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https://brg.me.uk/?p=7325 | ### by Howard Williams
#### Published Sunday April 04 2021 (link)
My kitchen floor is tiled with identically-sized equilateral triangle tiles while the floor of the bathroom is tiled with identically-sized regular hexagon tiles, the tiles being less than 1m across. In both cases the gaps between tiles are negligible. After much experimenting I found that a circular disc dropped at random onto either the kitchen or bathroom floor had exactly the same (non-zero) chance of landing on just one tile.
The length of each side of the triangular tiles and the length of each side of the hexagon tiles are both even triangular numbers of mm (ie, of the form 1+2+3+…).
What are the lengths of the sides of the triangular and hexagonal tiles?
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If the radius of the circular disc is $$r$$ and the sides of the regular hexagon and equilateral triangle are $$h$$ and $$t$$ respectively, inspection of the small lower right triangles in each shape shows that the side lengths of the inner hexagon and the triangle are respectively $$h\,-\, 2r/\sqrt{3}$$ and $$t\,-\,2\sqrt{3}r$$. With $$p_h$$ and $$p_t$$ as the probabilities of the disc falling entirely within a tile, these become:$p_h = \left(\frac{h \;- 2r/\sqrt{3}}{h}\right)^2=\left(1-\frac{2r}{\sqrt{3}h}\right)^2$
$p_t = \left(\frac{t\; – 2\sqrt{3}r}{t}\right)^2=\left(1-\frac{2\sqrt{3}r}{t}\right)^2$
Equating these shows that equal probabilities requires that $$t=3h$$. (see [1])
Since the sides are triangular numbers, let the hexagon’s and the triangle’s numbers be $$h(h+1)/2$$ and $$t(t+1)/2$$ respectively (this is a a change of notation from that above). We then have: $t(t+1) = 3h(h+1)$ which, after rearrangement, can be put in the form:$(2t+1)^2\; -\; 3(2h+1)^2=-2$This is a quadratic Diophantine equation for which there are standard techniques for solution which can be used to show that, if we can find one solution, an infinite series of solutions can be produced using two recursions:$t_{n+1}=2 t_{n}+3h_{n}+2$$h_{n+1}=t_{n}+2h_{n}+1$ Since the equation has the solution $$(t, h) = (0, 0)$$ the sequence of solutions can easily be generated $(t,h) = (0,0),(2,1),(9,5),(35,20),(132,76), …$ from which the possible side lengths are $(0,0),(3,1),(45,15),(630,210),(8778,2926), …$The solution is hence equilateral triangle tiles with sides of 630 mm and regular hexagon tiles with sides of 210 mm.
[1] More generally for regular polyhedra with $$m$$ and $$n$$ sides $$(s)$$ equal probabilities requires that $\tan(180/m)/s_m = \tan(180/n)/s_n$Here is a programmed solution using the quadratic Diophantine equation solver in my number theory library (available here). | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8289439082145691, "perplexity": 576.2301007261582}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943747.51/warc/CC-MAIN-20230321225117-20230322015117-00673.warc.gz"} |
https://brilliant.org/problems/hydrogen-spectral-series/ | # Hydrogen spectral series
From the above, [A] is the spectral series of hydrogen, on a logarithmic scale, and [B] is a schematic diagram of the energy levels of a hydrogen atom, where $$\lambda_1$$, $$\lambda_2$$ and $$\lambda_3$$ are the wavelengths of the electromagnetic waves released when the electron transition from $$n=3$$ to $$n=2,$$ or from $$n=2$$ to $$n=1,$$ or from $$n=3$$ to $$n=1,$$ occurs, respectively, according to the Bohr's atomic model. Which of the following statements is correct?
a) The wavelength $$\lambda_1$$ in [B] belongs to the visible light range in [A].
b) The wavelength $$\lambda_3$$ in [B] belongs to the Paschen series in [A].
c) The energy of photon with the wavelength of $$\lambda_1$$ in [B] is larger than the energy of photon with the wavelength of $$\lambda_2$$ in [B]
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http://www.wjetech.cl/e/ | A simple but interesting experiment.
Using the same classroom elements employed to exhibit the conservation of linear momentum, we can demonstrate a simple method that in a VERY NON-INTUITIVE manner allows a closed system to apparently disregard said law of conservation of linear momentum.
The results of the experiment open a thought-provoking discussion with interesting practical applications.
What we need.
A low friction track and cart (fig 1) common in many physics classrooms (air track type not suitable for this particular experiment).
Fig 1
The low friction track and its cart must itself be on a low friction surface (fig 2) so that both the track and the cart can move horizontally independently with minimum friction between them.
Fig 2
The desired effect can also be observed simply by putting the low friction track on a skateboard (or other low friction surface figs 3 and 4).
Fig 3
Fig 4
The experiment can also be performed on low friction rails, water (messy) or dry ice plucks.
Description of the setup.
We have the low friction track (1) on a low friction surface (2) with a low friction chart (3) on top.
There is a tensed spring (4) between the chart (3) and the track’s (1) +X side, the spring will be released by a trigger mechanism on command (Fig 5).
Fig 5 (system to be tested)
Test 1, release the spring with no air brake
Fig 6
When the spring is released (fig 6) it pushes with force F1 against the low friction track or mass 1 (M1) giving it an acceleration in the +X direction so that it gains a +X velocity of V1, an equal force F2 is exerted against the cart or mass 2 (M2) so that it gains a velocity (V2) in the –X direction.
Fig 7
When the cart (M2) collides with the track’s –X end (5) it exerts a force F4 that stops the track’s horizontal velocity, an equal and opposed force F3 stops the cart and all elements return to their original state (0 velocity).
F4 (and therefore F3) are equal to forces F1 and F2 because cart M2 travels from +X to –X at a CONSTANT velocity, as cart M1 travels from –X to +X also at a constant velocity.
Test 2, release the spring with an air brake attached to mass M2 (the cart).
Fig 8
We now place an air brake (6) (small parachute) on cart M2 and repeat the experiment.
Fig 9
Again when the spring is released it pushes with force F1 against the low friction track/ mass 1 (M1) giving it an acceleration in the +X direction so that it gains a +X velocity of V1 that is CONSTANT as mass M1 travels in the +X direction.
The cart/mass 2 (M2) gains an INITIAL velocity (V2i) in the –X direction that DECREASES (because of the air brake’s drag) without affecting the +X velocity of mass M1.
So while M1’s +X velocity is constant, M2’s –X velocity decreases.
Fig 10
F3= RV1/M1
Because the cart’s (M2) final velocity (V2f) is LESS than its initial velocity (V2i), force F4 (F4=V2f/M2) is not equal to the force that gave M2 its initial velocity (V2i), therefore M1’s momentum is greater than M2’s momentum and the track/M1 will retain a +X velocity (RV1), it has accelerated.
Up to now there is nothing unexpected in the experiment’s results.
Test 3, enclose masses M1 and M2 in an aerodynamic airtight cover.
Fig 11
Now comes the interesting part, the apparently controversial part, we enclose the low friction track/ mass 1 (M1) with an airtight covering (1) and repeat the experiment with and without the air brake
Without air brake.
Fig 12
When we execute the experiment without an air brake attached to mass M2 (figs 5, 6 and 7) mass M1 will go thru 3 distinct stages:
(1) At rest (fig 5)
(2) Spring is released and mass M1 gains velocity in the +X direction (fig 6)
(3) M1 stops when chart M2 collides against the inner –X side of M1 (fig 7).
With the air brake
Fig 13
When we execute the experiment with an air brake attached to mass M2 (figs 8, 9 and 10) mass M1 will behave a little differently:
(1) At rest (fig 8)
(2) Spring is released and mass M1 gains velocity in the +X direction (fig 9)
(3) As M2 (the chart) travels in the –X direction, drag on the air brake (little parachute) slows it’s –X velocity, the random collisions of air molecules do NOT interact directly on the inner surface of the airtight covering ( as the result of the experiment demonstrates)
(4) M2 collides against the inner –X side of M1 (fig 10) but as its velocity (therefore its momentum) has decreased it is not sufficient to stop the system and M1 continues moving in the +X
direction until friction stops it’s +X velocity.
But in a 0g and airless environment M1 will retain its velocity change in the +X direction and may increment that acceleration in every cycle:
For a bettor description of the cycle and practical application please see:
A simple description of the proposal using two astronauts and batman (Adam West version).
OR
A step by step explanation of the principals involved (For the serious minded, no kids, no skateboard, no batman, no puns).
OR
Article on Fortean Times magazine by David Hambling.
ALSO SEE:
Using metrology similar to Henry Cavendish’s 1706 experiment to measure the force of gravity between
Note 1:
The aerodynamic airtight cover (box) must be of sufficient width so that the air brake has plenty of space for the air in the box to behave in a turbulent manner, if not the air brake will act as a piston compressing the air against the –X side of the box and the experiment will not give the described results.
We have found that the larger the box the better (fig a), a 3m (meter) x 0.5m x 0.5m with a 0.4m x 0.4m air brake (fig b) gets the describe results without problems when tested on low friction rails, water (messy) or dry ice plucks. (although the experiment described here is adequate for demonstration purposes , a more complex torsion balance apparatus using metrology similar to Henry Cavendish’s 1706 experiment to measure the force of gravity between masses is recommended for evaluating true thrust output).
Fig a
Fig b
Main page: http://www.wjetech.cl/
Contact Information
William John Elliott S.
(Exit code) 56-2-2042863
(Exit code) 56-9-85530114
Exit codes (also known as international access codes or IDD International Direct Dialing codes) | {"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": 0.86709064245224, "perplexity": 3806.5235260651607}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607245.69/warc/CC-MAIN-20170523005639-20170523025639-00591.warc.gz"} |
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bobbym
2013-05-13 08:58:08
Hi;
#### Helsaint wrote:
Sin^2(t) = 1/2 - cos2t
That is not correct.
Helsaint
2013-05-13 08:41:54
Shouldn't this be
Sin^2(t) = 1/2 - cos2t
L{sin^2(t)} = 1/2s - s/(s^2+4)
bobbym
2012-10-26 15:38:46
Hi;
Welcome to the forum. That is not correct.
JamesRook
2012-10-26 09:18:26
to be clear I am talking about (sin(t))^2
JamesRook
2012-10-26 09:17:22
Hello,
I get that the laplace transform of sin^2t = -(sin^2te^-st)/s + 2/s^3+4s evaluated from 0...infinity.
when I evaluate the limit from 0..infinity I get that the transform to equal 0. Did I evaluate that right?
bobbym
2011-08-10 22:52:21
Wolfram is going to put it in terms of the Dirac delta function, which I think is a step function.
There are diiferent definitions for a fourier transform, that page will partly explain that.
zetafunc.
2011-08-10 22:50:01
Thanks...
I just tried the Fourier transform of f(x) = 1 and got
... is that correct? I'll check on Wolfram.
bobbym
2011-08-10 22:43:50
Hi;
There are FFT and DFT's. Wikipedia can be a horror story at times. To me that is exactly what that is saying.
I have never seen their notation. They are using small f with a cap ( borrowed from statistics)
zetafunc.
2011-08-10 22:43:40
Also what is the notation for a Fourier transform? For Laplace it's a fancy L, is it a fancy F for Fourier transforms?
zetafunc
2011-08-10 22:38:37
Sorry if it's a bother but do you know how to compute Fourier transforms? I'm trying to learn how, I've seen the Wikipedia article and saw this:
for every real number ξ.
Does this mean that if I put in some function of x, such as sin(x), I'll get f(ξ) where ξ is a real number? Not sure, I'll post my working in a second. Sorry if I sound stupid...
bobbym
2011-08-10 22:21:32
Hi;
Zeroing the LHS will leave you with just the Laplace term. That should be your answer.
I was just asking to see what you thought about it. Since t approaches infinity it will drown out s no matter how small as long as s > 0.
That is nice, spotting the Laplace Transform there.
In addition zetafunc, welcome to the forum! Why not consider becoming a member here?
zetafunc.
2011-08-10 22:20:29
I wasn't given an interval for s, sorry. I am just waiting for my GCSE results (I turn 16 in August) and I'm just trying to extend my knowledge of calculus. I want to learn about Fourier transforms too hopefully but I need some practice with that.
zetafunc.
2011-08-10 22:18:46
Hi,
What I meant is that I get
Then evaluate RHS at 0 and subtract that from the evaluation at infinity. I got 0... so then we have
Therefore
assuming s > 0.
I also tried the Laplace transformation for sina(t) and got
.
bobbym
2011-08-10 22:11:21
Hi;
I am glad to help but we are not done yet.
The LHS has to be evaluated at infinity and then you subtract the evaluation of it at 0. The RHS is untouched.
How are you getting 0 for the LHS?
If s is very small then the LHS is not zero. Were you given some interval for s?
zetafunc.
2011-08-10 22:03:07
Hi,
Thanks for the response again and confirming that my IBP was correct -- I think I get it now - subtract the rightmost term from both sides to get y(s) - 2/(s3 + 4s), evaluate the RHS at 0 and infinity to get 0 (0 - 0 = 0), then add 2/(s3 + 4s) to both sides to get the completed Laplace transform? Is that correct? Phew, thanks for your help. | {"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": 0.942825973033905, "perplexity": 1453.546292316043}, "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-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00323-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://www.fixedpointtheoryandapplications.com/content/2013/1/146 | Research
# On best proximity points for pseudocontractions in the intermediate sense for non-cyclic and cyclic self-mappings in metric spaces
Manuel De la Sen
Author Affiliations
Institute of Research and Development of Processes, University of the Basque Country, Campus of Leioa (Bizkaia), P.O. Box 644, Bilbao, 48080, Spain
Fixed Point Theory and Applications 2013, 2013:146 doi:10.1186/1687-1812-2013-146
Received: 17 September 2012 Accepted: 17 May 2013 Published: 5 June 2013
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
### Abstract
This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.
### 1 Introduction
Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate sense formulated in the framework of Hilbert spaces have received a certain attention in the last years concerning their convergence properties and the existence of fixed points. See, for instance, [1-4] and references therein. Results about the existence of a fixed point are discussed in those papers. On the other hand, important attention has been paid during the last decades to the study of the convergence properties of distances in cyclic contractive self-mappings on p subsets A i X of a metric space ( X , d ) , or a Banach space ( X , ) . The cyclic self-mappings under study have been of standard contractive or weakly contractive types and of Meir-Keeler type. The convergence of sequences to fixed points and best proximity points of the involved sets has been investigated in the last years. See, for instance, [5-20] and references therein. It has to be noticed that every nonexpansive mapping [21,22] is a 0-strict pseudocontraction and also that strict pseudocontractions in the intermediate sense are asymptotically nonexpansive [2]. The uniqueness of the best proximity points to which all the sequences of iterations converge is proven in [6] for the extension of the contractive principle for cyclic self-mappings in either uniformly convex Banach spaces (then being strictly convex and reflexive [23]) or in reflexive Banach spaces [13]. The p subsets A i X of the metric space ( X , d ) , or the Banach space ( X , ) , where the cyclic self-mappings are defined, are supposed to be nonempty, convex and closed. If the involved subsets have nonempty intersections, then all best proximity points coincide, with a unique fixed point being allocated in the intersection of all the subsets, and framework can be simply given on complete metric spaces. The research in [6] is centered on the case of the 2-cyclic self-mapping being defined on the union of two subsets of the metric space. Those results are extended in [7] for Meir-Keeler cyclic contraction maps and, in general, with the p ( 2 ) -cyclic self-mapping T : i p ¯ A i i p ¯ A i defined on any number of subsets of the metric space with p ¯ : = { 1 , 2 , , p } . Other recent research which has been performed in the field of cyclic maps is related to the introduction and discussion of the so-called cyclic representation of a set M, as the union of a set of nonempty sets as M = i = 1 m M i , with respect to an operator f : M M [14]. Subsequently, cyclic representations have been used in [15] to investigate operators from M to M which are cyclic φ-contractions, where φ : R 0 + R 0 + is a given comparison function, M X and ( X , d ) is a metric space. The above cyclic representation has also been used in [16] to prove the existence of a fixed point for a self-mapping defined on a complete metric space which satisfies a cyclic weak φ-contraction. In [18], a characterization of best proximity points is studied for individual and pairs of non-self-mappings S , T : A B , where A and B are nonempty subsets of a metric space. The existence of common fixed points of self-mappings is investigated in [24] for a class of nonlinear integral equations, while fixed point theory is investigated in locally convex spaces and non-convex sets in [25-28]. More recently, the existence and uniqueness of best proximity points of more general cyclic contractions have been investigated in [29,30] and a study of best proximity points for generalized proximal contractions, a concept referred to non-self-mappings, has been proposed and reported in detail in [31]. Also, the study and characterization of best proximity points for cyclic weaker Meir-Keeler contractions have been performed in [32] and recent contributions on the study of best proximity and proximal points can be found in [33-38] and references therein. In general, best proximity points do not fulfill the usual ‘best proximity’ condition x = S x = T x under this framework. However, best proximity points are proven to jointly globally optimize the mappings from x to the distances d ( x , T x ) and d ( x , S x ) . Furthermore, a class of cyclic φ-contractions, which contains the cyclic contraction maps as a subclass, has been proposed in [18] in order to investigate the convergence and existence results of best proximity points in reflexive Banach spaces completing previous related results in [6]. Also, the existence and uniqueness of best proximity points of cyclic φ-contractive self-mappings in reflexive Banach spaces have been investigated in [19]. This paper is devoted to the convergence properties and the existence of fixed points of a generalized version of pseudocontractive, strict pseudocontractive and asymptotically pseudocontractive in the intermediate sense in the more general framework of metric spaces. The case of 2-cyclic pseudocontractive self-mappings is also considered. The combination of constants defining the contraction may be different on each of the subsets and only the product of all the constants is requested to be less than unity. It is assumed that the considered self-mapping can perform a number of iterations on each of the subsets before transferring its image to the next adjacent subset of the 2-cyclic self-mapping. The existence of a unique closed finite limiting sequence on any sequence of iterations from any initial point in the union of the subsets is proven if X is a uniformly convex Banach space and all the subsets of X are nonempty, convex and closed. Such a limiting sequence is of size q p (with the inequality being strict if there is at least one iteration with image in the same subset as its domain), where p of its elements (all of them if q = p ) are best proximity points between adjacent subsets. In the case that all the subsets A i X intersect, the above limit sequence reduces to a unique fixed point allocated within the intersection of all such subsets.
### 2 Asymptotic contractions and pseudocontractions in the intermediate sense in metric spaces
If H is a real Hilbert space with an inner product , and a norm and A is a nonempty closed convex subset of H, then T : A A is said to be an asymptotically β-strictly pseudocontractive self-mapping in the intermediate sense for some β [ 0 , 1 ) if
lim sup n sup x , y A ( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) 0 (2.1)
for some sequence { α n } [ 1 , ) , α n 1 as n [1-4,23]. Such a concept was firstly introduced in [1]. If (2.1) holds for β = 1 , then T : A A is said to be an asymptotically pseudocontractive self-mapping in the intermediate sense. Finally, if α n α [ 0 , 1 ) as n , then T : A A is asymptotically β-strictly contractive in the intermediate sense, respectively, asymptotically contractive in the intermediate sense if β = 1 . If (2.1) is changed to the stronger condition
( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) 0 ; x , y A , n N , (2.2)
then the above concepts translate into T : A A being an asymptotically β-strictly pseudocontractive self-mapping, an asymptotically pseudocontractive self-mapping and asymptotically contractive one, respectively. Note that (2.1) is equivalent to
T n x T n y 2 α n x y 2 + β ( I T n ) x ( I T n ) y 2 + ξ n ; x , y A , n N (2.3)
or, equivalently,
T n x T n y , x y 1 2 β [ ( α n + β ) x y 2 + ( β 1 ) T n x T n y 2 + ξ n ] ; x , y A , n N , (2.4)
where
ξ n : = max { 0 , sup x , y A ( T n x T n y 2 α n x y 2 β ( I T n ) x ( I T n ) y 2 ) } ; n N . (2.5)
Note that the high-right-hand-side term ( I T n ) x ( I T n ) y 2 of (2.3) is expanded as follows for any x , y A :
(2.6)
The objective of this paper is to discuss the various pseudocontractive in the intermediate sense concepts in the framework of metric spaces endowed with a homogeneous and translation-invariant metric and also to generalize them to the β-parameter to eventually be replaced with a sequence { β n } in ( 0 , 1 ) . Now, if instead of a real Hilbert space H endowed with an inner product , and a norm , we deal with any generic Banach space ( X , ) , then its norm induces a homogeneous and translation invariant metric d : X × X R 0 + defined by d ( x , y ) = d ( x y , 0 ) = x y 1 / 2 ; x , y A so that (2.6) takes the form
(2.7)
Define
μ n ( x , y ) : = min ( ρ [ 1 , 1 ] : d 2 ( x y , T n x T n y ) d 2 ( x , y ) + d 2 ( T n x , T n y ) + 2 ρ d ( x , y ) d ( T n x , T n y ) ) ; x , y A , n N , (2.8)
which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that
{ 1 } { ρ R : ( I T n ) x ( I T n ) y 2 d 2 ( x , y ) + d 2 ( T n x , T n y ) + 2 ρ d ( x , y ) d ( T n x , T n y ) } ( ) . (2.9)
The following result holds related to the discussion (2.7)-(2.9) in metric spaces.
Theorem 2.1Let ( X , d ) be a metric space and consider a self-mapping T : X X . Assume that the following constraint holds:
d 2 ( T n x , T n y ) α n ( x , y ) d 2 ( x , y ) + β n ( x , y ) ( d 2 ( x , y ) + d 2 ( T n x , T n y ) ) + 2 μ n ( x , y ) β n ( x , y ) d ( x , y ) d ( T n x , T n y ) + ξ n ( x , y ) ; x , y X , n N (2.10)
with
ξ n = ξ n ( x , y ) : = max ( 0 , ( 1 β n ( x , y ) ) d 2 ( T n x , T n y ) ( α n ( x , y ) + β n ( x , y ) ) d 2 ( x , y ) 2 μ n ( x , y ) β n ( x , y ) d ( x , y ) d ( T n x , T n y ) ) 0 ; x , y X as n (2.11)
for some parameterizing bounded real sequences { α n ( x , y ) } , { β n ( x , y ) } and { μ n ( x , y ) } of general terms α n = α n ( x , y ) , β n = β n ( x , y ) , μ n = μ n ( x , y ) satisfying the following constraints:
(2.12)
with lim sup n [ β n ( x , y ) max ( 1 , 1 + 2 μ n ( x , y ) ) ] < 1 and, furthermore, the following condition is satisfied:
( μ n ( x , y ) 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) ) 0 ; x , y X as n (2.13)
if and only if α n + 2 β n ( 1 + μ n ) 1 ; x , y X as n .
Then the following properties hold:
(i) lim n d ( T n x , T n y ) d ( x , y ) for any x , y X so that T : X X is asymptotically nonexpansive.
(ii) Let ( X , d ) be complete, d : X × X R 0 + be, in addition, a translation-invariant homogeneous norm and let ( X , ) ( X , d ) , with being the metric-induced norm from d : X × X R 0 + , be a uniformly convex Banach space. Assume also that T : X X is continuous. Then any sequence { T n x } ; x A is bounded and convergent to some point z x = z x ( x ) C , being in general dependent onx, in some nonempty bounded, closed and convex subsetCofA, whereAis any nonempty bounded subset ofX. Also, d ( T n x , T n + m x ) is bounded; n , m N , lim n d ( T n x , T n + m x ) = 0 ; x A , m N and z x = z x ( x ) = T z x C is a fixed point of the restricted self-mapping T : C C ; x A . Furthermore,
lim n ( d 2 ( T n + 1 x , T n + 1 y ) d 2 ( T n x , T n y ) ) = 0 ; x , y A . (2.14)
Proof Consider two possibilities for the constraint (2.10), subject to (2.11), to hold for each given x , y X and n N as follows:
(A) d ( T n x , T n y ) d ( x , y ) for any x , y X , n N . Then one gets from (2.10)
d 2 ( T n x , T n y ) ( α n + β n ) d 2 ( x , y ) + β n d 2 ( T n x , T n y ) + 2 μ n β n d 2 ( x , y ) + ξ n d ( T n x , T n y ) k a n d 2 ( x , y ) + ξ n 1 β n ; (2.15)
x , y A , n N , where
k a n = k a n ( x , y ) = α n + β n ( 1 + 2 μ n ) 1 β n 1 ; x , y X as n , (2.16)
which holds from (2.12)-(2.13) if lim sup n β n ( x , y ) < 1 since
( μ n ( x , y ) 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) ) 0 ;
x , y X as n in (2.13) is equivalent to (2.16). Note that 0 k a n 1 is ensured either with min ( α n + β n ( 1 + 2 μ n ) , 1 β n ) 0 or with max ( α n + β n ( 1 + 2 μ n ) , 1 β n ) 0 if
(2.17)
However, β n > 1 with ξ n > 0 has to be excluded because of the unboundedness or nonnegativity of the second right-hand-side term of (2.15).
(B) d ( T n x , T n y ) d ( x , y ) for some x , y X , n N . Then one gets from (2.10)
d ( T n x , T n y ) 2 ( α n + β n ) d 2 ( x , y ) + β n d 2 ( T n x , T n y ) + 2 μ n β n d 2 ( T n x , T n y ) + ξ n d ( T n x , T n y ) 2 k b n d 2 ( x , y ) + ξ n 1 β n ( 1 + 2 μ n ) , (2.18)
where
k b n = k b n ( x , y ) = α n + β n 1 β n ( 1 + 2 μ n ) 1 as n , (2.19)
which holds from (2.12) and k b n 1 if lim sup n [ β n ( x , y ) max ( 1 , 1 + 2 μ n ( x , y ) ) ] < 1 , and
μ n ( x , y ) [ 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) , 1 β n ( x , y ) 2 β n ( x , y ) ) . (2.20)
Thus, (2.15)-(2.16), with the second option in the logic disjunction being true if and only if ξ n = 0 together with (2.18)-(2.20), are equivalent to (2.12)-(2.13) by taking k n = k n ( x , y ) to be either k a n or k b n for each n N . It then follows that lim sup n ( d ( T n x , T n y ) d ( x , y ) ) 0 ; x , y X from (2.15)-(2.19) since 0 k n = k n ( x , y ) 1 and k n ( x , y ) 1 ; x , y X as n . Thus, T : X X is asymptotically nonexpansive. Thus, Property (i) has been proven. Property (ii) is proven as follows. Consider the metric-induced norm equivalent to the translation-invariant homogeneous metric d : X × X R 0 + . Such a norm exists since the metric is homogeneous and translation-invariant so that norm and metric are formally equivalent. Rename A 0 A and define a sequence of subsets A j : = { T j x : x A 0 } of X. From Property (i), { d ( T n x , T n y ) } is bounded; x , y X if d ( x , y ) is finite, since it is bounded for any finite n N and, furthermore, it has a finite limit as n . Thus, all the collections of subsets i = 1 k A i ; k N are bounded since A 0 is bounded. Define the set C = C ( A 0 ) : = cl [ convex ( i = 1 A k ) ] which is nonempty bounded, closed and convex by construction. Since ( X , d ) is complete, ( X , ) ( X , d ) is a uniformly convex Banach space and T : C C is asymptotically nonexpansive from Property (i), then it has a fixed point z = T z C [1,23]. Since the restricted self-mapping T : C C is also continuous, one gets from Property (i)
lim n d ( T n x , T n z ) = lim n d ( T n x , z ) = d ( lim n T n x , z ) d ( x , z ) < ; x A . (2.21)
Then any sequence { T n x } is convergent (otherwise, the above limit would not exist contradicting Property (i)), and then bounded in C; x A . This also implies d ( T n x , T n + m x ) is bounded; x A , n , m N and lim n d ( T n x , T n + m x ) = 0 ; x A , m N . This implies also T n x z x ( x ) as n ; x A such that z x ( x ) = T z x ; x A which is then a fixed point of T : C C (otherwise, the above property lim n d ( T n x , T n + m x ) = 0 ; x A , m N would be contradicted). Hence, Property (ii) is proven. □
First of all, note that Property (ii) of Theorem 2.1 applies to a uniformly convex space which is also a complete metric space. Since the metric is homogeneous and translation-invariant, a norm can be induced by such a metric. Alternatively, the property could be established on any uniformly convex Banach space by taking a norm-induced metric which always exists. Conceptually similar arguments are used in later parallel results throughout the paper. Note that the proof of Theorem 2.1(i) has two parts: Case (A) refers to an asymptotically nonexpansive self-mapping which is contractive for any number of finite iteration steps and Case (B) refers to an asymptotically nonexpansive self-mapping which is allowed to be expansive for a finite number of iteration steps. It has to be pointed out concerning such a Theorem 2.1(ii) that the given conditions guarantee the existence of at least a fixed point but not its uniqueness. Therefore, the proof is outlined with the existence of a z Fix ( T | C ) for any nonempty, bounded and closed subset A of X. Note that the set C, being in general dependent on the initial set A, is bounded, convex and closed by construction while any taken nonempty set of initial conditions A X is not required to be convex. However, the property that all the sequences converge to fixed points opens two potential possibilities depending on particular extra restrictions on the self-mapping T : C C , namely: (1) the fixed point is not unique so that z x z for any x A (and any A in X) so that some set Fix ( T | C ) for some C = C ( A ) X contains more than one point. In other words, d 2 ( T n x , T n y ) 0 as n ; x , y A has not been proven although it is true that lim n ( d 2 ( T n + 1 x , T n + 1 y ) d 2 ( T n x , T n y ) ) = 0 ; x , y A ; (2) there is only a fixed point in X. The following result extends Theorem 2.1 for a modification of the asymptotically nonexpansive condition (2.10).
Theorem 2.2Let ( X , d ) be a metric space and consider the self-mapping T : X X . Assume that the constraint below holds:
d 2 ( T n x , T n y ) α n ( x , y ) d 2 ( x , y ) + β n ( x , y ) ( d 2 ( x , y ) + d 2 ( T n x , T n y ) ) + 2 μ n ( x , y ) β n ( x , y ) d 2 ( T n x , T n y ) + ξ n ( x , y ) ; x , y X , n N (2.22)
with
ξ n = ξ n ( x , y ) : = max ( 0 , ( 1 β n ( x , y ) ) d 2 ( T n x , T n y ) ( α n ( x , y ) + β n ( x , y ) ) d 2 ( x , y ) 2 μ n ( x , y ) β n ( x , y ) d 2 ( T n x , T n y ) ) 0 ; x , y X as n (2.23)
for some parameterizing real sequences α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) satisfying, for any n N ,
{ α n ( x , y ) } [ 0 , ) , { μ n ( x , y ) } [ 1 , 1 β n ( x , y ) 2 β n ( x , y ) ) , { β n ( x , y ) } [ 0 , 1 ] ; x , y X , n N . (2.24)
Then the following properties hold:
(i) lim n d ( T n x , T n y ) d ( x , y ) so that T : X X is asymptotically nonexpansive, and then lim n d ( T n x , T n y ) d ( x , y ) ; x , y X if
α n ( x , y ) + β n ( x , y ) 1 β n ( x , y ) ( 1 + 2 μ n ( x , y ) ) 1 μ n ( x , y ) [ 1 α n ( x , y ) 2 β n ( x , y ) 2 β n ( x , y ) , 1 β n ( x , y ) 2 β n ( x , y ) ) ; x , y X , n N (2.25)
and the following limit exists:
α n ( x , y ) + 2 β n ( x , y ) ( 1 + μ n ( x , y ) ) 1 ; x , y X as n . (2.26)
(ii) Property (ii) of Theorem 2.1 if ( X , d ) is complete and ( X , ) ( X , d ) is a uniformly convex Banach space under the metric-induced norm .
Sketch of the proof Property (i) follows in the same way as the proof of Property (i) of Theorem 2.1 for Case (B). Using proving arguments similar to those used to prove Theorem 2.1, one proves Property (ii). □
The relevant part in Theorem 2.1 being of usefulness concerning the asymptotic pseudocontractions in the intermediate sense and the asymptotic strict contractions in the intermediate sense relies on Case (B) in the proof of Property (i) with the sequence of constants k n ( x , y ) 1 ; x , y X , n N and k n ( x , y ) 1 ; as n , x , y X . The concepts of an asymptotic pseudocontraction and an asymptotic strict pseudocontraction in the intermediate sense motivated in Theorem 2.1 by (2.7)-(2.9), under the asymptotically nonexpansive constraints (2.10) subject to (2.11) and in Theorem 2.2 by (2.22) subject to (2.23) are revisited as follows in the context of metric spaces.
Definition 2.3 Assume that ( X , d ) is a complete metric space with d : X × X R 0 + being a homogeneous translation-invariant metric. Thus, T : A A is asymptotically β-strictly pseudocontractive in the intermediate sense if
lim sup n ( ( 1 β n ( 1 + 2 μ n ) ) d 2 ( T n x , T n y ) ( α n + β n ) d 2 ( x , y ) ) 0 ; x , y A (2.27)
for β n = β [ 0 , 1 ) ; n N and some real sequences { α n } , { μ n } being, in general, dependent on the initial points, i.e., α n = α n ( x , y ) , μ n = μ n ( x , y ) and
{ μ n } [ 1 , 1 β 2 β ) and { α n } [ 1 , ) ; n N , α n 1 and μ n 1 as n ; x , y A , n N . (2.28)
Definition 2.4 T : A A is asymptotically pseudocontractive in the intermediate sense if (2.30) holds with { μ n } [ 1 , 1 β n 2 β n ) , { β n } [ 0 , 1 ] , { α n } [ 1 , ) , α n 1 , β n 1 , μ n 1 as n and the remaining conditions as in Definition 2.3 with α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) .
Definition 2.5 T : A A is asymptotically β-strictly contractive in the intermediate sense if α n [ 0 , ) , β n = β [ 0 , 1 ) , μ n [ 1 , 1 β 2 β ) ; n N , μ n μ [ 1 , 1 β 2 β min ( 1 , 1 α + β ) ) , α n α [ 0 , 1 ) as n , in Definition 2.3 with α n = α n ( x , y ) , μ n = μ n ( x , y ) .
Definition 2.6 T : A A is asymptotically contractive in the intermediate sense if α n [ 0 , ) , { β n } [ 0 , 1 ) , μ n [ 1 , 1 β n 2 β n ) ; n N , μ n μ [ 1 , 1 + α 2 ) , α n α [ 0 , 1 ) , and β n β = 1 as n in Definition 2.3 with α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) .
Remark 2.7 Note that Definitions 2.3-2.5 lead to direct interpretations of their role in the convergence properties under the constraint (2.22), subject to (2.23), by noting the following:
(1) If T : A A is asymptotically β-strictly pseudocontractive in the intermediate sense (Definition 2.3), then the real sequence { k 1 n } of asymptotically nonexpansive constants has a general term k 1 n : = ( α n + β 1 β ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β 1 + β ) 1 / 2 , ) [ 1 , ) ; n N , and it converges to a limit k 1 = 1 since ξ n 0 and α n 1 as n ; x , y A from (2.22) since μ n 1 from (2.27). Then T : A A is trivially asymptotically nonexpansive as expected.
(2) If T : A A is asymptotically pseudocontractive in the intermediate sense (Definition 2.4), then the sequence { k 2 n } of asymptotically nonexpansive constants has the general term: k 2 n : = ( α n + β n 1 β n ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β n 1 + β n ) 1 / 2 , ) [ 1 , ) ; n N , and it converges to a limit k 2 = 1 since α n 1 , β n 1 as n . Then T : A A is also trivially asymptotically nonexpansive as expected. Since α n 1 , note that β n > β k 2 n > k 1 n and β n < β k 2 n < k 1 n for any n N , while k 1 n 1 , k 2 n 1 as n since ξ n 0 as n ; x , y A from (2.22)-(2.23).
(3) If T : A A is asymptotically β-strictly contractive in the intermediate sense (Definition 2.5), then the sequence of asymptotically contractive constants is defined by k 3 n : = ( α n + β 1 β ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β 1 + β ) 1 / 2 , ) [ ( β 1 + β ) 1 / 2 , ) ; n N and k 3 n k 3 = ( α + β 1 β ( 1 + 2 μ ) ) 1 / 2 [ 0 , 1 ) as n for any μ [ 1 , 1 α 2 β 2 β ) such that μ n μ as n , since α + 2 β ( 1 + μ ) < 1 . Then T : A A is an asymptotically strict contraction as expected since ξ n 0 as n ; x , y A from (2.22)-(2.23). Note that the asymptotic convergence rate is arbitrarily fast as α and β are arbitrarily close to zero, since k 3 = O ( α + β ) = o ( α + β ) = o ( max ( α , β ) ) becomes also arbitrarily close to zero, and k 3 2 K ( α + β ) with K = K ( β , μ ) = 1 1 β ( 1 + 2 μ ) ( 0 , ) .
(4) If T : A A is asymptotically contractive in the intermediate sense (Definition 2.6), then the sequence of asymptotically contractive constants is defined by
k 4 n : = ( α n + β n 1 β n ( 1 + 2 μ n ) ) 1 / 2 [ ( α n + β n 1 + β n ) 1 / 2 , ) [ ( β n 1 + β n ) 1 / 2 , ) ; n N
with β n β = 1 and k 4 n k 4 = ( 1 + α 2 | μ | ) 1 / 2 [ 1 2 , 1 ) as n for some μ [ 1 , 1 + α 2 ) since μ < 0 with | μ | > 1 + α 2 so that k 4 [ ( 1 + α 2 ) 1 / 2 , 1 ) . Then T : A A is an asymptotically strict contraction as expected since ξ n 0 as n ; x , y A from (2.23). Note that k 3 = k 4 if μ < 0 and | μ | = 1 + α 2 α and k 4 2 1 2 + o ( α ) 1 2 . Note also that k 3 < k 4 if μ 0 and | μ | < 1 + α 2 α , while k 3 k 4 if μ 0 and | μ | 1 + α 2 α . In the first case, the convergence to fixed points (see Theorem 2.8 below) is guaranteed to be asymptotically faster if the self-mapping is asymptotically β-strictly contractive in the intermediate sense than if it is just asymptotically contractive in the intermediate sense if β n > β , n N . Note also that if the sequences { α n } and { μ n } are identical in both cases, then k 3 n < k 4 n for any n N such that β n > β and k 3 n k 4 n for any n N such that β n β .
(5) The above considerations could also be applied to Theorem 2.1 for the case d ( T n x , T n y ) d ( x , y ) (Case (B) in the proof of Property (i)) being asymptotically nonexpansive for the asymptotically nonexpansive condition (2.10) subject to (2.11).
The subsequent result, being supported by Theorem 2.2, relies on the concepts of asymptotically contractive and pseudocontractive self-mappings in the intermediate sense. Therefore, it is assumed that { α n ( x , y ) } [ 1 , ) .
Theorem 2.8Let ( X , d ) be a complete metric space endowed with a homogeneous translation-invariant metric d : X × X R 0 + and consider the self-mapping T : X X . Assume that ( X , ) ( X , d ) is a uniformly convex Banach space endowed with a metric-induced norm from the metric d : X × X R 0 + . Assume that the asymptotically nonexpansive condition (2.22), subject to (2.23), holds for some parameterizing real sequences α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) satisfying, for any n N ,
{ α n ( x , y ) } [ 1 , ) , { μ n ( x , y ) } [ 1 , 1 β n ( x , y ) 2 β n ( x , y ) ) , { β n ( x , y ) } [ 0 , β ) [ 0 , 1 ] ; (2.29)
x , y X , n N . Then lim n d ( T n x , T n y ) d ( x , y ) for any x , y X satisfying the conditions
α n ( x , y ) + β n ( x , y ) 1 β n ( x , y ) ( 1 + 2 μ n ( x , y ) d ( x , y ) ) 1 ; α n ( x , y ) + 2 β n ( x , y ) ( 1 + μ n ( x , y ) ) 1 ; x , y X as n . (2.30)
Furthermore, the following properties hold:
(i) T : C C is asymptoticallyβ-strictly pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set C = C ( A ) X and any given nonempty, bounded and closed subset A X of initial conditions if (2.29) hold with 0 β n = β < 1 , { μ n } [ 1 , 1 β 2 β ) , { α n } [ 1 , ) , α n 1 and μ n 1 as n ; x , y A , n N . Also, T : C C has a fixed point for any such setCif T : X X is continuous.
(ii) T : C C is asymptotically pseudocontractive in the intermediate sense for some nonempty, bounded, closed and convex set C = C ( A ) X and any given nonempty, bounded and closed subset A X of initial conditions if (2.29) hold with { β n } [ 0 , 1 ] , { μ n } [ 1 , 1 β n 2 β n ) , { α n } [ 1 , ) , β n 1 , α n 1 and μ n 1 as n ; x , y A , n N . Also, T : C C has a fixed point for any such setCif T : X X is continuous.
(iii) If (2.29) hold with α n [ 0 , ) , β n = β [ 0 , 1 ) , μ n [ 1 , 1 β 2 β ) , μ n μ [ 1 , 1 α 2 β 2 β ) ; n N and α n α [ 0 , 1 ) as n , then T : X X is asymptoticallyβ-strictly contractive in the intermediate sense. Also, T : X X has a unique fixed point.
(iv) If (2.29) hold with α n [ 0 , ) , { β n } [ 0 , 1 ) , μ n [ 1 , 1 β n 2 β n ) , μ n μ [ 1 , 1 + α 2 ) ; n N , β n 1 and α n α [ 0 , 1 ) as n , then T : X X is asymptotically strictly contractive in the intermediate sense. Also, T : X X has a unique fixed point.
Proof (i) It follows from Definition 2.3 and the fact that Theorem 2.2 holds under the particular nonexpansive condition (2.22), subject to (2.23), so that T : A A is asymptotically nonexpansive (see Remark 2.7(1)). Property (ii) follows in a similar way from Definition 2.4 (see Remark 2.7(2)). Properties (iii)-(iv) follow from Theorem 2.2 and Definitions 2.5-2.6 implying also that the asymptotically nonexpansive self-mapping T : X X is also a strict contraction, then continuous with a unique fixed point, since α + 2 β ( 1 + μ ) < 1 (see Remark 2.7(3)) and μ < 0 with | μ | > 1 + α 2 (see Remark 2.7(4)), respectively. (The above properties could also be got from Theorem 2.1 for Case (B) of the proof of Theorem 2.1(ii) - see Remark 2.7(5).) □
Example 2.9 Consider the time-varying pth order nonlinear discrete dynamic system
x k + n = T n x k x k + 1 + F n ( x k ) ( x k + 1 x k ) + η n ( x k ) ; x 1 A 1 R p , k N ; (2.31)
k , n N for some given nonempty bounded set A 1 , where { F n k } is a R p × p matrix sequence of elements F n k : R p R p × p with F n k = F n ( x k ) and η n k : R p R p with η n k = η n ( x k ) ; k N , and T : R p R p defines the state-sequence trajectory solution { x k ( x 1 , x 2 ) } . Equation (2.13) requires the consistency constraint F 1 1 to calculate x 2 . However, other discrete systems being dealt with in the same way as, for instance, that obtained by replacing x k + 1 x k 1 in (2.31) with the initial condition x 0 A 1 (and appropriate ad hoc re-definition of the mapping which generates the trajectory solution from given initial conditions) do not require such a consistency constraint. The dynamic system (2.31) is asymptotically linear if η n ( x ) 0 as n ; x R p . Note that for the Euclidean distance (and norm), d ( x k + n , x k + 1 ) F n k d ( x k , x k + 1 ) + η n k ; k N . Assume that the squared spectral norm of F n k is upper-bounded by k n k 2 = k n 2 ( x k ) = α n k + β n k 1 β n k ( 1 + 2 μ n k ) for some parameterizing scalar sequences { α n k } , { β n k } and { μ n } which can be dependent, in a more general case, on the state x k . This holds, for instance, if F n k = F n ( x k ) = 1 a n k ( α n k + β n k 1 β n k ( 1 + 2 μ n k ) ) 1 / 2 P n k , where { a n k } is a real positive sequence satisfying a n k P n k and P n k ( = P n ( x k ) ) : N × R p R p × p both being potentially dependent on the state as the rest of the parameterizing sequences. Since the spectral norm equalizes the spectral radius if the matrix is symmetric, then k n can be taken exactly as the spectral radius of F n k in such a case, i.e., it equalizes the absolute value of its dominant eigenvalue. We have to check the condition
lim sup n ( ( 1 + 2 μ n k β n k ) d 2 ( T n x k , T n y k ) ( α n k + β n k ) d 2 ( x k , y k ) ) 0 ; k N (2.32)
provided, for instance, that the distance is the Euclidean distance, induced by the Euclidean norm, then both being coincident, and provided also that we take the metric space ( R p , d ) which holds, in particular, if
(a) { | η n k | } R 0 + , { α n k } [ 1 , ) , β n k = β [ 0 , 1 ) , { μ n k } [ 1 , 1 β 2 β ) ; n , k N , α n k 1 and η n k 0 , μ n k 1 , as n ; k N . This implies that k 1 n k 2 = α n k + β 1 β n k ( 1 + 2 μ n k ) 1 ; n , k N and k 1 n k 1 as n ; k N . Thus, T : R p R p is asymptotically nonexpansive being also an asymptotic strict β-pseudocontraction in the intermediate sense. This also implies that (2.31) is globally stable as it is proven as follows. Assume the contrary so that there is an infinite subsequence L u of { x n k } which is unbounded, and then there is also an infinite subsequence L u a which is strictly increasing. Since η n k = η n ( x k ) 0 and k 1 n k = k 1 n ( x k ) 1 as n ; k N , one has that for x 1 A 1 , any given k N and some sufficiently large m 01 = m 01 ( x k ) , m 02 = m 02 ( x k ) N , ε 1 = ε 1 ( m 01 ) R + , ε 2 = ε 2 ( m 02 ) R + such that k m 1 = k m 1 ( x k ) 1 + ε 1 and η m 2 ( x k ) ε 2 ; m 1 m 01 , m 2 m 02 . Now, take m 0 = max ( m 01 , m 02 ) and ε = max ( ε 1 , ε 2 ) . Then x k + m / x k 1 + ε + ε / x k ; m ( N ) m 0 and any given k N . If x k 0 , then stability holds trivially. Assume not, and there are unbounded solutions. Thus, take x k ( 0 ) , x k + m L u a such that x k + m x k M for any given M R + , m ( N ) m ¯ and some m ¯ = m ¯ ( M ) m 0 . Note that since L u a is a strictly increasing real sequence { M ( m ¯ ) } implying M ( m ¯ ) as m ¯ , which leads to a contradiction to the inequality M 1 + ε ( 1 + 1 x k ) for ε 0 + for some sufficiently large m ¯ , then for some sufficiently large M, if such a strictly increasing sequence L u a exists. Hence, there is no such sequence, and then no unbounded sequence L u for any initial condition in A 0 . As a result, for any initial condition in any given subset A 1 of R p (even if it is unbounded), any solution sequence of (2.31) is bounded, and then (2.31) is globally stable. The above reasoning implies that there is an infinite collection of numerable nonempty bounded closed sets { A i R p : i N } , which are not necessarily connected, such that x k A k ; k N and any given x 0 A 0 . Assume that the set A 0 of initial conditions is bounded, convex and closed and consider the collection of convex envelopes { convex A i R p : i N } , define constructively the closure convex set C ( A 0 ) = cl ( convex ( i = 1 convex A i ) ) which is trivially bounded, convex and closed. Note that it is not guaranteed that i = 1 convex A i is either open or closed since there is a union of infinitely many closed sets involved. Note also that the convex hull of all the convex envelopes of the collection of sets is involved to ensure that A is convex since the union of convex sets is not necessarily convex (so that i = 1 convex A i is not guaranteed to be convex while A is convex). Consider now the self-mapping T ¯ : C C which defines exactly the same solution as T : R p R p for initial conditions in A 1 so that T ¯ is identified with the restricted self-mapping T : R p | C C from a nonempty bounded, convex and closed set to itself. Note that ( R p , d ) for the Euclidean distance is a convex metric space which is also complete since it is finite dimensional. Then F n k : A R p R p × p and η n k : A R p R p are both continuous, then T ¯ : C C is also continuous and has a fixed point in A from Theorem 2.8(i).
(b) If the self-mapping is asymptotically pseudocontractive in the intermediate sense, then the above conclusions still hold with the modification k 2 n k 2 = α n k + β n k 1 β n k ( 1 + 2 μ n k ) ( 1 ) 1 and μ n k α n k β n k 1 as n ; k N . From Remark 2.7(2), β n k > β k 2 n k > k 1 n k and β n k < β k 2 n k < k 1 n k for any n , k N . Thus the convergence is guaranteed to be faster for an asymptotic β-strict pseudocontraction in the intermediate sense than for an asymptotic pseudocontraction in the intermediate sense with a sequence { β n k } such that β n k > β ; n N with the remaining parameters and parametrical sequences being identical in both cases. If F n k : A R p R p × p and η n k : A R p R p ; n , k N are both continuous, then T ¯ : A A is continuous and has a fixed point in A from Theorem 2.8(ii).
(c) If T : X X is asymptotically β-strictly contractive in the intermediate sense, then k 3 n k = α n k + β n k 1 β n k ( 1 + 2 μ n k ) k 3 = o ( max ( α , β ) ) [ 0 , 1 ) ; k N so that it is asymptotically strictly contractive and has a unique fixed point from Theorem 2.8(iii).
(d) If T : X X is asymptotically contractive in the intermediate sense, k 4 n k = ( α n k + β n k 1 β n k ( 1 + 2 μ n k ) ) 1 / 2 k 4 1 2 + o ( α ) 1 2 ; k N . Thus, T : X X is an asymptotic strict contraction and has a unique fixed point from Theorem 2.8(iv).
Remark 2.10 Note that conditions like (2.32) can be tested on dynamic systems being different from (2.31) by redefining, in an appropriate way, the self-mapping which generates the solution sequence from given initial conditions. This allows to investigate the asymptotic properties of the self-mapping, the convergence of the solution to fixed points, then the system stability, etc. in a unified way for different dynamic systems. Close considerations can be discussed for different dynamic systems and convergence of the solutions generated by the different cyclic self-mappings defined on the union of several subsets to the best proximity points of each of the involved subsets.
### 3 Asymptotic contractions and pseudocontractions of cyclic self-mappings in the intermediate sense
Let A , B X be nonempty subsets of X. T : A B A B is a cyclic self-mapping if T ( A ) B and T ( B ) A . Assume that the asymptotically nonexpansive condition (2.10), subject to (2.11), is modified as follows:
(3.1)
(3.2)
with ( ξ n γ n ( x , y ) D 2 ) 0 ; x , y X as n , and that the asymptotically nonexpansive condition (2.22), subject to (2.23), is modified as follows:
(3.3)
(3.4)
with ( ξ n γ n ( x , y ) D 2 ) 0 ; x , y X as n , where { γ n ( x , y ) } [ 0 , ) and D = dist ( A , B ) 0 . If A B , then D = 0 and Theorems 2.1, 2.2 and 2.8 hold with the replacement A A B . Then if A and B are closed and convex, then there is a unique fixed point of T : A B A B in A B . In the following, we consider the case that A B = so that D > 0 . The subsequent result based on Theorems 2.1, 2.2 and 2.8 holds.
Theorem 3.1Let ( X , d ) be a metric space and let T : A B A B be a cyclic self-mapping, i.e., T ( A ) B and T ( B ) A , whereAandBare nonempty subsets ofX. Define the sequence { k n } n N [ 0 , ) of asymptotically nonexpansive iteration-dependent constants as follows:
( x , y ) ( A × B ) ( B × A ) , n N provided that T : A B A B satisfies the constraint (3.1), subject to (3.2), and
[ ( d ( T n x , T n y ) d ( x , y ) β n = 1 ) ( γ n = 0 ) , ( x , y ) ( A × B ) ( B × A ) , n N ] (3.6)
and
k n = k n ( x , y ) = α n + β n 1 β n ( 1 + 2 μ n ) 1 ; (3.7)
n N for x A ( y B ) and for x B ( y A ) provided that T : A B A B satisfies the constraint (3.3) subject to (3.4) provided that the parameterizing bounded real sequences { α n ( x , y ) } , { β n ( x , y ) } , { μ n ( x , y ) } and { γ n ( x , y ) } of general terms α n = α n ( x , y ) , β n = β n ( x , y ) and μ n = μ n ( x , y ) fulfill the following constraints:
(3.8)
γ n = γ n ( x , y ) max ( 0 , 1 k n ) and assuming that the following limits exist:
(3.9)
Then, the following properties hold:
(i) T : A B A B satisfies (3.3) subject to (3.4)-(3.9); ( x , y ) ( A × B ) ( B × A ) . Then
lim n d ( T n x , T n y ) [ D , d ( x , y ) ] ; ( x , y ) ( A × B ) ( B × A )
so that T : A B A B is a cyclic asymptotically nonexpansive self-mapping. If x A is a best proximity point ofAand y B is a best proximity point ofB, then lim n d ( T n x , T n y ) = D and T 2 n x z x = z ( x ) and T 2 n y z y = z ( y ) , which are best proximity points ofAandB (not being necessarily identical toxandy), respectively if T : A B A B is continuous.
(ii) Property (i) also holds if T : A B A B satisfies (3.1) subject to (3.2), (3.7), (3.8)-(3.9) and (3.5b) provided that d ( T n x , T n y ) d ( x , y ) ; ( x , y ) ( A × B ) ( B × A ) .
Proof The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)
(3.10)
and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)
d ( T n x , T n y ) 2 k b n d 2 ( x , y ) + ξ n + γ n D 2 1 β n ( 1 + 2 μ n ) lim n d ( T n x , T n y ) [ D , d ( x , y ) ] ; ( x , y ) ( A × B ) ( B × A ) (3.11)
since T n x , T n x A B ; n N since T ( A ) B and T ( B ) A , and k n 1 and γ n ( x , y ) ( 1 k n ) 0 as n ; ( x , y ) ( A × B ) ( B × A ) . Note that (3.8) implies that there is no division by zero in (3.11). Now, assume that (3.10) holds with β n = 1 . From (3.8) and (3.2), μ n [ 1 + α n 2 , 0 ] , equivalently, | μ n | 1 + α n 2 and d ( T n x , T n y ) > α n + 1 2 | μ n | d 2 ( x , y ) d 2 ( x , y ) , which contradicts (3.5a) if ξ n > 0 so that β n = 1 in (3.5a) under (3.7) implies that ξ n = 0 and, since γ n = 0 from (3.6), there is no division by zero on the right-hand side of (3.10) if β n = 1 .
Also, if T : A B A B is continuous, then lim n d ( T 2 n x , T 2 n y ) = d ( lim n T 2 n x , lim n T 2 n y ) = D so that T 2 n x A ; n N , lim n T 2 n x cl A , T 2 n y B and lim n T 2 n x cl B since T ( A ) B and T ( B ) A . This proves Properties (i)-(ii). □
Remark 3.2 Note that Theorem 3.1 does not guarantee the convergence of { T 2 n x } and { T 2 n y } to best proximity points if the initial points for the iterations x A and y B are not best proximity points if T : A B A B is not contractive.
The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings with k n = α n + β n ( 1 + 2 μ n ) 1 β n < 1 ; n N subject to lim n k n = k c 1 .
Theorem 3.3Let ( X , d ) be a metric space and let T : A B A B be a cyclic self-mapping which satisfies the asymptotically nonexpansive constraint (3.1), subject to (3.2), whereAandBare nonempty subsets ofX. Let the sequence { k n } n N [ 0 , 1 ) of asymptotically nonexpansive iteration-dependent constants be defined by a general term k n ( x , y ) = k n : = α n + β n ( 1 + 2 μ n ) 1 β n [ 0 , 1 ) under the constraints γ n ( x , y ) = γ n : = δ j ( 1 k n ) ( 1 β n ) = o ( 1 β n ) , β n 1 μ n 1 + α n 2 , n N and lim n k n = 1 . Then the subsequent properties hold:
(i) The following limits exist:
lim n d ( T n x , T n y ) = D ; ( x , y ) ( A × B ) ( B × A ) ; lim n d ( T n x , T n + 1 x ) = D ; x A B . (3.12)
(ii) Assume, furthermore, that ( X , d ) is complete, AandBare closed and convex and d : X × X is translation-invariant and homogeneous and ( X , d ) ( X , ) is uniformly convex where is the metric-induced norm. Then
lim n d ( T 2 n x , T 2 n + 2 x ) = lim n d ( T 2 n + 1 x , T 2 n + 3 x ) = 0 ; x A B , (3.13)
{ T 2 n x } z A , { T 2 n + 1 x } T z B ; x A , and { T 2 n y } T z B , { T 2 n + 1 x } z A ; x A , y B , wherezandTzare unique best proximity points inAandB, respectively. If A B , then z = T z is the unique fixed point of T : A B A B .
Proof Note from (3.9), under (3.6) and (3.7), that there is no division by zero on the right-hand side of (3.10) and ξ n = γ n = 0 if β n = 1 . Then one has from (3.1)-(3.2), (3.5a), (3.6) and (3.7) that
d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) k j n d 2 ( T j n x , T j n y ) + ( 1 k j n ) D 2 + ξ j n 1 β j n ; j , n N . (3.14)
There are several possible cases as follows.
Case A: { d ( T j n x , T j n y ) } is non-increasing. Then d ( T j n x , T j n y ) g = g ( x , y ) D as n ; ( x , y ) ( A × B ) ( B × A ) . Since { k n } [ 0 , 1 ) , one gets (3.12).
Case B: { d ( T j n x , T j n y ) } is non-decreasing. Then either { d ( T j n x , T j n y ) } d ( T j n x , T j n y ) g = g ( x , y ) D as n ; ( x , y ) ( A × B ) ( B × A ) or it is unbounded. Then it has a subsequence which diverges, from which a strictly increasing subsequence can be taken. But this contradicts lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 following from (3.14) subject to the given parametrical constraints. Thus, if { d ( T j n x , T j n y ) } is non-decreasing, it cannot have a strictly increasing subsequence so that it is bounded and has a finite limit as in Case A.
Case C: { d ( T j n x , T j n y ) } has an oscillating subsequence. It is proven that such a subsequence is finite. Assume not, then if lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 , there is an integer sequence { p n } of general term subject to p n ( n , 2 n ) such that
lim sup n ( d 2 ( T ( j + 1 ) n + p n x , T ( j + 1 ) n + p n y ) d 2 ( T j n + p n x , T j n + p n y ) ) > 0 ,
but the above expression is equivalent, for x p n = T p n x and y p n = T p n y which are in A B , but not jointly in either A or B, to
lim sup n ( d 2 ( T ( j + 1 ) n x p n , T ( j + 1 ) n y p n ) d 2 ( T j n + p n x p n , T j n + p n y p n ) ) > 0 ,
which contradicts lim sup n ( d 2 ( T ( j + 1 ) n x , T ( j + 1 ) n y ) d 2 ( T j n x , T j n y ) ) 0 since both sequences { T j n x } | {"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": 0.9774866700172424, "perplexity": 468.60810851172215}, "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/1427131296603.6/warc/CC-MAIN-20150323172136-00247-ip-10-168-14-71.ec2.internal.warc.gz"} |
http://clay6.com/qa/30495/which-of-the-following-aromatic-substitution-has-zero-dipolemoment- | # Which of the following aromatic substitution has zero dipolemoment?
$\begin{array}{1 1}(a)\;\text{Ortho-derivative}\\(b)\; \text{Para-derivative}\\(c)\;\text{Meta-derivative}\\(d)\;\text{None of the above}\end{array}$
Para derivative of the aromatic substitution has the zero dipole moment.
Hence (b) is the correct answer. | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 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": 0.9985944032669067, "perplexity": 2220.869973040504}, "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-2017-34/segments/1502886105451.99/warc/CC-MAIN-20170819124333-20170819144333-00280.warc.gz"} |
https://www.gradesaver.com/textbooks/math/geometry/geometry-common-core-15th-edition/chapter-10-area-10-1-areas-of-parallelograms-and-triangles-lesson-check-page-619/4 | ## Geometry: Common Core (15th Edition)
36 $in^{2}$
The formula for the area of a triangle is A=$\frac{1}{2}$bh From the drawing you can see the base is given as 9 in. and the height is 8 in. A=$\frac{1}{2}$bh A=$\frac{1}{2}$(9)(8) A=36$in^{2}$ | {"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": 0.898322582244873, "perplexity": 1506.8626984144455}, "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/1526794865863.76/warc/CC-MAIN-20180523235059-20180524015059-00419.warc.gz"} |
https://www.maplesoft.com/support/help/Maple/view.aspx?path=author | Author Information - Maple Programming Help
Author Information
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Referencing the Maple Manuals
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Referencing the Maple Product
Use the following statement to reference the use of the Maple product.
"Maple (specify release). Maplesoft, a division of Waterloo Maple Inc., Waterloo, Ontario." | {"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": 0.9992898106575012, "perplexity": 2449.091337082205}, "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-2019-22/segments/1558232258621.77/warc/CC-MAIN-20190526025014-20190526051014-00387.warc.gz"} |
https://yutsumura.com/nilpotent-ideal-and-surjective-module-homomorphisms/ | # Nilpotent Ideal and Surjective Module Homomorphisms
## Problem 431
Let $R$ be a commutative ring and let $I$ be a nilpotent ideal of $R$.
Let $M$ and $N$ be $R$-modules and let $\phi:M\to N$ be an $R$-module homomorphism.
Prove that if the induced homomorphism $\bar{\phi}: M/IM \to N/IN$ is surjective, then $\phi$ is surjective.
Contents
## Proof.
Since the homomorphism $\bar{\phi}:M/IM \to N/IN$ is surjective, for any $b\in N$ there exists $a\in M$ such that
$\bar{\phi}(\bar{a})=\bar{b}, \tag{*}$ where $\bar{a}=a+IM$ and $\bar{b}=b+IN$.
By definition of $\bar{\phi}:M/IM \to N/IN$, we have
$\bar{\phi}(\bar{a})=\overline{\phi(a)}=\phi(a)+IN.$ Thus, it follows from (*) that
$\phi(a)+IN=b+IN,$ or equivalently
$b-\phi(a)\in IN.$ Thus we have
$b\in \phi(M)+IN.$
Now we claim that for any $b\in N$ and any positive integer $k$, we have
$b\in \phi(M)+I^kN.$ We prove this claim by induction on $k$.
The base case $k=1$ is proved above.
Suppose that $b\in \phi(M)+I^nN$. Then we prove that $b\in \phi(M)+I^{n+1}N$.
Since $b\in \phi(M)+I^nN$, we have
$b=\phi(a)+\sum_{i}\alpha_i c_i,$ where the sum is finite and $\alpha_i\in I^n$ and $c_i\in N$.
Since each $c_i\in N$, we have $c_i \in \phi(M)+IN$ by the base case.
Hence we have
$c_i=\phi(a_i)+\sum_{j_i}\beta_{j_i}d_{j_i}$ for some finite pairs $(\beta_{j_i}, d_{j_i})\in (I, N)$.
It follows that we have
\begin{align*}
b&=\phi(a)+\sum_{i}\alpha_i c_i\\
&=\phi(a)+\sum_{i}\alpha_i \left(\, \phi(a_i)+\sum_{j_i}\beta_{j_i}d_{j_i} \,\right)\\
&=\phi(a)+\sum_i\alpha_i\phi(a_i)+\sum_{i}\sum_{j_i}\alpha_i\beta_{j_i}d_{j_i}\\
&=\phi(a)+\sum_i\phi(\alpha_ia_i)+\sum_{i, j_i}(\alpha_i\beta_{j_i})d_{j_i}\\
&=\phi\left(\, a+\sum_i\alpha_ia_i \,\right)+\sum_{i, j_i}(\alpha_i\beta_{j_i})d_{j_i},
\end{align*}
where the last two equalities follows since $\phi$ is an $R$-module homomorphism.
Since $\alpha_i\in I^n$ and $\beta_{j_i}\in I$, the product $\alpha_i\beta_{j_i}\in I^{n+1}$.
Hence the above expression of $b$ yields that
$b\in \phi(M)+I^{n+1}N,$ and this completes the induction step and the claim is proved.
Now, since $I$ is a nilpotent ideal by assumption, there is a positive integer $n$ such that $I^n$ is the zero ideal of $R$. Thus, it follows from the claim that for any $b\in N$ we have
\begin{align*}
b\in \phi(M)+I^nN=\phi(M).
\end{align*}
This implies that $\phi:M\to N$ is surjective as required.
### More from my site
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##### Difference Between Ring Homomorphisms and Module Homomorphisms
Let $R$ be a ring with $1$ and consider $R$ as a module over itself. (a) Determine whether every module...
Close | {"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": 0.9900394082069397, "perplexity": 135.18717649102584}, "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/1547584518983.95/warc/CC-MAIN-20190124035411-20190124061411-00223.warc.gz"} |
https://iwaponline.com/hr/article-abstract/36/4-5/349/1030/The-realism-of-the-ECHAM5-models-to-simulate-the?redirectedFrom=PDF | A new version of the ECHAM model is investigated in respect of the hydrological cycle in the Arctic and North European area. Several horizontal and two vertical resolution versions are studied. The higher-resolution ECHAM5 models are, in many respects, superior to the lower-resolution versions of the same model family and the older ECHAM4 model. The vertical resolution has a decisive impact but also increased horizontal resolution leads mostly to improvements. Here T106 (about 110 km) often gives the best results. The summer maxima of precipitation, surface temperature and latent heat flux are simulated too early by about a month for several river catchment areas. This shift is strongest in the T106 and T159 models. Another problem with the annual cycle of precipitation is a relative minimum in August to October, especially in the low-resolution ECHAM5 models. The precipitation of the ECHAM5 simulations over the Arctic region exceeds all observational estimates by 5–15 mm/month, strongest in May–June. The latent heat flux over the river catchments has a clear trend towards increased fluxes with higher horizontal and vertical resolution, which seems to reach a maximum with T106. In the comparison of annual mean P-E (precipitation minus evaporation) with observed river discharge only the horizontal resolution seems to be important, again giving best results for the high-resolution models. The year-by-year variability of the simulations is too high, which is more pronounced for the higher-resolution versions. Especially strong impacts are found from the vertical resolution. The interannual variability of the latent heat flux is much smaller than that of precipitation and therefore the results shown for precipitation apply also for the simulated river discharge. Some forcing of ocean temperature anomalies on the precipitation over the Rhine, Kolyma and Indigirka catchment areas have been found, from the northeastern Atlantic and from the Pacific with developing El Niños. Despite the increased random variability in the higher-resolution models, the signal could be detected in almost all simulations. On the whole the higher-resolution (horizontal and vertical) ECHAM5 model simulations are quite improved compared to the low-resolution version of the same model and an older T42 model version. Increasing the vertical resolution from 19 to 31 levels is decisive for this better performance.
This content is only available as a PDF. | {"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": 0.8057541847229004, "perplexity": 1959.9302028962288}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-26/segments/1560628000306.84/warc/CC-MAIN-20190626114215-20190626140215-00248.warc.gz"} |
http://www.cheenta.com/2015/09/24/equal-roots-tomato-subjective-70/ | # Equal Roots (Tomato subjective 70)
problem: Suppose that all roots of the polynomial equation
$${\displaystyle{x^4 – 4x^3 + ax^2 +bx + 1}}$$ = 0 are positive real numbers.
Show that all the roots of the polynomial are equal.
solution: $${\displaystyle{x^4 – 4x^3 + ax^2 +bx + 1}}$$ = 0
If the roots are $${\displaystyle{\alpha}}$$, $${\displaystyle{\beta}}$$, $${\displaystyle{\gamma}}$$ and $${\displaystyle{\lambda}}$$ .
then $${\displaystyle{\alpha}}$$, $${\displaystyle{\beta}}$$, $${\displaystyle{\gamma}}$$ and $${\displaystyle{\lambda}}$$ = 1
& $${\displaystyle{\alpha}}$$ + $${\displaystyle{\beta}}$$ + $${\displaystyle{\gamma}}$$ + $${\displaystyle{\lambda}}$$ = 4.
Now all of $${\displaystyle{\alpha}}$$, $${\displaystyle{\beta}}$$, $${\displaystyle{\gamma}}$$ and $${\displaystyle{\lambda}}$$ are positive so AM-GM inequality is applicable.
$${\displaystyle{\frac{\alpha + \beta + \gamma + \lambda}{4}}}{\ge}$$ $${(\alpha\beta\lambda)^{\frac{1}{4}}}$$
$${\Rightarrow}$$ $${\frac{4}{4}} {\ge}$$ $${1^{\frac{1}{4}}}$$
$${\Rightarrow}$$ 1 $${\ge}$$ 1
Now we know equality in AM-GM occours if all the numbers are equal.So $${\displaystyle{\alpha}}$$, $${\displaystyle{\beta}}$$, $${\displaystyle{\gamma}}$$ and $${\displaystyle{\lambda}}$$ are all equal. | {"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": 0.9744812250137329, "perplexity": 200.0830890516143}, "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/1500549426234.82/warc/CC-MAIN-20170726162158-20170726182158-00429.warc.gz"} |
http://export.arxiv.org/abs/1801.05395 | nlin.CD
(what is this?)
# Title: Noise-induced tipping under periodic forcing: preferred tipping phase in a non-adiabatic forcing regime
Abstract: We consider a periodically-forced 1-D Langevin equation that possesses two stable periodic solutions in the absence of noise. We ask the question: is there a most likely noise-induced transition path between these periodic solutions that allows us to identify a preferred phase of the forcing when tipping occurs? The quasistatic regime, where the forcing period is long compared to the adiabatic relaxation time, has been well studied; our work instead explores the case when these timescales are comparable. We compute optimal paths using the path integral method incorporating the Onsager-Machlup functional and validate results with Monte Carlo simulations. Results for the preferred tipping phase are compared with the deterministic aspects of the problem. We identify parameter regimes where nullclines, associated with the deterministic problem in a 2-D extended phase space, form passageways through which the optimal paths transit. As the nullclines are independent of the relaxation time and the noise strength, this leads to a robust deterministic predictor of preferred tipping phase in a regime where forcing is neither too fast, nor too slow.
Subjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS) MSC classes: 68Q25, 60G17, 37J50 DOI: 10.1063/1.5083973 Cite as: arXiv:1801.05395 [nlin.CD] (or arXiv:1801.05395v3 [nlin.CD] for this version)
## Submission history
From: Yuxin Chen [view email]
[v1] Tue, 16 Jan 2018 18:15:52 GMT (7923kb,D)
[v2] Wed, 11 Jul 2018 20:09:06 GMT (6581kb,D)
[v3] Fri, 7 Dec 2018 21:05:41 GMT (4777kb,D)
Link back to: arXiv, form interface, contact. | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8065634965896606, "perplexity": 2868.5473872702214}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176922.14/warc/CC-MAIN-20201124170142-20201124200142-00047.warc.gz"} |
http://math.stackexchange.com/questions/319114/i-cannot-solve-this-problem-about-surface-area-of-a-cone | # I cannot solve this problem about surface area of a cone
this is the question that i had problems with
A cone with a radius of 5cm has a surface area of 2000$\pi$ cm${}^2$. what is the perpendicular height of this cone?
-
– 1015 Mar 3 '13 at 2:18
Let surface area of cone be $S$.
$$S = \pi \cdot r \cdot l$$ where, $r$ = radius of base and $l$ = slant height.
Again, we have $l = \sqrt{r^2 + h^2}$, where $h$ will be the perpendicular height of cone. Thus,
$$l = \dfrac{S}{\pi r} = \dfrac{2000 \pi}{5\pi} \text{ cm} = 400 \text{ cm}$$
And, $$h = \sqrt{400^2 - 5^2} = 25 \sqrt{255} = 399.218 \text{ cm}$$
Alternatively, the area can also mean the base circular part of cone(normally I think of the cone as a conical hats, and hence, don't consider the bottom area).
In this case, we'll have(using same notations as above):
\begin{align} S &= \pi \cdot r \cdot ( r + l ) \\ 2000 \pi &= 5 \pi ( 5 + l ) \\ l &= \dfrac{2000}{5} - 5 \\ l &= 395 \text{ cm} \end{align}
And, we'll have the height as:
$$h = \sqrt{395^2 - 5^2} = 20 \sqrt{390} = 394.968 \text{ cm}$$
-
I think in most cases the surface area of a cone is intended to include the base, whereas the term "lateral area" is without the base. – Zev Chonoles Mar 3 '13 at 2:50
@ZevChonoles Updated reply with the reason, why the base wasn't included. – hjpotter92 Mar 3 '13 at 3:06
As user julien commented above, the formula for the surface area $S$ of a cone with radius $r\text{ cm}$ and lateral height (a.k.a. slant height) $l\text{ cm}$ is $$S=\pi r^2+\pi rl.$$ You know the value of $S$ and $r$, so we can solve for $l$: $$2000\pi\text{ cm}^2=25\pi\text{ cm}^2+5l\pi\text{ cm}^2 \implies l=\frac{2000\pi-25\pi}{5\pi}=395$$ However, you want to find the perpendicular height of the cone. Let's say it is $h\text{ cm}$. Then looking at a vertical cross-section of the cone, we'd see a right triangle with one leg whose length is $5\text{ cm}$, whose other leg is $h\text{ cm}$, and whose hypotenuse is $395\text{ cm}$. Now use the Pythagorean theorem to solve for $h$.
-
The formula for the surface area, S, of a cone is:
$$S = πr^2 + πrl$$
With r being the radius and l being the length of the cone. We are given the surface area and radius so we can plug those values into our formula:
$$2000pi = (5^2)pi + 5pi*l$$
Solve for l to get:
$$(2000pi-25pi)/(5pi) = l$$
$$395 = l$$
If you look at the shape of a cone you will notice that the perpendicular (i.e right angled) height forms a right triangle with the radius of the cone being the base and the length of the cone being the hypotenuse. Using the Pythagorean theorem and plugging in our known values we will be able to solve for h, the height of the cone.
$$r^2 + h^2 = l^2$$
$$5^2 + h^2 = 395^2$$
$$h^2 = 156025 - 25$$
$$h^2 = 156000$$
$$h = 394.97$$
-
Everything is fine, except your last step is wrong: we have $156000<1000000$, so $$\sqrt{156000}<\sqrt{1000000}=1000$$ so there's no way $\sqrt{156000}=1249$. In fact, using a calculator, I get that $$\sqrt{156000}\approx 394.968$$ – Zev Chonoles Mar 3 '13 at 3:02
Also, you can find some good starting points on how to format mathematics on the site here. This AMS reference is very useful. If you need to format more advanced things, there are many excellent references on LaTeX on the internet, including StackExchange's own TeX.SE site. – Zev Chonoles Mar 3 '13 at 3:04
Thanks and thanks! – Johanna Mar 3 '13 at 3:10 | {"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": 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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9980053901672363, "perplexity": 308.798361432679}, "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-2015-32/segments/1438042989142.82/warc/CC-MAIN-20150728002309-00244-ip-10-236-191-2.ec2.internal.warc.gz"} |
https://openmse.com/tutorial-reference-points/msy_ref_points/ | ## Averaged MSY Reference Points
From the calculation of annual maximum sustainable yield (MSY) reference points, an average value is taken and reported in the simulated Data object. Following a similar procedure to that described in the first section for $SB_0$, we use $A_{50}$ as an approximation of generation time, and average the annual MSY values over $A_{50}$ years around the last historical year. For example, if OM@nyears = 50 and $A_{50}=5$, $SB_{\text{MSY}}$ is calculated as:
$$SB_{\text{MSY}} = \dfrac{\sum_{y=48}^{52}{SB_y^{\text{MSY}}}}{A_{50}}$$ where $SB_y^{\text{MSY}}$ is the spawning biomass corresponding with maximum sustainable yield in year $y$. The logic behind this is, if estimates of MSY, $\text{SB}_\text{MSY}$, etc are available, they are likely calculated based on current life-history information, which would be estimated from data spanning several age classes.
The averaged MSY reference points in the Data object, i.e., in Data@OM and Hist@Ref$ReferencePoints, are not updated in the future projection years. The averaged MSY reference points and annual MSY values by year are returned in MSE@RefPoint and MSE@RefPoint$ByYear, respectively.
Note
The MSY metrics in the MSE object are always calculated using annual values. The slot MSE@SB_SBMSY returns the spawning biomass in the projections divided by $SB_\text{MSY}$ in each year of the projections. For alternative methods to calculate $\text{SB}/\text{SB}_{\text{MSY}}$, such as relative to the constant $\text{SB}_\text{MSY}$ described above, use MSE@SSB or MSE@B and the data stored in MSE@RefPoint. | {"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": 0.893593430519104, "perplexity": 1388.4886523276966}, "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/1634323585177.11/warc/CC-MAIN-20211017113503-20211017143503-00290.warc.gz"} |
http://stats.stackexchange.com/questions/66999/imputing-standard-deviations-for-changes-from-baseline | # Imputing standard deviations for changes from baseline
This is more of a further question on a previous topic, so my apologies if there was some way to link them that I haven't found. It is essentially the same problem as How to back-calculate change from baseline from a p-value for a paired t test . I.e. I need to calculate the standard deviation of the change. However in the study I have the two samples are indepedent, and therefore the authors have already calculated a p-value using the two-sided Wilcoxon rank sum test. Specifically:
Sample 1: n=86, mean=0.58, sd = +/- 0.12
Sample 2: n=69, mean = 0.41 sd = +/- 0.108
The authors have used a two-sided Wilcoxon rank sum test to get a p-value of 0.51.
Is is possible to impute the standard deviation of the change? This is necessary for a meta-analysis I am writing.
Many thanks for any pointers.
- | {"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": 0.9447511434555054, "perplexity": 612.5334198169522}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701146241.46/warc/CC-MAIN-20160205193906-00316-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/solving-a-problem-using-conservation-of-energy.595840/ | # Solving a Problem Using Conservation of Energy
1. Apr 12, 2012
### Nivoh
I apologize in advance for my lack of knowledge of "physics english".
First off, this is high school level. :)
I'm terribly sorry if this is confusing due to my lack of understand of the subject, the metric system, and my lack of grasp for the English language as well as terminology. What is μ?
1. The problem statement, all variables and given/known data
An object is moves up a flat board, the board forms an angle of 30° with the x-axis. During 0.76s, the object accelerates from 5m/s to 0. The only forces affecting the board is gravity and friction.
2. Relevant equations
ƩF=ma, R=μ*N, E=0.5mv^2+mgh, W=F*s.
3. The attempt at a solution
I initially solved this quite easily using ƩF=ma=mg*sin30°+μ*cos30°, where a=Δt/Δv.
Out of curiousity, I figured I could solve this using conservation of mechanical energy, as R is a constant force.
So I went ahead and used R=μ*N, R=E/s, where E is the mechanical energy lost due to friction. N=mg/cos30°, E=0.5mv^2-mgh, therefore μ=R/N=(0.5mv^2-mgh)*cos30°/mgs. s=1.9m, h=0.95m, g=9.81m/s^2, v=5m/s.
I seem however, to have failed, I wonder why? :)
Last edited: Apr 12, 2012
2. Apr 12, 2012
### Nivoh
I am sorry for being scum of the earth and bumping my own post, but I really cannot rest without knowing what I've done wrong. Please let me know if my question is unclear, my English not sufficient to describe the problem, or if I'm not following guidelines.
3. Apr 12, 2012
### Rokas_P
Is it only me or is there no actual mention of what we have to calculate in this problem?
4. Apr 12, 2012
### Nivoh
Duh, how silly of me, my apologies. Thank you for making me aware, was looking for μ, the friction constant.
5. Apr 13, 2012
### Rokas_P
The general outline of how I would approach this problem is this:
1. calculate acceleration
2. write down Newton's Second Law for this problem (here all the sines and cosines come in)
3. calculate μ (since μ is in R=μN)
Edit: I see that you're asking not about how to solve it but why you can't get the right answer when you try to solve it using another method. Hopefully someone can help you out with that :) | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8924341797828674, "perplexity": 901.4437169968126}, "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/1521257648226.72/warc/CC-MAIN-20180323122312-20180323142312-00144.warc.gz"} |
https://hpc.fau.de/systems-services/hpc-usage-reports/physics/hpc-user-report-from-s-fey-theoretical-physics/ | # HPC User Report from S. Fey (Chair of Theoretical Physics I)
## Quantum criticality of two-dimensional quantum magnets with long-range interactions
We investigate the criticality and universality classes of the frustrated and unfrustrated Ising model with long-range interactions in a transverse field. Today, very little is known for these systems in two dimensions, so that we aim to close this gap with our studies.
#### Motivation and problem definition
There are many areas in which long-range interactions become important, although in most studies they have been neglected so far. There are some results for the quantum critical points and universality classes of a one-dimensional quantum Ising chain. However, in two dimension almost nothing is known to date.
Our goal is to shed some light on the question where the phase transition between the polarized and the ordered Z2-symmetry-broken phase happens and what the corresponding universality class is – both for a square and triangular lattice in two dimensions.
#### Methods and codes
In a new approach we combine a inked cluster expansion method (PCUT) using “white graphs” with classical Monte Carlo calculations to obtain a high-order series of the one-quasiparticle gap. Afterwards Pade and DLog Pade extra-polations allow the extension of the sum’s convergence radius and an extraction of quantum-critical points and exponents.
The series expansion yields a large amount of high-dimensional infinite sums which need to be evaluated to obtain numerical data about the phase transition. These sums are evaluated using classical Markov-chain Monte Carlo integration where these sums can be computed for different seeds in parallel.
For all calculations internal codes are used.
#### Results
For the ferromagnetic model on the square and triangular lattice there are three different regimes: The regime around the nearest-neighbor limit shows the nearest-neighbor universality class with the corresponding exponent. For very long-range interactions a mean-field exponent is obtained. In between those areas the critical exponent z*\nu varies continuously when tuning the
range of the interaction.
For an antiferromagnetic interaction we find a different behavior. For the complete parameter range the phase transition seems to stay in the nearest-neighbor universality class.
#### Outreach
The work has been published in PhysRevLett (https://doi.org/10.1103/PhysRevLett.122.017203).
The topic of the project is also a part of the application for an SFB.
#### Researcher’s Bio and Affiliation
Sebastian Fey is currently affiliated with Prof. Dr. Kai P. Schmidt at “Lehrstuhl für Theoretische Physik I” at the FAU Erlangen-Nürnberg. He studied physics at TU Dortmund and got his Master’s degree on the topic “Field-driven Instabilities of the Non-Abelian Topological Phase in the Kitaev Model”. | {"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": 0.8383643627166748, "perplexity": 1179.9511358479629}, "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/1618039398307.76/warc/CC-MAIN-20210420122023-20210420152023-00013.warc.gz"} |
https://www.physicsforums.com/threads/linearly-changing-density.742801/ | # Linearly changing density
1. Mar 11, 2014
### rmfw
1. The problem statement, all variables and given/known data
I need to find the Kinetic energy of a bar rotating about its center of mass.
I know the bar as length 3b and it's center of mass is located at 2b, the bar density changes linearly along it's length.
2. Relevant equations
T=1/2 W^2 I
3. The attempt at a solution
So I was trying to find I for this setup, which requires me to find the equation for density (λ).
I know the density changes linearly along it's length so it must be similar to an equation of the type: λ = k x , with k being a constant and x being position.
Now to find k I did the following equations:
∫λ dx = ∫ k x dx = m/2 (limits of integration are from 0 to 2b)
∫λ dx = ∫ k x dx = m/2 (limits of integration are from 2b to 3b)
The problem is that I get a false statement this way, making it impossible to find the equation for density.
This is very basic stuff but it's giving me a headache since I need to move forward on the problem but I can't due to this niche, help? thanks
2. Mar 11, 2014
### TSny
In general, it is not true that half the mass will be on one side of the CM and the other half on the other side of the CM.
See here, near the bottom of the page, for finding the CM of a continuous distribution.
3. Mar 11, 2014
### TSny
Note, "varying linearly" could be interpreted more generally as saying that $λ = a + kx$ where $a$ is some constant. But, you should be able to use the integral formula for $x_{cm}$ and the fact that $x_{cm} = 2b$ to show $a = 0$.
4. Mar 11, 2014
### Staff: Mentor
What is your general equation for the center of mass if the linear density λ(x) varies with with x? First state your equation for the total mass in terms of λ(x).
Chet
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https://astarmathsandphysics.com/university-maths-notes/topology/2217-proof-that-any-element-in-the-complement-of-a-compact-subset-of-a-hausdorff-space-is-in-a-open-subset-of-the-complement.html?tmpl=component&print=1 | ## Proof That any Element in the Complement of a Compact Subset of a Hausdorff Space is in a Open Subset of the Complement
Theorem
Letbe a compact subset of a Hausdorff spaceIfthen there is an open set such that
Open setsandexist such thatand
Thereforeand
To proveis open, letso that
Sinceis compact - hence closed - an open setexists such thathenceandis open. | {"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": 0.9454846382141113, "perplexity": 1304.4626519760095}, "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-2018-43/segments/1539583510866.52/warc/CC-MAIN-20181016180631-20181016202131-00430.warc.gz"} |
http://www.jlconline.com/projects/energy-efficient/q-a-mildew-in-closets_o | A. Marc Rosenbaum responds: To control mildew growth, we must first look at the relationship between air, water vapor, and relative humidity (RH). Water vapor is water in its gas form, and is present in air. The warmer the air, the more water vapor it can hold; the colder the air, the less water vapor it can hold. RH is a measure of how much water vapor is in the air compared to the maximum amount of water vapor the air can hold at that temperature. As air cools, without changing its water vapor content, its RH goes up. As the air continues to cool, it reaches the point where the water vapor it contains is all that it can hold — this is 100% RH. Cooling the air any further will result in condensation as some of the water vapor changes to liquid.
Mildew can only grow on surfaces where the RH exceeds 70%. Closet surfaces tend to be colder than adjacent rooms, because of poor air circulation from the heated room to the closet, and because, relative to their size, they often have more exterior surface area for heat to escape. A corner or cold wall section lacking proper insulation is... | {"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": 0.9330855011940002, "perplexity": 884.9900582219866}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-48/segments/1448398460942.79/warc/CC-MAIN-20151124205420-00330-ip-10-71-132-137.ec2.internal.warc.gz"} |
http://www.thefullwiki.org/Radial_basis_function_network | # Radial basis function network: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
A radial basis function network is an artificial neural network that uses radial basis functions as activation functions. It is a linear combination of radial basis functions. They are used in function approximation, time series prediction, and control.
## Network architecture
Figure 1: Architecture of a radial basis function network. An input vector x is used as input to all radial basis functions, each with different parameters. The output of the network is a linear combination of the outputs from radial basis functions.
Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The output, $\varphi : \mathbb{R}^n \to \mathbb{R}$, of the network is thus
$\varphi(\mathbf{x}) = \sum_{i=1}^N a_i \rho(||\mathbf{x}-\mathbf{c}_i||)$
where N is the number of neurons in the hidden layer, $\mathbf c_i$ is the center vector for neuron i, and ai are the weights of the linear output neuron. In the basic form all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance and the basis function is taken to be Gaussian
$\rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) = \exp \left[ -\beta \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert ^2 \right]$.
The Gaussian basis functions are local in the sense that
$\lim_{||x|| \to \infty}\rho(\left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert) = 0$
i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron.
RBF networks are universal approximators on a compact subset of $\mathbb{R}^n$. This means that a RBF network with enough hidden neurons can approximate any continuous function with arbitrary precision.
The weights ai, $\mathbf{c}_i$, and β are determined in a manner that optimizes the fit between $\varphi$ and the data.
Figure 2: Two unnormalized radial basis functions in one input dimension. The basis function centers are located at c1 = 0.75 and c2 = 3.25.
### Normalized
#### Normalized architecture
In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is
$\varphi ( \mathbf{x} ) \ \stackrel{\mathrm{def}}{=}\ \frac { \sum_{i=1}^N a_i \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } { \sum_{i=1}^N \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } = \sum_{i=1}^N a_i u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big )$
where
$u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) \ \stackrel{\mathrm{def}}{=}\ \frac { \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } { \sum_{i=1}^N \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) }$
is known as a "normalized radial basis function".
Figure 3: Two normalized radial basis functions in one input dimension. The basis function centers are located at c1 = 0.75 and c2 = 3.25.
#### Theoretical motivation for normalization
There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density
$P\left ( \mathbf{x} \land y \right ) = {1 \over N} \sum_{i=1}^N \, \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) \, \sigma \big ( \left \vert y - e_i \right \vert \big )$
where the weights $\mathbf{c}_i$ and ei are exemplars from the data and we require the kernels to be normalized
$\int \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) \, d^n\mathbf{x} =1$
and
$\int \sigma \big ( \left \vert y - e_i \right \vert \big ) \, dy =1$.
Figure 4: Three normalized radial basis functions in one input dimension. The additional basis function has center at c3 = 2.75
The probability densities in the input and output spaces are
$P \left ( \mathbf{x} \right ) = \int P \left ( \mathbf{x} \land y \right ) \, dy = {1 \over N} \sum_{i=1}^N \, \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big )$
and
The expectation of y given an input $\mathbf{x}$ is
$\varphi \left ( \mathbf{x} \right ) \ \stackrel{\mathrm{def}}{=}\ E\left ( y \mid \mathbf{x} \right ) = \int y \, P\left ( y \mid \mathbf{x} \right ) dy$
where
$P\left ( y \mid \mathbf{x} \right )$
is the conditional probability of y given $\mathbf{x}$. The conditional probability is related to the joint probability through Bayes theorem
$P\left ( y \mid \mathbf{x} \right ) = \frac {P \left ( \mathbf{x} \land y \right )} {P \left ( \mathbf{x} \right )}$
which yields
$\varphi \left ( \mathbf{x} \right ) = \int y \, \frac {P \left ( \mathbf{x} \land y \right )} {P \left ( \mathbf{x} \right )} \, dy$.
This becomes
$\varphi \left ( \mathbf{x} \right ) = \frac { \sum_{i=1}^N e_i \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } { \sum_{i=1}^N \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } = \sum_{i=1}^N e_i u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big )$
when the integrations are performed.
Figure 5: Four normalized radial basis functions in one input dimension. The fourth basis function has center at c4 = 0. Note that the first basis function (dark blue) has become localized.
### Local linear models
It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order,
$\varphi \left ( \mathbf{x} \right ) = \sum_{i=1}^N \left ( a_i + \mathbf{b}_i \cdot \left ( \mathbf{x} - \mathbf{c}_i \right ) \right )\rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big )$
and
$\varphi \left ( \mathbf{x} \right ) = \sum_{i=1}^N \left ( a_i + \mathbf{b}_i \cdot \left ( \mathbf{x} - \mathbf{c}_i \right ) \right )u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big )$
in the unnormalized and normalized cases, respectively. Here $\mathbf{b}_i$ are weights to be determined. Higher order linear terms are also possible.
This result can be written
$\varphi \left ( \mathbf{x} \right ) = \sum_{i=1}^{2N} \sum_{j=1}^n e_{ij} v_{ij} \big ( \mathbf{x} - \mathbf{c}_i \big )$
where
$e_{ij} = \begin{cases} a_i, & \mbox{if } i \in [1,N] \\ b_{ij}, & \mbox{if }i \in [N+1,2N] \end{cases}$
and
$v_{ij}\big ( \mathbf{x} - \mathbf{c}_i \big ) \ \stackrel{\mathrm{def}}{=}\ \begin{cases} \delta_{ij} \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) , & \mbox{if } i \in [1,N] \\ \left ( x_{ij} - c_{ij} \right ) \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) , & \mbox{if }i \in [N+1,2N] \end{cases}$
in the unnormalized case and
$v_{ij}\big ( \mathbf{x} - \mathbf{c}_i \big ) \ \stackrel{\mathrm{def}}{=}\ \begin{cases} \delta_{ij} u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) , & \mbox{if } i \in [1,N] \\ \left ( x_{ij} - c_{ij} \right ) u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) , & \mbox{if }i \in [N+1,2N] \end{cases}$
in the normalized case.
Here δij is a Kronecker delta function defined as
$\delta_{ij} = \begin{cases} 1, & \mbox{if }i = j \\ 0, & \mbox{if }i \ne j \end{cases}$.
## Training
In a RBF network there are three types of parameters that need to be chosen to adapt the network for a particular task: the center vectors $\mathbf c_i$, the output weights wi, and the RBF width parameters βi. In the sequential training of the weights are updated at each time step as data streams in.
For some tasks it makes sense to define an objective function and select the parameter values that minimize its value. The most common objective function is the least squares function
$K( \mathbf{w} ) \ \stackrel{\mathrm{def}}{=}\ \sum_{t=1}^\infty K_t( \mathbf{w} )$
where
$K_t( \mathbf{w} ) \ \stackrel{\mathrm{def}}{=}\ \big [ y(t) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ]^2$.
We have explicitly included the dependence on the weights. Minimization of the least squares objective function by optimal choice of weights optimizes accuracy of fit.
There are occasions in which multiple objectives, such as smoothness as well as accuracy, must be optimized. In that case it is useful to optimize a regularized objective function such as
$H( \mathbf{w} ) \ \stackrel{\mathrm{def}}{=}\ K( \mathbf{w} ) + \lambda S( \mathbf{w} ) \ \stackrel{\mathrm{def}}{=}\ \sum_{t=1}^\infty H_t( \mathbf{w} )$
where
$S( \mathbf{w} ) \ \stackrel{\mathrm{def}}{=}\ \sum_{t=1}^\infty S_t( \mathbf{w} )$
and
$H_t( \mathbf{w} ) \ \stackrel{\mathrm{def}}{=}\ K_t ( \mathbf{w} ) + \lambda S_t ( \mathbf{w} )$
where optimization of S maximizes smoothness and λ is known as a regularization parameter.
### Interpolation
RBF networks can be used to interpolate a function $y: \mathbb{R}^n \to \mathbb{R}$ when the values of that function are known on finite number of points: $y(\mathbf x_i) = b_i, i=1, \ldots, N$. Taking the known points $\mathbf x_i$ to be the centers of the radial basis functions and evaluating the values of the basis functions at the same points $g_{ij} = \rho(|| \mathbf x_j - \mathbf x_i ||)$ the weights can be solved from the equation
$\left[ \begin{matrix} g_{11} & g_{12} & \cdots & g_{1N} \ g_{21} & g_{22} & \cdots & g_{2N} \ \vdots & & \ddots & \vdots \ g_{N1} & g_{N2} & \cdots & g_{NN} \end{matrix}\right] \left[ \begin{matrix} w_1 \ w_2 \ \vdots \ w_N \end{matrix} \right] = \left[ \begin{matrix} b_1 \ b_2 \ \vdots \ b_N \end{matrix} \right]$
It can be shown that the interpolation matrix in the above equation is non-singular, if the points $\mathbf x_i$ are distinct, and thus the weights w can be solved by simple linear algebra:
$\mathbf{w} = \mathbf{G}^{-1} \mathbf{b}$
### Function approximation
If the purpose is not to perform strict interpolation but instead more general function approximation or classification the optimization is somewhat more complex because there is no obvious choice for the centers. The training is typically done in two phases first fixing the width and centers and then the weights. This can be justified by considering the different nature of the non-linear hidden neurons versus the linear output neuron.
#### Training the basis function centers
Basis function centers can be randomly sampled among the input instances or obtained by Orthogonal Least Square Learning Algorithm or found by clustering the samples and choosing the cluster means as the centers.
The RBF widths are usually all fixed to same value which is proportional to the maximum distance between the chosen centers.
#### Pseudoinverse solution for the linear weights
After the centers ci have been fixed, the weights that minimize the error at the output are computed with a linear pseudoinverse solution:
$\mathbf{w} = \mathbf{G}^+ \mathbf{b}$,
where the entries of G are the values of the radial basis functions evaluated at the points xi: gji = ρ( | | xjci | | ).
The existence of this linear solution means that unlike Multi-Layer Perceptron (MLP) networks the RBF networks have a unique local minimum (when the centers are fixed).
#### Gradient descent training of the linear weights
Another possible training algorithm is gradient descent. In gradient descent training, the weights are adjusted at each time step by moving them in a direction opposite from the gradient of the objective function (thus allowing the minimum of the objective function to be found),
$\mathbf{w}(t+1) = \mathbf{w}(t) - \nu \frac {d} {d\mathbf{w}} H_t(\mathbf{w})$
where ν is a "learning parameter."
For the case of training the linear weights, ai, the algorithm becomes
$a_i (t+1) = a_i(t) + \nu \big [ y(t) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] \rho \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )$
in the unnormalized case and
$a_i (t+1) = a_i(t) + \nu \big [ y(t) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] u \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )$
in the normalized case.
For local-linear-architectures gradient-descent training is
$e_{ij} (t+1) = e_{ij}(t) + \nu \big [ y(t) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] v_{ij} \big ( \mathbf{x}(t) - \mathbf{c}_i \big )$
#### Projection operator training of the linear weights
For the case of training the linear weights, ai and eij, the algorithm becomes
$a_i (t+1) = a_i(t) + \nu \big [ y(t) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] \frac {\rho \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )} {\sum_{i=1}^N \rho^2 \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )}$
in the unnormalized case and
$a_i (t+1) = a_i(t) + \nu \big [ y(t) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] \frac {u \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )} {\sum_{i=1}^N u^2 \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )}$
in the normalized case and
$e_{ij} (t+1) = e_{ij}(t) + \nu \big [ y(t) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] \frac { v_{ij} \big ( \mathbf{x}(t) - \mathbf{c}_i \big ) } {\sum_{i=1}^N \sum_{j=1}^n v_{ij}^2 \big ( \mathbf{x}(t) - \mathbf{c}_i \big ) }$
in the local-linear case.
For one basis function, projection operator training reduces to Newton's method.
Figure 6: Logistic map time series. Repeated iteration of the logistic map generates a chaotic time series. The values lie between zero and one. Displayed here are the 100 training points used to train the examples in this section. The weights c are the first five points from this time series.
## Examples
### Logistic map
The basic properties of radial basis functions can be illustrated with a simple mathematical map, the logistic map, which maps the unit interval onto itself. It can be used to generate a convenient prototype data stream. The logistic map can be used to explore function approximation, time series prediction, and control theory. The map originated from the field of population dynamics and became the prototype chaotic time series. The map, in the fully chaotic regime, is given by
$x(t+1)\ \stackrel{\mathrm{def}}{=}\ f\left [ x(t)\right ] = 4 x(t) \left [ 1-x(t) \right ]$
where t is a time index. The value of x at time t+1 is a parabolic function of x at time t. This equation represents the underlying geometry of the chaotic time series generated by the logistic map.
Generation of the time series from this equation is the forward problem. The examples here illustrate the inverse problem; identification of the underlying dynamics, or fundamental equation, of the logistic map from exemplars of the time series. The goal is to find an estimate
$x(t+1) = f \left [ x(t) \right ] \approx \varphi(t) = \varphi \left [ x(t)\right ]$
for f.
### Function approximation
#### Unnormalized radial basis functions
The architecture is
Figure 7: Unnormalized basis functions. The Logistic map (blue) and the approximation to the logistic map (red) after one pass through the training set.
$\varphi ( \mathbf{x} ) \ \stackrel{\mathrm{def}}{=}\ \sum_{i=1}^N a_i \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big )$
where
$\rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) = \exp \left[ -\beta \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert ^2 \right] = \exp \left[ -\beta \left ( x(t) - c_i \right ) ^2 \right]$.
Since the input is a scalar rather than a vector, the input dimension is one. We choose the number of basis functions as N=5 and the size of the training set to be 100 exemplars generated by the chaotic time series. The weight β is taken to be a constant equal to 5. The weights ci are five exemplars from the time series. The weights ai are trained with projection operator training:
$a_i (t+1) = a_i(t) + \nu \big [ x(t+1) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] \frac {\rho \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )} {\sum_{i=1}^N \rho^2 \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )}$
where the learning rate ν is taken to be 0.3. The training is performed with one pass through the 100 training points. The rms error is 0.15.
Figure 8: Normalized basis functions. The Logistic map (blue) and the approximation to the logistic map (red) after one pass through the training set. Note the improvement over the unnormalized case.
#### Normalized radial basis functions
The normalized RBF architecture is
$\varphi ( \mathbf{x} ) \ \stackrel{\mathrm{def}}{=}\ \frac { \sum_{i=1}^N a_i \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } { \sum_{i=1}^N \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } = \sum_{i=1}^N a_i u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big )$
where
$u \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) \ \stackrel{\mathrm{def}}{=}\ \frac { \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) } { \sum_{i=1}^N \rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) }$.
Again:
$\rho \big ( \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert \big ) = \exp \left[ -\beta \left \Vert \mathbf{x} - \mathbf{c}_i \right \Vert ^2 \right] = \exp \left[ -\beta \left ( x(t) - c_i \right ) ^2 \right]$.
Again, we choose the number of basis functions as five and the size of the training set to be 100 exemplars generated by the chaotic time series. The weight β is taken to be a constant equal to 6. The weights ci are five exemplars from the time series. The weights ai are trained with projection operator training:
$a_i (t+1) = a_i(t) + \nu \big [ x(t+1) - \varphi \big ( \mathbf{x}(t), \mathbf{w} \big ) \big ] \frac {u \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )} {\sum_{i=1}^N u^2 \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )}$
where the learning rate ν is again taken to be 0.3. The training is performed with one pass through the 100 training points. The rms error on a test set of 100 exemplars is 0.084, smaller than the unnormalized error. Normalization yields accuracy improvement. Typically accuracy with normalized basis functions increases even more over unnormalized functions as input dimensionality increases.
Figure 9: Normalized basis functions. The Logistic map (blue) and the approximation to the logistic map (red) as a function of time. Note that the approximation is good for only a few time steps. This is a general characterisitc of chaotic time series.
### Time series prediction
Once the underlying geometry of the time series is estimated as in the previous examples, a prediction for the time series can be made by iteration:
$\varphi(0) = x(1)$
${x}(t) \approx \varphi(t-1)$
${x}(t+1) \approx \varphi(t)=\varphi [\varphi(t-1)]$.
A comparison of the actual and estimated time series is displayed in the figure. The estimated times series starts out at time zero with an exact knowledge of x(0). It then uses the estimate of the dynamics to update the time series estimate for several time steps.
Note that the estimate is accurate for only a few time steps. This is a general characteristic of chaotic time series. This is a property of the sensitive dependence on initial conditions common to chaotic time series. A small initial error is amplified with time. A measure of the divergence of time series with nearly identical initial conditions is known as the Lyapunov exponent.
### Control of a chaotic time series
Figure 10: Control of the logistic map. The system is allowed to evolve naturally for 49 time steps. At time 50 control is turned on. The desired trajectory for the time series is red. The system under control learns the underlying dynamics and drives the time series to the desired output. The architecture is the same as for the time series prediction example.
We assume the output of the logistic map can be manipulated through a control parameter c[x(t),t] such that
${x}^{ }_{ }(t+1) = 4 x(t) [1-x(t)] +c[x(t),t]$.
The goal is to choose the control parameter in such a way as to drive the time series to a desired output d(t). This can be done if we choose the control paramer to be
$c^{ }_{ }[x(t),t] \ \stackrel{\mathrm{def}}{=}\ -\varphi [x(t)] + d(t+1)$
where
$y[x(t)] \approx f[x(t)] = x(t+1)- c[x(t),t]$
is an approximation to the underlying natural dynamics of the system.
The learning algorithm is given by
$a_i (t+1) = a_i(t) + \nu \varepsilon \frac {u \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )} {\sum_{i=1}^N u^2 \big ( \left \Vert \mathbf{x}(t) - \mathbf{c}_i \right \Vert \big )}$
where
$\varepsilon \ \stackrel{\mathrm{def}}{=}\ f[x(t)] - \varphi [x(t)] = x(t+1)- c[x(t),t] - \varphi [x(t)] = x(t+1) - d(t+1)$.
## References
• J. Moody and C. J. Darken, "Fast learning in networks of locally tuned processing units," Neural Computation, 1, 281-294 (1989). Also see Radial basis function networks according to Moody and Darken
• T. Poggio and F. Girosi, "Networks for approximation and learning," Proc. IEEE 78(9), 1484-1487 (1990).
• Roger D. Jones, Y. C. Lee, C. W. Barnes, G. W. Flake, K. Lee, P. S. Lewis, and S. Qian, ?Function approximation and time series prediction with neural networks,? Proceedings of the International Joint Conference on Neural Networks, June 17-21, p. I-649 (1990).
• Martin D. Buhmann (2003). Radial Basis Functions: Theory and Implementations. Cambridge University. ISBN 0-521-63338-9.
• Yee, Paul V. and Haykin, Simon (2001). Regularized Radial Basis Function Networks: Theory and Applications. John Wiley. ISBN 0-471-35349-3.
• John R. Davies, Stephen V. Coggeshall, Roger D. Jones, and Daniel Schutzer, "Intelligent Security Systems," in Freedman, Roy S., Flein, Robert A., and Lederman, Jess, Editors (1995). Artificial Intelligence in the Capital Markets. Chicago: Irwin. ISBN 1-55738-811-3.
• Simon Haykin (1999). Neural Networks: A Comprehensive Foundation (2nd edition ed.). Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-908385-5.
• S. Chen, C. F. N. Cowan, and P. M. Grant, "Orthogonal Least Squares Learning Algorithm for Radial Basis Function Networks", IEEE Transactions on Neural Networks, Vol 2, No 2 (Mar) 1991. | {"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": 68, "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": 0.8779569864273071, "perplexity": 674.4400446394085}, "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-2018-34/segments/1534221210362.19/warc/CC-MAIN-20180815220136-20180816000136-00615.warc.gz"} |
https://mathvisions.wordpress.com/2017/04/28/ | # Trigonometry Homework #9 due 5-4
Graph the following functions.
1. $y=\sec\left[\frac{1}{2}\left(x+\frac{\pi}{3}\right)\right]$
2. $y=2\sec{x}+3$
3. $y=\frac{\csc(\pi{x})}{2}$
4. $y=\csc\left(x-\frac{\pi}{4}\right)$
5. $y=\cot(2\pi{x})-1$
1. $\lim_{x \to 1}\frac{\ln{x}}{x^2-1}$
2. $\lim_{t \to\pi}\frac{\sin^2{t}}{t-\pi}$
3. $\lim_{x \to 0}\left ( \frac{1}{x}-\frac{1}{\sin{x}} \right )$
4. $\lim_{x \to 0^+}x^a\ln{x},\;a>0$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 9, "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": 0.9434345960617065, "perplexity": 711.3598543176169}, "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/1505818689775.56/warc/CC-MAIN-20170923175524-20170923195524-00347.warc.gz"} |
https://physics.stackexchange.com/questions/453638/conceptual-understanding-of-quantum-harmonic-oscillators | # Conceptual understanding of quantum harmonic oscillators
The way I understand it is that we have the time-independent Schrödinger equation for a particle described by a wave function $$\psi$$ in a potential V(x)
$$-\frac{\hbar}{2m}\frac{d^2}{dx^2} \psi + V(x)\psi(x) = E\psi(x)\tag{1} \, .$$
If I were to approximate the potential $$V(x)$$ to be quadratic in $$x$$ (for example $$V(x) = \frac{1}{2} m \omega x^2$$) we know that this describes a harmonic oscillator.
If we then solve equation (1) for $$\psi$$ and $$E(x)$$ we will find that
$$E_n = \hbar\omega \left(n+\frac{1}{2} \right)$$ and the corresponding $$\psi_n$$.
Now it is mostly the sentences I can't really formulate. Like for example which of these statements is correct:
• Our particle described by $$\psi$$ is a harmonic oscillator.
• Our particle described by $$\psi$$ is modeled by a harmonic oscillator.
• Let us consider a particle described as a harmonic oscillator...
As you might see, I have the basic mathematical framework, but I can't really grasp the words/concepts surrounding it.
• If we know all eigenfunctions and eigenvalues for a 1D Schrodinger equation then from that information we should be able to reproduce the potential energy V(x) and find that it is quadratic in x, so we'd know that our particle is a harmonic oscillator indeed. – Maxim Umansky Jan 12 at 0:38
• This is a deep question and I recommend 1. Hartle, James B. "Quantum mechanics of individual systems." American Journal of Physics 36.8 (1968): 704-712, 2. Peres, Asher. "What is a state vector?." American Journal of Physics 52.7 (1984): 644-650. – ZeroTheHero Jan 12 at 2:16
There is no prior difference about naming the concept. IT depends on how you call things, this is just one more.
• A quantum free particle = a wavefunction that obeys the Schrödinger's equation with no potential energy.
• A particle enclosed in an infinite-well = a wavefunction that obeys the Schrödinger's equation with such infinite-well potential.
• A quantum Harmonic Oscillator = a wavefunction that obeys the Schrödinger's equation with the harmonic potential.
But there are two "problems" here. The first one is general: talking about "particle " at this level is "risky". How do you define "particle"? If you want to be clear, talk in terms of "wavefunctions" or "systems".
And now, here comes the problem. The quantum harmonic oscillator behaves completely differently from the classical one. In fact, the states with defined energies are stationary, so they do not change in time, so it cannot oscillate. Yep, states with a single energy do not change in time. So we've got here an oscillator which does not oscillate!
This means that, if your wavefunction if, for example, the one with $$n=0$$, it has only one possible energy, $$E_0=½\hbar\omega$$. Since it has only one possible energy, that wavefunction stays the same all the time. So it is not oscillating.
That's why saying "it's an harmonic oscillator" can be a little confusing. When we say "a quantum harmonic oscillator", we mean " a particle under the influence of a potential that classically would yield a harmonic oscillator", but we are lazy and we just say "harmonic oscillator". The thing is that quantum particles behave really differently from classical mechanics.
Your sentences are right, because everybody understands that it means "a wavefunction that obeys Schrödinger's equation with a quadratic potential". That quadratic potential would give a classical harmonic oscillator if we used classical mechanics and the scalar equation for energy, but this is quantum mechanics and it works differently. The name has been kept the same.
• Oh that was so helpful thank you! I was thinking about what you've said and came to the conclusion that if it's the potential V(x) (in our case the harmonic oscillator) which allows us to calculate $E_n$, there should be always one unique Hamiltonian to each V(x), no? I am asking because usually in most examples I am given the Hamilitonian is kind of 'god given' so this would be the first time for me to see how a Hamiltonian emerges out of the potential V(x) and the schrödinger equation. – Caito Jan 12 at 10:19
• Kind of, yes. Most problems just give you the potential, because the kinetic term is always there. – FGSUZ Jan 12 at 10:54
• Perfect! Finally things start to make sense. One last question. I assume that in quantum mechanics the Hamiltonian usually 'generates' the unitaries (which are allowed?) for a given system. I am currently working on a system in a harmonic potential, hence a quantum harmonic oscillator. The unitaries we are using there are 2-level rotations between the different eigenstates. My question: Are 2-level rotations THE unitary one can use given a harmonic potential? Or am I missing something here? – Caito Jan 12 at 11:55
• I have some comment to this answer. "particle = a wavefunction" The wave function is not the particle. The wave function describes it. For example an electron is for all we know a point particle, but its wavefunction can have any size. Then, a quantum harmonic oscillator behaves exactly like the classical one, except that it's energy, hence its amplitude, is quantized. And, if its ground state is not oscillating, why does it have energy and a varying, though on average zero momentum? We have to be careful when interpreting quantum mechanics. – my2cts Jan 12 at 12:00
• You're right, that's why I said "talking about particles is 'risky' here". Hm, maybe what I said can lead to some confussion if that aspect. Thank you. – FGSUZ Jan 12 at 12:19
The same problem arises classically. If you have a mass on a spring we have a harmonic oscillator. But do we say the mass is a harmonic oscillator? Perhaps. If so then we would say for your question that the particle is a harmonic oscillator
But the spring is an important part of the system too. If we really want to be picky I would say that that the spring mass system constitutes a harmonic oscillator system. In that case we would say the particle in the given potential can be treated as a harmonic oscillator system.
In that sense anything which has a Hamiltonian that looks like
$$H = ax^2+bp^2$$
can be called a harmonic oscillator.
In any case, all of the sentences you wrote down are reasonable and would be understood by anyone. | {"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": 14, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8421147465705872, "perplexity": 287.6949680742584}, "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-2019-13/segments/1552912201812.2/warc/CC-MAIN-20190318232014-20190319013534-00025.warc.gz"} |
http://mathhelpforum.com/discrete-math/57307-verifying-using-propositional-equivalences-help.html | # Math Help - Verifying using propositional equivalences - HELP
1. ## Verifying using propositional equivalences - HELP
Hey can anyone help me with this equation: (need to verify using equivalences)
not(p ↔ q )
The answer to this should be (p ↔ not q) but I dont know how to get there step by step.
Help any one?
Thanks
2. $\begin{gathered}
\neg \left( {p \leftrightarrow q} \right) \hfill \\
\neg \left[ {\left( {p \to q} \right) \wedge \left( {q \to p} \right)} \right] \hfill \\
\left( {p \wedge \neg q} \right) \vee \left( {q \wedge \neg p} \right) \hfill \\
\end{gathered}$
$
\begin{gathered}
\left[ {\left( {p \wedge \neg q} \right) \vee q} \right] \wedge \left[ {\left( {p \wedge \neg q} \right) \vee \neg p} \right] \hfill \\
\left( {p \vee q} \right) \wedge \left( {\neg p \vee \neg q} \right) \hfill \\
\left( {\neg q \to p} \right) \wedge \left( {p \to \neg q} \right) \hfill \\
\left( {p \leftrightarrow \neg q} \right) \hfill \\
\end{gathered}$
3. Thank you very much for that mate, helped alot. | {"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": 2, "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": 0.8896179795265198, "perplexity": 2246.3547294219425}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860111612.51/warc/CC-MAIN-20160428161511-00052-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/7566/sweepstakes-probability | # Sweepstakes Probability
Is there a way to determine the probability of winning a particular sweepstakes with only the following information:
-the estimated odds of winning are 1 in 13,000
-contestants must be owners (property holders) within the corporation running the sweepstakes and the total number of owners is 222,600
-each owner can enter the sweepstakes up to 6 times
-there is one grand prize and three runner-up prizes
-
What does "estimated odds of winning are 1 in 13,000" mean? Does it mean 1 in 13,000 per entrance? And does it mean the odds of winning at all, or the odds of winning the grand prize? – Justin L. Oct 23 '10 at 5:30
If the estimated odds of winning are $1$ in $13,000$ per entrance, then if you enter once, your odds of winning are $1:13000$, of course.
However, entering more than once, it's a bit more complicated. It's easier to calculate the odds of you not winning, and subtract it from $1$.
The odds of you not winning once is $12999/13000$. The odds of you not winning twice is $(12999/13000)^2$, and three times is $(12999/13000)^3$, etc.
We could then say we can calculate your odds of not winning $n(x)$, where $x$ is the number of entries, as $n(x) = (12999/13000)^x$.
However, you want to calculate the odds of winning $p(x)$
We know that $n(x) + p(x) = 1$ (Think about it. The sum of the odds of not winning and the odds of winning equal up to 1, because there is a 100% chance that one or the other will happen). So, $p(x) = 1 - n(x)$
So to calculate your odds of winning, you would use
$p(x) = 1 - (12999/13000)^x$
where $x$ is the amount of times you've entered.
If you enter 6 times, then your total odds are $\frac{2227329628937465077999}{4826809000000000000000000}$, which is about $0.0004614497$.
-
Thank you for your answer; it makes perfect sense. This question is based on real contest, and they literally state that the odds of winning are 1:13,000 without indicating whether that applies per entrance or to which prize(s). They probably should have been more specific, but again, thanks! – Anonymous Oct 24 '10 at 17:30
based on your math if you enter 13000 times you only have about 64% chance of winning which is not true. – Bianca Morton Jan 19 '15 at 19:28 | {"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": 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": 0.8532465100288391, "perplexity": 227.2005454004109}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701150206.8/warc/CC-MAIN-20160205193910-00020-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://sora.ink/archives/1238 | 1. \cfrac{premise}{conclusion}
is called inference rule, it stands for: if premise above the line is established, we can then derive the conclusion below the line.
2. There are several ways to define a set, inductively, concretely and by using inference rules. Inductive definition starts with a base case and then derive inductive rules. Concrete definition using mathematical notations such as \cup to form a set explicitly. The inference rules using the axiom and premise part as the base case and conclusion part as induction rules. for example, suppose \mathcal{T} is set of terms, following inference rules define a language L(\mathcal{T}) on \mathcal{T}
\begin{align*}
&\text{1. } \tt true\in\mathcal{T}\ \ \ \tt false\in\mathcal{T}\ \ \ \ \tt0\in\mathcal{T}\\
&\text{2. } \cfrac{t_1\in\mathcal{T}}{\tt succ\ t_1\in\mathcal{T}}\ \ \ \cfrac{t_1\in\mathcal{T}}{\tt pred\ t_1\in\mathcal{T}}\ \ \ \cfrac{t_1\in\mathcal{T}}{\tt iszero\ t_1\in\mathcal{T}}\\
&\text{3. } \cfrac{t_1\in\mathcal{T}\land t_2\in\mathcal{T}\land t_3\in\mathcal{T}}{\tt if\ t_1\ then\ t_2\ else\ t_3\in\mathcal{T}}
\end{align*}
which can also be defined by Backus-Naur Form as given below:
\begin{align*}
term \rightarrow\
&\tt true|\tt false|\\
&\tt if\ \textit{term}\ then\ \textit{term}\ else\ \textit{term}|\\
&\tt 0|\\
&\tt succ\ \textit{term}|\tt pred\ \textit{term}|\tt iszero\ \textit{term}\\
\end{align*}
3. Operational Semantics use an abstract state machine to describe the behavior of the program, the meaning of a program is the final state of its corresponding state machine's halt state.
4. Denotational Semantics use mathematical functions to represent the meaning of a program, e.g., a function on natural numbers.
5. Axiomatic Semantics use program statement as laws and the meaning of the program is what can be proved about it, generally speaking, both operational and denotational semantics use program's behavior to derive statements, which is opposed to the axiomatic semantics
6. Evaluation Relation is a binary relation on a language(or other structures, in this example, L(\mathcal{T}), written in \mathtt{t}\rightarrow \mathtt{t}', denotes "\tt t evalutes to \tt t'" in one step, which stands for a transition of the state mechine: if the state machine is in state t, we can change to state t'. it can be constructed by a set of inference rules, for example:
\begin{align*}
&\text{1. }\mathtt{if\ true\ then\ t_2\ else\ t_3\rightarrow t_2}\\
&\text{2. }\mathtt{if\ false\ then\ t_2\ else \ t_3\rightarrow t_3}\\
&\text{3. }\cfrac{\tt t_1\rightarrow t_{1}'}{\tt if\ t_1\ then\ t_2\ else\ t_3\rightarrow if\ t_{1}'\ then\ t_2\ else\ t_3}
\end{align*}
the above inference rules stated:
1. if the current state(specifically, the expression/term that are currently being evaluated) satisifies rule 1, we can replace its entire content to \tt t_2, and so for rule 2. If their is another machine that evaluates \tt t_1 to \tt t_{1}', then we can replace all the occurences of \tt t_1 to \tt t_{1}'
7. An instance of an inference rule is obtained by consistenly replacing each metavariable by the corresponding actual value in both premise(if exist) and conclusions(similar to substitution)
8. A rule is satisified by a relation if for all instance of the rule, either one of the premises is not in the relation or its conclusion is in the relation.
9. The one-step evaluation (denote \rightarrow) is the smallest binary relation sastisfies all of the inference rules, which, in this example, is rule 1, 2 and 3, pair (\tt t, t') is said to be derivable if it is in the relation. generally speaking, a statement \tt t \rightarrow t' is derivable if and only if it is either an instance of one of the computation rule in the inference rules, or it's the conclusion of an instance of congruence rules, who's premises are derivable.
10. A term \tt t is said to be in normal form if no rule can apply to it, which means there is no such a \tt t' that \tt t\rightarrow t' | {"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": 0.9418914318084717, "perplexity": 2776.9843028309488}, "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/1674764500339.37/warc/CC-MAIN-20230206113934-20230206143934-00568.warc.gz"} |
https://bitbucket.org/hsvieira/django-diario | # Overview
Django Diário is a pluggable weblog application for Django Web Framework initially write by Guilherme Gondim (semente). See the file AUTHORS for a complete authors list of this application.
