6. Originally Posted by goldenroll
hahaha, I guess we just found out were I got lost at. when you say I have only done the right side of the equation what do you mean.
Equations have right sides, and then they have left sides. Soooooo, generally when we "solve" equations, we isolate the variable. In this case, the variable would be x. So, your task is to move everything to one side of the equation except x.
The MVT says that there will be at least one place in an interval [a,b] where the slope of the tangent line to the graph will have the same slope as the secant line through [a,f(a)] and [b,f(b)]. You have found the slope of the secant line. So, you're not done.
Start pressing the button!
7. oh, lamo, well I finally got the right answer when I finished the other side lol. Oh and just realized where the thanks button was lol.
8. Originally Posted by goldenroll
oh, lamo, well I finally got the right answer when I finished the other side lol. Oh and just realized where the thanks button was lol.
Good. I hate asking people for thank yous, and if I didn't help you, don't thank me. But I think that you'll find that once you start clicking that thank you button, people will start busting down your door trying to help you. | {"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": 0.8138918876647949, "perplexity": 819.5380224006329}, "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-2014-23/segments/1405997877881.80/warc/CC-MAIN-20140722025757-00245-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://www.crm.umontreal.ca/physmath/archives/working-seminar-on-integrable-systems-random-matrices-and-random-processes/ | Home » Archives » Working Seminar on Integrable Systems, Random Matrices, and Random Processes
# Working Seminar on Integrable Systems, Random Matrices, and Random Processes
## Fall 2008 – Winter 2009
#### Time/Date/Heure: Tuesday, September 16; mardi le 16 septembre 2008, at / à 3:30 p.m. / 15h30 Title/titre: « Dimer models and Donaldson Thomas Invariants » Speaker/Conférencier: Benjamin Young (McGill and CRM) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
Donaldson-Thomas theory is a « curve-counting » theory: it gives a virtual count of subschemes of a Calabi-Yau threefold X. If X should happen to be a toric threefold, then the computation reduces to the combinatorics of three-dimensional Young diagrams placed at the toric fixed points of X.
A similar phenomenon happens when computing « non-commutative » Donaldson-Thomas invariants associated to the path algebra of certain quivers; there, the problem reduces to the dimer model on the square lattice.
In this first working seminar, I’ll review the combinatorial tools one can use to deal with these objects: the dimer model, height functions, vertex operators, etc.; I will also to address their relationship with Donaldson-Thomas theory. (to be continued)
#### Time/Date/Heure: Tuesday, October 21; mardi le 21 octobre 2008, at / à 12:30 p.m. / 12h30 Title/titre: « Vertex operators for combinatorics » Speaker/Conférencier: Benjamin Young (McGill and CRM) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
I will give an overview of vertex operators and the infinite wedge formalsim, with an application to enumeration of 3D Young diagrams or pyramid partitions. The textbook is the collected appendices of A. Okounkov’s papers.
#### Time/Date/Heure: Tuesday, October 25; mardi le 25 octobre 2008, at / à 12:30 p.m. / 12h30 Title/titre: « PNG droplet and airy process: part 2 » Speaker/Conférencier: Dong Wang (CRM) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
This is an introduction to the paper « Scale invariance of the PNG droplet and the Airy Process » by M. Praehofer and H. Spohn.