For installation instructions, see the file INSTALL. For more info, read docs/overview.txt''.
If you find any problems in the code or documentation, please take 30 seconds to fill out an issue here:
Code or translation contribution is MUCH APPRECIATED. Feel free to send patches.
Django Diário is free software; you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version.
Django Diário is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public License along with this program; see the file COPYING.LESSER. If not, see <http://www.gnu.org/licenses/>. | {"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": 0.849245011806488, "perplexity": 978.1699648407325}, "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-2018-09/segments/1518891813187.88/warc/CC-MAIN-20180221004620-20180221024620-00514.warc.gz"} |
https://simbad.u-strasbg.fr/simbad/sim-ref?bibcode=2017A%26A...608A..18G | other querymodes : Identifierquery Coordinatequery Criteriaquery Referencequery Basicquery Scriptsubmission TAP Outputoptions Help
2017A&A...608A..18G - Astronomy and Astrophysics, volume 608A, 18-18 (2017/12-1)
Observational calibration of the projection factor of Cepheids. IV. Period-projection factor relation of Galactic and Magellanic Cloud Cepheids.
GALLENNE A., KERVELLA P., MERAND A., PIETRZYNSKI G., GIEREN W., NARDETTO N. and TRAHIN B.
Abstract (from CDS):
Context. The Baade-Wesselink (BW) method, which combines linear and angular diameter variations, is the most common method to determine the distances to pulsating stars. However, the projection factor, p-factor, used to convert radial velocities into pulsation velocities, is still poorly calibrated. This parameter is critical on the use of this technique, and often leads to 5-10% uncertainties on the derived distances.
Aims. We focus on empirically measuring the p-factor of a homogeneous sample of 29 LMC and 10 SMC Cepheids for which an accurate average distances were estimated from eclipsing binary systems.
Methods. We used the SPIPS algorithm, which is an implementation of the BW technique. Unlike other conventional methods, SPIPS combines all observables, i.e. radial velocities, multi-band photometry and interferometry into a consistent physical modelling to estimate the parameters of the stars. The large number and their redundancy insure its robustness and improves the statistical precision.
Results. We successfully estimated the p-factor of several Magellanic Cloud Cepheids. Combined with our previous Galactic results, we find the following P-p relation: -0.08±0.04(logP-1.18)+1.24±0.02. We find no evidence of a metallicity dependent p-factor. We also derive a new calibration of the period-radius relation, logR=0.684±0.007(logP-0.517)+1.489±0.002, with an intrinsic dispersion of 0.020. We detect an infrared excess for all stars at 3.6µm and 4.5µm, which might be the signature of circumstellar dust. We measure a mean offset of Δm3.6=0.057±0.006mag and Δm4.5=0.065±0.008mag.
Conclusions. We provide a new P-p relation based on a multi-wavelength fit that can be used for the distance scale calibration from the BW method. The dispersion is due to the LMC and SMC width we took into account because individual Cepheids distances are unknown. The new P-R relation has a small intrinsic dispersion: 4.5% in radius. This precision will allow us to accurately apply the BW method to nearby galaxies. Finally, the infrared excesses we detect again raise the issue of using mid-IR wavelengths to derive period-luminosity relation and to calibrate the Hubble constant. These IR excesses might be the signature of circumstellar dust, and are never taken into account when applying the BW method at those wavelengths. Our measured offsets may give an average bias of ∼2.8% on the distances derived through mid-IR P-L relations. | {"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": 0.8664349317550659, "perplexity": 4137.8067494206825}, "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-2022-49/segments/1669446708046.99/warc/CC-MAIN-20221126180719-20221126210719-00817.warc.gz"} |
http://www.lofoya.com/Solved/24/a-vessel-is-filled-with-liquid-3-parts-of-which-are-water-and-5 | # Moderate Alligations or Mixtures Solved QuestionAptitude Discussion
Q. A vessel is filled with liquid, 3 parts of which are water and 5 parts syrup. What part of the mixture must be drawn off and replaced with water so that the mixture may be half water and half syrup?
✖ A. 7/11 ✖ B. 6/7 ✔ C. 1/5 ✖ D. 2/7
Solution:
Option(C) is correct
Suppose the vessel initially contains 8 litres of liquid.
Let $x$ litres of this liquid be replaced with water then quantity of water in new mixture
$=3-\dfrac{3x}{8}+x$ litres
Quantity of syrup in new mixture $=5-\dfrac{5x}{8}$ litres
After replacement, the quantity of water and syrup is same in the new mixture.
Therefore,
\begin{align*}
3-\dfrac{3x}{8}+x &= 5-\dfrac{5x}{8}\\
\Rightarrow 5x+24 &=40-5x\\
\Rightarrow 10x &=16\\
\Rightarrow x &=\dfrac{8}{5}
\end{align*}
So part of the mixture replaced,
$=\dfrac{8}{5}\times \dfrac{1}{8} = \dfrac{1}{5}$
## (6) Comment(s)
Chakarya
()
Syrup part is only getting added with water mixture. So,
3/8 + 5x/8 = 1/2
(Water in mix + Syrup getting added with water) = Half of water or syrup
Solving for x, we get x = 1/5
Kritarth
()
why the answer 8/5 has been multiplied with 1/8 ? please explain...thank u
Smart
()
8 by 5 litre mixure out of 8 litre initial solution is replaced that is 1 by 5 part of initial mixture
Div
()
Parul
()
plzz explain it...
Sim
()
not getting 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": 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": 0.8864352107048035, "perplexity": 4227.875878729831}, "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/1484560281332.92/warc/CC-MAIN-20170116095121-00049-ip-10-171-10-70.ec2.internal.warc.gz"} |
http://math.stackexchange.com/questions/258383/coordinate-transform-partials-question | # Coordinate Transform partials question
I wish to go from cartesian to cylindrical coordinates using the chain rule. I see here that
$x = rcos(\phi)$
$y = r sin(\phi)$
$r = \sqrt{x^2 + y^2}$
$\phi = arctan(\frac{y}{x})$
I am having difficulties interpreting this differential
$$\frac{ \partial}{\partial x} = \frac{ \partial r}{\partial x}\frac{ \partial}{\partial r} + \frac{ \partial \phi}{\partial x}\frac{ \partial}{\partial \phi}$$
Intuitively this makes sense as x is a function of $\rho$ and $\phi$ only. What I don't understand is where exactly the two terms on the right came from. Was it some intermediate operation that was performed? There must be some form of the product rule at work, but I do not see it. Thanks
-
check this out: en.wikipedia.org/wiki/Jacobian_matrix_and_determinant – Lucas Dec 14 '12 at 4:32 | {"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": 0.8586761951446533, "perplexity": 248.00474288347647}, "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-2014-23/segments/1405997889314.41/warc/CC-MAIN-20140722025809-00189-ip-10-33-131-23.ec2.internal.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/4147-sampling.html | # Math Help - Sampling
1. ## Sampling
An unbiased dice is thrown once. The var(x)=35/12. The same dice is thrown 70 times.
a) Find the probability that the mean score is less than 3.3.
b)Find the probability that the total score exceeds 260.
2. Originally Posted by kingkaisai2
An unbiased dice is thrown once. The var(x)=35/12. The same dice is thrown 70 times.
a) Find the probability that the mean score is less than 3.3.
b)Find the probability that the total score exceeds 260.
I will assume that this is a question where we use the central limit theorem to
justify treating the distribution of the mean score as normal with mean equal
to the mean for a single throw of the die $\mu=3.5$ and variance equal to the variance
of the result of a single throw divided by the sample size $\sigma^2=(35/12)/70=1/24$.
a) Now a mean of less than 3.3 means 230 or less for the sum, so to correct
for the continuity of the normal approximation we use a z-score for a mean
of 230.5/70, which is the z-score for for a mean of 3.2928 so:
$
z=\frac{3.2928-3.5}{1/\sqrt{25}}\approx -1.015
$
Looking this up in a normal table or using a normal calculator we have for
a normal distributed RV we have:
$
p(z<-1.015) \approx 0.155
$
b) There are at least two equivalent ways to do this, one another is to
observe that the event: "total score exceeds 260.5" is the same as "the mean
exceeds 260.5/70". I will use the latter, then the z-score for a mean of
260.5/70 (again the extra 0.5s are continuity corrections) is:
$
z=\frac{260.5/70-3.5}{1/\sqrt{24}}\approx 1.0848
$
Looking this up as before we find:
$
p(z<1.0848) \approx 0.861
$
so:
$
p(z>1.0848) = 1-p(z<1.0848) \approx 0.139
$
.
RonL
Notes:
i. A mean of less than 3.3 means the sum of the 70 throws is less than
3.3x70=231. So as the normal distribution is continuous and the actual
distribution of the sum and mean is discrete we use in the normal model a
sum less than 230.5 to represent an actual sum of 230 or less.
ii. Similarly in part b), a total of more than 260 is modelled as more than 260.5
in the normal approximation.
iii. Given the nature of the way problems are set, there is a significant chance
that the expected answers to the two parts of this problem are the same.
So please check the working to see if there is some slight modification which
would make this so. | {"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": 7, "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": 0.9625788331031799, "perplexity": 500.87421238800823}, "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-2016-22/segments/1464049277475.33/warc/CC-MAIN-20160524002117-00114-ip-10-185-217-139.ec2.internal.warc.gz"} |
https://www.inference.vc/unsupervised-learning-by-predicting-noise-an-information-maximization-view-2/ | April 21, 2017
# Unsupervised Learning by Predicting Noise: an Information Maximization View
This note is about a recent paper that was making rounds on the internet yesterday:
The method is simple yet powerful, and it is perhaps a bit surprising and conter-intuitive: it learns representations by mapping data to to randomly sampled noise vectors. The authors call the methods noise-as-target (NAT).
In this post, I'm going to assume you've at least skimread the paper and may have been left wondering what really is going on here. I will only add my perspective on it: I re-interpret the algorithm as a proxy to information maximisation. I think that if you start thinking about the algorithm it from this perspective, perhaps it becomes less surprising that the method actually works.
#### Summary
1. I start from infomax: how to learn optimal compact representations by maximising the amount of information retained about the input data.
2. I show why restricting the representation to the unit sphere - as it is done in the paper - makes sense in the Infomax framework.
3. It's easy to show that if there is a deterministic representation that achieves uniform distribution, it maximises the infomax criterion. Therefore, if we believe such a solution exists, it makes sense to directly seek deterministic mappings that produce a distribution close to uniform.
4. this is exactly what NAT does: it seeks a deterministic mapping whose marginal distribution is uniform on the unit sphere, and it does so by minimising a sort of empirical earth movers' distance from uniform samples.
5. finally, I talk a bit about convergence of the stochastic Hungarian algorithm used in the paper for updating the assignment matrix.
## Learning representations by maximising information
Let's say we are trying to learn a compact representation of data $x_n$ drawn i.i.d. from some distribution $p_X$. In general, the representation can be described as a random variable $z_n$ that is sampled conditionally i.i.d. fashion from some parametric conditional distribution $p_{Z\vert X, \theta}$:
\begin{align}
x_n &\sim p_{X}\\
z_n &\sim p_{Z\vert X=x_n, \theta}
\end{align}
In variational autoencoders, this $p_{Z\vert X, \theta}$ would be called the encoder or recognition model or amortized variational posterior. But, importantly, we now work directly with the encoder, without having to specify a generative distribution $p_{X\vert Z;\theta}$ explicitly.
The InfoMax principle says that a good representation is compact (in terms of it's entropy) while it simultaneously retains as much information about the input $X$ as possible. (see e.g. Chen et al, 2016 - InfoGAN, Barber and Agakov, 2003). This goal can be formalised as follows:
$$\theta^{\text{infomax}} = \operatorname{argmax}_{\theta:\mathbb{H}[p_{Z;\theta}] \leq h} \mathbb{I}[p_{X,Z;\theta}],$$
where $\mathbb{I}$ denotes mutual information and $\mathbb{H}$ the Shannon's entropy. I also introduced the following notation for distributions:
\begin{align}
p_{Z;\theta} &= \mathbb{E}_{x\sim p_X} p_{Z\vert X=x,\theta} \\
p_{X,Z;\theta}(x,z) &= p_X(x) p_{Z\vert X=x;\theta}(z\vert x)
\end{align}
In real-valued spaces, this optimisation problem can be ill-posed, and it has a few problems:
1. The marginal entropy of the representation is hard to estimate in the general case. We need a smart way to implement the restriction $\mathbb{H}[p_{Z;\theta}]\leq h$ without actually having to compute the entropy.
2. If the representation has deterministic and invertible components, the mutual information ends up infinite (in continuous spaces), and the optimisation problem becomes meaningless. To make the optimisation problem meaningful one needs to ensure that the pathological invertible case is never reached.
To address these problems, we can make the following changes:
1. Firstly, let's restrict the domain of $Z$ to a finite volume (finite Lebesgue measure) subset of $\mathbb{R}^d$. This way, the differential entropy $\mathbb{H}[p_{Z;\theta}]$ is always upper bounded by the entropy of the uniform distribution on that domain. To be consistent with the paper, we can limit the domain of the representation to the unit Euclidean sphere $|z|_2^2=1$.
2. Secondly, let's consider maximising the information between $x$ and the noise-corrupted representation $\tilde{z} = z + \epsilon$ where $\epsilon$ is i.i.d. noise. I'm going to assume that $\epsilon$ follows a spherical distribution for simplicity. What this added noise does in practice is it places an upper bound on the predictability of $x$ from any given $\tilde{z}$ (or the an upper bound on the invertibility of the representation) thereby also limiting the mutual information to well-behaved, finite values. Our optimization problem thus becomes:
$$\operatorname{argmax}_{\theta: \mathbb{P}_{Z;\theta}(|z|_2^2=1)=1} \mathbb{I}[p_{X,\tilde{Z};\theta}],$$
This loss function makes a intuitive sense: you are trying to map your inputs $x_n$ to values $z_n$ on the unit sphere, potentially in a stochastic fashion, but in such a way that the original datapoints $x_n$ are easily recoverable (under the log-loss) from $\tilde{z}_n$ which is a noisy version of $z_n$ . In other words, we seek a representation that is in some sense robust to additive noise.
## Deterministic and uniform representations
It is not hard to show that if there is at least a single representation $p_{Z\vert X;\theta}$ which has the following two properties:
1. $z_n$ is a deterministic function of $x_n$, and
2. $p_{Z;\theta}$ is the uniform distribution on the unit sphere,
then this $p_{Z\vert X;\theta}$ is a global maximizer of the InfoMax objective.
Crucially, this deterministic solution may not be unique, there may be an infinite number of equally good representations, especially when $\operatorname{dim}(z)\ll \operatorname{dim}(x)$. Consider the case when $X$ is a multivariate standard Gaussian, and the representation Z is a normalised orthogonal projection of $X$, i.e. for some orthogonal transformation $A$:
$$z_n = \frac{Ax_n}{\sqrt{x_n^TA^TAx_n}}$$
Marginally, Z will have uniform distribution on the unit sphere, and it is also a deterministic mapping, therefore $Z$ is a maximum information representation and it is equally good for any orthogonal $A$.
So, if we assume there is at least one deterministic and uniform representation of $p_X$, it makes sense to seek only deterministic representations, and to seek ones that map data to an approximately uniform distribution in representation space.
This is exactly what the noise as targets (NAT) objective is trying to achieve
The way NAT tries to achieve a uniform distribution in representation space is by minimising a kind of empirical earth mover distance: first, we randomly draw as many uniform random representations as we have datapoints - these are the $c_n$. Then we try to optimally pair each $c_n$ to a datapoint so that the mean squared distance between $c_n$ and the corresponding representation $z_m=f(x_m,\theta)$ is minimal. Once a pairing is found, the mean squared distance between paired representations and noise vectors can be thought of as a measure of uniformity - indeed it's a sort of empirical estimate of the Wasserstein distance between $p_Z$ and a uniform.
### Is an InfoMax representation a good representation?
I've had too many conversations of this nature in the last few days at DALI: what is a good representation? What does unsupervised representation learning even mean? You can ask the same question about InfoMAX: is it a good guiding principle for finding good representations?
I don't think it is enough, in itself. For starters: you can transform your representation in arbitrary ways, and as long as your transformation is invertible, the mutual information should stay the same. So you can arbitrarily entangle or disentangle any representation using invertible transformations, without the InfoMAX objective noticing. Hence, the InfoMax criterion alone can't be trusted to find you disentangled representations, for example.
It is far more likely that the empirical success we see is the combination of ConvNets with the InfoMax principle. We optimise information only among representations that are easy to represent by a ConvNet. And it may be that the intersection of ConvNet-friendly representations with InfoMax representations somehow end up pretty useful.
## Summary
I showed that the noise-as-targets (NAT) representation-learning principle can be interpreted as seeking InfoMax representations: representations of limited entropy that retain maximum information about the input data in a way that is robust to noise. A similar information-maximisation-based interpretation can be given to Exemplar-CNNs (Dosovitsky et al, 2014) which work by trying to guess the index of a datapoint from a noisy version of the datapoint. These algorithms often look very odd and counterintuitive when you first encounter them, reinterpreting them as information maximisation helps demistify them a bit. At least I think it helps me.
### Bonus track: Notes on Stochastic version of earth-mover-distance
The difficulty in implementing the earth mover's distance in this literal form is that you need to find the optimal assignment between the two empirical distributions and that scales as $O(N^3)$ in the number of samples $N$. To sidestep this, the authors propose a 'stochastic' update to the optimal assignment matrix whereby the pairings are only updated one minibatch at a time.
Not that I expect this to be a big deal, but it's worth pointing out that this algorithm is prone to getting stuck in local minima. Let's assume parameters of the representation $\theta$ are fixed, and the only thing we want to update is the assignments. Let's consider a toy situation illustrated in the figure below:
Consider three datapoints $x_1, x_2, x_3$, whose embeddings are equidistant and lie on the 2D unit sphere (my Mom - or any sane person - would probably call this a circle). Also consider three possible noise assignments $c_1, c_2, c_3$ which are also equidistant as shown. The optimal assignment would clearly pairing $x_1$ with $c_1$, $x_2$ with $c_2$ and $x_3$ with $c_3$. Let's assume our current mapping is suboptimal as shown by the blue arrows, and we are only allowed to update the assigmnent on minibatches of size 2. On a minibatch of size 2, there are only two possibilities to consider: either keep it as it was before, or swap things around, which is shown by the red arrows. In this toy example, keeping the previous assignment (blue arrows) is preferable to switchinig (red arrows) under the squared loss. Hence, the minibatch-based algorithm gets stuck in this "local minimum" with respect to minibatch updates.
This is not to say that the method doesn't work, especially as $\theta$ is also simultaneously updated, which may actually help getting the algorithm out of this local minimum anyway. Secondly, the larger the batchsize, the harder it may be to find such local minima.
The way to think about local minima of this stochastic Hungarian algorithm is to think of it as a graph: each node is a state of the assignment matrix (a permutation), and each edge corresponds to a valid update based on a minibatch. A local minimum is a node which has a lower cost than any node in its immediate neighbourhood. If we consider minibatches of size $B$, with a total sample size $N$, there will be $N!$ nodes in the graph and each node will have an out-degree of ${N \choose B}B!$. Therefore, the probability of any two nodes being connected will be $\frac{1}{(N-B)!}$. As we increase the batchsize $B$, the graph gets more and more connected, so local minima start to disappear. | {"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": 0.9157671928405762, "perplexity": 403.22313640531183}, "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/1620243988775.80/warc/CC-MAIN-20210507090724-20210507120724-00175.warc.gz"} |
http://mathhelpforum.com/discrete-math/87176-gcd-proofs.html | # Math Help - gcd proofs
1. ## gcd proofs
Hi all, have a couple of gcd proofs that I'm stuck with.
1=If gcd(a,c)=1 and gcd(b,c)=1 then gcd(ab,c)=1
thus far all I can figure is to do the following linear combos:
1- ax+cy=1
2-bu+cv=1
3-abz+ct=1
But then I don't know what to do. I try solving for c in one of them and then substituting, but this leads nowhere.
2- If a|c and b|c and gcd(a,b)=1 then ab|c.
This is similar to 1, so I'm also stuck with this one.
Any help would be appreciated. Thanks!
2. Originally Posted by jusstjoe
Hi all, have a couple of gcd proofs that I'm stuck with.
1=If gcd(a,c)=1 and gcd(b,c)=1 then gcd(ab,c)=1
thus far all I can figure is to do the following linear combos:
1- ax+cy=1
2-bu+cv=1
3-abz+ct=1
But then I don't know what to do. I try solving for c in one of them and then substituting, but this leads nowhere.
2- If a|c and b|c and gcd(a,b)=1 then ab|c.
This is similar to 1, so I'm also stuck with this one.
Any help would be appreciated. Thanks!
Hi jusstjoe.
(1) You have
$ax+cy\ =\ 1$
$bu+cv\ =\ 1$
Now multiply them together:
$1\ =\ (ax+cy)(bu+cv)\ =\ ab(ux)+c(avx+buy+cvy)$
Thus $\gcd(ab,c)=1.$
(2) If $\gcd(a,b)=1,$ then $\mathrm{lcm}(a,b)=ab.$ The rest follows from the definition of LCM. | {"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": 6, "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": 0.9567946195602417, "perplexity": 860.7798140898294}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535920849.16/warc/CC-MAIN-20140909052200-00381-ip-10-180-136-8.ec2.internal.warc.gz"} |
http://matholympiad.org.bd/forum/search.php?author_id=2348&sr=posts&sid=7ad49cc26d89d1b578940ec4696f153c | ## Search found 176 matches
Wed Jul 19, 2017 1:17 am
Topic: IMO 2017 Problem 2
Replies: 3
Views: 992
### Re: IMO 2017 Problem 2
Let $P(x,y)$ denote the FE. Note that if $f$ is a solution then $-f$ is also a solution. Now, $P(2,2)$ imply that $f(f(2)^2)=0$. Let $t=f(2)^2$. Case 1: $t \neq 1$ $P(x,t)$ implies $f(0)+f(x+t)=f(xt)$, and we can find $x$ so that $x+t=xt$. So $f(0)=0$. Now $P(x,0)$ implies $f(x)=0$ for all real $x$....
Wed Jul 19, 2017 12:36 am
Topic: IMO 2017 Problem 2
Replies: 3
Views: 992
### IMO 2017 Problem 2
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y:$
$f(f(x)f(y)) + f(x+y) = f(xy).$
Wed Apr 26, 2017 7:39 pm
Forum: Number Theory
Topic: IMO Shortlist 2011 N5
Replies: 4
Views: 442
### Re: IMO Shortlist 2011 N5
But $f(n)|f(0)$ and $f(0)>0$, doesn't that mean $f$ does achieve a maximal value? So your solution is correct I think.
Mon Apr 24, 2017 1:45 pm
Forum: Geometry
Topic: USA(J)MO 2017 #3
Replies: 6
Views: 421
### Re: USA(J)MO 2017 #3
Zawadx wrote: There's a typo in the determinant: zero for you~
Edited. Latexing a determinant is a pain in the first place, locating these typos are difficult
It was correct in my paper, so if I had submitted, it wouldn't have been a zero, rather a seven.
Sat Apr 22, 2017 6:23 pm
Forum: Geometry
Topic: USA(J)MO 2017 #3
Replies: 6
Views: 421
### Re: USA(J)MO 2017 #3
We use barycentric coordinates. Let $P\equiv (p:q:r)$. Now, we know that $pq+qr+rp=0$ [The equation of circumcircle for equilateral triangles]. Now, $D\equiv (0:q:r), E\equiv (p:0:r), F\equiv (p:q:0)$. So, the area of $\triangle DEF$ divided by the area of $\triangle ABC$ is: \dfrac{1}{(p+q)(q+r)(...
Sat Apr 22, 2017 3:19 pm
Forum: Number Theory
Topic: USAJMO/USAMO 2017 P1
Replies: 3
Views: 314
### Re: USAJMO/USAMO 2017 P1
aritra barua wrote:When $a,b$ € $N$ and it follows that $a^x$=$b^y$,there exists $t$ € $N$ such that $a$=$t^k$,$b$=$t^q$.This lemma can be quite handy in this problem.
Are you sure that helps? $a$ and $b$ need to be coprime.
Sat Apr 22, 2017 11:24 am
Forum: Algebra
Topic: At last an ineq USAMO '17 #6
Replies: 3
Views: 262
### Re: At last an ineq USAMO '17 #6
For calculating the tangent, See that the derivative of the function at the point $2$ is $-\dfrac{1}{12}$, and plugging $2$ gives us $\dfrac{1}{12}$. So, the equation of the tangent is $-\dfrac{x}{12}+c$, and it goes through $(2,\dfrac{1}{12})$. This lets us determine the equation.
Sat Apr 22, 2017 11:19 am
Forum: Algebra
Topic: At last an ineq USAMO '17 #6
Replies: 3
Views: 262
### Re: At last an ineq USAMO '17 #6
Here we use the well known tangent line trick. It is easy to see that the minimum is achieved at $(a,b,c,d)=(0,0,2,2)$ and its cyclic variants. Also, the most troublesome thing in this inequality is the quantities of the form $\dfrac{1}{x^3+4}$. So, we draw a tangent of $\dfrac{1}{x^3+4}$ at the poi...
Thu Apr 20, 2017 9:16 pm
Forum: Social Lounge
Topic: BDMO Forum Mafia #2
Replies: 27
Views: 1714 | {"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": 0.9626055359840393, "perplexity": 1840.8883737688384}, "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-2018-05/segments/1516084891530.91/warc/CC-MAIN-20180122173425-20180122193425-00252.warc.gz"} |
https://math.stackexchange.com/questions/3132064/maximizing-a-nested-sum-of-infinitely-many-variables-discrete-variational-calcu | # Maximizing a nested sum of infinitely many variables (discrete variational calculus)
I am wondering if there is a way of finding a neat closed form solution for a series of numbers $$r_t$$ that satisfies the following. This is supposed to give the optimal strategy of an economic agent with exponential discounting of utilities.
$$\max_{r_t=0...\infty } (\sum_{t=0}^{\infty}(\delta^t\sqrt{b^2r_t^2-\beta(\varepsilon _0-\alpha b \sum_{k=0}^{t}r_k)^2} ))$$
What kind of techniques would one use to approach such a problem?
Sorry if the LaTeX is a mess. I am new to this site and don't have a lot of advanced math background. | {"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": 2, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8460171222686768, "perplexity": 132.63639595165358}, "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-2019-30/segments/1563195525829.33/warc/CC-MAIN-20190718211312-20190718233312-00204.warc.gz"} |
https://www.themathcitadel.com/category/spines/topology/ | ### Browsed byCategory: Topology
Topologies and Sigma-Algebras
## Topologies and Sigma-Algebras
Both topologies and $\sigma-$algebras are collections of subsets of a set $X$. What exactly is the difference between the two, and is there a relationship? We explore these notions by noting the definitions first. Let $X$ be any set.
### Topology
A topology $\tau$ is a collection of subsets of a set $X$ (also called a topology in $X$) that satisfies the following properties:
(1) $\emptyset \in \tau$, and $X \in \tau$
(2) for any finite collection of sets in $\tau$, $\{V_{i}\}_{i=1}^{n}$, $\cap_{i=1}^{n}V_{i} \in \tau$
(3) for any arbitrary collection of sets $\{V_{\alpha}: \alpha \in I\}$ in $\tau$ (countable or uncountable index set $I$), $\cup_{\alpha}V_{\alpha} \in \tau$
A topology is therefore a collection of subsets of a set $X$ that contains the empty set, the set $X$ itself, all possible finite intersections of the subsets in the topology, and all possible unions of subsets in the topology.
The simplest topology is called the trivial topology, where for a set $X$, $\tau = \{\emptyset, X\}$. Notice that (1) above is satisfied by design. All intersections we can make with the sets in $\tau$ are finite ones. (There’s just one, $X \cap \emptyset = \emptyset$.) Thus, (2) is satisfied. Any union here gives $X$, which is in $\tau$, so this is a topology.
This isn’t a very interesting topology, so let’s create another one.
Let’s take $X = \{1,2,3,4\}$ as the set. Let’s give a collection of subsets $\tau = \{\{1\},\{2\}, \{1,3\}, \{2,4\}, \{1,2,3\}, \{1,2,4\}, \emptyset, X\}.$ Notice that I didn’t include every single possible subset of $X$. There are two singleton sets, 4 pairs, and 1 set of triples missing. This example will illustrate that you can leave out subsets of a set and still have a topology. Notice that (1) is met. You can check all possible finite intersections of sets inside $\tau$, and notice that you either end up with $\emptyset$ or another of the sets in $\tau$. For example, $\{1,3\} \cap \{1,2,3\} = \{1,3\}$, $\{2\} \cap \{1\} = \emptyset$, etc. Lastly, we can only have finite unions here, since $\tau$ only has a finite number of things. You can check all possible unions, and notice that all of them result in a set already in $\tau$. For example, $\{1,3\} \cup \{2,4\} = X$, $\emptyset \cup \{1,2,4\} = \{1,2,4\}$, $\{1\} \cup \{1,3\} \cup \{2,4\} = X$, etc. Thus, $\tau$ is a topology on this set $X$.
We’ll look at one final example that’s a bit more abstract. Let’s take a totally ordered set $X$ (like the real line $\mathbb{R}$). Then the order topology on $X$ is the collection of subsets that look like one of the following:
• $\{x : a < x\}$ for all $a$ in $X$
• $\{x : b > x \}$ for all $b \in X$
• $\{x : a < b < x\}$ for all $a,b \in X$
• any union of sets that look like the above
To put something concrete to this, let $X = \{1,2,3,4\}$, the same set as above. This is a totally ordered set, since we can write these numbers in increasing order. Then
• The sets that have the structure $\{x : a < x\}$ for all $a \in X$ are
• $\{x : 1 < x\} = \{2,3,4\}$
• $\{x : 2 < x\} = \{3,4\}$
• $\{x : 3 < x\} = \{4\}$
• $\{x : 4 < x \} = \emptyset$
• The sets that have the structure $\{x : b > x\}$ for all $b \in X$ are
• $\{x : 1 > x\} = \emptyset$ (which we already handled)
• $\{x : 2 > x\} = \{1\}$
• $\{x : 3 > x \} = \{1,2\}$
• $\{x : 4 > x\} = \{1,2,3\}$
• The sets that have the structure $\{x : a < x < b\}$ for all $a,b \in X$ are
• $\{x : 1 < x < 3\} = \{2\}$
• $\{x : 1 < x < 4\} = \{2,3\}$
• $\{x : 2 < x < 4\} = \{3\}$
• The remaining combinations yield $\emptyset$
• The sets that are a union of the above sets (that aren’t already listed) are
• $X = \{x : 1 < x\} \cup \{x : 2 > x\}$
• $\{1,2,4\} = \{x : 3 > x\} \cup \{x : 3< x \}$
• $\{1,3,4\} = \{x : 2 < x\} \cup \{x : 2 > x\}$
• $\{1,3\} = \{x : 2 > x\} \cup \{x : 2< x < 4\}$
• $\{1,4\} = \{x : 2 > x\} \cup \{x : 3 < x\}$
• $\{2,4\} = \{x : 1 < x < 3\} \cup \{x : 3 < x \}$
The astute reader will note that in this case, the order topology on $X = \{1,2,3,4\}$ ends up being the collections of all subsets of $X$, called the power set.
### Sigma-Algebra
Let $X$ be a set. Then we define a $\sigma$-algebra.
A $\sigma$-algebra is a collection $\mathfrak{M}$ of subsets of a set $X$ such that the following properties hold:
(1) $X \in \mathfrak{M}$
(2) If $A \in \mathfrak{M}$, then $A^{c} \in \mathfrak{M}$, where $A^{c}$ is the complement taken relative to the set $X$.
(3) For a countable collection $\{A_{i}\}_{i=1}^{\infty}$ of sets that’s in $\mathfrak{M}$, $\cup_{i=1}^{\infty}A_{i} \in \mathfrak{M}$.
Let’s look at some explicit examples:
Again, take $X = \{1,2,3,4\}$. Let $$\mathfrak{M} = \{\emptyset, X, \{1,2\}, \{3,4\}\}.$$ We’ll verify that this is a $\sigma-$algebra. First, $X \in \mathfrak{M}$. Then, for each set in $\mathfrak{M}$, the complement is also present. (Remember that $X^{c} = \emptyset$.) Finally, any countable union will yield $X$, which is present in $\mathfrak{M}$, so we indeed have a $\sigma-$algebra.
Taking another example, we’ll generate a $\sigma$-algebra over a set from a single subset. Keep $X = \{1,2,3,4\}$. Let’s generate a $\sigma-$algebra from the set $\{2\}$. $$\mathfrak{M}(\{2\}) = \{X, \emptyset, \{2\},\{1,3,4\}\}$$ The singleton $\{2\}$ and its complement must be in $\mathfrak{M}(\{2\})$, and we also require $X$ and its complement $\emptyset$ to be present. Any countable union here results in the entire set $X$.
### What’s the difference between a topology and a $\sigma-$algebra?
Looking carefully at the definitions for each of a topology and a $\sigma-$algebra, we notice some similarities:
1. Both are collections of subsets of a given set $X$.
2. Both require the entire set $X$ and the empty set $\emptyset$ to be inside the collection. (The topology explicitly requires it, and the $\sigma-$algebra requires it implicitly by requiring the presence of $X$, and the presence of all complements.)
3. Both will hold all possible finite intersections. The topology explicitly requires this, and the $\sigma-$algebra requires this implicitly by requiring countable unions to be present (which includes finite ones), and their complements. (The complement of a finite union is a finite intersection.)
4. Both require countable unions. Here, the $\sigma-$algebra requires this explicitly, and the topology requires it implicitly, since all arbitrary unions–countable and uncountable–must be in the topology.
That seems to be a lot of similarities. Let’s look at the differences.
1. A $\sigma-$algebra requires only countable unions of elements of the collection be present. The topology puts a stricter requirement —all unions, even uncountable ones.
2. The $\sigma-$algebra requires that the complement of a set in the collection be present. The topology doesn’t require anything about complements.
3. The topology only requires the presence of all finite intersections of sets in the collection, whereas the $\sigma-$algebra requires all countable intersections (by combining the complement axiom and the countable union axiom).
It is with these differences we’ll exhibit examples of a topology that is not a $\sigma-$algebra, a $\sigma-$algebra that is not a topology, and a collection of subsets that is both a $\sigma-$algebra and a topology.
#### A topology that is not a $\sigma-$algebra
Let $X = \{1,2,3\}$, and $\tau = \{\emptyset, X, \{1,2\},\{2\}, \{2,3\}\}$. $\tau$ is a topology because
1. $\emptyset, X \in \tau$
2. Any finite intersection of elements in $\tau$ either yields the singleton $\{2\}$ or $\emptyset$.
3. Any union generates $X$, $\{1,2\}$, or $\{2,3\}$, all of which are already in $\tau$
However, because $\{2\}^{c} = \{1,3\} \not\in \tau$, we have a set in $\tau$ whose complement is not present, so $\tau$ cannot also be a $\sigma-$algebra. We used (2) in the list of differences to construct this example.
#### A $\sigma-$algebra that is not a topology
This example is a little trickier to construct. We need a $\sigma$-algebra, but not a topology, so we need to find a difference between the $\sigma-$algebra and the topology where the topology requirement is more strict than the $\sigma-$algebra’s version. We focus on difference (1) here.
Let $X = [0,1]$. Let $\mathfrak{M}$ be the collection of subsets of $X$ that are either themselves countable, or whose complements are countable. Some examples of things in $\mathfrak\{M\}:$
• all rational numbers between 0 and 1, represented as singleton sets. (countable)
• the entire collection of rational numbers between 0 and 1, represented as a set itself (countable)
• $\left\{\frac{1}{2^{x}}, x \in \mathbb{N}\right\}$. (countable)
• $[0,1] \setminus \{1/2, 1/4, 1/8\}$ (not countable, but its complement is $\{1/2, 1/4, 1/8\}$, which is countable)
• $\emptyset$ (countable)
• $X = [0,1]$ (not countable, but its complement $\emptyset$ is countable)
$\mathfrak{M}$ is a $\sigma-$algebra because:
• $X \in \mathfrak{M}$
• All complements of sets in $\mathfrak{M}$ are present, since we’ve designed the collection to be all pairs of countable sets with countable complements, and all uncountable sets with countable complements.
• Finally, all countable unions of countable sets are countable, so those are present. The countable union of uncountable sets with countable complements will have a countable complement1, and thus all countable unions of elements of $\mathfrak{M}$ are also in $\mathfrak{M}$, so we have a $\sigma-$algebra.
n particular, every single point of $[0,1]$ is in $\mathfrak{M}$ as a singleton set. To be a topology, any arbitrary union of elements of $\mathfrak{M}$ must also be in $\mathfrak{M}$. Take every real number between 0 and 1/2, inclusive. Then the union of all these singleton points is the interval $[0,1/2]$. However, $[0,1/2]^{c} = (1/2,1]$, which is uncountable. Thus, we have an uncountable set with an uncountable complement, so $[0,1/2] \notin \mathfrak{M}$. Since it can be represented as the arbitrary union of sets in $\mathfrak{M}$, $\mathfrak{M}$ is not a topology.
#### A collection that is both a $\sigma-$algebra and a topology
Take any set $X$ that is countable, and let $2^{X}$ be the power set on $X$ (the collection of all subsets of $X$). Then all subsets of $X$ are countable. We have that $\emptyset$ and $X$ are present, since both are subsets of $X$. Since the finite intersection of some subcollection of subsets of $X$ is a subset of $X$, it is in the collection. The arbitrary union of subsets of $X$ is either a proper subset of $X$ or $X$ itself. Thus $2^{X}$ is a topology (called the discrete topology).
The complement of a subset of $X$ is still a subset, and if all arbitrary unions are in $2^{X}$, then certainly countable unions are. Thus $2^{X}$ is also a $\sigma-$algebra.
To be explicit, return to the above where $X = \{1,2,3,4\}$. Write out all possible subsets of $X$, including singletons, $\emptyset$, and $X$ itself, and notice that all axioms in both the $\sigma-$algebra and the topology definitions are satisfied.
Sequences & Tendency: Topology Basics Pt. 2
## Introduction
In my previous post I presented abstract topological spaces by way of two special characteristics. These properties are enough to endow a given set with vast possibilities for analysis. Fundamental to mathematical analysis of all kinds (real, complex, functional, etc.) is the sequence.
We have covered the concept of sequences in some of our other posts here at The Math Citadel. As Rachel pointed out in her post on Cauchy sequences, one of the most important aspects of the character of a given sequence is convergence.
In spaces like the real numbers, there is convenient framework available to quantify closeness and proximity, and which allows naturally for a definition of limit or tendency for sequences. In a general topological space missing this skeletal feature, convergence must be defined.
This post will assume only some familiarity with sequences as mathematical objects and, of course, the concepts mentioned in Part 1. For a thorough treatment of sequences, I recommend Mathematical Analysis by Tom M. Apostol.
## Neighborhoods
Suppose $(X,\mathscr{T})$ is a given topological space, and nothing more is known. At our disposal so far are only open sets (elements of $\mathscr{T}$), and so it is on these a concept of vicinity relies.
Definition. Given a topological space $(X,\mathscr{T})$, a neighborhood of a point $x\in X$ is an open set which contains $x$.
That is, we say an element $T\in\mathscr{T}$ such that $x\in T$ is a neighborhood1 of $x$. To illustrate, take the examples from my previous post.
#### The Trivial Topology
When the topology in question is the trivial one: $\{\emptyset,X\}$, the only nonempty open set is $X$ itself, hence it is the only neighborhood of any point $x\in X$.
#### The Discrete Topology
Take $X=\{2,3,5\}$ and $\mathscr{T}$ to be the collection of all subsets of $X$:
$\emptyset$ $\{2\}$ $\{3\}$ $\{5\}$ $\{2,3\}$ $\{2,5\}$ $\{3,5\}$ $\{2,3,5\}$
Then, for, say $x=5$, neighborhoods include $\{5\}$, $\{2,5\}$, $\{3,5\}$, and $\{2,3,5\}$.
#### The Standard Topology on $\mathbb{R}$
The standard topology on $\mathbb{R}$ is defined to be the family of all sets of real numbers containing an open interval around each of its points. In this case, there are infinitely2 many neighborhoods of every real number. Taking $x=\pi$ for instance, then $(3,4)$, $(-2\pi,2\pi)$, and even
$$\bigcup_{n=1}^{\infty}\left(\pi-\frac{1}{n},\pi+\frac{1}{n}\right)$$
are all neighborhoods of $\pi$.
Remark. A special type of neighborhood in the standard topology is the symmetric open interval. Given a point $x$ and a radius $r>0$, the set
$$(x-r,x+r)=\{y\in\mathbb{R}\mathrel{:}|x-y|
is a neighborhood of $x$. These sets form what is referred to as a basis for the standard topology and are important to definition of convergence in $\mathbb{R}$ as a metric space.
## Convergence
“…the topology of a space can be described completely in terms of convergence.” —John L. Kelley, General Topology
At this point in our discussion of topological spaces, the only objects available for use are open sets and neighborhoods, and so it is with these that convergence of a sequence are built3.
Definition. A sequence $(\alpha_n)$ in a topological space $(X,\mathscr{T})$ converges to a point $L\in X$ if for every neighborhood $U$ of $L$, there exists an index $N\in\mathbb{N}$ such that $\alpha_n\in U$ whenever $n\geq N$. The point $L$ is referred to as the limit of the sequence $(\alpha_n)$.
Visually, this definition can be thought of as demanding the points of the sequence cluster around the limit point $L$. In order for the sequence $(\alpha_n)$ to converge to $L$, it must be the case that after finitely many terms, every one that follows is contained in the arbitrarily posed neighborhood $U$.
As you might expect, the class of neighborhoods available has a dramatic effect on which sequences converge, as well as where they tend. Just how close to $L$ are the neighborhoods built around it in the topology?
We will use the example topologies brought up so far to exhibit the key characteristics of this definition, and what these parameters permit of sequences.
#### The Trivial Topology
In case it was to this point hazy just how useful the trivial topology is, sequences illustrate the issue nicely. For the sake of this presentation, take the trivial topology on $\mathbb{R}$. There is precisely one neighborhood of any point, namely $\mathbb{R}$ itself. As a result, any sequence of real numbers converges, since every term belongs to $\mathbb{R}$. Moreover, every real number is a limit of any sequence. So, yes, the sequence $(5,5,5,\ldots)$ of all $5$‘s converges to $\sqrt{2}$ here.
#### The Discrete Topology
Whereas with the trivial topology a single neighborhood exists, the discrete topology is as packed with neighborhoods as can be. So, as the trivial topology allows every sequence to converge to everything, we can expect the discrete topology to be comparatively restrictive. Taking the set $\{2,3,5\}$ with the discrete topology as mentioned above, we can pinpoint the new limitation: every set containing exactly one point is a neighborhood of that point. Notice the sets4 $\{2\}$, $\{3\}$, and $\{5\}$ are all open sets.
What does this mean? Any sequence that converges to one of these points, say $3$, must eventually have all its terms in the neighborhood $\{3\}$. But that requires all convergent sequences to be eventually constant! This seems to be a minor issue with the finite set $\{2,3,5\}$, but it presents an undesirable, counter-intuitive problem in other sets.
Take $\mathbb{R}$ with the discrete topology, for example. Under these rules, the sequence
$$(\alpha_n)=\left(\frac{1}{n}\right)=\left(1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots\right),$$
though expected to converge to $0$, does not converge at all.
So, the discrete topology is too restrictive, and the trivial topology lets us get away with anything. Fortunately, a happy middle ground exists by being a little more selective with neighborhoods.
#### The Standard Topology
By requiring an open set to contain an open interval around each of its points, it is impossible that a singleton be an open set. Therefore a singleton cannot be a neighborhood, and we eliminate the trouble posed by the discrete topology. Yet every open interval around a real number $L$ contains a smaller one, and each of these is a neighborhood.
This effectively corrals the points of any convergent sequence, requiring the distance between the terms and the limit to vanish as $n$ increases. Take again the sequence
$$(\alpha_n)=\left(\frac{1}{n}\right)=\left(1,\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots\right).$$
We suspect $(\alpha_n)$ converges to $0$, but this requires proof. Therefore, we must consider an arbitrary neighborhood of $0$, and expose the index $N\in\mathbb{N}$ such that all terms, from the $N$th onward, exist in that neighborhood.
Suppose $U$ is a given neighborhood of $0$, so that $U$ contains an open interval surrounding $0$. Without loss of generality, we may assume this interval is symmetric; that is, the interval has the form $(-r,r)$ for some radius $r>0$. Take $N$ to be any integer greater than $\tfrac{1}{r}$. Then, whenever $n\geq N$,
$$\alpha_n = \frac{1}{n} \leq \frac{1}{N} < \frac{1}{1/r} = r.$$
But this means $\alpha_n\in(-r,r)\subset U$ so long as $n\geq N$. Since we chose $U$ arbitrarily, it follows $(\alpha_n)$ converges to $0$.
## Conclusion
The behavior of a sequence in a given set can change rather drastically depending on the network of neighborhoods the topology allows. However, with careful construction, it is possible to have all the sequential comforts of metric spaces covered under a briefly put definition.
My next post in this series will push the generalization of these concepts much further, by relaxing a single requirement. In order to define convergence in the preceding discussion, the set of indices $\mathbb{N}$ was important not for guaranteeing infinitely many terms, but instead for providing order. This allows us to speak of all terms subsequent to one in particular. It turns out that if we simply hold on to order, we can loosen the nature of the set on which it is defined. That is the key to Moore-Smith Convergence, to be visited next.
Building a Ground Floor: Topology Basics Pt. 1
## Building a Ground Floor: Topology Basics Pt. 1
Like some other terms in mathematics (“algebra” comes to mind), topology is both a discipline and a mathematical object. Moreover like algebra, topology as a subject of study is at heart an artful mathematical branch devoted to generalizing existing structures like the field of real numbers for their most convenient properties. It is also a favorite subject of mine, ever since my first introduction to it. This is due in large part to its exceedingly simple first principles, which make the wealth of expansion they allow all the more impressive.
It is my intent to discuss some of these starting points here, in the first of a short series of posts toward the goal of presenting one of my favorite results arising in topology: Moore-Smith convergence, a vast extension of the notion of the limit of a sequence. My representation here follows the explanation given by John L. Kelley in his classic text General Topology, which I recommend to all curious readers.
## What is a topology?
Definition. By topology is meant any collection $\mathscr{T}$ of sets satisfying two conditions:
$$\begin{array}{lrcl}\text{(1)}&A,B\in\mathscr{T}&\Rightarrow&A\cap B\in\mathscr{T};\\\text{(2)}&\mathscr{C}\subset\mathscr{T}&\Rightarrow&\bigcup\{C\in\mathscr{C}\}\in\mathscr{T}\end{array}$$
It is worthwhile to break this definition down. Condition $(1)$ requires that the intersection of any two elements of the collection $\mathscr{T}$ must itself be a member of $\mathscr{T}$. Condition $(2)$ states that the union of any subcollection of $\mathscr{T}$ must also belong to $\mathscr{T}$. These are referred to as closure to finite intersection and closure to arbitrary union, respectively, in some texts.
Notably, the definition speaks only of a collection of sets with no specification beyond the two conditions. Yet, even with these, one can deduce some further characteristic properties.
Corollary. If $\mathscr{T}$ is a topology, then
$$\begin{array}{ll}\text{(i)}&\emptyset\in\mathscr{T};\\\text{(ii)}&\bigcup\{T\in\mathscr{T}\}\in\mathscr{T}.\end{array}$$
Since $\emptyset\subset S$ for every set $S$, and $\mathscr{T}\subset\mathscr{T}$, it is enough to apply $(2)$ to both of these cases to prove the corollary. In fact, many texts make the definition $X\mathrel{:=}\bigcup\{T\in\mathscr{T}\}$, and refer to the pair $(X,\mathscr{T})$ as the topological space defined by $\mathscr{T}$.
This way, the space is given its character by way of the scheme that builds $\mathscr{T}$, rather than the set $X$. It is an important distinction, for many topologies are possible on a given set. With that, we can look at some examples.
## From Trivial to Complicated
### 1. The Trivial Topology
Based on the corollary just presented, it is enough to gather a given set $X$ and the empty set $\emptyset$ into a collection $\{\emptyset,X\}$ and have created a topology on $X$. Because $X$ and $\emptyset$ are its only members, the collection is easily closed to arbitrary union and finite intersection of its elements. This is known as the trivial or indiscrete topology, and it is somewhat uninteresting, as its name suggests, but it is important as an instance of how simple a topology may be. As per the corollary, every topology on $X$ must contain $\emptyset$ and $X$, and so will feature the trivial topology as a subcollection.
### 2. The Discrete Topology
For this example, one can start with an arbitrary set, but in order to better illustrate, take the set of the first three primes: $\{2,3,5\}$. Suppose we consider the collection of all possible subsets of $\{2,3,5\}$. This is also referred to as the power set of $\{2,3,5\}$, and denoted $\wp(\{2,3,5\})$. Fortunately, the set is small enough to list exhaustively1. Here they are listed from top-to-bottom in order of increasing inclusion:
$\emptyset$ $\{2\}$ $\{3\}$ $\{5\}$ $\{2,3\}$ $\{2,5\}$ $\{3,5\}$ $\{2,3,5\}$
Note these are all possible subsets of $\{2,3,5\}$. It is clear any union or intersection of the pieces in the table above exists as an entry, and so this meets criteria $(1)$ and $(2)$. This is a special example, known as the discrete topology. Because the discrete topology collects every existing subset, any topology on $\{2,3,5\}$ is a subcollection of this one.
For example, taking the sets
$$\emptyset,\quad\{5\},\quad\{2,3\},\quad\{2,3,5\}$$
from the collection in the table is enough to produce a topology2.
Remark. Given a topological space $(X,\mathscr{T})$, the elements of $\mathscr{T}$ are referred to as open sets. This nomenclature is motivated in the next example.
### 3. $\mathbb{R}$ and Open Intervals
This example will be more constructive than the previous ones. Consider the set of real numbers, $\mathbb{R}$. Let us define a special collection $\mathscr{T}$ of subsets of real numbers the following way: a set $T$ belongs to $\mathscr{T}$ if, and only if, for every $x\in T$, there exist real numbers $a$ and $b$ such that $x\in(a,b)$ and $(a,b)\subset T.$ That is, we say $T\in\mathscr{T}$ to mean $T$ contains an open interval around each of its elements.
It is good practice to take the time to prove this collection defines a topology on $\mathbb{R}$. To do so, it must be shown that $\bigcup\{T\in\mathscr{T}\}=\mathbb{R}$, and that $\mathscr{T}$ meets conditions $(1)$ and $(2)$.
Proof. To show $\bigcup\{T\in\mathscr{T}\}=\mathbb{R}$, it must be verified that $\bigcup\{T\in\mathscr{T}\}\subset\mathbb{R}$ and $\mathbb{R}\subset\bigcup\{T\in\mathscr{T}\}$. The first containment follows by defining every $T\in\mathscr{T}$ as a subset of $\mathbb{R}$ to begin with, so the reverse containment is all that is left. Let $x\in\mathbb{R}$ be given. Then certainly $x\in(x-1,x+1)$, and surely $(x-1,x+1)\in\mathscr{T}$, as it contains an open interval around all its points by its very design. Thus $x\in\bigcup\{T\in\mathscr{T}\}$.
On to proving $\mathscr{T}$ satisfies $(1)$ and $(2)$. For $(1)$, let $A,B\in\mathscr{T}$ be given and suppose3 $x\in A\cap B$. This holds if, and only if, $x\in A$ and $x\in B$. Since $A$ and $B$ both belong to $\mathscr{T}$, there exist real numbers $a$, $b$, $c$, and $d$ such that $x\in(a,b)\subset A$, and $x\in(c,d)\subset B$. But this means $x\in(a,b)\cap(c,d)$. Fortunately, two intervals of real numbers may only overlap in one way: this means either $c or $a. Without loss of generality, suppose it is the former case, that $c. Then $(a,b)\cap(c,d)=(c,b)$, and it is so that $x\in(c,b)$, an open interval contained in $A\cap B$ (precisely as desired), and it follows $A\cap B\in\mathscr{T}$.
To show $(2)$ is much easier. Let $\{T_\alpha\}_{\alpha\in\mathscr{A}}$ be a collection4 of sets belonging to $\mathscr{T}$, and suppose $x\in\bigcup_{\alpha\in\mathscr{A}}T_\alpha$. Then there exists an index, say $\alpha_0\in\mathscr{A}$, such that $x\in T_{\alpha_0}$. Since $T_{\alpha_0}\in\mathscr{T}$, there exist real numbers $a$ and $b$ such that $x\in(a,b)\subset T_{\alpha_0}$. But this means $x\in(a,b)\subset\bigcup_{\alpha\in\mathscr{A}}T_\alpha$. Since $x$ was chosen arbitrarily, it follows $\bigcup_{\alpha\in\mathscr{A}}T_\alpha\in\mathscr{T}$.
The proof above shows $(\mathbb{R},\mathscr{T})$ is a topological space; the collection $\mathscr{T}$ is referred to as the standard topology on $\mathbb{R}$. The open sets in this space are all the subsets of real numbers contained in open intervals. Fittingly, then, open intervals are open sets in the standard topology.
## Conclusion
This first post is meant to express the simple starting points of topology as a subject of study. It only takes the two criteria mentioned here to define a topology of sets, and yet an entire realm of theory rests upon them. This is a recurring theme in topology, algebra, and mathematics in general. Building the fully-featured universes that hold the answers for more specific inquiry: the complete ordered field of real numbers5 $\mathbb{R}$, the space $\mathcal{C}^{\infty}(\mathbb{C})$ of infinitely differentiable functions $f\mathrel{:}\mathbb{C}\to\mathbb{C}$, the class of all real-valued Lebesgue-integrable functions on $\mathbb{R}$, each of these requires a well-made foundation.
The next post in this series will cover the nature of sequences in topological spaces, particularly those instances where the convenient features afforded by the real numbers are no longer available. With the metric space structure stripped away, how does one define convergence and limit of sequences? What does it mean for elements in topological spaces to be close when distance is otherwise without definition? | {"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": 0, "wp-katex-eq": 404, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9632673859596252, "perplexity": 225.40473035794093}, "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-2021-31/segments/1627046153803.69/warc/CC-MAIN-20210728220634-20210729010634-00572.warc.gz"} |
https://socratic.org/questions/how-do-you-calculate-18-12-7-9 | Algebra
Topics
# How do you calculate 18/12 - 7/9?
Jan 21, 2015
The answer is $\frac{13}{18}$.
Here's how to calculate it:
You'll notice that $\frac{18}{12}$ and $\frac{7}{9}$ have different denominators.
In order to subtract these fractions, we need to find equivalent fractions ( i.e. fractions that are the same as our originals) that have the same denominator as one another.
To do this, let's find the least common denominator, or the smallest number that can be divided by both denominators.
One way to do this to write out the multiples of both denominators until a number repeats in the the list:
Multiples of 12: 12, 24, 36 , 48, 60
Multiples of 9: 9, 18, 27, 36
Since 36 appears in both lists, we can make both denominators 36 to create equivalent fractions.
Since $12 \cdot 3 = 36$, we can multiply $\frac{18}{12} \cdot \frac{3}{3}$ (since $\frac{3}{3}$ = 1) ...
$\frac{18}{12} \cdot \frac{3}{3} = \frac{54}{36}$
And since $9 \cdot 4 = 36$, we can multiply $\frac{7}{9} \cdot \frac{4}{4}$ ...
$\frac{7}{9} \cdot \frac{4}{4} = \frac{28}{36}$
Now, we have equivalent fractions with a denominator of 36 and can subtract:
$\frac{54}{36} - \frac{28}{36} = \frac{26}{36}$
And then divide the numerator and the denominator by 2 to simplify:
$\frac{26}{36} = \frac{13}{18}$ | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 12, "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": 0.9953534007072449, "perplexity": 494.65521744602415}, "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-20200126043644-20200126073644-00182.warc.gz"} |
http://mathhelpforum.com/trigonometry/121980-multiple-angle-formula-help.html | # Math Help - Multiple Angle Formula Help
1. ## Multiple Angle Formula Help
I have started a new chapter and I am having problems solving #6. I'm not even sure what I should do to start. Any help would be greatly appreciated.
2. Originally Posted by whiteforlife
I have started a new chapter and I am having problems solving #6. I'm not even sure what I should do to start. Any help would be greatly appreciated.
Recall that $\sin\theta=\frac{opposite}{hypotenuse}$
Well, you have the opposite, but not the hypotenuse. To find the hypotenuse, use the pythagorean theorem.
After you have computed sin, cos, and tan... You can make use of double angle Identities to find the values of the others.
3. Originally Posted by VonNemo19
Recall that $\sin\theta=\frac{opposite}{hypotenuse}$
Well, you have the opposite, but not the hypotenuse. To find the hypotenuse, use the pythagorean theorem.
After you have computed sin, cos, and tan... You can make use of double angle Identities to find the values of the others.
I was able to solve 1 & 2 but I do not recall Double Angle Identities. Want to steer me in the right direction?
4. Originally Posted by whiteforlife
I was able to solve 1 & 2 but I do not recall Double Angle Identities. Want to steer me in the right direction?
Check this out
Double-Angle and Half-Angle Formulas
5. Originally Posted by whiteforlife
I was able to solve 1 & 2 but I do not recall Double Angle Identities. Want to steer me in the right direction?
Double-Angle and Half-Angle Formulas
6. Thanks guys that helped alot, I have it all figured out now. | {"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": 2, "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": 0.8464183211326599, "perplexity": 412.69474687933314}, "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-2014-15/segments/1397609521558.37/warc/CC-MAIN-20140416005201-00115-ip-10-147-4-33.ec2.internal.warc.gz"} |
http://mathhelpforum.com/algebra/130723-algebra-problem.html | 1. ## Algebra problem
I am completely stuck on this, please explain clearly.
Many thanks
In this question x * y means (x+2y)/3 if x < y, and (2x + y)/3 if x is larger or equal to y.
a) Find the value of
i) 5 * 8
ii) 8 * 5
b) Solve
iii) x * 7 = 8
iv) x * (x-1) = 4 2/3 (that's 4 followed by 2/3 as a fraction)
c) In terms of p and q, find the possible values of (p * q) *p, explaining carefully how you choose which calculations to make.
2. In this question x * y means $\frac{x+2y}{3}$ if x < y, and $\frac {2x+y}{3}$ if x is larger or equal to y.
a) Find the value of
i) 5 * 8
So x = 5 and y = 8. Clearly, x<y, so just substitute the values 'x=5' and 'y=8' into the first equation to give:
$\frac{5+(2)(8)}{3}$
This simplifies to give an answer of 7.
Can you apply this to part ii)?
Then for part b) iii)
We know that y=7 and the answer is 8. Substituting that into the original equation:
$\frac{x+2y}{3}$
Gives:
$\frac{x+2(7)}{3}=8$
$\frac{x+14}{3}=8$
$x+14=24$
$x=10$
As this value of x is greater than our value of y, however, this is incorrect as we can only use this equation 'if x < y'
So you'll have to substitute the values into the second equation.
$\frac {2x+(7)}{3}=8$
$2x+7=24$
$2x=17$
$x=\frac{17}{2}=8.5$
Can you apply this to part ii?
FInally, part c. I don't understand part c as it seems to me that there could be almost infinite values for that equation. Is it not just trial and error?
3. Originally Posted by Natasha1
I am completely stuck on this, please explain clearly.
Many thanks
In this question x * y means (x+2y)/3 if x < y, and (2x + y)/3 if x is larger or equal to y.
c) In terms of p and q, find the possible values of (p * q) *p, explaining carefully how you choose which calculations to make.
I think there are two possibilities:
1. P<q and (P * q)>p
2. p>q and (p * q)<p
1. $[p * q] * p = [(p+2q)/3]* p$ {In this case I have used ,(x+2y)/3 if x < y, by letting p=x and q=y}
$= [2([(p+2q)/3])+p]/3$. { In this case I have used, (2x + y)/3 if x is larger or equal to y, by substituting p=y and [(p+2q)/3] = x}
you can try the second part!! | {"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": 14, "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": 0.8102995753288269, "perplexity": 844.5059683017369}, "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-2016-40/segments/1474738660548.33/warc/CC-MAIN-20160924173740-00129-ip-10-143-35-109.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/linear-algebra-eigenvalue-character-polynomials-proof.559335/ | # Homework Help: Linear algebra: eigenvalue & character polynomials proof
1. Dec 11, 2011
### ISuckAtMath
we are given B = CAC^-1
Prove that A and B have the same characteristic polynomial
given a hint: explain why ƛIn = CƛInC^-1
what I did was:
B = CAC^-1
BC = CA
Det(BC) = Det(CA)
Det(B) Det(C) = Det(C) Det(A)
Now they’re just numbers so I divide both sides by Det(C)
Det(B) = Det(A)
not im stuck
2. Dec 11, 2011
### AlephZero
Re: need help with a eigenvalue determinant proof
The determinant is one term in the characteristic polynomial, but you need more than that to show the characteristic polynomials are the same.
The eigenvalues are the roots of the characteristic polynomial so you need to show that every eigenvalue of B is also eigenvalue of A, and vice versa.
If λ is an eigenvalue of B, then there is a vector x such that Bx = λx.
So you need to find a vector y such that Ay = λy.
You are only given one thing that connects A and B, so just plug and chug...
3. Dec 11, 2011
### ISuckAtMath
1. The problem statement, all variables and given/known data
Suppose that C is an invertible matrix, and you are told that
B = CA(C^-1)
prove that A and B have exactly the same characteristic polynomial
do not assume A and B are triangular or diagonal matrices
2. Relevant equations
given hint: explain why ƛIn = CƛIn(C^-1)
3. The attempt at a solution
B = CA(C^-1)
BC = CA
Det(BC) = Det(CA)
Det(B) Det(C) = Det(C) Det(A)
now that they're just numbers, i divded both sides by Det(C)
Det(B) = Det(A)
i don't know what to do next
4. Dec 11, 2011
### Dick
That's fine so far. But it doesn't help. What expression involving Det gives you the characteristic polynomial?
5. Dec 11, 2011
### ISuckAtMath
well i know det(ƛIn - A)= 0
thus giving the characteristic polynomial (ƛ-ƛ1)(ƛ-ƛ2)....(ƛ-ƛn)
do i set set det(ƛIn - A) = det(ƛIn - B)?
with the given vector v: Av = ƛv and Bv = ƛv
therefore Av = Bv?
6. Dec 11, 2011
### Dick
Ok, so the characteristic polynomial for B=CAC^(-1) is det(CAC^(-1)-lambda*I). Now pay attention to the hint.
7. Dec 11, 2011
### Rockoz
You would want to start with det(λI - A) and conclude that it equals det(λI-B), i.e. their characteristic polynomials are the same. Starting with det(λI-A), think about the relationship between A and B, then the hint, and recall that matrice have a distributive property which allows you to factor.
Last edited: Dec 11, 2011
8. Dec 12, 2011
### micromass
Re: need help with a eigenvalue determinant proof
Note that proving that the eigenvalues are thesame for A and B isn't enough. You also need to prove that the multiplicities are the same.
9. Dec 12, 2011
### ISuckAtMath
is this correct?
B = CAC^(-1)
B-λIn = CAC^(-1) - λIn
using the hint: λIn = CλInC^(-1)
B-λIn = CAC(^-1) - CλInC^(-1)
B-λIn = C[ AC^(-1) - λInC^(-1) ] factored out C
B-λIn = CC^(-1)( A-λIn)
B-λIn = A-λIn, CC^(-1) cancel each other out
therefore det(B-λIn) = det(A-λIn)
so they're the same characteristic polynomial
10. Dec 12, 2011
### Deveno
Re: need help with a eigenvalue determinant proof
what you want to do is compare:
det(xI - A) and det(xI - B) = det(xI - CAC-1).
here is a hint:
CIC-1 = I
11. Dec 12, 2011
### Dick
No! The only way B-lambda*I=A-lambda*I is if A=B!. You can't prove that. You went wrong when you changed AC^(-1) into C^(-1)A in the third line. You can't do that. Matrices don't necessarily commute. Just factor the C^(-1) out on the right and the C on the left. Then take the det.
Last edited: Dec 12, 2011
12. Dec 12, 2011
### AlephZero
Re: need help with a eigenvalue determinant proof
Oops. But that is easily fixed up. Just start the argument by saying
If λ is a diagonal matrix of all the eigenvalues of B, then there is a matrix x such that Bx = λx.
13. Dec 12, 2011
### taxidriverhk
Re: need help with a eigenvalue determinant proof
You are trying to prove that det(λI - B) = det(λI - A)
and there is basically no way to prove that B = A because they are really not equal to each other
You are given a hint that λI = CλIC-1
and B = CAC-1
then you substitute these two into det(λI - B)
you should be able to prove that det(λI - B) = det(λI - A)
*you will need to use the properties of determinants, and also matrix multiplication is associative, be careful when factoring any matrix out...
Last edited: Dec 12, 2011 | {"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": 0.9307956695556641, "perplexity": 2224.0826776504787}, "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-2018-34/segments/1534221215858.81/warc/CC-MAIN-20180820062343-20180820082343-00172.warc.gz"} |
https://brilliant.org/problems/where-are-you-nene/ | # Where are you, Nene?
Geometry Level 1
The circle above has center $E$, diameter $RK = 24 \text{ m}$ and $\angle REM = 60^\circ$. Find the length of the line segment $NE$ in meters.
× | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 4, "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": 0.9446072578430176, "perplexity": 808.8139843809315}, "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-2021-31/segments/1627046154099.21/warc/CC-MAIN-20210731172305-20210731202305-00315.warc.gz"} |
http://mathhelpforum.com/number-theory/49578-defining-equation-golden-ratio-quadratic-fields.html | # Math Help - Defining equation for golden ratio; quadratic fields
1. ## Defining equation for golden ratio; quadratic fields
Definition: If $\alpha$ is an irrational number in $\mathbf{Q}\left(\sqrt{d}\right)$, then the equation $ax^{2}+bx+c=0$ is called the defining equation for $\alpha$ if $\alpha$ satisfies the equation and a, b, and c are integers, $(a,b,c)=1$, and $a>0$.
Find a defining equation for the golden ratio $\dfrac{1+\sqrt{5}}{2}$.
How do you approach this problem?
2. Hello,
Originally Posted by Pn0yS0ld13r
Definition: If $\alpha$ is an irrational number in $\mathbf{Q}\left(\sqrt{d}\right)$, then the equation $ax^{2}+bx+c=0$ is called the defining equation for $\alpha$ if $\alpha$ satisfies the equation and a, b, and c are integers, $(a,b,c)=1$, and $a>0$.
Find a defining equation for the golden ratio $\dfrac{1+\sqrt{5}}{2}$.
How do you approach this problem?
The solutions to such an equation are :
$x=\frac{-b {\color{red}\pm} \sqrt{b^2-4ac}}{2a}$
Since a and b are integers, the only way to get this $\sqrt{5}$ is $\sqrt{b^2-4ac}$.
So you can see that $\frac{1{\color{red}-}\sqrt{5}}{2}$ will also be a root. (The reasoning may be similar to the ones involving complex roots).
Develop $\left(x-\frac{1+\sqrt{5}}{2}\right)\left(x-\frac{1-\sqrt{5}}{2}\right)$ and identify a,b and c
3. Thank you Moo.
I can't believe I didn't get this before... | {"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": 21, "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": 0.9612906575202942, "perplexity": 174.04530751387628}, "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/1443737931434.67/warc/CC-MAIN-20151001221851-00132-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://dayline.info/relationship-between-and/relationship-between-charge-and-current-formula.php | # Relationship between charge and current formula
### BBC Bitesize - GCSE Combined Science - Electric circuits - AQA - Revision 2
This formula is derived from Ohm's law. Where we have: V: voltage. I: current If rate of flow of charge is not constant then the current at any instant is given by the emf of the source is exactly equal to the potential difference between its ends. Electric power basic formulas calculator voltage current mathematical . For a temporally constant power, the relationship between the charge and current. Charge, Current & Potential Difference in circuits. Conventional current flows around a circuit from the positive (+) side of the potential difference equation.
## Electric current
The direction of current is taken as the direction in which positive charges move. In conduction although the current is only due to electrons, the current was earlier assumed to be due to positive charges flowing from the positive of the battery to the negative.
The direction of current therefore is taken as opposite to the flow of electrons. If current is constant: This law is based on the principle of conservation of charge and states that in an electrical circuit or network of wires the algebraic sum of currents meeting at a point is zero.
• Electric circuits - AQA
• Calculating electrical power
The arrow head marked in circuit represents the direction of conventional current i. The algebraic sum of the product of the current and resistance in any closed loop of a circuit is equal to the algebraic sum of electromotive forces acting in that loop. The average flow of electrons in a conductor not connected to battery is zero i. Thus no current flows through the conductor until it is connected to the battery.
Drift velocity of free electrons in a metallic conductor In the absence of an electric field, the free electrons in a metal randomly in all directions and therefore their average velocity are zero. When an electric field is applied, they are accelerated opposite to the direction of the field and therefore they have a net drift in that direction.
However, due to frequent collisions with the atoms, their average velocity is very small. This average velocity with which the electrons move in a conductor under a potential difference is called the drift velocity.
It is characterized by the Meissner effectthe complete ejection of magnetic field lines from the interior of the superconductor as it transitions into the superconducting state. The occurrence of the Meissner effect indicates that superconductivity cannot be understood simply as the idealization of perfect conductivity in classical physics.
Semiconductor In a semiconductor it is sometimes useful to think of the current as due to the flow of positive " holes " the mobile positive charge carriers that are places where the semiconductor crystal is missing a valence electron. This is the case in a p-type semiconductor. A semiconductor has electrical conductivity intermediate in magnitude between that of a conductor and an insulator.
In the classic crystalline semiconductors, electrons can have energies only within certain bands i.
Energetically, these bands are located between the energy of the ground state, the state in which electrons are tightly bound to the atomic nuclei of the material, and the free electron energy, the latter describing the energy required for an electron to escape entirely from the material. The energy bands each correspond to a large number of discrete quantum states of the electrons, and most of the states with low energy closer to the nucleus are occupied, up to a particular band called the valence band.
### Electric current - Wikipedia
Semiconductors and insulators are distinguished from metals because the valence band in any given metal is nearly filled with electrons under usual operating conditions, while very few semiconductor or virtually none insulator of them are available in the conduction band, the band immediately above the valence band. The ease of exciting electrons in the semiconductor from the valence band to the conduction band depends on the band gap between the bands.
The size of this energy band gap serves as an arbitrary dividing line roughly 4 eV between semiconductors and insulators. With covalent bonds, an electron moves by hopping to a neighboring bond. The Pauli exclusion principle requires that the electron be lifted into the higher anti-bonding state of that bond.
For delocalized states, for example in one dimension — that is in a nanowirefor every energy there is a state with electrons flowing in one direction and another state with the electrons flowing in the other. For a net current to flow, more states for one direction than for the other direction must be occupied.
Charge, Current and Voltage - GCSE Physics - Doodle Science
For this to occur, energy is required, as in the semiconductor the next higher states lie above the band gap. Often this is stated as: However, as a semiconductor's temperature rises above absolute zerothere is more energy in the semiconductor to spend on lattice vibration and on exciting electrons into the conduction band.
The current-carrying electrons in the conduction band are known as free electrons, though they are often simply called electrons if that is clear in context.
## Charge, Current & Potential Difference
Current density and Ohm's law Main article: Current density Current density is a measure of the density of an electric current. It is defined as a vector whose magnitude is the electric current per cross-sectional area. In SI unitsthe current density is measured in amperes per square metre. | {"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": 0.9063414931297302, "perplexity": 330.29377978232765}, "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-2019-35/segments/1566027322170.99/warc/CC-MAIN-20190825021120-20190825043120-00376.warc.gz"} |
https://www.physicsforums.com/threads/what-does-over-determination-mean.464539/ | # What does over determination mean?
• Start date
• #1
666
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## Main Question or Discussion Point
What does "over determination" mean?
I have come across instances of the phrase "over determination" in discussing Einstein's attempt at a unified field theory.
For example:
"Again and again, Einstein returned to the question of whether one could obtain the quantum of action from an over determination of the field variables."
-- Jagdish Mehra, "Einstein, Hilbert and the Theory of Gravitation"
Related Differential Equations News on Phys.org
• #2
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pellman:
I'm not certain what the quote you gave necessarily means, exactly. However, in general, "over determination", or an "over determined" system is where one has more equations to satisfy than unknowns to be determined. (From a linear perspective, at least. Of course, non-linear systems can have cases where one has exactly the same number of equations and unknowns, while still not having solutions---at least not solutions within the prescribed space, such as real numbers, etc.)
In this case, one has the very real possibility that the system is not solvable (no solutions exist). Another possibility is that the system of equations is redundant, in the sense that the system can actually be reduced to a simpler system that more closely matches the number of unknowns to be determined.
However, in some cases, the system of equations may not be reducible to be commensurate with the number of unknowns, and, yet, solutions do exist. (Incidentally, this can only occur in non-linear systems.) This is where one ends up having to look to prove the existence of solutions, before wasting time trying to solve a potentially unsolvable system. (Einstein was faced with this issue when he developed General Relativity.)
One potential application of such a concept would be to nail down the mass spectrum of fundamental particles with an over determined system. Since the present Standard Model can accommodate arbitrary masses, because the particle masses are simply parameters of the model, one could envision having some kind of overdetermined system that would only yield solutions for a specific mass spectrum. (Hopefully one that matches our observations, of course.)
• #3
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Thanks, Halliday.
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https://nool.uoit.ca/mathematics/basic/factorial.php | # Factorial
## Introduction
The factorial of a nonnegative integer, n, is denoted by: n! and represents the product of all the positive integers less than or equal to n. By definition, 0!=1 and 1!=1.
Example: Find 5!
Solution:
5! = (5)(4)(3)(2)(1)
= 120
Example: Find 10!
Solution:
10! = (10)(9)(8)...(3)(2)(1)
= 3628800
Why is this important?
You might encounter factorials when studying probability, Taylor series (in a calculus class), or in a discrete mathematics course. | {"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": 0.8738150596618652, "perplexity": 2205.6597737997777}, "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/1563195526536.46/warc/CC-MAIN-20190720153215-20190720175215-00253.warc.gz"} |
https://scicomp.stackexchange.com/questions/20321/convex-optimization-problem-with-sum-of-absolute-value-constraints | # Convex Optimization problem with sum of absolute value constraints
How to solve the optimization problem written below?
\begin{align} &\operatorname{argmax}\limits_{a}\; a^T b - \frac{1}{2} a^T X a\\ &\text{subject to } \sum_i |a_i|=4,\; \sum_i a_i = 0 \end{align}
where $a$, $b$ are $n$-vectors and $X$ is a $n\times n$ matrix. Also, $b$ and $X$ are constants.
My main issue is about the absolute values. Without absolute values, there is actually an analytic solution. I guess with absolute values, I have to use iterative approach such as quadratic programming but still not sure how to express the problem to call relevant optimization procedures.
Unfortunately, your problem isn't a convex optimization problem because the constraint $\Sigma_{i} | a_{i}|=4$ describes a non-convex feasible region. If you could change this to $\Sigma_{i} | a_{i} | \leq 4$, you'd have a convex constraint.
If the constraint were $\Sigma_{i} | a_{i} | \leq 4$, then you can introduce auxiliary variables $t_{i}$, and add the constraints
$\Sigma_{i} t_{i} \leq 4$
$t_{i} \geq a_{i}$ for all $i$
$t_{i} \geq -a_{i}$ for all $i$
This is a standard reformulation technique in convex optimization.
Another issue with your original problem statement is that $X$ must be positive semidefinite to ensure concavity of the objective function.
Assuming that $X$ is positive semidefinite, you've now got a linear constrained convex quadratic programming problem which is solvable by lots of solvers.
• I see your point. Let's say the constraint is changed to Σi|ai|≤4, how then it should be solved? – Bill Z Jul 30 '15 at 3:48
• see my expanded answer. – Brian Borchers Jul 30 '15 at 4:11
• what if you had another constraint sum(abs(a))>=2? – citynorman Sep 10 '17 at 13:24
• That set described by $\sum | a_{i} | \geq 2$ isn't a convex set, and this your problem couldn't be formulated as a convex optimization problem. – Brian Borchers Sep 10 '17 at 16:47
• @BrianBorchers Thanks for the answer. Out of interest, do you have a text book reference for this? – spinkus Feb 13 at 1:40 | {"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8242778182029724, "perplexity": 375.7857894051752}, "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/1570986692126.27/warc/CC-MAIN-20191019063516-20191019091016-00477.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/12164-ring-help.html | # Thread: Ring help
1. ## Ring help
Two questions:
1. If R is a ring then is every element of R either a unit or zero-divisor.
COnversely
2. IF R is a ring, then there is no elements of R which is both a unit and a zero-divisor.
2. Originally Posted by chadlyter
Two questions:
1. If R is a ring then is every element of R either a unit or zero-divisor.
Is R a finite ring?
Because, if it is then we can prove it as follows:
If x is a zero-divisor the proof is complete.
If x is not a zero-divisor then consider the finite set,
{a_1x,a_2x,...,a_nx} where a_i are the non-zero elements of the ring (assuming it is non-trivial).
Then, we can show none are equal to each other.
(Excercise).
Then by Dirichlet's Pigeonhole Principle, is an enumeration of all non-zero elements in R. Hence we can find an a_i such that a_i x = 1. Thus, x is a unit.
3. Originally Posted by TheNextDirichlet
Is R a finite ring?
It has to be.
Because, for example, consider, Z.
The integers under addition and multiplication.
The element 2 in Z, it neither a unit nor zero divisor
(However, for all finite rings this is true).
4. ## RIng
The Ring is not defined as a finite ring. These are true/false questions I have encountered in a abstract algebra class. If false I have to give a counterexample. For example, If R is a ring, then every element of R is either a unit or zero-divisor. In my mind this is false but the counterexample is what is messing with my head.
5. Originally Posted by chadlyter
If false I have to give a counterexample. For example,
It is false consdier the example I gave you. | {"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": 0.9260757565498352, "perplexity": 499.38643013670514}, "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-2018-05/segments/1516084886830.8/warc/CC-MAIN-20180117063030-20180117083030-00439.warc.gz"} |
https://abhcompta.be/accu3/is-a-irrational-number-aff619 | We could write 1 as 12 (so it has an even exponent), and then we have: But there is still an odd exponent (on the 2). If the decimal form of a number The Square Root of 2. If the rational number is a/b, then that becomes a 2 /b 2 when squared. An irrational number isn't as scary as it sounds; it's just a number that can't be expressed as a simple fraction or, to put it another way, an irrational number is a never-ending decimal that continues an infinite number of places past the decimal point. Let me explain ... Squaring a Rational Number. Any non-terminating and non-recurring decimal is an irrational number. So it contradicts. Irrational number definition: any real number that cannot be expressed as the ratio of two integers , such as π | Meaning, pronunciation, translations and examples T HE JOB OF ARITHMETIC when confronted with geometry, that is, with things that are continuous-- length, area, time -- is to come up with the name of a number to be its measure.For if we say that a length is 3½ meters, Identifying Rational and Irrational Numbers Irrational numbers are a subset of real numbers which are not rational numbers but can be represented on the number line. Irrational number, any real number that cannot be expressed as the quotient of two integers. Let p be a prime number and a be a positive integer. e.g.\ (\sqrt {2},\sqrt {3},\Pi \\)……………. Irrational Numbers. It can be expressed in the form 9/1, i.e. Our mission is to provide a free, world-class education to anyone, anywhere. Rational numbers are denoted QQ. While an irrational number cannot be written in a fraction. By recalling the Pythagorean theorem, we can see that irrational numbers are necessary. (number under radical is less than 20) The 3 has an exponent of 2 (32) and the 2 has an exponent of 4 (24). ⅔ is an example of rational numbers whereas √2 is an irrational number. This means that the value that was squared to make 2 (ie the square root of 2) cannot be a rational number. We say therefore that is an irrational number. 12. Since 9 = 9/1 it meets the definition. An irrational number can be written as a decimal, but not as a fraction. But the 3 has an exponent of 1, so 3 could not have been made by squaring a rational number, either. A "Rational" Number can be written as a "Ratio", or fraction. The invention of irrational numbers. In rational numbers, both numerator and denominator are whole numbers, where the denominator is not equal to zero. Irrational numbers include √ 2, π, e, and φ. An irrational number is the opposite of a rational number. So, all irrational numbers have non-terminating, non-repeating decimals, when simplified. An irrational number cannot be expressed as a ratio between two numbers and it cannot be written as a simple fraction because there is not a finite number of numbers when written as a decimal. But to do this properly we should really break the numbers down into their prime factors (any whole number above 1 is prime or can be made by multiplying prime numbers): Notice that the exponents are still even numbers. So we can see that when we square a rational number, the result is made up of prime numbers whose exponents are all even numbers. A number is irrational if and only if its decimal representation is non-terminating and non-repeating. Learn its properties, examples, symbol and list at BYJU'S. Since 3, a and b are integers a/3b be a irrational number. From this, we come to know that a and b have common divisor other than 1. But an irrational number cannot be written in the form of simple fractions. Irrational Numbers (Definition, List, Properties, and Examples) Irrational numbers are numbers that are neither terminating nor recurring and cannot be expressed as a ratio of integers. Example: 1.5 is rational, because it can be written as the ratio 3/2, Example: 7 is rational, because it can be written as the ratio 7/1, Example 0.317 is rational, because it can be written as the ratio 317/1000. To find if the square root of a number is irrational or not, check to see if its prime factors all have even exponents. Which is 21/11 ,and that has odd exponents! Proof: there's an irrational number between any two rational numbers (Opens a modal) About this unit. | {"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": 0.9358344078063965, "perplexity": 238.49825247356696}, "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-2023-14/segments/1679296945433.92/warc/CC-MAIN-20230326044821-20230326074821-00505.warc.gz"} |
https://ohhaskme.com/8410/q5b-is-there-any-way-to-solve-it-without-using-trial-and-error | Q5b, is there any way to solve it without using trial and error?
There are various successive approximations you can use to solve 2t - cos t = 0. I don't know if you'd call them "trial and error".
For instance, Newton's method, where from each guess t you get the next guess t\_new by
t\_new = t - $f(t) / f'(t)$= t - $(2t - cos t) /(2 + sin t)$
If you start with the guess t = 1.0, this converges really rapidly:
t = 1.0
t = 0.486288017
t = 0.450418605
t = 0.450183622
t = 0.450183611
Differentiating the equation of motion for x and setting it to be greater than zero shows that the time must be smaller than half a second.
The question comes down to finding the intercept between the functions y=2x and y=cos(x). As mentioned in previous answers Newton's method is a reliable way of calculating this.
As a physicist I would truncate the Taylor expansion of cos x and then solve a horrible polynomial on the computer to approximate the answer which is a little more than 0.45
0 like 0 dislike | {"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": 0.8736720681190491, "perplexity": 441.11107524006417}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945333.53/warc/CC-MAIN-20230325130029-20230325160029-00313.warc.gz"} |
https://brilliant.org/discussions/thread/intrig3gral/ | # INTrig3GRAL
$\int \frac{ \tan^3 x } { \sin^3 x + \cos^3 x } \, dx$
If you give up, you can look below.
5 years, 8 months ago
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$\displaystyle \int \frac{\tan^3 x}{\sin^3 x +\cos^3 x} dx= \int \frac{\tan^3 x +1}{\sin^3 x +\cos^3 x} dx - \int \frac{1}{\sin^3 x +\cos^3 x} dx$
Now working on the first one, take $\displaystyle \cos^3 x$ as a common factor in the denominator : $\displaystyle \int \frac{\tan^3 x +1}{\sin^3 x +\cos^3 x} dx = \int \frac{\tan^3 x +1}{\cos^3 x (\tan^3 x +1)} dx = \int \sec^3 x dx$
Let $\displaystyle I = \int \sec^3 x dx$ , Integrating by parts with $\displaystyle u=\sec x , dv= \sec^2 x dx$ :
$\displaystyle I= \sec x \tan x - \int \sec x \tan^2 x dx= \sec x \tan x - \int \sec x (\sec^2 x -1) dx = \sec x \tan x - \int \sec^3 x dx + \int \sec x dx$
$\displaystyle => I= \sec x \tan x - I + \ln(\sec x + \tan x)$
$\displaystyle => I= \frac{ \sec x \tan x + \ln(\sec x + \tan x)}{2}$
Then we have :
$\displaystyle \int \frac{\tan^3 x}{\sin^3 x +\cos^3 x} dx = \frac{ \sec x \tan x + \ln(\sec x + \tan x)}{2} - \int \frac{1}{\sin^3 x +\cos^3 x} dx$
You can see the solution to the last integral from This note. Thus:
$\displaystyle \boxed{\int \frac{\tan^3 x}{\sin^3 x +\cos^3 x} dx = \frac{ \sec x \tan x + \ln(\sec x + \tan x)}{2} - \frac{\sqrt{2}}{6} \ln(\frac{\cos x -\sin x -\sqrt{2}}{\cos x -\sin x +\sqrt{2}}) + \frac{2}{3} \arctan(\cos x -\sin x ) + C}$
- 5 years, 8 months ago
Ahha very tricky! I see what you did there. Very good.
Hmm... Which one looks more threatening, one with tangent or one with just sin and cos on bottom? Sayin' cause it seems like tan can be used to simplify sin and cos, and plus when I usually see many things I think they can be manipulated in a certain way to be less complex (like this). And when there's less stuff it seems like you have less options to work with and it all looks scary lol... But this is cause I'm experienced, I guess. What about for those who ain't that pro at math? DUnno... Lol.
Well, King Integral, I've got more stuff coming up for ya. Meanwhile, maintain that streak (we're head to head)! Whoever breaks first is a dweeb.
Now get back to being awesome
salsa
- 5 years, 8 months ago
Yes u are right , integrals with less expressions are more confusing more threatening.
Btw I will try to keep my streak alive although my school began today :)
- 5 years, 8 months ago
Hehe well I kinda cheat: go to practice, algebra, pattern recognition. takes 5 seconds xD
the toughest part is not to forget. but yeh, go with that if ya want :D
- 5 years, 8 months ago
Cheating huh?!
heheh me too I solve level one algeba problem to keep my streak :))
- 5 years, 8 months ago
O_O oooomg no way did you actually do this?? I got no time to check but WHOA and for some reason your answer looks waay shorter than and of WolframAlpha's answers. I'ma check later but great job if you got it right
- 5 years, 8 months ago
yes I checked it several times,also I will try to find a way to prove that wolfram alpha answers equal this.
- 5 years, 8 months ago
Heh if you do that that'd be totally on hook. Also try to do it without like just differentiating it, that's kinda wack.
right now me gotta speed off somewhere, but I shall address this feat when I return. Think of any cool names for yourself by then ;) I thought of Intergral Master or King Integral lol...
- 5 years, 8 months ago
Hehehehe :)
- 5 years, 8 months ago
how did you type the integral sign
- 5 years, 8 months ago
\displaystyle \int
between $and$
- 5 years, 8 months ago
here ya go:
s
And yeah this is doable...
sa
Whoever solves this gets to call itself whatever it wants.
SOLVE ON
Here's some extra booze if you need some:
aga
- 5 years, 8 months ago | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 20, "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": 0.9685857892036438, "perplexity": 3879.5302497857806}, "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-20200528170840-20200528200840-00190.warc.gz"} |
http://www.talkstats.com/members/charliemurphy.100854/recent-content | # Recent content by CharlieMurphy
1. ### Predictions after mean-centred regression
Hi, I wonder if anyone can help me with a problem I'm having. I just performed an ordinary (LS) regression in which both the dependent and independent variables were logged (the model is log-linear) and grand-mean-centred. The results look good. However, for my purposes, I need to be able to... | {"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": 0.8976665139198303, "perplexity": 901.1444555472666}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400192783.34/warc/CC-MAIN-20200919173334-20200919203334-00452.warc.gz"} |
https://scipy.github.io/devdocs/reference/generated/scipy.spatial.distance.correlation.html | # scipy.spatial.distance.correlation#
scipy.spatial.distance.correlation(u, v, w=None, centered=True)[source]#
Compute the correlation distance between two 1-D arrays.
The correlation distance between u and v, is defined as
$1 - \frac{(u - \bar{u}) \cdot (v - \bar{v})} {{\|(u - \bar{u})\|}_2 {\|(v - \bar{v})\|}_2}$
where $$\bar{u}$$ is the mean of the elements of u and $$x \cdot y$$ is the dot product of $$x$$ and $$y$$.
Parameters:
u(N,) array_like
Input array.
v(N,) array_like
Input array.
w(N,) array_like, optional
The weights for each value in u and v. Default is None, which gives each value a weight of 1.0
centeredbool, optional
If True, u and v will be centered. Default is True.
Returns:
correlationdouble
The correlation distance between 1-D array u and v. | {"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": 0.9076612591743469, "perplexity": 1911.5781249956683}, "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/1674764500339.37/warc/CC-MAIN-20230206113934-20230206143934-00317.warc.gz"} |
http://www.ipam.ucla.edu/abstract/?tid=8843&pcode=CMAWS3 | ## Existence and Counting
#### Jarik NesetrilCharles University, Prague
We introduce the dichotomy between classes of graphs (and more generally of
structures): Nowhere Dense and Somewhere Dense classes.
We show how this dichotomy naturally arises and show its several characterizations. Particularly, we show how these classes can be defined by the limit frequencies of its subgraphs (which are integral and can be interpreted as degrees of freedom).
This is a joint work with Patrice Ossona de Mendez (EHESS Paris).
Back to Workshop III: Topics in Graphs and Hypergraphs | {"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": 0.9435983896255493, "perplexity": 1535.4107909750585}, "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/1531676593586.54/warc/CC-MAIN-20180722194125-20180722214125-00534.warc.gz"} |
https://www.physicsforums.com/threads/what-does-an-infinte-series.563175/ | # What does an infinte series =?
1. Dec 27, 2011
### TheKracken
1. The problem statement, all variables and given/known data
This isint really homework...but some kid at my school was telling me about the series 1/2+1/4+1/8+1/16+1/32+1/64.....going on forever...now thinking about this it approaches 1....but apperantly is also infinte??? could someone explain this? is this kid just full of himself?
2. Relevant equations
3. The attempt at a solution
2. Dec 27, 2011
### jgens
Suppose $0 \leq r < 1$ and fix $k \in \mathbb{N}$. Then we have the following:
$$(1-r)\sum_{n=0}^k r^n = \sum_{n=0}^k r^n - r\sum_{n=0}^k r^n = 1-r^{k+1}$$
This allows us to evaluate the sum in the finite case. That is,
$$\sum_{n=0}^k r^n = \frac{1-r^{k+1}}{1-r}$$
Evaluating the limit as $k \rightarrow \infty$ tells us how to evaluate the infinite sum. In particular, we have
$$\sum_{n=0}^{\infty} r^n = \frac{1}{1-r}$$
The sum that you listed is the case when $r = 2^{-1}$. This means we have
$$\sum_{n=0}^{\infty} 2^{-n} = 1 + \sum_{n=1}^{\infty} 2^{-n} = 2$$
This tells you that
$$\sum_{n=1}^{\infty} 2^{-n} = 1$$
So the sum converges to 1.
3. Dec 27, 2011
### Curious3141
From the way this post is worded, I'm assuming you don't want to hear a lot of technical jargon. So I'll keep it simple.
Just in case you weren't aware of the nomenclature (because I'll make reference to that):
Sequence = ordering of terms.
Series = Sum of terms in a sequence.
This is called a geometric series. This is the sum of a sequence of terms where the next term is derived by multiplying the current term by a fixed number (called the common ratio). So by knowing the first term and the common ratio, you can get all the terms of the sequence, and by then summing them, find the sum of the series.
Here the first term is 1/2 and the common ratio is also 1/2. Meaning the terms progressively get smaller. They will never completely vanish (become zero), but they will become arbitrary close as you take more and more terms.
This is called an infinite series because you "let" the series grow to an "infinite" length i.e. you don't stipulate an end to it. Practically, you will never be able to write down all the terms, but the continuation (...) at the end means that you're theoretically considering an infinite number of terms. As the sequence goes on, the terms get closer to zero, and this is important for something known as convergence, which means that the sum actually exists and is finite. If the series doesn't converge, it just "blows up" to arbitrarily larger numbers (this is called diverging to infinity).
In this case, because of the common ratio of 1/2, the terms get smaller and smaller and the series converges. We can work out the sum of the infinite geometric series as jgens stated (here it's $\frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1$ as your friend said). This essentially means that if you take more and more terms and sum them up (i.e. do the "partial sums" for a greater and greater number of successive terms), you will get arbitrarily close to 1, but never quite hit it. Here the partial sums will always be strictly less than one because the terms are all positive.
Hope this has made things very clear to you.
4. Dec 27, 2011
### cragar
think of it as taking a piece of paper of area 1, and then cutting this piece of paper in half then cut the remaining half in half again, and do this forever and then you will have an infinite number of rectangles and when you add them back up you get the original piece of paper.
5. Dec 27, 2011
### micromass
Staff Emeritus
This is not really a proof. But it might convince you.
Let
$$x=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...,$$
then
$$2x=1+\frac{1}{2}+\frac{1}{}+\frac{1}{8}+\frac{1}{16}+...,$$
so
$$2x-1=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...$$
But the left-hand side is equal to x!!
So $2x-1=x$. Or equivalently $x=1$.
6. Dec 28, 2011
### TheKracken
Great explanation~! I accually understood what you were saying :P Thank you very much, and to the poster that posted all of those #'s and stuff, I honestly had no idea what they meant, but I do appreciate the help, maybe explain all the mathematical stuff in a little detail??? I know I have seen Ʃ before because I am currently self studying calculus and I belive that is called a piece wise function? yes? like it gives you 2 different things for a function...but not sure if that is the case here.
7. Dec 28, 2011
### cragar
interesting trick micromass for the series.
8. Dec 28, 2011
### epenguin
This square of area 1
should convince you that
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
as we loosely say, and clarify what it means, that as we add more and more terms we get closer and closer to 1. That however many terms we include in the series we shall never get exactly to 1, but we shall get as close as we wish to 1 by taking enough terms. Therefore we can use this and similar seemingly ineffable statements for perfectly practical uses.
Last edited: Dec 28, 2011
9. Dec 28, 2011
### Benn
I got really excited when I saw your post, Jgens. I have wanted to know how the formula for evaluating infinity sums was derived for a while. I don't understand the first part though.
How are you getting $\sum_{n=0}^k r^n$ to equal 1?
Last edited: Dec 28, 2011 | {"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": 0.9329732060432434, "perplexity": 411.576545749476}, "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-2017-47/segments/1510934805881.65/warc/CC-MAIN-20171119234824-20171120014824-00310.warc.gz"} |
https://indico.cern.ch/event/733738/ | Particle and Astro-Particle Physics Seminars
# Factorisation in Colour Singlet Production
## by Jonathan Richard Gaunt (CERN)
Europe/Zurich
4/3-006 - TH Conference Room (CERN)
### 4/3-006 - TH Conference Room
#### CERN
100
Show room on map
Description
We rely heavily on factorisation formulae to make useful predictions at colliders such as the LHC. But when do such formulae work, and why? When do they not work? I will discuss this issue in the context of the production of a colour singlet V at hadronic colliders. Observables discussed include the total cross section for the production of V, transverse momentum of V (including spin-dependent observables), and hadronic transverse energy E_T accompanying V. Based on arXiv:1405.2080 and arXiv:1709.04935. | {"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": 0.8688886761665344, "perplexity": 4451.635947866121}, "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/1563195526401.41/warc/CC-MAIN-20190720004131-20190720030131-00499.warc.gz"} |
https://mathematics21.org/2018/05/18/funcoids-and-topologies/?replytocom=1911 | # Mappings between endofuncoids and topological spaces
I started research of mappings between endofuncoids and topological spaces.
Currently the draft is located in volume 2 draft of my online book.
I define mappings back and forth between endofuncoids and topologies.
The main result is a representation of an endofuncoid induced by a topological space.
The formula is $f\mapsto 1\sqcup\mathrm{Compl}\, f\sqcup(\mathrm{Compl}\, f)^2\sqcup \dots$.
However I proved this theorem only for the special case if every singleton is a closed set. Also the proof is not yet checked for errors.
## One comment
1. The proof was with a fatal error. I have removed the said theorem from my drafts. Instead I’ve added a counterexample. | {"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": 0.9182915091514587, "perplexity": 1127.379776551101}, "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/1590347402457.55/warc/CC-MAIN-20200529054758-20200529084758-00370.warc.gz"} |
https://encyclopediaofmath.org:443/index.php?title=Luzin_space | # Luzin space
An uncountable topological $T_1$-space without isolated points in which every nowhere-dense subset is countable. The existence of a Luzin space on the real line follows from the continuum hypothesis. From the negation of the continuum hypothesis and Martin's axiom (cf. Suslin hypothesis) together it follows that no Luzin space exists. In particular, this is compatible with the Zermelo–Fraenkel system of axioms of set theory and the axiom of choice. The existence of metrizable Luzin spaces has been proved under very general assumptions about the place of the cardinality of the continuum in the scale of alephs. Any Luzin space $X$ that lies in a separable metric space $Y$ has the following property: For any sequence $\{\lambda_n\}$ of positive numbers there is a sequence of sets $\{A_n\}$ such that $X=\bigcup_{n=1}^\infty A_n$ and $\delta(A_n)<\lambda_n$, where $\delta(A)$ is the diameter of the set $A$. This property is invariant under continuous mappings. Any continuous image of a Luzin space lying in $Y$ has Lebesgue measure zero and dimension zero. Moreover, it is totally imperfect, that is, it does not contain a Cantor set. The continuum hypothesis implies that there is a regular hereditarily-separable, hereditarily Lindelöf, extremally-disconnected Luzin space of countable $\pi$-weight and with the cardinality of the continuum.
#### References
[1] N.N. [N.N. Luzin] Lusin, "Sur un problème de M. Baire" C.R. Acad. Sci. Paris , 158 (1914) pp. 1258–1261 [2] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
Three slightly different definitions of Luzin space are still in use (apart from whether they must be $T_1$, $T_2$ or $T_3$): An uncountable space all of whose nowhere-dense sets are countable, with 1) no isolated points; or 2) at most $\omega$ isolated points; or 3) any number of isolated points.
[a1] K. Kunen, "Luzin spaces" Topology Proceedings , 1 (1977) pp. 191–199 [a2] J. Roitman, "Basic $S$ and $L$" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 295–326 [a3] W. Weiss, "Versions of Martin's axiom" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 827–886 [a4] A.W. Miller, "Special subsets of the real line" K. Kunen (ed.) J.E. Vaughan (ed.) , Handbook of Set-Theoretic Topology , North-Holland (1984) pp. 201–233 | {"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": 0.9799773097038269, "perplexity": 609.2790151437791}, "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/1659882573744.90/warc/CC-MAIN-20220819161440-20220819191440-00132.warc.gz"} |
http://www.ck12.org/earth-science/Geothermal-Power/lesson/Geothermal-Power/ | <meta http-equiv="refresh" content="1; url=/nojavascript/">
# Geothermal Power
## Geothermal energy is a clean, renewable energy source.
%
Progress
Practice Geothermal Power
Progress
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Geothermal Power
How could geothermal energy be used just about anywhere?
Geothermal energy comes from heat deep below the surface of the Earth. That heat may come to the surface naturally or it may be available through drilling. Nothing must be done to the geothermal energy. It is a resource that can be used without processing.
### Geothermal Energy
The heat that is used for geothermal power may come to the surface naturally as hot springs or geysers, like The Geysers in northern California. Where water does not naturally come to the surface, engineers may pump cool water into the ground. The water is heated by the hot rock and then pumped back to the surface for use. The hot water or steam from a geothermal well spins a turbine to make electricity.
Geothermal energy is clean and safe. The energy source is renewable since hot rock is found everywhere in the Earth, although in many parts of the world the hot rock is not close enough to the surface for building geothermal power plants. In some areas, geothermal power is common ( Figure below ).
A geothermal energy plant in Iceland. Iceland gets about one fourth of its electricity from geothermal sources.
In the United States, California is a leader in producing geothermal energy. The largest geothermal power plant in the state is in the Geysers Geothermal Resource Area in Napa and Sonoma Counties. The source of heat is thought to be a large magma chamber lying beneath the area.
Where Earth's internal heat gets close to the surface, geothermal power is a clean source of energy. In California, The Geysers supplies energy for many nearby homes and businesses.
Find out more at http://www.kqed.org/quest/television/geothermal-heats-up .
### Summary
• Most geothermal energy being used now is in regions where hot material comes to the surface.
• Hot rocks are everywhere below Earth's surface so geothermal energy could be used anywhere with drilling.
• Geothermal energy is clean and does not release greenhouse gases.
### Practice
Use this resource to answer the questions that follow.
1. How does geothermal energy work?
2. Where were geothermal plans in the past?
3. How does dry steam geothermal work?
4. How does a flash steam power plant work?
### Review
1. How is geothermal energy harnessed?
2. How would it be possible for a geothermal plant to gather energy if the hot material was not located at the surface?
3. Why is geothermal energy becoming more popular? | {"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": 0.8133968114852905, "perplexity": 2325.593182682939}, "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-2015-27/segments/1435375095671.53/warc/CC-MAIN-20150627031815-00273-ip-10-179-60-89.ec2.internal.warc.gz"} |
http://physics.stackexchange.com/questions/64680/gyro-vs-accelero-why-is-that-assessment-false | Gyro vs Accelero : Why is that assessment false?
Can we say that the Accelerometer's values are somewhat the integral of the Gyroscope's values ?
I think that because I know that the velocity is the derivative of the position with respect to time and the acceleration is the derivative of the velocity with respect to time.
Let's say I turn in circles. If my assessment is true, then my accelerometer should tell me my velocity is getting bigger and bigger. And it will, even if my velocity doesn't change.
So, I know my assessment is false, but I don't understand why ...
Would you happen to know ?
-
Velocity is the derivative of position. Acceleration is the derivative of velocity. If you turn in a perfect constant velocity circle the magnitude of velocity (speed) is constant however the direction is changing. Because the direction is changing there will be a constant magnitude acceleration pointing towards the center of the circle. – OSE May 15 '13 at 13:47 | {"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": 0.9509974718093872, "perplexity": 230.78590902287294}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701161946.96/warc/CC-MAIN-20160205193921-00066-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/arc-length-problem.292872/ | # Homework Help: Arc length problem.
1. Feb 16, 2009
### negatifzeo
1. The problem statement, all variables and given/known data
Find the arclength of the function $$y=x^2$$ when x is between 0 and 10.
2. Relevant equations
Arclength here is $$\int_{0}^{10} \sqrt{1+(2x)^2} dx$$
(It's intended to be the integral from 0 to 10 of the quare root of 1+(2x)^2. My latex skills suck.)
3. The attempt at a solution
Trig substitution. 2x= tan (theta)
Integrate secant theta
Which is ln(sec(theta)+tan(theta))
Substituting theta back in for arcsin(2x)
ln(sec(arctan(2x))+tan(arctan(2x))) evaluated from 10 to 0. Solving for arctan(2x) my final answer is
ln(sqrt(1+4x^2)+2x) evaluated from 0 to 10. I know this is wrong as it is much to short of an arc length but I don't know where I went wrong. Any clues?
3. The attempt at a solution
Last edited by a moderator: Feb 16, 2009
2. Feb 16, 2009
### Tom Mattson
Staff Emeritus
That's fine.
That's not fine. Why would you integrate $\sec (\theta )$?
3. Feb 16, 2009
### negatifzeo
Well, when I use trig substitution here I change 2x to tan(theta). This changes the integrand to sqrt(1+tan^2(theta)). The trig identity says that 1 + tangent squared equals secant squared, and since it is the quare root of that it just becomes secant.
4. Feb 16, 2009
### Tom Mattson
Staff Emeritus
But you've forgotten about the $dx$.
5. Feb 16, 2009
### negatifzeo
Oh, duh. Wait, why am I having a hard time remembering how to get the dx here? Does dx here equal 1/2*sec^2(theta) d(theta)?
6. Feb 16, 2009
### Tom Mattson
Staff Emeritus
Yes, that's right. | {"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": 0.9227173924446106, "perplexity": 2520.116357491898}, "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-2018-43/segments/1539583514708.24/warc/CC-MAIN-20181022050544-20181022072044-00380.warc.gz"} |
https://math.stackexchange.com/questions/2896147/dissecting-the-complexity-of-prime-numbers | # Dissecting the complexity of prime numbers
Each prime number greater than $9$, written in base $10$, ends with one of the four digits $1,3,7,9$. Therefore, each ten can be classified according to which of these four digits, summed to the ten, yields to a prime number.
For example, for the first ten we have $1 \rightarrow \{1,3,7,9\}$. In fact, $10+1$, $10+3$, $10+7$ and $10+9$ are all primes. Conversely, for the twentieth ten the association reads $20 \rightarrow \{\}$, since there are no primes between $200$ and $209$.
It is easy to see that each ten is associated to one (and only one) group of symbols, chosen among the following $16$ distinct alternatives: $\{\}$, $\{1\}$, $\{3\}$, $\{7\}$, $\{9\}$, $\{1,3\}$, $\{1,7\}$, $\{1,9\}$, $\{3,7\}$, $\{3,9\}$, $\{7,9\}$, $\{1,3,7\}$, $\{1,3,9\}$, $\{1,7,9\}$, $\{3,7,9\}$, $\{1,3,7,9\}$.
For the sake of simplicity, we can identify each of these $16$ distinct groups of symbols with a single symbol, or with a single color, as illustrated below:
Each of these colors represents how many prime numbers there are in one ten (and which ones). In practice, we have just split the complexity of primes into tens and colors.
This allows us to rearrange the colors within the Pascal's triangle, by means of the associated ten, obtaining the following scheme (the numbers in the squares represent the tens):
The complexity of the primes sequence has been now split into rows, diagonals and colors.
An advantage of such representation is that it mixes groups of primes related to far tens, allowing maybe to identify patterns and/or to dig out connections among already known integer sequences. The clear disadvantage is that the patterns on this triangle depend on which base we use.
I partially introduced this representation of prime numbers here, but I am not sure whether these further developments overlap with some very well known technique (e.g. Sieve of Eratosthenes?).
In conclusion, just not to re-invent the wheel, my question is:
Do you know if such representation has been already devised? In that case, could you please give me some reference?
Sorry for naivety and incorrectness, and thank you very much for your suggestions and comments!
NOTE: The conjecture contained in the following EDIT is FALSE, and there is a mistake in the code! Please, if you have a good software to produce the picture, please tell me! Thanks! (Thanks also to Paul!)
I produced the following plot (omitting the first ten)
I hope that there is not some mistake in my code! However,
It is conjectural that, for very big numbers, there cannot be colored squares other than on the outer edge of the triangle,
which means that, beyond a certain integer $N$, the primes will all fall in tens (powers of ten) that can be written in the form $t=\binom{n}{k}$, where $k$ is $2,3,4$ maximum. This may lead to interesting consequences, considering that this property should not change much according to the base, and primes are infinite.
As mentioned in the NOTE, this picture is actually incorrect. It should look like this:
Still working on it!
• The most interesting thing is, it looks symmetric! – tarit goswami Aug 27 '18 at 12:58
• @taritgoswami Well, this is a property of Pascal's triangle. – user559615 Aug 27 '18 at 13:00
• When there are no primes in a block of $10$, like between $200$ and $209$, you should denote this $20 \to \{\}$, not $20 \to \{0\}$. The notation you use suggests that $200$ is prime. The former notation, $20 \to \{\}$ correctly suggests that the set of primes is the empty set, i.e., there are no primes. – 6005 Aug 27 '18 at 19:52
• @AndreaPrunotto: This is uncannily complete in many languages: rosettacode.org/wiki/Ulam_spiral_(for_primes) So instead of highlighting primes, highlight the ten's with your gradations. – Alex R. Aug 28 '18 at 18:30
• @AndreaPrunotto I tried to reproduce the image and I don't have the big white area. I think your code doesn't handle big numbers (numbers above $2^{31}-1$, the maximum value of a signed 32-bit integer) well. Which programming language are you using? You must use some big number library, which may be already available in the language you're using, or otherwise you can probably find it online somewhere. – Paul Aug 28 '18 at 21:19 | {"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": 0.905180811882019, "perplexity": 381.8586820274679}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-39/segments/1568514572934.73/warc/CC-MAIN-20190916200355-20190916222355-00436.warc.gz"} |
https://blockchain-techminers.com/blog/article.php?page=0ed5b8-how-to-calculate-gamma-function | # how to calculate gamma function
Assume if the number is a ‘s’ and also it is a positive integer, then the gamma function will be the factorial of the number. It is to be noted that the value of gamma approaches zero as the option goes either deeper into the money or deeper out of the money. In mathematics, the Gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers. It is represented by S. Step 2: Next, determine the strike price of the underlying asset from the details of the option. Using techniques of integration, it can be shown that Γ(1) = 1. Similarly, using a technique from calculus known as integration by parts, it can be proved that the gamma function has the following recursive property: if x > 0, then Γ(x + 1) = xΓ(x). By signing up for this email, you are agreeing to news, offers, and information from Encyclopaedia Britannica. Mathematically, the gamma function formula of an underlying asset is represented as. (a) Prove that Gamma(x + 1) = x Gamma (x) for all positive x. CFA® And Chartered Financial Analyst® Are Registered Trademarks Owned By CFA Institute.Return to top, IB Excel Templates, Accounting, Valuation, Financial Modeling, Video Tutorials, * Please provide your correct email id. )}{s^{(n+1)}} This article was most recently revised and updated by, https://www.britannica.com/science/gamma-function. For large arguments the GammaLn function can be used. | {{course.flashcardSetCount}} Use the method of substitution and the Gamma function to evaluate the following integral. You can see that when n = 1, we get the square root of π. Be on the lookout for your Britannica newsletter to get trusted stories delivered right to your inbox. Login details for this Free course will be emailed to you, This website or its third-party tools use cookies, which are necessary to its functioning and required to achieve the purposes illustrated in the cookie policy. While you learn integration you'll work with two main types of integrals, definite integrals and indefinite integrals. This method is much nicer to work with than the Euler integral when possible, because it's generally quicker to solve a simple factorial than an integral. Finally, we complete the proof by substituting in the results for Γ(1) into the Γ(n) formula. The Gamma Function Calculator is used to calculate the Gamma function Γ(x) of a given positive number x. Gamma Function. for every n \in N. 1)Evaluate \int_0^\infty x^4e^{-x/2}dx 2) Evaluate \int_0^\infty x^2e^{-\sqrt{x}/2}dx 3) Show \Gamma(1/2) = \sqrt{\pi} }] Hint: use1= \int_\infty^\infty \phi(z)dz. Third, the double integral is evaluated by While it's standard to define the gamma function in integral form by Euler's integral of the second kind, it can also be viewed as an extension of the factorial function when n is a positive integer. Praxis Psychology (5391): Practice & Study Guide, DSST Physical Geology: Study Guide & Test Prep, Common Core ELA Grade 8 - Writing: Standards, American Imperialism (1890-1919) Lesson Plans, Quiz & Worksheet - Importance of Mitotic Cell Division, Quiz & Worksheet - Figurative Language in I Have a Dream, Quiz & Worksheet - Happy in Death of a Salesman, Charley in Death of a Salesman: Character Analysis, Research-Based Instructional Strategies for Math, Texas Teacher Certification Test Limit Waiver, Tech and Engineering - Questions & Answers, Health and Medicine - Questions & Answers, For an integer n greater than or equal to 1. It's very common to see upper limits of integration set to infinity, and lower limits set to negative infinity. $\begingroup$ If what you desire are closed forms, Wikipedia mentions the existence of such are known, though they are in terms complete elliptic integrals of the first kind. While you learn integration you'll work with two main types of integrals, definite integrals and indefinite integrals. The integral representation of the gamma function is as follows where x > 0. Create your account, Already registered? [1] 2020/07/21 10:28 Male / 50 years old level / A teacher / A researcher / Useful /, [2] 2020/06/11 04:26 Male / Under 20 years old / High-school/ University/ Grad student / Very /, [3] 2020/05/19 19:50 Male / Under 20 years old / High-school/ University/ Grad student / Not at All /, [4] 2020/05/19 16:55 - / 60 years old level or over / High-school/ University/ Grad student / Very /, [5] 2020/05/07 06:40 Male / 20 years old level / High-school/ University/ Grad student / Very /, [6] 2020/04/20 20:14 Female / Under 20 years old / High-school/ University/ Grad student / Very /, [7] 2020/04/16 02:20 Male / 20 years old level / High-school/ University/ Grad student / Useful /, [8] 2020/04/15 21:58 Male / 20 years old level / High-school/ University/ Grad student / Useful /, [9] 2020/04/07 17:54 Female / 20 years old level / High-school/ University/ Grad student / Very /, [10] 2020/03/15 19:38 Male / Under 20 years old / High-school/ University/ Grad student / Useful /. First is Euler's reflection formula. In this lesson we'll look at the properties of one famous function defined by an improper integral known as the gamma function. One important example of an improper integral is Euler's integral of the second kind; otherwise known as the gamma function. …scale, respectively, applied to the gamma function. Gamma can be expressed as the second derivative of the premium of the option with respect to the price of the underlying asset. The reason why this is true is a direct result of the formula for the gamma function. Along with the integral representation, the gamma function can also be represented in factorial form when n is a positive integer. Finally, the last property we'll cover is known as the duplication formula. It is that the following is true for n > 1. imaginable degree, area of CFA Institute Does Not Endorse, Promote, Or Warrant The Accuracy Or Quality Of WallStreetMojo. Then dive deeper into the gamma function's properties by looking at several examples of them. Did you know… We have over 220 college We can see an example of this property working through a simple comparison of it to the standard factorial form of the gamma function with n = 4. As the name implies, there is also a Euler's integral of the first kind. | 18 Gamma(n) is defined as: When x is a real number Gamma(x) is defined by the integral: If n is larger than 171.62, the function overflows and returns an error. Before we do this, there are a few things from calculus that we must know, such as how to integrate a type I improper integral, and that e is a mathematical constant. credit by exam that is accepted by over 1,500 colleges and universities. First we find, The Gamma function is denoted by \Gamma (p) and is defined by the integral: \Gamma(p + 1)= \int_0^\infty e^{-x} x^p dx The integral converges as x \rightarrow \infty for all p. For p < 0 is it imprope, The gamma function is denoted by Gamma (p) and is defined by the integral Gamma (p + 1) = integral^infinity_0 e^{-x} x^p dx. Many times the gamma functions show up behind the scenes. One way to answer this question is by looking at several sample calculations with the gamma function. This is found by setting z = 1 in the above formula: We calculate the above integral in two steps: The next example calculation that we will consider is similar to the last example, but we increase the value of z by 1. For x > 0, the Gamma function Γ(x) is defined as: The Gamma function is important in pure and applied mathematics, science, and engineering, including applications involving heat conduction in lasers and human tissues. The next three properties will show the gamma function operating on fractions and imaginary numbers, starting with the property that tells us what we get when solving for Γ(1/2). Formula for the calculation of an option's gamma. These come up within the gamma and beta distributions that you'll work with often there. Let’s calculate Γ(4.8) using a calculator that is implemented already. study An example of where you might see both the gamma and beta functions is within the field of statistics. Any definite integral tha… \Gamma(x)= \int_{0}^{\infty} t^{x - 1} e^{-t}dt a) Show that \Gamma(x) = (x - 1)\Gamma(x - 1). You can download this Gamma Function Formula Excel Template from here – Gamma Function Formula Excel Template, This has been a guide to Gamma of an Option and its definition. Gamma is the amplitude of the change of an option's delta subsequently to a change in the price of the option's underlying. Our editors will review what you’ve submitted and determine whether to revise the article. When = 1 2, 1 2 simpli es as 1 2 = 2 Z 1 0 e u2du To derive the value for 1 2, the following steps are used. Gamma distributions occur frequently in models used in engineering (such as time to failure of equipment and load levels for telecommunication services), meteorology (rainfall), and business (insurance claims and loan defaults) for which the variables are always positive and the results are skewed….
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http://mathhelpforum.com/advanced-statistics/87387-obtain-maximum-likelihoob-estimate-average-arrival-rate.html | # Math Help - Obtain the maximum likelihoob estimate of the average arrival rate
1. ## Obtain the maximum likelihoob estimate of the average arrival rate
It is assumed that the arrival of the number of calls X per hour follows Poisson distribution with parameter $\lambda$. A random sample $X_1 \; = \; x_1$ , $X_2 \; = \; x_2$ , ... $X_n \; = \; x_n$ is taken . Obtain the maximum likelihood estimate of the average arrival rate.
2. Originally Posted by Stats
It is assumed that the arrival of the number of calls X per hour follows Poisson distribution with parameter $\lambda$. A random sample $X_1 \; = \; x_1$ , $X_2 \; = \; x_2$ , ... $X_n \; = \; x_n$ is taken . Obtain the maximum likelihood estimate of the average arrival rate.
Can you construct the likelihood function? | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 8, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9991360306739807, "perplexity": 372.44331255840757}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701146241.46/warc/CC-MAIN-20160205193906-00226-ip-10-236-182-209.ec2.internal.warc.gz"} |
https://verification.asmedigitalcollection.asme.org/ICNMM/proceedings-abstract/ICNMM2007/4272X/41/318413?searchresult=1 | Effects of flow fluctuation at the test section inlet on flow boiling heat transfer were investigated for FC72 by the use of a horizontal circular tube with a diameter of 0.51mm. Flow fluctuation was minimized by employing a high-powered syringe pump in one experiment, while it was intensified in another experiment by the connection of an auxiliary tank exposed to atmosphere allowing the reverse flow. In the experiments of strict inlet flow rate regulation, heat transfer characteristics were similar to those observed in normal size tubes. But if the flow rate was fluctuated up to ± 20% of the total under the weak inlet flow rate regulation, the trend that the heat transfer coefficient was increased with increasing vapor quality in moderate vapor quality region was disappeared and heat transfer deterioration due to partial dryout started at lower vapor quality. Boiling heat transfer characteristics in minichannels could be changed considerably by the existence of flow fluctuation caused by the rapid axial growth of elongated bubbles at low vapor quality. The regulation of inlet flow rate seems to be one of key parameters to reduce the scattering in heat transfer data encountered in flow boiling of minichannels. Experimental data obtained by using a pump of weak power or data using a liquid reservoir can never be inherently consistent with those obtained for the constant inlet flow rate conditions. Although such weak restriction of inlet conditions are actually encountered in application systems, the difference in heat transfer characteristics between the normal and mini tubes should be clarified, as the first objective of the research, under the same inlet conditions without flow fluctuation.
This content is only available via PDF. | {"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": 0.9294742345809937, "perplexity": 1317.9257983484472}, "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/1656103943339.53/warc/CC-MAIN-20220701155803-20220701185803-00330.warc.gz"} |
http://math.stackexchange.com/questions/280481/proof-of-the-identity-sum-k-0-minp-qp-choose-kq-choose-knk-choose | # Proof of the identity $\sum_{k=0}^{\min[p,q]}{p\choose k}{q\choose k}{n+k\choose p+q}={n\choose p}{n\choose q}$
Prove the identity: $$\sum_{k=0}^{\min[p,q]}{p\choose k}{q\choose k}{n+k\choose p+q}={n\choose p}{n\choose q}.$$
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@Euler....IS_ALIVE I believe OP is talking about the error message one receives for posting a very short question. – Austin Mohr Jan 17 at 3:46
It does fail to meet our following quality suggestions: You seem to be new to math.stackexchange. Welcome! (1) Please refrain from adding extraneous tags. This question has nothing to do with multinomial coefficients. If this is a homework problem, then you should also add the [homework] tag. (2) It would be nice of you to put questions in full sentence form, and to avoid using the imperative mood when asking questions. (3) It is proper stackexchange etiquette to describe what serious attempts you have made to answer your question. Please read the faq (upper right corner) for details. – proximal Jan 17 at 3:51
@JavaMan: Definitely. Or $q \leq p$. – gnometorule Jan 17 at 4:27
@JavaMan: I only just now see the $p$ as a summation index too, and am looking for a brown bag. – gnometorule Jan 17 at 4:30
$p\le q$ or $q\le p$ wasn't important! – user58554 Jan 17 at 4:45
\begin{align} \sum_k\binom{p}{k}\binom{q}{k}\binom{n+k}{p+q} &=\sum_{j,k}\binom{p}{k}\binom{q}{k}\binom{n}{p+q-j}\binom{k}{j}\tag{1}\\ &=\sum_{j,k}\binom{p}{k}\binom{n}{p+q-j}\binom{q}{j}\binom{q-j}{q-k}\tag{2}\\ &=\sum_{j}\binom{p+q-j}{q}\binom{n}{p+q-j}\binom{q}{j}\tag{3}\\ &=\sum_{j}\binom{n-q}{n-p-q+j}\binom{n}{q}\binom{q}{j}\tag{4}\\ &=\sum_{j}\binom{n-q}{p-j}\binom{n}{q}\binom{q}{j}\tag{5}\\ &=\binom{n}{p}\binom{n}{q}\tag{6} \end{align} Explanation
$(1)$ $\displaystyle\binom{n+k}{p+q}=\sum_j\binom{n}{p+q-j}\binom{k}{j}$
$(2)$ $\displaystyle\binom{q}{k}\binom{k}{j}=\binom{q}{j}\binom{q-j}{q-k}$
$(3)$ $\displaystyle\sum_k\binom{p}{k}\binom{q-j}{q-k}=\binom{p+q-j}{q}$
$(4)$ $\displaystyle\binom{p+q-j}{q}\binom{n}{p+q-j}=\binom{n-q}{n-p-q+j}\binom{n}{q}$
$(5)$ $\displaystyle\binom{n-q}{n-p-q+j}=\binom{n-q}{p-j}$
$(6)$ $\displaystyle\sum_j\binom{n-q}{p-j}\binom{q}{j}=\binom{n}{p}$
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+1: This was the same approach I was trying to use, but I made an error when applying Identity (2) and got myself into a mess that didn't simplify. – Mike Spivey Jan 17 at 17:46
Very nice. (This one word "Explanation" sort of spoils the fun of an answer containing "only" a calculation. Wouldn't a horizontal line do?) – Martin Jan 17 at 19:20
Since $$[x^i] (1-x)^{-(r+1)} =\binom{r+i}{i}=\binom{r+i}{r},\qquad (*)$$ we can write $$\binom{n}{p} \binom{n}{q} = [x^{n-p}y^{n-q}] (1-x)^{-(p+1)} (1-y)^{-(q+1)}.$$ This can be rewritten as a complex integral $$\frac{1}{(2\pi i)^2} \int x^{p-n-1} y^{q-n-1} (1-x)^{-(p+1)} (1-y)^{-(q+1)} dx dy,$$ where $x$ and $y$ both traverse small counterclockwise circles around the origin. Now, make the substitution $$x = z\frac{1+w}{1+z}, \qquad y=w\frac{1+z}{1+w}.$$ Since $$dx \wedge dy = \frac{1-zw}{(1+z)(1+w)} dz \wedge dw$$ this changes the integral to $$\frac{1}{(2\pi i)^2} \int z^{p-n-1} w^{q-n-1} (1+w)^p (1+z)^q (1-zw)^{-(p+q+1)} dz\, dw.$$ On the surface of integration, $z$ and $w$ will now remain in small annuli around the origin. Remaining inside the region where the integrand is holomorphic, we can deform the surface of integration until $z$ and $w$ move counterclockwise around small circles around the origin. This does not change the value of the integral, so it will equal $$[z^{n-p} w^{n-q}] (1+w)^p (1+z)^q (1-zw)^{-(p+q+1)}.$$ Using (*) and the binomial theorem, this is equal to $$\sum_{\ell\in{\Bbb Z}} \binom{p}{n-q-\ell} \binom{q}{n-p-\ell} \binom{p+q+\ell}{p+q},$$ where the binomial coefficient $\binom{i}{j}$ is taken to vanish if $j$ is not in $\{0,1,\dots,i\}$. Substituting $\ell:=n-p-q+k$ now gives the desired result.
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Nice contour integration (+1). I've rarely, if ever, used contour integration to prove a binomial identity. I tried multiplying both sides by $x^py^q$, summing, and comparing the coefficients in $(1+x)^n(1+y)^n=(1+x+y+xy)^n$, but couldn't get it to work. – robjohn♦ Jan 17 at 15:51 | {"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": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9289059042930603, "perplexity": 515.8245871160964}, "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-2013-20/segments/1368703057881/warc/CC-MAIN-20130516111737-00087-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/molar-mass-of-acetone.233989/ | Molar mass of acetone
• Start date
• #1
Hammertong
2
0
A 125mL flask contains 0.239g of acetone vapor at 766mm Hg and 100*C. Calculate the molar mass of acetone.
766x125/373Kx273K/760=92.210mL
.239g/92.210/1000=2.592g/mL
Where am I going wrong? I have no idea if I'm anywhere close.
• #2
Staff Emeritus
Homework Helper
12,151
170
Those equations aren't really what we want.
Hint: molar mass is the number of grams per moles.
• #3
Hammertong
2
0
I know what molar mass is, but I don't understand how to get it from what I'm given. My teacher goes over the equations once then moves on. Then he expects us to be able to do it backwards, forwards, and sideways.
I figured it out, I just needed to multiply my end answer by 22.4L to get weight of one mole at STP.
Last edited:
• #4
Homework Helper
Gold Member
6,979
1,596
Not sure anymore how all the units work; "R" is the complicated part, but the question seems like an ideal gas type. PV=nRT. You can determine the number of moles from that, "n". Next, you compare the number of moles to the number of grams of mass, which you are given.
• #5
Homework Helper
Gold Member
3,961
1,003
These things are basically simple proportions.
A mole as you say occupies 22.4 l under standard conditions (still called STP or NTP?) . Got a texbook to check out how those are defined?
The excercise is not under standard conditions, but a higher temperature.
Again it is simple proportions - you have to know what absolute temperature means.
You do not need to use explicitly that equation - if you use the Charles temperature proportionality law you are using it implicitly anyway .
You can actually work out R from the above information.
Hope these thing become clear and simple.
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http://mathhelpforum.com/calculus/21702-one-more-system.html | ## One More System
Determine the general solution of the following system:
$\frac{dx}{dt} = 2x + y + 2z$
$\frac{dy}{dt} = 3x + 6z$
$\frac{dz}{dt} = -4x - 3z$ | {"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": 0.874957799911499, "perplexity": 2494.5611664358803}, "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-2014-41/segments/1410657136963.94/warc/CC-MAIN-20140914011216-00086-ip-10-234-18-248.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/437/sums-of-cubes-and-more?sort=votes | # Sums of cubes and more
It's well-known that every natural number can be written as a sum of 4 squares of integers.
Has there been any recent progress about the similar problem for the cubes, 4-th powers and so on? I believe this was proven to be representable using some N — that depends on the power — and what's the story about it?
-
Arguably, the problem "what is the least g=g(k) such that every integer can be written as the sum of g k-th powers" is less interesting than the version that ignores the random stuff that's happening with a finite number of special cases.
Namely, the "real" question should be "what is the least G=G(k) such that for some N, every integer greater than N can be represented as the sum of G k-th powers".
For example, every number is the sum of 19 4th powers, but every number greater than 13,792 is actually the sum of just 16 4th powers. The "16" was known for quite some time; the verification that 13,792 is the last offender is quite recent (I found the value on Wikipedia, btw).
Evaluating G(k) is harder than evaluating g(k), and most of the actual values are still not known. I don't think there was tremendous recent progress on this front, although there certainly is progress on things like bounds, number of representations, etc.
You should look at Wooley's survey here (I haven't read it yet).
-
Indeed, 16 is more interesting, especially the probability theory says the bound could be 5. – Ilya Nikokoshev Oct 14 '09 at 4:55
I don't know, but I attended a talk by John Conway on this sort of stuff at Berkeley last year. Unfortunately I didn't take notes, but below is the abstract from the talk.
I’m rather proud of having proved the “Fifteen Theorem,” according to which one can check that a positive definite quadratic form with integral matrix represents all positive integers provided only that it represents those up to 15. I’m almost as proud of having made the “290 Conjecture,” now proved by Bhargava and Hanke, that performs the similar service for integer valued forms. Why? – because together these two theorems end more than 200 years of work on universal quadratic forms, the first theorem in that subject being Lagrange’s 1770 Four Squares Theorem.
It’s now time to try to do the same for near-universality of three-dimensional forms, which subject was begun by Legendre’s much subtler Three Squares Theorem of 1798.
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This is called "Waring's problem."
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https://www.physicsforums.com/threads/complementarity-of-spin.638070/ | # Complementarity of Spin?
1. Sep 22, 2012
I’m a layman who has reviewed some popular literature on QM. Apologies in advance for my naivety.
I’m of the impression that the quantum spin (±½) of an electron can be determined as it travels through a Stern-Gerlach apparatus (SGA) for the single direction in which the SGA is aligned (say, along the X-axis). Spin along other axes is a superposition of those states, with amplitudes corresponding to the angle departing from X. Spin is also said to be complementary. Spin measurement along different axes is mutually exclusive. For example, “A particle cannot have definite values for both X and Z at the same time.[ref]
It seems possible, at least in theory, to obtain values for both X and Z spin at the same time by the following method. Please let me know where it goes wrong.
1). Two electrons are prepared in an entangled, total-spin-zero state (each has spin opposite the other).
2). Two SGAs are oriented perpendicularly, one X-axis aligned and the other Z-axis aligned.
3). Each entangled electron is sent simultaneously through one of the perpendicular SGAs.
4). Spin X is determined for one electron while spin Z is determined for the other, at the same time.
5). Each electron had spin opposite the other, so spin X and spin Z of both electrons were simultaneously determined.
Last edited: Sep 23, 2012
2. Sep 23, 2012
### Dali
You are opening up an interesting track here... This is more or less exactly the same reasoning as in the original EPR-setup. Einstein, Podolsky and Rosen then used it to argue that QM had to be incomplete because it seemed possible to simultaneously measure definite values of two non-commuting observables (which should not be possible according to QM as you point out).
Your setup is also very close to the original setup proposed by J.S. Bell when deriving his inequalities. Thanks to Bell and later experiments confirming the strangness of entanglement, we now know what is wrong in the argument.
You (and EPR!) assume that the measurement of one particle does not change the state of the other particle. This does seem very reasonably indeed, but is not true for entangled particles! This is precisely the strange "spooky action at a distance"-behavior of entanglement that is clearly proven in the many experimental verifications of Bells theorem.
3. Sep 23, 2012
I admit that I was rooting for EPR in the chapters leading up to Bell’s theorem (which I am now reviewing again.) Initially, I was curious whether spin complementarity was truly a property of the particle or simply a limitation of the measuring device (SGA). But as the given quote, “A particle cannot have definite values for both X and Z [spin] at the same time.[ref] indicates, QM insists that complementarity is a property of the particle.
Direct simultaneous measurements of a single particle may indeed be impossible. However, two particles sharing a single quantum state would seem to offer the opportunity to examine that state in different (even complementary) ways at once.
Last edited: Sep 23, 2012
4. Sep 23, 2012
### Bill_K
Your reasoning founders on the use of the word "simultaneous".
4a) Suppose spin X is determined for particle A first. This determines the spin X for particle B. Then spin Z for particle B is measured. Particle A is not influenced by this. The two particles wind up with independent values.
4b) Suppose spin Z is determined for particle B first. This determines the spin Z for particle A. Then spin X for particle A is measured. Particle B is not influenced by this. The two particles wind up with independent values.
Same results, you can't tell 4a from 4b. And either way, you have independent values, no entanglement, no contradiction.
5. Sep 23, 2012
Are you saying that complementarity prohibits "simultaneous" measurement of two different (though entangled) particles?
I didn’t propose measuring one electron spin first, then the other. They are to each proceed through a separate Stern-Gerlach apparatus for simultaneous X-spin and Z-spin assessments.
Is that not even theoretically possible? What mechanism prevents it?
Last edited: Sep 23, 2012
6. Sep 23, 2012
### Bill_K
No, I am saying that you are confusing yourself by imagining measurements at two different locations to be precisely simultaneous. The term has no operational meaning. The slightest departure from simultaneity in either direction, even one attosecond, spoils the paradox.
7. Sep 23, 2012
Then we might as well drop the term “simultaneous” from our language. But I said, “theoretically possible”, not operationally possible.
Special Relativity makes clear that for any pair of events deemed simultaneous in one inertial reference frame, there are an infinite number of other frames in which they are not simultaneous. Conversely, if we find that two spin measurements are not quite simultaneous in our reference frame, we can be assured that there always exist inertial frames in which they are. Simultaneity is guaranteed, both theoretically and operationally, in this sense.
Last edited: Sep 23, 2012
8. Sep 23, 2012
### Bill_K
No, special relativity guarantees lack of simultaneity. Any argument that depends on the attainment of precise simultaneity fails, because the events are nonsimultaneous in all other rest frames.
However I assume we are talking about Schrodinger quantum mechanics, which is nonrelativistic. If you perform the stated measurements on particles A and B, and find that particle A has a definite value of Z spin, while particle B has a definite value of X spin, how can you claim that a basic principle has been violated?
Answer: you would have to show that the measurements took place at precisely the same time. PRECISELY the same time. There is no way to do this. Measurement of a time interval can only be done to some finite precision, and regardless of whether it is an attosecond or a millionth of an attosecond (whatever that is called!) it cannot be zero. And if it is not zero, quantum mechanics is saved!
9. Sep 23, 2012
Thanks Bill, but I don’t think QM has much to fear from me!
I would suggest that for any two separate events in space-time (not having an event horizon between them), there exists a set of locations having equal intervals from both events. Observers occupying these will interpret the two events as simultaneous. However, I concede that QM is often claimed to be non-relativistic. (Notwithstanding that the Dirac equation improved upon Schrodinger’s by applying relativistic considerations to the electron.)
It seems inconsistent to, on the one hand deny simultaneity (of measurements) and on the other hand, require it. What is entanglement if not particles sharing a state simultaneously?
Last edited: Sep 23, 2012
10. Sep 24, 2012
Entanglement is still fairly mysterious. Perhaps it was not the best initial approach to the complementarity of quantum spin (a.k.a. intrinsic spin). Switching to a single particle would avoid the relativity of simultaneity, for now. My question might be restated this way (pertaining to an electron):
Quantum spin is not “classical” spin. The same spin magnitude is always observed on every axis measured. If no device can measure quantum spin on more than a single axis at a time, how can a claim that full quantum spin occurs on every observable axis at once, be refuted?
11. Sep 24, 2012
### DrChinese
You can perform this experiment, and simultaneity is irrelevant to the results. You now know Alice's X and Bob's Z. Cool.
Now the \$64,000 question is whether you also know Alice's Z and Bob's X. The answer to that question is NO. And it would need to be YES for your idea to be useful. Learning Alice's X resets her Y and Z to indeterminate values. So you have accomplished nothing. It is simply a philosophical exercise to assert you know both Alice's X and Z components when no experiment can confirm that. (And further that experiments will show the Z to match the "expected" value only randomly.)
12. Sep 24, 2012
### Bill_K
Thank you, DrChinese. This, believe it or not, is what I was struggling to say. The result is the same regardless of the time sequence.
13. Sep 24, 2012
### Dali
So, also just elaborating on DrChinese's answer above, the argument in the original post goes wrong at:
This is not true, as DrChinese says. But I think the confusion here is about the non-local aspects of entanglement. In the original EPR paper they argued (just as you do!) that since a measurement at Alice's place could not possibly affect what they called "the physical reality" at Bob's place, Alice's X-measurement should also let us know Bob's X. But it is exactly this assumption - that a Alice measurement does not affect outcomes at Bill's - that leads to the various Bells inequalities. And they are clearly violated by QM predictions and in real experiments. So the (strange!) conclusion is that we can't assume that, and hence Alice's X-measurement does not determine Bob's X.
Entanglement is difficult to grasp. It is not really two particles "sharing a state", but rather a two-particle system that can only be viewed as one single QM system. In the case of two spin-half particles, we have to view them as one single 4-state system. (Which actually has one more degree of freedom than two separate 2-state systems!) And all predictions about measurements on the two individual particles and their correlations are independent of the order or simultaneousness in which measurements are done.
You have to be carefull with the word "occurs" here. If there is no way to measure it, how could you conclude that "it" occurs? I would say that it is the very statement that there is no conceivable measurement device that could (even in principle) measure more then one spin component at a time that is the reason for refuting the idea of simultaneous spin components of a single particle.
14. Sep 24, 2012
### DrChinese
Yes, this part is hard. Made extra hard because we don't at all understand the process by which the 2 individual particles emerge from a 2 particle system. When and how do the 2 individual particles appear? There is no single answer that is really completely satisfying.
15. Sep 24, 2012
### DrChinese
Your words were good, thought I might put a different way to help the OP follow the line of reasoning provided. Sometimes a reader picks up something said one way, while another picks up on a different analogy.
16. Sep 24, 2012
Thanks for the clarification.
It seems the very essence of entanglement, that learning Alice’s Z instantly gives us Bob’s entangled Z. (If not, what is it that we are claiming is entangled in a "total-spin-zero" state?) Conversely, learning Bob’s X instantly gives us Alice’s entangled X. And you have graciously conceded the simultaneity of these! With all due respect (having read several of your other, quite helpful posts), all caps doesn’t make “NO” correct.
Now that’s “irrelevant”. In the moment before resetting*, we knew both X and Z, for Alice and Bob, at once.
*Do we actually believe “resetting” is a real physical process (as opposed to a model)?
Exactly! It seems that the Principle of Complementarity boils down to denial of falsifiablity. That’s a serious shortcoming for any model.
QM asserts that “A particle cannot have definite values for both X and Z at the same time.”[ref]. The Principle of Complementarity conveniently prevents us from disproving this.
Under the same guise, it may be irrefutably claimed that quantum spin occurs in all observable directions at once. No matter which direction is measured, full spin is found.
Last edited: Sep 24, 2012
17. Sep 24, 2012
Thanks for your trouble Dali. That’s why, in post #10 and the end of #16, I'm focusing instead on a single particle. After I reread Bell’s inequality about six more times, I’m sure I’ll have something to say about it in a separate thread. As you have guessed, despite overwhelming verification of Bell, I’m still rooting for EPR! And yet I’m not denying QM, I just feel it's still growing.
The evidence is as I say in post #16, “No matter which direction is measured, full spin is found.” It has already been established that quantum spin is not “classical” spin, so we should expect something to be different.
18. Sep 24, 2012
### DrChinese
Ah, sorry, you are making basic mistake after basic mistake. You should follow the entire line of reasoning before making statements like the above.
QM does not assert something which is not falsifiable here, you are. QM states that you cannot predict both Alice's X and Z components simultaneously. Therefore, QM does not assert they are both simultaneously real. It is the realist (that would be you) who asserts this, and then points out that QM does not support it - which is somewhat humorous you must admit. It is you who is asserting something which cannot be demonstrated, that you DO know both Alice's X and Z. And yet, a subsequent experiment will show one of those to be the same and the other to only randomly correlate. Not really the same thing now, is it?
At any rate, the concept that you know Alice's X and Z was precisely the ingredient which makes Bell's Theorem. Extending your idea that you know X and Z, you get to the idea that Y exists too. Which ultimately leads to the Bell contradiction. So the point is, your starting position is that of EPR (where there was a disagreement about the simultaneity issue, as Bill_K referenced earlier). But we now know EPR is demonstrably incorrect.
A little late for you to be rooting for EPR! They didn't have the benefit, as you do, of knowing about Bell. Rosen (the R in EPR) lived long enough to learn of Bell, so his position naturally changed. Sadly, E and P did not.
Last edited: Sep 24, 2012
19. Sep 25, 2012
That’s what beginners do! I find it preferable to not beginning.
By “rooting for EPR”, I mean that at my early stage of learning, and presuming Bell (thus, Bohr) to be correct, I adopt the EPR position as strongly as I can, in order to force the most out of Bell’s argument. Clearly, I’m not there yet.
That said, I would point out, “[Bell’s] inequality is experimentally testable, and there have been numerous relevant experiments,… They have all shown agreement with quantum mechanics rather than the principle of local realism. However, the issue is not finally settled, for each of these experimental tests has left open at least one loophole by which it is possible to question the validity of the results.[ref]
I hope you'll try to understand why a beginner might feel that QM implies simultaneous spin about all observable axes.
I found, “One consequence of the generalized uncertainty principle is that the spin projection operators (which measure the spin along a given direction like x, y, or z), cannot be measured simultaneously. Physically, this means that it is ill defined what axis a particle is spinning about. A measurement of the z-component of spin destroys any information about the x and y components that might previously have been obtained.[ref]
It speaks of an “ill defined” axis which seems akin to the ill-defined position of an electron occupying an orbital. We're taught to consider the electron as spread out over the probability cloud, effectively occupying a set of locations simultaneously.
Further, it speaks of “spin components” about other axes. Though they “cannot be measured simultaneously”, multiple components are clearly implied to exist (about every observable axis).
I also found, “…Schrödinger equation, angular momentum is quantized… so that total spin angular momentum [is] √(3/2)ħ. However, … when the electron is observed along one axis, such as the Z-axis, is quantized in terms of a magnetic quantum number, which can be viewed as a quantization of a vector component of this total angular momentum, which can have only the values of ±½ħ.[ref]
The spin½ we refer to is treated as "a component" of some larger “total angular momentum”. I’m given the impression of a more classical total spin about some axis which distributes over all observable axes. Is that so unreasonable?
What I found was, “the spin projection operators (which measure the spin along a given direction like x, y, or z), cannot be measured simultaneously[ref]
Not really the same thing now, is it?
When I suggest the Principle of Complementarity seems to deny falsifiability, that applies equally to models where spin occurs about all observable axes or where spin occurs about any one measured axis. If we can’t measure two axes simultaneously, we can’t falsify either model.
Last edited: Sep 25, 2012 | {"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": 0.8785601854324341, "perplexity": 973.745621857874}, "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/1537267155702.33/warc/CC-MAIN-20180918205149-20180918225149-00053.warc.gz"} |
https://cs.stackexchange.com/questions/87812/comparison-between-ma-and-pnp | # Comparison between MA and $P^{NP}$
I am aware that $MA \subseteq ZPP^{NP} \subseteq \Sigma_2^{P}\cap \Pi_{2}^{P}$ and also $NP^{BPP} \subseteq MA \subseteq AM \subseteq \Pi_{2}^{P}$.
My question is: In this whole thing, where does $P^{NP}$ lie? (Obviously, $P^{NP} \subseteq ZPP^{NP}$) But, what about the relationship between $MA$ and $P^{NP}$? Which is contained in which one?
Intuitively, It is known that $MA_{EXP} \not\subset P/poly$, but for $EXP^{NP}$, no polynomial size circuit lower bound is known. Does that mean $P^{NP}$ is contained in $MA$?
Apparently, no relationship is known between $MA$ and $P^{NP}$. Even $BPP$ ($\subseteq MA$) vs $P^{NP}$ problem is still open. It is conjectured though that $MA = NP$ (Under standard hardness assumptions). So, it is likely that $MA \subseteq P^{NP}$. | {"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": 0.9527740478515625, "perplexity": 253.62870295916937}, "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-2021-39/segments/1631780060877.21/warc/CC-MAIN-20210928153533-20210928183533-00278.warc.gz"} |
http://mathhelpforum.com/differential-equations/72513-separable-differential-equation-problems.html | # Math Help - Separable Differential Equation problems
1. ## Separable Differential Equation problems
I'm having trouble with the two following separable differential equation equation problems. I'm able to separate the variables, and even do the individual integration (I think the left side in both cases has to be integrated using partial fractions. If there is a different way, I would appreciate any guidance/suggestions). I'm just having trouble getting from the results of the initial integration to the final solution of the differential equations. Any help would be appreciated. Thank you very much in advance.
1. Solve the given differential equation:
2. Solve the given initial-value problem:
,
2. Originally Posted by rgriss1
I'm having trouble with the two following separable differential equation equation problems. I'm able to separate the variables, and even do the individual integration (I think the left side in both cases has to be integrated using partial fractions. If there is a different way, I would appreciate any guidance/suggestions). I'm just having trouble getting from the results of the initial integration to the final solution of the differential equations. Any help would be appreciated. Thank you very much in advance.
1. Solve the given differential equation:
2. Solve the given initial-value problem:
,
It's not clear exactly what your difficulty is. Yes, those are separable equations:
The first is $\frac{dy}{y^2- 1}= \frac{dx}{x^3}$
or $\frac{dy}{(y-1)(y+1)}= x^{-3}dx$
Yes, the left side can be integrated using integration by parts.
The second is $\frac{dy}{y^2- 4}= dx$
or $\frac{dy}{(y-2)(y+2)}= dx$
with condition y(0)= -6
and the left side can be integrated using integration by parts.
Perhaps you have having trouble solving for y as a function of x? That normally is not required for problems like these but can be done. For example, integrating the second problem, you get (1/4)ln(|y- 2|)- (1/4)ln(|y+2|)= x+ C. Setting y= -6 when x= 0 gives (1/4)ln(8)- (1/4)ln(4)= (1/4)(ln(8)- ln(4))= (1/4)ln(8/4)= (1/4)ln(2)= C so we have (1/4)ln(|y-2|)- (1/4)ln(|y+2|)= x+ (1/4)ln(2). which we can write as ln|(y-2)/(y+2)|= 4x+ ln 2. Taking the exponential of both sides, |(y-2)/(y+2)|= $e^{4x+ ln 2}$= 2 $e^{4x}$. Since the initial value is at y= -6 the solution can only be valid for y< -2 which makes both numerator and denominator negative, so the fraction is positive, so we can remove the absolute values and solve $\frac{y-2}{y+2}= 2e^{4x}$ for y. | {"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": 7, "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": 0.9722508788108826, "perplexity": 376.57345051690817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931009179.34/warc/CC-MAIN-20141125155649-00136-ip-10-235-23-156.ec2.internal.warc.gz"} |
http://mathoverflow.net/questions/157587/2f0-sup-z-w-in-d-fz-fw-if-and-only-if-f-is-linear | # $2|f^{'}(0)| = \sup_{z, w \in D} |f(z)-f(w)|$ if and only if $f$ is linear
I know the following is a well-known result.
Let $D = B(0,1) \subset \mathbb{C}$ a disc, $f$ holomorphic on $D$. Show that $$2|f^{'}(0)| \le \sup_{z, w \in D} |f(z)-f(w)|$$ Furthermore, there is equality if and only if $f$ is linear.
I need some reference about the second part, i.e. there is equality if and only if $f$ is linear.
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Out of curiosity, would not it be simpler to give a direct proof to the fact rather than looking for the reference? – Alvin Feb 14 '14 at 20:39
@Alvin: if you know how to prove that equality if possible only for linear $f$, then tell us the proof. – Alexandre Eremenko Feb 14 '14 at 21:38
This was first proved by Landau and Toeplitz in 1907. A reference for the proof (and for generalizations) is the paper Area, capacity and diameter versions of Schwarz's lemma by Burckel, Marshall, Minda, Poggi-Corradini and Ransford.
See Theorem 1.3 here
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From the reference you give, I conclude that the statement about extremal functions was not in Landau and Toepliz paper; it is due to Gehring and Hag. Inequality itself has one-line proof (see my answer). – Alexandre Eremenko Feb 16 '14 at 21:17
@AlexandreEremenko : After Theorem 1.3 of the paper mentioned in my answer, it is written The main contribution of the Landau-Toeplitz paper is perhaps its elucidation of the extremal case. – Kalim Feb 17 '14 at 19:40
Unfortunately, I hardly read German:-( But anyway, we have several proofs of the extremal case now. – Alexandre Eremenko Feb 17 '14 at 20:56
Edit.
Let us first prove the inequality: $$\sup_{z,w}|f(z)-f(w)|\geq\sup_z|f(z)-f(-z)|=\sup|g(z)|\geq|g'(0)|=2|f'(0)|,$$ where the Schwarz Lemma was applied to $g(z)=f(z)-f(-z)$. Equality in Schwarz lemma can happen only if $g(z)=kz$, thus $f(z)=kz+\phi(z)$, where $\phi$ is even.
Now let us see when equality is possible in the first inequality. We must have $$\sup_{|z|<1,|w|<1}|k(z-w)+\phi(z)-\phi(w)|=2|k|,$$ which an even function $\phi$ analytic in the unit disc. We have to derive from here that $\phi$ is constant.
In the reference http://www.math.wustl.edu/~geknese/schwarzpoly.pdf, where extremal functions for the Schwarz lemma in the polydisc are described. But those functions are extremal at every point. And our function is extremal at only one point, the origin.
I can prove that $\phi$ is constant only under the additonal restriction that $\phi$ is differentiable in the closed disc. WLOG $k=1$. Let $|z|=1$ and put $w=-ze^{it}$, where $t$ is small. Then, neglecting the high powers of $t$, we must have $$|2+it-\phi'(z)it|\leq 2$$ This implies that $\phi'(z)$ must be real for all $z$ on the unit circle. But such function must be constant, and as $\phi$ is even, we conclude that $\phi$ is constant.
To get rid of the additional assumption of differentiability, one may combine this and and this.
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Why can you apply the Schwarz Lemma to $g(z)$ ? – user33122 Feb 15 '14 at 7:38
And I don't understand why $\sup_{|z|<1,|w|<1}|k(z-w)+\phi(z)-\phi(w)|\leq 2|k|$ – user33122 Feb 15 '14 at 7:41
1. Because $g$ is analytic and $g(0)=0$. 2. This is the statement that the first inequality in the first line becomes equality. – Alexandre Eremenko Feb 15 '14 at 14:44 | {"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": 0.9851559400558472, "perplexity": 286.1340688653974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860110805.57/warc/CC-MAIN-20160428161510-00154-ip-10-239-7-51.ec2.internal.warc.gz"} |
http://www.mathjournals.org/jot/2013-069-002/2013-069-002-008.html | Previous issue · Next issue · Most recent issue · All issues
# Journal of Operator Theory
Volume 69, Issue 2, Spring 2013 pp. 463-481.
Compact composition operators on the Hardy-Orlicz and weighted Bergman-Orlicz spaces on the ball
Authors Stephane Charpentier
Author institution: Departement de Mathematiques, Batiment 425, Universite Paris-Sud, F-91405, Orsay, France
Summary: Using recent characterizations of the compactness of composition operators on the Hardy--Orlicz and Bergman--Orlicz spaces on the ball \cite{Charp1}, \cite{Charp2}, we first show that a composition operator which is compact on every Hardy--Orlicz (or Bergman--Orlicz) space has to be compact on $H^{\infty}$. Then, although it is well-known that a map whose range is contained in some nice Kor\'anyi approach region induces a compact composition operator on $H^{p} (\mathbb{B}_{N} )$ or on $A_{\alpha}^{p} (\mathbb{B}_{N} )$, we prove that, for each Kor\'anyi region $\Gamma$, there exists a map $\phi:\mathbb{B}_{N} \to \Gamma$ such that $C_{\phi}$ is not compact on $H^{\psi} (\mathbb{B}_{N} )$, when $\psi$ grows fast. Finally, we extend (and simplify the proof of) a result by K. Zhu for the classical weighted Bergman spaces, by showing that, under reasonable conditions, a composition operator $C_{\phi}$ is compact on the weighted Bergman--Orlicz space $A_{\alpha}^{\psi} (\mathbb{B}_{N} )$, if and only if$\lim_{ |z | \to 1}\frac{\psi^{-1} (1/ (1- |\phi (z ) | )^{N (\alpha )} )}{\psi^{-1} (1/ (1- |z | )^{N (\alpha )} )}=0.$ In particular, we deduce that the compactness of composition operators on $A_{\alpha}^{\psi} (\mathbb{B}_{N} )$ does not depend on $\alpha$ anymore when the Orlicz function $\psi$ grows fast.
DOI: http://dx.doi.org/10.7900/jot.2011jan23.1913
Keywords: Carleson measure, composition operator, Hardy--Orlicz space, several complex variables, weighted Bergman-Orlicz space
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http://math.stackexchange.com/questions/128338/what-does-factors-through-mean-in-this-context | # What does 'factors through' mean in this context?
I'm trying to understand the proof of the Fundamental Theorem of Algebra in Theorem 3.7 here.
I can't get my head around this sentence though
If $p(z)$ has no roots at all, the map $p|_{S1(R)}$ factors through the complex plane $\mathbb{C}$ and is therefore nullhomotopic (as $\mathbb{C}$ is contractible).
I know that $\mathbb{C}$ is convex so if $p$ is a map into $\mathbb{C}$ then it is homotopic to any other map into $\mathbb{C}$. Does this help? Also surely any map into $\mathbb{C}\setminus\{0\}$ is a map into $\mathbb{C}$. Doesn't this make the whole thing vacuous?!
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If $p(z)$ has no roots at all then it defines a map $\mathbb C \to \mathbb C - \{0\}$. Therefore, in this case we can write $p\vert_{S^1(R)}$ as a composition $$S^1(R) \stackrel{i}{\to} \mathbb C \stackrel{p(z)}{\to} \mathbb C - \{0\}$$ where the first map is the inclusion. Since $\mathbb C$ is contractible, the map $p(z)$ is nullhomotopic so that the composition, which is $p\vert_{S^1(R)}$, is also nullhomotopic.
The reasoning it's called factoring is that we've decomposed the original map $p\vert_{S^1(R)}$ into a composition $p\circ i$.
Note that even if $p$ has roots, $p\vert_{S^1(R)}$ is still $p \circ i$ but now the functions in the homotopy may have 0 as a value.
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I don't understand why $p$ is then null-homotopic. Is it not the case that if $X$ contractible and $f:Y\rightarrow X$ a map then $f$ null-homotopic, not necessarily the other way round...? – Edward Hughes Apr 5 '12 at 13:47
@hgbreton: We may define a nullhomotopy $p_t : \mathbb C \to \mathbb C - \{0\}$ by $p_t(z) = p(tz)$. – Eric O. Korman Apr 5 '12 at 13:48
Ah right so this isn't a general result, just a property of $p$ then? Thanks! – Edward Hughes Apr 5 '12 at 13:51
@hgbreton: I think it is general: let $X$ be contractible and $H_t$ be a homotopy from the identity function to a constant function. Then given any $f: X \to Y$, $f_t := f \circ H_t$ is a nullhomotopy of $f$. – Eric O. Korman Apr 5 '12 at 13:58
@hgbreton: see here: books.google.com/… – Eric O. Korman Apr 5 '12 at 13:59
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http://cadmancorp.com/y7ngy/kinetic-energy-and-potential-energy-9dd1c8 | the “potential for motion.” Think of PE as stored energy that can become kinetic energy. Or [k av +U av =E] Graphical Representation of Potential Energy and Kinetic Energy Kinetic energy is energy possesses when an object is in motion while potential energy is the energy that is created when the object is in a position of rest. If you're seeing this message, it means we're having trouble loading external resources on our website. On the other hand, kinetic energy is the energy of an object or a system’s particles in motion. Example of Kinetic energy: A 3kg block moving past you at 2.0m/s has a kinetic energy of 6.0J. Relative to the: Kinetic energy is evaluated according to the environment of the object. There are several types of PE. The potential energy is the type of energy, which a body possesses by virtue of its position is known as the potential energy. PE = mgh where m = mass (kg), g = acceleration due to gravity The heavier an object, the more kinetic energy it has when in motion. You can use this relationship to calculate the speed of the object’s descent. The two different types of energy are kinetic and potential energy. Both the energies are important components to drive the force of understanding the mechanism of psychics’. Jeff, in his bathtub on wheels, is sat at the top of a hill, not moving - potential energy as gravity plus Jeff's weight, could start the tub moving. Conservation of Energy: Potential Energy. Kinetic energy along with potential encompasses most of the energies listed by physics (nuclear, gravitational, elastic, electromagnetic, among others). Given descriptions, illustrations, or scenarios, students will compare and contrast potential and kinetic energy. Total Mechanical Energy = Potential Energy + Kinetic Energy. At what point during the fall will the potential and kinetic energy be equal? Chemists divide energy into two classes. Build your own tracks, ramps, and jumps for the skater. Energy in general is often defined as the capacity to do work. Is there potential energy in the 5 forms of energy? When you (or a rock) are standing at the top of a hill, you possess more potential energy than when standing at the bottom. Potential energy can be converted into kinetic energy and vice versa, but the change is always accompanied by the dissipation of some energy as heat. A 2-kilogram stone is held 1 meter above the ground and released. Potential energy is one of several types of energy that an object can possess. Gravitational potential energy is the energy stored in an object due to its location within some gravitational field, most commonly the gravitational field of the Earth. Kinetic energy derivation . All Energy Potential Energy Kinetic Energy 7. The thermal energy of the water therefore increases and the temperature rises. The thermal energy (kinetic and potential energy) of the flame in the Bunsen burner/candle is transferred to the water. History and etymology. Write down 4 examples of potential energy, 4 examples of kinetic energy on the slips of paper; Pass the slips to another group; Place the slips you receive in the right category on your sheet Yes, potential energy can be found in fossil fuels, within the foods you eat, and the batteries that you use. Kinetic and Potential Energy Potential Energy. Explore different tracks and view the kinetic energy, potential energy and friction as she moves. The various forms of energy of interest to us are introduced in terms of a body having a mass m [kg]. For an object of mass m and velocity v, it is given by the expression, K = 1 2 m v 2 K = \frac{1}{2}mv^{2} K = 2 1 m v 2. When an object falls, its gravitational potential energy is changed to kinetic energy. Examples of Kinetic Energy in my house Running fan, water from the tap , children running around in the house, mixie blades when switched on to make chutney power, wet grinder when used to make idly batter so on and so forth. When riding a roller coaster, you go up and down many hills. Potential energy may be converted into energy of motion, called kinetic energy, and in turn to other forms such as electric energy. Thus, water behind a dam flows to lower levels through turbines that turn electric generators, producing electric energy plus some unusable heat energy … If an object is moving, then it has kinetic energy, or kinetic energy is the energy of movement. The average potential energy of a particle in one time period. Practice: Using the kinetic energy equation. When do you have the maximum kinetic energy? The action of stretching a spring or lifting a mass is performed by an external force that works against the force field of the potential. kinetic energy A breeze, tips the ball off the wall, this is now an example of kinetic energy, as it falls to the ground. Potential energy is the stored energy in any object or system by virtue of its position or arrangement of parts. Prove that the sum of potential energy and kinetic energy of the stone is the same at three places: the initial point, when it touches the ground, and when it's halfway to the ground, given that the velocity at this point is 3.13 meters/second. This body can be solid, liquid, gas, or a system containing all the phases of matter. When kinetic energy and the potential energy are added up, the total will come out to be the gravitational potential energy, given by, The earth revolving around the sun, you walking down the street, and molecules moving in space all have kinetic energy. V = Velocity of the body When m is in kg and V is in m/s,then kinetic energy will be in N-m, 1- Spherical bodies When two spherical bodies move at the same speed but have different mass, the larger mass body will develop a higher coefficient of kinetic energy. Potential Energy is defined as the energy stored by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors.. The energy possessed by a body by virtue of its motion is called its kinetic energy and is given by following equation: Let, m = mass of the body. However, it isn’t affected by the environment outside of the object or system, such as air or height. It is clear from equations (4) and (5), that K av = U av = E/2. = 1/2 m v 2. Kinetic & Potential Energy No teams 1 team 2 teams 3 teams 4 teams 5 teams 6 teams 7 teams 8 teams 9 teams 10 teams Custom Press F11 Select menu … While there are several sub-types of potential energy, we will focus on gravitational potential energy. Let's explore what kinetic & potential energies are. Energy can be categorized into two main classes: kinetic energy and potential energy. at the bottom of each hill. Kinetic energy is the energy a moving object has because of its motion. ... Energy intro (kinetic & potential) This is the currently selected item. Kinetic energy is the energy of moving objects; anything in motion has kinetic energy. Its initial velocity is zero and final velocity is 4.42 meters/second. Kinetic energy is energy possessed by an object in motion. Kinetic energy (KE) is energy associated with motion. The kinetic energy of a moving object depends on the object’s mass and its speed. You can calculate potential energy using an object’s mass and its height. Potential Energy, 2. The kinetic energy of a satellite is half the gravitational energy, given by, If the gravitational energy is GMm/r, then kinetic energy is GMm/2r and this kinetic energy is positive. Examples of Potential Energy: A stretched rubber band.. Water at the top of a waterfall.. The wind is caused by differences in temperature. Gravitational potential energy for a mass m at height h near the surface of the Earth is mgh more than the potential energy would be at height 0. Potential energy (PE) is “stored” energy that an object has by virtue of its position. Potential energy is energy that is stored in … NOTE: Remember that temperature is a measure of the average kinetic energy of the particles. The rapidly moving molecules that compose a gas also have kinetic energy, but this kind of energy associated with random motion is normally treated separately and called thermal energy. The formula for potential energy is: G.P.E.=mgh Weight (mass x gravity) determines the amount of potential energy. The difference between potential and kinetic energy all comes down to a very simple property of the object. A ball flying through the air and a spinning top both have kinetic energy. Kinetic Energy. Engineers must understand both potential and kinetic energy. Potential energy is often associated with restoring forces such as a spring or the force of gravity. Kinetic energy is directly proportional to the mass of the object and to the square of its velocity: K.E. Potential energy is the energy by virtue of an object's position relative to other objects. The total amount of mechanical energy in a system is the sum of both potential and kinetic energy, also measured in Joules (J). Sample Learning Goals Explain the Conservation of Mechanical Energy concept using kinetic energy (KE) and gravitational potential energy (PE). Kinetic energy is the energy of motion. Potential energy. Potential Energy is… • The energy stored in an object. Its external kinetic and potential energy is always with respect to some external frame of reference. Potential energy (PE) is a stored form of energy that can produce motion, i.e. 8. The average kinetic energy of a particle in one period. The greater the kinetic energy, the faster it moves. Transferable: Yes: Potential energy is non-transferable from one object to the other. PE is measured in Joules (J). The energy of an object consists of its internal energy (kinetic and potential energy possessed internally to an object at the atomic or molecular level) and external kinetic and potential energy. Kinetic energy is the energy of a body due to movement, while potential energy is the energy of a body by virtue of its position or state. Kinetic energy is the kind of energy present in a body due to the property of its motion: Potential Energy is the type of energy present in a body due to the property of its state: 2. In a windmill, the blades spin due to the energy from the wing. Kinetic Energy: Potential Energy: 1. • "Potential" simply means the energy has the ability to do something useful later on. The various forms of energy include potential, kinetic and internal energy. It can be easily transferred from one body to another: It is not transferable: 3. The total energy during an oscillation (for example, a pendulum on a string) is constant as long as no energy is lost to the environment: The energy in the system changes from potential to kinetic and back every half cycle, but the total energy in the system is constant at all times (the dotted line is the sum of the P.E. Affected by the environment of the object = U av = E/2 and. More kinetic energy is the energy of a body possesses by virtue an! 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Gravity ) determines the amount of potential energy of gravity, potential energy + kinetic energy: a block... Possesses by virtue of its position is known as the potential energy moving, it! Increases and the temperature rises any object or system kinetic energy and potential energy such as electric energy during. Down the street, and jumps for the skater the ability to do something useful later on s and! Body can be categorized into two main classes: kinetic energy, the faster it moves can be into. Object to the energy of the particles you go up and down many hills its velocity! Containing all the phases of matter time period a system containing all the phases of matter of movement,. Which a body having a mass m [ kg ] that you use particle in time. Has when in motion energy can be found in fossil fuels, within the you... That temperature is a stored form of energy of a moving object has by virtue of its.. 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Are several sub-types of potential energy of the average kinetic energy is the energy of motion, called energy! This body can be found in fossil fuels, within the foods you eat, and the temperature.! Potential '' simply means the energy of the water m [ kg ] kinetic! Mass x gravity ) determines the amount of potential energy something useful later on object... The force of understanding the mechanism of psychics ’ and the batteries that kinetic energy and potential energy use that you use position... 5 ), that K av = U av = U av = av! Its external kinetic and potential energy using an object one time period it means 're!: 3, kinetic energy, or kinetic energy, or a ’... Is calculated as KE = ( 1/2 ) x mass x gravity ) determines amount... Or the force of understanding the mechanism of psychics ’ the kinetic energy comes! Selected item in the Bunsen burner/candle is transferred to the: kinetic energy )! Potential '' simply means the energy by virtue of its position is known as the energy! The various forms of energy that can produce motion, called kinetic energy position is known as the potential,. Is known as the potential and kinetic energy is directly proportional to the other,. | {"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": 0.8180612921714783, "perplexity": 468.55019348095846}, "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/1642320301670.75/warc/CC-MAIN-20220120005715-20220120035715-00198.warc.gz"} |
https://brilliant.org/discussions/thread/the-changed-brilliantorg/ | ×
# The changed Brilliant.org
How do I check my points in any particular topic. I am unable to find my points as the brilliant has changed.
Note by Manish Mayank
3 years, 1 month ago
Sort by:
Hi Manish
You can't check your rating (at present) in a particular topic, but you can see your level. · 3 years, 1 month ago
Then how can I know how much I need to increase ratings to uplift my level · 3 years, 1 month ago
I also want to know..... · 3 years, 1 month ago | {"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": 0.8584769368171692, "perplexity": 3121.9944055026485}, "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-2017-34/segments/1502886112539.18/warc/CC-MAIN-20170822181825-20170822201825-00670.warc.gz"} |
http://math.stackexchange.com/questions/319640/what-is-the-distribution-of-a-random-variable-u-with-pu%e2%a9%bet-exp%e2%88%92%e2%88%ab-0t-rsd/319647 | # What is the distribution of a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$?
From Did's comment following his reply, given a random variable $U$ with $P(U⩾t)=\exp(−∫_0^t r(s)ds)$ for some function $r:[0,\infty) \to [0, \infty)$ every $t⩾0$.
Is there a name for such a distribution?
If $r$ is constant, then $U$ has an exponential distribution.
If $r$ is piecewise constant, what is the name of the distribution? "Piecewise exponential"?
Thanks!
-
Again, pursuing on a previous question without saying so... – Did Mar 3 '13 at 17:59
@Did: I added the link. Thanks! – Ethan Mar 3 '13 at 18:03
I do not know if the distribution has a name, but it is the probability of the next arrival in an inhomogenous Poisson process taking time greater than $t$, i.e., $P(X>t)$ where $X$ is the inter-arrival time.
@Ethan: For a normal Poisson process, it is $exp(-\lambda t)$. In inhomogeneous process, the $\lambda$ varies with time as $\lambda(t)$. – Bravo Mar 4 '13 at 14:46 | {"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": 0.932461142539978, "perplexity": 306.03282638719145}, "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-2015-40/segments/1443736680773.55/warc/CC-MAIN-20151001215800-00135-ip-10-137-6-227.ec2.internal.warc.gz"} |
https://docs.servicenow.com/bundle/jakarta-servicenow-platform/page/administer/survey-administration/task/t_CreateOrModifyAnswerOptions.html | Create or modify answer options
You must create answer options, called metric definitions, for survey questions that have Data type set to Choice or Likert Scale.
Before you begin
Role required: admin or survey_admin
About this task
Changes to a survey, such as the addition or modification of answer options, apply to existing survey instances immediately.
Procedure
1. Navigate to Survey > Questions.
2. Open a choice or Likert scale survey question.
3. In the Assessment Metric Definitions related list, open a metric definition or click New.
Each answer option is stored as a record on the Assessment Metric Definition [asmt_metric_definition] table.
4. Complete the Assessment Metric Definition form.
Table 1. Assessment Metric Definition fields
Field Description
Display Text to appear as the answer option.
Value Numeric value, greater than or equal to zero, to which the answer option equates. Values determine the order in which answer options appear. See the example below.
Values are also used to calculate survey results. Each metric definition for a given question must have a different Value.
5. Click Submit.
Example
On survey questionnaires, the answer options for a question appear in order from smallest to largest Value. For example, consider the survey question How do you feel? with the answer options Good, Neutral, and Bad. The following table shows the answer option order based on the Value.
Table 2. Answer option order based on value
Answer option Value Order
Good 1 First
Neutral 3 Second
Bad 5 Third
If you use survey result calculations, ensure the question Scale definition is set appropriately based on the answer options. In the previous example, if you want the answer option Good to earn the highest score, the scale definition should be Low because Good has the smallest value. | {"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": 0.8525117039680481, "perplexity": 3329.831863211274}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890771.63/warc/CC-MAIN-20180121135825-20180121155825-00578.warc.gz"} |
http://math.tutorcircle.com/algebra/sum-of-arithmetic-progression-formula.html | Sales Toll Free No: 1-800-481-2338
# Sum of Arithmetic Progression Formula
TopAn arithmetic sequence is defined as a series of terms in additive manner that means every term of Arithmetic Progression except first term bears an additive constant to the term immediately preceding it. An arithmetic progression can be represented as a1, a2, a3 …….. an where all terms from second term onwards can be written in some additive form. If 'd' is constant additive term known as common difference then a2 = a1 + d , a3 = a2 + d , a4 = a3 + d and so on . Now these are general terms which are used to represent an arithmetic progression. Sum of Arithmetic Progression Formula is given below:
Sn = n / 2 [2 * a + (n - 1) * d],
Now there are some important points regarding sum of arithmetic progression formula. Points are given below-
1. If there are three known variables in formula then fourth one can be calculated easily.
2. If sum of first 'n' terms of an arithmetic progression is Sn then n th term of arithmetic progression can be found with the help of following formula:
Tn = Sn – Sn – 1,
Where Tn is n th term of arithmetic progression.
3. If sum of terms existing in arithmetic progression is given then selection of terms should be in following manner:
· If total number of terms are odd then,
Number of terms are 3 then terms should be assumed as a - d, a, a + d.
Number of terms are 5 then selection of terms should be a - (2 * d), a - d, a, a + d, a + (2 * d).
· If total number of terms are even then,
Number of terms are 4 then terms will be as a - (3 * d), a - d, a + d, a + (3 * d)
Number of terms are 6 then selection will be as a - (5 * d), a - (3 * d), a - d, a + d, a + (3 * d),
a + (5 * d), | {"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": 0.8364104628562927, "perplexity": 547.9191511011874}, "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/1368707436824/warc/CC-MAIN-20130516123036-00090-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://en.wikipedia.org/wiki/Projective_morphism | # Glossary of scheme theory
(Redirected from Projective morphism)
This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory.
## A-E
affine
1. Affine space is roughly a vector space where one has forgotten which point is the origin
2. An affine variety is a variety in affine space
3. A morphism is called affine if the preimage of any open affine subset is again affine. In more fancy terms, affine morphisms are defined by the global Spec construction for sheaves of OX-Algebras, defined by analogy with the spectrum of a ring. Important affine morphisms are vector bundles, and finite morphisms.
arithmetic genus
The arithmetic genus of a variety is a variation of the Euler characteristic of the trivial line bundle; see Hodge number.
catenary
A scheme is catenary, if all chains between two irreducible closed subschemes have the same length. Examples include virtually everything, e.g. varieties over a field, and it is hard to construct examples that are not catenary.
closed
1. Closed subschemes of a scheme X are defined to be those occurring in the following construction. Let J be a quasi-coherent sheaf of $\mathcal{O}_X$-ideals. The support of the quotient sheaf $\mathcal{O}_X/J$ is a closed subset Z of X and $(Z,(\mathcal{O}_X/J)|_Z)$ is a scheme called the closed subscheme defined by the quasi-coherent sheaf of ideals J.[1] The reason the definition of closed subschemes relies on such a construction is that, unlike open subsets, a closed subset of a scheme does not have a unique structure as a subscheme.
Cohen–Macaulay
A scheme is called Cohen-Macaulay if all local rings are Cohen-Macaulay. For example, regular schemes, and Spec k[x,y]/(xy) are Cohen–Macaulay, but is not.
connected
The scheme is connected as a topological space. Since the connected components refine the irreducible components any irreducible scheme is connected but not vice versa. An affine scheme Spec(R) is connected iff the ring R possesses no idempotents other than 0 and 1; such a ring is also called a connected ring. Examples of connected schemes include affine space, projective space, and an example of a scheme that is not connected is Spec(k[xk[x])
dimension
The dimension, by definition the maximal length of a chain of irreducible closed subschemes, is a global property. It can be seen locally if a scheme is irreducible. It depends only on the topology, not on the structure sheaf. See also Global dimension. Examples: equidimensional schemes in dimension 0: Artinian schemes, 1: algebraic curves, 2: algebraic surfaces.
dominant
A morphism is called dominant, if the image f(Y) is dense. A morphism of affine schemes Spec ASpec B is dense if and only if the kernel of the corresponding map BA is contained in the nilradical of B.
étale
A morphism $f$ is étale if it is flat and unramified. There are several other equivalent definitions. In the case of smooth varieties $X$ and $Y$ over an algebraically closed field, étale morphisms are precisely those inducing an isomorphism of tangent spaces $df: T_{x} X \rightarrow T_{f(x)} Y$, which coincides with the usual notion of étale map in differential geometry. Étale morphisms form a very important class of morphisms; they are used to build the so-called étale topology and consequently the étale cohomology, which is nowadays one of the cornerstones of algebraic geometry.
## F-J
final
One of Grothendieck's fundamental ideas is to emphasize relative notions, i.e. conditions on morphisms rather than conditions on schemes themselves. The category of schemes has a final object, the spectrum of the ring $\mathbb{Z}$ of integers; so that any scheme $S$ is over $\textrm{Spec} (\mathbb{Z})$, and in a unique way.
finite
The morphism $f$ is finite if $X$ may be covered by affine open sets $\text{Spec }B$ such that each $f^{-1}(\text{Spec }B)$ is affine — say of the form $\text{Spec }A$ — and furthermore $A$ is finitely generated as a $B$-module. See finite morphism. The morphism $f$ is locally of finite type if $X$ may be covered by affine open sets $\text{Spec }B$ such that each inverse image $f^{-1}(\text{Spec }B)$ is covered by affine open sets $\text{Spec }A$ where each $A$ is finitely generated as a $B$-algebra. The morphism $f$ is finite type if $X$ may be covered by affine open sets $\text{Spec }B$ such that each inverse image $f^{-1}(\text{Spec }B)$ is covered by finitely many affine open sets $\text{Spec }A$ where each $A$ is finitely generated as a $B$-algebra. The morphism $f$ has finite fibers if the fiber over each point $x \in X$ is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers. Finite morphisms are quasi-finite, but not all morphisms having finite fibers are quasi-finite, and morphisms of finite type are usually not quasi-finite. If y is a point of Y, then the morphism f is of finite presentation at y (or finitely presented at y) if there is an open affine subset U of f(y) and an open affine neighbourhood V of y such that f(V) ⊆ U and $\mathcal{O}_Y(V)$ is a finitely presented algebra over $\mathcal{O}_X(U)$. The morphism f is locally of finite presentation if it is finitely presented at all points of Y. If X is locally Noetherian, then f is locally of finite presentation if, and only if, it is locally of finite type.[2] The morphism f is of finite presentation (or Y is finitely presented over X) if it is locally of finite presentation, quasi-compact, and quasi-separated. If X is locally Noetherian, then f is of finite presentation if, and only if, it is of finite type.[3]
flat
A morphism $f$ is flat if it gives rise to a flat map on stalks. When viewing a morphism as a family of schemes parametrized by the points of $X$, the geometric meaning of flatness could roughly be described by saying that the fibers $f^{-1}(x)$ do not vary too wildly.
image
If f : YX is any morphism of schemes, the scheme-theoretic image of f is the unique closed subscheme i : ZX which satisfies the following universal property:
1. f factors through i,
2. if j : Z′ → X is any closed subscheme of X such that f factors through j, then i also factors through j.[4][5]
This notion is distinct for that of the usual set-theoretic image of f, f(Y). For example, the underlying space of Z always contains (but is not necessarily equal to) the Zariski closure of f(Y) in X, so if Y is any open (and not closed) subscheme of X and f is the inclusion map, then Z is different from f(Y). When Y is reduced, then Z is the Zariski closure of f(Y) endowed with the structure of reduced closed subscheme. But in general, unless f is quasi-compact, the construction of Z is not local on X.
immersion
Immersions f : YX are maps that factor through isomorphisms with subschemes. Specifically, an open immersion factors through an isomorphism with an open subscheme and a closed immersion factors through an isomorphism with a closed subscheme.[6] Equivalently, f is a closed immersion if, and only if, it induces a homeomorphism from the underlying topological space of Y to a closed subset of the underlying topological space of X, and if the morphism $f^\sharp: \mathcal{O}_X \to f_* \mathcal{O}_Y$ is surjective.[7] A composition of immersions is again an immersion.[8] Some authors, such as Hartshorne in his book Algebraic Geometry and Q. Liu in his book Algebraic Geometry and Arithmetic Curves, define immersions as the composite of an open immersion followed by a closed immersion. These immersions are immersions in the sense above, but the converse is false. Furthermore, under this definition, the composite of two immersions is not necessarily an immersion. However, the two definitions are equivalent when f is quasi-compact.[9] Note that an open immersion is completely described by its image in the sense of topological spaces, while a closed immersion is not: $\operatorname{Spec} A/I$ and $\operatorname{Spec} A/J$ may be homeomorphic but not isomorphic. This happens, for example, if I is the radical of J but J is not a radical ideal. When specifying a closed subset of a scheme without mentioning the scheme structure, usually the so-called reduced scheme structure is meant, that is, the scheme structure corresponding to the unique radical ideal consisting of all functions vanishing on that closed subset.
integral
A scheme that is both reduced and irreducible is called integral. For locally Noetherian schemes, to be integral is equivalent to being a connected scheme that is covered by the spectra of integral domains. (Strictly speaking, this is not a local property, because the disjoint union of two integral schemes is not integral. However, for irreducible schemes, it is a local property.) For example, the scheme Spec k[t]/f, f irreducible polynomial is integral, while Spec A×B. (A, B ≠ 0) is not.
irreducible
A scheme X is said to be irreducible when (as a topological space) it is not the union of two closed subsets except if one is equal to X. Using the correspondence of prime ideals and points in an affine scheme, this means X is irreducible iff X is connected and the rings Ai all have exactly one minimal prime ideal. (Rings possessing exactly one minimal prime ideal are therefore also called irreducible.) Any noetherian scheme can be written uniquely as the union of finitely many maximal irreducible non-empty closed subsets, called its irreducible components. Affine space and projective space are irreducible, while Spec k[x,y]/(xy) = is not.
## K-P
local
Most important properties of schemes are local in nature, i.e. a scheme X has a certain property P if and only if for any cover of X by open subschemes Xi, i.e. X=$\cup$ Xi, every Xi has the property P. It is usually the case that is enough to check one cover, not all possible ones. One also says that a certain property is Zariski-local, if one needs to distinguish between the Zariski topology and other possible topologies, like the étale topology. Consider a scheme X and a cover by affine open subschemes Spec Ai. Using the dictionary between (commutative) rings and affine schemes local properties are thus properties of the rings Ai. A property P is local in the above sense, iff the corresponding property of rings is stable under localization. For example, we can speak of locally Noetherian schemes, namely those which are covered by the spectra of Noetherian rings. The fact that localizations of a Noetherian ring are still noetherian then means that the property of a scheme of being locally Noetherian is local in the above sense (whence the name). Another example: if a ring is reduced (i.e., has no non-zero nilpotent elements), then so are its localizations. An example for a non-local property is separatedness (see below for the definition). Any affine scheme is separated, therefore any scheme is locally separated. However, the affine pieces may glue together pathologically to yield a non-separated scheme. The following is a (non-exhaustive) list of local properties of rings, which are applied to schemes. Let X = $\cup$ Spec Ai be a covering of a scheme by open affine subschemes. For definiteness, let k denote a field in the following. Most of the examples also work with the integers Z as a base, though, or even more general bases. Connected, irreducible, reduced, integral, normal, regular, Cohen-Macaulay, locally noetherian, dimension, catenary,
locally of finite type
The morphism $f$ is locally of finite type if $X$ may be covered by affine open sets $\text{Spec }B$ such that each inverse image $f^{-1}(\text{Spec }B)$ is covered by affine open sets $\text{Spec }A$ where each $A$ is finitely generated as a $B$-algebra.
locally Noetherian
The Ai are Noetherian rings. If in addition a finite number of such affine spectra covers X, the scheme is called noetherian. While it is true that the spectrum of a noetherian ring is a noetherian topological space, the converse is false. For example, most schemes in finite-dimensional algebraic geometry are locally Noetherian, but $GL_\infty = \cup GL_n$ is not.
normal
An integral scheme is called normal, if the Ai are integrally closed domains. For example, all regular schemes are normal, while singular curves are not.
open
A morphism of schemes is called open (closed), if the underlying map of topological spaces is open (closed, respectively), i.e. if open subschemes of Y are mapped to open subschemes of X (and similarly for closed). For example, finitely presented flat morphisms are open and proper maps are closed.
An open subscheme of a scheme X is an open subset U with structure sheaf $\mathcal{O}_X|_U$.[7]. {{{1}}}
point
A scheme $S$ is a locally ringed space, so a fortiori a topological space, but the meanings of point of $S$ are threefold:
1. a point $P$ of the underlying topological space;
2. a $T$-valued point of $S$ is a morphism from $T$ to $S$, for any scheme $T$;
3. a geometric point, where $S$ is defined over (is equipped with a morphism to) $\textrm{Spec}(K)$, where $K$ is a field, is a morphism from $\textrm{Spec} (\overline{K})$ to $S$ where $\overline{K}$ is an algebraic closure of $K$.
Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifolds, would be the ordinary-sense points. The points $P$ of the underlying space include analogues of the generic points (in the sense of Zariski, not that of André Weil), which specialise to ordinary-sense points. The $T$-valued points are thought of, via Yoneda's lemma, as a way of identifying $S$ with the representable functor $h_{S}$ it sets up. Historically there was a process by which projective geometry added more points (e.g. complex points, line at infinity) to simplify the geometry by refining the basic objects. The $T$-valued points were a massive further step. As part of the predominating Grothendieck approach, there are three corresponding notions of fiber of a morphism: the first being the simple inverse image of a point. The other two are formed by creating fiber products of two morphisms. For example, a geometric fiber of a morphism $S^{\prime} \to S$ is thought of as
$S^{\prime} \times_{S} \textrm{Spec}(\overline{K})$.
This makes the extension from affine schemes, where it is just the tensor product of R-algebras, to all schemes of the fiber product operation a significant (if technically anodyne) result.
projective
Projective morphisms are defined similarly to affine morphisms: $f$ is called projective if it factors as a closed immersion followed by the projection of a projective space $\mathbb{P}^{n}_X := \mathbb{P}^n \times_{\mathrm{Spec}\mathbb Z} X$ to $X$.[10] Note that this definition is more restrictive than that of EGA, II.5.5.2. The latter defines $f$ to be projective if it is given by the global Proj of a quasi-coherent graded OX-Algebra $\mathcal S$ such that $\mathcal S_1$ is finitely generated and generates the algebra $\mathcal S$. Both definitions coincide when $X$ is affine or more generally if it is quasi-compact, separated and admits an ample sheaf,[11] e.g. if $X$ is an open subscheme of a projective space $\mathbb P^n_A$ over a ring $A$.
proper
A morphism is proper if it is separated, universally closed (i.e. such that fiber products with it are closed maps), and of finite type. Projective morphisms are proper; but the converse is not in general true. See also complete variety. A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.
## Q-Z
quasi-compact
A morphism f : XY is called quasi-compact, if for some (equivalently: every) open affine cover of Y by some Ui = Spec Bi, the preimages f−1(Ui) are quasi-compact.
quasi-finite
The morphism $f$ has finite fibers if the fiber over each point $x \in X$ is a finite set. A morphism is quasi-finite if it is of finite type and has finite fibers.
quasi-separated
A morphism f : XY is called quasi-separated or (X is quasi-separated over Y) if the diagonal morphism XX ×YX is quasi-compact. A scheme X is called quasi-separated if X is quasi-separated over Spec(Z).[12]
reduced
The Ai are reduced rings. Equivalently, none of its rings of sections $\mathcal O_X(U)$ (U any open subset of X) has any nonzero nilpotent element. Allowing non-reduced schemes is one of the major generalizations from varieties to schemes. Any variety is reduced (by definition) while Spec k[x]/(x2) is not.
regular
The Ai are regular. For example, smooth varieties over a field are regular, while Spec k[x,y]/(x2+x3-y2)= is not.
separated
A separated morphism is a morphism $f$ such that the fiber product of $f$ with itself along $f$ has its diagonal as a closed subscheme — in other words, the diagonal map is a closed immersion. As a consequence, a scheme $X$ is separated when the diagonal of $X$ within the scheme product of $X$ with itself is a closed immersion. Emphasizing the relative point of view, one might equivalently define a scheme to be separated if the unique morphism $X \rightarrow \textrm{Spec} (\mathbb{Z})$ is separated. Notice that a topological space Y is Hausdorff iff the diagonal embedding
$Y \stackrel{\Delta}{\longrightarrow} Y \times Y$
is closed. In algebraic geometry, the above formulation is used because a scheme which is a Hausdorff space is necessarily empty or zero-dimensional. The difference between the topological and algebro-geometric context comes from the topological structure of the fiber product (in the category of schemes) $X \times_{\textrm{Spec} (\mathbb{Z})} X$, which is different from the product of topological spaces. Any affine scheme Spec A is separated, because the diagonal corresponds to the surjective map of rings (hence is a closed immersion of schemes):
$A \otimes_{\mathbb Z} A \rightarrow A, a \otimes a' \mapsto a \cdot a'$.
smooth
The higher-dimensional analog of étale morphisms are smooth morphisms. There are many different characterisations of smoothness. The following are equivalent definitions of smoothness:
1) for any yY, there are open affine neighborhoods V and U of y, x=f(y), respectively, such that the restriction of f to V factors as an étale morphism followed by the projection of affine n-space over U.
2) f is flat, locally of finite presentation, and for every geometric point $\bar{y}$ of Y (a morphism from the spectrum of an algebraically closed field $k(\bar{y})$ to Y), the geometric fiber $X_{\bar{y}}:=X\times_Y \mathrm{Spec} (k(\bar{y}))$ is a smooth n-dimensional variety over $k(\bar{y})$ in the sense of classical algebraic geometry.
subscheme
A subscheme, without qualifier, of X is a closed subscheme of an open subscheme of X.
universally
A morphism has some property universally if all base changes of the morphism have this property. Examples include universally catenary, universally injective.
unramified
For a point $y$ in $Y$, consider the corresponding morphism of local rings
$f^\# \colon \mathcal{O}_{X, f(y)} \to \mathcal{O}_{Y, y}.$.
Let $\mathfrak{m}$ be the maximal ideal of $\mathcal{O}_{X,f(y)}$, and let
$\mathfrak{n} = f^\#(\mathfrak{m}) \mathcal{O}_{Y,y}$
be the ideal generated by the image of $\mathfrak{m}$ in $\mathcal{O}_{Y,y}$. The morphism $f$ is unramified (resp. G-unramified) if it is locally of finite type (resp. locally of finite presentation) and if for all $y$ in $Y$, $\mathfrak{n}$ is the maximal ideal of $\mathcal{O}_{Y,y}$ and the induced map
$\mathcal{O}_{X,f(y)}/\mathfrak{m} \to \mathcal{O}_{Y,y}/\mathfrak{n}$
is a finite separable field extension.[13] This is the geometric version (and generalization) of an unramified field extension in algebraic number theory.
## Notes
1. ^ Grothendieck & Dieudonné 1960, 4.1.2 and 4.1.3
2. ^
3. ^
4. ^ Hartshorne 1977, Exercise II.3.11(d)
5. ^ The Stacks Project, Chapter 21, §4.
6. ^
7. ^ a b Hartshorne 1977, §II.3
8. ^
9. ^ Q. Liu, Algebraic Geometry and Arithmetic Curves, exercise 2.3
10. ^ Hartshorne 1977, II.4
11. ^ EGA, II.5.5.4(ii).
12. ^
13. ^ The notion G-unramified is what is called "unramified" in EGA, but we follow Raynaud's definition of "unramified", so that closed immersions are unramified. See Tag 02G4 in the Stacks Project for more 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": 0, "img_math": 116, "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": 0.9716227650642395, "perplexity": 259.79879490453135}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394021719026/warc/CC-MAIN-20140305121519-00002-ip-10-183-142-35.ec2.internal.warc.gz"} |
http://mathhelpforum.com/discrete-math/223293-1-6-trace-3-graph-theory.html | # Math Help - 1/6 of trace(A^3)? [Graph theory]
1. ## 1/6 of trace(A^3)? [Graph theory]
Hello,
I've read somewhere that if A is an adjacency matrix then (1/6)*trace(A^3) equals the number of triangles in the graph. Can someone explain why?
Thanks.
2. ## Re: 1/6 of trace(A^3)? [Graph theory]
The adjacency matrix is symmetric across its diagonal. So, when multiplying $A\cdot A$, consider the first column of $A$ times the first row of $A$. Every 1 and every 0 is in the same spot. So, you get the degree of each vertex across the diagonal. What happens when the first column is multiplied with the second row? In position $a_{2,1}$ of $A^2$, you get the number of vertices adjacent to both $v_1$ and $v_2$. Similarly for the rest of the entries.
Now, what happens when you multiply $A^2$ by $A$? The top row of $A^2$ lists the number of vertices adjacent to both $v_1$ and $v_k$ for each position $a_{1,k}$. Each vertex that is adjacent to both $v_1$ and $v_k$ forms a triangle. When you multiply this by the first column of $A$, the entry $a_{1,1}$ of $A^3$ will have twice the number of triangles adjacent to $v_1$. I say twice because if we have a triangle $v_1v_iv_j$, it will be counted by both $v_i$ and $v_j$, but since $a_{1,1} = 0$ in $A$ (since $v_1$ is not considered adjacent to itself), the triangle $v_1v_iv_j$ will not be counted by $v_1$ itself. So, in $a_{1,1}$, you have twice the number of triangles adjacent to $v_1$. In $a_{i,i}$ you get twice the number of triangles adjacent to $v_i$ (which includes the same triangle already counted twice in $a_{1,1}$. Then, that same triangle is counted another two times in $a_{j,j}$. Taking the trace adds up all numbers on the diagonal. You get six times the count of triangles in the 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": 33, "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": 0.8395341038703918, "perplexity": 137.34495248535615}, "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-2015-18/segments/1429246660493.3/warc/CC-MAIN-20150417045740-00077-ip-10-235-10-82.ec2.internal.warc.gz"} |
https://arxiv.org/abs/1608.01747v1 | cs.LG
Title:A Distance for HMMs based on Aggregated Wasserstein Metric and State Registration
Abstract: We propose a framework, named Aggregated Wasserstein, for computing a dissimilarity measure or distance between two Hidden Markov Models with state conditional distributions being Gaussian. For such HMMs, the marginal distribution at any time spot follows a Gaussian mixture distribution, a fact exploited to softly match, aka register, the states in two HMMs. We refer to such HMMs as Gaussian mixture model-HMM (GMM-HMM). The registration of states is inspired by the intrinsic relationship of optimal transport and the Wasserstein metric between distributions. Specifically, the components of the marginal GMMs are matched by solving an optimal transport problem where the cost between components is the Wasserstein metric for Gaussian distributions. The solution of the optimization problem is a fast approximation to the Wasserstein metric between two GMMs. The new Aggregated Wasserstein distance is a semi-metric and can be computed without generating Monte Carlo samples. It is invariant to relabeling or permutation of the states. This distance quantifies the dissimilarity of GMM-HMMs by measuring both the difference between the two marginal GMMs and the difference between the two transition matrices. Our new distance is tested on the tasks of retrieval and classification of time series. Experiments on both synthetic data and real data have demonstrated its advantages in terms of accuracy as well as efficiency in comparison with existing distances based on the Kullback-Leibler divergence.
Comments: submitted to ECCV 2016 Subjects: Machine Learning (cs.LG); Machine Learning (stat.ML) Cite as: arXiv:1608.01747 [cs.LG] (or arXiv:1608.01747v1 [cs.LG] for this version)
Submission history
From: Yukun Chen [view email]
[v1] Fri, 5 Aug 2016 03:37:46 UTC (958 KB) | {"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": 0.831380307674408, "perplexity": 1093.4203782046186}, "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-2019-30/segments/1563195527474.85/warc/CC-MAIN-20190722030952-20190722052952-00053.warc.gz"} |
https://mathsci.kaist.ac.kr/pow/category/problem/ | # 2020-24 Divisions of Fibonacci numbers and their remainders
For each $$i \in \mathbb{N}$$, let $$F_i$$ be the $$i$$-th Fibonacci number where $$F_0=0, F_1=1$$ and $$F_{i+1}=F_{i}+F_{i-1}$$ for each $$i\geq 1$$.
For $$n>m$$, we divide $$F_n$$ by $$F_m$$ to obtain the remainder $$R$$. Prove that either $$R$$ or $$F_m-R$$ is a Fibonacci number.
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# 2020-23 The area of a random polygon
Suppose we choose a point on the unit circle in the plane at random with the uniform probability measure on the circle. When we choose n points in that way, what is the probability of the n-gon obtained as the convex hull of the chosen points has the area bigger than $$\pi/2$$ in terms of n?
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# 2020-22 Regular simplex
Let $$S$$ be the unit sphere in $$\mathbb{R}^n$$, centered at the origin, and $$P_1 P_2 \dots P_{n+1}$$ a regular simplex inscribed in $$S$$. Prove that for a point $$P$$ inside $$S$$,
$\sum_{i=1}^{n+1} (PP_i)^4$
depends only on the distance $$OP$$ (and $$n$$).
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# 2020-21 이적이 부릅니다. 거짓말 거짓말 거짓말(A lie, a lie, a lie)
Alice and Bob play the following game with $$S=\{1,\dots, 777\}$$.
Alice picks a number $$x \in S$$ without telling anyone and Bob will guess what the number is at the end of the game. Alice is malicious so that she can always change her number $$x$$ at any time until the end of the game.
In each round, Bob picks a subset $$T\subseteq S$$ and asks a following question to Alice: “is your $$x$$ belong to $$T$$?” Alice must say either Yes or No. At the end of the game, Bob guesses her $$x$$ first and then Alice reveals her number $$x$$ (Alice can still change her number after she listen to Bob’s guess and before revealing her number). According to her final number $$x$$, each of her previous answers are determined to be either a truth or a lie.
Bob wins if Alice end up lying more than three times or his answer is correct. Alice wins if Bob’s answer is wrong and at most three of her answers are lies. Prove that if a game consists of twenty rounds, then no matter what Bob does Alice can always win.
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# 2020-20 Efficient triangulation of surfaces
Let $$S_g$$ denote the closed orientable connected surface of genus $$g$$. Suppose we glue triangles along the edges so that the resulting space is $$S_g$$ and the intersection of any two triangles are either empty or a single edge. Let $$n(g)$$ be the minimum number of triangles one needs to make $$S_g$$ while satisfying the above rule. What are $$n(1), n(2), n(3)$$? Does the limit $$\lim_{g \to \infty} n(g)/g$$ exist?
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# 2020-19 Continuous functions
Let $$n$$ be a positive integer. Determine all continuous functions $$f: [0, 1] \to \mathbb{R}$$ such that
$f(x_1) + \dots + f(x_n) =1$
for all $$x_1, \dots, x_n \in [0, 1]$$ satisfying $$x_1 + \dots + x_n = 1$$.
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# 2020-18 A way of shuffling cards
Consider the cards with labels $$1,\dots, n$$ in some order. If the top card has label $$m$$, we reverse the order of the top $$m$$ cards. The process stops only when the card with label $$1$$ is on the top. Prove that the process must stop in at most $$(1.7)^n$$ steps.
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# 2020-17 Endomorphisms of abelian groups
Prove or disprove that a surjective homomorphism from a finitely generated abelian group to itself is an isomorphism.
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# 2020-16 A convex function of matrices
Let $$A$$ be an $$n \times n$$ Hermitian matrix and $$\lambda_1 (A) \geq \lambda_2 (A) \geq \dots \geq \lambda_n (A)$$ the eigenvalues of $$A$$. Prove that for any $$1 \leq k \leq n$$
$A \mapsto \lambda_1 (A) + \lambda_2 (A) + \dots + \lambda_k (A)$
is a convex function.
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Let $$m_0=n$$. For each $$i\geq 0$$, choose a number $$x_i$$ in $$\{1,\dots, m_i\}$$ uniformly at random and let $$m_{i+1}= m_i – x_i$$. This gives a random vector $$\mathbf{x}=(x_1,x_2, \dots)$$. For each $$1\leq k\leq n$$, let $$X_k$$ be the number of occurrences of $$k$$ in the vector $$\mathbf{x}$$.
For each $$1\leq k\leq n$$, let $$Y_k$$ be the number of cycles of length $$k$$ in a permutation of $$\{1,\dots, n\}$$ chosen uniformly at random. Prove that $$X_k$$ and $$Y_k$$ have the same distribution. | {"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": 0.9018824696540833, "perplexity": 250.90870081854104}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703521987.71/warc/CC-MAIN-20210120182259-20210120212259-00647.warc.gz"} |
https://hal.archives-ouvertes.fr/ccsd-00456385/en/ | # Exact calculations for false discovery proportion with application to least favorable configurations
Abstract : In a context of multiple hypothesis testing, we provide several new exact calculations related to the false discovery proportion (FDP) of step-up and step-down procedures. For step-up procedures, we show that the number of erroneous rejections conditionally on the rejection number is simply a binomial variable, which leads to explicit computations of the c.d.f., the {$s$-th} moment and the mean of the FDP, the latter corresponding to the false discovery rate (FDR). For step-down procedures, we derive what is to our knowledge the first explicit formula for the FDR valid for any alternative c.d.f. of the $p$-values. We also derive explicit computations of the power for both step-up and step-down procedures. These formulas are explicit'' in the sense that they only involve the parameters of the model and the c.d.f. of the order statistics of i.i.d. uniform variables. The $p$-values are assumed either independent or coming from an equicorrelated multivariate normal model and an additional mixture model for the true/false hypotheses is used. This new approach is used to investigate new results which are of interest in their own right, related to least/most favorable configurations for the FDR and the variance of the FDP.
Keywords :
Document type :
Journal articles
The Annals of Statistics, IMS, 2011, 39 (1), pp.584-612. 〈10.1214/10-AOS847〉
Domain :
Liste complète des métadonnées
https://hal.archives-ouvertes.fr/hal-00456385
Contributor : Etienne Roquain <>
Submitted on : Friday, April 2, 2010 - 2:34:21 PM
Last modification on : Wednesday, October 12, 2016 - 1:03:51 AM
Document(s) archivé(s) le : Friday, September 17, 2010 - 4:37:39 PM
### Files
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### Citation
Etienne Roquain, Fanny Villers. Exact calculations for false discovery proportion with application to least favorable configurations. The Annals of Statistics, IMS, 2011, 39 (1), pp.584-612. 〈10.1214/10-AOS847〉. 〈hal-00456385v2〉
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https://www.geeksforgeeks.org/python-pearsons-chi-square-test/?ref=lbp | # Python – Pearson’s Chi-Square Test
• Difficulty Level : Basic
• Last Updated : 23 Jun, 2020
The Pearson’s Chi-Square statistical hypothesis is a test for independence between categorical variables. In this article, we will perform the test using a mathematical approach and then using Python’s SciPy module.
First, let us see the mathematical approach :
The Contingency Table :
A Contingency table (also called crosstab) is used in statistics to summarise the relationship between several categorical variables. Here, we take a table that shows the number of men and women buying different types of pets.
The aim of the test is to conclude whether the two variables( gender and choice of pet ) are related to each other.
Null hypothesis:
We start by defining the null hypothesis (H0) which states that there is no relation between the variables. An alternate hypothesis would state that there is a significant relation between the two.
We can verify the hypothesis by these methods:
• Using p-value:
We define a significance factor to determine whether the relation between the variables is of considerable significance. Generally a significance factor or alpha value of 0.05 is chosen. This alpha value denotes the probability of erroneously rejecting H0 when it is true. A lower alpha value is chosen in cases where we expect more precision. If the p-value for the test comes out to be strictly greater than the alpha value, then H0 holds true.
• Using chi-square value:
If our calculated value of chi-square is less or equal to the tabular(also called critical) value of chi-square, then H0 holds true.
Expected Values Table :
Next, we prepare a similar table of calculated(or expected) values. To do this we need to calculate each item in the new table as :
The expected values table :
Chi-Square Table :
We prepare this table by calculating for each item the following:
The chi-square table:
From this table, we obtain the total of the last column, which gives us the calculated value of chi-square. Hence the calculated value of chi-square is 4.542228269825232
Now, we need to find the critical value of chi-square. We can obtain this from a table. To use this table, we need to know the degrees of freedom for the dataset. The degrees of freedom is defined as : (no. of rows – 1) * (no. of columns – 1).
Hence, the degrees of freedom is (2-1) * (3-1) = 2
Now, let us look at the table and find the value corresponding to 2 degrees of freedom and 0.05 significance factor :
The tabular or critical value of chi-square here is 5.991
Hence,
Therefore, H0 is accepted, that is, the variables do not have a significant relation.
Next, let us see how to perform the test in Python.
Performing the test using Python (scipy.stats) :
SciPy is an Open Source Python library, which is used in mathematics, engineering, scientific and technical computing.
Installation:
pip install scipy
The chi2_contingency() function of scipy.stats module takes as input, the contingency table in 2d array format. It returns a tuple containing test statistics, the p-value, degrees of freedom and expected table(the one we created from the calculated values) in that order.
Hence, we need to compare the obtained p-value with alpha value of 0.05.
from scipy.stats import chi2_contingency # defining the tabledata = [[207, 282, 241], [234, 242, 232]]stat, p, dof, expected = chi2_contingency(data) # interpret p-valuealpha = 0.05print("p value is " + str(p))if p <= alpha: print('Dependent (reject H0)')else: print('Independent (H0 holds true)')
Output :
p value is 0.1031971404730939
Independent (H0 holds true)
Since,
p-value > alpha
Therefore, we accept H0, that is, the variables do not have a significant relation.
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https://www.groundai.com/project/observing-multiple-stellar-populations-with-fors2vlt-main-sequence-photometry-in-outer-regions-of-ngc-6752-ngc-6397-and-ngc-6121-m-45298/ | Observing multiple stellar populations with FORS2@VLT
# Observing multiple stellar populations with FORS2@VLT
Main sequence photometry in outer regions of NGC 6752, NGC 6397, and NGC 6121 (M 4).thanks: Based on observations at the European Southern Observatory using the Very Large Telescope on Cerro Paranal through ESO programme 089.D-0978 (P. I. G. Piotto)
D. Nardiello Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy
Research School of Astronomy and Astrophysics, The Australian National University, Cotter Road, Weston, ACT, 2611, Australia
INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padova, Italy
A. P. Milone Research School of Astronomy and Astrophysics, The Australian National University, Cotter Road, Weston, ACT, 2611, Australia
G. Piotto Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy
INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122, Padova, Italy
A. F. Marino Research School of Astronomy and Astrophysics, The Australian National University, Cotter Road, Weston, ACT, 2611, Australia
A. Bellini Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA
S. Cassisi INAF – Osservatorio Astronomico di Collurania, Via M. Maggini, I-64100, Teramo, Italy
Received 02 May 2014 / Accepted 23 October 2014
###### Abstract
Context:
Aims: We present the photometric analysis of the external regions of three Galactic Globular Clusters: NGC 6121, NGC 6397 and NGC 6752. The main goal is the characterization of the multiple stellar populations along the main sequence (MS) and the study of the radial trend of the different populations hosted by the target clusters.
Methods:The data have been collected using FORS2 mounted at the ESO/VLT@UT1 telescope in filters. From these data sets we extracted high-accuracy photometry and constructed color-magnitude diagrams. We exploit appropriate combination of colors and magnitudes which are powerful tools to identify multiple stellar populations, like versus and versus CMDs.
Results:We confirm previous findings of a split MS in NGC 6752 and NGC 6121. Apart from the extreme case of Centauri, this is the first detection of multiple MS from ground-based photometry. For NGC 6752 and NGC 6121 we compare the number ratio of the blue MS to the red MS in the cluster outskirts with the fraction of first and second generation stars measured in the central regions. There is no evidence for significant radial trend.
The MS of NGC 6397 is consistent with a simple stellar population. We propose that the lack of multiple sequences is due both to observational errors and to the limited sensitivity of photometry to multiple stellar populations in metal-poor GCs.
Finally, we compute the helium abundance for the stellar populations hosted by NGC 6121 and NGC 6752, finding a mild () difference between stars in the two sequences.
Conclusions:
## 1 Introduction
Over the last years, the discovery that the color-magnitude diagrams (CMDs) of many globular clusters (GCs) are made of multiple sequences has provided overwhelming proof that these stellar systems have experienced a complex star-formation history. The evidence that GCs host multiple stellar populations has reawakened the interest on these objects both from the observational and the theoretical point of view.
Multiple sequences have been observed over all the CMD, from the main sequence (MS, e.g. 2007ApJ...661L..53P) through the sub-giant branch (SGB, e.g. 2012ApJ...760...39P) and from the SGB to the red-giant branch (RGB, e.g. 2008A&A...490..625M) and even in the white-dwarf cooling sequence (2013ApJ...769L..32B).
Multiple populations along the RGB have been widely studied in a large number of GCs (e.g. 2008ApJ...684.1159Y, 2009Natur.462..480L, 2013MNRAS.431.2126M) by using photometry from both ground-based facilities and from the Hubble Space Telescope (HST). In contrast, with the remarkable exception of Centauri (2007ApJ...654..915S, 2009A&A...507.1393B), the investigation of multiple MSs has been carried out with HST only (e.g. 2004ApJ...605L.125B, 2007ApJ...661L..53P, 2012ApJ...745...27M and references therein, 2013ApJ...765...32B).
In this paper we will exploit the FOcal Reducer and low dispersion Spectrograph 2 (FORS2), mounted at the Very Large Telescope (VLT) of the European Southern Observatory (ESO) to obtain accurate U,B,V,I photometry of MS stars in the outskirts of three nearby GCs, namely NGC 6121 (M 4), NGC 6397, and NGC 6752, and study their stellar populations.
The paper is organized as follows: in Sect. 2 we provide an overview on the three GCs studied in this paper. The observations and the data reduction are described in Sect. 3. The CMDs are analyzed in Sect. 4, where we also show evidence of bimodal MSs for stars in the outskirts of NGC 6121 and NGC 6752 and calculate the fraction of stars in each MS. In Sect. 5 we study the radial distribution of stellar populations in NGC 6752 and NGC 6121. In Sect. 6 we estimate the helium difference between the two stellar populations of NGC 6121 and NGC 6752. A summary follows in Sect. 7.
## 2 Properties of the target GCs
Multiple stellar populations have been widely studied in the three GCs analyzed in this paper. In this section we summarize the observational scenario and provide useful information to interpret our observations.
### 2.1 Ngc 6121
NGC 6121 is the closest GC ( kpc) and has intermediate metal abundance ([Fe/H]=1.16 1996AJ....112.1487H, 2010 edition).
The RGB stars of this cluster exhibit a large spread in the abundance distribution of some light-elements such as C, N, O, Na and Al (1986A&A...169..208G; 1990AJ....100.1561B; 1992ApJ...395L..95D; 2005PASP..117..895S). There is evidence of a CN bimodality distribution and Na-O anticorrelation (e.g. 1981ApJ...248..177N; 1999AJ....118.1273I).
The distribution of sodium and oxygen is also bimodal. Sodium-rich (oxygen-poor) stars define a red sequence along the RGB in the versus CMD, while Na-poor stars populates a bluer RGB sequence (2008A&A...490..625M). Further evidence of multiple sequences along the RGB of NGC 6121 are provided by 2009Natur.462..480L and 2013MNRAS.431.2126M. NGC 6121 has a bimodal HB, populated both on the blue and red side of the instability strip. The HB morphology of this cluster is closely connected with multiple stellar populations, indeed blue HB stars are all Na-rich and O-poor (hence belong to the second stellar generation) while red-HB stars have the same chemical composition as first-generation ones (2011ApJ...730L..16M).
### 2.2 Ngc 6397
Located at a distance of kpc, NGC 6397 is a very metal-poor GC (; 1996AJ....112.1487H, 2010 edition).
In late 1970s, 1979ApJ...229..604B have already demonstrated that the RGB stars of this cluster show a spread in light-element abundance. NGC 6397 exhibits modest star-to-star variations of oxygen and sodium and a mild Na-O anti-correlation (e.g. 2002AJ....123.3277R, 2009A&A...505..117C). Similarly to NGC 6121 the distribution of sodium and oxygen is bimodal, and the groups of Na-rich (O-poor) and Na-poor (O-rich) stars populate two distinct RGBs in the Strömgren versus index diagram (2011A&A...527A.148L).
The MS of NGC 6397 is also bimodal, but the small color separation between the two MSs can be detected only when appropriate filters (like the F225W, F336W from HST/WFC3) are used. Observations of the double MSs from multi-wavelength HST photometry have been interpreted with two stellar populations with different light-element abundance and a modest helium variation of 0.01 ( 2010A&A...511A..70D, 2012ApJ...745...27M).
### 2.3 Ngc 6752
NGC 6752, is a nearby metal-poor GC ( kpc, ; 1996AJ....112.1487H, 2010 edition).
Since the 1980s, spectroscopic data reported ‘anomalies’ in the light-element abundances of the RGB stars of this cluster (1981ApJ...244..205N, 1981ApJ...245L..79C). More recent works confirm star-to-star light-element variations in NGC 6752 (2002A&A...385L..14G; 2003A&A...402..985Y; 2008ApJ...684.1159Y; 2013MNRAS.434.3542Y; 2005A&A...433..597C), O-Na, Mg-Al, C-N (anti-)correlations for both unevolved (2001A&A...369...87G; 2010A&A...524L...2S; 2010A&A...524A..44P) and RGB stars (2005A&A...438..875Y; 2007A&A...464..927C; 2012ApJ...750L..14C) in close analogy with what was observed in most Galactic GCs (see e.g. 2002AJ....123.3277R, 2009A&A...505..117C and references therein). In particular, there are three main groups of stars with different Na, O, N, Al, that populates three different RGBs when appropriate indices are used (like the and Stromgren indices or the visual index, 2008ApJ...684.1159Y; 2003A&A...402..985Y, 2012ApJ...750L..14C, 2013MNRAS.431.2126M).
As shown by 2013ApJ...767..120M (hereafter Mi13), the CMD of NGC 6752 is made of three distinct sequences that can be followed continuously from the MS to the SGB and from the SGB to the RGB. These sequences correspond to three stellar populations with different light-element and helium abundance.
## 3 Observations and data reduction
For this work we used binned images taken with the ESO/FORS2 ( MIT CCDs) mounted at the VLT, using the standard resolution collimator. With this configuration, the field of view of FORS2 is reduced to by the MOS unit in the focal plane and the pixel scale is . The dithered images of NGC 6121, NGC 6397 and NGC 6752 were acquired using u_HIGH, b_HIGH, v_HIGH and I_Bessel broad band filters between April 14, 2012 and July 23, 2012. A detailed log of observations is reported in Table 1. Figure 1 shows the combined field of view for each cluster.
For the data reduction we used a modified version of the software described in 2006A&A...454.1029A. Briefly, for each image we obtained a grid of 18 spatially varying empirical point spread functions (PSFs, an array of PSFs for each chip of FORS2) using the most isolated, bright and not saturated stars. In this way, to each pixel of the image corresponds a PSF that is a bi-linear interpolation of the closest four PSFs of the grid. This makes it possible to measure star positions and fluxes in each individual exposure using an appropriate PSF and to obtain a catalog of stars for each frame. We registered, for each cluster and for each filter, all star positions and magnitudes of each catalog into a common frame (master-frame) using linear transformations. The final result is a list of stars (master list) for each cluster. For each filter and each star measured in NGC 6121, in Fig. 2 we plot the rms of the photometric residual and of the position as a function of the mean magnitude. In the case of NGC 6752 and NGC 6397 the distributions are similar. As required by the referee, we specify here that we used magnitude to express the luminosities of stars in this paper.
We noted that all the CMDs of the three GCs showed unusually spread out sequences (see panel (a) of Fig. 3 for an example of the versus CMD of NGC 6397). Part of this spread is due to differential reddening, but this is not the only cause. In fact, we found that, selecting stars in different regions of the master list, we obtained shifted MSs. As an example, in panel (d) of Fig. 3 we show the variation of the color for NGC 6397. This plot shows that there is an important gradient of along the x-axis. We selected in this plot two groups of stars: the stars with and that with . We plotted these two sub-samples in the versus CMDs, respectively in black crosses () and red circles (): panel (a) of Fig. 3 shows the result. The two groups form two shifted MSs. This effect is present in all the CMD of the three GCs, even if with different extent levels.
2007ASPC..364..113F showed that there is an illumination gradient in the FORS2 flats produced by the twilight sky. This gradient changes with time and with the position of the Sun relative to the pointing of the telescope and could produce a photometric zero point variation across the FORS2 detectors.
The illumination gradient in the flat fields could contribute to the observed enlargement of the CMDs.
We obtained star fluxes using local sky values, and therefore it is expected that these systematic effects are negligible. If the gradient in the flat-field images is not properly removed during the pre-reduction procedure, the pixel quantum efficiency correction will be wrong. The consequence is that the luminosity of a star measured in a given location of the CCD will be underestimated (or overestimated) with respect to the luminosity of the same star measured in another location of the CCD.
Using the measured star positions and fluxes, we performed a correction in a similar way to what described by 2009A&A...493..959B. It is a self-consistent auto-calibration of the illumination map and takes advantage of the fact that the images are well dithered.
For each cluster and for each filter, the best image (characterized by lower airmass and best seeing) is defined as reference frame. We considered the measured raw magnitude of each star in each image . Using common stars between the image and the reference frame we computed the average magnitude shift:
Δj=1MM∑i=1(mij−mi,ref)
where M is the number of stars in common between the reference frame and the single image . For each star, that is centered in a different pixel in each dithered frame, we computed its average magnitude in the reference system:
¯m=1NN∑j=1(mij−Δj)
where is the number of images in which the star appears. Then we computed the residual:
δij=mij−Δj−¯mj.
We divided each FORS2 chip in a spatial grid of boxes, and, for each box, we computed the average of the residuals from the stars located in that region in each single image. This provides a first spatial correction to our photometry. To obtain the best correction, we iterate until the residual average became smaller than 1 mmag. To guarantee convergence we applied, for each star, half of the correction calculated in each box. Moreover, to obtain the best correction at any location of the camera, we computed a bi-linear interpolation of the closest 4 grid points. At the edges of the detectors the correction is less efficient, because the corresponding grid-points have been moved toward the external borders of the grid to allow the bi-linear interpolation to be computed all across the CCDs. In panel (e) of Fig. 3 we show our final correction grid for the v_HIGH filter.
The final correction grids are different for each filter and for each set of images. In particular, the patterns are different from filter to filter, as well as the size of the zero point variations. This quantity also changes using different data-sets. The maximum amplitudes of our corrections are tabulated in Table 2.
We corrected the spread of the CMD due to differential reddening using the procedure as described by 2012A&A...540A..16M. Briefly, we defined a fiducial line for the MS of the cluster. Then, for each star, we considered a set of neighbors (usually 30, selected anew for each filter combination) and estimated the median offset relative to the fiducial sequence. These systematic color and magnitude offsets, measured along the reddening line, represent an estimate of the local differential reddening. With this procedure we also mitigated the photometric zero-point residuals left by the illumination correction (especially close to the corners of the field of view). Panel (c) of Fig. 3 shows the CMD after all the corrections are applied.
The photometric calibration of FORS2 data for Johnson and Cousins bands was obtained using the photometric Secondary Standards star catalog by 2000PASP..112..925S. We matched our final catalogs to the Stetson standard ones, and derived calibration equations by means of least squares fitting of straight lines using magnitudes and colors.
### 3.1 Proper motions
Since NGC 6121 (l,b=35097,1597) and NGC 6397 (l,b=33817;1196) are projected at low Galactic latitude, their CMDs are both dramatically contaminated by Disk and Bulge stars, contrary to NGC 6752 (l,b=33649;2563) that presents low field contamination. The average proper motions of NGC 6121, NGC 6397 and NGC 6752 strongly differ from that of these field stars (e.g. 2003AJ....126..247B, 2006A&A...456..517M). Therefore, to minimize the contamination from field stars we identified cluster members on the basis of stellar proper motions.
In order to get information on the cluster membership, we estimated the displacement between the stellar positions obtained from our FORS2 data-set and those in the ground-based data taken from the image archive maintained by 2000PASP..112..925S and also used in 2013MNRAS.431.2126M. These observations include images from different observing runs with the Max Planck 2.2m telescope, the CTIO 4m, 1.5m and 0.9m telescopes and the Dutch 0.9m telescope on La Silla. To obtain the displacement, we used six-parameter local transformations based on a sample of likely cluster members, in close analogy to what was done by 2003AJ....126..247B and 2006A&A...454.1029A to calculate stellar proper motions.
Results are shown in Fig. 4. The figure shows the versus CMDs for NGC 6121, NGC 6397 and NGC 6752. The insets show the vector-point diagrams of the stellar displacements for the same stars shown in the CMDs: the cluster-field separation is evident. Likely cluster members are plotted in black both in the CMDs and in the vector-point diagrams, in gray the rejected stars.
## 4 The CMDs of the three GCs
Previous studies on multiple stellar populations have demonstrated that the color is very efficient in detecting multiple RGBs (see 2008A&A...490..625M and 2010ApJ...709.1183M for the cases of NGC 6121 and NGC 6752), and multiple MSs (see 2012A&A...540A..16M, Mi13 for the cases of NGC 6397 and NGC 6752). As discussed by 2011A&A...534A...9S, CNO abundance variations affect wavelengths shorter than nm owing to the rise of molecular absorption bands in cooler atmospheres. The consequences are that the CMDs in filters show enlarged sequences, mainly due to variations in the N abundance, with the largest variations affecting the RGB and the lower MS.
Motivated by these results, we started our analysis from the versus CMDs shown in Fig. 5; the inset of each panel is a zoom in of the upper MS, between and magnitudes below the turn off.
A visual inspection at these CMDs reveals that the color broadening of MS stars in both NGC 6121 and NGC 6752 is larger than that of NGC 6397. A small fraction of MS stars in NGC 6121 and NGC 6752 defines an additional sequence on the blue side of the most-populous MS.
To investigate whether the widening of the MSs is due to the presence of multiple populations we identified in the versus CMD of each cluster two groups of red-MS (rMS) and blue-MS (bMS) stars, as shown in panels (a) of Fig. 6. We colored the two MSs in red and blue respectively, and these colors are used consistently hereafter. In the case of NGC 6121 and NGC 6752, where there is some hint of a split MS, we identified by eye the fiducial that divide the rMS and bMS. In the case of NGC 6397 we have considered as rMS (or bMS), the stars that are redder (or bluer) than the MS fiducial line. Each inset shows the color distribution of for the two MSs, where is obtained by subtracting the color of the fiducial that divide the two MSs to the color of the rMS and bMS stars. In the cases of NGC 6121 and NGC 6752 the color distributions show a double peak that could be due to the presence of two populations; we fitted them with a sum of Gaussians (in red and blue respectively for the rMS and bMS). We applied a moving box procedure to further verify that the distribution of in the case of NGC6121 is bimodal. We changed the binsize, ranging from 0.005 to 0.02 (approximately the error in color) with steps of 0.001. Furthermore, from our dataset we determined a kernel-density distribution by assuming a Gaussian kernel with 0.02 mag. I all cases, we consistently found that the distribution can only be reproduced by two Gaussians.
A multiple sequence in NGC 6752 had already been identified by Mi13 using HST data. Very recently, we had the first F275W, F336W, F438W WFC3 images of NGC 6121 from the HST GO-13297 UV Large Legacy Program (P.I. Piotto). Even a preliminary reduction of the data shows a clear separation of the MS into two branches in the F438W vs F336W-F438W CMD, fully confirming what we anticipate here in the equivalent, groundbased vs diagram of Fig. 6.
In the case of NGC 6397, it is possible to fit the distribution with a single Gaussian.
As an additional check for the presence of multiple populations, we investigated whether the widening of the MSs of all the GCs is intrinsic or if it is entirely due to photometric errors. We have compared two CMDs, versus with versus , obtained using independent datasets. We considered the rMS and bMS defined previously: if the color spread is entirely due to photometric errors, a star which is red (or blue) in the versus CMD will have the same chance of being either red or blue in the vs . By contrast, the fact that the two sequences identified in the first CMD have systematically different colors in the second one, would be a proof that the color broadening of the MS is intrinsic. In panels (b) of Fig. 6 we plotted the rMS and bMS in versus CMDs. The fact that the rMS stars of both NGC 6121 and NGC 6752 have, on average, different than bMS stars demonstrates that the color broadening of their MS in the versus CMDs is intrinsic. This is the first evidence that the MS of NGC 6121 is not consistent with a simple stellar population. In the case of NGC 6397 rMS and bMS stars share almost the same thus suggesting that most of the colors broadening is due to photometric errors.
As last test, we plotted the two MSs in the versus CMD. The index, which is defined as the color difference , is a very efficient tool to identify multiple sequences in GCs. Indeed it maximizes the color separation between the stellar populations that is due to both helium and light-element variations (2013MNRAS.431.2126M). Panels (c) of Fig. 6 confirm the previous results: rMS and bMS of NGC 6121 and NGC 6752 are well defined in the versus CMDs, but this is less evident for NGC 6397.
As a last prof, we plotted in Fig. 7 the versus diagrams for each cluster: in red and in blue are plotted the rMS and bMS defined previously. The figure shows that for both NGC 6121 and NGC 6752 the two MSs are well defined, while for NGC 6397 the rMS and bMS stars are mixed
Figure 8 shows the versus diagram for the three GCs studied in this paper. In their analysis of multiple stellar populations in 22 GCs, 2013MNRAS.431.2126M found that all the analyzed clusters show a multimodal or spread RGB in the versus diagram, and the value of each star depends on its light-element abundance. The -index width of the RGB () correlates with the cluster metallicity, with the more metal rich GCs having also the largest values of . In order to compare the MSs of the three GCs studied in this paper, we introduce a quantity, , which is akin of , but is indicative of the -index broadening of the MS. The procedure to determine is illustrated in Fig. 9 for NGC 6121 and is the same for the other clusters. We have considered the magnitude of the TOs as magnitude of reference: in the case of NGC 6397, for NGC 6752, and for NGC 6121. Panel (a) of Fig. 9 shows the versus diagram for NGC 6121 in a range of magnitudes from to . In this range of magnitudes, we obtained the fiducial line for the MSs computing the -clipped median of the color in interval of 0.35 mag and interpolated these points with a spline. In our analysis we used only MS stars with where the MS split is visible for both NGC 6121 and NGC 6752. This magnitude interval is delimited by the two dashed lines of Fig. 9a. The thick line is the fiducial in the considered magnitude interval. The verticalized versus diagrams is plotted in panel (b) of Fig. 9, while panel (c) shows the histogram distribution of . The MS width, , is defined as the extension of the histogram and is obtained by rejecting the 5% of the reddest and the bluest stars on the extreme sides. To account for photometric error, we have subtracted from the observed the average error in in the same magnitude interval, i.e. .
We found that the most metal-rich GC, NGC 6121, exhibits the largest index width for MS stars ().
The spread in is smaller in the case of NGC 6752 () and drops down to in the most metal-poor GC NGC 6397. To estimate the statistical uncertainty in measuring , we used the bootstrap resampling of the data to generate 10000 samples drawn from the original data sets. We computed the standard deviation from the mean of the simulated and adopted this as uncertainty of the observed .
These findings make it tempting to speculate that the index width of the MS could be correlated with the cluster metallicity, in close analogy with what observed for RGB stars. An analysis of a large sample of GCs is mandatory to infer any conclusion on the relation between and [Fe/H].
Mi13 have identified three stellar populations in NGC 6752 that they have named as ‘a’, ‘b’, and ‘c’. Population ‘a’ has a chemical composition similar to field halo stars of the same metallicity, population ‘c’ is enhanced in sodium and nitrogen, depleted in carbon and oxygen and enhanced in helium (0.03), while population ‘b’ has an intermediate chemical composition between ‘a’ and ‘c’ and is slightly helium enhanced (0.01). However, the MSs of populations ‘b’ and ‘c’ are nearly coincident in the versus CMD, while population ‘a’ stars have bluer colors (Mi13, see their Fig. 8). The three MSs exhibit a similar behavior also in the and colors. Since these colors are similar to , the less populous MS identified in this paper should correspond to the population ‘a’ identified by Mi13, while the rMS hosts both population ‘b’ and population ‘c’ stars.
### 4.1 The fraction of rMS and bMS in NGC 6121 and NGC 6752
In order to measure the fraction of stars in each MS we followed the procedure illustrated in Fig. 10 for NGC 6752, which we already used in several previous papers (e.g. 2007ApJ...661L..53P, Mi13).
Panel (a) shows the versus CMD of the MS stars in the magnitude interval , where the MS split is most evident. We verticalized the selected MS by subtracting the color of the stars to the color of the fiducial line of the rMS, obtaining . The fiducial line is obtained by hand selecting the stars of the rMS, dividing them in bins of magnitude, computing the median colors of the stars within each bin and interpolating these median points with a spline. The verticalized versus diagram is plotted in panel (b).
The color distribution of the stars for three magnitude intervals is shown in panels (c). Each histogram clearly shows two peaks and has been simultaneously fitted with a double Gaussian, whose single components are shown in blue and in red for the bMS and the rMS, respectively.
For each magnitude interval, from the area under the Gaussians we infer the fraction of bMS and rMS stars. The errors () associated to the fraction of stars are estimated as , where is derived from binomial statistics and is the uncertainty introduced by the histogram binning and is derived as in 2014A&A...563A..80L. Briefly, we have derived times the population ratio as described above, but by varying the binning and starting/ending point in the histogram. We assumed as the rms scatter of these determinations.
We computed the weighted mean of the bMS and rMS fractions of the three magnitude intervals, using as weight . In the case of versus , we obtained that the rMS and bMS contain respectively and of MS stars. In panels (d), (e) and (f) of Fig. 10 we applied the same procedure in the versus using the stars with . We obtained that the blue MS contains of the total number of MS stars, and the red MS is made of the remaining stars. We also calculated the weighted mean of the results of the two CMDs of NGC 6752, obtaining that in the rMS there are the of stars, and in the bMS the remaining .
We have already demonstrated that the distribution, in the versus CMD of NGC 6121, shows a double peak, proving the presence of multiple populations (see panel a of Fig. 6). We performed a detailed analysis of the MS of NGC 6121, applying the same procedure described for NGC 6752. The procedure and the results are shown in Fig. 11. We find from the analysis of the versus CMD that the bMS contains of MS stars, and the rMS includes the remaining . In the case of the versus diagram we infer that rMS and bMS contain and of the total number of MS stars, respectively. We computed the fraction of rMS and bMS for NGC6121 using different histogram binsizes and changing starting/ending points. We used binsizes with values between 0.005 and 0.025 (larger than color error), starting points between -0.5 and -0.15 and ending points between 0.15 and 0.5. I all cases, the resulting fraction of stars are in agreement with the values we quote above within the errors The results from the two CMDs imply that in the rMS and bMS there are, respectively, and of MS stars.
## 5 The radial distribution of stellar populations in NGC 6752 and NGC 6121
The analysis of the radial distribution of rMS and bMS stars in NGC 6121 and NGC 6752 is an important ingredient to shed light on the formation and the evolution of multiple stellar populations in these GCs. Indeed, theoretical models predict that, when the GC forms, second-generation stars should be more centrally concentrated than first-generation ones, and many GCs could still keep memory of the primordial radial distribution of their stellar populations (e.g. 2008MNRAS.391..825D, 2011MNRAS.412.2241B, 2013MNRAS.429.1913V).
The radial distribution of stellar populations in NGC 6752 is still controversial. 2011A&A...527L...9K determined wide-field multi-band photometry of NGC 6752 and studied the distribution of its stellar populations across the field of view. They have concluded that there is a strong difference in the radial distribution between the populations of RGB stars that are bluer (bRGB) and redder (rRGB) in color, and obtained similar findings from the study of the SGB. Specifically, at a radial distance close to the half-mass radius (; 1996AJ....112.1487H, 2010 edition) the fraction of rRGB stars abruptly decreases. These results are in disagreement with the conclusions by Mi13 who showed that the three stellar populations identified in their paper share almost the same radial distribution. Kravtsov and collaborators analyzed stars with a radial distance from the center of NGC 6752 out to , while the study by Mi13 is limited to the innermost arcmin. In this paper we extend the analysis to larger radii111We assume that stars in the fields of NGC6752 and NGC6121 are representative of stellar populations at the studied radial distance; we are not able to investigate any dependency on the angular position using the dataset presented in this work..
As already mentioned in Sect. 4, we suggest that the bMS of NGC 6752 corresponds to the population ‘a’ identified by Mi13, while the most populous rMS hosts both population ‘b’ and population ‘c’ stars of Mi13. For this reason, in this section, we rename the bMS in MSa and the rMS in MSbc.
In order to investigate the radial distribution of stellar populations within the field of view analyzed in this paper, we divided the catalog of NGC 6752 stars into two groups at different radial distance from the cluster center, each containing almost the same total number of stars.
The inner sample of stars (inner field) lies between and from the cluster center. The outer group of stars (outer field) is between and from the center. We estimated the fraction of stars in each group by following the same procedure described in Sect. 4.1.
The results are illustrated in Fig. 12. In the left panels we show the verticalized versus and the versus diagrams for stars in the inner field. In this region the MSa contains and the MSbc hosts the remaining of the total number of MS stars. In the outer field (right panels of Fig. 12) the MSa and the MSbc are made of the and of MS stars, respectively. We conclude that there is no evidence for a gradient within the field of view studied in this paper.
To further investigate the radial distribution of stellar populations in NGC 6752 we compare the results obtained in this paper for stars with distance from the cluster center larger than arcmin and the fraction of stars that have been estimated by Mi13 in the internal regions by using the same method.
Since the MSbc contains both populations ‘b’ and population ‘c’ stars, we have added together the fractions of population ‘b’ () and population ‘c’ stars () listed by 2013ApJ...767..120M and calculated the fraction of stars in these two populations: . As aforementioned in Sect. 4, we further compare the fractions of population ‘a’ stars by Mi13, with the fractions of MSa stars derived in this paper. The values of and are listed in Tab. 3.
Results are shown in Fig 13, where the top panels show the distribution of the fraction of population ‘a’ (in blue) and the fraction of population ‘b’+‘c’ (in red) as a function of the radial distance from the cluster center, while the bottom panels show the radial trend of the ratio between the fraction of population ‘a’ and the fraction of population ‘b’+‘c’. In the left panels we show both the above described distributions considering single radial intervals for each set of data, while in the right panels we divided each radial interval in different bins. Our findings suggest that there is no evidence for a radial gradient among population ‘a’ and population ‘b’+‘c’ of NGC 6752.
In order to investigate the radial distribution of stellar populations in NGC 6121, we divided the field of view analyzed in this paper into two regions, with radial distance from the cluster center (inner field) and (outer field). Each region contains almost the same number of stars. We determined the fraction of rMS and bMS stars by following the same recipe described in detail for NGC 6752. The results are shown in Fig. 14. We found that in the inner field the fraction of bMS is and the fraction of rMS is . For the outer field we obtain that the bMS and the rMS contains respectively the and the of the total number of the considered MS. Also for NGC 6121 we found no evidence of population gradients.
## 6 The helium content of stellar populations in NGC 6121 and NGC 6752.
The ultraviolet pass-band is very efficient to separate multiple sequences due to its sensitivity to difference in C, N, O abundance (2008A&A...490..625M, 2011A&A...534A...9S). In contrast, and colors are marginally affected by light-element variations, but are very sensitive to the helium abundance of the stellar populations (e.g. 2002A&A...395...69D, 2007ApJ...661L..53P, 2011A&A...534A...9S, 2012AJ....144....5K, 2013MmSAI..84...91C), thus providing us with an efficient tool to infer the helium content.
### 6.1 Ngc 6121
The procedure to estimate the average helium difference between bMS and rMS stars for NGC 6121 is illustrated in Fig. 15 and is already used in several papers by our group (Mi13, 2012ApJ...745...27M; 2012ApJ...744...58M). Since we have already extracted the stellar populations in NGC 6121 by using the versus CMD of Fig. 11, we can now follow them in any other CMD. By combining photometry in four filters, we can construct three CMDs with versus (), where . The fiducial lines of bMS and rMS in these CMDs are plotted in the upper panels of Fig. 15. rMS is redder than bMS in color, whereas it is bluer in and colors.
We measured the color distance between the two MSs at a reference magnitude (), and repeated this procedure for =17.3, 17.5, 17.7, 17.9, and 18.1 (corresponding to the magnitude interval where the two MS separations is maximal, cf Fig. 6). The color difference () is plotted in the lower panel of Fig. 15 as a function of the central wavelength of the filter (gray dots), for the case of =17.7.
We estimated effective temperatures () and gravities () at different = for the two MS stars and for the different helium contents by using BaSTI isochrones (2004ApJ...612..168P; 2009ApJ...697..275P).
We assumed a primordial helium abundance for the bMS, , and used for the rMS different helium content, with varying from 0.248 to 0.400 in steps of =0.001. To account for the appropriate chemical composition of the two stellar populations of NGC 6121 we assumed for the bMS and the rMS the abundances of C, N, O, Mg, Al, and Na as measured for first and second-generation RGB stars listed by 2008A&A...490..625M.
We used the ATLAS12 program and the SYNTHE code (2005MSAIS...8...14K,2005MSAIS...8...25C, 2007IAUS..239...71S) to generate synthetic spectra for the adopted chemical compositions, from 2,500 Å to 10,000 Å. Synthetic spectra have been integrated over the transmission curves of the filters, and, we calculated the color difference for each value of helium of our grid.
The best-fitting model is determined by means of chi-square minimization. Since the magnitude is strongly affected by the abundance of light elements we used and colors only to estimate . The helium difference corresponding to the best-fit models are listed in Table 4 for each value of .
We derived that the rMS is slightly helium enhanced with respect to the bMS (which has ), with an average helium abundance of This is the internal error estimated as the rms scatter of the independent measurements divided by the square root of . Results are shown in Fig. 15 for the case of =17.7, where we represented the synthetic colors corresponding to the best-fitting model as red asterisks.
For completeness we also calculated synthetic colors of two MS stars with the and the same chemical composition (same abundance of light elements). We assumed for bMS primordial helium and for rMS the helium abundance of the best-fitting model. Results are represented as blue squares in Fig. 15 and confirm that the abundance of light elements assumed in the model does not affect our conclusion on the helium abundance of the two MSs, which are based on the optical colors. Instead the different CNO content strongly affect the band.
In principle, the He content of stellar populations in GCs can also be estimated using He lines in HB star spectra (e.g. the HeI line at line, 2009A&A...499..755V; 2012ApJ...748...62V; 2014MNRAS.437.1609M ). However, spectroscopic measurement of He in GC stars has many limitations. First of all, He can only be measured for stars in a very limited temperature interval ( K). In fact, stars with K are not sufficiently hot to form He lines, while stars bluer than the Grundahl jump (1999ApJ...524..242G, K) are affected by He sedimentation and metal levitation which alter the original surface abundance. The HB of NGC 6121 is populated both on the red and the blue side of the RR Lyrae instability strip. Spectroscopic investigation by 2011ApJ...730L..16M reveals that the blue HB is made of second population Na-rich and O-poor stars, while red HB stars belong to the first population. In NGC 6121, the HB segment with K corresponds to the blue HB, and therefore it only provides partial information only. In this cluster (as in many others) it is not possible to spectroscopically measure the He content of the first population.
2012ApJ...748...62V have used the HeI line at to estimate the helium content of six blue-HB stars in the blue HB of NGC 6121. All of them are second-population stars. They derived a mean value of =0.290.01 (random) 0.01 (systematic) and conclude that second-population stars would be enhanced in helium by 0.04-0.05 dex. This estimate of the He content has been made by assuming LTE approximation. However, the HeI line at is affected by NLTE effect, which can cause an error in the estimate as large as =0.10 (see 2014MNRAS.437.1609M for the case of NGC 2808). Appropriate NLTE analysis is required to infer reliable He abundances from spectroscopy of HB stars in NGC 6121. In contrast, the He difference between red- and blue-MS stars in NGC 6121 comes from the colors of the fiducial lines, which have small color uncertainties.
### 6.2 Ngc 6752
We followed the same procedure to estimate the average helium difference between MSa and MSbc stars. We measured the color distance between the two fiducial lines of MSa and MSbc in the versus CMDs (Fig. 16), where , at reference magnitudes and . The color difference at is plotted in the bottom panel of Fig. 16 as a function of the central wavelength of the filter.
We used BaSTI isochrones (2004ApJ...612..168P; 2009ApJ...697..275P) to estimate and at different .
We assumed that MSa has primordial helium abundance, , and varied the helium content of the MSbc between 0.248 and 0.400 in steps of . We assumed for the MSa the same C, N, O, Mg, Al and Na abundances of the population ‘a’ of Mi13; for the chemical composition of the MSbc we considered the average of the abundances of the population ‘b’ and ‘c’ listed by Mi13.
As mentioned above, we obtained synthetic spectra for the adopted chemical compositions, integrated them over the transmission curves of the , , , filters, and computed the helium difference using the best-fitting model.
We obtained that the MSbc is helium enhanced with respect to the MSa of . As for NGC 6121, since the magnitude is affected by the abundance of light elements, we used and colors only to estimate . Note that the abundance of light elements assumed in the model does not affect our conclusion on the helium abundance of the two MSs, as already proved in the case of NGC 6121.
### 6.3 Relation between HB morphology and Helium abundance
In their work, 2014ApJ...785...21M have sought correlations between HB morphology indicators and physical and morphological GC parameters. Among these parameters there is also the maximum helium difference between stellar populations hosted by GCs.
They introduced two different parameters to describe the HB morphology: , that is the color difference between the RGB and the coolest border the HB, and , that is the color extension of the HB (for more details, see Fig. 1 of 2014ApJ...785...21M).
They divided the sample of 74 GCs in three groups: in the first group, G1, there are GCs with [Fe/H]; the second group, G2, includes GCs with [Fe/H] and ; the third group, G3 contains GCs with .
They found a tight correlation between and the maximum internal helium difference (, measured on the MS) for the group G2+G3 (see Fig. 8 of their paper).
In our work we add two more points to their data-set, the Helium difference between the two populations of NGC 6752 and NGC 6121, as computed in this work. In the case of NGC 6752, the added point constitutes a lower limit because the Helium difference between Pop and Pop, (Pop-Pop), is the average value between (Pop-Pop) and (Pop-Pop).
The result is in Fig. 17: in black there are the points of 2014ApJ...785...21M and in grey the points added in this work. In analogy to the work of 2014ApJ...785...21M, the crosses refer to the G1 GCs, triangles to G2 group and dots to G3 clusters. Our data points confirm the tight correlation between L2 and . We found a Spearman’s rank correlation coefficient (to be compared to found by 2014ApJ...785...21M), with (the uncertainty in is estimated by means of bootstrapping statistic, as in 2014ApJ...785...21M).