#### Time/Date/Heure : Tuesday, December 9; mardi le 9 octobre 2008 at / à : Part I : 11:00 a.m. / 11h – Part II : 12:30 a.m. / 12h30 Title/titre : « Quantum subgroups of Lie groups and modular invariance in conformal field theories » Speaker/Conférencier: Robert Coquereaux (Directeur de recherche, CNRS. CPT, Luminy-Marseille) Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
« For quantum groups at roots of unity, one can construct a monoidal category of representations that admits, for special values of the chosen root, module-categories, ie additive categories on which the previous one acts. In the case of quantum SU2, those « quantum subgroups » are classified by the usual ADE Dynkin diagrams. This classification is equivalent to another problem solved long ago in the case of SU2 by theoretical physicists, in the context of conformal field theories with boundaries, namely the classification of modular-invariant sesquilinear forms, for the Hurwitz – Verlinde representations of SL(2,Z). Each such quantum subgroup is associated with a weak Hopf algebra of a special kind (an Ocneanu quantum groupoid) that admits two, usually distinct, representations theories whose multiplicative structures can be encoded by graphs: the fusion graph and the graph of quantum symmetries. The purpose of the seminar is to provide a general introduction to the above ideas and to describe what happens when SU2 is replaced by more general Lie groups. This leads in particular to higher analogues of Coxeter-Dynkin diagrams (that will be presented for SU3 and SU4) and to higher graphs of quantum symmetries. »
#### Time/Date/Heure : Friday, January 9; vendredi le 9 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « Hermitian matrix model with external source » Speaker/Conférencier: Seung-Yeop Lee, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
« I will introduce my on-going project with Robert Buckingham and Virgil Pierce about the hermitian matrix model with external source. I will describe the phenomena of « jumping outliers. »
#### Time/Date/Heure : Friday, January 16; vendredi le 16 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « Tensor networks in graph combinatorics » Speaker/Conférencier: Peter Zograf Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
A tensor network is an assignment of a tensor to each vertex of a graph together with a one-to-one correspondence between tensor indices and half-edges incident to the vertex. We will show how tensor networks can be applied to some of the well-known combinatorial problems like the dimer problem, edge coloring, the Hamiltonian cycle problem, etc.
#### Time/Date/Heure : Friday, January 23; vendredi le 23 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « KdV equation and computational geometry of moduli spaces of curves » Speaker/Conférencier: Peter Zograf Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
We present a fast algorithm for computing intersection numbers on moduli spaces of complex algebraic curves in terms of the coefficients of special solutions of the KdV equation. As an application, we derive conjectural large genus asymptotics of these numbers (in particular, Weil-Petersson volumes).
#### Time/Date/Heure : Friday, January 29; vendredi le 29 janvier 2009 at / à 2:00 p.m. / 14h Title/titre : « Convolution symmetries of integrable hierarchies, matrix models and tau-functions (Part 3) » Speaker/Conférencier: John Harnad Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
Les symétries convolutives généralisées des hiérarchies intégrables du type KP-Toda et 2KP-Toda ont l’effet de multiplier les coefficients de Fourier de la fonction de Baker-Akhiezer par une suite de constantes donnée. L’action induite sur l’espace de Fock fermionique associé est diagonale dans la base orthonormale standard déterminée par les sites occupés et étiquetés par les partitions. Les coefficients dans les développements simples et doubles en fonctions de Schur de la fonction-tau associée, qui sont les coordonnées de Pluecker d’un élément décomposable, sont multipliés par des facteurs diagonaux correspondants. Appliquant de telles transformations aux intégrales de matrices, nous obtenons des nouveaux modèles de matrices du type extérieurement couplé qui sont également des fonctions-tau KP-Toda ou 2KP-Toda. Des représentations intégrales multiples plus générales des fonctions-tau sont également obtenues, aussi bien que des expressions déterminantales finies pour elles.
Generalized convolution symmetries of integrable hierarchies of KP-Toda and 2KP-Toda type have the effect of multiplying the Fourier coefficients of the Baker-Akhiezer function by a specified sequence of constants. The induced action on the associated fermionic Fock space is diagonal in the standard orthonormal base determined by occupation sites and labeled by partitions. The coefficients in the single and double Schur function expansions of the associated tau-functions, which are the Pluecker coordinates of a decomposable element, are multiplied by the corresponding diagonal factors. Applying such transformations to matrix integrals, we obtain new matrix models of externally coupled type which are also KP-Toda or 2KP-Toda tau-functions. More general multiple integral representations of tau functions are similarly obtained, as well as finite determinantal expressions for them (e.g. determinental correlation functions). (The language of the presentation will determined at the time of the lecture.)