This result is a further proof that the helium-enhanced stellar populations are likely related to the HB extension, as predicted by theory.
## 7 Summary
The photometric analysis of ESO/FORS2 data of the external regions of the three nearby Galactic GCs NGC 6121 (M 4), NGC 6752 and NGC 6397 has confirmed that the first two GCs host multiple stellar populations. Indeed, the versus and versus CMDs of NGC 6752 and NGC 6121 show a split of the MS in two components. Excluding the unique case of Cen, this is the first time that a split of the MS is observed using ground-based facilities.
The multiple stellar populations of NGC 6397 was investigated by 2012ApJ...745...27M using HST data. They found two stellar populations characterized by a modest helium variation . Unfortunately, in this work, it was not possible to analyze these populations, because of the size of our photometric errors is comparable to the small color separation between the MSs.
Using HST data, Mi13 have already demonstrated that NGC 6752 host three stellar populations. They computed the radial trend of the ratio between the number of stars of different populations out a radial distance from the center of . Because of larger photometric errors, we have resolved only two MSs. Comparing them with the work of Mi13, we found that the less populous MS corresponds to their population ‘a’, while the most populous MS hosts both their populations ‘b’ and ‘c’. In average we found that the MSa contains about 26% of the total number of stars and the MSbc host about 74% of the MS stars. The most straightforward interpretation is that the MSa is formed by stars of the first generation with chemical abundances similar to that of the Galactic halo field stars with the same metallicity; the MSbc hosts stars of second generation, formed out of material processed through first-generations stars. This population is characterized by stars enhanced in helium, with . Our measurement of the helium enhancement is in agreement with the average of the populations ‘b’ () and ‘c’ () obtained by Mi13. We extended the study of the radial trend of the populations of NGC 6752 to more external regions, confirming the results of Mi13, of a flat distribution. Therefore we cannot confirm the results by 2011A&A...527L...9K and 2014ApJ...783...56K; they found that the two populations show a strong gradient at a radial distance close to the half-mass radius.
In a recent work on NGC 6121, 2014MNRAS.439.1588M investigate the bottom of the MS of this cluster using HST near-infrared photometry. They found that the MS splits into two sequences below the MS knee. In particular they identified two MSs: a MS that contains of stars and MS formed by the remaining . They show that the split of the MS is mainly due to the effect of molecules, present in the atmospheres of M-dwarfs, on their near-infrared color, and that it is possible to associate the MS to a first generation of stars and the MS to a second one. 2008A&A...490..625M, analyzing spectra of RGB stars, found that of stars are Na-rich and O-poor and the remaining have chemical abundances similar to those of Halo-field stars with the same metallicity. All these results are in agreement with what we have obtained in this work: the MS of NGC 6121 splits in the versus and versus CMDs. We found two MSs: a less populous MS that contains of MS stars and which constitutes the first generation of stars and a more populous second generation MS that contains of stars. 2012ApJ...748...62V, using spectroscopic measurements of blue HB (bHB) stars, obtained that the difference in helium abundance between these stars and the red HB (rHB) stars is . A spectroscopic analysis of 2011ApJ...730L..16M revealed that the rHB stars have solar-scaled [Na/Fe], while bHB stars are Na enhanced. In contrast to the results of 2012ApJ...748...62V, a lower constraint to the level of He enhancement is set by 2014ApJ...782...85V, founding a maximum between bHB and rHB stars. Analyzing how the two MS of NGC 6121 behave in different CMDs, we computed the helium abundance difference between them. Our result is , in agreement with that obtained by 2012ApJ...748...62V. Also in the case of NGC 6121, we did not find evidence of changes in the fraction of bMS/rMS stars in the radial range between .
2014ApJ...785...21M found a correlation between the HB morphological parameter and the maximum helium difference among stellar populations in GCs. Using the helium abundances computed in this work for NGC 6121 and NGC 6752, we confirm this correlation and the theoretical indications that helium enhanced stellar populations are responsible of the HB extension.
###### Acknowledgements.
DN is supported by a grant “Borsa di studio per lâestero, bando 2013” awarded by “Fondazione Ing. Aldo Gini” in Padua (Italy). APM acknowledges the financial support from the Australian Research Council through Discovery Project grant DP120100475. AFM has been supported by grants FL110100012 and DP120100991.
## References
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http://www.physicsforums.com/showthread.php?t=485503&page=3 | # If light has no weight, how can it push objects?
by Jarfi
Tags: light, mass, momentum, wtf
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P: 6,187
Quote by Dickfore So, what's the 'real mass' of a photon?
I'm going with DaleSpam here.
What we have is 'rest mass' which is 0.
And 'relativistic mass' which is E/c2.
P: 3,014
Quote by I like Serena I'm going with DaleSpam here. What we have is 'rest mass' which is 0.
Exactly, and when we have 0 of something, we say we have none. Hence, photons are massless.
P: 38 Can somebody explain to an ignorant lay man: where is gone mass of electron and positron in their proces of so called " annihilation "? plus what about magnetic property of electron, from where it come? I will be very gratefull. mquirce
Mentor
P: 15,055
Quote by I like Serena I'm going with DaleSpam here. What we have is 'rest mass' which is 0. And 'relativistic mass' which is E/c2.
If you are going with DaleSpam, why didn't you take note of that key phrase "which is rarely"?
Relativistic mass is just a synonym for energy. So what value does the concept add?
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P: 16,942
Quote by mquirce Can somebody explain to an ignorant lay man: where is gone mass of electron and positron in their proces of so called " annihilation "?
It hasn't gone anywhere. The invariant mass of the electron-positron system is equal to the invariant mass of the photon-photon system.
http://www.physicsforums.com/showpos...3&postcount=56
Note that the mass of a system is not equal to the sum of the masses of the consitutent particles.
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Quote by D H If you are going with DaleSpam, why didn't you take note of that key phrase "which is rarely"? Relativistic mass is just a synonym for energy. So what value does the concept add?
I get the feeling you feel affronted by me.
P: 38 Sorry for my stupidity, but i don,t understand sentence ' invariant mass of photon photon, and the meaning of summa of masses. If i bother you live with no more. mquirce.
C. Spirit
Thanks
P: 5,401
Quote by mquirce Sorry for my stupidity, but i don,t understand sentence ' invariant mass of photon photon, and the meaning of summa of masses. If i bother you live with no more. mquirce.
Basically there are certain measurable quantities that have a defined global meaning over a manifold. There are other quantities like energy which depend on the metric being used to describe the geometry of the space - time and one cannot say that the same metric field covers the entire manifold. There can be different metrics being used to describe different parts of the manifold. Quantities that don't depend on the coordinate system being used are invariant.
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Quote by D H If you are going with DaleSpam, why didn't you take note of that key phrase "which is rarely"? Relativistic mass is just a synonym for energy. So what value does the concept add?
I don't really believe in what people "usually" mean, or what "all accounts" say, or "which is rarely".
If it matters what you mean, it needs to be specified if it's not clear from the context.
In non-relativistic contexts it doesn't matter what you mean when you mention "mass".
Otherwise it needs to be specified.
I guess I've been neglecting to be clear myself, using the word "mass" for "relativistic mass". My apologies for that - I'll be more careful.
What does the concept of "relativistic mass" add?
Well, for instance, you can predict the effect of a gravitational lens on photons.
Or you can predict how much photons add to dark matter mass.
Or you can say something about conservation of momentum in relativistic situations since you can't really ignore "mass-less" photons in quantum physical reactions.
If I've insulted you somehow, please enlighten me.
Otherwise accept my apologies for it was not my intention to insult anyone.
P: 3,014
Quote by I like Serena What does the concept of "relativistic mass" add?
Quote by I like Serena Well, for instance, you can predict the effect of a gravitational lens on photons.
Photons travel along null geodesics in cuverd spacetime (since they are massless), which we interpret as bending of their trajectory. No need for relativstic mass. In fact, if you try to describe it with relativistic mass, you will encounter trouble because Newton's Law of Universal Gravitation is not relativistically covariant.
Quote by I like Serena Or you can predict how much photons add to dark matter mass. '
Dark matter is defined as that part of the matter content responsible for the extra gravitational field of galaxies, but which does not interact via the electromagnetic interaction, hence it's dark. Photons do not satisfy this criterion.
Quote by I like Serena Or you can say something about conservation of momentum in relativistic situations since you can't really ignore "mass-less" photons in quantum physical reactions.
What does momentum have to do with mass? Those are distinct physical quantities. In fact, the OP made a non-sequitur. Photons DO push objects because they carry MOMENTUM (and transfer it), not because of their mass.
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P: 15,055
Quote by Dickfore What does momentum have to do with mass? Those are distinct physical quantities. In fact, the OP made a non-sequitur. Photons DO push objects because they carry MOMENTUM (and transfer it), not because of their mass.
What he said.
This discussion of the intrinsic mass of a collection of photons has diverted this thread from the original question. The intrinsic mass of a single photon is identically zero. The momentum of a single photon is not zero. It is instead given by p=hf/c=h/λ where h is Planck's constant, f is the photon's frequency, and λ is its wavelength.
I like Serena, you have used E=mc2 multiple times in this thread. A much better version of this equation is
$$E^2 = (m_0c)^2 + (pc)^2$$
where m0 is the rest mass and p is momentum. For photons, which have a rest mass of zero, this reduces to E=pc. No mention of mass whatsoever.
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P: 6,187
Quote by Dickfore Photons travel along null geodesics in cuverd spacetime (since they are massless), which we interpret as bending of their trajectory. No need for relativstic mass. In fact, if you try to describe it with relativistic mass, you will encounter trouble because Newton's Law of Universal Gravitation is not relativistically covariant.
Interesting.
Can you say how much Newton is "off"?
Quote by Dickfore Dark matter is defined as that part of the matter content responsible for the extra gravitational field of galaxies, but which does not interact via the electromagnetic interaction, hence it's dark. Photons do not satisfy this criterion.
True enough.
So how about the contribution of photons to the known mass in the universe?
Quote by Dickfore What does momentum have to do with mass? Those are distinct physical quantities. In fact, the OP made a non-sequitur. Photons DO push objects because they carry MOMENTUM (and transfer it), not because of their mass.
Doesn't the equation p = mrelativistic v = γ m0 v still hold?
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P: 6,187
Quote by D H A much better version of this equation is $$E^2 = (m_0c)^2 + (pc)^2$$ where m0 is the rest mass and p is momentum. For photons, which have a rest mass of zero, this reduces to E=pc.
I agree that this is a better version.
My point is mainly that it is a matter of perspective.
Afaik both forms are true, or in other words, it's "relative", which is what relativity theory is all about.
PF Gold P: 60 In plain english a photon has momentum because of the energy it carries. If it loses ALL of that energy then there is no photon, therefore it is massless. Some may argue that a virtual photon remains, but it is still massless.
P: 9 "Light, photons, is an electromagnetic field which exerts force on charges --> Push objects." OK, I understand this. The light is an electromagnetic field, and the electromagnetic field interacts with the electrons, and the protons of the material, pushing them back and forth. But why would that push the solar sail away from the light source? The electrons (and protons) are going to vibrate perpendicularly to the direction of the ray of light that shines on the sail, right?, because the magnetic and electric fields of the light ray are perpendicular to the direction the light ray is traveling. The 'pressure' on the electron would be back and forth perpendicularly to the direction the light ray is traveling. I see that causing heat, warming up the sail. But, how does that cause the sail to move away from the light source?
P: 9
Quote by Jarfi And also when I shine a light on paper and it heats the paper(gives it momentum?) does it loose energy and change wavelength?
Does heating up the paper cause the photon to change wavelength, or to change amplitude?
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P: 15,055
Quote by dlr "Light, photons, is an electromagnetic field which exerts force on charges --> Push objects." OK, I understand this. The light is an electromagnetic field, and the electromagnetic field interacts with the electrons, and the protons of the material, pushing them back and forth. But why would that push the solar sail away from the light source? The electrons (and protons) are going to vibrate perpendicularly to the direction of the ray of light that shines on the sail, right?, because the magnetic and electric fields of the light ray are perpendicular to the direction the light ray is traveling. The 'pressure' on the electron would be back and forth perpendicularly to the direction the light ray is traveling. I see that causing heat, warming up the sail. But, how does that cause the sail to move away from the light source?
You are ignoring that while photons are massless particles, they have non-zero momentum.
When a photon interacts with a surface, one of three things will happen.
• Specular reflection.
The photon (or another one just like the incoming photon) will bounce off the surface as if the surface were a mirror. "Angle of incidence = angle of reflection". The imparted momentum will be twice that of the photon if the photon hits the surface squarely, less than that for the incidence angle less than 90 degrees.
• Diffuse reflection.
If the surface is microscopically rough, the reflection can be diffuse. Modeling diffuse reflection is a bit ad hoc. One widely used scheme is that the incoming photons impart all their momentum to the object, and the outgoing photons are distributed uniformly over the hemisphere pointing away from the object.
• Absorption / thermal emission.
Some photons are absorbed rather than reflected. The absorbed photons impart all of their momentum to the object. This heats the object. The heated object will emit thermal photons based on the local temperature. If the object is at a uniform temperature, the momentum transfer due to this thermal emission averages out to zero. If the object is not at a uniform temperature there will be a non-zero force on the object pointing roughly in the direction of the coolest point on the object. The hotter side emits more photons, and more energetic photons, than does the cooler side.
P: 9
Quote by D H You are ignoring that while photons are massless particles, they have non-zero momentum. When a photon interacts with a surface, one of three things will happen.Specular reflection. The photon (or another one just like the incoming photon) will bounce off the surface as if the surface were a mirror. "Angle of incidence = angle of reflection". The imparted momentum will be twice that of the photon if the photon hits the surface squarely, less than that for the incidence angle less than 90 degrees. Diffuse reflection. If the surface is microscopically rough, the reflection can be diffuse. Modeling diffuse reflection is a bit ad hoc. One widely used scheme is that the incoming photons impart all their momentum to the object, and the outgoing photons are distributed uniformly over the hemisphere pointing away from the object. Absorption / thermal emission. Some photons are absorbed rather than reflected. The absorbed photons impart all of their momentum to the object. This heats the object. The heated object will emit thermal photons based on the local temperature. If the object is at a uniform temperature, the momentum transfer due to this thermal emission averages out to zero. If the object is not at a uniform temperature there will be a non-zero force on the object pointing roughly in the direction of the coolest point on the object. The hotter side emits more photons, and more energetic photons, than does the cooler side.
But what is actually happening, physically, at the molecular level?
1) Specular Reflection When the photon "bounces off the surface" that means the photon was absorbed by an electron, and then emitted again, in some random direction, right? Same wavelength, so same energy, so how could any momentum (energy) have been imparted to the sail?
2) Thermal What I don't see is how heating translates to linear motion at the molecular level, with individual electrons, protons and photons. The energetic photon comes in and it's electromagnetic field causes all of the electrons (and protons?) it passes to begin vibrating -- but back and forth, not in any continuous direction. The electrons are accelerating/decelerating as they bounce back and forth, so they are giving off photons. But how does that process of giving off photons cause the molecule move? And why consistently in one direction, instead of randomly? I mean, what is going on physically, at the level of the individual electron or molecule?
Related Discussions General Physics 10 General Physics 27 Classical Physics 4 Introductory Physics Homework 1 General Physics 33 | {"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": 0.816623330116272, "perplexity": 616.2583843850758}, "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-2014-23/segments/1405997876165.43/warc/CC-MAIN-20140722025756-00172-ip-10-33-131-23.ec2.internal.warc.gz"} |
https://www.bartleby.com/solution-answer/chapter-155-problem-22e-calculus-early-transcendentals-8th-edition/9781285741550/if-you-attempt-to-use-formula-2-to-find-the-area-of-the-top-half-of-the-sphere-x2-y2-z2-a2/0162a75b-52f4-11e9-8385-02ee952b546e | Chapter 15.5, Problem 22E
### Calculus: Early Transcendentals
8th Edition
James Stewart
ISBN: 9781285741550
Chapter
Section
### Calculus: Early Transcendentals
8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem
# If you attempt to use Formula 2 to find the area of the top half of the sphere x2 + y2 + z2 = a2, you have a slight problem because the double integral is improper. In fact, the integrand has an infinite discontinuity at every point of the boundary circle x2 + y2 = a2. However, the integral can be computed as the limit of the integral over the disk x2 + y2 ≤ t2 as t → a-. Use this method to show that the area of a sphere of radius a is 4πa2.
To determine
To show: The area of the sphere of radius a is equal to 4πa2 .
Explanation
Formula used:
The surface area with equation z=f(x,y),(x,y)D , where fx and fy are continuous, is A(S)=D[fx(x,y)]2+[fy(x,y)]2+1dA .
Here, D is the given region.
If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)
If g(x) is the function of x and h(y) is the function of y then,
abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)
Given:
The sphere, x2+y2+z2=a2 .
Calculation:
Solve the given equation as shown below.
x2+y2+z2=a2z2=a2x2y2z=a2x2y2
The partial derivatives fx and fy are,
fx=2x2a2x2y2=xa2x2y2fy=2y2a2x2y2=ya2x2y2
Then, by the formula mentioned above, the area of the surface is given by,
A(S)=D(xa2x2y2)2+(ya2x2y2)2+1dA=Dx2a2x2y2+y2a2x2y2+1dA=Dx2+y2+a2x2y2a2x2y2dA=Da2a2x2y2dA
Use polar coordinates to solve
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#### Evaluate the indefinite integral. cos(/x)x2dx
Single Variable Calculus: Early Transcendentals
#### True or False: converges mean exists.
Study Guide for Stewart's Multivariable Calculus, 8th | {"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": 0.983718991279602, "perplexity": 1403.7292157698869}, "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-2019-43/segments/1570986657949.34/warc/CC-MAIN-20191015082202-20191015105702-00436.warc.gz"} |
http://perfdynamics.blogspot.com/2008/11/what-harmonic-mean-means.html | Wednesday, November 19, 2008
What the Harmonic Mean Means
As I discuss in Chapter 1 of The Practical Performance Analyst, time is the fundamental performance metric. Computer system performance metrics are therefore either direct measures of time, e.g, seconds, hours, minutes, or they are rates. All rate metrics have their units of time in the denominator, e.g., GB/s, MIPS, TPS, IOPS.
A conceptual difficulty can arise when we try to summarize a set of performance numbers as a single number; especially if they're rates.The usual approach is to employ the average value to express a single number, but there is more than one way to calculate an average or mean value. In particular, there is the arithmetic mean (A); the one you learnt in school, the geometric mean (G); used in SPEC CINT2006 benchmark ratings (rightly or wrongly), and the lesser-known harmonic mean (H). For example, the harmonic mean of three quantities: $a, b$, and $c$ is: $$H = \dfrac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} = \dfrac{3 \, abc}{a + b + c}$$ We discuss each of these means at length in the GDAT class. We also point out that the magnitude of these means are always ranked according to: H < G < A, when applied to the same data. In other words, except for special cases, H is always the least of them and A is always the greatest.
That said, almost all of us (including me) promptly ignore these distinctions in practice; especially when it comes to the application of the harmonic mean. One reason for this avoidance is the unintuitive nature of the harmonic mean. However, when it comes to averaging rate metrics, we really do need to apply the harmonic mean. Here's why.
All averaging involves addition. Because rate metrics are inverted-time quantities, adding rates is analogous to adding fractions with different denominators. For example, you can't add 5/3 + 1/5 by simply adding the top numbers together and the bottom numbers together (Well, you can but those are called Farey fractions, and that's a whole different topic). When adding fractions that have different denominators (in ordinary arithmetic), you need to find the Lowest Common Denominator or LCD. No doubt you remember from school, how unintuitive the LCD notion was. We need to do the same kind of thing to determine the harmonic mean, and you can see that going on in the equation above.
Because the harmonic mean is unintuitive, it is hard to see how to apply it in practical situations. So, now I'm going to present two examples where the harmonic mean is absolutely necessary: (1) variable-speed processors and (2) load balancing servers.
Let's start with a very simple example involving just two speed settings and then I'll generalize it.
1. Variable-speed Processing
1. Suppose we process 1000 writes at 400 IOPS followed by 1000 reads at 600 IOPS. Naturally, the two types of IO operations will take different elapsed times to complete. Overall, the total time to process 2000 IOs is 4.167 seconds. We could ask the question: What average IO rate would allow us to complete 2000 IOs is 4.167 seconds? If we assume the that arithmetic mean (A) of the rates represents the average IO rate, we get: $$A = \dfrac{1}{2} (400 + 600) = 500 ~\text{IOPS}$$ But that would result in completing the IOs in just 2000 IOs/500 IOPS = 4 seconds. On the other hand, if we assume the harmonic mean instead, then using the definition above for 2 quantities (i.e., a and b), we find: $$H = \dfrac{2}{\frac{1}{400} + \frac{1}{600}} = \dfrac{2 (400 × 600)}{1000} = 480 ~\text{IOPS}$$ and 2000 IOs /480 IOPS = 4.167 seconds; which is correct.
Note that H is computed by adding the inverse IOPS rates. That's so we are effectively averaging time-based quantities (which is the same as inverse rates). From this example, it should be clear that the harmonic mean is the correct type of average to use when it comes to summarizing rates.
2. Now let's generalize this idea to the case where we have a truly variable-speed "green" processor that ramps up its service rate (e.g., as measured by its SPEC value) to meet demand. Otherwise, it cycles at low speed in order to consume less electrical power. The plot below shows the respective processor speed in SPEC (rate) units as it processes 100 tasks in each of 6 increments. The time to process all 600 tasks is 176.88 seconds. The harmonic mean H = 3.39 TPS (lower horizontal line), while the corresponding arithmetic mean A = 10.02 TPS (upper horizontal line); a significant difference! This large difference is a direct consequence of inverting the rates in the harmonic sum, which ends up giving more weight to the first 4 columns in the plot.
Consider application requests arriving at a rate of λ = 100 TPS to be serviced by 2 application servers; one of which is faster than the other. The fast server handles 60 TPS and takes 10 milliseconds to process each request (on average), while the slow server only handles 40 TPS and takes 20 milliseconds per request. If we measured the utilization of each server, we would find the fast server is 60% busy, while the slower server is 80% busy, i.e., they are unbalanced. What service speed will balance them? To achieve peace and harmony, we compute the harmonic mean of the service times: $$S_H = \dfrac{2}{\frac{1}{10} + \frac{1}{20}} = 13.33 ~\text{ms}$$ and split the incoming transactions equally across the 2 servers so that each one gets requests at a rate of $\lambda/2 = 50$ TPS. We can use Little's law $(\rho = \lambda S_H/2)$ to confirm that each server will be balanced at $\rho = 50 ~\text{TPS} \times 0.0133333 ~\text{s} = 0.6667$ or 67% busy.
I'll probably present more on this topic during my Capacity Planning Boot Camp sessions at CMG 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": 4, "x-ck12": 0, "texerror": 0, "math_score": 0.8271216750144958, "perplexity": 715.6672387615704}, "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-2014-42/segments/1414119645898.42/warc/CC-MAIN-20141024030045-00304-ip-10-16-133-185.ec2.internal.warc.gz"} |
http://www.powerelectricalblog.com/2007/03/two-port-networks-z-y-h-g-abcd.html | ## 06 March, 2007
### Two Port Networks - Z , Y , h , g , ABCD Parameters
A pair of terminals at which a signal (voltage or current) may enter or leave is called a port.
A network having only one such pair of terminals is called a one port network.
A two-port network (or four-terminal network, or quadripole) is an electrical circuit or device with two pairs of terminals.Examples include transistors, filters and matching networks. The analysis of two-port networks was pioneered in the 1920s by Franz Breisig, a German mathematician.
A two-port network basically consists in isolating either a complete circuit or part of it and finding its characteristic parameters. Once this is done, the isolated part of the circuit becomes a "black box" with a set of distinctive properties, enabling us to abstract away its specific physical buildup, thus simplifying analysis. Any circuit can be transformed into a two-port network provided that it does not contain an independent source.
A two-port network is represented by four external variables: voltage and current at the input port, and voltage and current at the output port, so that the two-port network can be treated as a black box modeled by the the relationships between the four variables , , and . There exist six different ways to describe the relationships between these variables, depending on which two of the four variables are given, while the other two can always be derived.
Note: All voltages and currents below are complex variables and represented by phasors containing both magnitude and phase angle. However, for convenience the phasor notation and are replaced by V and I respectively.
The parameters used in order to describe a two-port network are the following: Z, Y, A , h, g. They are usually expressed in matrix notation and they establish relations between the following parameters:
Input voltage V1
Output voltage V2
Input current I1
Output current I2
Z-model : In the Z-model or impedance model, the two currents I1 and I2 are assumed to be known, and the voltages V1and V2can be found by:
where
Here all four parameters Z11,Z12 ,Z21 , and Z22 represent impedance. In particular, Z21 and Z12 are transfer impedances, defined as the ratio of a voltage V1(or V2) in one part of a network to a current I2(or I1 ) in another part . Z12 = V1 / I2 . Z is a 2 by 2 matrix containing all four parameters.
Y-model : In the Y-model or admittance model, the two voltages V1 and V2 are assumed to be known, and the currents I1 and I2 can be found by:
where
Here all four parameters Y11,Y12 ,Y21 , and Y22 represent admittance. In particular, Y21 and Y12 are transfer admittances. Y is the corresponding parameter matrix.
ABCD -model : In the A-model or transmission model, we assume V1 and I1 are known, and find V2 and I2 by:
where
Here A and D are dimensionless coefficients, B is impedance and C is admittance. A negative sign is added to the output current I2 in the model, so that the direction of the current is out-ward, for easy analysis of a cascade of multiple network models.
H-model : In the H-model or hybrid model, we assume V2 and I1 are known, and find V1 and I2 by:
where
Here h12 and h21 are dimensionless coefficients, h11 is impedance and h22 is admittance.
g model :In g model or inverse hybrid model, we assume V1 and I2 are known, and find V2 and I1 by :
where
Here g12 and g21 are dimensionless coefficients, g22 is impedance and g11 is admittance.
For more details on this topic visit the following links :
http://fourier.eng.hmc.edu/e84/lectures/ch3/node11.html
http://en.wikipedia.org/wiki/Two-port_network
http://web.cecs.pdx.edu/~ece2xx/ECE222/Slides/TwoPortsx4.pdf | {"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": 0.8723812699317932, "perplexity": 1448.6240358650066}, "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/1440644064869.18/warc/CC-MAIN-20150827025424-00147-ip-10-171-96-226.ec2.internal.warc.gz"} |
http://www.physicsforums.com/showthread.php?t=238517 | ODE on the plane
by wofsy
Tags: plane
P: 707 Can someone tell me how to solve the ODE, d2x/dt2 = -cosx d2y/dt2 = -cosy in the plane?
P: 659 You can use seperation of variables and then integrate twice (with an integration constant!)
P: 707 thanks
Math
Emeritus
Thanks
PF Gold
P: 38,705
ODE on the plane
Those two equations are completely independent- so it is not necessary to "separate variables". You are just solving two separate second order differential equations. And, in fact you just need to integrate each twice.
HW Helper
P: 1,391
Quote by HallsofIvy Those two equations are completely independent- so it is not necessary to "separate variables". You are just solving two separate second order differential equations. And, in fact you just need to integrate each twice.
Hrm? They don't look independent to me. The derivatives are with respect to t, not x or y. Integrating the x equation would give, for instance,
$$\dot{x(t)} - \dot{x(t_0)} = \int_{t_0}^{t}d\tau~\cos y(\tau)$$
which isn't so useful if you can't solve for what y(t) is.
Math Emeritus Sci Advisor Thanks PF Gold P: 38,705 If d2x/dt2= cos(x) is a single equation in the dependent variable x as a function of t. There is absolutely no reason to introduce y. Since the independent variable "t" does not appear in the equation, I would use "quadrature": Let v= dx/dt so that d2x/dt2= dv/dt= (dv/dx)(dx/dt)= v dv/dx. Now you have vdv/dx= cos(x) or vdv= cos(x)dx. Integrating, (1/2)v2= sin(x)+ C. dx/dt= v= $\sqrt{2(sin(x)+ C)}$ or $$\frac{dx}{\sqrt{2(sin(x)+ C)}}= dt$$ That left side is an "elliptical integral". Of course, y will be exactly the same, though possibly with different constants of integration.
HW Helper
P: 1,391
Quote by HallsofIvy If d2x/dt2= cos(x) is a single equation in the dependent variable x as a function of t. There is absolutely no reason to introduce y. Since the independent variable "t" does not appear in the equation, I would use "quadrature": Let v= dx/dt so that d2x/dt2= dv/dt= (dv/dx)(dx/dt)= v dv/dx. Now you have vdv/dx= cos(x) or vdv= cos(x)dx. Integrating, (1/2)v2= sin(x)+ C. dx/dt= v= $\sqrt{2(sin(x)+ C)}$ or $$\frac{dx}{\sqrt{2(sin(x)+ C)}}= dt$$ That left side is an "elliptical integral". Of course, y will be exactly the same, though possibly with different constants of integration.
Ah, I see, I didn't parse the problem the way it was intended to be read. I read it as
$$\frac{d^2x}{dt^2} = -\cos x \frac{d^2y}{dt^2} = -\cos y$$
i.e.,
$$\frac{d^2x}{dt^2} = -\cos y$$
and
$$\cos x \frac{d^2y}{dt^2} = \cos y$$
This is why I always use some sort of punctuation in between separate equations written on the same line. =P
Math Emeritus Sci Advisor Thanks PF Gold P: 38,705 And how do you know that was how it was "intended to be read"?
P: 707 I was able to get to the elliptic integral. But I have no idea what it looks like. Further, if x and y are both the same elliptic integral then the orbits in the plane should be fairly simple. But what do they look like?
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https://mathsci.kaist.ac.kr/pow/2021/09/17/ | # 2021-15 Triangles with integer side lengths
For a natural number $$n$$, let $$a_n$$ be the number of congruence classes of triangles whose all three sides have integer length and its perimeter is $$n$$. Obtain a formula for $$a_n$$.
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https://www.jstage.jst.go.jp/article/vss/63/3/63_20180508/_article/-char/en | Vacuum and Surface Science
Online ISSN : 2433-5843
Print ISSN : 2433-5835
Special Feature : A New Kind of Advanced Computing : Evolution of Ising Machine
Principle and Applications of Simulated Bifurcation Machine
Hayato GOTO
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JOURNALS RESTRICTED ACCESS
2020 Volume 63 Issue 3 Pages 129-133
Details
Abstract
Recently, we have proposed a new heuristic algorithm for combinatorial optimization inspired by our proposed quantum computer. We named the algorithm “simulated bifurcation (SB) algorithm,” because it is based on the numerical simulation of nonlinear Hamiltonian systems exhibiting bifurcations. Here we present the principle, performance, and applications of our SB machine.
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この記事はクリエイティブ・コモンズ [表示 - 非営利 4.0 国際]ライセンスの下に提供されています。 | {"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": 0.8538155555725098, "perplexity": 4926.698131881562}, "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/1606141746320.91/warc/CC-MAIN-20201205044004-20201205074004-00095.warc.gz"} |
http://docplayer.net/27148-Getting-started-with-l-a-tex.html | # Getting Started with L A TEX
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## Transcription
1 Getting Started with L A TEX David R. Wilkins 2nd Edition Copyright c David R. Wilkins 1995 Contents 1 Introduction to L A TEX What is L A TEX? A Typical L A TEX Input File Characters and Control Sequences Producing Simple Documents using L A TEX Producing a L A TEX Input File Producing Ordinary Text using L A TEX Blank Spaces and Carriage Returns in the Input File Quotation Marks and Dashes Section Headings in L A TEX Changing Fonts in Text Mode Accents used in Text Active Characters and Special Symbols in Text Producing Mathematical Formulae using L A TEX Mathematics Mode Characters in Mathematics Mode Superscripts and Subscripts Greek Letters Mathematical Symbols Changing Fonts in Mathematics Mode Standard Functions (sin, cos etc.) Text Embedded in Displayed Equations Fractions and Roots Ellipsis (i.e., three dots ) Accents in Mathematics Mode
2 3.12 Brackets and Norms Multiline Formulae in L A TEX Matrices and other arrays in L A TEX Derivatives, Limits, Sums and Integrals Further Features of L A TEX Producing White Space in L A TEX Lists Displayed Quotations Tables The Preamble of the L A TEX Input file Defining your own Control Sequences in L A TEX Introduction to L A TEX 1.1 What is L A TEX? L A TEX is a computer program for typesetting documents. It takes a computer file, prepared according to the rules of L A TEX and converts it to a form that may be printed on a high-quality printer, such as a laser writer, to produce a printed document of a quality comparable with good quality books and journals. Simple documents, which do not contain mathematical formulae or tables may be produced very easily: effectively all one has to do is to type the text straight in (though observing certain rules relating to quotation marks and punctuation dashes). Typesetting mathematics is somewhat more complicated, but even here L A TEX is comparatively straightforward to use when one considers the complexity of some of the formulae that it has to produce and the large number of mathematical symbols which it has to produce. L A TEX is one of a number of dialects of TEX, all based on the version of TEX created by D. E. Knuth which is known as Plain TEX. L A TEX (created by L. B. Lamport) is one of these dialects. It is particularly suited to the production of long articles and books, since it has facilities for the automatic numbering of chapters, sections, theorems, equations etc., and also has facilities for cross-referencing. It is probably one of the most suitable version of L A TEX for beginners to use. 1.2 A Typical L A TEX Input File In order to produce a document using L A TEX, we must first create a suitable input file on the computer. We apply the L A TEX program to the input file and then use the printer to print out the so-called DVI file produced by the L A TEX program (after first using another program to translate the DVI file into a form that the printer can understand). Here is an example of a typical L A TEX input file: 2
3 \documentclass[a4paper,12pt]{article} \begin{document} The foundations of the rigorous study of \textit{analysis} were laid in the nineteenth century, notably by the mathematicians Cauchy and Weierstrass. Central to the study of this subject are the formal definitions of \textit{limits} and \textit{continuity}. Let $D$ be a subset of $\bf R$ and let $f \colon D \to \textbf{r}$ be a real-valued function on $D$. The function $f$ is said to be \textit{continuous} on $D$ if, for all $\epsilon > 0$ and for all $x \in D$, there exists some $\delta > 0$ (which may depend on $x$) such that if $y \in D$ satisfies $y - x < \delta$ then $f(y) - f(x) < \epsilon.$ One may readily verify that if $f$ and $g$ are continuous functions on $D$ then the functions $f+g$, $f-g$ and $f.g$ are continuous. If in addition $g$ is everywhere non-zero then $f/g$ is continuous. \end{document} When we apply L A TEX to these paragraphs we produce the text The foundations of the rigorous study of analysis were laid in the nineteenth century, notably by the mathematicians Cauchy and Weierstrass. Central to the study of this subject are the formal definitions of limits and continuity. Let D be a subset of R and let f: D R be a real-valued function on D. The function f is said to be continuous on D if, for all ɛ > 0 and for all x D, there exists some δ > 0 (which may depend on x) such that if y D satisfies y x < δ then f(y) f(x) < ɛ. 3
4 One may readily verify that if f and g are continuous functions on D then the functions f + g, f g and f.g are continuous. If in addition g is everywhere non-zero then f/g is continuous. This example illustrates various features of L A TEX. Note that the lines \documentclass[a4paper,12pt]{article} \begin{document} are placed at the beginning of the input file. These are followed by the main body of the text, followed by the concluding line \end{document} Note also that, although most characters occurring in this file have their usual meaning, yet there are special characters such as \, $, { and } which have special meanings within L A TEX. Note in particular that there are sequences of characters which begin with a backslash \ which are used to produce mathematical symbols and Greek letters and to accomplish tasks such as changing fonts. These sequences of characters are known as control sequences. 1.3 Characters and Control Sequences We now describe in more detail some of the features of L A TEX illustrated in the above example. Most characters on the keyboard, such as letters and numbers, have their usual meaning. However the characters \ { }$ ^ _ % ~ # & are used for special purposes within L A TEX. Thus typing one of these characters will not produce the corresponding character in the final document. Of course these characters are very rarely used in ordinary text, and there are methods of producing them when they are required in the final document. In order to typeset a mathematical document it is necessary to produce a considerable number of special mathematical symbols. One also needs to be able to change fonts. Also mathematical documents often contain arrays of numbers or symbols (matrices) and other complicated expressions. These are produced in L A TEX using control sequences. Most control sequences consist of a backslash \ followed by a string of (upper or lower case) letters. For example, \alpha, \textit and \sum are control sequences. In the example above we used the control sequences \textit and \textbf to change the font to italic and boldface respectively. Also we used the control sequences \to, 4
5 \in, \delta and \epsilon to produce the mathematical symbols and and the Greek letters δ and ɛ There is another variety of control sequence which consists of a backslash followed by a single character that is not a letter. Examples of control sequences of this sort are \{, \" and \$. The special characters { and } are used for grouping purposes. Everything enclosed within matching pair of such brackets is treated as a single unit. We have applied these brackets in the example above whenever we changed fonts. We shall see other instances where one needs to use { and } in L A TEX to group words and symbols together (e.g., when we need to produce superscripts and subscripts which contain more than one symbol). The special character$ is used when one is changing from ordinary text to a mathematical expression and when one is changing back to ordinary text. Thus we used for all $\epsilon > 0$ and for all $x \in D$, to produce the phrase for all ɛ > 0 and for all x D, in the example given above. Note also that we used $and$ in the example above to mark the beginning and end respectively of a mathematical formula that is displayed on a separate line. The remaining special characters ^ _ % ~ # & have special purposes within L A TEX that we shall discuss later. 2 Producing Simple Documents using L A TEX 2.1 Producing a L A TEX Input File We describe the structure of a typical L A TEX input file. The first line of the input file should consist of a \documentclass command. The recommended such \documentclass command for mathematical articles and similar documents has the form \documentclass[a4paper,12pt]{article} 5
6 (You do not have to worry about what this command means when first learning to use L A TEX: its effect is to ensure that the final document is correctly positioned on A4 size paper and that the text is of a size that is easy to read.) There are variants of this \documentclass command which are appropriate for letters or for books. The documentstyle command may be followed by certain other optional commands, such as the \pagestyle command. It is not necessary to find out about these commands when first learning to use L A TEX. After the \documentclass command and these other optional commands, we place the command \begin{document} This command is then followed by the main body of the text, in the format prescribed by the rules of L A TEX. Finally, we end the input file with a line containing the command \end{document} 2.2 Producing Ordinary Text using L A TEX To produce a simple document using L A TEX one should create a L A TEX input file, beginning with a \documentclass command and the \begin{document} command, as described above. The input file should end with the \end{document} command, and the text of the document should be sandwiched between the \begin{document} and \end{document} commands in the manner described below. If one merely wishes to type in ordinary text, without complicated mathematical formulae or special effects such as font changes, then one merely has to type it in as it is, leaving a completely blank line between successive paragraphs. You do not have to worry about paragraph indentation: L A TEX will automatically indent all paragraphs with the exception of the first paragraph of a new section (unless you take special action to override the conventions adopted by L A TEX) For example, suppose that we wish to create a document containing the following paragraphs: If one merely wishes to type in ordinary text, without complicated mathematical formulae or special effects such as font changes, then one merely has to type it in as it is, leaving a completely blank line between successive paragraphs. You do not have to worry about paragraph indentation: all paragraphs will be indented with the exception of the first paragraph of a new section. One must take care to distinguish between the left quote and the right quote on the computer terminal. Also, one should use two single quote 6
7 characters in succession if one requires double quotes. One should never use the (undirected) double quote character on the computer terminal, since the computer is unable to tell whether it is a left quote or a right quote. One also has to take care with dashes: a single dash is used for hyphenation, whereas three dashes in succession are required to produce a dash of the sort used for punctuation such as the one used in this sentence. To create this document using L A TEX we use the following input file: \documentclass[a4paper,12pt]{article} \begin{document} If one merely wishes to type in ordinary text, without complicated mathematical formulae or special effects such as font changes, then one merely has to type it in as it is, leaving a completely blank line between successive paragraphs. You do not have to worry about paragraph indentation: all paragraphs will be indented with the exception of the first paragraph of a new section. One must take care to distinguish between the left quote and the right quote on the computer terminal. Also, one should use two single quote characters in succession if one requires double quotes. One should never use the (undirected) double quote character on the computer terminal, since the computer is unable to tell whether it is a left quote or a right quote. One also has to take care with dashes: a single dash is used for hyphenation, whereas three dashes in succession are required to produce a dash of the sort used for punctuation---such as the one used in this sentence. \end{document} Having created the input file, one then has to run it through the L A TEX program and then print it out the resulting output file (known as a DVI file). 7
8 2.3 Blank Spaces and Carriage Returns in the Input File L A TEX treats the carriage return at the end of a line as though it were a blank space. Similarly L A TEX treats tab characters as blank spaces. Moreover, L A TEX regards a sequence of blank spaces as though it were a single space, and similarly it will ignore blank spaces at the beginning or end of a line in the input file. Thus, for example, if we type This is a silly example of a file with many spaces. This is the beginning of a new paragraph. then we obtain This is a silly example of a file with many spaces. This is the beginning of a new paragraph. It follows immediately from this that one will obtain the same results whether one types one space or two spaces after a full stop: L A TEX does not distinguish between the two cases. Any spaces which follow a control sequence will be ignored by L A TEX. If you really need a blank space in the final document following whatever is produced by the control sequence, then you must precede this blank by a backslash \. Thus in order to obtain the sentence we must type L A TEX is a very powerful computer typesetting program. \LaTeX\ is a very powerful computer typesetting program. (Here the control sequence TeX is used to produce the L A TEX logo.) In general, preceding a blank space by a backslash forces L A TEX to include the blank space in the final document. As a general rule, you should never put a blank space after a left parenthesis or before a right parenthesis. If you were to put a blank space in these places, then you run the risk that L A TEX might start a new line immediately after the left parenthesis or before the right parenthesis, leaving the parenthesis marooned at the beginning or end of a line. 8
9 2.4 Quotation Marks and Dashes Single quotation marks are produced in L A TEX using and. Double quotation marks are produced by typing and. (The undirected double quote character " produces double right quotation marks: it should never be used where left quotation marks are required.) L A TEX allows you to produce dashes of various length, known as hyphens, endashes and em-dashes. Hyphens are obtained in L A TEX by typing -, en-dashes by typing -- and em-dashes by typing ---. One normally uses en-dashes when specifying a range of numbers. Thus for example, to specify a range of page numbers, one would type on pages Dashes used for punctuating are often typeset as em-dashes, especially in older books. These are obtained by typing ---. The dialogue You were a little grave, said Alice. Well just then I was inventing a new way of getting over a gate would you like to hear it? Very much indeed, Alice said politely. I ll tell you how I came to think of it, said the Knight. You see, I said to myself The only difficulty is with the feet: the head is high enough already. Now, first I put my head on the top of the gate then the head s high enough then I stand on my head then the feet are high enough, you see then I m over, you see. (taken from Alice through the Looking Glass, by Lewis Carroll) illustrates the use of quotation marks and dashes. It is obtained in L A TEX from the following input: You \emph{were} a little grave, said Alice. Well just then I was inventing a new way of getting over a gate---would you like to hear it? Very much indeed, Alice said politely. I ll tell you how I came to think of it, said the Knight. You see, I said to myself The only difficulty is with the feet: the \emph{head} is high enough already. Now, first I put my head on the top of the gate---then the head s high 9
10 enough---then I stand on my head---then the feet are high enough, you see---then I m over, you see. Sometimes you need single quotes immediately following double quotes, or vica versa, as in I regard computer typesetting as being reasonably straightforward he said. The way to typeset this correctly in L A TEX is to use the control sequence \, between the quotation marks, so as to obtain the necessary amount of separation. The above example is thus produced with the input I regard computer typesetting as being reasonably straightforward \, he said. 2.5 Section Headings in L A TEX Section headings of various sizes are produced (in the article document style) using the commands \section,\subsection and \subsubsection commands. L A TEX will number the sections and subsections automatically. The title of the section should be surrounded by curly brackets and placed immediately after the relevant command. Thus if we type \section{section Headings} We explain in this section how to obtain headings for the various sections and subsections of our document. \subsection{headings in the article Document Style} In the article style, the document may be divided up into sections, subsections and subsubsections, and each can be given a title, printed in a boldface font, simply by issuing the appropriate command. then the title of the section and that of the subsection will be printed in a large boldface font, and will be numbered accordingly. Other document styles (such as the book and letter styles) have other sectioning commands available (for example, the book style has a \chapter command for beginning a new chapter). 10
11 Sometimes one wishes to suppress the automatic numbering provided by L A TEX. This can be done by placing an asterisk before the title of the section or subsection. Thus, for example, the section numbers in the above example could be suppressed by typing \section*{section Headings} We explain in this section how to obtain headings for the various sections and subsections of our document. \subsection*{headings in the article Document Style} In the article style, the document may be divided up into sections, subsections and subsubsections, and each can be given a title, printed in a boldface font, simply by issuing the appropriate command. 2.6 Changing Fonts in Text Mode L A TEX has numerous commands for changing the typestyle. The most useful of these is \emph{text} which emphasizes some piece of text, setting it usually in an italic font (unless the surrounding text is already italicized). Thus for example, the text The basic results and techniques of Calculus were discovered and developed by Newton and Leibniz, though many of the basic ideas can be traced to earlier work of Cavalieri, Fermat, Barrow and others. is obtained by typing The basic results and techniques of \emph{calculus} were discovered and developed by \emph{newton} and \emph{leibniz}, though many of the basic ideas can be traced to earlier work of \emph{cavalieri}, \emph{fermat}, \emph{barrow} and others. Another useful font-changing command is \textbf{text}, which typesets the specified portion of text in boldface. A font family or typeface in L A TEX consists of a collection of related fonts characterized by size, shape and series. The font families available in L A TEX include roman, sans serif and typewriter: Roman is normally the default family and includes upright, italic, slanted, small caps and boldface fonts of various sizes. 11
12 There is a sans serif family with upright, slanted and boldface fonts of various sizes. There is a typewriter family with upright, italic, slanted and small caps fonts of various sizes. The sizes of fonts used in L A TEX are can be determined and changed by means of the control sequences \tiny, \scriptsize, \footnotesize, \small, \normalsize, \large, \Large, \LARGE, \huge and \HUGE: This text is tiny. This text is scriptsize. This text is footnotesize. This text is small. This text is normalsize. This text is large. This text is Large. This text is LARGE. This text is huge. This text is Huge. The shape of a font can be upright, italic, slanted or small caps: The LaTeX command \textup{text} typesets the specified text with an upright shape: this is normally the default shape. The LaTeX command \textit{text} typesets the specified text with an italic shape. The LaTeX command \textsl{text} typesets the specified text with a slanted shape: slanted text is similar to italic. The LaTeX command \textsc{text} typesets the specified text with a small caps shape in which all letters are capitals (with uppercase letters taller than lowercase letters). The series of a font can be medium (the default) or boldface: The LaTeX command \textmd{text} typesets the specified text with a medium series font. 12
13 The LaTeX command \textbf{text} typesets the specified text with a boldface series font. If the necessary fonts are available, one can combine changes to the size, shape and series of a font, for example producing boldface slanted text by typing \textbf{\textsl{boldface slanted text}}. There are in L A TEX font declarations corresponding to the the font-changing commands described above. When included in the L A TEX input such declarations determine the type-style of the subsequent text (till the next font declaration or the end of the current group delimited by curly brackets or by appropriate \begin and \end commands). Here is a list of font-changing commands and declarations in text mode: Command Declaration \textrm \rmfamily Roman family \textsf \sffamily Sans serif family \texttt \ttfamily Typewriter family \textup \upshape Upright shape \textit \itshape Italic shape \textsl \slshape Slanted shape \textsc \scshape Small caps shape \textmd \mdseries Medium series \textbf \bfseries Boldface series 2.7 Accents used in Text There are a variety of control sequences for producing accents. For example, the control sequence \ {o} produces an acute accent on the letter o. Thus typing produces Se\ {a}n \ {O} Cinn\ {e}ide. Seán Ó Cinnéide. Similarly we use the control sequence \ to produce the grave accent in algèbre and we use \" to produce the umlaut in Universität. The accents provided by L A TEX include the following: 13
14 \ {e} é e.g., math\ {e}matique yields mathématique \ {e} è e.g., alg\ {e}bre yields algèbre \^{e} ê e.g., h\^{o}te yields hôte \"{o} ö e.g., H\"{o}lder yields Hölder \~{n} ñ e.g., ma\~{n}ana yields mañana \={o} ō \.{o} ȯ \u{o} ŏ \v{c} č e.g., \v{c}ech yields Čech \H{o} ő \t{oo} oo \c{c} ç e.g., gar\c{c}on yields garçon \d{o} ọ \b{o} ō These accents are for use in ordinary text. They cannot be used within mathematical formulae, since different control sequences are used to produce accents within mathematics. The control sequences \i and \j produce dotless i and j. These are required when placing an accent on the letter. Thus í is produced by typing \ {\i}. 2.8 Active Characters and Special Symbols in Text The active characters # $% & \ ^ _ { } ~ have special purposes within L A TEX. Thus they cannot be produced in the final document simply by typing them directly. On the rare occasions when one needs to use the special characters #$ % & { } in the final document, they can be produced by typing the control sequences \# \$\% \& \_ \{ \} respectively. However the characters \, ^ and ~ cannot be produced simply by preceding them with a backslash. They can however be produced using \char92 (in the \texttt font only), \char94 and \char126 respectively. (The decimal numbers 92, 94 and 126 are the ASCII codes of these characters.) Other special symbols can be introduced into text using the appropriate control sequences: 14 15 Symbol Control Sequence œ, Œ \oe, \OE æ, Æ \ae, \AE å, Å \aa, \AA ø, Ø \o, \O l, L \l, \L ß \ss?! \dag \ddag \S \P c \copyright \pounds ı \i j \j 3 Producing Mathematical Formulae using L A TEX 3.1 Mathematics Mode In order to obtain a mathematical formula using L A TEX, one must enter mathematics mode before the formula and leave it afterwards. Mathematical formulae can occur either embedded in text or else displayed between lines of text. When a formula occurs within the text of a paragraph one should place a$ sign before and after the formula, in order to enter and leave mathematics mode. Thus to obtain a sentence like Let f be the function defined by f(x) = 3x + 7, and let a be a positive real number. one should type Let $f$ be the function defined by $f(x) = 3x + 7$, and let $a$ be a positive real number. In particular, note that even mathematical expressions consisting of a single character, like f and a in the example above, are placed within $signs. This is to ensure that they are set in italic type, as is customary in mathematical typesetting. L A TEX also allows you to use $$and$$ to mark the beginning and the end respectively of a mathematical formula embedded in text. Thus Let f be the function defined by f(x) = 3x 16 may be produced by typing Let $$f$$ be the function defined by $$f(x) = 3x + 7$$. However this use of $$...$$ is only permitted in L A TEX: other dialects of TeX such as Plain TEX$$and$$AmSTeX use$... $. In order to obtain an mathematical formula or equation which is displayed on a line by itself, one places $before$$and$$$ after the formula. Thus to obtain If f(x) = 3x + 7$$and$$g(x) = x + 4 then f(x) + g(x) = 4x + 11$$and$$f(x)g(x) = 3x x one would type If$f(x) = 3x + 7$$$and$$$g(x) = x + 4$then $$$f(x) + g(x) = 4x + 11$$$$$and$$$$$f(x)g(x) = 3x^2 + 19x +28.$$$ (Here the character ^ is used to obtain a superscript.) L A TEX provides facilities for the automatic numbering of displayed equations. If you want an numbered equation then you use $$and$$ instead of using $$$and$$$. Thus produces If$f(x) = 3x + 7$$$and$$$g(x) = x + 4$then $$f(x) + g(x) = 4x + 11$$$$and$$$$f(x)g(x) = 3x^2 + 19x +28.$$ If f(x) = 3x + 7$$and$$g(x) = x + 4 then$$f(x) + g(x) = 4x + 11$$(1)$$and$$f(x)g(x) = 3x x (2) 16 17 3.2 Characters in Mathematics Mode All the characters on the keyboard have their standard meaning in mathematics mode, with the exception of the characters #$ % & ~ _ ^ \ { } Letters are set in italic type. In mathematics mode the character has a special meaning: typing $u + v$ produces u + v When in mathematics mode the spaces you type between letters and other symbols do not affect the spacing of the final result, since L A TEX determines the spacing of characters in formulae by its own internal rules. Thus $u v + w = x$ and $uv+w=x$ both produce uv + w = x You can also type carriage returns where necessary in your input file (e.g., if you are typing in a complicated formula with many Greek characters and funny symbols) and this will have no effect on the final result if you are in mathematics mode. To obtain the characters # $% & _ { } in mathematics mode, one should type \# \$ \% \& \_ \{ \}. To obtain in mathematics mode, one may type \backslash. 3.3 Superscripts and Subscripts Subscripts and superscripts are obtained using the special characters _ and ^ respectively. Thus the identity is obtained by typing ds 2 = dx dx dx 2 3 c 2 dt 2 $ds^2 = dx_1^2 + dx_2^2 + dx_3^2 - c^2 dt^2$ It can also be obtained by typing $ds^2 = dx^2_1 + dx^2_2 + dx^2_3 - c^2 dt^2$ since, when a superscript is to appear above a subscript, it is immaterial whether the superscript or subscript is the first to be specified. Where more than one character occurs in a superscript or subscript, the characters involved should be enclosed in curly brackets. For example, the polynomial x 17 1 is obtained by typing $x^{17} - 1$. 17
18 One may not type expressions such as $s^n^j$ since this is ambiguous and could be interpreted either as s nj or as s nj The first of these alternatives is obtained by typing $s^{n j}$, the second by typing $s^{n^j}$. A similar remark applies to subscripts. Note that one can obtain in this way double superscripts (where a superscript is placed on a superscript) and double subscripts. It is sometimes necessary to obtain expressions in which the horizontal ordering of the subscripts is significant. One can use an empty group {} to separate superscripts and subscripts that must follow one another. For example, the identity can be obtained by typing R i j kl = g jm R imkl = g jm R mikl = R j ikl $R_i{}^j{}_{kl} = g^{jm} R_{imkl} = - g^{jm} R_{mikl} = - R^j{}_{ikl}$ 3.4 Greek Letters Greek letters are produced in mathematics mode by preceding the name of the letter by a backslash \. Thus to obtain the formula A = πr 2 one types A = \pi r^2. Here are the control sequences for the standard forms of the lowercase Greek letters:- α \alpha ι \iota ρ \rho β \beta κ \kappa σ \sigma γ \gamma λ \lambda τ \tau δ \delta µ \mu υ \upsilon ɛ \epsilon ν \nu φ \phi ζ \zeta ξ \xi χ \chi η \eta o o ψ \psi θ \theta π \pi ω \omega There is no special command for omicron: just use o. Some Greek letters occur in variant forms. The variant forms are obtained by preceding the name of the Greek letter by var. The following table lists the usual form of these letters and the variant forms:- ɛ \epsilon ε \varepsilon θ \theta ϑ \vartheta π \pi ϖ \varpi ρ \rho ϱ \varrho σ \sigma ς \varsigma φ \phi ϕ \varphi 18
19 Upper case Greek letters are obtained by making the first character of the name upper case. Here are the control sequence for the uppercase letters: Γ \Gamma Ξ \Xi Φ \Phi \Delta Π \Pi Ψ \Psi Θ \Theta Σ \Sigma Ω \Omega Λ \Lambda Υ \Upsilon 3.5 Mathematical Symbols There are numerous mathematical symbols that can be used in mathematics mode. These are obtained by typing an appropriate control sequence. Miscellaneous Symbols: ℵ \aleph \prime \forall h \hbar \emptyset \exists ı \imath \nabla \neg j \jmath \surd \flat l \ell \top \natural \wp \bot \sharp R \Re \ \clubsuit I \Im \angle \diamondsuit \partial \triangle \heartsuit \infty \ \backslash \spadesuit Large Operators: \sum \bigcap \bigodot \prod \bigcup \bigotimes \coprod \bigsqcup \bigoplus Binary Operations: \int \oint \bigvee \biguplus \bigwedge 19
20 Relations: ± \pm \cap \vee \mp \cup \wedge \ \setminus \uplus \oplus \cdot \sqcap \ominus \times \sqcup \otimes \ast \triangleleft \oslash \star \triangleright \odot \diamond \wr \dagger \circ \bigcirc \ddagger \bullet \bigtriangleup \amalg \div \bigtriangledown Negated Relations: \leq \geq \equiv \prec \succ \sim \preceq \succeq \simeq \ll \gg \asymp \subset \supset \approx \subseteq \supseteq = \cong \sqsubseteq \sqsupseteq \bowtie \in \ni \propto \vdash \dashv = \models. \smile \mid = \doteq \frown \parallel \perp \not< \not> \not= \not\leq \not\geq \not\equiv \not\prec \not\succ \not\sim \not\preceq \not\succeq \not\simeq \not\subset \not\supset \not\approx \not\subseteq \not\supseteq = \not\cong \not\sqsubseteq \not\sqsupseteq \not\asymp Arrows: 20
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https://worldwidescience.org/topicpages/i/interstellar+boundary+explorer.html | #### Sample records for interstellar boundary explorer
1. The Interstellar Boundary Explorer (IBEX) - Time to Launch!
Science.gov (United States)
McComas, David
The Interstellar Boundary Explorer (IBEX) mission is scheduled to launch in mid-July 2008, right around the time of this COSPAR meeting. IBEX will make the first global observations of the heliosphere's interaction with the interstellar medium. IBEX achieves these breakthrough observations by traveling outside of the Earth's magnetosphere in a highly elliptical orbit and taking global Energetic Neutral Atoms (ENA) images with two very large aperture single pixel ENA cameras. IBEX-Lo makes measurements in 8 contiguous energy pass bands covering from ˜10 eV to 2 keV; IBEX-Hi similarly covers from ˜300 eV to 6 keV in 6 contiguous pass bands. IBEX's high-apogee (˜50RE ) orbit enables heliospheric ENA measurements by providing viewing from far outside the earth's relatively bright magnetospheric ENA emissions. The IBEX cameras view perpendicular to the spacecraft's sun-pointed spin axis. Each six months, the spacecraft spin and progression of the sun-pointing spin axis as the Earth moves around the Sun lead naturally to global, all-sky images. IBEX is the first mission to achieve a high altitude from a standard Pegasus launch vehicle. We accomplish this by adding the propulsion from an IBEX-supplied solid rocket motor and the spacecraft's hydrazine propulsion system. Additional information on IBEX is available at www.ibex.swri.edu. This talk, on behalf of the IBEX science and engineering teams, will summarize the IBEX science and mission and will provide an up-to-the-minute update on the status of the mission, including any new information on the launch and commissioning status.
2. INTERSTELLAR GAS FLOW PARAMETERS DERIVED FROM INTERSTELLAR BOUNDARY EXPLORER-Lo OBSERVATIONS IN 2009 AND 2010: ANALYTICAL ANALYSIS
International Nuclear Information System (INIS)
Möbius, E.; Bochsler, P.; Heirtzler, D.; Kucharek, H.; Lee, M. A.; Leonard, T.; Schwadron, N. A.; Wu, X.; Petersen, L.; Valovcin, D.; Wurz, P.; Bzowski, M.; Kubiak, M. A.; Fuselier, S. A.; Crew, G.; Vanderspek, R.; McComas, D. J.; Saul, L.
2012-01-01
Neutral atom imaging of the interstellar gas flow in the inner heliosphere provides the most detailed information on physical conditions of the surrounding interstellar medium (ISM) and its interaction with the heliosphere. The Interstellar Boundary Explorer (IBEX) measured neutral H, He, O, and Ne for three years. We compare the He and combined O+Ne flow distributions for two interstellar flow passages in 2009 and 2010 with an analytical calculation, which is simplified because the IBEX orientation provides observations at almost exactly the perihelion of the gas trajectories. This method allows separate determination of the key ISM parameters: inflow speed, longitude, and latitude, as well as temperature. A combined optimization, as in complementary approaches, is thus not necessary. Based on the observed peak position and width in longitude and latitude, inflow speed, latitude, and temperature are found as a function of inflow longitude. The latter is then constrained by the variation of the observed flow latitude as a function of observer longitude and by the ratio of the widths of the distribution in longitude and latitude. Identical results are found for 2009 and 2010: an He flow vector somewhat outside previous determinations (λ ISM∞ = 79. 0 0+3. 0 0(–3. 0 5), β ISM∞ = –4. 0 9 ± 0. 0 2, V ISM∞ 23.5 + 3.0(–2.0) km s –1 , T He = 5000-8200 K), suggesting a larger inflow longitude and lower speed. The O+Ne temperature range, T O+Ne = 5300-9000 K, is found to be close to the upper range for He and consistent with an isothermal medium for all species within current uncertainties.
3. Summative Evaluation Findings from the Interstellar Boundary Explorer (IBEX) Education and Public Outreach Program
Science.gov (United States)
Bartolone, L.; Nichols-Yehling, M.; Davis, H. B.; Davey, B.
2014-07-01
The Interstellar Boundary Explorer mission includes a comprehensive Education and Public Outreach (EPO) program in heliophysics that is overseen and implemented by the Adler Planetarium and evaluated by Technology for Learning Consortium, Inc. Several components of the IBEX EPO program were developed during the prime phase of the mission that were specifically designed for use in informal institutions, especially museums and planetaria. The program included a widely distributed planetarium show with accompanying informal education activities, printed posters, lithographs and other resources, funding for the development of the GEMS Space Science Sequence for Grades 6-8 curriculum materials, development of the IBEX mission website, development of materials for people with special needs, participation in the Heliophysics Educator Ambassador program, and support for the Space Explorers Afterschool Science Club for Chicago Public Schools. In this paper, we present an overview of the IBEX EPO program summative evaluation techniques and results for 2008 through 2012.
4. A Test of the Interstellar Boundary EXplorer Ribbon Formation in the Outer Heliosheath
Energy Technology Data Exchange (ETDEWEB)
Gamayunov, Konstantin V.; Rassoul, Hamid [Department of Physics and Space Sciences, Florida Institute of Technology, Melbourne, FL 32901 (United States); Heerikhuisen, Jacob, E-mail: [email protected] [Department of Space Science, University of Alabama in Huntsville, Huntsville, AL 35899 (United States)
2017-08-10
NASA’s Interstellar Boundary EXplorer ( IBEX ) mission is imaging energetic neutral atoms (ENAs) propagating to Earth from the outer heliosphere and local interstellar medium (LISM). A dominant feature in all ENA maps is a ribbon of enhanced fluxes that was not predicted before IBEX . While more than a dozen models of the ribbon formation have been proposed, consensus has gathered around the so-called secondary ENA model. Two classes of secondary ENA models have been proposed; the first class assumes weak scattering of the energetic pickup protons in the LISM, and the second class assumes strong but spatially localized scattering. Here we present a numerical test of the “weak scattering” version of the secondary ENA model using our gyro-averaged kinetic model for the evolution of the phase-space distribution of protons in the outer heliosheath. As input for our test, we use distributions of the primary ENAs from our MHD-plasma/kinetic-neutral model of the heliosphere-LISM interaction. The magnetic field spectrum for the large-scale interstellar turbulence and an upper limit for the amplitude of small-scale local turbulence (SSLT) generated by protons are taken from observations by Voyager 1 in the LISM. The hybrid simulations of energetic protons are also used to set the bounding wavenumbers for the spectrum of SSLT. Our test supports the “weak scattering” version. This makes an additional solid step on the way to understanding the origin and formation of the IBEX ribbon and thus to improving our understanding of the interaction between the heliosphere and the LISM.
5. The interstellar boundary explorer (IBEX): Update at the end of phase B
International Nuclear Information System (INIS)
McComas, D. J.; Allegrini, F.; Pope, S.; Scherrer, J.; Bartolone, L.; Knappenberger, P.; Bochsler, P.; Wurz, P.; Bzowski, M.; Collier, M.; Moore, T.; Fahr, H.; Fichtner, H.; Frisch, P.; Funsten, H.; Fuselier, Steve; Gloeckler, G.; Gruntman, M.; Izmodenov, V.; Lee, M.
2006-01-01
The Interstellar Boundary Explorer (IBEX) mission will make the first global observations of the heliosphere's interaction with the interstellar medium. IBEX achieves these breakthrough observations by traveling outside of the Earth's magnetosphere in a highly elliptical orbit and taking global Energetic Neutral Atoms (ENA) images over energies from ∼10 eV to 6 keV. IBEX's high-apogee (∼50 RE) orbit enables heliospheric ENA measurements by providing viewing from far above the Earth's relatively bright magnetospheric ENA emissions. This high energy orbit is achieved from a Pegasus XL launch vehicle by adding the propulsion from an IBEX-supplied solid rocket motor and the spacecraft's hydrazine propulsion system. IBEX carries two very large-aperture, single-pixel ENA cameras that view perpendicular to the spacecraft's Sun-pointed spin axis. Each six months, the continuous spinning of the spacecraft and periodic re-pointing to maintain the sun-pointing spin axis naturally lead to global, all-sky images. Over the course of our NASA Phase B program, the IBEX team optimized the designs of all subsystems. In this paper we summarize several significant advances in both IBEX sensors, our expected signal to noise (and background), and our groundbreaking approach to achieve a very high-altitude orbit from a Pegasus launch vehicle for the first time. IBEX is in full scale development and on track for launch in June of 2008
6. The interstellar boundary explorer (IBEX): Update at the end of phase B
Science.gov (United States)
McComas, D. J.; Allegrini, F.; Bartolone, L.; Bochsler, P.; Bzowski, M.; Collier, M.; Fahr, H.; Fichtner, H.; Frisch, P.; Funsten, H.; Fuselier, Steve; Gloeckler, G.; Gruntman, M.; Izmodenov, V.; Knappenberger, P.; Lee, M.; Livi, S.; Mitchell, D.; Möbius, E.; Moore, T.; Pope, S.; Reisenfeld, D.; Roelof, E.; Runge, H.; Scherrer, J.; Schwadron, N.; Tyler, R.; Wieser, M.; Witte, M.; Wurz, P.; Zank, G.
2006-09-01
The Interstellar Boundary Explorer (IBEX) mission will make the first global observations of the heliosphere's interaction with the interstellar medium. IBEX achieves these breakthrough observations by traveling outside of the Earth's magnetosphere in a highly elliptical orbit and taking global Energetic Neutral Atoms (ENA) images over energies from ~10 eV to 6 keV. IBEX's high-apogee (~50 RE) orbit enables heliospheric ENA measurements by providing viewing from far above the Earth's relatively bright magnetospheric ENA emissions. This high energy orbit is achieved from a Pegasus XL launch vehicle by adding the propulsion from an IBEX-supplied solid rocket motor and the spacecraft's hydrazine propulsion system. IBEX carries two very large-aperture, single-pixel ENA cameras that view perpendicular to the spacecraft's Sun-pointed spin axis. Each six months, the continuous spinning of the spacecraft and periodic re-pointing to maintain the sun-pointing spin axis naturally lead to global, all-sky images. Over the course of our NASA Phase B program, the IBEX team optimized the designs of all subsystems. In this paper we summarize several significant advances in both IBEX sensors, our expected signal to noise (and background), and our groundbreaking approach to achieve a very high-altitude orbit from a Pegasus launch vehicle for the first time. IBEX is in full scale development and on track for launch in June of 2008.
7. GEOMETRY AND CHARACTERISTICS OF THE HELIOSHEATH REVEALED IN THE FIRST FIVE YEARS OF INTERSTELLAR BOUNDARY EXPLORER OBSERVATIONS
Energy Technology Data Exchange (ETDEWEB)
Zirnstein, E. J.; McComas, D. J.; Schwadron, N. A. [Southwest Research Institute, San Antonio, TX 78228 (United States); Funsten, H. O. [Los Alamos National Laboratory, Los Alamos, NM 87545 (United States); Heerikhuisen, J.; Zank, G. P., E-mail: [email protected] [Department of Space Science, University of Alabama in Huntsville, Huntsville, AL 35899 (United States)
2016-07-20
We investigate and interpret the geometry and characteristics of the inner heliosheath (IHS) plasma and their impact on the heliotail structure as observed in energetic neutral atom (ENA) maps acquired during the first 5 yr of Interstellar Boundary Explorer ( IBEX ) observations. In particular, IBEX observations of the heliotail reveal distinct, localized emission features (lobes) that provide a rich set of information about the properties and evolution of the heliosheath plasma downstream of the termination shock (TS). We analyze the geometry of the heliotail lobes and find that the plane intersecting the port and starboard heliotail lobe centers is ∼6° from the solar equatorial plane, and the plane intersecting the north and south heliotail lobe centers is ∼90° from the solar equatorial plane, both indicating strong correlation with the fast–slow solar wind asymmetry, and thus reflecting the structure of the IHS flow around the Sun. We also analyze the key parameters and processes that form and shape the port and starboard lobes, which are distinctly different from the north and south lobes. By comparing IBEX ENA observations with results from a simplistic flow model of the heliosphere and a multicomponent description for pickup ions (PUIs) in the IHS, we find that the port and starboard lobe formation is driven by a thin IHS, large nose–tail asymmetry of the distance to the TS (and consequently, a large nose–tail asymmetry of the relative abundance of PUIs at the TS) and the energy-dependent removal of PUIs by charge exchange in the IHS.
8. Structure of the Heliotail from Interstellar Boundary Explorer Observations: Implications for the 11-year Solar Cycle and Pickup Ions in the Heliosheath
Energy Technology Data Exchange (ETDEWEB)
Zirnstein, E. J.; McComas, D. J. [Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 (United States); Heerikhuisen, J.; Zank, G. P.; Pogorelov, N. V. [Department of Space Science, University of Alabama in Huntsville, Huntsville, AL 35899 (United States); Funsten, H. O. [Los Alamos National Laboratory, Los Alamos, NM 87545 (United States); Reisenfeld, D. B. [Department of Physics and Astronomy, University of Montana, Missoula, MT 59812 (United States); Schwadron, N. A., E-mail: [email protected] [Southwest Research Institute, San Antonio, TX 78228 (United States)
2017-02-20
Interstellar Boundary Explorer ( IBEX ) measurements of energetic neutral atoms (ENAs) from the heliotail show a multi-lobe structure of ENA fluxes as a function of energy between ∼0.71 and 4.29 keV. Below ∼2 keV, there is a single structure of enhanced ENA fluxes centered near the downwind direction. Above ∼2 keV, this structure separates into two lobes, one north and one south of the solar equatorial plane. ENA flux from these two lobes can be interpreted as originating from the fast solar wind (SW) propagating through the inner heliosheath (IHS). Alternatively, a recently published model of the heliosphere suggests that the heliotail may split into a “croissant-like” shape, and that such a geometry could be responsible for the heliotail ENA feature. Here we present results from a time-dependent simulation of the heliosphere that produces a comet-like heliotail, and show that the 11-year solar cycle leads to the formation of ENA lobes with properties remarkably similar to those observed by IBEX . The ENA energy at which the north and south lobes appear suggests that the pickup ion (PUI) temperature in the slow SW of the IHS is ∼10{sup 7} K. Moreover, we demonstrate that the extinction of PUIs by charge-exchange is an essential process required to create the observed global ENA structure. While the shape and locations of the ENA lobes as a function of energy are well reproduced by PUIs that cross the termination shock, the results appear to be sensitive to the form of the distribution of PUIs injected in the IHS.
9. The ESO Diffuse Interstellar Band Large Exploration Survey (EDIBLES)
Science.gov (United States)
Cami, J.; Cox, N. L.; Farhang, A.; Smoker, J.; Elyajouri, M.; Lallement, R.; Bacalla, X.; Bhatt, N. H.; Bron, E.; Cordiner, M. A.; de Koter, A..; Ehrenfreund, P.; Evans, C.; Foing, B. H.; Javadi, A.; Joblin, C.; Kaper, L.; Khosroshahi, H. G.; Laverick, M.; Le Petit, F..; Linnartz, H.; Marshall, C. C.; Monreal-Ibero, A.; Mulas, G.; Roueff, E.; Royer, P.; Salama, F.; Sarre, P. J.; Smith, K. T.; Spaans, M.; van Loon, J. T..; Wade, G.
2018-03-01
The ESO Diffuse Interstellar Band Large Exploration Survey (EDIBLES) is a Large Programme that is collecting high-signal-to-noise (S/N) spectra with UVES of a large sample of O and B-type stars covering a large spectral range. The goal of the programme is to extract a unique sample of high-quality interstellar spectra from these data, representing different physical and chemical environments, and to characterise these environments in great detail. An important component of interstellar spectra is the diffuse interstellar bands (DIBs), a set of hundreds of unidentified interstellar absorption lines. With the detailed line-of-sight information and the high-quality spectra, EDIBLES will derive strong constraints on the potential DIB carrier molecules. EDIBLES will thus guide the laboratory experiments necessary to identify these interstellar “mystery molecules”, and turn DIBs into powerful diagnostics of their environments in our Milky Way Galaxy and beyond. We present some preliminary results showing the unique capabilities of the EDIBLES programme.
10. A scenario for interstellar exploration and its financing
CERN Document Server
Bignami, Giovanni F
2013-01-01
This book develops a credible scenario for interstellar exploration and colonization. In so doing, it examines: • the present situation and prospects for interstellar exploration technologies; • where to go: the search for habitable planets; • the motivations for space travel and colonization; • the financial mechanisms required to fund such enterprises. The final section of the book analyzes the uncertainties surrounding the presented scenario. The purpose of building a scenario is not only to pinpoint future events but also to highlight the uncertainties that may propel the future in different directions. Interstellar travel and colonization requires a civilization in which human beings see themselves as inhabitants of a single planet and in which global governance of these processes is conducted on a cooperative basis. The key question is, then, whether our present civilization is ready for such an endeavor, reflecting the fact that the critical uncertainties are political and cultural in nature. I...
11. Project Icarus: Stakeholder Scenarios for an Interstellar Exploration Program
Science.gov (United States)
Hein, A. M.; Tziolas, A. C.; Osborne, R.
The Project Icarus Study Group's objective is to design a mainly fusion-propelled interstellar probe. The starting point are the results of the Daedalus study, which was conducted by the British Interplanetary Society during the 1970's. As the Daedalus study already indicated, interstellar probes will be the result of a large scale, decade-long development program. To sustain a program over such long periods, the commitment of key stakeholders is vital. Although previous publications identified political and societal preconditions to an interstellar exploration program, there is a lack of more specific scientific and political stakeholder scenarios. This paper develops stakeholder scenarios which allow for a more detailed sustainability assessment of future programs. For this purpose, key stakeholder groups and their needs are identified and scientific and political scenarios derived. Political scenarios are based on patterns of past space programs but unprecedented scenarios are considered as well. Although it is very difficult to sustain an interstellar exploration program, there are scenarios in which this seems to be possible, e.g. the discovery of life within the solar system and on an exoplanet, a global technology development program, and dual-use of technologies for defence and security purposes. This is a submission of the Project Icarus Study Group.
12. EXPLORING THE POSSIBILITY OF O AND Ne CONTAMINATION IN ULYSSES OBSERVATIONS OF INTERSTELLAR HELIUM
International Nuclear Information System (INIS)
Wood, Brian E.; Müller, Hans-Reinhard; Bzowski, Maciej; Sokół, Justyna M.; Möbius, Eberhard; Witte, Manfred; McComas, David J.
2015-01-01
We explore the possibility that interstellar O and Ne may be contributing to the particle signal from the GAS instrument on Ulysses, which is generally assumed to be entirely He. Motivating this study is the recognition that an interstellar temperature higher than any previously estimated from Ulysses data could potentially resolve a discrepancy between Ulysses He measurements and those from the Interstellar Boundary Explorer (IBEX). Contamination by O and Ne could lead to Ulysses temperature measurements that are too low. We estimate the degree of O and Ne contamination necessary to increase the inferred Ulysses temperature to 8500 K, which would be consistent with both the Ulysses and IBEX data given the same interstellar flow speed. We find that producing the desired effect requires a heavy element contamination level of ∼9% of the total Ulysses/GAS signal. However, this degree of heavy element contribution is about an order of magnitude higher than expected based on our best estimates of detection efficiencies, ISM abundances, and heliospheric survival probabilities, making it unlikely that heavy element contamination is significantly affecting temperatures derived from Ulysses data
13. EXPLORING THE POSSIBILITY OF O AND Ne CONTAMINATION IN ULYSSES OBSERVATIONS OF INTERSTELLAR HELIUM
Energy Technology Data Exchange (ETDEWEB)
Wood, Brian E. [Naval Research Laboratory, Space Science Division, Washington, DC 20375 (United States); Müller, Hans-Reinhard [Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755 (United States); Bzowski, Maciej; Sokół, Justyna M. [Space Research Centre of the Polish Academy of Sciences, Ul. Bartycka 18 A, 00-716 Warsaw (Poland); Möbius, Eberhard [Space Science Center and Department of Physics, University of New Hampshire, Durham, NH 03824 (United States); Witte, Manfred [Max-Planck-Institute for Solar System Research, Katlenburg-Lindau (Germany); McComas, David J., E-mail: [email protected] [Southwest Research Institute, San Antonio, TX 78228 (United States)
2015-10-15
We explore the possibility that interstellar O and Ne may be contributing to the particle signal from the GAS instrument on Ulysses, which is generally assumed to be entirely He. Motivating this study is the recognition that an interstellar temperature higher than any previously estimated from Ulysses data could potentially resolve a discrepancy between Ulysses He measurements and those from the Interstellar Boundary Explorer (IBEX). Contamination by O and Ne could lead to Ulysses temperature measurements that are too low. We estimate the degree of O and Ne contamination necessary to increase the inferred Ulysses temperature to 8500 K, which would be consistent with both the Ulysses and IBEX data given the same interstellar flow speed. We find that producing the desired effect requires a heavy element contamination level of ∼9% of the total Ulysses/GAS signal. However, this degree of heavy element contribution is about an order of magnitude higher than expected based on our best estimates of detection efficiencies, ISM abundances, and heliospheric survival probabilities, making it unlikely that heavy element contamination is significantly affecting temperatures derived from Ulysses data.
14. Small Body Exploration Technologies as Precursors for Interstellar Robotics
Energy Technology Data Exchange (ETDEWEB)
Noble, Robert; /SLAC; Sykes, Mark V.; /PSI, Tucson
2012-02-15
The scientific activities undertaken to explore our Solar System will be the same as required someday at other stars. The systematic exploration of primitive small bodies throughout our Solar System requires new technologies for autonomous robotic spacecraft. These diverse celestial bodies contain clues to the early stages of the Solar System's evolution as well as information about the origin and transport of water-rich and organic material, the essential building blocks for life. They will be among the first objects studied at distant star systems. The technologies developed to address small body and outer planet exploration will form much of the technical basis for designing interstellar robotic explorers. The Small Bodies Assessment Group, which reports to NASA, initiated a Technology Forum in 2011 that brought together scientists and technologists to discuss the needs and opportunities for small body robotic exploration in the Solar System. Presentations and discussions occurred in the areas of mission and spacecraft design, electric power, propulsion, avionics, communications, autonomous navigation, remote sensing and surface instruments, sampling, intelligent event recognition, and command and sequencing software. In this paper, the major technology themes from the Technology Forum are reviewed, and suggestions are made for developments that will have the largest impact on realizing autonomous robotic vehicles capable of exploring other star systems.
15. Interstellar Explorer Observations of the Solar System's Debris Disks
Science.gov (United States)
Lisse, C. M.; McNutt, R. L., Jr.; Brandt, P. C.
2017-12-01
Planetesimal belts and debris disks full of dust are known as the "signposts of planet formation" in exosystems. The overall brightness of a disk provides information on the amount of sourcing planetesimal material, while asymmetries in the shape of the disk can be used to search for perturbing planets. The solar system is known to house two such belts, the Asteroid belt and the Kuiper Belt; and at least one debris cloud, the Zodiacal Cloud, sourced by planetisimal collisions and Kuiper Belt comet evaporative sublimation. However these are poorly understood in toto because we live inside of them. E.g., while we know of the two planetesimal belt systems, it is not clear how much, if any, dust is produced from the Kuiper belt since the near-Sun comet contributions dominate near-Earth space. Understanding how much dust is produced in the Kuiper belt would give us a much better idea of the total number of bodies in the belt, especially the smallest ones, and their dynamical collisional state. Even for the close in Zodiacal cloud, questions remain concerning its overall shape and orientation with respect to the ecliptic and invariable planes of the solar system - they aren't explainable from the perturbations caused by the known planets alone. In this paper we explore the possibilities of using an Interstellar Explorer telescope placed at 200 AU from the sun to observe the brightness, shape, and extent of the solar system's debris disk(s). We should be able to measure the entire extent of the inner, near-earth zodiacal cloud; whether it connects smoothly into an outer cloud, or if there is a second outer cloud sourced by the Kuiper belt and isolated by the outer planets, as predicted by Stark & Kuchner (2009, 2010) and Poppe et al. (2012, 2016; Figure 1). VISNIR imagery will inform about the dust cloud's density, while MIR cameras will provide thermal imaging photometry related to the cloud's dust particle size and composition. Observing at high phase angle by looking
16. Exploring the magnetospheric boundary layer
International Nuclear Information System (INIS)
Hapgood, M.A.; Bryant, D.A.
1992-01-01
We show how, for most crossings of the boundary layer, one can construct a 'transition parameter', based on electron density and temperature, which orders independent plasma measurements into well-defined patterns which are consistent from case to case. We conclude that there is a gradual change in the balance of processes which determine the structure of the layer and suggest that there is no advantage in dividing the layer into different regions. We further conclude that the mixing processes in layer act in an organised way to give the consistent patterns revealed by the transition parameter. More active processes must sometimes take to give the extreme values (e.g. in velocity) which are seen in some crossings
17. NASA's interstellar probe mission
International Nuclear Information System (INIS)
Liewer, P.C.; Ayon, J.A.; Wallace, R.A.; Mewaldt, R.A.
2000-01-01
NASA's Interstellar Probe will be the first spacecraft designed to explore the nearby interstellar medium and its interaction with our solar system. As envisioned by NASA's Interstellar Probe Science and Technology Definition Team, the spacecraft will be propelled by a solar sail to reach >200 AU in 15 years. Interstellar Probe will investigate how the Sun interacts with its environment and will directly measure the properties and composition of the dust, neutrals and plasma of the local interstellar material which surrounds the solar system. In the mission concept developed in the spring of 1999, a 400-m diameter solar sail accelerates the spacecraft to ∼15 AU/year, roughly 5 times the speed of Voyager 1 and 2. The sail is used to first bring the spacecraft to ∼0.25 AU to increase the radiation pressure before heading out in the interstellar upwind direction. After jettisoning the sail at ∼5 AU, the spacecraft coasts to 200-400 AU, exploring the Kuiper Belt, the boundaries of the heliosphere, and the nearby interstellar medium
18. The Ultimate Destination: Choice of Interplanetary Exploration Path can define Future of Interstellar Spaceflight
Science.gov (United States)
Silin, D. V.
Manned interstellar spaceflight is facing multiple challenges of great magnitude; among them are extremely large distances and the lack of known habitable planets other than Earth. Many of these challenges are applicable to manned space exploration within the Solar System to the same or lesser degree. If these issues are resolved on an interplanetary scale, better position to pursue interstellar exploration can be reached. However, very little progress (if any) was achieved in manned space exploration since the end of Space Race. There is no lack of proposed missions, but all of them require considerable technological and financial efforts to implement while yielding no tangible benefits that would justify their costs. To overcome this obstacle highest priority in future space exploration plans should be assigned to the creation of added value in outer space. This goal can be reached if reductions in space transportation, construction and maintenance of space-based structures costs are achieved. In order to achieve these requirements several key technologies have to be mastered, such as near-Earth object mining, space- based manufacturing, agriculture and structure assembly. To keep cost and difficulty under control next exploration steps can be limited to nearby destinations such as geostationary orbit, low lunar orbit, Moon surface and Sun-Earth L1 vicinity. Completion of such a program will create a solid foundation for further exploration and colonization of the Solar System, solve common challenges of interplanetary and interstellar spaceflight and create useful results for the majority of human population. Another important result is that perception of suitable destinations for interstellar missions will change significantly. If it becomes possible to create habitable and self-sufficient artificial environments in the nearby interplanetary space, Earth-like habitable planets will be no longer required to expand beyond our Solar System. Large fraction of the
19. Boundary Conditions for the Paleoenvironment: Chemical and Physical Processes in Dense Interstellar Clouds: Summary of Research
Science.gov (United States)
Irvine, William M.
1999-01-01
The basic theme of this program was the study of molecular complexity and evolution for the biogenic elements and compounds in interstellar clouds and in primitive solar system objects. Research included the detection and study of new interstellar and cometary molecules and investigation of reaction pathways for astrochemistry from a comparison of theory and observed molecular abundances. The latter includes studies of cold, dark clouds in which ion-molecule chemistry should predominate, searches for the effects of interchange of material between the gas and solid phases in interstellar clouds, unbiased spectral surveys of particular sources, and systematic investigation of the interlinked chemistry and physics of dense interstellar clouds. In addition, the study of comets has allowed a comparison between the chemistry of such minimally thermally processed objects and that of interstellar clouds, shedding light on the evolution of the biogenic elements during the process of solar system formation. One PhD dissertation on this research was completed by a graduate student at the University of Massachusetts. An additional 4 graduate students at the University of Massachusetts and 5 graduate students from other institutions participated in research supported by this grant, with 6 of these thus far receiving PhD degrees from the University of Massachusetts or their home institutions. Four postdoctoral research associates at the University of Massachusetts also participated in research supported by this grant, receiving valuable training.
20. First exploration of a single thermal interface between the two dominant phases of the interstellar medium
Science.gov (United States)
Gry, Cecile
2017-08-01
Two phases of the interstellar medium, the Warm Neutral Medium (WNM) and the Hot Ionized Medium (HIM) occupy most the volume of space in the plane of our Galaxy. Because the boundaries between these phases are important sources of energy loss for the hot gas, they are supposed to play an important role in the thermal structure and evolution of the ISM and of galaxies.Many theorists have created descriptions of the nature of such boundaries and have derived two fundamental concepts: (1) a conductive interface and (2) a turbulent mixing layer.We have yet to observe in detail either kind of boundary. This is achieved by using UV absorption lines of moderately high ionization stages of heavy elements. Yet, over most lines of sight the diagnostics are blurred out by the superposition of different regions with vastly different physical conditions, making them difficult to interpret. To characterize the nature of the physical processes at a boundary one must observe along a sight line that penetrates just one such region. The simplest configuration is the outer boundary of the Local Cloud, the WNM ((T 7000 K) that surrounds the Sun and which is embedded in a very low density, soft X-ray emitting hot medium ( 10^6 K) that fills a cavity ( 200 pc in diameter) called the Local Bubble.We propose to observe an ideal target: a nearby, bright B9V star (i.e. hot enough to provide a high-SNR continuum, but not enough to contaminate it with absorptions from circumstellar high-ionization species), located in a direction where the relative orientation of the magnetic field and the cloud boundary does not quench thermal conduction and thus favors a full extent of the interface.
1. An Exploration of Boundaries and Solidarity in Counseling Relationships
Science.gov (United States)
Speight, Suzette L.
2012-01-01
This article explores the boundaries between clinicians and clients in light of the construct of solidarity. A universal conception of boundaries is critiqued and a culturally congruent view of boundaries is examined, rooted in the concept of solidarity. The article includes case illustrations of the connection between boundaries and solidarity…
2. Exploring the planetary boundary for chemical pollution
DEFF Research Database (Denmark)
Diamond, Miriam L.; de Wit, Cynthia A.; Molander, Sverker
2015-01-01
Rockström et al. (2009a, 2009b) have warned that humanity must reduce anthropogenic impacts defined by nine planetary boundaries if "unacceptable global change" is to be avoided. Chemical pollution was identified as one of those boundaries for which continued impacts could erode the resilience...... of ecosystems and humanity. The central concept of the planetary boundary (or boundaries) for chemical pollution (PBCP or PBCPs) is that the Earth has a finite assimilative capacity for chemical pollution, which includes persistent, as well as readily degradable chemicals released at local to regional scales......, which in aggregate threaten ecosystem and human viability. The PBCP allows humanity to explicitly address the increasingly global aspects of chemical pollution throughout a chemical's life cycle and the need for a global response of internationally coordinated control measures. We submit that sufficient...
3. Three-dimensional Features of the Outer Heliosphere Due to Coupling between the Interstellar and Heliospheric Magnetic Field. V. The Bow Wave, Heliospheric Boundary Layer, Instabilities, and Magnetic Reconnection
Energy Technology Data Exchange (ETDEWEB)
Pogorelov, N. V.; Heerikhuisen, J. [Department of Space Science, The University of Alabama in Huntsville, AL 35805 (United States); Roytershteyn, V. [Space Science Institute, Boulder, CO 80301 (United States); Burlaga, L. F. [NASA Goddard Space Flight Center, Greenbelt, MD 20771 (United States); Gurnett, D. A.; Kurth, W. S., E-mail: [email protected] [Department of Physics and Astronomy, The University of Iowa, Iowa City, IA 52242 (United States)
2017-08-10
The heliosphere is formed due to interaction between the solar wind (SW) and local interstellar medium (LISM). The shape and position of the heliospheric boundary, the heliopause, in space depend on the parameters of interacting plasma flows. The interplay between the asymmetrizing effect of the interstellar magnetic field and charge exchange between ions and neutral atoms plays an important role in the SW–LISM interaction. By performing three-dimensional, MHD plasma/kinetic neutral atom simulations, we determine the width of the outer heliosheath—the LISM plasma region affected by the presence of the heliosphere—and analyze quantitatively the distributions in front of the heliopause. It is shown that charge exchange modifies the LISM plasma to such extent that the contribution of a shock transition to the total variation of plasma parameters becomes small even if the LISM velocity exceeds the fast magnetosonic speed in the unperturbed medium. By performing adaptive mesh refinement simulations, we show that a distinct boundary layer of decreased plasma density and enhanced magnetic field should be observed on the interstellar side of the heliopause. We show that this behavior is in agreement with the plasma oscillations of increasing frequency observed by the plasma wave instrument onboard Voyager 1. We also demonstrate that Voyager observations in the inner heliosheath between the heliospheric termination shock and the heliopause are consistent with dissipation of the heliospheric magnetic field. The choice of LISM parameters in this analysis is based on the simulations that fit observations of energetic neutral atoms performed by Interstellar Boundary Explorer .
4. Exploring exotic states with twisted boundary conditions
International Nuclear Information System (INIS)
2017-01-01
he goal of this thesis is to develop methods to study the nature and properties of exotic hadrons from lattice simulations. The main focus lies in the application of twisted boundary conditions. The thesis consists of a general introduction and the collection of three papers, represented respectively in three chapters. The introduction of the thesis reviews the theoretical background, which is further used in the rest of the thesis. Further implementing partially twisted boundary conditions in the scalar sector of lattice QCD is studied. Then we develop a method to study the content of the exotic hadrons by determining the wave function renormalization constant from lattice simulations, exploiting the dependence of the spectrum on the twisted boundary conditions. The final chapter deals with a novel method to study the multi-channel scattering problem in a finite volume, which is relevant for exotic states. Its key idea is to extract the complex hadron-hadron optical potential, avoiding the difficulties, associated with the solution of the multi-channel Luescher equation.
5. Exploring exotic states with twisted boundary conditions
Energy Technology Data Exchange (ETDEWEB)
2017-09-11
he goal of this thesis is to develop methods to study the nature and properties of exotic hadrons from lattice simulations. The main focus lies in the application of twisted boundary conditions. The thesis consists of a general introduction and the collection of three papers, represented respectively in three chapters. The introduction of the thesis reviews the theoretical background, which is further used in the rest of the thesis. Further implementing partially twisted boundary conditions in the scalar sector of lattice QCD is studied. Then we develop a method to study the content of the exotic hadrons by determining the wave function renormalization constant from lattice simulations, exploiting the dependence of the spectrum on the twisted boundary conditions. The final chapter deals with a novel method to study the multi-channel scattering problem in a finite volume, which is relevant for exotic states. Its key idea is to extract the complex hadron-hadron optical potential, avoiding the difficulties, associated with the solution of the multi-channel Luescher equation.
6. Exploring the planetary boundary for chemical pollution.
Science.gov (United States)
Diamond, Miriam L; de Wit, Cynthia A; Molander, Sverker; Scheringer, Martin; Backhaus, Thomas; Lohmann, Rainer; Arvidsson, Rickard; Bergman, Åke; Hauschild, Michael; Holoubek, Ivan; Persson, Linn; Suzuki, Noriyuki; Vighi, Marco; Zetzsch, Cornelius
2015-05-01
Rockström et al. (2009a, 2009b) have warned that humanity must reduce anthropogenic impacts defined by nine planetary boundaries if "unacceptable global change" is to be avoided. Chemical pollution was identified as one of those boundaries for which continued impacts could erode the resilience of ecosystems and humanity. The central concept of the planetary boundary (or boundaries) for chemical pollution (PBCP or PBCPs) is that the Earth has a finite assimilative capacity for chemical pollution, which includes persistent, as well as readily degradable chemicals released at local to regional scales, which in aggregate threaten ecosystem and human viability. The PBCP allows humanity to explicitly address the increasingly global aspects of chemical pollution throughout a chemical's life cycle and the need for a global response of internationally coordinated control measures. We submit that sufficient evidence shows stresses on ecosystem and human health at local to global scales, suggesting that conditions are transgressing the safe operating space delimited by a PBCP. As such, current local to global pollution control measures are insufficient. However, while the PBCP is an important conceptual step forward, at this point single or multiple PBCPs are challenging to operationalize due to the extremely large number of commercial chemicals or mixtures of chemicals that cause myriad adverse effects to innumerable species and ecosystems, and the complex linkages between emissions, environmental concentrations, exposures and adverse effects. As well, the normative nature of a PBCP presents challenges of negotiating pollution limits amongst societal groups with differing viewpoints. Thus, a combination of approaches is recommended as follows: develop indicators of chemical pollution, for both control and response variables, that will aid in quantifying a PBCP(s) and gauging progress towards reducing chemical pollution; develop new technologies and technical and social
7. The First Interstellar Explorer: What should it do when it Arrives at its Destination?
Science.gov (United States)
Freeman, A.; Alkalai, L.
2017-12-01
Imagine that we have decided to embark on mankind's most ambitious project: a 40-year duration mission to visit a habitable-zone planet orbiting one of our nearest stellar neighbors. To plan our mission we must consider altogether 6 mission phases: I. Accelerate out of our Solar System; II. Survive Cruise to Proxima Centauri; III. Decelerate on Approach; IV. Adjust Trajectory for Close Encounter; V. Acquire Data; VI. Return Information to Earth. Most papers on this topic address only the first two phases. This paper addresses Phases III-VI - what would we want our interstellar spacecraft to do when it arrives at its destination, and how should it be configured when it gets there? Should the mission be a simple flyby, collecting data on the planetary system as it swings by in a few short days? Or should it attempt orbit insertion around the target star, so that it can spend longer in the system? Categories of information that we might want returned to Earth include: images; spectral signatures from the surface; detailed atmospheric composition; a moon count; the magnetosphere characteristics. These will only be of interest if we have not been able to discern this information remotely, i.e. observing from our own solar system during the 40 years it takes to arrive at the destination. This means that the questions we seek to answer may be refined en route from basic discovery questions to perhaps more process-oriented ones. This brings us to a central point of this paper - that the spacecraft that leaves our solar system will not be configured appropriately for mission Phases III-VI, especially given that those phases may occur nearly 40 years after solar system escape. The ability to reconfigure itself, perhaps even cannibalize itself, should be built into the design of an interstellar explorer from the start, which would allow us to send hardware and software upgrades that mirror technologies developed on Earth during the long cruise. Should the spacecraft carry a 3
8. Interstellar Probe: First Step to the Stars
Science.gov (United States)
McNutt, R. L., Jr.
2017-12-01
The idea of an "Interstellar Probe," a robotic spacecraft traveling into the nearby interstellar medium for the purpose of scientific investigation, dates to the mid-1960s. The Voyager Interstellar Mission (VIM), an "accidental" 40-year-old by-product of the Grand Tour of the solar system, has provided initial answers to the problem of the global heliospheric configuration and the details of its interface with interstellar space. But the twin Voyager spacecraft have, at most, only another decade of lifetime, and only Voyager 1 has emerged from the heliosheath interaction region. To understand the nature of the interaction, a near-term mission to the "near-by" interstellar medium with modern and focused instrumentation remains a compelling priority. Imaging of energetic neutral atoms (ENAs) by the Ion Neutral CAmera (INCA) on Cassini and from the Interstellar Boundary Explorer (IBEX) in Earth orbit have provided significant new insights into the global interaction region but point to discrepancies with our current understanding. Exploring "as far as possible" into "pristine" interstellar space can resolve these. Hence, reaching large heliocentric distances rapidly is a driver for an Interstellar Probe. Such a mission is timely; understanding the interstellar context of exoplanet systems - and perhaps the context for the emergence of life both here and there - hinges upon what we can discover within our own stellar neighborhood. With current spacecraft technology and high-capability launch vehicles, such as the Space Launch System (SLS), a small, but extremely capable spacecraft, could be dispatched to the near-by interstellar medium with at least twice the speed of the Voyagers. Challenges remain with payload mass and power constraints for optimized science measurements. Mission longevity, as experienced by, but not designed into, the Voyagers, communications capability, and radioisotope power system performance and lifetime are solvable engineering challenges. Such
9. Exploring boundary-spanning practices among creativity managers
DEFF Research Database (Denmark)
Andersen, Poul Houman; Kragh, Hanne
2015-01-01
Purpose – External inputs are critical for organisational creativity. In order to bridge different thought worlds and cross-organisational barriers, managers must initiate and motivate boundary spanning processes. The purpose of this paper is to explore how boundary spanners manage creativity...... and observation. Findings – Three meta-practices used by managers to manage boundary-spanning creative projects are presented: defining the creative space, making space for creativity and acting in the creative space. These practices are detailed in seven case studies of creative projects. Research limitations...
10. The ESO Diffuse Interstellar Bands Large Exploration Survey (EDIBLES) . I. Project description, survey sample, and quality assessment
Science.gov (United States)
Cox, Nick L. J.; Cami, Jan; Farhang, Amin; Smoker, Jonathan; Monreal-Ibero, Ana; Lallement, Rosine; Sarre, Peter J.; Marshall, Charlotte C. M.; Smith, Keith T.; Evans, Christopher J.; Royer, Pierre; Linnartz, Harold; Cordiner, Martin A.; Joblin, Christine; van Loon, Jacco Th.; Foing, Bernard H.; Bhatt, Neil H.; Bron, Emeric; Elyajouri, Meriem; de Koter, Alex; Ehrenfreund, Pascale; Javadi, Atefeh; Kaper, Lex; Khosroshadi, Habib G.; Laverick, Mike; Le Petit, Franck; Mulas, Giacomo; Roueff, Evelyne; Salama, Farid; Spaans, Marco
2017-10-01
The carriers of the diffuse interstellar bands (DIBs) are largely unidentified molecules ubiquitously present in the interstellar medium (ISM). After decades of study, two strong and possibly three weak near-infrared DIBs have recently been attributed to the C60^+ fullerene based on observational and laboratory measurements. There is great promise for the identification of the over 400 other known DIBs, as this result could provide chemical hints towards other possible carriers. In an effort tosystematically study the properties of the DIB carriers, we have initiated a new large-scale observational survey: the ESO Diffuse Interstellar Bands Large Exploration Survey (EDIBLES). The main objective is to build on and extend existing DIB surveys to make a major step forward in characterising the physical and chemical conditions for a statistically significant sample of interstellar lines-of-sight, with the goal to reverse-engineer key molecular properties of the DIB carriers. EDIBLES is a filler Large Programme using the Ultraviolet and Visual Echelle Spectrograph at the Very Large Telescope at Paranal, Chile. It is designed to provide an observationally unbiased view of the presence and behaviour of the DIBs towards early-spectral-type stars whose lines-of-sight probe the diffuse-to-translucent ISM. Such a complete dataset will provide a deep census of the atomic and molecular content, physical conditions, chemical abundances and elemental depletion levels for each sightline. Achieving these goals requires a homogeneous set of high-quality data in terms of resolution (R 70 000-100 000), sensitivity (S/N up to 1000 per resolution element), and spectral coverage (305-1042 nm), as well as a large sample size (100+ sightlines). In this first paper the goals, objectives and methodology of the EDIBLES programme are described and an initial assessment of the data is provided.
11. Exploring Sulfur & Argon Abundances in Planetary Nebulae as Metallicity- Indicator Surrogates for Iron in the Interstellar Medium
Science.gov (United States)
Kwitter, Karen B.; Henry, Richard C.
1999-02-01
Our primary motivation for studying S and Ar distributions in planetary nebulae (PNe) across the Galactic disk is to explore the possibility of a surrogacy between (S+Ar)/O and Fe/O for use as a metallicity indicator in the interstellar medium. The chemical history of the Galaxy is usually studied through O and Fe distributions among objects of different ages. Historically, though, Fe and O have not been measured in the same systems: Fe is easily seen in stars but hard to detect in nebulae; the reverse is true for O. We know that S and Ar abundances are not affected by PN progenitor evolution, and we therefore seek to exploit both their unaltered abundances and ease of detectability in PNe to explore their surrogacy for Fe. If proven valid, this surrogacy carries broad and important ramifications for bridging the gap between stellar and interstellar abundances in the Galaxy, and potentially beyond. Observed S/O and Ar/O gradients will also provide constraints on theoretical stellar yields of S and Ar, since they can be compared with chemical evolution models (which incorporate theoretically-predicted stellar yields, an initial mass function, and rates of star formation and infall) to help place constraints on model parameters.
12. EXPLORING THE POTENTIAL FORMATION OF ORGANIC SOLIDS IN CHONDRITES AND COMETS THROUGH POLYMERIZATION OF INTERSTELLAR FORMALDEHYDE
International Nuclear Information System (INIS)
Kebukawa, Yoko; Cody, George D.; David Kilcoyne, A. L.
2013-01-01
Polymerization of interstellar formaldehyde, first through the formose reaction and then through subsequent condensation reactions, provides a plausible explanation for how abundant and highly chemically complex organic solids may have come to exist in primitive solar system objects. In order to gain better insight on the reaction, a systematic study of the relationship of synthesis temperature with resultant molecular structure was performed. In addition, the effect of the presence of ammonia on the reaction rate and molecular structure of the product was studied. The synthesized formaldehyde polymer is directly compared to chondritic insoluble organic matter (IOM) isolated from primitive meteorites using solid-state 13 C nuclear magnetic resonance, Fourier transform infrared, and X-ray absorption near edge structure spectroscopy. The molecular structure of the formaldehyde polymer is shown to exhibit considerable similarity at the functional group level with primitive chondritic IOM. The addition of ammonia to the solution enhances the rate of polymerization reaction at lower temperatures and results in substantial incorporation of nitrogen into the polymer. Morphologically, the formaldehyde polymer exists as submicron to micron-sized spheroidal particles and spheroidal particle aggregates that bare considerable similarity to the organic nanoglobules commonly observed in chondritic IOM. These spectroscopic and morphological data support the hypothesis that IOM in chondrites and refractory organic carbon in comets may have formed through the polymerization of interstellar formaldehyde after planetesimal accretion, in the presence of liquid water, early in the history of the solar system.
13. EXPLORING THE POTENTIAL FORMATION OF ORGANIC SOLIDS IN CHONDRITES AND COMETS THROUGH POLYMERIZATION OF INTERSTELLAR FORMALDEHYDE
Energy Technology Data Exchange (ETDEWEB)
Kebukawa, Yoko; Cody, George D. [Geophysical Laboratory, Carnegie Institution of Washington, 5251 Broad Branch Road NW, Washington, DC 20015 (United States); David Kilcoyne, A. L., E-mail: [email protected], E-mail: [email protected] [Advanced Light Source, Lawrence Berkeley National Laboratory, Mail Stop 7R0222, 1 Cyclotron Road, Berkeley, CA 94720 (United States)
2013-07-01
Polymerization of interstellar formaldehyde, first through the formose reaction and then through subsequent condensation reactions, provides a plausible explanation for how abundant and highly chemically complex organic solids may have come to exist in primitive solar system objects. In order to gain better insight on the reaction, a systematic study of the relationship of synthesis temperature with resultant molecular structure was performed. In addition, the effect of the presence of ammonia on the reaction rate and molecular structure of the product was studied. The synthesized formaldehyde polymer is directly compared to chondritic insoluble organic matter (IOM) isolated from primitive meteorites using solid-state {sup 13}C nuclear magnetic resonance, Fourier transform infrared, and X-ray absorption near edge structure spectroscopy. The molecular structure of the formaldehyde polymer is shown to exhibit considerable similarity at the functional group level with primitive chondritic IOM. The addition of ammonia to the solution enhances the rate of polymerization reaction at lower temperatures and results in substantial incorporation of nitrogen into the polymer. Morphologically, the formaldehyde polymer exists as submicron to micron-sized spheroidal particles and spheroidal particle aggregates that bare considerable similarity to the organic nanoglobules commonly observed in chondritic IOM. These spectroscopic and morphological data support the hypothesis that IOM in chondrites and refractory organic carbon in comets may have formed through the polymerization of interstellar formaldehyde after planetesimal accretion, in the presence of liquid water, early in the history of the solar system.
14. Exploring the Potential Formation of Organic Solids in Chondrites and Comets through Polymerization of Interstellar Formaldehyde
Science.gov (United States)
Kebukawa, Yoko; Kilcoyne, A. L. David; Cody, George D.
2013-07-01
Polymerization of interstellar formaldehyde, first through the formose reaction and then through subsequent condensation reactions, provides a plausible explanation for how abundant and highly chemically complex organic solids may have come to exist in primitive solar system objects. In order to gain better insight on the reaction, a systematic study of the relationship of synthesis temperature with resultant molecular structure was performed. In addition, the effect of the presence of ammonia on the reaction rate and molecular structure of the product was studied. The synthesized formaldehyde polymer is directly compared to chondritic insoluble organic matter (IOM) isolated from primitive meteorites using solid-state 13C nuclear magnetic resonance, Fourier transform infrared, and X-ray absorption near edge structure spectroscopy. The molecular structure of the formaldehyde polymer is shown to exhibit considerable similarity at the functional group level with primitive chondritic IOM. The addition of ammonia to the solution enhances the rate of polymerization reaction at lower temperatures and results in substantial incorporation of nitrogen into the polymer. Morphologically, the formaldehyde polymer exists as submicron to micron-sized spheroidal particles and spheroidal particle aggregates that bare considerable similarity to the organic nanoglobules commonly observed in chondritic IOM. These spectroscopic and morphological data support the hypothesis that IOM in chondrites and refractory organic carbon in comets may have formed through the polymerization of interstellar formaldehyde after planetesimal accretion, in the presence of liquid water, early in the history of the solar system.
15. Interstellar Deuterium, Nitrogen and Oxygen Towards HZ43A: Results from the Far Ultraviolet Spectroscopic Explorer (FUSE) Mission
Science.gov (United States)
Kruk, J. W.; Howk, J. C.; Andre, M.; Moos, H. W.; Oegerle, William R.; Oliveira, C.; Sembach, K. R.; Chayer, P.; Linsky, J. L.; Wood, B. E.
2002-01-01
We present an analysis of interstellar absorption along the line of sight to the nearby white dwarf star HZ43A. The distance to this star is 68+/-13 pc, and the line of sight extends toward the north Galactic pole. Column densities of O(I), N(I), and N(II) were derived from spectra obtained by the Far Ultraviolet Spectroscopic Explorer (FUSE), the column density of D(I) was derived from a combination of our FUSE spectra and an archival HST GARDENS spectrum, and the column density of H(I) was derived from a combination of the GARDENS spectrum and values derived from EUVE data obtained from the literature. We find the following abundance ratios (with 2 sigma uncertainties): D(I)/H(I)=(1.66+/-0.28)x10(exp -5), O(I)/H(I)=(3.63+/-0.84)x10(exp -4), and N(I)/H(I)=(3.80+/-0.74)x10(exp -5). The N(II) column density was slightly greater than that of N(I), indicating that ionization corrections are important when deriving nitrogen abundances. Other interstellar species detected along the line of sight were C(II), C(III), O(VI), Si(II), Ar(I), Mg(II) and Fe(II); an upper limit was determined for N(III). No elements other than H(I) were detected in the stellar photosphere.
16. Exploring the diffuse interstellar bands with the Sloan Digital Sky Survey
Science.gov (United States)
Lan, Ting-Wen; Ménard, Brice; Zhu, Guangtun
2015-10-01
We use star, galaxy and quasar spectra taken by the Sloan Digital Sky Survey to map out the distribution of diffuse interstellar bands (DIBs) induced by the Milky Way. After carefully removing the intrinsic spectral energy distribution of each source, we show that by stacking thousands of spectra, it is possible to measure statistical flux fluctuations at the 10-3 level, detect more than 20 DIBs and measure their strength as a function of position on the sky. We create a map of DIB absorption covering about 5000 deg2 and measure correlations with various tracers of the interstellar medium: atomic and molecular hydrogen, dust and polycyclic aromatic hydrocarbons (PAHs). After recovering known correlations, we show that each DIB has a different dependence on atomic and molecular hydrogen: while they are all positively correlated with N_{H I}, they exhibit a range of behaviours with N_{H_2} showing positive, negative or no correlation. We show that a simple parametrization involving only N_{H I} and N_{H_2} applied to all the DIBs is sufficient to reproduce a large collection of observational results reported in the literature: it allows us to naturally describe the relations between DIB strength and dust reddening (including the so-called skin effect), the related scatter, DIB pair-wise correlations and families, the affinity for σ/ζ-type environments and other correlations related to molecules. Our approach allows us to characterize DIB dependencies in a simple manner and provides us with a metric to characterize the similarity between different DIBs.
17. Acoustic explorations of the upper ocean boundary layer
Science.gov (United States)
Vagle, Svein
2005-04-01
The upper ocean boundary layer is an important but difficult to probe part of the ocean. A better understanding of small scale processes at the air-sea interface, including the vertical transfer of gases, heat, mass and momentum, are crucial to improving our understanding of the coupling between atmosphere and ocean. Also, this part of the ocean contains a significant part of the total biomass at all trophic levels and is therefore of great interest to researchers in a range of different fields. Innovative measurement plays a critical role in developing our understanding of the processes involved in the boundary layer, and the availability of low-cost, compact, digital signal processors and sonar technology in self-contained and cabled configurations has led to a number of exciting developments. This talk summarizes some recent explorations of this dynamic boundary layer using both active and passive acoustics. The resonant behavior of upper ocean bubbles combined with single and multi-frequency broad band active and passive devices are now giving us invaluable information on air-sea gas transfer, estimation of biological production, marine mammal behavior, wind speed and precipitation, surface and internal waves, turbulence, and acoustic communication in the surf zone.
18. The Galactic Club or Galactic Cliques? Exploring the limits of interstellar hegemony and the Zoo Hypothesis
Science.gov (United States)
Forgan, Duncan H.
2017-10-01
The Zoo solution to Fermi's Paradox proposes that extraterrestrial intelligences (ETIs) have agreed to not contact the Earth. The strength of this solution depends on the ability for ETIs to come to agreement, and establish/police treaties as part of a so-called Galactic Club'. These activities are principally limited by the causal connectivity of a civilization to its neighbours at its inception, i.e. whether it comes to prominence being aware of other ETIs and any treaties or agreements in place. If even one civilization is not causally connected to the other members of a treaty, then they are free to operate beyond it and contact the Earth if wished, which makes the Zoo solution soft'. We should therefore consider how likely this scenario is, as this will give us a sense of the Zoo solution's softness, or general validity. We implement a simple toy model of ETIs arising in a Galactic Habitable Zone, and calculate the properties of the groups of culturally connected civilizations established therein. We show that for most choices of civilization parameters, the number of culturally connected groups is >1, meaning that the Galaxy is composed of multiple Galactic Cliques rather than a single Galactic Club. We find in our models for a single Galactic Club to establish interstellar hegemony, the number of civilizations must be relatively large, the mean civilization lifetime must be several millions of years, and the inter-arrival time between civilizations must be a few million years or less.
19. Interstellar Extinction
OpenAIRE
Gontcharov, George
2017-01-01
This review describes our current understanding of interstellar extinction. This differ substantially from the ideas of the 20th century. With infrared surveys of hundreds of millions of stars over the entire sky, such as 2MASS, SPITZER-IRAC, and WISE, we have looked at the densest and most rarefied regions of the interstellar medium at distances of a few kpc from the sun. Observations at infrared and microwave wavelengths, where the bulk of the interstellar dust absorbs and radiates, have br...
20. Doctoral research on architecture in Nigeria: Exploring domains, extending boundaries
Directory of Open Access Journals (Sweden)
2016-03-01
Full Text Available This paper explored through a literature review, the domains of research in Architecture and the nature of doctoral research, with a view to contributing to the evolving research agenda in the Nigerian context. The research method involved a descriptive and thematic analysis of the titles and abstracts of completed doctoral theses in Architecture in Nigeria, in the last 26 years (1990–2015, complemented by semi-structured interviews with six key informants. The study revealed an emphasis on Housing-related topics (34% relative to other research modules, such as׳ History and Theory׳ (20% and ׳Design and Production׳ (18%. It also reflected the limited coverage and scope of current research, relative to the global terrain, as evidenced in the article titles and contents of 45 Architecture-related Journals. The results of the interviews indicated the strong influence of supervisors׳ areas of interest in the choices of thesis titles. It highlighted reasons for the perceived focus on Housing, which reflect its unique place and multi-disciplinary nature. It concluded that extending the boundaries of architectural research at the doctoral level could be beneficial to the discipline and profession in Nigeria in order to align with global trends, while keeping cognizance of the local contexts.
1. Planetary Boundaries: Exploring the Safe Operating Space for Humanity
DEFF Research Database (Denmark)
Richardson, Katherine; Rockström, Johan; Steffen, Will
2009-01-01
boundaries are rough, first estimates only, surrounded by large uncertainties and knowledge gaps. Filling these gaps will require major advancements in Earth System and resilience science. The proposed concept of "planetary boundaries" lays the groundwork for shifting our approach to governance...... and management, away from the essentially sectoral analyses of limits to growth aimed at minimizing negative externalities, toward the estimation of the safe space for human development. Planetary boundaries define, as it were, the boundaries of the "planetary playing field" for humanity if we want to be sure...
2. Planetary boundaries: exploring the safe operating space for humanity
Science.gov (United States)
Johan Rockström; Will Steffen; Kevin Noone; Asa Persson; F. Stuart Chapin; Eric Lambin; Timothy M. Lenton; Marten Scheffer; Carl Folke; Hans Joachim Schellnhuber; Björn Nykvist; Cynthia A. de Wit; Terry Hughes; Sander van der Leeuw; Henning Rodhe; Sverker Sörlin; Peter K. Snyder; Robert Costanza; Uno Svedin; Malin Falkenmark; Louise Karlberg; Robert W. Corell; Victoria J. Fabry; James Hansen; Brian Walker; Diana Liverman; Katherine Richardson; Paul Crutzen; Jonathan Foley
2009-01-01
Anthropogenic pressures on the Earth System have reached a scale where abrupt global environmental change can no longer be excluded. We propose a new approach to global sustainability in which we define planetary boundaries within which we expect that humanity can operate safely. Transgressing one or more planetary boundaries may be deleterious or even catastrophic due...
3. REVISITING ULYSSES OBSERVATIONS OF INTERSTELLAR HELIUM
International Nuclear Information System (INIS)
Wood, Brian E.; Müller, Hans-Reinhard; Witte, Manfred
2015-01-01
We report the results of a comprehensive reanalysis of Ulysses observations of interstellar He atoms flowing through the solar system, the goal being to reassess the interstellar He flow vector and to search for evidence of variability in this vector. We find no evidence that the He beam seen by Ulysses changes at all from 1994-2007. The direction of flow changes by no more than ∼0.°3 and the speed by no more than ∼0.3 km s –1 . A global fit to all acceptable He beam maps from 1994-2007 yields the following He flow parameters: V ISM = 26.08 ± 0.21 km s –1 , λ = 75.54 ± 0.°19, β = –5.44 ± 0.°24, and T = 7260 ± 270 K; where λ and β are the ecliptic longitude and latitude direction in J2000 coordinates. The flow vector is consistent with the original analysis of the Ulysses team, but our temperature is significantly higher. The higher temperature somewhat mitigates a discrepancy that exists in the He flow parameters measured by Ulysses and the Interstellar Boundary Explorer, but does not resolve it entirely. Using a novel technique to infer photoionization loss rates directly from Ulysses data, we estimate a density of n He = 0.0196 ± 0.0033 cm –3 in the interstellar medium
4. REVISITING ULYSSES OBSERVATIONS OF INTERSTELLAR HELIUM
Energy Technology Data Exchange (ETDEWEB)
Wood, Brian E. [Naval Research Laboratory, Space Science Division, Washington, DC 20375 (United States); Müller, Hans-Reinhard [Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755 (United States); Witte, Manfred, E-mail: [email protected] [Max-Planck-Institute for Solar System Research, Katlenburg-Lindau D-37191 (Germany)
2015-03-01
We report the results of a comprehensive reanalysis of Ulysses observations of interstellar He atoms flowing through the solar system, the goal being to reassess the interstellar He flow vector and to search for evidence of variability in this vector. We find no evidence that the He beam seen by Ulysses changes at all from 1994-2007. The direction of flow changes by no more than ∼0.°3 and the speed by no more than ∼0.3 km s{sup –1}. A global fit to all acceptable He beam maps from 1994-2007 yields the following He flow parameters: V {sub ISM} = 26.08 ± 0.21 km s{sup –1}, λ = 75.54 ± 0.°19, β = –5.44 ± 0.°24, and T = 7260 ± 270 K; where λ and β are the ecliptic longitude and latitude direction in J2000 coordinates. The flow vector is consistent with the original analysis of the Ulysses team, but our temperature is significantly higher. The higher temperature somewhat mitigates a discrepancy that exists in the He flow parameters measured by Ulysses and the Interstellar Boundary Explorer, but does not resolve it entirely. Using a novel technique to infer photoionization loss rates directly from Ulysses data, we estimate a density of n {sub He} = 0.0196 ± 0.0033 cm{sup –3} in the interstellar medium.
5. Defensible Spaces in Philadelphia: Exploring Neighborhood Boundaries Through Spatial Analysis
Directory of Open Access Journals (Sweden)
Rory Kramer
2017-02-01
Full Text Available Few spatial scales are as important to individual outcomes as the neighborhood. However, it is nearly impossible to define neighborhoods in a generalizable way. This article proposes that by shifting the focus to measuring neighborhood boundaries rather than neighborhoods, scholars can avoid the problem of the indefinable neighborhood and better approach questions of what predicts racial segregation across areas. By quantifying an externality space theory of neighborhood boundaries, this article introduces a novel form of spatial analysis to test where potential physical markers of neighborhood boundaries (major roads, rivers, railroads, and the like are associated with persistent racial boundaries between 1990 and 2010. Using Philadelphia as a case study, the paper identifies neighborhoods with persistent racial boundaries. It theorizes that local histories of white reactions to black in-migration explain which boundaries persistently resisted racial turnover, unlike the majority of Philadelphia’s neighborhoods, and that those racial boundaries shape the location, progress, and reaction to new residential development in those neighborhoods.