Abstract
#### Time/Date/Heure : Friday, February 13; vendredi le 13 février 2009 at / à 2:00 p.m. / 14h Title/titre : « On (iso)monodromic deformations » Speaker/Conférencier: Marco Bertola, Concordia Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
Given a system of first order ODEs in the complex plane with rational coefficients one can (univocally) associate other data, which I will call « Birkhoff data ». They consist in the « isomonodromic times » and the « generalized monodromy data », namely, the monodromy representation of the ODE together with the datum of the Stokes’ matrices.
Solving this forward problem amounts to solving the ODE. The inverse problem consists in reconstructing the ODE (or its solution) starting from the Birkhoff data: this is an inherently harder problem, which involves the solution of a Riemann–Hilbert problem (i.e. a singular-integral equation).
It is known that for generic Birkhoff data, the inverse problem admits a (unique) solution; however there are special data, called the « Malgrange Theta divisor » where this inversion problem fails.
The invertibility of the problem hinges on the invertibility of a suitable Toeplitz operator and hence the Malgrange divisor could be thought of as the locus of data where the « determinant » of the operator vanishes (with the nasty detail that the operator in question does not have a notion of determinant).
In a parallel approach, in the eighties, Jimbo-Miwa-Ueno gave a computable definition of a closed differential of the isomonodromic times which involved purely algebraic manipulations of the coefficients of the ODE. The (locally defined) function with this differential has the property that it vanishes precisely and only on the Malgrange divisor, as proved by Malgrange for the case of Fuchsian systems and Palmer for systems with irregular singularities.
The goal of this seminar is to show how to transform JMU’s differential into an object more closely related to the Riemann–Hilbert problem and hence extend in a natural way its definition so as to allow us to compute the derivatives of the tau function with respect to the {\em monodromy} data (Stokes etc.)
I will try to give a no-nonsense approach with details on the construction and properties of the Birkhoff data.
Some applications to random matrices and Painleve’ equations will be outlined.
The seminar is based on ongoing research.
#### Time/Date/Heure : Friday, February 20; vendredi le 20 février 2009 at / à 2:00 p.m. / 14h Title/titre : « Riemann-Hilbert problems associated to Frobenius manifold structures on Hurwitz spaces. » Speaker/Conférencier: Vasilisa Shramchenko Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336
There are two (dual to each other) Riemann-Hilbert problems naturally associated to every Frobenius manifold. In the case of Frobenius structures on Hurwitz spaces, the corresponding Riemann-Hilbert problems turn out to be solvable in terms of meromorphic bidifferentials defined on the underlying surface. In this talk, I will present these solutions and discuss their monodromy groups. The monodromy transformations for both solutions are related to the monodromy in the appropriate spaces of contours on the Riemann surface.
#### Time/Date/Heure : Friday, February 27; vendredi le 27 février 2009 at / à 2:00 p.m. / 14h Title/titre : « Riemann–Hilbert problems: Schlesinger and Sato » Speaker/Conférencier: Marco Bertola Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
Given any (sufficiently well-behaved) family of Remann–Hilbert problems where the jump matrices depend arbitrarily on deformation parameters, we can construct a one-form $\Omega$ on the deformation space (Malgrange’s differential).
Such a one–form has a pole where the deformation family meets the Malgrange Theta divisor, namely, the set of unsolvable RHP. I will try to review this fact reading through Malgrange and Palmer’s papers.
The differential $\Omega$ fails to be closed in general, but when it does the formula $\tau := {\rm e}^{\int \Omega}$ defines locally a function that vanishes precisely on $\Theta$.