6. TRIANGULATION OF THE INTERSTELLAR MAGNETIC FIELD
Energy Technology Data Exchange (ETDEWEB)
Schwadron, N. A.; Moebius, E. [University of New Hampshire, Durham, NH 03824 (United States); Richardson, J. D. [Massachusetts Institute of Technology, Cambridge, MA 02139 (United States); Burlaga, L. F. [Goddard Space Flight Center, Greenbelt, MD 20771 (United States); McComas, D. J. [Southwest Research Institute, San Antonio, TX 78228 (United States)
2015-11-01
Determining the direction of the local interstellar magnetic field (LISMF) is important for understanding the heliosphere’s global structure, the properties of the interstellar medium, and the propagation of cosmic rays in the local galactic medium. Measurements of interstellar neutral atoms by Ulysses for He and by SOHO/SWAN for H provided some of the first observational insights into the LISMF direction. Because secondary neutral H is partially deflected by the interstellar flow in the outer heliosheath and this deflection is influenced by the LISMF, the relative deflection of H versus He provides a plane—the so-called B–V plane in which the LISMF direction should lie. Interstellar Boundary Explorer (IBEX) subsequently discovered a ribbon, the center of which is conjectured to be the LISMF direction. The most recent He velocity measurements from IBEX and those from Ulysses yield a B–V plane with uncertainty limits that contain the centers of the IBEX ribbon at 0.7–2.7 keV. The possibility that Voyager 1 has moved into the outer heliosheath now suggests that Voyager 1's direct observations provide another independent determination of the LISMF. We show that LISMF direction measured by Voyager 1 is >40° off from the IBEX ribbon center and the B–V plane. Taking into account the temporal gradient of the field direction measured by Voyager 1, we extrapolate to a field direction that passes directly through the IBEX ribbon center (0.7–2.7 keV) and the B–V plane, allowing us to triangulate the LISMF direction and estimate the gradient scale size of the magnetic field.
7. Interstellar ammonia
International Nuclear Information System (INIS)
Ho, P.T.P.; Townes, C.H.
1983-01-01
Investigations and results on interstellar NH3 are discussed. The physics of the molecule, its interstellar excitation, and its formation and dissociation mechanisms are reviewed. The observing techniques and instruments, including single-antenna facilities, infrared and submillimeter techniques, and interferometric studies using the Very Large Array are briefly considered. Spectral data analysis is discussed, including the derivation of optical depths, excitation measurements, ortho-para measurements, and cross sections. Progress achieved in understanding the properties and evolution of the interstellar medium through NH3 studies is reviewed, including observations of nearby dark clouds and of clumping effects in molecular clouds, as well as interferometric observations of hot molecular cores in Orion, W51, and Sagittarius A. Research results on extragalactic NH3, far-infrared, submillimeter, and midinfrared NH3 observations are described. 101 references
8. Horses for courses: analytical tools to explore planetary boundaries
Science.gov (United States)
van Vuuren, Detlef P.; Lucas, Paul L.; Häyhä, Tiina; Cornell, Sarah E.; Stafford-Smith, Mark
2016-03-01
There is a need for more integrated research on sustainable development and global environmental change. In this paper, we focus on the planetary boundaries framework to provide a systematic categorization of key research questions in relation to avoiding severe global environmental degradation. The four categories of key questions are those that relate to (1) the underlying processes and selection of key indicators for planetary boundaries, (2) understanding the impacts of environmental pressure and connections between different types of impacts, (3) better understanding of different response strategies to avoid further degradation, and (4) the available instruments to implement such strategies. Clearly, different categories of scientific disciplines and associated model types exist that can accommodate answering these questions. We identify the strength and weaknesses of different research areas in relation to the question categories, focusing specifically on different types of models. We discuss that more interdisciplinary research is need to increase our understanding by better linking human drivers and social and biophysical impacts. This requires better collaboration between relevant disciplines (associated with the model types), either by exchanging information or by fully linking or integrating them. As fully integrated models can become too complex, the appropriate type of model (the racehorse) should be applied for answering the target research question (the race course).
9. Interstellar holography
NARCIS (Netherlands)
Walker, M. A.; Koopmans, L. V. E.; Stinebring, D. R.; van Straten, W.
2008-01-01
The dynamic spectrum of a radio pulsar is an in-line digital hologram of the ionized interstellar medium. It has previously been demonstrated that such holograms permit image reconstruction, in the sense that one can determine an approximation to the complex electric field values as a function of
10. Interstellar matter
International Nuclear Information System (INIS)
Mezger, P.G.
1978-01-01
An overview of the formation of our galaxy is presented followed by a summary of recent work in star formation and related topics. Selected discussions are given on interstellar matter including absorption characteristics of dust, the fully ionised component of the ISM and the energy density of lyc-photons in the solar neighbourhood and the diffuse galactic IR radiation
11. SECONDARY POPULATION OF INTERSTELLAR NEUTRALS seems deflected to the side
Science.gov (United States)
Nakagawa, H.; Bzowski, M.; Yamazaki, A.; Fukunishi, H.; Watanabe, S.; Takahashi, Y.; Taguchi, M.
Recently the neutral hydrogen flow in the inner heliosphere was found to be deflected relative to the helium flow by about 4 degrees Lallement et al 2005 The explanation of this delfection offered was a distortion of the heliosphere under the action of an ambient interstellar magnetic field In a separate study a number of data sets pertaining to interstellar neutral atoms obtained with various techniques were compiled and interpreted as due to an inflow of interstellar gas from an ecliptic longitude shifted by 10 - 40 degrees from the canonical upstream interstellar neutral flow direction at 254 degrees Collier et al 2004 The origin and properties of such a flow is still under debate We have performed a cross-experiment analysis of the heliospheric hydrogen and helium photometric observations performed simltaneously by the Nozomi spacecraft between the Earth and Mars orbit and explored possible deflection of hydrogen and helium flows with respect to the canonical upwind direction For the interpretation we used predictions of a state of the art 3D and fully time-dependent model of the neutral gas in the heliosphere with the boundary conditions ionization rates and radiation pressure taken from literature The model includes two populations of the thermal interstellar hydrogen predicted by the highly-reputed Moscow Monte Carlo model of the heliosphere The agreement between the data and simulations is not satifactory when one assumes that the upwind direction is the same for both populations and identical with the direction derived from inerstellar helium
12. Ionization of Interstellar Hydrogen
Science.gov (United States)
Whang, Y. C.
1996-09-01
Interstellar hydrogen can penetrate through the heliopause, enter the heliosphere, and may become ionized by photoionization and by charge exchange with solar wind protons. A fluid model is introduced to study the flow of interstellar hydrogen in the heliosphere. The flow is governed by moment equations obtained from integration of the Boltzmann equation over the velocity space. Under the assumption that the flow is steady axisymmetric and the pressure is isotropic, we develop a method of solution for this fluid model. This model and the method of solution can be used to study the flow of neutral hydrogen with various forms of ionization rate β and boundary conditions for the flow on the upwind side. We study the solution of a special case in which the ionization rate β is inversely proportional to R2 and the interstellar hydrogen flow is uniform at infinity on the upwind side. We solve the moment equations directly for the normalized density NH/NN∞, bulk velocity VH/VN∞, and temperature TH/TN∞ of interstellar hydrogen as functions of r/λ and z/λ, where λ is the ionization scale length. The solution is compared with the kinetic theory solution of Lallement et al. The fluid solution is much less time-consuming than the kinetic theory solutions. Since the ionization rate for production of pickup protons is directly proportional to the local density of neutral hydrogen, the high-resolution solution of interstellar neutral hydrogen obtained here will be used to study the global distribution of pickup protons.
13. AN ESTIMATE OF THE NEARBY INTERSTELLAR MAGNETIC FIELD USING NEUTRAL ATOMS
International Nuclear Information System (INIS)
Heerikhuisen, J.; Pogorelov, N. V.
2011-01-01
The strength and orientation of the magnetic field in the nearby interstellar medium have remained elusive, despite continual improvements in observations and models. Data from NASA's Voyager mission and the Solar Wind ANisotropies (SWAN) experiment on board Solar and Heliospheric Observatory (SOHO) have placed observational constraints on the magnetic field, and the more recent Interstellar Boundary Explorer (IBEX) data appear to also bear an imprint of the interstellar magnetic field (ISMF). In this paper, we combine computational models of the heliosphere with data from Voyager, SOHO/SWAN, and IBEX to estimate both the strength and direction of the nearby ISMF. On the basis of our simulations, we find that a field strength of 2-3 μG pointing from ecliptic coordinates (220-224, 39-44), combined with an interstellar hydrogen density of ∼0.15 cm -3 , produces results most consistent with observations.
14. Exploring the boundaries of corporate social responsibility and innovation
DEFF Research Database (Denmark)
Maier, Maximilian; Brem, Alexander; Kauke, Matthias
2016-01-01
into the corporate strategy and its linkages to innovation. On the basis of a systematic literature review, a conceptual framework is developed. This framework categorises socio-political stakeholders and identifies other relevant stakeholders on the basis of a theoretical typology. Finally, dialogue strategies...... are examined regarding their fit within the corporate innovation process. In this regard, stakeholder collaboration is suggested as the appropriate strategy of engaging strategically significant stakeholders. The paper concludes with implications, limitations and further research suggestions.......Corporate social responsibility (CSR) and innovation are common keywords in management research and practice. Both of them are understood in different ways. To discover the boundaries of CSR and innovation, this paper sheds light on the traditional view of CSR, with a focus on its (CSR) integration...
15. Interstellar Gas Flow Vector and Temperature Determination over 5 Years of IBEX Observations
International Nuclear Information System (INIS)
Möbius, E; Heirtzler, D; Kucharek, H; Lee, M A; Leonard, T; Schwadron, N; Bzowski, M; Kubiak, M A; Sokół, J M; Fuselier, S A; McComas, D J; Wurz, P
2015-01-01
The Interstellar Boundary Explorer (IBEX) observes the interstellar neutral gas flow trajectories at their perihelion in Earth's orbit every year from December through early April, when the Earth's orbital motion is into the oncoming flow. These observations have defined a narrow region of possible, but very tightly coupled interstellar neutral flow parameters, with inflow speed, latitude, and temperature as well-defined functions of inflow longitude. The best- fit flow vector is different by ≈ 3° and lower by ≈ 3 km/s than obtained previously with Ulysses GAS, but the temperature is comparable. The possible coupled parameter space reaches to the previous flow vector, but only for a substantially higher temperature (by ≈ 2000 K). Along with recent pickup ion observations and including historical observations of the interstellar gas, these findings have led to a discussion, whether the interstellar gas flow into the solar system has been stable or variable over time. These intriguing possibilities call for more detailed analysis and a longer database. IBEX has accumulated observations over six interstellar flow seasons. We review key observations and refinements in the analysis, in particular, towards narrowing the uncertainties in the temperature determination. We also address ongoing attempts to optimize the flow vector determination through varying the IBEX spacecraft pointing and discuss related implications for the local interstellar cloud and its interaction with the heliosphere
16. Anisotropies in TeV Cosmic Rays Related to the Local Interstellar Magnetic Field from the IBEX Ribbon
International Nuclear Information System (INIS)
Schwadron, N A; Moebius, E; Adams, F C; Christian, E; Desiati, P; Frisch, P; Funsten, H O; Jokipii, J R; McComas, D J; Zank, G P
2015-01-01
The Interstellar Boundary Explorer (IBEX) observes enhanced Energetic Neutral Atoms (ENAs) emission in the keV energy range from a narrow (∼20° wide) ''ribbon'' in the sky that appears to be centered on the direction of the local interstellar (LIS) magnetic field. The Milagro collaboration, the Asγ collaboration and the IceCube observatory have recently made global maps of cosmic ray fluxes in the TeV energy range, revealing anisotropic structures ordered in part by the local interstellar magnetic field and the interstellar flow. This paper following from a recent publication in Science makes the link between these disparate observations by developing a simple model of the magnetic structure surrounding the heliosphere in the Local Interstellar Medium (LISM) that is consistent with both IBEX ENA fluxes and TeV cosmic ray anisotropies. The model also employs the revised velocity direction of the LIC derived from neutral He observations by IBEX. By modeling the propagation of cosmic rays through this magnetic field structure, we specifically show that (1) the large-scale TeV anisotropy provides a roughly consistent orientation for the local interstellar magnetic field at the center of the IBEX Ribbon and corroborates the ∼ 3 μG magnitude of the local interstellar magnetic field derived from IBEX observations of the global heliosphere; (2) and small-scale structures in cosmic rays (over < 30° angular scales) are influenced by the interstellar field interaction with the heliosphere at energies < 10 TeV. Thus, we provide a link between IBEX ENA observations, IBEX neutral observations of interstellar He, and TeV cosmic ray anisotropies, which are strongly influenced by the interactions between the local interstellar magnetic field, the flow of the local interstellar plasma, and the global heliosphere
17. Challenges in the determination of the interstellar flow longitude from the pickup ion cutoff
Science.gov (United States)
Taut, A.; Berger, L.; Möbius, E.; Drews, C.; Heidrich-Meisner, V.; Keilbach, D.; Lee, M. A.; Wimmer-Schweingruber, R. F.
2018-03-01
Context. The interstellar flow longitude corresponds to the Sun's direction of movement relative to the local interstellar medium. Thus, it constitutes a fundamental parameter for our understanding of the heliosphere and, in particular, its interaction with its surroundings, which is currently investigated by the Interstellar Boundary EXplorer (IBEX). One possibility to derive this parameter is based on pickup ions (PUIs) that are former neutral ions that have been ionized in the inner heliosphere. The neutrals enter the heliosphere as an interstellar wind from the direction of the Sun's movement against the partially ionized interstellar medium. PUIs carry information about the spatial variation of their neutral parent population (density and flow vector field) in their velocity distribution function. From the symmetry of the longitudinal flow velocity distribution, the interstellar flow longitude can be derived. Aim. The aim of this paper is to identify and eliminate systematic errors that are connected to this approach of measuring the interstellar flow longitude; we want to minimize any systematic influences on the result of this analysis and give a reasonable estimate for the uncertainty. Methods: We use He+ data measured by the PLAsma and SupraThermal Ion Composition (PLASTIC) sensor on the Solar TErrestrial RElations Observatory Ahead (STEREO A) spacecraft. We analyze a recent approach, identify sources of systematic errors, and propose solutions to eliminate them. Furthermore, a method is introduced to estimate the error associated with this approach. Additionally, we investigate how the selection of interplanetary magnetic field angles, which is closely connected to the pickup ion velocity distribution function, affects the result for the interstellar flow longitude. Results: We find that the revised analysis used to address part of the expected systematic effects obtains significantly different results than presented in the previous study. In particular
18. Interstellar grains
Energy Technology Data Exchange (ETDEWEB)
Hoyle, F.; Wickramasinghe, N.C.
1980-11-01
Interstellar extinction of starlight was observed and plotted as a function of inverse wavelength. Agreement with the calculated effects of the particle distribution is shown. The main kinds of grain distinguished are: (1) graphite spheres of radius 0.02 microns, making up 10% of the total grain mass (2) small dielectric spheres of radius 0.04 microns making up 25% and (3) hollow dielectric cylinders containing metallic iron, with diameters of 2/3 microns making up 45%. The remaining 20% consists of other metals, metal oxides, and polysiloxanes. Absorption factor evidence suggests that the main dielectric component of the grains is organic material.
19. Interstellar chemistry.
Science.gov (United States)
Klemperer, William
2006-08-15
In the past half century, radioastronomy has changed our perception and understanding of the universe. In this issue of PNAS, the molecular chemistry directly observed within the galaxy is discussed. For the most part, the description of the molecular transformations requires specific kinetic schemes rather than chemical thermodynamics. Ionization of the very abundant molecular hydrogen and atomic helium followed by their secondary reactions is discussed. The rich variety of organic species observed is a challenge for complete understanding. The role and nature of reactions involving grain surfaces as well as new spectroscopic observations of interstellar and circumstellar regions are topics presented in this special feature.
20. The existence and nature of the interstellar bow shock
Energy Technology Data Exchange (ETDEWEB)
Ben-Jaffel, Lotfi [UPMC Univ Paris 06, UMR7095, Institut d' Astrophysique de Paris, F-75014, Paris (France); Strumik, M.; Ratkiewicz, R.; Grygorczuk, J., E-mail: [email protected] [Space Research Centre, Polish Academy of Sciences, Bartycka 18A, 00-716 Warsaw (Poland)
2013-12-20
We report a new diagnosis of two different states of the local interstellar medium (LISM) near our solar system by using a sensitivity study constrained by several distinct and complementary observations of the LISM, solar wind, and inner heliosphere. Assuming the Interstellar Boundary Explorer (IBEX) He flow parameters for the LISM, we obtain a strength of ∼2.7 ± 0.2 μG and a direction pointing away from galactic coordinates (28, 52) ± 3° for the interstellar magnetic field as a result of fitting Voyager 1 and Voyager 2 in situ plasma measurements and IBEX energetic neutral atoms ribbon. When using Ulysses parameters for the LISM He flow, we recently reported the same direction but with a strength of 2.2 ± 0.1 μG. First, we notice that with Ulysses He flow, our solution is in the expected hydrogen deflection plane (HDP). In contrast, for the IBEX He flow, the solution is ∼20° away from the corresponding HDP plane. Second, the long-term monitoring of the interplanetary H I flow speed shows a value of ∼26 km s{sup –1} measured upwind from the Doppler shift in the strong Lyα sky background emission line. All elements of the diagnosis seem therefore to support Ulysses He flow parameters for the interstellar state. In that frame, we argue that reliable discrimination between superfast, subfast, or superslow states of the interstellar flow should be based on most existing in situ and remote observations used together with global modeling of the heliosphere. For commonly accepted LISM ionization rates, we show that a fast interstellar bow shock should be standing off upstream of the heliopause.
1. Visualizing Interstellar's Wormhole
Science.gov (United States)
James, Oliver; von Tunzelmann, Eugénie; Franklin, Paul; Thorne, Kip S.
2015-06-01
Christopher Nolan's science fiction movie Interstellar offers a variety of opportunities for students in elementary courses on general relativity theory. This paper describes such opportunities, including: (i) At the motivational level, the manner in which elementary relativity concepts underlie the wormhole visualizations seen in the movie; (ii) At the briefest computational level, instructive calculations with simple but intriguing wormhole metrics, including, e.g., constructing embedding diagrams for the three-parameter wormhole that was used by our visual effects team and Christopher Nolan in scoping out possible wormhole geometries for the movie; (iii) Combining the proper reference frame of a camera with solutions of the geodesic equation, to construct a light-ray-tracing map backward in time from a camera's local sky to a wormhole's two celestial spheres; (iv) Implementing this map, for example, in Mathematica, Maple or Matlab, and using that implementation to construct images of what a camera sees when near or inside a wormhole; (v) With the student's implementation, exploring how the wormhole's three parameters influence what the camera sees—which is precisely how Christopher Nolan, using our implementation, chose the parameters for Interstellar's wormhole; (vi) Using the student's implementation, exploring the wormhole's Einstein ring and particularly the peculiar motions of star images near the ring, and exploring what it looks like to travel through a wormhole.
2. SBME : Exploring boundaries between formal, non-formal, and informal learning
OpenAIRE
Shahoumian, Armineh; Parchoma, Gale; Saunders, Murray; Hanson, Jacky; Dickinson, Mike; Pimblett, Mark
2013-01-01
In medical education learning extends beyond university settings into practice. Non-formal and informal learning support learners’ efforts to meet externally set and learner-identified objectives. In SBME research, boundaries between formal, non-formal, and informal learning have not been widely explored. Whether SBME fits within or challenges these categories can make a contribution. Formal learning is described in relation to educational settings, planning, assessment, and accreditation. In...
3. The loop I superbubble and the local interstellar magnetic field
International Nuclear Information System (INIS)
Frisch, Priscilla Chapman
2014-01-01
Recent data on the interstellar magnetic field in the low density nearby interstellar medium suggest a new perspective for understanding interstellar clouds within 40 pc. The directions of the local interstellar magnetic field found from measurements of optically polarized starlight and the very local field found from the Ribbon of energetic neutral atoms discovered by IBEX nearly agree. The geometrical relation between the local magnetic field, the positions and kinematics of local interstellar clouds, and the Loop I S1 superbubble, suggest that the Sun is located in the boundary of this evolved superbubble. The quasiperpendicular angle between the bulk kinematics and magnetic field of the local ISM indicates that a complete picture of low density interstellar clouds needs to include information on the interstellar magnetic field.
4. The influence of the interstellar medium on climate and life
International Nuclear Information System (INIS)
Talbot, R.J. Jr.
1980-01-01
Recent studies of the gas and dust between the stars, the interstellar medium, reveal a complex chemistry which indicates that prebiotic organic chemistry is ubiquitous. The relationship between this interstellar chemistry and the organic chemistry of the early solar system and the Earth is explored. The interstellar medium is also considered as likely to have a continuing influence upon the climate of the Earth and other planets. Life forms as known are not only descendants of the organic evolution begun in the interstellar medium, but their continuing evolution is also molded through occasional interactions between the interstellar medium, the Sun and the climate on Earth. (author)
5. The Interstellar Medium
CERN Document Server
Lequeux, James
2005-01-01
Describing interstellar matter in our galaxy in all of its various forms, this book also considers the physical and chemical processes that are occurring within this matter. The first seven chapters present the various components making up the interstellar matter and detail the ways that we are able to study them. The following seven chapters are devoted to the physical, chemical and dynamical processes that control the behaviour of interstellar matter. These include the instabilities and cloud collapse processes that lead to the formation of stars. The last chapter summarizes the transformations that can occur between the different phases of the interstellar medium. Emphasizing methods over results, "The Interstellar Medium" is written for graduate students, for young astronomers, and also for any researchers who have developed an interest in the interstellar medium.
6. Exploring telicity and transitivity in primordial thought language and body boundary imagery.
Science.gov (United States)
Cariola, Laura A
2014-12-01
Linguistics research on 'unconscious knowledge' related to the right brain-hemisphere represents a shift from the prevalent scientific investigation of the linguistic processes of grammatical structures associated with the dominant 'verbal' left brain-hemisphere. This study explores the relationship among primordial thought language, body boundary awareness and syntactic features--i.e., telicity, perfectivity and transitivity-in autobiographical narratives of everyday and dream memories. The results showed that event descriptions with atelic predicates and intransitive structures were more frequent in dream recall than in narratives of everyday memories. Primordial thought language and body boundary awareness, however, decreased with atelic predicates and transitive structures, which might indicate both the tendency of events to describe result states, such as achievements and accomplishments, as a means to bring about an unconscious wish fulfilment and the emphasis on event arguments to be realised without the inclusion of an external object. In narratives of everyday memories, penetration imagery increased with imperfective verb forms and decreased with perfective verb forms, and emotion lexis increased with atelic predicates and transitive structures, but not in dream memories.
7. Exploring the boundaries of quantum mechanics: advances in satellite quantum communications.
Science.gov (United States)
Agnesi, Costantino; Vedovato, Francesco; Schiavon, Matteo; Dequal, Daniele; Calderaro, Luca; Tomasin, Marco; Marangon, Davide G; Stanco, Andrea; Luceri, Vincenza; Bianco, Giuseppe; Vallone, Giuseppe; Villoresi, Paolo
2018-07-13
Recent interest in quantum communications has stimulated great technological progress in satellite quantum technologies. These advances have rendered the aforesaid technologies mature enough to support the realization of experiments that test the foundations of quantum theory at unprecedented scales and in the unexplored space environment. Such experiments, in fact, could explore the boundaries of quantum theory and may provide new insights to investigate phenomena where gravity affects quantum objects. Here, we review recent results in satellite quantum communications and discuss possible phenomena that could be observable with current technologies. Furthermore, stressing the fact that space represents an incredible resource to realize new experiments aimed at highlighting some physical effects, we challenge the community to propose new experiments that unveil the interplay between quantum mechanics and gravity that could be realizable in the near future.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'. © 2018 The Author(s).
8. Nature of interstellar turbulence
International Nuclear Information System (INIS)
Altunin, V.
1981-01-01
A significant role in producing the pattern of interstellar scintillation observed in discrete radio sources may be played by the magnetoacoustic turbulence that will be generated as shock waves are propagated at velocity V/sub sh/roughly-equal 20--100 km/sec through the interstellar medium, as well as by irregularities in stellar wind emanating from type OB stars
9. Interstellar hydrogen bonding
Science.gov (United States)
Etim, Emmanuel E.; Gorai, Prasanta; Das, Ankan; Chakrabarti, Sandip K.; Arunan, Elangannan
2018-06-01
This paper reports the first extensive study of the existence and effects of interstellar hydrogen bonding. The reactions that occur on the surface of the interstellar dust grains are the dominant processes by which interstellar molecules are formed. Water molecules constitute about 70% of the interstellar ice. These water molecules serve as the platform for hydrogen bonding. High level quantum chemical simulations for the hydrogen bond interaction between 20 interstellar molecules (known and possible) and water are carried out using different ab-intio methods. It is evident that if the formation of these species is mainly governed by the ice phase reactions, there is a direct correlation between the binding energies of these complexes and the gas phase abundances of these interstellar molecules. Interstellar hydrogen bonding may cause lower gas abundance of the complex organic molecules (COMs) at the low temperature. From these results, ketenes whose less stable isomers that are more strongly bonded to the surface of the interstellar dust grains have been observed are proposed as suitable candidates for astronomical observations.
10. LOCAL INTERSTELLAR HYDROGEN'S DISAPPEARANCE AT 1 AU: FOUR YEARS OF IBEX IN THE RISING SOLAR CYCLE
International Nuclear Information System (INIS)
Saul, Lukas; Rodríguez, Diego; Scheer, Juergen; Wurz, Peter; Bzowski, Maciej; Kubiak, Marzena; Sokół, Justina; Fuselier, Stephen; McComas, Dave; Möbius, Eberhard
2013-01-01
NASA's Interstellar Boundary Explorer (IBEX) mission has recently opened a new window on the interstellar medium (ISM) by imaging neutral atoms. One ''bright'' feature in the sky is the interstellar wind flowing into the solar system. Composed of remnants of stellar explosions as well as primordial gas and plasma, the ISM is by no means uniform. The interaction of the local ISM with the solar wind shapes our heliospheric environment with hydrogen being the dominant component of the very local ISM. In this paper, we report on direct sampling of the neutral hydrogen of the local ISM over four years of IBEX observations. The hydrogen wind observed at 1 AU has decreased and nearly disappeared as the solar activity has increased over the last four years; the signal at 1 AU has dropped off in 2012 by a factor of ∼8 to near background levels. The longitudinal offset has also increased with time presumably due to greater radiation pressure deflecting the interstellar wind. We present longitudinal and latitudinal arrival direction measurements of the bulk flow as measured over four years beginning at near solar minimum conditions. The H distribution we observe at 1 AU is expected to be different from that outside the heliopause due to ionization, photon pressure, gravity, and filtration by interactions with heliospheric plasma populations. These observations provide an important benchmark for modeling of the global heliospheric interaction. Based on these observations we suggest a further course of scientific action to observe neutral hydrogen over a full solar cycle with IBEX.
11. DRAPING OF THE INTERSTELLAR MAGNETIC FIELD OVER THE HELIOPAUSE: A PASSIVE FIELD MODEL
International Nuclear Information System (INIS)
Isenberg, Philip A.; Forbes, Terry G.; Möbius, Eberhard
2015-01-01
As the local interstellar plasma flows past our heliosphere, it is slowed and deflected around the magnetic obstacle of the heliopause. The interstellar magnetic field, frozen into this plasma, then becomes draped around the heliopause in a characteristic manner. We derive the analytical solution for this draped magnetic field in the limit of weak field intensity, assuming an ideal potential flow around the heliopause, which we model as a Rankine half-body. We compare the structure of the model magnetic field with observed properties of the Interstellar Boundary Explorer (IBEX) ribbon and with in situ observations at the Voyager 1 spacecraft. We find reasonable qualitative agreement, given the idealizations of the model. This agreement lends support to the secondary ENA model of the IBEX ribbon and to the interpretation that Voyager 1 has crossed the heliopause. We also predict that the magnetic field measured by Voyager 2 after it crosses the heliopause will not be significantly rotated away from the direction of the undisturbed interstellar field
12. Representing culture in interstellar messages
Science.gov (United States)
Vakoch, Douglas A.
2008-09-01
As scholars involved with the Search for Extraterrestrial Intelligence (SETI) have contemplated how we might portray humankind in any messages sent to civilizations beyond Earth, one of the challenges they face is adequately representing the diversity of human cultures. For example, in a 2003 workshop in Paris sponsored by the SETI Institute, the International Academy of Astronautics (IAA) SETI Permanent Study Group, the International Society for the Arts, Sciences and Technology (ISAST), and the John Templeton Foundation, a varied group of artists, scientists, and scholars from the humanities considered how to encode notions of altruism in interstellar messages . Though the group represented 10 countries, most were from Europe and North America, leading to the group's recommendation that subsequent discussions on the topic should include more globally representative perspectives. As a result, the IAA Study Group on Interstellar Message Construction and the SETI Institute sponsored a follow-up workshop in Santa Fe, New Mexico, USA in February 2005. The Santa Fe workshop brought together scholars from a range of disciplines including anthropology, archaeology, chemistry, communication science, philosophy, and psychology. Participants included scholars familiar with interstellar message design as well as specialists in cross-cultural research who had participated in the Symposium on Altruism in Cross-cultural Perspective, held just prior to the workshop during the annual conference of the Society for Cross-cultural Research . The workshop included discussion of how cultural understandings of altruism can complement and critique the more biologically based models of altruism proposed for interstellar messages at the 2003 Paris workshop. This paper, written by the chair of both the Paris and Santa Fe workshops, will explore the challenges of communicating concepts of altruism that draw on both biological and cultural models.
13. The galactic interstellar medium
CERN Document Server
Burton, WB; Genzel, R
1992-01-01
This volume contains the papers of three extended lectures addressing advanced topics in astronomy and astrophysics. The topics discussed include the most recent observational data on interstellar matter outside our galaxy and the physics and chemistry of molecular clouds.
14. Dynamics of interstellar matter
International Nuclear Information System (INIS)
Kahn, F.D.
1975-01-01
A review of the dynamics of interstellar matter is presented, considering the basic equations of fluid flow, plane waves, shock waves, spiral structure, thermal instabilities and early star cocoons. (B.R.H.)
15. Interstellar organic chemistry.
Science.gov (United States)
Sagan, C.
1972-01-01
Most of the interstellar organic molecules have been found in the large radio source Sagittarius B2 toward the galactic center, and in such regions as W51 and the IR source in the Orion nebula. Questions of the reliability of molecular identifications are discussed together with aspects of organic synthesis in condensing clouds, degradational origin, synthesis on grains, UV natural selection, interstellar biology, and contributions to planetary biology.
16. On Graphene in the Interstellar Medium
Science.gov (United States)
Chen, X. H.; Li, Aigen; Zhang, Ke
2017-11-01
The possible detection of C24, a planar graphene that was recently reported to be in several planetary nebulae by García-Hernández et al., inspires us to explore whether and how much graphene could exist in the interstellar medium (ISM) and how it would reveal its presence through its ultraviolet (UV) extinction and infrared (IR) emission. In principle, interstellar graphene could arise from the photochemical processing of polycyclic aromatic hydrocarbon (PAH) molecules, which are abundant in the ISM, due to the complete loss of their hydrogen atoms, and/or from graphite, which is thought to be a major dust species in the ISM, via fragmentation caused by grain–grain collisional shattering. Both quantum-chemical computations and laboratory experiments have shown that the exciton-dominated electronic transitions in graphene cause a strong absorption band near 2755 \\mathringA . We calculate the UV absorption of graphene and place an upper limit of ∼5 ppm of C/H (i.e., ∼1.9% of the total interstellar C) on the interstellar graphene abundance. We also model the stochastic heating of graphene C24 in the ISM, excited by single starlight photons of the interstellar radiation field and calculate its IR emission spectra. We also derive the abundance of graphene in the ISM to be <5 ppm of C/H by comparing the model emission spectra with that observed in the ISM.
17. INTERSTELLAR MAGNETIC FIELD SURROUNDING THE HELIOPAUSE
International Nuclear Information System (INIS)
Whang, Y. C.
2010-01-01
This paper presents a three-dimensional analytical solution, in the limit of very low plasma β-ratio, for the distortion of the interstellar magnetic field surrounding the heliopause. The solution is obtained using a line dipole method that is the integration of point dipole along a semi-infinite line; it represents the magnetic field caused by the presence of the heliopause. The solution allows the variation of the undisturbed magnetic field at any inclination angle. The heliosphere is considered as having blunt-nosed geometry on the upwind side and it asymptotically approaches a cylindrical geometry having an open exit for the continuous outflow of the solar wind on the downwind side. The heliopause is treated as a magnetohydrodynamic tangential discontinuity; the interstellar magnetic field lines at the boundary are tangential to the heliopause. The interstellar magnetic field is substantially distorted due to the presence of the heliopause. The solution shows the draping of the field lines around the heliopause. The magnetic field strength varies substantially near the surface of the heliopause. The effect on the magnetic field due to the presence of the heliopause penetrates very deep into the interstellar space; the depth of penetration is of the same order of magnitude as the scale length of the heliosphere.
18. Diffuse interstellar clouds
International Nuclear Information System (INIS)
Black, J.H.
1987-01-01
The author defines and discusses the nature of diffuse interstellar clouds. He discusses how they contribute to the general extinction of starlight. The atomic and molecular species that have been identified in the ultraviolet, visible, and near infrared regions of the spectrum of a diffuse cloud are presented. The author illustrates some of the practical considerations that affect absorption line observations of interstellar atoms and molecules. Various aspects of the theoretical description of diffuse clouds required for a full interpretation of the observations are discussed
19. Infrared diffuse interstellar bands
Science.gov (United States)
Galazutdinov, G. A.; Lee, Jae-Joon; Han, Inwoo; Lee, Byeong-Cheol; Valyavin, G.; Krełowski, J.
2017-05-01
We present high-resolution (R ˜ 45 000) profiles of 14 diffuse interstellar bands in the ˜1.45 to ˜2.45 μm range based on spectra obtained with the Immersion Grating INfrared Spectrograph at the McDonald Observatory. The revised list of diffuse bands with accurately estimated rest wavelengths includes six new features. The diffuse band at 15 268.2 Å demonstrates a very symmetric profile shape and thus can serve as a reference for finding the 'interstellar correction' to the rest wavelength frame in the H range, which suffers from a lack of known atomic/molecular lines.
20. Parameterizing the interstellar dust temperature
Science.gov (United States)
Hocuk, S.; Szűcs, L.; Caselli, P.; Cazaux, S.; Spaans, M.; Esplugues, G. B.
2017-08-01
The temperature of interstellar dust particles is of great importance to astronomers. It plays a crucial role in the thermodynamics of interstellar clouds, because of the gas-dust collisional coupling. It is also a key parameter in astrochemical studies that governs the rate at which molecules form on dust. In 3D (magneto)hydrodynamic simulations often a simple expression for the dust temperature is adopted, because of computational constraints, while astrochemical modelers tend to keep the dust temperature constant over a large range of parameter space. Our aim is to provide an easy-to-use parametric expression for the dust temperature as a function of visual extinction (AV) and to shed light on the critical dependencies of the dust temperature on the grain composition. We obtain an expression for the dust temperature by semi-analytically solving the dust thermal balance for different types of grains and compare to a collection of recent observational measurements. We also explore the effect of ices on the dust temperature. Our results show that a mixed carbonaceous-silicate type dust with a high carbon volume fraction matches the observations best. We find that ice formation allows the dust to be warmer by up to 15% at high optical depths (AV> 20 mag) in the interstellar medium. Our parametric expression for the dust temperature is presented as Td = [ 11 + 5.7 × tanh(0.61 - log 10(AV) ]χuv1/5.9, where χuv is in units of the Draine (1978, ApJS, 36, 595) UV field.
1. Transcending Organizational Boundaries:Exploring intra- and inter-organizational processes of business model innovation in a port authority
OpenAIRE
Kringelum, Louise Tina Brøns
2017-01-01
This thesis explores how processes of business model innovation can unfoldin a port authority by transcending organizational boundaries throughinter-organizational collaboration. The findings contribute to two fields ofacademic inquiry: the study of business model innovation and the study ofhow the roles of port authorities evolve. This contribution is made by combiningthe two fields, where the study of business model innovation is usedas an analytical concept for understanding the evolution ...
2. Engaging boundary objects in OMS and STS? Exploring the subtleties of layered engagement
NARCIS (Netherlands)
Zeiss, R.; Groenewegen, P.
2009-01-01
This paper considers STS aspirations to engage with the field of Organization and Management Studies (OMS). It does so by investigating the employability of the concept of boundary object in OMS. Through an extensive literature review, the paper shows that rather than a simple engagement between STS
3. Nebulae and interstellar matter
International Nuclear Information System (INIS)
1987-01-01
The South African Astronomical Observatory (SAAO) has investigated the IRAS source 1912+172. This source appears to be a young planetary nebula with a binary central star. During 1986 SAAO has also studied the following: hydrogen deficient planetary nebulae; high speed flows in HII regions, and the wavelength dependence of interstellar polarization. 2 figs
4. Influence of the interstellar medium on climate and life: the Black Cloud revisited
Energy Technology Data Exchange (ETDEWEB)
Talbot, Jr, R J
1980-06-01
Recent studies of the gas and dust between the stars, the interstellar medium, reveal a complex chemistry which indicates that prebiotic organic chemistry is ubiquitous. The relationship between this interstellar chemistry and the organic chemistry of the early solar system and the earth is explored. The interstellar medium is also considered as likely to have a continuing influence upon the climate of the earth and other planets. Life forms as we know them are not only descendants of the organic evolution begun in the interstellar medium, but their continuing evolution is also molded through occasional interactions between the interstellar medium, the sun and the climate on earth.
5. Influence of the interstellar medium on climate and life. The black cloud revisited
Energy Technology Data Exchange (ETDEWEB)
Talbot, Jr, R J [Rice Univ., Houston, TX (USA). Dept. of Space Physics and Astronomy
1980-06-01
Recent studies of the gas and dust between the stars, the interstellar medium, reveal a complex chemistry which indicates that prebiotic organic chemistry is ubiquitous. The relationship between this interstellar chemistry and the organic chemistry of the early solar system and the Earth is explored. The interstellar medium is also considered as likely to have a continuing influence upon the climate of the Earth and other planets. Life forms as known are not only descendants of the organic evolution begun in the interstellar medium, but their continuing evolution is also molded through occasional interactions between the interstellar medium, the Sun and the climate on Earth.
6. Neutral interstellar helium parameters based on Ulysses/GAS and IBEX-LO observations: What are the reasons for the differences?
International Nuclear Information System (INIS)
Katushkina, O. A.; Izmodenov, V. V.; Wood, B. E.; McMullin, D. R.
2014-01-01
Recent analysis of the interstellar helium fluxes measured in 2009-2010 at Earth's orbit by the Interstellar Boundary Explorer (IBEX) has suggested that the interstellar velocity (both direction and magnitude) is inconsistent with that derived previously from Ulysses/GAS observations made in the period from 1990 to 2002 at 1.5-5.5 AU from the Sun. Both results are model dependent, and models that were used in the analyses are different. In this paper, we perform an analysis of the Ulysses/GAS and IBEX-Lo data using our state-of-the-art three-dimensional time-dependent kinetic model of interstellar atoms in the heliosphere. For the first time, we analyze Ulysses/GAS data from year 2007, the closest available Ulysses/GAS observations in time to the IBEX observations. We show that the interstellar velocity derived from the Ulysses 2007 data is consistent with previous Ulysses results and does not agree with the velocity derived from IBEX. This conclusion is very robust since, as is shown in the paper, it does not depend on the ionization rates adopted in theoretical models. We conclude that Ulysses data are not consistent with the new local interstellar medium (LISM) velocity vector from IBEX. In contrast, IBEX data, in principle, could be explained with the LISM velocity vector derived from the Ulysses data. This is possible for the models where the interstellar temperature increased from 6300 K to 9000 K. There is a need to perform further studies of possible reasons for the broadening of the helium signal core measured by IBEX, which could be an instrumental effect or could be due to unconsidered physical processes.
7. Neutral interstellar helium parameters based on Ulysses/GAS and IBEX-LO observations: What are the reasons for the differences?
Energy Technology Data Exchange (ETDEWEB)
Katushkina, O. A.; Izmodenov, V. V. [Space Research Institute, Russian Academy of Sciences, Moscow (Russian Federation); Wood, B. E. [Naval Research Laboratory, Space Science Division, Washington, DC 20375 (United States); McMullin, D. R., E-mail: [email protected] [Space Systems Research Corporation, Alexandria, VA 22314 (United States)
2014-07-01
Recent analysis of the interstellar helium fluxes measured in 2009-2010 at Earth's orbit by the Interstellar Boundary Explorer (IBEX) has suggested that the interstellar velocity (both direction and magnitude) is inconsistent with that derived previously from Ulysses/GAS observations made in the period from 1990 to 2002 at 1.5-5.5 AU from the Sun. Both results are model dependent, and models that were used in the analyses are different. In this paper, we perform an analysis of the Ulysses/GAS and IBEX-Lo data using our state-of-the-art three-dimensional time-dependent kinetic model of interstellar atoms in the heliosphere. For the first time, we analyze Ulysses/GAS data from year 2007, the closest available Ulysses/GAS observations in time to the IBEX observations. We show that the interstellar velocity derived from the Ulysses 2007 data is consistent with previous Ulysses results and does not agree with the velocity derived from IBEX. This conclusion is very robust since, as is shown in the paper, it does not depend on the ionization rates adopted in theoretical models. We conclude that Ulysses data are not consistent with the new local interstellar medium (LISM) velocity vector from IBEX. In contrast, IBEX data, in principle, could be explained with the LISM velocity vector derived from the Ulysses data. This is possible for the models where the interstellar temperature increased from 6300 K to 9000 K. There is a need to perform further studies of possible reasons for the broadening of the helium signal core measured by IBEX, which could be an instrumental effect or could be due to unconsidered physical processes.
8. Boundary Layer Transition During the Orion Exploration Flight Test 1 (EFT-1)
Science.gov (United States)
Kirk, Lindsay C.
2016-01-01
Boundary layer transition was observed in the thermocouple data on the windside backshell of the Orion reentry capsule. Sensors along the windside centerline, as well as off-centerline, indicated transition late in the flight at approximately Mach 4 conditions. Transition progressed as expected, beginning at the sensors closest to the forward bay cover (FBC) and moving towards the heatshield. Sensors placed in off-centerline locations did not follow streamlines, so the progression of transition observed in these sensors is less intuitive. Future analysis will include comparisons to pre-flight predictions and expected transitional behavior will be investigated. Sensors located within the centerline and off-centerline launch abort system (LAS) attach well cavities on the FBC also showed indications of boundary layer transition. The transition within the centerline cavity was observed in the temperature traces prior to transition onset on the sensors upstream of the cavity. Transition behavior within the off centerline LAS attach well cavity will also be investigated. Heatshield thermocouples were placed within Avcoat plugs to attempt to capture transitional behavior as well as better understand the aerothermal environments. Thermocouples were placed in stacks of two or five vertically within the plugs, but the temperature data obtained at the sensors closest to the surface did not immediately indicate transitional behavior. Efforts to use the in depth thermocouple temperatures to reconstruct the surface heat flux are ongoing and any results showing the onset of boundary layer transition obtained from those reconstructions will also be included in this paper. Transition on additional features of interest, including compression pad ramps, will be included if it becomes available.
9. THE AGE OF THE LOCAL INTERSTELLAR BUBBLE
International Nuclear Information System (INIS)
Abt, Helmut A.
2011-01-01
The Local Interstellar Bubble is an irregular region from 50 to 150 pc from the Sun in which the interstellar gas density is 10 -2 -10 -3 of that outside the bubble and the interstellar temperature is 10 6 K. Evidently most of the gas was swept out by one or more supernovae. I explored the stellar contents and ages of the region from visual double stars, spectroscopic doubles, single stars, open clusters, emission regions, X-ray stars, planetary nebulae, and pulsars. The bubble has three sub-regions. The region toward the galactic center has stars as early as O9.5 V and with ages of 2-4 M yr. It also has a pulsar (PSRJ1856-3754) with a spin-down age of 3.76 Myr. That pulsar is likely to be the remnant of the supernova that drove away most of the gas. The central lobe has stars as early as B7 V and therefore an age of about 160 Myr or less. The Pleiades lobe has stars as early as B3 and therefore an age of about 50 Myr. There are no obvious pulsars that resulted from the supernovae that cleared out those areas. As found previously by Welsh and Lallement, the bubble has five B stars along its perimeter that show high-temperature ions of O VI and C II along their lines of sight, confirming its high interstellar temperature.
10. Interstellar extinction correlations
International Nuclear Information System (INIS)
Jones, A.P.; Williams, D.A.; Duley, W.W.
1987-01-01
A recently proposed model for interstellar grains in which the extinction arises from small silicate cores with mantles of hydrogenated amorphous carbon (HAC or α-C:H), and large, but thinly coated, silicate grains can successfully explain many of the observed properties of interstellar dust. The small silicate cores give rise to the 2200 A extinction feature. The extinction in the visual is produced by the large silicates and the HAC mantles on the small cores, whilst the far UV extinction arises in the HAC mantles with a small contribution form the silicate grains. The grain model requires that the silicate material is the more resilient component and that variations in the observed extinction from region to region are due to the nature and depletion of the carbon in the HAC mantles. (author)
11. Who Reaps the Benefits of Social Change? Exploration and Its Socioecological Boundaries.
Science.gov (United States)
Lechner, Clemens M; Obschonka, Martin; Silbereisen, Rainer K
2017-04-01
We investigated the interplay between the personality trait exploration and objective socioecological conditions in shaping individual differences in the experience of two individual-level benefits of current social change: new lifestyle options, which arise from the societal trend toward individualization, and new learning opportunities, which accrue from the societal trend toward lifelong learning. We hypothesized that people with higher trait exploration experience a greater increase in lifestyle options and learning opportunities--but more so in social ecologies in which individualization and lifelong learning are stronger, thus offering greater latitude for exploring the benefits of these trends. We employed structural equation modeling in two parallel adult samples from Germany (N = 2,448) and Poland (N = 2,571), using regional divorce rates as a proxy for individualization and Internet domain registration rates as a proxy for lifelong learning. Higher exploration was related to a greater perceived increase in lifestyle options and in learning opportunities over the past 5 years. These associations were stronger in regions in which the trends toward individualization and lifelong learning, respectively, were more prominent. Individuals higher in exploration are better equipped to reap the benefits of current social change--but the effects of exploration are bounded by the conditions in the social ecology. © 2015 Wiley Periodicals, Inc.
12. Evolution of interstellar grains
International Nuclear Information System (INIS)
Greenberg, J.M.
1984-01-01
The principal aim of this chapter is to derive the properties of interstellar grains as a probe of local physical conditions and as a basis for predicting such properties as related to infrared emissivity and radiative transfer which can affect the evolution of dense clouds. The first sections will develop the criteria for grain models based directly on observations of gas and dust. A summary of the chemical evolution of grains and gas in diffuse and dense clouds follows. (author)
13. The Interstellar Mapping and Acceleration Probe - A Mission to Discover the Origin of Particle Acceleration and its Fundamental Connection to the Global Interstellar Interaction
Science.gov (United States)
2017-12-01
Our piece of cosmic real-estate, the heliosphere, is the domain of all human existence - an astrophysical case-history of the successful evolution of life in a habitable system. The Interstellar Boundary Explorer (IBEX) was the first mission to explore the global heliosphere and in concert with Voyager 1 and Voyager 2 is discovering a fundamentally new and uncharted physical domain of the outer heliosphere. In parallel, Cassini/INCA maps the global heliosphere at energies ( 5-55 keV) above those measured by IBEX. The enigmatic IBEX ribbon and the INCA belt were unanticipated discoveries demonstrating that much of what we know or think we understand about the outer heliosphere needs to be revised. The global structure of the heliosphere is highly complex and influenced by competing factors ranging from the local interstellar magnetic field, suprathermal populations both within and beyond the heliopause, and the detailed flow properties of the LISM. Global heliospheric structure and microphysics in turn influences the acceleration of energetic particles and creates feedbacks that modify the interstellar interaction as a whole. The next quantum leap enabled by IMAP will open new windows on the frontier of Heliophysics and probe the acceleration of suprathermal and higher energy particles at a time when the space environment is rapidly evolving. IMAP ultimately connects the acceleration processes observed directly at 1 AU with unprecedented sensitivity and temporal resolution with the global structure of our heliosphere. The remarkable synergy between IMAP, Voyager 1 and Voyager 2 will remain for at least the next decade as Voyager 1 pushes further into the interstellar domain and Voyager 2 moves through the heliosheath. IMAP, like ACE before it, will be a keystone of the Heliophysics System Observatory by providing comprehensive energetic particle, pickup ion, suprathermal ion, neutral atom, solar wind, solar wind heavy ion, and magnetic field observations to diagnose
14. Exploring the Contribution of Primary Marine Organic Matter to the Arctic Boundary Layer
Science.gov (United States)
Collins, D. B.; Chang, R. Y. W.; Boyer, M.; Abbatt, J.
2016-12-01
The ocean is a significant source of aerosol to the atmosphere, and contributes significantly to the aerosol population especially in remote locations. Both primary and secondary processes connect the ocean to ambient aerosol loadings, but the extent to which the ocean is a source of organic material to the atmosphere is a current topic of scientific debate. The contribution of primary marine aerosol to atmospheric organic matter may have an influence on the water uptake properties and chemical reactivity of primary marine aerosol particles, influencing their climate-relevant properties. In this study, we characterize the contribution of primary marine aerosol to the arctic marine boundary layer using coincident quantitative measurements of freshly-produced sea spray aerosol and ambient marine aerosol to the arctic boundary layer during an expedition aboard the CCGS Amundsen. Sea spray production experiments were conducted during the cruise using a tank fitted with a plunging waterfall apparatus, a technique which has been recently shown to closely mimic the aerosol production behavior of controlled breaking waves. Comparison of the chemical composition of sea spray particles generated from water samples in various locations throughout the Canadian Archipelago will be presented. A tracer analysis of specific compounds known to be important contributors to primary marine organic material are tracked using GC/MS, along with those known to be tracers of biological aerosol and other organic matter sources. Size-segregated trends in tracer concentrations and ratios with inorganic components will be discussed in the context of understanding the contribution of primary organics to the Arctic atmosphere and in comparison with other sources of organic material observed during the ship-board campaign.
15. Demonstration of Planet Labs web explorer combined with data from danish field boundaries
DEFF Research Database (Denmark)
2018-01-01
Exploring planet labs satellite data using Land-parcel identification system (LPIS) data from Denmark. The video is intended as a short demo to show how one can manually find the cloud-free satellite images for a specific agricultural field. Afterward, the relevant satellite images can be download...
16. Martian Surface Boundary Layer Characterization: Enabling Environmental Data for Science, Engineering and Human Exploration
Science.gov (United States)
England, C.
2000-01-01
For human or large robotic exploration of Mars, engineering devices such as power sources will be utilized that interact closely with the Martian environment. Heat sources for power production, for example, will use the low ambient temperature for efficient heat rejection. The Martian ambient, however, is highly variable, and will have a first order influence on the efficiency and operation of all large-scale equipment. Diurnal changes in temperature, for example, can vary the theoretical efficiency of power production by 15% and affect the choice of equipment, working fluids, and operating parameters. As part of the Mars Exploration program, missions must acquire the environmental data needed for design, operation and maintenance of engineering equipment including the transportation devices. The information should focus on the variability of the environment, and on the differences among locations including latitudes, altitudes, and seasons. This paper outlines some of the WHY's, WHAT's and WHERE's of the needed data, as well as some examples of how this data will be used. Environmental data for engineering design should be considered a priority in Mars Exploration planning. The Mars Thermal Environment Radiator Characterization (MTERC), and Dust Accumulation and Removal Technology (DART) experiments planned for early Mars landers are examples of information needed for even small robotic missions. Large missions will require proportionately more accurate data that encompass larger samples of the Martian surface conditions. In achieving this goal, the Mars Exploration program will also acquire primary data needed for understanding Martian weather, surface evolution, and ground-atmosphere interrelationships.
17. Thinking beyond the Obvious Boundaries in Mathematics: An Exploration of Joyous Discovery.
Science.gov (United States)
de Vries, Marianne E.
1992-01-01
Ideas for the development of creative exploration in mathematics are offered, including games to play in class (e.g., card games and tangrams), competitions (sample problems), clubs and math evenings (math relays and treasure hunts), projects (possible topics in patchwork quilting, art, and music), and math camps. (DB)
18. Inner core boundary topography explored with reflected and diffracted P waves
Science.gov (United States)
deSilva, Susini; Cormier, Vernon F.; Zheng, Yingcai
2018-03-01
The existence of topography of the inner core boundary (ICB) can affect the amplitude, phase, and coda of body waves incident on the inner core. By applying pseudospectral and boundary element methods to synthesize compressional waves interacting with the ICB, these effects are predicted and compared with waveform observations in pre-critical, critical, post-critical, and diffraction ranges of the PKiKP wave reflected from the ICB. These data sample overlapping regions of the inner core beneath the circum-Pacific belt and the Eurasian, North American, and Australian continents, but exclude large areas beneath the Pacific and Indian Oceans and the poles. In the pre-critical range, PKiKP waveforms require an upper bound of 2 km at 1-20 km wavelength for any ICB topography. Higher topography sharply reduces PKiKP amplitude and produces time-extended coda not observed in PKiKP waveforms. The existence of topography of this scale smooths over minima and zeros in the pre-critical ICB reflection coefficient predicted from standard earth models. In the range surrounding critical incidence (108-130 °), this upper bound of topography does not strongly affect the amplitude and waveform behavior of PKIKP + PKiKP at 1.5 Hz, which is relatively insensitive to 10-20 km wavelength topography height approaching 5 km. These data, however, have a strong overlap in the regions of the ICB sampled by pre-critical PKiKP that require a 2 km upper bound to topography height. In the diffracted range (>152°), topography as high as 5 km attenuates the peak amplitudes of PKIKP and PKPCdiff by similar amounts, leaving the PKPCdiff/PKIKP amplitude ratio unchanged from that predicted by a smooth ICB. The observed decay of PKPCdiff into the inner core shadow and the PKIKP-PKPCdiff differential travel time are consistent with a flattening of the outer core P velocity gradient near the ICB and iron enrichment at the bottom of the outer core.
19. Growing interstellar molecules with ion-molecule reactions
International Nuclear Information System (INIS)
Bohme, D.K.
1989-01-01
Laboratory measurements of gas-phase ion-molecule reactions continue to provide important insights into the chemistry of molecular growth in interstellar environments. It is also true that the measurements are becoming more demanding as larger molecules capture our interest. While some of these measurements are motivated by current developments in chemical models of interstellar environments or by new molecular observations by astronomers, others explore novel chemistry which can lead to predictions of new interstellar molecules. Here the author views the results of some recent measurements, taken in the Ion Chemistry Laboratory at York University with the SIFT technique, which address some of the current needs of modellers and observers and which also provide some new fundamental insight into molecular growth, particularly when it occurs in the presence of large molecules such as PAH molecules which are now thought to have a major influence on the chemistry of interstellar environments in which they are present
20. Molecular diagnostics of interstellar shocks
International Nuclear Information System (INIS)
Hartquist, T.W.; Oppenheimer, M.; Dalgarno, A.
1980-01-01
The chemistry of molecules in shocked regions of the interstellar gas is considered and calculations are carried out for a region subjected to a shock at a velocity of 8 km s -1 Substantial enhancements are predicted in the concentrations of the molecules H 2 S, SO, and SiO compared to those anticipated in cold interstellar clouds
1. Molecular diagnostics of interstellar shocks
Science.gov (United States)
Hartquist, T. W.; Dalgarno, A.; Oppenheimer, M.
1980-02-01
The chemistry of molecules in shocked regions of the interstellar gas is considered and calculations are carried out for a region subjected to a shock at a velocity of 8 km/sec. Substantial enhancements are predicted in the concentrations of the molecules H2S, SO, and SiO compared to those anticipated in cold interstellar clouds.
2. Observational constraints on interstellar chemistry
International Nuclear Information System (INIS)
Winnewisser, G.
1984-01-01
The author points out presently existing observational constraints in the detection of interstellar molecular species and the limits they may cast on our knowledge of interstellar chemistry. The constraints which arise from the molecular side are summarised and some technical difficulties encountered in detecting new species are discussed. Some implications for our understanding of molecular formation processes are considered. (Auth.)
3. Molecular diagnostics of interstellar shocks
Science.gov (United States)
Hartquist, T. W.; Dalgarno, A.; Oppenheimer, M.
1980-01-01
The chemistry of molecules in shocked regions of the interstellar gas is considered and calculations are carried out for a region subjected to a shock at a velocity of 8 km/sec. Substantial enhancements are predicted in the concentrations of the molecules H2S, SO, and SiO compared to those anticipated in cold interstellar clouds.
4. THE INTERSTELLAR MAGNETIC FIELD CLOSE TO THE SUN. II
International Nuclear Information System (INIS)
Frisch, P. C.; Andersson, B-G; Berdyugin, A.; Piirola, V.; DeMajistre, R.; Funsten, H. O.; Magalhaes, A. M.; Seriacopi, D. B.; McComas, D. J.; Schwadron, N. A.; Slavin, J. D.; Wiktorowicz, S. J.
2012-01-01
The magnetic field in the local interstellar medium (ISM) provides a key indicator of the galactic environment of the Sun and influences the shape of the heliosphere. We have studied the interstellar magnetic field (ISMF) in the solar vicinity using polarized starlight for stars within 40 pc of the Sun and 90° of the heliosphere nose. In Frisch et al. (Paper I), we developed a method for determining the local ISMF direction by finding the best match to a group of interstellar polarization position angles obtained toward nearby stars, based on the assumption that the polarization is parallel to the ISMF. In this paper, we extend the analysis by utilizing weighted fits to the position angles and by including new observations acquired for this study. We find that the local ISMF is pointed toward the galactic coordinates l, b =47° ± 20°, 25° ± 20°. This direction is close to the direction of the ISMF that shapes the heliosphere, l, b =33° ± 4°, 55° ± 4°, as traced by the center of the 'Ribbon' of energetic neutral atoms discovered by the Interstellar Boundary Explorer (IBEX) mission. Both the magnetic field direction and the kinematics of the local ISM are consistent with a scenario where the local ISM is a fragment of the Loop I superbubble. A nearby ordered component of the local ISMF has been identified in the region l ≈0° → 80° and b ≈0° → 30°, where PlanetPol data show a distance-dependent increase of polarization strength. The ordered component extends to within 8 pc of the Sun and implies a weak curvature in the nearby ISMF of ∼0. 0 25 pc –1 . This conclusion is conditioned on the small sample of stars available for defining this rotation. Variations from the ordered component suggest a turbulent component of ∼23°. The ordered component and standard relations between polarization, color excess, and H o column density predict a reasonable increase of N(H) with distance in the local ISM. The similarity of the ISMF directions traced
5. THE INTERSTELLAR MAGNETIC FIELD CLOSE TO THE SUN. II
Energy Technology Data Exchange (ETDEWEB)
Frisch, P. C. [Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637 (United States); Andersson, B-G [SOFIA Science Center, Universities Space Research Association, NASA Ames Research Center, M.S. N232-12 Moffett Field, CA 94035 (United States); Berdyugin, A.; Piirola, V. [Finnish Centre for Astronomy with ESO, University of Turku (Finland); DeMajistre, R. [The Johns Hopkins University Applied Physics Laboratory, Laurel, MD (United States); Funsten, H. O. [Los Alamos National Laboratory, Los Alamos, NM (United States); Magalhaes, A. M.; Seriacopi, D. B. [Inst. de Astronomia, Geofisica e Ciencias Atmosfericas, Universidade de Sao Paulo (Brazil); McComas, D. J. [Southwest Research Institute, San Antonio, TX (United States); Schwadron, N. A. [Space Science Center, University of New Hampshire, Durham, NH (United States); Slavin, J. D. [Harvard-Smithsonian Center for Astrophysics, Cambridge, MA (United States); Wiktorowicz, S. J. [Department of Astronomy, University of California at Santa Cruz, Santa Cruz, CA (United States)
2012-12-01
The magnetic field in the local interstellar medium (ISM) provides a key indicator of the galactic environment of the Sun and influences the shape of the heliosphere. We have studied the interstellar magnetic field (ISMF) in the solar vicinity using polarized starlight for stars within 40 pc of the Sun and 90 Degree-Sign of the heliosphere nose. In Frisch et al. (Paper I), we developed a method for determining the local ISMF direction by finding the best match to a group of interstellar polarization position angles obtained toward nearby stars, based on the assumption that the polarization is parallel to the ISMF. In this paper, we extend the analysis by utilizing weighted fits to the position angles and by including new observations acquired for this study. We find that the local ISMF is pointed toward the galactic coordinates l, b =47 Degree-Sign {+-} 20 Degree-Sign , 25 Degree-Sign {+-} 20 Degree-Sign . This direction is close to the direction of the ISMF that shapes the heliosphere, l, b =33 Degree-Sign {+-} 4 Degree-Sign , 55 Degree-Sign {+-} 4 Degree-Sign , as traced by the center of the 'Ribbon' of energetic neutral atoms discovered by the Interstellar Boundary Explorer (IBEX) mission. Both the magnetic field direction and the kinematics of the local ISM are consistent with a scenario where the local ISM is a fragment of the Loop I superbubble. A nearby ordered component of the local ISMF has been identified in the region l Almost-Equal-To 0 Degree-Sign {yields} 80 Degree-Sign and b Almost-Equal-To 0 Degree-Sign {yields} 30 Degree-Sign , where PlanetPol data show a distance-dependent increase of polarization strength. The ordered component extends to within 8 pc of the Sun and implies a weak curvature in the nearby ISMF of {approx}0.{sup 0}25 pc{sup -1}. This conclusion is conditioned on the small sample of stars available for defining this rotation. Variations from the ordered component suggest a turbulent component of {approx}23 Degree-Sign . The
6. Interstellar molecules and masers
International Nuclear Information System (INIS)
Nguyen-Q-Rieu; Guibert, J.
1978-01-01
The study of dense and dark clouds, in which hydrogen is mostly in molecular form, became possible since the discovery of interstellar molecules, emitting in the centimeter and millimeter wavelengths. The molecular lines are generally not in local thermal equilibrium (LTE). Their intensity can often be explained by invoking a population inversion mechanism. Maser emission lines due to OH, H 2 O and SiO molecules are among the most intense molecular lines. The H 2 CO molecule, detected in absorption in front of the cold cosmic background radiation of 2.7 K, illustrates the inverse phenomenon, the antimaser absorption. For a radio transition of frequency v, the inversion rate Δn (relative population difference between the upper and lower level) as well as the maser gain can be determined from the radio observations. In the case of the OH lines in the 2 PIsub(3/2), J=3/2 state, the inversion rates approximately 1 to 2% derived from the observations, are comparable with those obtained in the laboratory. The determination of the excitation mechanisms of the masers, through the statistical equilibrium and radiative transfer equations, implies the knowledge of collisional and radiative transition probabilities. A pumping model, which can satisfactorily explain the radio observations of some interstellar OH clouds, will be discussed [fr
7. Interstellar Neutral Helium in the Heliosphere from IBEX Observations. V. Observations in IBEX-Lo ESA Steps 1, 2, and 3
Science.gov (United States)
Swaczyna, Paweł; Bzowski, Maciej; Kubiak, Marzena A.; Sokół, Justyna M.; Fuselier, Stephen A.; Galli, André; Heirtzler, David; Kucharek, Harald; McComas, David J.; Möbius, Eberhard; Schwadron, Nathan A.; Wurz, P.
2018-02-01
Direct-sampling observations of interstellar neutral (ISN) He by the Interstellar Boundary Explorer (IBEX) provide valuable insight into the physical state of and processes operating in the interstellar medium ahead of the heliosphere. The ISN He atom signals are observed at the four lowest ESA steps of the IBEX-Lo sensor. The observed signal is a mixture of the primary and secondary components of ISN He and H. Previously, only data from one of the ESA steps have been used. Here, we extend the analysis to data collected in the three lowest ESA steps with the strongest ISN He signal, for the observation seasons 2009–2015. The instrument sensitivity is modeled as a linear function of the atom impact speed onto the sensor’s conversion surface separately for each ESA step of the instrument. We find that the sensitivity increases from lower to higher ESA steps, but within each of the ESA steps it is a decreasing function of the atom impact speed. This result may be influenced by the hydrogen contribution, which was not included in the adopted model, but seems to exist in the signal. We conclude that the currently accepted temperature of ISN He and velocity of the Sun through the interstellar medium do not need a revision, and we sketch a plan of further data analysis aiming at investigating ISN H and a better understanding of the population of ISN He originating in the outer heliosheath.
8. Interstellar dust and extinction
International Nuclear Information System (INIS)
Mathis, J.S.
1990-01-01
It is noted that the term interstellar dust refers to materials with rather different properties, and that the mean extinction law of Seaton (1979) or Savage and Mathis (1979) should be replaced by the expression given by Cardelli et al. (1989), using the appropriate value of total-to-selective extinction. The older laws were appropriate for the diffuse ISM but dust in clouds differs dramatically in its extinction law. Dust is heavily processed while in the ISM by being included within clouds and cycled back into the diffuse ISM many times during its lifetime. Hence, grains probably reflect only a trace of their origin, although meteoritic inclusions with isotopic anomalies demonstrate that some tiny particles survive intact from a supernova origin to the present. 186 refs
9. The diffuse interstellar medium
Science.gov (United States)
Cox, Donald P.
1990-01-01
The last 20 years of the efforts to understand the diffuse ISM are reviewed, with recent changes of fundamental aspects being highlighted. Attention is given to the interstellar pressure and its components, the weight of the ISM, the midplane pressure contributions, and pressure contributions at 1 kpc. What velocity dispersions, cosmic ray pressure, and magnetic field pressure that can be expected for a gas in a high magnetic field environment is addressed. The intercloud medium is described, with reference to the work of Cox and Slavin (1989). Various caveats are discussed and a number of areas for future investigation are identified. Steps that could be taken toward a successful phase segregation model are discussed.
10. Update on an Interstellar Asteroid
Science.gov (United States)
Kohler, Susanna
2018-01-01
Lowell Observatorys 4.3-m Discovery Channel Telescope. The data indicate that the asteroids period is at least 3 hours in length,and most likely more than 5 hours. Assuming the light curves variation is caused by the tumbling asteroids changing cross-section, Oumuamuamust be a minimum of3 times as long as it is wide. Knight and collaborators seeno signs in their images of a coma or tail emitted from Oumuamua, suggesting there isno volatile material sublimating from its surface under the heat of the Sun.No coma is visible around Oumuamua. [Knight et al. 2017]A study of the asteroids photometry, led by Michele Bannister (Queens University Belfast, UK), usedthe Gemini-North telescope in Hawaii and the William Herschel Telescope in Spainto explore the asteroids shape and color. Bannister and collaborators refined the estimate of the asteroids shape to be at least 5.3 times as long as it is wide, which requiresthis body to have significant internal cohesion to hold together as it tumbles. Their measured color for Oumuamua is largely neutral.What Does This Visitor Imply?Masses and semimajor axes of known exoplanets. Colors correspond to the ratio of escape velocity to circular velocity. The presence of Oumuamua implies a vast and cool, stillundetected population of planets. [Laughlin Batygin, 2017]Gregory Laughlinof Yale University and Konstantin Batyginof Caltech(andPlanet Nine fame) explore some of the consequences of Oumuamuas parameters. They arguethat its current passage, if its not a fluke, suggests the presence ofan enormous number (1027) ofsuch objects in our galaxy alone enough to account for two Earth-masses of material for every star in the galaxy. Flinging asteroids like Oumuamuaout into interstellar space isnteasy, though; the necessary multi-body interaction requires the system to containa giant and long-period planet like our Neptune or Jupiter. Taken together, this information suggests that every star in the galaxy may host a Neptune-like planet at a Neptune
11. Emerging boundaries
DEFF Research Database (Denmark)
Løvschal, Mette
2014-01-01
of temporal and material variables have been applied as a means of exploring the processes leading to their socioconceptual anchorage. The outcome of this analysis is a series of interrelated, generative boundary principles, including boundaries as markers, articulations, process-related devices, and fixation...
12. Interstellar scattering and resolution limitations
International Nuclear Information System (INIS)
Dennison, B.
1987-01-01
Density irregularities in both the interplanetary medium and the ionized component of the interstellar medium scatter radio waves, resulting in limitations on the achievable resolution. Interplanetary scattering (IPS) is weak for most observational situations, and in principle the resulting phase corruption can be corrected for when observing with sufficiently many array elements. Interstellar scattering (ISS), on the other hand, is usually strong at frequencies below about 8 GHz, in which case intrinsic structure information over a range of angular scales is irretrievably lost. With the earth-space baselines now planned, it will be possible to search directly for interstellar refraction, which is suspected of modulating the fluxes of background sources. 14 references
13. The distribution of interstellar dust
International Nuclear Information System (INIS)
Clocchiatti, A.; Marraco, H.G.
1986-01-01
We propose the interstellar matter structural function as a tool to derive the features of the interstellar dust distribution. We study that function resolving some ideal dust distribution models. Later we describe the method used to find a reliable computing algorithm for the observational case. Finally, we describe the steps to build a model for the interstellar matter composed by spherically symmetrical clouds. The density distribution for each of these clouds is D(r) = D 0 .esup(-r/r 0 ) 2 . The preliminary results obtained are summarised. (author)
14. Recent interstellar molecular line work
International Nuclear Information System (INIS)
Winnewisser, G.
1975-01-01
A summary of recent interstellar molecular line work is presented. Transitions of the following molecules have been detected in Sgr B2: Vinylcyanide, H 2 C 2 HCN, formic acid, HCOOH, dimethyl ether (CH 3 ) 2 O and isotopically labelled cyanoacetylene- 13 C,HC 13 CCN and HCC 13 CN. The data on cyanoacetylene give an upper limit to the abundance ratio 12 C/ 13 C of 36 +- 5. A short discussion of the interstellar chemistry leads to the conclusion that hydrocarbons such as acetylene, HCCH, ethylen, H 2 CCH 2 and ethane H 3 CCH 3 should be present in interstellar clouds. 13 refs
15. Four Interstellar Dust Candidates from the Stardust Interstellar Dust Collector
Science.gov (United States)
Westphal, A. J.; Allen, C.; Bajt, S.; Bechtel, H. A.; Borg, J.; Brenker, F.; Bridges, J.; Brownlee, D. E.; Burchell, M.; Burghammer, M.;
2011-01-01
In January 2006, the Stardust sample return capsule returned to Earth bearing the first solid samples from a primitive solar system body, Comet 81P/Wild2, and a collector dedicated to the capture and return of contemporary interstellar dust. Both collectors were approx. 0.1 sq m in area and were composed of aerogel tiles (85% of the collecting area) and aluminum foils. The Stardust Interstellar Dust Collector (SIDC) was exposed to the interstellar dust stream for a total exposure factor of 20 sq m/day. The Stardust Interstellar Preliminary Examination (ISPE) is a consortium-based project to characterize the collection using nondestructive techniques. The goals and restrictions of the ISPE are described . A summary of analytical techniques is described.
16. OpenAIRE
Demyk K.
2012-01-01
Cosmic dust is omnipresent in the Universe. Its presence influences the evolution of the astronomical objects which in turn modify its physical and chemical properties. The nature of cosmic dust, its intimate coupling with its environment, constitute a rich field of research based on observations, modelling and experimental work. This review presents the observations of the different components of interstellar dust and discusses their evolution during the life cycle of the interstellar medium.
17. OBSERVATIONS OF THE INTERPLANETARY HYDROGEN DURING SOLAR CYCLES 23 AND 24. WHAT CAN WE DEDUCE ABOUT THE LOCAL INTERSTELLAR MEDIUM?
International Nuclear Information System (INIS)
Vincent, Frédéric E.; Quémerais, Eric; Koutroumpa, Dimitra; Katushkina, Olga; Izmodenov, Vladislav; Ben-Jaffel, Lotfi; Harris, Walter M.; Clarke, John
2014-01-01
Observations of interstellar helium atoms by the Interstellar Boundary Explorer (IBEX) spacecraft in 2009 reported a local interstellar medium (LISM) velocity vector different from the results of the Ulysses spacecraft between 1991 and 2002. The interplanetary hydrogen (IPH), a population of neutrals that fills the space between planets inside the heliosphere, carries the signatures of the LISM and its interaction with the solar wind. More than 40 yr of space-based studies of the backscattered solar Lyα emission from the IPH provided limited access to the velocity distribution, with the first temporal evolution map of the IPH line-shift during solar cycle 23. This work presents the results of the latest IPH observations made by the Hubble Space Telescope's Space Telescope Imaging Spectrograph during solar cycle 24. These results have been compiled with previous measurements, including data from the Solar Wind Anisotropies instrument on the Solar and Heliospheric Observatory. The whole set has been compared to physically realistic models to test both sets of LISM physical parameters as measured by Ulysses and IBEX, respectively. This comparison shows that the LISM velocity vector has not changed significantly since Ulysses measurements
18. OBSERVATIONS OF THE INTERPLANETARY HYDROGEN DURING SOLAR CYCLES 23 AND 24. WHAT CAN WE DEDUCE ABOUT THE LOCAL INTERSTELLAR MEDIUM?
Energy Technology Data Exchange (ETDEWEB)
Vincent, Frédéric E.; Quémerais, Eric; Koutroumpa, Dimitra [Université Versailles St.-Quentin, Sorbonne Universités, UPMC Univ. Paris 06, CRNS/INSU, LATMOS-IPSL, 11 boulevard d' Alembert, 78280 Guyancourt (France); Katushkina, Olga; Izmodenov, Vladislav [Space Research Institute of Russian Academy of Sciences, Moscow (Russian Federation); Ben-Jaffel, Lotfi [UPMC Univ. Paris 06, UMR7095, Institut d' Astrophysique de Paris, F-75014, Paris (France); Harris, Walter M. [University of Arizona, Lunar and Planetary Laboratory, 1629 E. University Blvd., Tucson, AZ 85721 (United States); Clarke, John [Center for Space Physics, Boston University, 725 Commonwealth Avenue, Boston, MA 02215 (United States)
2014-06-20
Observations of interstellar helium atoms by the Interstellar Boundary Explorer (IBEX) spacecraft in 2009 reported a local interstellar medium (LISM) velocity vector different from the results of the Ulysses spacecraft between 1991 and 2002. The interplanetary hydrogen (IPH), a population of neutrals that fills the space between planets inside the heliosphere, carries the signatures of the LISM and its interaction with the solar wind. More than 40 yr of space-based studies of the backscattered solar Lyα emission from the IPH provided limited access to the velocity distribution, with the first temporal evolution map of the IPH line-shift during solar cycle 23. This work presents the results of the latest IPH observations made by the Hubble Space Telescope's Space Telescope Imaging Spectrograph during solar cycle 24. These results have been compiled with previous measurements, including data from the Solar Wind Anisotropies instrument on the Solar and Heliospheric Observatory. The whole set has been compared to physically realistic models to test both sets of LISM physical parameters as measured by Ulysses and IBEX, respectively. This comparison shows that the LISM velocity vector has not changed significantly since Ulysses measurements.
19. Enabling the First Interstellar Missions
Science.gov (United States)
Lubin, P.
2017-12-01
All propulsion systems that leave the Earth are based on chemical reactions. Chemical reactions, at best, have an efficiency compared to rest mass of 10-10 (or about 1eV per bond). All the mass in the universe converted to chemical reactions would not propel even a single proton to relativistic speeds. While chemistry will get us to Mars it will not allow interstellar capability in any reasonable mission time. Barring new physics we are left with few realistic solutions. None of our current propulsion systems, including nuclear, are capable of the relativistic speeds needed for exploring the many nearby stellar systems and exo-planets. However recent advances in photonics and directed energy systems now allow us to realize what was only a decade ago, simply science fiction, namely the ability to seriously conceive of and plan for relativistic flight. From fully-functional gram-level wafer-scale spacecraft capable of speeds greater than c/4 that could reach the nearest star in 20 years to spacecraft for large missions capable of supporting human life with masses more than 105 kg (100 tons) for rapid interplanetary transit that could reach speeds of greater than 1000 km/s can be realized. With this technology spacecraft can be propelled to speeds currently unimaginable. Photonics, like electronics, and unlike chemical propulsion is an exponential technology with a current double time of about 20 months. This is the key. The cost of such a system is amortized over the essentially unlimited number of launches. In addition, the same photon driver can be used for many other purposes including beamed energy to power high Isp ion engines, remote asteroid composition analysis and planetary defense. This would be a profound change in human capability with enormous implications. Known as Starlight we are now in a NASA Phase II study. The FY 2017 congressional appropriations request directs NASA to study the feasibility of an interstellar mission to coincide with the 100th
20. Riddling bifurcation and interstellar journeys
International Nuclear Information System (INIS)
Kapitaniak, Tomasz
2005-01-01
We show that riddling bifurcation which is characteristic for low-dimensional attractors embedded in higher-dimensional phase space can give physical mechanism explaining interstellar journeys described in science-fiction literature
1. Negotiating boundaries
DEFF Research Database (Denmark)
Aarhus, Rikke; Ballegaard, Stinne Aaløkke
2010-01-01
to maintain the order of the home when managing disease and adopting new healthcare technology. In our analysis we relate this boundary work to two continuums of visibility-invisibility and integration-segmentation in disease management. We explore five factors that affect the boundary work: objects......, activities, places, character of disease, and collaboration. Furthermore, the processes are explored of how boundary objects move between social worlds pushing and shaping boundaries. From this we discuss design implications for future healthcare technologies for the home.......To move treatment successfully from the hospital to that of technology assisted self-care at home, it is vital in the design of such technologies to understand the setting in which the health IT should be used. Based on qualitative studies we find that people engage in elaborate boundary work...
2. The Interstellar Conspiracy
Science.gov (United States)
Johnson, Les; Matloff, Gregory L.
2005-01-01
If we were designing a human-carrying starship that could be launched in the not-too-distant future, it would almost certainly not use a warp drive to instantaneously bounce around the universe, as is done in Isaac Asimov's classic Foundation series or in episodes of Star Trek or Star Wars. Sadly, those starships that seem to be within technological reach could not even travel at high relativistic speeds, as does the interstellar ramjet in Poul Anderson's Tau Zero. Warp-speeds seem to be well outside the realm of currently understood physical law; proton-fusing ramjets may never be technologically feasible. Perhaps fortunately in our terrorist-plagued world, the economics of antimatter may never be attractive for large-scale starship propulsion. But interstellar travel will be possible within a few centuries, although it will certainly not be as fast as we might prefer. If humans learn how to hibernate, perhaps we will sleep our way to the stars, as do the crew in A. E. van Vogt's Far Centaurus. However, as discussed in a landmark paper in The Journal of the British Interplanetary Society, the most feasible approach to transporting a small human population to the planets (if any) of Alpha Centauri is the worldship. Such craft have often been featured in science fiction. See for example Arthur C. Clarke's Rendezvous with Rama, and Robert A. Heinlein's Orphans of the Sky. Worldships are essentially mobile versions of the O Neill free-space habitats. Constructed mostly from lunar and/or asteroidal materials, these solar-powered, multi-kilometer-dimension structures could house 10,000 to 100,000 humans in Earth-approximating environments. Artificial gravity would be provided by habitat rotation, and cosmic ray shielding would be provided by passive methods, such as habitat atmosphere and mass shielding, or magnetic fields. A late 21st century space-habitat venture might support itself economically by constructing large solar-powered satellites to beam energy back to
3. A dirty window diffuse and translucent molecular gas in the interstellar medium
CERN Document Server
Magnani, Loris
2017-01-01
This book provides an introduction to the physics of interstellar gas in the Galaxy. It deals with the diffuse interstellar medium which supplies a complex environment for exploring the neutral gas content of a galaxy like the Milky Way and the techniques necessary for studying this non-stellar component. After an initial exposition of the phases of the interstellar medium and the role of gas in a spiral galaxy, the authors discuss the transition from atomic to molecular gas. They then consider basic radiative transfer and molecular spectroscopy with particular emphasis on the molecules useful for studying low-density molecular gas. Observational techniques for investigating the gas and the dust component of the diffuse interstellar medium throughout the electromagnetic spectrum are explored emphasizing results from the recent Herschel and Planck missions. A brief exposition on dust in the diffuse interstellar medium is followed by a discussion of molecular clouds in general and high-latitude molecular clouds...
4. Search for interstellar methane
International Nuclear Information System (INIS)
Knacke, R.F.; Kim, Y.H.; Noll, K.S.; Geballe, T.R.
1990-01-01
Researchers searched for interstellar methane in the spectra of infrared sources embedded in molecular clouds. New observations of several lines of the P and R branches of the nu 3 band of CH4 near 3.3 microns give column densities in the range N less than 1(-2) times 10 to the minus 16th power cm(-2). Resulting abundance ratios are (CH4)/(CO) less than 3.3 times 10 to the minus 2nd power toward GL961 in NGC 2244 and less than 2.4 times 10 to the minus 3rd power toward GL989 in the NGC 2264 molecular cloud. The limits, and those determined in earlier observations of BN in Orion and GL490, suggest that there is little methane in molecular clouds. The result agrees with predictions of chemical models. Exceptions could occur in clouds where oxygen may be depleted, for example by H2O freezing on grains. The present observations probably did not sample such regions
5. Detection of interstellar methylcyanoacetylene
International Nuclear Information System (INIS)
Broten, N.W.; MacLeod, J.M.; Avery, L.W.; Irvine, W.M.; Hoeglund, B.; Friberg, P.; Hjalmarson
1984-01-01
A new interstellar molecule, methylcyanoacetylene (CH 3 C 3 N), has been detected in the molecular cloud TMC-1. The J = 8 → 7, J = 7 → 6, J = 6 → 5, and J = 5 → 4 transitions have been observed. For the first three of these, both the K = 0 and K = 1 components are present, while for J = 5 → 4, only the K = 0 line has been detected. The observed frequencies were calculated by assuming a value of radial velocity V/sub lSR/ = 5.8 km s -1 for TMC-1, typical of other molecules in the cloud. All Observed frequencies are within 10 kHz of the calculated frequencies, which are based on the 1982 laboratory constants of Moises et al., so the identification is secure. The lines are broadened by hyperfine splitting, and the J = 5 → 4, K = 0 transition shows incipient resolution into three hyperfine components. The rotational temperature determined from these observations is quite low, with 2.7 K 12 cm -2
6. Interstellar He Flow Analysis over the Past 9 Years with Observations over the Full IBEX-Lo Energy Range
Science.gov (United States)
Moebius, E.; Bower, E.; Bzowski, M.; Fuselier, S. A.; Heirtzler, D.; Kubiak, M. A.; Kucharek, H.; Lee, M. A.; McComas, D. J.; Schwadron, N.; Swaczyna, P.; Sokol, J. M.; Wurz, P.
2017-12-01
The Sun's motion relative to the surrounding interstellar medium leads to an interstellar neutral (ISN) wind through the heliosphere. This wind is moderately depleted by ionization and can be analyzed in-situ with pickup ions and direct neutral atom imaging. Since 2009, observations of the ISN wind at 1 AU with the Interstellar Boundary Explorer (IBEX) have returned a very precise 4-dimensional parameter tube for the flow vector (speed VISN, longitude λISN, and latitude βISN) and temperature TISN of interstellar He in the local cloud, which organizes VISN, βISN, and TISN as a function of λISN, and the local flow Mach number (VThISN/VISN). Typically, the uncertainties along this functional dependence are larger than across it. Here we present important refinements of the determination of this parameter tube by analyzing the spin-integrated ISN flux for its maximum as a function of ecliptic longitude for each year through 2017. In particular, we include a weak energy dependence of the sensor efficiency by comparing the response in all four energy steps that record the ISN He flow. In addition, a recent operational extension of letting the spin axis pointing of IBEX drift to the maximum offset west of the Sun, results in an additional constraint that helps breaking the degeneracy of the ISN parameters along the 4D tube. This constraint is part of the complement of drivers for the determination of all four ISN parameters effective in the full χ2-minimization by comparing the observed count rate distribution with detailed modeling of the ISN flow (e.g. Bzowski et al., 2015, ApJS, 220:28; Schwadron et al., 2015, ApJS, 220:25) and is complementary to the independent determination of λISN using the longitude dependence of the He+ pickup ion cut-off speed with STEREO PLASTIC and ACE SWICS (Möbius et al., 2015, ApJ 815:20).
7. The Interstellar Ethics of Self-Replicating Probes
Science.gov (United States)
Cooper, K.
Robotic spacecraft have been our primary means of exploring the Universe for over 50 years. Should interstellar travel become reality it seems unlikely that humankind will stop using robotic probes. These probes will be able to replicate themselves ad infinitum by extracting raw materials from the space resources around them and reconfiguring them into replicas of themselves, using technology such as 3D printing. This will create a colonising wave of probes across the Galaxy. However, such probes could have negative as well as positive consequences and it is incumbent upon us to factor self-replicating probes into our interstellar philosophies and to take responsibility for their actions.
8. SOLAR PHOTOIONIZATION RATES FOR INTERSTELLAR NEUTRALS IN THE INNER HELIOSPHERE: H, He, O, AND Ne
Energy Technology Data Exchange (ETDEWEB)
Bochsler, P.; Kucharek, H.; Möbius, E. [Space Science Center and Department of Physics, University of New Hampshire, Durham, NH 03824 (United States); Bzowski, Maciej; Sokół, Justyna M. [Space Research Center of the Polish Academy of Sciences, Ul. Bartycka 18A, 00-716 Warsaw (Poland); Didkovsky, Leonid; Wieman, Seth, E-mail: [email protected] [Space Sciences Center, University of Southern California, Los Angeles, CA 90089-1341 (United States)
2014-01-01
Extreme UV (EUV) spectra from the Thermosphere Ionosphere Mesosphere Energetics and Dynamics (TIMED)/Solar EUV Experiment are used to infer photoionization rates in the inner heliosphere. Relating these rates to various proxies describing the solar EUV radiation, we construct a multi-linear model which allows us to extrapolate ionization rates back to periods when no routine measurements of the solar EUV spectral distribution have been available. Such information is important, e.g., for comparing conditions of the interstellar neutral particles in the inner heliosphere at the time of Ulysses/GAS observations with conditions during the more recent observations of the Interstellar Boundary Explorer. From a period of 11 yr when detailed spectra from both TIMED and three proxies—Solar and Heliospheric Observatory/CELIAS/SEM-rates, F10.7 radio flux, and Mg II core-to-wing indices—have been available, we conclude that the simple model is able to reproduce the photoionization rates with an uncertainty of typically 5%.
9. Interstellar Sweat Equity
Science.gov (United States)
Cohen, M. H.; Becker, R. E.; O'Donnell, D. J.; Brody, A. R.
So, you have just launched aboard the Starship, headed to an exoplanet light years from Earth. You will spend the rest of your natural life on this journey in the expectation and hope that your grandchildren will arrive safely, land, and build a new settlement. You will need to govern the community onboard the Starship. This system of governance must meet unique requirements for participation, representation, and decision-making. On a spaceship that can fly and operate by itself, what will the crewmembers do for their generations in transit? Certainly, they will train and train again to practice the skills they will need upon arrival at a new world. However, this vicarious practice neither suffices to prepare the future pioneers for their destiny at a new star nor will it provide them with the satisfaction in their own work. To hone the crewmembers' inventive and technical skills, to challenge and prepare them for pioneering, the crew would build and expand the interstellar ship in transit. This transstellar sweat equity'' gives a stake in the enterprise to all the people, providing meaningful and useful activity to the new generations of crewmembers. They build all the new segments of the vessel from raw materials - including atmosphere - stored on board. Construction of new pressure shell modules would be one option, but they also reconstruct or fill-in existing pressurized volumes. The crew makes new life support system components and develops new agricultural modules in anticipation of their future needs. Upon arrival at the new star or planet, the crew shall apply these robustly developed skills and self-sufficient spirit to their new home.
10. Comet Halley and interstellar chemistry
International Nuclear Information System (INIS)
Snyder, L.E.
1989-01-01
How complex is the chemistry of the interstellar medium? How far does it evolve and how has it interacted with the chemistry of the solar system? Are the galactic chemical processes destroyed, preserved, or even enhanced in comets? Are biogenic molecules formed in space and have the formation mechanisms interacted in any way with prebiotic organic chemical processes on the early earth? Radio molecular studies of comets are important for probing deep into the coma and nuclear region and thus may help answer these questions. Comets are believed to be pristine samples of the debris left from the formation of the solar system and may have been the carrier between interstellar and terrestrial prebiotic chemistries. Recent observations of Comet Halley and subsequent comets have given the author an excellent opportunity to study the relationship between interstellar molecular chemistry and cometary chemistry
11. Near-Local Interstellar Medium (LISM): What we know from the Voyagers and ENA and what an Interstellar Probe (ISP) can do
Science.gov (United States)
Krimigis, S. M.
2017-12-01
In situ measurements by the two Voyagers over the past 13 years have revealed the presence of the long-predicted termination shock (TS) and heliopause (HP), albeit not where theory had placed them. Further, the advent of energetic neutral atom (ENA) imaging by Cassini/INCA since 2003 and IBEX since 2009 have provided images of the global heliosphere that have challenged our long-held views of its shape and the processes that are dominant in its formation (Krimigis et al, and McComas et al, 2009; Dialynas et al, 2017). In addition, continuing measurements from Voyager 1 beyond the HP, now at 140 AU (1 AU=1.5x108 km, the Sun-Earth distance), have shown that the influence of the Sun extends well beyond the HP (at 122 AU). This influence is manifested through the occasional appearance of anisotropies in the galactic cosmic rays (GCR) where none were predicted, that last as long as a year and are accompanied by electron plasma oscillations in the vicinity of the spacecraft. Thus, an ISP mission with a fast ( 15 AU per year compared to Voyager's 3.6) trajectory would traverse a dynamic region near and beyond the TS and HP, and enable imaging the shape of the heliosphere from beyond its boundaries through ENA. In situ measurements should include the ISM magnetic field (ISMF), plasma density and distribution function, plasma waves, and neutral atom density and composition, as a minimum. All of these measurements would be new information with modern instrumentation that would place the very limited Voyager measurements into the proper context. References: Krimigis, S. M., D. G. Mitchell, E. C. Roelof, K. C. Hsieh and D. J. McComas, Imaging the Interaction of the Heliosphere with the Interstellar Medium from Saturn with Cassini, Science, 326, 5955, p. 971, doi: 10.1126/science.1181079, 2009 McComas, D. J., et al, Global Observations of the Interstellar Interaction from the Interstellar Boundary Explorer (IBEX), Science, 326, 5955, pp. 959, doi: 10.1126/science.1180906, 2009
12. Boundary Spanning
DEFF Research Database (Denmark)
Zølner, Mette
The paper explores how locals span boundaries between corporate and local levels. The aim is to better comprehend potentialities and challenges when MNCs draws on locals’ culture specific knowledge. The study is based on an in-depth, interpretive case study of boundary spanning by local actors in...... approach with pattern matching is a way to shed light on the tacit local knowledge that organizational actors cannot articulate and that an exclusively inductive research is not likely to unveil....
13. Formation of interstellar anions
Science.gov (United States)
Senent, Maria Luisa
2012-05-01
Formation of interstellar anions: M.L. Senent. The recent detection of negative charged species in the ISM1 has instigated enthusiasm for anions in the astrophysical community2. Many of these species are new and entail characterization. How they are formed in astrophysical sources is a question of major relevance. The anion presence in ISM was first predicted theoretically on the basis of electron affinities and on the negative linear chain molecular stabilities. Although very early, they were considered in astrochemical models3-4, their discovery is so recent because their abundances seem to be relatively low. These have to be understood in terms of molecular stabilities, reaction probabilities and radiative and collisional excitations. Then, we present our theoretical work on even carbon chains type Cn and CnH (n=2,4,6) focused to the understanding of anion abundances. We use highly correlated ab initio methods. We performed spectroscopic studies of various isomers that can play important roles as intermediates5-8. In previous papers9-10, we compared C2H and C2H- collisional rates responsible for observed line intensities. Actually, we study hydrogen attachment (Cn +H → CnH and Cn- +H → CnH-) and associative detachment processes (Cn- +H → CnH +e-) for 2, 4 and 6 carbon atom chains11. [1] M.C.McCarthy, C.A.Gottlieb, H.Gupta, P.Thaddeus, Astrophys.J, 652, L141 (2006) [2] V.M.Bierbaum, J.Cernicharo, R.Bachiller, eds., 2011, pp 383-389. [3] A. Dalgarno, R.A. Mc Cray, Astrophys.J,, 181, 95 (1973) [4] E. Herbst E., Nature, 289, 656 (1981); [5] H.Massó, M.L.Senent, P.Rosmus, M.Hochlaf, J.Chem.Phys., 124, 234304 (2006) [6] M.L.Senent, M.Hochlaf, Astrophys. J. , 708, 1452(2010) [7] H.Massó, M.L.Senent, J.Phys.Chem.A, 113, 12404 (2009) [8] D. Hammoutene, M.Hochlaf, M.L.Senent, submitted. [9] A. Spielfiedel, N. Feautrier, F. Najar, D. ben Abdallah, F. Dayou, M.L. Senent, F. Lique, Mon.Not.R.Astron.Soc., 421, 1891 (2012) [10] F.Dumouchel, A, Spielfieldel , M
14. Chemisputtering of interstellar graphite grains
International Nuclear Information System (INIS)
Draine, B.T.
1979-01-01
The rate of erosion of interstellar graphite grains as a result of chemical reaction with H, N, and O is estimated using the available experiment evidence. It is argued that ''chemical sputtering'' yields for interstellar graphite grains will be much less than unity, contrary to earlier estimates by Barlow and Silk. Chemical sputtering of graphite grains in evolving H II regions is found to be unimportant, except in extremely compact (n/sub H/> or approx. =10 5 cm -3 ) H II regions. Alternative explanations are considered for the apparent weakness of the lambda=2175 A extinction ''bump'' in the direction of several early type stars
15. Interstellar Initiative Web Page Design
Science.gov (United States)
Mehta, Alkesh
1999-01-01
This summer at NASA/MSFC, I have contributed to two projects: Interstellar Initiative Web Page Design and Lenz's Law Relative Motion Demonstration. In the Web Design Project, I worked on an Outline. The Web Design Outline was developed to provide a foundation for a Hierarchy Tree Structure. The Outline would help design a Website information base for future and near-term missions. The Website would give in-depth information on Propulsion Systems and Interstellar Travel. The Lenz's Law Relative Motion Demonstrator is discussed in this volume by Russell Lee.
16. Interstellar matter within elliptical galaxies
Science.gov (United States)
Jura, Michael
1988-01-01
Multiwavelength observations of elliptical galaxies are reviewed, with an emphasis on their implications for theoretical models proposed to explain the origin and evolution of the interstellar matter. Particular attention is given to interstellar matter at T less than 100 K (atomic and molecular gas and dust), gas at T = about 10,000 K, and gas at T = 10 to the 6th K or greater. The data are shown to confirm the occurrence of mass loss from evolved stars, significant accretion from companion galaxies, and cooling inflows; no evidence is found for large mass outflow from elliptical galaxies.
17. Experimental interstellar organic chemistry - Preliminary findings
Science.gov (United States)
Khare, B. N.; Sagan, C.
1973-01-01
Review of the results of some explicit experimental simulation of interstellar organic chemistry consisting in low-temperature high-vacuum UV irradiation of condensed simple gases known or suspected to be present in the interstellar medium. The results include the finding that acetonitrile may be present in the interstellar medium. The implication of this and other findings are discussed.
18. Interstellar turbulence and shock waves
International Nuclear Information System (INIS)
Bykov, A.M.
1982-01-01
Random deflections of shock fronts propagated through the turbulent interstellar medium can produce the strong electro-density fluctuations on scales l> or approx. =10 13 cm inferred from pulsar radio scintillations. The development of turbulence in the hot-phase ISM is discussed
19. Stardust Interstellar Preliminary Examination (ISPE)
Science.gov (United States)
Westphal, A. J.; Allen, C.; Bajt, S.; Basset, R.; Bastien, R.; Bechtel, H.; Bleuet, P.; Borg, J.; Brenker F.; Bridges, J.
2009-01-01
In January 2006 the Stardust sample return capsule returned to Earth bearing the first solid samples from a primitive solar system body, C omet 81P/Wild2, and a collector dedicated to the capture and return o f contemporary interstellar dust. Both collectors were approximately 0.1m(exp 2) in area and were composed of aerogel tiles (85% of the co llecting area) and aluminum foils. The Stardust Interstellar Dust Col lector (SIDC) was exposed to the interstellar dust stream for a total exposure factor of 20 m(exp 2-) day during two periods before the co metary encounter. The Stardust Interstellar Preliminary Examination ( ISPE) is a three-year effort to characterize the collection using no ndestructive techniques. The ISPE consists of six interdependent proj ects: (1) Candidate identification through automated digital microsco py and a massively distributed, calibrated search (2) Candidate extr action and photodocumentation (3) Characterization of candidates thro ugh synchrotronbased FourierTranform Infrared Spectroscopy (FTIR), S canning XRay Fluoresence Microscopy (SXRF), and Scanning Transmission Xray Microscopy (STXM) (4) Search for and analysis of craters in f oils through FESEM scanning, Auger Spectroscopy and synchrotronbased Photoemission Electron Microscopy (PEEM) (5) Modeling of interstell ar dust transport in the solar system (6) Laboratory simulations of h ypervelocity dust impacts into the collecting media
20. Magnetite and the interstellar medium
International Nuclear Information System (INIS)
Landaberry, S.C.; Magalhaes, A.M.
1976-01-01
Recent observations concerning interstellar circular polarization are explained by a simple two-cloud model using magnetite (Fe 3 O 4 ) grains as polarizing agents. Three stars covering a wide range of linear polarization spectral shapes were selected. Reasonably low column densities are required in order to interpret polarization data [pt
1. Interstellar Propulsion Research: Realistic Possibilities and Idealistic Dreams
Science.gov (United States)
Johnson, Les
2009-01-01
Though physically possible, interstellar travel will be exceedingly difficult. Both the known laws of physics and the limits of our current understanding of engineering place extreme limits on what may actually be possible. Our remote ancestors looked at the night sky and assumed those tiny points of light were campfires around which other tribes were gathered -- and they dreamed of someday making the trip to visit them. In our modern era, we've grown accustomed to humans regularly traveling into space and our robots voyaging ever-deeper into the outer edges of our solar system. Traveling to those distant campfires (stars) has been made to look easy by the likes of Captains Kirk and Picard as well as Han Solo and Commander Adama. Our understanding of physics and engineering has not kept up with our imaginations and many are becoming frustrated with the current pace at which we are exploring the universe. Fortunately, there are ideas that may one day lead to new physical theories about how the universe works and thus potentially make rapid interstellar travel possible -- but many of these are just ideas and are not even close to being considered a scientific theory or hypothesis. Absent any scientific breakthroughs, we should not give up hope. Nature does allow for interstellar travel, albeit slowly and requiring an engineering capability far beyond what we now possess. Antimatter, fusion and photon sail propulsion are all candidates for relatively near-term interstellar missions. The plenary lecture will discuss the dreams and challenges of interstellar travel, our current understanding of what may be possible and some of the "out of the box" ideas that may allow us to become an interstellar species someday in the future.
2. Interstellar dehydrogenated PAH anions: vibrational spectra
Science.gov (United States)
Buragohain, Mridusmita; Pathak, Amit; Sarre, Peter; Gour, Nand Kishor
2018-03-01
Interstellar polycyclic aromatic hydrocarbon (PAH) molecules exist in diverse forms depending on the local physical environment. Formation of ionized PAHs (anions and cations) is favourable in the extreme conditions of the interstellar medium (ISM). Besides in their pure form, PAHs are also likely to exist in substituted forms; for example, PAHs with functional groups, dehydrogenated PAHs etc. A dehydrogenated PAH molecule might subsequently form fullerenes in the ISM as a result of ongoing chemical processes. This work presents a density functional theory (DFT) calculation on dehydrogenated PAH anions to explore the infrared emission spectra of these molecules and discuss any possible contribution towards observed IR features in the ISM. The results suggest that dehydrogenated PAH anions might be significantly contributing to the 3.3 μm region. Spectroscopic features unique to dehydrogenated PAH anions are highlighted that may be used for their possible identification in the ISM. A comparison has also been made to see the size effect on spectra of these PAHs.
3. Physical processes in the interstellar medium
CERN Document Server
Spitzer, Lyman
2008-01-01
Physical Processes in the Interstellar Medium discusses the nature of interstellar matter, with a strong emphasis on basic physical principles, and summarizes the present state of knowledge about the interstellar medium by providing the latest observational data. Physics and chemistry of the interstellar medium are treated, with frequent references to observational results. The overall equilibrium and dynamical state of the interstellar gas are described, with discussions of explosions produced by star birth and star death and the initial phases of cloud collapse leading to star formation.
4. PAHs in Translucent Interstellar Clouds
Science.gov (United States)
Salama, Farid; Galazutdinov, G.; Krelowski, J.; Biennier, L.; Beletsky, Y.; Song, I.
2011-05-01
We discuss the proposal of relating the origin of some of the diffuse interstellar bands (DIBs) to neutral polycyclic aromatic hydrocarbons (PAHs) present in translucent interstellar clouds. The spectra of several cold, isolated gas-phase PAHs have been measured in the laboratory under experimental conditions that mimic the interstellar conditions and are compared with an extensive set of astronomical spectra of reddened, early type stars. This comparison provides - for the first time - accurate upper limits for the abundances of specific PAH molecules along specific lines-of-sight. Something that is not attainable from IR observations alone. The comparison of these unique laboratory data with high resolution, high S/N ratio astronomical observations leads to two major findings: (1) a finding specific to the individual molecules that were probed in this study and, which leads to the clear and unambiguous conclusion that the abundance of these specific neutral PAHs must be very low in the individual translucent interstellar clouds that were probed in this survey (PAH features remain below the level of detection) and, (2) a general finding that neutral PAHs exhibit intrinsic band profiles that are similar to the profile of the narrow DIBs indicating that the carriers of the narrow DIBs must have close molecular structure and characteristics. This study is the first quantitative survey of neutral PAHs in the optical range and it opens the way for unambiguous quantitative searches of PAHs in a variety of interstellar and circumstellar environments. // Reference: F. Salama et al. (2011) ApJ. 728 (1), 154 // Acknowledgements: F.S. acknowledges the support of the NASA's Space Mission Directorate APRA Program. J.K. acknowledges the financial support of the Polish State (grant N203 012 32/1550). The authors are deeply grateful to the ESO archive as well as to the ESO staff members for their active support.
5. Molecular Diagnostics of the Interstellar Medium and Star Forming Regions
Science.gov (United States)
Hartquist, T. W.; Dalgarno, A.
1996-03-01
Selected examples of the use of observationally inferred molecular level populations and chemical compositions in the diagnosis of interstellar sources and processes important in them (and in other diffuse astrophysical sources) are given. The sources considered include the interclump medium of a giant molecular cloud, dark cores which are the progenitors of star formation, material responding to recent star formation and which may form further stars, and stellar ejecta (including those of supernovae) about to merge with the interstellar medium. The measurement of the microwave background, mixing of material between different nuclear burning zones in evolved stars and turbulent boundary layers (which are present in and influence the structures and evolution of all diffuse astrophysical sources) are treated.
6. Exploring the boundaries of individual and collective land use management: institutional arrangements in the PAE Chico Mendes (Acre, Brazil
Directory of Open Access Journals (Sweden)
François-Michel Le Tourneau
2017-03-01
Full Text Available The economic modernization of the Amazon fostered by the Brazilian military government during the 1960s and 1970s was largely realized without taking into consideration the presence of local households which lived from the extraction of forest products (mainly non-timber. When they began to be expulsed, a political resistance, often guided by the Catholic Church, appeared as well as the creation of unions based on traditional identities, especially that of rubber tappers. During the 1980s, these unions made a strategic alliance with the ecologist movement which started to consider traditional populations, whose lifestyle depended on the forest, as allies for the protection of the Amazon rainforest. The movement gained a decisive momentum at the end of the decade by putting forward new proposals of land tenure for traditional populations, grounded on collective land rights. This strategy has been very efficient during the 1990s and 2000s, during which about 1,300,000 km2 of rainforest were set apart and reserved for the use of “traditional communities” under a variety of legal status. But it has also led to mix under the same “collective” etiquette and principles a number of different ways of using and managing land and natural resources. This assumption however should be nuanced by a careful analysis of the resource management systems existing in each case, for they are in general complex and mix varying proportions of individual and collective decisions. The aim of this paper is to explore this question using the example of the Chico Mendes agroextractive settlement (PAE-CM, inhabited by about 100 rubber tapper families and symbolic of the political struggle of traditional populations in the Amazon for being the home of the rubber tapper leader Chico Mendes assassinated in 1988. Applying Ostrom “design principles”, we try to catch what are the local institutional arrangements and to see if they suggest
7. Interstellar Mapping and Acceleration Probe (IMAP)
International Nuclear Information System (INIS)
Schwadron, N. A.; Moebius, E.; Spence, H. E.; Opher, M.; Kasper, J.; Zurbuchen, T. H.; Mewaldt, R.
2016-01-01
Our piece of cosmic real estate, the heliosphere, is the domain of all human existence – an astrophysical case history of the successful evolution of life in a habitable system. By exploring our global heliosphere and its myriad interactions, we develop key physical knowledge of the interstellar interactions that influence exoplanetary habitability as well as the distant history and destiny of our solar system and world. IBEX is the first mission to explore the global heliosphere and in concert with Voyager 1 and Voyager 2 is discovering a fundamentally new and uncharted physical domain of the outer heliosphere. In parallel, Cassini/INCA maps the global heliosphere at energies (∼5-55 keV) above those measured by IBEX. The enigmatic IBEX ribbon and the INCA belt were unanticipated discoveries demonstrating that much of what we know or think we understand about the outer heliosphere needs to be revised. This paper summarizes the next quantum leap enabled by IMAP that will open new windows on the frontier of Heliophysics at a time when the space environment is rapidly evolving. IMAP with 100 times the combined resolution and sensitivity of IBEX and INCA will discover the substructure of the IBEX ribbon and will reveal, with unprecedented resolution, global maps of our heliosphere. The remarkable synergy between IMAP, Voyager 1 and Voyager 2 will remain for at least the next decade as Voyager 1 pushes further into the interstellar domain and Voyager 2 moves through the heliosheath. Voyager 2 moves outward in the same region of sky covered by a portion of the IBEX ribbon. Voyager 2’s plasma measurements will create singular opportunities for discovery in the context of IMAP's global measurements. IMAP, like ACE before, will be a keystone of the Heliophysics System Observatory by providing comprehensive measurements of interstellar neutral atoms and pickup ions, the solar wind distribution, composition, and magnetic field, as well as suprathermal ion
8. Modelling interstellar extinction: Pt. 1
International Nuclear Information System (INIS)
Jones, A.P.
1988-01-01
Several methods of calculating the extinction of porous silicate grains are discussed, these include effective medium theories and hollow spherical shells. Porous silicate grains are shown to produce enhanced infrared, ultraviolet and far-ultraviolet extinction and this effect can be used to reduce the abundance of carbon required to match the average interstellar extinction, however, matching the visual extinction is rather more problematical. We have shown that the enhanced extinction at long and short wavelengths have different origins, and have explained why the visual extinction is little affected by porosity. The implications of porous grains in the interstellar medium are discussed with particular reference to surface chemistry, the polarization of starlight, and their dynamical evolution. (author)
9. Interstellar Grains: 50 Years on
Science.gov (United States)
Wickramasinghe, N. C.
Our understanding of the nature of interstellar grains has evolved considerably over the past half century with the present author and Fred Hoyle being intimately involved at several key stages of progress. The currently fashionable graphite-silicate-organic grain model has all its essential aspects unequivocally traceable to original peer-reviewed publications by the author and/or Fred Hoyle. The prevailing reluctance to accept these clear-cut priorities may be linked to our further work that argued for interstellar grains and organics to have a biological provenance -- a position perceived as heretical. The biological model, however, continues to provide a powerful unifying hypothesis for a vast amount of otherwise disconnected and disparate astronomical data.
10. Why do interstellar grains exist
International Nuclear Information System (INIS)
Seab, C.G.; Hollenbach, D.J.; Mckee, C.F.; Tielens, A.G.G.M.
1986-01-01
There exists a discrepancy between calculated destruction rates of grains in the interstellar medium and postulated sources of new grains. This problem was examined by modelling the global life cycle of grains in the galaxy. The model includes: grain destruction due to supernovae shock waves; grain injection from cool stars, planetary nebulae, star formation, novae, and supernovae; grain growth by accretion in dark clouds; and a mixing scheme between phases of the interstellar medium. Grain growth in molecular clouds is considered as a mechanism or increasing the formation rate. To decrease the shock destruction rate, several new physical processes, such as partial vaporization effects in grain-grain collisions, breakdown of the small Larmor radius approximation for betatron acceleration, and relaxation of the steady-state shock assumption are included
11. Shifting contours of boundaries: an exploration of inter-agency integration between hospital and community interprofessional diabetes programs.
Science.gov (United States)
Wong, Rene; Breiner, Petra; Mylopoulos, Maria
2014-09-01
This article reports on research into the relationships that emerged between hospital-based and community-based interprofessional diabetes programs involved in inter-agency care. Using constructivist grounded theory methodology we interviewed a purposive theoretical sample of 21 clinicians and administrators from both types of programs. Emergent themes were identified through a process of constant comparative analysis. Initial boundaries were constructed based on contrasts in beliefs, practices and expertise. In response to bureaucratic and social pressures, boundaries were redefined in a way that created role uncertainty and disempowered community programs, ultimately preventing collaboration. We illustrate the dynamic and multi-dimensional nature of social and symbolic boundaries in inter-agency diabetes care and the tacit ways in which hospitals can maintain a power position at the expense of other actors in the field. As efforts continue in Canada and elsewhere to move knowledge and resources into community sectors, we highlight the importance of hospitals seeing beyond their own interests and adopting more altruistic models of inter-agency integration.
12. Origins of amorphous interstellar grains
International Nuclear Information System (INIS)
Hasegawa, H.
1984-01-01
The existence of amorphous interstellar grains has been suggested from infrared observations. Some carbon stars show the far infrared emission with a lambda -1 wavelength dependence. Far infrared emission supposed to be due to silicate grains often show the lambda -1 wavelength dependence. Mid infrared spectra around 10 μm have broad structure. These may be due to the amorphous silicate grains. The condition that the condensed grains from the cosmic gas are amorphous is discussed. (author)
13. Interstellar Grains: 50 Years On
OpenAIRE
Wickramasinghe, N. Chandra
2011-01-01
Our understanding of the nature of interstellar grains has evolved considerably over the past half century with the present author and Fred Hoyle being intimately involved at several key stages of progress. The currently fashionable graphite-silicate-organic grain model has all its essential aspects unequivocally traceable to original peer-reviewed publications by the author and/or Fred Hoyle. The prevailing reluctance to accept these clear-cut priorities may be linked to our further work tha...
14. Interstellar space: the astrochemist's laboratory
International Nuclear Information System (INIS)
Allen, M.A.
1976-01-01
A mechanism for the formation of molecules on small (radius less than or equal to 0.04 μ) interstellar grains is proposed. A simplified H 2 formation model is then presented that utilizes this surface reaction mechanism. This approach is further developed into an ab initio chemical model for dense interstellar clouds that incorporates 598 grain surface reactions, with small grains again providing the key reaction area. Gas-phase molecules are depleted through collisions with grains. The abundances of 372 chemical species are calculated as a function of time and are found to be of sufficient magnitude to explain most observations. The reaction rates for ion-molecule chemistry are approximately the same, therefore indicating that surface and gas-phase chemistry may be coupled in certain regions. The composition of grain mantles is shown to be a function of grain radius. In certain grain size ranges, large molecules containing two or more heavy atoms are more predominant than lighter ''ices''--H 2 O, NH 3 , and CH 4 . It is possible that absorption due to these large molecules in the mantles may contribute to the observed 3μ band in astronomical spectra. The second part of this thesis is an account of a radio astronomy observational program to detect new transitions of both previously observed and yet undetected interstellar molecules. The negative results yield order ofmagnitude upper limits to the column densities of the lower transition states of the various molecules. One special project was the search for the Λ-doublet transitions of the 2 H/sub 3 / 2 /, J = 3 / 2 state of OD. The resulting upper limit for the OD/OH column density ratio towards the galactic center is 1/400 and is discussed with reference to theories about deuterium enrichment in interstellar molecules
15. On the ionization of interstellar magnesium
International Nuclear Information System (INIS)
1977-01-01
It has been shown that two concentric ionization zones of interstellar magnesium must exist around each star: internal, with a radius coinciding with that of the zone of hydrogen ionization Ssub(H); and external, with a radius greater than Ssub(H), by one order. Unlike interstellar hydrogen, interstellar magnesium is ionized throughout the Galaxy. It also transpires that the ionizing radiation of ordinary hot stars cannot provide for the observed high degree of ionization of interstellar magnesium. The discrepance can be eliminated by assuming the existence of circumstellar clouds or additional ionization sources of interstellar magnesium (X-ray background radiation, high-energy particles, etc.). Stars of the B5 and BO class play the main role in the formation of ionization zones of interstellar magnesium; the contribution of O class stars is negligible (<1%). (Auth.)
16. A chemical model for the interstellar medium in galaxies
OpenAIRE
Bovino, S.; Grassi, Tommaso; Capelo, P. R.; Schleicher, D. R. G.; Banerjee, R.
2016-01-01
Aims: We present and test chemical models for three-dimensional hydrodynamical simulations of galaxies. We explore the effect of changing key parameters such as metallicity, radiation, and non-equilibrium versus equilibrium metal cooling approximations on the transition between the gas phases in the interstellar medium. Methods: The microphysics was modelled by employing the public chemistry package KROME, and the chemical networks were tested to work in a wide range of densities and temp...
17. Mechanisms of heating the interstellar matter
International Nuclear Information System (INIS)
Lequeux, J.
1975-01-01
The knowledge of the interstellar medium has been considerably improved in the recent years, thanks in particular to Radioastronomy and Ultraviolet Space Astronomy. This medium is a natural laboratory where the conditions and various and very different to what can be realised in terrestrial laboratories. To illustrate its interest for physicists here one of the most interesting but controversial points of interstellar astronomy is discussed: the mechanisms for heating and cooling the interstellar medium [fr
18. A Data-driven Model of the Solar Wind, Interstellar Pickup Ions, and Turbulence Throughout the Interplanetary Space
Science.gov (United States)
Kim, T. K.; Kryukov, I.; Pogorelov, N. V.; Elliott, H. A.; Zank, G. P.
2017-12-01
The outer heliosphere is an interesting region characterized by the interaction between the solar wind and the interstellar neutral atoms. Having accomplished the mission to Pluto in 2015 and currently on the way to the Kuiper Belt, the New Horizons spacecraft is following the footsteps of the two Voyager spacecraft that previously explored this region lying roughly beyond 30 AU from the Sun. We model the three-dimensional, time-dependent solar wind plasma flow to the outer heliosphere using our own software Multi-Scale Fluid-Kinetic Simulation Suite (MS-FLUKSS), which, in addition to the thermal solar wind plasma, takes into account charge exchange of the solar wind protons with interstellar neutral atoms and treats nonthermal ions (i.e., pickup ions) born during this process as a separate fluid. Additionally, MS-FLUKSS allows us to model turbulence generated by pickup ions. We use MS-FLUKSS to investigate the evolution of plasma and turbulent fluctuations along the trajectory of the New Horizons spacecraft using plasma and turbulence parameters from OMNI data as time-dependent boundary conditions at 1 AU for the Reynolds-averaged MHD equations. We compare the model with in situ plasma observations by New Horizons, Voyager 2, and Ulysses. We also compare the model pickup proton parameters with those derived from the Ulysses-SWICS data.
19. Interstellar Silicon Depletion and the Ultraviolet Extinction
Science.gov (United States)
Mishra, Ajay; Li, Aigen
2018-01-01
Spinning small silicate grains were recently invoked to account for the Galactic foreground anomalous microwave emission. These grains, if present, will absorb starlight in the far ultraviolet (UV). There is also renewed interest in attributing the enigmatic 2175 Å interstellar extinction bump to small silicates. To probe the role of silicon in the UV extinction, we explore the relations between the amount of silicon required to be locked up in silicates [Si/H]dust and the 2175 Å bump or the far-UV extinction rise, based on an analysis of the extinction curves along 46 Galactic sightlines for which the gas-phase silicon abundance [Si/H]gas is known. We derive [Si/H]dust either from [Si/H]ISM - [Si/H]gas or from the Kramers- Kronig relation which relates the wavelength-integrated extinction to the total dust volume, where [Si/H]ISM is the interstellar silicon reference abundance and taken to be that of proto-Sun or B stars. We also derive [Si/H]dust from fi�tting the observed extinction curves with a mixture of amorphous silicates and graphitic grains. We fi�nd that in all three cases [Si/H]dust shows no correlation with the 2175 Å bump, while the carbon depletion [C/H]dust tends to correlate with the 2175 Å bump. This supports carbon grains instead of silicates as the possible carrier of the 2175 Å bump. We also �find that neither [Si/H]dust nor [C/H]dust alone correlates with the far-UV extinction, suggesting that the far-UV extinction is a combined effect of small carbon grains and silicates.
20. Probing the diffuse interstellar medium with diffuse interstellar bands
Science.gov (United States)
Theodorus van Loon, Jacco; Bailey, Mandy; Farhang, Amin; Javadi, Atefeh; Khosroshahi, Habib
2015-08-01
For a century already, a large number of absorption bands have been known at optical wavelengths, called the diffuse interstellar bands (DIBs). While their carriers remain unidentified, the relative strengths of these bands in various environments make them interesting new probes of the diffuse interstellar medium (ISM). We present the results from two large, dedicated campaigns to map the ISM using DIBs measured in the high signal-to-noise spectra of hundreds of early-type stars: [1] in and around the Local Bubble using ESO's New Technology Telescope and the Isaac Newton Telescope, and [2] across both Magellanic Clouds using the Very Large Telescope and the Anglo-Australian Telescope. We discuss the implications for the structure and dynamics of the ISM, as well as the constraints these maps place on the nature of the carriers of the DIBs. Partial results have appeared in the recent literature (van Loon et al. 2013; Farhang et al. 2015a,b; Bailey, PhD thesis 2014) with the remainder being prepared for publication now.
1. Grain destruction in interstellar shocks
International Nuclear Information System (INIS)
Seab, C.G.; Shull, J.M.
1984-01-01
One of the principal methods for removing grains from the Interstellar Medium is to destroy them in shock waves. Previous theoretical studies of shock destruction have generally assumed only a single size and type of grain; most do not account for the effect of the grain destruction on the structure of the shock. Earlier calculations have been improved in three ways: first, by using a ''complete'' grain model including a distribution of sizes and types of grains; second, by using a self-consistent shock structure that incorporates the changing elemental depletions as the grains are destroyed; and third, by calculating the shock-processed ultraviolet extinction curves for comparison with observations. (author)
2. Crossing Boundaries: Exploring Black Middle and Upper Class Preservice Teachers' Perceptions of Teaching and Learning in High Poverty Urban Schools
Science.gov (United States)
Lewis, Andrea D.
2012-01-01
The intent of this study was to explore the perceptions of Black middle and upper class preservice teachers as they relate to teaching and learning in high poverty urban schools. Participants included 11 senior early childhood education preservice teachers at a historically Black college in the southeast region of the United States. The study was…
3. Using Drawings of the Brain Cell to Exhibit Expertise in Neuroscience: Exploring the Boundaries of Experimental Culture
Science.gov (United States)
Hay, David B.; Williams, Darren; Stahl, Daniel; Wingate, Richard J.
2013-01-01
This paper explores the research perspective of neuroscience by documenting the brain cell (neuron) drawings of undergraduates, trainee scientists, and leading neuroscience researchers in a single research-intensive university. Qualitative analysis, drawing-sorting exercises, and hierarchical cluster analysis are used to answer two related…
4. Science From Beyond: NASA's Pioneer Plaque and the History of Interstellar Communication, 1957- 1972
Science.gov (United States)
Macauley, William
2012-05-01
In the late twentieth century, science and technology facilitated exploration beyond the Solar System and extended human knowledge through messages comprised of pictures and mathematical symbols, transmitted from radio telescopes and inscribed on material artifacts attached to spacecraft. ‘Interstellar communication' refers to collective efforts by scientists and co-workers to detect and transmit intelligible messages between humans and supposed extraterrestrial intelligence in remote star systems. Interstellar messages are designed to communicate universal knowledge without recourse to text, human linguistic systems or anthropomorphic content because it is assumed that recipients have no prior knowledge of humankind or the planet we inhabit. Scientists must therefore imagine how extraterrestrials will relate to human knowledge and culture. The production and transmission of interstellar messages became interdisciplinary design problems that involved collaboration and exchange of ideas between scientists, visual artists, and others. My proposed paper will review sociocultural aspects of interstellar communication since the late 1950s and focus on key issues regarding conception, design and production of a specific interstellar message launched into space during the early 1970s - NASA's Pioneer plaque. The paper will explore how research on the history of interstellar communication relates to previous historical and sociological studies on rhetorical aspects of visual representation and mathematics in scientific practice. In particular, I will explain how the notion of ‘inscription' is an appropriate conceptual tool for analyzing how scientists have used pictures to articulate and validate knowledge claims and scientific facts. I argue that scientific knowledge carried on interstellar messages such as the Pioneer plaque is constituted in material practices and inscription technologies that translate natural objects, agency and culture into legible forms
5. Identifying specific interstellar polycyclic aromatic hydrocarbons
International Nuclear Information System (INIS)
Mulas, Giacomo; Malloci, Giuliano; Porceddu, Ignazio
2005-01-01
Interstellar Polycyclic Aromatic Hydrocarbons (PAHs) have been thought to be ubiquitous for more than twenty years, yet no single species in this class has been identified in the Interstellar Medium (ISM) to date. The unprecedented sensitivity and resolution of present Infrared Space Observatory (ISO) and forthcoming Herschel observations in the far infrared spectral range will offer a unique way out of this embarrassing impasse
6. Can spores survive in interstellar space
Energy Technology Data Exchange (ETDEWEB)
Weber, P.; Greenberg, J.M.
1985-08-01
Inactivation of spores (Bacillus subtilis) has been investigated in the laboratory by vacuum ultraviolet radiation in simulated interstellar conditions. Damage produced at the normal interstellar particle temperature of 10 K is less than at higher temperatures: the major damage being produced by radiation in the 2,000-3,000 A range. The results place constraints on the panspermia hypothesis. (author).
7. MEASURING THE FRACTAL STRUCTURE OF INTERSTELLAR CLOUDS
NARCIS (Netherlands)
VOGELAAR, MGR; WAKKER, BP
To study the structure of interstellar matter we have applied the concept of fractal curves to the brightness contours of maps of interstellar clouds and from these estimated the fractal dimension for some of them. We used the so-called perimeter-area relation as the basis for these estimates. We
8. MEASURING THE FRACTAL STRUCTURE OF INTERSTELLAR CLOUDS
NARCIS (Netherlands)
VOGELAAR, MGR; WAKKER, BP
1994-01-01
To study the structure of interstellar matter we have applied the concept of fractal curves to the brightness contours of maps of interstellar clouds and from these estimated the fractal dimension for some of them. We used the so-called perimeter-area relation as the basis for these estimates. We
9. Interstellar grains - the 75th anniversary
International Nuclear Information System (INIS)
Li Aigen
2005-01-01
The year of 2005 marks the 75th anniversary since Trumpler (1930) provided the first definitive proof of interstellar grains by demonstrating the existence of general absorption and reddening of starlight in the galactic plane. This article reviews our progressive understanding of the nature of interstellar dust
10. THERMODYNAMICS AND CHARGING OF INTERSTELLAR IRON NANOPARTICLES
Energy Technology Data Exchange (ETDEWEB)
Hensley, Brandon S. [Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109 (United States); Draine, B. T., E-mail: [email protected] [Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 (United States)
2017-01-10
Interstellar iron in the form of metallic iron nanoparticles may constitute a component of the interstellar dust. We compute the stability of iron nanoparticles to sublimation in the interstellar radiation field, finding that iron clusters can persist down to a radius of ≃4.5 Å, and perhaps smaller. We employ laboratory data on small iron clusters to compute the photoelectric yields as a function of grain size and the resulting grain charge distribution in various interstellar environments, finding that iron nanoparticles can acquire negative charges, particularly in regions with high gas temperatures and ionization fractions. If ≳10% of the interstellar iron is in the form of ultrasmall iron clusters, the photoelectric heating rate from dust may be increased by up to tens of percent relative to dust models with only carbonaceous and silicate grains.
11. Exploring bainite formation kinetics distinguishing grain-boundary and autocatalytic nucleation in high and low-Si steels
International Nuclear Information System (INIS)
Ravi, Ashwath M.; Sietsma, Jilt; Santofimia, Maria J.
2016-01-01
Bainite formation in steels begins with nucleation of bainitic ferrite at austenite grain boundaries (γ/γ interfaces). This leads to creation of bainitic ferrite/austenite interfaces (α/γ interfaces). Bainite formation continues through autocatalysis with nucleation of bainitic ferrite at these newly created α/γ interfaces. The displacive theory of bainite formation suggests that the formation of bainitic ferrite is accompanied by carbon enrichment of surrounding austenite. This carbon enrichment generally leads to carbide precipitation unless such a reaction is thermodynamically or kinetically unfavourable. Each bainitic ferrite nucleation event is governed by an activation energy. Depending upon the interface at which nucleation occurs, a specific activation energy would be related to a specific nucleation mechanism. On the basis of this concept, a model has been developed to understand the kinetics of bainite formation during isothermal treatments. This model is derived under the assumptions of displacive mechanism of bainite formation. The fitting parameters used in this model are physical entities related to nucleation and microstructural dimensions. The model is designed in such a way that the carbon redistribution during bainite formation is accounted for, leading to prediction of transformation kinetics both with and without of carbide precipitation during bainite formation. Furthermore, the model is validated using two different sets of kinetic data published in the literature.
12. Zeppelin NT - Measurement Platform for the Exploration of Atmospheric Chemistry and Dynamics in the Planetary Boundary Layer
Science.gov (United States)
Hofzumahaus, Andreas; Holland, Frank; Oebel, Andreas; Rohrer, Franz; Mentel, Thomas; Kiendler-Scharr, Astrid; Wahner, Andreas; Brauchle, Artur; Steinlein, Klaus; Gritzbach, Robert
2014-05-01
The planetary boundary layer (PBL) is the chemically most active and complex part of the atmosphere where freshly emitted reactive trace gases, tropospheric radicals, atmospheric oxidation products and aerosols exhibit a large variability and spatial gradients. In order to investigate the chemical degradation of trace gases and the formation of secondary pollutants in the PBL, a commercial Zeppelin NT was modified to be used as an airborne measurement platform for chemical and physical observations with high spatial resolution. The Zeppelin NT was developed by Zeppelin Luftschifftechnik (ZLT) and is operated by Deutsche Zeppelin Reederei (DZR) in Friedrichshafen, Germany. The modification was performed in cooperation between Forschungszentrum Jülich and ZLT. The airship has a length of 75 m, can lift about 1 ton of scientific payload and can be manoeuvered with high precision by propeller engines. The modified Zeppelin can carry measurement instruments mounted on a platform on top of the Zeppelin, or inside the gondola beneath the airship. Three different instrument packages were developed to investigate a. gas-phase oxidation processes involving free radicals (OH, HO2) b. formation of secondary organic aerosols (SOA) c. new particle formation (nucleation) The presentation will describe the modified airship and provide an overview of its technical performance. Examples of its application during the recent PEGASOS flight campaigns in Europe will be given.
13. Working memory capacity and the top-down control of visual search: Exploring the boundaries of "executive attention".
Science.gov (United States)
Kane, Michael J; Poole, Bradley J; Tuholski, Stephen W; Engle, Randall W
2006-07-01
The executive attention theory of working memory capacity (WMC) proposes that measures of WMC broadly predict higher order cognitive abilities because they tap important and general attention capabilities (R. W. Engle & M. J. Kane, 2004). Previous research demonstrated WMC-related differences in attention tasks that required restraint of habitual responses or constraint of conscious focus. To further specify the executive attention construct, the present experiments sought boundary conditions of the WMC-attention relation. Three experiments correlated individual differences in WMC, as measured by complex span tasks, and executive control of visual search. In feature-absence search, conjunction search, and spatial configuration search, WMC was unrelated to search slopes, although they were large and reliably measured. Even in a search task designed to require the volitional movement of attention (J. M. Wolfe, G. A. Alvarez, & T. S. Horowitz, 2000), WMC was irrelevant to performance. Thus, WMC is not associated with all demanding or controlled attention processes, which poses problems for some general theories of WMC. Copyright 2006 APA, all rights reserved.
14. Interstellar rendezvous missions employing fission propulsion systems
International Nuclear Information System (INIS)
Lenard, Roger X.; Lipinski, Ronald J.
2000-01-01
There has been a conventionally held nostrum that fission system specific power and energy content is insufficient to provide the requisite high accelerations and velocities to enable interstellar rendezvous missions within a reasonable fraction of a human lifetime. As a consequence, all forms of alternative mechanisms that are not yet, and may never be technologically feasible, have been proposed, including laser light sails, fusion and antimatter propulsion systems. In previous efforts, [Lenard and Lipinski, 1999] the authors developed an architecture that employs fission power to propel two different concepts: one, an unmanned probe, the other a crewed vehicle to Alpha Centauri within mission times of 47 to 60 years. The first portion of this paper discusses employing a variant of the ''Forward Resupply Runway'' utilizing fission systems to enable both high accelerations and high final velocities necessary for this type of travel. The authors argue that such an architecture, while expensive, is considerably less expensive and technologically risky than other technologically advanced concepts, and, further, provides the ability to explore near-Earth stellar systems out to distances of 8 light years or so. This enables the ability to establish independent human societies which can later expand the domain of human exploration in roughly eight light-year increments even presuming that no further physics or technology breakthroughs or advances occur. In the second portion of the paper, a technology requirement assessment is performed. The authors argue that reasonable to extensive extensions to known technology could enable this revolutionary capability
15. Exploration
International Nuclear Information System (INIS)
Lohrenz, J.
1992-01-01
Oil and gas exploration is a unique kind of business. Businesses providing a vast and ever-changing panoply of products to markets are a focus of several disciplines' energetic study and analysis. The product inventory problem is robust, pertinent, and meaningful, and it merits the voluminous and protracted attention received from keen business practitioners. Prototypical business practitioners, be they trained by years of business hurly-burly, or sophisticated MBAs with arrays of mathematical algorithms and computers, are not normally prepared, however, to recognize the unique nature of exploration's inventories. Put together such a business practitioner with an explorationist and misunderstandings, hidden and open, are inevitable and predictably rife. The first purpose of this paper is to articulate the inherited inventory handling paradigms of business practitioners in relation to exploration's inventories. To do so, standard pedagogy in business administration is used and a case study of an exploration venture is presented. A second purpose is to show the burdens that the misunderstandings create. The result is not just business plans that go awry, but public policies that have effects opposite from those intended
16. Radio propagation through the turbulent interstellar plasma
International Nuclear Information System (INIS)
Rickett, B.J.
1990-01-01
The current understanding of interstellar scattering is reviewed, and its impact on radio astronomy is examined. The features of interstellar plasma turbulence are also discussed. It is concluded that methods involving the investigation of the flux variability of pulsars and extragalactic sources and the VLBI visibility curves constitute new techniques for probing the ISM. However, scattering causes a seeing limitation in radio observations. It is now clear that variation due to RISS (refractive interstellar scintillations) is likely to be important for several classes of variable sources, especially low-frequency variables and centimeter-wave flickering. 168 refs
17. Physics of the galaxy and interstellar matter
International Nuclear Information System (INIS)
Scheffler, H.; Elsasser, H.
1988-01-01
This book is based on the authors' long standing experience in teaching astronomy courses. It presents in a modern and complete way our present picture of the physics of the Milky Way system. The first part of the book deals with topics of more empirical character, such as the positions and motions of stars, the structure and kinetics of the stellar systems and interstellar phenomena. The more advanced second part is devoted to the interpretation of observational results, i.e. to the physics of interstellar gas and dust, to stellar dynamics, to the theory of spiral structures and the dynamics of interstellar gas
18. Structure and evolution of the interstellar medium
International Nuclear Information System (INIS)
Chieze, J.P.
1985-10-01
We give a two dimensional hydrodynamical analysis of HI clouds collisions in order to determine the mass spectrum of diffuse interstellar clouds. We have taken into account evaporation and abrasion by supernovae blast waves. The conditions for cloud merging or fragmentation are precised. Applications to the model of the interstellar medium of Mc Kee and Ostriker are also discussed. On the other hand, we show that molecular clouds belong to a one parameter family which can be identified to the sequence of the gravitationally unstable states of clouds bounded by the uniform pressure of the coronal phase of the interstellar medium. Hierarchical fragmentation of molecular clouds is analysed in this context [fr
19. Interstellar clouds and the formation of stars
Energy Technology Data Exchange (ETDEWEB)
Alfven, H; Carlqvist, P [Kungliga Tekniska Hoegskolan, Stockholm (Sweden). Institutionen foer Plasmafysik
1978-05-01
Part I gives a survey of the drastic revision of cosmic plasma physics which is precipitated by the exploration of the magnetosphere through in situ measurements. The 'pseudo-plasma formalism', which until now has almost completely dominated theoretical astrophysics, must be replaced by an experimentally based approach involving the introduction of a number of neglected plasma phenomena, such as electric double layers, critical velocity, and pinch effect. The general belief that star light is the main ionizer is shown to be doubtful; hydromagnetic conversion of gravitational and kinetic energy may often be much more important. In Part II the revised plasma physics is applied to dark clouds and star formation. Magnetic fields do not necessarily counteract the contraction of a cloud; they may just as well 'pinch' the cloud. Magnetic compression may be the main mechanism for forming interstellar clouds and keeping them together. Part III treats the formation of stars in a dusty cosmic plasma cloud. Star formation is due to an instability, but it is very unlikely that it has anything to do with the Jeans instability. A reasonable mechanism is that the sedimentation of 'dust' (including solid bodies of different size) is triggering off a gravitationally assisted accretion. A 'stellesimal' accretion analogous to the planetesimal accretion leads to the formation of a star surrounded by a very low density hollow in the cloud. Matter falling in from the cloud towards the star is the raw material for the formation of planets and satellites.
20. Abundances in the diffuse interstellar medium
International Nuclear Information System (INIS)
Harris, A.W.
1988-04-01
The wealth of interstellar absorption line data obtained with the Copernicus and IUE satellites has opened up a new era in studies of the interstellar gas. It is now well established that certain elements, generally those with high condensation temperatures, are substantially under-abundant in the gas-phase relative to total solar or cosmic abundances. This depletion of elements is due to the existence of solid material in the form of dust grains in the interstellar medium. Surprisingly, however, recent surveys indicate that even volatile elements such as Zn and S are significantly depleted in many sight lines. Developments in this field which have been made possible by the large base of UV interstellar absorption line data built up over recent years are reviewed and the implications of the results for our understanding of the physical processes governing depletion are discussed. (author)
1. The composition of circumstellar and interstellar dust
NARCIS (Netherlands)
Tielens, AGGM; Woodward, CE; Biscay, MD; Shull, JM
2001-01-01
A large number of solid dust components have been identified through analysis of stardust recovered from meteorites, and analysis of IR observations of circumstellar shells and the interstellar medium. These include graphite, hydrogenated amorphous carbon, diamond, PAHs, silicon-, iron-, and
2. Experimental interstellar organic chemistry: Preliminary findings
Science.gov (United States)
Khare, B. N.; Sagan, C.
1971-01-01
In a simulation of interstellar organic chemistry in dense interstellar clouds or on grain surfaces, formaldehyde, water vapor, ammonia and ethane are deposited on a quartz cold finger and ultraviolet-irradiated in high vacuum at 77K. The HCHO photolytic pathway which produces an aldehyde radical and a superthermal hydrogen atom initiates solid phase chain reactions leading to a range of new compounds, including methanol, ethanol, acetaldehyde, acetonitrile, acetone, methyl formate, and possibly formic acid. Higher nitriles are anticipated. Genetic relations among these interstellar organic molecules (e.g., the Cannizzaro and Tischenko reactions) must exist. Some of them, rather than being synthesized from smaller molecules, may be degradation products of larger organic molecules, such as hexamethylene tetramine, which are candidate consitituents of the interstellar grains. The experiments reported here may also be relevant to cometary chemistry.
3. Newly detected molecules in dense interstellar clouds
Science.gov (United States)
Irvine, William M.; Avery, L. W.; Friberg, P.; Matthews, H. E.; Ziurys, L. M.
Several new interstellar molecules have been identified including C2S, C3S, C5H, C6H and (probably) HC2CHO in the cold, dark cloud TMC-1; and the discovery of the first interstellar phosphorus-containing molecule, PN, in the Orion "plateau" source. Further results include the observations of 13C3H2 and C3HD, and the first detection of HCOOH (formic acid) in a cold cloud.
4. Carbon chain molecules in interstellar clouds
International Nuclear Information System (INIS)
Winnewisser, G.; Walmsley, C.M.
1979-01-01
A survey of the distribution of long carbon chain molecules in interstellar clouds shows that their abundance is correlated. The various formation schemes for these molecules are discussed. It is concluded that the ion-molecule type formation mechanisms are more promising than their competitors. They have also the advantage of allowing predictions which can be tested by observations. Acetylene C 2 H 2 and diacetylene HCCCCH, may be very abundant in interstellar clouds. (Auth.)
5. Historical Reveiw of Interstellar Probe Concepts and Examination of Payload Mass Considerations for Different System Architectures
Science.gov (United States)
Long, K.
2017-12-01
The ability to send a space probe beyond the Voyager probes, through the interstellar medium and towardsthe distant stars, has long been the ambition of both the science ction literature but also a small community ofadvocates that have argued for a broader and deeper vision of space exploration that goes outside of our SolarSystem. In this paper we discuss some of the historical interstellar probe concepts which are propelled usingdierent types of propulsion technology, from energetic reaction engines to directed energy beaming, and considerthe payload mass associated with such concepts. We compare and contrast the dierent design concepts, payloadmass fractions, powers and energies and discuss the implications for robotic space exploration within the stellarneighbourhood. Finally, we consider the Breakthrough Starshot initiative, which proposes to send a Gram-scalelaser driven spacecraft to the Alpha Centauri system in a 20 year mission travelling at v 0.2c. We show howthis is a good start in pushing our robotic probes towards interstellar destinations, but also discuss the potentialfor scaling up this systems architecture to missions closer at home, or higher mass missions wider aeld. This is apresentation for the American Geophysical Union at the AGU Fall meeting, New Orleans, 11-15 December 2017,Special Session on the Interstellar Probe Missions.Keywords: Interstellar Probe, Breakthrough Starshot
6. Strategic Roadmap for the Development of an Interstellar Space Program
Science.gov (United States)
Gifra, M.; Peeters, W.
Recent technological advances and scientific discoveries, particularly in astronomy and space technology, are opening our minds into the deepest realms of the universe, and also they are bringing a new era of space exploration and development. This sense of entering into a new era of space exploration is being boosted by the permanent discovery of new planets - to date, there are 684 confirmed extrasolar planets [1] - outside our solar system. The possibility that astronomers may soon find a habitable extrasolar planet near Earth and the recent advances in space propulsion that could reduce travel times have stimulated the space community to consider the development of an interstellar manned mission. But this scenario of entering into a new era of space development is ultimately contingent on the outcome of the actual world's economic crisis. The current financial crisis, on top of recent national and sovereign debts problems, could have serious consequences for space exploration and development as the national budgets for space activities are to freeze [2].This paper proposes a multi-decade space program for an interstellar manned mission. It designs a roadmap for the achievement of interstellar flight capability within a timeframe of 40 years, and also considers different scenarios where various technological and economical constraints are taken into account in order to know if such a space endeavour could be viable. It combines macro-level scenarios with a strategic roadmap to provide a framework for condensing all information in one map and timeframe, thus linking decision-making with plausible scenarios. The paper also explores the state of the art of space technologies 20 to 40 years in the future and its potential economic impact. It estimates the funding requirements, possible sources of funds, and the potential returns.The Interstellar Space Program proposed in this paper has the potential to help solve the global crisis by bringing a new landscape of
7. Components in the interstellar medium
International Nuclear Information System (INIS)
Martin, E.R.
1981-01-01
An analysis is made of the lines of sight toward 32 stars with a procedure that gives velocity components for various interstellar ions. The column densities found for species expected to be relatively undepleted are used to estimate the column density of neutral hydrogen in each component. Whenever possible, the molecular hydrogen excitation temperature, abundances (relative to S II), electron density, and hydrogen volume density are calculated for each component. The results for each star are combined to give total HI column density as a function of (LSR) velocity. The derived velocities correspond well with those found in optical studies. The mean electron density is found to be approximately constant with velocity, but the mean hydrogen volume density is found to vary. The data presented here are consistent with the assumption that some of the velocity components are due to circumstellar material. The total HI column density toward a given star is generally in agreement with Lyman alpha measurements, but ionization and abundance effects are important toward some stars. The total HI column density is found to vary exponentially with velocity (for N(HI)> 10 17 cm -2 ), with an indication that the velocity dispersion at low column densities (N(HI) 17 cm -2 ) is approximately constant. An estimate is made of the kinetic energy density due to cloud motion which depends only on the total HI column density as a function of velocity. The value of 9 x 10 42 erg/pc 3 is in good agreement with a theoretical prediction
8. Characterization of Interstellar Organic Molecules
International Nuclear Information System (INIS)
Gencaga, Deniz; Knuth, Kevin H.; Carbon, Duane F.
2008-01-01
Understanding the origins of life has been one of the greatest dreams throughout history. It is now known that star-forming regions contain complex organic molecules, known as Polycyclic Aromatic Hydrocarbons (PAHs), each of which has particular infrared spectral characteristics. By understanding which PAH species are found in specific star-forming regions, we can better understand the biochemistry that takes place in interstellar clouds. Identifying and classifying PAHs is not an easy task: we can only observe a single superposition of PAH spectra at any given astrophysical site, with the PAH species perhaps numbering in the hundreds or even thousands. This is a challenging source separation problem since we have only one observation composed of numerous mixed sources. However, it is made easier with the help of a library of hundreds of PAH spectra. In order to separate PAH molecules from their mixture, we need to identify the specific species and their unique concentrations that would provide the given mixture. We develop a Bayesian approach for this problem where sources are separated from their mixture by Metropolis Hastings algorithm. Separated PAH concentrations are provided with their error bars, illustrating the uncertainties involved in the estimation process. The approach is demonstrated on synthetic spectral mixtures using spectral resolutions from the Infrared Space Observatory (ISO). Performance of the method is tested for different noise levels.
9. The photoevaporation of interstellar clouds
International Nuclear Information System (INIS)
Bertoldi, F.
1989-01-01
The dynamics of the photoevaporation of interstellar clouds and its consequences for the structure and evolution of H II regions are studied. An approximate analytical solution for the evolution of photoevaporating clouds is derived under the realistic assumption of axisymmetry. The effects of magnetic fields are taken into account in an approximate way. The evolution of a neutral cloud subjected to the ionizing radiation of an OB star has two distinct stages. When a cloud is first exposed to the radiation, the increase in pressure due to the ionization at the surface of the cloud leads to a radiation-driven implosion: an ionization front drives a shock into the cloud, ionizes part of it and compresses the remaining into a dense globule. The initial implosion is followed by an equilibrium cometary stage, in which the cloud maintains a semistationary comet-shaped configuration; it slowly evaporates while accelerating away from the ionizing star until the cloud has been completely ionized, reaches the edge of the H II region, or dies. Expressions are derived for the cloud mass-loss rate and acceleration. To investigate the effect of the cloud photoevaporation on the structure of H II regions, the evolution of an ensemble of clouds of a given mass distribution is studied. It is shown that the compressive effect of the ionizing radiation can induce star formation in clouds that were initially gravitationally stable, both for thermally and magnetically supported clouds
10. The interstellar medium in galaxies
CERN Document Server
1997-01-01
It has been more than five decades ago that Henk van de Hulst predicted the observability of the 21-cm line of neutral hydrogen (HI ). Since then use of the 21-cm line has greatly improved our knowledge in many fields and has been used for galactic structure studies, studies of the interstellar medium (ISM) in the Milky Way and other galaxies, studies of the mass distribution of the Milky Way and other galaxies, studies of spiral struc ture, studies of high velocity gas in the Milky Way and other galaxies, for measuring distances using the Tully-Fisher relation etc. Regarding studies of the ISM, there have been a number of instrumen tal developments over the past decade: large CCD's became available on optical telescopes, radio synthesis offered sensitive imaging capabilities, not only in the classical 21-cm HI line but also in the mm-transitions of CO and other molecules, and X-ray imaging capabilities became available to measure the hot component of the ISM. These developments meant that Milky Way was n...
11. Wavelength dependence of interstellar polarization
International Nuclear Information System (INIS)
Mavko, G.E.
1974-01-01
The wavelength dependence of interstellar polarization was measured for twelve stars in three regions of the Milky Way. A 120A bandpass was used to measure the polarization at a maximum of sixteen wavelengths evenly spaced between 2.78μ -1 (3600A) and 1.28μ -1 (7800A). For such a wide wavelength range, the wavelength resolution is superior to that of any previously reported polarization measurements. The new scanning polarimeter built by W. A. Hiltner of the University of Michigan was used for the observations. Very broad structure was found in the wavelength dependence of the polarization. Extensive investigations were carried out to show that the structure was not caused by instrumental effects. The broad structure observed is shown to be in agreement with concurrent extinction measurements for the same stars. Also, the observed structure is of the type predicted when a homogeneous silicate grain model is fitted to the observed extinction. The results are in agreement with the hypothesis that the very broad band structure seen in the extinction is produced by the grains. (Diss. Abstr. Int., B)
12. Comprehensive models of diffuse interstellar clouds : physical conditions and molecular abundances
NARCIS (Netherlands)
Dishoeck, van E.F.; Black, J.H.
1986-01-01
The limitations of steady state models of interstellar clouds are explored by means of comparison with observational data corresponding to clouds in front of Zeta Per, Zeta Oph, Chi Oph, and Omicron Per. The improved cloud models were constructed to reproduce the observed H and H2(J) column
13. Chemistry and photophysics of polycyclic aromatic hydrocarbons in the interstellar medium
NARCIS (Netherlands)
Boschman, Leon
2017-01-01
Polycyclic aromatic hydrocarbons (PAHs) are ubiquitous in the interstellar medium, and it is thought that they are a key factor in the formation of molecular hydrogen at high gas and dust grain temperatures. We have explored how PAHs can contribute to the formation of H2 by taking a small PAH
14. Organic chemistry and biology of the interstellar medium
Science.gov (United States)
Sagan, C.
1973-01-01
Interstellar organic chemistry is discussed as the field of study emerging from the discovery of microwave lines of formaldehyde and of hydrogen cyanide in the interstellar medium. The reliability of molecular identifications and comparisons of interstellar and cometary compounds are considered, along with the degradational origin of simple organics. It is pointed out that the contribution of interstellar organic chemistry to problems in biology is not substantive but analogical. The interstellar medium reveals the operation of chemical processes which, on earth and perhaps on vast numbers of planets throughout the universe, led to the origin of life, but the actual molecules of the interstellar medium are unlikely to play any significant biological role.
15. INTERSTELLAR NEUTRAL ATOMS AT 1 AU OBSERVED BY THE IMAGE/LENA IMAGER
International Nuclear Information System (INIS)
Fuselier, S. A.; Ghielmetti, A. G.; Wurz, P.
2009-01-01
Observations from the Imager for Magnetopause to Aurora: Global Exploration (IMAGE) Low Energy Neutral Atom (LENA) imager from 2005 are used to investigate characteristics of interstellar neutrals in the inner solar system. The LENA imager detected an interstellar neutral signal starting in 2004 December and extending to early 2005 April. Using the orientation of the field of view of the imager and the date of the loss of the interstellar neutral signal, it is concluded that the signal is consistent with a relatively compact (several degrees wide in ecliptic latitude and longitude) source of neutral helium and/or energetic (>150 eV) hydrogen originating from the solar apex direction. Observations later in 2005 are used to distinguish the composition and conclude that the relatively compact source likely contains some energetic hydrogen (in addition to the helium).
16. A theoretical quantum chemical study of alanine formation in interstellar medium
Science.gov (United States)
Shivani; Pandey, Parmanad; Misra, Alka; Tandon, Poonam
2017-08-01
The interstellar medium, the vast space between the stars, is a rich reservoir of molecular material ranging from simple diatomic molecules to more complex, astrobiologically important molecules such as amino acids, nucleobases, and other organic species. Radical-radical and radical-neutral interaction schemes are very important for the formation of comparatively complex molecules in low temperature chemistry. An attempt has been made to explore the possibility of formation of complex organic molecules in interstellar medium, through detected interstellar molecules like CH3CN and HCOOH. The gas phase reactions are theoretically studied using quantum chemical techniques. We used the density functional theory (DFT) at the B3LYP/6-311G( d, p) level. The reaction energies, potential barrier and optimized structures of all the geometries, involved in the reaction path, has been discussed. We report the potential energy surfaces for the reactions considered in this work.
17. Photodissociation and excitation of interstellar molecules
International Nuclear Information System (INIS)
Dishoeck, E.F. van.
1984-01-01
Apart from a rather long introduction containing some elementary astrophysics, quantum chemistry and spectroscopy and an incomplete, historical review of molecular observations, this thesis is divided into three sections. In part A, a rigorous quantum chemical and dynamical study is made of the photodissociation processes in the OH and HCl molecules. In part B, the cross sections obtained in part A are used in various astrophysical problems such as the study of the abundances of the OH and HCl molecules in interstellar clouds, the use of the OH abundance as a measure of the cosmic ray ionization rate, the lifetime of the OH radical in comets and the abundance of OH in the solar photosphere. Part C discusses the excitation of the C 2 molecule under interstellar conditions, its use as a diagnostic probe of the temperature, density and strength of the radiation field in interstellar clouds. Quadrupole moments and oscillator strengths are analyzed. (Auth.)
18. On the nature of interstellar turbulence
International Nuclear Information System (INIS)
Altunin, V.I.
1981-01-01
Possible reasons of interstellar medium turbulence manifested in pulsar scintillation and radio-frequency emission scattering of extragalactic sources near by the Galaxy plane, are discussed. Sources and conditions of turbulence emergence in HII region shells, supernova, residue and in stellar wind giving observed scattering effects are considered. It is shown that in the formation of the interstellar scintillation pattern of discrete radio-frequency emission sources a certain role can be played by magnetosound turbulence, which arises due to shock-waves propagating in the interstellar medium at a velocity Vsub(sh) approximately 20-100 km/s as well as by stellar-wind inhomogeneity of OB classes stars [ru
19. Physics of the interstellar and intergalactic medium
CERN Document Server
Draine, Bruce T
2010-01-01
This is a comprehensive and richly illustrated textbook on the astrophysics of the interstellar and intergalactic medium--the gas and dust, as well as the electromagnetic radiation, cosmic rays, and magnetic and gravitational fields, present between the stars in a galaxy and also between galaxies themselves. Topics include radiative processes across the electromagnetic spectrum; radiative transfer; ionization; heating and cooling; astrochemistry; interstellar dust; fluid dynamics, including ionization fronts and shock waves; cosmic rays; distribution and evolution of the interstellar medium; and star formation. While it is assumed that the reader has a background in undergraduate-level physics, including some prior exposure to atomic and molecular physics, statistical mechanics, and electromagnetism, the first six chapters of the book include a review of the basic physics that is used in later chapters. This graduate-level textbook includes references for further reading, and serves as an invaluable resourc...
20. Investigating nearby exoplanets via interstellar radar
Science.gov (United States)
Scheffer, Louis K.
2014-01-01
Interstellar radar is a potential intermediate step between passive observation of exoplanets and interstellar exploratory missions. Compared with passive observation, it has the traditional advantages of radar astronomy. It can measure surface characteristics, determine spin rates and axes, provide extremely accurate ranges, construct maps of planets, distinguish liquid from solid surfaces, find rings and moons, and penetrate clouds. It can do this even for planets close to the parent star. Compared with interstellar travel or probes, it also offers significant advantages. The technology required to build such a radar already exists, radar can return results within a human lifetime, and a single facility can investigate thousands of planetary systems. The cost, although too high for current implementation, is within the reach of Earth's economy.
1. Experiments on chemical and physical evolution of interstellar grain mantles
International Nuclear Information System (INIS)
Greenberg, J.M.
1984-01-01
The Astrophysical Laboratory at the University of Leiden is the first to succeed in simulating the essential conditions in interstellar space as they affect the evolution of interstellar grains. (author)
2. Energetic neutral atom and interstellar flow observations with IBEX: Implications for the global heliosphere
Energy Technology Data Exchange (ETDEWEB)
Schwadron, N. A., E-mail: [email protected] [University of New Hampshire, Durham NH, 03824 (United States); Southwest Research Institute, San Antonio, TX, 78238 (United States); McComas, D. J.; Desai, M. I.; Fuselier, S. A. [Southwest Research Institute, San Antonio, TX, 78238 (United States); University of Texas, San Antonio, TX, 78249 (United States); Christian, E. R. [Goddard Space Flight Center, Greenbelt, MD 20771 (United States); Funsten, H. O. [Los Alamos National Laboratory, Los Alamos, NM 87545 (United States); Moebius, E. [University of New Hampshire, Durham NH, 03824 (United States); Reno, M.; Scherrer, J.; Zirnstein, E. [Southwest Research Institute, San Antonio, TX, 78238 (United States)
2016-03-25
Since launch in Oct. 2008, IBEX, with its two energetic neutral atom (ENA) cameras, has provided humankind with the first-ever global images of the complex boundary separating the heliosphere from the local interstellar medium (LISM). IBEX’s energy-resolved all-sky maps, collected every six months, are yielding remarkable new insights into the heliospheres structure as it is shaped by the combined forces of the local interstellar flow, the local interstellar magnetic field (LISMF), and the evolving solar wind. IBEX has also acquired the first images of ENAs backscattered from the surface of the moon as well as global images of the magnetospheric response to solar wind disturbances. IBEX thus addresses all three Heliophysics science objectives set forth in the 2014 Science Plan for NASAs Science Mission Directorate (SMD) as well as the goals in the recent Solar and Space Physics Decadal Survey (NRC 2012). In addition, with the information it provides on the properties of the LISM and the LISMF, IBEX represents a unique bridge between heliophysics and astrophysics, and fills in critical knowledge for understanding the habitability of exoplanetary systems and the future habitability of Earth and the solar system. Because of the few-year time lag due to solar wind and ENA transport, IBEX observed the solar wind/ LISM interaction characteristic of declining phase/solar minimum conditions. In the continuing mission, IBEX captures the response of the interstellar boundaries to the changing structure of the solar wind in its transition toward the “mini” solar maximum and possibly the decline into the next solar minimum. The continuing IBEX mission affords never-to-be-repeated opportunities to coordinate global imaging of the heliospheric boundary with in-situ measurements by the Voyagers as they pass beyond the heliopause and start to directly sample the LISM.
3. Surface chemistry on interstellar oxide grains
International Nuclear Information System (INIS)
Denison, P.; Williams, D.A.
1981-01-01
Detailed calculations are made to test the predictions of Duley, Millar and Williams (1978) concerning the chemical reactivity of interstellar oxide grains. A method is established for calculating interaction energies between atoms and the perfect crystal with or without surface vacancy sites. The possibility of reactions between incident atoms and absorbed atoms is investigated. It is concluded that H 2 formation can occur on the perfect crystal surfaces, and that for other diatomic molecules the important formation sites are the Fsub(s)- and V 2- sub(s)-centres. The outline by Duley, Millar and Williams (1979) of interstellar oxide grain growth and destruction is justified by these calculations. (author)
4. Rocket and satellite observations of the local interstellar medium
International Nuclear Information System (INIS)
Jelinsky, P.N.
1988-01-01
The purpose of the study described in this thesis was to obtained new information on the structure of the local interstellar medium (ISM). Two separate experiments using different instruments were used in this study. The first experiment employed a spectrometer with a spectral bandpass from 350-1150 angstrom which was placed at the focus of a 95 cm, f/2.8 normal incidence telescope flown on an Aries sounding rocket. The purpose of this experiment was to measure the interstellar absorption edges, due to neutral helium and neutral hydrogen, in the spectrum of a hot white dwarf. The hot white dwarf G191-B2B was observed for 87 seconds during the flight. Unfortunately, due to high pressure in the rocket, no scientifically useful data was obtained during the flight. The second experiment utilized the high resolution spectrometer on the International Ultraviolet Explorer satellite. The purpose of the experiment was to observe interstellar absorption lines in the spectrum of hot white dwarfs. A new method of determining the equivalent widths of absorption lines and their uncertainties was developed. The neutral hydrogen column density is estimated from the N I, Si II, and C II columns. Unfortunately, the uncertainties in the neutral hydrogen columns are very large, only two are constrained to better than an order of magnitude. High ionization species (N V, Si IV, and C IV) are seen in five of the stars. Upper limits to the temperature of the ISM are determined from the velocity dispersions. The temperature of the low ionization gas toward four of the stars is constrained to be less than 50,000 K
5. Exploring the role of wave drag in the stable stratified oceanic and atmospheric bottom boundary layer in the cnrs-toulouse (cnrm-game) large stratified water flume
NARCIS (Netherlands)
Kleczek, M.; Steeneveld, G.J.; Paci, A.; Calmer, R.; Belleudy, A.; Canonici, J.C.; Murguet, F.; Valette, V.
2014-01-01
This paper reports on a laboratory experiment in the CNRM-GAME (Toulouse) stratified water flume of a stably stratified boundary layer, in order to quantify the momentum transfer due to orographically induced gravity waves by gently undulating hills in a boundary layer flow. In a stratified fluid, a
6. Interstellar propagation of low energy cosmic rays
International Nuclear Information System (INIS)
Cesarsky, C.J.
1975-01-01
Wave particles interactions prevent low energy cosmic rays from propagating at velocities much faster than the Alfven velocity, reducing their range by a factor of order 50. Therefore, supernovae remnants cannot fill the neutral portions of the interstellar medium with 2 MeV cosmic rays [fr
7. SILICATE COMPOSITION OF THE INTERSTELLAR MEDIUM
Energy Technology Data Exchange (ETDEWEB)
Fogerty, S.; Forrest, W.; Watson, D. M.; Koch, I. [Department of Physics and Astronomy, University of Rochester, Rochester, NY 14627 (United States); Sargent, B. A., E-mail: [email protected] [Center for Imaging Science and Laboratory for Multiwavelength Astrophysics, Rochester Institute of Technology, 54 Lomb Memorial Drive, Rochester, NY 14623 (United States)
2016-10-20
The composition of silicate dust in the diffuse interstellar medium and in protoplanetary disks around young stars informs our understanding of the processing and evolution of the dust grains leading up to planet formation. An analysis of the well-known 9.7 μ m feature indicates that small amorphous silicate grains represent a significant fraction of interstellar dust and are also major components of protoplanetary disks. However, this feature is typically modeled assuming amorphous silicate dust of olivine and pyroxene stoichiometries. Here, we analyze interstellar dust with models of silicate dust that include non-stoichiometric amorphous silicate grains. Modeling the optical depth along lines of sight toward the extinguished objects Cyg OB2 No. 12 and ζ Ophiuchi, we find evidence for interstellar amorphous silicate dust with stoichiometry intermediate between olivine and pyroxene, which we simply refer to as “polivene.” Finally, we compare these results to models of silicate emission from the Trapezium and protoplanetary disks in Taurus.
8. Fluorescent excitation of interstellar H2
NARCIS (Netherlands)
Black, J.H.; Dishoeck, van E.F.
1987-01-01
The infrared emission spectrum of H2 excited by ultraviolet absorption, followed by fluorescence, was investigated using comprehensive models of interstellar clouds for computing the spectrum and to assess the effects on the intensity to various cloud properties, such as density, size, temperature,
9. Organics in meteorites - Solar or interstellar?
Science.gov (United States)
Alexander, Conel M. O'D.; Cody, George D.; Fogel, Marilyn; Yabuta, Hikaru
2008-10-01
The insoluble organic material (IOM) in primitive meteorites is related to the organic material in interplanetary dust particles and comets, and is probably related to the refractory organic material in the diffuse interstellar medium. If the IOM is representative of refractory ISM organics, models for how and from what it formed will have to be revised.
10. Optical observations of nearby interstellar gas
Science.gov (United States)
Frisch, P. C.; York, D. G.
1984-11-01
Observations indicated that a cloud with a heliocentric velocity of approximately -28 km/s and a hydrogen column density that possibly could be on the order of, or greater than, 5 x 10 to the 19 power/square cm is located within the nearest 50 to 80 parsecs in the direction of Ophiuchus. This is a surprisingly large column density of material for this distance range. The patchy nature of the absorption from the cloud indicates that it may not be a feature with uniform properties, but rather one with small scale structure which includes local enhancements in the column density. This cloud is probably associated with the interstellar cloud at about the same velocity in front of the 20 parsec distant star alpha Oph (Frisch 1981, Crutcher 1982), and the weak interstellar polarization found in stars as near as 35 parsecs in this general region (Tinbergen 1982). These data also indicate that some portion of the -14 km/s cloud also must lie within the 100 parsec region. Similar observations of both Na1 and Ca2 interstellar absorption features were performed in other lines of sight. Similar interstellar absorption features were found in a dozen stars between 20 and 100 parsecs of the Sun.
11. SILICATE COMPOSITION OF THE INTERSTELLAR MEDIUM
International Nuclear Information System (INIS)
Fogerty, S.; Forrest, W.; Watson, D. M.; Koch, I.; Sargent, B. A.
2016-01-01
The composition of silicate dust in the diffuse interstellar medium and in protoplanetary disks around young stars informs our understanding of the processing and evolution of the dust grains leading up to planet formation. An analysis of the well-known 9.7 μ m feature indicates that small amorphous silicate grains represent a significant fraction of interstellar dust and are also major components of protoplanetary disks. However, this feature is typically modeled assuming amorphous silicate dust of olivine and pyroxene stoichiometries. Here, we analyze interstellar dust with models of silicate dust that include non-stoichiometric amorphous silicate grains. Modeling the optical depth along lines of sight toward the extinguished objects Cyg OB2 No. 12 and ζ Ophiuchi, we find evidence for interstellar amorphous silicate dust with stoichiometry intermediate between olivine and pyroxene, which we simply refer to as “polivene.” Finally, we compare these results to models of silicate emission from the Trapezium and protoplanetary disks in Taurus.
12. Interstellar Extinction in the Gaia Photometric Systems
Directory of Open Access Journals (Sweden)
Bridžius A.
2003-12-01
Full Text Available Three medium-band photometric systems proposed for the Gaia space mission are intercompared in determining color excesses for stars of spectral classes from O to M at V = 18 mag. A possibility of obtaining a three-dimensional map of the interstellar extinction is discussed.
13. MEASURING THE FRACTAL STRUCTURE OF INTERSTELLAR CLOUDS
NARCIS (Netherlands)
VOGELAAR, MGR; WAKKER, BP; SCHWARZ, UJ
1991-01-01
To study the structure of interstellar clouds we used the so-called perimeter-area relation to estimate fractal dimensions. We studied the reliability of the method by applying it to artificial fractals and discuss some of the problems and pitfalls. Results for two different cloud types
14. Modeling Shocks Detected by Voyager 1 in the Local Interstellar Medium
Energy Technology Data Exchange (ETDEWEB)
Kim, T. K.; Pogorelov, N. V. [Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, AL 35805 (United States); Burlaga, L. F. [NASA Goddard Space Flight Center, Code 673, Greenbelt, MD 20771 (United States)
2017-07-10
The magnetometer (MAG) on Voyager 1 ( V1 ) has been sampling the interstellar magnetic field (ISMF) since 2012 August. The V1 MAG observations have shown draped ISMF in the very local interstellar medium disturbed occasionally by significant enhancements in magnetic field strength. Using a three-dimensional, data-driven, multi-fluid model, we investigated these magnetic field enhancements beyond the heliopause that are supposedly associated with solar transients. To introduce time-dependent effects at the inner boundary at 1 au, we used daily averages of the solar wind parameters from the OMNI data set. The model ISMF strength, direction, and proton number density are compared with V1 data beyond the heliopause. The model reproduced the large-scale fluctuations between 2012.652 and 2016.652, including major events around 2012.9 and 2014.6. The model also predicts shocks arriving at V1 around 2017.395 and 2019.502. Another model driven by OMNI data with interplanetary coronal mass ejections (ICMEs) removed at the inner boundary suggests that ICMEs may play a significant role in the propagation of shocks into the interstellar medium.
15. RUSTY OLD STARS: A SOURCE OF THE MISSING INTERSTELLAR IRON?
International Nuclear Information System (INIS)
McDonald, I.; Zijlstra, A. A.; Markwick, A. J.; Sloan, G. C.; Bernard-Salas, J.; Matsunaga, N.; Matsuura, M.; Kraemer, K. E.
2010-01-01
Iron, the universe's most abundant refractory element, is highly depleted in both circumstellar and interstellar environments, meaning it exists in solid form. The nature of this solid is unknown. In this Letter, we provide evidence that metallic iron grains are present around oxygen-rich asymptotic giant branch stars, where it is observationally manifest as a featureless mid-infrared excess. This identification is made using Spitzer Space Telescope observations of evolved globular cluster stars, where iron dust production appears ubiquitous and in some cases can be modeled as the only observed dust product. In this context, FeO is examined as the likely carrier for the 20 μm feature observed in some of these stars. Metallic iron appears to be an important part of the dust condensation sequence at low metallicity, and subsequently plays an influential role in the interstellar medium. We explore the stellar metallicities and luminosities at which iron formation is observed, and how the presence of iron affects the outflow and its chemistry. The conditions under which iron can provide sufficient opacity to drive a wind remain unclear.
16. Dust in the Diffuse Neutral Interstellar Medium
Science.gov (United States)
Sofia, Ulysses J.
2008-05-01
Studies of interstellar dust have always relied heavily upon Laboratory Astrophysics for interpretation. Laboratory values, in the broad sense that includes theory, are needed for the most basic act of measuring interstellar abundances, to the more complex determination of what grains are responsible for particular extinction. The symbiotic relationship between astronomical observations and Laboratory Astrophysics has prompted both fields to move forward, especially in the era of high-resolution ultraviolet spectroscopy when new elemental species could be interpreted and observations were able to show the limits of laboratory determinations. Thanks to this synergy, we currently have a good idea of the quantity of the most abundant elements incorporated into dust in diffuse neutral interstellar clouds: carbon, oxygen, iron, silicon and magnesium. Now the task is to figure out how, chemically and physically, those elements are integrated into interstellar grains. We can do this by comparing extinction curves to grain populations in radiative transfer models. The limitation at the present time is the availability of optical constants in the infrared through ultraviolet for species that are likely to exist in dust, i.e., those that are easy to form in the physical environments around stars and in molecular clouds. Extinction in some lines of sight can be fit within current abundance limits and with the optical constants that are available. However the inability to reproduce other extinction curves suggests that optical constants can be improved, either in quality for compounds that have been measured, or quantity in the sense of providing data for more materials. This talk will address the current state and the future of dust studies in the diffuse neutral interstellar medium. This work is supported by the grant HST-AR-10979.01-A from the Space Telescope Science Institute to Whitman College.
17. Boundary issues
Science.gov (United States)
Townsend, Alan R.; Porder, Stephen
2011-03-01
the one for oceans. Encouragingly, while they argue that we've already crossed one key boundary in the P cycle, they also suggest it's not a Rubicon moment. The inefficiencies in P use that motivate these boundary debates are also clear targets for improvement, and some world regions may be on a trajectory towards greater P use efficiency (Vitousek et al 2009). This is a critical step for society, because even absent concerns over freshwater eutrophication or marine anoxic events, accelerating rates of P mining and inefficiencies in agricultural P use would still pose very real threats. There is legitimate debate over when readily accessible P reserves may run out (Cordell et al 2009, Van Kauenbergh 2010), but nobody argues with their finite nature. Sooner or later, we will be forced to keep P out of our waterways, if only because we will have to keep it on our farms. Without such a shift, we may face severe P constraints to food security within just a few human generations. As current P reserves decline, rising economic values of low concentration P stores may catalyze their harvest, but without considerable policy interventions, that price hike would exacerbate already strong global inequities in the distribution and use of chemical fertilizers (Sanchez and Swaminathan 2005). The harvest of low concentration P reserves would also create substantial collateral damage to the surrounding environment. Furthermore, even without exhaustion of high-concentration P reserves, their location in only a few countries creates geopolitical risks from the demand for an increasingly valuable resource (Cordell et al 2009). Policies aimed at lowering P inputs to aquatic environments will not only reduce the eutrophication risks explored by Carpenter and Bennett, they will increase P retention in agricultural landscapes and slow the decline of finite P reserves. Shifts in human diets can also make a profound difference in the amount of P (and N) required to meet caloric needs. Society
18. EXPLORING THE INTERSTELLAR MEDIA OF OPTICALLY COMPACT DWARF GALAXIES
Energy Technology Data Exchange (ETDEWEB)
Most, Hans P.; Cannon, John M.; Engstrom, Eric; Fliss, Palmer [Department of Physics and Astronomy, Macalester College, 1600 Grand Avenue, Saint Paul, MN 55105 (United States); Salzer, John J. [Department of Astronomy, Indiana University, 727 East Third Street, Bloomington, IN 47405 (United States); Rosenberg, Jessica L., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [School of Physics, Astronomy, and Computational Science, George Mason University, Fairfax, VA 22030 (United States)
2013-06-15
We present new Very Large Array H I spectral line, archival Sloan Digital Sky Survey, and archival Spitzer Space Telescope imaging of eight star-forming blue compact dwarf galaxies that were selected to be optically compact (optical radii <1 kpc). These systems have faint blue absolute magnitudes (M{sub B} {approx}> -17), ongoing star formation (based on emission-line selection by the H{alpha} or [O III] lines), and are nearby (mean velocity = 3315 km s{sup -1} {approx_equal} 45 Mpc). One galaxy in the sample, ADBS 113845+2008, is found to have an H I halo that extends 58 r-band scale lengths from its stellar body. In contrast, the rest of the sample galaxies have H I radii to optical-scale-length ratios ranging from 9.3 to 26. The size of the H I disk in the 'giant disk' dwarf galaxy ADBS 113845+2008 appears to be unusual as compared with similarly compact stellar populations.
19. THE INTERSTELLAR MEDIUM IN THE KEPLER SEARCH VOLUME
Energy Technology Data Exchange (ETDEWEB)
Johnson, Marshall C. [Department of Astronomy, University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712 (United States); Redfield, Seth [Astronomy Department, Van Vleck Observatory, Wesleyan University, Middletown, CT 06459 (United States); Jensen, Adam G., E-mail: [email protected] [Department of Physics and Physical Science, University of Nebraska-Kearney, Bruner Hall of Science, 2401 11th Ave, Kearney, NE 68849 (United States)
2015-07-10
The properties of the interstellar medium (ISM) surrounding a planetary system can impact planetary climate through a number of mechanisms, including changing the size of the astrosphere (one of the major shields for cosmic rays) as well as direct deposition of material into planetary atmospheres. In order to constrain the ambient ISM conditions for exoplanetary systems, we present observations of interstellar Na i and K i absorption toward seventeen early type stars in the Kepler prime mission field of view (FOV). We identify 39 Na i and 8 K i velocity components, and attribute these to 11 ISM clouds. Six of these are detected toward more than one star, and for these clouds we put limits on the cloud properties, including distance and hydrogen number density. We identify one cloud with significant (≳1.5 cm{sup −3}) hydrogen number density located within the nominal ∼100 pc boundary of the Local Bubble. We identify systems with confirmed planets within the Kepler FOV that could lie within these ISM clouds, and estimate upper limits on the astrosphere sizes of these systems under the assumption that they do lie within these clouds. Under this condition, the Kepler-20, 42, and 445 multiplanet systems could have compressed astrospheres much smaller than the present-day heliosphere. Among the known habitable zone planet hosts, Kepler-186 could have an astrosphere somewhat smaller than the heliosphere, while Kepler-437 and KOI-4427 could have astrospheres much larger than the heliosphere. The thick disk star Kepler-444 may have an astrosphere just a few AU in radius.
20. Detection of organic matter in interstellar grains.
Science.gov (United States)
Pendleton, Y J
1997-06-01
Star formation and the subsequent evolution of planetary systems occurs in dense molecular clouds, which are comprised, in part, of interstellar dust grains gathered from the diffuse interstellar medium (DISM). Radio observations of the interstellar medium reveal the presence of organic molecules in the gas phase and infrared observational studies provide details concerning the solid-state features in dust grains. In particular, a series of absorption bands have been observed near 3.4 microns (approximately 2940 cm-1) towards bright infrared objects which are seen through large column densities of interstellar dust. Comparisons of organic residues, produced under a variety of laboratory conditions, to the diffuse interstellar medium observations have shown that aliphatic hydrocarbon grains are responsible for the spectral absorption features observed near 3.4 microns (approximately 2940 cm-1). These hydrocarbons appear to carry the -CH2- and -CH3 functional groups in the abundance ratio CH2/CH3 approximately 2.5, and the amount of carbon tied up in this component is greater than 4% of the cosmic carbon available. On a galactic scale, the strength of the 3.4 microns band does not scale linearly with visual extinction, but instead increases more rapidly for objects near the Galactic Center. A similar trend is noted in the strength of the Si-O absorption band near 9.7 microns. The similar behavior of the C-H and Si-O stretching bands suggests that these two components may be coupled, perhaps in the form of grains with silicate cores and refractory organic mantles. The ubiquity of the hydrocarbon features seen in the near infrared near 3.4 microns throughout out Galaxy and in other galaxies demonstrates the widespread availability of such material for incorporation into the many newly forming planetary systems. The similarity of the 3.4 microns features in any organic material with aliphatic hydrocarbons underscores the need for complete astronomical observational
1. Organic Synthesis in Simulated Interstellar Ice Analogs
Science.gov (United States)
Dworkin, Jason P.; Bernstein, Max P.; Sandford, Scott A.; Allamandola, Louis J.; Deamer, David W.; Elsila, Jamie; Zare, Richard N.
2001-01-01
Comets and carbonaceous micrometeorites may have been significant sources of organic compounds on the early Earth. Ices on grains in interstellar dense molecular clouds contain a variety of simple molecules as well as aromatic molecules of various sizes. While in these clouds the icy grains are processed by ultraviolet light and cosmic radiation which produces more complex organic molecules. We have run laboratory simulations to identify the types of molecules which could have been generated photolytically in pre-cometary ices. Experiments were conducted by forming various realistic interstellar mixed-molecular ices with and without polycyclic aromatic hydrocarbons (PAHs) at approx. 10 K under high vacuum irradiated with UV light from a hydrogen plasma lamp. The residue that remained after warming to room temperature was analyzed by HPLC, and by laser desorption mass spectrometry. The residue contains several classes of compounds which may be of prebiotic significance.
2. Diffuse interstellar gas in disk galaxies
International Nuclear Information System (INIS)
1989-01-01
The physical properties of the diffuse gas in our Galaxy are reviewed and considered as a starting point for interstellar (IS) studies of disk galaxies. Attention is focussed on the atomic and ionic component, detected through radio, optical, ultraviolet (UV) and X-ray observations. The cooling and heating processes in the IS gas are briefly recalled in order to introduce current models of disk and halo gas. Observations of nearby galaxies critical to test IS models are considered, including 21-cm surveys, optical and UV absorptions of bright, extragalactic sources, and X-ray emission from hot halos. Finally, further steps necessary to develop a global model for the structure and evolution of the interstellar medium are indicated. (author)
3. Glaciations and dense interstellar clouds; and reply
Energy Technology Data Exchange (ETDEWEB)
McCrea, W H [Sussex Univ., Brighton (UK); Dennison, B; Mansfield, V N
1976-09-16
Reference is made to Dennison and Mansfield (Nature 261:32 (1976)) who offered comments on a previous paper by the author (Nature 255:607 (1975)), in which he suggested that a possible cause of an ice age on the Earth was the passage of the solar system through an interstellar matter compression region bordering a spiral arm of the Galaxy. Dennison and Mansfield criticised this suggestion because it led them to expect to find a dense cloud of interstellar matter still very close to the Earth, whereas no such cloud is known. It is stated here that this criticism ignores the structure of the Galaxy, that provided the basis of the suggestion. A reply by Dennison and Mansfield is appended.
4. Fast Neutral reactions in cold interstellar clouds
International Nuclear Information System (INIS)
Graff, M.M.
1989-01-01
The dynamics of exothermic neutral reactions between radical species have been examined, with particular attention to reactivity at the very low energies characteristic of cold interstellar clouds. Long-range interactions (electrostatic and spin-orbit) were considered within in the adiabatic capture-infinite order sudden approximation (ACIOSA). Analytic expressions have been developed for cross sections and rate constants of exothermic reactions between atoms and dipolar radicals at low temperatures. A method for approximating the adiabatic potential surface for the reactive state will be presented. The reaction systems O+OH and O+CH are both predicted to be fast at low temperatures. The systems C+CH and C+OH are expected to be nonreactive at low temperatures, and upper limits of rate constants for these reactions have been estimated. General predictions are made for other reaction systems. Implications for interstellar chemistry will be discussed
5. Identification of interstellar polysaccharides and related hydrocarbons
International Nuclear Information System (INIS)
Hoyle, F.; Olavesen, A.H.; Wickramasinghe, N.C.
1978-01-01
A discussion is presented on the infrared transmittance spectra of several polysaccharides that may be of interest as possible interstellar candidates. It is stated that a 2.5 to 15 μm spectrum computed from the author's measurements is remarkably close to that required to explain a wide range of astronomical data, except for two points. First the required relative opacity at the 3 μm absorption dip is a factor of about 1.5 lower than was found in laboratory measurements; this difference may arise from the presence of water in terrestrial polysaccharide samples. Secondly, in the 9.5 to 12 μm waveband an additional source of opacity appears to be necessary. Close agreement between the spectrum of this additional opacity and the absorption spectrum of propene, C 3 H 6 , points strongly to the presence of hydrocarbons of this type, which may be associated with polysaccharide grains in interstellar space. (U.K.)
6. Polarization of submillimetre lines from interstellar medium
Science.gov (United States)
Zhang, Heshou; Yan, Huirong
2018-04-01
Magnetic fields play important roles in many astrophysical processes. However, there is no universal diagnostic for the magnetic fields in the interstellar medium (ISM) and each magnetic tracer has its limitation. Any new detection method is thus valuable. Theoretical studies have shown that submillimetre fine-structure lines are polarized due to atomic alignment by ultraviolet photon-excitation, which opens up a new avenue to probe interstellar magnetic fields. We will, for the first time, perform synthetic observations on the simulated three-dimensional ISM to demonstrate the measurability of the polarization of submillimetre atomic lines. The maximum polarization for different absorption and emission lines expected from various sources, including star-forming regions are provided. Our results demonstrate that the polarization of submillimetre atomic lines is a powerful magnetic tracer and add great value to the observational studies of the submilimetre astronomy.
7. Absorption and emission characteristics of interstellar dust
International Nuclear Information System (INIS)
Allamandola, L.J.
1984-01-01
Molecular transitions which occur in the middle infrared region of the spectrum correspond with the characteristic frequencies of molecular vibrations. Thus, moderate resolution spectroscopy of the interstellar medium offers unique evidence about the molecules in the condensed and gaseous phases and their distribution. The author discusses the spectral properties of the condensed phase. However, in the astrophysical literature, it is difficult to find a qualitative description of the effects the solid state has on molecular vibrations, and since it is these which largely determine the spectroscopic properties of the interstellar dust, this discussion begins with a general description of these effects and then is directed toward describing the optical characteristics of the molecular ice component of the dust. The properties of this component of the dust are stressed, rather than those expected from more homogeneous components such as silicates, graphite, or amorphous carbon since these have been discussed in considerable detail elsewhere. (Auth.)
8. CN radical in diffuse interstellar clouds
International Nuclear Information System (INIS)
Federman, S.R.; Danks, A.C.; Lambert, D.L.
1984-01-01
A survey of 15 lines of sight for the CN B 2 Σ + --X 2 Σ + interstellar absorption lines shows that the CN column density in diffuse interstellar clouds follows the relation log N(CN)proportionalm log N(H 2 ), where mroughly-equal3. This result is reproduced by a reaction network in which CN is produced primarily from C 2 by the neutral-neutral reaction C 2 +N → CN+C, and photodissociation is the main destruction pathway for the neutral molecules CH, C 2 , and CN. The CN radical is the first molecular species observed in diffuse clouds that requires a neutral-neutral reaction for its formation in the gas phase. The network also reproduces the observed ratio N(CN)/N(H 2 )
9. The Rosseland mean opacity of interstellar grain
International Nuclear Information System (INIS)
Ali, A.; El Shalaby, M.A.; El-Nawawy, M.S.
1990-10-01
We have calculated the opacity of interstellar grains in the temperature range 10 deg. K - 1500 deg. K. Two composite grain models have been considered. One of them consists of silicate coated with ice mantle and the second has a graphite core coated also with ice mantle. These models are compared with isolated grain models. An exact analytical and computational development of Guettler's formulae for composite grain models has been used to calculate the extinction coefficient. It has been found that the thickness of the mantle affects the opacity of the interstellar grains. The opacity of composite models differs from that of the isolated models. The effect of the different species (ice, silicate and graphite) is also clear. (author). 22 refs, 4 figs, 1 tab
10. Human factors issues for interstellar spacecraft
Science.gov (United States)
Cohen, Marc M.; Brody, Adam R.
1991-01-01
Developments in research on space human factors are reviewed in the context of a self-sustaining interstellar spacecraft based on the notion of traveling space settlements. Assumptions about interstellar travel are set forth addressing costs, mission durations, and the need for multigenerational space colonies. The model of human motivation by Maslow (1970) is examined and directly related to the design of space habitat architecture. Human-factors technology issues encompass the human-machine interface, crew selection and training, and the development of spaceship infrastructure during transtellar flight. A scenario for feasible instellar travel is based on a speed of 0.5c, a timeframe of about 100 yr, and an expandable multigenerational crew of about 100 members. Crew training is identified as a critical human-factors issue requiring the development of perceptual and cognitive aids such as expert systems and virtual reality.
11. The composition of interstellar grain mantles
International Nuclear Information System (INIS)
Tielens, A.G.G.M.
1984-01-01
The molecular composition of interstellar grain mantles employing gas phase as well as grain surface reactions has been calculated. The calculated mixtures consist mainly of the molecules H 2 O H 2 CO, N 2 , CO, O 2 , CO 2 , H 2 O 2 , NH 3 , and their deuterated counterparts in varying ratios. The exact compositions depend strongly on the physical conditions in the gas phase. The calculated mixtures are compared to the observations by using laboratory spectra of grain mantle analogs. (author)
12. Kinetic chemistry of dense interstellar clouds
International Nuclear Information System (INIS)
Graedel, T.E.; Langer, W.D.; Frerking, M.A.
1982-01-01
A detailed model of the time-dependent chemistry of dense interstellar clouds has been developed to study the dominant chemical processes in carbon and oxygen isotope fractionation, formation of nitrogen-containing molecules, evolution of product molecules as a function of cloud density and temperature, and other topics of interest. The full computation involves 328 individual reactions (expanded to 1067 to study carbon and oxygen isotope chemistry); photodegradation processes are unimportant in these dense clouds and are excluded
13. An investigation of the interstellar extinction
International Nuclear Information System (INIS)
Roche, P.F.; Aitken, D.K.; Melbourne Univ., Point Cook
1984-01-01
The 10 μm extinction towards six WC8 or WC9 Wolf-Rayet stars is investigated. All objects show smooth dust emission suffering silicate absorption with depths well correlated with the extinction in the visible. The de-reddened spectra are well represented by emission from featureless grain components, possibly from iron or carbon grains. The extinction to the stars is found to be dominantly interstellar in origin with little extinction from the circumstellar shell. (author)
14. Stochastic histories of refractory interstellar dust
International Nuclear Information System (INIS)
Liffman, K.; Chayton, D.D.
1988-01-01
The authors calculate histories for refractory dust particles in the interstellar medium. The double purposes are to learn something of the properties of interstellar dust as a system and to evaluate with specific assumptions the cosmic chemical memory interpretation of a specific class of isotopic anomalies. They assemble the profile of a particle population from a large number of stochastic, or Monte Carlo, histories of single particles, which are necessarily taken to be independent with this approach. They specify probabilities for each of the events that may befall a given particle and unfold its history by a sequence of random numbers. They assume that refractory particles are created only by thermal condensation within stellar material during its ejection from stars, and that these refractory particles can be destroyed only by being sputtered to a size too small for stability or by being incorporated into the formation of new stars. In order to record chemical detail, the authors take each new refractory particle to consist of a superrefractory core plus a more massive refractory mantle. They demonstrate that these superrefractory cores have effective lifetimes much longer than the turnover time of dust mass against sputtering. As examples of cosmic chemical memory they evaluate the 16 O-richness of interstellar aluminum and mechanisms for the 48 Ca/ 50 Ti correlation. Several related consequences of this approach are discussed
15. Design for minimum energy in interstellar communication
Science.gov (United States)
Messerschmitt, David G.
2015-02-01
Microwave digital communication at interstellar distances is the foundation of extraterrestrial civilization (SETI and METI) communication of information-bearing signals. Large distances demand large transmitted power and/or large antennas, while the propagation is transparent over a wide bandwidth. Recognizing a fundamental tradeoff, reduced energy delivered to the receiver at the expense of wide bandwidth (the opposite of terrestrial objectives) is advantageous. Wide bandwidth also results in simpler design and implementation, allowing circumvention of dispersion and scattering arising in the interstellar medium and motion effects and obviating any related processing. The minimum energy delivered to the receiver per bit of information is determined by cosmic microwave background alone. By mapping a single bit onto a carrier burst, the Morse code invented for the telegraph in 1836 comes closer to this minimum energy than approaches used in modern terrestrial radio. Rather than the terrestrial approach of adding phases and amplitudes increases information capacity while minimizing bandwidth, adding multiple time-frequency locations for carrier bursts increases capacity while minimizing energy per information bit. The resulting location code is simple and yet can approach the minimum energy as bandwidth is expanded. It is consistent with easy discovery, since carrier bursts are energetic and straightforward modifications to post-detection pattern recognition can identify burst patterns. Time and frequency coherence constraints leading to simple signal discovery are addressed, and observations of the interstellar medium by transmitter and receiver constrain the burst parameters and limit the search scope.
16. Chemical reactivities of some interstellar molecules
Energy Technology Data Exchange (ETDEWEB)
1980-01-01
Work in the area of chemical evolution during the last 25 years has revealed the formation of a large number of biologically important molecules produced from simple starting materials under relatively simple experimental conditions. Much of this work has resulted from studies under atmospheres simulating that of the primitive earth or other planets. During the last decade, progress has also been made in the identification of chemical constituents of interstellar medium. A number of these molecules are the same as those identified in laboratory experiments. Even though the conditions of the laboratory experiments are vastly different from those of the cool, low-density interstellar medium, some of the similarities in composition are too obvious to go unnoticed. The present paper highlights some of the similarities in the composition of prebiotic molecules and those discovered in the interstellar medium. Also the chemical reactions which some of the common molecules e.g., NH3, HCN, H2CO, HC(triple bond)-C-CN etc. can undergo are surveyed.
17. Gamma rays from the interstellar medium
International Nuclear Information System (INIS)
Bloemen, J.B.G.M.
1985-01-01
This thesis describes new gamma-ray views on cosmic rays and the interstellar medium. The author describes the COS-B data base and the pre-launch and in-flight calibration data used for all analyses. Diffuse galactic gamma radiation (> 50 MeV) may be either a result of cosmic-ray-matter interactions, or of the cosmic-ray electrons with the interstellar radiation field (mainly at optical and infrared wavelengths), through the inverse-Compton process. A detailed comparison between the gamma-ray observations of the large complex of interstellar clouds in Orion and Monoceros and the CO and HI surveys of this region is given. It gives insight into the cloud penetration of cosmic rays and in the relation between CO detections and molecular hydrogen column densities. Next, the radial distribution of gamma rays in the Galaxy is studied, as well as the galactic centre (more precisely, the central 400 pc), which contains a large concentration of CO molecules. The H 2 /CO abundance and the cosmic-ray density in the galactic centre are discussed and compared to the findings for the galactic disk. In various analyses in this thesis a likelihood-ratio method is applied for parameter estimation and hypothesis testing. A general description of this method is added as an appendix. (Auth.)
18. Properties of interstellar dust in reflection nebulae
International Nuclear Information System (INIS)
Sellgren, K.
1988-01-01
Observations of interstellar dust in reflection nebulae are the closest analog in the interstellar medium to studies of cometary dust in our solar system. The presence of a bright star near the reflection nebula dust provides the opportunity to study both the reflection and emission characteristics of interstellar dust. At 0.1 to 1 micrometer, the reflection nebula emission is due to starlight scattered by dust. The albedo and scattering phase function of the dust is determined from observations of the scattered light. At 50 to 200 micrometers, thermal emission from the dust in equilibrium with the stellar radiation field is observed. The derived dust temperature determines the relative values of the absorption coefficient of the dust at wavelengths where the stellar energy is absorbed and at far infrared wavelengths where the absorbed energy is reradiated. These emission mechanisms directly relate to those seen in the near and mid infrared spectra of comets. In a reflection nebula the dust is observed at much larger distances from the star than in our solar system, so that the equilibrium dust temperature is 50 K rather than 300 K. Thus, in reflection nebulae, thermal emission from dust is emitted at 50 to 200 micrometer
19. Interstellar matrices: the chemical composition and evolution of interstellar ices as observed by ISO.
Science.gov (United States)
d'Hendecourt, L; Dartois, E
2001-03-15
Matrix isolation techniques have been developed in the early sixties as a tool for studying the spectroscopic properties of out of equilibrium species (atoms, radicals, ions, reactive molecules), embedded in rare gas inert matrices at low temperatures. Cold interstellar grains surfaces are able to condense out gas phase molecules, routinely observed by radioastronomy. These grain 'mantles' can be considered as 'interstellar matrices'. However, these matrices are not clean and unreactive. They are made principally of dirty ices whose composition must be determined carefully to assess the importance of the solid state chemistry that takes place in the Interstellar Medium. Infrared spectroscopy, both in astronomy and in the laboratory, is the unique tool to determine the chemical composition of these ices. Astronomical spectra can directly be compared with laboratory ones obtained using classical matrix isolation techniques. Furthermore, dedicated experiments may be undertaken to further improve the understanding of the basic physico-chemical processes that take place in cosmic ices.
20. The Ingenious Theory of Interstellar Trade
Science.gov (United States)
This paper extends interplanetary trade theory to an interstellar setting. It is chiefly concerned with the following question: How should interest charges on goods in transit be computed when the goods travel at speeds close to the actual speed of light? This is a problem because the time taken in transit will appear less to an observer travelling with the goods than to a stationary observer. An innovative and ingenious solution is derived from the economic theory, and two useless but TRUE theorems are proved. The interstellar trade would happen in such a way that two time frames must be considered namely that of the stationary observer whose time runs faster compared to the time frame of the observer in transit The interest in a given trade is purely based on the time taken for the debtor to pay the amount, once the goods have been delivered by the seller. But, in case of interstellar trade, the interest to be calculated in between two time frames would lead to the question of which time frame to be considered and moreover, the time taken for the goods to reach the destination is signicantly prolonged compared to the interplanetary trade, which means, even the slightest variations in the interest rate would be magnied. Apart from this, various new factors arise while calculating the interest. The factors include the time value of money, and the risk of variation in demand for goods, the risk of interspace accidents causing loss of the goods and the rate of perish-ability in case of organic goods. The first two factors considered, for which the time frame of the stationary observer is considered and the factors such as the risk of accidents and the rate of perish-ability of the goods are considered based on the time frame of the observer in transit's point of view. The reasons for such considerations and various assumptions on these concepts are dealt in this paper. The theorems that are formulated in this paper would provide the interstellar traders a basic
1. Interstellar propulsion using a pellet stream for momentum transfer
International Nuclear Information System (INIS)
Singer, C.E.
1979-10-01
A pellet-stream concept for interstellar propulsion is described. Small pellets are accelerated in the solar system and accurately guided to an interstellar probe where they are intercepted and transfer momentum. This propulsion system appears to offer orders-of-magnitude improvements in terms of engineering simplicity and power requirements over any other known feasible system for transport over interstellar distance in a time comparable to a human lifespan
2. Changing Boundaries
DEFF Research Database (Denmark)
Brodkin, Evelyn; Larsen, Flemming
2013-01-01
project that is altering the boundary between the democratic welfare state and the market economy. We see workfare policies as boundary-changing with potentially profound implications both for individuals disadvantaged by market arrangements and for societies seeking to grapple with the increasing...
3. An introduction to the physics of interstellar dust
CERN Document Server
Krugel, Endrik
2007-01-01
Streamlining the extensive information from the original, highly acclaimed monograph, this new An Introduction to the Physics of Interstellar Dust provides a concise reference and overview of interstellar dust and the interstellar medium. Drawn from a graduate course taught by the author, a highly regarded figure in the field, this all-in-one book emphasizes astronomical formulae and astronomical problems to give a solid foundation for the further study of interstellar medium. Covering all phenomena associated with cosmic dust, this inclusive text eliminates the need to consult special physica
4. Synthesis of molecules in interstellar clouds and star formation
International Nuclear Information System (INIS)
Ghosh, K.K.; Ghosh, S.N.
1981-01-01
Study of the formation and destruction processes of interstellar molecules may throw certain light on interstellar medium. Formation and destruction processes of some interstellar molecules are proposed on the basis of laboratory data. The abundances of these molecules are calculated under steady-state condition. The calculated values are then compared with the observed values, obtained by different investigators. It appears that gas phase ion-neutral reactions are capable of synthesizing most interstellar molecules. The role of ion-neutral reactions to star formation has also been discussed. (author)
5. Analysis of "Midnight" Tracks in the Stardust Interstellar Dust Collector: Possible Discovery of a Contemporary Interstellar Dust Grain
Science.gov (United States)
Westphal, A. J.; Allen, C.; Bajit, S.; Bastien, R.; Bechtel, H.; Bleuet, P.; Borg, J.; Brenker, F.; Bridges, J.; Brownlee, D. E.;
2010-01-01
In January 2006, the Stardust sample return capsule returned to Earth bearing the first solid samples from a primitive solar system body, Comet 81P/Wild2, and a collector dedicated to the capture and return of contemporary interstellar dust. Both collectors were approximately 0.1m(exp 2) in area and were composed of aerogel tiles (85% of the collecting area) and aluminum foils. The Stardust Interstellar Dust Collector (SIDC) was exposed to the interstellar dust stream for a total exposure factor of 20 m(exp 2) day. The Stardust Interstellar Preliminary Examination (ISPE) is a three-year effort to characterize the collection using nondestructive techniques.
6. International Nuclear Information System (INIS)
Dwek, E.; Scalo, J.M.
1979-01-01
We have examined theoretically the evolution of refractory interstellar grain abundances and corresponding metal deplections in the solar neighborhood. The calculations include a self-consistent treatment of red-giant winds, planetary nebulae, protostellar nebulae, and suprnovae as sources of grains and star formation, and of encounters with supernova blast waves as sinks. We find that in the standard two-phase model for the interstellar medium (ISM), grain destruction is very efficient, and the abundance of refractory grains should be negligible, contrary to observations. In a cloudy three-phase ISM most grains reside in the warm and cold phases of the medium. Supernova blast waves expand predominantly in the hot and tenuous phase of the medium and are showed down as they propagate through a cloud. In order to obtain significant (approx.3) depletions of metals presubably locked up in refractory grain cores, the destruction of grains that reside in the clouds must be minimal. This requires that (a) the density contrast between the cloud and intercloud medium be sufficiently high, and (b) the filling factor of the hot and tenuous gas of the interstellar medium, which presumably gives rise to the O VI absorption and soft X-ray emission, be nearly unity. Much larger depletions (> or approx. =10) must reflect accretion of mantles within interstellar clouds
7. Magnetic Fields in the Interstellar Medium
Science.gov (United States)
Clark, Susan
2017-01-01
The Milky Way is magnetized. Invisible magnetic fields thread the Galaxy on all scales and play a vital but still poorly understood role in regulating flows of gas in the interstellar medium and the formation of stars. I will present highlights from my thesis work on magnetic fields in the diffuse interstellar gas and in accretion disks. At high Galactic latitudes, diffuse neutral hydrogen is organized into an intricate network of slender linear features. I will show that these neutral hydrogen “fibers” are extremely well aligned with the ambient magnetic field as traced by both starlight polarization (Clark et al. 2014) and Planck 353 GHz polarized dust emission (Clark et al. 2015). The structure of the neutral interstellar medium is more tightly coupled to the magnetic field than previously known. Because the orientation of neutral hydrogen is an independent predictor of the local dust polarization angle, our work provides a new tool in the search for inflationary gravitational wave B-mode polarization in the cosmic microwave background, which is currently limited by dust foreground contamination. Magnetic fields also drive accretion in astrophysical disks via the magnetorotational instability (MRI). I analytically derive the behavior of this instability in the weakly nonlinear regime and show that the saturated state of the instability depends on the geometry of the background magnetic field. The analytical model describes the behavior of the MRI in a Taylor-Couette flow, a set-up used by experimentalists in the ongoing quest to observe MRI in the laboratory (Clark & Oishi 2016a, 2016b).
8. Observing Interstellar and Intergalactic Magnetic Fields
Science.gov (United States)
Han, J. L.
2017-08-01
Observational results of interstellar and intergalactic magnetic fields are reviewed, including the fields in supernova remnants and loops, interstellar filaments and clouds, Hii regions and bubbles, the Milky Way and nearby galaxies, galaxy clusters, and the cosmic web. A variety of approaches are used to investigate these fields. The orientations of magnetic fields in interstellar filaments and molecular clouds are traced by polarized thermal dust emission and starlight polarization. The field strengths and directions along the line of sight in dense clouds and cores are measured by Zeeman splitting of emission or absorption lines. The large-scale magnetic fields in the Milky Way have been best probed by Faraday rotation measures of a large number of pulsars and extragalactic radio sources. The coherent Galactic magnetic fields are found to follow the spiral arms and have their direction reversals in arms and interarm regions in the disk. The azimuthal fields in the halo reverse their directions below and above the Galactic plane. The orientations of organized magnetic fields in nearby galaxies have been observed through polarized synchrotron emission. Magnetic fields in the intracluster medium have been indicated by diffuse radio halos, polarized radio relics, and Faraday rotations of embedded radio galaxies and background sources. Sparse evidence for very weak magnetic fields in the cosmic web is the detection of the faint radio bridge between the Coma cluster and A1367. Future observations should aim at the 3D tomography of the large-scale coherent magnetic fields in our Galaxy and nearby galaxies, a better description of intracluster field properties, and firm detections of intergalactic magnetic fields in the cosmic web.
9. Boundaries of the universe
CERN Document Server
Glasby, John S
2013-01-01
The boundaries of space exploration are being pushed back constantly, but the realm of the partially understood and the totally unknown is as great as ever. Among other things this book deals with astronomical instruments and their application, recent discoveries in the solar system, stellar evolution, the exploding starts, the galaxies, quasars, pulsars, the possibilities of extraterrestrial life and relativity.
10. Long Term Perspective On Interstellar Flight
Science.gov (United States)
Millis, M. G.
2017-12-01
The process and interim findings of a broad interstellar flight assessment is presented. In contrast to precursor mission studies, this assessment takes a longer view and also considers factors that have been underrepresented in prior studies. The goal is to chart a conceptual roadmap for interstellar flight development that takes all the factors into account and ultimately identifies which research options, today, might have the greatest overall impact on future progress. Three envisioned flight eras are examined, the "era of precursors," the "era of infrastructure," and the "unforeseeable future." Several influential factors have typically been missing from prior studies that will now be assessed; a) the impact of different, often implicit, motivations, b) the interdependency of infrastructure with vehicle design, c) the pace of different developments, and d) the enormous energy required for any interstellar mission. Regarding motivations for example, if the driving motivation is to launch soon, then the emphasis is on existing technologies. In contrast, if the motivation is the survival of humanity, then the emphasis would be on 'world ships.' Infrastructure considerations are included in a broader system-level context. Future infrastructure will support multiple in-space activities, not just one mission-vehicle development. Though it may be too difficult to successfully assess, the study will attempt to compare the rates of different developments, such as the pace of Earth-based astronomy, miniaturization, artificial intelligence, infrastructure development, transhumanism, and others. For example, what new information could be acquired after 30 years of further advances in astronomy compared to a space probe with current technology and a 30 year flight time? The final factor of the study is to assess the pace and risks of the enormous energy levels required for interstellar flight. To compare disparate methods, a set of 'meta measures' will be defined and
11. Planetary nebulae and the interstellar magnetic field
International Nuclear Information System (INIS)
Heiligman, G.M.
1980-01-01
Previous workers have found a statistical correlation between the projected directions of the interstellar magnetic field and the major axes of planetary nebulae. This result has been examined theoretically using a numerical hydromagnetic model of a cold plasma nebula expanding into a uniform vacuum magnetic field, with nebular gas accreting on the surface. It is found that magnetic pressure alone is probably not sufficient to shape most planetary nebulae to the observed degree. Phenomena are discussed which could amplify simple magnetic pressure, alter nebular morphology and account for the observed correlation. (author)
12. Interstellar extinction in the Taurus dark clouds
International Nuclear Information System (INIS)
Meistas, E.; Straizys, V.
1981-01-01
The results of photoelectric photometry of 89 stars in the Vilnius seven-color system in the area of the Taurus dark clouds with corrdinates (1950) 4sup(h)16sup(m)-4sup(h)33sup(m), +16 0 -+20 0 are presented. Photometric spectral types, absolute magnitude, color excesses, interstellar extinctions and distances of the stars are determined. The distance of the dark nebula is found to be 140 pc and is in a good agreement with the distance determined for the dark nebula Khavtassi 286, 278. The average extinction Asub(v) in the investigated area is of the order of 1.4. (author)
13. Interstellar colonization and the zoo hypothesis
International Nuclear Information System (INIS)
Jones, E.M.
1978-01-01
Michael Hart and others have pointed out that current estimates of the number of technological civilizations arisen in the Galaxy since its formation is in fundamental conflict with the expectation that such a civilization could colonize and utilize the entire Galaxy in 10 to 20 million years. This dilemma can be called Hart's paradox. Resolution of the paradox requires that one or more of the following are true: we are the Galaxy's first technical civilization; interstellar travel is immensely impractical or simply impossible; technological civilizations are very short-lived; or we inhabit a wildnerness preserve. The latter is the zoo hypothesis
14. The interstellar medium in galaxies - An overview
Science.gov (United States)
Knapp, G. R.
1990-01-01
Recent observational developments on the subject of the interstellar medium in galaxies are summarized, with emphasis placed on global properties. The properties and distribution of the ISM in the solar neighborhood and in the Galactic plane are examined and a number of results from the most important observational probes (HI, CO, and infrared) are described. A recent development is the observation of the ISM in galaxies of all morphological types, early to late. These developments are summarized and the properties of different types of galaxies are compared to one another. The origin of radio galaxies, the effect of environment, and the prospects for direct observations of ISM evolution in galaxies are discussed.
15. OH radiation from the interstellar cloud medium
Energy Technology Data Exchange (ETDEWEB)
Nguyen-Q-Rieu,; Winnberg, A [Max-Planck-Institut fuer Radioastronomie, Bonn (F.R. Germany); Guibert, J [Observatoire de Paris, Section de Meudon, 92 (France); Lepine, J R.D. [Universidade Mackenzie, Sao Paulo (Brazil). Centro de Radio-Astronomia et Astrofisica; Johansson, L E.B. [Rymdobservatoriet, Onsala (Sweden); Goss, W M [Commonwealth Scientific and Industrial Research Organization, Epping (Australia). Div. of Radiophysics
1976-02-01
We have detected OH in the direction of about 50% of the continuum sources investigated. The OH abundance is one order of magnitude less than usually found in dust clouds. Most of the OH features have HI counterparts. This suggests that the OH radiation arises from the HI interstellar cold clouds. Our observations allowed in some cases the determination of the excitation temperatures in all four lines. A pumping model involving far-infrared radiation and collisions with neutral and charged particles has been proposed. It explains the observed excitation temperatures.
16. IMAGINE: Interstellar MAGnetic field INference Engine
Science.gov (United States)
Steininger, Theo
2018-03-01
IMAGINE (Interstellar MAGnetic field INference Engine) performs inference on generic parametric models of the Galaxy. The modular open source framework uses highly optimized tools and technology such as the MultiNest sampler (ascl:1109.006) and the information field theory framework NIFTy (ascl:1302.013) to create an instance of the Milky Way based on a set of parameters for physical observables, using Bayesian statistics to judge the mismatch between measured data and model prediction. The flexibility of the IMAGINE framework allows for simple refitting for newly available data sets and makes state-of-the-art Bayesian methods easily accessible particularly for random components of the Galactic magnetic field.
17. Chemical equilibrium models of interstellar gas clouds
International Nuclear Information System (INIS)
Freeman, A.
1982-10-01
This thesis contains work which helps towards our understanding of the chemical processes and astrophysical conditions in interstellar clouds, across the whole range of cloud types. The object of the exercise is to construct a mathematical model representing a large system of two-body chemical reactions in order to deduce astrophysical parameters and predict molecular abundances and chemical pathways. Comparison with observations shows that this type of model is valid but also indicates that our knowledge of some chemical reactions is incomplete. (author)
18. Interstellar extinction in the Large Magellanic Cloud
International Nuclear Information System (INIS)
Nandy, K.; Morgan, D.H.; Willis, A.J.; Wilson, R.; Gondhalekar, P.M.; Houziaux, L.
1980-01-01
Recent UV observations together with complementary visible data of several reddened and comparison stars of similar spectral types in the Large Magellanic Cloud have been used to study the interstellar extinction in that galaxy. Most of the reddened stars studied here are located within 2 0 of 30 Doradus and show remarkably high extinction in the far UV, suggesting a large abundance of small particles. From the optical wavelength to 2,600 A the normalised extinction curves of the LMC stars are similar to the mean galactic extinction law. (author)
19. PROPERTIES OF DIFFUSE INTERSTELLAR BANDS AT DIFFERENT PHYSICAL CONDITIONS OF THE INTERSTELLAR MEDIUM
International Nuclear Information System (INIS)
Kos, J.; Zwitter, T.
2013-01-01
Diffuse interstellar bands (DIBs) can trace different conditions of the interstellar medium (ISM) along the sightline toward the observed stars. A small survey was made in optical wavelengths, producing high-resolution and high signal-to-noise spectra. We present measurements of 19 DIBs' properties in 50 sightlines toward hot stars, distributed at a variety of galactic coordinates and interstellar reddening. Equivalent widths were obtained by fitting asymmetric Gaussian and variable continua to DIBs. Conditions of the ISM were calculated from eight atomic and molecular interstellar lines. Two distinctly different types of DIBs were identified by carefully comparing correlation coefficients between DIBs and reddening and by different behavior in UV-shielded (ζ) and nonshielded (σ) sightlines. A ratio of DIBs at 5780 Å and 5797 Å proved to be reliable enough to distinguish between two different sightline types. Based on the linear relations between DIB equivalent width and reddening for σ and ζ sightlines, we divide DIBs into type I (where both linear relations are similar) and type II (where they are significantly different). The linear relation for ζ type sightlines always shows a higher slope and larger x-intercept parameter than the relation for σ sightlines. Scatter around the linear relation is reduced after the separation, but it does not vanish completely. This means that UV shielding is the dominant factor of the DIB equivalent width versus reddening relation shape for ζ sightlines, but in σ sightlines other physical parameters play a major role. No similar dependency on gas density, electron density, or turbulence was observed. A catalog of all observed interstellar lines is made public
20. Structural, chemical and isotopic examinations of interstellar organic matter extracted from meteorites and interstellar dust particles
Science.gov (United States)
Busemann, Henner; Alexander, Conel M. O'D.; Nittler, Larry R.; Stroud, Rhonda M.; Zega, Tom J.; Cody, George D.; Yabuta, Hikaru; Kilcoyne, A. L. David
2008-10-01
Meteorites and Interplanetary Dust Particles (IDPs) are supposed to originate from asteroids and comets, sampling the most primitive bodies in the Solar System. They contain abundant carbonaceous material. Some of this, mostly insoluble organic matter (IOM), likely originated in the protosolar molecular cloud, based on spectral properties and H and N isotope characteristics. Together with cometary material returned with the Stardust mission, these samples provide a benchmark for models aiming to understand organic chemistry in the interstellar medium, as well as for mechanisms that secured the survival of these fragile molecules during Solar System formation. The carrier molecules of the isotope anomalies are largely unknown, although amorphous carbonaceous spheres, so-called nanoglobules, have been identified as carriers. We are using Secondary Ion Mass Spectrometry to identify isotopically anomalous material in meteoritic IOM and IDPs at a ~100-200 nm scale. Organics of most likely interstellar origin are then extracted with the Focused-Ion-Beam technique and prepared for synchrotron X-ray and Transmission Electron Microscopy. These experiments yield information on the character of the H- and N-bearing interstellar molecules: While the association of H and N isotope anomalies with nanoglobules could be confirmed, we have also identified amorphous, micron-sized monolithic grains. D-enrichments in meteoritic IOM appear not to be systematically associated with any specific functional groups, whereas 15N-rich material can be related to imine and nitrile functionality. The large 15N- enrichments observed here (δ15N > 1000 ‰) cannot be reconciled with models using interstellar ammonia ice reactions, and hence, provide new constraints for understanding the chemistry in cold interstellar clouds.
1. DEVELOPMENT OF THE MODEL OF GALACTIC INTERSTELLAR EMISSION FOR STANDARD POINT-SOURCE ANALYSIS OF FERMI LARGE AREA TELESCOPE DATA
Energy Technology Data Exchange (ETDEWEB)
Acero, F.; Ballet, J. [Laboratoire AIM, CEA-IRFU/CNRS/Université Paris Diderot, Service d’Astrophysique, CEA Saclay, F-91191 Gif sur Yvette (France); Ackermann, M.; Buehler, R. [Deutsches Elektronen Synchrotron DESY, D-15738 Zeuthen (Germany); Ajello, M. [Department of Physics and Astronomy, Clemson University, Kinard Lab of Physics, Clemson, SC 29634-0978 (United States); Albert, A.; Baldini, L.; Bloom, E. D.; Bottacini, E.; Caliandro, G. A.; Cameron, R. A. [W. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94305 (United States); Barbiellini, G. [Istituto Nazionale di Fisica Nucleare, Sezione di Trieste, I-34127 Trieste (Italy); Bastieri, D. [Istituto Nazionale di Fisica Nucleare, Sezione di Padova, I-35131 Padova (Italy); Bellazzini, R. [Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, I-56127 Pisa (Italy); Bissaldi, E. [Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari (Italy); Bonino, R. [Istituto Nazionale di Fisica Nucleare, Sezione di Torino, I-10125 Torino (Italy); Brandt, T. J.; Buson, S. [NASA Goddard Space Flight Center, Greenbelt, MD 20771 (United States); Bregeon, J. [Laboratoire Univers et Particules de Montpellier, Université Montpellier, CNRS/IN2P3, Montpellier (France); Bruel, P., E-mail: [email protected], E-mail: [email protected] [Laboratoire Leprince-Ringuet, École polytechnique, CNRS/IN2P3, Palaiseau (France); and others
2016-04-01
Most of the celestial γ rays detected by the Large Area Telescope (LAT) on board the Fermi Gamma-ray Space Telescope originate from the interstellar medium when energetic cosmic rays interact with interstellar nucleons and photons. Conventional point-source and extended-source studies rely on the modeling of this diffuse emission for accurate characterization. Here, we describe the development of the Galactic Interstellar Emission Model (GIEM), which is the standard adopted by the LAT Collaboration and is publicly available. This model is based on a linear combination of maps for interstellar gas column density in Galactocentric annuli and for the inverse-Compton emission produced in the Galaxy. In the GIEM, we also include large-scale structures like Loop I and the Fermi bubbles. The measured gas emissivity spectra confirm that the cosmic-ray proton density decreases with Galactocentric distance beyond 5 kpc from the Galactic Center. The measurements also suggest a softening of the proton spectrum with Galactocentric distance. We observe that the Fermi bubbles have boundaries with a shape similar to a catenary at latitudes below 20° and we observe an enhanced emission toward their base extending in the north and south Galactic directions and located within ∼4° of the Galactic Center.
2. Distribution of Interstellar Reddening Material in the Galactic Plane
Directory of Open Access Journals (Sweden)
Chulhee Kim
1987-12-01
Full Text Available By using the recently determined color excess and distance data of classical cepheids by Kim(1985, the distribution of interstellar reddening material was studied to see the general picture of the average rate of interstellar absorption out to about 7-8kpc in the Galactic plane in various directions from the sun.
3. Streaming of interstellar grains in the solar system
Science.gov (United States)
Gustafson, B. A. S.; Misconi, N. Y.
1979-01-01
Results of a theoretical study of the interactions between interstellar grains streaming through the solar system and the solar wind are presented. It is shown that although elongated core-mantle interstellar particles of a characteristic radius of about 0.12 microns are subject to a greater force due to radiation pressure than to gravitational attraction, they are still able to penetrate deep inside the solar system. Calculations of particle trajectories within the solar system indicate substantial effects of the solar activity cycle as reflected in the interplanetary magnetic field on the distribution of 0.12- and 0.0005-micron interstellar grains streaming through the solar system, leading to a 50-fold increase in interstellar grain densities 3 to 4 AU ahead of the sun during years 8 to 17 of the solar cycle. It is noted that during the Solar Polar Mission, concentrations are expected which will offer the opportunity of detecting interstellar grains in the solar system.
4. A Search for Interstellar Monohydric Thiols
Energy Technology Data Exchange (ETDEWEB)
Gorai, Prasanta; Das, Ankan; Das, Amaresh; Chakrabarti, Sandip K. [Indian Centre for Space Physics, 43 Chalantika, Garia Station Rd., Kolkata, 700084 (India); Sivaraman, Bhalamurugan [Atomic Molecular and Optical Physics Division, Physical Research Laboratory, Ahmedabad, 380009 (India); Etim, Emmanuel E., E-mail: [email protected] [Indian Institute of Science Bangalore, 560012 (India)
2017-02-10
It has been pointed out by various astronomers that a very interesting relationship exists between interstellar alcohols and the corresponding thiols (sulfur analog of alcohols) as far as the spectroscopic properties and chemical abundances are concerned. Monohydric alcohols such as methanol and ethanol are widely observed and 1-propanol was recently claimed to have been seen in Orion KL. Among the monohydric thiols, methanethiol (chemical analog of methanol) has been firmly detected in Orion KL and Sgr B2(N2) and ethanethiol (chemical analog of ethanol) has been observed in Sgr B2(N2), though the confirmation of this detection is yet to come. It is very likely that higher order thiols could be observed in these regions. In this paper, we study the formation of monohydric alcohols and their thiol analogs. Based on our quantum chemical calculation and chemical modeling, we find that the Tg conformer of 1-propanethiol is a good candidate of astronomical interest. We present various spectroscopically relevant parameters of this molecule to assist in its future detection in the interstellar medium.
5. 26Al in the interstellar medium
International Nuclear Information System (INIS)
Clayton, D.D.; Leising, M.D.
1987-01-01
Several different lines of physical reasoning have converged on the importance of the radioactive nucleus 26 Al. The sciences of meteoritics, nucleosynthesis, gamma-ray astronomy, galactic chemical evolution, solar system formation, and interstellar chemistry all place this nucleus in a central position with possible profound implications. Perhaps more importantly the study of this radioactivity can unite these diverse fields in a complicated framework which will benefit all of them. This review traces the evolution of ideas concerning 26 Al in the context of these disciplines. 26 Al was first discussed for the possibility that its decay energy could melt meteorite parent bodies, and its daughter, 26 Mg, was later found in meteorites with enhanced abundance. It was also among the first radioactivities expected to be synthesized in interestingly large quantities in nulceosynthetic events. The first definitive detection of gamma-rays from an interstellar radioactivity is that of 1.809 MeV gamma-rays from 26 Al. This discovery has many implications, some of which are outlined here. The whole problem of isotopic anomalies in meteorites is greatly influenced by the specific issues surrounding excess 26 Mg, whether it represents in situ decay of 26 Al or memory of conditions of the ISM. The relationships among these ideas and their implications are examined. (orig.)
6. UV observations of local interstellar medium.
Science.gov (United States)
Kurt, V.; Mironova, E.; Fadeev, E.
2008-12-01
The methods of the interstellar matter study are described. The brief information of space missions aimed at observations in the unreachable for ground based telescopes UV spectral range (IUE, As- tron, HST and GALEX.) is presented. The history of discovery of H and He atoms entering the Solar System from the local interstellar medium (LISM) is given in brief. The results of observations performed by the group from Stern- berg Astronomical Institute (SAI MSU) and Space Research Institute (IKI RAS) performed with the help of the missions Prognoz-5, Prognoz-6 and the stations Zond-1, Venera and Mars and aimed at estimation of all basic LISM parameters (the velocity of the Sun in relation to LISM, directions of movement, densities of H and He atoms, LISM temperature) are presented. We also describe the present-day investigations of LISM performed with SOHO and ULYSSES mis- sions including the direct registration of He atoms entering the Solar System. The problem of interaction between the incoming flow of the ISM atoms ("in- terstellar wind") and the area of two shocks at the heliopause border (100-200 AU) is discussed. The LISM parameters obtained using the available data are presented in two tables.
7. PRECURSORS TO INTERSTELLAR SHOCKS OF SOLAR ORIGIN
Energy Technology Data Exchange (ETDEWEB)
Gurnett, D. A.; Kurth, W. S. [University of Iowa, Department of Physics and Astronomy, Iowa City, IA 52242 (United States); Stone, E. C.; Cummings, A. C. [California Institute of Technology, 1200 East California Boulevard, Pasadena, CA 91125 (United States); Krimigis, S. M.; Decker, R. B. [Applied Physics Laboratory/JHU, 11100 Johns Hopkins Road, Laurel, MD 20723 (United States); Ness, N. F. [Catholic University of America, 620 Michigan Avenue NE, Washington, DC 20064 (United States); Burlaga, L. F., E-mail: [email protected] [NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771 (United States)
2015-08-20
On or about 2012 August 25, the Voyager 1 spacecraft crossed the heliopause into the nearby interstellar plasma. In the nearly three years that the spacecraft has been in interstellar space, three notable particle and field disturbances have been observed, each apparently associated with a shock wave propagating outward from the Sun. Here, we present a detailed analysis of the third and most impressive of these disturbances, with brief comparisons to the two previous events, both of which have been previously reported. The shock responsible for the third event was first detected on 2014 February 17 by the onset of narrowband radio emissions from the approaching shock, followed on 2014 May 13 by the abrupt appearance of intense electron plasma oscillations generated by electrons streaming outward ahead of the shock. Finally, the shock arrived on 2014 August 25, as indicated by a jump in the magnetic field strength and the plasma density. Various disturbances in the intensity and anisotropy of galactic cosmic rays were also observed ahead of the shock, some of which are believed to be caused by the reflection and acceleration of cosmic rays by the magnetic field jump at the shock, and/or by interactions with upstream plasma waves. Comparisons to the two previous weaker events show somewhat similar precursor effects, although differing in certain details. Many of these effects are very similar to those observed in the region called the “foreshock” that occurs upstream of planetary bow shocks, only on a vastly larger spatial scale.
8. Exploring the boundary of a specialist service for adults with intellectual disabilities using a Delphi study: a quantification of stakeholder participation.
Science.gov (United States)
Hempe, Eva-Maria; Morrison, Cecily; Holland, Anthony
2015-10-01
There are arguments that a specialist service for adults with intellectual disabilities is needed to address the health inequalities that this group experiences. The boundary of such a specialist service however is unclear, and definition is difficult, given the varying experiences of the multiple stakeholder groups. The study reported here quantitatively investigates divergence in stakeholders' views of what constitutes a good specialist service for people with intellectual disabilities. It is the first step of a larger project that aims to investigate the purpose, function and design of such a specialist service. The results are intended to support policy and service development. A Delphi study was carried out to elicit the requirements of this new specialist service from stakeholder groups. It consisted of three panels (carers, frontline health professionals, researchers and policymakers) and had three rounds. The quantification of stakeholder participation covers the number of unique ideas per panel, the value of these ideas as determined by the other panels and the level of agreement within and between panels. There is some overlap of ideas about of what should constitute this specialist service, but both carers and frontline health professionals contributed unique ideas. Many of these were valued by the researchers and policymakers. Interestingly, carers generated more ideas regarding how to deliver services than what services to deliver. Regarding whether ideas are considered appropriate, the variation both within and between groups is small. On the other hand, the feasibility of solutions is much more contested, with large variations among carers. This study provides a quantified representation of the diversity of ideas among stakeholder groups regarding where the boundary of a specialist service for adults with learning disabilities should sit. The results can be used as a starting point for the design process. The study also offers one way to measure the
9. Grant Proposal for the Continuation of the Voyager Interstellar Mission: LECP Investigation
Science.gov (United States)
Krimigis, Stamatios M.; Armstrong, Thomas P.; Lanzerotti, Louis J.; Ip, Wing-H.; Decker, Robert B.; Keath, Edwin P.; Mauk, Barry H.; McNutt, Ralph L., Jr.; Gloeckler, George; Hamilton, Douglas C.
1996-01-01
This proposal documents the plans of the Low Energy Charged Particle (LECP) investigation team for participation in NASA's Voyager Interstellar Mission (VIM) as the Voyager 1 and 2 spacecraft explore the outer reaches of the heliosphere and search for the termination shock and the heliopause. The proposal covers the four year period from 1 January 1997 to 31 December 2000. The LECP instruments on Voyager 1 and 2 measure in situ intensities of charged particles with energies from about 30 keV to 100 MeV for ions, and about 20 keV to greater than 10 MeV for electrons. The instruments provide detailed spectral, angular, and compositional information about the particles. Composition is available for greater than 200 keV/nuc using multi-parameter measurements. Angular information is obtained by a mechanically scanned platform that rotates at various commanded rates. Measurements of low energy ion and electron intensities versus time and spatial location within the heliosphere contain an abundance of information regarding various transport and acceleration processes on both local (approx. 1 hr, approx. 0.01 AU) and global (approx. 11 yrs, approx. 100 AU) scales. The LECP instruments provide unique observations of such dynamical processes, and we anticipate that it will return critical information regarding the boundaries of the heliosphere. Several recent and exciting discoveries based on LECP measurements emphasize the important role that low energy charged particle distributions play in physical processes in the interplanetary medium. Yet, at the same time, these discoveries also underscore the fact that our understanding of processes in the outer heliosphere is, in most cases, incomplete, and in others, only rudimentary at best. Among the discoveries referred to above are the following: (1) Shocks: Examination of greater than 30 keV ion intensities have revealed: (a) a total absence of acceleration beyond only -100-200 keV at a strong transient shock in May 1991 at 35
10. Uv spectra of nearby white dwarfs and the nature of the local interstellar medium
International Nuclear Information System (INIS)
Bruhweiler, F.C.; Kondo, Y.
1982-01-01
We have investigated the local interstellar medium in the directions of four white dwarfs, G191-B2B, W1346, HD 149499B, and Sirius B. All the observational data were obtained at the high-resolution mode (lambda/Δlambdaroughly-equal10 4 ) in the spectral range from about 1150 to 3200 A with the International Ultraviolet Explorer (IUE). Interstellar absorption lines of several elements in various stages of ionization are seen against the continuum of the white dwarfs. Low average hydrogen number densities (n-bar/sub HtsI/) are found. They range from n-bar/sub HtsI/ = 0.08 cm -3 for Sirius B, the nearest white dwarf (2.7 pc), to n-bar/sub HtsI/ = 0.006 cm -3 for G191-B2B, the most distant white dwarf (48 pc) studied. The results show, when combined with other recent ultraviolet, EUV, and diffuse X-ray observations, that: (a) the Sun is located inside a low-density (n-bar/sub HtsI/roughly-equal0.1 cm -3 ) cloud; (b) beyond 2--3 pc from the Sun, this cloud is surrounded, at least in most directions, by an extended region of hot (Troughly-equal10/sup 5en-dash6/ K) thin (nroughly-equal10 -2 to 10 -3 cm -3 ) interstellar plasma with no evidence for additional clouds in the lines of sight studied; (c) the elemental depletions of C, N, O, Si, Mg, and possibly Fe are low in the solar vicinity as previously found toward α Vir, (d) the Sun is moving through this cloud at a relative velocity of about 20 km s -1 ; and (e) the current results, which are quite consistent with previous ultraviolet, EUV, and diffuse X-ray observations, have significant bearings on the theoretical modeling of the interstellar medium. Subject headings: interstellar: abundances: interstellar: matter: stars: white dwarfs: ultraviolet: spectra
11. Blurring Boundaries
DEFF Research Database (Denmark)
Neergaard, Ulla; Nielsen, Ruth
2010-01-01
of welfare functions into EU law both from an internal market law and a constitutional law perspective. The main problem areas covered by the Blurring Boundaries project were studied in sub-projects on: 1) Internal market law and welfare services; 2) Fundamental rights and non-discrimination law aspects......; and 3) Services of general interest. In the Blurring Boundaries project, three aspects of the European Social Model have been particularly highlighted: the constitutionalisation of the European Social Model, its multi-level legal character, and the clash between market access justice at EU level...... and distributive justice at national level....
12. Interstellar depletion anomalies and ionization potentials
International Nuclear Information System (INIS)
Tabak, R.G.
1979-01-01
Satellite observations indicate that (1) most elements are depleted from the gas phase when compared to cosmic abundances, (2) some elements are several orders of magnitude more depleted than others, and (3) these depletions vary from cloud to cloud. Since the most likely possibility is that the 'missing' atoms are locked into grains, depletions occur either by accretion onto core particles in interstellar clouds or earlier, during the period of primary grain formation. If the latter mechanism is dominant, then the most important depletion parameter is the condensation temperature of the elements and their various compounds. However, this alone is not sufficient to explain all the observed anomalies. It is shown that electrostatic effects - under a wide variety of conditions- can enormously enhance the capture cross-section of the grain. It is suggested that this mechanism can also account for such anomalies as the apparent 'overabundance' of the alkali metals in the gas phase. (orig.)
13. Interstellar scattering of pulsar radiation. Pt. 1
International Nuclear Information System (INIS)
Backer, D.C.
1975-01-01
An investigation of the intensity fluctuations of 28 pulsars near 0.4 GHz indicates that spectra of interstellar scintillation are consistent with a gaussian shape, that scintillation indices are near unity, and that scintillation bandwidth depends linearly on dispersion measure. Observations at cm wavelengths show that the observer is in the near field of the scattering medium for objects with the lowest dispersion measures, and confirm the step dependence of correlation bandwidth on dispersion measure found by Sutton (1971). The variation of scattering parameters with dispersion measure may indicate that the rms deviation of thermal electron density on the scale of 10 11 cm grows with path length through the galaxy. (orig.) [de
14. Hot interstellar matter in elliptical galaxies
CERN Document Server
Kim, Dong-Woo
2012-01-01
Based on a number of new discoveries resulting from 10 years of Chandra and XMM-Newton observations and corresponding theoretical works, this is the first book to address significant progress in the research of the Hot Interstellar Matter in Elliptical Galaxies. A fundamental understanding of the physical properties of the hot ISM in elliptical galaxies is critical, because they are directly related to the formation and evolution of elliptical galaxies via star formation episodes, environmental effects such as stripping, infall, and mergers, and the growth of super-massive black holes. Thanks to the outstanding spatial resolution of Chandra and the large collecting area of XMM-Newton, various fine structures of the hot gas have been imaged in detail and key physical quantities have been accurately measured, allowing theoretical interpretations/predictions to be compared and tested against observational results. This book will bring all readers up-to-date on this essential field of research.
15. The mass spectrum of interstellar clouds
International Nuclear Information System (INIS)
Dickey, J.M.; Garwood, R.W.
1989-01-01
The abundances of diffuse clouds and molecular clouds in the inner Galaxy and at the solar circle are compared. Using results of recent low-latitude 21 cm absorption studies, the number of diffuse clouds per kiloparsec along the line of sight is derived as a function of the cloud column density, under two assumptions relating cloud densities and temperatures. The density of clouds is derived as a function of cloud mass. The results are consistent with a single, continuous mass spectrum for interstellar clouds from less than 1 solar mass to 1,000,000 solar masses, with perhaps a change of slope at masses where the atomic and molecular mass fractions are roughly equal. 36 refs
16. Structure and characteristics of diffuse interstellar clouds
International Nuclear Information System (INIS)
Arshutkin, L.N.; Kolesnik, I.G.
1978-01-01
The results of model calculations for spherically symmetrical interstellar clouds being under external pressure are given. Thermal balance of gas clouds is considered. Ultraviolet radiation fields in clouds and equilibrium for chemical elements are calculated for this purpose. Calculations were carried out in the case when cooling is under way mainly by carbon atoms and ions. The clouds with mass up to 700 Msub(sun) under external pressure from 800 to 3000 K cm -3 are considered. In typical for Galactic disk conditions, clouds have dense n > or approximately 200 cm -3 , and cold T approximately 20-30 K state clouds depending on external pressure is given. The critical mass for clouds at the Galactic disk is approximately 500-600 Msub(sun). It is less than the isothermal solution by a factor of approximately 1.5. The massive gas-dust cloud formation problem is discussed
17. Ionization of Local Interstellar Gas Based on STIS and FUSE spectra of Nearby Stars
Science.gov (United States)
Redfield, Seth; Linsky, J. L.
2009-01-01
The ultraviolet contains many resonance line transitions that are sensitive to a range of ionization stages of ions present in the local interstellar medium (LISM). We couple observations of high resolution ultraviolet spectrographs, STIS and GHRS on the Hubble Space Telescope (HST) and the Far-Ultraviolet Spectroscopic Explorer (FUSE) in order to make a comprehensive survey of the ionization structure of the local interstellar medium. In particular, we focus on the sight line toward G191-B2B, a nearby (69 pc) white dwarf. We present interstellar detections of highly ionized elements (e.g., SiIII, CIII, CIV, etc) and compare them directly to neutral or singly ionized LISM detections (e.g., SiII, CII, etc). The extensive observations of G191-B2B provides an opportunity for a broad study of ionization stages of several elements, while a survey of several sight lines provides a comprehensive look at the ionization structure of the LISM. We acknowledge support for this project through NASA FUSE Grant NNX06AD33G.
18. Turbulence in nearly incompressible fluids: density spectrum, flows, correlations and implication to the interstellar medium
Directory of Open Access Journals (Sweden)
S. Dastgeer
2005-01-01
Full Text Available Interstellar scintillation and angular radio wave broadening measurements show that interstellar and solar wind (electron density fluctuations exhibit a Kolmogorov-like k-5/3 power spectrum extending over many decades in wavenumber space. The ubiquity of the Kolmogorov-like interstellar medium (ISM density spectrum led to an explanation based on coupling incompressible magnetohydrodynamic (MHD fluctuations to density fluctuations through a 'pseudosound' relation within the context of 'nearly incompressible' (NI hydrodynamics (HD and MHD models. The NI theory provides a fundamentally different explanation for the observed ISM density spectrum in that the density fluctuations can be a consequence of passive scalar convection due to background incompressible fluctuations. The theory further predicts generation of long-scale structures and various correlations between the density, temperature and the (magneto acoustic as well as convective pressure fluctuations in the compressible ISM fluids in different thermal regimes that are determined purely by the thermal fluctuation level. In this paper, we present the results of our two dimensional nonlinear fluid simulations, exploring various nonlinear aspects that lead to inertial range ISM turbulence within the context of a NI hydrodymanics model. In qualitative agreement with the NI predictions and the in-situ observations, we find that i the density fluctuations exhibit a Kolmogorov-like spectrum via a passive convection in the field of the background incompressible fluctuations, ii the compressible ISM fluctuations form long scale flows and structures, and iii the density and the temperature fluctuations are anti-correlated.
19. Searching for Cost-Optimized Interstellar Beacons
Science.gov (United States)
Benford, Gregory; Benford, James; Benford, Dominic
2010-06-01
What would SETI beacon transmitters be like if built by civilizations that had a variety of motives but cared about cost? In a companion paper, we presented how, for fixed power density in the far field, a cost-optimum interstellar beacon system could be built. Here, we consider how we should search for a beacon if it were produced by a civilization similar to ours. High-power transmitters could be built for a wide variety of motives other than the need for two-way communication; this would include beacons built to be seen over thousands of light-years. Extraterrestrial beacon builders would likely have to contend with economic pressures just as their terrestrial counterparts do. Cost, spectral lines near 1 GHz, and interstellar scintillation favor radiating frequencies substantially above the classic "water hole." Therefore, the transmission strategy for a distant, cost-conscious beacon would be a rapid scan of the galactic plane with the intent to cover the angular space. Such pulses would be infrequent events for the receiver. Such beacons built by distant, advanced, wealthy societies would have very different characteristics from what SETI researchers seek. Future searches should pay special attention to areas along the galactic disk where SETI searches have seen coherent signals that have not recurred on the limited listening time intervals we have used. We will need to wait for recurring events that may arriarrive in intermittent bursts. Several new SETI search strategies have emerged from these ideas. We propose a new test for beacons that is based on the Life Plane hypotheses.
20. Studies of interstellar vibrationally-excited molecules
International Nuclear Information System (INIS)
Ziurys, L.M.; Snell, R.L.; Erickson, N.R.
1986-01-01
Several molecules thus far have been detected in the ISM in vibrationally-excited states, including H 2 , SiO, HC 3 N, and CH 3 CN. In order for vibrational-excitation to occur, these species must be present in unusually hot and dense gas and/or where strong infrared radiation is present. In order to do a more thorough investigation of vibrational excitation in the interstellar medium (ISM), studies were done of several mm-wave transitions originating in excited vibrational modes of HCN, an abundant interstellar molecule. Vibrationally-excited HCN was recently detected toward Orion-KL and IRC+10216, using a 12 meter antenna. The J=3-2 rotational transitions were detected in the molecule's lowest vibrational state, the bending mode, which is split into two separate levels, due to l-type doubling. This bending mode lies 1025K above ground state, with an Einstein A coefficient of 3.6/s. The J=3-2 line mode of HCN, which lies 2050K above ground state, was also observed toward IRC+10216, and subsequently in Orion-KL. Further measurements of vibrationally-excited HCN were done using a 14 meter telescope, which include the observations of the (0,1,0) and (0,2,0) modes towards Orion-KL, via their J=3-2 transitions at 265-267 GHz. The spectrum of the J=3-2 line in Orion taken with the 14 meter telescope, is shown, along with a map, which indicates that emission from vibrationally-excited HCN arises from a region probably smaller than the 14 meter telescope's 20 arcsec beam
1. Gitting of infrared data to the interstellar polarization law
Energy Technology Data Exchange (ETDEWEB)
Clarke, D
1984-02-15
The ability of Serkowski's law describing the wavelength dependence of interstellar polarization to encompass new infrared measurements in combination with optical data has been examined. Fitting by least-squares procedures reveals departures from the law in various wavelength zones or at specific wavelength points across the optical and infrared spectrum. These structures may be caused by a combination of effects such as normal experimental noise, complex interstellar clouds or systematic errors in the polarimetry but the possibility remains that some, particularly in the infrared, reflect the scattering properties of interstellar grains. 8 references.
2. Fitting of infrared data to the interstellar polarization law
Energy Technology Data Exchange (ETDEWEB)
Clarke, D [Glasgow Univ., Great Britain
1984-02-15
The ability of Serkowski's law describing the wavelength dependence of interstellar polarization to encompass new infrared measurements in combination with optical data has been examined. Fitting by least-squares procedures reveals departures from the law in various wavelength zones or at specific wavelength points across the optical and infrared spectrum. These structures may be caused by a combination of effects such as normal experimental noise, complex interstellar clouds or systematic errors in the polarimetry but the possibility remains that some, particularly in the infrared, reflect the scattering properties of interstellar grains.
3. Interstellar Ices and Radiation-induced Oxidations of Alcohols
Science.gov (United States)
Hudson, R. L.; Moore, M. H.
2018-04-01
Infrared spectra of ices containing alcohols that are known or potential interstellar molecules are examined before and after irradiation with 1 MeV protons at ∼20 K. The low-temperature oxidation (hydrogen loss) of six alcohols is followed, and conclusions are drawn based on the results. The formation of reaction products is discussed in terms of the literature on the radiation chemistry of alcohols and a systematic variation in their structures. The results from these new laboratory measurements are then applied to a recent study of propargyl alcohol. Connections are drawn between known interstellar molecules, and several new reaction products in interstellar ices are predicted.
4. Chemistry in interstellar space. [environment characteristics influencing reaction dynamics
Science.gov (United States)
Donn, B.
1973-01-01
The particular characteristics of chemistry in interstellar space are determined by the unique environmental conditions involved. Interstellar matter is present at extremely low densities. Large deviations from thermodynamic equilibrium are, therefore, to be expected. A relatively intense ultraviolet radiation is present in many regions. The temperatures are in the range from 5 to 200 K. Data concerning the inhibiting effect of small activation energies in interstellar clouds are presented in a table. A summary of measured activation energies or barrier heights for exothermic exchange reactions is also provided. Problems of molecule formation are discussed, taking into account gas phase reactions and surface catalyzed processes.
5. SEARCHING FOR NAPHTHALENE CATION ABSORPTION IN THE INTERSTELLAR MEDIUM
International Nuclear Information System (INIS)
Searles, Justin M.; Destree, Joshua D.; Snow, Theodore P.; Salama, Farid; York, Donald G.; Dahlstrom, Julie
2011-01-01
Interstellar naphthalene cations (C 10 H + 8 ) have been proposed by a study to be the carriers of a small number of diffuse interstellar bands (DIBs). Using an archive of high signal-to-noise spectra obtained at the Apache Point Observatory, we used two methods to test the hypothesis. Both methods failed to detect significant absorption at lab wavelengths of interstellar spectra with laboratory spectra. We thereby conclude that C 10 H + 8 is not a DIB carrier in typical reddened sight lines.
6. Atomic and molecular excitation mechanisms in the interstellar medium
International Nuclear Information System (INIS)
Sternberg, A.
1986-01-01
The detailed infrared response of dense molecular hydrogen gas to intense ultraviolet radiation fields in photodissociation regions is presented. The thermal and chemical structures of photodissociation regions are analyzed, and the relationship between the emission by molecular hydrogen and trace atomic and molecular species is explored. The ultraviolet spectrum of radiation generated by cosmic rays inside dense molecular clouds is presented, and the resulting rates of photodissociation for a variety of interstellar molecules are calculated. Effects of this radiation on the chemistry of dense molecular clouds are discussed, and it is argued that the cosmic ray induced photons will significantly inhibit the production of complex molecular species. It is argued that the annihilation of electrons and positrons at the galactic center may result in observable infrared line emission by atomic hydrogen. A correlation between the intensity variations of the 511 keV line and the hydrogen infrared lines emitted by the annihilation region is predicted. The observed infrared fluxes from compact infrared sources at the galactic center may be used to constrain theories of pair production there
7. Grain Boundary Complexions
Science.gov (United States)
2014-05-01
Cantwell et al. / Acta Materialia 62 (2014) 1–48 challenging from a scientific perspective, but it can also be very technologically rewarding , given the...energy) is a competing explanation that remains to be explored. Strategies to drive the grain boundary energy toward zero have produced some success...Thompson AM, Soni KK, Chan HM, Harmer MP, Williams DB, Chabala JM, et al. J Am Ceram Soc 1997;80:373. [172] Behera SK. PhD dissertation, Materials Science
8. The Interstellar Medium in External Galaxies: Summaries of contributed papers
Science.gov (United States)
Hollenbach, David J. (Editor); Thronson, Harley A., Jr. (Editor)
1990-01-01
The Second Wyoming Conference entitled, The Interstellar Medium in External Galaxies, was held on July 3 to 7, 1989, to discuss the current understanding of the interstellar medium in external galaxies and to analyze the basic physical processes underlying interstellar phenomena. The papers covered a broad range of research on the gas and dust in external galaxies and focused on such topics as the distribution and morphology of the atomic, molecular, and dust components; the dynamics of the gas and the role of the magnetic field in the dynamics; elemental abundances and gas depletions in the atomic and ionized components; cooling flows; star formation; the correlation of the nonthermal radio continuum with the cool component of the interstellar medium; the origin and effect of hot galactic halos; the absorption line systems seen in distant quasars; and the effect of galactic collisions.