We then introduce the notion of Schlesinger discrete transformation: it means that we allow the solution of the RHP to have poles (or zeros) at prescribed point(s). Interestingly, even if $\Omega$ is not closed, the difference of $\Omega$ evaluated along solution of the original RHP and the Schlesinger transformed RHP is closed off $\Theta$; in fact, such difference is shown to be the logarithmic differential (on the deformation space) of a function -say- $H$.
I will show how this function $H$, as a function of the position of the points of the Schlesinger transformation, yields a natural generalization of Sato formula for the Baker–Akhiezer vector even in the absence of a tau function, and it realizes the solution of the RHP as such BA vector.
#### Time/Date/Heure : Friday, March 6; vendredi le 6 mars 2009 at / à 2:00 p.m. / 14h Title/titre : « Asymptotic analysis of random Hermitian matrices with a small-rank external source. » Speaker/Conférencier: Robert Buckingham Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
We will discuss ongoing work on the behavior of the eigenvalues of a large Hermitian matrix drawn from a random ensemble with an external source. We will consider three cases depending on the strength of the source, refered to as supercritical, subcritical, and critical. This is joint work with Seung Yeop Lee and Virgil Pierce and is a continuation of a previous talk by Lee earlier this semester.
#### Time/Date/Heure : Friday, March 20; vendredi le 20 mars 2009 at / à 2:00 p.m. / 14h Title/titre : « A (conjectural) PDE satisfied by the gap probability of the Gaussian matrix model with external source. » Speaker/Conférencier: Dong Wang, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
In this talk I show an attempt to apply the Adler-van Moerbeke method on the Gaussian matrix model with external source.
#### Time/Date/Heure : Friday, March 27; vendredi le 27 mars 2009 at / à 2:00 p.m. / 14h Title/titre : » r-Airy parametrix in Hermitian matrix model with external source » Speaker/Conférencier: Seung-Yeop Lee, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
In this talk, I consider the hermitian matrix model with external source at the criticality. It is already known that, for Gaussian potential, the kernel is given by r-Airy kernel. We derive this through 3 by 3 Riemann-Hilbert problem hence the result applies to more general potentials. This is work in progress with Marco Bertola, Robert Buckingham and Virgil Pierce.Â
#### Time/Date/Heure : Friday, April 3; vendredi le 3 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Rank 1 real external source model » Speaker/Conférencier: Dong Wang, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
In this talk I will explain how a trick (partially) solves the rank 1 real external source model. The trick resembles the convolution symmetry.
#### Time/Date/Heure : Tuesday, April 7; Mardi le 7 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Rank 1 real external source model (Part II) » Speaker/Conférencier : Dong Wang, CRM Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
In this talk I will explain how a trick (partially) solves the rank 1 real external source model. The trick resembles the convolution symmetry.
#### Time/Date/Heure : Friday, April 10; vendredi le 10 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Hydrodynamics of Calogero-Sutherland model: Bidirectional Benjamin-Ono equation » Speaker/Conférencier: Alexander G. Abanov, Stony Brook University Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
Calogero-Sutherland model is the model of particles in one-dimension interacting through the inverse square potential. Both classical and quantum versions of the model are known to be integrable. I consider the hydrodynamic description of this model. The hydrodynamic equations derived in the limit of infinitely many particles are also integrable. They present a bidirectional analogue of classical Benjamin-Ono equation which first appeared as an equation for interface waves in deep stratified fluids. I discuss hydrodynamic soliton solutions and dispersive shock waves for this equation.
#### Time/Date/Heure : Friday, April 17; vendredi le 17 avril 2009 at / à 2:00 p.m. / 14h Title/titre : « Multiple orthogonal polynomials and nonlinear automorphisms of the Heisenberg-Weyl algebra. » Speaker/Conférencier: Alexei Zhedanov, Donetsk Institute for Physics and Technology Room/Salle : CRM, UdeM, Pavillon André Aisenstadt, 2920, ch. de la Tour, salle 4336.
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