9. Electromagnetic Forces on a Relativistic Spacecraft in the Interstellar Medium
Energy Technology Data Exchange (ETDEWEB)
Hoang, Thiem [Korea Astronomy and Space Science Institute, Daejeon 34055 (Korea, Republic of); Loeb, Abraham, E-mail: [email protected], E-mail: [email protected] [Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA (United States)
2017-10-10
A relativistic spacecraft of the type envisioned by the Breakthrough Starshot initiative will inevitably become charged through collisions with interstellar particles and UV photons. Interstellar magnetic fields would therefore deflect the trajectory of the spacecraft. We calculate the expected deflection for typical interstellar conditions. We also find that the charge distribution of the spacecraft is asymmetric, producing an electric dipole moment. The interaction between the moving electric dipole and the interstellar magnetic field is found to produce a large torque, which can result in fast oscillation of the spacecraft around the axis perpendicular to the direction of motion, with a period of ∼0.5 hr. We then study the spacecraft rotation arising from impulsive torques by dust bombardment. Finally, we discuss the effect of the spacecraft rotation and suggest several methods to mitigate it.
10. The local interstellar medium and gamma-ray astronomy
International Nuclear Information System (INIS)
Lebrun, F.; Paul, J.
1985-08-01
The recent improvement of the calibration of the galaxy counts used as an interstellar-absorption tracer modifies significantly the picture of the local interstellar medium (ISM). Consequently, previous analyses of the γ-ray emission from the local ISM involving galaxy counts have to be revised. In this paper, we consider the implications regarding the cosmic-ray (CR) density in the local ISM, and in particular within Loop I, a nearby supernova remnant (SNR)
11. Stellar and interstellar K lines - Gamma Pegasi and iota Herculis.
Science.gov (United States)
Hobbs, L. M.
1973-01-01
High-resolution scans show that the relatively strong (about 90 mA) K lines of Ca II in the early B stars gamma-Peg and iota-Her are almost entirely stellar in origin, although the latter case includes a small interstellar contribution. Such stellar lines can be of great importance in augmenting the interstellar absorption, up through the earliest of the B stars.
12. The Turbulent Interstellar Medium: Insights and Questions from Numerical Models
OpenAIRE
Mac Low, Mordecai-Mark; de Avillez, Miguel A.; Korpi, Maarit J.
2003-01-01
"The purpose of numerical models is not numbers but insight." (Hamming) In the spirit of this adage, and of Don Cox's approach to scientific speaking, we discuss the questions that the latest generation of numerical models of the interstellar medium raise, at least for us. The energy source for the interstellar turbulence is still under discussion. We review the argument for supernovae dominating in star forming regions. Magnetorotational instability has been suggested as a way of coupling di...
13. New look at radiative association in dense interstellar clouds
International Nuclear Information System (INIS)
Herbst, E.
1980-01-01
A corrected statistical theory of radiative association reactions is presented and discussed. Calculations are undertaken to determine the rate coefficients of a variety of radiative association reactions of possible importance in dense interstellar clouds. Our results confirm the suggestion of Smith and Adams that certain radiative association reactions occur quite rapidly at low temperature and are probably important in the synthesis of complex interstellar molecules
14. Interstellar medium structure and content and gamma ray astronomy
International Nuclear Information System (INIS)
Lebrun, F.
1982-05-01
A general description of gamma-ray astronomy is presented with special emphasis on the study of diffuse gamma-ray emission. This is followed by a collection of reflections and observations on the structure and the gas and dust content of the local interstellar medium. Results of gamma-ray observations on the local interstellar medium are given. The last part is devoted to the whole of the galactic gamma-ray emission and its interpretation [fr
15. The hydrogen coverage of interstellar PAHs [Polycyclic Aromatic Hydrocarbons
International Nuclear Information System (INIS)
Tielens, A.G.G.M.; Allamandola, L.J.; Barker, J.R.; Cohen, M.
1986-02-01
The rate at which the CH bond in interstellar Polycyclic Aromatic Hydrocarbons (PAHs) rupture due to the absorption of a uv photon has been calculated. The results show that small PAHs (less than or equal to 25 carbon atoms) are expected to be partially dehydrogenated in regions with intense uv fields, while large PAHs (greater than or equal to 25 atoms) are expected to be completely hydrogenated in those regions. Because estimate of the carbon content of interstellar PAHs lie in the range of 20 to 25 carbon atoms, dehydrogenation is probably not very important. Because of the absence of other emission features besides the 11.3 micrometer feature in ground-based 8 to 13 micrometer spectra, it has been suggested that interstellar PAHs are partially dehydrogenated. However, IRAS 8 to 22 micrometer spectra of most sources that show strong 7.7 and 11.2 micrometer emission features also show a plateau of emission extending from about 11.3 to 14 micrometer. Like the 11.3 micrometer feature, this new feature is attributed to the CH out of plane bending mode in PAHs. This new feature shows that interstellar PAHs are not as dehydrogenated as estimated from ground-based 8 to 13 micrometer spectra. It also constrains the molecular structure of interstellar PAHs. In particular, it seems that very condensed PAHs, such as coronene and circumcoronene, dominate the interstellar PAH mixture as expected from stability arguments
16. Small-scale structure in the diffuse interstellar medium
International Nuclear Information System (INIS)
Meyer, D.M.
1990-01-01
The initial results of a study to probe the small-scale structure in the diffuse interstellar medium (ISM) through IUE and optical observations of interstellar absorption lines toward both components of resolvable binary stars is reported. The binaries (Kappa CrA, 57 Aql, 59 And, HR 1609/10, 19 Lyn, and Theta Ser) observed with IUE have projected linear separations ranging from 5700 to 700 Au. Except for Kappa CrA, the strengths of the interstellar absorption lines toward both components of these binaries agree to within 10 percent. In the case of Kappa CrA, the optically thin interstellar Mg I and Mn II lines are about 50 percent stronger toward Kappa-2 CrA than Kappa-1 CrA. Higher resolution observations of interstellar Ca II show that this difference is concentrated in the main interstellar component at V(LSR) = 9 + or - 2 km/s. Interestingly, this velocity corresponds to an intervening cloud that may be associated with the prominent Loop I shell in the local ISM. Given the separation (23 arcsec) and distance (120 pc) of Kappa CrA, the line strength variations indicate that this cloud has structure on scales of 2800 AU or less. 21 refs
17. Matrix isolation as a tool for studying interstellar chemical reactions
Science.gov (United States)
Ball, David W.; Ortman, Bryan J.; Hauge, Robert H.; Margrave, John L.
1989-01-01
Since the identification of the OH radical as an interstellar species, over 50 molecular species were identified as interstellar denizens. While identification of new species appears straightforward, an explanation for their mechanisms of formation is not. Most astronomers concede that large bodies like interstellar dust grains are necessary for adsorption of molecules and their energies of reactions, but many of the mechanistic steps are unknown and speculative. It is proposed that data from matrix isolation experiments involving the reactions of refractory materials (especially C, Si, and Fe atoms and clusters) with small molecules (mainly H2, H2O, CO, CO2) are particularly applicable to explaining mechanistic details of likely interstellar chemical reactions. In many cases, matrix isolation techniques are the sole method of studying such reactions; also in many cases, complexations and bond rearrangements yield molecules never before observed. The study of these reactions thus provides a logical basis for the mechanisms of interstellar reactions. A list of reactions is presented that would simulate interstellar chemical reactions. These reactions were studied using FTIR-matrix isolation techniques.
18. Absorption of X-rays in the interstellar medium
International Nuclear Information System (INIS)
Ride, S.K.; Stanford Univ., Calif.; Walker, A.B.C. Jr.; Stanford Univ., Calif.
1977-01-01
In order to interpret soft X-ray spectra of cosmic X-ray sources, it is necessary to know the photoabsorption cross-section of the intervening interstellar material. Current models suggest that the interstellar medium contains two phases which make a substantial contribution to the X-ray opacity: cool, relatively dense clouds that exist in pressure equilibrium with hot, tenuous intercloud regions. We have computed the soft X-ray photoabsorption cross-section (per hydrogen atom) of each of these two phases. The calculation are based on a model of the interstellar medium which includes chemical evolution of the galaxy, the formation of molecules and grains, and the ionization structure of each of each phase. These cross-sections of clouds and of intercloud regions can be combined to yield the total soft X-ray photoabsorption cross-section of the interstellar medium. By choosing the appropriate linear combination of cloud and intercloud cross-sections, we can tailor the total cross-section to a particular line-of-sight. This approach, coupled with our interstellar model, enables us to better describe a wide range of interstellar features such as H II regions, dense (molecular) clouds, or the ionized clouds which may surround binary X-ray sources. (orig.) [de
19. Starry messages: Searching for signatures of interstellar archaeology
Energy Technology Data Exchange (ETDEWEB)
Carrigan, Richard A., Jr.; /Fermilab
2009-12-01
Searching for signatures of cosmic-scale archaeological artifacts such as Dyson spheres or Kardashev civilizations is an interesting alternative to conventional SETI. Uncovering such an artifact does not require the intentional transmission of a signal on the part of the original civilization. This type of search is called interstellar archaeology or sometimes cosmic archaeology. The detection of intelligence elsewhere in the Universe with interstellar archaeology or SETI would have broad implications for science. For example, the constraints of the anthropic principle would have to be loosened if a different type of intelligence was discovered elsewhere. A variety of interstellar archaeology signatures are discussed including non-natural planetary atmospheric constituents, stellar doping with isotopes of nuclear wastes, Dyson spheres, as well as signatures of stellar and galactic-scale engineering. The concept of a Fermi bubble due to interstellar migration is introduced in the discussion of galactic signatures. These potential interstellar archaeological signatures are classified using the Kardashev scale. A modified Drake equation is used to evaluate the relative challenges of finding various sources. With few exceptions interstellar archaeological signatures are clouded and beyond current technological capabilities. However SETI for so-called cultural transmissions and planetary atmosphere signatures are within reach.
20. Infrared spectra of interstellar deuteronated PAHs
Science.gov (United States)
Buragohain, Mridusmita; Pathak, Amit; Sarre, Peter
2015-08-01
Polycyclic Aromatic Hydrocarbon (PAH) molecules have emerged as a potential constituent of the ISM that emit strong features at 3.3, 6.2, 7.7, 8.6, 11.2 and 12.7 μm with weaker and blended features in the 3-20μm region. These features are proposed to arise from the vibrational relaxation of PAH molecules on absorption of background UV photons (Tielens 2008). These IR features have been observed towards almost all types of astronomical objects; say H II regions, photodissociation regions, reflection nebulae, planetary nebulae, young star forming regions, external galaxies, etc. A recent observation has proposed that interstellar PAHs are major reservoir for interstellar deuterium (D) (Peeters et al. 2004). According to the deuterium depletion model' as suggested by Draine (2006), some of the Ds formed in the big bang are depleted in PAHs, which can account for the present value of D/H in the ISM. Hence, study of deuterated PAHs (PADs) is essential in order to measure D/H in the ISM.In this work, we consider another probable category of the large PAH family, i.e. Deuteronated PAHs (DPAH+). Onaka et al. have proposed a D/H ratio which is an order of magnitude smaller than the proposed value of D/H by Draine suggesting that if Ds are depleted in PAHs, they might be accommodated in large PAHs (Onaka et al. 2014). This work reports a Density Functional Theory' calculation of large deuteronated PAHs (coronene, ovalene, circumcoronene and circumcircumcoronene) to determine the expected region of emission features and to find a D/H ratio that is comparable to the observational results. We present a detailed analysis of the IR spectra of these molecules and discuss the possible astrophysical implications.ReferencesDraine B. T. 2006, in ASP Conf. Ser. 348, Proc. Astrophysics in the Far Ultraviolet: Five Years of Discovery with FUSE, ed. G. Sonneborn, H. Moos, B-G Andersson (San Francisco, CA:ASP) 58Onaka T., Mori T. I., Sakon I., Ohsawa R., Kaneda H., Okada Y., Tanaka M
1. boundary dissipation
Directory of Open Access Journals (Sweden)
Mehmet Camurdan
1998-01-01
are coupled by appropriate trace operators. This overall model differs from those previously studied in the literature in that the elastic chamber floor is here more realistically modeled by a hyperbolic Kirchoff equation, rather than by a parabolic Euler-Bernoulli equation with Kelvin-Voight structural damping, as in past literature. Thus, the hyperbolic/parabolic coupled system of past literature is replaced here by a hyperbolic/hyperbolic coupled model. The main result of this paper is a uniform stabilization of the coupled PDE system by a (physically appealing boundary dissipation.
2. Interstellar clouds and the formation of stars
International Nuclear Information System (INIS)
Alfen, H.; Carlqvist, P.
1977-12-01
The 'pseudo-plasma formalism' which up to now has almost completely dominated theoretical astrophysics must be replaced by an experimentally based approach, involving the introduction of a number of neglected plasma phenomena, such as electric double layers, critical velocity, and pinch effect. The general belief that star light is the main ionizer is shown to be doubtful; hydromagnetic conversion of gravitational and kinetic energy may often be much more important. The revised plasma physics is applied to dark clouds and star formation. Magnetic fields do not necessarily counteract the contraction of a cloud, they may just as well 'pinch' the cloud. Magnetic compression may be the main mechanism for forming interstellar clouds and keeping them together. Star formation is due to an instability, but it is very unlikely that it has anything to do with the Jeans instablility. A reasonable mechanism is that the sedimentation of 'dust' (including solid bodies of different size) is triggering off a gravitationally assisted accretion. The study of the evolution of a dark cloud leads to a scenario of planet formation which is reconcilable with the results obtained from studies based on solar system data. This means that the new approach to cosmical plasma physics discussed logically leads to a consistent picture of the evolution of dark clouds and the formation of solar systems
3. Stability of interstellar clouds containing magnetic fields
International Nuclear Information System (INIS)
Langer, W.D.; and Bell Laboratories, Crawford Hill Laboratory, Holmdel, NJ)
1978-01-01
The stability of interstellar clouds against gravitational collapse and fragmentation in the presence of magnetic fields is investigated. A magnetic field can provide pressure support against collapse if it is strongly coupled to the neutral gas; this coupling is mediated by ion-neutral collisions in the gas. The time scale for the growth of perturbations in the gas is found to be a sensitive function of the fractional ion abundance of the gas. For a relatively large fractional ion abundance, corresponding to strong coupling, the collapse of the gas is retarded. Star formation is inhibited in dense clouds and the collapse time for diffuse clouds cn exceed the limit on their lifetime set by disruptive processes. For a small fractional ion abundance, the magnetic fields do not inhibit collapse and the distribution of the masses of collapsing fragments are likely to be quite different in regions of differing ion abundance. The solutions also predict the existence of large-scale density waves corresponding to two gravitational-magnetoacoustic modes. The conditions which best support these modes correspond to those found in the giant molecular clouds
4. VIBRONIC PROGRESSIONS IN SEVERAL DIFFUSE INTERSTELLAR BANDS
International Nuclear Information System (INIS)
Duley, W. W.; Kuzmin, Stanislav
2010-01-01
A number of vibronic progressions based on low-energy vibrational modes of a large molecule have been found in the diffuse interstellar band (DIB) spectrum of HD 183143. Four active vibrational modes have been identified with energies at 5.18 cm -1 , 21.41 cm -1 , 31.55 cm -1 , and 34.02 cm -1 . The mode at 34.02 cm -1 was previously recognized by Herbig. Four bands are associated with this molecule, with origins at 6862.61 A, 6843.64 A, 6203.14 A, and 5545.11 A (14589.1 cm -1 , 14608.08 cm -1 , 16116.41 cm -1 , and 18028.9 cm -1 , respectively). The progressions are harmonic and combination bands are observed involving all modes. The appearance of harmonic, rather than anharmonic, terms in these vibronic progressions is consistent with torsional motion of pendant rings, suggesting that the carrier is a 'floppy' molecule. Some constraints on the type and size of the molecule producing these bands are discussed.
5. Three-Dimensional Messages for Interstellar Communication
Science.gov (United States)
Vakoch, Douglas A.
One of the challenges facing independently evolved civilizations separated by interstellar distances is to communicate information unique to one civilization. One commonly proposed solution is to begin with two-dimensional pictorial representations of mathematical concepts and physical objects, in the hope that this will provide a foundation for overcoming linguistic barriers. However, significant aspects of such representations are highly conventional, and may not be readily intelligible to a civilization with different conventions. The process of teaching conventions of representation may be facilitated by the use of three-dimensional representations redundantly encoded in multiple formats (e.g., as both vectors and as rasters). After having illustrated specific conventions for representing mathematical objects in a three-dimensional space, this method can be used to describe a physical environment shared by transmitter and receiver: a three-dimensional space defined by the transmitter--receiver axis, and containing stars within that space. This method can be extended to show three-dimensional representations varying over time. Having clarified conventions for representing objects potentially familiar to both sender and receiver, novel objects can subsequently be depicted. This is illustrated through sequences showing interactions between human beings, which provide information about human behavior and personality. Extensions of this method may allow the communication of such culture-specific features as aesthetic judgments and religious beliefs. Limitations of this approach will be noted, with specific reference to ETI who are not primarily visual.
6. X-ray scattering by interstellar dust
International Nuclear Information System (INIS)
Rolf, D.
1980-10-01
This thesis reports work carried out to make a first observation of x-rays scattered by interstellar dust grains. Data about the dust, obtained at wavelengths ranging from the infrared to ultra-violet spectral regions, are discussed in order to establish a useful description of the grains themselves. This is then used to estimate the magnitude and form of the expected x-ray scattering effect which is shown to manifest itself as a diffuse halo accompanying the image of a celestial x-ray source. Two x-ray imaging experiments are then discussed. The first, specifically proposed to look for this effect surrounding a point x-ray source, was the Skylark 1611 project, and comprised an imaging proportional counter coupled to an x-ray mirror. This is described up to its final calibration when the basis for a concise model of its point response function was established. The experiment was not carried out but its objective and the experience gained during its testing were transferred to the second of the x-ray imaging experiments, the Einstein Observatory. The new instrumental characteristics are described and a model for its point response function is developed. Using this, image data for the point x-ray source GX339-4 is shown to exhibit the sought after scattering phenomenon. (author)
7. Interstellar Extinction in 20 Open Star Clusters
Science.gov (United States)
Rangwal, Geeta; Yadav, R. K. S.; Durgapal, Alok K.; Bisht, D.
2017-12-01
The interstellar extinction law in 20 open star clusters namely, Berkeley 7, Collinder 69, Hogg 10, NGC 2362, Czernik 43, NGC 6530, NGC 6871, Bochum 10, Haffner 18, IC 4996, NGC 2384, NGC 6193, NGC 6618, NGC 7160, Collinder 232, Haffner 19, NGC 2401, NGC 6231, NGC 6823, and NGC 7380 have been studied in the optical and near-IR wavelength ranges. The difference between maximum and minimum values of E(B - V) indicates the presence of non-uniform extinction in all the clusters except Collinder 69, NGC 2362, and NGC 2384. The colour excess ratios are consistent with a normal extinction law for the clusters NGC 6823, Haffner 18, Haffner 19, NGC 7160, NGC 6193, NGC 2401, NGC 2384, NGC 6871, NGC 7380, Berkeley 7, Collinder 69, and IC 4996. We have found that the differential colour-excess ΔE(B - V), which may be due to the occurrence of dust and gas inside the clusters, decreases with the age of the clusters. A spatial variation of colour excess is found in NGC 6193 in the sense that it decreases from east to west in the cluster region. For the clusters Berkeley 7, NGC 7380, and NGC 6871, a dependence of colour excess E(B - V) with spectral class and luminosity is observed. Eight stars in Collinder 232, four stars in NGC 6530, and one star in NGC 6231 have excess flux in near-IR. This indicates that these stars may have circumstellar material around them.
8. DYNAMIC SPECTRAL MAPPING OF INTERSTELLAR PLASMA LENSES
Energy Technology Data Exchange (ETDEWEB)
Tuntsov, Artem V.; Walker, Mark A. [Manly Astrophysics, 3/22 Cliff Street, Manly 2095 (Australia); Koopmans, Leon V. E. [Kapteyn Astronomical Institute, University of Groningen, P.O. Box 800, NL-9700 AV Groningen (Netherlands); Bannister, Keith W.; Stevens, Jamie; Johnston, Simon [CSIRO Astronomy and Space Science, P.O. Box 76, Epping NSW 1710 (Australia); Reynolds, Cormac; Bignall, Hayley E., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [International Centre for Radio Astronomy Research—Curtin University, Perth (Australia)
2016-02-01
Compact radio sources sometimes exhibit intervals of large, rapid changes in their flux density, due to lensing by interstellar plasma crossing the line of sight. A novel survey program has made it possible to discover these “Extreme Scattering Events” (ESEs) in real time, resulting in a high-quality dynamic spectrum of an ESE observed in PKS 1939–315. Here we present a method for determining the column-density profile of a plasma lens, given only the dynamic radio spectrum of the lensed source, under the assumption that the lens is either axisymmetric or totally anisotropic. Our technique relies on the known, strong frequency dependence of the plasma refractive index in order to determine how points in the dynamic spectrum map to positions on the lens. We apply our method to high-frequency (4.2–10.8 GHz) data from the Australia Telescope Compact Array of the PKS 1939–315 ESE. The derived electron column-density profiles are very similar for the two geometries we consider, and both yield a good visual match to the data. However, the fit residuals are substantially above the noise level, and deficiencies are evident when we compare the predictions of our model to lower-frequency (1.6–3.1 GHz) data on the same ESE, thus motivating future development of more sophisticated inversion techniques.
9. Detection of interstellar vibrationally excited HCN
International Nuclear Information System (INIS)
Ziurys, L.M.; Turner, B.E.
1986-01-01
Vibrationally excited HCN has been observed for the first time in the interstellar medium. The J = 3-2 rotational transitions of the l-doubled (0,1/sup 1d/,1c, 0) bending mode of HCN have been detected toward Orion-KL and IRC +10216. In Orion, the overall column density in the (0,1,0) mode, which exclusively samples the ''hot core,'' is 1.7-10 16 cm -2 and can be understood in terms of the ''doughnut'' model for Orion. The ground-state HCN column density implied by the excited-state observations is 2.3 x 10 18 cm -2 in the hot core, at least one order of magnitude greater than the column densities derived for HCN in its spike and plateau/doughnut components. Radiative excitation by 14 μm flux from IRc2 accounts for the (0,1,0) population provided the hot core is approx.6-7 x 10 16 cm distant from IRc2, in agreement with the ''cavity'' model for KL. Toward IRC +10216 we have detected J = 3-2 transitions of both (0,1/sup 1c/,/sup 1d/,0) and (0,2 0 ,0) excited states. The spectral profiles have been modeled to yield abundances and excitation conditions throughout the expanding envelope
10. Interstellar extinction in the Large Magellanic Cloud
International Nuclear Information System (INIS)
Nandy, K.; Morgan, D.H.; Willis, A.J.; Wilson, R.; Gondhalekar, P.M.
1981-01-01
A systematic investigation of interstellar extinction in the ultraviolet as a function of position in the Large Magellanic Cloud has been made from an enlarged sample of reddened and comparison stars distributed throughout the cloud. Except for one star SK-69-108, the most reddened star of our sample, the shape of the extinction curves for the LMC stars do not show significant variations. All curves show an increase in extinction towards 2200 A, but some have maxima near 2200 A, some near 1900 A. It has been shown that the feature of the extinction curve near 1900 A is caused by the mismatch of the stellar F III 1920 A feature. The strength of this 1920 A feature as a function of luminosity and spectral type has been determined. The extinction curves have been corrected for the mismatch of the 1920 feature and a single mean extinction curve for the LMC normalized to Asub(V) = 0 and Esub(B-V) = 1 is presented. For the same value of Esub(B-V) the LMC stars show the 2200 A feature weaker by a factor 2 as compared with the galactic stars. Higher extinction shortward of 2000 A in the LMC extinction curves than that in our Galaxy, as reported in earlier papers, is confirmed. (author)
11. Interstellar extinction and polarization in the infrared
International Nuclear Information System (INIS)
Martin, P.G.; Whittet, D.C.B.
1990-01-01
The wavelength dependences of interstellar continuum extinction and polarization in the range 0.35-5 microns are examined. The existence of a universal extinction curve with power law index of about 1.8 extending from the near-IR to at least 5 microns appears to be established for both diffuse and dense cloud dust. The polarization yields evidence for some degree of universality in the 1.6-5 micron regime which may be represented by a power law with index 1.5-2.0, encompassing that for extinction. The form of the polarization curve in the IR seems independent of the wavelength at which the degree of polarization peaks in the optical, implying that variations in that wavelength are caused by changes in the optical properties of the particle at blue-visible rather than IR wavelengths. It is argued that the more significant alterations of the grain size distribution from one environment to another occur for the smaller particles. 47 refs
12. Modelling interstellar structures around Vela X-1
Science.gov (United States)
Gvaramadze, V. V.; Alexashov, D. B.; Katushkina, O. A.; Kniazev, A. Y.
2018-03-01
We report the discovery of filamentary structures stretched behind the bow-shock-producing high-mass X-ray binary Vela X-1 using the SuperCOSMOS H-alpha Survey and present the results of optical spectroscopy of the bow shock carried out with the Southern African Large Telescope. The geometry of the detected structures suggests that Vela X-1 has encountered a wedge-like layer of enhanced density on its way and that the shocked material of the layer partially outlines a wake downstream of Vela X-1. To substantiate this suggestion, we carried out 3D magnetohydrodynamic simulations of interaction between Vela X-1 and the layer for three limiting cases. Namely, we run simulations in which (i) the stellar wind and the interstellar medium (ISM) were treated as pure hydrodynamic flows, (ii) a homogeneous magnetic field was added to the ISM, while the stellar wind was assumed to be unmagnetized, and (iii) the stellar wind was assumed to possess a helical magnetic field, while there was no magnetic field in the ISM. We found that although the first two simulations can provide a rough agreement with the observations, only the third one allowed us to reproduce not only the wake behind Vela X-1, but also the general geometry of the bow shock ahead of it.
13. Deuterium fractionation in dense interstellar clouds
International Nuclear Information System (INIS)
Millar, T.J.; Bennett, A.; Herbst, E.
1989-01-01
The time-dependent gas-phase chemistry of deuterium fractionation in dense interstellar clouds ranging in temperature between 10 and 70 K was investigated using a pseudo-time-dependent model similar to that of Brown and Rice (1986). The present approach, however, considers much more complex species, uses more deuterium fractionation reactions, and includes the use of new branching ratios for dissociative recombinations reactions. Results indicate that, in cold clouds, the major and most global source of deuterium fractionation is H2D(+) and ions derived from it, such as DCO(+) and H2DO(+). In warmer clouds, reactions of CH2D(+), C2HD(+), and associated species lead to significant fractionation even at 70 K, which is the assumed Orion temperature. The deuterium abundance ratios calculated at 10 K are consistent with those observed in TMC-1 for most species. However, a comparison between theory and observatiom for Orion, indicates that, for species in the ambient molecular cloud, the early-time results obtained with the old dissociative recombination branching ratios are superior if a temperature of 70 K is utilized. 60 refs
14. Deuterium fractionation in dense interstellar clouds
Science.gov (United States)
Millar, T. J.; Bennett, A.; Herbst, Eric
1989-05-01
The time-dependent gas-phase chemistry of deuterium fractionation in dense interstellar clouds ranging in temperature between 10 and 70 K was investigated using a pseudo-time-dependent model similar to that of Brown and Rice (1986). The present approach, however, considers much more complex species, uses more deuterium fractionation reactions, and includes the use of new branching ratios for dissociative recombinations reactions. Results indicate that, in cold clouds, the major and most global source of deuterium fractionation is H2D(+) and ions derived from it, such as DCO(+) and H2DO(+). In warmer clouds, reactions of CH2D(+), C2HD(+), and associated species lead to significant fractionation even at 70 K, which is the assumed Orion temperature. The deuterium abundance ratios calculated at 10 K are consistent with those observed in TMC-1 for most species. However, a comparison between theory and observatiom for Orion, indicates that, for species in the ambient molecular cloud, the early-time results obtained with the old dissociative recombination branching ratios are superior if a temperature of 70 K is utilized.
15. Hydrocarbons on Saturns Satellites: Relationship to Interstellar Dust and the Solar Nebula
Science.gov (United States)
Cruikshank, D. P.
2012-01-01
To understand the origin and evolution of our Solar System, and the basic components that led to life on Earth, we study interstellar and planetary spectroscopic signatures. The possible relationship of organic material detected in carbonaceous meteorites, interplanetary dust particles (IDPs), comets and the interstellar medium have been the source of speculation over the years as the composition and processes that governed the early solar nebula have been explored to understand the extent to which primitive material survived or became processed. The Cassini VIMS has provided new data relevant to this problem. Three of Saturn's satellites, Phoebe, Iapetus, and Hyperion, are found to have aromatic and aliphatic hydrocarbons on their surfaces. The aromatic hydrocarbon signature (C-H stretching mode at 3.28 micrometers) is proportionally significantly stronger (relative to the aliphatic bands) than that seen in other Solar System bodies (e.g., comets) and materials (Stardust samples, IDPs, meteorites) and the distinctive sub-features of the 3.4 micrometer aliphatic band (CH2 and CH3 groups) are reminiscent of those widely detected throughout the diffuse ISM. Phoebe may be a captured object that originated in the region beyond the present orbit of Neptune, where the solar nebula contained a large fraction of original interstellar ice and dust that was less processed than material closer to the Sun. Debris from Phoebe now resident on Iapetus and Hyperion, as well as o Phoebe itself, thus presents a unique blend of hydrocarbons, amenable to comparisons with interstellar hydrocarbons and other Solar System materials. The dust ring surrounding Saturn, in which Phoebe is embedded, probably originated from a collision with Phoebe. Dust ring particles are the likely source of the organic-bearing materials, and perhaps the recently identified small particles of Fe detected on Saturn's satellites. Lab measurements of the absolute band strengths of representative aliphatic and
16. Status of Solar Sail Propulsion Within NASA - Moving Toward Interstellar Travel
Science.gov (United States)
Johnson, Les
2015-01-01
NASA is developing solar sail propulsion for two near-term missions and laying the groundwork for their future use in deep space and interstellar precursor missions. Solar sails use sunlight to propel vehicles through space by reflecting solar photons from a large, mirror-like sail made of a lightweight, highly reflective material. This continuous photon pressure provides propellantless thrust, allowing for very high (Delta)V maneuvers on long-duration, deep space exploration. Since reflected light produces thrust, solar sails require no onboard propellant. The Near Earth Asteroid (NEA) Scout mission, managed by MSFC, will use the sail as primary propulsion allowing it to survey and image one or more NEA's of interest for possible future human exploration. Lunar Flashlight, managed by JPL, will search for and map volatiles in permanently shadowed Lunar craters using a solar sail as a gigantic mirror to steer sunlight into the shaded craters. The Lunar Flashlight spacecraft will also use the propulsive solar sail to maneuver into a lunar polar orbit. Both missions use a 6U cubesat architecture, a common an 85 sq m solar sail, and will weigh less than 12 kilograms. Both missions will be launched on the first flight of the Space Launch System in 2018. NEA Scout and Lunar Flashlight will serve as important milestones in the development of solar sail propulsion technology for future, more ambitious missions including the Interstellar Probe - a mission long desired by the space science community which would send a robotic probe beyond the edge of the solar system to a distance of 250 Astronomical Units or more. This paper will summarize the development status of NEA Scout and Lunar Flashlight and describe the next steps required to enable an interstellar solar sail capability.
17. Distributed Tuning of Boundary Resources
DEFF Research Database (Denmark)
Eaton, Ben; Elaluf-Calderwood, Silvia; Sørensen, Carsten
2015-01-01
in the context of a paradoxical tension between the logic of generative and democratic innovations and the logic of infrastructural control. Boundary resources play a critical role in managing the tension as a firm that owns the infrastructure can secure its control over the service system while independent...... firms can participate in the service system. In this study, we explore the evolution of boundary resources. Drawing on Pickering’s (1993) and Barrett et al.’s (2012) conceptualizations of tuning, the paper seeks to forward our understanding of how heterogeneous actors engage in the tuning of boundary...
18. INTERSTELLAR ABUNDANCES TOWARD X Per, REVISITED
International Nuclear Information System (INIS)
Valencic, Lynne A.; Smith, Randall K.
2013-01-01
The nearby X-ray binary X Per (HD 24534) provides a useful beacon with which to examine dust grain types and measure elemental abundances in the local interstellar medium (ISM). The absorption features of O, Fe, Mg, and Si along this line of sight were measured using spectra from the Chandra X-Ray Observatory's LETG/ACIS-S and XMM-Newton's RGS instruments, and the Spex software package. The spectra were fit with dust analogs measured in the laboratory. The O, Mg, and Si abundances were compared to those from standard references, and the O abundance was compared to that along lines of sight toward other X-ray binaries. The results are as follows. First, it was found that a combination of MgSiO 3 (enstatite) and Mg 1.6 Fe 0.4 SiO 4 (olivine) provided the best fit to the O K edge, with N(MgSiO 3 )/N(Mg 1.6 Fe 0.4 SiO 4 ) = 3.4. Second, the Fe L edge could be fit with models that included metallic iron, but it was not well described by the laboratory spectra currently available. Third, the total abundances of O, Mg, and Si were in very good agreement with that of recently re-analyzed B stars, suggesting that they are good indicators of abundances in the local ISM, and the depletions were also in agreement with expected values for the diffuse ISM. Finally, the O abundances found from X-ray binary absorption spectra show a similar correlation with Galactocentric distances as seen in other objects.
19. Surfatron accelerator in the local interstellar cloud
Energy Technology Data Exchange (ETDEWEB)
Loznikov, V. M., E-mail: [email protected]; Erokhin, N. S.; Zol’nikova, N. N.; Mikhailovskaya, L. A. [Russian Academy of Sciences, Space Research Institute (Russian Federation)
2017-01-15
Taking into account results of numerous experiments, the variability of the energy spectra of cosmic rays (protons and helium nuclei) in the energy range of 10 GeV to ~10{sup 7} GeV is explained on the basis of a hypothesis of the existence of two variable sources close to the Sun. The first (soft) surfatron source (with a size of ~100 AU) is located at the periphery of the heliosphere. The second (hard) surfatron source (with a size of ~1 pc) is situated in the Local Interstellar Cloud (LIC) at a distance of <1 pc. The constant background is described by a power-law spectrum with a slope of ~2.75. The variable heliospheric surfatron source is described by a power-law spectrum with a variable amplitude, slope, and cutoff energy, the maximum cutoff energy being in the range of E{sub CH}/Z < 1000 GeV. The variable surfatron source in the LIC is described by a power-law spectrum with a variable amplitude, slope, and cut-off energy, the maximum cut-off energy being E{sub Ð}¡{sub L}/Z ≤ 3 × 10{sup 6} GeV. The proposed model is used to approximate data from several experiments performed at close times. The energy of each cosmic-ray component is calculated. The possibility of surfatron acceleration of Fe nuclei (Z = 26) in the LIC up to an energy of E{sub CL} ~ 10{sup 17} eV and electron and positrons to the “knee” in the energy spectrum is predicted. By numerically solving a system of nonlinear equations describing the interaction between an electromagnetic wave and a charged particle with an energy of up to E/Z ~ 3 × 10{sup 6} GeV, the possibility of trapping, confinement, and acceleration of charged cosmic-ray particles by a quasi-longitudinal plasma wave is demonstrated.
20. Organic compounds in circumstellar and interstellar environments.
Science.gov (United States)
Kwok, Sun
2015-06-01
Recent research has discovered that complex organic matter is prevalent throughout the Universe. In the Solar System, it is found in meteorites, comets, interplanetary dust particles, and planetary satellites. Spectroscopic signatures of organics with aromatic/aliphatic structures are also found in stellar ejecta, diffuse interstellar medium, and external galaxies. From space infrared spectroscopic observations, we have found that complex organics can be synthesized in the late stages of stellar evolution. Shortly after the nuclear synthesis of the element carbon, organic gas-phase molecules are formed in the stellar winds, which later condense into solid organic particles. This organic synthesis occurs over very short time scales of about a thousand years. In order to determine the chemical structures of these stellar organics, comparisons are made with particles produced in the laboratory. Using the technique of chemical vapor deposition, artificial organic particles have been created by injecting energy into gas-phase hydrocarbon molecules. These comparisons led us to believe that the stellar organics are best described as amorphous carbonaceous nanoparticles with mixed aromatic and aliphatic components. The chemical structures of the stellar organics show strong similarity to the insoluble organic matter found in meteorites. Isotopic analysis of meteorites and interplanetary dust collected in the upper atmospheres have revealed the presence of pre-solar grains similar to those formed in old stars. This provides a direct link between star dust and the Solar System and raises the possibility that the early Solar System was chemically enriched by stellar ejecta with the potential of influencing the origin of life on Earth.
1. Solid H2 in the interstellar medium
Science.gov (United States)
Füglistaler, A.; Pfenniger, D.
2018-06-01
Context. Condensation of H2 in the interstellar medium (ISM) has long been seen as a possibility, either by deposition on dust grains or thanks to a phase transition combined with self-gravity. H2 condensation might explain the observed low efficiency of star formation and might help to hide baryons in spiral galaxies. Aims: Our aim is to quantify the solid fraction of H2 in the ISM due to a phase transition including self-gravity for different densities and temperatures in order to use the results in more complex simulations of the ISM as subgrid physics. Methods: We used molecular dynamics simulations of fluids at different temperatures and densities to study the formation of solids. Once the simulations reached a steady state, we calculated the solid mass fraction, energy increase, and timescales. By determining the power laws measured over several orders of magnitude, we extrapolated to lower densities the higher density fluids that can be simulated with current computers. Results: The solid fraction and energy increase of fluids in a phase transition are above 0.1 and do not follow a power law. Fluids out of a phase transition are still forming a small amount of solids due to chance encounters of molecules. The solid mass fraction and energy increase of these fluids are linearly dependent on density and can easily be extrapolated. The timescale is below one second, the condensation can be considered instantaneous. Conclusions: The presence of solid H2 grains has important dynamic implications on the ISM as they may be the building blocks for larger solid bodies when gravity is included. We provide the solid mass fraction, energy increase, and timescales for high density fluids and extrapolation laws for lower densities.
2. THE YOUNG INTERSTELLAR BUBBLE WITHIN THE ROSETTE NEBULA
International Nuclear Information System (INIS)
Bruhweiler, F. C.; Bourdin, M. O.; Freire Ferrero, R.; Gull, T. R.
2010-01-01
We use high-resolution International Ultraviolet Explorer (IUE) data and the interstellar (IS) features of highly ionized Si IV and C IV seen toward the young, bright OB stars of NGC 2244 in the core of the Rosette Nebula to study the physics of young IS bubbles. Two discrete velocity components in Si IV and C IV are seen toward stars in the 6.2 pc radius central cavity, while only a single velocity component is seen toward those stars in the surrounding H II region, at the perimeter and external to this cavity. The central region shows characteristics of a very young, windblown bubble. The shell around the central hot cavity is expanding at 56 km s -1 with respect to the embedded OB stars, while the surrounding H II region of the Rosette is expanding at ∼13 km s -1 . Even though these stars are quite young (∼2-4 Myr), both the radius and expansion velocity of the 6.2 pc inner shell point to a far younger age; t age ∼ 6.4 x 10 4 years. These results represent a strong contradiction to theory and present modeling, where much larger bubbles are predicted around individual O stars and O associations. Specifically, the results for this small bubble and its deduced age extend the 'missing wind luminosity problem' to young evolving bubbles. These results indicate that OB star winds mix the surrounding H II regions and the wind kinetic energy is converted to turbulence and radiated away in the dense H II regions. These winds do not form hot, adiabatically expanding cavities. True IS bubbles appear only to form at later evolutionary times, perhaps triggered by increased mass loss rates or discrete ejection events. Means for rectifying discrepancies between theory and observations are discussed.
3. Dust clouds in Orion and the interstellar neutral hydrogen distribution
International Nuclear Information System (INIS)
Bystrova, N.V.
1989-01-01
According to published examples of the far IR observations in the Orion and its surroundings, several well defined dust clouds of different sizes and structure are present. For comparison of these clouds with the neutral hydrogen distribution on the area of approx. 1000 sq degs, the data from Pulkovo Sky Survey in the interstellar neutral Hydrogen Radio Line as well as special observations with the RATAN-600 telescope in 21 cm line were used. From the materials of Pulkovo HI Survey, the data were taken near the line emission at ten velocities between -21.8 and +25.6 km/s LSR for the structural component of the interstellar hydrogen emission. The results given concern mainly the Orion's Great Dust Cloud and the Lambda Orionis region where the information about the situation with the dust and interstellar hydrogen is very essential for interpretation
4. Cosmic ray diffusion in a violent interstellar medium
International Nuclear Information System (INIS)
Bykov, A.M.; Toptygin, I.N.
1985-01-01
A variety of the avaiable observational data on the cosmic ray (CR) spectrum, anisotropy and composition are in good agreement with a suggestion on the diffusion propagation of CR with energy below 10(15) eV in the interstellar medium. The magnitude of the CR diffusion coefficient and its energy dependence are determined by interstellar medium (ISM) magnetic field spectra. Direct observational data on magnetic field spectra are still absent. A theoretical model to the turbulence generation in the multiphase ISM is resented. The model is based on the multiple generation of secondary shocks and concomitant large-scale rarefactions due to supernova shock interactions with interstellar clouds. The distribution function for ISM shocks are derived to include supernova statistics, diffuse cloud distribution, and various shock wave propagation regimes. This permits calculation of the ISM magnetic field fluctuation spectrum and CR diffusion coefficient for the hot phase of ISM
5. Magnetic seismology of interstellar gas clouds: Unveiling a hidden dimension.
Science.gov (United States)
Tritsis, Aris; Tassis, Konstantinos
2018-05-11
Stars and planets are formed inside dense interstellar molecular clouds by processes imprinted on the three-dimensional (3D) morphology of the clouds. Determining the 3D structure of interstellar clouds remains challenging because of projection effects and difficulties measuring the extent of the clouds along the line of sight. We report the detection of normal vibrational modes in the isolated interstellar cloud Musca, allowing determination of the 3D physical dimensions of the cloud. We found that Musca is vibrating globally, with the characteristic modes of a sheet viewed edge on, not the characteristics of a filament as previously supposed. We reconstructed the physical properties of Musca through 3D magnetohydrodynamic simulations, reproducing the observed normal modes and confirming a sheetlike morphology. Copyright © 2018 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works.
6. Necessity for non-standard models of interstellar turbulence. The 'Champagne bottle' model
Energy Technology Data Exchange (ETDEWEB)
Bonazzola, S; Celnikier, L M; Chevreton, M [Observatoire de Paris, Section de Meudon, 92 (France)
1978-01-01
A complete treatment of interstellar pulsar scintillation by the Physically Thin Screen phase changing model allows one to obtain better agreement with observation and thereby extract new information about the turbulence structure of the interstellar plasma.
7. On the necessity for non-standard models of interstellar turbulence. The 'Champagne bottle' model
International Nuclear Information System (INIS)
Bonazzola, S.; Celnikier, L.M.; Chevreton, M.
1978-01-01
A complete treatment of interstellar pulsar scintillation by the Physically Thin Screen phase changing model allows one to obtain better agreement with observation and thereby extract new information about the turbulence structure of the interstellar plasma
8. Optimal boundary control and boundary stabilization of hyperbolic systems
CERN Document Server
Gugat, Martin
2015-01-01
This brief considers recent results on optimal control and stabilization of systems governed by hyperbolic partial differential equations, specifically those in which the control action takes place at the boundary. The wave equation is used as a typical example of a linear system, through which the author explores initial boundary value problems, concepts of exact controllability, optimal exact control, and boundary stabilization. Nonlinear systems are also covered, with the Korteweg-de Vries and Burgers Equations serving as standard examples. To keep the presentation as accessible as possible, the author uses the case of a system with a state that is defined on a finite space interval, so that there are only two boundary points where the system can be controlled. Graduate and post-graduate students as well as researchers in the field will find this to be an accessible introduction to problems of optimal control and stabilization.
9. Brain response to prosodic boundary cues depends on boundary position
Directory of Open Access Journals (Sweden)
Julia eHolzgrefe
2013-07-01
Full Text Available Prosodic information is crucial for spoken language comprehension and especially for syntactic parsing, because prosodic cues guide the hearer’s syntactic analysis. The time course and mechanisms of this interplay of prosody and syntax are not yet well understood. In particular, there is an ongoing debate whether local prosodic cues are taken into account automatically or whether they are processed in relation to the global prosodic context in which they appear. The present study explores whether the perception of a prosodic boundary is affected by its position within an utterance. In an event-related potential (ERP study we tested if the brain response evoked by the prosodic boundary differs when the boundary occurs early in a list of three names connected by conjunctions (i.e., after the first name as compared to later in the utterance (i.e., after the second name. A closure positive shift (CPS — marking the processing of a prosodic phrase boundary — was elicited only for stimuli with a late boundary, but not for stimuli with an early boundary. This result is further evidence for an immediate integration of prosodic information into the parsing of an utterance. In addition, it shows that the processing of prosodic boundary cues depends on the previously processed information from the preceding prosodic context.
10. Interstellar Scintillation and Scattering of Micro-arc-second AGN
Directory of Open Access Journals (Sweden)
David L. Jauncey
2016-11-01
Full Text Available The discovery of the first quasar 3C 273 led directly to the discovery of their variability at optical and radio wavelengths. We review the radio variability observations, in particular the variability found at frequencies below 1 GHz, as well as those exhibiting intra-day variability (IDV at cm wavelengths. Observations have shown that IDV arises principally from scintillation caused by scattering in the ionized interstellar medium of our Galaxy. The sensitivity of interstellar scintillation towards source angular sizes has provided a powerful tool for studying the most compact components of radio-loud AGN at microarcsecond and milliarcsecond scale resolution.
11. Thermoluminescence of Simulated Interstellar Matter after Gamma-ray Irradiation
OpenAIRE
Koike, K.; Nakagawa, M.; Koike, C.; Okada, M.; Chihara, H.
2002-01-01
Interstellar matter is known to be strongly irradiated by radiation and several types of cosmic ray particles. Simulated interstellar matter, such as forsterite $\\rm Mg_{2}SiO_{4}$, enstatite $\\rm MgSiO_{3}$ and magnesite $\\rm MgCO_{3}$ has been irradiated with the $\\rm ^{60}Co$ gamma-rays in liquid nitrogen, and also irradiated with fast neutrons at 10 K and 70 K by making use of the low-temperature irradiation facility of Kyoto University Reactor (KUR-LTL. Maximum fast neutron dose is \$10^{...
12. Quenched carbonaceous composite (QCC): a likely candidate for interstellar grains
International Nuclear Information System (INIS)
Sakata, A.; Wada, S.; Tanabe, T.; Onaka, T.
1984-01-01
The authors have recently reported that a carbonaceous composite synthesized from a hydrocarbon plasma shows an extinction property quite resembling the observed average interstellar extinction curve around the 220 nm hump. This composite is synthesized by quenching the excited gas ejecting from a plasma of methane gas, so it is called 'quenched carbonaceous composite' or 'QCC'. A recent study of QCC in the infrared region has shown that QCC can also account for some of the unidentified bands in the infrared region detected in several celestial objects. These results suggest that most of the pronounced features of the interstellar grains originate from substances whose major constituent is carbon. (author)
13. A photometric map of interstellar reddening within 100 PC
Science.gov (United States)
Perry, C. L.; Johnston, L.; Crawford, D. L.
1982-12-01
Color excesses and distances are calculated for 300 bright, northern, late F stars using uvby beta photometric indices. The data allow an extension of the earlier maps by Perry and Johnston of the spatial distribution of interstellar reddening into the local (r less than 100 pc) solar neighborhood. Some definite conclusions are made regarding the distribution of interstellar dust in the northern hemisphere and within 300 pc of the sun by merging these results and the polarimetric observations by Tinbergen (1982) for 180 stars within 35 pc of the sun.
14. Rotational Spectra in 29 Vibrationally Excited States of Interstellar Aminoacetonitrile
Energy Technology Data Exchange (ETDEWEB)
Kolesniková, L.; Alonso, E. R.; Mata, S.; Alonso, J. L. [Grupo de Espectroscopia Molecular (GEM), Edificio Quifima, Área de Química-Física, Laboratorios de Espectroscopia y Bioespectroscopia, Parque Científico UVa, Unidad Asociada CSIC, Universidad de Valladolid, E-47011 Valladolid (Spain)
2017-04-01
We report a detailed spectroscopic investigation of the interstellar aminoacetonitrile, a possible precursor molecule of glycine. Using a combination of Stark and frequency-modulation microwave and millimeter wave spectroscopies, we observed and analyzed the room-temperature rotational spectra of 29 excited states with energies up to 1000 cm{sup −1}. We also observed the {sup 13}C isotopologues in the ground vibrational state in natural abundance (1.1%). The extensive data set of more than 2000 new rotational transitions will support further identifications of aminoacetonitrile in the interstellar medium.
15. UV IRRADIATION OF AROMATIC NITROGEN HETEROCYCLES IN INTERSTELLAR ICE ANALOGS
Science.gov (United States)
Elsila, J. E.; Bernstein, M. P.; Sanford, S. A.
2005-01-01
Here, we present information on the properties of the ANH quinoline frozen in interstellar water-ice analogs. Quinoline is a two-ring compound structurally analogous to the PAH naphthalene. In this work, binary mixtures of water and quinoline were frozen to create interstellar ice analogs, which were then subjected to ultraviolet photolysis. We will present the infrared spectra of the resulting ices at various temperatures, as well as chromatographic analysis of the residues remaining upon warm-up of these ices to room temperature.
16. Interstellar gas near and within the solar system
International Nuclear Information System (INIS)
Burgin, M.S.
1981-01-01
The picture of the interaction between the local interstellar medium (LISM) and the solar environment developed in recent years is described, and prospects are discussed for obtaining complete information about the LISM. Special attention is given to the neutral component of the LISM, particularly to the results of observations of the uv radiation scattered from hydrogen and helium atoms penetrating the solar system from interstellar space. The properties of the LISM plasma are considered only as they pertain to the interaction with the neutral component
17. Mechanical heating of the interstellar medium. I. The source and rate
International Nuclear Information System (INIS)
Cox, D.P.
1979-01-01
A simple model is presented for the evolution of a supernova disturbance in the very low density, high temperature, interstellar matrix in order to explore consequences of such disturbances on the interstellar clouds. It is assumed that higher density material is sufficiently common to impede the velocity field. It is further assumed that thermal conduction is magnetically quenched between the matrix and H I regions. The individual disturbances evolve very rapidly (tauapprox.3 x 10 5 years) to very large sizes (Rapprox.140 pc) without appreciable radiative cooling before the interior pressure becomes comparable to the ambient pressure. The net effect of the overlapping of ancient disturbances is then shown to be capable of determining this ambient presure.The work done by such blast waves in compressing interstellar clouds is estimated. An individual disturbance is found to lose at least a modest fraction of its energy in this way. The calculated power input to individual clouds is very large, resulting in large-amplitude vibrations similar to what is observed. The heating is partly impulsive (most clouds should contain at least one shock of modest strength at any time) and partly quasi-steady due to vibrational dissipation. Within large uncertainties and variations, the material temperatures are expected to be less than 100 K for n> or approx. =6 cm -3 and approach 10 4 K for n -3 . Between these densities, the temperature depends sensitively on density, elemental depletions, and fractional ionization. Thus the power input is of the magnitude required to provide a cloud, intercloud segregation of material. Unlike earlier models, however, the heating is not intrinsically accompanied by ionization. Finally, the net acceleration of clouds by these blast waves is found to be small unless the clouds initially have n -3
18. Stochastic evolution of refractory interstellar dust during the chemical evolution of a two-phase interstellar medium
International Nuclear Information System (INIS)
Liffman, K.; Clayton, D.D.
1989-01-01
The evolution course of refractory interstellar dust during the chemical evolution of a two-phase interstellar medium (ISM) is studied using a simple model of the chemical evolution of ISM. It is assumed that, in this medium, the stars are born in molecular clouds, but new nucleosynthesis products and stellar return are entered into a complementary diffuse medium; the well-mixed matter of each interstellar phase is repeatedly cycled stochastically through the complementary phase and back. The dust is studied on a particle-by-particle bases as it is sputtered by shock waves in the diffuse medium, accretes an amorphous mantle of gaseous refractory atoms while its local medium joins the molecular cloud medium, and encounters the possibility of astration within molecular clouds. Results are presented relevant to the size spectrum of accreted mantles, its age spectrum and the distinction among its several lifetimes, depletion factors of refractory atoms in the diffuse gas, and isotopic anomalies. 26 refs
19. Problems of matter-antimatter boundary layers
International Nuclear Information System (INIS)
Lehnert, B.
1975-01-01
This paper outlines the problems of the quasi-steady matter-antimatter boundary layers discussed in Klein-Alfven's cosmological theory, and a crude model of the corresponding ambiplasma balance is presented: (i) at interstellar particle densities, no well-defined boundary layer can exist in presence of neutral gas, nor can such a layer be sustained in an unmagnetized fully ionized ambiplasma. (ii) Within the limits of applicability of the present model, sharply defined boundary layers are under certain conditions found to exist in a magnetized ambiplasma. Thus, at beta values less than unity, a steep pressure drop of the low-energy components of matter and antimatter can be balanced by a magnetic field and the electric currents in the ambiplasma. (iii) The boundary layer thickness is of the order of 2x 0 approximately 10/BT 0 sup(1/4) meters, where B is the magnetic field strength in MKS units and T 0 the characteristic temperature of the low-energy components in the layer. (Auth.)
20. Technology for Boundaries
DEFF Research Database (Denmark)
Bødker, Susanne; Kristensen, Jannie Friis; Nielsen, Christina
2003-01-01
.After analysing the history and the current boundary work, the paper will propose new technological support for boundary work. In particular the paper will suggest means of supporting boundaries when these are productive and for changing boundaries when this seems more appropriate. In total, flexible technologies......This paper presents a study of an organisation, which is undergoing a process transforming organisational and technological boundaries. In particular, we shall look at three kinds of boundaries: the work to maintain and change the boundary between the organisation and its customers; boundaries...... seem a core issue when dealing with technology for boundaries....
1. Pushing Boundaries while Maintaining Stability
DEFF Research Database (Denmark)
Lippke, Lena; Wegener, Charlotte
at the same time. The exploration of transforming practices shows how disturbances in relation to the ‘normal’ practices are created and thus makes invisible conventions which are taken for granted visible. Thus, this paper addresses two types of invisibility: the unnoticed boundary-pushing practices...
2. Surface science studies of ethene containing model interstellar ices
Science.gov (United States)
Puletti, F.; Whelan, M.; Brown, W. A.
2011-05-01
The formation of saturated hydrocarbons in the interstellar medium (ISM) is difficult to explain only by taking into account gas phase reactions. This is mostly due to the fact that carbonium ions only react with H_2 to make unsaturated hydrocarbons, and hence no viable route to saturated hydrocarbons has been postulated to date. It is therefore likely that saturation processes occur via surface reactions that take place on interstellar dust grains. One of the species of interest in this family of reactions is C_2H_4 (ethene) which is an intermediate in several molecular formation routes (e.g. C_2H_2 → C_2H_6). To help to understand some of the surface processes involving ethene, a study of ethene deposited on a dust grain analogue surface (highly oriented pyrolytic graphite) held under ultra-high vacuum at 20 K has been performed. The adsorption and desorption of ethene has been studied both in water-free and water-dominated model interstellar ices. A combination of temperature programmed desorption (TPD) and reflection absorption infrared spectroscopy (RAIRS) have been used to identify the adsorbed and trapped species and to determine the kinetics of the desorption processes. In all cases, ethene is found to physisorb on the carbonaceous surface. As expected water has a very strong influence on the desorption of ethene, as previously observed for other model interstellar ice systems.
3. Interstellar C2, CH, and CN in translucent molecular clouds
NARCIS (Netherlands)
Dishoeck, van E.F.; Black, J.H.
1989-01-01
Optical absorption-line techniques have been applied to the study of a number of translucent molecular clouds in which the total column densities are large enough that substantial molecular abundances can be maintained. Results are presented for a survey of absorption lines of interstellar C2, CH,
4. Three-Component Dust Models for Interstellar Extinction C ...
without standard' method were used to constrain the dust characteristics in the mean ISM (RV = 3.1), ... Interstellar dust models have evolved as the observational data have advanced, and the most popular dust ... distribution comes from the IRAS observation which shows an excess of 12 μ and. 25 μ emission from the ISM ...
5. The Stardust Interstellar Dust Collector and Stardust@home
Science.gov (United States)
Westphal, A. J.; Anderson, D.; Bastien, R.; Butterworth, A.; Frank, D.; Gainsforth, Z.; Kelley, N.; Lettieri, R.; Mendez, B.; Prasad, R.; Tsitrin, S.; von Korff, J.; Warren, J.; Wertheimer, D.; Zhang, A.; Zolensky, M.
2006-12-01
The Stardust sample return mission is effectively two missions in one. Stardust brought back to earth for analytical study the first solid samples from a known solar system body beyond the moon, comet Wild2. The first results of the analyses of these samples are reported elsewhere in this session. In a separate aerogel collector, Stardust also captured and has returned the first samples of contemporary interstellar dust. Landgraf et al. [1] has estimated that ~ 50 interstellar dust particles in the micron size range have been captured in the Stardust Interstellar Dust Collector. Their state after capture is unknown. Before analysis of these particles can begin, they must be located in the collector. Here we describe the current status of Stardust@home, the massively distributed public search for these tiny interstellar dust particles. So far more than 13,000 volunteers have collectively performed more than 10,000,000 searches in stacks of digital images of ~10% of the collector. We report new estimates of the flux of interplanetary dust at ~2 AU based on the results of this search, and will compare with extant models[2]. References: [1] Landgraf et al., (1999) Planet. Spac. Sci. 47, 1029. [2] Staubach et al. (2001) in Interplanetary Dust, E. Grün, ed., Astron. &Astro. Library, Springer, 2001.
6. Rapid interstellar scintillation of quasar PKS 1257-326
NARCIS (Netherlands)
Bignall, Hayley E.; Jauncey, David L.; Lovell, James E. J.; Tzioumis, Anastasios K.; Macquart, Jean-Pierre; Kedziora-Chudczer, Lucyna; Engvold, O
2005-01-01
PKS 1257-326 is one of three quasars known to show unusually large and rapid, intra-hour intensity variations, as a result of scintillation in the turbulent Galactic interstellar medium. We have measured time delays in the variability pattern arrival times at the VLA and the ATCA, as well as an
7. Interstellar Matters: Neutral Hydrogen and the Galactic Magnetic Field
Science.gov (United States)
Verschuur, Gerrit; Schmelz, Joan T.; Asgari-Targhi asgari-Targhi, M.
2018-01-01
The physics of the interstellar medium was revolutionized by the observations of the Galactic Arecibo L-Band Feed Array (GALFA) HI survey done at the Arecibo Observatory. The high-resolution, high-sensitivity, high-dynamic- range images show complex, tangled, extended filaments, and reveal that the fabric of the neutral interstellar medium is deeply tied to the structure of the ambient magnetic field. This discovery prompts an obvious question – how exactly is the interstellar {\\it neutral} hydrogen being affected by the galactic magnetic field? We look into this question by examining a set of GALFA-HI data in great detail. We have chosen a long, straight filament in the southern galactic sky. This structure is both close by and isolated in velocity space. Gaussian analysis of profiles both along and across the filament reveal internal structure – braided strands that can be traced through the simplest part, but become tangled in more complex segments. These braids do not resemble in any way the old spherical HI clouds and rudimentary pressure balance models that were used to explain the pre-GALFA- HI interstellar medium. It is clear that these structures are created, constrained, and dominated by magnetic fields. Like many subfields of astronomy before it, e.g., physics of the solar coronal, extragalactic radio jets, and pulsar environment, scientists are confronted with observations that simply cannot be explained by simple hydrodynamics and are forced to consider magneto-hydrodynamics.
8. Radiation-pressure-driven dust waves inside bursting interstellar bubbles
NARCIS (Netherlands)
Ochsendorf, B.B.; Verdolini, S.; Cox, N.L.J.; Berné, O.; Kaper, L.; Tielens, A.G.G.M.
2014-01-01
Massive stars drive the evolution of the interstellar medium through their radiative and mechanical energy input. After their birth, they form "bubbles" of hot gas surrounded by a dense shell. Traditionally, the formation of bubbles is explained through the input of a powerful stellar wind, even
9. Deep, Broadband Spectral Line Surveys of Molecule-rich Interstellar Clouds
Energy Technology Data Exchange (ETDEWEB)
Widicus Weaver, Susanna L.; Laas, Jacob C.; Zou, Luyao; Kroll, Jay A.; Rad, Mary L.; Hays, Brian M.; Sanders, James L.; Cross, Trevor N.; Wehres, Nadine; McGuire, Brett A. [Department of Chemistry, Emory University, Atlanta, GA 30322 (United States); Lis, Dariusz C.; Sumner, Matthew C., E-mail: [email protected] [California Institute of Technology, Cahill Center for Astronomy and Astrophysics 301-17, Pasadena, CA 91125 (United States)
2017-09-01
Spectral line surveys are an indispensable tool for exploring the physical and chemical evolution of astrophysical environments due to the vast amount of data that can be obtained in a relatively short amount of time. We present deep, broadband spectral line surveys of 30 interstellar clouds using two broadband λ = 1.3 mm receivers at the Caltech Submillimeter Observatory. This information can be used to probe the influence of physical environment on molecular complexity. We observed a wide variety of sources to examine the relative abundances of organic molecules as they relate to the physical properties of the source (i.e., temperature, density, dynamics, etc.). The spectra are highly sensitive, with noise levels ≤25 mK at a velocity resolution of ∼0.35 km s{sup −1}. In the initial analysis presented here, column densities and rotational temperatures have been determined for the molecular species that contribute significantly to the spectral line density in this wavelength regime. We present these results and discuss their implications for complex molecule formation in the interstellar medium.
10. MHD STABILITY OF INTERSTELLAR MEDIUM PHASE TRANSITION LAYERS. I. MAGNETIC FIELD ORTHOGONAL TO FRONT
International Nuclear Information System (INIS)
Stone, Jennifer M.; Zweibel, Ellen G.
2009-01-01
We consider the scenario of a magnetic field orthogonal to a front separating two media of different temperatures and densities, such as cold and warm neutral interstellar gas, in a two-dimensional plane-parallel geometry. A linear stability analysis is performed to assess the behavior of both evaporation and condensation fronts when subject to incompressible, corrugational perturbations with wavelengths larger than the thickness of the front. We discuss the behavior of fronts in both super-Alfvenic and sub-Alfvenic flows. Since the propagation speed of fronts is slow in the interstellar medium (ISM), it is the sub-Alfvenic regime that is relevant, and magnetic fields are a significant influence on front dynamics. In this case, we find that evaporation fronts, which are unstable in the hydrodynamic regime, are stabilized. Condensation fronts are unstable, but for parameters typical of the neutral ISM the growth rates are so slow that steady-state fronts are effectively stable. However, the instability may become important if condensation proceeds at a sufficiently fast rate. This paper is the first in a series exploring the linear and nonlinear effects of magnetic field strength and orientation on the corrugational instability, with the ultimate goal of addressing outstanding questions about small-scale ISM structure.
11. Stardust Interstellar Preliminary Examination X: Impact Speeds and Directions of Interstellar Grains on the Stardust Dust Collector
Science.gov (United States)
Sterken, Veerle J.; Westphal, Andrew J.; Altobelli, Nicolas; Grun, Eberhard; Hillier, Jon K.; Postberg, Frank; Allen, Carlton; Stroud, Rhonda M.; Sandford, S. A.; Zolensky, Michael E.
2014-01-01
On the basis of an interstellar dust model compatible with Ulysses and Galileo observations, we calculate and predict the trajectories of interstellar dust (ISD) in the solar system and the distribution of the impact speeds, directions, and flux of ISD particles on the Stardust Interstellar Dust Collector during the two collection periods of the mission. We find that the expected impact velocities are generally low (less than 10 km per second) for particles with the ratio of the solar radiation pressure force to the solar gravitational force beta greater than 1, and that some of the particles will impact on the cometary side of the collector. If we assume astronomical silicates for particle material and a density of 2 grams per cubic centimeter, and use the Ulysses measurements and the ISD trajectory simulations, we conclude that the total number of (detectable) captured ISD particles may be on the order of 50. In companion papers in this volume, we report the discovery of three interstellar dust candidates in the Stardust aerogel tiles. The impact directions and speeds of these candidates are consistent with those calculated from our ISD propagation model, within the uncertainties of the model and of the observations.
12. The VLT-FLAMES Tarantula Survey. IX. The interstellar medium seen through diffuse interstellar bands and neutral sodium
NARCIS (Netherlands)
van Loon, J.Th.; Bailey, M.; Tatton, B.L.; Maíz Apellániz, J.; Crowther, P.A.; de Koter, A.; Evans, C.J.; Hénault-Brunet, V.; Howarth, I.D.; Richter, P.; Sana, H.; Simón-Díaz, S.; Taylor, W.; Walborn, N.R.
2013-01-01
Context. The Tarantula Nebula (a.k.a. 30 Dor) is a spectacular star-forming region in the Large Magellanic Cloud (LMC), seen through gas in the Galactic disc and halo. Diffuse interstellar bands (DIBs) offer a unique probe of the diffuse, cool-warm gas in these regions. Aims. The aim is to use DIBs
13. Interstellar Organics, the Solar Nebula, and Saturn's Satellite Phoebe
Science.gov (United States)
Pendleton, Y. J.; Cruikshank, D. P.
2014-01-01
The diffuse interstellar medium inventory of organic material (Pendleton et al. 1994, Pendleton & Allamandola 2002) was likely incorporated into the molecular cloud in which the solar nebula condensed. This provided the feedstock for the formation of the Sun, major planets, and the smaller icy bodies in the region outside Neptune's orbit (transneptunian objects, or TNOs). Saturn's satellites Phoebe, Iapetus, and Hyperion open a window to the composition of one class of TNO as revealed by the near-infrared mapping spectrometer (VIMS) on the Cassini spacecraft at Saturn. Phoebe (mean diameter 213 km) is a former TNO now orbiting Saurn. VIMS spaectral maps of PHoebe's surface reveal a complex organic spectral signature consisting of prominent aromatic (CH) and alophatic hydrocarbon (CH2, CH3) absorption bands (3.2-3.6 micrometers). Phoebe is the source of a huge debris ring encircling Saturn, and from which particles (approximately 5-20 micrometer size) spiral inward toward Saturn. They encounter Iapetus and Hperion where they mix with and blanket the native H2O ice of those two bodies. Quantitative analysis of the hydrocarbon bands on Iapetus demonstrates that aromatic CH is approximately 10 times as abundant as aliphatic CH2+CH3, significantly exceeding the strength of the aromatic signature in interplanetary dust particles, comet particles, ad in carbonaceous meteorites (Cruikshank et al. 2013). A similar excess of aromatics over aliphatics is seen in the qualitative analysis of Hyperion and Phoebe itself (Dalle Ore et al. 2012). The Iapetus aliphatic hydrocarbons show CH2/CH3 approximately 4, which is larger than the value found in the diffuse ISM (approximately 2-2.5). In so far as Phoebe is a primitive body that formed in the outer regions of the solar nebula and has preserved some of the original nebula inventory, it can be key to understanding the content and degree of procesing of the nebular material. There are other Phoebe-like TNOs that are presently
14. Observations of Carbon Isotopic Fractionation in Interstellar Formaldehyde
Science.gov (United States)
Wirstrom, E. S.; Charnley, S. B.; Geppert, W. D.; Persson, C. M.
2012-01-01
Primitive Solar System materials (e.g. chondrites. IDPs, the Stardust sample) show large variations in isotopic composition of the major volatiles (H, C, N, and O ) even within samples, witnessing to various degrees of processing in the protosolar nebula. For ex ample. the very pronounced D enhancements observed in IDPs [I] . are only generated in the cold. dense component of the interstellar medium (ISM), or protoplanetary disks, through ion-molecule reactions in the presence of interstellar dust. If this isotopic anomaly has an interstellar origin, this leaves open the possibility for preservation of other isotopic signatures throughout the form ation of the Solar System. The most common form of carbon in the ISM is CO molecules, and there are two potential sources of C-13 fractionation in this reservoir: low temperature chemistry and selective photodissociation. While gas-phase chemistry in cold interstellar clouds preferentially incorporates C-13 into CO [2], the effect of self-shielding in the presence of UV radiation instead leads to a relative enhancement of the more abundant isotopologue, 12CO. Solar System organic material exhibit rather small fluctuations in delta C-13 as compared to delta N-15 and delta D [3][1], the reason for which is still unclear. However, the fact that both C-13 depleted and enhanced material exists could indicate an interstellar origin where the two fractionation processes have both played a part. Formaldehyde (H2CO) is observed in the gas-phase in a wide range of interstellar environments, as well as in cometary comae. It is proposed as an important reactant in the formation of more complex organic molecules in the heated environments around young stars, and formaldehyde polymers have been suggested as the common origin of chondritic insoluable organic matter (IOM) and cometary refractory organic solids [4]. The relatively high gas-phase abundance of H2CO observed in molecular clouds (10(exp- 9) - 10(exp- 8) relative to H2) makes
15. Challenging the Boundaries
DEFF Research Database (Denmark)
Nørgaard, Nina
2004-01-01
To many people, challenging the boundaries between the traditional disciplines in foreign language studies means doing cultural studies. The aim of this article is to pull in a different direction by suggesting how the interface between linguistics and literature may be another fertile field...... to explore in the study and teaching of foreign languages. Not only may linguistics and literature be employed to shed light on each other, the insights gained may furthermore prove useful in a broader context in our foreign language studies. The article begins with a brief introduction to literary...... linguistics in general and to Hallidayan linguistics in particular. The theoretical framework thus laid out, it is exemplified how Halliday's theory of language may be employed in the analysis of literature. The article concludes by considering the possible status of literary linguistics in a broader...
16. Transcending Organizational Boundaries
DEFF Research Database (Denmark)
Kringelum, Louise Tina Brøns
by applying the engaged scholarship approach, thereby providing a methodological contribution to both port and business model research. Emphasizing the interplay of intra- and inter-organizational business model innovation, the thesis adds insight into the roles of port authorities, business model trends......This thesis explores how processes of business model innovation can unfold in a port authority by transcending organizational boundaries through inter-organizational collaboration. The findings contribute to two fields of academic inquiry: the study of business model innovation and the study of how...... the roles of port authorities evolve. This contribution is made by combining the two fields, where the study of business model innovation is used as an analytical concept for understanding the evolution of port authorities, and where the study of port authorities is used as a contextual setting...
17. The Primordial Inflation Explorer
Science.gov (United States)
Kogut, Alan J.
2012-01-01
The Primordial Inflation Explorer is an Explorer-class mission to measure the gravity-wave signature of primordial inflation through its distinctive imprint on the linear polarization of the cosmic microwave background. PIXIE uses an innovative optical design to achieve background-limited sensitivity in 400 spectral channels spanning 2.5 decades in frequency from 30 GHz to 6 THz (1 cm to 50 micron wavelength). The principal science goal is the detection and characterization of linear polarization from an inflationary epoch in the early universe, with tensor-to-scalar ratio r < 10(exp -3) at 5 standard deviations. The rich PIXIE data set will also constrain physical processes ranging from Big Bang cosmology to the nature of the first stars to physical conditions within the interstellar medium of the Galaxy. I describe the PIXIE instrument and mission architecture needed to detect the inflationary signature using only 4 semiconductor bolometers.
18. Rigid supersymmetry with boundaries
Energy Technology Data Exchange (ETDEWEB)
Belyaev, D.V. [Deutsches Elektronen-Synchrotron (DESY), Hamburg (Germany); Van Nieuwenhuizen, P. [State Univ. of New York, Stony Brook, NY (United States). C.N. Yang Inst. for Theoretical Physics
2008-01-15
We construct rigidly supersymmetric bulk-plus-boundary actions, both in x-space and in superspace. For each standard supersymmetric bulk action a minimal supersymmetric bulk-plus-boundary action follows from an extended F- or D-term formula. Additional separately supersymmetric boundary actions can be systematically constructed using co-dimension one multiplets (boundary superfields). We also discuss the orbit of boundary conditions which follow from the Euler-Lagrange variational principle. (orig.)
19. Interstellar extinction in the dark Taurus clouds. Pt. 1
International Nuclear Information System (INIS)
Straizys, V.; Meistas, E.
1980-01-01
The results of photoelectric photometry of 74 stars in the Vilnius seven-color system in the area of Taurus dark clouds with coordinates (1950) 4sup(h)20sup(m)-4sup(h)48sup(m)+24 0 .5-+27 0 are presented. Photometric spectral types, absolute magnitudes, color excesses, interstellar extinctions and distances of the stars are determined. The dark cloud Khavtassi 286, 278 and the surrounding absorbing nebulae are found to extend from 140 to 175 pc from the sun. The average interstellar extinction Asub(V) on both sides of the dark cloud is of the order of 1sup(m).5. We find no evidence of the existence of several absorbing clouds situated at various distances. (author)
20. Optical Polarization as a Probe of the Local Interstellar Medium
Science.gov (United States)
Tinbergen, J.
1984-01-01
The use of interstellar polarization as a tool for measuring interstellar dust is discussed. Problems resulting from dust and magnetic field configurations becoming mixed up are discussed, as is the availability of sufficiently bright stars to obtain the photons needed for precision measurements. It is proposed that: (1) on the scale of several hundred parsec, there is a preferential magnetic field direction, as evidenced by observations at the Galactic poles and selected longitudes in the Galactic plane; (2) the local (r 50 pc) region is devoid of dust, as evidenced by the mean square degree of polarization as a function of distance; and, less certainly, that (3) at a distance of less than 5 pc, there is a patch of dust which may be of interest in connection with cloud models.
1. The synthesis of complex molecules in interstellar clouds
Science.gov (United States)
Huntress, W. T., Jr.; Mitchell, G. F.
1979-01-01
The abundances of polyatomic molecules that may be formed by CH3(+) radiative association reactions in dense interstellar molecular clouds are reevaluated. The formation of a number of complex interstellar molecules via radiative association reactions involving ionic precursors other than CH3(+) is also investigated; these additional precursors include CH3O(+), CH3CO(+), CH5(+), HCO(+), NO(+), H2CN(+), C2H2(+), and NH3(+). The results indicate that the postulated gas-phase ion-molecule radiative association reactions could potentially explain the synthesis of most of the more complex species observed in dense molecular clouds such as Sgr B2. It is concluded, however, that in order to be conclusive, laboratory data are needed to show whether or not these reactions proceed at the required rates at low temperatures.
2. On the carbon enrichment of the interstellar medium
International Nuclear Information System (INIS)
Sarmiento, A.; Peimbert, M.
1985-01-01
The contribution of novae, IMS, and massive stars to the 12 C and 13 C enrichment of the interstellar medium is evaluated. The following results are obtained: a) novae are not important contributors to the 12 C abundance but contribute significantly to 13 C, b) limits to the ratio of the mixing length to the pressure scale height,α, and to the mass loss rate parameter, eta, are derived for IMS, c) IMS are the main contributors to the 12 C and 13 C enrichment of the interstellar medium, d) it is easier to explain the solar vicinity 12 C/ 13 C ratio than the solar system ratio, e) to explain the 12 C/ 13 C ratio in the ISM the mass ejected per nova outburst has to be approx. 1 x 10 -5 M sub(sun). (author)
3. The nature of interstellar dust as revealed by light scattering
Directory of Open Access Journals (Sweden)
D. A. Williams
2011-09-01
Full Text Available Interstellar dust was first identified through the extinction that it causes of optical starlight. Initially, observational and theoretical studies of extinction were made to identify simple ways of removing the effect of extinction. Over the last few decades it has become clear that dust has a number of very important roles in interstellar physics and chemistry, and that through these roles dust affects quite fundamentally the evolution of the Milky Way and other galaxies. However, our detailed knowledge of the actual material of dust remains relatively poor. The use of accurate models for the interaction of electromagnetic radiation with particles of arbitrary shape and composition remains vital, if our description of dust is to improve.
4. Fission-Based Electric Propulsion for Interstellar Precursor Missions
International Nuclear Information System (INIS)
HOUTS, MICHAEL G.; LENARD, ROGER X.; LIPINSKI, RONALD J.; PATTON, BRUCE; POSTON, DAVID; WRIGHT, STEVEN A.
1999-01-01
This paper reviews the technology options for a fission-based electric propulsion system for interstellar precursor missions. To achieve a total ΔV of more than 100 km/s in less than a decade of thrusting with an electric propulsion system of 10,000s Isp requires a specific mass for the power system of less than 35 kg/kWe. Three possible configurations are described: (1) a UZrH-fueled,NaK-cooled reactor with a steam Rankine conversion system,(2) a UN-fueled gas-cooled reactor with a recuperated Brayton conversion system, and (3) a UN-fueled heat pipe-cooled reactor with a recuperated Brayton conversion system. All three of these systems have the potential to meet the specific mass requirements for interstellar precursor missions in the near term. Advanced versions of a fission-based electric propulsion system might travel as much as several light years in 200 years
5. Interstellar material in front of chi ophiuchi. I. Optical observations
International Nuclear Information System (INIS)
Frisch, P.C.
1979-01-01
Optical observations of the interstellar material in front of chi Oph are discussed. The main interstellar cloud is made up of several regions with velocities between -6 and -12 km s -1 (heliocentric). Both CH and CH + are found within this feature, but with central velocities which differ by 2 km s -1 . Another cloud, with a velocity of -26 km s -1 , contains relatively strong Ca + lines. It has a ratio between Ca + and Na 0 column densities that is appropriate for ''high-velocity'' clouds. Calcium, iron, and sodium column densities are used to estimate an average electron density for the line of sight as well as for each cloud. The abundances of CH and CH + , and the absence of CN, are analyzed in terms of current theories about their origin
6. Energetic Processing of Interstellar Silicate Grains by Cosmic Rays
Energy Technology Data Exchange (ETDEWEB)
Bringa, E M; Kucheyev, S O; Loeffler, M J; Baragiola, R A; Tielens, A G Q M; Dai, Z R; Graham, G; Bajt, S; Bradley, J; Dukes, C A; Felter, T E; Torres, D F; van Breugel, W
2007-03-28
While a significant fraction of silicate dust in stellar winds has a crystalline structure, in the interstellar medium nearly all of it is amorphous. One possible explanation for this observation is the amorphization of crystalline silicates by relatively 'low' energy, heavy ion cosmic rays. Here we present the results of multiple laboratory experiments showing that single-crystal synthetic forsterite (Mg{sub 2}SiO{sub 4}) amorphizes when irradiated by 10 MeV Xe{sup ++} ions at large enough fluences. Using modeling, we extrapolate these results to show that 0.1-5.0 GeV heavy ion cosmic rays can rapidly ({approx}70 Million yrs) amorphize crystalline silicate grains ejected by stars into the interstellar medium.
Science.gov (United States)
Löhmer, O.; Mitra, D.; Gupta, Y.; Kramer, M.; Ahuja, A.
2004-10-01
Using radio pulsars as probes of the interstellar medium (ISM) we study the frequency evolution of interstellar scattering. The frequency dependence of scatter broadening times, τsc, for most of the pulsars with low and intermediate dispersion measures (DM ≲ 400 pc cm-3) is consistent with the Kolmogorov spectrum of electron density fluctuations in a turbulent medium. In contrast, the measured τsc's for highly dispersed pulsars in the central region of the Galaxy are larger than expected and show a spectrum which is flatter than the Kolmogorov law. We analyse the first measurements of spectral indices of scatter broadening over the full known DM range and discuss possible explanations for the anomalous scattering behaviour along peculiar lines of sight (LOS).
8. Spiral arms and a supernova-dominated interstellar medium
International Nuclear Information System (INIS)
Brand, P.W.J.L.; Heathcote, S.R.
1982-01-01
Models of the interstellar medium (ISM) utilizing the large energy output of supernovae to determine the average kinematical properties of the gas, are subjected to an imposed (spiral) density wave. The consequent appearance of the ISM is considered. In particular the McKee-Ostriker model with cloud evaporation is used, but it is shown that the overall appearance of the galaxy model does not change significantly if a modification of Cox's mechanism, with no cloud evaporation, is incorporated. It is found that a spiral density wave shock can only be self-sustaining if quite restrictive conditions are imposed on the values of the galactic supernova rate and the mean interstellar gas density. (author)
9. Tholins - Organic chemistry of interstellar grains and gas
Science.gov (United States)
Sagan, C.; Khare, B. N.
1979-01-01
The paper discusses tholins, defined as complex organic solids formed by the interaction of energy - for example, UV light or spark discharge - with various mixtures of cosmically abundant gases - CH4, C2H6, NH3, H2O, HCHO, and H2S. It is suggested that tholins occur in the interstellar medium and are responsible for some of the properties of the interstellar grains and gas. Additional occurrences of tholins are considered. Tholins have been produced experimentally; 50 or so pyrolytic fragments of the brown, sometimes sticky substances have been identified by gas chromatography-mass spectrometry, and the incidence of these fragments in tholins produced by different procedures is reported.
10. Chemical Evolution in the Interstellar Medium: From Astrochemistry to Astrobiology
Science.gov (United States)
Allamandola, Louis J.
2009-01-01
Great strides have been made in our understanding of interstellar material thanks to advances in infrared astronomy and laboratory astrophysics. Ionized polycyclic aromatic hydrocarbons (PAHs), shockingly large molecules by earlier astrochemical standards, are widespread and very abundant throughout much of the Universe. In cold molecular clouds, the birthplace of planets and stars, interstellar molecules freeze onto dust and ice particles forming mixed molecular ices dominated by simple species such as water, methanol, ammonia, and carbon monoxide. Within these clouds, and especially in the vicinity of star and planet forming regions, these ices and PAHs are processed by ultraviolet light and cosmic rays forming hundreds of far more complex species, some of biogenic interest. Eventually, these are delivered to primordial planets by comets and meteorites. Astrochemical evolution, highlights of this field from a chemist's perspective, and the astronomer's infrared toolbox will be reviewed.
11. Solar Sails: Sneaking up on Interstellar Travel
Science.gov (United States)
Johnson, L.
Throughout the world, government agencies, universities and private companies are developing solar sail propulsion systems to more efficiently explore the solar system and to enable science and exploration missions that are simply impossible to accomplish by any other means. Solar sail technology is rapidly advancing to support these demonstrations and missions, and in the process, is incrementally advancing one of the few approaches allowed by physics that may one day take humanity to the stars. Continuous solar pressure provides solar sails with propellantless thrust, potentially enabling them to propel a spacecraft to tremendous speeds theoretically much faster than any present-day propulsion system. The next generation of sails will enable us to take our first real steps beyond the edge of the solar system, sending spacecraft out to distances of 1000 Astronomical Units, or more. In the farther term, the descendants of these first and second generation sails will augment their thrust by using high power lasers and enable travel to nearby stellar systems with flight times less than 500 years a tremendous improvement over what is possible with conventional chemical rockets. By fielding these first solar sail systems, we are sneaking up on a capability to reach the stars.
12. SYSTEMATIC THEORETICAL STUDY ON THE INTERSTELLAR CARBON CHAIN MOLECULES
Energy Technology Data Exchange (ETDEWEB)
Etim, Emmanuel E.; Arunan, Elangannan [Inorganic and Physical Chemistry Department, Indian Institute of Science Bangalore, 560012 (India); Gorai, Prasanta; Das, Ankan [Indian Centre for Space Physics, 43 Chalantika, Garia Station Road, Kolkata 700 084 (India); Chakrabarti, Sandip K., E-mail: [email protected] [Department of Chemical Sciences, Federal University Wukari, Katsina-Ala Road, P.M.B. 1020 Wukari, Taraba State (Nigeria)
2016-12-01
In an effort to further our interest in understanding the basic chemistry of interstellar molecules, here we carry out an extensive investigation of the stabilities of interstellar carbon chains; C{sub n}, H{sub 2}C{sub n}, HC{sub n}N and C{sub n}X (X = N, O, Si, S, H, P, H{sup −}, N{sup −}). These sets of molecules account for about 20% of all the known interstellar and circumstellar molecules. Their high abundances, therefore, demand serious attention. High-level ab initio quantum chemical calculations are employed to accurately estimate the enthalpy of formation, chemical reactivity indices, global hardness and softness, and other chemical parameters of these molecules. Chemical modeling of the abundances of these molecular species has also been performed. Of the 89 molecules considered from these groups, 47 have been astronomically observed, and these observed molecules are found to be more stable with respect to other members of the group. Of the 47 observed molecules, 60% are odd-numbered carbon chains. Interstellar chemistry is not actually driven by thermodynamics, but it is primarily dependent on various kinetic parameters. However, we found that the detectability of the odd-numbered carbon chains could be correlated due to the fact that they are more stable than the corresponding even-numbered carbon chains. Based on this aspect, the next possible carbon chain molecule for astronomical observation in each group is proposed. The effect of kinetics in the formation of some of these carbon chain molecules is also discussed.
13. Superconducting ion scoop and its application to interstellar flight
Energy Technology Data Exchange (ETDEWEB)
Matloff, G L; Fennelly, A J
1974-09-01
Physical and engineering aspects of a superconducting ion scoop with an effective field radius of 10/sup 4/ km are discussed. Application of the system to interstellar ramjet travel is considered. Used in conjunction with a large boron sail towed behind the spacecraft, the scoop could be applied as a deceleration mechanism for thermonuclear-rocket-boosted vehicles moving at least as fast as 0.2C.
14. Hot interstellar tunnels. I. Simulation of interacting supernova remnants
International Nuclear Information System (INIS)
Smith, B.W.
1977-01-01
Reexamining a suggestion of Cox and Smith, we find that intersecting supernova remnants can indeed generate and maintain hot interstellar regions with napproximately-less-than10 -2 cm -3 and Tapprox.10 6 K. These regions are likely to occupy at least 30% of the volume of a spiral arm near the midplane of the gaseous disk if the local supernova rate there is greater than 1.5 x 10 -7 Myr -1 pc -3 . Their presence in the interstellar medium is supported by observations of the soft X-ray background. The theory required to build a numerical simulation of interacting supernova remnants is developed. The hot cavities within a population of remnants will become connected for a variety of assumed conditions in the outer shells of old remnants. Extensive hot cavity regions or tunnels are built and enlarged by supernovae occurring in relatively dense gas which produce connections, but tunnels are kept hot primarily by supernovae occurring within the tunnels. The latter supernovae initiate fast shock waves which apparently reheat tunnels faster than they are destroyed by thermal conduction in a galactic magnetic field or by radiative cooling. However, the dispersal of these rejuvenating shocks over a wide volume is inhibited by motions of cooler interstellar gas in the interval between shocks. These motions disrupt the contiguity of the component cavities of a tunnel and may cause its death.The Monte Carlo simulations indicate that a quasi-equilibrium is reached within 10 7 years of the first supernova in a spiral arm. This equilibrium is characterized by a constant average filling fraction for cavities in the interstellar volume. Aspects of the equilibrium are discussed for a range of supernova rates. Two predictions of Cox and Smith are not confirmed within this range: critical growth of hot regions to encompass the entire medium, and the efficient quenching of a remnant's expansion by interaction with other cavities
15. The Abundance of Mg in the Interstellar Medium
Science.gov (United States)
Fitzpatrick, Edward L.
1997-06-01
An empirical determination of the f-values of the far-UV Mg II λλ1239, 1240 lines is reported. The strong near-UV Mg II λλ2796, 2803 lines are generally highly saturated along most interstellar sight lines outside the local interstellar medium (ISM) and usually yield extremely uncertain estimates of Mg+ column densities in interstellar gas. Since Mg+ is the dominant form of Mg in the neutral ISM, and since Mg is expected to be a significant constituent of interstellar dust grains, the far-UV lines are critical for assessing the role of this important element in the ISM. This study consists of complete component analyses of the absorption along the lines of sight toward HD 93521 in the Galactic halo and ξ Persei and ζ Ophiuchi in the Galactic disk, including all four UV Mg+ lines and numerous other transitions. The three analyses yield consistent determinations of the λλ1239, 1240 f-values, with weighted means of (6.4 +/- 0.4) × 10-4 and (3.2 +/- 0.2) × 10-4, respectively. These results are a factor of ~2.4 larger than a commonly used theoretical estimate, and a factor of ~2 smaller than a recently suggested empirical revision. The effects of this result on gas- and dust-phase abundance measurements of Mg are discussed. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, under NASA contract NAS5-2655. This Letter is dedicated to the memory of Professor Lyman Spitzer Jr. He was a great guy.
16. Plasma Diagnostics of the Interstellar Medium with Radio Astronomy
OpenAIRE
Haverkorn, Marijke; Spangler, Steven R.
2013-01-01
We discuss the degree to which radio propagation measurements diagnose conditions in the ionized gas of the interstellar medium (ISM). The "signal generators" of the radio waves of interest are extragalactic radio sources (quasars and radio galaxies), as well as Galactic sources, primarily pulsars. The polarized synchrotron radiation of the Galactic non-thermal radiation also serves to probe the ISM, including space between the emitting regions and the solar system. Radio propagation measurem...
17. The variation of interstellar element abundances with hydrogen density
International Nuclear Information System (INIS)
Keenan, F.P.; Hibbert, A.; Dufton, P.L.; Murray, M.J.
1986-01-01
The variation of the interstellar nitrogen, oxygen and magnesium abundances with mean line-of-sight hydrogen density is analysed in terms of a two-component model, which consists of warm, low-density neutral gas and cold clouds. In all cases the gas-phase abundances have been deduced using reliable oscillator strengths specifically calculated for this purpose. Depletions in the warm and cold gas, are derived from non-linear least-squares fits to the data. (author)
18. HERSCHEL/HIFI DISCOVERY OF HCL+ IN THE INTERSTELLAR MEDIUM
International Nuclear Information System (INIS)
De Luca, M.; Gerin, M.; Falgarone, E.; Gupta, H.; Drouin, B. J.; Pearson, J. C.; Neufeld, D.; Teyssier, D.; Lis, D. C.; Monje, R.; Phillips, T. G.; Goicoechea, J. R.; Godard, B.; Bell, T. A.; Coutens, A.
2012-01-01
The radical ion HCl + , a key intermediate in the chlorine chemistry of the interstellar gas, has been identified for the first time in the interstellar medium with the Herschel Space Observatory's Heterodyne Instrument for the Far-Infrared. The ground-state rotational transition of H 35 Cl + , 2 Π 3/2 J = 5/2-3/2, showing Λ-doubling and hyperfine structure, is detected in absorption toward the Galactic star-forming regions W31C (G10.6-0.4) and W49N. The complex interstellar absorption features are modeled by convolving in velocity space the opacity profiles of other molecular tracers toward the same sources with the fine and hyperfine structure of HCl + . This structure is derived from a combined analysis of optical data from the literature and new laboratory measurements of pure rotational transitions, reported in the accompanying Letter by Gupta et al. The models reproduce well the interstellar absorption, and the frequencies inferred from the astronomical observations are in exact agreement with those calculated using spectroscopic constants derived from the laboratory data. The detection of H 37 Cl + toward W31C, with a column density consistent with the expected 35 Cl/ 37 Cl isotopic ratio, provides additional evidence for the identification. A comparison with the chemically related molecules HCl and H 2 Cl + yields an abundance ratio of unity with both species (HCl + : H 2 Cl + : HCl ∼ 1). These observations also yield the unexpected result that HCl + accounts for 3%-5% of the gas-phase chlorine toward W49N and W31C, values several times larger than the maximum fraction (∼1%) predicted by chemical models.
19. Source of the 26Al observed in the interstellar medium
International Nuclear Information System (INIS)
Dearborn, D.S.P.; Blake, J.B.
1985-01-01
Recent HEAO 3 observations have been interpreted by Mahoney and colleagues as requiring approximately 3 M/sub sun/ of 26 Al alive in the interstellar medium. Calculations briefly discussed in this Letter indicate that there is substantial production and dispersal of 26 Al in the stellar winds of O and W-R stars and suggest that the stellar winds of very massive stars are a significant source of 26 Al
20. Numerical study of rotating interstellar clouds: equilibrium and collapse
International Nuclear Information System (INIS)
Norman, M.L.
1980-06-01
Equilibrium and collapse of rotating, axisymmetric, idealized interstellar gas clouds is calculated with a 2D hydrodynamics code. The hydrodynamics features an improved angular momentum advection algorithm. Angular momentum is advected consistently with mass by deriving angular momentum fluxes from mass fluxes and the local distribution of specific angular momentum. Local conservation is checked by a graph of mass versus specific angular momentum for the cloud as a whole
1. Skating on thin ice: surface chemistry under interstellar conditions
Science.gov (United States)
Fraser, H.; van Dishoeck, E.; Tielens, X.
Solid CO2 has been observed towards both active star forming regions and quiescent clouds (Gerakines et. al. (1999)). The high abundance of CO2 in the solid phase, and its low abundance in the gas phase, support the idea that CO2 is almost exclusively formed in the solid state. Several possible formation mechanisms have been postulated (Ruffle &Herbst (2001): Charnley &Kaufman (2000)), and the detection of CO2 towards quiescent sources such as Elias 16 (Whittet et. al. (1998)) clearly suggests that CO2 can be produced in the absence of UV or electron mediated processes. The most likely route is via the surface reactions between O atoms, or OH radicals, and CO. The tools of modern surface- science offer us the potential to determine many of the physical and chemical attributes of icy interstellar grain mantles under highly controlled conditions, that closely mimic interstellar environments. The Leiden Surface Reaction Simulation Device ( urfreside) combines UHV (UltraS High Vacuum) surface science techniques with an atomic beam to study chemical reactions occurring on the SURFACE and in the BULK of interstellar ice grain mimics. By simultaneously combining two or more surface analysis techniques, the chemical kinetics, reaction mechanisms and activation energies can be determined directly. The experiment is aimed at identifying the key barrierless reactions and desorption pathways on and in H2 O and CO ices under interstellar conditions. The results from traditional HV (high vacuum) and UHV studies of the CO + O and CO + OH reactions will be presented in this paper. Charnley, S.B., & Kaufman, M.J., 2000, ApJ, 529, L111 Gerakines, P.A., 1999, ApJ, 522, 357 Ruffle, D.P., & Herbst, E., 2001, MNRAS, 324, 1054 Whittet, D.C.B., et.al., 1998, ApJ, 498, L159
2. Optical properties of likely constituents of interstellar dust
International Nuclear Information System (INIS)
Dayawansa, I.J.
1977-07-01
Optical properties of polyoxymethylene (POM) at room temperature have been measured from the near ultra-violet to infrared as an initial stage of a link between interstellar dust and organic matter, and the results, which are particularly relevant to interstellar extinction, are reported. There is a strong possibility of a more complex organic component which could significantly contribute to the interstellar extinction. Measurements have also been made of the effect of fast neutron bombardment on the optical properties of quartz (SiO 2 ). At a high total flux of neutrons the crystalline quartz will change to its amorphous form which has extinction properties that resemble the interstellar extinction. Extinction due to small particles of several forms of SiO 2 has been measured and among them the hydrated mineral, opal, behaved like an amorphous silica. Neutron irradiated olivine showed a stronger and a broader 10μm band in addition to weaker bands towards the longer wavelengths which indicated that atomic damage has been produced. At high fluxes more atomic damage is expected to change the crystalline structure and thereby cause changes in the infrared absorption properties. Extinction measurements were also made for smoke particles of MgO in the infrared. When the measurements were made with the particles deposited on substrates, in addition to a very broad surface mode absorption feature around 20μm an extinction maximum was observed typical of the bulk mode at 25μm. Extinction measurements for MgO smoke particles in air also showed similar results. However when the particles were dispersed in a non-absorbing medium, the bulk absorption mode was not observed. This implies that the appearance of the bulk mode is due to clumping. (author)
3. Dissociative recombination of interstellar ions: electronic structure calculations for HCO+
International Nuclear Information System (INIS)
Kraemer, W.P.; Hazi, A.U.
1985-01-01
The present study of the interstellar formyl ion HCO + is the first attempt to investigate dissociative recombination for a triatomic molecular ion using an entirely theoretical approach. We describe a number of fairly extensive electronic structure calculations that were performed to determine the reaction mechanism of the e-HCO + process. Similar calculations for the isoelectronic ions HOC + and HN 2 + are in progress. 60 refs
4. Interstellar scattering in the inner parts of the galaxy
International Nuclear Information System (INIS)
Rao, A.P.; Ananthakrishnan, S.
1984-01-01
A new survey of the galactic plane for sources with size less than 1 arc s at 327 MHz shows that towards the inner parts of the galaxy for galactic latitudes less than 5deg, interstellar scattering is much larger than expected from data at higher latitudes. The enhanced scattering varies both with galactic latitude and longitude. A two-component model for the distribution of scattering matter in the Galaxy is proposed to interpret the observations. (author)
5. Interaction of Interstellar Shocks with Dense Obstacles: Formation of Bullets''
Science.gov (United States)
The so-called cumulative effect take place in converging conical shock waves arising behind dense obstacles overtaken by incident interstellar shock. A significant part of energy of converging flow of matter swept-up by a radiative conical shock can be transferred to a dense jet-like ejection (bullet'') directed along the cone axis. Possible applications of this effect for star-forming regions (e.g., OMC-1) and supernova remnants (e.g., Vela SNR) are discussed.
6. THz Time-Domain Spectroscopy of Interstellar Ice Analogs
Science.gov (United States)
Ioppolo, Sergio; McGuire, Brett A.; de Vries, Xander; Carroll, Brandon; Allodi, Marco; Blake, Geoffrey
2015-08-01
The unambiguous identification of nearly 200 molecular species in different astronomical environments proves that our cosmos is a ‘Molecular Universe’. The cumulative outcome of recent observations, laboratory studies, and astrochemical models indicates that there is a strong interplay between the gas and the solid phase throughout the process of forming molecules in space. Observations of interstellar ices are generally limited to lines-of-sight along which infrared absorption spectroscopy is possible. Therefore, the identification of more complex prebiotic molecules in the mid-IR is difficult because of their low expected interstellar abundances and the overlap of their absorption features with those from the more abundant species. In the THz region, telescopes can detect Interstellar ices in emission or absorption against dust continuum. Thus, THz searches do not require a background point source. Moreover, since THz spectra are the fingerprint of inter- and intramolecular forces, complex species can present unique modes that do not overlap with those from simpler, more abundant molecules. THz modes are also sensitive to temperature and phase changes in the ice. Therefore, spectroscopy at THz frequencies has the potential to better characterize the physics and chemistry of the ISM. Currently, the Herschel Space Telescope, SOFIA, and ALMA databases contain a vast amount of new THz spectral data that require THz laboratory spectra for interpretation. The latter, however, are largely lacking. We have recently constructed a new THz time-domain spectroscopy system operating in the range between 0.3 - 7.5 THz. This work focuses on the laboratory investigation of the composition and structure of the most abundant interstellar ice analogs compared to some more complex species. Different temperatures, mixing ratios, and matrix isolation experiments will be shown. The ultimate goal of this research is to provide the scientific community with an extensive THz ice
KAUST Repository
Peng, Chi-Han; Barton, Michael; Jiang, Caigui; Wonka, Peter
2014-01-01
Here we presented a framework to explore quad mesh topologies. The core of our work is a systematic enumeration algorithm that can generate all possible quadrangular meshes inside a defined boundary with an upper limit of v3-v5 pairs. The algorithm is orders of magnitude more efficient than previous work. The combination of topological enumeration and shape-space exploration demonstrates that mesh topology has a powerful influence on geometry. The Fig. 18. A gallery of different quadrilateral meshes for a Shuriken. The quadrilaterals of the model were colored in a postprocess. Topological variations have distinctive, interesting patterns of mesh lines. © 2014 ACM 0730-0301/2014/01-ART3 15.00.
KAUST Repository
Peng, Chi-Han
2014-02-04
Here we presented a framework to explore quad mesh topologies. The core of our work is a systematic enumeration algorithm that can generate all possible quadrangular meshes inside a defined boundary with an upper limit of v3-v5 pairs. The algorithm is orders of magnitude more efficient than previous work. The combination of topological enumeration and shape-space exploration demonstrates that mesh topology has a powerful influence on geometry. The Fig. 18. A gallery of different quadrilateral meshes for a Shuriken. The quadrilaterals of the model were colored in a postprocess. Topological variations have distinctive, interesting patterns of mesh lines. © 2014 ACM 0730-0301/2014/01-ART3 15.00.
9. Quantum walk with one variable absorbing boundary
International Nuclear Information System (INIS)
Wang, Feiran; Zhang, Pei; Wang, Yunlong; Liu, Ruifeng; Gao, Hong; Li, Fuli
2017-01-01
Quantum walks constitute a promising ingredient in the research on quantum algorithms; consequently, exploring different types of quantum walks is of great significance for quantum information and quantum computation. In this study, we investigate the progress of quantum walks with a variable absorbing boundary and provide an analytical solution for the escape probability (the probability of a walker that is not absorbed by the boundary). We simulate the behavior of escape probability under different conditions, including the reflection coefficient, boundary location, and initial state. Moreover, it is also meaningful to extend our research to the situation of continuous-time and high-dimensional quantum walks. - Highlights: • A novel scheme about quantum walk with variable boundary is proposed. • The analytical results of the survival probability from the absorbing boundary. • The behavior of survival probability under different boundary conditions. • The influence of different initial coin states on the survival probability.
10. INTERSTELLAR PICKUP ION PRODUCTION IN THE GLOBAL HELIOSPHERE AND HELIOSHEATH
Energy Technology Data Exchange (ETDEWEB)
Wu, Y.; Florinski, V.; Guo, X., E-mail: [email protected] [Center for Space Plasma and Aeronomic Research, University of Alabama in Huntsville, Huntsville, AL 35805 (United States)
2016-11-20
Interstellar pickup ions (PUIs) play a significant part in mediating the solar wind (SW) interaction with the interstellar medium. In this paper, we examine the details of spatial variation of the PUI velocity distribution function (VDF) in the SW by solving the PUI transport equation. We assume the PUI distribution is isotropic resulting from strong pitch-angle scattering by wave–particle interaction. A three-dimensional model combining the MHD treatment of the background SW and neutrals with a kinetic treatment of PUIs throughout the heliosphere and the surrounding local interstellar medium has been developed. The model generates PUI power-law tails via second-order Fermi process. We analyze how PUIs transform across the heliospheric termination shock and obtain the PUI phase space distribution in the inner heliosheath including continuing velocity diffusion. Our simulated PUI spectra are compared with observations made by New Horizons , Ulysses , Voyager 1, 2 , and Cassini , and a satisfactory agreement is demonstrated. Some specific features in the observations, for example, a cutoff of PUI VDF at v = V {sub SW} and a f ∝ v {sup -5} tail in the reference frame of the SW, are well represented by the model.
11. The Starflight Handbook: A Pioneer's Guide to Interstellar Travel
Science.gov (United States)
Mallove, Eugene F.; Matloff, Gregory L.
1989-06-01
The Starflight Handbook A Pioneer's Guide to Interstellar Travel "The Starflight Handbook is an indispensable compendium of the many and varied methods for traversing the vast interstellar gulf--don't leave the Solar System without it!" --Robert Forward "Very sensible, very complete and useful. Its good use of references and technical `sidebars' adds to the book and allows the nontechnical text to be used by ordinary readers in an easy fashion. I certainly would recommend this book to anyone doing any thinking at all about interstellar flight or the notion of possibilities of contacts between hypothetical civilizations in different stat systems." --Louis Friedman Executive Director, The Planetary Society The Starflight Handbook is the first and only compendium on planet Earth of the radical new technologies now on the drawing boards of some of our smartest and most imaginative space scientists and engineers. Scientists and engineers as well as general readers will be captivated by its: In-depth discussions of everything from nuclear pulse propulsion engines to in-flight navigation, in flowing, non-technical language Sidebars and appendices cover technical and mathematical concepts in detail Seventy-five elegant and enlightening illustrations depicting starships and their hardware
12. Laboratory studies of ion-molecule reactions and interstellar chemistry
International Nuclear Information System (INIS)
Koyano, Inosuke
1989-01-01
Several types of laboratory studies have been performed on ion-molecule reactions relevant to the formation of the interstellar molecules. Special emphasis is placed on the formation, structure, and reactivity of the C 3 H 3 + ions, which are believed to play a key role in interstellar chemistry. When these ions are produced by the reaction of C 3 H 4+ with C 3 H 4 in a beam-gas arrangement, their times-of-flight (TOF) show abnormally broad distributions regardless of the sources of the reactant C 3 H 4 + ion (photoionization of allene, propyne, the cyclopropene) and the nature of the neutral reactant, while all other product ions from the same reaction show sharp TOF distributions. On the other hand, all C 3 H 3 + ions produced by unimolecular decomposition of energetic C 3 H 4 + ions show sharp TOF distribution. The peculiarity of the C 3 H 3 + ions manifested in these and other experiments is discussed in conjunction with interstellar chemistry
13. Exploring the dusty Universe
Directory of Open Access Journals (Sweden)
Borghese, F
2005-11-01
Full Text Available Dust is an ubiquitous inhabitant of the interstellar medium, and leaves an unmistakable signature in its optical properties, and physico-chemical evolution. Although there is little direct knowledge of the true nature of interstellar dust grains, strong evidences point toward the possibility that such grains are composites of many small monomers (mainly made of silicates and carbonaceous materials. We consider two different models of fluffy dust aggregates, occurring as result of ballistic particle-cluster and cluster-cluster aggregation, and a cluster with a Gaussian-like sphere size distribution. We study the optical properties of such composite structures through the multipole fields and the Transition Matrix approach. Our results show the severe limits of applicability of the effective medium theories. By comparing radiation and gravitational forces, we also infer some relevant insights into the dynamical evolution of composite grains in the Solar System. We finally explore the possible role of composite fluffy dust grains in igniting an extraterrestrial prebiotic chemistry.
14. A review of interstellar rocketry fundamentals
Science.gov (United States)
Oliver, B. M.
1990-01-01
To reach even the nearest stars in a human lifetime requires a ship speed that is a substantial fraction of the speed of light. This means an enormous kinetic energy investment in the ship and suggests that maximizing the efficiency may be more important than minimizing the mass ratio. This paper develops the pertinent relativistic rocket equations and finds the conditions for high kinematic efficiency. Using the limiting efficiency of percent, the minimum energy needed for one-way and for round trip voyages, and to explore all 'good' suns out to a given distance, using prefueled rockets, is determined. Savings due to refueling the rocket (and reloading propellant) at the destination and in flight are both somewhat greater than 2:1.
15. The interstellar medium and star formation of galactic disks. I. Interstellar medium and giant molecular cloud properties with diffuse far-ultraviolet and cosmic-ray backgrounds
Science.gov (United States)
Li, Qi; Tan, Jonathan C.; Christie, Duncan; Bisbas, Thomas G.; Wu, Benjamin
2018-01-01
We present a series of adaptive mesh refinement hydrodynamic simulations of flat rotation curve galactic gas disks, with a detailed treatment of the interstellar medium (ISM) physics of the atomic to molecular phase transition under the influence of diffuse far-ultraviolet (FUV) radiation fields and cosmic-ray backgrounds. We explore the effects of different FUV intensities, including a model with a radial gradient designed to mimic the Milky Way. The effects of cosmic rays, including radial gradients in their heating and ionization rates, are also explored. The final simulations in this series achieve 4 pc resolution across the ˜20 kpc global disk diameter, with heating and cooling followed down to temperatures of ˜10 K. The disks are evolved for 300 Myr, which is enough time for the ISM to achieve a quasi-statistical equilibrium. In particular, the mass fraction of molecular gas is stabilized by ˜200 Myr. Additional global ISM properties are analyzed. Giant molecular clouds (GMCs) are also identified and the statistical properties of their populations are examined. GMCs are tracked as the disks evolve. GMC collisions, which may be a means of triggering star cluster formation, are counted and their rates are compared with analytic models. Relatively frequent GMC collision rates are seen in these simulations, and their implications for understanding GMC properties, including the driving of internal turbulence, are discussed.
16. The interstellar medium and star formation of galactic disks. I. Interstellar medium and giant molecular cloud properties with diffuse far-ultraviolet and cosmic-ray backgrounds
Science.gov (United States)
Li, Qi; Tan, Jonathan C.; Christie, Duncan; Bisbas, Thomas G.; Wu, Benjamin
2018-05-01
We present a series of adaptive mesh refinement hydrodynamic simulations of flat rotation curve galactic gas disks, with a detailed treatment of the interstellar medium (ISM) physics of the atomic to molecular phase transition under the influence of diffuse far-ultraviolet (FUV) radiation fields and cosmic-ray backgrounds. We explore the effects of different FUV intensities, including a model with a radial gradient designed to mimic the Milky Way. The effects of cosmic rays, including radial gradients in their heating and ionization rates, are also explored. The final simulations in this series achieve 4 pc resolution across the ˜20 kpc global disk diameter, with heating and cooling followed down to temperatures of ˜10 K. The disks are evolved for 300 Myr, which is enough time for the ISM to achieve a quasi-statistical equilibrium. In particular, the mass fraction of molecular gas is stabilized by ˜200 Myr. Additional global ISM properties are analyzed. Giant molecular clouds (GMCs) are also identified and the statistical properties of their populations are examined. GMCs are tracked as the disks evolve. GMC collisions, which may be a means of triggering star cluster formation, are counted and their rates are compared with analytic models. Relatively frequent GMC collision rates are seen in these simulations, and their implications for understanding GMC properties, including the driving of internal turbulence, are discussed.
17. Event boundaries and anaphoric reference.
Science.gov (United States)
Thompson, Alexis N; Radvansky, Gabriel A
2016-06-01
The current study explored the finding that parsing a narrative into separate events impairs anaphor resolution. According to the Event Horizon Model, when a narrative event boundary is encountered, a new event model is created. Information associated with the prior event model is removed from working memory. So long as the event model containing the anaphor referent is currently being processed, this information should still be available when there is no narrative event boundary, even if reading has been disrupted by a working-memory-clearing distractor task. In those cases, readers may reactivate their prior event model, and anaphor resolution would not be affected. Alternatively, comprehension may not be as event oriented as this account suggests. Instead, any disruption of the contents of working memory during comprehension, event related or not, may be sufficient to disrupt anaphor resolution. In this case, reading comprehension would be more strongly guided by other, more basic language processing mechanisms and the event structure of the described events would play a more minor role. In the current experiments, participants were given stories to read in which we included, between the anaphor and its referent, either the presence of a narrative event boundary (Experiment 1) or a narrative event boundary along with a working-memory-clearing distractor task (Experiment 2). The results showed that anaphor resolution was affected by narrative event boundaries but not by a working-memory-clearing distractor task. This is interpreted as being consistent with the Event Horizon Model of event cognition.
18. Interstellar and Solar Nebula Materials in Cometary Dust
Science.gov (United States)
Messenger, Scott; Nakamura-Messenger, Keiko; Keller, Lindsay; Nguyen, Ann; Clemett, Simon
2017-01-01
Laboratory studies of cometary dust collected in the stratosphere and returned from comet 81P/Wild 2 by the Stardust spacecraft have revealed ancient interstellar grains and molecular cloud organic matter that record a range of astrophysical processes and the first steps of planetary formation. Presolar materials are rarer meteorites owing to high temperature processing in the solar nebula and hydrothermal alteration on their asteroidal parent bodies. The greater preservation of presolar materials in comets is attributed to their low accretion temperatures and limited planetary processing. Yet, comets also contain a large complement of high temperature materials from the inner Solar System. Owing to the limited and biased sampling of comets to date, the proportions of interstellar and Solar System materials within them remains highly uncertain. Interstellar materials are identified by coordinated isotopic, mineralogical, and chemical measurements at the scale of individual grains. Chondritic porous interplanetary dust particles (CP IDPs) that likely derive from comets are made up of 0.1 - 10 micron-sized silicates, Fe-Ni-sulfides, oxides, and other phases bound by organic material. As much as 1% of the silicates are interstellar grains that have exotic isotopic compositions imparted by nucleosynthetic processes in their parent stars. Crystalline silicates in CP IDPs dominantly have normal isotopic compositions and probably formed in the Solar System. 81P samples include isotopically normal refractory minerals that resemble Ca-Al rich inclusions and chondrules common in meteorites. The origins of sub-micron amorphous silicates in IDPs are not certain, but at least a few % of them are interstellar grains. The remainder have isotopic compositions consistent with Solar System origins and elemental compositions that are inconsistent with interstellar grain properties, thus favoring formation in the solar nebula [4]. The organic component in comets and primitive
19. The structure and statistics of interstellar turbulence
International Nuclear Information System (INIS)
Kritsuk, A G; Norman, M L; Ustyugov, S D
2017-01-01
We explore the structure and statistics of multiphase, magnetized ISM turbulence in the local Milky Way by means of driven periodic box numerical MHD simulations. Using the higher order-accurate piecewise-parabolic method on a local stencil (PPML), we carry out a small parameter survey varying the mean magnetic field strength and density while fixing the rms velocity to observed values. We quantify numerous characteristics of the transient and steady-state turbulence, including its thermodynamics and phase structure, kinetic and magnetic energy power spectra, structure functions, and distribution functions of density, column density, pressure, and magnetic field strength. The simulations reproduce many observables of the local ISM, including molecular clouds, such as the ratio of turbulent to mean magnetic field at 100 pc scale, the mass and volume fractions of thermally stable Hi, the lognormal distribution of column densities, the mass-weighted distribution of thermal pressure, and the linewidth-size relationship for molecular clouds. Our models predict the shape of magnetic field probability density functions (PDFs), which are strongly non-Gaussian, and the relative alignment of magnetic field and density structures. Finally, our models show how the observed low rates of star formation per free-fall time are controlled by the multiphase thermodynamics and large-scale turbulence. (paper)
20. Political State Boundary (National)
Data.gov (United States)
Department of Transportation — State boundaries with political limit - boundaries extending into the ocean (NTAD). The TIGER/Line Files are shapefiles and related database files (.dbf) that are an...
1. Allegheny County Municipal Boundaries
Data.gov (United States)
Allegheny County / City of Pittsburgh / Western PA Regional Data Center — This dataset demarcates the municipal boundaries in Allegheny County. Data was created to portray the boundaries of the 130 Municipalities in Allegheny County the...
2. HUD GIS Boundary Files
Data.gov (United States)
Department of Housing and Urban Development — The HUD GIS Boundary Files are intended to supplement boundary files available from the U.S. Census Bureau. The files are for community planners interested in...
Data.gov (United States)
Kansas Data Access and Support Center — This database comprises 28 State agency boundaries and point of contact. The Kansas Geological Survey collected legal descriptions of the boundaries for various...
4. Stardust Interstellar Preliminary Examination VII: Synchrotron X-Ray Fluorescence Analysis of Six Stardust Interstellar Candidates Measured with the Advanced Photon Source 2-ID-D Microprobe
Science.gov (United States)
Allen, Carlton C.; Anderson, David; Bastien, Ron K.; Brenker, Frank E.; Flynn, George J.; Frank, David; Gainsforth, Zack; Sandford, Scott A.; Simionovici, Alexandre S.; Zolensky, Michael E.
2014-01-01
The NASA Stardust spacecraft exposed an aerogel collector to the interstellar dust passing through the solar system. We performed X-ray fluorescence element mapping and abundance measurements, for elements 19 < or = Z < or = 30, on six "interstellar candidates," potential interstellar impacts identified by Stardust@Home and extracted for analyses in picokeystones. One, I1044,3,33, showed no element hot-spots within the designated search area. However, we identified a nearby surface feature, consistent with the impact of a weak, high-speed particle having an approximately chondritic (CI) element abundance pattern, except for factor-of-ten enrichments in K and Zn and an S depletion. This hot-spot, containing approximately 10 fg of Fe, corresponds to an approximately 350 nm chondritic particle, small enough to be missed by Stardust@Home, indicating that other techniques may be necessary to identify all interstellar candidates. Only one interstellar candidate, I1004,1,2, showed a track. The terminal particle has large enrichments in S, Ti, Cr, Mn, Ni, Cu, and Zn relative to Fe-normalized CI values. It has high Al/Fe, but does not match the Ni/Fe range measured for samples of Al-deck material from the Stardust sample return capsule, which was within the field-of-view of the interstellar collector. A third interstellar candidate, I1075,1,25, showed an Al-rich surface feature that has a composition generally consistent with the Al-deck material, suggesting that it is a secondary particle. The other three interstellar candidates, I1001,1,16, I1001,2,17, and I1044,2,32, showed no impact features or tracks, but allowed assessment of submicron contamination in this aerogel, including Fe hot-spots having CI-like Ni/Fe ratios, complicating the search for CI-like interstellar/interplanetary dust.
5. On boundary superalgebras
International Nuclear Information System (INIS)
Doikou, Anastasia
2010-01-01
We examine the symmetry breaking of superalgebras due to the presence of appropriate integrable boundary conditions. We investigate the boundary breaking symmetry associated with both reflection algebras and twisted super-Yangians. We extract the generators of the resulting boundary symmetry as well as we provide explicit expressions of the associated Casimir operators.
6. Dust in the small Magellanic cloud. 1: Interstellar polarization and extinction data
Science.gov (United States)
Magalhaes, A. M.; Rodrigues, C. V.; Coyne, C. V.; Piirola, V.
1996-01-01
The typical extinction curve for the Small Magellanic Cloud (SMC), in contrast to that for the Galaxy, has no bump at 2175 A and has a steeper rise into the far ultraviolet. For the Galaxy the interpretation of the extinction and, therefore, the dust content of the interstellar medium has been greatly assisted by measurements of the wavelength dependence of the polarization. For the SMC no such measurements existed. Therefore, to further elucidate the dust properties in the SMC we have for the first time measured linear polarization with five colors in the optical region of the spectrum for a sample of reddened stars. For two of these stars, for which there were no existing UV spectrophotometric measurements, but for which we measured a relatively large polarization, we have also obtained data from the International Ultraviolet Explorer (IUE) in order to study the extinction. We also attempt to correlate the SMC extinction and polarization data. The main results are: the wavelength of maximum polarization, lambda(sub max), in the SMC is typically smaller than that in the Galaxy; however, AZC 456, which shows the UV extinction bump, has a lambda(sub max) typical of that in the Galaxy, but its polarization curve is narrower and its bump is shifted to shorter wavelengths as compared to the Galaxy; and from an analysis of both the extinction and polarization data it appears that the SMC has typically smaller grains than those in the Galaxy. The absence of the extinction bump in the SMC has generally been thought to imply a lower carbon abundance in the SMC compared to the Galaxy. We interpret our results to mean that te size distribution of the interstellar grains, and not only the carbon abundance, is different in the SMC as compared to the Galaxy. In Paper 2 we present dust model fits to these observations.
7. SEARCH FOR INTERSTELLAR METHOXYACETONITRILE AND CYANOETHANOL: INSIGHTS INTO COUPLING OF CYANO- TO METHANOL AND AMMONIA CHEMISTRY
International Nuclear Information System (INIS)
Braakman, R.; Belloche, A.; Menten, K. M.; Blake, G. A.
2010-01-01
As part of an effort to study gas-grain chemical models in star-forming regions as they relate to molecules containing cyanide (-C≡N) groups, we present here a search for the molecules 2-cyanoethanol (OHCH 2 CH 2 CN) and methoxyacetonitrile (CH 3 OCH 2 CN) in the galactic center region SgrB2. These species are structural isomers of each other and are targeted to investigate the cross-coupling of pathways emanating from the photolysis products of methanol and ammonia with pathways involving cyano-containing molecules. Methanol and ammonia ices are two of the main repositories of the elements C, O, and N in cold clouds and understanding their link to cyanide chemistry could give important insights into prebiotic molecular evolution. Neither species was positively detected, but the upper limits we determined allow comparison to the general patterns gleaned from chemical models. Our results indicate the need for an expansion of the model networks to better deal with cyano-chemistry, in particular with respect to pathways including products of methanol photolysis. In addition to these results, the two main observational routes for detecting new interstellar molecules are discussed. One route is by decreasing detection limits at millimeter wavelength through spatial filtering with interferometric studies at the Atacama Large Millimeter Array (ALMA), and the second is by searching for intense torsional states at THz frequencies using the Herschel Space Observatory. 2-cyanoethanol and methoxyacetonitrile would both be good test beds for exploring the capabilities of ALMA and Herschel in the study of complex interstellar chemistry.
8. Voyager observations of the interaction of the heliosphere with the interstellar medium
Directory of Open Access Journals (Sweden)
John D. Richardson
2013-05-01
Full Text Available This paper provides a brief review and update on the Voyager observations of the interaction of the heliosphere with the interstellar medium. Voyager has found many surprises: (1 a new energetic particle component which is accelerated at the termination shock (TS and leaks into the outer heliosphere forming a foreshock region; (2 a termination shock which is modulated by energetic particles and which transfers most of the solar wind flow energy to the pickup ions (not the thermal ions; (3 the heliosphere is asymmetric; (4 the TS does not accelerate anomalous cosmic rays at the Voyager locations; and (5 the plasma flow in the Voyagers 1 (V1 and 2 (V2 directions are very different. At V1 the flow was small after the TS and has recently slowed to near zero, whereas at V2 the speed has remained constant while the flow direction has turned tailward. V1 may have entered an extended boundary region in front of the heliopause (HP in 2010 in which the plasma flow speeds are near zero.
9. Voyager observations of the interaction of the heliosphere with the interstellar medium.
Science.gov (United States)
Richardson, John D
2013-05-01
This paper provides a brief review and update on the Voyager observations of the interaction of the heliosphere with the interstellar medium. Voyager has found many surprises: (1) a new energetic particle component which is accelerated at the termination shock (TS) and leaks into the outer heliosphere forming a foreshock region; (2) a termination shock which is modulated by energetic particles and which transfers most of the solar wind flow energy to the pickup ions (not the thermal ions); (3) the heliosphere is asymmetric; (4) the TS does not accelerate anomalous cosmic rays at the Voyager locations; and (5) the plasma flow in the Voyagers 1 (V1) and 2 (V2) directions are very different. At V1 the flow was small after the TS and has recently slowed to near zero, whereas at V2 the speed has remained constant while the flow direction has turned tailward. V1 may have entered an extended boundary region in front of the heliopause (HP) in 2010 in which the plasma flow speeds are near zero.
10. The distribution of interstellar dust in CALIFA edge-on galaxies via oligochromatic radiative transfer fitting
Science.gov (United States)
De Geyter, Gert; Baes, Maarten; Camps, Peter; Fritz, Jacopo; De Looze, Ilse; Hughes, Thomas M.; Viaene, Sébastien; Gentile, Gianfranco
2014-06-01
We investigate the amount and spatial distribution of interstellar dust in edge-on spiral galaxies, using detailed radiative transfer modelling of a homogeneous sample of 12 galaxies selected from the Calar Alto Legacy Integral Field Area survey. Our automated fitting routine, FITSKIRT, was first validated against artificial data. This is done by simultaneously reproducing the Sloan Digital Sky Survey g-, r-, i- and z-band observations of a toy model in order to combine the information present in the different bands. We show that this combined, oligochromatic fitting has clear advantages over standard monochromatic fitting especially regarding constraints on the dust properties. We model all galaxies in our sample using a three-component model, consisting of a double-exponential disc to describe the stellar and dust discs and using a Sérsic profile to describe the central bulge. The full model contains 19 free parameters, and we are able to constrain all these parameters to a satisfactory level of accuracy without human intervention or strong boundary conditions. Apart from two galaxies, the entire sample can be accurately reproduced by our model. We find that the dust disc is about 75 per cent more extended but only half as high as the stellar disc. The average face-on optical depth in the V band is 0.76 and the spread of 0.60 within our sample is quite substantial, which indicates that some spiral galaxies are relatively opaque even when seen face-on.
11. Interstellar Chemistry Gets More Complex With New Charged-Molecule Discovery
Science.gov (United States)
2007-07-01
Astronomers using data from the National Science Foundation's Robert C. Byrd Green Bank Telescope (GBT) have found the largest negatively-charged molecule yet seen in space. The discovery of the third negatively-charged molecule, called an anion, in less than a year and the size of the latest anion will force a drastic revision of theoretical models of interstellar chemistry, the astronomers say. Molecule formation Formation Process of Large, Negatively-Charged Molecule in Interstellar Space CREDIT: Bill Saxton, NRAO/AUI/NSF Click on image for page of graphics and detailed information "This discovery continues to add to the diversity and complexity that is already seen in the chemistry of interstellar space," said Anthony J. Remijan of the National Radio Astronomy Observatory (NRAO). "It also adds to the number of paths available for making the complex organic molecules and other large molecular species that may be precursors to life in the giant clouds from which stars and planets are formed," he added. Two teams of scientists found negatively-charged octatetraynyl, a chain of eight carbon atoms and one hydrogen atom, in the envelope of gas around an old, evolved star and in a cold, dark cloud of molecular gas. In both cases, the molecule had an extra electron, giving it a negative charge. About 130 neutral and about a dozen positively-charged molecules have been discovered in space, but the first negatively-charged molecule was not discovered until late last year. The largest previously-discovered negative ion found in space has six carbon atoms and one hydrogen atom. "Until recently, many theoretical models of how chemical reactions evolve in interstellar space have largely neglected the presence of anions. This can no longer be the case, and this means that there are many more ways to build large organic molecules in cosmic environments than have been explored," said Jan M. Hollis of NASA's Goddard Space Flight Center (GSFC). Ultraviolet light from stars can
12. Consequences of the Solar System passage through dense interstellar clouds
Directory of Open Access Journals (Sweden)
A. G. Yeghikyan
2003-06-01
Full Text Available Several consequences of the passage of the solar system through dense interstellar molecular clouds are discussed. These clouds, dense (more than 100 cm-3, cold (10–50 K and extended (larger than 1 pc, are characterized by a gas-to-dust mass ratio of about 100, by a specific power grain size spectrum (grain radii usually cover the range 0.001–3 micron and by an average dust-to-gas number density ratio of about 10-12. Frequently these clouds contain small-scale (10–100 AU condensations with gas concentrations ranging up to 10 5 cm-3. At their casual passage over the solar system they exert pressures very much enhanced with respect to today’s standards. Under these conditions it will occur that the Earth is exposed directly to the interstellar flow. It is shown first that even close to the Sun, at 1 AU, the cloud’s matter is only partly ionized and should mainly interact with the solar wind by charge exchange processes. Dust particles of the cloud serve as a source of neutrals, generated by the solar UV irradiation of dust grains, causing the evaporation of icy materials. The release of neutral atoms from dust grains is then followed by strong influences on the solar wind plasma flow. The behavior of the neutral gas inflow parameters is investigated by a 2-D hydrodynamic approach to model the interaction processes. Because of a reduction of the heliospheric dimension down to 1 AU, direct influence of the cloud’s matter to the terrestrial environment and atmosphere could be envisaged.Key words. Interplanetary physics (heliopause and solar wind termination; interplanetary dust; interstellar gas
13. Consequences of the Solar System passage through dense interstellar clouds
Directory of Open Access Journals (Sweden)
A. G. Yeghikyan
Full Text Available Several consequences of the passage of the solar system through dense interstellar molecular clouds are discussed. These clouds, dense (more than 100 cm-3, cold (10–50 K and extended (larger than 1 pc, are characterized by a gas-to-dust mass ratio of about 100, by a specific power grain size spectrum (grain radii usually cover the range 0.001–3 micron and by an average dust-to-gas number density ratio of about 10-12. Frequently these clouds contain small-scale (10–100 AU condensations with gas concentrations ranging up to 10 5 cm-3. At their casual passage over the solar system they exert pressures very much enhanced with respect to today’s standards. Under these conditions it will occur that the Earth is exposed directly to the interstellar flow. It is shown first that even close to the Sun, at 1 AU, the cloud’s matter is only partly ionized and should mainly interact with the solar wind by charge exchange processes. Dust particles of the cloud serve as a source of neutrals, generated by the solar UV irradiation of dust grains, causing the evaporation of icy materials. The release of neutral atoms from dust grains is then followed by strong influences on the solar wind plasma flow. The behavior of the neutral gas inflow parameters is investigated by a 2-D hydrodynamic approach to model the interaction processes. Because of a reduction of the heliospheric dimension down to 1 AU, direct influence of the cloud’s matter to the terrestrial environment and atmosphere could be envisaged.
Key words. Interplanetary physics (heliopause and solar wind termination; interplanetary dust; interstellar gas
14. Stochastic histories of dust grains in the interstellar medium
International Nuclear Information System (INIS)
Liffman, K.; Clayton, D.D.
1989-01-01
The purpose is to study an evolving system of refractory dust grains within the Interstellar Medium (ISM). This is done via a combination of Monte Carlo processes and a system of partial differential equations, where refractory dust grains formed within supernova remnants and ejecta from high mass loss stars are subjected to the processes of sputtering and collisional fragmentation in the diffuse media and accretion within the cold molecular clouds. In order to record chemical detail, the authors take each new particle to consist of a superrefractory core plus a more massive refractory mantle. The particles are allowed to transfer to and fro between the different phases of the interstellar medium (ISM) - on a time scale of 10(exp 8) years - until either the particles are destroyed or the program finishes at a Galaxy time of 6x10(exp 9) years. The resulting chemical and size spectrum(s) are then applied to various astrophysical problems with the following results. For an ISM which has no collisional fragmentation of the dust grains, roughly 10 percent by mass of the most refractory material survives the rigors of the ISM intact, which leaves open the possibility that fossilized isotopically anomalous material may have been present within the primordial solar nebula. Stuctured or layered refractory dust grains within the model cannot explain the observed interstellar depletions of refractory material. Fragmentation due to grain-grain collisions in the diffuse phase plus the accretion of material in the molecular cloud phase can under certain circumstances cause a bimodal distribution in grain size
15. OBSERVATIONAL CONSTRAINTS ON METHANOL PRODUCTION IN INTERSTELLAR AND PREPLANETARY ICES
International Nuclear Information System (INIS)
Whittet, D. C. B.; Cook, A. M.; Herbst, Eric; Chiar, J. E.; Shenoy, S. S.
2011-01-01
Methanol (CH 3 OH) is thought to be an important link in the chain of chemical evolution that leads from simple diatomic interstellar molecules to complex organic species in protoplanetary disks that may be delivered to the surfaces of Earthlike planets. Previous research has shown that CH 3 OH forms in the interstellar medium predominantly on the surfaces of dust grains. To enhance our understanding of the conditions that lead to its efficient production, we assemble a homogenized catalog of published detections and limiting values in interstellar and preplanetary ices for both CH 3 OH and the other commonly observed C- and O-bearing species, H 2 O, CO, and CO 2 . We use this catalog to investigate the abundance of ice-phase CH 3 OH in environments ranging from dense molecular clouds to circumstellar envelopes around newly born stars of low and high mass. Results show that CH 3 OH production arises during the CO freezeout phase of ice-mantle growth in the clouds, after an ice layer rich in H 2 O and CO 2 is already in place on the dust, in agreement with current astrochemical models. The abundance of solid-phase CH 3 OH in this environment is sufficient to account for observed gas-phase abundances when the ices are subsequently desorbed in the vicinity of embedded stars. CH 3 OH concentrations in the ices toward embedded stars show order-of-magnitude object-to-object variations, even in a sample restricted to stars of low mass associated with ices lacking evidence of thermal processing. We hypothesize that the efficiency of CH 3 OH production in dense cores and protostellar envelopes is mediated by the degree of prior CO depletion.
16. Evolution of interstellar organic compounds under asteroidal hydrothermal conditions
Science.gov (United States)
Vinogradoff, V.; Bernard, S.; Le Guillou, C.; Remusat, L.
2018-05-01
Carbonaceous chondrites (CC) contain a diversity of organic compounds. No definitive evidence for a genetic relationship between these complex organic molecules and the simple organic molecules detected in the interstellar medium (ISM) has yet been reported. One of the many difficulties arises from the transformations of organic compounds during accretion and hydrothermal alteration on asteroids. Here, we report results of hydrothermal alteration experiments conducted on a common constituent of interstellar ice analogs, Hexamethylenetetramine (HMT - C6H12N4). We submitted HMT to asteroidal hydrothermal conditions at 150 °C, for various durations (up to 31 days) and under alkaline pH. Organic products were characterized by gas chromatography mass spectrometry, infrared spectroscopy and synchrotron-based X-ray absorption near edge structure spectroscopy. Results show that, within a few days, HMT has evolved into (1) a very diverse suite of soluble compounds dominated by N-bearing aromatic compounds (> 150 species after 31 days), including for instance formamide, pyridine, pyrrole and their polymers (2) an aromatic and N-rich insoluble material that forms after only 7 days of experiment and then remains stable through time. The reaction pathways leading to the soluble compounds likely include HMT dissociation, formose and Maillard-type reactions, e.g. reactions of sugar derivatives with amines. The present study demonstrates that, if interstellar organic compounds such as HMT had been accreted by chondrite parent bodies, they would have undergone chemical transformations during hydrothermal alteration, potentially leading to the formation of high molecular weight insoluble organic molecules. Some of the diversity of soluble and insoluble organic compounds found in CC may thus result from asteroidal hydrothermal alteration.
17. Ketene Formation in Interstellar Ices: A Laboratory Study
Science.gov (United States)
Hudson, Reggie L.; Loeffler, Mark Josiah
2013-01-01
The formation of ketene (H2CCO, ethenone) in polar and apolar ices was studied with in situ 0.8 MeV proton irradiation, far-UVphotolysis, and infrared spectroscopic analyses at 10-20 K. Using isotopically enriched reagents, unequivocal evidencewas obtained for ketene synthesis in H2O-rich and CO2-rich ices, and several reaction products were identified. Results from scavenging experiments suggested that ketene was formed by free-radical pathways, as opposed to acid-base processes or redox reactions. Finally, we use our results to draw conclusions about the formation and stability of ketene in the interstellar medium.
18. Use of Laboratory Data to Model Interstellar Chemistry
Science.gov (United States)
Vidali, Gianfranco; Roser, J. E.; Manico, G.; Pirronello, V.
2006-01-01
Our laboratory research program is about the formation of molecules on dust grains analogues in conditions mimicking interstellar medium environments. Using surface science techniques, in the last ten years we have investigated the formation of molecular hydrogen and other molecules on different types of dust grain analogues. We analyzed the results to extract quantitative information on the processes of molecule formation on and ejection from dust grain analogues. The usefulness of these data lies in the fact that these results have been employed by theoreticians in models of the chemical evolution of ISM environments.
19. Recommended rest frequencies for observed interstellar molecular transitions
International Nuclear Information System (INIS)
Lovas, F.J.; Snyder, I.E.; Johnson, D.R.
1979-01-01
The most accurate values presently available for the rest frequencies of all known interstellar molecular transitions are presented and recommended for reference in future astronomical observations in the radio and microwave regions. The recommended values have been carefully selected after critical evaluation of the spectroscopic literature. Probable error limits along with the proper molecular and quantum mechanical labels are presented for each observed transition. Representative line antenna temperatures are also presented for a typical source as a convenience to users. References are cited to both the astronomical and the laboratory literature
20. Study of the interstellar medium towards RCW 103
OpenAIRE
Paron, Sergio Ariel; Reynoso, Estela Marta; Dubner, Gloria Mabel; Castelletti, Gabriela Marta
2017-01-01
RCW 103 is a shell type supernova remnant (SNR) that, according to near infrared observations, is interacting with a molecular cloud, specially to the south. In this paper we report on the study of the interstellar medium in an extended region towards RCW 103 based on HI 21 cm data acquired with the ATCA radiotelescope. Also, we report on the detection of HCO+ and CO emission in the rotational transition J=1-0 associated with the remnant. These observations were carried out with the millimete...
1. Interstellar ice grains in the Taurus molecular clouds
International Nuclear Information System (INIS)
Whittet, D.C.B.; Bode, M.F.; Baines, D.W.T.; Evans, A.
1983-01-01
Observations made in November 1981 using the United Kingdom Infrared Telescope (UKIRT) at Mauna Kea of the 3 μm ice absorption feature in the spectra of several obscured stars in the Taurus interstellar clouds are reported. The feature correlated in strength with extinction at visual wavelengths (Asub(v)), and is present in stars with Asub(v) as low as 4-6 mag. Ice may be widespread in the Taurus clouds, vindicating ideas on grain composition and growth first reported nearly 50 yr ago. (author)
2. Dense interstellar cloud chemistry: Basic issues and possible dynamical solution
International Nuclear Information System (INIS)
Prasad, S.S.; Heere, K.R.; Tarafdar, S.P.
1989-01-01
Standing at crossroad of enthusiasm and frustration, dense intertellar cloud chemistry has a squarely posed fundamental problem: Why do the grains appear to play at best a minor role in the chemistry? Grain surface chemistry creates considerable difficulties when the authors treat dense clouds as static objects and ignore the implications of the processes by which the clouds became dense in the first place. A new generation of models which treat chemical and dynamical evolutions concurrently are therefore presented as possible solution to the current frustrations. The proposed modeling philosophy and agenda could make the next decade quite exciting for interstellar chemistry
3. Spectroscopy of the earth's atmosphere and interstellar medium
CERN Document Server
Rao, KN
1992-01-01
Spectroscopy of the Earth's Atmosphere and Interstellar Medium focuses on the characteristics of the electromagnetic spectrum of the Earth's atmosphere in the far-infrared and microwave regions. It discusses the modes of observation in field measurements and reviews the two techniques used in the spectral region. Organized into six chapters, this volume begins with an overview of the effect of water-vapor absorption, followed by a discussion on the two frequently used method for deriving atmospheric parameters from high-resolution infrared atmospheric spectra, namely, the equivalent width
4. Physical conditions in CaFe interstellar clouds
OpenAIRE
Gnacinski, P.; Krogulec, M.
2007-01-01
Interstellar clouds that exhibit strong Ca I and Fe I lines were called CaFe clouds. The ionisation equilibrium equations were used to model the column densities of Ca II, Ca I, K I, Na I, Fe I and Ti II in CaFe clouds. The chemical composition of CaFe clouds is that of the Solar System and no depletion of elements onto dust grains is seen. The CaFe clouds have high electron densities n=1 cm^-3 that leads to high column densities of neutral Ca and Fe.
5. Future spacecraft propulsion systems. Enabling technologies for space exploration. 2. ed.
Energy Technology Data Exchange (ETDEWEB)
Czysz, Paul A. [St. Louis Univ., MO (United States). Oliver L. Parks Endowed Chair in Aerospace Engineering; Bruno, Claudio [Univ. degli Studi di Roma (Italy). Dipt. di Meccanica e Aeronautica
2009-07-01
In this second edition of Future Spacecraft Propulsion Systems, the authors demonstrate the need to break free from the old established concepts of expendable rockets, using chemical propulsion, and to develop new breeds of launch vehicle capable of both launching payloads into orbit at a dramatically reduced cost and for sustained operations in low-Earth orbit. The next steps to establishing a permanent 'presence' in the Solar System beyond Earth are the commercialisation of sustained operations on the Moon and the development of advanced nuclear or high-energy space propulsion systems for Solar System exploration out to the boundary of interstellar space. In the future, high-energy particle research facilities may one day yield a very high-energy propulsion system that will take us to the nearby stars, or even beyond. Space is not quiet: it is a continuous series of nuclear explosions that provide the material for new star systems to form and provide the challenge to explore. This book provides an assessment of the industrial capability required to construct and operate the necessary spacecraft. Time and distance communication and control limitations impose robotic constraints. Space environments restrict human sustained presence and put high demands on electronic, control and materials systems. This comprehensive and authoritative book puts spacecraft propulsion systems in perspective, from earth orbit launchers to astronomical/space exploration vehicles. It includes new material on fusion propulsion, new figures and updates and expands the information given in the first edition. (orig.)
6. Knowledge Sharing Across Global-Local Boundaries
DEFF Research Database (Denmark)
Zølner, Mette
The paper explores how locals span boundaries between corporate and local levels. The aim is to better comprehend potentialities and challenges when MNCs draws on locals’ culture specific knowledge. The study is based on an in-depth, interpretive case study of boundary spanning by local actors in...... approach with pattern matching is a way to shed light on the tacit local knowledge that organizational actors cannot articulate and that an exclusively inductive research is not likely to unveil....
7. Research, Boundaries, and Policy in Networked Learning
DEFF Research Database (Denmark)
This book presents cutting-edge, peer reviewed research on networked learning organized by three themes: policy in networked learning, researching networked learning, and boundaries in networked learning. The "policy in networked learning" section explores networked learning in relation to policy...... networks, spaces of algorithmic governance and more. The "boundaries in networked learning" section investigates frameworks of students' digital literacy practices, among other important frameworks in digital learning. Lastly, the "research in networked learning" section delves into new research methods...
8. Equation of Motion of an Interstellar Bussard Ramjet with Radiation and Mass Losses
Science.gov (United States)
Semay, Claude; Silvestre-Brac, Bernard
2008-01-01
An interstellar Bussard ramjet is a spaceship using the protons of the interstellar medium in a fusion engine to produce thrust. In recent papers, it was shown that the relativistic equation of motion of an ideal ramjet and that of a ramjet with radiation loss are analytical. When a mass loss appears, the limit speed of the ramjet is more strongly…
9. Longitudinal vortices in a transitioning boundary layer
International Nuclear Information System (INIS)
Anders, J.B.; Backwelder, R.F.
1980-01-01
Naturally occurring spanwise variations of the streamwise velocity component, characteristic of longitudinal vortices embedded in a transitioning boundary layer were explored using hot-wire anemometers. A vibrating ribbon introduced stable or unstable Tollmien-Schlichting waves into the laminar boundary layer. These damped or growing disturbances always developed a strong three-dimensional pattern even though no spanwise perturbations were artificially induced. Changing the radius of the leading edge and other modifications to the flat plate, wind tunnel and boundary layer did not alter the spanwise wavelength of the vortices. (orig.)
10. Modelling ultraviolet-line diagnostics of stars, the ionized and the neutral interstellar medium in star-forming galaxies
Science.gov (United States)
Vidal-García, A.; Charlot, S.; Bruzual, G.; Hubeny, I.
2017-09-01
We combine state-of-the-art models for the production of stellar radiation and its transfer through the interstellar medium (ISM) to investigate ultraviolet-line diagnostics of stars, the ionized and the neutral ISM in star-forming galaxies. We start by assessing the reliability of our stellar population synthesis modelling by fitting absorption-line indices in the ISM-free ultraviolet spectra of 10 Large Magellanic Cloud clusters. In doing so, we find that neglecting stochastic sampling of the stellar initial mass function in these young (∼10-100 Myr), low-mass clusters affects negligibly ultraviolet-based age and metallicity estimates but can lead to significant overestimates of stellar mass. Then, we proceed and develop a simple approach, based on an idealized description of the main features of the ISM, to compute in a physically consistent way the combined influence of nebular emission and interstellar absorption on ultraviolet spectra of star-forming galaxies. Our model accounts for the transfer of radiation through the ionized interiors and outer neutral envelopes of short-lived stellar birth clouds, as well as for radiative transfer through a diffuse intercloud medium. We use this approach to explore the entangled signatures of stars, the ionized and the neutral ISM in ultraviolet spectra of star-forming galaxies. We find that, aside from a few notable exceptions, most standard ultraviolet indices defined in the spectra of ISM-free stellar populations are prone to significant contamination by the ISM, which increases with metallicity. We also identify several nebular-emission and interstellar-absorption features, which stand out as particularly clean tracers of the different phases of the ISM.
11. The effect of catastrophic collisional fragmentation and diffuse medium accretion on a computational interstellar dust system
Science.gov (United States)
Liffman, Kurt
1990-01-01
The effects of catastrophic collisional fragmentation and diffuse medium accretion on a the interstellar dust system are computed using a Monte Carlo computer model. The Monte Carlo code has as its basis an analytic solution of the bulk chemical evolution of a two-phase interstellar medium, described by Liffman and Clayton (1989). The model is subjected to numerous different interstellar processes as it transfers from one interstellar phase to another. Collisional fragmentation was found to be the dominant physical process that shapes the size spectrum of interstellar dust. It was found that, in the diffuse cloud phase, 90 percent of the refractory material is locked up in the dust grains, primarily due to accretion in the molecular medium. This result is consistent with the observed depletions of silicon. Depletions were found to be affected only slightly by diffuse cloud accretion.
12. Determination of interstellar pickup ion distributions in the solar wind with SOHO and Cluster
Directory of Open Access Journals (Sweden)
E. Möbius
1996-05-01
Full Text Available Over the last 10 years, the experimental basis for the study of the local interstellar medium has been substantially enhanced by the direct detection of interstellar pickup ions and of interstellar neutral helium within the heliosphere. Pickup ions can be studied for a wide range of interstellar species. However, currently the accuracy of the method to determine the parameters of the interstellar medium, namely neutral density, temperature and relative velocity, is hampered by two problems: (1 In most cases the crucial ionization rates are not available from simultaneous measurements and (2 the transport of the pickup ions in the interplanetary medium substantially modifies the measured spatial distribution of the ions. In this study we will discuss how the enhanced capabilities of the instrumentation on SOHO and Cluster in combination with ongoing efforts to model the pickup ion distributions will lead to a significant improvement over the coming years.
13. BENZENE FORMATION ON INTERSTELLAR ICY MANTLES CONTAINING PROPARGYL ALCOHOL
Energy Technology Data Exchange (ETDEWEB)
Sivaraman, B.; Mukherjee, R.; Subramanian, K. P.; Banerjee, S. B., E-mail: [email protected] [Space and Atmospheric Sciences Division, Physical Research Laboratory, Ahmedabad (India)
2015-01-10
Propargyl alcohol (CHCCH{sub 2}OH) is a known stable isomer of the propenal (CH{sub 2}CHCHO) molecule that was reported to be present in the interstellar medium (ISM). At astrochemical conditions in the laboratory, icy layers of propargyl alcohol grown at 85 K were irradiated by 2 keV electrons and probed by a Fourier Transform InfraRed spectrometer in the mid-infrared (IR) region, 4000-500 cm{sup –1}. Propargyl alcohol ice under astrochemical conditions was studied for the first time; therefore, IR spectra of reported amorphous (85 K) and crystalline (180 K) propargyl alcohol ices can be used to detect its presence in the ISM. Moreover, our experiments clearly show benzene (C{sub 6}H{sub 6}) formation to be the major product from propargyl alcohol irradiation, confirming the role of propargyl radicals (C{sub 3}H{sub 3}) formed from propargyl alcohol dissociation that was long expected based on theoretical modeling to effectively synthesize C{sub 6}H{sub 6} in the interstellar icy mantles.
14. THE STRUCTURE, ORIGIN, AND EVOLUTION OF INTERSTELLAR HYDROCARBON GRAINS
Energy Technology Data Exchange (ETDEWEB)
Chiar, J. E.; Ricca, A. [SETI Institute, Carl Sagan Center, 189 Bernardo Avenue, Mountain View, CA 94043 (United States); Tielens, A. G. G. M. [Leiden Observatory, P.O. Box 9513, NL-2300 RA Leiden (Netherlands); Adamson, A. J., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [Gemini Observatory, Northern Operations Center, 670 North A' ohoku Place, Hilo, HI 96729 (United States)
2013-06-10
Many materials have been considered for the carrier of the hydrocarbon absorption bands observed in the diffuse interstellar medium (ISM). In order to refine the model for ISM hydrocarbon grains, we analyze the observed aromatic (3.28, 6.2 {mu}m) and aliphatic (3.4 {mu}m) hydrocarbon absorption features in the diffuse ISM along the line of sight toward the Galactic center Quintuplet Cluster. Observationally, sp {sup 2} bonds can be measured in astronomical spectra using the 6.2 {mu}m CC aromatic stretch feature, whereas the 3.4 {mu}m aliphatic feature can be used to quantify the fraction of sp {sup 3} bonds. The fractional abundance of these components allows us to place the Galactic diffuse ISM hydrocarbons on a ternary phase diagram. We conclude that the Galactic hydrocarbon dust has, on average, a low H/C ratio and sp {sup 3} content and is highly aromatic. We have placed the results of our analysis within the context of the evolution of carbon dust in the ISM. We argue that interstellar carbon dust consists of a large core of aromatic carbon surrounded by a thin mantle of hydrogenated amorphous carbon (a-C:H), a structure that is a natural consequence of the processing of stardust grains in the ISM.
15. THE STRUCTURE, ORIGIN, AND EVOLUTION OF INTERSTELLAR HYDROCARBON GRAINS
International Nuclear Information System (INIS)
Chiar, J. E.; Ricca, A.; Tielens, A. G. G. M.; Adamson, A. J.
2013-01-01
Many materials have been considered for the carrier of the hydrocarbon absorption bands observed in the diffuse interstellar medium (ISM). In order to refine the model for ISM hydrocarbon grains, we analyze the observed aromatic (3.28, 6.2 μm) and aliphatic (3.4 μm) hydrocarbon absorption features in the diffuse ISM along the line of sight toward the Galactic center Quintuplet Cluster. Observationally, sp 2 bonds can be measured in astronomical spectra using the 6.2 μm CC aromatic stretch feature, whereas the 3.4 μm aliphatic feature can be used to quantify the fraction of sp 3 bonds. The fractional abundance of these components allows us to place the Galactic diffuse ISM hydrocarbons on a ternary phase diagram. We conclude that the Galactic hydrocarbon dust has, on average, a low H/C ratio and sp 3 content and is highly aromatic. We have placed the results of our analysis within the context of the evolution of carbon dust in the ISM. We argue that interstellar carbon dust consists of a large core of aromatic carbon surrounded by a thin mantle of hydrogenated amorphous carbon (a-C:H), a structure that is a natural consequence of the processing of stardust grains in the ISM.
16. Effects of time-dependent photoionization on interstellar pickup atoms
International Nuclear Information System (INIS)
Isenberg, P.A.; Lee, M.A.
1995-01-01
We present an analytical model for the density variations of interstellar pickup ions in the solar wind due to a time-dependent variation in the photoionization rate, our model predicts a pickup ion density enhancement lasting for a time of the order of the duration of the increase plus the solar wind convection time to the observation point. If the photoionization rate returns to its initial value, this enhancement is followed by a decreased pickup ion density resulting from a depleted interstellar neutral particle density. In the absence of further variations in the photoionization rate, the pickup ion density recovers on a time which scales as the radial position of the observation point divided by the inflow speed of the neutral particles. Gradual variations in the photoionization rate result in a pickup ion density which tends to track the ionization rate, though the density variations are smoothed and delayed in time due to the solar wind convection of ions picked up at points closer to the Sun. 27 refs., 4 figs
17. TRAJECTORIES AND DISTRIBUTION OF INTERSTELLAR DUST GRAINS IN THE HELIOSPHERE
International Nuclear Information System (INIS)
Slavin, Jonathan D.; Frisch, Priscilla C.; Müller, Hans-Reinhard; Heerikhuisen, Jacob; Pogorelov, Nikolai V.; Reach, William T.; Zank, Gary
2012-01-01
The solar wind carves a bubble in the surrounding interstellar medium (ISM) known as the heliosphere. Charged interstellar dust grains (ISDG) encountering the heliosphere may be diverted around the heliopause or penetrate it depending on their charge-to-mass ratio. We present new calculations of trajectories of ISDG in the heliosphere, and the dust density distributions that result. We include up-to-date grain charging calculations using a realistic UV radiation field and full three-dimensional magnetohydrodynamic fluid + kinetic models for the heliosphere. Models with two different (constant) polarities for the solar wind magnetic field (SWMF) are used, with the grain trajectory calculations done separately for each polarity. Small grains a gr ∼ gr ∼> 1.0 μm, pass into the inner solar system and are concentrated near the Sun by its gravity. Trajectories of intermediate size grains depend strongly on the SWMF polarity. When the field has magnetic north pointing to ecliptic north, the field de-focuses the grains resulting in low densities in the inner heliosphere, while for the opposite polarity the dust is focused near the Sun. The ISDG density outside the heliosphere inferred from applying the model results to in situ dust measurements is inconsistent with local ISM depletion data for both SWMF polarities but is bracketed by them. This result points to the need to include the time variation in the SWMF polarity during grain propagation. Our results provide valuable insights for interpretation of the in situ dust observations from Ulysses.
18. ACCURATE MODELING OF X-RAY EXTINCTION BY INTERSTELLAR GRAINS
International Nuclear Information System (INIS)
Hoffman, John; Draine, B. T.
2016-01-01
Interstellar abundance determinations from fits to X-ray absorption edges often rely on the incorrect assumption that scattering is insignificant and can be ignored. We show instead that scattering contributes significantly to the attenuation of X-rays for realistic dust grain size distributions and substantially modifies the spectrum near absorption edges of elements present in grains. The dust attenuation modules used in major X-ray spectral fitting programs do not take this into account. We show that the consequences of neglecting scattering on the determination of interstellar elemental abundances are modest; however, scattering (along with uncertainties in the grain size distribution) must be taken into account when near-edge extinction fine structure is used to infer dust mineralogy. We advertise the benefits and accuracy of anomalous diffraction theory for both X-ray halo analysis and near edge absorption studies. We present an open source Fortran suite, General Geometry Anomalous Diffraction Theory (GGADT), that calculates X-ray absorption, scattering, and differential scattering cross sections for grains of arbitrary geometry and composition
19. ACCURATE MODELING OF X-RAY EXTINCTION BY INTERSTELLAR GRAINS
Energy Technology Data Exchange (ETDEWEB)
Hoffman, John; Draine, B. T., E-mail: [email protected], E-mail: [email protected] [Princeton University Observatory, Peyton Hall, Princeton, NJ 08544-1001 (United States)
2016-02-01
Interstellar abundance determinations from fits to X-ray absorption edges often rely on the incorrect assumption that scattering is insignificant and can be ignored. We show instead that scattering contributes significantly to the attenuation of X-rays for realistic dust grain size distributions and substantially modifies the spectrum near absorption edges of elements present in grains. The dust attenuation modules used in major X-ray spectral fitting programs do not take this into account. We show that the consequences of neglecting scattering on the determination of interstellar elemental abundances are modest; however, scattering (along with uncertainties in the grain size distribution) must be taken into account when near-edge extinction fine structure is used to infer dust mineralogy. We advertise the benefits and accuracy of anomalous diffraction theory for both X-ray halo analysis and near edge absorption studies. We present an open source Fortran suite, General Geometry Anomalous Diffraction Theory (GGADT), that calculates X-ray absorption, scattering, and differential scattering cross sections for grains of arbitrary geometry and composition.
20. Stochastic histories of dust grains in the interstellar medium
International Nuclear Information System (INIS)
Liffman, K.
1988-01-01
The purpose of this thesis is to study an evolving system of SU-perNOva CONdensateS (SUNOCONS) within the Interstellar Medium (ISM). This is done via a Monte Carlo process where refractory dust grains formed within supernova remnants are subjected to the processes of sputtering and collisional fragmentation in the diffuse phase and accretion within the cold molecular cloud phase. In order to record chemical detail, we take each new particle to consist of a superrefractory core plus a more massive refractory mantle. The particles are allowed to transfer to and from between the different phases of the ISM until either the particles are destroyed or the program finishes. The resulting chemical and size spectrum(s) are then applied to various astrophysical problems with the following results: (1) after six thousand million years roughly 10 to 20% by mass of the most refractory material (Al 2 O 3 ) survives the rigors of the ISM intact, which leaves open the possibility that fossilized isotopically anomalous material may have been present within the primordial solar nebula. (2) structured or layered refractory dust grains within our model cannot explain the observed interstellar depletions of refractory material. (3) fragmentation due to grain-grain collisions in the diffuse phase plus the accretion of material in the molecular cloud phase can under certain circumstances cause a biomodal distribution in grain size
1. From interstellar dust to comets - A unification of observational constraints
International Nuclear Information System (INIS)
Greenberg, J.M.; Hage, J.I.
1990-01-01
The interstellar dust model of comets is numerically worked out to satisfy simultaneously several basic constraints provided by observations of Comet Halley, and to derive the porosity of coma dust. The observational constraints are (1) the strengths of the 3.4 and 9.7 micron emission bands, (2) the shape of the 9.7 micron band, (3) the amount of silicates relative to organic materials, and (4) the mass distribution of the dust. The method used involves precise calculations of temperatures and the emission characteristics of porous aggregates of interstellar dust as a function of their mass, porosity, and distance to the sun and the wavelength. The results indicate that coma dust has a porosity in the range 0.93-0.975, i.e., a packing factor of 0.07 or less, consistent with independent observations of comet densities of 0.6 to 0.26 g/cu cm and meteor densities of less than 0.2 g/cu cm. 63 refs
2. Nuclear abundances and evolution of the interstellar medium
International Nuclear Information System (INIS)
Wannier, P.G.
1980-01-01
Observations of molecular and elemental abundances in the interstellar medium (ISM) are reviewed, with special attention given to isotope ratios. The derivation of molecular isotope abundances for the ISM is discussed, along with H and C fractionation. Millimeter- and centimeter-wave spectra of giant clouds are examined with respect to isotope abundances of C, O, N, Si, S, and D. Evidence for the current enrichment of the ISM by mass loss from evolved stars is considered, together with chemical abundance gradients in H II regions and planetary nebulae. Cosmic-ray observations pertaining to abundances in the ISM are summarized, with emphasis on available results for Ne, Mg, Si, Fe, and Ni. The observations reviewed are shown to support arguments in favor of: (1) the cosmological production of D and He-3 (2) the production of the CNO elements by hydrostatic hydrogen burning (3) the nucleosynthesis of Ne, Mg, Si, S, Fe, and Ni as a result of He burning (4) solar abundances of interstellar S, Fe, and Ni and (5) a direct association between observed inhomogeneities in the ISM and mass loss from evolved stellar objects
3. An infrared measurement of chemical desorption from interstellar ice analogues
Science.gov (United States)
Oba, Y.; Tomaru, T.; Lamberts, T.; Kouchi, A.; Watanabe, N.
2018-03-01
In molecular clouds at temperatures as low as 10 K, all species except hydrogen and helium should be locked in the heterogeneous ice on dust grain surfaces. Nevertheless, astronomical observations have detected over 150 different species in the gas phase in these clouds. The mechanism by which molecules are released from the dust surface below thermal desorption temperatures to be detectable in the gas phase is crucial for understanding the chemical evolution in such cold clouds. Chemical desorption, caused by the excess energy of an exothermic reaction, was first proposed as a key molecular release mechanism almost 50 years ago1. Chemical desorption can, in principle, take place at any temperature, even below the thermal desorption temperature. Therefore, astrochemical network models commonly include this process2,3. Although there have been a few previous experimental efforts4-6, no infrared measurement of the surface (which has a strong advantage to quantify chemical desorption) has been performed. Here, we report the first infrared in situ measurement of chemical desorption during the reactions H + H2S → HS + H2 (reaction 1) and HS + H → H2S (reaction 2), which are key to interstellar sulphur chemistry2,3. The present study clearly demonstrates that chemical desorption is a more efficient process for releasing H2S into the gas phase than was previously believed. The obtained effective cross-section for chemical desorption indicates that the chemical desorption rate exceeds the photodesorption rate in typical interstellar environments.
4. Magnetic field amplification in interstellar collisionless shock waves
International Nuclear Information System (INIS)
Chevalier, R.A.
1977-01-01
It is stated that it is commonly assumed that a simple compression of the magnetic field occurs in interstellar shock waves. Recent space observations of the Earth's bow shock have shown that turbulent amplification of the magnetic field can occur in a collisionless shock. It is shown here that radio observations of Tycho's supernova remnant indicate the presence of a shock wave with such magnetic field amplification. There is at present no theory for the microinstabilities that give rise to turbulent amplification of the magnetic field. Despite the lack of theoretical understanding the possibility of field amplification in interstellar shock waves is here considered. In Tycho's supernova remnant there is evidence for the presence of a collisionless shock, and this is discussed. On the basis of observations of the Earth's bow shock, it is expected that turbulent magnetic field amplification occurs in the shock wave of this remnant, and this is supported by radio observations of the remnant. Consideration is given as to what extent the magnetic field is amplified in the shock wave on the basis of the non-thermal radio flux. (U.K.)
5. Polarimetric study of the interstellar medium in Taurus Dark Clouds
International Nuclear Information System (INIS)
Hsu, J.
1985-01-01
An optical linear polarimetric survey was completed for more than 300 stars in an area of 6.5 0 x 10 0 toward the Taurus Dark Clouds Complex. It was found that the orientation of the magnetic field is roughly perpendicular to the elongation direction of the dust lanes, indicating cloud contraction along the magnetic field lines. The distance to the front edge of the dark clouds in Taurus is determined to be 126 pc. There is only insignificant amount of obscuring material between the cloud complex and the Sun. Besides the polarization data, the reddenings of about 250 stars were also obtained from the UBV photometry. The mean polarization to reddening ratio in the Taurus region is 4.6, which is similar to that of the general interstellar matter. The wavelengths of maximum polarization were determined for 30 stars in Taurus. They show an average value of lambda/sub max/ = 0.57 μm, which is only slightly higher than the mean value of the general interstellar medium, lambda/sub max/ = 0.55 μm. A few stars that show higher values of lambda/sub max/ are found near the small isolated regions of very high extinction. One such highly obscured small region where very complex long chain molecules have been discovered in the ratio spectra, is the Taurus Molecular Cloud 1
6. VARIATIONS BETWEEN DUST AND GAS IN THE DIFFUSE INTERSTELLAR MEDIUM
International Nuclear Information System (INIS)
Reach, William T.; Heiles, Carl; Bernard, Jean-Philippe
2015-01-01
Using the Planck far-infrared and Arecibo GALFA 21 cm line surveys, we identified a set of isolated interstellar clouds (approximately degree-sized on the sky and comprising 100 solar masses) and assessed the ratio of gas mass to dust mass. Significant variations of the gas/dust ratio are found both from cloud to cloud and within regions of individual clouds; within the clouds, the atomic gas per unit dust decreases by more than a factor of 3 compared with the standard gas/dust ratio. Three hypotheses are considered. First, the apparently low gas/dust ratio could be due to molecular gas. Comparing to Planck CO maps, the brightest clouds have a H 2 /CO ratio comparable to Galactic plane clouds, but a strong lower limit is placed on the ratio for other clouds, such that the required amount of molecular gas is far higher than would be expected based on the CO upper limits. Second, we consider self-absorbed 21 cm lines and find that the optical depth must be ∼3, significantly higher than found from surveys of radio sources. Third, grain properties may change within the clouds: they become more emissive when they are colder, while not utilizing heavy elements that already have their cosmic abundance fully locked into grains. It is possible that all three processes are active, and follow-up studies will be required to disentangle them and measure the true total gas and dust content of interstellar clouds
7. ON THE FORMATION OF DIPEPTIDES IN INTERSTELLAR MODEL ICES
International Nuclear Information System (INIS)
Kaiser, R. I.; Kim, Y. S.; Stockton, A. M.; Jensen, E. C.; Mathies, R. A.
2013-01-01
The hypothesis of an exogenous origin and delivery of biologically important molecules to early Earth presents an alternative route to their terrestrial in situ formation. Dipeptides like Gly-Gly detected in the Murchison meteorite are considered as key molecules in prebiotic chemistry because biofunctional dipeptides present the vital link in the evolutionary transition from prebiotic amino acids to early proteins. However, the processes that could lead to the exogenous abiotic synthesis of dipeptides are unknown. Here, we report the identification of two proteinogenic dipeptides—Gly-Gly and Leu-Ala—formed via electron-irradiation of interstellar model ices followed by annealing the irradiated samples to 300 K. Our results indicate that the radiation-induced, non-enzymatic formation of proteinogenic dipeptides in interstellar ice analogs is facile. Once synthesized and incorporated into the ''building material'' of solar systems, biomolecules at least as complex as dipeptides could have been delivered to habitable planets such as early Earth by meteorites and comets, thus seeding the beginning of life as we know it.
8. Protostellar formation in rotation interstellar clouds. III. Nonaxisymmetric collapse
International Nuclear Information System (INIS)
Boss, A.P.
1980-01-01
A full three spatial-dimension gravitational hydrodynamics code has been used to follow the collapse of isothermal rotating clouds subjected to various nonaxialy symmetric perturbations (NAP). An initially axially symmetric cloud collapsed to form a ring which then fragmented into a binary protostellar system. A low thermal energy cloud with a large bar-shaped NAP collapsed and fragmented directly into a binary; higher thermal energy clouds damp out such NAPs while higher rotational rotational energy clouds produce binaries with wider separations. Fragmentation into single and binary systems has been seen. The tidal effects of other nearby protostellar clouds are shown to have an important effect upon the collapse and should not be neglected. The three-dimensional calculations indicate that isothermal interstellar clouds may fragment (with or without passing through a transitory ring phase) into protostellar objects while still in the isothermal regime. The fragments obtained have masses and specific spin angular momenta roughly a 10th that of the original cloud. Interstellar clouds and their fragments may pass through successive collapse phases with fragmentation and reduction of spin angular momentum (by conversion to orbital angular momentum and preferential accretion of low angular momentum matter) terminating in the formation of pre--main-sequence stars with the observed pre--main-sequence rotation rates
9. Dark matter properties implied by gamma ray interstellar emission models
Energy Technology Data Exchange (ETDEWEB)
Balázs, Csaba; Li, Tong, E-mail: [email protected], E-mail: [email protected] [ARC Centre of Excellence for Particle Physics at the Tera-scale, School of Physics and Astronomy, Monash University, Melbourne, Victoria 3800 (Australia)
2017-02-01
We infer dark matter properties from gamma ray residuals extracted using eight different interstellar emission scenarios proposed by the Fermi-LAT Collaboration to explain the Galactic Center gamma ray excess. Adopting the most plausible simplified ansatz, we assume that the dark matter particle is a Majorana fermion interacting with standard fermions via a scalar mediator. To trivially respect flavor constraints, we only couple the mediator to third generation fermions. Using this theoretical hypothesis, and the Fermi residuals, we calculate Bayesian evidences, including Fermi-LAT exclusion limits from 15 dwarf spheroidal galaxies as well. Our evidence ratios single out one of the Fermi scenarios as most compatible with the simplified dark matter model. In this scenario the dark matter (mediator) mass is in the 25-200 (1-1000) GeV range and its annihilation is dominated by bottom quark final state. Our conclusion is that the properties of dark matter extracted from gamma ray data are highly sensitive to the modeling of the interstellar emission.
10. MAD with aliens? Interstellar deterrence and its implications
Science.gov (United States)
Korhonen, Janne M.
2013-05-01
The possibility that extraterrestrial intelligences (ETIs) could be hostile to humanity has been raised as a reason to avoid even trying to contact ETIs. However, there is a distinct shortage of analytical discussion about the risks of an attack, perhaps because of an implicit premise that we cannot analyze the decision making of an alien civilization. This paper argues that we can draw some inferences from the history of the Cold War and nuclear deterrence in order to show that at least some attack scenarios are likely to be exaggerated. In particular, it would seem to be unlikely that the humanity would be attacked simply because it might, sometime in the future, present a threat to the ETI. Even if communication proves to be difficult, rational decision-makers should avoid unprovoked attacks, because their success would be very difficult to assure. In general, it seems believable that interstellar conflicts between civilizations would remain rare. The findings advise caution for proposed interstellar missions, however, as starfaring capability itself might be seen as a threat. On the other hand, attempting to contact ETIs seems to be a relatively low-risk strategy: paranoid ETIs must also consider the possibility that the messages are a deception designed to lure out hostile civilizations and preemptively destroy them.
11. Scientists Toast the Discovery of Vinyl Alcohol in Interstellar Space
Science.gov (United States)
2001-10-01
Astronomers using the National Science Foundation's 12 Meter Telescope at Kitt Peak, AZ, have discovered the complex organic molecule vinyl alcohol in an interstellar cloud of dust and gas near the center of the Milky Way Galaxy. The discovery of this long-sought compound could reveal tantalizing clues to the mysterious origin of complex organic molecules in space. Vinyl Alcohol and its fellow isomers "The discovery of vinyl alcohol is significant," said Barry Turner, a scientist at the National Radio Astronomy Observatory (NRAO) in Charlottesville, Va., "because it gives us an important tool for understanding the formation of complex organic compounds in interstellar space. It may also help us better understand how life might arise elsewhere in the Cosmos." Vinyl alcohol is an important intermediary in many organic chemistry reactions on Earth, and the last of the three stable members of the C2H4O group of isomers (molecules with the same atoms, but in different arrangements) to be discovered in interstellar space. Turner and his colleague A. J. Apponi of the University of Arizona's Steward Observatory in Tucson detected the vinyl alcohol in Sagittarius B -- a massive molecular cloud located some 26,000 light-years from Earth near the center of our Galaxy. The astronomers were able to detect the specific radio signature of vinyl alcohol during the observational period of May and June of 2001. Their results have been accepted for publication in the Astrophysical Journal Letters. Of the approximately 125 molecules detected in interstellar space, scientists believe that most are formed by gas-phase chemistry, in which smaller molecules (and occasionally atoms) manage to "lock horns" when they collide in space. This process, though efficient at creating simple molecules, cannot explain how vinyl alcohol and other complex chemicals are formed in detectable amounts. For many years now, scientists have been searching for the right mechanism to explain how the building
12. Interstellar communication. II. Application to the solar gravitational lens
Science.gov (United States)
Hippke, Michael
2018-01-01
We have shown in paper I of this series [1] that interstellar communication to nearby (pc) stars is possible at data rates of bits per second per Watt between a 1 m sized probe and a large receiving telescope (E-ELT, 39 m), when optimizing all parameters such as frequency at 300-400 nm. We now apply our framework of interstellar extinction and quantum state calculations for photon encoding to the solar gravitational lens (SGL), which enlarges the aperture (and thus the photon flux) of the receiving telescope by a factor of >109 . For the first time, we show that the use of the SGL for communication purposes is possible. This was previously unclear because the Einstein ring is placed inside the solar coronal noise, and contributing factors are difficult to determine. We calculate point-spread functions, aperture sizes, heliocentric distance, and optimum communication frequency. The best wavelength for nearby (meter-sized telescopes, an improvement of 107 compared to using the same receiving telescope without the SGL. A 1 m telescope in the SGL can receive data at rates comparable to a km-class "normal" telescope.
13. Effects of noise on the interstellar polarization law
International Nuclear Information System (INIS)
Clarke, D.; Al-Roubaie, A.
1983-01-01
A re-appraisal has been made of catalogued four- and seven-colour polarimetric data in terms of the Serkowski law P(lambda)/Psub(max)=exp(-K ln 2 (lambdasub(max)/lambda)) for the wavelength dependence of interstellar polarization. It has been found that the parameter (K) controlling the peakiness of the p(lambda) curve is not a constant at 1.15 but that it is correlated with the value of lambdasub(max), the wavelength corresponding to the maximum value of p(lambda). It has also been found the the form of the correlation depends significantly on the choice of the wavelength values used to measure p(lambda). A numerical exercise involving data simulation shows that the correlations found in the real data could be an artifact of the random noise on the p(lambda) measurements. It is also suggested that a recent proposal to refine the interstellar law reflects, at least partly, the effects of random noise associated with polarimetric measurements. (author)
14. Analysis of ensembles of moderately saturated interstellar lines
International Nuclear Information System (INIS)
Jenkins, E.B.
1986-01-01
It is shown that the combined equivalent widths for a large population of Gaussian-like interstellar line components, each with different central optical depths tau(0) and velocity dispersions b, exhibit a curve of growth (COG) which closely mimics that of a single, pure Gaussian distribution in velocity. Two parametric distributions functions for the line populations are considered: a bivariate Gaussian for tau(0) and b and a power law distribution for tau(0) combined with a Gaussian dispersion for b. First, COGs for populations having an extremely large number of nonoverlapping components are derived, and the implications are shown by focusing on the doublet-ratio analysis for a pair of lines whose f-values differ by a factor of two. The consequences of having, instead of an almost infinite number of lines, a relatively small collection of components added together for each member of a doublet are examined. The theory of how the equivalent widths grow for populations of overlapping Gaussian profiles is developed. Examples of the composite COG analysis applied to existing collections of high-resolution interstellar line data are presented. 39 references
15. Interstellar scintillation of the double pulsar J0737–3039
Energy Technology Data Exchange (ETDEWEB)
Rickett, B. J.; Coles, W. A.; Nava, C. F. [ECE Dept., University of California San Diego, La Jolla, CA 92093-0407 (United States); McLaughlin, M. A. [West Virginia University, Morgantown, WV 26505 (United States); Ransom, S. M. [National Radio Astronomy Observatory, Charlottesville, VA 22903 (United States); Camilo, F. [National Astronomy and Ionosphere Center, Arecibo, PR 00612-8346 (United States); Ferdman, R. D.; Kramer, M.; Lyne, A. G. [Jodrell Bank Center for Astrophysics, University of Manchester, M13 9PL (United Kingdom); Freire, P. C. C. [Dept. of Physics, McGill University, Montréal, QC H3A 2T8 (Canada); Stairs, I. H., E-mail: [email protected] [Dept. of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1 (Canada)
2014-06-01
We report a series of observations of the interstellar scintillation (ISS) of the double pulsar J0737–3039 over the course of 18 months. As in earlier work, the basic phenomenon is the variation in the ISS caused by the changing transverse velocities of each pulsar, the ionized interstellar medium (IISM), and the Earth. The transverse velocity of the binary system can be determined both by very long baseline interferometry and timing observations. The orbital velocity and inclination is almost completely determined from timing observations, but the direction of the orbital angular momentum is not known. Since the Earth's velocity is known, and can be compared with the orbital velocity by its effect on the timescale of the ISS, we can determine the orientation Ω of the pulsar orbit with respect to equatorial coordinates (Ω = 65 ± 2°). We also resolve the ambiguity (i = 88.°7 or 91.°3) in the inclination of the orbit deduced from the measured Shapiro delay by our estimate i = 88.°1 ± 0.°5. This relies on the analysis of the ISS over both frequency and time, and provides a model for the location, anisotropy, turbulence level, and transverse phase gradient of the IISM. We find that the IISM can be well-modeled during each observation, typically of a few orbital periods, but its turbulence level and mean velocity vary significantly over the 18 months.
16. Solar neutrinos and solar accretion of interstellar matter
International Nuclear Information System (INIS)
Newman, M.J.; Talbot, R.J. Jr.
1976-01-01
It is argued that if the Hoyle-Lyttleton mass accretion rate applies (Proc. Camb. Phil. Soc., Math. Phys. Sci. 35: 405 (1939)) the accretion of interstellar matter by the Sun is sufficient to enhance the surface heavy element abundances. This will also apply to other solar-type stars. The enhancement may be sufficient to allow the construction of consistent solar models with an interior heavy element abundance significantly lower than the observed surface abundance. This state of affairs lowers the predicted solar neutrino flux. It has been suggested that a similar enhancement of surface abundances might occur due to accretion of 'planetesimals' left over after formation of the solar system, and both processes may occur, thereby increasing the effect. The simple accretion model of Hoyle and Lyttleton is discussed mathematically. A crucial question to be answered by future research, however, is whether or not accretion on to the solar surface actually occurs. One of the most obvious obstacles is the outward flowing solar wind, and this is discussed. It appears that the outward flow can be reversed to an inward flow for certain interstellar cloud densities. (U.K.)
17. Simulating STARDUST: Reproducing Impacts of Interstellar Dust in the Laboratory
Science.gov (United States)
Postberg, F.; Srama, R.; Hillier, J. K.; Sestak, S.; Green, S. F.; Trieloff, M.; Grün, E.
2008-09-01
Our experiments are carried out to support the analysis of interstellar dust grains, ISDGs, brought to earth by the STARDUST mission. Since the very first investigations, it has turned out that the major problem of STARDUST particle analysis is the modification (partly even the destruction) during capture when particles impact the spacecraft collectors with a velocity of up to 20 km/s. While it is possible to identify, extract, and analyse cometary grains larger than a few microns in aerogel and on metal collector plates, the STARDUST team is not yet ready for the identification, extraction, and analysis of sub-micron sized ISDGs with impact speeds of up to 20 km/s. Reconstructing the original particle properties requires a simulation of this impact capture process. Moreover, due to the lack of laboratory studies of high speed impacts of micron scale dust into interstellar STARDUST flight spares, the selection of criteria for the identification of track candidates is entirely subjective. Simulation of such impact processes is attempted with funds of the FRONTIER program within the framework of the Heidelberg University initiative of excellence. The dust accelerator at the MPI Kernphysik is a facility unique in the world to perform such experiments. A critical point is the production of cometary and interstellar dust analogue material and its acceleration to very high speeds of 20 km/s, which has never before been performed in laboratory experiments. Up to now only conductive material was successfully accelerated by the 2 MV Van de Graaf generator of the dust accelerator facility. Typical projectile materials are Iron, Aluminium, Carbon, Copper, Silver, and the conducting hydrocarbon Latex. Ongoing research now enables the acceleration of any kind of rocky planetary and interstellar dust analogues (Hillier et al. 2008, in prep.). The first batch of dust samples produced with the new method consists of micron and submicron SiO2 grains. Those were successfully
18. Grain boundary migration
International Nuclear Information System (INIS)
Dimitrov, O.
1975-01-01
Well-established aspects of grain-boundary migration are first briefly reviewed (influences of driving force, temperature, orientation and foreign atoms). Recent developments of the experimental methods and results are then examined, by considering the various driving of resistive forces acting on grain boundaries. Finally, the evolution in the theoretical models of grain-boundary motion is described, on the one hand for ideally pure metals and, on the other hand, in the presence of solute impurity atoms [fr
19. Stable Boundary Layer Issues
OpenAIRE
Steeneveld, G.J.
2012-01-01
Understanding and prediction of the stable atmospheric boundary layer is a challenging task. Many physical processes are relevant in the stable boundary layer, i.e. turbulence, radiation, land surface coupling, orographic turbulent and gravity wave drag, and land surface heterogeneity. The development of robust stable boundary layer parameterizations for use in NWP and climate models is hampered by the multiplicity of processes and their unknown interactions. As a result, these models suffer ...
20. Ultraviolet interstellar absorption toward stars in the small Magellanic Cloud. III. THe structure and kinematics of Small Magellanic Cloud
International Nuclear Information System (INIS)
Fitzpatrick, A.L.
1985-01-01
The structure and kinematical properties of the Small Magellanic Cloud (SMC) are investigated by combining ultraviolet data obtained from the International Ultraviolet Explorer (IUE) satellite with existing optical and radio data. The SMC structure is complicated, undoubtedly a result of gravitational interaction with the Milky Way and the Large Magellanic Cloud, and is poorly understood. It has been known for some time that most of the H I in the SMC is concentrated in two complexes at velocities of approximately 134 and 167 km s -1 (heliocentric). Recent 21 cm emission surveys have revealed two additional, widespread H I components at approx.100 and approx.200 km s -1 . With the radio data alone, however, the relative line-of-sight locations of those complexes cannot be determined, nor can the associations of stars with the complexes be deduced. By using the ultraviolet interstellar absorption-line data in conjunction with radio and optical data, the stellar-interstellar kinematical and morphological relationships can be established. We find that in the southwest region of the SMC, most of the stars observed by IUE, including a group with only low-dispersion IUE spectra, are associated with the 134 km s -1 H I complex
1. Stable Boundary Layer Issues
NARCIS (Netherlands)
Steeneveld, G.J.
2012-01-01
Understanding and prediction of the stable atmospheric boundary layer is a challenging task. Many physical processes are relevant in the stable boundary layer, i.e. turbulence, radiation, land surface coupling, orographic turbulent and gravity wave drag, and land surface heterogeneity. The
2. Development of boundary layers
International Nuclear Information System (INIS)
Herbst, R.
1980-01-01
Boundary layers develop along the blade surfaces on both the pressure and the suction side in a non-stationary flow field. This is due to the fact that there is a strongly fluctuating flow on the downstream blade row, especially as a result of the wakes of the upstream blade row. The author investigates the formation of boundary layers under non-stationary flow conditions and tries to establish a model describing the non-stationary boundary layer. For this purpose, plate boundary layers are measured, at constant flow rates but different interferent frequency and variable pressure gradients. By introducing the sample technique, measurements of the non-stationary boundary layer become possible, and the flow rate fluctuation can be divided in its components, i.e. stochastic turbulence and periodical fluctuation. (GL) [de
3. KETENE FORMATION IN INTERSTELLAR ICES: A LABORATORY STUDY
Energy Technology Data Exchange (ETDEWEB)
Hudson, Reggie L.; Loeffler, Mark J., E-mail: [email protected] [Astrochemistry Laboratory, NASA Goddard Space Flight Center, Greenbelt, MD 20771 (United States)
2013-08-20
The formation of ketene (H{sub 2}CCO, ethenone) in polar and apolar ices was studied with in situ 0.8 MeV proton irradiation, far-UV photolysis, and infrared spectroscopic analyses at 10-20 K. Using isotopically enriched reagents, unequivocal evidence was obtained for ketene synthesis in H{sub 2}O-rich and CO{sub 2}-rich ices, and several reaction products were identified. Results from scavenging experiments suggested that ketene was formed by free-radical pathways, as opposed to acid-base processes or redox reactions. Finally, we use our results to draw conclusions about the formation and stability of ketene in the interstellar medium.
4. An interstellar origin for Jupiter's retrograde co-orbital asteroid
Science.gov (United States)
Namouni, F.; Morais, M. H. M.
2018-06-01
Asteroid (514107) 2015 BZ509 was discovered recently in Jupiter's co-orbital region with a retrograde motion around the Sun. The known chaotic dynamics of the outer Solar system have so far precluded the identification of its origin. Here, we perform a high-resolution statistical search for stable orbits and show that asteroid (514107) 2015 BZ509 has been in its current orbital state since the formation of the Solar system. This result indicates that (514107) 2015 BZ509 was captured from the interstellar medium 4.5 billion years in the past as planet formation models cannot produce such a primordial large-inclination orbit with the planets on nearly coplanar orbits interacting with a coplanar debris disc that must produce the low-inclination small-body reservoirs of the Solar system such as the asteroid and Kuiper belts. This result also implies that more extrasolar asteroids are currently present in the Solar system on nearly polar orbits.
5. Characteristics of old neutron stars in dense interstellar clouds
International Nuclear Information System (INIS)
Boehringer, H.; Morfill, G.E.; Zimmermann, H.U.
1987-01-01
The forms observable radiation will assume as old neutron stars pass through interstellar clouds and accrete material are examined theoretically. The radiation, mainly X-rays and gamma rays, will be partially absorbed by the surrounding dust and gas, which in turn produces far-IR radiation from warm dust and line radiation from the gas. Adiabatic compression of the accretion flow and the accretion shock are expected to produce cosmic rays, while gamma rays will be emitted by interaction of the energetic particles with the cloud material. The calculations indicate that the stars will then be identified as X-ray sources, some of which may be unidentified sources in the COS-B database. 37 references
6. The formation of molecules in contracting interstellar clouds
International Nuclear Information System (INIS)
Suzuki, Hiroko; Miki, Satoshi; Sato, Katsuhiko; Kiguchi, Masayoshi; Nakagawa, Yoshitsugu
1976-01-01
The abundances of atoms, molecules and ions in contracting interstellar clouds are investigated in the wide ranges of density (from 10 cm -3 to 10 7 cm -3 ) and optical depth. Abundances of molecules are not in a steady state in optically thick stages because their reaction time scales are very long (10sup(12.5)-10sup(13.5) sec) compared with the contraction time scales. At some stage of contraction the abundances of neutral molecules become frozen, and the frozen abundances are considerably different from the steady-state abundances. The frozen abundances are mainly determined by the contraction time scale of the cloud. Especially, molecules containing carbon except for CO are less abundant for the cloud contracting more slowly. (auth.)
7. Investigating the dynamical history of the interstellar object 'Oumuamua
Science.gov (United States)
Dybczyński, Piotr A.; Królikowska, Małgorzata
2018-02-01
Here we try to find the origin of 1I/2017 U1 'Oumuamua, the first interstellar object recorded inside the solar system. To this aim, we searched for close encounters between 'Oumuamua and all nearby stars with known kinematic data during their past motion. We had checked over 200 thousand stars and found just a handful of candidates. If we limit our investigation to within a 60 pc sphere surrounding the Sun, then the most probable candidate for the 'Oumuamua parent stellar habitat is the star UCAC4 535-065571. However GJ 876 is also a favourable candidate. However, the origin of 'Oumuamua from a much more distant source is still an open question. Additionally, we found that the quality of the original orbit of 'Oumuamua is accurate enough for such a study and that none of the checked stars had perturbed its motion significantly. All numerical results of this research are available in the appendix.
8. Interstellar clouds toward 3C 154 and 3C 353
International Nuclear Information System (INIS)
Federman, S.R.; Evans, N.J. II; Willson, R.F.; Falgarone, E.; Combes, F.; Texas Univ., Austin; Tufts Univ., Medford, MA; Meudon, Observatoire, France)
1987-01-01
Molecular observations of the interstellar clouds toward the radio sources 3C 154 and 3C 353 were obtained in order to elucidate the physical conditions within the clouds. Maps of (C-12)O emission in the J = 1-0 and J = 2-1 lines were compared with observations of the (C-13)O, CH, and OH molecules. The peak emission in the (C-12)O transitions does not occur in the direction of the continuum sources, and thus, an incomplete picture arises when only one line of sight in the two clouds is analyzed. The cloud toward 3C 154 appears to have a low extinction, but a relatively high CO abundance, suggesting that it is similar to high-latitude clouds and CO-rich diffuse clouds. The cloud toward 3C 353 is considerably denser than that toward 3C 154 and may be more like a dark cloud. 32 references
9. Formation of buckminsterfullerene (C60) in interstellar space
Science.gov (United States)
Berné, Olivier; Tielens, Alexander G. G. M.
2012-01-01
Buckminsterfullerene (C60) was recently confirmed to be the largest molecule identified in space. However, it remains unclear how, and where this molecule is formed. It is generally believed that C60 is formed from the build up of small carbonaceous compounds, in the hot and dense envelopes of evolved stars. Analyzing infrared observations, obtained by Spitzer and Herschel, we found that C60 is efficiently formed in the tenuous and cold environment of an interstellar cloud illuminated by strong ultraviolet (UV) radiation fields. This implies that another formation pathway, efficient at low densities, must exist. Based on recent laboratory and theoretical studies, we argue that Polycyclic Aromatic Hydrocarbons are converted into graphene, and subsequently C60, under UV irradiation from massive stars. This shows that alternative - top-down - routes are key to understanding the organic inventory in space.
10. Physical model for the 2175 A interstellar extinction feature
International Nuclear Information System (INIS)
Hecht, J.H.
1986-01-01
Recent IUE observations have shown that the 2175 A interstellar extinction feature is constant in wavelength but varies in width. A model has been constructed to explain these results. It is proposed that the 2175 A feature will only be seen when there is extinction due to carbon grains which have lost their hydrogen. In particular, the feature is caused by a separate population of small (less than 50 A radius), hydrogen-free carbon grains. The variations in width would be due to differences in either their temperature, size distribution, or impurity content. All other carbon grains retain hydrogen, which causes the feature to be suppressed. If this model is correct, then it implies that the grains responsible for the unidentified IR emission features would not generally cause the 2175 A feature. 53 references
11. Thermal emission from interstellar dust in and near the Pleiades
International Nuclear Information System (INIS)
White, R.E.
1989-01-01
IRAS survey coadds for a 8.7 deg x 4.3 deg field near the Pleiades provide evidence for dynamical interaction between the cluster and the surrounding interstellar medium. The far-infrared images show large region of faint emission with bright rims east of the cluster, suggestive of a wake. Images of the far-infrared color temperature and 100 micron optical depth reveal temperature maxima and optical depth minima near the bright cluster stars, as well as a strong optical depth peak at the core of the adjacent CO cloud. Models for thermal dust emission near the stars indicate that most of the apparent optical depth minima near stars are illusory, but also provide indirect evidence for small interaction between the stars and the encroaching dust cloud
12. Thermal emission from interstellar dust in and near the Pleiades
Science.gov (United States)
White, Richard E.
1989-01-01
IRAS survey coadds for a 8.7 deg x 4.3 deg field near the Pleiades provide evidence for dynamical interaction between the cluster and the surrounding interstellar medium. The far-infrared images show large region of faint emission with bright rims east of the cluster, suggestive of a wake. Images of the far-infrared color temperature and 100 micron optical depth reveal temperature maxima and optical depth minima near the bright cluster stars, as well as a strong optical depth peak at the core of the adjacent CO cloud. Models for thermal dust emission near the stars indicate that most of the apparent optical depth minima near stars are illusory, but also provide indirect evidence for small interaction between the stars and the encroaching dust cloud.
13. Study of the diffuse interstellar gas near the Pleiades
International Nuclear Information System (INIS)
Federman, S.R.
1982-01-01
The interstellar gas toward the Pleiades was studied by observing lines of CH, CH + , and K I. New detections of CH and K I, and of CH + and K I in the directions of 20 Tau and eta Tau, respectively, are reported. Evidence for a moderately strong shock of velocity 10--15 km -1 was found for the line of sight toward 20 Tau, where the CH line is blueshifted by 3--4 km s -1 was respect to the CH + line. The relative weakness of the K I features, as well as the weakness of the other previously observed atomic species, requires the gas to be approx.0.3 pc from the stars. A reexamination of the observed distribution of H 2 among its rotational levels indicates that collisions occurring in the shock are largely responsible for populating levels with J>2
14. Rocket propulsion by nuclear microexplosions and the interstellar paradox
Energy Technology Data Exchange (ETDEWEB)
Winterberg, F
1979-11-01
Magnetic insulation is discussed with regard to generating ultra-intense ion beams (IIBs) for thermonuclear microexplosion ignition. With energies up to 10 to the 9th Joule reached by IIB pulses or target staging, the ignition of the hydrogen/boron-11 (HB-11) thermonuclear reaction by the addition of DT and fissionable material is considered. In addition, the possibility of HB-11 as a rocket propulsion system utilizing a magnetic mirror whose magnetic field is generated with high field superconductors is discussed in terms of interstellar travel at up to 1/10 the velocity of light. Attention is also given to the possibility of a relatively unique advanced civilization on earth caused by a rare, near-Roche limit capture of the moon and the subsequent tidal effects resulting in a land/water combination favorable for rapid evolution of life forms.
15. Organic Chemistry: From the Interstellar Medium to the Solar System
Science.gov (United States)
Sandford, Scott; Witteborn, Fred C. (Technical Monitor)
1997-01-01
This talk will review the various types of organic materials observed in different environments in the interstellar medium, discuss the processes by which these materials may have formed and been modified, and present the evidence supporting the contention that at least a fraction of this material survived incorporation, substantially unaltered, into our Solar System during its formation. The nature of this organic material is of direct interest to issues associated with the origin of life, both because this material represents a large fraction of the Solar System inventory of the biogenically-important elements, and because many of the compounds in this inventory have biogenic implications. Several specific examples of such molecules will be briefly discussed.
16. On the effects of rotation on interstellar molecular line profiles
International Nuclear Information System (INIS)
1988-01-01
Theoretical models are constructed to study the effects of systematic gas rotation on the emergent profiles of interstellar molecular lines, in particular the effects of optical depth and different velocity laws. Both rotational and radial motions (expansion or contraction) may produce similar asymmetric profiles, but the behaviour of the velocity centroid of the emergent profile over the whole cloud (iso-centroid maps) can be used to distinguish between these motions. Iso-centroid maps can also be used to determine the location and orientation of the rotation axis and of the equatorial axis. For clouds undergoing both radial and rotational motion, the component of the centroid due to the rotational motion can be separated from that due to the radial motion. Information on the form of the rotational velocity law can also be derived. (author)
17. PRESSURE PULSES AT VOYAGER 2 : DRIVERS OF INTERSTELLAR TRANSIENTS?
Energy Technology Data Exchange (ETDEWEB)
Richardson, J. D. [Kavli Center for Astrophysics and Space Science, Massachusetts Institute of Technology, Cambridge, MA 02139 (United States); Wang, C.; Liu, Y. D. [State Key Laboratory for Space Weather, Chinese Academy of Sciences, Beijing (China); Šafránková, J.; Němeček, Z. [Charles University, Faculty of Mathematics and Physics, V Holešovičkách 2, 180 00 Prague 8 (Czech Republic); Kurth, W. S., E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected], E-mail: [email protected] [University of Iowa, Iowa City, IA 52242 (United States)
2017-01-10
Voyager 1 ( V1 ) crossed the heliopause into the local interstellar medium (LISM) in 2012. The LISM is a dynamic region periodically disturbed by solar transients with outward-propagating shocks, cosmic-ray intensity changes and anisotropies, and plasma wave oscillations. Voyager 2 ( V2 ) trails V1 and thus may observe the solar transients that are later observed at V1. V2 crossed the termination shock in 2007 and is now in the heliosheath. Starting in 2012, when solar maximum conditions reached V2 , five possible merged interaction regions (MIRs) have been observed by V2 in the heliosheath. The timing is consistent with these MIRs driving the transients observed by V1 in the LISM. The largest heliosheath MIR was observed by V2 in late 2015 and should reach V1 in 2018.
18. Event boundaries and memory improvement.
Science.gov (United States)
Pettijohn, Kyle A; Thompson, Alexis N; Tamplin, Andrea K; Krawietz, Sabine A; Radvansky, Gabriel A
2016-03-01
The structure of events can influence later memory for information that is embedded in them, with evidence indicating that event boundaries can both impair and enhance memory. The current study explored whether the presence of event boundaries during encoding can structure information to improve memory. In Experiment 1, memory for a list of words was tested in which event structure was manipulated by having participants walk through a doorway, or not, halfway through the word list. In Experiment 2, memory for lists of words was tested in which event structure was manipulated using computer windows. Finally, in Experiments 3 and 4, event structure was manipulated by having event shifts described in narrative texts. The consistent finding across all of these methods and materials was that memory was better when the information was distributed across two events rather than combined into a single event. Moreover, Experiment 4 demonstrated that increasing the number of event boundaries from one to two increased the memory benefit. These results are interpreted in the context of the Event Horizon Model of event cognition. Copyright © 2015 Elsevier B.V. All rights reserved.
19. The interstellar medium and the highly ionized species observed in the spectrum of the nearby white dwarf G191-B2B
Science.gov (United States)
Bruhweiler, F. C.; Kondo, Y.
1981-01-01
High-resolution spectra of the nearby (48 pc) white dwarf G191-B2B, obtained with the International Ultraviolet Explorer, reveal sharp resonance lines of N V, C IV, and Si IV. The origin of these features is most likely linked to the white dwarf, possibly being formed in an expanding halo around the star. Interstellar lines of C II, N I, Mg II, Si II, and Fe II are also seen in the spectrum. Analysis of these features indicates an average neutral hydrogen number density of 0.064 for this line of sight. In combination with the recent EUV and soft X-ray results, this is interpreted to mean that the interstellar medium in the most immediate solar vicinity is of the normal density n approximately equal to 0.1/cu cm of lower ionization, while just beyond it, at least in some directions, is a hot lower density plasma. These results are apparently in conflict with the model of the interstellar medium by McKee and Ostriker (1977) in its present form.
20. Interstellar medium and the highly ionized species observed in the spectrum of the nearby white dwarf G191-B2B
International Nuclear Information System (INIS)
Bruhweiler, F.C.; Kondo, Y.
1981-01-01
High-resolution spectra of the neargy (48 pc) white dwarf G191-B2B obtained with the International Ultraviolet Explorer (IUE) reveal sharp resonance lines of N V, C IV, and Si IV. The origin of these features is most likely linked to the white dwarf, possibly being formed in an expanding halo around the star. Interstellar lines of C II, N I, Mg II, Si II, Fe II are also seen in the spectrum. Analysis of these features indicates an average neutral hydrogen number density, n/sub Htsi/ = 6.4 x 10 -3 , for this line of sight. In combination with the recent EUV and soft X-ray results, we interpret this to mean that the interstellar medium in the most immediate solar vicinity is of the ''normal'' density (nroughly-equal0.1 cm -3 ) of lower ionization, while just beyond it, at least in some directions, is a hot, lower density plasma. These results are apparently in conflict with the model of the interstellar medium by McKee and Ostriker in its present form
1. ON THE FORMATION OF CO2 AND OTHER INTERSTELLAR ICES
International Nuclear Information System (INIS)
Garrod, R. T.; Pauly, T.
2011-01-01
We investigate the formation and evolution of interstellar dust-grain ices under dark-cloud conditions, with a particular emphasis on CO 2 . We use a three-phase model (gas/surface/mantle) to simulate the coupled gas-grain chemistry, allowing the distinction of the chemically active surface from the ice layers preserved in the mantle beneath. The model includes a treatment of the competition between barrier-mediated surface reactions and thermal-hopping processes. The results show excellent agreement with the observed behavior of CO 2 , CO, and water ice in the interstellar medium. The reaction of the OH radical with CO is found to be efficient enough to account for CO 2 ice production in dark clouds. At low visual extinctions, with dust temperatures ∼>12 K, CO 2 is formed by direct diffusion and reaction of CO with OH; we associate the resultant CO 2 -rich ice with the observational polar CO 2 signature. CH 4 ice is well correlated with this component. At higher extinctions, with lower dust temperatures, CO is relatively immobile and thus abundant; however, the reaction of H and O atop a CO molecule allows OH and CO to meet rapidly enough to produce a CO:CO 2 ratio in the range ∼2-4, which we associate with apolar signatures. We suggest that the observational apolar CO 2 /CO ice signatures in dark clouds result from a strongly segregated CO:H 2 O ice, in which CO 2 resides almost exclusively within the CO component. Observed visual-extinction thresholds for CO 2 , CO, and H 2 O are well reproduced by depth-dependent models. Methanol formation is found to be strongly sensitive to dynamical timescales and dust temperatures.
2. TRAJECTORIES AND DISTRIBUTION OF INTERSTELLAR DUST GRAINS IN THE HELIOSPHERE
Energy Technology Data Exchange (ETDEWEB)
Slavin, Jonathan D. [Harvard-Smithsonian Center for Astrophysics, MS 83, 60 Garden Street, Cambridge, MA 02138 (United States); Frisch, Priscilla C. [Department of Astronomy and Astrophysics, University of Chicago, 5460 S. Ellis Avenue, Chicago, IL 60637 (United States); Mueller, Hans-Reinhard [Department of Physics and Astronomy, Dartmouth College, Hanover, NH 03755 (United States); Heerikhuisen, Jacob; Pogorelov, Nikolai V. [Department of Physics and Center for Space Physics and Aeronomic Research, University of Alabama, Huntsville, AL 35899 (United States); Reach, William T. [Universities Space Research Association, MS 211-3, Moffett Field, CA 94035 (United States); Zank, Gary [Department of Physics and Center for Space Plasma and Aeronomic Research, University of Alabama, Huntsville, AL 35805 (United States)
2012-11-20
The solar wind carves a bubble in the surrounding interstellar medium (ISM) known as the heliosphere. Charged interstellar dust grains (ISDG) encountering the heliosphere may be diverted around the heliopause or penetrate it depending on their charge-to-mass ratio. We present new calculations of trajectories of ISDG in the heliosphere, and the dust density distributions that result. We include up-to-date grain charging calculations using a realistic UV radiation field and full three-dimensional magnetohydrodynamic fluid + kinetic models for the heliosphere. Models with two different (constant) polarities for the solar wind magnetic field (SWMF) are used, with the grain trajectory calculations done separately for each polarity. Small grains a {sub gr} {approx}< 0.01 {mu}m are completely excluded from the inner heliosphere. Large grains, a {sub gr} {approx}> 1.0 {mu}m, pass into the inner solar system and are concentrated near the Sun by its gravity. Trajectories of intermediate size grains depend strongly on the SWMF polarity. When the field has magnetic north pointing to ecliptic north, the field de-focuses the grains resulting in low densities in the inner heliosphere, while for the opposite polarity the dust is focused near the Sun. The ISDG density outside the heliosphere inferred from applying the model results to in situ dust measurements is inconsistent with local ISM depletion data for both SWMF polarities but is bracketed by them. This result points to the need to include the time variation in the SWMF polarity during grain propagation. Our results provide valuable insights for interpretation of the in situ dust observations from Ulysses.
3. Are CO Observations of Interstellar Clouds Tracing the H2?
Science.gov (United States)
Federrath, Christoph; Glover, S. C. O.; Klessen, R. S.; Mac Low, M.
2010-01-01
Interstellar clouds are commonly observed through the emission of rotational transitions from carbon monoxide (CO). However, the abundance ratio of CO to molecular hydrogen (H2), which is the most abundant molecule in molecular clouds is only about 10-4. This raises the important question of whether the observed CO emission is actually tracing the bulk of the gas in these clouds, and whether it can be used to derive quantities like the total mass of the cloud, the gas density distribution function, the fractal dimension, and the velocity dispersion--size relation. To evaluate the usability and accuracy of CO as a tracer for H2 gas, we generate synthetic observations of hydrodynamical models that include a detailed chemical network to follow the formation and photo-dissociation of H2 and CO. These three-dimensional models of turbulent interstellar cloud formation self-consistently follow the coupled thermal, dynamical and chemical evolution of 32 species, with a particular focus on H2 and CO (Glover et al. 2009). We find that CO primarily traces the dense gas in the clouds, however, with a significant scatter due to turbulent mixing and self-shielding of H2 and CO. The H2 probability distribution function (PDF) is well-described by a log-normal distribution. In contrast, the CO column density PDF has a strongly non-Gaussian low-density wing, not at all consistent with a log-normal distribution. Centroid velocity statistics show that CO is more intermittent than H2, leading to an overestimate of the velocity scaling exponent in the velocity dispersion--size relation. With our systematic comparison of H2 and CO data from the numerical models, we hope to provide a statistical formula to correct for the bias of CO observations. CF acknowledges financial support from a Kade Fellowship of the American Museum of Natural History.
4. Let the Right One In: Ethnic Boundaries in a Colombian Immigrant Youth Program
Science.gov (United States)
Pineda, Claudia G.
2017-01-01
Although research on minority youth has established the value of coethnic spaces for safe ethnic identity exploration, research has seldom examined how youth in these spaces draw ethnic boundaries or offered appropriate frameworks addressing boundary-setting. This study uses Berry's acculturation framework to explore ethnic boundary-setting within…
5. Relative amounts of stars and interstellar matter in the local Milky Way
International Nuclear Information System (INIS)
Jura, M.
1987-01-01
This paper considers the balance between star formation and mass loss from evolved stars in the region within 1 kpc of the sun. There is considerably more mass in stars than in the interstellar medium, and more material is being incorporated into new stars than is being returned by evolved stars. In the simplest interpretation of the data, it appears that unless there is some infall of new interstellar gas, the era of substantial star formation out of interstellar gas will be over in a few (perhaps 3) billion years. 34 references
6. Cometary and interstellar dust grains - Analysis by ion microprobe mass spectrometry and other techniques
Science.gov (United States)
Zinner, Ernst
1991-01-01
A survey of microanalytical measurements on interplanetary dust particles (IDPs) and interstellar dust grains from primitive meteorites is presented. Ion-microprobe mass spectrometry with its capability to determine isotopic compositions of many elements on a micron spatial scale has played a special role. Examples are measurements of H, N, and O isotopes and refractory trace elements in IDPs; C, N, Mg, and Si isotopes in interstellar SiC grains; and C and N isotopes and H, N, Al, and Si concentrations in interstellar graphite grains.
7. A Polarimetric Investigation on Interstellar Dust Within 50-PARSECS from the Sun
Science.gov (United States)
Leroy, J. L.
1993-07-01
We have analyzed the polarization catalogue, for 1000 stars closer than 50 Pc, which has been presented in a companion paper. Although the accuracy of the measurements is generally very good (better than 0.02% for more than 700 stars), the catalogue contains essentially unpolarized stars, which is to be related to the well known depletion of dust in the Sun's vicinity. However, setting apart some stars which are known to display intrinsic polarization (e.g. several Ap magnetic variables), we have been able to select 25 stars whose polarization is indisputable. But, looking in more detail at the (presently available) data on the distance of this sample reveals that, in almost all the cases, the distances have been under-estimated. While this result will require a confirmation with the help of the Hipparcos parallaxes, we can already say that we fully confirm the main result of the preceding investigation on this topic by Tinbergen (1982), namely the complete depletion of dust within 35 pc from the Sun. Actually, we find that the dust signature begins at 40-50 pc in a few cases, but more much frequently at 70 to 100 pc, which seems to be the boundary of the local Bubble, as far as dust is concerned. We cannot confirm the previous detection by Tinbergen of a faint, near-by dusty region around l = 0°, b = -20°. Altogether, the picture given by the polarization analysis is consistent (although more sensitive) with the data derived from reddening measurements, and, to a lesser extent, with the investigations based on the measurements of interstellar absorption lines. Extending the polarization measurements to the 50-100 pc zone would provide a very precise picture of the location of those dust clouds which appear, here and there, as one gets out from the local Bubble.
8. Measuring the Alfvénic nature of the interstellar medium: Velocity anisotropy revisited
International Nuclear Information System (INIS)
Burkhart, Blakesley; Lazarian, A.; Leão, I. C.; De Medeiros, J. R.; Esquivel, A.
2014-01-01
The dynamics of the interstellar medium (ISM) are strongly affected by turbulence, which shows increased anisotropy in the presence of a magnetic field. We expand upon the Esquivel and Lazarian method to estimate the Alfvén Mach number using the structure function anisotropy in velocity centroid data from Position-Position-Velocity maps. We utilize three-dimensional magnetohydrodynamic simulations of fully developed turbulence, with a large range of sonic and Alfvénic Mach numbers, to produce synthetic observations of velocity centroids with observational characteristics such as thermal broadening, cloud boundaries, noise, and radiative transfer effects of carbon monoxide. In addition, we investigate how the resulting anisotropy-Alfvén Mach number dependency found in Esquivel and Lazarian might change when taking the second moment of the Position-Position-Velocity cube or when using different expressions to calculate the velocity centroids. We find that the degree of anisotropy is related primarily to the magnetic field strength (i.e., Alfvén Mach number) and the line-of-sight orientation, with a secondary effect on sonic Mach number. If the line of sight is parallel to up to ≈45 deg off of the mean field direction, the velocity centroid anisotropy is not prominent enough to distinguish different Alfvénic regimes. The observed anisotropy is not strongly affected by including radiative transfer, although future studies should include additional tests for opacity effects. These results open up the possibility of studying the magnetic nature of the ISM using statistical methods in addition to existing observational techniques.
9. Administrative Area Boundaries 2 (State Boundaries), Region 9, 2010, NAVTEQ
Data.gov (United States)
U.S. Environmental Protection Agency — NAVTEQ Administrative Area Boundaries 2 (State Boundaries) for Region 9. There are five Administrative Area Boundaries layers (1, 2, 3, 4, 5). These layers contain...
10. Administrative Area Boundaries 4 (City Boundaries), Region 9, 2010, NAVTEQ
Data.gov (United States)
U.S. Environmental Protection Agency — NAVTEQ Administrative Area Boundaries 4 (City Boundaries) for Region 9. There are five Administrative Area Boundaries layers (1, 2, 3, 4, 5). These layers contain...
11. Probing the Origin and Evolution of Interstellar and Protoplanetary Biogenic Ices with SPHEREx
Science.gov (United States)
Melnick, Gary; SPHEREx Science Team
2018-01-01
Many of the most important building blocks of life are locked in interstellar and protoplanetary ices. Examples include H2O, CO, CO2, and CH3OH, among others. There is growing evidence that within the cores of dense molecular clouds and the mid-plane of protoplanetary disks the abundance of these species in ices far exceeds that in the gas phase. As a result, collisions between ice-bearing bodies and newly forming planets are thought to be a major means of delivering these key species to young planets. There currently exist fewer than 250 ice absorption spectra toward Galactic molecular clouds, which is insufficient to reliably trace the ice content of clouds through the various evolutionary stages of collapse to form stars and planets. Likewise, the current number of spectra is inadequate to assess the effects of environment, such as cloud density and temperature, presence or absence of embedded sources, external FUV and X-ray radiation, gas-phase composition, or cosmic-ray ionization rate, on the ice composition of clouds at similar stages of evolution. Ultimately, our goal is to understand how these findings connect to our own Solar System.SPHEREx will be a game changer for the study of interstellar, circumstellar, and protoplanetary disk ices. SPHEREx will obtain spectra over the entire sky in the optical and near-IR, including the 2.5 to 5.0 micron region, which contains the above biogenic ice features. SPHEREx will detect millions of potential background continuum point sources already catalogued by NASA’s Wide-field Infrared Survey Explorer (WISE) at 3.4 and 4.6 microns for which there is evidence for intervening gas and dust based on the 2MASS+WISE colors with sufficient sensitivity to yield ice absorption spectra with SNR ≥ 100 per spectral resolution element. The resulting > 100-fold increase in the number of high-quality ice absorption spectra toward a wide variety of regions distributed throughout the Galaxy will reveal correlations between ice
12. Tax Unit Boundaries
Data.gov (United States)
Kansas Data Access and Support Center — The Statewide GIS Tax Unit boundary file was created through a collaborative partnership between the State of Kansas Department of Revenue Property Valuation...
13. 500 Cities: City Boundaries
Data.gov (United States)
U.S. Department of Health & Human Services — This city boundary shapefile was extracted from Esri Data and Maps for ArcGIS 2014 - U.S. Populated Place Areas. This shapefile can be joined to 500 Cities...
14. National Forest Boundaries
Data.gov (United States)
Minnesota Department of Natural Resources — This theme shows the USFS national forest boundaries in the state. This data was acquired from the GIS coordinators at both the Chippewa National Forest and the...
15. Allegheny County Parcel Boundaries
Data.gov (United States)
Allegheny County / City of Pittsburgh / Western PA Regional Data Center — This dataset contains parcel boundaries attributed with county block and lot number. Use the Property Information Extractor for more control downloading a filtered...
16. Boundary representation modelling techniques
CERN Document Server
2006-01-01
Provides the most complete presentation of boundary representation solid modelling yet publishedOffers basic reference information for software developers, application developers and users Includes a historical perspective as well as giving a background for modern research.
17. NM School District Boundaries
Data.gov (United States)
Earth Data Analysis Center, University of New Mexico — The dataset represents the boundaries of all public school districts in the state of New Mexico. The source for the data layer is the New Mexico Public Education...
18. Site Area Boundaries
Data.gov (United States)
U.S. Environmental Protection Agency — This dataset consists of site boundaries from multiple Superfund sites in U.S. EPA Region 8. These data were acquired from multiple sources at different times and...
19. HUC 8 Boundaries
Data.gov (United States)
Kansas Data Access and Support Center — This data set is a digital hydrologic unit boundary that is at the 4-digit, 6-digit, 8-digit, and 11-digit level. The data set was developed by delineating the...
20. State Park Statutory Boundaries
Data.gov (United States)
Minnesota Department of Natural Resources — Legislative statutory boundaries for sixty six state parks, six state recreation areas, and eight state waysides. These data are derived principally from DNR's... | {"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": 0.8426264524459839, "perplexity": 3215.1189381300187}, "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/1537267158450.41/warc/CC-MAIN-20180922142831-20180922163231-00325.warc.gz"} |
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## Identify quaternion group as a subgroup of the general linear group of dimension 2 over complex numbers
Let $i$ and $j$ be the generators of $Q_8 = \langle i, j \ |\ i^4 = j^4 = 1, i^2 = j^2, ji = i^3 j \rangle$. Prove that the map $\varphi : Q_8 \rightarrow GL_2(\mathbb{C})$ defined on generators by $\varphi(i) = A = \left[ {\sqrt{-1} \atop 0} {0 \atop -\sqrt{-1}} \right]$ and $\varphi(j) = B = \left[ {0 \atop 1} {-1 \atop 0} \right]$ extends to a homomorphism. Prove that $\varphi$ is injective.
Solution: First we prove a lemma.
Lemma. Let $S = \{ i,j \}$ and $G$ be a group. If $\overline{\varphi} : S \rightarrow G$ such that $\overline{\varphi}(i)^4 = \overline{\varphi}(j)^4 = 1$, $\overline{\varphi}(i)^2 = \overline{\varphi}(j)^2$, and $\overline{\varphi}(j) \overline{\varphi}(i) = \overline{\varphi}(i)^3 \overline{\varphi}(j)$, then $\overline{\varphi}$ extends to a homomorphism $\varphi : Q_8 \rightarrow G$. Moreover, if $\overline{\varphi}(j)$ is not a power of $\overline{\varphi}(i)$ and the powers $\overline{\varphi}(i)^k$ are distinct for $0 \leq k < 4$, then $\varphi$ is injective.
Proof: We have that $\overline{\varphi}(i)$ and $\overline{\varphi}(j)$ satisfy the relations of $Q_8$. Now every element of $Q_8$ can be written uniquely in the form $i^aj^b$ for some $0 \leq a < 4$ and $0 \leq b < 2$; define $\varphi(i^aj^b) = \overline{\varphi}(i)^a \overline{\varphi}(j)^b$. It is straightforward to see that $\varphi$ is in fact a homomorphism. To see injectivity, let $0 \leq a,c < 4$ and $0 \leq b,d < 2$ and suppose $\varphi(i^aj^b) = \varphi(i^cj^d)$ but that $i^aj^b \neq i^cj^d$. Then either $a \neq c$ or $b \neq d$. If $b \neq d$, then without loss of generality we have $b = 1$ and $d = 0$. Thus $\overline{\varphi}(j) = \overline{\varphi}(i)^{c-a}$, a contradiction. If $b = d$ and $a \neq c$, then we have $\overline{\varphi}(i)^a = \overline{\varphi}(i)^c$, a contradiction. Thus $\varphi$ is injective. $\blacksquare$
By the lemma, it suffices to show that (1) $A^4 = B^4 = I$, (2) $A^2 = B^2$, (3) $BA = A^3B$, (4) $B \neq A^k$ for all integers $k$, and (5) $A^k$ are distinct for $k \in \{ 0,1,2,3 \}$. All but (4) are established by a simple calculation. To see (4), note that A is diagonal; so all powers of $A$ are diagonal. But $B$ is not diagonal. Thus the mapping $\varphi : Q_8 \rightarrow GL_2(\mathbb{C})$ is an injective homomorphism. | {"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": 0.9971245527267456, "perplexity": 29.05119437054245}, "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/1642320304954.18/warc/CC-MAIN-20220126131707-20220126161707-00693.warc.gz"} |
https://www.physicsforums.com/threads/linear-algebra-solve-linear-system-with-complex-constants.187993/ | # Linear algebra - solve linear system with complex constants
1. Sep 30, 2007
Solve the following linear system:
ix + (1+i)y = i
(1-i)x + y - iz = 1
iy + z = 1
I am getting nowhere with this.
is there a trick to do these? I keep getting more and more variations of i. like i^2-1, and (1-i^2)-1
ix + (1+i)y=i
(1-i)x + y-iz=1
y + z = 1
ix + (1+i)y = i
i(1-i)x - (i^2-1)z = i-1 [iR2-r3]
[(1-i)(1+i)-i]x - (1+i)iz = (1+i)-i [(1+i)R2-R1]
okay.. and i simplified this, and got stuck.
ix + (1+i) = i
i(1-i)x + (1-i^2)z = i-1
[(1-i^2)-i]x - (1+i)iz = 1
any help would be great, thanks.
Last edited: Sep 30, 2007
2. Sep 30, 2007
### Staff: Mentor
Since i is the sqrt(-1) then i^2 = -1. That will simplify all of your powers of i down to simple complex numbers. Other than that there is nothing different or unusual in solving this system.
3. Sep 30, 2007
### HallsofIvy
Staff Emeritus
Looks to me like it's straight forward. If y+ z= 1 then clearly z= 1- y. Put that into the second equation and you have (1-i)x+ y- i- iy= (1-i)x+ (1-i)y= 1- i so x+ y= 1. From that, y= 1-x so the first equation becomes ix+ (1+i)(1-x)= ix+ 1-x+i- ix= x+ 1-i= i. That should be easy to solve.
Similar Discussions: Linear algebra - solve linear system with complex constants | {"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": 0.8477340340614319, "perplexity": 4873.567445739814}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463607120.76/warc/CC-MAIN-20170522211031-20170522231031-00115.warc.gz"} |
https://tex.stackexchange.com/questions/241118/autoref-to-dmath-from-breqn-package-generates-wrong-link-text | # autoref to dmath (from breqn package) generates wrong link text
In the following minimal example, what is wrong with my dmath equation? Why does hyperref call it a Figure, while autoref gets the link text correct for an equation defined with
\documentclass{amsart}
\usepackage{breqn}
\begin{document}
$$\label{multiplication} 2 * 1 = 2$$
\begin{figure}
\caption{Something strange}
\label{fig}
\end{figure}
1 + 1 = 2
\end{dmath}
$$\label{subtraction} 2 - 1 = 1$$
And \autoref{multiplication} is not very interesting. Let me speak of \autoref{addition}.
\end{document}
The document has three hyperlinks to equations and one to the figure. But the output says "Figure" three times and "Equation" only once. All the links to breqn environments are wrong.
It is using the correct number, but the wrong text, combined to form "Figure 2" that doesn't actually exist in the document.
Swapping the order of \usepackage lines makes no difference.
What's the right way to use \label on \begin{dmath} and \begin{dgroup} environments?
Environment dmath of package breqn does not use \refstepcounter for the incrementation of the counter equation. Package hyperref hooks into \refstepcounter to add a hyper anchor as link target. Therefore your example uses the anchor before (figure.1) and you get the previous figure as link target.
Workaround: The following example patches \eq@setnumber to use \refstepcounter instead of \stepcounter:
\documentclass{amsart}
\usepackage{breqn}
\usepackage{etoolbox}
\makeatletter
\patchcmd\eq@setnumber{\stepcounter}{\refstepcounter}{}{%
\errmessage{Patching \noexpand\eq@setnumber failed}%
}
\makeatother
\begin{document}
\begin{figure}
\caption{Something strange}
\label{fig}
\end{figure}
\begin{figure}
\caption{Something strange}
\label{fig2}
\end{figure}
1 + 1 = 2
\end{dmath}
$$\label{subtraction} 2 - 1 = 1$$
• dmaht sets \@currrentlabel manually instead of using \refstepcounter. Thus it ignores hyperref completely. \autoref uses the anchor name to get the counter, the last anchor name was figure.1, thus \autoref sets the prefix for figures. – Heiko Oberdiek Apr 29 '15 at 9:54
• On further investigation, it looks like you're right that the hyperlink actually was going to the figure prior to the equation (usually they were on the same page making it hard to detect). Your patchcmd fixes both the text and the behavior. Thanks. – Ben Voigt Apr 29 '15 at 15:53 | {"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": 3, "x-ck12": 0, "texerror": 0, "math_score": 0.9249559044837952, "perplexity": 4384.994600322379}, "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/1614178369553.75/warc/CC-MAIN-20210304235759-20210305025759-00442.warc.gz"} |
http://var.scholarpedia.org/article/Spontaneous_symmetry_breaking_in_classical_systems | # Spontaneous symmetry breaking in classical systems
Post-publication activity
Curator: Franco Strocchi
The mechanism of spontaneous symmetry breaking in classical systems amounts to the fact that a symmetric dynamics, typically described by a symmetric Hamiltonian or Lagrangian, may allow a non-symmetric behavior of the system.
## Contents
### Introduction
The mechanism of spontaneous symmetry breaking has become one of the corner stones of modern theoretical physics, with crucial applications to many body theory and to elementary particle physics. It amounts to the fact that a symmetric dynamics, typically described by a symmetric Hamiltonian or Lagrangian, may allow a non-symmetric behavior of the system. Since the mechanism is very subtle, its precise definition requires some care. The literature on the meaning of spontaneous symmetry breaking, especially from the point of view of general philosophy, is rather rich also because it is somewhat intriguing to understand how a symmetry of the dynamics may get lost.
In the standard popular accounts, spontaneous symmetry breaking is identified with the existence of a non-symmetric lowest energy configuration or state, but this explanation without further qualification may be misleading, since it does not in general account for the dramatic loss of symmetry which accompanies spontaneous symmetry breaking. E.g., for a particle on a horizontal plane subject to a constant gravitational field perpendicular to the plane, each point of the plane defines a lowest energy configuration, which is not invariant under translations on the plane, but it would be improper to speak of symmetry breaking, since the behavior of the system does not display a loss of symmetry.
Even less satisfactory is the identification of spontaneous symmetry breaking with the existence of non-symmetric solutions in a theory described by symmetric equations; e.g. in the two-body problem with a central potential the existence of orbits which are not circles have not been (and should not be) regarded as the breaking of rotational symmetry.
As a matter of fact, a non-symmetric behavior of a system described by a symmetric Hamiltonian seems to run into conflict with the general wisdom of classical mechanics.
### The "Noether theorem"
The classical "Noether theorem" for Hamiltonian systems with a finite number $$2 s$$ of degrees of freedom, described by the canonical variables $$q_k, p_k\ ,$$ $$k = 1, \cdots s\ ,$$ states that the symmetry of the Hamiltonian under an $$N$$-dimensional Lie group $$G$$ implies the existence of $$N$$ functions of the canonical variables, $$Q^i\ ,$$ $$i = 1, \cdots N\ ,$$ which are constants under time evolution (briefly constants of motion) and which are the generators of the symmetry transformations of the canonical variables $\delta^i (q_k) = \{q_k, \,Q^i\}, \,\,\,\,\delta^i (p_k) = \{p_k, \,Q^i\},$
where $$\delta^i$$ denotes the infinitesimal variation under the $$i$$-th subgroup of $$G\ ,$$ and $$\{\,, \,\}$$ the Poisson bracket $\{ f, g \} = \frac{\partial f}{\partial q_k} \, \frac{ \partial g }{ \partial p_k } - \frac { \partial f }{ \partial p_k } \, \frac{ \partial g }{ \partial q_k }$ (a sum over repeated indices being always understood). The time independence of $$Q^i$$ implies that $\delta^i ( q_k(t) ) = \{ q_k(t), \, Q^i \} = \{ q_k(t), \, Q^i(t) \} = ( \delta^i q_k )(t)$ $\delta^i ( p_k(t) ) = ( \delta^i p_k )(t),$
i.e. the symmetry transformation commutes with the time evolution.
For continuous classical systems described by a set of fields $$\phi_a(\mathbf x,t) \ ,$$ $$a = 1, \cdots n\ ,$$ $$\mathbf x \in \mathbf R^s \ ,$$ $$s$$ being the space dimensions, Noether theorem (Noether 1918) states that the invariance of the Lagrangian under an $$N$$-dimensional Lie group $$G$$ implies the existence of $$N$$ conserved currents $$J_\mu^i(\mathbf x, t) \ ,$$ $$\mu = 0, 1, 2, \cdots s,$$ $$i = 1, \cdots N,$$ $\partial^\mu J^i_\mu(x) = \partial_t J_0^i(x, t) + \mbox{div} \mathbf J^i(x, t)= 0 ,$
and of $$N$$ time independent "charges" $$Q^i\ ,$$ $Q^i(t) = \int d^s x \, J_0^i(\mathbf x, t) = Q^i(0) = Q^i.$
Furthermore, as functionals of the canonical variables $$\phi_a, \pi_a\ ,$$ ($$\pi_a$$ denotes the field canonically conjugated to $$\phi_a$$), the charges $$Q^i$$ are the generators of the symmetry transformations of the canonical variables, with the Poisson brackets defined by functional derivatives in place of partial derivatives, see e.g. (Schmid 1987). Hence, the time evolution of the system does not display a loss of symmetry.
### Symmetry breaking
To understand how a possible loss of symmetry may arise, we recall that a symmetry $$g$$ of a classical physical system is an invertible transformation of the set coordinates $$\gamma$$ which identify its configurations $$S_\gamma\ ,$$ $$g: \gamma \rightarrow g \, \gamma\ ,$$ such that it commutes with the time evolution $$a_t\ ,$$ $$t \in \mathbf R$$ $a_t g S_\gamma = a_t S_{g \gamma} = S_{(g \gamma)(t)} = g a_t S_\gamma ,$ (for simplicity, we do not consider the more general case in which the dynamics transforms covariantly under the symmetry). For classical canonical systems, such a commutation is equivalent to the invariance of the Hamiltonian under the action of $$g\ .$$
As we learned from the foundations of special relativity and from the principles of quantum mechanics, a good theoretical concept must always be confronted with its operational counterpart. Therefore, the symmetric behavior of the system has an operational meaning, i.e. it can be detected, if both the configuration $$S_\gamma$$ and its transformed one $$S_{g \gamma}$$ are accessible by the same physical observer.
For this purpose, it is convenient to introduce the following structure, namely the partition of the set of configurations into classes or phases $$\Gamma\ ,$$ each being characterized by the property that any two configurations $$S_{\gamma_1}, \,S_{\gamma_2} \in \Gamma$$ may be obtained one from the other by physically realizable operations, so that they are both accessible by the same physical observer. For example, this is not the case if the two configurations are separated by an infinite potential barrier, since infinite energy would be required for going from one to the other.
Thus, different phases describe physically disjoint realizations of the system.
A symmetry $$g$$ is physically realized or unbroken in a phase $$\Gamma$$ if $$g$$ leaves $$\Gamma$$ stable (i.e. it maps configurations of $$\Gamma$$ into configurations of $$\Gamma)\ ,$$ and it is broken in $$\Gamma$$ otherwise.
This definition of unbroken symmetry accounts for the fact that not only the dynamics is symmetric, but, more crucially, the symmetric behavior of the system can be checked by confronting the time evolution of any given configuration $$S_{\gamma} \in \Gamma$$ and of its transformed one $$S_{g \gamma}\ ,$$ since both belong to $$\Gamma$$ and are therefore accessible by the same observer.
It is worthwhile to apply the above definition of phases to the popular mechanical examples of the particle in a (finite) one-dimensional double well, corresponding to the potential energy $$V(q) = \lambda (q^2 - a^2)^2\ ,$$ as well as that of the particle in a Mexican hat potential (in three space dimensions), $$V(q) = \lambda (\mathbf q^2 - a^2)^2$$ ($$a>0$$ is assumed in both cases).
In the first case, unless one puts an artificial bound on the kinetic energy, all the configurations fall in only one phase, since there is no physical obstruction for going from one configuration to another. In the second case, the degenerate ground states (defined by the points at the bottom of the Mexican hat) have all the same energy and one cannot envisage any physical obstruction which prevents from going from one ground state to the other; hence, there is only one phase. Then, according to the above definition, no symmetry can be broken and such mechanical examples are not good examples of spontaneous symmetry breaking. On the other hand, if one lets the local maximum of the above one-dimensional double well potential to go to infinity, e.g. by adding to the potential the function $$v(q) = q^{-2} - (a/2)^{-2}\ ,$$ for $$|q| \leq a/2\ ,$$ $$v(q) = 0\ ,$$ for $$|q| > a/2 \ ,$$ $$a > 0\ ,$$ then two phases arise, $$\Gamma_1 = \{q > 0, \, p \in \mathbf R \}\ ,$$ $$\Gamma_2 = \{ q < 0, \,p \in \mathbf R \}\ ,$$ and the discrete symmetry $$g: q \rightarrow - q$$ is broken in each phase.
It is clear from these examples that in classical Hamiltonian systems with a finite number of degrees of freedom it is difficult to have spontaneous symmetry breaking: discrete symmetries can be broken in very artificial cases while continuous symmetries can never be broken.
The situation changes substantially if one considers infinitely extended systems. The new situation is well illustrated by the familiar prototypic system described by the (hyperbolic) field equation $\tag{1} \square \phi(x,t) = U'(\phi(x, t))$
with $$\phi$$ an $$n$$-component field ($$\phi: \mathbf R^s \rightarrow \mathbf R^n$$), and $$U(\phi)$$ a potential satisfying suitable regularity conditions (Parenti, Strocchi and Velo 1975) in $$s = 1$$ space dimensions $$U$$ is required to be an entire function, in $$s = 2$$ space dimensions if $$U(\phi) = \sum_{a \in \mathbf N^n}^{\infty} C_a \phi^a\ ,$$ $$a$$ being a multi index $$\phi^a = \phi_1^{a_1}\cdots \phi_n^{a_n}\ ,$$ $$a_i \in \mathbf N\ ,$$ then the following condition is required$\sum_{a \in \mathbf N^n}^{\infty} |C_a| |a|^{|a|/2} |\phi|^{|a|} < \infty$ and finally in $$s = 3$$ space dimensions $$U$$ is required to be a twice differentiable real function satisfying$\mbox{sup}_{\phi}\,(1 + |\phi|^2)^{-1}\,| U''(\phi)| < \infty.$ As we shall see, the relevant point is the emergence of a natural non-trivial structure of phases, i.e. of physically disjoint realizations of the system described by the above field equation.
The crucial physical property is the inevitable localization in space of any physical operation. Thus, for infinitely extended systems a condition for the physical realizability of two initial conditions (by the same observer) is that they have the same behavior at space infinity, since no physically realizable operation can change the boundary condition of the phase or universe in which the observer is living. Moreover, since the time evolution is one of the realizable operations, if an initial data is realizable by one observer, so is also its time translated one.
The above considerations are technically formalized by the following notion: a sector $$S$$ is a set of initial data such that i) any pair of them $$u =( \phi, \,\psi)\ ,$$ $$\psi = \partial_t \phi \ ,$$ $$u' = ( \phi', \,\psi')$$ differ by elements of $$(H^1(\mathbf R^s), L^2(\mathbf R^s))\ ,$$ i.e. $$\phi' - \phi$$ and $$\partial_j (\phi' - \phi)\ ,$$ $$j = 1, \cdots s$$ are square integrable and $$\psi' - \psi \in L^2(\mathbf R^s)\ ,$$ (a motivation for the choice of such a functional space is that it leads to a Hilbert space structure stable under time evolution, see below), ii) if $$u$$ belongs to $$S\ ,$$ also its time translated one $$u_\tau = u(\tau)$$ belongs to $$S\ .$$ Clearly, the sector $$S_u\ ,$$ to which $$u$$ belongs, is a subset of the Hilbert space $$H_u = u + (H^1(\mathbf R^s), L^2(\mathbf R^s))\ .$$ A distinguished class of sectors are those of the form $$S_u = H_u\ ;$$ they are isomorphic to Hilbert spaces and are called Hilbert sectors.
A necessary and sufficient condition for the stability of a sector $$H_u\ ,$$ $$u = (\phi, \psi)\ ,$$ under space translation is that $$\partial_j \phi \in L^2(\mathbf R^s)\ ,$$ $$j = 1, \cdots s$$ (Strocchi 2008).
The realization and relevance of the above definitions is provided by the following theorem (Parenti, Strocchi and Velo 1977, Strocchi 2008)
A constant initial data $$u_0 = (\phi_0, \,0)\ ,$$ corresponding to an absolute minimum of the potential $$U\ ,$$ defines an Hilbert sector, i.e. a set of configurations $$H_{u_0} = u_0 + (H^1(\mathbf R^s), L^2(\mathbf R^s))\ ,$$ with the following properties: i) $$H_{u_0}$$ is stable under time evolution and under spaces translations, ii) all the solutions corresponding to initial data $$u = (\phi, \psi) \in H_{u_0}$$ have the same asymptotic limit $$\phi_0$$ for $$|x| \rightarrow \infty\ ,$$ for any time. Furthermore the energy of the configurations $$u \in H_{u_0}$$ is bounded from below.
The property of energy bounded from below guarantees a sort of stability, since otherwise small perturbations may lead to solutions with lower and lower energy, i.e. to a collapse. Thus, different (absolute) minima of the potential define disjoint Hilbert sectors, which may be interpreted as disjoint stable realizations of the system (like the different thermodynamical phases of a complex system in the infinite volume limit), since two configurations belonging to different Hilbert sectors cannot be both physically accessible by the same observer.
It is worthwhile to remark that also initial data corresponding to relative minima of the potential define Hilbert sectors, but the energy is not bounded from below; for the general characterization of the initial data which define Hilbert sectors see (Parenti, Strocchi and Velo 1977).
A symmetry $$g$$ of the system described by the above field equation is an invertible mapping of the initial data which commutes with the time evolution. The symmetry is broken in the realization of the system described by the Hilbert sector $$H_{u_0}$$ if it does not leave $$H_{u_0}$$ stable.
In the following, for simplicity, we shall consider groups $$G$$ of internal symmetries, i.e. of the form $$g(\phi) = A \phi\ ,$$ with $$A$$ an invertible matrix satisfying $$A^T \, A = \lambda \mathrm I\ ,$$ $$\lambda \in \mathbf R\ ,$$ $$A^T$$ the transpose of $$A\ ,$$ and $$U(A \phi ) = \lambda U(\phi)$$ (Parenti, Velo and Strocchi 1977).
Clearly, an internal symmetry of the Lagrangian maps Hilbert sectors into Hilbert sectors and if $$G$$ is the group of internal symmetries of the Lagrangian each Hilbert sector $$H_{u_0}$$ determines the subgroup $$G_{u_0}$$ of $$G\ ,$$ called the stability group of $$H_{u_0}\ ,$$ which leaves the sector invariant and therefore describes the unbroken subgroup of $$G\ .$$
A time independent functional $$Q$$ generates a transformation of the solutions $$u\ ,$$ $$u_k = \phi_k\ ,$$ for $$k = 1, \cdots n\ ,$$ $$u_k = \psi_{k - n}\ ,$$ for $$k = n + 1, \cdots 2 n\ ,$$ through the Poisson brackets $\{u_k(x,t), Q\}=\sum_{a=1}^n \int d^3 \mathbf{y} \left[\frac{\delta u_k(x,t)}{\delta \phi_a(y,t)} \frac{\delta Q} {\delta \psi_a(y,t)} - \frac{\delta u_k(x, t)}{\delta \psi_a(y,t)} \frac{\delta Q}{\delta \phi_a(y,t)}\right],$ where $$\delta F/ \delta \phi_a(y,t)$$ denotes the functional derivative of $$F$$ with respect to $$\phi_a(y,t)\ ,$$ see (Schmid 1987, Marsden and Ratiu 1994).
Then, one has the following generalization of Noether theorem which takes into account the possibility of symmetry breaking
Let $$G$$ be an $$N$$-parameter Lie group of internal symmetries of (the Lagrangian of) eq. (1), then there exist $$N$$ currents $$J_\mu^i(u(x, t))\ ,$$ $$i = 1, \cdots N\ ,$$ which obey the continuity equation $$\partial^\mu J_\mu^i = 0$$ (local conservation law). Given a Hilbert sector $$H_{u_0}$$ and a one-parameter subgroup $$G^i$$ of $$G\ ,$$ the Noether charge$Q^i(u) = \int d^3 x \, J_0^i(u(x, t))$ exists and it is independent of time for all solutions $$u(x, t) \in H_{u_0}\ ,$$ equivalently it defines a linear operator $$\tilde{Q}^i: H_{u_0} \rightarrow H_{u_0}$$ acting as the generator of the corresponding transformation$(\tilde{Q}^i u)_k = \{ u_k, \,Q^i\} = (\delta^i u)_k,$ if and only if $$G^i$$ is a subgroup of the stability or unbroken group $$G_{u_0}$$ of $$H_{u_0}\ .$$ In this case, $$G^i$$ is represented by unitary operators in $$H_{u_0}\ .$$
### Example. Non-linear scalar field in three space dimensions
The model is defined by the following field equation$\square \phi = - 2 \lambda \phi (\phi^2 - \mu^2)^2, \,\,\,\,\phi: \mathbf R^3 \rightarrow \mathbf R^n.$ It displays some similarity with the mechanical model of a particle in $$\mathbf R^n$$ with a Mexican hat potential $$U(q) = \lambda (q^2 - \mu^2)^2\ .$$ However, as explained above, the difference is substantial: in the field case, each point $$q$$ has become infinite dimensional and the absolute minima $$\phi_0$$ identify disjoint Hilbert sectors. Whereas in the finite dimensional case there is no physical obstruction or barrier which prevents the motion from one minimum to the other, in the field case there is no physically realizable operation which leads from one sector to the other, since this would require to change the asymptotic limit of the configurations and this is not possible by physically realizable operations, which are necessarily localized.
More generally, infinitely extended systems described by hyperbolic equations which admit solutions with non-trivial behaviors at space infinity, have disjoint realizations described by sectors stable under time evolution. Thus, the mechanism of spontaneous symmetry breaking discussed above applies.
Another wide class of infinitely extended systems, which display a structure of phases and spontaneous symmetry breaking is given by complex systems in the thermodynamical limit. A few comments are in order about the realization of spontaneous symmetry breaking in complex systems, some examples of which are used as prototypic examples for explaining the phenomenon of spontaneous symmetry breaking, which, strictly speaking, requires the infinite volume limit. Even if one is always dealing with finitely extended systems, the use of the thermodynamical limit has proved to be very useful for the theoretical description of systems with a very large number of degrees of freedom and with a very large size. The theory of phase transitions is one of the most successful achievements of such a strategy. In the thermodynamical limit different ground states define disjoint thermodynamical phases; they are selected by the boundary conditions imposed in taking the infinite volume limit, as discussed by Ruelle in his general treatment of Statistical Mechanics (Ruelle 2004). Thus, the general philosophy discussed above applies, since different phases describe disjoint realizations of the system, and a non-symmetric ground state implies the breaking of the symmetry in the corresponding phase.
## References
• H. Goldstein, Classical Mechanics, 2nd. ed., Addison-Wesley 1980
• J. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, Springer 1994
• E. Noether, Invariante Variations Probleme, Nachr. d. Kgl. Ges. d. Wiss. Göttingen (1918), p.235-257; English translation in N.A. Tavel, Milestones in mathematical physics, Transport Theory and Statistical Mechanics, 1, 183-207 (1971)
• C. Parenti, F. Strocchi and G. Velo, Solutions of Classical Field Equations with Local Finite Kinetic Energy, Phys. Lett. 59 B, 157-158 (1975)
• C. Parenti, F. Strocchi and G. Velo, Hilbert space sectors for solutions of non-linear relativistic field equations, Comm. Math. Phys. 53, 65-96 (1977)
• D. Ruelle, Thermodynamic Formalism: The Mathematical Structure of Equilibrium Statistical Mechanics, Cambridge University Press 2004
• R. Schmid, Infinite Dimensional Hamiltonian Systems, Bibliopolis 1987
• F. Strocchi, Symmetry Breaking, 2nd ed., Springer 2008; there one can find a more extensive discussion, the proofs of the propositions stated in this note and further references.
• A.S. Wightman, Constructive Field Theory, in Fundamental Interactions in Physics and Astrophysics, G. Iverson et al. eds., Plenum 1972 | {"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": 0.941501259803772, "perplexity": 285.416892456346}, "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/1674764499646.23/warc/CC-MAIN-20230128153513-20230128183513-00796.warc.gz"} |
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